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Investigation of Electron-Nuclear Spin Interactions in Two-Dimensional Electron Systems via Magnetoresistively Detected Magnetic Resonance

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Investigation of Electron-Nuclear Spin Interactions in Two-Dimensional Electron Systems via Magnetoresistively Detected Magnetic Resonance
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CALDWELL, JOSHUA D. ( Author, Primary )
Copyright Date:
2008

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Electric fields ( jstor )
Electron paramagnetic resonance ( jstor )
Electrons ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnets ( jstor )
Microwaves ( jstor )
Quadrupoles ( jstor )
Signals ( jstor )
Temperature dependence ( jstor )

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University of Florida
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University of Florida
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Copyright Joshua D. Caldwell. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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12/31/2005
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436098632 ( OCLC )

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INVESTIGATIONS OF ELECTRON-NUCL EAR SPIN INTERACTIONS IN TWODIMENSIONAL ELECTRON SYSTEM S VIA MAGNETORESISTIVELY DETECTED MAGNETIC RESONANCE By JOSHUA D. CALDWELL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Joshua David Caldwell

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To my parents, who allowed me to grow and made me desire this. To my family, who has instilled in me streng th and integrity, which made this possible. And to my wife Betsy, whose love, support, an d companionship have made this a reality.

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iv ACKNOWLEDGMENTS The work presented within this dissertati on has required the help of many people. The direction and support that I have received from my research advisor, Dr. C. R. Bowers, have been instrumental in my deve lopment as a scientist and person. I have thoroughly enjoyed working within his research group and appreciate the many friendships that I have deve loped during my time at the Un iversity of Florida. I must thank Greg Labbe and John Graham of the Cryogenic Services Group. Their continued patience with my many requests for liquid helium and questions allowed for the experiments presented within this di ssertation to be completed. Todd Prox of the UF Chemistry Machine Shop has provided a leve l of expertise and support that has made my time at UF much easier. His work on the optical pumping probe design and construction was very much appreciated. Finally, Steve Miles of the UF Chemistry Electronic Shop has provided his amazing knowle dge of electronics toward the design of circuitry, the repairing of instrumentation, and for my own general information. The friendship that both Todd and Steve have offered also contributed to my many enjoyable memories of my time at UF. I have had the opportunity to work alongs ide some truly amazing scientists during my graduate schooling. I would like specifica lly like to mention my appreciation to Dr. Eugene Olshanetsky, Dr. Alexey Kovalev, and Dr. Anil Patel, all of whom provided me with numerous helpful discussions and work ed along side me on any one of my many experimental endeavors.

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v Of course the experiments discussed here in required samples that were provided from numerous sources. I have had the oppor tunity to work with some of the best samples available, which allowed me to pe rform a great number of experiments. The majority of this work was completed using the EA series samples, which were grown by Drs. Jerry Simmons and John Reno of Sandi a National Laboratory. The parabolic quantum well work utilized samples (AG662) grown by Dr. G. M. Gusev of the University of Sao Paolo, Brazil. Finally, th e extremely low density systems (HM series) were fabricated by F. Capotondi, G. Biasiol, and L. Sorba of the Laboratorio Nazionale TASC (INFM) Strada in Italy. Without the love and support of my wife Betsy none of this would have been even a possibility. The level of sacrifice and suppor t that she has offered me over the past few years can never be fully repaid. Her unders tanding and companionship have allowed me to maintain my steady effort towards graduati on. She is an unending source of strength, support, humor and inspiration. I would also like to thank my parents. Their efforts to in still a solid work ethic and strong morals within me, I only hope are viewed as successes. They, as with my wife, have sacrificed more for me than I could ever begin to acknowle dge. Finally, I would like to thank my brother Jason. Over the year s, his humor and friendship have helped me grow to be the man I am today.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...............................................................................................................x LIST OF FIGURES...........................................................................................................xi ABSTRACT....................................................................................................................xvi i CHAPTER 1 INTRODUCTION TO SPIN, MAGNETIC RESONANCE AND THE OVERHAUSER EFFECT............................................................................................1 Spin Angular Momentum and Its Implications............................................................1 Magnetic Resonance.....................................................................................................4 The CW ESR Experiment.............................................................................................6 Basic Theory..........................................................................................................7 Experimental method.............................................................................................8 Overhauser Effect.........................................................................................................9 Electric Quadrupole Interaction..................................................................................15 2 TWO-DIMENSIONAL ELECTRON SY STEMS AND THE QUANTUM HALL EFFECT......................................................................................................................19 Two-Dimensional Electron Systems..........................................................................19 The Integer Quantum Hall Effect...............................................................................21 Historical Perspective..........................................................................................21 Physical Description............................................................................................23 Electron-Nuclear Interactions in Quantum Hall Systems...........................................31 3 MAGNETORESISTIVELY DETECTED ELECTRON SPIN RESONANCE (MDESR) IN HIGH MOBILITY/DENSITY SAMPLES..........................................34 Initial Experiments......................................................................................................35 First Detection of MDESR..................................................................................35 Theory of MDESR..............................................................................................36

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vii Selected Further Electron Spin Experiments.......................................................39 Zero-field spin-splitting and field dependence of * eg...................................39 Evidence for long-range spin excitations.....................................................40 Evidence of thermally excited Skyrmionic spin excitations........................42 Evidence for single spin excitations in MDESR..........................................45 Experimental...............................................................................................................49 Apparatus.............................................................................................................49 Magnet and cryostat.....................................................................................49 Sample mounting..........................................................................................50 Transport measurements xx R via single lock-in detection........................51 Differential resistance measurements xx R via double lock-in detection..................................................................................................52 Samples................................................................................................................53 Experimental Results..................................................................................................54 Magnetoresitively Detected ESR (MDESR).......................................................54 Temperature Dependence of MDESR.................................................................56 MDESR characteristic features....................................................................57 Discussion of MDESR excitation a nd its temperature dependence.............60 Concluding remarks regarding heating model and temperature dependence..............................................................................................72 Filling Factor Dependence of MDESR...............................................................73 Concluding Remarks..................................................................................................80 4 ELECTRON-NUCLEAR HYPERFINE IN TERACTION AND ITS EFFECT ON MDESR AND MDENDOR........................................................................................82 Hyperfine Interactions in 2DESs................................................................................84 First Observation of MDENDOR........................................................................86 Direct Absorption Detection of ENDOR in n -Type GaAs..................................89 First Observation of MDENDOR Peaks.............................................................91 MDNMR in 2DESs.............................................................................................93 Experimental Results: Influence of Hyperfine Coupling on MDESR Spectra...........96 Sweep Rate Dependence of MDESR..................................................................96 Microwave Power Dependence of MDESR......................................................103 Temperature Dependence of Over hauser Broadening of MDESR...................105 Measurement of the Overhauser Shift Decay....................................................107 Experimental..............................................................................................107 Results........................................................................................................108 DNP Induced by DC Injection..........................................................................109 Experimental..............................................................................................109 Results........................................................................................................110 Experimental Results: MDENDOR..........................................................................111 Experimental......................................................................................................113 Experimental Results.........................................................................................117

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viii Microwave power and temperat ure dependence of MDENDOR response.................................................................................................117 RF dependence...........................................................................................118 RF power dependence................................................................................121 Concluding Remarks................................................................................................123 5 OBSERVATION OF NUCLEAR Q UADRUPOLE SPLITTI NG AND DOUBLE QUANTUM EXCITATIONS IN THE MDMR SPECTRA OF GaAs/AlGaAs QUANTUM WELLS................................................................................................125 Nuclear Quadrupole Splitting of 75As MDENDOR.................................................125 Previous Observations.......................................................................................126 Observation of Quadrupole Splitting.................................................................126 RF Sweep Rate Dependence of Quadrupole Splitting......................................128 RF Power Dependence of Quadrupole Splitting...............................................129 Tilt Angle Dependence of Quadrupole Split ting; Investigation into Strain Effects............................................................................................................131 Current Dependence of the Quadrupole Splitting.............................................134 Filling Factor Dependence of Quadrupole Splitting.........................................136 Overhauser Shift Dependence of Quadrupole Splitting....................................138 Discussion of the Origin of the Quadrupole Splitting.......................................139 Double Quantum/Overtone Excitations....................................................................140 Previously Reported Observations....................................................................141 Observation of 2Im Excitations...............................................................141 Discussion..........................................................................................................145 Concluding Remarks................................................................................................147 6 MDESR/MDENDOR NUMERICAL SIMULATIONS..........................................149 MDESR Simulations................................................................................................149 Algorithm..........................................................................................................149 Simulation Results.............................................................................................156 Field Swept MDENDOR Simulations......................................................................161 Algorithm..........................................................................................................161 Simulation Results.............................................................................................162 Radiofrequency Swept MDENDOR Simulations....................................................166 Algorithm..........................................................................................................166 Simulation Results.............................................................................................167 Concluding Remarks................................................................................................168 7 CHANGE IN EDESR BEHAVIOR A SSOCIATED WITH THE EVOLUTION FROM A 2DES TO 3DES WITHIN WIDE PARABOLIC QUANTUM WELL (WPQW) STRUCTURES........................................................................................170 Properties of a 3DES................................................................................................171 Parabolic Quantum Wells.........................................................................................171 Effect of Sample Rotation.................................................................................174

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ix Study of Spin Properties and Dynamics within WPQW Systems.....................178 Experimental Results................................................................................................180 Sample Characteristics......................................................................................180 Results...............................................................................................................180 Angle dependence of the bare electron g-factor.........................................180 Temperature dependence of EDESR within a 3DES.................................183 Dependence of the Overhauser shift decay on temperature.......................191 Sweep rate dependence of the EDESR response.......................................192 Frequency swept MDENDOR response....................................................198 Concluding Remarks................................................................................................199 8 MDESR IN LOW DENSITY 2DES IN GaAs/AlGaAs QUANTUM WELLS.......202 Observation and Characterization of MD ESR in Low Electron Density Systems...202 Introduction and Motivation..............................................................................202 Description of Samples......................................................................................203 Experimental Results.........................................................................................204 Microwave power, channel current a nd g-factor characterization of MDESR.................................................................................................205 Temperature dependence of MDESR response.........................................207 Observation of 2eg Resonance............................................................................208 Description of 2eg Peak Structure................................................................208 Effect of Resonant RF Irra diation upon ESR Structure....................................211 Temperature Dependence of 2eg EDESR....................................................211 Microwave Power Dependence.........................................................................213 Concluding Remarks................................................................................................213 9 CONCLUDING DISCUSSION...............................................................................215 APPENDIX A SYMBOLS LIST......................................................................................................222 B ACRONYM LIST....................................................................................................225 LIST OF REFERENCES.................................................................................................226 BIOGRAPHICAL SKETCH...........................................................................................234

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x LIST OF TABLES Table page 3-1 Tabulated results from the seven samp les discussed within the text of this dissertation...............................................................................................................59 4-1 Collection of important properties of the three spin-bea ring isotopes in GaAs.........................................................................................................................83

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xi LIST OF FIGURES Figure page 1-1 Diagram displaying the relative energy levels of a coupled electron and nucleus through the FC hfi, both with a spin of ½................................................................11 1-2 Diagram representation of Overhauser effect.........................................................15 1-3 Energy levels of a 3/2 I system subject to a quadrupole interaction.................18 2-1 Schematic of a -doped, single GaAs/AlxGa1-xAs quantum well structure.............20 2-2 Schematic for a GaAs/AlxGa1-xAs single heterostructure........................................21 2-3 Illustration of a single quantum well samp le patterned into th e standard Hall bar geometry...................................................................................................................24 2-4 D.O.S. diagram for an ideal 2DES in an externally applied magnetic field and the corresponding longitudinal and Hall resistances......................................................25 2-5 The effect of sample tilting on the QHE..................................................................26 2-6 Illustration of the concept of Landau level filling....................................................29 2-7 Magnetoand Hall resistance traces for sample EA124B........................................30 3-1 Theoretical calculation of the dispersion curve for a 2DES with spin-waves as the dominant electron spin excitation............................................................................38 3-2 Pictorial representation of three main electron spin excitations discussed in 2DES in the regime of the integer QHE...................................................................44 3-3 Photo of probe used for MDESR measurements.....................................................51 3-4 Schematic diagram of experimental setup for single and double lock-in measurement techniques..........................................................................................53 3-5 (a) Typical xx R vs. 0 B trace for sample EA124 at T=1.5 K. (b) Typical xx R vs. 0 B response............................................................................55 3-6 Measurement of bare electron g-factor....................................................................56

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xii 3-7 Temperature depe ndence of MDESR at 1 .........................................................59 3-8 Temperature dependence of (a) MDESR peak amplitude and (b) MDESR line width.............................................................................................60 3-9 (a) Temperature dependence of the magnetoresistance around =1 filling factor. (b) Arrhenius plot of lnxx R vs. 1/ T.....................................................................62 3-10 Temperature dependence of electron spin polarization at =1 and 05.7 T B based on the theoretical models of spin -waves (eq. (3.23)) and itinerant electron systems (eq. (3.24)) for a 2DES...............................................................................69 3-11 Schematic representation of the propos ed heating model used to explain the temperature dependence of MDESR amplitude.......................................................69 3-12 Temperature dependenc e of MDESR amplitude.....................................................71 3-13 (a) Multiple magnetoresistance traces for calibration of the filling factor (b) The MDESR amplitude as a function of filling factor in the region of =1......75 3-14 (a) Representative spectr a of the MDESR exhibiting the anomalous line shape. (b) MDESR peaks at multiple frequencies with in the region of the critical points in the temperature dependence.................................................................................76 3-15 The magnetoresistance taken in the region where the anomalous line shape in the EDESR was observed...............................................................................................78 3-16 Observation of the anomalous line shape within a 69Ga MDENDOR spectrum.....79 3-17 (a) Filling factor dependence of th e activation gap (b) The dependence of magnetoresistance prefactor on filling factor...........................................................79 4-1 Sweep rate dependence of MDESR in EA129.......................................................100 4-2 Sweep Rate dependence of MDES R in (a) EA124B and in (b) EA124................101 4-3 Sweep rate dependence of MDESR amplitude (solid blue circle s) and line width (open black squares)...............................................................................................104 4-4 Microwave power dependence of MDESR response in sample EA129................105 4-5 Temperature dependence of the Over hauser broadened EDESR response ...........106 4-6 Activation plot in EA129.......................................................................................107 4-7 Time dependence of the Overhauser shift decay...................................................109

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xiii 4-8 (a) Current induced heating effect on th e magnetoresistance (b) Investigation of DC injection on the MDESR spectral features.......................................................112 4-9 The effect of increased DC inj ection time upon the shift in the MDESR..............113 4-10 Field swept MDENDOR spectra of 71Ga...............................................................116 4-11 Field swept MDENDOR sp ectrum for all three isotopes at three different RF frequencies.............................................................................................................116 4-12 Determination of the gyromagnetic ratio of 75As MDENDOR..............................117 4-13 The three graphs correspond to the RF swept MDENDOR experiment measured as a function of (a) time, (b) magnetic field and (c) frequency.............117 4-14 75As field swept MDENDOR response in sa mple at two different temperatures and microwave powers...........................................................................................119 4-15 71Ga field swept MDENDOR response at a variety of RF frequencies after a slow down-sweep the ESR condition.....................................................................120 4-16 (a) field swept MDENDOR response from Figure 4-15. (b) field swept MDENDOR response at the same conditions at half of the RF power of (a). (c) MDESR under the same conditions, but under application of non-resonant RF.....................................................................................................122 4-17 (a) MDENDOR amplitude as a function of RF (b) The resu ltant loss of the MDESR signal following MDENDOR excitation as a function of RF.................123 5-1 field swept MDENDOR spectra for all three NMR active nuclei within GaAs.......................................................................................................................127 5-2 75As MDENDOR spectrum obtained via the (a) field-swept and (b) RF-swept methods..................................................................................................................128 5-3 RF sweep rate dependence of the 75As RF swept MDENDOR response..............129 5-4 75As RF swept MDENDOR spectra taken at the maximum and 50% of the maximum RF power output from the RF source....................................................130 5-5 RF Power dependence of the central (in black), left satellit e (in blue) and right satellite (in red) peaks of the 75As RF swept MDENDOR spectra........................131 5-6 75As RF swept MDENDOR spectra taken at approximately =1 within sample EA124B at 6.92 (blue) and 63.920 (red).................................................................133 5-7 Angle dependence of the quadrupole splitting of the 75As RF swept MDENDOR response..................................................................................................................134

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xiv 5-8 Current dependence of the quadrupole splitting of the 75As RF swept MDENDOR response..................................................................................................................136 5-9 Filling factor dependence of the 75As RF swept MDENDOR quadrupole splitting...................................................................................................................138 5-10 75As RF swept MDENDOR spectra taken at Overhauser shifts of 11.00 and 47.71 mT................................................................................................................139 5-11 Single (black) and double (blue) quantum excitations of 75As RF swept MDENDOR at 53 (a) and 610 (b)...........................................................................143 5-12 Schematic diagram of the Zeeman energy levels without (left) and with (right) a quadrupole splitting.............................................................................................144 5-13 Single (black) and double (blue) quantum excitations of 69Ga RF swept MDENDOR at 610.................................................................................................144 6-1 Experimental and simulated MDES R spectra at various sweep rates....................159 6-2 Experimental (blue squares) and simu lated (black circles) dependence of the MDESR response on magnetic field sweep rate....................................................160 6-3 Numerical simulations of the nuclear spin polarization pr ofiles during a (a) 100 mT/min and (b) 4.6 mT/min down-field sweep incorporating spin diffusion effects.....................................................................................................................160 6-4 Simulated field swept MDENDOR a nd corresponding experimental MDENDOR response..................................................................................................................162 6-5 Simulation of the production of the nuc lear field during the slow down-sweep due to DNP, and the effect upon meeting the NMR condition..............................164 6-6 Pictorial representation of the MDENDOR response at a) crit nn B B and critESR nnFWHMBB.............................................................................................165 6-7 Pictorial representation of the MDENDOR response at a) crit nn B B and crit nn B B...........................................................................................................166 6-8 Simulations of the RF swept MDENDOR results as a function of (a) time, (b) magnetic field and (c) frequency............................................................................168 7-1 Comparison between a square (left) and parabolic (righ t) quantum well..............173 7-2 a) Magnetoresistance traces at 900 (3DES), 800 (both a 3D and 2DES), 570 (=1 minimum within range of system) and 00 (2DES) b) Magnetoresistance trace at 00 c) Magnetoresistance trace at 900...................................................................176

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xv 7-3 Used with the permission of G. M. Gusev.106 As the angle is tilted from a perpendicular (00) to a parallel (900) orientation, a clear co llapse of the energy levels of the different electr onic subbands is observed..........................................178 7-4 Position of the EDESR response at 36.65 GHz at four different angles within a WPQW...................................................................................................................184 7-5 EDESR condition as a func tion of the applied microwav e frequency at a number of different angles...................................................................................................185 7-6 Dependence of the bare electron g-factor upon the sample tilt angle....................185 7-7 Magnetoresistance traces at 0.42, 1.42 and 10.0 K within a WPQW....................188 7-8 a) Magnetoresistance traces within th e range where EDESR was observed at many different temperatures b) The temperature dependence of the magnetoresistance at three different magnetic fields with in the range where EDESR was observed..188 7-9 Processed and unprocessed MDESR sp ectra observed at 2.0 (a) and 10.0 K (b) in a WPQW oriented at 900..............................................................................189 7-10 Processed and unprocessed MDESR spect ra observed at 1.67 (a) and 5.09 K (b) in a WPQW oriented at 00................................................................................189 7-11 Temperature dependence of the EDESR amplitude and line width in a WPQW in the (a) parallel and (b) pe rpendicular orientations.................................................190 7-12 Temperature dependence of the MDESR position.................................................190 7-13 Measurements of the Overhauser shift decay within a WPQW in the parallel orientation at T=1.6 (a) and 5.0 K (b).......................................................193 7-14 Measurements of the Overhauser shift decay within a WPQW in the perpendicular orientation at T=1.4 (a), 2.0 (b) and 3.0 K (c).................................193 7-15 Sweep rate dependence of the ED/M DESR response within a (a) 3D and (b) 2DES in a WPQW............................................................................................195 7-16 EDESR response at a) 492, b) 400, c) 300, d) 200, e) 150, f) 100, g) 50, h) 25 and i) 10 mT/min at 1.6 K in a WPQW.................................................................197 7-17 Sweep rate Dependence of the EDES R line width (FWHM) within a WPQW at the parallel orientatio n at a variety of temperatures for both the down (a) and up (b) sweeps in magnetic field.................................................................197 7-18 Sweep rate dependence of the EDESR am plitude within a WPQW in the parallel orientation at a variety of temperatur es, for both the down (a) and up (b) sweeps in magnetic field.....................................................................................................198

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xvi 7-19 (a) 75As, (b) 71Ga and (c) 69Ga RF swept MDENDOR spectra recorded in a WPQW...................................................................................................................199 8-1 MDESR peak in a low density 2DES....................................................................205 8-2 Microwave power depende nce of the MDESR response in a low density 2DES..206 8-3 The microwave frequency dependence of the MDESR resonant field in a low density 2DES..........................................................................................................206 8-4 Temperature dependence of the MDESR response in a low density 2DES......................................................................................................................207 8-5 Observation of the EDESR at 2eg in a low density 2DES................................209 8-6 Determination of the g-factor for the cen tral peak of the lo w frequency resonance in a low density 2DES............................................................................................210 8-7 EDESR of 1.97eg resonance at two temperat ures in the low density 2DES sample HM0455...........................................................................................212 8-8 Temperature dependence of the 1.97eg EDESR resonance in low density 2DES sample HM0461...........................................................................................212 8-9 Microwave power dependence of the 1.97eg resonance in a low density 2DES......................................................................................................................213

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xvii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INVESTIGATION OF ELECTRON-NUC LEAR INTERACTIONS IN TWODIMENSIONAL ELECTRON SYSTEM S VIA MAGNETORESISTIVELY DETECTED MAGNETIC RESONANCE By Joshua D. Caldwell December 2004 Chair: Clifford R. Bowers Major Department: Chemistry The polarization of the electr on spins and their in teractions with the local nuclei are of considerable interest within two-dime nsional electron systems (2DES) within the regime of the quantum Hall effect (QHE). ESR is an ideal experi mental technique to probe both of these features. Unfortunately, due to the low electron densities, it is not usually possible to detect the ESR spectra vi a typical absorption techniques. However, the resonant microwave absorption by the conduction electrons may be observed as a sharp change in the magnetoresistivity within these systems, allowing for the observation of the ESR spectra via the magneto resistive detection (MD) method. Through the hyperfine interaction (hfi) be tween the conduction electrons and the local nuclei, the determination of the sign of the electronic g-f actor and a further understanding of the couplings present were obtained. The broade ning of the ESR during slow down-sweeps in magnetic field, which occu rs due to the Overhauser shift of the ESR to lower field, has provided an id eal method for obtaining large nuclear

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xviii polarizations within these systems. Through resonant RF irradiation of the sample, a sharp change in the Overhauser broadened MD ESR spectrum is observed. This change is directly associated with the NMR excitation of the lattice nuclei and is referred to as MDENDOR spectroscopy. The temperature dependence of the MDESR within the =1 minimum was measured and allowed for the formulation of a theoretical model of the MD mechanism. This model is based on the heating of the 2D ES via resonant absorption of microwaves and was utilized to provide numerical simu lations of the sweep-ra te dependence of the MDESR and the overall features of the MD ENDOR spectra. The qualitative agreement between these simulations and the experime ntal measurements has provided further evidence in support of the model. The MD method was used to observe ESR within extremely low electron density and wide parabolic quantum well samples. Bo th of these observations were the first of their kind. These experiments provide exampl es for two distinct systems where this method may be used to obtain direct measurem ents of the spin transitions and electronnuclear spin interactions associated with the conduction electrons.

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1 CHAPTER 1 INTRODUCTION TO SPIN, MAGNETI C RESONANCE AND THE OVERHAUSER EFFECT To fully understand the implications of the re search presented in this dissertation, a brief introduction into magnetic resonance is necessary. In this chapter, the basic principles of magnetic resonance spectroscopy will be discussed. Furthermore, due to the predominance of electron nuclear interactions, specifically the Overhauser effect, on this research, the discussion will encompass an introduction into this topic as well. Spin Angular Momentum and Its Implications There are many properties that determine th e nature of electrons and nuclei, though none is more abstract than that of spin angular moment um. Unlike rotational angular momentum, there is no macroscopic analogy to explain spin. While the mathematics tends to the understanding of sp in as an electron or nucleus spinning on its own axis, this is not correct. Also, a species with spin maintains this property even at T=0 K. However, the rules by which we explain spin are the same as those used to explain rotational angular momentum.1 Spin angular momentum is an intrinsic quantum mechanical property. As such, spin is quantized, having values of 1/21 II with each energy level consisting of 21 I sublevels, which are degenerate in the absence of an external magnetic field. Here, I is the total spin angular momentum. Fo r the purpose of clarity, from this point further the spin quantum number will be denoted as Spertaining to electron spin, while

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2 I will be used for nuclei. The spin of all free electrons is 1 2 S , while the spin of nuclei is more complex to determine. Since nuclei are made up of a varying number of protons and neutrons, dependent on the element and isot ope, the spin of the nuclei depends on the total spin of all the protons and neutrons within that nucleus. I for a given nucleus is given as the gr ound nuclear spin state. It may exhibit integer or half integer values . Nuclei which have a ground nu clear spin state greater than 1/2 are referred to as quadrupol ar nuclei and exhibit many differ ent spectral effects. This will be discussed in further detail in Chap ter 5. While the ground nuclear spin state cannot be predicted exactly for every nucle us, it has been determined empirically. Excited spin states of nuclei do of course exist, but for t ypical experimental conditions, including those presented here, the energy difference is quite large with respect to the thermally available energy BkT and thus the excited spin state may be ignored. Nonzero values for I orS infer the presence of a magnetic moment. The relationship between the magnetic moment and the spin quantum number for the nucleus is as follows: ˆ ˆn I (1.1) where n is the gyromagnetic ratio, a constant specifi c to each nucleus, typically stated in units of rad s-1T-1. For electrons, the magnetic moment is typically defined in terms of eg : ˆ ˆeBgS (1.2) where eg is the g-factor of the electron, a cons tant dependent on the environment of the electron, and B is the Bohr magneton. While nnng (n is the nuclear magneton),

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3 and eBeg , the differences are not limited to the exchange of the Bohr and the nuclear magnetons. The gyromagnetic ratio of both th e nuclei and the electron are sensitive to the environment, but the effect on the nuclear gyromagnetic ratio is small, while the effect on the electron is quite large. For in stance, the electronic gfactor can vary by a very large amount with a value of -7.4 in InAs2 to 2.0023 for a free, unbound electron to -0.443 for an electron in GaAs; hence, the di fferent notation in comparison to nuclear species. Bulk magnetization is induced by th e application of a magnetic field 1 00inducedVB (1.3) where -7 04 x 10 H/m, Vis the volume, is the magnetic susceptibility and 0 B is the applied magnetic field. The implications of this fact are that the individual magnetic moment of these species may only be measured in the presence of an externally applied magnetic field, and that the greater the fiel d, the larger the observed moment. The interaction has a potential energy as follows: 0 E B (1.4) with the negative sign indicating that the energy is lowest when the magnetic moment and external field align in the same direction. This implies that species with negative gyromagnetic ratios or g-factors will see their spins align anti-parallel to 0 B and , and vice-versa for those with positive values.1 The direction of the vector sum of the individual magnetic moments is referred to as the spin polarization axis. In the absence of an applied magnetic field, the magnetic moments point in all directions in space and th erefore lead to a zero sum. However, if a magnetic field is applied, the moments begi n to precess about the field, although they do

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4 not immediately align with it; instead, the spins simply precess about the magnetic field at the original angle (in reference to0 B ) they were in when 0 B was first applied. The frequency of precession about 0 B is named the Larmor frequency and is given by 00 B (1.5) Due to relaxation, as the spins precess, they begin to align with the external field. The realignment is due to the presence of local fluctuating fields. As this occurs, the magnetic moments begin to align and theref ore the vector sum grows. Thus a net macroscopic magnetization is create d parallel to the direction of 0 B . This relaxation process is referred to as the spin-lattice rela xation, and the time constant for this process is denoted as 1nT.1 Magnetic Resonance Magnetic resonance refers to the excitati on of nuclear (NMR) or electronic (ESR) magnetic moments by electromagnetic radiation. While the application of these two different resonance methods involves differen t experimental setups , the theory governing both is essentially the same. In both ESR and NMR, a static magnetic fi eld is applied to the sample along the zaxis. For a spin I =1/2 system, the Hamiltonian for this interaction is 0ˆˆzzHI (1.6) where ˆz I is the component of the spin angular momentum operator that is aligned with 0 B . The eigenstates of ˆz I for this system are found to be 11 , 22Im The Hamiltonian, when applied to the tw o eigenstates gives the following:

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5 0 0111 ˆ 222 111 ˆ 222z zH H (1.7) The total polarization of a sy stem of electron or nuclear spins may be defined as ˆ 1m z m m mmN I p I IN (1.8) where mis the spin eigenvalue and mN is the population of the thmstate. In the case of the I =1/2 system, there are only two states; thus, the polarization is defined as the difference between the higher and lower energy spin states divided by the total number of spins. Next, an electromagnetic field 1 appB is applied in the direction transverse to 0 B (along the x or y-axis). If the magnetizati on is observed in a frame of reference rotating about 0 B at a frequency app , the field interacting with the magnetic moments appears to be an effective field made up of two components, as follows: 10ˆ ˆapp eff B BxBz (1.9) If the 1 B field is applied at the Larmor frequency 0 app , the term 0 appB disappears and only the 1ˆ B x component remains. This is called the res onance condition. Quantum mechanically, magnetic dipole tran sitions are stimulat ed by the oscillating electromagnetic field. Referring to the 1/2 and 1/2 states as and , respectively, the difference between the energy levels is determined to be

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6 00000,11 ˆˆ , for nuclei 22 11 ˆ , for electrons 22zNMRNMRNMRNMRNMR zeBeBeBeBeEII ESgBgBgBgB (1.10) where NMR and 0, e represent the Larmor frequenc ies of the nuclei and electrons, respectively. This energy difference is referred to as the Zeeman energy and is equal to the total energy required for magnetic resonance excitation.4 At high temperatures, the thermal equilibrium populations for nuclear and electronic spin states are governed by the Boltzmann distribution. Magnetic resonance transitions change the populations of the energy levels until a st eady-state condition is achieved. The population difference between th e two states is expressed as follows: 0expBpp kT (1.11) The CW ESR Experiment While both ESR and NMR excitation measurem ents are explored in the research presented in this dissertation, the main exci tation technique is base d quite rigorously on the continuous wave (CW) detection of ESR and electron nuclear double resonance (ENDOR), respectively. Therefor e, a brief introduction into these two techniques will be provided. ESR, also referred to as electron parama gnetic resonance (EPR) is a spectroscopic technique that detects magneti c resonance transitions of unpaired electrons. Such electrons are found as free radicals, in some tr ansition metal ions, defect sites in solids, and in the conduction band of metals and se miconductors. While unpaired electrons in most cases tend to be short lived, they have a strong effect upon ma ny processes such as

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7 photosynthesis, oxidation, catalysis and polymerization reactions, and play a distinct role in the use of electronic devices. ESR is the only technique currently availa ble that directly de tects the electronÂ’s presence and their interactions with the ot her magnetic moments within their proximity. This method may be used to identify different electronic species and is sensitive to the local environment, therefore pr oviding insight into the molecu lar or lattice structure in the vicinity of the excited el ectron. Studies of changes to the ESR lineshape may also provide evidence of kinetic pro cesses such as molecular motion.5 Basic Theory As evidenced from eq. (1.5), in order to observe ESR transitions, the application of an external magnetic field along with the elect romagnetic irradiation that is necessary in spectroscopic measurements, is required. Furt hermore, the relationshi p dictates that the excitation frequency is linearly dependent upon the chosen magnetic field strength. Since the Zeeman energy associated with the ESR transitions is dependent upon both app and 0 B , we cannot use either of these as the de fining trait for the re sonant condition of a specific species. Therefor e, the electronic g-factor, e e Bg , is typically the recorded feature and provides a wealth of informati on about the electron and its environment and spin lifetime, but does not provide inform ation on molecular structure like NMR. However, are sensitive to their immediate surroundings and may couple to local nuclei through the hyperfine intera ction (hfi). For instance, if an unpaired electron is coupled to an 1/2I nucleus, a splitting of the single ESR transition line into a doublet, with both peaks equally spaced from the original ESR position will be observed. Electrons may be

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8 coupled to more than one nucleus at a time and therefore complex spectra, providing a good deal of insight into the local structure of the system can be obtained. The hfi will be described in more detail in an upcoming section. Experimental Method While conventional spectros copic techniques typically employ a sweep of the irradiation frequency, allowing for transitions to be recorded in the frequency domain, due to the requirement for an external magne tic field and their pr oportional relationship, ESR may be observed through the application of a constant frequency while the magnetic field is swept through the resonance cond ition. Due to limitations in microwave electronics and the typical use of resonant microwave cavities, the second detection mechanism is preferred. A typical ESR spectrometer employs the use of both a source and a detector for the electromagnetic radiation, both of which are typically house d within a device called the bridge. The sample is placed within a micr owave cavity, which is simply a hollow metal enclosure that has been designed of specifi c dimensions so as to minimize microwave reflection within a specified frequency range when the ESR condition is not met. This allows for standing waves within the cavity to be maintained with minimal losses in microwave power. Since ESR involves magnetic interactions of the electronic spins, the cavities are designed to maximize the magne tic field component of the microwave radiation, while minimizing the electric field component that is typically responsible for non-resonant heating of the system. Use of a cavity greatly minimizes losses due to poor impedance matching and therefore provide s a maximum ESR response. However, because the cavities are only in resonan ce within a limited range of microwave frequencies, they limit the range of experiments possible.

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9 CW ESR typically utilizes a phase sensit ive detection method, which incorporates the modulation of the applied magnetic field for improved sensitivity.6 Because of this the ESR spectra are typically presented as the first derivative of the signal and are recorded not as a function of the transmission of the microwaves, but instead as a measure of the reflectance. Other possi ble methods can involve modulation of the microwave power during a magnetic field sweep through the ESR resona nce. This latter method is utilized for the research discussed within this dissertation. Overhauser Effect The Overhauser effect may be observed in a system of strongly correlated electrons that are in strong Fermi contact (FC) with the local nuclei. This is typically observed in metals and highly doped semiconductors and can lead to a shifting of the ESR condition (Overhauser shift) or of the NMR condition (Kni ght shift) or may be used to create large non-equilibrium nuclear spin polarizations. The consequence of this effect is that under strong electron spin excitations (approaching saturation), a large degree of nuclear spin polarization may be induced.4 The Overhauser effect is of central importance in this dissertation. For simplicity, we will refer to a two sp in system, consisting of one nucleus, 1 2I coupled to one electron spin, 1 2S , in an external magnetic field, 0 B , applied along the z-axis. The Hamiltonian for such a system is as follows: 00ˆˆ ˆˆˆeznzHBSAISBI (1.12)

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10 where the first and last terms pertain to th e electronic and nuclear Zeeman interactions respectively, and the middle term represents the FC hfi between the two spins, where A is the hyperfine coupling constant. Under the strong field approximation 0eBgBA, zSessentially commutes with ˆ H. This allows for the use of Smas a good quantum number. If we presume that Im may be utilized as a good quantum number as well (this is not strictly correct, however, the solution this assumption provides is the sa me), we may rewrite the Hamiltonian in eq. (1.12) as4 00ˆeBzzznzHgBSAISBI (1.13) From this Hamiltonian, the energy eigenvalues may be determined to be 0011 , 22zSeBISIn SIEmgBAmmmB mm (1.14) The correlated states and wave function may be denoted as 2,2SImm . Therefore the eigenstates are determined to be , , and . The selection rules for excitations due to electromagnetic irradiation are 1, 0SImm for ESR and 1, 0ISmm for NMR. From this we may no w determine the resonant frequencies for this coupled spin system to be 0,0 0, for electrons , for nucleieeBI NMRnSA gBm A Bm (1.15) Therefore we are left with four transitions that may be excited by the absorption of electromagnetic radiation.

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11 Figure 1-1 displays the energy level diag ram of this two spin system, assuming 0e and 0n , as is the case in GaAs. It should be mentioned that this figure is not to scale, as the electron Zeeman energy is a c ouple orders of magnitude larger than the associated nuclear Zeeman energies. Th e transition labeled ‘f orbidden’ is a double quantum transition and is referred to as the ‘f lip-flop’ transition. This involves electronnuclear spin exchange and is the cross-rela xation pathway required for the Overhauser effect. nmr nmresr esr ‘ F o rb i d d en ’ 2 4 3 1 hfien nmr nmresr esr ‘ F o rb i d d en ’ 2 4 3 1 hfien Figure 1-1: Diagram displayi ng the relative energy levels of a coupled electron and nucleus through the FC hfi, both with a spin of ½. As shown in the key at the bottom, the wave functions are labeled according to the convention 2 2 s Imm. The NMR, ESR and forb idden ‘flip-flop’ transition are displayed. The thermal equilibrium populations of the energy levels are determined by Boltzmann statistics. As shown in Figure 1-2 a, at sufficiently low temperatures a large difference between the upper and lower ener gy states associated with the electronic Zeeman splitting is observed leading to a ve ry large population difference across the ESR transition. However, due to the much smaller Zeeman splitting of the nuclei, only a small (we will assume negligible for the purpose of this exercise) polarization is attained.

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12 Continuous microwave excitation at 0, appe results in the saturation of the ESR transition as shown in Figure 1-2 (b). Due to the fluctuations in the FC hyperfine field, electron-nuclear cross-relaxati on occurs via the ‘flip-flop’ transition. Because the ESR excitation is continuous, the transition remains saturated and thus the ‘flip-flop’ process facilitates the build-up of nuclear polarization, as depicted in Figure 1-2 (c). The saturation of the ESR allows the large thermal equilibrium electron spin polarization to be converted into a large nuclear spin polarization. To describe this more completely, if we apply an electromagnetic field at 0, appe , we will induce transitions across the ESR gap at a rate of eW . For simplicity, we will refer to the different energy levels , , and as levels 1, 2, 3 and 4, respectively, and the relaxation rates between each level by the notation ABW , where A and B refer to the starting and final state involved in the relaxation pathway. The populations of each level may then be expr essed as the following partial differential equations:4 1 22111221 2 11222133222312 3 223332443334 4 334443 e ep pWpWppW t p p WpWpWpWppW t p pWpWpWpW t p pWpW t (1.16) where the sum of the four probabilities must equal one. Under conditions of complete saturation of the ESR condition, a steady-state is attained where 12 p p . This allows for us to solve for 4323 342 3432WW ppp WW . From this we can clearly understand that while

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13 saturation will change all of the populations, the only ratio that is changed is between states 1 and 2. The probabilities of a pair of levels in thermal equilibrium may be expressed as follows: exp/ABBABBBAppEEkTp (1.17) This allows us to represent the probabilities for each energy level as follows: 12 2324 23 3 2324 24 4 23241 2 2 2pp p p (1.18) From here we can now calculate the change in the nuclear polarization due to the Overhauser effect described above. For a gi ven nuclear spin stat e, we can right the expectation value of the nu clear spin polarization as 2324 23242 1 22zAz A B B IpAIA B B (1.19) If we observe the high temperature approximation 1AB AB BEE kT , we can solve the expression for z I as follows: 3242 001 24 /22 1 24z B en BEEEE I kT B AB kT (1.20) Because eq. (1.20) is dominated by the elec tronic Zeeman term, we may approximate the nuclear spin polarization to be

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14 01 24e z B B I kT (1.21) The thermal equilibrium nuclear spin polarization may be written as 212324 2123241 1 21z eq B BB I B BB (1.22) which can be written as 01 22n z eq b B I kT (1.23) Taking the ratio between the dynamically enhanced and the thermal equilibrium nuclear spin polarization, we can see that under saturation of the ESR transition we are left with the 2z e zn eqI I (1.24) Since we will be exciting both ESR transitions at the same time, the overall enhancement, in reference to thermal equilibrium, may be determined by taking the ratio of the enhanced and thermal equilibrium nuclear spin polarizations as follows: z e zn eqI I (1.25) It should also be noted that complete sa turation of the electron spin system is not necessary. Any perturbation of the electron spin under such conditions will lead to an increased nuclear spin polarizat ion. A saturation parameter s, ranging from 0 to 1, may be instituted to define the degree of ESR satura tion. This parameter may be expressed as

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15 |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e na. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e nb. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e nc. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e na. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e nb. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e nb. |+ – |–– nmr nmresr esr |+ + |–+ Fo r b i d d e nc. Figure 1-2: Pictorial representa tion of Overhauser effect. In a, the system is in thermal equilibrium. Due to the very large el ectron Zeeman splitting, a large electron spin polarization is realized. On th e contrary, the relatively small nuclear Zeeman splitting leads to minimal spin pol arization. Under the application of continuous resonant microwave irradiat ion of the electron system, the ESR transition may be saturated as shown in b. In systems where there is strong FC between the electrons a nd nuclei, the typically forbidden, or ‘flip-flop’ transitions become active pathways fo r electron spin relaxation. Thus polarization is transferred fr om the electronic to the nuclear systems. This is presented in c. 22 112 2 222 20,1121zz eq eeee z eq eappeeeeeSS BTT s S TBTT (1.26) where 1 e B refers to the strength of the applied microwave field in Tesla and 1 eT and 2 eT are the electron spin-lattice a nd spin-spin relaxation times.7 This saturation parameter may be input into eq. (1 .26) giving the following: z e zn thermI s I (1.27) Electric Quadrupole Interaction To this point, the discussion has focused on the simple 1 2 I system and has not included anything but magnetic interac tions of the spins. But, if 1/2 I , the electric quadrupole interaction may appear in the magnetic resonance spectra.

