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PAGE 1 1 DENSE QUANTIZED MAGNETOPLASMAS IN HIGH MAGNETIC FIELDS PROBED BY ULTRAFAST LASERS BY JINHO LEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR T HE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Jinho Lee PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS First and foremost, I would like to express my deepest appreciation for my advisor, Professo r David H. Reitze whose patient supervision, instruction and encouragement, tremendous support and friendship during my Ph.D allowed me to pursue my research to the best of my ability. This study would never have been achieved without his patient guidance and it has been a great pleasure to work with him. I am very grateful to the member of my dissertation committee, Prof. David Micha, Prof Andrew Rinzler, Prof. Guido Mueller, and Prof. Valeria Kleiman for their valuable advice. I would also like to thank to Professor Junichiro Kono from Rice University, Prof. Alexey Belyanin from Texas A&M University, and Prof. Christopher Stanton from University of Florida for their experimental and theoretical supervision I received during my research project. Appreciat ion is extended to Dr. Youngdahl Cho, Dr. Xiaoming Wang and Dr. Steve McGill. During my Ph.D. at the National High Magnetic Field Laboratory, I received the great help from them and this work would not be completed without their tremendous contributions. My sincere gratitude also goes to my friends in Korean Baptist Church who made life in Tallahassee all the more exciting. They always cheered me up and helped me relax during my tough life in the laboratory. Last but not least, I want to send my warmest thanks to my parent for their encouragement, support, and their endless love throughout my career. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 8 LIST OF FIGURES .............................................................................................................................. 9 LIST OF ABBREVIATIONS ............................................................................................................ 17 ABSTRACT ........................................................................................................................................ 18 CHAPTER 1 INTRODUCTION AND OVERVIEW ..................................................................................... 20 2 MAGNETO OPTICS OF SEMICONDUCTOR QUANTUM WELLS ................................. 27 2.1 Introduction ........................................................................................................................... 27 2.2 Excitonic Optical effects in Semiconductor Quantum Well .............................................. 28 2.2.1 Symmetry Properties and Excitonic effect in Quantum Wells ................................ 28 2.2.2 Energy Expression near Point and Heavy and Light Hole ................................... 31 2.2.3 Excitons in Two Dimensions ..................................................................................... 32 2.3 Magnetospectroscopy of Semiconductor ........................................................................... 34 2.4 Optical Resonance of the Two Level Dipole System ......................................................... 37 2.4.1 Line shape and Line Width ........................................................................................ 38 2.4.2 Optical Bloch Equation and Two Level System ...................................................... 39 2.4.3 Phenomenological Decay Constants ......................................................................... 42 3 ULTRAFAST FACILITIES AT NATIONAL HIGH MAGNETIC FIELD LABORATORY .......................................................................................................................... 51 3.1 Introduction ........................................................................................................................... 51 3.1.1 Magnet system for Magnet optic Experiment .......................................................... 51 3.1.2 Cryogenic System for Magnet Optic Experiment .................................................... 52 3.1.3 Laser Light Sources .................................................................................................... 5 4 4 CONTINUOUS WAVE SPECTROSCOPY OF INGAAS MQWS IN HIGH MAGNETIC FIELD ................................................................................................................... 65 4.1 Background ........................................................................................................................... 65 4.2 Basic Theory of Magnetooptics .......................................................................................... 67 4.2.1 Introduction to Emission Proces s of Simple Two Level Dipole System ................ 68 4.2.2 Strong Field Limit ...................................................................................................... 72 4.2.3 Band Mixing ............................................................................................................... 73 PAGE 6 6 4.3 Experimental Motivation ...................................................................................................... 75 4.4 Sample Structure and Experimental Methods ..................................................................... 76 4.5 Prior Study of Photo luminescence Spectroscopy of 2 D Quantum Well .......................... 77 4.5.1 Photoluminescence Spectroscopy at Weak Field Region ........................................ 77 4.5.2 Prior Study of InxGa1xAs Photoluminescence Spectroscopy .................................. 79 4.6 Magnetooptical Characterization of InxGa1 xAs Quantum Wells at High Fields and Carrier Densities: Absorption and PL Spectroscopy ...................................................... 81 4.6.1 Sample Characteristics ............................................................................................... 