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16 Nuclei with 1 2 I have a quadrupole moment Q. For quadrupolar nuclei, there will be a non-vanishing electric quadrupole interact ion if there is an electric field gradient (EFG) at the nucleus. This EFG is creat ed by a distortion in the electric charge distribution from a spherical di stribution at the nuclear site. Such charge distributions can occur for many reasons. One such possibility is if a nuclear site, within a crystal, lacks inversion symmetry. The nuclear quadrupole interaction results from the coupling between the nuclear quadrupole moment and the EFG. The Hamiltonian describing the interaction is as follows:8 2221 ˆ 31 4212QZZZeQ HVIIIII II (1.28) where eis the charge of the electron, zzV is the value of the EFG tensor ijV along the zaxis in the principle axis system and XXYY ZZVV V ZZYYXXVVV . We have defined the z-axis as being aligned with 0 B , therefore if we assume that the quadrupole gradient has cylindrical symmetry or iented in the z-axis, we find that 0 and we may use the transformation cossinZzxIII (1.29) to switch from the principle to laboratory fram e of reference. This transformation causes the Hamiltonian presented as eq. (1.28) to be expanded to 22 2221 ˆ 3cos131 4212 33 sincossin 24 } {Qz zzeQ HIII II I IIIIIII (1.30)

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17 where xy I IiI . To first order, when QNMR 8 2 23cos1 31 4212ImINMRzzIeQ EmVmII II (1.31) Due to the quadrupole interac tion, the normally degenerate energy levels are split into 21 I levels. For a 3/2I nucleus, there will be four equally spaced energy levels, which are defined by the following expression: 2215 3cos13 122ImINMRzzIeQ EmVm (1.32) If we define 23cos1 2zz QeQV , it can be determined that the energies associated with 3/2Im will be increased by 1 2Q , while those with 1/2Im will be reduced by the same amount. This results in three single-quantum excita tion frequencies where the central transition is remains unchanged. These are presented as follows: 3/21/2 1/21/2 1/23/2 NMRQ NMR NMRQ (1.33) Hence, by measuring the angle dependence of Q , the magnitude of zzV may be inferred. The effect of a quadrupole splitting on the ener gy level diagram is shown in Figure 1-3. The electric quadrupole interaction will be revisited in Chapter 5.

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18 3 2Im 1 2Im 1 2Im 3 2Im 31 22INMRm 11 22INMRm 13 22INMRm NMRQ NMR NMRQ 3 2Im 1 2Im 1 2Im 3 2Im 31 22INMRm 11 22INMRm 13 22INMRm NMRQ NMR NMRQ Figure 1-3: Energy levels of a 3/2I system subject to a quadr upole interaction. This coupling causes an increase in energy of the 3/2Im levels by the amount 2Q , and a corresponding decrease by the same amount to the 1/2Im energy levels. This leaves the system with three distinct transition frequencies, with the position of the hi gher and lower energy transitions being symmetric about the central, whic h is unaffected to first-order.

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19 CHAPTER 2 TWO-DIMENSIONAL ELECTRON SY STEMS AND THE QUANTUM HALL EFFECT In this chapter, a discussion of twodimensional electron systems (2DES), the quantum Hall effect (QHE), and the interactio ns pertaining to the tr ansport, electronic and nuclear spin properties, will be provided. Two-Dimensional Electron Systems In 1966, A.B. Fowler et al.9 realized the first true 2D ES at the interface of the 100 surface of p-type silicon at temperatur es of 1.4 K and gate voltages around 15 V, Shubnikov-de Haas oscillations in the magnetoconductivity were . Since this discovery, research in this area has led to the fabric ation of such devices as semiconductor quantum wells, heterostructures, metal oxide field e ffect transistors (MO SFETs), quantum dots and quantum wires. The realization of a 2DES re quires a structure in which el ectrons are confined to a plane. For example, electrons may be contained to a potential well is formed by ‘sandwiching’ a thin layer of semiconducting material, such as GaAs, in between two thicker layers of a larger band gap semiconductor, such as AlxGa1-xAs (referred to as the barriers), as is depicted in Fi gure 2-1. Provided the thicknes s of the GaAs layer is less than the delocalization length of the elec tronic wave function (10-30 nm in the bulk material, defined by the de-Broglie wavele ngth), the conduction elect ron’s translational motion will be confined to a plane.

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20 AlxGa1-xAs Barrier GaAs QW Sidoping Si epoxy bond AlxGa1-xAs Barrier Sidoping support V(z) z 10-30 nm AlxGa1-xAs Barrier GaAs QW Sidoping Si epoxy bond AlxGa1-xAs Barrier Sidoping support AlxGa1-xAs Barrier GaAs QW Sidoping Si epoxy bond AlxGa1-xAs Barrier Sidoping support V(z) z 10-30 nm Figure 2-1: Schematic of a -doped, single GaAs/AlxGa1-xAs quantum well structure. In order for the conduction electrons, whic h as seen in th e potential energy diagram to the right, to be confined to a plane, the width of the quantum well must be smaller than the delocalization length of the electronic wave functione . Silicon -doping is used to provide additional conduction electrons in order to in crease the electron density. Heterostructure devices are in many respects similar to quantum wells. In a heterostructure, electrons are confined at the interface be tween two different semiconductor layers of differing band gap. A schematic and corresponding energy level diagram for such a device is presented in Figure 2-2. The drop in potential energy found at the interface serves as a tr ap for the conduction electrons, thus confining the electrons to two dimensions in a direction pa rallel to the semic onductor interface. Because the transport features associ ated with the QHE are most strongly developed at low temperatures (T<5 K), good lattice matching between the two semiconductor layers must be obtained. In the absence of such matching, thermally induced strain to the sample will diminish th e electron mobility and thus adversely affect the transport properties. In the samples st udied in this dissert ation, Al has been substituted for Ga in amounts of 1-30%, depending on the sample. The substitution of Ga sites with Al sufficiently raises the band gap of the material to meet the requirements

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21 for quantum confinement. However, GaAs and AlxGa1-xAs are closely lattice matched over a wide range of x, so no adverse effects on the transport properties are observed. GaAs 2DES Interface Sidoping Si epoxy bond AlxGa1-xAs Sidoping support V(z) z 0 E 1 E 2 E Conduction Band Electron Distribution 10-30 nm GaAs 2DES Interface Sidoping Si epoxy bond AlxGa1-xAs Sidoping support V(z) z 0 E 1 E 2 E Conduction Band Electron Distribution 10-30 nm Figure 2-2: Schematic for a GaAs/AlxGa1-xAs single heterostructure. Note the potential energy minimum observed at the interface between the GaAs and AlxGa1-xAs layers. The electron density penetrates into the AlxGa1-xAs barrier region very minimally. The three lowest electron ic subbands of the conduction band are shown schematically to the right. The Integer Quantum Hall Effect Historical Perspective The QHE pertains to the series of pl ateaus observed in the Hall resistance and conductance of a 2DES as a function of the pe rpendicularly (in refere nce to the plane of the 2DES) applied magnetic field. The term Hall effect refers to the classical counterpart discovered by Dr. Edwin Hall in 1879.10 Following the realization of the 2DES by Fowler et al.,9 it wasnÂ’t long before the first observation of Shubnikov-de Haas oscillatio ns were made within 2DES in other low dimensional conductors. Perhaps the largest contribution to the realization of the QHE, short of the discovery itself, came in the form of a theory paper written by Tsuneya Ando et al. in

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22 1975. In this paper, the authors predicte d almost every major feature involving the interactions of 2D electrons with an applied transverse field, including the oscillating magnetoconductivity xx and magnetoresistivity xx R dependence on both the Landau quantum number and the factor2/e h. In the early 1970s, silicon-MOSFETS were at the forefront of 2DES research, and utilization of the electronic gates incorporat ed in these devices allowed for the complete investigation of the c onductivity and resis tivity dependence at a cons tant magnetic field. Under such experimental design, Thomas Englert, in collaboration with Klaus von Klitzing, observed the oscillating minima in xx R and xx , as well as plateaus in the Hall resistance xy R and conductivity xy that define the QHE. It would be another 2 ½ years before von Klitzing would perform these experiments at higher magnetic fields where the plateaus in the Hall resistance became more pronounced. Upon resolution of the plateaus at hi gh magnetic fields in 1980, von Klitzing observed that the 4th plateau (which corresponds to the 4th minimum in xx R ) in the Hall resistance occurred at exactly 6450 independent of the sample investigated. He determined that this numbe r pertained to the factor 24 h e , exactly 1/4th the value expected for free electrons, and quickly determined that this was a sufficient method for measuring the fine structure constant. La ter, this observation was used as a new standard of measure for the ohm.11 For this discovery, Dr. Klaus von Klitzing was awarded the Nobel Prize in Physics in 1985.

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23 Physical Description As stated previously, application of a ma gnetic field perpendicu lar to the plane of the 2DES results in the induc tion of minima and quantized plateaus in the longitudinal and Hall (transverse) resistivities respec tively. A schematic diagram for a single quantum well patterned into a Hall bar is pr esented in Figure 2-3 and gives a clearer picture of the physical difference between th e two measurements. Due to the quantum confinement of the conduction electrons, the ki netic energy is quantized in units of 1 2NcEN (2.1) withN being the Landau quantum number, and eceB m the cyclotron frequency. The degree of broadening in the density of stat es (D.O.S.) of the conduction electrons is directly dependent upon the degr ee of electron scattering off of defects within the system. The time scale for this scattering, s cat , and the cyclotron freque ncy define the degree of D.O.S. or Landau level broa dening. If the condition 1scatc is met, no overlap between Landau Levels exists, and gaps in the D.O.S. will be observed.12 The D.O.S. for such a high mobility 2DES is shown in Figure 2-4 a. While it was mentioned earlier that the kinetic energy of the conduction electrons in a 2DES is quantized, the degeneracy of these energy levels is dependent on the perpendicular magnetic field. This is due to the quantum confinemen t of the 2DES. As stated earlier, electrons, when placed in an external magnetic field, will move in cyclotron orbits, the size of these orbits being indirectly proportional to the size of the perpendicular component of the applied magnetic field B . In classical terms, if given

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24 a 2D plane of area 0A , only a finite number of circles (orbits) of a specific radius cr can be placed within the confines of the plane w ithout overlap. This is analogous to the case of a 2DES. Due to the Pauli Exclusion Pr inciple, no two electrons may maintain the same set of quantum numbers in the same sp ace at the same time. Therefore, since the Landau quantum number is a good quantum number, no two electr ons in a specific Landau Level may have their cyclotron orbits coincide. Thus th e degeneracy of the Landau Level will increase with an increase in B and conversely, the cyclotron radius will be reduced. Vxx VxyIAC Vxx VxyIAC Figure 2-3: Illustration of a single quantum we ll sample patterned into the standard Hall bar geometry. The geometry utilizes a standard four point probe technique for measuring the magnetotransport properties . An alternating current is applied to the sample through one contact (s ource) as shown, with the opposing contact connected to an electrical gro und (drain). The l ongitudinal voltage drop (Vxx), which is proportional to the longitudinal resistance xx R , and conductivity xx , is determined through measurement along the path of the current. The voltage drop measured pe rpendicular to the current path (Vxy), is associated with the Hall resistivity xy R and conductivityxy . It was stated that the size of the cyclotron orbits is inversely proportional to B , not 0 B . Thus, if the sample is rotated with respect to 0 B , such that the 2D conduction plane is at some angle less than 90º with respect to 0 B , only the perpendicular component of the magnetic field, 0cos BB , defines the orbital radii and therefore controls the

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25 integer QHE features. Here, is defined as the angle be tween the normal direction to the 2DES and 0 B . Thus the features associated with the QHE will be shifted to higher0 B . This effect is represented in Figure 2-5. density of states delocalized states localized statesE EF EF 2/xxxxFRVIDE 2/ ,1,2,3,xyxyRVI h e (a) (b) (c)xxRxyRdensity of states delocalized states localized statesEdensity of states delocalized states localized statesE EF EF 2/xxxxFRVIDE 2/ ,1,2,3,xyxyRVI h e 2/xxxxFRVIDE 2/ ,1,2,3,xyxyRVI h e (a) (b) (c)xxRxyR Figure 2-4: (a) D.O.S. diagram for an ideal 2DES in an externally applied magnetic field. As the perpendicular compone nt of the magnetic field is increased, the Fermi energy FEdrops, passing into and out of the delocalized and localized states of the D.O.S. Fully localized states, corresponding to the FE positioned between two peaks in the D.O.S. diagram, minima and plateaus are observed in the xx R and xy R as shown in (b) and (c), respectively. Landau level broadening reduces the size of the fully localized states, therefore adversely affecting the QHE properties. The Fermi level is defined as the energy at which all lower energy states are full and all higher states are empty. Since the el ectron density in a specific sample is constant (barring adjustments via gate voltage bias or illumination of elec tron donors within the

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26 sample), FE , within a given sample, is dependent on ly on the degeneracy of the different energy levels. The degeneracy of the ener gy levels is dependent on the size of the cyclotron orbits. Leading to the following expression regard ing the filling of the Landau levels, where the filling ratio is referred to as the filling factor : 0# #/sNh electrons eBstateslevel (2.2) where s N represents the electron density. Thus, as B is increased, the size of the cyclotron orbits is reduced, a llowing more orbits to fit within the 2D space allotted and thus, the total number of energy states per le vel is increased. This is observed as a reduction in FE with increasing B . z x y 0BB 0cos BB z x y 0BB z x y 0BB 0cos BB 0cos BB Figure 2-5: The effect of sample tilting with respect to the direction of the externally applied magnetic field. Because the elec trons are confined to two dimensions, so are their cyclotron or bits. Thus as the sample is tilted, only the perpendicular component of 0 B , denoted B , determines the size of the cyclotron orbits. Since the quantum Ha ll filling factor is determined by the electron density and the degeneracy of electronic states per energy level, and this degeneracy is dependent upon the si ze of the cyclotron orbit, the filling factor may be tuned to a specific value of 0 B , through sample tilting.

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27 When the FE falls between two peaks in the D.O.S., a minimum in xx R and a corresponding plateau in thexy R are observed, as shown in Figure 2-4 (b) and (c), respectively. This condition occurs when the Fermi energy lies in the gap in between two peaks in the D.O.S. At temp eratures fulfilling the condition BckT , all electrons are confined to their current position in space as th e entirety of the 2D plane is occupied. One other property of the conduction elec trons that manifests itself in the magnetoresistance spectrum, is that of the el ectron spin. Since the spin quantum number is also a good quantum number, two electrons may occupy the same Landau level provided they are of opposing spin. This in teraction introduces another term into eq. (2.1) as follows: 1 2ncspinsplitENE (2.3) with ** 0spinspliteB E gB, where B is the Bohr magneton and ** egis the exchange enhanced electron g factor, which is made up of a Zeeman and a Coul ombic term that is associated with the electron-electron inte ractions under quantum Hall conditions. The implication of these two terms will be discussed further in Chapter 3. In order for the Zeeman effect to be realized as a gap in th e D.O.S., the degree of electron scattering and the temperature* 0BeBkTgBmust be sufficiently low, thereby narrowing the peaks in the D.O.S. In Figure 2-6, we can see a schemati c diagram illustrating the mechanism for Landau level filling described above, correspond ing to the position of the Fermi level in the D.O.S. For simplicity, an 8 electron sy stem placed within an increasing externally applied transverse magnetic fiel d will be assumed. At low 0 B , a point is reached where

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28 exactly two cyclotron orbits can fit within the 2D space availabl e. At this point, we have four times as many electrons as degenerate states in the spin-split Landau level, thus in accordance with eq. (2.2) we refer to this st ate as filling factor 4, which corresponds to both spin states of Landau level 1N being entirely filled. At a field0422* B BB, the cyclotron radii have been halved. At this field, there are exactly twice as many electrons as available stat es, so the electrons fill both spin states of the lowest Landau level, 0N. Finally, upon doubling 0 B again, we reach 1 , where the number of available states and the number of conduction el ectrons are equivalent. It should be noted that at 1 , a 100% polarized electron spin state is realized. This fact lends itself to the experiments which will be discussed in the upcoming chapters. Presented in Figure 27 are the longitudinal xx R and Hall xy R traces in sample EA124B at 1.3 K displaying the features asso ciated with the filling factors discussed above. By plotting xx R vs. 01/ B with 00 , the electron density may be extracted by the determined positions of the differe nt filling factors using eq. (2.2). Taking a step back, it is noted that when FE lies between two non-overlapping peaks in the D.O.S., a minimum in the xx R as well as inxx is observed. These two parameters are related in quantum Hall systems by the following expression13: 22xxxxxxxy R RR (2.4)

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29 =4 N=0 {N=1 { Energy 0#electrons/area degeneracy/area / s NN Filling factor: =2 B0 B0 =1 c * 0 ** 0 eBc eBgBEE EgB =4 N=0 {N=1 { Energy 0#electrons/area degeneracy/area / s NN Filling factor: =4 N=0 {N=1 { Energy 0#electrons/area degeneracy/area / s NN Filling factor: =4 N=0 {N=1 { Energy 0#electrons/area degeneracy/area / s NN Filling factor: Energy 0#electrons/area degeneracy/area / s NN Filling factor: =2 B0 =2 B0 =2 B0 B0 =1 c * 0 ** 0 eBc eBgBEE EgB Figure 2-6: Illustration of the Landau level filling based on increases in0 B . As 0 B is increased, the radius of the cyclotron or bits is reduced, as the Lorentz forces acting on the electrons increase proportionally with0 B . The filling of the Landau levels depends upon the ratio of the electron density (constant under magnetic field dependence experiments) and the degeneracy of allowed cyclotron orbits within the allowed 2D space. When an integer ratio is reached, a minimum in xx R and a plateau in xy R is observed. Note the 100% polarization of the 2DES at =1. Now, under the conditions expressed above for integer filling, it is quite obvious that if the carriers are trapped in locali zed states, they cannot contribute to the conductivity, thus the dip in xx comes as no surprise. Not so intuitive are the minima in xx R . Theory has determined that these mini ma, along with the corresponding plateaus in the xy R , are dependent on the presence of impurities and/or defects within the sample, allowing for some free electrons to contribute to the transpor t during these states of high electron localization.11 These free or itinerant electrons are scattered at a very low rate,

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30 in large part due to the localization of the other electrons. Therefore, a minimum in the longitudinal resistance is observe d. The plateaus, as stated previously, occur at integral fractions of 2h e, which is exactly the value expected for itinerant electrons. Thus with localized electrons no longer participating in transport, the Hall resistivity is entirely dependent on that of the free electrons, creating plateaus at values pertaining to 2h ie. 0123456 0 100 200 300 400 500 Rxy (k) R xx ()B(T)0 10 20 30 40 50 =1 =2 =3 =4 =6 0123456 0 100 200 300 400 500 Rxy (k) R xx ()B(T)0 10 20 30 40 50 =1 =2 =3 =4 =6 Figure 2-7: Magnetoand Hall resistance traces for sample EA124B at T=1.3 K with the sample oriented perpendicu lar to the direction of 0 B . Note the clear minima at the even filling factors 8 , and the appearance of the spin splitting of the Landau levels at 3 It should be noted here th at such transport effects, although due to distinctly different physics, are found at certain fractional intervals as well. This discovery, made by D.C. Tsui, H.L. Stormer and A.C. Gossard, was published in 198214 and resulted in these scientists being awarde d the Nobel Prize in Physics for this discovery in 1997.

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31 None of the experiments presented in this di ssertation were performed in the regime of the fractional QHE, so no further atten tion will be paid to this phenomenon. Electron-Nuclear Interactions in Quantum Hall Systems The FC hfi and its consequences on the conduction electron system, most importantly the Overhauser effect, were introduced in Chapter 1. A review of this effect, in particular how it is manifested within a 2DES, along with the cons equences this effect has upon the characteristics of the electron and nuclear spin systems, will be explored in this section. It has been stated that th e FC hfi provides a relaxation mechanism for electronic spins. The consequence of this mechanism is a polarization transfer from the electronic to the nuclear spin system following electron spin excitation.15 This results in a buildup of nuclear spin polarization, which creates an additional nuclear field n B . This field, along with 0 B , forms an effective field, eff B , which determines the position of the ESR condition, ESR B . The sign of the nuclear and electr onic gyromagnetic ratios determines the relative direction of n B with respect to 0 B . In GaAs, n B is positive and therefore is additive to 0 B , thus the effective field is defined as 0effn B BB (2.5) To this point, little mention of the other hf i, the dipole-dipole hfi, has been made. This interaction refers to the interaction between nuclei and elect rons which possess nonspherical orbital angular momentum distri butions and will also further increase the Zeeman splitting of the electronic system. Determining the dominant relaxation mechanism within GaAs/AlGaAs quantum wells and heterostructures is important for

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32 understanding the electron-nuclear interactions within such systems and how they affect observable quantities such as the nuc lear spin lattice relaxation time, 1nT. Nuclear spin-lattice relaxation measurements have long stood as an efficient intrinsic probe of the electr onic properties and electro n-nuclear couplings within conductors. In 2DES, Landau level quan tization under the application of high 0 B , manifests itself as magnetoquantum oscillations in transport and physical properties, most notably xx R , xx and magnetization. It was suggested16 and then observed17, 18 that the nuclear spin-lattice relaxation time also exhibits such oscillations. This is contrary to the typical magnetic field inde pendent Korringa relaxation4 which is typically observed in 3 dimensional conductors. The mechanism for the magnetoquantum oscillations of 1nT offers some insight into the dominant type of hfi between th e electronic and nuclear spin systems. Since the Zeeman splitting of the electronic and nuclear spin levels is directly dependent on the size of 0 B , it should be observed that di rect FC hyperfine coupling, which involves an energy transfer from the electronic to the nuclear spin, would be extremely limited, as the large difference in the Zeeman gaps are inefficient in maintaining energy conservation. Thus, as th e electronic Zeeman energy is discrete, an activation type of dependence of 1nT on 0 B is observed (similar to that of xx R ) in the regime of the QHE.19 Finite values for 1nT can only be observed if some external potentials, such as impurities19, 20 or edge states,21 are present to reduce the discreteness of the electronic Zeeman energy. Thus, in the absence or severe suppression of such

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33 potentials, (e.g. in extremely clean samples) , the dominant nuclear relaxation pathway is through the dipole-di pole interaction.22 The dipole-dipole interaction does not require conservation of the total spin and is not sensitive to the electronic Zeeman splitti ng, because the spin angular momentum of the nuclei is converted to elec tronic orbital angular momentum.21 Thus, in most samples, such as those presented here, the presence of impurities and edge states allow for the observed FC hfi to dominate. This is supporte d by the fact that th e 2DES typically has spherical or s-type orbital angular momentum, and therefore has minimal dipole moments to allow for substantial dipoledipole hyperfine coupling. The implication of this is the dominance of the FC hfi in de termining the electron-nuclear interactions within a 2DES.

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34 CHAPTER 3 MAGNETORESISTIVELY DETECTED ELECTRON SPIN RESONANCE (MDESR) IN HIGH MOBILITY/DENSITY SAMPLES There is a wealth of interesting properties to explore within 2DES. While such parameters as the enhanced electronic ma ss and activation gap measurements may be determined via cyclotron resonance, transport measurements23 and temperature dependence data, the Zeeman spin splitting and single electron g-factor* eg, may only be determined via resonance experiments as first discussed theoretically by Janak24 in 1969. Furthermore, as the spin splitting between Landau levels is due to both a Zeeman and exchange energy contribution (enhanced spin-splitting), resonant experiments alone can explore excitations across the bare spin gap, thus removing the exchange interactions and isolating those associated with the Zeeman energy.25 Finally, ESR experiments may also serve as a sensitive tool for studying inter-ba nd effects caused by inte rfacial interactions in heterostructures,26 as a method for probing the type and mechanism of electron spin excitations within 2DES and to measure th e influence of defect s on the transport and magnetic properties of electroni c and opto-elect ronic devices.27 Thus pure resonant experiments of the 2DES serve an integral role in completing our understanding of the QHE and 2DESs. While ESR is known to be a useful tool in studying electron spin dynamics within 2DES, pure absorption ESR on such systems presents many difficulties. First of all, direct observation inherently has a major problem with sensitivity, as typical 2DES densities (on the order of 1011 electrons/cm2) are insufficient for observation of

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35 absorption ESR under most conditions.26 Furthermore, issues with exclusivity persist, as all unbound electrons within the sample, includi ng those associated with shallow donor sites, impurity traps, -doping layers (paralle l conduction) along with those in the 2DES will display an ESR response. Therefore, a nother mechanism for observing the resonant response of the 2DES is required. Because of the role that th e electron spin splitting play s within the energy spectrum of the 2DES, it should be clear that pertur bations to the ground electronic spin state should be directly observable as changes to the transport properties, most notably the magnetoconductivity and resis tivity. Therefore, microwave excitations of ESR transitions within the integer QHE , especially around filling factor =1 where ~100% polarization of the 2DES is realized, should be observ able as changes to the magnetoresistance. Initial Experiments First Detection of MDESR The first observation of ESR within a 2DES was obtained via the magnetoresistively (electrically ) detected (MD) method, i.e. as a rapid spike in the magnetoresistance vs. 0 B trace by Stein, von Klitzing and Weimann in 1983.26 Notable in this, is the continued success in experime ntation within the QHE by von Klitzing, who later would win the Nobel Pri ze for his discovery of the in teger QHE. This experiment utilized a double lock-in technique, which is still utilized in most MD magnetic resonance (MDMR) experiments today. This techni que will be explored further within the experimental section of this chapter.

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36 The MD method, often referred to as electri cal detection (ED), differs slightly from this more general moniker. MD refers explic itly to the observation of excitations within the minima associated with the QHE. ED, more generally, refers to observations of excitations as a function of re sistivity or conductivity, irregardless of the presence of the QHE. Numerous observations were made in this initial report. First of all, they only observed MDESR in samples with mobilities greater than 100,000 cm2/Vs and when 0 B was set such that FE lay between two spin-split Landau levels (odd integer filling). Since the Zeeman energy alone is dependent upon 0 B , rotating the sample, allowed the authors the capability to measure th e MDESR peak at numerous values of0 B , while keeping B , and therefore the filling factor , constant. In doing so, the 0 B dependence of the MDESR frequency was determined ove r a broad range of magnetic fields, thus leading to precise determination of the bare-electronic g-factor, *eg. Finally, it was determined that the magnitude of the spin -splittings measured from magnetotransport were much larger than those measured from MDESR experiments, proving that resonant experimentation truly investigat es the Zeeman splitting only. Theory of MDESR As stated above, the spin-splittings extracted from analysis of the 0 B dependence of the Shubnikov-de Haas oscillations inco rporates both the Zeeman and exchange energies, and therefore this splitting is defined by an enhanced g-factor, **eg. This enhanced g-factor has been measured as high as **13eg , and typically is found to be on the order of 20-30 times larger than that of *eg .28 Ando and Uemura derived an

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37 expression for calculating the exchange enha ncement and its dependence on the electronelectron interactions to be25 2 *** 00' '' ',0eBeBNN NN qNVq EgBgBJqnn q (3.1) where Vqis the Fourier transform of the Coulomb interaction, ,0 q is the static dielectric function of the electron, ' Nn and ' Nnare the populations of the two spin-split levels within the thNLandau level and ' NNJqbeing a function describing the wave function overlap betwee n electrons in the thNand the thN Landau levels. With experiments being performed at odd integer f illing factors, the hi gh field approximation may be taken, thus Landau level overlap may be ignored and' NN . This leaves E to be dominated by the spin splitting and NNJq. This equation defines the energy required to excite a well separated (fully ionized) elect ron-hole pair that is necessary for a change in the resistivity to occur, and is represented by the large k limit in the dispersion function.25 The dispersion relationship for spin-w ave excitations (presumed to be the dominant type of resonant electron spin excitation at the ferromagnetic1 ) within a 2DES was calculated in dependently by both Bychov et al.29 and Kallin and Halperin30 22 01/2 22 2 /4 * 0 00 001 424kl eBkl e EkgBeI l (3.2) Here the electron-electron exch ange energy is described by, 2 00170 K 4ce E l31 at 10 T, * 0 eBgBis the single electron Zeeman energy, kis the wave vector of the excited

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38 spin-wave mode, 0/ leB is the magnetic length and 0 I is the modified Bessel function of the first kind. While a kexcitation is required for the ESR to induce a change in resistance, according to KohnÂ’s theorem,32 electromagnetic radiation such as microwaves may only couple to center of mass or 0k excitations, as Coulomb interactions between electrons do not participate in determining the Zeeman splitting energy. Therefore, while theory predicted and explained many of the magnetic resonance effects within the integer QHE, prior to the work presented in this diss ertation, no comprehensive theory for the mechanism of MDESR had been proposed. A th eoretical calculation of the dispersion relationship calculated from eq. (3.2) is shown in Figure 3-1. Note the relative positions of the bare electron Zeeman and the enha nced (ionization energy) spin gaps. 1 2 3 4 5 6 5 10 15 20 E( klc)(K)klc Ionization Gap, k=0 ESR gap k asymptote 1 2 3 4 5 6 5 10 15 20 E( klc)(K)klc Ionization Gap, k=0 ESR gap k asymptote* 0, B g B 1 2 3 4 5 6 5 10 15 20 E( klc)(K)klc Ionization Gap, k=0 ESR gap k asymptote 1 2 3 4 5 6 5 10 15 20 E( klc)(K)klc Ionization Gap, k=0 ESR gap k asymptote* 0, B g B Figure 3-1: Theoretical calculation of the disp ersion curve for a 2DES with spin-waves as the dominant electron spin excitation. Note the position of the 0k and k gaps in k-space, and their associated spin-splitting energies.

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39 Selected Further Electr on Spin Experiments Zero-field spin-splitting and field dependence of * eg In 1988, members of K. von KlitzingÂ’s res earch team, continuing on their initial MDESR experiments, reported further data determining the Zeeman spin splitting at magnetic fields of up to 14.5 T, within 1, 3, and 5 in two different samples.33 Even in the absence of 0 B , electronic motion within 2D ESs remains quantized into electronic subbands. Under the a pplication of such a field, these subbands are then split into energy levels, which in the limit of high0 B , and s N , are further split into Landau levels of opposing spin states (odd intege r QHE). To understand the energy of these subbands, it is necessary to determine the dependence of * eg on0 B . The basic expression utilized for defining the ener gy of the different subbands, * 00 *1 2neBe EENBgB m (3.3) was split into three terms, 0E the energy splitting of the electric subbands at00 B , the cyclotron energy that is defined by the filling factor index and B , and the Zeeman energy. The studies were again observed utiliz ing the double lock-in technique that was previously described. The authors determined that the xx R spectra presented the periodicity of the Shubnikov-de Haas oscillations present in the xx R vs. 0 B trace outside of the electron resonance condition. This re sponse had been mentioned as being due to non-resonant heating of the sa mple in previous literature26, 34, 35 and was stated to be related to the derivative of the magnetores istivity with respect to the temperature, xxdR dT .33

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40 The experimentation showed the spin sp litting of the Landau levels did not vary linearly with 0, B on the contrary it was defined by a quadratic polynomial function, dependent on the filling factor . Extrapolation of this to 00 B showed that while the spin splitting became smaller with decreasing magnetic field, a zero-field splitting was realized. This had been re ported previously by Uemura,36 as well as Marques and Sham.37 From examining these effects, along with the values of the zero field electric subbands, the relationship for the dependence of *eg on 0 B was determined33 to be * 0001 , 2egBNgcNB (3.4) where 00.4 g and c are sample specific constants and Nis the Landau level index (1, 2, 3Â…). The effect of tunneling was fu rther explored by Jiang and Yablonovich,38 where the dependence of * eg on either front or back gate voltage at constant 0 B was explored. They determined that when a bias was applie d such that the 2DES was pulled towards the AlGaAs side of the heterostructures used, the g-factor decreased in value, while the reverse was observed (*eg increased, approaching the value for bulk GaAs), if the opposing bias was applied. This is consis tent with the reported g-factors for AlxGa1-xAs (x=0.3) of eg =+0.4 and for bulk GaAs, where eg =-0.44.3 Evidence for long-range spin excitations Much research has been devoted to obse rving and determining the type of spin excitations present within the pseudo-ferroma gnetic electronic ground state of a 2DES at odd integer filling factors. Because of the strong electron-electron in teractions within a 2DES within the regime of the integer QHE, long-range collective spin excitations such as spin-waves or Skyrmions are presumed to be the dominant spin excitations as opposed

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41 to single spin flips. The firs t experimental investigation into the type of thermally excited spin excitations was reported by Usher et al.31 They stated that the total electronic energy spectrum * 00,cSeBckmmgBEkB (3.5) of a 2DES may be described by three ma in contributions, the cyclotron energy cm , the exchange interaction energy 0,c E kB and the Zeeman spin splitting * 0 SeBmgB. Usher et al., stated that the elementary electron excitations within quantum Hall systems should be magnetoplasmons, with1m , and spin-waves with 0m and1sm , due to the large exchange energies associated with these systems. By measuring the magnetic field dependence of the exchange energy, the magnitude of the electron-electron interactions can be determined providing further evidence of co llective excitations.31 Within a limited temperature range, the enhanced spin-splitting energy cZ E EE is activated in nature. Th erefore, at high temperatures BkTE , thermal excitations over the activation ba rrier may readily occur, while at low temperatures, the mechanism for conduction is thought to change to one of a “hopping” nature. Since ionization of the electronic excitations must occur for a change in the resistivity to be observed, excitations over the thermal activation barrier must be induced. The magnitude of E may be determined by measuri ng the temperature dependence of the minima in xx R associated with the filling factor of interest. This measurement probes the large2 0/krl , limit of the dispersion relationship or the ionization energy. By measuring the 0 B dependence of the exchange en ergy through the ut ilization of an

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42 external gate, the enhanced g-factor, **egcan also be determined. In the samples discussed in these experiments, **7.3eg which is approximately 20 times that of * eg.31 The authors presumed that the excha nge energy was dominated by the Coulomb energy. Therefore, in measuring 0 E B , it was assumed a1/2 0 B dependence would be observed, due to the dominance of the Coul omb energy. Oddly, the dependence was found to be directly proportional. This was pr oposed to be due in part to residual Landau level overlap from disorder within the system. Thus, as 0 B was increased the exchange energy played an even more important role in the electron-electron interactions. It was theorized that at the higher fields associated with the =1 filling factor, that spin-waves or other long-range spin excitations would be lower in energy than single spin flips. Evidence of thermally excited Skyrmionic spin excitations Sondhi, et al., theorized that even in the absence of the Zeeman interaction, odd integer filling factor s would persist. This would o ccur due to the development of spontaneous ferromagnetic ordering of the conduction electrons as 0 KT, based on the large exchange interactions present.39 Spin excitations within such systems have been described as long-range sp in-waves or Skyrmions.39, 40 A pictorial representation of the three main spin excitations discussed is presen ted in Figure 3.2. Skyrmions, described as “a large smooth distortion of the spin field, in which many spins are flipped”41 (shown in Figure 3.2c), cost an increased amount of Zeeman energy due to the large number of spins involved in such excitations. However, when coulomb interactions between electrons are significant, such as in ferromagne tically ordered systems, the flipping of one spin costs an increased amount of excha nge energy due to the increased coulomb repulsion between the flipped spin and its neighbors, therefore, as one spin is flipped, the

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43 neighboring electrons may collapse upon the pos ition in space occupi ed by the flipped spin, which no longer maintains an identical set of quantum numbers; thus, the Pauli exclusion principle no longer appl ies. As the electrons move closer to one another an increase in the electron-electron repulsion is observed. Thus, the dominant type of spin excitation, within quantum Hall systems, is dependent upon the competition between these two energies. The total change in spinS, associated with a single Skyrmion excitation, is also dependent upon the competition between the ex change and the Zeeman energies. In contrast, the change in spin associated with a single spin-wave excitation, like that of a single spin flip is one. Thus, as the ratio between the Zeeman and the exchange energies, denoted as g, is decreased, the lowest excited spin state shifts from that of a single spin flip or spin-wave, to that of a Skyrmion (as 0ZE ). As previously discussed, the activation ga p is the sum of the exchange and Zeeman contributions, * 0ceBEEBSgB (3.6) The magnetic field couples only to the Zeeman contribution of E , while cE is theoretically dependent upon B . Therefore, by measuring o E B , while maintaining B constant, the value 2 0 eBEE Sg B , may be extracted. By determining the value for * eg, the slope of the plot of E vs. 0 B may be used to determine the value of S. Schmeller et al.,41 after performing the above mentioned experiments in four modulation doped GaAs/AlGaAs heterostructur es of varying mob ilities and electron

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44 densities, determined that at =1 and =3 and 5. At all three filling factors, exchange interactions should dominate, although the ma gnitude of the spin-splitting will diminish as the filling factor increases. As stated previously, single spin flips or spin-wave excitations consist of 1S, so while this holds true at 3 and 5, a large departure from this value is observed at =1 providing strong evidence for thermally excited Skyrmions within these systems. Th e total spin of a Skyrmion as stated is dependent upon g, with large values leading to lower values of S, while as 0g , 1 22 E , or exactly one half the energy required to flip a single spin. Spin-Wave/Magnon long range interactions =1 S Single Spin Flip Itinerant Electrons =1S Skyrmion 2D long Range interactions >1SRepresentation by Steve Girvin, U. Indiana Uniform k =0 mode(a) (b) (c) Spin-Wave/Magnon long range interactions =1 S Single Spin Flip Itinerant Electrons =1S Skyrmion 2D long Range interactions >1SRepresentation by Steve Girvin, U. Indiana Uniform k =0 mode Spin-Wave/Magnon long range interactions =1 S Single Spin Flip Itinerant Electrons =1S Single Spin Flip Itinerant Electrons =1S Skyrmion 2D long Range interactions >1SRepresentation by Steve Girvin, U. Indiana Skyrmion 2D long Range interactions >1SSkyrmion 2D long Range interactions >1SRepresentation by Steve Girvin, U. Indiana Uniform k =0 mode(a) (b) (c) Figure 3-2: Pictorial representa tion of three main electron sp in excitations discussed in 2DES in the regime of =1 in the integer QHE. A simple spin flip is depicted in a. While this type of excitation (a long with the spin-waves as depicted in b), require the lowest Zeeman energy * 0eBSgB, due to the singular change in the spin Sper excitation, also will incorporat e the largest increase in energy due to the large exchange interaction th at will be observed. Spin-waves and Skyrmions (shown in c), both utiliz e the tilting of multiple spins simultaneously to incorporate the overall change in Swith minimal costs in exchange energy. Skyrmions, typically being associated with large changes in Sper excitation, will inherently cost the most Zeeman energy. The competition between the Zeeman and exch ange energies of the 2D electronic spin system defines the brand of spin excitation.

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45 Evidence for single spin excitations in MDESR While many theoretical and experimental reports on spin excitations of a 2DES in the regime of filling factor 1 have focused on Skyrmion excitations, most have focused on experiments where the Skyrmions are not ex cited directly, but ra ther through indirect methods such as optically pumped NMR,42 electron-hole recombination luminescence,43 and thermal activation of transport41 as discussed previously. Experiments of this type do not isolate the Zeeman influence on the spin excitation and allow the very large exchange energy to dominate. In an effort to unde rstand the mechanism of MDESR, where as stated previously, the excitation cannot inco rporate this large exchange energy due to KohnÂ’s theorem, it is important to determine the type of spin excitations present under true electron spin excitati on. In a very unique and eloquent experiment, Vitkalov et al.,44 were successful in doing just this. As stated previously, the relevant parameter governing th e type of excitations is the number of spin flips per excitation,S. Sis strictly dependent upon the ratio between the Zeeman and exchange energies44 * 0 2 0/eBgB g el (3.7) which according to Hartree-Fock calculations in corporating real sample effects, such as Landau level mixing and finite thickness interactions, will lead to 1 S if 0.025g .45 This value has not been corroborated by experi ments however, and varied degrees of spin polarization at =141-43 and a factor of two differen ce between these theoretical and numerous experimental resu lts for the activation gap31 lead to questions of the actual

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46 value for gwhere this transition should be observe d. Thus, experimental measurements to determine Sunder MDESR conditions are necessary. The experiment at hand utilized the eff ect of the enhanced nuclear field upon the MDESR condition in determining the total sp in change per excitation. At thermal equilibrium, these effects may be incorporated into eq. (3.6), so we are left with * 0 eq ceBn E EBSgBB (3.8) where ,eqeq nizi i B AI (3.9) is the local nuclear field acting upon the 2DES under equilibrium conditions, iA is the hyperfine coupling constant and ,eq zi I is the magnitude of the nuclear spin polarization in the z-direction for nucleus i. This variation of the activation energy equivalence expression was first reported by Dobers et al. in 1988.46 Ordinarily, eq n B may be neglected as the thermal equilibrium nuclear sp in polarization is quite small, even at T=1 K, but if significant enhancement of z I is achieved, the effect of n B on the electron spin system cannot be ignored. If one assumes that the effect of n B is uniform upon the 2DES, a good approximation due to the high de gree of symmetry associated with the cubic GaAs lattice, the summation in eq. (3.9) may be replaced by a weighted average over all nuclear spin isotopes within the vicinity of the electrons, iz i I I . Leading to the following expression: ,eqeq niziz i B AIAI (3.10)

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47 As in most studies in the literature, th e samples utilized in these experiments exhibited activated transport with in the temperature region studied, 2.5 4.2 KT. Therefore, as all experiments were performed at 2.5 K T , the magnetoconductivity may be described by the Arrenhius expression: 0exp/2eq xxBeq E kT (3.11) If eq.s (3.8) and (3.11) are combined, we can derive an expression for the dependence on the normalized change in the magnetoconductivity on the change in the magnitude of the effective nuclear field from thermal equilibrium and the sample temperature as shown: *2eq eBn xx eq xxBeqSgB kT (3.12) where eqeqDNP xxxxxx and eqDNP nESRESRBBB . Here eq xx and eq ESR B are the thermal equilibrium magnetoconductivity and MDESR position in Tesla, while D NP xxand D NP ESRB are the dynamic nuclear pol arization (DNP) enhanced magnetoconductivity and Overhauser shifted MDESR position in Tesla re spectively. Therefore, if the normalized change in magnetoconductivity under DNP conditions eq xx eq xx is plotted against *2eBn BeqgB kT at varying values forn B , the slope of the fit will be equal to S.44 The dependence should be linear, provided xx is not dramatically affected by spin-exchange scattering effects due to the hf i, where under such conditions, 2 2 /xxzeq I T.47

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48 However, at temperatures like those studied in this report, the spin-exchange scattering mechanism is relatively weak, and did not affect the experiment. The experimental method discussed offe red many advantages over previous experiments other than the determination of Sunder electron spin excitation conditions. In the test, both 0 B and B were held constant, removing any Landau level mixing or changes in the electric subband energy th at can occur under such variations. Furthermore, within any one experimental r un, the temperature was not changed, this removing any effects related to an unknown temperature dependence of the exchange energy. In the experiments they performed, z I was enhanced through the creation of DNP via a slow down-field sweep in magnetic field 0dB dt ~0.5 mT/s that is begun at0eq ESR B B . The slow sweep through eq ESR B , allows for a high rate of cross-polarization of the nuclear spin system through the Overha user effect, as discussed in Chapter 1, creating an additional nuclear field which acts on the electron system. This field n B causes an Overhauser shift of the MDESR c ondition to lower magnetic field (due to the negative value for * eg and positive values for all threen ). The position of the Overhauser-shifted MDESR is referred to as D NP ESRB . By stopping at different values of D NP ESRB and measuring D NP xx, the plot eq xx eq xx vs. *2eBn BeqgB kT discussed above can be created.44 The results yielded a slope of 1.1 0.15 S at 1 . Therefore, under direct excitation of the electron spin system, the ex citations may be desc ribed as single spin

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49 flips or single spin waves, and do not pe rtain to Skyrmions. This is supported by theoretical predictions which state that in the absence of the spin-orbit interaction, as in the conduction band of GaAs,48, 49 the electron-nuclear hfi acts only on the electronic spin state (Zeeman spin splitting) as the Hamiltonian for the interaction between the electron spin and the static or transverse microwave field commutes with th e orbital Hamiltonian, which includes the electron-electron interact ion. Microwave scr eening effects, which may remove this commutation relationship, may be neglected in this instance as electrons interacting within this sample only do so within a length scale associated with the magnetic length (015 nm l at05.5 T B ), and thus no long range effects of the microwave screening are created. Therefore, it is valid to consider MDESR as a pure 1Sphenomenon.44 Experimental Apparatus Magnet and cryostat Unless noted, all experiments were performe d using an Oxford variable field, high homogeneity superconducting magnet which is capable of magnetic fields up to 10 T. Sweeping of the magnetic field is made possible by a non-superconducting shunt coupled with continuously connected copper leads from an IPS120-10 superconducting magnet power supply. The magnetic field sweepi ng rate is dependent upon the magnetic field range being swept in. At 0 B < 7.165 T, the field may be swep t at rates up to 0.492 T/min. Above this field, but below 9.4 T, the sweep ra te must be reduced to 0.223 T/min, and at 0 B > 9.4 T, the sweep rate cannot exceed 0.134 T/min.