81 4.6.2 Dark Transition and Anti Crossing ........................................................................... 82 4.6.3 Diamagnetic and Landau Shifts in InGaAs Quantum Wells ................................... 83 4.6.4 Photoluminescence Spectra in High Magnetic Fields .............................................. 84 4. 7 Energy States in In0.2Ga0.8As/GaAs Quantum Wells at High Magnetic Fields: Excitons, Magnetoplasmas, and Mott Transitions ............................................................... 87 4.7.1 Bandgap Renormalization and Landau Fan Diagram .............................................. 87 4.7.2 Coexistence of Exciton and Landau Levels .............................................................. 89 4.7.3 Fitting Method for Landau Levels ............................................................................ 91 4.7.4 Binding Energy .......................................................................................................... 93 4.7.5 Mott Transition and Luminescence from Higher Subbands .................................... 94 4.7.6 Reduced Mass ............................................................................................................. 95 4.7.7 Summary ..................................................................................................................... 96 5 TIME RESOLVED ABSORPTION MEASUREMENTS OF INGAAS/GAAS QUANTUM WELLS IN HIGH MAGNETIC FIELDS ......................................................... 114 5.1 Introduction ......................................................................................................................... 114 5.2 Relaxation Process in Semiconductors .............................................................................. 114 5.3 Motivat ion ........................................................................................................................... 118 5.4 Experimental Method ......................................................................................................... 119 5.4.1 Femtosecond Pump Probe Spectroscopy ................................................................ 119 5.4.2 Experimental Setup .................................................................................................. 120 5.5 Experimental Results and Discussion ................................................................................ 122 5.5.1 Time Transient Absorption Measurement at 0 Tesla ............................................. 122 5.5.2 Time Transient Absorption Measurement at 17.5 Tesla ........................................ 125 5.6 Summary.............................................................................................................................. 129 6 TIME RESOLVED CHARACTERIZATION OF ULTRASHORT PHOTOLUMINESCENCE BURSTS: DIRECT OBSERVATION OF INTER LANDAU LEVEL RECOMBINATION ................................................................................ 143 6.1 Introduction ......................................................................................................................... 143 6.2 Photoluminescence of Simple 2 Level System ................................................................. 144 6.3 Motivation ........................................................................................................................... 146 6.4 Experimental Methods ........................................................................................................ 147 6.4.1 Installation of Streak Camera at NHMFL .............................................................. 147 6.4.2 Experimental Methods ............................................................................................. 150 6.5 Experimental Results and Discussion ................................................................................ 152 6.6 Summary.............................................................................................................................. 157 PAGE 7 7 7 CONCLUSION AND FU TURE WORKS .............................................................................. 172 APPENDIX A HARMONIC OSCILLATOR AND OPTICAL SUSCEPTIBILITY .................................... 175 B ABSORPTION AND REFRACTION ..................................................................................... 177 C DENSITY OF STATES FOR D DIMENSIONAL SYSTEM ............................................... 182 D FITTING METHOD OF PHOTOLUMINESENCE .............................................................. 185 LIST OF REFERENCES ................................................................................................................. 187 BIOGRAPHICAL SKETCH ........................................................................................................... 194 PAGE 8 8 LIST OF TABLES Table page 2 1 Examples of Luttinger param eter [ 45]. ................................................................................. 44 4 1. Material parameter for InAs and GaAs [ 68] ......................................................................... 98 4.2 Critica l magnetic field cH ..................................................................................................... 99 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Calculated band structure of the dire ct gap semiconductor for GaAs [ 22]. Reprint with permission from J. R. Chelikowsky, M. L. Cohen, PRB 14 556 (1976).( 1976 American Physical Society) ................................................................................................... 25 1 2 Energy band versus lattice co nstant for a number of se miconductor materials [ 23]. Reprint with permission from 1994 IEEE ( 1994 IEEE) .................................................. 25 1 3 Chemical lattice image, a technique based on transmission electron microscopy, of a GaAs/AlGaAs heterostructure [ 5 ]. Reprint with permission from A. Ourmazd, D. W. Taylor,et. al., PRL 62 933 (1989) .( 1989 American Physical Society) ................................................................................................... 