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50 A sorb pumped, liquid 3He cryostat is placed within the magnet for experimentation at low temperatures. Under normal temp erature operation, a SOGEVAC SV65, single stage, oil-sealed rotary vane pump is used to siphon 4He gas from the dewar housing the superconducting coil through the 1 K pot. A solenoid controlle d needle valve regulates the flow of 4He gas into the pot. This pumping cool s the gas to approximately 1.4 K. A sorb is placed above the 1 K pot and is meant to absorb 3He gas. A heater, in thermal contact with the sorb, is used to increase the temperature to approximately 30 K. P-I-D control of the heater curre nt is supplied by the ITC503 temperature controller. At this temperature, the sorb is extremely inefficient at absorbing 3He gas and thus all gas is released. Upon coming into contact with the 1 K pot, the 3He gas is condensed and collects in the sample space. This proce dure allows for continuous use at a sample temperature of about 1.5 K. Lower temperatur es may be attained, after an initial period spent condensing 3He, by removing the current supply to the sorb heater. As the sorb cools down, it will begin to abso rb the gas from the boiling liquid 3He and the sample temperature will drop. This will allow fo r temperatures of approximately 300-400 mK to be obtained for around 12 hours depending on the heat load of the probe. Another heater is placed at the bottom of the probe, as shown in Figure 3-3. By application of a small current to this heater , sample temperatures up to 30-40 K may be attained. As with the sorb heater, temperat ure control is achieved through the use of the temperature controller. Sample mounting Presented in Figure 3-4 is a schematic fo r the apparatus used in the experiments discussed in this chapter. The sample is mounted via an 8 pin male-type header connector, which is fitted into its mate th at is mounted on a goni ometric rotation stage

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51 capable of 0-90° rotation of the sample in reference to the 0 B . For the purpose of carrier excitation from silicon -doped layers within the sample s, a red LED may be placed in close proximity to the sample, but requires th e utilization of one of the 8 electrical connections from the header connections. An optical fiber, attached to a single Coherent FAP laser (~795 nm), is also placed within cl ose proximity to the sample for optically pumped experiments and to serve as a backup for carrier illumination. rotation stage w antenna High Freq coax rotator shaft region immersed In liquid helium-3 heater & sensor QW sample Goniometric Rotation stage Not pictured: RF coaxial feedthrough, RF coil, LED, optical fiber rotation stage w antenna High Freq coax rotator shaft region immersed In liquid helium-3 heater & sensor QW sample Goniometric Rotation stage Not pictured: RF coaxial feedthrough, RF coil, LED, optical fiber Figure 3-3: Photo of probe us ed for MDESR measurements. Shown in the picture is the rotator shaft used for cha nging sample orientation via the goniometric rotation stage. The heater in conjunction with the liquid 3He cryostat (not shown), allow for variable temperature measurements. Transport measurements xx R via single lock-in detection MDESR is observed as microwave i nduced peaks, superimposed upon the magnetoresistance trace, similar to the results reported by Stein et al.26 As mentioned in Chapter 2 the samples are patterned into a Hall bar type geometry for standard four point probe measurements. As shown in Figure 23, an AC current (0.1-5V, 100-537 Hz) from

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52 Lock-in #2 is passed through an ex ternal resistor such that externalsampleRR(typically 1-10 MW), thus attaining a pseudo DC current through the samp le. Resistance changes are measured by a standard four point probe tech nique where the voltage drop is evaluated between two points along the current channel as shown, with the signal being input to Lock-in #2 where the difference between the tw o voltage readings is determined. During standard single lock-in detection, i.e. transport measurements of xx R , the typical time constant used on the lock-in is 300 ms. Differential resistance measurements xx R via double lock-in detection Microwaves are created via an yttrium-irongarnet (YIG) oscillator (Micro Lambda model MLOS 1392PA) with a tunable output of 7-18.33 GHz (determined by application of a DC bias to the YIG in the range of approximately -3 to 10.5 VDC). A doubling amplifier is utilized to increase the fre quency range to 14-36.65 GHz needed for the majority of experiments. An absorptive p-i-n diode modulator (G eneral Microwave model D1959) is used for th e square-wave modulation of the microwave field at a frequency mod315 Hz, f for high sensitivity xx R measurements. The modulation is controlled by the 0 (no attenua tion to microwaves) to 5 V (f ull attenuation) TTL output from Lock-in #1. A 30 MHz synthesized function generator (Stanford Research Instruments model DS345) may be used in pl ace of the TTL output to vary the degree of attenuation and thus adjust the microwave power. A s econd doubling amplifier (Spacek Labs Inc. model AV-2XW), fitted to a waveguide to 3.5 mm adapter (Anritsu model 35WR19KF) fitted at the output, may be used to increase the frequency range to 28-72 GHz. All microwave compone nts are connected together using 3.5-mm coaxial connectors and adapters (Anrits u K-connectors) and are carr ied to the sample via a 50 ,

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53 3 mm-o.d. semi rigid coax terminated approxi mately 5 mm above the sample with either a dual arm or simple loop antenna. 10-18 GHz 20-36 GHz 40-72GHz,12Hz B0 In 12Hz Ref out Output Lockin#1 20-36 GHz,12 Hz. YIG Oscillator Doubling amp In 1V 330Hz Out Output Lockin#2 modulator Doubling amp coax 1-10M Heliox3He Probe Hall bar sample on Goniometer Rotation stage 10-18 GHz 20-36 GHz 40-72GHz,12Hz B0 In 12Hz Ref out Output Lockin#1 20-36 GHz,12 Hz. YIG Oscillator Doubling amp In 1V 330Hz Out Output Lockin#2 modulator Doubling amp coax 1-10M Heliox3He Probe Hall bar sample on Goniometer Rotation stage Figure 3-4: Schematic diagra m of experimental setup fo r single and double lock-in measurement techniques used for magnetoresistance xx R and differential magnetoresistance xx R measurements. The apparatus enclosed within the dotted box on the left are external and t hose on the right inte rnal to the liquid 3He cryostat which is housed within a variable field high homogeneity Oxford magnet capable of magnetic fields up to 10 T. A Labview based program is utilized to collect all data. The output from Lock-in #2, is a signal proportional to xx R and contains an oscillatory component at mod f due to the effect of the mi crowave field upon the signal, which is proportional to the differential resistance, xx R . The time constant of Lock-in #2 is typically set to 3 ms while the time cons tant of Lock-in #1 is set to either 300 ms or 1 s. Samples The samples utilized for the experiments di scussed within the next three chapters were all grown by Drs. Jerry Simmons and John Reno of Sandia Na tional Laboratory in

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54 Albuquerque, New Mexico. The signals were observed in both multiple AlxGa1-xAs/GaAs quantum well and single heterostructure samp les. The samples exhibit mobilities between =4.4X105 and 5.3X106 cm2/Vs and electron densities per layer between s N = (7-25)X1010 cm-2. All samples were grown by mol ecular beam epitaxy. As stated before, the samples are patterned into a Hall bar geometry with 0.2 mm channel widths and voltage probe separations on the orde r of 1.5 mm. All samples exhibited qualitatively the same behavior in the experime nts to be discussed. Data presented were performed using samples EA124 and EA124B ( 21 X 300Å wide GaAs wells, x=0.1), two samples from the same growth wafer, but patterned separately, and EA129 (single layer heterostructure sample, x=0.3). Both qua ntum well samples exhibit mobilities of =4.4X105 cm2/Vs and electron densities of s N =6.9X1010 cm-2, while the single heterostructure displays a mobility =5.3X106 cm2/Vs and s N =1.4X1011 cm-2. Experimental Results Magnetoresitively Detected ESR (MDESR) As alluded to in the previous sections, th e effect of the microwave field, including the ESR, typically may be observed in either single xx R or double xx R lock-in measurements. Figure 35 displays a typical ,xx R and corresponding ,xx R traces acquired at 1.5 K, with the former taken unde r the application of microwaves at 32 GHz during a magnetic field sweep rate of 0dB dt=0.448 T/min. The sample was tilted such that =60º. This angle allows for the =1 minimum to be brought to a field of 0 B =5.7 T. Notice the sharp peak associ ated with the ESR excitation found at 0 B =5.47 T. The inset shows an expanded view of the MDESR feature. The dependence of the

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55 microwave frequency on0 B , gives the bare-electronic g-factors of * eg= 0.418 and * eg=0.396 in sample EA124 and EA124B, respect ively. The microwave frequency vs. 0 B dependence for EA124B is presented in Fi gure 3-6. From sweep rate dependence data, which will be discussed in the next chapte r, the sign is determined to be negative. This value is consistent with those reported in the literature, and is typical for the samples being studied. The values for *gin selected samples to be di scussed within the text of this dissertation, along with other character istics, are presented in Table 3-1. 0 100 200 300 Rxx(A.U.) B0 0 1 2 3 4 5 6 Rxx( ) 0 1 2 3 4 5 6 B04 2 3 1 1 2 3 4 MDESR 0 100 200 300 Rxx(A.U.) B0 0 1 2 3 4 5 6 Rxx(A.U.) B0 0 1 2 3 4 5 6 Rxx( ) 0 1 2 3 4 5 6 B0 Rxx( ) Rxx( ) 0 1 2 3 4 5 6 B04 2 3 1 1 2 3 4 MDESR Figure 3-5: (a) Typical xx R vs. 0 B trace for sample EA124 at T=1.5 K. The =1 minimum is found at 0 B =5.7 T. The sample is tilted such that the plane of the 2DES is at an angle of =60 in reference to0 B . The numbers correspond to the relative filling factors. (b) Typical xx R vs. 0 B response. A sharp peak is observed at 5.47 T, which is associat ed with MDESR. The inset shows the MDESR feature on an expanded scale. The microwave frequency is 32 GHz in (b). The MDESR signals obtained at sweep rates equal to 0.448 and 0.492 T/min in samples EA124 and EA124B respectively, displa y no dependence in either the amplitude or line width on the direction of the magnetic fi eld sweep. This is evidence of a lack of

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56 DNP interactions under these experimental c onditions. For slower sweep rates, as is consistent with prior reports,16, 44, 50 DNP and Overhauser shifts and broadening (narrowing) were significant in spectra observed during a down-sweep (up-sweep) through the ESR condition. All spectra within this section were taken at 0.448 T/min, unless otherwise noted in the text to avoid the influence of DNP on the MDESR lineshape and resonant position. 4.24.44.64.85.05.25.45.65.86.06.26.46.6 24 26 28 30 32 34 36 38 ESR(GHz)=(g*B/h)B0 ESR + 00=1.62279 GHz +/0.06582 g*=0.3966 +/0.00084ESR (GHz)B0 ESR (T) Figure 3-6: g-factor dete rmination through frequency dependence of the MDESR resonant position in Tesla. Data was taken at two different sample tilt angles (solid, blue squares 57.33° and open, re d circles 64.34º) in sample EA124B at T=1.5 K. The linear fit to both data sets leads to a determin ed bare electron gfactor of * eg =-0.3966 +/0.00084. A similar data set for EA124 determined the g-factor to be -0.418. Temperature Dependence of MDESR As evidenced in the preceding sections of this chapter, the ability to determine a comprehensive theory for spin excitations w ithin the regime of the integer QHE is of

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57 intense interest both experimentally and th eoretically. Utilization of MDESR as a method for discriminating between compe ting theories of magnetic ordering and excitations of a 2DES in this regime was the primary motivation behind the temperature dependence measurements to be disc ussed within this section. MDESR characteristic features As stated previously, the non -resonant features of the xx R spectra are observed to be an oscillating function of0 B , whose main features co rrelate to those of the magnetoresistance trace. Depicted in Figur e 3-7 are typical MDES R spectra taken at different temperatures within the range 0.44.3 KT and with 05.375 TESRB, while the =1 minimum may be found at approximately 5.7 T. There are two things of importance to take note of. First of all, a definitive temperature dependent maximum in the MDESR amplitude is observed. Secondly, this maximum not only corresponds to the maximum in the MDESR, but also to a ma ximum in the non-resonant response in the area directly around the MDESR peak. This last observation is in dire ct contrast to the relationship between the MD ESR and non-resonant responses reported by Meisels et al.51 where it was observed that a disappearance in MDESR is observed as temperature is decreased below that of the maximum respons e, but this temperat ure decrease is also accompanied by a continued increase in the si ze of the non-resonant background. This observation was interpreted as elec tron spin depolarization of the =1 quantum Hall state at low temperatures due to electron -electron interaction effects. In order to make any quantitative measur ements of the MDESR features such as line width and amplitude, both of which are needed for understanding the temperature dependence of the MDESR, the non-resonant component must be removed. This

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58 removal was performed through the following procedure. First, the part of the xx R spectra in the region of the MDESR peak wa s deleted. Several sample points were inserted into this missing re gion to facilitate the interpol ation of the background signal. A standard 6th order polynomial was fitted to the background and then the fit was subtracted from the original raw data, leavi ng only the component of the spectra due to the resonant microwave absorption associated with the MDESR response. The observed features of the MDESR, were found to be rela tively insensitive to the placement of the inserted points. A standard Gaussian f it was then used to extract the necessary parameters associated with th e peak. Shown in Figure 3-8 is the temperature dependence of the MDESR amplitude (a) and line width, taken as the full-width at half-maximum (FWHM) (b) after removal of the non-resonant background for the raw data presented in Figure 3-7. From the dependence of the MDESR peak integral on temperature, it is observed that the maximum occurs at appr oximately 2.2 K, with sharp decreases observed at higher and lower temperatures. Unlike the amplitude, the FWHM shows no apparent temperature dependence. Therefore, assuming the line width is proportiona l to the transverse relaxation time of the electronic spin system2eT , as in conventional ESR detecti on, then it may be stated that within the temperatur e range studied that 2eT remains relatively constant. Taking the observed FWHM of approximately 1/2 B =18.5 mT (11.3 mT) in EA124 (EA124B) we can estimate the value for 2eT ~4 ns (5 ns).

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59 Table 3-1: Tabulated results from the seven samples discussed within the text of this dissertation. Note the wide range in electron densities, mobilities and *eg . The similarities in MDESR response despit e the wide variety in sample design speaks to the robustness of th is method for experimentation Sample # Description Mobility Density s N *eg EA124 Multiple QW X=0.1 21 X 300 Ã… Wells 4.4 X 105 cm2/Vs 6.9 X 1010 cm-2 -0.418 EA124B Same as EA124 4.4 X 105 cm2/Vs 6.9 X 1010 cm-2 -0.397 EA129 Single Heterostructure X=0.3 5.3 X 106 cm2/Vs 1.4 X 1011 cm-2 -0.41 HM0455 Single 300 Ã… QW X=.11 2.37 X 106 cm2/Vs 9.0 X 1010 cm-2 -0.42 HM0459 Single 300 Ã… QW X=.05 1.41 X 106 cm2/Vs 4.5 X 1010 cm-2 -0.42 HM0461 Single 300 Ã… QW X=.03 1.41 X 105 cm2/Vs 1.77 X 1010 cm-2 -0.44 AG662 Single Parabolic QW 1.5 X 106 cm2/Vs 1.2 X 1010 cm-2 -.399, 090 -.427, 00 Figure 3-7: Temperature dependence of MDESR at 060rotation in sample EA124 taken at a sweep rate of 0.448 T/min. Note the maximum in both MDESR and non-resonant background observed at a te mperature of 2.2 K. This data is taken does not correspond to the dependence at 1 (5.7T).

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60 T(K) T(K) Linewidth (mT) Rxx(arb. Units) a) b) T(K) T(K) Linewidth (mT) Rxx(arb. Units) a) b) Figure 3-8: Temperature dependence of (a) MDESR peak amplitude and (b) MDESR line width at 0dB dt =448 mT/min, 05.375 TESRBand =1 at 5.7 T 060. Both data sets were taken from Gaussian fits to the data after removal of the nonresonant background via the procedure listed in the text. As can be seen, there is a true maximum in the amplitude found at 2.2 K, while the line width, within this range of temperatures, appears relatively independent of temperature. Discussion of MDESR excitation and its temperature dependence In explaining the temperature dependen ce, a mechanism for the observation of MDESR was proposed. In the simple D.O.S. model for a 2DES at =1, a minimum in the magnetoresistivity is observed when FE is located between the spin up and spin down states of the lowest Landau level (N=0). The energy difference between these two spin states (filling factors one and two), is typically measured via activated transport experimentation as discussed by Schmeller et al.41 As the magnetoresistivity within the temperature region measured in this experiment is within the activat ed region of electron transport, the Arrhenius relationship between xx and Tintroduced in eq. (3.11) may be modified through the use eq. (2.4 ) to give us the following: 0exp 2xx BeqE RR kT (3.13)

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61 and therefore E may be extracted from the slope of the lnxx R vs. 1/T plot. The temperature dependence of xx R at 0 B =5.7T and =60 û , along with the corresponding Arrhenius plot, in sample EA124 is shown in Figure 3-9. From the plot, an enhanced spin-splitting of /7 KBEk was obtained. This is in comparison to the single-electron spin Zeeman interaction * 0/1.6 KeBBgBk at this field. The close proximity of the latter energy gap to that of the maximum in the temperature dependence of the MDESR should be noted. The corresponding gaps in EA124B are /9 KBEk and * 0/1.5 K.eBBgBk As discussed previously, the energy gap a ssociated with MDESR transitions is not the activation gap, but instead is associated with the smaller, bare-electron spin-splitting energy * 0/eBBgBk due to Larmor’s and Kohn’s theorems regarding the freedom of electron spin Larmor frequency from electr on correlation effects and the inability of electromagnetic radiation (such as microwaves) to couple to 0 k spin-wave modes. Instead, the activated transport measurements probe the large kspin-wave excitations which produce charge carriers via ionizat ion processes that correspond to a wellseparated electron-hole pair. At =1 in the samples discussed here, the Coulomb term in the klimit is much larger than the Zeeman term. While * 0/1.6 KeBBgBk at 05.7 T B the corresponding theoretical Coul omb interaction energy is /162 KBEkk . While the experimentally meas ured activation gap of /7 KBEk is significantly smaller than the theoretical gap of an ideal 2DES, this may possibly be explained by

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62 incorporation of real-world e ffects not taken into account in the ideal model, such as finite z extent, sample disorder, and well to well variations in el ectron density that may be found in multiple quantum well samples su ch as those taken from the EA124 wafer.31 Despite the inconsistencies between the theoretical and experimentally determined activation gaps, our gap is still significantly larger than that of the bare-electron Zeeman spin-splitting. Therefore, excitation of the system at the smaller energy associated with the Zeeman gap cannot be directly attributed to changes in the magnetoresistance which require the larger energy k ionization excitations. To bridge the gap between the spin excitations and thei r corresponding changes in xx R , electron correlation effects must be incorporated within our proposed mechanism. 5.05.56.06.5 0 100 200 300 400 Rxx ()B (T) 0.00.40.81.21.6 -2 0 2 E=7 Kln(Rxx)1/T (K-1)(a) (b)5.05.56.06.5 0 100 200 300 400 Rxx ()B (T) 0.00.40.81.21.6 -2 0 2 E=7 Kln(Rxx)1/T (K-1)(a) (b) Figure 3-9: (a) Temperature depe ndence of the magnetoresistance0 xx R B around =1 filling factor. (b) Arrhenius plot of lnxx R vs. 1/ T. These data provide a measurement of the activation gap, /7 KBEk . The scatter in the data points around 0.6 to 0.4 T is assumed to be due to errors in the temperature calibration at the high and low ends of the two temperature sensors, respectively.

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63 In the literature, the elementary neutral spin excitations have been analyzed as spinwaves (i.e. magnons), which are also referred to as spin excitons.29, 52 As stated previously, excitation of a spin-wave mode, co rresponds to exactly on e electron spin flip distributed over many neighbori ng electron spins. Not mentio ned is the feature of these excitations to conserve wave vectork . The proposal put forth from our investigation is depicted as a three step process. First, the 2DES absorbs resonant photons of microwave energy. This absorption causes an increase in the internal energy of the electron system. This heating enables the excitation of higher energy kmodes through thermal processes, thus eventually ionizing some spin-wave excitations, resulting in char ge carriers, which are observed by change in the magnetoresistance, xx R . Our model, in being based on the idea of spin-waves as the elementary excitations of the 2D electron sp in system, requires the incorporation of the dispersion relationship for such excitations introduced as eq . (3.2). For computational purposes, this equation may be simplified to 1/2 00/0.27681x BExkBBeIx (3.14) where 22/4cxkl and /cleB . At thermal equilibrium, the average number of spin-waves of mode k excited may be described by the Planck distribution 1 exp/1k BN EkkT (3.15) Taking the number of excited spin-waves of mode ,k per unit area, to be equivalent with the number of spins per unit area that contri bute to the 2DES magnetization, the number of spin-waves of mode kmay be defined as

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64 2 2 2 0 001111 exp/exp/1 2k k BBNdkdx l EkkTExkT (3.16) Since our model describes a resonant heating of the system, the temperature change of the 2DES is due only to absorp tion of microwave energy into the 0kspin-wave modes. To explain the observed changes in xx R under such excitati ons, a relationship between the uniform mode 0 k excitations and xx R at =1 must be defined. At low microwave fields, the magnetizati on of a ferromagnetically ordered system may be described by the classical torque due to an effective magnetic fieldeff B , acting upon it.53 *dissipative termeB effg dM MB dt (3.17) Under ferromagnetic resonan ce conditions, the electron spin-lattice relaxation time 1,eT may be described as the rate of energy transfer from 0 k to 0 k spin-wave modes.54, 55 This energy dissipation may occur via di rect spin relaxation to the lattice or to the short wavelength spin-wave modes, wh ich in turn may relax to higher wavelength spin-wave modes or to the lattice. This process is a dynamic one which will eventually result in the full ionization of so me of the spin wave excitations. As discussed in the experimental section of this chapter, a linearly polarized fixed frequency, microwave field of amplitude 1, b is applied to the sample, resultant in the MDESR observation. This field is composed of both left an d right circular ly polarized components, which are resonant with the pr ecessing magnetization. The magnitude of this microwave field is defined by the amplit ude of the field and the frequency of the precession as follows:

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65 11111 cos,sin,0 22 Bbtbt (3.18) Since the microwave field is also being square wave modulated at a frequency mod12 Hz, f a de-modulated signal will only be obs erved at the output of the second lock-in if the xx R response follows the square-wave modulation. Therefore, the appearance of an MDESR signal indicates that a steady-state conditi on is obtained on a time scale that is short compared to mod1/83 ms. f Another important parameter is that of th e energy transfer rate from the 2DES to the surroundings, henceforth referred to as th e bath. The time constant for this process will be dependent upon the heat capacity of the 2DES, as well as the thermal conductance between the 2DES and the bath . This is depicted as /extsbathC (3.19) From heat capacity measurements performed by V. Bayot et al.,56 we can determine that under these experimental conditions, mod1/ext f . In performing these experiments, a wide range of m odulation frequencies were used mod 1 Hz 30 Hz f with no qualitative differences in the temper ature dependences observed. There were slight variations in the MDESR amplitude, as well as in the form of the non-resonant background at a specific temperature, but these did not influence the results being discussed. In the continuation of ou r description for the mechanism of MDESR and its corresponding dependence on the te mperature, some assumptions must be made. First, it must be assumed that a “slow-passage” 0 dM dt condition is maintained during the

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66 magnetic field sweep. As the timescales for such a steady-state to be obtained are defined by the relaxation times 1eT and2eT , knowledge of these values would be especially beneficial in determining if this assumption is valid. Unfortunately, these times have not been measured directly for a 2DES at =1, however high-field and low temperature timeresolved ESR measurements have measured 1eT values on the order of 10-9 s.57 Such a short value for 1eT provides evidence that the electron spin relaxation occurs on a time scale several orders of magnit ude faster than that of the modulation of the microwaves. The symmetry and relative independence of the MDESR lineshape and amplitude on slight variations to the sweep rate provide d further supporting eviden ce for the validity of this slow-passage assumption. The transverse, complex magnetization of the electron spin system under steadystate conditions is described by * 1 0 2 * 0,1/2/2 /2eBz appeeBgMb M igB (3.20) This magnetization is related to the magnetic susceptibility of the 2DES through the expression, 01//2 iMb . Here, describes the loss by the sample and is defined through the following expression: * 1/2 2 2 2 * 0,1/2/2 /2eBz app appeeBgMB gB (3.21) Assuming low microwave power 2 * 1121eBeegbTT , the approximation eq zz M Mmay be made. Under the steady-st ate condition, the microwave power absorbed by the sample per unit area may be expressed as

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67 2 11 4appdQ p b dt (3.22) which on resonance simplifies to 2 10,1/2/2ze p MbB. The magnetization of the 2DES is be st described by the electron spin polarization /zPTNNNN, where N and N correspond to the number of electron spins in the up and down state respectively. Therefore, the total magnetization per unit area may be defined as *1 2eq zeBsz M TgNPT. In continuation of our model, two types of spin excitations will be c onsidered. First, the thermal equilibrium electron spin polarization of a spin-wave system may be determined as58 2 1zk k sPTN N (3.23) Second, for comparison purposes, we consider the thermal equilibrium electron spin polarization for a non-interacting itinerant electron system * 0tanh 4eB z BgB PT kT (3.24) The relative temperature dependences of these two functions are shown in Figure 3-10. On resonance, the power absorbed by the 2DES is 2 * 011/2/4szeB p BNPgbB , with the temperature dependence of this stemming exclusively fromzP . Taking a rough estimate of the amplitude of the microwave field to be 2 mT, the power absorbed by the system at 5.7 T and 1.5 K is 28.56 and 7.36 mW/m2 for a spin-wave and for a non-interacti ng electron system, respectively.

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68 Taking the expression for the dissipated mi crowave power, an expression for the heat flow may be written as ss s bsbdUdT CpTT dtdt (3.25) This heat flow process is pictorially repr esented in Figure 3-11. Under steady-state conditions 0sdU dt , so the expression may be rearranged to / s bbTTTp (3.26) In this form, the expression defines the change in temperature of the spin system under resonant microwave irradiation a nd the power absorbed by the 2DES. According to eq. (3.13), any change in th e magnetoresistance, like those associated with MDESR, must be due to a resultant change in the temperature of the current carriers, which as stated earlier are magnetic excitations withk . Under an infinitesimally small change in temperature dT, the change in the magnetoresistance may be described as 0 2exp 22xx BbBbEE R RT kTkT (3.27) or under more general conditions as 0expexp 22xx BsBbEE RR kTkT (3.28) where s T is the temperature of the 2D electron spin system and bT is the temperature of the bath, which may be considered equivale nt to the temperature of the sample.

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69 Figure 3-10: Temperature dependenc e of electron spin polarizationzPT, at =1 and 05.7 T, Bbased on the theoretical models of spin-waves (eq. (3.23)) and itinerant electron systems (eq. (3.24)) for a 2DES. 2DES, all k Ts, C2D kbath Bath, T b /esrPdQdt 2DES, k=0 microwave excitation 2DES, all k Ts, C2D kbath Bath, T b /esrPdQdt 2DES, k=0 microwave excitation Figure 3-11: Schematic representation of the proposed heating model used to explain the temperature dependence of MDESR amplitude . As stated in the text, resonant microwaves are absorbed by the 2DES, thereby exciting 0kspin-wave transitions. These uniform mode exc itations may relax via a transfer of energy to higher 0 k spin-wave modes. The 2DES is thus heated in reference to the sample (bath) and even tually exciting full ionizations of the spin-waves, providing the well-separate d electron and hole pair required for changes to be observed in the ma gnetotransport properties, such asxx R . A second assumption must be made at this poi nt to continue this discussion. Under this assumption, a single temperature s T must be ascribed to all spin-wave modes. This

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70 assumption, unlike the first, is very difficult to ensure. Even at steady-state conditions, some temperature variations, throughout th e many spin-wave modes, may be observed and the energy distribution among these diffe rent modes will be determined by the competition between the rate of energy transfer from the spin-waves of mode 1k to those of mode 1kk and the rate of energy flow fr om the spin-wave system and the surrounding bath. The analysis of this comp etition is an extremely complex theoretical problem that has not been addressed in the lit erature and is outside of the scope of this dissertation. So despite our inability to en sure the validity of this assumption, this assumption will be accepted as valid in order to continue this discussion. A major stipulation of the proposed heating model is the ability of the absorbed resonant microwave energy to be redistributed to 0k spin-wave modes. Such a mechanism, involving scattering proce sses was discussed by Haas and Callen,59 and provides a natural extension to the heating mechanism for the xx R detection of ESR. Under this model, an increase in xx R is predicted if an increase in the spin temperature is created. This is in agreement with our experimental data. Taking the model a step further, we can use it to calculate the te mperature dependence of the MDESR response. From eq. (3.13), it can be determined that a maximum in the slope of xx R dT is observed when the condition /4BbkTE (3.29) is met. Subsequent decreases in the slope are observed as either 0 KbT or bT. Presented in Figure 3-12 is the temperatur e dependence of the MDESR amplitude with 0 ESR B set to the =1 minimum (5.7 T) for both up (ope n circles) and down-sweeps (filled

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71 circles) in 0 B , along with the corresponding temper ature dependence traces for the noninteracting (solid line) and spin-wave (das hed line) models. The maximum in MDESR amplitude at 1.7 KbT is correctly predicted by the spin-wave model and corresponds very closely to the maximum MDESR c ondition defined by (3.29) based on the experimentally determ ined activation gap /41.75 K E . 024681012 0.0 0.2 0.4 0.6 0.8 1.0 non-interacting electrons spin-wavesESR amplitude, (a.u.)T, K T(K) MDESR Amplitude (arb. units)024681012 0.0 0.2 0.4 0.6 0.8 1.0 non-interacting electrons spin-wavesESR amplitude, (a.u.)T, K T(K) MDESR Amplitude (arb. units) Figure 3-12: Temperature depe ndence of MDESR amplitude at05.7 TESRB, which corresponds to the =1 minimum 060 for data recorded during up (open circles) and down-sweeps (solid circles) in 0 B . The curves represent theoretical calculations for the te mperature dependence of the MDESR amplitude based on a spin-wave (solid line) and itinerant electrons (dashed line) models. While the itinerant electron model simulates the fast drop off in amplitude with increasing temperatur e more closely, the spin-wave model predicts the position of the MDESR ma ximum and the low temperature decay in signal amplitude. One last assumption must be made and veri fied prior to acceptance of this heating model. The possibility of el ectron spin saturation must be ruled out under the conditions of this experiment. As will be discussed in the next chapter, a linear relationship between the microwave power and the MDESR amplitude was observed (as in Figure 4-4). This clearly indicates the absence of microwave saturation effects.

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72 To calculate an estimate for the steady-sta te temperature increase of the 2DES from eq. (3.26), an approximate value for the ther mal conductance must be determined. This may be determined from a self-consistent comparison with the experimentally observed increase in magnetoresistance at 1.5 K and 5.7 T. The increase in xx R under the application of the micr owave irradiation was 10 mTT . This, along with an estimate for the dissipated power, allows for a calculation ofb . For the spin-wave model, a value of 2/0.66 W/KmBESRpT was calculated. Provided there is minimal temperature dependence of bwithin the temperature range studie d, the temperature dependence of MDESR amplitude may be calculated. This theoretically calculated temperature dependence, based on our proposed heating m odel, is presented in Figure 3-12 for both spin-waves and for itinerant electrons. Further evidence in suppor t of this heating mechanism may be found in the xy R or the derivative Hall resistance signal. Under re sonant excitation of th e electron spins, no change in the Hall resistance within the plat eau associated with the different filling factors is observed provided no diagonal conductance is reco rded (this indicates a mixing of the xx R and xy R signals and therefore does depict ex citations of diminished amplitude due to the limited xx R component). This further supports the connection between the temperature dependence of the ESR and magne toresistance, therefore offering support for the heating mechanism described herein. Concluding remarks regarding heatin g model and temperature dependence The theoretical model proposed here to de scribe the temperature dependence of the MDESR response (both line width and amp litude) shows good qualitative agreement

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73 with the experimentally acquired data. Our model correctly predicts the position of the maximum in the amplitude of MDESR and corre ctly indicates that the position of this maximum is dependent upon E . In addition, the model al so predicts the MDESR will vanish as temperature is increased or decrea sed from the temperature associated with the maximum in MDESR amplitude. Thus, while the position of this maximum in MDESR amplitude is sensitive to the nature of th e spin excitations under resonant microwave absorption by the 2DES, the occurrence of a maximum and the disappearance of the signal as 0 KT are predicted by the heating model utilizing either the itinerant electron or spin-wave models for the spin exc itations of the 2DES. This is in direct contrast to the conclusi on reported by Meisels et al. , where, as stated previously, they reported a decrease in the MDESR amplitude with decreasing temperature that they interpreted as being due to depolariza tion of the electron spin system at =1 at low temperatures. This depolarization was stated to be due to electron correlation effects. The data and model shown in this study certain ly demonstrate that this interpretation is not necessarily concrete. Finally, the results of this study have indicated the possibl e use of MDESR to discriminate between competing theories for the magnetic ordering and excitations within 2DES in the regime of the QHE. Filling Factor Dependence of MDESR Understanding the full impact of the comp ression of the conduction electrons into two dimensions and therefore the influe nce of the QHE upon the electronic spin excitations is clearly necessary for the crea tion of a model of MDESR within the integer QHE. As stated by Vagner and Maniv,21 at filling factor =1, many direct and indirect

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74 properties of the transport electrons reach associative minimums, such as xx R and xx or maximums, as in 11/nT , and magnetization. Therefore, understanding the filling factor dependence of MDESR should provide further clues as to the relationship between the MDESR excitations and the QHE governing the electronic motion. The filling factor dependence of the MDESR response, within the vicinity of =1 0.8751.125, was taken in sample EA124 at a temperature of 1.5 K. Outside of this filling factor range, no MDESR peaks were observed. This includes filling factor 3, where MDESR has been observed by a number of groups, but due to the very small minima in this sample associated with this fi lling factor at 1.5 K (s ee figure 2-7) this is not surprising, as temperature dependence data shows the MDESR being dependent upon the magnitude of the activation gap. The filling factor was varied by incrementally rotating the sample at a fixed microwave power and frequency. Thus, only the effect of the integer QHE filling factor changes will be observed. An array of magnetoresistance traces, used to determine the f illing factor at each angle, is presented in Figure 3-13 (a). A maximum in the MDESR amplitude is observed at approximately =1. This is to be expected due to the 100% polarized electronic spin state at exactly =1. An interesting observation is made when the filling factor is raised above 1.08 or below 0.94. At these points, no MDESR signal is observed. Continuing to move away from =1 in both directions, it is clear ly seen that the MDESR peak cha nges in sign. Throughout the study, it was found that the MDESR peak had the same phase as the slope of()xx R T . As the filling factor is moved away from =1, where the phase is positive, the amplitude decreases, passing through zero at the critical points on both sides of the minima, where

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75 the magnetoresistance is temperature indepe ndent. These critical points are associated with the crossing points in th e temperature dependence of xx R presented in Figure 3-8 (a). After passing through the critical point s, the MDESR switches sign and continues as such until the signal disappears. The filling factor dependence of the MDESR is shown in Figure 3-13 (b). Although not observed in EA124, an odd ch ange in the signal is observed in EA124B around the critical points discussed above. In this study, the MDESR was observed while varying the microwave frequency, leaving the sample tilt angle constant. Due to differences in the microwave absorp tion at different frequencies, the amplitude between the different peaks is not comparable. a) b)Filling Factor Rxx(arb. units) Rxx( )0() B T a) b)Filling Factor Rxx(arb. units) Rxx( )0() B T Figure 3-13: (a) Multip le magnetoresistance traces, take n at each different angle for calibration of the filling factor for the experiment. (b) The MDESR amplitude as a function of filling factor in the region of =1. As stated in the text, a change in phase of th e signal is observed at <0.94 and >1.08. It was determined that the phase of the MDESR signal is correlated to the temperature dependence of xx R at that filling factor . A maximum in the MDESR amplitude is obtained at approximately =1 as expected. As a critical point in xx R T is approached, it was obs erved that the signal turns from positive in sign to an anomalous lines hape, where the signal shows both positive and negative components, then approaches a fully negative phase. An MDESR spectrum, taken at a microwave frequency of 35.40 GHz a nd a temperature of 1.6 K, is presented in

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76 Figure 3-14 (a). In (b), MDESR spectra take n at multiple microwave frequencies within this region are presented agains t a backdrop of the correspondingxx R trace. Note the spectrum at the highest microwave frequency is entirely positive in phase. A shift towards the anomalous lineshape is continue d as the microwave frequency is decreased, until at the last microwave frequency observable, we see a mostly negatively phased MDESR peak. Unfortunately, the complete transition to a negativ ely phased signal was not reached as the signal to noi se at lower frequencies was too poor for the signal to be observed. 5.905.956.006.056.106.156.20 100 150 200 250 300 350 400 35.40 GHz0 B T Rxx( ) 0 B T5.45.65.86.06.26.4 50 100 150 200 250 300 350 400 FildB (T) Rxx (=1 @ 6.5T) MW=35.75 GHz MW=35.40 GHz MW=35.00 GHz MW=34.60 GHz MW=34.00 GHz MW=33.50 GHz MW=33.00 GHz MW=32.50 GHza) b)5.905.956.006.056.106.156.20 100 150 200 250 300 350 400 35.40 GHz0 B T Rxx( ) 0 B T5.45.65.86.06.26.4 50 100 150 200 250 300 350 400 FildB (T) Rxx (=1 @ 6.5T) MW=35.75 GHz MW=35.40 GHz MW=35.00 GHz MW=34.60 GHz MW=34.00 GHz MW=33.50 GHz MW=33.00 GHz MW=32.50 GHz5.905.956.006.056.106.156.20 100 150 200 250 300 350 400 35.40 GHz0 B T Rxx( ) 0 B T5.45.65.86.06.26.4 50 100 150 200 250 300 350 400 FildB (T) Rxx (=1 @ 6.5T) MW=35.75 GHz MW=35.40 GHz MW=35.00 GHz MW=34.60 GHz MW=34.00 GHz MW=33.50 GHz MW=33.00 GHz MW=32.50 GHza) b) Figure 3-14: (a) MDESR spectrum taken at app =35.40 GHz, T=1.6 K, and the sample tilted to an angle of 063.92(=1 at approximately 6.52T). During the down-sweep, the signal first begins nega tive and then approximately half way across the line width of the peak it changes to a positive phase. (b) MDESR peaks at multiple frequencies within the region of the critical points in the temperature dependence against the background magnetoresistance trace for comparison. Upon closer evaluation, it was determined that this transition from positive phase to the anomalous signal occurs at a magnetic field of 6.175 T (35.75 GHz) under these conditions. As seen in the inset of Figure 3-14, this transition is cl ear within this range and the magnetic field at which this transition oc curs correlates very closely to the critical

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77 point in xx R T, as seen in Figure 3-14. The xx R traces presented in this figure depict the magnetoresistance in the absence (red curve) and in the presence (blue curve) of microwave radiation on the sample. The crossi ng point occurs at approximately 6.18 T. The difference between these two traces, xx R =xx R (no microwaves) -xx R (with microwaves), is presente d in green and crosses 0xxR at approximately06.1 T B. Due to residual parallel conductance, the true phase change probably correlates to a small positive value of xx R , so this field should not be ta ken as exact, although the close proximity of the crossing points in xx R T and the crossing of 0xxR to that of the behavior change of the MDESR, lends str ong evidence that this change is dependent upon the slope of xx R Tand occurs once the temperature independent filling factor is past. As a side note, the MDENDOR peak, wh ich will be discussed in detail in the next chapter, also exhibits this anomalous lineshape in this re gion. This is presented for 69Ga MDENDOR in Figure 3-16, where the disper sion shape is only observed on the downsweep in RF. It is presumed that as the sample is a multiple quantum well, the dispersion signals pertain to different wells being at different stages of the phase transition at each frequency measured. The filling factor also plays a strong role in the magnitude of the enhanced spinsplitting gap and the value of0 R . The xx R T was observed for EA124 (black squares), EA124B (red circles), and EA129 (blue tr iangles). The ac tivation gap and 0 R were determined at each filling fact or within the vicinity of =1 through the fitting of xx R Tto the Arrhenius expression introduced as eq. (3.13). As shown in Figure 3-17 (a) and (b), both the activation gap (deter mined in samples EA124, EA124B, EA129) and

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78 0 R (determined only in sample EA124), respectively, reach a maximum at =1 as expected. The activation gaps at =1 vary from 16.5 K for EA129 to 7.2 K for EA124, with EA124B having a gap of 8.9 K. The gap qui ckly drops to zero as the filling factor is moved away from =1. The same can be seen for0 R , where a maximum 378 is observed at =1, while the value drops off quickly to 266 and 214 at filling factors below and above =1. 5.05.25.45.65.86.06.26.46.66.8 250 200 150 100 -50 0 50 100 150 200 250 300 350 No microwaves With microwaves RxxPositive Phased Signal Anomalous Lineshape Limits of TemperatureB0(T) 5.956.006.056.106.156.206.256.306.35 -4 -3 -2 -1 0 1 6.175T p Rxx()B0(T) B0 ( T ) Rxx ) Rxx=Rxx(noMW)-Rxx(withMW)5.05.25.45.65.86.06.26.46.66.8 250 200 150 100 -50 0 50 100 150 200 250 300 350 No microwaves With microwaves RxxPositive Phased Signal Anomalous Lineshape Limits of TemperatureB0(T) 5.956.006.056.106.156.206.256.306.35 -4 -3 -2 -1 0 1 6.175T p Rxx()B0(T) B0 ( T ) Rxx )5.956.006.056.106.156.206.256.306.35 -4 -3 -2 -1 0 1 6.175T p Rxx()B0(T) B0 ( T ) Rxx ) Rxx=Rxx(noMW)-Rxx(withMW) Figure 3-15: The xx R taken at1.6 K T , 063.92 and with microwaves off (red curve) and at35.40 GHzapp . The change in the magnetoresistance between the two curves is due to non-resonant mi crowave heating of the sample. A vertical and horizontal line are draw n at the MDESR phase transition point 06.175 T B and at0 xxR , respectively. Inset is an array of MDESR spectra in the vicinity of the phase transition.

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79 61.7061.7561.8061.8561.90 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 p Upsweep in RF Downsweep in RFRF Frequency (MHz ) Rxx( )61.7061.7561.8061.8561.90 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 p Upsweep in RF Downsweep in RFRF Frequency (MHz ) Rxx( ) Figure 3-16: 69Ga MDENDOR taken during a sweep of RF at stop field conditions 6.048 TstopB after a DNP producing down-sweep at 10 mT/min. CW microwaves were applied at 35.40 GHz. Note only the down-sweep in RF shows the anomalous lineshape. 0.850.900.951.001.051.101.15 -4 -2 0 2 4 6 8 10 12 14 16 0.850.900.951.001.051.101.15 200 220 240 260 280 300 320 340 360 380 400 0 Filling Factor Filling FactorActivation Gap, (K) R0 ( )a)b)0.850.900.951.001.051.101.15 -4 -2 0 2 4 6 8 10 12 14 16 0.850.900.951.001.051.101.15 200 220 240 260 280 300 320 340 360 380 400 0 Filling Factor Filling FactorActivation Gap, (K) R0 ( )0.850.900.951.001.051.101.15 -4 -2 0 2 4 6 8 10 12 14 16 0.850.900.951.001.051.101.15 200 220 240 260 280 300 320 340 360 380 400 0 Filling Factor Filling FactorActivation Gap, (K) R0 ( )a)b) Figure 3-17: (a) Filling factor dependence of the activation gap in samples EA124 (black squares, =1 at 5.7 T), EA124B (red circles, =1 at 6.5 T), and EA129 (blue triangles, =1 at 5.8 T). All three samples exhibit a maximum in E at =1 with a sharp decrease, that approaches zero, as the filling factor is moved away in either direction from =1. (b) The dependence of 0 R on filling factor in sample EA124 at05.7 T B . Again a maximum value is obtained at =1 with sharp decreases occurring to either side, although the value at high filling factor (lower magnetic field) approaches a smaller resistance value than that of the lower filling factor (higher magnetic field).

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80 Concluding Remarks The temperature 0.34.3 K T and filling factor dependence (in vicinity of =1) of the MDESR response, along with th e filling factor dependences of E and 0, R were presented. These data were measured using two multiple quantum well and one heterostructure samples, EA124, EA124B a nd EA129, respectively. The MDESR signal is observed as a sharp peak in the ,xx R induced by photons of resonant microwave energy associated with the bare-electron Zeeman energy, which are absorbed by the 2DES. There appears to be no measurable temperature dependence of the MDESR line width, leading to the conclusion that the electron spin-spin relaxation time2 eT , is temperature independent within the range of te mperatures associated with this study. A maximum in the MDESR is observed at 1.7 K T at =1 in sample EA124 tilted to 060, 015.7 T B. This correlates very well w ith the calculated bare-electron spin gap of 1.6 K at this condition, a nd the experimentally measured activation gap/7 KBEk. This maximum is found at a temperature /4BkTE, as is predicted using our spin mode l for spin-wave excitations. The position of the maximum in the temperature dependence is sensitive to the magnitude of the activation gap, although the occurrence of a maximum a nd the disappearance of MDESR as 0 K T are observed theoretically, utilizing our heating model for both spin-waves and independent electrons. As stated previously, this is in contrast to the results presented by Meisels et al.51 Our heating model correctly predicted the location of the maximum in the observed temperature dependence of MDES R amplitude and the disappearance of the signal at both higher and lower temperatures.

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81 The filling factor dependence of MDESR peak amplitude, activation energy and0 R , all exhibit maxima at approximately =1 and are drastically reduced as the filling factor is either increased or decreased. The results discussed in this chapte r demonstrate the ability of MDESR experiments to discriminate between comp eting theories of magnetic ordering and excitations within a 2DES in the regime of the integer QHE.

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82 CHAPTER 4 ELECTRON-NUCLEAR HYPERFINE IN TERACTION AND ITS EFFECT ON MDESR AND MDENDOR The FC hfi between the electr ons in a 2DES and the latti ce nuclei in their vicinity, is associated with a remarkable variety of phenomena in quantum Hall systems. As discussed in Chapter 1, electronnuclear cross-relaxation, together with either complete or partial saturation of the ESR transition, can l ead to a large non-equi librium nuclear spin polarization via the Overhauser effect. Enhanced nuclear spin polarization in the vicinity of the 2DES has been demonstrated to be extremely useful as a tool for studying the correlated electronic states. Some examples include controlling the effective g-factor with the enhanced local hyperfine field,60 measurement of the nuclear spin-lattice relaxation time 1 nT , which yields information on the D.O.S. at the Fermi level,17, 18 and sensitivity enhancement for NMR Knight Shif t studies of the elec tron spin polarization and dynamics.61 Another important aspect of the Overhauser effect is the influence the increased nuclear spin polarization has upon the ES R condition. Due to the relatively long relaxation times of the nuclei and the corres pondingly short relaxation times associated with the electrons, if the mi crowave irradiation is remove d, the nuclear polarization will persist long after the electr on spin system has completely relaxed. Under the ESR saturation conditions mentioned above, a large nuclear hyperfi ne field develops as the nuclear polarization increases. Due to the c oupling between the electrons and nuclei, this

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83 field acts back upon the electroni c system. The magnitude of the change of the electronic Zeeman energy due to this coupling may be expressed as 2 ,8 3lattice hfi i Z eBiieBn i eB ieBniA EgrIgB g Ag (4.1) where ir and n B are the electronic wave function, (within the nuclear radius of isotope i ), and the total nuclear field due to the Overhauser effect acting upon the electronic spin system, respectively.62 A compilation of important nucleus specific values for the AlGaAs/GaAs quantum wells and he terostructures is listed in Table 4.1. Table 4-1: Collection of important properties of the three spin-bearing isotopes in GaAs. Nuclei Natural abundance (ia ), zi I iA62max()n B T63 69Ga 60.4% 33 22i zI 0.91 9.1z I 71Ga 39.6% 33 22i zI 0.78 7.8z I 75As 100% 33 22i zI 1.84 18.4z I Thus, the maximum nuclear field that may be induced within bulk GaAs is max5.3n B T. Depending on the relative signs of the gyromagnetic ratios of the electrons and nuclei, n B , will act in addition or subtraction to 0 B . Thus, a total effective field 0 effn B BB (4.2) is said to be acting upon the el ectronic spin system. Thus the resonant frequency of the electrons undergoes an Overhauser shift and is defined as follows: 0,0 eeBngBB (4.3)

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84 Previous experiments studying the electronnuclear correlations in these systems have involved MDENDOR measur ements at filling factor =3,50, 63 low temperature MDNMR at 3/54 ,64 optically pumped NMR in GaAs/AlxGa1-xAs quantum wells,65, 66 and specially patterned samples invest igating the spin-spin scattering of 2D edge state electrons with the local nuclei.67, 68 In this chapter, microwave induced DNP effects associated with slower magnetic fi eld sweep rates on the MDESR spectra as well as MDENDOR at varying nuclear polarization levels, within high mobility GaAs/AlGaAs quantum well and heterostruct ure samples will be discussed. All measurements were performed within the vicinity of the =1 filling factor. Numerical simulations, based on the heating model discusse d in Chapter 3, will be introduced in the Chapter 6. Hyperfine Interactions in 2DESs As first observed by Stein et al.,26 the electron spin-splitti ng of the Landau levels, in the absence of exchange interactions, can be measured via MDESR. But due to the influence of the nuclear spin polarization upon the electron system via the FC hfi, as discussed in Chapter 1, the “bare” spin-spl itting is not due enti rely to the electron Zeeman effect. The Hamiltonian for an electron of spin , S interacting with a nucleus of spin, I may be described by * 0ˆˆ ˆˆeBHgBSAIS (4.4) which is similar to eq. (1.12), with only the nuclear Zeeman contribu tion removed, as this does not affect the electronic energy. Taking all interacting nuclear species as having a total weighted average effect upon the electronic system, th e energy of this system may be written as

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85 2 * 008 ˆ 0 3zeBBnEgBSgI (4.5) where 0g is the free-electron g-factor, n is the weighted average of the nuclear gyromagnetic ratios, 20is the electronic wave func tion at the nuclear site, and I is the average nuclear spin angular momentum. At thermal equilibrium, even at the low temperatures (~1.5 K) and high fields 0314.5 T B associated with these experiments, I is small (on the order of 3X10-4 for 75As at 6 T and 1.5 K) and therefore has little to no effect on the electron spin system. This is supported by our observation, discussed in the previous chap ter, that at the highest sw eep rates, no change in the MDESR lineshape is observed. Thus without DNP enhancement of , I the second term in eq. (4.5) may be neglected. DNP occurs if the hfi (Overhauser e ffect) is the dominant mechanism of electronic spin relaxation. This second term in eq. (4.4) may be expanded as 0.5zzAISAISISIS (4.6) The spectral density of the fluctuations of the flip-flop term (in parentheses) is responsible for the transitions associated with electron-nuc lear cross-relaxation. As previously stated, the hfi is also responsible for induci ng the local nuclear field, n B which shifts the electron spin Larmor frequency. If the electrons in a conduction band described by s-type Bloch func tions, as they can be in GaAs, the FC hfi Hamiltonian is written as 216 ˆ ˆˆ 3BnHrIS (4.7)

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86 Since there are a large number of nuclei contribu ting to the Hamiltonian, we may rewrite eq. (4.7) in terms of the average nu clear spin polarization as follows: ˆ ˆˆlocHASI (4.8) This describes the interaction between th e electrons and the local nuclear field, n B ˆ ˆlocenHBS (4.9) The observed ESR position is theref ore shifted as follows: * 0 zeBn E gBB (4.10) where 2 08 0 3nBn B gI. This effect had been observed in ESR measurements in metals,8, 69 semiconductors70 and organic materials71 and was first measured in 2DES within GaAs/AlxGa1-xAs quantum wells by Dobers et al.63 in 1988 and was used to observe MDENDOR spectra within the =3 minimum of the integer QHE. First Observation of MDENDOR Through the RF saturation of NMR tran sitions of the dynamically polarized nuclei in FC with the 2DES, sharp drop-o ffs of the Overhaus er broadened MDESR spectra were observed, thus proving the nuclear origin of this broadening/shift. The authors studied the effects of the hfi upon the partially polarized 2DES within the vicinity of the =3 filling factor. During an up-sweep in 0 B of 02.35 T/min dB dt, a sharp MDESR peak in xx R was observed, while during a down-sweep, a broad, smeared out feature beginning at the equilibrium ESR position 0, e B , and ending at some point further down-field, was observed. As stated a bove, this broadening occurs when the ESRinduced DNP creates a local nuclear magnetic fieldn B , which acts in addition to 0 B upon

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87 the 2DES, thereby shifting the ESR condition to lower 0 B . The Overhauser shifted ESR condition will be denoted as D NP ESRB . The shifted ESR condition may be determined through the following equation: eq ESRESRn B BB (4.11) The implication of this relationship is that in creases in the nuclear spin polari zation (and therefore inn B ) will result in a down-field shift in the MDESR condition.63 The authors stated that at the microwav e powers utilized (100 mW at the source), the ESR position shifted at a rate faster than the down-sweep in external magnetic field, thus the MDESR condition was always slightly “out of reach” of full resonant microwave absorption, thus explaining the broad, smearedout feature. A further explanation of this broadening will be presented in the discussion se ction of this chapter. Furthermore, after the microwaves were turned off, a drop in the xx R to the non-excited value associated with the absence of microwave irradiation was observed. A similar drop in the MDESR spectra was observed when radiofrequency (RF), tuned to the nuclear resonance condition of one of the local nuclear species (effects were observed for 75As and 69Ga nuclei) at a field eq NMRESR B B.63 The important parameters of the nuclei found within the GaAs quantum well region are presented in Table 1-1. NMR induced changes in the MDESR spectra are referred to within this dissertation as MDENDOR features. It was determined that the decay of the nuclear spin polarization, and therefore the Overhauser shift of the MDESR was on the order of 20 min at =3. The authors state that the nuclear polarization, due to the presence of free electrons, will relax via Korringa relaxation and displays a 1 nT much shorter than that obser ved in high-purity bulk GaAs.72

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88 In a later paper,17 the authors went on to measure the magnetic field (filling factor) dependence of 11/nT and determined that, as predicted by Vagner et al.,16 11/nT went through magnetoquantum oscillations, hitting mini mums at the magnetic fields associated with minimums in the xx R , and a maximum 1820nTs was measured at =3, (=1 was not attainable in the samples explored by th eir apparatus). This was later verified by Tycko et al,17 through the use of optically pumped NMR measurements of the nuclear relaxation time. In this later experiment, measurements of 1 nT at =1 were made, and it are determined that a dramatic increase in 1 nT was observed at this point. Berg et al.17 also completed simulations of the 11/nT dependence, in the range 24 , assuming Korringa relaxation and a stri ctly 2D case, were calculated and compared to the experimental values . In doing so, an adaptation to incorporate essential features of a 2DES to SlichterÂ’s formula for the relaxation time in metals4 was made, producing the following relationship: 4 22 11 1nAzDDffE T (4.12) where is the volume of the unit cell ( =45.2X10-30 m3 in GaAs), z is the envelope portion of the electr onic wave function defining the width of the 2DES in the z direction (zzuz , where uz is the Block function in the z direction), f E is the Fermi function and D and D are the D.O.S. for the spin-p and down electrons, respectively. Very good qualitative ag reement between the simulation and the experimental data, which supports the idea that Korringa relaxation is the dominant mechanism of nuclear spin relaxation for nucle i in the vicinity of a 2DES. Furthermore,

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89 upon increasing the length of the well region in the z-direction in th e simulations, thereby removing the electron confinement, the magnetoquantum oscillations 11/nT became less pronounced. Thus it appears that the exchange interaction is necessary for understanding the filling factor dependence of 1 nT at odd integer filling factors. Direct Absorption Detection of ENDOR in n -Type GaAs Prior to this study, performe d by Seck, Potemski and Wyder,15 no observation of direct absorption detection of ESR of electrons bound to shallow donors within n -type GaAs had been reported. This study represented the first de tection of DNP effects from the associated FC hfi in the conduction band of GaAs. The authors observed asymmetric lineshapes, dependent upon the magnetic fiel d sweep direction, with an out of phase component of the ESR signal occurring at sl ow microwave modulation frequencies. The presence of this component suggests the contribution of the nuclear spin system upon the observed ESR, thus providi ng evidence of DNP effects. The ESR signal appeared as a sharp increas e in signal at the resonant condition of the electrons, followed by a slow decay to the non-excited level (zer o). The peaks were observed to have increased amplitude on the down-sweeps in 0 B , and with reduced temperature, at least to the lowest temperatur es measured (1.4 K). Interestingly, the ESR susceptibility was observed to decrease and narrow at low microwave powers, this in contrast to what is typically observed due to reduction in saturation effects at such powers. It was also noted that no dependence of the signal amplitude or line width on the magnetic field sweep rate was observed within the range of 00.050.5 T/min dB dt.

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90 To prove the contribution of DNP eff ects upon the observed ESR lineshape, CW RF was applied at different frequencies asso ciated with NMR conditions for one of the isotopes within the proximity of the ESR resonance. They observed that if the NMR condition was set prior to the onset of ESR, or outside of the ESR line width, that there was no response. If the NMR condition was se t within the ESR line width, one of two things occurred, if the condition was set at a magnetic field close to the eq ESR B , a small dip in the ESR signal was observed, followed by a re covery of the ESR signal to its original lineshape. If the NMR condition was set furt her down-field from the ESR, but still within the lineshape, a drop in the ESR to zero signal was seen and no recovery of the ESR was recorded. There were three main features which the authors explained. First, the asymmetry was said to be due to dynamic microwave i nduced Overhauser shifts of the ESR, thus causing a broadening of the ESR signal. Secondly, they state that the observed broadening of the ESR at high microwave power s is associated with increased DNP rates thereby increasing the Overhauser shift of the ESR condition and correspondingly increased broadening of the ESR peak. Fina lly, the RF induced depolarization of the dynamically polarized nuclei causes a reducti on in the Overhauser shift back toward eq ESR B . If the condition is set su ch that a sufficient Overhaus er shift in the ESR condition is created, this destruction of nuclear polari zation will lead to a dr amatic reduction in the Overhauser shift, thereby losing the ESR c ondition. The presence of this change in behavior, based on the relative posit ion of the NMR and ESR conditions ESRNMRB, leads to the inferred presence of a critical condition or fiel d where the transition from a dip to the destruction in the ESR line occurs.

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91 Computer simulations modeling the effect s of the Overhauser effect upon the ESR were performed and provided good qualitati ve agreement to the ESR peaks observed under the conditions presented. No simulati ons of the NMR depolarization effects were performed, however. These simulations provi de the first tested model for DNP effects upon observed ESR associated with conduction el ectrons in GaAs and were used as the basis for MDENDOR simulations performed by Hillman and Jiang50 which will be discussed in further detail in Chapter 6. First Observation of MDENDOR Peaks A later study, performed by the team of Hillman and Jiang,50 observed MDENDOR excitations, not as losses of the MDESR condition as observed by Berg et al.,17 but instead as sharp peaks, superimposed upon the Overhauser xx R trace within the vicinity of =3 filling factor. Similar to the experiments reported by Berg et al.,17 the MDESR and MDENDOR spectra were recorded as a function of the magnetoresistance xx R , using a the single lock-in techni que similar to that which was described in the experimental section of Chapter 3. The experiments were performed through the use of a slow DN P producing down-sweep through eq ESR B , while CW RF was applied at a frequency associat ed with the NMR condition of one of the local nuclei, such that eq NMRESR B B. Upon passing through the NMR condition, a sharp peak in the magnetoresistance was observed. As in the pr evious experiments, the authors determined that the MDESR peak in the absence of RF was broadened with slower down-field sweep rates and correspondingly narrowe d during up-sweeps. It was also determined that the position of the MDESR peak shifted to lowe r magnetic field slightly with increased

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92 microwave power, leading to the assumpti on that higher microwave powers yields increased DNP production rates. The position of the narrow peak in th e MDESR associated with MDENDOR is dependent entirely on the RF applied, and af ter experiments involvi ng all three nuclei (75As, 71Ga, and 69Ga) the position of the MDENDOR p eaks were plotted against the RF and as predicted, the slopes from the linea r fits produced the gyromagnetic ratio of the nuclei in question, removing all doubt of the nuclear origin of these peaks. Under their experimental conditions, the slow 030 mT/min dB dtdown-sweeps used to obtain the MDENDOR spectra did not show an observable change in the xx R associated with the Overhauser broadened MDESR, thus the MDENDOR were observed as a single peak superimposed on the visibly unchanged xx R trace. Considerable DNP was assured through the observed broadening under fast er down-field sweep rates and through subsequent fast up-sweeps following th e slow DNP producing down-sweep, thereby measuring the time decay of the Overhauser shift of the MDESR.50 In another experiment disc ussed in this article, the dependence of the MDENDOR peak amplitude was measured as a functi on of decreasing RF (t hereby moving the NMR condition to lower magnetic fields ) while maintaining a continuous eq ESR B . In doing so, it was observed that the MDENDOR peak increases in amplitude as ESRNMRB was increased throughout the experime ntal dataset provided. Finally, the authors presented the magnetic field sweep rate dependence of the MDENDOR peak. As the sweep rate was slowed from 360 mT/min to 18 mT/min, the

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93 MDENDOR peak narrowed and increased in size dramatically, while the overall amplitude of the Overhauser broadene d MDESR background reduced in size. Simulations based on a theoretical model first proposed by Seck et al.15 were performed and simulated the sweep rate depe ndence data to good qualitative agreement. They stated that the appear ance of the MDENDOR peak is due to a “consequence of the competition between the ESR-induced DNP and the NMR-induced nuclear polarization”.50 Further analysis of this model is presented in the discussion presented later in the chapter. MDNMR in 2DESs Previous experiments reporting the huge longitudinal resistance induced by extremely slow down-sweeps in the regime of =2/3,73 determined such effects were of a nuclear origin,74 providing more insight into the cl ose relationship between the hyperfine coupled nuclei and 2DES. The first observa tion of the influence of RF induced NMR transitions upon the conduction electron system , in the absence of DNP, was reported in 2002 by Desrat et al.75 This report represents the first measurements of the electronnuclear couplings at a large range of filling factors using only the thermal equilibrium nuclear spin polarization within GaAs/AlGaAs heterojunctions. Since only the thermal polarization of the nuclear system was utilized, a dilution refrigerator was necessary to attain appropriate nuclear polarization levels. It is estimated by the authors that at 50 mK and 010 T, B that the nuclear spin system is approximately 10% polarized. As shown in their experiments, this is sufficient for the effects of the nuclear system on the conduction electrons to be realized.

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94 In these experiments, a constant 0 B , associated with the desired filling factor, was applied to the sample. RF was swept through the resonant condition of a nuclear isotope, NMR B , and the effect upon xx R was recorded. At the resonant conditions of 69Ga, 71Ga, and 75As, a sharp drop in xx R occurred, followed by a slow er relaxation back to the unperturbed value. They state this relaxa tion time was a measurement of the nuclear relaxation time 1 nT , and measured times on the order of 30s outside of the =1 minimum. The MDNMR signals were measured for all three nuclei with in the well (although as expected, 27Al was not observed), at odd (=1 and 3), even (=2 and 4) and fractional (=3/5, 2/3, 4/3, and 5/3) filling factors. When measurements were recorded about the =1 minimum, the MDNMR amplitudes were en hanced by two orders of magnitude and a dramatic drop in the measured 13 snT was observed. These measured relaxation times are consistent with measurements by Hashimoto et al.,76 who utilized a similar detection method, although both of these are in direct contradiction to those determined by the more accepted methods of time depe ndence of Overhauser shift relaxation17, 77 and optically pumped NMR18 where relaxation times on the or der of hundreds of seconds are observed. The reason for this discrepancy is still unclear. One possibility is the presence of gapless Skyrmions at =1 within the samples presented in this and HashimotoÂ’s results, which should lead to shortened relaxa tion times for the nuclear system. Another possibility is that the rela xation mechanism involved in such a measurement is not a direct measure of 1 nT but rather is a time constant asso ciated with the influence of the NMR depolarization upon the necessary k mode excitations for changes in electrical transport. Finally, it was noted that if the sweep rate of the RF was reduced,

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95 that more symmetric lineshapes were observe d, thereby changing the relaxation time of the signal to the non-excited level. The observed normalized changes in resistance xx xx R R , at filling factors other than =1, were measured to be approximately 0.5% . This corresponds to a change in the electronic Zeeman gap of approximately 2.55 eV, provided the thermally available nuclear spin polarization of 10% is accepted. Such a change to the Zeeman gap was calculated to create a corresponding change in xx R of between 0.6-1.0 percent, remarkably close to the experimental observatio ns. Furthermore, the observed values for 1 nT outside of the =1 minimum, follow the exp ected Korringa relationship 11 F nTE T (4.13) where Tis the temperature and F E is the density of states at the Fermi level. Within the =1 minima, a drastic increase in MDNMR amplitude, a dramatic drop in 1 nT , and an “anomalous lineshape” similar to a dispersion lineshape, are observed. The authors state that these three features, only associated with =1, are evidence for the presence of gapless spin-wave excitations of a Skyrme crystal. They went on to measure the magnitude of the MDNMR amplitude at another sample tilt angle 014.5 T B , where g no longer favored Skyrmion formation at =1, and they observed a dramatic loss in signal amplitude. This is contrary to the expected increas e in signal amplitude typically associated with increased 0 B , thus providing further support for their hypothesis. Finally, measurements under a tilted field such that =3 was observed at

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96 approximately the same 0 B as that where =1 is observed at 00 , did not display the anomalous, dispersion-like shape. The evid ence supporting the presence of Skyrmions at zero tilt angle in this sample could be the reason for the dramatic difference in the filling factor dependence of 1 nT around =1 from that measured in the samples presented in this dissertation as well as in previous reports. Experimental Results: Influence of Hyperfine Coupling on MDESR Spectra Experiments probing the FC hfi between the 2DES and local nuclear lattice will be discussed in this chapter. These experi ments measured the influence of DNP on the MDESR and the observation of MDENDOR and ot her associated spectr al features. All experiments were performed using sample s EA124, EA124B and/or EA129 which were introduced in Chapter 3. The double lock-in technique, also previously discussed, was utilized for obtaining the spectra. Sweep Rate Dependence of MDESR Presented in Figure 4-1 are the MDESR spectra ta ken in sample EA129 under microwave irradiation of 32.85 GHz at 1.7 K for three different magnetic field sweep rates, 0dB dt =448, 100, and 50 mT/min for (a), (b) and (c) respectively on both the up (in red) and down-sweep (in black). It is obser ved that the highest sweep rates exhibit no distinct differences between the up and down sw ept spectra. As eluded to previously in this text, upon slowing the sweep rate, the down-sweeps in magnetic field cause a broadening of the signal, while the opposite is true for the up-sweep. This is consistent with previous reports in the literature.17, 50, 66, 75 This broadening is due not to a broadening of the ESR absorption itself, but rather to a dynamic shifting of the ESR

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97 condition, ,eq ESRB to lower 0 B due to the Overhauser effect. At the end of the sweep, the new Overhauser shifted ESR condition is denoted as D NP ESRB . The root cause of this Overhauser shifting is the DNP that is cr eated under electron-nuc lear cross-relaxation following microwave excitation of ESR. The mechanism for the Overhauser effect is discussed in detail in Chapter 1. A more complete depiction (492 mT/m in – 2 mT/min) of the sweep rate dependence of the MDESR signal in both EA 124 and EA124B are pres ented in Figure 42 (a) and (b), respectively. The clear broade ning associated with slower sweep rates is present, but also it may be observed that th e amplitude of the MDESR appears to initially increase in magnitude then reach a maximum and then slowly decay as the broadening is continued. This dependence wi ll be discussed in further detail later in the chapter. To refresh, the resonant condition of the electrons is determined through the following equation: * 0/ESReBngBB (4.14) where n B is the local nuclear field acting on the 2DES due to DNP. The negative sign of * eg and positive signs for n for all three nuclei, cause n B to be positive and therefore adds constructively to0 B . Thus, an effective field acting on the electrons, 0 effn B BB, is created that now defines the resonant conditio n of the electrons as in eq. (4.14). Since no other quantity involved in the equation is changed, the total 0 B required to meet the ESR condition is less, as n B compensates for some of the total field that is necessary. In other words, the DNP will lead to an Over hauser shift of the ESR condition to lower 0 B . If CW microwave irradiation is applied, unde r DNP conditions, the ESR is seen to move

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98 down-field away from eq ESR B . Since this is a dynamic pro cess, this is observed as a broadening in the ESR line, rather than di stinct Overhauser shifted ESR peaks. In describing the DNP created via resona nt absorption of CW microwaves during a magnetic field down-sweep, the following equati on, slightly modified from the one given by Abragam,8 should be consulted 2 00 2 11ˆ 11 ˆˆi z iiiii e zz iii nnnI I IsIDI tTTz (4.15) where ˆˆ ˆeq zz eq zSS s S (4.16) is the saturation parameter determining the degree of electron spin saturation due to the resonant absorption of microwaves, iD is the nuclear spin diffusion constant, * eeBg is the gyromagnetic ratio for the electrons, and 0 i I is the thermal equilibrium nuclear spin polarization for nucleus i. In order to utilize eq. (4.16), a functional form using known or measurable quantities must be established. If the a ssumption is made that the Bloch-Bloembergen equation for the induction of electron spin magnetization at low mi crowave powers, via resonantly absorbed microwave radiation, is valid 2 1121eeeTT, 2 112 2 2 21eee eappESRTT s T (4.17) where 1 eT and 2 eT are the longitudinal and transverse electron spin relaxation times, ESR is the resonant frequency of the electrons (0, ESRe when no DNP has occurred, where

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99 0, e is the Larmor frequency of the elec tron spins in the absence of DNP) and * 11 eeBegB is the electron spin Rabi frequenc y pertaining to the precession in the transverse microwave field of frequency 2ESRESR with an amplitude of 1 e B . In order to utilize eq. (4. 15), a useable expression for 1 nT must be determined. Using eq. (4.12), it may be stated th at at odd integer filling factors, 4 1222 1 1nneTzDDffdE (4.18) where2 ,,2 exp/2nDEEE , with referring to the standard deviation of the Landau level broadening, and is a Bloch correction factor which is dependent upon the nucleus. Assuming BkT, the integrand in expression (4.18) may be removed and the expression may be expressed as the following proportionality: 4 1222 1 nneBTzDDkT (4.19) Therefore, the Korringa relations hip (expression (4.13)), applicab le in metals, is predicted to be obeyed within the 2DES. The process of the Overhauser effect a nd its influence on the MDESR spectra, may be thought of in terms of a competition between the tendency of D NP zI to relax to 0 I , the continued polarizatio n of the nuclear spin system vi a electron-nuclear cross relaxation under pseudo ESR saturation conditions, and the diffusion of the induced nuclear spin polarization throughout the sample. This compe tition is presented clear ly in eq. (4.15). For an arbitrary electr on spin saturation level 01 s , a steady-state nuclear spin

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100 polarization is attained, 0 1e z nIIs . Therefore, the maximum nuclear spin polarization that may be achieved is determ ined by the ratio betw een the electronic and nuclear gyromagnetic ratio 5.655.705.755.805.85 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 Downsweep Upsweep .448 T/min(a)xx()B (T)5.655.705.755.805.85 500 1000 1500 2000 2500 3000 3500 4000 4500 .10 T/min(b) 5.655.705.755.805.85 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 .05T/min(c) Rxx B0(T)5.655.705.755.805.85 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 Downsweep Upsweep .448 T/min(a)xx()B (T)5.655.705.755.805.85 500 1000 1500 2000 2500 3000 3500 4000 4500 .10 T/min(b) 5.655.705.755.805.85 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 .05T/min(c) Rxx B0(T) Figure 4-1: Sweep rate dependence of MDESR in sample EA129 at T=1.7 K. (a) Corresponds to the fastest sw eep rate recorded, 448 mT /min. At this rate, it should be observed that there is no si gnificant difference between the up (red) and down (black) swept spectra, so DNP effects can be assumed to be negligible. (b) MDESR spect ra at a sweep rate of 100 mT/min. At this rate a significant change is observed between the two sweep rate directions. Increased amplitude and line width are observed on the down-sweep, while the exact opposite is obs erved on the up-sweep. (c) MDESR spectra at 50 mT/min. At this rate the MD ESR is broadened further.

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101 5.75.85.9 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124B 492.0 mT/min 400.0 mT/min 300.0 mT/min 200.0 mT/min 150.0 mT/min 100.0 mT/min 50.0 mT/min 10.0 mT/min 4.6 mT/minxx() B0 (T) 5.705.755.805.85 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124 448 mT/min 50 mT/min 10 mT/min 8 mT/min 5 mT/min 2 mT/minxx()B, T EA124B EA124 (a) (b) Rxx B0(T) B0(T)5.75.85.9 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124B 492.0 mT/min 400.0 mT/min 300.0 mT/min 200.0 mT/min 150.0 mT/min 100.0 mT/min 50.0 mT/min 10.0 mT/min 4.6 mT/minxx() B0 (T) 5.705.755.805.85 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124 448 mT/min 50 mT/min 10 mT/min 8 mT/min 5 mT/min 2 mT/minxx()B, T EA124B EA124 (a) (b)5.75.85.9 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124B 492.0 mT/min 400.0 mT/min 300.0 mT/min 200.0 mT/min 150.0 mT/min 100.0 mT/min 50.0 mT/min 10.0 mT/min 4.6 mT/minxx() B0 (T) 5.705.755.805.85 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124 448 mT/min 50 mT/min 10 mT/min 8 mT/min 5 mT/min 2 mT/minxx()B, T EA124B EA1245.75.85.9 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124B 492.0 mT/min 400.0 mT/min 300.0 mT/min 200.0 mT/min 150.0 mT/min 100.0 mT/min 50.0 mT/min 10.0 mT/min 4.6 mT/minxx() B0 (T) 5.705.755.805.85 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 EA124 448 mT/min 50 mT/min 10 mT/min 8 mT/min 5 mT/min 2 mT/minxx()B, T EA124B EA124 (a) (b) Rxx B0(T) B0(T) Figure 4-2: Sweep Rate dependence of MD ESR in (a) EA124B under irradiation from CW microwaves at a frequency of 34. 25 GHz while tilted to approximately =690 and T=1.7 K and in (b) EA124 under application of 33.64 GHz microwaves at T=1.5 K and =610. Both show the same general behavior as presented in EA129, except in (a) the appearance of a doublet structure, not observed in any other samples, is clear within the range of 0100200 mT/min dB dt. 716975445, 566, and 793 for Ga, Ga, and Ase i n and the saturation parameter, s. The DNP created in a 2DES under re sonant microwave absorption causes an Overhauser shift of eq ESR B to lower 0 B , creating a new resonant condition D NP ESRB , which moves dynamically with 0 B . The resonant condition D NP ESRB is pushed below 0 B , thereby reducing the saturation level achieved until a new steady-state z I is attained. If 0 B is reduced, it will approach D NP ESRB and an increase in microwave saturation will be observed. This will establish a new, larger steady-state z I . Under experimental conditions with a sufficiently slow enough magnetic sweep rate an d the irradiation of the sample by a CW microwave field, a continuous increase in the z I will be observed until the maximum

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102 polarization max zI , is established. Since the ratio between e and n is fixed, max zI is determined exclusively by the level of microwave saturation obtained. According to eq. (4.17), when the system in on resonance appESR, the maximum saturation 2 112 M AXeeesTT. Therefore the maximum nuclear po larization may be expressed as max 223 01120112ˆ 1/1/zeeeeneeeenIITTIBTT (4.20) A few assumptions lead us to a further understanding of the Ov erhauser shift. First, the shift will be negative (to lower magnetic field) presuming that 00 I, which has been determined to be a good assumption at the temperatures investigated. Secondly, from eq. (4.20), it is clear that MAX 2 1ˆ zeI , provided the microwave power is well below the level required for full saturation of the ESR. Finally, if we assume that the electron spin relaxation is governed by fluctu ations occurring on a short time scale, defined by a correlation time 2/cESR , then relaxation theory leads us to the conclusion that 12eeTT. Assuming that at 448 mT /min the MDESR peak is homogeneously broadened, the ESR line width 15 mTFWHMmay be used to estimate the value of 22/3.7 nsESR eeFWHMT. Under experimental conditions, non-resonant heating and the magnitude of the exchanged enhanced spin-splitti ng of the Landau level of interest may also limit MAXˆzI. Looking closer at Figure 4-2 (a), we again see the sweep rate dependence 04.6492 mT/min dB dt of MDESR at T =1.7 K. This is the most complete study of

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103 the sweep rate dependence and therefore was u tilized to determine the dependence of the MDESR line width, at FWHM, and amplitude on the sweep rate; this is presented in Figure 4-3. At high sweep rates, the line widt h stays relatively constant, until the rate is reduced below 150 mT/min. At this point, the MDESR peak broadens, with the line width increasing monotonically within the sweep rates attainable by our magnet system. However, the amplitude displays quite a diffe rent dependence. In itially, upon reducing the sweep rate from the fastest rate of 492 mT/min, the MDESR begins to increase in magnitude until it reaches a maximum at 150 mT/min. Continued reduction below this point, causes the MDESR amplitude to decr ease. This reduction appears to be exponential. The MDESR peaks appear roughly symmetric at all sweep rates studied. Within the range of 0100200 mT/min dB dt, a doublet like feature is observed in the MDESR spectra. The appearance of the doublet feature might po ssibly be explained by heterogeneity among the different quantum we ll layers or from distinct, inequivalent sets of electrons within the many layers. Microwave Power Dependence of MDESR According to eq. (4.20), the nuclear spin pol arization is directly proportional to the square of the microwave field strength 1 e B . In order for our previous assumptions to be valid, it is essential that we ensure the e xperiments are run within the low microwave excitation power regime where 1s. The linear relationship between the microwave power and the MDESR amplitude in sample EA129 at a sweep rate of 50 mT/min, is presented in the inset of Figure 4-4. These data were extracted from the stacked plot within the same figure. This clearly displays the validity of the low power assumption in this sample. Similar results are observed in the other samples investigated, solidifying

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104 the validation of this assumption. Additiona lly, it is apparent that increased microwave power is also accompanied by an increase in the MDESR FWHM. Using the equation for the maxi mum electron spin polarization 2 2 112/MAXeeeesBTT, our estimate for 123.7 nseeTT , and estimating the microwave power to be approximately 11 mTeB , we can estimate 0.034MAXs. This gives an upper limit to the nuclear spin polarization of maxˆ 0.236zI . 0100200300400500 10 20 30 40 50 60 70 80 Sweep Rate (T/min)FWHM (mT) Width Amplitude0 2 Peak Amplitude ( ) Figure 4-3: Sweep rate depende nce of MDESR amplitude (solid blue circles) and line width (open black squares) in EA124B taken from the data presented in Figure 4-2 (a). The amplitude re mains unchanged until approximately 150 mT/min, whereupon the signal increases dramatically until about 50 mT/min is attained. At this point the amplitude decreases quickly. The line width, taken as the FWHM of the MDESR peak , also remains unchanged until it is dropped below 150 mT/min, where a fast in crease in line width is observed.

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105 0100200300400 5.645.665.685.705.725.745.765.78 232 mV 300 mV 364 mV sweep rate = 50 mT/minRxxB0(T)Rxx-wave Power (mV on detector) Figure 4-4: Microwave power dependence of MDESR response in sample EA129. The response increases in both line width a nd amplitude as the microwave power is increased. All powers are shown in mV of output from our microwave power detector. Inset: Plot of MDESR amplitude vs. microwave power. It is clear that the response is linear within the range of our microwave system, thereby validating the low power assumption. Temperature Dependence of Overhauser Broadening of MDESR The Korringa relationship, which governs the nuclear spin relaxa tion in metals, is believed to be the dominant relaxation mechanism of nuclei in FC with a 2DES as well. This relationship, introduced in eq. (4. 13), states that in such materials, 1 nT is inversely proportional to T and the D.O.S. FE. By measuring the temperature dependence on the MDESR response under slow, DNP produci ng, sweep rates, the validity of the Korringa relationship to 2DESs can be explored. The reason for the decrease in signal breadth at T >3.5 K is unclear. As shown in Chapter 3, the MDESR signal response was quite minimal at temperatures above 4 K, so the loss of signal may be attributed to decrea sed sensitivity. As evidenced by the large amplitude of the MDESR signal at the conditi ons shown here, this seems unlikely. A second possible explanation pertains to the deviation of the temperature dependence of the magnetoresistance at =1 from an activated transport mechanism above T =4.5 K.

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106 This can be observed in Figur e 4-6, which displays the temperature dependence of the magnetoresistance (inset), al ong with the corresponding activ ation plot. Only those points which fell within a linear region were included in the calculation of the activation gap, /17.29 KBEk . At such a change in the xx R T, it is possible that the change in the conduction mechanism could cause the re laxation to deviate from the observed Korringa behavior, which also shows a dependence on D.O.S. FxxER. 5.655.705.755.805.8 5 downsweep upsweepB 0 (T) Figure 4-5: MDESR signals in EA129 for both up (red) and down-sweeps in 0 B at variable temperatures at 0dB dt =50 mT. The signal exhibits a broadening in the down swept spectra as the temperatur e is increased, this being consistent with the Korringa relationship between 1 nT and temperature.