26 1 4 Schematic representation of a A) type I semiconductor quantum well and B) type II structure. ................................................................................................................................. 26 2 1 Schematics of a single quantum well. ................................................................................... 45 2 2 Single quantum wells and associated ground state and first excited state wave functions for A) infinite and B) finite quantum barriers. ..................................................... 45 2 3 Schematic band structure in the vicinity of 0 k for a bulk zinc blende semiconductor, such as GaAs ............................................................................................. 46 2 4 Schematic representation of the absorptio n lines in A) 3dimensions and B) 2 dimensions. The dashed line indicates the spectrum neglecting the Coulomb interaction and the solid line is with the Coulomb interaction. ........................................... 47 2 5 A): Energy wave vector relation for electrons in a magnetic field. B): In a magnetic field, the continuous energy levels in the range c cn E to c cn E ) 1 ( evolve into discrete Landau levels, c cn E ) 2 / 1 ( ....................................................................... 48 2 6 The individual Lorentzian emission lines (homogeneous line width) associated with different atomic dipoles oscillating at five distinct frequencies. ......................................... 49 2 7 Energy level of an atomic system, showing two levels connected by near resonance transition. Line broadening mechanisms are not shown for simplicity. ............................ 49 2 8 Pseudospin vector s traces out orbit on the unit sphere. ...................................................... 50 3 1 Schematic technical drawing of the 31 T resistive magnet in cell 5 at NHMFL. Reprint with permission National High Magnetic Fiel d Laboratory ( 2009 NHMFL) ................................................................................................................................. 57 PAGE 10 10 3 2 Schematic technical drawing of the 17.5 T superconducting magnet (SCM3) located in Cell 3 at NHMFL [ 46]. ...................................................................................................... 58 3 3 Experimental setup in A) Cell 3 and B) Cell 5 in NHMFL. C): Laser from Cell 3 goes through the hole in the wall between Cell 3 and Cell 5 and is introduced to the mirror mounted under the 3 1T r esistive magnet in Cell 5[ 46]. ........................................... 59 3 4 Technical drawing of cryostat and optical probe for A) 31 T resistive magnet in Cell 3 and B) 17T superconductor magnet in Cell 3 at NHMFL ................................................ 60 3 5 Block diagram of CW spectroscopy of the magneto optical experiment. Main apple computer controls magnet, temperature and spectrometer. ................................................. 61 3 6 Schematic diagram of Faraday geometry. ............................................................................ 62 3 7 Tungsten lamp spectrum for CW spectroscopy. .................................................................. 62 3 8 Schematic diagram of Chirped pulsed amplifier system (Coherent Legend F). Evolution is pumping source of CPA. .................................................................................. 63 3 9 Principle of chirped pulsed amplification. Input 130fs pulse is stretched to 100ps and then goes to amplifier stage. Final pulse after compresso r is 1kHz, 150fs and pulse energy is 2mJ .......................................................................................................................... 63 3 10 Schematic diagram of optical parametric amplifier (TOPAS OPA). .................................. 64 3 11 Schemati c diagram for Optical Parametric Amplifier system. Solid line denotes the real quantum mechanical state and the dashed is for the virtual state. ............................... 64 4 1 Simplified picture of A) spontaneous emi ssion and B) amplified spontaneous emission process of a simple two level atomic system. ..................................................... 100 4 2 The ASE and SF processes. ASE: There is no interaction between atoms on excited states, thus no coherence between the emitted photons. Omnidirectional emission results. SF: Atoms in the excited state interact with vacuum electromagnetic field fluctuations and establish a coherent ensemble dipole, giving rise to strong emission. .. 100 4 3 Energy dispersion relations for hole states in a quantum well. The solid lines represent the mixing of heavy and light hole valance bands from Eq. 4 19 and the dashed lines shows the energies neglecting b and mixing. ................................................. 101 4 4 Sample structure of In0.2 Ga0.8As/GaAs multiple quantum well. ................................... 101 4 5 Simplified schematic setup of InGaAs m ultiple quantum well absorption and photoluminescence spectroscopy. ....................................................................................... 102 4 6 Schematic diagram of collection of the PL emission from Faraday geometry. One right angle prism is located at the ed ge of the sample to collect in plane PL. ................. 102 PAGE 11 11 4 7 Landau fan diagram for interband transition in 2D electron hole plasma with carrier density 1.7 10 12 /cm2 of InxGa1xAs multiple quantum well [15]. Reprint with permission from L. V. Butov., et al., Phys. Rev. B 46, 15156 (1992).( 1992 American Physical Society) ....................................................................................... 103 4 8 The reduced carrier effective mass as a fun ction of square wave vector for different electron hole plasma densities. The dependence is shown for both strained 7.5nm thick In0.28Ga0.72As/GaAs and an unstrained 15nm thick In0.53Ga0. 