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107 0.100.150.200.250.300.350.400.450.500.550.600.65 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 ln Rxx = 8.26036 8.64482(1/T) = 17.290 Kln Rxx1/T ( K-1 ) 5.45.65.86.06. 26.4 0 250 500 750 1000 1250 1500 EA129Rxx()B0 (T) Rxx( ) B0(T) 1/T (K-1) ln(Rxx) E 0.100.150.200.250.300.350.400.450.500.550.600.65 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 ln Rxx = 8.26036 8.64482(1/T) = 17.290 Kln Rxx1/T ( K-1 ) 5.45.65.86.06. 26.4 0 250 500 750 1000 1250 1500 EA129Rxx()B0 (T) Rxx( ) B0(T) 1/T (K-1) ln(Rxx) E Figure 4-6: Activation plot in EA129. E , determined from the linear region of the plot, reveals an activation gap of 17.290 K. At temperatures above 4.5K and below 2.75 K the dependence appears to deviate from the activated transport mechanism. Measurement of the Overhauser Shift Decay As discussed in the introduction to both Ch apters 3 and 4, a preferred method for estimating the1 nT , within the odd integers of the integer QHE, is the measurement of the relaxation of the Overhauser shift foll owing a slow, DNP producing, down-sweep through eq ESR B . In this method, the relaxation of the Overhauser shifted ESR condition, D NP ESRB , back to eq ESR B , is a direct method of measuring ndB dt . The time constant for this decay, OS , may be used as an upper limit estimation for the 1.nT Experimental The experiment is started by applying CW microwave radiation which corresponds to an equilibrium ESR condition eq ESR B , in the vicinity of =1 or another desired filling factor. Starting at 0 eq ESR B B , a slow down-sweep 010 mT/min, dB dt capable of creating sufficient DNP, is initiated. As the field is swept through eq ESR B , electron-nuclear

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108 cross-relaxation causes DNP and therefore a buildup of n B . This increaseseff B . As shown in eq. (4.14), this effective field cau ses an Overhauser shift of the ESR condition from eq ESR B to D NP ESRB . As 0 B is continuously decreas ed, provided the condition 0 ndBdB dtdt is met, the ESR condition will be essentially “locked” to 0 B and the DNP production is continued throughout the trace un til the sweep is stopped, the sweep is passes outside of the =1 minima, or a slight mismatch in the relationship 0 ndBdB dtdt causes the signal to be lost. At this point, a timer is started. Periodically, fast up and down-sweeps through the region between ,max DNP ESRstop B B, and eq ESR B are performed to determine the position of the ESR as a function of time until n B has completely relaxed. D NP ESRB vs. time is then plotted and is fitted to an exponential decay. The time constant of this decay is related to 1 nT , as the decay of the Overhauser shift will be dependent upon the nuclear spin relaxation along with other effects such as nuclear spin diffusion. Therefore we may use this time constant as an estimate of the upper limit for 1 nT . Results Previous experimentation in EA 124 within our lab reported 214 sOS at =1 and at =600.77 Presented in Figure 4-7 are two measurements of OS in sample EA124B at =1 at =00 (2.83T) and 430 (3.90T). The corresponding decay times measured were 38722 s and 46210. s, respectively. At this time, the dependence of OS on 0 B has not been performed within these samples, however, reports discu ssing this dependence

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109 via application of a gate voltage31 and the tilted field method41 are presented in the literature. 0500100015002000 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Bn=0.0704*exp(Time/OS)OS= 471 +/11 s B0=3.90T MW=24.3404 GHzTime(s) Downsweeps Upsweeps05001000150020002500 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 B0=2.86T MW=17.590GHzBn=0.057*exp(-Time/OS)OS=372 +/-22 sBn (T)Time (s)0500100015002000 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Bn=0.0704*exp(Time/OS)OS= 471 +/11 s B0=3.90T MW=24.3404 GHzTime(s) Downsweeps Upsweeps05001000150020002500 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 B0=2.86T MW=17.590GHzBn=0.057*exp(-Time/OS)OS=372 +/-22 sBn (T)Time (s) Figure 4-7: Measurement of the Overhauser shift decay of the ESR following a slow (10 mT/min) down-sweep at T=1.5 K in sample EA124B at sample tilt angles of 00 (2.83 T) and 430 (3.90 T). Decay times of 38722 s and 46210. s were measured at these angles, resp ectively, within the vicinity of =1. DNP Induced by DC Injection Using the heating mechanism for MDESR detection and the understanding of the FC hfi, we have explained the resonant microwave absorption induced DNP and Overhauser shifts at slow magnetic field sweep rates. From this understanding, other methods of exclusively heating the 2DES system may be predic ted to create DNP as well. The injection of a DC current through the 2DES was one such mechanism investigated. Experimental The MDESR was first observed under norma l conditions at T=1.6 K. Prior to making a second measurement, a DC current D C I , is applied across the sample for a period of time t , in an effort to induce DNP and an Overhauser shift of the ESR condition. Directly following this time period, the magnetic field is swept down at a fast sweep rate, under conditions where no DNP was induced, and the position of the ESR

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110 was determined. The relaxation of the ESR back to the equilibrium position, from any Overhauser shift, , may be measured via a method similar to that utilized for measurements of 1 nT . Results The DC injection induced DNP experi ment was performed in both EA124 and EA124B with similar results observed in both sa mples, therefore, we will concentrate on experiments in the latter. Pr esented in Figure 4-8 (a) are th e magnetoresistance traces at a variety of currents. As observed from the gra ph, current heating of the system occurs at 2 ADCI , with a complete breakdown of the QHE occurring somewhere in the vicinity of 25 ADCI . A series of down swept MDESR spectra ta ken after 10 minutes of exposure to DC injections of 5 A50 ADCI are presented in Figure 4-8 (b). The initial trace, in the absence of DC injection, is shown in black . The inset shows the overall shift dependent on current. There was minimal reproducibility and no defined dependence. The largest shift observed was approximately 5 mT at both 5 and 7 A within EA124B, while a shift of 10 mT was recorded in samp le EA124. No shifts larger than this were observed and there appears to be no relation between the si ze of the shift and the current used. In EA124, it was observed that currents greater than 10 A destroyed the MDESR peak, this was not the case in EA124B. During the up-sw eep (not shown), it was observed that the shift is to higher field, in direct contra diction of the hypothesis of DC induced DNP. In Figure 4-9 we see the time dependent effects of current injection upon the MDESR response at two different currents (spectra offset for cl arity). Shown in black is the initial MDESR trace prior to DC injection. The solid blac k vertical line is set to the

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111 equilibrium ESR condition. Note the minimal shift and the lack of dependence on time. At low current, it was observed that doubli ng the time caused no additional shift in the MDESR position, while at higher current, an in crease in injection time of 2.5 times, had minimal effect, actually i nducing a smaller shift. From the data obtained from the two samples, the best that can be said is the experiment was inconclusive. While a sma ll shift in the position of the MDESR was observed within a small window of applie d currents and appli cation times, no clear dependence may be extracted. Furthermore, th e finding that the shift is towards higher 0 B during an up-sweep brings into doubt the DC induced DNP hypot hesis. It is fair to say that the effect being investigated is minima l at best, and if any dependence on current or injection time exists, that they are within the error of the measurement. Experimental Results: MDENDOR ENDOR was first detected in the magnetoresistance of a 2DES in a GaAs/AlGaAs heterostructure by Dobers et al . 63 In this section we will discuss experimental results pertaining to the such observat ions and the use of this met hod to investigate the coupling of the conduction electrons and the lattice nuc lei as well as quadrupole splittings and double quantum excitations within the MDENDO R spectra. Such experimental evidence explores not only the effect the increasing nuclear field has upon the electronic spin system through the Overhauser effect, but al so gives insight into the mechanism and strength of those interactions.

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112 4.85.05.25.45.65.86.06.26.46.6 0 50 100 150 200 250 300 500 nA, 1A 2A 3A 4A 5A 25Axx()B0(T)5.255.305.355.405.455.505.555.605.655.70 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Initial Trace 5 A 7 A 10 A 50 A01020304050 0 1 2 3 4 5 6 10 min Exposure to CurrentESR Shift (mT)Current (A)(a) (b) Rxx Rxx B0(T) 4.85.05.25.45.65.86.06.26.46.6 0 50 100 150 200 250 300 500 nA, 1A 2A 3A 4A 5A 25Axx()B0(T)5.255.305.355.405.455.505.555.605.655.70 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Initial Trace 5 A 7 A 10 A 50 A01020304050 0 1 2 3 4 5 6 10 min Exposure to CurrentESR Shift (mT)Current (A)(a) (b) Rxx Rxx B0(T) Figure 4-8: (a) Current induced heating effect on the magnetoresistance. Initial heating effects are observed at sour ce-drain currents as low as2 A I , with significant heating at 5 A I and a complete breakdown of the QHE observed around 25 A I . (b) Investigation of DC current injection and the effect on the MDESR spectral features. A minor shift of at most 5 mT, is observed under application of DC fo r 10 min prior to MDESR observation, but as presented in the inset graph, there appears to be no effective dependence of the shift on th e amplitude of the current.

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113 5.405.455.505.555.60 -6 -4 -2 0 0 A, initial Trace 5 A 10 A xx()B 0 (T) Initial Trace 5 min 5 A 10 min 5 A 10 min 10 A 25 min 10 A(Offset for clarity) Rxx( ) 5.405.455.505.555.60 -6 -4 -2 0 0 A, initial Trace 5 A 10 A xx()B 0 (T) Initial Trace 5 min 5 A 10 min 5 A 10 min 10 A 25 min 10 A(Offset for clarity) 5.405.455.505.555.60 -6 -4 -2 0 0 A, initial Trace 5 A 10 A xx()B 0 (T) Initial Trace 5 min 5 A 10 min 5 A 10 min 10 A 25 min 10 A(Offset for clarity) Rxx( ) Figure 4-9: The effect of in creased DC injection time upon the shift in the MDESR. Traces have been offset for clarity. The vertical line corresponds to the equilibrium ESR condition as shown in the black curve. At 5 A, no difference in the shift is observed with increased or decreased application time. At increased currents, the shift decreases as the current is increased. The shift itself is moderately repr oducible, but the error in the MDESR position under these conditions approaches the size of the shift itself, bringing the result into doubt. Experimental We have detected MDENDOR by the fiel d swept method, and we have also developed a new method in which the field is held constant wh ile the RF field is swept. We call our new method, RF swept MDENDO R. While the field swept method allows for simpler instrumentation setup and pr ovides MDENDOR peaks within the MDESR spectrum, the RF swept method allows detect ion at a fixed magneti c field and specific Overhauser shift. This has the advantage of allowing efficient signal averaging, or efficient variation of experimental parame ters such as current, temperature and microwave/RF power without the need to repolarize the nuclei ev ery time with a DNP down-sweep or the full relaxation of the nuclear spin system.

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114 The field swept method is very similar to that used in the Overhauser shift decay experiments discussed earlier. CW microwaves are applied to the sample at a frequency corresponding to a resonant condition, eq ESR B . CW RF is also app lied, corresponding to an NMR condition for nucleus iat a magnetic field ieq NMRESR B B . The RF is applied to the sample via a multiple turn coil placed such that the RF field is perpendicular to 0 B and parallel to the conduction path in the 2DES when the sample is in the horizontal position. The coil is fastened to a semi rigid, copper coax and is attached via BNC connections to the RF source, a PTS 310 RF generator. While the exact RF power is unknown and varies with the frequency, it is clear that the power is sufficient to detect MDENDOR. Care to ensure minimal non-resonant h eating from both the microwaves and RF application is essential. With the magnetic field set such that 0 eq ESR B B , a slow 010 mT/min dB dt down-field sweep is initiated, passing through the ESR resonant condition, creating DNP and thus causing an Ov erhauser shift (and therefore broadening) of the ESR. The DNP production is continued until 0 i NMR B B , where the resonant absorption of RF depolarizes the nuclear system to some extent and as depicted in Figure 4-10 a sharp peak, in this case a ssociated with excitation of the 71Ga nuclei at 50.492 MHzNMR , is observed, superimposed upon the MDESR spectrum. As discussed in previously re ported MDENDOR experiments, MDENDOR associated with all three nuclei within the well region we re observed as shown in Figure 4-11, where MDENDOR of all three nuclei were observed during the same down-sweep, with the RF applied being changed after each MDENDOR p eak was recorded. A splitting, due to the coupling of the electric quadrupole moment to an EFG at the nucleus, is observed for

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115 75As in sample EA124B only. Further inve stigation into this quadrupole splitting is discussed in Chapter 5. The magnetic field where the MDENDOR is observed is correlated to NMR B very closely. A plot of the 75As MDENDOR peak position as a function of RF in sample EA124 is presented in Figure 4-12. A linear fit to the data provides a slope, which provide s the gyromagnetic ratio of 75As, therefore proving the origin of the MDENDOR peak as being associated with the NMR excitation. Similar experiments provided the same re sult for the two Ga nuclei. The RF swept experiment utilizes what we refer to as the stop-field method. This experiment can be understood more completely through the use of the spectra presented in Figure 4-13. Here, the expe rimental results of such an experiment in sample EA124B at T=1.5 K are presented as a function of (a ) time, (b) field and (c) radiofrequency. Within the traces, the different coordinates, ma rked in red, correspond to different events and their corresponding effect upon the different spectra. Initially the experiment is exactly the same as the field swept method, w ith the distinct exception of the application of RF energy. Upon coming into resonance with 33.70 GHz irradiation (a) , the slow down-sweep (10 mT/min) in magnetic field cau ses the typical broa dening of the MDESR associated with the field locking under DNP conditions. The fiel d sweep is stopped at s top B once a desired Overhauser shift 5.735 T, D NPeqDNP ESRStopnESRESRBBBBB is attained (b) . This stop disrupts the ESR excitation, and thus the DNP rate falls. Eventually n B relaxes, bringing the MDESR and DNP into a steady-state (c) . At this point, an RF sweep is begun, sweeping through the 75As (d) , 71Ga (f) , or 69Ga (not shown) resonance. After the sweep, the RF is turn ed off and the MDESR response returns to the

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116 steady-state value (e) . It is clear from points (c) and (e) that some non-resonant heating due to the RF irradiation is present. 3.873.883.893.903.913.923.93 26.5 27.0 27.5 28.0 28.5 29.0 Ga71 (50.492 MHz) xx()B0(T) Rxx( ) B0(T)3.873.883.893.903.913.923.93 26.5 27.0 27.5 28.0 28.5 29.0 Ga71 (50.492 MHz) xx()B0(T) Rxx( ) B0(T) Figure 4-10: MDENDOR spectra of 71Ga. Microwaves were a pplied at 23.34 GHz and RF at 50.492 MHz to sample EA124B. A 10 mT/min magnetic field sweep rate was utilized to obtain the spectrum with the sample tilted to 440 at a temperature of 1.5 K. The peak is ob served as a narrow feature superimposed on the MDESR background. 3.853.863.873.883.893.903.913.923.93 24 25 26 27 28 29 Ga71 (50.168 MHz) Ga69 (39.615 MHz) As75 (28.358 MHz) R xxB(T) B0(T) Rxx( )3.853.863.873.883.893.903.913.923.93 24 25 26 27 28 29 Ga71 (50.168 MHz) Ga69 (39.615 MHz) As75 (28.358 MHz) R xxB(T) B0(T) Rxx( ) Figure 4-11: MDENDOR spectrum for all th ree isotopes at th ree different RF frequencies, as indicated. All other pa rameters were the same as in Figure 410. Note the quadrupole splitting of the 75As signal.

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117 5.645.665.685.705.725.74 41.1 41.2 41.3 41.4 41.5 41.6 41.7 41.8 q =7.28686BNMR (T)NMR(MHz) 757.287 MHz/TnAs 5.645.665.685.705.725.74 41.1 41.2 41.3 41.4 41.5 41.6 41.7 41.8 q =7.28686BNMR (T)NMR(MHz) 757.287 MHz/TnAs Figure 4-12: Position of NMR B for 75As MDENDOR as a function of applied RF in sample EA124. The slope is used to extract th e gyromagnetic ratio, which in this case was determined to be 7.287 MHz/T, which is consistent with the values in the literature for 75As. 5.725.745.765.785.805.82 2 3 4 5 6 7 (a) (b) B0(T)02004006008001000120014001600 2 4 6 8 (f) (e) (d) (c) (b) (a) xx()Time (s)74.4074.4574.5074.5574.60 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (f) Frequency (MHz)(a) (b)(c) B0(T) Rxx( ) Time (s) Freq (MHz)5.725.745.765.785.805.82 2 3 4 5 6 7 (a) (b) B0(T)02004006008001000120014001600 2 4 6 8 (f) (e) (d) (c) (b) (a) xx()Time (s)74.4074.4574.5074.5574.60 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (f) Frequency (MHz)(a) (b)(c)5.725.745.765.785.805.82 2 3 4 5 6 7 (a) (b) B0(T)02004006008001000120014001600 2 4 6 8 (f) (e) (d) (c) (b) (a) xx()Time (s)74.4074.4574.5074.5574.60 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (f) Frequency (MHz)5.725.745.765.785.805.82 2 3 4 5 6 7 (a) (b) B0(T)02004006008001000120014001600 2 4 6 8 (f) (e) (d) (c) (b) (a) xx()Time (s)74.4074.4574.5074.5574.60 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (f) Frequency (MHz)(a) (b)(c) B0(T) Rxx( ) Time (s) Freq (MHz) Figure 4-13: The three graphs correspond to the RF swept MDENDOR experiment measured as a function of (a) time, (b) magnetic field and (c) frequency in sample EA124B at T=1.5 K. The corre sponding events and their effect upon the spectra, are presented as (a) (f) , and are described in detail within the text above. Experimental Results Microwave power and temperature dependence of MDENDOR response The 75As MDENDOR response in sample EA12 4 is presented in Figure 4-14 under the same excitation conditions with th e exception of the microwave power and temperature. In this sample, the quadrupole sp litting was not observed. It is clear from

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118 comparing (a) and (b), which were taken at the same microw ave power (364 mV on detector) but at T=2.5 and 1.4 K. The MDENDOR response was observed to mirror the response of the MDESR as a function of temperat ure, and the increase in amplitude of the MDENDOR from 1.4 to 2.5 K is consistent w ith this observation in sample EA124. The same can be said of the MDENDOR depende nce on microwave power. As seen in (b) and (c), which were excited using microwave powers measured as 364 mV and 232 mV at the detector, the signal scales with in creasing microwave power, again mirroring the dependence observed in the MDESR response. RF dependence A series of 71Ga MDENDOR spectra, obtained us ing the field swept method at varied RF frequencies, we re acquired at T=1.5 K and app=34.00 GHz 5.84 Teq ESRB , and are presented in Figure 4-15 as a function of both magnetic field and filling factor. It is observed, that within a limited range of RF frequencies (traces h-e, 75.644 MHz75.170 MHz), that as the RF is reduced, thereby moving NMR B further below eq ESR B (i.e. ESRNMRB is increased), that the amplitude of the MDENDOR increases while the xx R returns to the level observed prior to th e NMR excitation. As the frequency is reduced further (traces d-a, 74.994 MHz-74.513 MHz), the MDENDOR becomes increasingly narrow and is followed by a large drop in xx R to a value below that of the pre-NMR MDESR. The amplitude of the peak and the amount of MDESR retained following the NMR depolarization decrease, as the frequency is reduced. Eventually a point is reached, as in trace a, where no p eak is observed and a complete loss of the MDESR condition to its non-resona nt value is realized.

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119 5.655.705.755.8 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 75As MDENDOR364 mV, 2.5 K 232 mV, 1.4 K 364 mV, 1.4 KRxx (arb. units)B0 (T) Figure 4-14: 75As MDENDOR response in sample EA 124 at two different temperatures, 2.5 K (blue) and 1.4 K (black) and microwave powers 364 mV (black) and 232 mV (red) read by the microwave de tector. The MDENDOR response has been observed to mirror that of th e MDESR response as a function of temperature and microwave power. The change of the MDENDOR response, as the NMR condition is moved further down-field from eq ESR B , from a peak, at small Overhauser shifts, to a step-shaped drop off, at large Overhauser shifts, co rresponds to the presence of a critical field, beyond which, a de-pinning of the ESR condition to 0 B is observed. This de-pinning occurs when the depolarization of the nuclei due to NMR ex citation causes a drop in the nuclear field n B that surpasses a critical value crit n B . A qualitative explanation of these observations is discussed below.

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120 5.685.725.765.805.845.8 8 0 1 2 3 4 5 a b c d e f g h (h) 75.644 (g) 75.486 (f) 75.328 (e) 75.170 (d) 74.994 (c) 74.734 (b) 74.578 (a) 74.513Rxx()B0(T)1.0451.0401.0351.0301.0251.0201.015 Figure 4-15: 71Ga MDENDOR response at a variety of RF frequencies after a slow (4.6 mT/min) down-sweep through eq ESR B =5.84 T ( =34.00 GHz). Starting with spectrum (h), the response is observed as a very small spike superimposed upon the MDESR background. Foll owing the excitation, the xx R returns to its value prior to the MD ENDOR excitation. At sm all Overhauser shifts, upon reducing the RF, the MDENDOR peak amp litude is seen to increase (traces he). Once the loss in nuclear polarization n B surpasses a critical value crit n B , the MDENDOR peak becomes increasingly narrow, decreases in amplitude and is accompanied by a loss of the MDESR condition (traces da). At sufficiently large Overhauser shifts, the presence of a peak is no longer observed and is replaced by a complete loss of the MDESR condition to the non-resonant value. If the NMR condition, and therefore the MDENDOR excitation, occurs such that crit nn B B , the MDENDOR is observed as a sudden spike inxx R . This change in nuclear field shifts the system closer to the maximum ESR absorption and therefore is due to a sudden increase in microwave absorp tion. This increased absorption is due to the reduction in n B and the Overhauser shift from the NMR depolarization associated with the MDENDOR excitation. This brings the ESR condition closer to its absorption maximum. The increased absorption is conc urrent with an increase in the DNP rate.

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121 This causes a rapid repolarization, which compensates for the NMR depolarization, returning n B to its pre-NMR value. This allows the pinning of the ESR condition to 0 B to be re-established. If the NMR is excited at suffici ently large Overhauser shifts crit nn B B , the NMR condition causes a large nuclear depolari zation, destroying a significant portion of n B such that the entire ESR line is rapidly swept through the resona nt condition over the range 0 n B B to 0 nn B BB . Due to the short time sp ent on resonance during this sweep, an extremely narrow peak is observed. As ESRNMRB is increased, so too is n B , and the time spent on resonance is reduced, therefore smaller peaks are recorded. The minimal time spent on resonance is also accomp anied by a lack of nuclear repolarization, resulting in the ESR pinning condition to be lo st completely and a drop in the MDESR is registered. The remaining MD ESR signal decays as a function 1 nT and the magnetic field sweep rate. Roughly speaking, th e critical loss in nuclear fiel d is equivalent to the line width of the ESR absorption, critESR nFWHMB . Numerical simulations based on our heating model for MDESR, testing this theory , are discussed in detail in the Chapter 6. RF power dependence Presented in Figure 4-16 are two series of MDENDOR spectra at varied RF frequencies under the same conditions but at (a) the maximum RF power output (same as Figure 4-15) by the RF source and (b) half of the maximum output. Figure 4-16 (c) displays the MDESR under the same conditions, but with non-re sonant RF applied. It should be noticed that the RF , and therefore the magnetic fi eld, associated with the critical transition crit nnBB , is lower (pushed down-field ) in the series obtained at

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122 half power, but the overall featur es and qualitative behavior remain the same. Two main points can be drawn from this observation. Firs t, it is clear that since there is a distinct difference between the two RF powers, the NMR excitation is not completely depolarizing the nucleus of interest, otherw ise power changes would have no effect. Secondly, this supports our hypothesis, sin ce at lower powers, le ss overall nuclei are excited by the application of the RF, and therefor e less of the nuclear field is destroyed. This means that larger overall nuclear pol arizations must be established for the corresponding NMR depolarization to meet the critical condition. 5.685.705.725.745.765.785.805.825.845.865.88 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 74.734 MHz 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 75.170 MHz 75.644 MHz 75.486 MHz 75.328 MHz 75.170 MHz 74.994 MHz 74.734 MHz 74.578 MHz 74.513 MHz Rxx( )B0(T)(a) (b)(c)5.685.705.725.745.765.785.805.825.845.865.88 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 74.734 MHz 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 75.170 MHz 75.644 MHz 75.486 MHz 75.328 MHz 75.170 MHz 74.994 MHz 74.734 MHz 74.578 MHz 74.513 MHz5.685.705.725.745.765.785.805.825.845.865.88 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 74.734 MHz 5.685.705.725.745.765.785.805.825.845.865.88 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 75.170 MHz 75.644 MHz 75.486 MHz 75.328 MHz 75.170 MHz 74.994 MHz 74.734 MHz 74.578 MHz 74.513 MHz Rxx( )B0(T)(a) (b)(c) Figure 4-16: (a) MDENDOR response at a series of RF frequencies taken from Figure 415. (b) MDENDOR response at the same conditions as (a), but under a broader range of RF freque ncies applied at half of the RF power. Reduction in RF power is associated with a shift in the critical field to lower 0 B . (c) MDESR under the same conditions, but under application of non-resonant RF. The MDENDOR amplitude and the loss of the MDESR condition following MDENDOR excitation as a functi on of the Overhauser effect ESRNMRB at both RF powers are presented in Figures 417 (a) and (b) respectively. In this figure, the increase in ESRNMRB necessary to see the drop in amplitude under lower RF power irradiation is clear. It is not so clear from Figure 4-16 that at lower RF powers the MDENDOR maximum amplitude attained is smaller as well. The drop off in MDENDOR peak amplitude appears to be exponential with incr easing Overhauser shift, once the critical

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123 field is surpassed. Furthermore, from (b), it is observed that the magnitude of the loss in MDESR following the MDENDOR excitation is less dramatic at lower RF powers. 0.000.020.040.060.080.100.120.14 0.0 0.5 1.0 1.5 2.0 2 . 5 Amp ()Besr-nmr Max RF Power Half Power 0.000.020.040.060.080.100.120.14 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 Amplitude of loss ()Besr-nmr(a) (b) BESR-NMR BESR-NMR MDENDOR Amplitude ( ) MDESR Loss ( )0.000.020.040.060.080.100.120.14 0.0 0.5 1.0 1.5 2.0 2 . 5 Amp ()Besr-nmr Max RF Power Half Power 0.000.020.040.060.080.100.120.14 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 Amplitude of loss ()Besr-nmr(a) (b) BESR-NMR BESR-NMR MDENDOR Amplitude ( ) MDESR Loss ( ) Figure 4-17: (a) MDENDOR amplitude as a function of ESRNMRB at both maximum (black squares) and half of the maxi mum (red circles) RF powers in sample EA124B at T=1.5 K. A reduction in power leads to a smaller maximum MDENDOR amplitude attained as well as pushing the critical condition to lower 0 B . The decay in signal after surpa ssing the critical field appears to occur on a faster time scale as well. (b) The resultant loss of the MDESR signal following MDENDOR excitation. Similar to (a), a reduction in RF power led to a faster transition from minimal to maximum loss of the ESR condition, along with pushing the critical condition to lower 0 B . Concluding Remarks Presented within this chapter were the ma gnetic field sweep rate dependence of the MDESR response and the corresponding microw ave power and temperature dependences of this response at slower sweep rates. The reduction is sweep rate brings about a broadening of the MDESR signal response due to dynamic Overhauser shifting of the ESR condition. The microwave power depe ndence provides evidence that the MDESR excitations are occurring in the low power regi me and therefore saturation effects of the MDESR can be ignored. Increases in MDESR amplitude and line width, with increased microwave powers, were also observed. Th e temperature dependence of the MDESR at

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124 50 mT/min presented a dependence that is c onsistent with the Korringa relation between 11/nT and temperature. The nuclear spin relaxation time was determined at =1 within our samples via the relaxation of the Overha user shifted ESR condition following a slow DNP producing down-sweep in magnetic field a nd the results obtained were close to the values previously observed under such conditions. Along with the MDESR results, the MDENDOR spectra for all three nuclei within the well region were observed through both a field swept a nd RF swept method. The dependence of the MDENDOR response upon temperature and microwave power was determined to mirror that of the MDESR respons e. A distinct change in behavior of the MDENDOR response was observed to occur at RF powers and Overhauser shifts such that the reduction in the nuclear field due to NMR depolarization, n B , is greater than a critical nuclear field, crit n B . This critical nuclear field is approximately equivalent to the FWHM of the ESR line width. The observations presented in this chapter will be utilized within Chapter 6 in discussing numerical simulations that were performed to test our theoretical model for MDESR/MDENDOR. These simu lations are based on the heating model discussed in Chapter 3 and the descriptions of the hfi and their eff ect upon the spectra discussed within this chapter. The data and simulations together provide a cl ear, broad picture of the mechanism behind MDMR and offer insight into the complex interactions between the conduction electrons and the nuclei of the host lattice.

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125 CHAPTER 5 OBSERVATION OF NUCLEAR QUAD RUPOLE SPLITTING AND DOUBLE QUANTUM EXCITATIONS IN THE MDMR SPECTRA OF GaAs/AlGaAs QUANTUM WELLS Nuclear Quadrupole Splitting of 75As MDENDOR If a solid lattice lacks inversion symmetry at a nuclear site with 1/2 I a quadrupole splitting of the NMR transitions may be observed. Such splittings may also be observed in a crystal with i nversion symmetry (such as a cubi c lattice structure) if that symmetry is removed due to a distortion of the lattice. This distortion may be realized via an inherent defect or if a force causes such distortions, either via internal (lattice mismatch within superlattice structures) or external (induced strain via a bending or twisting of the sample) sources. The lifting of this symmetry causes a distortion of the naturally symmetric electronic orbital motion, thereby creating an internal EFG about the nucleus. As discussed in chapter 1, this EFG will couple to th e nuclear quadrupole moment, thereby causing a splitting of the NMR spectrum. NMR, which is known to be a sensitive technique for measuring many structural and chemical environments, is also sensitive to variations in the lattice constants, as is observed at the interfaces within semiconducto r superlattices, for instance within GaAs and AlxGa1-xAs quantum well devices, providi ng sensitivity of less than 10-5.78 Observation of a quadrupole splitting within the NMR or ENDOR spectrum of such a device indicates the presence of a strain upon the sample that will influence many of the electrical properties of the device. Exploring the quadr upole splitting can therefore provide insight into various for ces that are acting upon the sample.

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126 Previous Observations Quadrupole splittings of the NM R resonance lines in GaAs/AlxGa1-xAs quantum wells and heterostructures have been prev iously reported in the literature. This quadrupole splitting of the 75As resonance is larger than the other isotopes because this nucleus has the largest quadr upole coupling constant of the nuclear isotopes within GaAs. Quadrupole splitting has been observed via optic ally detected (observation of changes to the electron photoluminescence spectrum),78-82 optically pumped,66 and optical Larmor beat detection83 of NMR, within the MDENDOR spectra associated with edge states of a quantum well84 and within the MDNMR spectra75 of GaAs/ AlxGa1-xAs quantum wells and heterostructures. Observation of Quadrupole Splitting As is apparent in the CW NMR spectr um acquired via the field swept MDENDOR method using three different RF frequencies pertaining to the thr ee different NMR active nuclei within GaAs, which is presented in Figure 5-1, no quadrupole splitting is observed on isotopes other than 75As. The magnitude of the splitt ing is approximately 5.1 mT in this spectrum. Using the quadrupole c oupling constants for the three nuclei (75AsQ =2.9X10-29 m2, 71GaQ =1.2X10-29 m2, and 69GaQ =1.9X10-29 m2 85), the expected splittings for 71Ga and 69Ga can be inferred to be approximately 2.1 and 3.3 mT, respectively. Since the expected splitti ngs exceed the observed MDENDOR line widths of 1.4 and 1.6 mT, for 71Ga and 69Ga respectively, we would expect that the splitting should be resolved for all three isotopes. Brun et al.86 have reported that the EFG centered at the As sites is a f actor of 1.3 times greater than the EFG at the Ga nuclei. In that study, the magnitude of the EFG was inferred from 1 i nT measurements. However,

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127 even after correcting for this difference, the splittings of the 71Ga should still be resolved. The absence of a splitting on the Ga NMR tran sitions could be due to the inability to resolve them. The 69Ga and 71Ga lines do appear to be broader than the individual transitions of the 75As spectrum, so it is entirely possibl e that this is due to an unresolved quadrupole splitting. 3.853.863.873.883.893.903.913.923.93 24 25 26 27 28 29 Ga71 (50.168 MHz) Ga69 (39.615 MHz) As75 (28.358 MHz) Rxx( )B0(T)3.853.863.873.883.893.903.913.923.93 24 25 26 27 28 29 Ga71 (50.168 MHz) Ga69 (39.615 MHz) As75 (28.358 MHz) Rxx( )B0(T) Figure 5-1: MDENDOR spectra for all three NMR active nuclei within GaAs obtained using the field swept method. All thr ee NMR transitions associated with the lattice nuclei of GaAs were obser ved. A quadrupole splitting of the 75As MDENDOR spectrum was observed that was not present in the spectra of the other two nuclei. As shown in Figure 5-2, the quadrupole sp litting is observed in both the field and RF swept 75As MDENDOR. If the 75As gyromagnetic ratio of 7.29 MHz/T is used to convert the splitting from units of frequency to units of field, we find that the splitting is approximately the same as in the field swep t spectrum. The advantage of the frequency swept method is that it allows us to obtain multiple spectra within the same DNP down-

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128 sweep, thereby permitting efficient signal aver aging. For this reason, all further measurements have utilized the freque ncy swept experimental technique. 29.4029.4529.5029.5529.60 3.0 3.2 3.4 3.6 3.8 4.0 Frequency (MHz)3.973.983.994.00 0.0 0.5 xx ()B0 (T) 5.1 mT 39.4 kHz=5.4 mT(a) (b) B0(T) Freq. (MHz) Rxx( )29.4029.4529.5029.5529.60 3.0 3.2 3.4 3.6 3.8 4.0 Frequency (MHz)3.973.983.994.00 0.0 0.5 xx ()B0 (T) 5.1 mT 39.4 kHz=5.4 mT(a) (b)29.4029.4529.5029.5529.60 3.0 3.2 3.4 3.6 3.8 4.0 Frequency (MHz)3.973.983.994.00 0.0 0.5 xx ()B0 (T) 5.1 mT 39.4 kHz=5.4 mT(a) (b) B0(T) Freq. (MHz) Rxx( ) Figure 5-2: 75As MDENDOR spectrum obtained via the (a) field-swept and (b) RF-swept methods. As shown in the plots, the overall splitting is approximately the same when slight differences in the angle (43 and 410, respectively), microwave frequency (24.3 and 29.1 GHz, respectively) and filling factor, which are present between the scans, are incorporated. RF Sweep Rate Dependence of Quadrupole Splitting In order to obtain accurate measurements of the 75As splitting, it is necessary to determine if any artificial changes to this va lue are created based on the sweep rate of the RF. Shown in Figure 5-3 are the 75As MDENDOR spectra obta ined in sample EA124B at 1.5 K, 00, using a microwave frequency of 17.590 GHz and s top B =2.8 T pertaining to positive (RF is swept from low to high fre quency) RF sweep rates of 1 (in red), 2 (in green), and 5 (in blue) kHz/s. The faster sweep rates tend to asymmetrically broaden the peaks and shift them towards the side opposite the RF star ting point. Close inspection however shows there is no overall change in the quadrupole splitting. Therefore, the RF sweep rate can be ruled out as a factor in this study.

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129 20.3820.4020.4220.4420.4620.4820.50 -0.02 0.00 0.02 0.04 0.06 0.08 Shift of Central Transition = 2.5 kHzxxRF Freq (MHz) 1 kHz/s 2 kHz/s 5 kHz/s Rxx( ) Freq. (MHz)20.3820.4020.4220.4420.4620.4820.50 -0.02 0.00 0.02 0.04 0.06 0.08 Shift of Central Transition = 2.5 kHzxxRF Freq (MHz) 1 kHz/s 2 kHz/s 5 kHz/s Rxx( ) Freq. (MHz) Figure 5-3: RF sweep rate dependence of the 75As MDENDOR response. The spectra were obtained following the sa me stop-field initialization, s top B =2.8 T, using a microwave frequency of 17.59 GHz and at temperature of 1.5 K. A shift of 2.5 kHz to higher frequency (during upsweeps in RF), in the position of the central transition is observed when the sweep rate is increased from 1 to 5 kHz/s. Also associated with this cha nge in rate is a decrease in the overall amplitude of the signals. RF Power Dependence of Quadrupole Splitting One possible source of the quadrupole spli tting is the Linear Quadrupole Stark Effect (LQSE), first theorized by Bloembergen87 and then experimentally observed by Kushida and Saiki88 in 1961. In this effect, an exte rnal electric field, while unable to couple directly to the quadr upole moment, may distort the el ectronic wave function about the nucleus, thereby creating an EFG at the nuclear site, this in turn will couple to the quadrupolar moment of the nucleus.80, 87-90 One possible source of this electric field is from the applied RF irradiation as observed by Brun et al.86 If the electric field associated with the RF excitation influen ces the splitting of the MDENDOR spectrum, then the splitting should be de pendent upon the total RF power.

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130 Shown in Figure 5-4 are two MDENDOR spectra for 75As at the maximum output power (in red) and 50% of the maximum power (in blue). It is clearly evident that no change in the observed splitting occurs. In order to determine that the change in RF power is sufficient to create a ch ange in the MDENDOR response, the 75As MDENDOR amplitude, at a sweep rate of 1 kHz/s, for all three quadrupole split peaks, was measured as a function of the RF Power, and is presente d in Figure 5-5. All three peaks show very similar power dependence and a clear trend away from a linear dependence leads to the clear understanding that the RF power does significantly a ffect the MDENDOR features, but not the quadrupole splitting. Therefore, while the source of th is splitting may be affected by an interaction with an external el ectric field via the LQSE , it is clear that the source of this electric field is not from the applied RF radiation. 20.3820.4020.4220.4420.4620.4820.50 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Freq. (MHz) Maximum RF Power 50% Max RF Power Rxx( , Offset for clarity )20.3820.4020.4220.4420.4620.4820.50 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Freq. (MHz) Maximum RF Power 50% Max RF Power Rxx( , Offset for clarity ) Figure 5-4: 75As MDENDOR spectra taken at the maximum and 50% of the maximum RF power output from the RF source. An offset of 0.117 was added to the red trace.

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131 123456 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 Peak Amplitude (Rxx)RF Synthesizer output (VPP) Left Satellite Central Transition Right Satellite Figure 5-5: RF Power dependence of the central (in black), left sa tellite (in blue) and right satellite (in red) peaks of the spectra. Notice the increase in amplitude between the central and satellite peaks along with the overall similarities in the dependence on RF Power. Tilt Angle Dependence of Quadrupole Split ting; Investigation into Strain Effects As stated in the Chapter 1, one possible source of the quadrupole splitting is from a uniaxial strain at the interface between the quantum well a nd barrier layers. This strain originates from a slight mismatch between the usually nominally lattice matched semiconductor layers. The Hamiltonian for the quadrupole interaction was first introduced in Chapter 1 and may be expressed as4 22ˆˆˆ 3 ˆˆ 421QzzZeQ HVII II (5.1) As discussed in chapter 1, the quadrupole splitting (in Hz) may then be expressed as follows:

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132 23cos1 2zz QeQV h (5.2) So from eq. (5.2), it is clear that the overa ll splitting of the magnetic resonance spectra, under a uniaxial strain applied along the z-axis, is not only due to zzV , but is also dependent upon , the tilt angle of the sample. Ther efore, the splitting should exhibit a 23cos1 dependence provided the splitting is due to a cylindrically symmetric EFG aligned along the z-axis. This implies that th e splitting will disapp ear at a tilt angle of 54.70. In sample EA124, no quadrupole splitting of the 75As MDENDOR spectra was observed under normal conditions, but a uniaxia l strain was induced within the sample when it was fastened to a silicon substrat e with epoxy. This was manifested as a quadrupole splitting of the OPNMR spectrum.66 A comprehensive study of the dependence of the splitting on th e tilt angle showed the expected dependence, therefore assuring that the source of this splitting was due to the lattice mismatch from the attachment to the silicon base. As stated, no change to sample EA1 24B was required to induce the observed quadrupole splitting, and amazingly, this sample did not display the expected tilt angle dependence. Presented in Figure 5-6 are two 75As MDENDOR spectra taken at tilt angles of 6.92 and 63.920, both at approximately =1. A clear increase in the overall splitting is seen at the larger tilt angle.

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133 20.7020.7520.8020.8520.9020.9521.00 0 1 =63.91 6 =6.920 As Splitting vs Theta Vertical Offsets used fo r RF Freq Shifted -24 for comparison pur p xx ( ) RF Freq (kHz) 0.00 0.05 0.10 0.15 -0.05 -0.10 -0.15 63.916 06.92Center Freq 20.85 MHz 63.920 6.920Center Freq 45.04 MHz RF Freq (kHz) Rxx( , Offset for Clarity) Freq. Offset (kHz)20.7020.7520.8020.8520.9020.9521.00 0 1 =63.91 6 =6.920 As Splitting vs Theta Vertical Offsets used fo r RF Freq Shifted -24 for comparison pur p xx ( ) RF Freq (kHz) 0.00 0.05 0.10 0.15 -0.05 -0.10 -0.15 63.916 06.92Center Freq 20.85 MHz 63.920 6.920Center Freq 45.04 MHz RF Freq (kHz) Rxx( , Offset for Clarity) Freq. Offset (kHz) Figure 5-6: 75As MDENDOR spectra taken at approximately =1 within sample EA124B at 6.92 (blue) and 63.920 (red). The center frequencies have been noted in the plot. Notice the large increase in the splitting associated with such a change in the tilt angle. An offset of 0.75 was added to the blue trace. A full study of the angle dependence of th e quadrupole splitting was performed and is presented in Figure 5-7. The sp ectra were all taken at filling factor =1 0.15 to remove any possible effects of the filling factor upon the splitting. Due to the necessity of measuring within the =1 minimum, it was not possible to measure the splitting at different angles while maintaining the magnetic field and microwave frequency constant. The magnetic field should not have any affect upon the splitting however, as is evidenced by its absence in eq. (5.1). From this plot, it is clear that the expected 23cos1 is not observed. On the contrary, the splitting appears to increase almost linearly with increasing tilt angle. This is clear eviden ce that the source of the splitting is most assuredly not due to a uniaxial strain from a simple lattice mi smatch, as described above.

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134 -10010203040506070 20 25 30 35 40 45 50 pgp T~1.5K75As Splitting (kHz)Normal Angle (deg) Sample Tilt Angle (degrees) -10010203040506070 20 25 30 35 40 45 50 pgp T~1.5K75As Splitting (kHz)Normal Angle (deg) Sample Tilt Angle (degrees) Figure 5-7: Angle dependence of the quadrupole splitting of the 75As MDENDOR response. Irregardless of th e large scatter in the data , a clear increase (~250% from 0 to 63.920) in the splitting is observed upon increasing the tilt angle away from the perpendicular orientati on. All spectra were obtained at =1 within sample EA124B at a temperature of approximately 1.5 K. Current Dependence of the Quadrupole Splitting The electrical contacts conn ecting the quantum well with the external electrical leads may also be a source of an electric fiel d that may induce a quadrupole splitting. As electrons travel through the semiconductor-met al interface where th e well and contacts are connected at the drain, a Schottky barrier may be created. A Schottky barrier is created when an electron passes through an interface such as the one described above, leaving an ionized state behi nd within the semiconductor laye r. An EFG, due to the creation of a Schottky barrier, will be created in a direction parallel to the growth axis (zaxis).80

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135 The strength of the EFG asso ciated with such an inte raction should be dependent upon the number of ionized sites left by th e exiting electrons, which in turn should depend upon the source-drain current, SD I , through the sample. Measurements of this type were made following a stop-field experi ment. Each experiment was made following a re-establishment of the steady-state ESR c ondition in the absence of RF excitation and in the presence of the new value of the applie d current. The RF sweep is then initiated and the MDENDOR spectra recorded. The quadrupole splittings observed in the MDENDOR spectra taken at multiple SD I are presented in Figure 5-8. It is clear that no dependence upon the current is observed. The reduction in amplitude with an increase in SD I from 2 to 5 V is presumably due to current heating of the sample. From these observations, the hypothesis fo r the origin of the quadrupole splitting being from the creation of a Schottky ba rrier at the semiconductor-metal interface between the quantum well and the electrical contact appears to be lacking. This idea is further eroded due to the tilt angle dependence discussed in the previous section. As the EFG from this effect would be oriented along the growth axis, this would be analogous to a uniaxial strain upon the sample, th erefore should al so exhibit the 23cos1 dependence, which was not observed.

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136 Frequency (MHz) Rxx(Offsetfor clarity)41.9542.0042.0542.1042.15 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 0.002 A 0.005 A 0.20 A 2.00 A 5.00 A Frequency (MHz) Rxx(Offsetfor clarity)41.9542.0042.0542.1042.15 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 0.002 A 0.005 A 0.20 A 2.00 A 5.00 A Figure 5-8: Current dependence of the quadrupole splitting of the 75As MDENDOR taken at =62.10 in sample EA124B at =1 and s top B =6.05 T. The currents used are noted in the figure. Note the lack of dependence upon the current. The overall trend shows an increase in the response with increasing current, but a decrease is seen when the current is increased above 2.00 A, presumably due to current heating of the sample. The 5.0, 2.0, 0.2 and 0.005 A traces were offset by 8.0, 6.0, 4.5 and 2.0 from the 0.002 A, respectively. All traces were initially offset to 0 . Filling Factor Dependence of Quadrupole Splitting A full filling factor dependence of the qua drupole splitting at multiple angles was performed. The experiment was performed utilizing the RF swept MDENDOR method. Initially the system was allowed to relax to thermal equilibrium. A field sweep to the desired stop field was initiated with the RF off. After the steady-state was attained, an RF sweep through the NMR condition was started. Three sweeps were performed for

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137 each position. Following the completion of th e measurement at that filling factor, the system was allowed to re-equilibrate before another measurement was performed. Each data point represents the average of the thre e sweeps acquired. For each measurement, it was required to change the microwave fr equency (and therefore microwave power), stopping field and RF. A separate study inve stigating the dependence of the quadrupole splitting on the magnitude of the Overhauser shift was inconclusive, therefore each experimental trial consisted of an Overhaus er shift of approximately 30 mT +/6 mT. Presented in Figure 5-9 is the fi lling factor dependence of the 75As quadrupole splitting within the =1 minimum at an angle of =57.60. The splitting, while showing a larger variation than the error in the measur ement, (+/1.3 kHz), the experiment when performed at other angles was not reproducib le. Looking at the re presentative spectra shown in the inset of Figure 5-9, it is appare nt that the variation in the line width of the MDENDOR measurement may account for the irre producible results. Finally, inherent within the experiment itself was the effect of the Overhauser shift, microwave frequency and power upon the splitting, therefore any imp act of these parameters upon the splitting could not be separated. Therefore, it may be concluded that while a dependence upon the filling factor cannot be ruled out, it appears th at there is no resolvable dependence of the quadrupole splitting upon th e filling factor.