47As/InP single quantum well [ 15]. Reprint with permission from L. V. Butov., et al., Phys. Rev. B 46, 15156 (1992).( 1992 American Physical Society) ....................................................................................... 103 4 9 Magnetic field dependence of the high excitation luminescence peaks (open circles) and of the absorption peaks (small closed circles) [ 50]. Reprint with permission from M. Potemski, J. C. Maan, K. Ploog and G. Weimann, Properties of a dense quasi two dimensional electron hole gas at high magnetic fields Solid State Communication 75, 185 (1990).( 1990 Elsevier) ........................................................... 104 4 10 A) Experimental schematic showing single shot excitation and collection. B) Four representative emission spectra from edge 1(black) and edge 2(red) fibers exited from a single laser pulse and measured simultaneously. Normalized emission strength from 0th LL versus shot number in the C) SF regime and D) ASE regime [4]. Reprin t with permission from, Y. D., Wang, X Reitze, D. H. and et. al.., Phys. Rev. Lett. 96, 237401(2006) .( 2006 American Physical Society) ...................... 105 4 11 A) Absorption Spectra and B) low excitation PL for four different samples. Arrows in A) represent emission from the dark states which are not allowed. Each peak in the figure is corresponding to e1h1, e1 l1 and e2 h2 transitions for all samples. The energy separations between e1 h1 and e2 h2 in fig ure A) are 75.3meV, 77.8meV, 76.8meV and 76.3meV for S324, S322, M507 and M508. ............................................... 106 4 12 A) Heavy and light hole transitions as a result of confinement in quantum wells and B) possible allowed electron and hole transitions (solid arrows, 0 h ej j j ) and forbidden transitions (dark transition dashed arrows, h ej j )....................................... 106 4 13 A) White light transmission spectrum for sample M508 and B) the inferred absorption spectrum obtained using Eq. 3 1. A dark state starts to appear at 11.5T. EGa in A) indicates the band energy of GaAs ................................................................... 107 4 14 A) PL emi ssion dependent on magnetic field (He Ne CW laser used with 2.1W/cm2 ) and B) peak position for the 00 LL (black squares). Linear fitting (solid red line) of Landau shift from Eq. 4.25 and quadratic fitting (solid blue line) for diamagnetic shift from Eq. 4.24 are also shown in B). The data was taken using sample M508. ....... 108 4 15 Photoluminescence spectra of M507 at 5K and 0T. He Ne laser and CPA were used for the low (green) and high excitation (red) and for comparison with white light absorption, absorption spectrum(blue) from Tungsten lamp is also shown in the PAGE 12 12 figure. All photoluminescence is collected from the center of the samples. PL is shown up to 1.48 eV only for the comparison of InGaAs PL. PL from GaAs (1.49eV) is not shown. ......................................................................................................... 109 4 16 Photoluminescence spectra of M507 at 5.5T A) and 17.5T B) are taken from the inplane direction. Landau levels from e1 h1 and e1 l1 are also s hown in the figure. ........ 109 4 17 Emission strength (black squares) and linewidth (blue squares) of the Lorentzian high energy peak for the 00 LL, 11 LL and 22 LL versus laser fluence at 17.5 T for sample M507. Fitting method for Gaussian for lower energy side and Loretzian for higher energy side for 00LL are used and only Lorentzian function is used for 11LL and 22LL. All linewidth analysis is based on Appendix D. .............................................. 110 4 18 00LL is shown in A) for white light absorption with Tungsten lamp and PL from low excitation (2.1 w/cm2) with He Ne laser and high excitaion( 3 1010 w/cm2 ) with CPA. Figure B) display the Landau fan diagrams for white light absorption and high excitation. .............................................................................................................................. 111 4 19 Plots of the 2D energies for the Coulomb interaction dominated regime (Eq. 4 30, dots) and the magnetic field dominated regime, (Eq. 4 31, so lid lines). The energies are plotted in terms the of 3 dimensional Rydberg energy (1 Rydbergy energy is corresponding to 4.23T for In0.2Ga0.8As) ........................................................................ 111 4 20 A) Absorption spectrum at different m agnetic field and B) time integrated photo luminescence of InGaAs multiple quantum wells. ............................................................ 112 4 21 Landau fan diagram for A) low intensity white light absorption and B) excitation with ultrafa st laser (1.53mJ/cm2). Red lines are fitting results based on Eq. 4 32 and Eq. 4 33. Open circles are experimentally obtained data points and filled circles are points for fitting (anti crossing from the dark transition is not used for fitting (open circle s: Section 4.6.2)). Black, blue and yellow circles represent hh1, lh1 and hh2 for each. ...................................................................................................................................... 112 4 22 Reduced mass dependence on the Landau level index for A) the first heavy hole, hh1, B) the firs t light hole, lh1 and C) the second heavy hole, hh2,. Here m0 is the bare free electron mass. Reduced mass for hh1, lh1 and hh2 calculated for 10 LLs, 4LLs and 5LLs for each because as shown Fig.4 21 (B), hh1, lh1 and hh2 spectrum shows 10 LLs, 4 LLs and 5LLs. .......................................................................................... 113 5 1 Schematic diagram of pump probe experiment of 2D quantum well in magnetic fields. A series of LLs in conduction and valance band are shown in the figure. Pump laser is tuned 150m eV above the band edge and probe light is tuned at the emission wavelengths of allowed transition. 800nm central wavelength of pump pulse excites carriers in quantum well system and probe pulse(OPA) is tuned at the selective wavelengths which are carrier s tates ................................................................... 130 5 2 Relaxation processes and time constants for each process. ............................................... 130 PAGE 13 13 5 3 A) Scattering of exciton by phonons. Acoustic (l eft) and LO (right) phonon scattering. B) Schematic drawing of the dispersion relation of excitons and of the main time constants .............................................................................................................. 131 5 4 Simplified picture of relaxation process in 2D semicon ductors in magnetic fields: A) laser excitation with femtosecond (very fast process), B) phonon and electron electron scattering (~1ps) C) intra Landau level transition (~5 ps) and D) inter Landau level transition(~100ps) .......................................................................................... 131 5 5 Schematic of femtosecond spectroscopy. Pump pulse and probe pulse are delayed by tP with respect to each other. The transmission of the probe pulse is measured with and without the presence of the pump to obtain DTS. ....................................................... 132 5 6 Schematic diagram of pump probe experiment with Lock in. The chopping frequency of the pump is 271Hz and the dT/T is fluctuation normalized by the OPA reference signal. .................................................................................................................... 132 5 7 Schematic diagram of the pump probe experiment with McPHERSON spectrometer, Two inset show the fiber coupler and 2 dimensional image on CCD camera. ................ 133 5 8 Spectra of transmission with Tungsten lamp(Black), PL with CPA (Red) and probe (blue) for A) 0T and B,C and D) 17.5T. Probe pulses are tuned at transmission spectra minimum assuming carriers live in transmission peaks (or absorption pea k). .... 133 5 9 Normalized transient absorption at 0T. Probe was tuned at the peak of Tungsten lamp white light transmission minimum (Sample M507). ................................................ 134 5 10 Two dimensional pump probe measurement at 0T (refer to the setting, Fig. 5 7). Exictation power is 0.8mJ/cm2 and M507 was used for the measurement ...................... 134 5 11 T/T(blue) at 0 time delay with high excitation PL spectrum(red) and white light absorption (green) based on Fig. 5 10. ............................................................................... 135 5 12 Photoluminescence with 0.047mJ/cm2 excitation power at 0T and exponential fitting. Fitting function follows the Boltzmann distribution function,. ............................. 135 5 13 Transient absorption signals at various time delays. .......................................................... 136 5 14 Normalized transient absorption at 17.5T. Probe was tuned at white light absorption peak of each LL (Sample M507 and excitation power=2.2mJ/cm2). ............................... 137 5 15 Normalized transient absorption at 17.5T. Probe was tuned at PL peaks of each LL. (Sample M507 and excitation power=2.2mJ/cm2). ............................................................ 138 5 16 Decay time for each Landau levels. We used TA data for PL maximum. 0T represe nt first heavy hole at 0T and 00LL, 11LL and 22LL measured at corresponding LL at 17.5T (Sample M507) .......................................................................................................... 138 PAGE 14 14 5 17 The function of reduced dimensionality. A): By reducing the separation of confine ment, separation between adjacent energy level increases and result in reduction of density of state. B): Function of applying magnetic files in quantum well reduces dimensionality and increase the density of state. ......................................... 139 5 18 Two dimensional pump probe measurement at 17.5T (refer to the setting, Fig. 5 7). Excitation power is 4.6mJ/cm2 and M507 was used for the measurement ...................... 139 5 19 T/T(blue) at 0 time delay with high excitation PL spectrum(4.6mJ/cm2 red) and white light absorption (green) based on Fig. 5 18. There are the second maximum around the first maximum of T/T and the arrows represent position of this second maximum. ............................................................................................................................. 140 5 20 Maximum PL peaks(red), white light absorption peaks(green) and time transient absorption peak(blue).0T represent first heavy hole at 0T and 00LL, 11LL and 22LL measured at corresponding L L at 17.5T (Sample M507). Black star is the position of the second peaks around LL as shown in Fig. 5 19. .......................................................... 140 5 21 Transient absorption signals at various time delays. Red arrow displays for g uiding of main peaks (from Fig. 518). .......................................................................................... 141 5 22 Transient absorption signals for M508 at various time delays (excitation power=4.8mJ/cm2. Red arrow displays for guiding of main peaks and red d ot and green dot for guiding of LL peaks for PL and white light absorption. Black arrows represent the position of the second peak. High excitation PL(solid red) and white light absorption(solid green) are plotted for the comparison. ........................................... 142 6 1 Different recombination processes in A) direct and B indirect gap materials in k space and C)real space. Five different recombination processes in C are shown (1) donor to accepter, (2) conduction band to valance band, (3) conduction band to acceptor, (4) donor to valance band and (5)conduction band to donor. ........................... 158 6 2 Schematic diagram of a simple two level system, showing radiative and nonradia tive recombination channels. The parameters 21 and 21T denote the time constants for nonradiative channel and radiative channel from level 2 to 1 ....................................... 