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138 0.900.920.940.960.981.001.021.041.061.08 40 42 44 46 48 50 52 54 75As Splitting vs Filling Factor in Frequency Units75As Splitting (kHz)Filling Factor -0.10-0.050.000.050.10 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Filling Factor of 0.978 Scaling Factor 7.48 Filling Factor of 1.071 R xx (Arb. Units)Fre q Offset ( MHz ) 0.900.920.940.960.981.001.021.041.061.08 40 42 44 46 48 50 52 54 75As Splitting vs Filling Factor in Frequency Units75As Splitting (kHz)Filling Factor -0.10-0.050.000.050.10 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Filling Factor of 0.978 Scaling Factor 7.48 Filling Factor of 1.071 R xx (Arb. Units)Fre q Offset ( MHz ) -0.10-0.050.000.050.10 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 Filling Factor of 0.978 Scaling Factor 7.48 Filling Factor of 1.071 R xx (Arb. Units)Fre q Offset ( MHz ) Figure 5-9: De pendence of 75As quadrupole splitting on filling factor at an angle of 57.60 and a temperature of 1.5 K. The data presented are representative of the dependence observed at this angle. Th is dependence was not reproducible at other angles. Inset: spectra taken at 1. 071 (red, with a scaling factor of 7.48) and 0.978. A clear increase is seen as the filling factor is reduced at these conditions. Overhauser Shift Dependence of Quadrupole Splitting The dependence of the quadrupole splitting on the magnitude of the Overhauser shift during the stop field preparation fo r the MDENDOR experiments should provide insight into whether the origin of the split ting is associated with the FC between the quadrupolar nuclei and the resonant ly excited electron spins. Presented in Figure 5-10 is the 75As MDENDOR spectra after a stop field preparation of n B =11.0 (red) and 47.7

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139 (blue) mT, s top B =5.547 T and a microwave frequency of 32.406 GHz. Clear from these spectra, is not only the improvement in signal to noise with increased n B , but also the absence of any clear dependence of the splitt ing upon the degree of the Overhauser shift. Therefore, it is clear that the origin of the splitting is not directly associated with the microwave excitation of the el ectron spins within the 2DES. -0.10-0.050.000.050.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 Scaling Factor = 9.81 Center Freq. 40.35 MHz Center Freq. 40.07 MHz 11.00 mT Shift 47.71 mT Shiftxx (arb. units)Frequency offset (kHz) Rxx( , offset for clarity) Freq. Offset (MHz) -0.10-0.050.000.050.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 Scaling Factor = 9.81 Center Freq. 40.35 MHz Center Freq. 40.07 MHz 11.00 mT Shift 47.71 mT Shiftxx (arb. units)Frequency offset (kHz) Rxx( , offset for clarity) Freq. Offset (MHz) Figure 5-10: 75As MDENDOR spectra taken after a slow DNP producing down-sweep (4.6 mT/min) through the ESR condition determined by the 32.40 GHz microwaves used. A stop field of 5.5355 T (red) and 5.4988 T (black) were used pertaining to Overhauser shifts of 11.00 and 47.71 mT, respectively. A clear increase in the peak amplitude is observed as the overall Overhauser shift is increased (similarly to the RF dependence of the MDENDOR amplitude discussed in Chapter 4), but the quadrupole splitting is unaffected. The center frequencies for the spectra are depicted in the plot. The two traces were initially offset to 0 . Discussion of the Origin of the Quadrupole Splitting From the experiments discussed within this ch apter to this point, it is clear that the origin of the quadrupole splitting is not due to one of the mechanisms introduced thus far.

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140 In this study, the idea of a uniaxial strain due to either a simple lattice mismatch, deformation of the sample, or from formation of a Schottky barrier ha ve been ruled out. The interaction of the electric field from the applied RF has also been eliminated. Thus, while many possible mechanisms have been ru led out, we are no closer to determining the origin of this splitting. On the other hand, the possibility of this splitting being due to an externally applied or induced electric fi eld, via the LQSE, has not been eliminated. Studies investigating the eff ect of an externally app lied strain upon the degree of quadrupole splitting within the 75As ODNMR spectrum of a GaAs/AlGaAs quantum well sample78, 80 determined a clear linear dependenc e of the splitting upon the size of the strain. Another possibility is associated with an EFG due to electron migration from the doping layers and barrie rs to the well region.80 However, no such observations have been reported in the literature. No reports of a linear dependence of the splitting on the sample tilt angle have been observed either. Theref ore, no definitive conclusion about the origin of the quadrupole splitting within this sample can be made. Further investigation into this effect is required. Double Quantum/Overtone Excitations The observation of double quantum NMR, pertaining to 2Im excitations of quadrupolar nuclei, may be due to one of a few different mechanisms. These mechanisms include overtone NMR excitations at 2appNMR ,91 coupling of an external electric field with the quad rupolar moment via the LQSE,89, 92 double absorption of energetic photons of 1IEEm by the same nuclei,91 through acoustic resonance,93, 94 or via single photon 1Im excitations within systems involving

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141 induced dipolar order via optical pumping.95, 96 The purpose of this section is to introduce and discuss such excitations obs erved in GaAs/AlGaAs quantum wells via MDENDOR spectroscopy. Previously Reported Observations Double quantum 2Im excitations via the re sonant absorption of electromagnetic radiation at twice the Larmor frequency have been re ported within both bulk GaAs single crystals and within GaAs /AlGaAs quantum wells. In most cases, a splitting of the NMR line was observed by si ngle-photon, double-quant um excitations. These excitations have been reported in bul k GaAs single crystals under the application of an external electri c field via the LQSE,90, 92 and within GaAs quantum wells via ODNMR79, 81, 82 through detection of the photolumin escence polarization via the Larmor beat detection method,89 and by means of Time Resolved Faraday Rotation (TRFR) spectroscopy.97 Within this last report, observations of excitations at 1/2 and 2/3 of the Larmor frequency were also weakly observed. This obs ervation was also reported by Vega et al.98 within an oriented crystal of perdeuterated oxal ic acid dihydrate, 2 2COOD2DO. No determination of the origin of these excitations was reported. Observation of 2Im Excitations A clearly detected single-photon, double-quantum excitation of the 75As nuclei was observed in sample EA124B and is presente d in Figure 5-11 at tw o different sets of microwave frequencies, stopping field values, and angles. This is the first observation of 2Im transitions via the MD method. As shown in Figure 11 a, the double quantum (in blue) excitations appear at twice the resonant fr equency as the observed 1Im (in black) excitations. Two peaks are observ ed due to the two possible excitations,

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142 3/21/2Im and 1/23/2Im , which occur at frequencies of 2appNMRQ , respectively. These transitions ar e indicated in Figure 5-12. From the positions of these peaks, it is easy to see that the separation (splitting) of the peaks is equivalent to the separation of each satell ite and the central transitions in the 1Im spectrum (shown in black in Figure 5-11). The overall amplitude of the 2Im excitations was much smaller than that of the corresponding 1Im spectra. Due to variations of the RF power associated with changes in the frequency, an exact correla tion between the overal l magnitude of the absorptions at the two different frequencies wasnÂ’t possible, but it is clear that the single quantum spectra are more strongly excited. The lower signal-to-noise of the double quantum signal required signal av eraging. It is clear from the minor differences in the changes in xx R due to RF heating of the sample, that the powers associated with the two different frequencies are not su fficiently different to explain the large difference in signal intensities. Such a reduction in signal intensities was also reported by Eickhoff et al.79 in RF induced changes within the Hanle curve under photoluminescence polarization measurements.

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143 a) b)83.9083.9584.0084.0584.1084.1584.2084.2584.3084.35 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 scans Center Freq. 42.06 MHz Scaling Factor = 25 4 scans Center Freq. 84.12 MHzxx (arb. units)RF Freq (MHz) mI=+/-1 mI=+/-2 0.00 0.050.100.150.20 -0.05 -0.10-0.15 -0.20-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 mI=+/-1 mI=+/-2 2 scans Center Freq. 34.10 MHz Scaling Factor = 10 10 scans Center Freq. 68.21 MHzxx (arb. units)0.000.100.20 -0.10 -0.20 -0.30 Freq. Offset (MHz) Rxx( , offset for clarity)a) b)83.9083.9584.0084.0584.1084.1584.2084.2584.3084.35 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 scans Center Freq. 42.06 MHz Scaling Factor = 25 4 scans Center Freq. 84.12 MHzxx (arb. units)RF Freq (MHz) mI=+/-1 mI=+/-2 0.00 0.050.100.150.20 -0.05 -0.10-0.15 -0.20-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 mI=+/-1 mI=+/-2 2 scans Center Freq. 34.10 MHz Scaling Factor = 10 10 scans Center Freq. 68.21 MHzxx (arb. units)0.000.100.20 -0.10 -0.20 -0.30a) b)83.9083.9584.0084.0584.1084.1584.2084.2584.3084.35 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 scans Center Freq. 42.06 MHz Scaling Factor = 25 4 scans Center Freq. 84.12 MHzxx (arb. units)RF Freq (MHz) mI=+/-1 mI=+/-2 0.00 0.050.100.150.20 -0.05 -0.10-0.15 -0.20 0.00 0.050.100.150.20 -0.05 -0.10-0.15 -0.20-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 mI=+/-1 mI=+/-2 2 scans Center Freq. 34.10 MHz Scaling Factor = 10 10 scans Center Freq. 68.21 MHzxx (arb. units)0.000.100.20 -0.10 -0.20 -0.30 Freq. Offset (MHz) Rxx( , offset for clarity) Figure 5-11: Single (black) and double (blue) quantum excitations of 75As at 53 (a) and 610 (b). The associated stopping fiel ds and microwave frequencies were 4.680 T and 27.906 GHz for (a) and 5.772 T and 34.00 GHz for (b). The double quantum/Overtone spectra were obt ained via an RF field sweep at twice the Larmor frequency. The large difference in amplitudes between the single and double quantum spectra re quired a scaling factor and signal averaging to be used. The number of scans and the scaling factor are denoted in the plots. All spectra were initially offset to 0 .

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144 3 2Im 1 2Im 1 2Im 3 2Im 31 22INMRm 11 22INMRm 13 22INMRm 1Im 2Im 2NMRQh 2NMRQh 3 2Im 1 2Im 1 2Im 3 2Im 31 22INMRm 11 22INMRm 13 22INMRm 1Im 2Im 2NMRQh 2NMRQh Figure 5-12: Schematic diagram of the Zeeman energy levels without (left) and with (right) a quadrupole splitting. The si ngle and double quantum transitions are denoted on the right in black and blue, respectively. -2 -1 0 1 2 3 4 5 2 scans Center Freq. 58.96 MHz Scaling Factor = 40 20 scans Center Freq. 117.92 MHz mI=+/-1 mI=+/-2xx (arb. units)RF Freq (MHz)0.00 0.05 0.100.150.20 -0.05 -0.10 -0.15 -0.20 -0.25 Rxx( , offset for clarity) Freq. Offset (MHz)-2 -1 0 1 2 3 4 5 2 scans Center Freq. 58.96 MHz Scaling Factor = 40 20 scans Center Freq. 117.92 MHz mI=+/-1 mI=+/-2xx (arb. units)RF Freq (MHz)0.00 0.05 0.100.150.20 -0.05 -0.10 -0.15 -0.20 -0.25-2 -1 0 1 2 3 4 5 2 scans Center Freq. 58.96 MHz Scaling Factor = 40 20 scans Center Freq. 117.92 MHz mI=+/-1 mI=+/-2xx (arb. units)RF Freq (MHz)0.00 0.05 0.100.150.20 -0.05 -0.10 -0.15 -0.20 -0.25 Rxx( , offset for clarity) Freq. Offset (MHz) Figure 5-13: Single (black) and double (blue) quantum excitations of 69Ga at 610. The stopping field and d microwave freque ncy were 5.772 T and 34.00 GHz. The double quantum spectra observed here was even smaller in amplitude in comparison to those obtained for 75As. The observation of this transition supports the hypothesis that a quadrupole splitting of the 69Ga MDENDOR spectrum is present, but unresolvable. Again, the number of signal averaged scans and the scaling factor are denoted in the plots. All spectra were initially offset to 0 .

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145 Single-photon double-quantum excitation of the 69Ga isotope was also observed, although this transition was or ders of magnitude smaller than those detected in 75As. This spectrum is presented in Figure 5-13. Here, a singl e broad peak was observed in both the single (in black) and double (in bl ue) quantum excitati on spectra, with the double quantum excitation occurring at a fre quency double that of the single quantum excitation. Since the detection of such double quantum spectra typically requires a removal of the degeneracy of the 21 I quadrupole energy leve ls, this observation provides proof that a quadrupole splitting of the NMR transi tions associated with the 69Ga isotope are present, and low resolution and/or lower EFGs at the Ga sites are the reason for the absence of the quadrupole split triplet. Discussion One investigation into the origin of these si ngle-photon, double-quantum excitations was pe rformed by Eickhoff et al.,79 where the effect of the application of resonant RF upon the Hanle curve (which depicts the luminescence polarization by a transverse magnetic field, was studied. Here , the luminescence polarization of the 2DES system was measured via op tical excitation of the 2D electrons at a wavelength associated with the band gap of GaAs, as is typical in optically detected and optically pumped NMR experiments, during a sweep of the magnetic field. After of the Hanle curve was recorded in the absence of RF, another scan was made, this time under application of 9 MHz i rradiation. The field was swept from about 1.4 to 0 T, therefore passing through the reso nant conditions of all three nuclei. When the resonant condition of the nuclei was achie ved, a drop in the Ha nle curve, followed by a time dependent re-establishment of the preNMR polarization level was observed. The

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146 field was swept at a slow enough rate so th at the Hanle trace was allowed to be reestablished prior to the excite ment of the next nuclei. In recording this spectra, drops in the Hanle curve were recorded not only for the 1Im , but also for 2Im transitions. The signals pertaining to each excitation were deconvoluted, allowing for the corres ponding NMR spectrum to be extracted. Similar to the results reported in this dissertation, the 75As spectrum was split into a triplet, with broad si nglets for the two Ga isotopes. The double quantum excitations of 75As, 69Ga and 71Ga were observed as doublet s. However, while the spectrum associated with the 75As nuclear excitation exhibited a broad splitting, where two distinct transitions were observed, both Ga spectra displayed very small splittings, such that they appeared as two overlapping singlets. In order to determine the origin of these excitations, two experi ments were set up to determine if the transitions were due to the magnetic dipole or electric quadrupole interactions. The first experiment involved the modulation of the la ser intensity of the pump (excitation) laser beam, thereby modulat ing magnetic interacti ons and the electric field. In the second, the polarization of the pump laser was modulated between and , which does not affect the charge density (the refore will not affect the electric field), but still modulates the magnetic dipole intera ctions. The comparison of these two results clearly differentiates between the transitions. While the 1Im transitions were observed in both experiments, the double quant um excitations were only observed under the intensity modulation experiment. This cl early shows the dependence of this transition on the presence of an electric field, a nd therefore, on the presence of a quadrupole splitting of the nuclei. They stated, as is consistent with current theories, that the

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147 magnetic dipole transitions are dominated by the hfi between the photo-excited electrons and the nuclei. On the other hand, the 2Im excitations were dependent upon the charge distribution of the phot o-excited carriers, and the c oupling between these electrons and the nuclei via the electr ic quadrupole interaction.79 Unlike conventional NMR methods, doubl e quantum excitations under such conditions do not require tw o RF photons; instead it re quires only a single photon through the coupling with the el ectric quadrupole interaction. This is quite similar to Overtone spectroscopy, which is us ed quite often in the NMR of 14N in order to narrow the required excitation bandwidths.91 Concluding Remarks In this chapter, the quadrupole splitt ing and double quantum excitations of the MDENDOR spectra of GaAs/AlxGa1-xAs quantum wells has been determined. Such effects have been observed within samples of this type by a variety of different methods, but investigations aimed at determining their orig in have to this point been limited. In the case of the quadrupole splitting, typically a uniaxial strain of the quantum well is assumed. The quadrupole splitting of the 75As MDENDOR spectra was observed by both the field and RF swept methods and the corres ponding splittings were determined to be relatively unchanged between the two met hods. The dependence of the splitting was determined to be insensitive to the RF sw eep rate and power, source-drain current, and the degree of Overhauser shift. A dependen ce upon the filling fact or is possible, but appears to be small and could not be extrac ted from the data acquired. As expected, a strong dependence upon the tilt angle was evident. Unexpectedly, the simple 23cos1

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148 dependence associated with either a simple uniaxial strain applied along the growth axis due to the lattice mismatching at the well-barrier interface or from the creation of a Schottky barrier at the semiconductor-metal interface as the conduction electrons exit the well region. Further investigations have ruled out many other possible mechanisms as well. It may be theorized that the splitting is due to either a strain or iented at an odd (not aligned with one of the Cart esian axes) axis, from an in teraction betwee n the electric quadrupole moment and an extern al electric field through th e LQSE, the creation of an EFG due to inhomogeneities within the elec tron distribution thr oughout the quantum well sample, or from some other, as of now, unknown interaction. The observation of single-photon, doubl e-quantum excitations within the 75As and 69Ga MDENDOR spectra are most likely due to a coupling through the electric quadrupole interaction, as fi rst reported by Eickhoff, Lenzmann, Flinn, and Süter.79 The observation of the double quantum 69Ga transition provides s upport for the idea that a quadrupole splitting of the associated MDEN DOR exists, but is not resolvable. Further experiments in this sample in vestigating the quadr upole splitting and double quantum excitation, perhaps involvi ng low temperature MDNMR as reported by Desrat et al.,75 over the entire field range at constant tilt angle, are necessary to determine a definitive mechanism for the origin of th e quadrupole splitting. Understanding of this origin may provide further understanding of the many complex intera ctions that exist between the conduction electrons and the local lattice nucle i within such systems.

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149 CHAPTER 6 MDESR/MDENDOR NUMERICAL SIMULATIONS Using the mechanism for MDESR based on resonant microwave heating of the electron spin system discussed in Ch apter 3 and presented by Olshanetsky et al.,99 we have performed numerical simulations for the magnetoresistance under MDMR conditions. The simulations were used to replicate the dependence of the MDESR response on the magnetic field sweep rate along with the MDENDOR spectra and its dependence upon the NMR frequency. These simulations provided insight into the mechanism and the overall importance of the many parameters associated with these systems. MDESR Simulations Algorithm As first presented in Chapter 4, the fre quency of the ESR condition is dependent upon the total effective field, 0 effn B BB and is expressed as */ESReBeffgBh (6.1) The Overhauser shift may be expressed as the difference between the thermal equilibrium nuclear field, eq n B , and the DNP enhanced nuclear magnetic field, D NP nB eqDNP nnnBBB (6.2) As stated, due to the relative signs of th e bare electron g-factor and the nuclear gyromagnetic ratios, the Overhauser effect will le ad to a nuclear field that is parallel with 0 B and therefore will be added to th is field. During a down-sweep in 0 B , the creation of

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150 n B will compensate for the loss of 0 B . If the sweep rate is set slow enough, such that 0 ndBdB dtdt , the ESR condition will be “locked” to0 B . This effectively causes an Overhauser shift of the ESR condition to lower 0 B . The rate equation describing the nuclear spin polarization due to DNP production from the resonant absorption of CW microwaves by the conduction electrons and o ffset by the nuclear relaxation and nuclear spin diffusion, as introduced in eq. (4.15) is8 2 00 2 11ˆ 11 ˆˆi z iiiii e zz ii nnnI I IsIDI tTTz (6.3) where s is the saturation parameter that was introduced in general form as eq. (4.16) and in the functional form utilized in the simulation algorithm as eq. (4.17). The nuclear spin lattice relaxation time is written as a function of temperature and the D.O.S. FE was introduced in eq.s (4.18) and (4.19). Eq. (6 .3), neglecting nuclear spin diffusion effects, depicts the competition between the desire of the nuclear spin system to return to the thermal equilibrium polarization 0I and the rate of DNP production via electron-nuclear cross relaxation. Incorporating nuclear spin diffusion however is very important. Effects of this nature are presumed to be very appreciable, with reductions in sweep rate increasing their significance. The distance th e nuclear magnetization will di ffuse through the lattice may be determined through the following expression: 1 ndDT (6.4) Taking the nuclear spin relaxation time a nd nuclear spin diffusion constants for 75As to be 120 s and 10 nm2/s,100 respectively, the magnetization can be expected to diffuse a

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151 distance of approximately 35 nm. Therefor e, the nuclear spin polarization can be expected to diffuse into the barrier regi ons, independent of where the nuclear spin polarization originated. As the relaxation times in the barriers can be orders of magnitude longer than those for nuclei within the well, diffusion into the barriers will have a significant effect on the overall relaxation time of n B , and therefore will play a significant role in the simulations. For the effect of nuclear spin diffusi on to be included into the numerical simulations, a functional form of the third term in eq. (6.3) must be derived. As the sample was grown along the 100 direction, the nuclei, and therefore the nuclear spins, exist within parallel planes of area 0A . These planes are separated by a distance equivalent to the lattice constant for GaAs, 0a . Knowing this, a di fference equation may be created to describe the nuclear spin diffu sion between adjacent layers of nuclei (for simplicity, interactions between next neares t neighbors have been neglected). We may describe the total angular mo mentum along the z-axis as ,zkI , where /kkNV is the density of spins within plane j . Here, jN is the total number of nuclei in the th j plane and 00jVAa is the volume of that plane. Using FickÂ’s Law, the flux of the tota l angular momentum per unit area per unit time , ijJ from layer 1 j to j is ,,1 , , 0 ijij ij zz z ijiiII dI JDD dza (6.5)

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152 Similarly, the flux from layer j to layer 1 j may be written as ,,1 ,1 0 ijij zz ijiII JD a . Assuming the density of the sample is constant, we may write the net flux of the z-component of the spin angular momentum as , ,1,1, 2 02ij i z ijijij zzzdI D III dta (6.6) If we take measurements at a small enough increment in time, we may solve for the change in nuclear spin polarization as follows: ,,1,1, 2 02i ijijijij zzzzD I IIIt a (6.7) The change in nuclear spin polarization due to spin diffusion during each time increment may be calculated from eq. (6.7) and the change due to DNP in the th j layer during this same time increment may be determined exactly. ,,,,, 11exp/11exp/ijijijijij e zznzn eq nItIttTIstT (6.8) Under these small time steps, it is valid to determine the nuclear spin polarization as the sum of the differential contributions gi ven by eq.s (6.3) and (6.8). This criterion will be met provided the stability condition, 2 02iDta , is not violated for any nuclei. Taking the lattice constant of 0a =0.56 nm and the diffusion constant for 75As, we can estimate this stability condition will me maintained provided 15.6 ms t . For the sake of the simulations, it was necessary to ensure that at our slowest sweep rate, where the time increment per step 0 0/ dB tB dt would be the longest , this condition would

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153 be maintained. During the numerical simulations the 6.05.7 T field sweep was divided equally into 106 increments. Taking the slowest sweep rate of 4.6 mT/min, max6.5 mst . Therefore, throughout th e range of sweep rates n eeded to simulate our experimental data this stab ility criterion is met. Using these expressions, along with the heating mechanism for MDESR discussed in Chapter 3, we form the basic premise for the numerical simulations. The simulations are performed by calculating the imaginary part of the electronic magnetic susceptibility, which is defined as 02 2 2 * 1121 2 1ESRe app appESReBeeeT gBTT (6.9) where * 0 0/2eq s eBzNgP B is the static magnetic suscep tibility of the 2DES. Here tanh 4eq z eq eE P T is the equilibrium electron spin polarization †. Under our heating model, resonant microwave energy is absorbed by the 0 k spin-wave modes. This energy is then di ssipated to non-zero spin-wave modes through a thermal heat flow process, eventually exci ting a well separated el ectron-hole ionization pair (k spin-wave modes) that is require d to facilitate changes in the magnetoresistivity of the 2DES. The power dissipated by the electron spin system during this process is written as follows: † It was determined through our simulations that the qualitative results remain the same, independent of whether a system of itinerant (single spin flips) or str ongly correlated (spin-waves) electrons is assumed. Therefore, the simpler expression for eq zPpertaining to the itinerant electron system was utilized to greatly simplify the calculations.

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154 2 11 "() 2appePBA (6.10) with A being the area of the 2DES. From th is, the temperature of the 2DES under resonant microwave absorption may be ascertained eq se bP TT (6.11) Here, eq eT is the temperature of the sample lattice and b is the thermal conductance between the 2DES and the lattice. This change in the spin temperat ure of the 2DES leads to a change in the magnetoresistance, xx R which is written as 0expexp 22xx eq BsBeEE RR kTkT (6.12) The simulations are performed by calculating "app at each increment of 0 B . It is important to choose an increment size, 0 B , such that the time spent at each magnetic field step is short compared to 1 nT , 0 01/ s tepndB B T dt . By assuring this, it can be definite that 0zdI dt during each field step. From eq.s (4.18) and (4.19), it is clear that 1 nT is a function of z, which is in turn dependent upon the displacement of the electron from the center of the well, z . While within the ground state electronic subband, we may write z as 1/22/cos/ zwzw , where w is the width of the quantum well and 22 ww z . Using eq. (6.3), i z I z is calculated for each plane of nuclei within the quantum well. Within sample EA124 and EA124B, the quantum

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155 wells are of a width w=30 nm in the 100 growth direction and therefore consist of //2 mwL planes of Ga and As atoms. Negl ecting spin diffusion, these equations may be combined and used to derive an expr ession for the sum of the nuclear field over all nuclear planes, j n B , where j refers to the center plane of nuclei, as follows: 2, 1 2 1 1cos// cos/m iiji jz m ji j nn m j j jzwAII BB zw (6.13) Here, iA is the hyperfine coupli ng constant for nucleus, i . The values for iA were previously measured in an unpatterned sample of EA124 via optically pumped Overhauser shifts and were determined to be 690.727 TA, 710.621 TA, and 751.47 T A .66 The value of n B is then input into eq . (6.1) to determine the Overhauser shifted ESR frequency, which will then be used in the next increment of 0 B for the calculation of "app, P , s T , and xx R . As stated previously,63 the effects of spin diffusion upon the Overhauser shift decay are presumed to be significant, as this should allow for the replenishment of nuclear polarization to the faster rela xing well nuclei from those more within the barriers, which relax much slower. To further complicate matters, the nuclea r spin relaxation time, along with the activation gap and the magnetoresistivity pref actor, are known to be very dependent on filling factor within the =1 minimum. To compensate for this, attempts to introduce the filling factor dependence of these parameters were included within the simulation code and had minimal impact on the qualitative featur es of the simulations, presumably due to

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156 the small range of filling factors covered during any of the sweeps. The data for the dependence of E on filling factor were extracted from datasets obtained in EA124B, EA124 and EA129 at =1, while EA124 was utilized for obtaining the dependence of 0 R . From data reported by Tycko et al.18, the filling factor dependence of 1 nT was estimated. In the case of the dependence of 1 nT , it should be noted that the data presented at =1 by this report was obtained via optical pumping experiments and obviously incorporate the influence of nuclear spin diffusion as well, so it is unclear how well such data correlates to those obtained via the MD mechanism. However, observations of the filling factor dependence of 1 nT at >1 performed by Berg et al.,17 which were completed utilizing the previously discussed relaxation of the Over hauser shift using th e MD method, produced comparable datasets to those obtained by Tycko et al.18 Because both results were consistent with the theory put forth by Vagner et al.,16 it appears that the use of this data for the simulation of the qualitative behavior was justified. However, due to the limited impact on the simulations by the inclusion of these effects, they were removed for simplicity. Simulation Results As shown in Figure 6-1 (a)(f), simulations (dotted lines) of the dependence of the MDESR response upon the magnetic field sw eep rate can be compared to the experimental MDESR traces (solid lines) that were presented in Figure 4-2 (b). The shift of the peak to lower magnetic field with re duced sweep rates and th e overall lineshape of the MDESR peaks at faster sweep rates (i n trace a through d) are simulated with good qualitative agreement. Another success of th e model is the simulation of the increase in the MDESR amplitude as the sweep rate is in creased from the slowest sweep rate of 4.6

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157 mT/min (trace f) to 100 mT/min (trace e). This is shown in Figure 6-2 (black squares) in comparison to the experimental data presente d as Figure 4-3 (blue circles). However, there is a total failure of the model in si mulating either the fall in MDESR amplitude upon further reduction to the sweep rate (traces a and b) or in simulating the symmetry of the MDESR lineshape at the slowest sweep rates (traces e and f). The increase in the ESR absorption as the sweep rate is reduced may be due to any number of reasons. One explan ation may be that due to an increased time on resonance, an increased saturation of the ESR transition may occur. While saturation effects were excluded, the total photons of resonant energy absorbed is dependent on the time spent under resonance conditions. Since the power absorbed is a measur e of the number of photons of the resonant energy absorbed per un it time, this would a ppear analogous to the microwave power dependence discussed in Chap ter 4 and presented in Figure 4-4. It could be hypothesized that the increase in amplitude will be maximized at a sweep rate where the rate of DNP producti on is still slower than the rate of resonant microwave absorption. At this point, DNP effect s would not have created a large enough n B to significantly shift the MDESR response, wh ich would lead to a large Overhauser broadening and therefore decrease the amplitude as is seen as the sweep rate is reduced below 100 mT/min. Other possibilities could be due to nuclear spin diffusion or another as yet undetermined mechanism. The asymmetry in the numerical simulations at the slowest sweep rates (Figure 6-1 e and f) studied is in definite contrast to the experimental results, which appear to be quite symmetric. In the simulations, a sharp decrease on the low-field side is observed, which can be associated with the de-pinning of the ESR condition from 0 B . The reason

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158 for the failure of the simulations in this rega rd is still of concern and other improvements to the model to remedy this are being consid ered. Finally, the split ting that is readily apparent in trace b and residually present in sp ectrum c was not possible to simulate. As this splitting was not observed in any other samp le, this is not assumed to be a failure of the model, but instead an issue associated with the sample itself, perhaps from heterogeneity among the quantum well layers within the superla ttice or due to in equivalent electrons, which e xhibit different characteristic interactions with the local nuclei. From the simulations, the influence of nuc lear spin diffusion may be realized. Shown in Figure 6-3 a and b ar e the profiles of the nuclear spin polarization along the zaxis for 100 and 4.6 mT/min, respectively, at numerous positions in magnetic field during the sweep. At the beginning of the sweep, no significant nuclear polar ization exists. As the ESR condition is approached, the nuclear polar ization begins to increase in the center of the well due to the Overhauser effect. A maximum in the nuclear polarization at the center is attained 10 mT earli er for the 100 mT/min sweep th an it is in the 4.6 mT/min sweep. Also, it is clear during the slower sweep that nuclear spin diffusion plays a much more significant role, as the nuclear spin pol arization profile is mu ch broader across the well. Furthermore, once the ESR field lock ing condition is lost and the polarization begins to decrease at the center, the polarizat ion profile is broadened in both cases, where in the case of the slower sweep, the maximum in the nuclear polarizat ion is shifted away from the center of the well once 05.7 T B is reached.

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159 5.7 5.85.9B0 (T)1.0 1.02 1.03 1.04 1.05 Rxx( )(f) (e) (d) (c) (b) (a) 5.7 5.85.9B0 (T)1.0 1.02 1.03 1.04 1.05 Rxx( )(f) (e) (d) (c) (b) (a) Figure 6-1: The different boxes refer to the experimental (solid line) and simulated (dotted line) at (a) 492, (b) 150, (c) 100, (d) 50, (e) 10, and (f) 4.6 mT/min.

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160 0100200300400500 0.5 1.0 1.5 2.0 2.5 3.0 xx ()dB 0 /dt (mT/min) exp. sim. Rxx( )0100200300400500 0.5 1.0 1.5 2.0 2.5 3.0 xx ()dB 0 /dt (mT/min) exp. sim. Rxx( ) Figure 6-2: Experimentally co llected (blue squares) and co rresponding simulations (black circles) of the dependence of the MDESR peak amplitude on the magnetic field sweep rate. The increase in peak amplitude that occurs when the sweep rate is increased from th e slowest sweep rates to a sweep rate of 150 mT/min, is simulated quite well. Beyond this poi nt, the simulations fail to predict the drop in the amplitude with continue d increases in sweep rate beyond 150 mT/min. 05001000150020002500 1.0 1.2 1.4 1.6 1.8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 TDistance from Center (Angstroms)Enhancement Factor100 mT/min Downsweep4.6 mT/min Downsweep05001000150020002500 1 2 3 4 5 6 7 8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 T(a) (b)05001000150020002500 1.0 1.2 1.4 1.6 1.8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 TDistance from Center (Angstroms)Enhancement Factor100 mT/min Downsweep4.6 mT/min Downsweep05001000150020002500 1 2 3 4 5 6 7 8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 T05001000150020002500 1.0 1.2 1.4 1.6 1.8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 TDistance from Center (Angstroms)Enhancement Factor100 mT/min Downsweep4.6 mT/min Downsweep05001000150020002500 1 2 3 4 5 6 7 8 initial 5.90 T 5.85 T 5.80 T 5.75 T 5.70 T(a) (b) Figure 6-3: Numerical simulations of the nuclear spin polarization profiles during a (a) 100 mT/min and (b) 4.6 mT/min down-fiel d sweep at a variety of different magnetic fields during the sweep. It can be seen that the nuclear spin polarization initially increases in ma gnitude as we pass through the ESR condition. Once this maximum is pa ssed through, the ESR field locking condition is no longer maintained and nuc lear spin diffusion allows for the polarization to expand into th e rest of the quantum well.

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161 Field Swept MDENDOR Simulations Algorithm In order to test the hypothesi s presented in Chapter 4 to explain the appearance and RF dependence of the MDENDOR spectra, th e simulations were performed using the algorithm reported above with one distinct change. In order to simulate these spectra, it is obviously necessary to inco rporate the nuclear depolariz ation due to the resonant absorption of RF when the NMR condition is met for one of the nuclear isotopes. The rate equation for this depolari zation may be taken from Slichter4 ˆ ˆ 2i z ii zdI WI dt (6.14) where 2 11 4ii nrfWg is the transition rate of the nuclear spin excitation, which is defined by an NMR lineshape function 2 21 exp/ 2 2i rfNMR iii rf ig . In this expression the line width is 22log2ii FWHM. Adding this to the rate eq. (6.3), we get the total, overall rate equation ˆˆ exp1expi iiii zz iB I tItAttAtt A (6.15) where 11 2ii i nAW T and 1ˆ 1i z eq i e ii n nI B s T . From the optically pumped 71Ga NMR spectrum discussed in the report by,66 for sample EA124, a 3 kHzNMR FWHM line width is observed. Within this sample, a value of 1214 snT was observed via relaxation of the Overhauser shift following a slow, DNP producing down-sweep in 0 B .44

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162 Simulation Results In a remarkable improvement from the mixed results obtained in the sweep rate simulations, all major features of the MD ENDOR and its dependen ce upon the relative RF, were simulated with incredible agreemen t as shown in Figure 6-4. The evolution from a small peak at small separations betw een the ESR and NMR, to a large peak at intermediate separations, (but at prior to passing the critical fiel d), to a step-shaped response at even larger separations, were reproduced with extr aordinary similarity. 5.755.805.85 0 2 0 2 5.755.805.85 0 2 0 2 5.755.805.85 0 2 0 2 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 1 2 3 0 1 2 3 5.755.805.85 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Rxx( )B0(a) 74.513 (b) 74.578(c) 74.734(d) 74.994 (e) 75.170(f) 75.328(g) 75.486(h) 75.644 5.755.805.85 0 2 0 2 5.755.805.85 0 2 0 2 5.755.805.85 0 2 0 2 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 2 4 0 2 4 5.755.805.85 0 1 2 3 0 1 2 3 5.755.805.85 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Rxx( )B0(a) 74.513 (b) 74.578(c) 74.734(d) 74.994 (e) 75.170(f) 75.328(g) 75.486(h) 75.644 Figure 6-4: Simulated MDENDOR spectra (top) and corresponding experimentally observed (bottom) MDENDOR response. Remarkable agreement between the two sets was observed. The transition from the pure peak-like structure, superimposed upon the otherwise unpert urbed MDESR background, to that of the diminished MDENDOR peak and lo ss of the ESR field-locking condition is simulated quite accurately. Another success of this model is the simulation of the increased MDENDOR peak amp litude observed as we move from spectrum (h) to (e), and the following decrease in trac es (d) to (a).

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163 The transition in the MDENDOR respons e observed upon increas ing the separation between the nuclear and ESR c onditions was observed by Seck et al.,15 utilizing traditional ENDOR techniques, within n-type GaAs. The effect of the excitation of 75As NMR, within the line width of the field-swept CW-ESR, was remarkably similar to the effects observed in the experiments pr esented within this dissertation. Simulations of the dependence of the creation ofn B on decreasing 0 B at 75.328 MHz (trace a, which is associated with Fi gure 6-4 f), and 74.734 MHz (trace b which is associated with Figure 6-4 c), are presented in Figure 6-5. This figure provides insight into the origin of the critical field tran sition observed in the RF dependence of the MDENDOR response. As discussed, when the NMR position is set prior to this critical field, the reduction in the nuclear field, n B , is less than this critical decrease, crit n B . As the NMR condition is moved further down-field from eq ESR B , we see that the magnitude of n B increases. As shown between these two spectra, when critESR nnFWHMBB , the transition from the peak-like structure and continued ESR-locking condition to the steplike drop off and the destruction of the locking condition occurs. A physical picture of this phenomenon may be described using the next two figures. As shown in Figur e 6-5 a, at the condition critESR nnFWHMBB, the shift in the ESR line following the NMR depolarization is sm all. This size of this shift is much smaller than the line width of the ESR, and so it only brings the ESR s lightly closer to the maximum microwave absorption condition, which occurs when D NP appESR . With the increased absorption of the microwaves co mes an increased DNP. This in turn

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164 repolarizes the NMR-excited nu clei, thereby reestablishing th e field-locking condition to its pre-NMR level. 0 20 40 60 80 100 5.755.805.855.900 20 40 60 80 100 16 mT 6.5 mT B0(T)Bn(mT) Bn(mT)(a) rf=75.328 MHz (b) rf=74.734 MHz0 20 40 60 80 100 5.755.805.855.900 20 40 60 80 100 16 mT 6.5 mT B0(T)Bn(mT) Bn(mT)(a) rf=75.328 MHz (b) rf=74.734 MHz Figure 6-5: Simulation of the production of n B during the slow down-sweep due to DNP, and the effect upon meeting the NMR condition. As discussed in the text, when critESR nnFWHMBB , the transition in the MDENDOR behavior is observed.

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165 As the NMR condition is shifted away from eq ESR B , eventually the condition crit nn B B is established. This is shown in Fi gure 6-6 b. At this point, following the NMR excitation, D NP ESRB is shifted to a point where the condition D NP appESR is met. This leads to the maximum absorption of the reso nant microwaves and therefore the highest nuclear repolarization rate. This is observed as a maximum in the MDENDOR amplitude, while the magnetic field-locking co ndition of the ESR is still reasserted. Upon further decreases in the RF, as de picted in Figure 6-7 a, the shift of D NP ESRB , following NMR, is such that the maximum c ondition is rapidly passed due to the initial nuclear depolarization that is a ssociated with the NMR excitation crit nnBB . Following the NMR excitation, we find that D NP appESR , therefore, while a peak is observed in the MDENDOR spectra, the nuclear repolarization rate is not sufficient to compensate for the large shift. This leads to the loss of the field-locking condition. ENDORB0 a)Bn B0 ENDOR ENDOR B0Bn ENDORB0 b)Bn B0 ENDOR ENDOR B0Bn BnBn * L * L * L * L * L * L app app app app app app ENDORB0 a)Bn B0 ENDOR ENDOR B0Bn ENDORB0 b)Bn B0 ENDOR ENDOR B0Bn BnBn * L * L * L * L * L * L app app app app app app Figure 6-6: Pictorial re presentation of the MD ENDOR response at a) crit nn B B and critESR nnFWHMBB .