158 6 3 Calculated time evolution of the 2n population in a two level system. Level 2 has been assumed to be populated by a () t pulse at t = 0. ...................................................... 159 6 4 Operating principle of the streak tube. Light pulse at t1, t2 and t3 is converted to three electrons by photocathode. These three electrons deflected by time varying voltage bias and impinge against a phosphor screen at different positions. .................................. 159 6 5 Sample mount for time resolved PL collections. Sample is attached to sapphire substrate and multimode fibers butted up against it. Micro prism redirects PL from the edge of the sample and transfers the signal to the multimode fiber. ........................... 160 PAGE 15 15 6 6 Top panel: raw 2 dimensional streak camera image of the CPA laser pu lse. For this measurement, the laser was attenuated and sent directly into the streak camera. The measured duration was 2 ps, corresponding to the minimum achievable resolution of the streak camera. ................................................................................................................. 161 6 7 (a) Measurement of the dispersion (relative time delay) as a function of wavelength after propagation through a 7m long graded index multi mode fiber (GIMF). (b) Measurement of the dispersion (relative time delay) including the Acton spectr ometer in the beamline. The OPA was used to vary the wavelength center wavelength. ........................................................................................................................... 162 6 8 A) Raw 2 dimensional streak image of 150fs and 800 CPA laser pulse without mu metal shielding (50 Gau ss magnetic field at the streak camera and B) 2 dimensional streak image with mu metal covered .................................................................................. 162 6 9 Schematic diagram of the experimental setup for the time resolved PL measurement. The las er beam is injected into the 17.5 T superconducting magnet (in Cell 3 at the NHMFL). A graded index multimode fiber (GIMF) collects the PL from the sample and directs it to the streak camera injection optics. The injection optics, Acton spectrometer, an d Hamamatsu streak camera are on the roof of Cell 3 to reduce the magnetic field effect and minimize the length of the GIMF. ............................................ 163 6 10 Time integrated photoluminescence spectra of In0.2Ga0.8As/ GaAs multiple quantum wells (sample M508) at 0T (dash) and 17.5T (solid) with 7.13mJ/cm2 excitation intensity. Signal at 17.5 Tesla is 10 times stronger than that of 0 Tesla (for comparison normalization of signals are used) .................................................................. 164 6 11 The time resolved photoluminescence signal for sample M508 at 0 T for pump fluences with three different excitation powers for A),B) and C) and for samples M508 with and M507 ( A) 0.4456 mJ/cm2 ,B) 1.9502 mJ/cm2 C) 6.2652 mJ/cm2 and D) 1.4413 mJ/cm2). The signal is collected from the edge. Exponential fitting is used to determine the rise time, 232ps, and decay time, 233 ps of the PL for A), B), C) and D) with 3ps. Data is normalized for visualization. Signal int ensities of B) and C) are almost same and intensity of A) is three times weaker than B) and C). All the data is taken at 0 Tesla and at 1.317eV for A), B) and C) and 1.326ev for D) corresponding to e1 h1 transition of sample M508 and M507. ........................................ 165 6 12 TA( time transient absorption, red) and TR(time resolved photoluminescence, blue) of M507 with 4 mJ/cm2 excitation power at 0 Tesla. ........................................................ 166 6 13 Time resolved photoluminescence for sample M507 at A) 0 T, B) 00LL C) 11LL and D) 22LL from different magnetic fields with 4.9042 mJ/cm2 excitation fluence. PL was collected from the in plane (edge collection). ...................................................... 166 6 14 Time resolved photoluminescence for sample M508 at A) 0 T, B) 00LL C) 11LL and D) 22LL from different magnetic fields with 6.265 mJ/cm2 excitation fluence. PL was collected from the in plane (edge collection). ...................................................... 167 PAGE 16 16 6 15 A) M507 spectrum at different magnetic fields from the edge collection with 4.9042 mJ/cm2 excitation power and B) magnetic field dependent of 00LL peak intensity ( black circle) and linear fit(solid red). .................................................................................. 167 6 16 M508 time resolved photoluminescence from the edge collection at different excitation power and Landau level. All measurements are carried at 17.5 Tesla ............ 168 6 17 Time position of multiple bursts along excitation power at 17.5T (based on Figure 6 16) .......................................................................................................................................... 168 6 18 Time separation between first two adjacent burst of signals with 3.7679 mJ/cm2 excitation intensity based on Fig. 6 18. .............................................................................. 169 6 19 Schematic diagram of multi level system. Because of high carrier density, the relaxation bottleneck oc curs. Carriers are reloaded after T1 and T2 As a result multiple bursts in emission appear with time interval T1 and T2. To simplify the picture, higher energy levels are represented by dot line. ................................................. 169 6 20 Correlation between time transient absorption (blue) and time resolved photoluminescence (red) for sample M507. This plot compares the transient absorption signal probed at the center sample in transmission with the edge emission for tim e resolved PL. Excitation intensity is 4.1 mJ/cm2 and 4.9 mJ/cm2 for TA and TR .......................................................................................................................................... 170 6 21 Decay time (blue circle) at each LL and position of maximum peak of time resolved photoluminescence (red cross). For the comparison, 0 Tesla data from e1 h1 transition is plotted. 22LL at 17.5 Tesla shows the emission of photoluminescence right after TA signal decrease. (Data is taken from Fig. 6 20) ......................................... 171 B1 Dispersion of the real and imaginary part of the dielectric function of Eq. B 5 and Eq. B 6, respectively where 1 0 /0 and 0 2 max "2 / pl .......................................... 181 C1 Schema tic representation of the density of state for each dimension ............................... 184 D 1 Fitting method using a Gaussian ( broad peak) and Lorentzian( sharp peak).Black dot is the spectrum of the first LL at 17. 5T (sample M507 with 5.17 1010 W/cm2 excitation power).Gaussian(red) function for the lower energy side of spectrum and Lorentzian(Green) for the higher erergy sharp peak are shown and the fitting result (sum of these two functions(blue)) give the good agr eement to the data. ........................ 186 PAGE 17 17 LIST OF ABBREVIATIONS LL Landau level e Electron h Hole OPA Optical parametric amplifier CCD Charged coupled device CW Continuous wave MQW Multiple quantum well NHMFL National High Magnetic F ield Laboratory PL Photoluminescence SF Superflurescence SCM Superconductor magnet LL Landau MBE Molecular beam epitaxy MOCVD Metal Organic Chemical Vapor Deposition 3D 3 dimension 2D 2 dimension 1D 1 dimension 0D 0 dimension EMA Effective mass approx imation SHG Second harmonic generation FHG Fourth harmonic generation DFG Difference frequency generation PAGE 18 18 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy D ENSE Q UANTIZED M AGNETO P LASMAS IN HIGH M AGNETIC F IELDS P ROBED BY ULTRAFAST LASERS By Jinho Lee May 2009 Chair: David H. Reitze Major: Physics Investigations of the energy states and dynamics of highly excited In0. 2Ga0.8As/GaAs quantum wells in strong magnetic fields have been performed at the National High Magnetic Field Laboratory (NHMFL). Time integrated and time resolved magnetospectroscopy reveal significant new information about the nature of electronic corr elations, energy states, and dynamical relaxation processes of photo induced magneto excitonic and magneto plasmonic states in this system. T ime integrated spectroscopy reveals much about the character of the electron hole magneto plasma at high densities Time resolved investigations are crucial because they reveal the mechanisms by which a macroscopic polarization can form in dense e lectron h ole magneto plasma s. In addition, they allow us to map out the complex evolution of the relaxation dynamics of p hoto excited carriers excited by ultrafast laser pulses in a highly quantized Landau level system. Using the unique Fast Optics facility that we have developed at the NHMFL in Tallahassee, we perform three distinct but interrelated investigations. A fem tosecond chirped pulse amplified system (CPA) is used to generate carrier densities of 1013/cm2 and greater in the samples. First, by measuring the absorption spectra and emitted photoluminescence (PL) spectra and fitting the spectra to determine the energ y states as a function of carrier density, we are able PAGE 19 19 to clearly identify interband Landau level (LL) transitions and resolve the transformation from a steplike 2 dimensional density of states (DOS) at zero field to a function like 0 dimensional DOS. S ignificantly, we find a novel excitonic metal insulator transition (Mott transition) for the lowest lying LL induced by the presence of the magnetic field. Second, we experimentally characterize the inter and intra LL relaxation dynamics of the In0.2Ga0.8As/GaAs system using time resolved transient absorption. The multi level system consisting of many Landau levels results in complicated and nonexponential relaxation dynamics. At high magnetic fields, we observe the fast initial increase of transmittance in all LLs, however 100 ps after the initial excitation we observed the abrupt, non exponential reduction of the transmittance signal The abrupt dynamics result from rapid inter LL electron hole recombination. A third experiment probing the time structure of the PL emissions reveal that rapid bursts of emission are correlated with the temporal signature of the absorption recovery. In addition, for high excitations multiple bursts are observed which result from re loading of the LLs through intra LL sub band relaxation. PAGE 20 20 CHAPTER 1 INTRODUCTION AND OVERVIEW In this thesis, we will investigate how a dense photoexcited population of electrons and holes rapidly relax and recombine in quantum well systems in the presence of a strong magnetic field. The com bination of ultrafast spectroscopic techniques and high magnetic field research has become recently available through the Fast Optics facility at the National High Magnetic Field Laboratory in Tallahassee, Florida. The ability to create and probe carrie rs in nanostructures in ultrahigh magnetic fields (> 20T) is a new and totally unexplored regime, encompassing topics ranging from Bose Einstein condensation to the quantum optics of two level systems. Before delving into the details of this research, we begin with a brief introduction of the basic concepts needed to understand our results compound semiconductors and quantum wells, ultrafast lasers, and high magnetic fields. Semiconductors are defined as having an energy gap gE betw een the lowest fully occupied band (the valance band) and the higher energy unfilled bands (the conduction bands). The presence of this energy gap makes semiconductors fundamentally different from metals in both their electrical and optical properties. As an example, the computed band structure of one of the most common semiconductors gallium arsenide (GaAs) is shown in Fig. 1 1. GaAs is a direct gap semiconductor in which the extrema of the conduction and valence band are located at the same point in k spa ce. Combinations of Ga with As (From group III and V, respectively), Cd with S (from groups II and VI), or Cu with Cl (from groups I and