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166 Finally, when crit nn B B, as in Figure 6-7 b, the loss in nuclear polarization is quite large and therefore the shift in D NP ESR is such that no resonant microwave absorption is continued. This is seen as the absence of the MDENDOR peak, as in Figure 6-4 a, and a dramatic loss of the aforementioned ESR field-locking condition. ENDOR B0B0Bn ENDOR B0Bn a) B0B0Bn ENDOR B0Bn b) * L * L * L * L * L * L app app app app app app ENDOR B0B0Bn ENDOR B0Bn a) B0B0Bn ENDOR B0Bn b) * L * L * L * L * L * L app app app app app app ENDOR B0B0Bn ENDOR B0Bn a) B0B0Bn ENDOR B0Bn b) * L * L * L * L * L * L app app app app app app ENDOR B0B0Bn ENDOR B0Bn a) B0B0Bn ENDOR B0Bn b) * L * L * L * L * L * L app app app app app app Figure 6-7: Pictorial re presentation of the MD ENDOR response at a) crit nn B B and crit nn B B. Radiofrequency Swept MDENDOR Simulations Algorithm The algorithm utilized for simulating the RF swept MDENDOR results using the field stop method, were begun through the same program used for the field swept results discussed above. The similarities were continued until a predetermined magnetic field, denoted as s top B , is attained. At this point, the down-field sweep in magnetic field is suspended. In order for the simulations to closely match the laboratory experiments, a time delay of approximately equal to the time of the field sweep is included at this point

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167 to ensure that the system has come to the steady-state condition. During this wait, the simulations continue to calculate the nuc lear spin polarization and the corresponding features"app, P , s T , and xx R at the same time intervals. Following the delay, a sweep through RF, beginning at RF_start a nd continuing until RF _stop is met, is initiated. The rate of the sweep is determ ined through the parameter RF_swrate. During this sweep, the calculation of the above menti oned parameters is continued, in this case incorporating the effects of th e RF irradiation, at each equa lly spaced time interval. Once the RF sweep is completed, the simulation is stopped. Simulation Results Shown in Figure 6-8 is the RF swept MDENDOR for 71Ga at the same conditions used for the sweep rate dependence of MDESR and the field swept MDENDOR simulations above. The figure shows the sweep as a function of (a) time, (b) field and (c) RF. Through comparison with Figure 4-13 (a)-( c) it can be seen that the similarities between the simulations and the experiment ar e extraordinary, with all major qualitative features being correctly simulate d. The scan is begun at 6.2 T and data is recorded after 05.925 T B . As the down-field sweep is continued, the system approaches ESR resonance (position (a) ), and the typically observed increase in xx R and broadened MDESR signal are observed. Th e field sweep continues until 05.753 TstopBB , this point is denoted as (b) in Figure 6-8. At this point, the system is allowed to relax to the steady-state condition, which is attained approximately 5800 s after the field stop. After this point, the RF is turned on, and a sh arp step, due to non-resonant effects, in xx R is immediately seen, (c) . Here, the RF sweep is initiated and is swept until the resonance

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168 condition is approached (shown in (d) ), where the resonant abso rption of RF irradiation causes a destruction of some of n B , temporarily shifting the system closer to the maximum ESR absorption, this is seen as an increase in xx R . This sharp increase is coupled with a rapid repolarization of the nuclei, causing the stea dy-state, ESR fieldlocking condition to be re-established. Th e RF sweep is continued until RF_stop is attained, whereupon the simu lations are stopped. 2000400060008000100001200014000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) (b) (a) 58.558.658.758.858.959.059.159.259.359.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) 5.765.785.805.825.845.865.885.905.925.94 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (b) (a) Time (s) B0(T) Freq (MHz) Rxx( )(a)(b) (c)2000400060008000100001200014000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) (b) (a) 58.558.658.758.858.959.059.159.259.359.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) 5.765.785.805.825.845.865.885.905.925.94 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (b) (a) 2000400060008000100001200014000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) (b) (a) 58.558.658.758.858.959.059.159.259.359.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (d) (c) 5.765.785.805.825.845.865.885.905.925.94 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (b) (a) Time (s) B0(T) Freq (MHz) Rxx( )(a)(b) (c) Figure 6-8: Simulations of the RF swept MDE NDOR results as a function of (a) time, (b) magnetic field and (c) frequency. An e xplanation of the re sults and algorithm is available in the text. Concluding Remarks We have presented a model based on th e resonant microwave heating of the 2DES99 and have used it to perform numerical simulations of the magnetic field sweep rate dependence of the MDESR response, the MDENDOR spectral features, in either the field or RF swept experiment, and the pr esence of the critical transition in the MDENDOR response as a function of increased separation between eq ESR B and NMR B , ESRNMRB . The overall appearance and qualita tive behavior of the MDESR brought about by reducing the magnetic field sweep rate were simulated quite well. The model reproduced the increase in the MDESR line wi dth and the overall decrease in amplitude, observed along with this increa sed line width, at the slowes t sweep rates. This model has

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169 provided insight into the mechanism for th e dramatic transition in the MDENDOR response and has allowed for a qualitative physical picture of the mechanism for the MDENDOR observation. The simulations of th e qualitative spectral features for both the field and RF swept experiments were quite remarkable. In a clear improvement upon the simulations of Seck et al.,15 in which no attempt to simulate the ENDOR effects were made, and upon those of Hillman and Jiang,50 which only simulated the appearance of the peak in the MDESR spectrum associated with MDENDOR and the sweep rate dependence of the MDENDOR response, this effort was able to simulate essentially all main qualita tive spectral features, with two exceptions. The first is the inability of the model to simulate the symmetry of the MDESR at the slowest sweep rates. Secondly, the model does not simulate the initial in crease in MDESR amplitude as the magnetic field sweep rate is reduced from the maximum rate, 492 mT/min, to a rate of 150 mT/m in. We feel that a continue d effort to provide better estimates of relaxation parameters, the microw ave field strength, and the introduction of other mechanisms that may participate in this effect, such as nuclear spin diffusion, will improve the simulations. Hopefully, improveme nts of this model will lead to a more complete understanding of th e electron-nuclear spin dynamics within 2DESs. This knowledge is of importance in the continued fundam ental study of these systems and in the future applications of such solid state semiconductor devices.

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170 CHAPTER 7 CHANGE IN EDESR BEHAVIOR ASSOCIAT ED WITH THE EVOLUTION FROM A 2DES TO 3DES WITHIN WIDE PARABOLIC QUANTUM WELL (WPQW) STRUCTURES Throughout this dissertation, the discussi on has centered upon magnetic resonance interactions within 2DESs. It should be noted that many in teresting and exotic properties are thought to be associated with 3DESs, in the presence of a large magnetic field, as well.101-106 In the presence of an applied magneti c field, the symmetry of the ground state of a 3DES may be broken. Th e ground state of the electron system may exist as either a spin-density-wave (SDW) at lower 0 B 104-106 or a charge-density-wave (CDW) or Wigner crystal101-103 at larger magnetic fields. While th e existence of such states has been theorized for quite some time now, the intera ctions between electr ons and neutralizing impurity ions within most samples causes the st ability of such exotic ground states to be lacking within 3D doped semiconductor structures.107 Reduction in the doping concentration may serve to diminish the interactions between the electrons and the nearby ions, in effect spa tially separating the involved particles and lowering the energy of their attraction. While 3D semiconductor devices with low enough impurity levels have not ye t been realized, parabolic quantum wells (PQW) have provided the capability for experi mental study of such 3D effects. These samples exhibit 2D properties when the sample is oriented such that 0 B is perpendicular to the electron system (00 ,referred to as the perpendi cular orientation), but they display 3D properties when the samp le is rotated with respect to 0 B by 900 (referred to as

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171 the parallel orientation).107 In this chapter, a compar ison between the ESR response and the spin dynamics within these samples associated with the change from a three to two dimensional electron system will be explored. This should provide some insight into the different ground electronic spin states, as well as the overall electronic and nuclear properties of these samples. Properties of a 3DES According to Brey,107 under the app lication of a 0 B , the ground electronic spin state of the 3DES is predicted to be a SDW at low magnetic fields, while as 0 B is increased, a transition to a CDW or Wigner crystal ground st ate is anticipated. The magnetic field where this transition should occur will depend upon the sample characteristics such as the electron density and the sample channel size. Furthermore, Brey explains that excitations of a SDW (S DE) involve the excitati on of an electron and hole of different spin orient ations, which correspond to 0k transitions that obey LarmorÂ’s theorem, therefore the energy of these excitations is simply the electronic Zeeman energy * 0.zeBEESDEgB CDW (CDE) excitations on the other hand pertain to an excitation of an electron and hole with the same spin orientation, and correspond to k excitations. Parabolic Quantum Wells Within a square potential well sample, such as those discussed previously in this dissertation, if the width of the well is incr eased, the confinement of the electrons, and therefore the associated QHE, will disappear.108 Instead of confini ng the electrons to a 2D layer by a square well structure, if we periodically increase the Al concentration within the well as a function of z (with z=0 at the center of the well), such that the

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172 energy of the conduction band varies periodically as 2 0az , this loss of quantum confinement does not occur. The structure just described is referred to as a PQW.109, 110 Due to constraints in the t echnology of the preferred sample growth technique, molecular beam epitaxy, the periodic adjustment of Al concentration within the well is performed step-wise. For example, the Al concentrati on in the barrier regions of sample AG662 is 0.29. As the well region is grown, this am ount is reduced after an integer number of atomic layers have been grown, for instan ce to 0.27. Following another defined growth at this concentration level, a nother drop in the Al concentrat ion is performed for the same number of atomic layers. This step-wise decrease continues until the center of the well, where the Al concentration is zero, is reached . At this point, the prescribed number of atomic layers of pure GaAs is grown, follo wed by the initiation of the reverse of the above growth process until the concentration of 0.29 is reached once again. The difference between these two structures is illustrated in Figure 7-1. In 1983, with the creation of th e first PQW structure by Gossard,111 investigations of the evolution from 2D to 3DESs, within the same sample, became possible. A wide, partially filled, PQW is an id eal system for studying this evol ution, because the electronic energy spectrum depends upon the angle between 0 B and the growth axis of the sample.112 In a strong parallel magnetic field, th e cyclotron radius is reduced to a size smaller than the width of the PQW, therefore the quantum confinement of the electrons to 2D is no longer valid. As discovered by MacDonald et al.113 through Hartree-Fock calculations, this evolution in dimensionality of the electron system will also depend upon changes to the electron density. Since the electron density within the well may be increased via optical illumination, the widt h of the electronic wave function profile

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173 within the well may be increas ed, thereby providing a method for control of the transition in the dimensionality of the electron system. Another method is thr ough the utilization of an externally applied gate voltage. (b) Parabolic Quantum Well AlxGa1-xAs x=.3 x=.2 x=.1 x=0 x=.1 x=.2 x=.3 AlxGa1-xAs (a) Quantum Well Al0.1Ga0.9As GaAs Al0.1Ga0.9As Si V (z)E0E1 30 nm E0E1 400 nm Si V (z) (b) Parabolic Quantum Well AlxGa1-xAs x=.3 x=.2 x=.1 x=0 x=.1 x=.2 x=.3 AlxGa1-xAs (a) Quantum Well Al0.1Ga0.9As GaAs Al0.1Ga0.9As Si V (z)E0E1 30 nm E0E1 400 nm Si V (z) Figure 7-1: Comparison between a square (left) and parabo lic (right) quantum well. Notice in the PQW the concentration of the Al within the well is incrementally increased as you move away from the center of the well, (there is no Al content within the well of the square quantum well samples). This incremental change causes a corresp onding incremental increase in the potential energy as a function of z. This forms an approximate parabolic potential as shown. For comparison, if a sample with conti nuous Al concentration within a wide well, similar in size to that descri bed above, but comparable in concentration profile to those discussed earlier in this dissertation, were grown, no such transition would occur. Instead, a drop in the potential energy would occur at the interfaces between the well and barrier regions, therefore a sample similar to a double quantum hete rostructure would be realized, with minimal electron density found within the bulk of the wide well region. By creating a WPQW as described above, a uniform charge density is created within the well

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174 causing the potential energy to be minimized at approximately the center of the well under application of a strong perp endicular magnetic field. Effect of Sample Rotation As mentioned earlier, when the sample is in the perpendicula r orientation, the electron system is confined to a plane and the associated physics, notably the QHE, are readily observed. While in this orientat ion, each electronic subband represents Landau levels with an energy of ,,SNmE, which are dependent upon the quantum numbers 1 ,,0,1, 2SNm where N is the Landau level and is the integer filling factor index for the highest completely filled La ndau sublevel. The electronic energy for the 2DES, under the conditions mentioned above, has been derived107 and is presented in the following: ,,,1 2SSN NmmcEEN (7.1) As seen in eq. (7.1), the energy is not dependent upon the wave vector yk , and therefore, each energy level consists of a degeneracy 22 0/2 Ll , with L referring to the width of the well in the z-direction, which is associated with the different possible locations of the minimum of the harmonic oscillator potential al ong the z-axis. This energy is the total electronic energy of the system, made up of both the Zeeman and exchange interactions. At high magnetic field and with the sample tilted to 900, the electron system will no longer be confined to two dimensions sin ce the cyclotron orbit is reduced to a size smaller than the width of the quantum well. However, the limit of quantum behavior is still met, and thus the system is govern ed by 3D quantum physics. Under such

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175 conditions, the quantization of the Landau en ergy levels is removed and an energy continuum is realized at each electronic subband. This results in th e absence of the QHE, with the associated oscillations in the magnetoresistance being replaced by low field oscillations, which are inde pendent of the tilt angle.108 This absence of QHE features is expected as the cyclotron or bits are no longer forced to fit within a two-dimensional plane, thereby removing the usage of the QHE filling factor as a good quantum number. Magnetoresistance traces taken at a vari ety of angles, in sample AG662, are displayed in Figure 7-2 a. The sample is initially tilted to 00 (in blue and also displayed in Figure 7-2 b). Here, the presence of the QHE is clear, with =1 present at approximately 4.23 T. As the sample is tilted away from this position, the =1 minimum shifts to higher magnetic field. The la st angle where this minimum was observable within the range of our microw ave system is displayed in Fi gure 7-2 a in light blue. If the sample is tilted further, the QHE features continually shift to higher magnetic field and begin to shallow as the system approach es a three-dimensional quantum condition. Eventually, the sample is tilted such that th e QHE features are no longer present. Again, this is due to the cyclotron orbits, which are decreasing in diameter with increasing field, becoming smaller than the sample width. The last angle at which the QHE was still observable was 800 and is displayed in red in Figure 7-2 a. Finally, upon rotating the sample further, the aforementioned low field oscillations, associated with a 3DES, are observed. The magnetoresistance trace taken at 900 is displayed in green in Figures 7-2 a and c, with the oscillations clearly present in the latter figure. The position of these features, once present, is indepe ndent of the tilt angle. On the contrary, the minima in the

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176 magnetoresistivity associated with the QHE (as shown in b), shift with the cosine of the orientation angle as th ey are dependent only on B , as shown in the following:114 0 000,11cos BBB (7.2) 012345 0 500 1000 1500 2000 2500 3000 3500 4000 0 2 4 6 8 200 400 600 800 1000 1200 xx ( ) Rxx( )(a) (b) (c)-1012345678910 0 1000 2000 3000 4000 5000 6000 =2=1=1 Sample 4000a Magnetoresistance vs Tilt AngleRxx()B0(T) 900 800 570 00 B0 (T) 012345 0 500 1000 1500 2000 2500 3000 3500 4000 0 2 4 6 8 200 400 600 800 1000 1200 xx ( ) Rxx( )(a) (b) (c)-1012345678910 0 1000 2000 3000 4000 5000 6000 =2=1=1 Sample 4000a Magnetoresistance vs Tilt AngleRxx()B0(T) 900 800 570 00 B0 (T) Figure 7-2: a) Magnetore sistance traces at 900 (green) where the system is entirely that of a 3DES, 800 (red) where features of both a 3D and 2DES are observed, 570 (light blue) where the =1 minimum is lowered in 0 B such that it is within the range of our microwave system, and 00 (blue) where the system is that of a 2DES. b) Magnetoresistance trace at 00 . Note the presence of the =1 minimum at approximately 4.4 T. c) Magnetoresistance trace at 090 . Here, the QHE is notably absent, although the presence of thr ee oscillations at low 0 B , which are due to the magnetic depolarization of the electronic subbands, are evident. As discussed above, a slow evolution from two to three-dimensional behavior is observed, with the sample initially displaying features associated w ith the QHE at angles less than 800 and then beginning to show the low field oscillations associated with a 3DES at larger angles; thus, th e study of the differences in the electronic behavior under these two very distinct, situations is possible.115 The electronic stat es associated with these intermediate orientations are referred to as oblique states and consist of plane waves along the direction of 0 B , with a localization length that grows with increasing tilt angle, as /tanxW , where xW is the width of the current channel in the x direction

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177 (perpendicular to both the external magnetic field and the current flow), and have characteristics similar to those of a one-dim ensional “particle in a box” that are defined by strong Coulomb interactions.112 As the sample is tilted toward the pa rallel orientation, th e 2D Landau levels, associated with each subband, decrease in energy (i.e. decreasing B ). Figure 7-3 presents a calculation of the tilt angle dependence of the electronic energy spectrum.107 The first 10 cyclotron levels are shown for the two lowest subbands of a 200 nm wide PQW. As the sample is tilted towards 900 the spacing between the cyclotron levels decreases and becomes unresolved when the separation is less than the Landau level broadening. As the degree of confinement of the electrons is dependent upon the tilt angle, and this confinement contributes to th e magnitude of the Coul ombic interactions, it is not surprising that not onl y is the Zeeman contribution to the electronic energy dependent upon the tilt angle, but so is th e exchange interaction as shown in the following equation:112 * ,0cosSN meB E EgB (7.3) The energy of the electrons in the oblique states of a WPQW, may be described as follows: 2 22 ,, *1 22SNmxycEkkN m (7.4) where 22 0/c , * 002/ am , 222 0 c and *m is the effective mass. In a WPQW placed within a strong 0 B , 0 c , therefore, upon the full rotation of the sample to the parallel orientation the energy simplifies to

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178 22 , *1 22x Nck EN m (7.5) This is the energy of the thre e-dimensional Landau subbands.112 Since the type of spin excitation (e.g. single spin flip, spin-wave, skyrmion, SDW or CDW) instigated by the application of resonant microwaves is dependent upon the ratio of the Zeeman and exchange energies, eq.s (7.4) and (7.5) imply that the type of spin excitation may be dependent upon the tilt angle as well. Figure 7-3: Used with the pe rmission of G. M. Gusev.106 As the angle is tilted from a perpendicular (00) to a parallel (900) orientation, a clear collapse of the energy levels of the different electronic subbands is observed. Finally, upon completion of the full rotation, the subbands will be completely collapsed into a single energy level per subband as shown. Study of Spin Properties and Dynamics within WPQW Systems Up to this point in time, the only report ed experimental studies probing the spin dynamics of a 3DES or oblique/2DES within a WPQW were performed through the use

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179 of photoluminescence or TRFR measurements. Wang and coworkers116 performed the first such experiments via photoluminescence polarization measuremen ts in 1993. In a later study by Sinyavski and Sokovnich,117 it was determined that the energy levels explored by luminescence within WPQW sample s, are quantized into levels separated by steps of 1.46/d, where dis the length in the z-dimension of the quantum well in Angstroms. It was further determined that these energy levels can influence the kinetic energy of the electron system at 100 K T . Under such conditions and with the sample in the parallel orientation, the now perpendicular electric field component from 0 B , will cause a shift in the dispersion relationshi p, effectively causing the valence and conduction bands to move in opposite directions in k-space, thereby changing the usual direct band gap into an indirect one. Poggio et al.97 used TRFR in a study of the angl e and gate voltage dependences of the bare electron g-factor in a WPQW. It was determined that under conditions of sample rotation and applied gate voltages, a change in *eg will occur. As *eg depends upon the concentration of Al within the well region, th e change in the g-factor is presumed to be associated with a shift in the position of th e electronic wave functi on within the quantum well along the z-axis. As the center of the electr onic wave function is shifted from the center of the well into the areas of higher Al concentrations, * eg should increase in value (the absolute value will d ecrease) towards zero. If a large enough shift occurs, * eg will pass through zero and then will begin to incr ease in both absolute and true value. Calculations by Brey 107 have led to great insight in to the spin dynamics of these systems. One such result states that at 0 k in the dispersion curve, intra-Landau level (spin excitations) and inter-Landau level excita tions (crossing of th e activation gap), are

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180 completely decoupled. However, upon shifting the value of the wave vector to non-zero values, some mixing of these two different types of excitations may occur. Experimental Results The data presented here are the first su ccessful magnetic resonance measurements of a PQW system. In this study, NMR a nd ESR transitions were observed via the photoconductivity signal th rough the method discussed in chapter 3. Sample Characteristics All experiments within this chapter were completed using sample AG662, an AlxGa1-xAs WPQW with x varied from 0 to 0.29 fr om the center to the outer regions of the 4000Ã… well. This well region is bound between two undoped AlyGa1-yAs spacers (y=0.310). The height of the well parabola is 1202.5 meV and the height of the Al0.31Ga0.69As barriers is 275 meV . Silicon -doping layers are placed on either side of the well within the barriers. The 1.5X1011 cm-2 electrons exhibit a mobility of 1.2X105 cm2/Vs in the dark. Saturation of the sample via illumination, such that the PQW is filled with conduction electrons, leads to an increa se in both the mobility and density to 2.4X105 cm2/Vs and 3.5X1011 cm-2, respectively. Results Angle dependence of the bare electron g-factor The average * eg within the well may be written as *21ee eggzdz W (7.6) where eW is the effective width of the electron slab within the well. As discussed earlier, the electron density may be adjusted by optic al illumination via an LED, allowing for a

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181 change in the g-factor to be observed. This is due to th e broadening of the electronic wave function within the well, thereby causi ng it to extend into areas of higher Al concentration, which due to the lower g-fact ors associated with these positions, will cause the average * eg to decrease. Another method for tuning * eg within these samples is through the tilting of the sample with respect to 0 B . As discussed earlier, if the sample is tilted so that the magnetic field is parallel to the plane of the electron system and the magnetic field is large enough such that the cy clotron diameter is smaller than the width of the quantum well, the spectrum will be that of a three-dimensional electron system in the quantum limit. Again the g-factor will re main variable along the z-axis. This will lead to a broadening of the Zeem an energy as in the following: * 0/ Z eseEzgznzBNW (7.7) Here nz refers to the electron concentration profile, which because of the exchange interaction, may differ from the profile of the electronic wave function with the sample placed within a perpendicular magnetic field. In measuring the dependence of the bare electronic g-factor on tilt angle, it is necessary to observe the ESR resonance at a variety of magnetic fields. Within ESR measurements in classical systems, the g-fact or is found to be inde pendent of the applied magnetic field. However, results from two studies reported by Dobers et al.32 and Meisels et al. ,50 it was determined that within the re gime of the integer and the fractional QHEs, respectively that the g-factor actually varies linearly with the applied magnetic field as follows: * 001 2eggcNB (7.8)

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182 where 0g and c are sample dependent constants and Nrepresents the Landau level, which in the case of the experiments presented here (all measured at =1) is zero. In order to determine the true dependence of the g-factor upon the tilt angle, the field dependence must also be accounted for. Displayed in Figure 7-4 a-d are EDES R spectra taken at 90, 83, 60 and 440, respectively, all at 1.5 K T under the application of 36.65 GHzapp resonant microwaves. Trace (a) pertains to the EDES R of an approximate 3DES, this being the first time that ESR has been observed in the absence of the QHE with in a quantum well. As the sample is tilted, a clear shift in the position of the ESR is observed. Systematic measurement of the g-factor at seven different angles (E SR was not observable at all angles where the 3DES was apparent, or wh ere 2D characteristics were present, but =1 was found at 06.5 T B ), ranging from 0-900, are presented as Figure 7-5. From these data sets the bare electronic g-factor was extr acted, and this is pl otted as a function of angle in Figure 7-6. The g-f actor is observed to decrease as the angle is changed from 0900. The effective g-factor, that was extracte d from the frequency/field dependence data presented in Figure 7-5, exhi bited the linear dependence on magnetic field presented in eq. (7.8), which taking into account 0N , may be simplified to the following:32 * 001 2eggcB (7.9) From this we find that 0g ranges from -0.43 to -0.46 as the sample is rotated from the perpendicular to the parallel orie ntation. This small change implies that the electrons are located within the center of the parabo lic well throughout the entire series of measurements, however, evidence of a true dependence upon the tilt angle is apparent.

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183 Further experiments with a higher degree of LED illumination, thereby broadening the electronic wave function within the well, may serve to increase the effect the tilt angle has upon the sample characteristics, most notably the g-factor. An effect similar to that observed in the TRFR measurements97 is most likely the reason for such an observation in our sample. The drastic difference in g-factors between their sample and ours is presumably due to the application of a gate voltage in their experiments, allowing for the wave function to be shifted into areas of higher Al concentration by both the tilt angle and the applied bias. The overall change in * eg is relatively small in our sample and therefore provides evidence that only a minimal shift in the position of the electronic wave function from the center of the quantum well is created. It is possible however that this change in g-factor is due to the anisotropy observed under tilted fields within square potential quant um wells. Further study to separate these two mechanisms is necessary for a more convincing explanation for this behavior to be obtained. Temperature dependence of EDESR within a 3DES Upon observation of EDESR within a 3DES of a WPQW sample in the parallel orientation, a clear characterization of th ese spectral features was completed for comparison to a similar study performed in this sample at a tilt angle of 00 . If these excitations are created through the same m echanism as the MDESR excitations of a 2DES, then a similar temperature depende nce of the EDESR peak amplitude, and a correspondence with the associated temperatur e dependence of the ma gnetoresistance is expected.

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184 6.406.456.506.556.606.65 -0.02 0.00 0.02 0.04 0.06 0.08 6.406.456.506.556.606.65 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 6.406.456.506.556.606.65 -0.5 0.0 0.5 1.0 1.5 2.0 XAiTitl 6.406.456.506.556.606.65 -0.2 0.0 0.2 0.4 0.6 0.8 900 Parallel orientation830600All At 36.65 GHz440Within =1 minimum Rxx( )B0 (T)6.406.456.506.556.606.65 -0.02 0.00 0.02 0.04 0.06 0.08 6.406.456.506.556.606.65 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 6.406.456.506.556.606.65 -0.5 0.0 0.5 1.0 1.5 2.0 XAiTitl 6.406.456.506.556.606.65 -0.2 0.0 0.2 0.4 0.6 0.8 900 Parallel orientation830600All At 36.65 GHz440Within =1 minimum Rxx( )B0 (T) Figure 7-4: EDESR peaks ta ken at 4 different angles at T=1.5 K and 36.65 GHzapp . The top two peaks pertain to observations within a 3DES, in the absence of the QHE. The bottom two peaks were observed within the =1 minimum of the magnetoresistance trace, therefore th ey pertain to measurements within a 2DES. Note the shift of the resonance position to lower field as the angle is increased.

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185 4567 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Sample 4000a g-factor measurement at all anglesResonant Frequency (GHz)Resonant Field (T) 10 turns 00 8 turns 120 7 turns 230 6 turns 350 5 turns 460 1 turns 830 0 turns 900 B0(T)4567 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Sample 4000a g-factor measurement at all anglesResonant Frequency (GHz)Resonant Field (T) 10 turns 00 8 turns 120 7 turns 230 6 turns 350 5 turns 460 1 turns 830 0 turns 900 B0(T) Figure 7-5: Plot of the pos ition of the EDESR condition as a function of the applied microwave frequency at a number of differe nt angles. Note the change in the slope of the line be tween the many different angles and the wide range of magnetic fields where the ESR was observed. -20020406080100 0.370 0.375 0.380 0.385 0.390 0.395 0.400 0.405 bare electron g-factor, g*Sample Tilt Angle (degrees) Figure 7-6: Dependence of th e bare electron g-factor, * eg , upon the sample tilt angle. Despite the scatter in the da ta (partially due to the low sensitivity at high tilt angles), a clear decrease is observed as the angle is decreased from 00 (perpendicular) to 900 (parallel orientation).

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186 Displayed in Figure 7-7 are the full magnetoresistance tr aces of the 3DES at 420 mK (blue line), 1.4 K (black line) and 10 K (red line). While esse ntially no change is observed between the spectra recorded at the two lower temperatures, a dramatic loss of the low field oscillations, which are associat ed with the magnetic depolarization of the electric subbands106, is observed. The temperature de pendence of the ma gnetoresistance, spanning the entire range where the EDESR response was observable 0.42010.0 KT , is shown in Figure 7-8 (a). With in this range, the magnetoresistance appears to follow an approxima te linear increase with 0 B . It can be seen in (b) that the temperature dependence is approximat ely the same at all values of 0 B within this region and displays an overall increase in magnetoresistance with decreasing temperature, although an exact dependence is difficult to decipher. It is clear however, that the temperature dependence of the magnetoresistance within this range is clearly not the same as the activated dependence observed with in the minima associat ed with the integer QHE. Two representative EDESR spectra, record ed with the sample in the parallel orientation at 2.0 (temperature of strongest response) and 10.0 K (maximum temperature where response was observed) are presented in Figure 7-9 (a) and (b). The spectra are shown as unprocessed (as reco rded; displayed as black, ope n squares) and processed (with the non-resonant background removed via the method discussed in chapter 3; displayed as blue, closed circles). For comparison, two analogous spectra, taken at T=1.67 (temperature of strongest response) and 5.09 K (maximum temperature where response was observed) while the sample wa s in the perpendicular orientation are presented as Figure 7-10 (a) and (b); these spectra were observed at approximately =1

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187 (4.31 T). The approximate 50% decrease in the observable temperature range, along with the dramatic increase in the signal to noise ratio at 00 , suggests that the mechanism for ED in the parallel orientation might not be due to simple heating. A comparison between the temperature dependence of the ESR amplitude and line widths taken at 900 and 00 are displayed in Figure 7-11 (a) and (b), re spectively. While it is clear that in both cases a maximum in the amplitude is observed at max1.72.1 K T (similar to the behavior observed in sample EA124), the trend in the ESR line widths directly contradict one another, with an increase in the line widt h occurring as the temperature is reduced in the 2DES and the exact opposite occurring in the 3DES. Both of these results are at odds with the observed behavior in square quantum well samples where there is essentially no dependence of the ESR line width on temper ature within the observable temperature range. The observed EDESR signals are also approximately a factor of two broader in the 3DES than those associated with the 2DES in either sample AG662 or EA124B. Finally, while an approximate four-fold incr ease in the EDESR amplitude is observed at 00 , only half that increase is observed at 090 . Finally, the overall resonance position of the electrons as a function of temperature at 090is shown in Figure 7-12. Although the overa ll scatter is rather large, a shift of approximately 40-50 mT is observed upon loweri ng the temperature from 10 to 0.4 K. No such shift was observed in this sample at 00 or in EA124. An understanding of this effect is still not cl ear, although one possible mechan ism would infer a temperature dependence of the position or breadth of the electronic wave functi on along the z-axis. This would cause an effect similar to the tilt angle dependence discussed previously. However, evidence for this mechanism is lacking and is purely speculation.

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188 -1012345678 200 400 600 800 1000 1200 xx()B0 (T) T= 0.42K T= 1.42K T=10.00K Rxx( )-1012345678 200 400 600 800 1000 1200 xx()B0 (T) T= 0.42K T= 1.42K T=10.00K Rxx( ) Figure 7-7: Magnetoresistance tr aces at 0.42, 1.42 and 10.0 K within sample AG662. Notice the minimal change between the traces recorded at the two lowest temperatures and the almost complete lo ss of the low field oscillations at the highest temperature. 6.06.46.87.2 1020 1040 1060 1080 1100 1120 1140 1160 1180 1200 xx()B0 (T) 0.416K 1.416K 2.008K 3.015K 4.002K 5.002K 6.003K 7.002K 8.002K 9.004K 10.004K0246810 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 Temp (K) Rxx at 6.548T (MDESR) Rxx at 7T Rxx at 6.05T(a) (b) Rxx( ) B0 (T) Temp (K)6.06.46.87.2 1020 1040 1060 1080 1100 1120 1140 1160 1180 1200 xx()B0 (T) 0.416K 1.416K 2.008K 3.015K 4.002K 5.002K 6.003K 7.002K 8.002K 9.004K 10.004K0246810 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 Temp (K) Rxx at 6.548T (MDESR) Rxx at 7T Rxx at 6.05T(a) (b)6.06.46.87.2 1020 1040 1060 1080 1100 1120 1140 1160 1180 1200 xx()B0 (T) 0.416K 1.416K 2.008K 3.015K 4.002K 5.002K 6.003K 7.002K 8.002K 9.004K 10.004K0246810 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 Temp (K) Rxx at 6.548T (MDESR) Rxx at 7T Rxx at 6.05T(a) (b) Rxx( ) B0 (T) Temp (K) Figure 7-8: a) Magnetoresistan ce traces within the range where EDESR was observed at many different temperatures. While there was some scatter between the different traces, the overall trend show s an increase in the magnetoresistance with decreasing temperature. b) The temperature dependence of the magnetoresistance at three different magnetic fields within the range. Note the overall dependence is very reproduc ible between the three positions and the apparent maximum at 1.4 K.

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189 6.466.486.506.526.546.566.586.606.626.64 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 B0 (T)xx () unprocessed-0.0 1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 T=2.0 K 6.486.506.526.546.566.586.606.626.646.66 1.74 1.76 1.78 1.80 1.82 T=10.0 KB0 (T)-0.01 0.00 0.01 0.02 0.03 0.04 xx() Processed(a) (b) Rxx Rxx B0(T) B0(T) T=2.0 K T=10.0 K Unprocessed Processed 6.466.486.506.526.546.566.586.606.626.64 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 B0 (T)xx () unprocessed-0.0 1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 T=2.0 K 6.486.506.526.546.566.586.606.626.646.66 1.74 1.76 1.78 1.80 1.82 T=10.0 KB0 (T)-0.01 0.00 0.01 0.02 0.03 0.04 xx() Processed(a) (b) Rxx Rxx B0(T) B0(T) T=2.0 K T=10.0 K Unprocessed Processed Unprocessed Processed Figure 7-9: Processed (using th e processing method discussed in chapter 3; displayed as solid blue circles) and unprocessed (open black squares) of the EDESR observed at 2.0 (a) and 10.0 K (b) taken in sample AG662 in the parallel orientation. In both cases the peak was clearly observable prior to processing, but the noise level at 10.0 K was much higher. Note the much larger peak amplitude associated with the EDESR spectrum recorded at 2.0 K. Rxx Rxx B0(T) B0(T) 4.244.264.284.304.324.344.364.384.404.42 -0.2 -0.1 0.0 0.1 0.2 0.3 T=5.09 K0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4.244.264.284.304.324.344.364.384.404.42 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T=1.67 K(a)(b) Unprocessed Processed T=1.67 K T=5.09 K Rxx Rxx B0(T) B0(T) 4.244.264.284.304.324.344.364.384.404.42 -0.2 -0.1 0.0 0.1 0.2 0.3 T=5.09 K0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4.244.264.284.304.324.344.364.384.404.42 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 T=1.67 K(a)(b) Unprocessed Processed T=1.67 K T=5.09 K Figure 7-10: Processed (solid blue diamonds ) and unprocessed (open black squares) of the MDESR observed at 1.67 (a) and 5.09 K (b) in sample AG662 in the perpendicular orientation at =1. Both signals were obtained under irradiation with 25.69 GHz microwaves. In both cas es the peak was observable prior to processing, but the noise le vel at 5.09 K is significantly increased. Note the approximate five-fold increase in peak amplitude associated with the MDESR spectrum recorded at 2.0 K.

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190 0246810 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 Amplitude ()Temp (K)0246810 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 pp FWHM (T)Temperature (K)(a) EDESR Amp ( ) Temp (K) Temp (K) Linewidth(T)012345 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 012345 4 6 8 10 12 14 16 (b)0246810 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 Amplitude ()Temp (K)0246810 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 pp FWHM (T)Temperature (K)(a) EDESR Amp ( ) Temp (K) Temp (K) Linewidth(T)012345 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 012345 4 6 8 10 12 14 16 (b) Figure 7-11: Temperature dependence of the EDESR amplitude and line width in sample AG662 in the (a) parallel and (b) perpendicular orientations. 0246810 6.52 6.53 6.54 6.55 6.56 6.57 6.58 6.59 MDESR Position (T)Temperature (K) Figure 7-12: Temperature dependence of the MDESR position at a microwave frequency of 36.65 GHz with the sample in the parallel orientat ion. Although there is a good deal of scatter in the da taset, it appears that a steady shift to lower field is observed (higher g-factor). No such effect was observed at 00 .

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191 Dependence of the Overhauser shift decay on temperature Due to the difference in the xx R response between the two orientations, the temperature dependence of the decay of th e Overhauser shift was measured at both angles following a slow DNP-producing down-sw eep. Shown in Figure 7-13 is the time dependent decay of the Overhauser shifte d ESR peak measured during both up (red circles) and down-sweeps (black squares) at 0492 mT/min dB dt and at T=1.6 and 5.0 K following a 10 mT/min down-sweep with the sample in the parallel orientation. Attempts to measure the decay time at higher or lowe r temperatures in this orientation were not possible due to the decreased signal to noise, especially as the sweep rate is reduced. Single exponential fits to the recorded data provided the values of the decay time as 785 120.5 sand 737 81.8 s, for the T=1.6 and 5.0 K, respectively. While the error in the datasets is rather large, it is still clear from the similarity in the observed time constants that no significant differences are observed between the two temperatures. The lack of temperature dependence of the decay time within the 3DES directly contradicts the expected decrease with increasing temper ature associated with Korringa relaxation. Shown in 7-14 a-c are the corresponding decays of the Overhauser shift at T=1.4, 2.0, and 3.0 K in sample AG662 at the perpendicula r orientation. Nota bly, not only is the expected decrease in decay time with increasing temperature observed at 00, but a dramatic four-fold decrease in the time cons tant in comparison to those recorded at 090 is found. This provides further eviden ce that the various relaxation pathways (Korringa, nuclear spin diffusion, etcÂ…) are mu ch more active in the 2DES and that the mechanism for the electron-nuclear interactions is quite different.

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192 Sweep rate dependence of the EDESR response Due to the incorporation of Al into the well region, a change in sign of the bare electron g-factor is quite possible in the limit of an extremely broad electronic distribution (this stems from the positive value for the g-factor in Al0.3Ga0.7As). Through measurement of the dependence of the xx R response upon the magnetic field sweep rate and direction at both 0 and 900 will provide a comparison between the response in both the two and three-dimensional electron sy stems. Also, comparisons with previous measurements in sample EA124B will provide insight into th e relative strengt h of the FC hfi in PQWs with respect to that found in square quantum well samples. Shown in Figure 7-15 is the dependen ce of the EDESR response upon the downfield sweep rate throughout the entire range of our magnet system at approximately 1.5 K in AG662 at (a) 090 and (b) 00 . While both show a clear broadening with decreasing sweep rate, it is quite puzzling why a smaller Overhauser shift is observed within the 2DES where the decay time of the Overhauser shift and presumably the nuclear spin relaxation time are much faster . The shorter relaxati on time should lead to broader signals (larger Overhauser shifts at the same sweep rate s), although the exact opposite is observed. So wh ile a similar dependence on 0dB dt is present, and the broadening of the signal duri ng the down-sweep leaves no question that the sign of * eg is negative in sample AG662 at both orientations , an explanation for th e differences in the overall broadening between the two orientations is lacking and will require further study.

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193 050010001500200025003000350040004500 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 1.6 KOS=785 +/120.5 sBnTime (s)050010001500200025003000 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T1n=737 +/81.8 s T1n at 5.0 K upsweep downsweepTime (s)(a) (b) Bn(T) Time (s) Time (s) 1.67 K5.00 K OS=785 +/-120.5 s OS=737 +/-81.8 s050010001500200025003000350040004500 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 1.6 KOS=785 +/120.5 sBnTime (s)050010001500200025003000 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T1n=737 +/81.8 s T1n at 5.0 K upsweep downsweepTime (s)(a) (b) Bn(T) Time (s) Time (s) 1.67 K5.00 K050010001500200025003000350040004500 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 1.6 KOS=785 +/120.5 sBnTime (s)050010001500200025003000 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T1n=737 +/81.8 s T1n at 5.0 K upsweep downsweepTime (s)(a) (b)050010001500200025003000350040004500 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 1.6 KOS=785 +/120.5 sBnTime (s)050010001500200025003000 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T1n=737 +/81.8 s T1n at 5.0 K upsweep downsweepTime (s)(a) (b) Bn(T) Time (s) Time (s) 1.67 K5.00 K OS=785 +/-120.5 s OS=737 +/-81.8 s Figure 7-13: Measurements of the Overhauser shift decay within sample AG662 in the parallel orientation at T=1.6 (a) and 5.0 K (b). Note that the relaxation times within this range are unchanged within the experimental error. 0200400600800100012001400 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T=1.4 K 193.671 +/9.097 s 213.510 +/16.888 s downsweep upsweep OS=204 +/-16 s (a)02004006008001000120014001600 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T=2.0 K OS=188 +/-20 s-200020040060080010001200140016001800 0.000 0.005 0.010 0.015 0.020 0.025 T=3.0 K(b)(c) T=1.4 K T=2.0 K T=3.0 K OS=147 +/-28 s0200400600800100012001400 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T=1.4 K 193.671 +/9.097 s 213.510 +/16.888 s downsweep upsweep OS=204 +/-16 s (a)02004006008001000120014001600 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 T=2.0 K OS=188 +/-20 s-200020040060080010001200140016001800 0.000 0.005 0.010 0.015 0.020 0.025 T=3.0 K(b)(c) T=1.4 K T=2.0 K T=3.0 K OS=147 +/-28 s Figure 7-14: Measurements of the Overhauser shift decay within sample AG662 in the perpendicular orientation at T=1.4 (a), 2.0 (b) and 3.0 K (c). The decay times not only appear to follow the dependen ce expected for Korringa behavior, but also are approximately four times shor ter than those found in the associated 3DES. A comparison between the EDESR respons e during an up (black) and down (red) field sweep at multiple sweep rates, is show n in Figure 7-16 a-i with the sample in the parallel orientation. As in the 2DESs disc ussed in Chapter 4 and the results observed at 00 in sample AG662, the general trend shows the EDESR peak going through a broadening (narrowing) during the down (up) field sweeps as the rate is reduced. This is clearly evident in Fi gure 7-17, where the dependence of the EDESR line width upon the sweep rate is displayed at a variety of temperatures dur ing both down (a) and up (b) sweeps in magnetic field. The line wi dth appears to remain unchanged until

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194 approximately 50 mT/min, whereupon a drasti c increase in the broadening of the xx R signal is observed. During the up-sweeps in magnetic field, a steady decrease in the line width is recorded throughout the entire range, as the sweep rate is reduced. Both of these dependences appear to be independent of the temperature. The dependence of the EDESR amplitude upon the magnetic field sweep rate for sample AG662 in the parallel orientation at a variety of temperatures is displayed in Figure 7-18 for both the down (a) and up (b) fi eld sweeps. It app ears that the EDESR amplitude, during a down-field sweep, initiall y increases in magnit ude, passes through a maximum, and then begins to diminish. This is similar to the results obtained in the previously explored square quantum well samp les and is illustrated quite nicely by Figure 7-18 a. While a significant decrease in th e overall amplitude was seen at increased temperatures, the same general increasing tre nd with reduced sweep rates is observed. A distinct maximum in the amplitude is recorded at 025 mT/min dB dt for 1.6 KT . A similar maximum was observed at 5.0 K, while a continued increase in the amplitude was seen for the other temperatures measured. It is unclear if this is a true shift in the position of the maximum at 5.0 KT and 1.6 KT , or if this is merely an artifact due to a reduction in the number of sweep rates measured and the significant reduction in signal to noise at higher temperatures.

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195 6.446.466.486.506.526.546.566.586.606.626.646.66 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 D own Fi e ld S weep R a t e D epen d ence MD ESR Sample 4000a T=1.62K Parallel orientationxx()B0(T) 492 mT/min 400 mT/min 300 mT/min 200 mT/min 150 mT/min 100 mT/min 50 mT/min 25 mT/min 10 mT/min (a) (b) Rxx( ) B0(T) B0(T) 4.254.264.274.284.294.30 4.314.324.334.344.354.36 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 492 mT/min 400 mT/min 300 mT/min 200 mT/min 150 mT/min 100 mT/min 75 mT/min 50 mT/min 25 mT/min 10 mT/min 4.6 mT/min 6.446.466.486.506.526.546.566.586.606.626.646.66 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 D own Fi e ld S weep R a t e D epen d ence MD ESR Sample 4000a T=1.62K Parallel orientationxx()B0(T) 492 mT/min 400 mT/min 300 mT/min 200 mT/min 150 mT/min 100 mT/min 50 mT/min 25 mT/min 10 mT/min (a) (b) Rxx( ) B0(T) B0(T) 4.254.264.274.284.294.30 4.314.324.334.344.354.36 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 492 mT/min 400 mT/min 300 mT/min 200 mT/min 150 mT/min 100 mT/min 75 mT/min 50 mT/min 25 mT/min 10 mT/min 4.6 mT/min Figure 7-15: Sweep rate dependence of the ED /MDESR response within a (a) 3D and (b) 2DES in sample AG662. Both sets of data were obtained at approximately 1.6 K. The microwave frequencies used were 36.65 GHz for the spectra in (a) and 25.69 GHz for the spectra presented in (b). Taking the drastic differences between the two systems, the simila rities between these dependences are pretty extraordinary. The minimal broadening associated with the spectra presented in (b) is not clear. The dependence of the EDESR amplitude upon sweep rate during an up-sweep in magnetic field is shown in Figure 7-18 (b). At temperatures ot her than 1.6 K, the amplitude passes through a maximum at a pproximately 50 mT/min, while at 1.6 K a steady increase in amplitude is observed. Ag ain, whether the differences are due to real temperature effects is unknown. A more comp lete study of the sweep rate dependence is required for conclusive resu lts to be determined. The increasing trend in the amplitude during the up-sweep is in direct contradiction to that which is seen in the previously st udied square quantum well samples however. In those samples, as the sweep rate was redu ced, a reduction in the up-sweep amplitude was recorded. The dependence within the 2DESs ma y be explained by the following. First, as the up-sweep in magnetic field brings the system into ESR, polarization of the nuclei via the cross-relaxation with th e electrons occurs. This cause s an Overhauser shift of the

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196 ESR toward lower 0 B . This shift effectively brings D NP ESR closer to app , thereby increasing the absorption of resonant microw aves dramatically. This in turn causes a more rapid nuclear polarization, which causes an increased Overhauser shift of the ESR to even lower 0 B . Quickly the ESR frequency dr ops below the applied frequency DNP ESRapp , and because the magnetic field is bei ng swept to higher fields, a loss of the ESR condition is quickly realized. This eff ect will be exacerbated with decreasing sweep rates. Therefore, it makes sense that the reduction in sweep rates would show a reduction in the EDESR amplitude during the up-sweep. Seeing the opposite effect in the 3DES is very puzzling. One possible explan ation is due to the broader ESR 25 mTESR FWHM of the 3DES excitations relative to the ESR obser ved previously. The DNP associated with the reduced sweep rates could cause the line to be initially narrowed, with the more rapid nuclear polarization “pulling” the ESR line to ward one point. Eventually a slow enough sweep rate would be attained where the line would be narrowed as much as possible and after this point, furthe r reduction in the rate would lead to a reduction in the amplitude, this being the exact opposite effect observed during the down-sweep. This would require a monotonic reduction in the line width during the same reduction in the sweep rate, and possibly the observance of a maximum amp litude when the peak had reached the maximum possible narrowing. This is ex actly what is observed in the line width dependence on sweep rate that was introduced in Figure 7-17 b. The appearance of a maximum in the up-sweep rate dependence of the EDESR amplitude in Figure 7-18 b at all temperatures other than 1.6 K, also lends support to this hypothe sis. The absence of such increases in the square well samples co uld be due to the narrower line widths, which

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197 would lead to the maximum in the depende nce occurring at higher overall sweep rates, perhaps corresponding to the maximum ra te attainable by our magnet system. 6.466. 486.506.526. 546.566.586. 606.626.64 0.0 0.5 0.025 T/min Down SweepRxx()B0(T) 6.446.466.486. 506.526.546.566. 586. 606.62 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.010 T/ minRxx() B (T) Downsweep Upsweep 6.546. 566. 586. 606.62 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.050 T/min Down Sweep Up SweepRxx() B (T) 6. 486.506. 526.546.566. 586.606. 626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.100 T/min Down Sweep Up SweepRxx()B0(T) 6. 486.506. 526.546. 566.586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.150 T/min Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200 T/min Down S weep Up SweepRxx()B0(T) 6. 486.506. 526.546.566. 586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0 . 400 T/ m in Down Sweep Up SweepRxx()B0(T) 6. 486.506.526.546.566.586.606.626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 B0(T) Down Sweep Up Sweep (a)(b) (c) (d)(e) (f) (g) (h) (i) B0 (T) Rxx( )6.466. 486.506.526. 546.566.586. 606.626.64 0.0 0.5 0.025 T/min Down SweepRxx()B0(T) 6.446.466.486. 506.526.546.566. 586. 606.62 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.010 T/ minRxx() B (T) Downsweep Upsweep 6.546. 566. 586. 606.62 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.050 T/min Down Sweep Up SweepRxx() B (T) 6. 486.506. 526.546.566. 586.606. 626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.100 T/min Down Sweep Up SweepRxx()B0(T) 6. 486.506. 526.546. 566.586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.150 T/min Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200 T/min Down S weep Up SweepRxx()B0(T) 6. 486.506. 526.546.566. 586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0 . 400 T/ m in Down Sweep Up SweepRxx()B0(T) 6. 486.506.526.546.566.586.606.626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 B0(T) Down Sweep Up Sweep (a)(b) (c) (d)(e) (f) (g) (h) (i)6.466. 486.506.526. 546.566.586. 606.626.64 0.0 0.5 0.025 T/min Down SweepRxx()B0(T) 6.446.466.486. 506.526.546.566. 586. 606.62 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.010 T/ minRxx() B (T) Downsweep Upsweep 6.546. 566. 586. 606.62 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.050 T/min Down Sweep Up SweepRxx() B (T) 6. 486.506. 526.546.566. 586.606. 626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.100 T/min Down Sweep Up SweepRxx()B0(T) 6. 486.506. 526.546. 566.586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.150 T/min Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200 T/min Down S weep Up SweepRxx()B0(T) 6. 486.506. 526.546.566. 586.606. 626.646. 66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Down S weep Up SweepRxx()B0(T) 6.486. 506.526. 546.566.586. 606.626. 646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0 . 400 T/ m in Down Sweep Up SweepRxx()B0(T) 6. 486.506.526.546.566.586.606.626.646.66 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 B0(T) Down Sweep Up Sweep (a)(b) (c) (d)(e) (f) (g) (h) (i) B0 (T) Rxx( ) Figure 7-16: EDESR response at a) 492, b) 400, c) 300, d) 200, e) 150, f) 100, g) 50, h) 25 and i) 10 mT/min at 1.6 K in samp le AG662 rotated to the parallel orientation, for both up (in black) and dow n (in red) sweeps in magnetic field. 0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)Temp0.00.10.20.30.40.5 0.00 0.01 0.02 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)TempDown sweepUp sweep (a)(b) Linewidth(T) Temp (K) Temp (K)0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)Temp0.00.10.20.30.40.5 0.00 0.01 0.02 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)TempDown sweepUp sweep (a)(b)0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)Temp0.00.10.20.30.40.5 0.00 0.01 0.02 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)TempDown sweepUp sweep 0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)Temp0.00.10.20.30.40.5 0.00 0.01 0.02 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)TempDown sweepUp sweep0.00.10.20.30.40.5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)Temp0.00.10.20.30.40.5 0.00 0.01 0.02 10.00 K 8.00 K 5.00 K 1.60 K 0.43 KFWHM (T)TempDown sweepUp sweep (a)(b) Linewidth(T) Temp (K) Temp (K) Figure 7-17: Dependence of the EDESR line width (FWHM), within sample AG662 at the parallel orientation, on the sweep rate at a variet y of temperatures for both the down (a) and up (b) sweeps in magnetic field.

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198 0.00.10.20.30.40.5 0.0 0.1 0.2 0.3 0.4 0.5 AmplitudeSweep Rate 10.00K 8.00K 5.00K 1.60K 0.43K0.00.10.20.30.40.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 AmplitudeSwee p Rate 10.00K 8.00K 5.00K 1.60K 0.43KDown sweepUp sweep(a) (b) EDESR Amp ( ) Sweep Rate (T/min) Sweep Rate (T/min)0.00.10.20.30.40.5 0.0 0.1 0.2 0.3 0.4 0.5 AmplitudeSweep Rate 10.00K 8.00K 5.00K 1.60K 0.43K0.00.10.20.30.40.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 AmplitudeSwee p Rate 10.00K 8.00K 5.00K 1.60K 0.43KDown sweepUp sweep0.00.10.20.30.40.5 0.0 0.1 0.2 0.3 0.4 0.5 AmplitudeSweep Rate 10.00K 8.00K 5.00K 1.60K 0.43K0.00.10.20.30.40.5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 AmplitudeSwee p Rate 10.00K 8.00K 5.00K 1.60K 0.43KDown sweepUp sweep(a) (b) EDESR Amp ( ) Sweep Rate (T/min) Sweep Rate (T/min) Figure 7-18: Dependence of the EDESR amplit ude, within sample AG662 at the parallel orientation, on the sweep rate, at a vari ety of temperatures, for both the down (a) and up (b) sweeps in magnetic field. Frequency swept MDENDOR response Shown in Figure 7-19 (a)-(c) are the 75As, 71Ga and 69Ga MDENDOR spectra obtained at approximately 016 and T=1.6 K within sample AG662 via the RF swept method discussed in chapter 4. It is notable that despite the presence of Al within the parabolic well, no 27Al resonance is observed. Th is clearly shows that around the perpendicular orientation an in significant portion of the nucle ar hyperfine field is due to the Al nuclei. Similar measurements taken within the 3DES associated with the parallel orientation were not possible due to th e dramatic loss of sensitivity under these conditions. Furthermore, a minor splitting in the 75As MDENDOR spectra may be attributable to a small quadrupole splitting, again not resolvable or present within the other nuclei. Further experime ntation to confirm the existence of this splitting has not been successful.

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199 42.5042.5542.6042.6542.7042.7542.80 3.2 3.4 3.6 3.8 4.0 4.2 4.4 75As MDENDOR75.8075.8575.9075.9576.0076.05 3.2 3.4 3.6 3.8 4.0 4.2 71Ga MDENDOR59.6559.7059.7559.8059.85 2.8 3.0 3.2 3.4 3.6 3.8 69Ga MDENDOR Rxx( )(a)(b)(c) B0(T)42.5042.5542.6042.6542.7042.7542.80 3.2 3.4 3.6 3.8 4.0 4.2 4.4 75As MDENDOR75.8075.8575.9075.9576.0076.05 3.2 3.4 3.6 3.8 4.0 4.2 71Ga MDENDOR59.6559.7059.7559.8059.85 2.8 3.0 3.2 3.4 3.6 3.8 69Ga MDENDOR42.5042.5542.6042.6542.7042.7542.80 3.2 3.4 3.6 3.8 4.0 4.2 4.4 75As MDENDOR75.8075.8575.9075.9576.0076.05 3.2 3.4 3.6 3.8 4.0 4.2 71Ga MDENDOR59.6559.7059.7559.8059.85 2.8 3.0 3.2 3.4 3.6 3.8 69Ga MDENDOR Rxx( )(a)(b)(c) B0(T) Figure 7-19: (a) 75As, (b) 71Ga and (c) 69Ga MDENDOR recorded via the RF swept method in WPQW sample AG662 at T=1.6 K. No resonance for the 27Al transition was recorded. Note the splitti ng features in the As spectrum. It is possible that a small quadrupole splitti ng is present within this nuclear transition, but further experime nts have yet to confirm this. Concluding Remarks Within this chapter, the first ever re ported observation of magnetic resonance within a 3DES associated with a WPQW was made. The dependence of the EDESR upon tilt angle, temperature, and sweep rate, along with the temper ature dependence of the decay of the Overhauser shift follow ing a slow DNP producing down-sweep were reported. These results were compared to analogous measurements made at =1 in the corresponding 2DES with the sample tilted to 00 . Many similarities in the behavior of the ESR in these two orientations exist, but many distinct differences are present as well. A clear shift from a 2DES, where features in the magnetoresistance associated with the integer QHE were observed, to a 3DES , where the integer QHE features were replaced with angle independent low field os cillations in the magnetoresistance trace, was observed within the 4000 wide PQW sample AG662. Duri ng this transition, a drop in the bare-electronic g-factor was observed. The small change in the g-factor lends itself to the understanding that the electr onic wave function is not sh ifted significantly from the

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200 center of the well at any angl e studied. One possible explan ation for this change, which is based upon an angle dependent shift of th e electronic wave function along the z-axis, was provided. A shift in the position of the ESR towards lower magnetic fields was also observed in the parallel orientation as the temperatur e was reduced from the highest temperature where EDESR was observable, 10 K. No such shift was observed in the perpendicular orientation. Also associated with the perpendicular orient ation, was an approximate 50% reduction in the temperature range where th e EDESR signal was observable. While a reason for the observed shift was provided, ev idence for why such a shift should occur is clearly lacking. Therefore, a more in -depth investigation is necessary for a comprehensive understanding of th is effect to be obtained. Measurements of the EDESR, within the two and three-dimensional electron systems associated with the different sample orientations, found similar temperature and down-field sweep rate dependences as our pr evious measurements in square quantum well samples, although the line width of the EDESR, at the fastest sweep rate, was increased by about a factor of 2 in the case of the 3DES. A dramatic difference in the upfield sweep rate dependence for the 3DES wa s observed, and a possible explanation for this difference was provided. Finally, RF swept MDENDOR spectra associ ated with the three NMR active nuclei of GaAs were observed. Despite the Al c oncentration within the quantum well region in this sample, no 27Al resonance was observed, therefor e leading to the conclusion that these nuclei do not play a signifi cant role in the formation of the nuclear hyperfine field responsible for the Overhauser shift of the ESR condition. This implies a narrow

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201 electronic wave function, centered at z=0 within the well, at the perpendicular orientation. Further experimentation within this and other PQW samples is necessary for a more complete understanding of the electron-nuclear interacti ons and the effect of the 23D transition upon these interact ions to be obtained. Experimentation within other samples, which exhibit similar behavior, su ch as double quantum we lls, would also be helpful in achieving this goal.

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202 CHAPTER 8 MDESR IN LOW DENSITY 2DES IN GaAs/AlGaAs QUANTUM WELLS Observation and Characterization of MDESR in Low Electron Density Systems Introduction and Motivation Within recent years, many developments within the field of quantum computing have been attained. Numerous different sy stems and methods have been established as possible candidates for use in quantum in formation processing, but the use of semiconductors devices,118-120 through the appl ication of nuclear121, 122 and/or electronic38 spins as qubits, has created increased concentr ation in the study of such systems and the use of MD magnetic resonance (MDMR) techni ques as methods for manipulating single electron spins. MDESR and MDENDOR have been shown with in the text of this dissertation as well as in the literature as valid methods for probing electron -nuclear interactions within low dimensional systems. In quantum inform ation processing, quantum bits, or qubits, must be of an increasingly small size. Th erefore MDMR is a natu ral candidate for qubit manipulation, but its usage in lower density systems must be proven, since for this method to be appropriate, it must be capable of sufficient sensitivity in the limit of single electron manipulation. The complementary na ture of MDESR with other techniques such as the coincidence method,123 measurements of gvia the magnetoresistivity and its dependence on nuclear spin orientation,44 and its usage within t uned g-factor devices via an external gate,38 provide further support for its usefulness in this role.

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203 The experiments discussed within this ch apter may aid in the realization of new spintronic devices based on quantum dots with an extremely low number of electrons (1100). In such systems, changes in electro nic transport due to resonant microwave irradiation could provide a mechanism for spin transitions between adjacent dots and the observation of such transitions. The work presented in this chapter, pe rtains to MDESR results in quantum well samples with intrinsically low electron densities that appear to be s calable to the size of such quantum dot arrays. Description of Samples The samples used for the observations disc ussed within this chapter were three single quantum well samples with electron densities around 1010 cm-2 grown at the Laboratorio Nazionale TASC (IN FM) in Trieste, Italy by the team of Drs. F. Capotondi, G. Biasiol, and L. Sorba. Estimation from the signal response yields an approximation of 2X106 electrons within the sample exhibiting th e lowest density. At the time of this writing, this is the lowest density sample in which MDESR has been successfully observed. The samples used were grown using mo lecular beam epitaxy and consisted on a single, 30 nm wide, GaAs quant um well, surrounded by the AlxGa1-xAs barrier regions, where x=0.11, 0.05, and 0.03 for samples HM0455, HM0459, and HM0461, respectively. Three silicon -doping layers were grown within each sample. Two were placed at positions nearly symmetric to one another about the GaAs well, these providing the conductive carriers for the sample. These we re grown and placed to compensate for silicon diffusion into the AlGaAs barrier regions.124 The third layer is placed in close

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204 proximity to the free surface of the quantum well and is used to reduce the depletion layer. The carrier concentrations were fu rther reduced in each successive sample via increased AlGaAs spacer layers. The samples were patterned into the typical Hall bar geometry, with a 60 m wide and 260 m long channel, using photolithography and wet chemical etch processes. The electron mobilities and densities within these sa mples, observed at 1.4 K, are displayed in Table 3-1. Illumination of the samples at T= 10 K, using a red LED in close proximity to the sample, was applied for a period of a c ouple seconds in order to provide sufficient numbers of conduction electrons to the well region from the silicon -doping layers. Following illumination, decreases in the el ectron density, on the order of 20%, were observed in samples HM0455 and HM0459. Th is reduction was presumably due to defect trapping. Experimental Results Displayed in Figure 8-1 is the MDESR sp ectrum obtained in sample HM0459 as a function of xx R , using the single lock-in detection te chnique, at T=1.4 K and within the =1 minimum, and is representative of the si gnals observed within the samples presented. All spectra were reco rded at sweep rates of 492 mT/m in, although slower sweeps were performed and in the higher density samples, similar DNP effects as those previously presented were observed. The MDESR response was measured in both HM0455 and HM0459 at a variety of channel curr ents within the range of 4 nASDI 1.2 A. No appreciable changes were obser ved within this range. Therefore, all experiments were performed at SD I =1.2 A.

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205 Microwave power, channel current and g -factor characterization of MDESR The 30.03 GHz MDESR response to increasi ng microwave power was measured at T=1.4 K and is displayed in Figure 8-2. As in our previous studies using this microwave setup, the signal response is linear with resp ect to microwave power. It should be noted that the dependence of the signal response upon microwave po wer does not extrapolate to zero. Thus the MDESR response does not app ear to be linear at very low microwave powers. B0 (T)Rxx( ) B0 (T)Rxx( ) Figure 8-1: MDESR peak as a function of xx R at T=1.4 K, 060 , and within the =1 minimum. Microwaves were applied to the sample at 23.00 GHz. Unlike the other two samples, a lowe r channel current was required in measurements of the lowest density sample, HM0461, to avoid current heating effects. Furthermore, MDESR detection was only possibl e at the lowest temperature attainable by our cryostat, (T 400 mK). All three samples exhibited bare electron g-factors, (*0.42eg in HM0455 and HM0459, and *0.44eg in HM0461, same as in bulk GaAs) slightly higher than those measured in previous samples. The resonance positions in all three samples were measured at both =1 (circles) and 3 (squares) and are displayed in Figure 8-3. Using the

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206 tilted field method, the =1 and 3 minima were shifted to higher magnetic fields, allowing for observation and characterizati on of the MDESR at a wide variety of magnetic fields. Power Detector Readout (mV) Rxx ) Power Detector Readout (mV) Rxx ) Figure 8-2: Microwave power dependence of the MDESR response in sample HM0455 at T=1.4 K. From the linearity of the plot, it is clear that no microwave saturation effects are observ ed. The source of the nonzero x-intercept is not understood. B0(T)ESR(GHz) B0(T)ESR(GHz) Figure 8-3: The microwave fr equency dependence of the MDESR resonant field is displayed for all three samples at =1 (squares) and 3 (circles) at numerous angles as shown in the graph. The temperature during the measurements was approximately 1.5 K. The lowest field resonance, pertaining to 8.5 GHz and a sample tilt angle of 058 , is the lowest magnetic field where the =1 MDESR has been measured in GaAs/A lGaAs quantum wells. The g-factors were determined to be -0.42 (HM0455 and HM0459) and -0.44 (HM0461).

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207 Temperature dependence of MDESR response The maximum in the MDESR amplitude was observed at T=1.5 K for the two higher density samples, while, as previous ly stated, the lower density sample only exhibited the resonant response at T=0.4 K. The temperature dependence of the MDESR response in sample HM0459 is presented in Figu re 8-4. The maximum at 1.5 K in this sample corresponds to approximately 1/4th of the activation gap 6 KE , as predicted by the temperature dependence model discu ssed in Chapter 3 and in Olshanetsky et al.99 The fact that an MDESR response was only observed in sample HM0461 at the lowest temperature attainable may be explained by the “small difference between the Fermi energy and the zero point energy of the well (0 .642 meV), the relativel y close values of FE and BkT (0.120 meV), and the fact that th e Landau level energy separation is comparable to BkT at 1.4 K”.125 T (K) Rxx( ) T (K) Rxx( ) Figure 8-4: Temperature de pendence of the MDESR response in sample HM0459. A maximum in the MDESR resonance wa s observed at approximately 1.5 K while 6 KE . This is consistent with the heating model discussed in Chapter 3, which predicts th e maximum MDESR response at /4BTEk .

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208 Observation of 2 g Resonance ESR has presented itself as an ideal met hod for probing electron in teractions within the conduction band of GaAs/AlGaAs quantum wells and heterostructures. Recently, ESR due to electrons with *0.4egg have been observed th rough both optical and ED methods within such samples. These ESR tran sitions have been associated with localized electrons, centered at intrinsic surface defects. Previously, the ED of these defect-bound electrons was explained through a trapping/un-tr apping mechanism of the electrons. This mechanism should cause temporary occupations of energies within the band gap, while resonant excitations would provi de the driving force for spin -dependent recombination. In order to create such occupations and ther efore allow the resonant observations, deep penetrating LED illumination is typically required. In this chapter, direct observat ion of an EDESR resonance with 2 g , is discussed. The detection of this resonan ce was attained without initial illuminati on of the sample. This resonance was observed and characterized within the temperat ure range of 0.9-6 K by the ED method, and under microwave ir radiation between 10 and 50 GHz. This observation was made in HM0455 and HM0461. No such resonance was observed within higher density samp les such as EA124 and AG662. Description of 2 g Peak Structure The EDESR signals were observed us ing both the single and double lock-in techniques discussed in Chapte r 3. The signals were char acterized within the 15-50 GHz range, at a wide array of applied magnetic fields . The signal appears as a triplet, with the splitting between the two satellites bei ng 378 mT in sample HM0461 and 391 mT in sample HM0455. The splitting of these satellit es is centered at a position within 1 mT of

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209 the central peak. The central transition is furt her split into a triplet, exhibiting splittings on the order of 140 mT in HM0461 and 45-80 mT in HM0455. The central triplet structure is shown in detail in Figure 8-5. A clear hysteresi s is observed, consisting of a smoothing of the far edge of the peaks with respect to the starting position of the sweep, (e.g. during a down-sweep of the field through the 2 g resonance, the smoothing of the peak will be observed on the low field, or left hand side of the peak). The triplet splitting of the resonance along with the observed triplet structure of the central transition, lends itself to the possibility of the resonance havi ng a triplet of triplet structure, however, poor resolution of the peak prevents a true description of the splitting. B0 (T)Rxx B0 (T)Rxx Figure 8-5: EDESR response in sample HM0455 at T=1.7 K, while irradiated with 36.02 GHz microwaves. The external satelli te peaks were removed for clearer observation of the triplet splitting of the central transition. The splitting between the two outer pe aks of this transition is approximately 45 mT. The magnetic field vs. microwave frequency dependence of the central peak of the central transition is displayed in Figure 8-6 and leads to an observed g-factor of 1.97. This dependence is observed to extrapolate to zero, which is inconsistent with the MDESR of the conduction electr ons within the quantum well, which typically exhibit a small zero-field splitting as re ported previously by Uemura,36 and the team of Marques

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210 and Sham.37 This is not true of the splitting of the central ESR transition line. This splitting displays essentially no magnetic fi eld dependence, therefore leading to the presence of a zero field splitting. B0 (T) (GHz) B0 (T) (GHz) Figure 8-6: Determination of the g-factor for the central peak of the low frequency resonance in sample HM0455. The lin ear fit extracts a g-factor for the EDESR of 1.97. Through a tilt angle experiment, the magne tic field dependence of the splitting between the two exterior lines of the central triplet was inve stigated. While a change in the splitting at 4.5 K was obs erved at two angles within HM0455, no clear dependence upon angle could be extracted. Within this sa me sample at T=1.4 K, no change in the splitting of 45 mT was observed, independent of the angle. This is consistent with the results in sample HM0461, where the peak-topeak distance of 140 mT was constant at all angles, independent of the temperature. The splitting of the MDESR peaks is not only independent of angle, but al so independent of the applie d magnetic field and microwave excitation frequency.

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211 Effect of Resonant RF I rradiation upon ESR Structure Resonant RF was used to irradiate the sample during field-swept EDESR experiments. Irradiation at frequencies pe rtaining to the resonant conditions of 71Ga, 69Ga, and 75As (found in well region), 27Al (barrier region), 29Si ( -doping layers), 197Au, 73Ge, and 61Ni (ohmic contacts) were performed dur ing the down-sweep of magnetic field such that the NMR would occur within the linewid th of the central or satellite transitions. No change in ESR lines was observed, indepe ndent of the position of the NMR condition. Slow magnetic field sweeps through eq ESR B , produced no observable DNP effects, regardless of the sweep direction. This indicates that the el ectrons associated with this resonance are localized and therefore the FC hfi is only observed between the excited electron and the small number of nuclei centered about the electron trapping site. Because of this fact, stop-field MDENDO R experimentation was not possible. Temperature Dependence of 2 g EDESR The 2 g EDESR was characterized in both HM0455 and HM0461 within the temperature range of 1.4-4.0 K and 0.9-6.0 K, respectively. These traces are displayed in Figures 8-7 (HM0455) and 8-8 (HM0461). In co ntrast to the resona nce associated with the *0.4eg , the g =1.97 resonance in HM0455 showed no significant temperature dependence. On the contrary, this resona nce decreased in amplitude with increasing temperature in HM0461, similar to the dependence observed for the *0.4eg resonance within this temperature range. The reason for these vastly different temperature dependences is not clear at this time.

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212 B0(T)Rxx 4.60 K 1.57 K B0(T)Rxx B0(T)Rxx 4.60 K 1.57 K Figure 8-7: EDESR of g =1.97 resonance at two temperatures in sample HM0455. No temperature dependence is appare nt within this sample. B0(T)Rxx( ) 6.0 K 1.0 K B0(T)Rxx( ) B0(T)Rxx( ) B0(T)Rxx( ) 6.0 K 1.0 K Figure 8-8: Temperatur e dependence of the g=1.97 EDESR resonance in sample HM0461. Unlike HM0455, a clear te mperature dependence is observed within this sample, showing a decrease in EDESR amplitude with increased temperature.

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213 Microwave Power Dependence Similar to experiments in higher density samples, determining if microwave saturation of this resonance is present is very important. A series of ESR spectra, taken at a variety of microwave powers, ranging from full to half power, are displayed in Figure 8-9. Unlike previous studies, the possibil ity of microwave saturation of this signal is present at high powers. This is evident from the lack of a power dependence in the last three traces. B0 (T)Rxx( )Max Half B0 (T)Rxx( )Max Half Figure 8-9: Microwave pow er dependence of the g =1.97 resonance at 50.07 GHzESR The presence of the two satellit e peaks is apparent at high microwave powers. The constant EDESR peak amplitudes at the higher microwave powers, indicates a saturation of the resonance. Concluding Remarks Experimentation involving the 1.97 g resonance is still very preliminary. It appears that interactions betw een the electrons and nuclear species are localized at the trapping site of the electron, therefore the hfi does not indu ce a large nuclear field that would be capable of causing an Overhauser shift of the ESR condition. This is in direct

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214 contrast to the observed interactions betw een the conduction electrons in the quantum well and well nuclei. The 1.97 g EDESR peak is observed as a triple t, with a triplet structure of the central transition apparent duri ng most observations. It is po ssible that the satellites are due to separate electron excitations entirely, although this does not seem likely due to their equal spacing about the central peak. While the splitting of the centr al peak is not dependent upon 0 B , which is consistent with a hyperfine coupling to a nucle i, the lack of an angle dependence, DNP broadening, and NMR excitation e ffects, casts doubt that such a coupling is present. Here, two possible mechanisms for this ex citation are presented. First, if the change in magnetoresistivity is due to a trap ping of the resonantly excited electron at a defect site followed by the release of the elect ron, such a splitting could be possible due to the hyperfine coupling between the electr on and the immediately local nuclei during the trapping period. Secondly, resonant abso rption of microwaves by the electrons within the GaAs capping layer (not associated w ith the well region), throughout the entire structure, or due to elect rons involved in parallel c onductance through the silicon doping layers ( g =1.98 for conductive electrons with in bulk silicon) could possibly account for this observation. Further expe riments to discern between these possible mechanisms, or any other not yet determined possibilities, must still be explored.

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215 CHAPTER 9 CONCLUDING DISCUSSION In conclusion of this dissertation, a summary of the research and its accomplishments is required. The temperatur e and filling factor dependence (within the vicinity of the =1 filling factor) of the MDESR re sponse was examined. The latter study was extended to include the effects of filling factor upon the activation gap E , and on the magnetoresistance prefactor 0 R . There appears to be no dependence of the MDESR line width upon temperature, while the MDESR amplitude appears to go through a maximum in the vicinity of 2 K in sample EA124 at 1 . The relative temperature independence of the MDESR lin e width led to the determination that 2 eT is also independent of temperature within this range. The maximum found in the dependence of the MDESR amplitude was found at a temperature /42 KBEk , which is predicted by the model that was develope d and presented for the MDESR mechanism. The filling factor study determin ed that the MDESR amplitude, E , and 0 R all go through a maximum at approximately =1. The data presented here led to the development of the aforementioned model, whic h is based on the resonant absorption of microwave irradiation leading to a heating of the 2DES. This heating model was tested for both an itinerant (non-i nteracting electrons) and spin -wave (highly correlated electrons) system of electr ons and in both cases predicted the appearance and the approximate position of the maximum in the MDESR amplitude as a function of temperature. The filling factor dependence of MDESR amplitude is consistent with the

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216 similar dependence found for MDNMR,75 which also exhibits a maximum as =1 is approached from either the high or low field side of the minimum. Using the strong FC hfi betw een the conduction electrons and local lattice nuclei, the first comprehensive studies of the sweep rate and temperature dependences of the MDESR response under the effect of the Over hauser effect, and the dependence of the MDENDOR response upon RF were performed. MDENDOR was observed for all three nuclei associated with the quantum well region. Due to the FC hf i, upon a reduction in sweep rate, a drastic increase in the breadth of the MDESR line width is observed. This was explained by a dynamic Overhauser shift of the ESR due to the increased nuclear polarization built up by the cross-relaxation of the electron spins to the nuclei. The opposite was observed on the up-sweep, wh ere a reduction in the line width corresponding to the reduction in sweep rate was evident. The temperature dependence of the MDESR at a sweep rate of 50 mT/min , presented an overall broadening in the MDESR line width with increased temperat ure that may be associated with a corresponding decrease in the nuclear spin lattice relaxation time, 1 nT . This dependence provides experimental evidence supporting th e theoretical hypothesi s for the Korringa relaxation mechanism within these systems. Finally, the dependence of the MDENDOR response on RF, determined that upon reduci ng the frequency corre sponding to the NMR excitation in respect to the ESR condition, ESRNMRB , a distinct change in behavior is observed. Initially, this reduction in fre quency leads to an increase in the observed MDENDOR peak amplitude, while the locking of the ESR condition to 0 B is maintained. Eventually, the peak passes through a maximum. After this point, the peak begins to diminish in amplitude and a corresponding loss of the MDESR condition is observed.

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217 This abrupt change in behavior is determin ed to occur at a critical field, which is dependent upon the RF power in the non-sa turation regime. The microwave power dependence of the MDESR and the nuclear spin lattice relaxation times at a sampling of magnetic fields and tilt a ngles were also explored. Furthermore, the first observati on of a quadrupole splitting of the 75As nuclei and double quantum excitations of 75As and 69Ga nuclei within the MDENDOR spectra, were reported. The quadrupole splitting was observed via both the field and RF-swept methods, while only the RF-swept met hod was attempted for observing the double quantum spectra due to the requirement of signal averaging. A comprehensive study aimed at determining the origin of the qua drupole splitting and its association to the double quantum excitations was performed. Incorporated within this study was the dependence of the quadrupole splitting upon th e magnetic field sweep rate, applied RF power, source-drain current, Overhauser shift, quantum Hall filling factor and tilt angle of the sample. Only the tilt angle had a cl ear effect. Upon tilting the sample away from the perpendicular orientation (whe re the plane of the 2DES is to 0 B ), the magnitude of the splitting increased almost linearly, w ith a definite absence of the expected 23cos1 dependence observed. The studies presented appear to rule out a number of possible explanations for the splitting, such as a lattice mismatch at th e interface between the quantum well and barrier regi ons or the formation of a Schottky barrier at the semiconductor-metal interface between the quantum well and the ohmic contacts. Possible mechanisms which should be explored are the presence of an internal strain on the sample aligned at an axis other than the z-axis, the LQSE, which allows for an interaction between an external electric fi eld and the nuclear qua drupole moment, or the

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218 presence of inhomogeneities within the el ectronic distribution throughout the quantum well, thereby creating an EFG within the well region. The observation of the single-photon, double quantum excitations of both 75As and 69Ga are thought to be due to a coupling through th e electric quadrupole interaction. This interaction was first reported by Eickhoff et al.79 This observation provides evidence for the unresolvable presence of a quadrupole splitting of the 69Ga MDENDOR spectra. The utilization of low temperature MDNMR at a wide magnetic field range would be very beneficial in exploring both the quadrupol e splitting and double quantum excitations. Using the heating model that was presen ted in Chapter 3 a nd first published by Olshanetsky et al.,99 an attempt at providing numerical simulations to explain the effect of the FC hfi upon the MDESR sp ectra and the appearance and ab rupt change in behavior of the MDENDOR response, were performed. These simulations reproduced the majority of the qualitative features of th e sweep rate dependence of the MDESR spectra, including the increase in breadth and loss of MDESR amplitude at the slowest sweep rates. Further corrections must be incor porated into the simulation program to account for the inability to simulate the overall symm etry in the line shape at the slowest sweep rates. On the other hand, the MDENDOR spectra, its dependence upon RF, and most importantly, the critical change in behavior , were reproduced extrao rdinarily well. The main observation taken from the simulations is the correlation betw een the appearance of the critical change in behavior with the a decrease in the nuclear field n B that is larger than the unperturbed ESR line width, ESR FWHM . The simulations provided a more comprehensive look at the behavior of th e MDESR and MDENDOR responses than the previous attempts made by Seck et al.15 and Hillman and Jiang.50

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219 Utilizing the methods and observations di scussed above, the detection of EDESR within a 3D and 2DES within a WPQW was realized for the first time. A full study of the behavior of the EDESR in the 3D state, al ong with a determination of the effect of the tilt angle upon the EDESR dependence, was pe rformed. From these observations it was determined that the EDESR of the 3DES is similar to that of the MDESR found in the QHE within 2DESs. The sweep rate determined that the sign of the bare electron g-factor was still negative, and that the dependen ce on the down-field sweep rate was quite similar. The most intriguing observation within the WPQW sample was related to the temperature dependence of the decay time of the Overhauser shift (and therefore probably the nuclear spin lattice relaxation tim e). This decay was observed to increase by 2-4 times in the 3DES. Furthermore, a l ack of temperature dependence of the decay time in this orientation, predicts that the Korringa relaxation mechanism is not dominating at this angle. This is in dire ct contrast with the observed decrease in OS found in the 2DES. Therefor e it appears that not only may the nuclear spin lattice relaxation time be tuned by the tilt angle, but the overall dominant relaxation mechanism may be changed as well. It was observed that the temperature had minimal effect upon the EDESR line width, although a maximum in the amplitude at about 2 K was observed, consistent observations in the 2DESs. However, a fe w notable differences between the 3D and 2DESs were detected. First of all, the ra nge of temperatures at which the EDESR was detected was much wider for the 3D case. Furthermore, the up-field0 B sweep rate dependence of the EDESR amplitude was in direct contradiction to that which is found in

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220 the 2DESs. Finally, the observation of a down-field shift in th e equilibrium EDESR position with decreasing temperature is some thing that has not been observed in any sample prior to this one and is only present with the sample in the parallel orientation. The bare electron g-factor also displayed a dependence upon the tilt angle. Here a clear increase in the g-factor was observed as the sample was tilted away from parallel orientation. This observation is coupled with the clear change in the dimensionality of the conduction electrons from a three dimens ional to a two dimensional system, although a direct correspondence betw een these two phenomena is still not definite. Finally, MDESR, within the regime of both the =1 and 3 filling factors, was observed in the lowest density samples repor ted to date. This observation provides preliminary steps in attaini ng the use of this detection method for qubit manipulation in quantum information processing. Furtherm ore, the ED of an ESR exhibiting 1.97 g , was also reported. This resonance was observe d as a triplet, with the central transition further split into a triplet as well. Explorati on into the origin of this resonance is still preliminary, although it is clear that the Overha user effect does not pl ay a significant role in the sweep rate dependence of this peak. There does not appear to be any definitive effect of the magnetic field, tilt angle or temperature upon the observed splitting. Two possible theories for the or igin of this peak were presented, although sufficient experimental evidence for ei ther is still lacking. In conclusion, the work presented within this dissertation has provided insights into the interactions betw een the conduction electrons and the lattice nuclei within 2DES exhibiting the QHE. A model fo r the mechanism of this detection was presented and has been supported by further experimental evid ence. This detection method has been

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221 utilized to investigate different varieties of sa mples and it is the opini on of the author that this method should provide a technique for e xploring the electronic and nuclear systems, and the interactions between them, in a wide variety of low-dimensional conductive systems, such as carbon nanotube arrays.

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222 APPENDIX A SYMBOLS LIST Symbol Description A Hyperfine coupling constant 0A Area 0a Lattice constant B Component of 0 B perpendicular to the 2DES 1b Microwave field amplitude eff B Effective magnetic field ESR B Magnetic field of ESR condition D NP ESRB Magnetic field of Overhauser shifted ESR B n B Nuclear field D NP nB Magnitude of nuclear field due to DNP NMR B Magnetic field of NMR condition s top B Stopping magnetic field during field sweep Magnetic susceptibility D Diffusion constant d Distance ESRNMRB Difference between ESR B and NMR B n B Change in nuclear field crit n B Critical change in nuclear field E Arrhenius activation gap of xx R T ESR FWHM ESR line width NMR FWHM NMR line width e Electronic charge cE Electron-electron exchange interaction energy FE Fermi energy Z E Zeeman energy mod f Modulation frequency of microwaves g Ratio of Zeeman and exchange energies eg Bare electron g-factor (general case) *eg Bare electron g-factor (2DES)

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223 ** eg Enhanced electron g-factor (2DES) e Electron gyromagnetic ratio n Nuclear gyromagnetic ratio I Nuclear spin angular momentum 0 I Thermal equilibrium nuclear spin polarization D C I Magnitude of injected DC current SD I Magnitude of source-drain current z I Average nuclear spin pol arization aligned with 0 B D NP zI Average nuclear spin pol arization aligned with 0 B due to DNP j Index of nuclear plane within well J Total angular momentum k Wave vector Bk Boltzmann constant B Constant of thermal conductance between 2DES and bath 0l Magnetic length ˆ Magnetic moment Mobility M Magnetization *m Effective electron mass em Bare electron mass B Bohr magneton n Nuclear magneton Im Eigenvalues of ˆ I Sm Eigenvalues of ˆ S Filling factor N Landau level index nz Electron concentration profile in well 0N Degeneracy of spin-split Landau levels RF Applied RF s N Electron density p Power dissipated from 2DES to bath P Spin polarization p Population Q Nuclear quadrupole coupling constant Sample tilt angle r Radius FE Density of states at the Fermi level

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224 0 R Resistance prefactor cr Cyclotron radius xx R Longitudinal resistance (along current path) xy R Hall resistance (perpendicular to current path) ˆ S Electron spin angular momentum S Number of electronic spin flips s Saturation parameter of ESR xx Longitudinal conductivity (along current path) xy Hall conductivity (perpendicular to current path) T Temperature t Time 1 eT Electron spin-lattice relaxation time 2 eT Electron spin-spin relaxation time 1 nT Nuclear spin-lattice relaxation time 2 nT Nuclear spin-spin relaxation time OS Overhauser shift decay time constant bT Bath temperature s T Spin temperature ijV Magnitude of the electric field gradient W Transition rate 0, e Electron Larmor freque ncy (absence of DNP) 1 e Electron Rabi frequency app Applied microwave frequency c Cyclotron frequency D NP ESR Resonant frequency of the electron spins (general) ESR Electron Larmor freque ncy (general case) NMR Nuclear Larmor frequency xW Width of current channel perpendi cular to current and growth axis Electronic wave function

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225 APPENDIX B ACRONYM LIST Acronym Description 2DES Two-dimensional electron system 3DES Three-dimensional electron system CDE Charge density excitation CDW Charge density wave CW Continuous wave D.O.S. FE Density of states at the Fermi level DNP Dynamic nuclear polarization ED Electrically detected /electrical detection EFG Electric field gradient ENDOR Electron nuclear double resonance ESR Electron spin resonance FC hfi Fermi contact hyperfine interaction FWHM Full-width half-maximum LQSE Linear quadrupole Stark effect MD Magnetoresistively detected/magnetoresistive detection MOSFET Metal oxide semiconducto r field effect transistor NMR Nuclear magnetic resonance QHE Quantum Hall effect RF Radiofrequency SDE Spin density excitation SDW Spin density wave TRFR Time resolved Faraday rotation WPQW Wide parabolic quantum well

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234 BIOGRAPHICAL SKETCH Joshua Caldwell was born in Attleboro, MA, on December 31, 1976. He grew up in Attleboro, attend ing Attleboro High School where he was a 3 year letterman and captain of the swim team. In August of 1995, Joshua moved to Blacksburg, VA, where he began his studies in chemistry at Virginia Polytechnic Institute and State University (Virginia Tech). While attending Virginia Tech, Joshua participated in the coop program, taking a position with ITT Industrie s-Night Vision in Roanoke, VA, where he spent 3 terms working as a process/manufactur ing engineer. During this time he was a member of the Virginia Tech Rugby Football Club and was asked to join the Chemistry Honors Fraternity. He graduated with a B.A. in chemistry and minor in History in May of 2000. After getting married on May 27, 2000, Joshua and his new bride Betsy moved to Micanopy, FL, where he entered the graduate program in the Chemistry Department at the University of Florida. During his st udies, he was awarded the Chemical Physics Fellowship for the Fall and Spring 2000/2001, the Liberal Arts and Sciences Aschoff Dissertation Fellowship for the Spring 2004 semester, and a Chemistry Departmental Teaching Award for his time as the TA in NMR Services.