VII) allow us to realize a variety of binary semiconductor compounds, and are termed III V, II VI, and I VII semiconduc tors, respectively. Besides the elementary group IV semiconductor materials Si and Ge, binary, ternary and quaternary semiconductors also can be realized. A combination of three elements such as Ga, Al and As makes a ternary compound and a combination of f our elements such as In, Ga, Al, and PAGE 21 21 As constitutes a quaternary compound. The fundamental physical and technological appeal of ternary and quaternary semiconductors lies in the underlying range of energies that can be achieved one can precisely tailor the bandgap energy of ternaries and quaternaries by changing the composition to obtain desired optical and electrical properties. As an example, InGaAs ternary alloys are the key components of h igh speed electronic devi ces [ 1 ], long wavelength quantum cas cade lasers, [ 2 ] and infrared lasers [ 3 ]. Fig.1 2 shows the accessible bandgap energies of different compositions of semiconductor materials. To a good approximation, a linear interpolation between two different materials is generally used, i.e. for A1xBxC (with x ranging from 0 to 1) Eg (A1xBxC)=(1 x) Eg(AC)+x Eg(BC). In addition, another powerful way of controlling the electronic and optical properties of semiconductors is via the epitaxial growth of two different semiconducting materials. The most frequently investigated semiconductor heterostructure is GaAs/Al1xGaxAs because of the almost perfect lattice matching for any choice of compositional fraction x. The semiconductor lasers used for todays CD players are made from this particular type of h eterostructure [ 4 ]. Fig. 1 3 shows a transmission electron microscopy showing the individual atoms position and demonstrates the amazing level of perfection to which epitaxial growth has advanced in this material syst em [ 5 ]. A spatially varying band ga p energy can be obtained by the choice of layers of different semiconductor materials (Fig. 1 4 ); these systems can be described by a spatially varying confining potential for the electron and holes within the structure. If a thin (< 20 nm) layer of a sma ller energy gap semiconductor is sandwiched between two larger gap materials, a quantum well is formed which can confine electrons and holes. When electrons and holes are located within the same material in this kind of potential structure, a type I heter ostructure is formed. If PAGE 22 22 they electrons and holes occupy different spatial positions, a type II heterostructure is formed. Our research will focus on type I heterostructures. A tremendous amount of research on reduced dimensionality systems such as quant um wells and quantum dots has been made possible by advances over the past thirty years in high quality layer growth techniques such as molecular beam epitaxy (MBE) and m eta l o rganic c hemical v apor d eposition (MOCVD) Both techniques can precisely control of the composition and the thickness of a particular structure. Semiconductor microstructures grown in this way make it possible to study how electronic systems evolve from bulk three dimensional (3D) geometries to two dimensional (2D) quantum well struct ures and, ultimately, to zero dimensional (0D) quantum dot structure. This revolution in growth technology together with parallel developments in fabrication and processing techniques has greatly enhanced our understanding of the fundamental physical prope rties of semiconductor materials. Indeed, these advances have made it possible to have insight into the fundamental dependence of physical properties on the dimensionality of the system. In addition to advances in semiconductor material growth, advances in ultrashort femtosescond pulsed laser sources such as chir ped pulsed amplifiers (CPA) [ 6 ] and optical parametric amplifiers (OPA) [ 7 ], have enabled research on physical phenomena occurring on femtosecond times scales in these systems. Many linear and n onlinear processes in semiconductors and quantum wells happen on femtosecond time scales, and to observe these kinds of processes, ultrashort dura tion laser puls es are needed [ 8 ]. Finally, the availability of very strong (> 20 T) constant (or DC) magnetic fields provide another important dimension in which to explore new physical phenomena. When a strong magnetic field is applied to a 2 dimensional sample, a discrete series of energy levels (Landau PAGE 23 23 levels) emerges, because of the quantization of the orbit al motion perpendicular to the field lines. This strong magnetic field transforms 2 dimensional system into a controllable 0 dimensional system. By controlling the magnetic field the length scale of the confinement can be tuned. The physical properties in this system can be changed from 2D to 0D by controlling the magnetic field. For this reason, quantum wells immersed in high magnetic field provide an ideal system to study the physical properties continuous evolution from 2D to 0D electronhole physics. Th e transport and optical properties of electron hole gas in 2 dimensional semiconductors in high magnetic field has be en studied since late 1980s [ 9 16, 1721]. However, most of this work have been carried out with long excitation pulses, low excitation laser powers (<1011/cm2) and relatively low magnetic fields (<12T). In this dissertation, we study dense electron hole plasmas in InxGa1xAs/GaAs multiple quantum wells (MQW) in high magnetic fields. The outline of this thesis is as follows. Chapter 2 de scribes the basic theory of magnetooptics such as the symmetry properties, excitons in 2dimensional quantum wells, and the basic recombination processes of simple two level systems. In order to investigate the ultrafast physical properties of semiconduct or in the strong field regime new facilities are required. We have developed an ultrafast spectroscopy laboratory at the National High Magnetic Field Laboratory combining strong DC magnetic fields (31 T Bitter type and 17.5 Tesla superconductor magnets) w ith powerful ultrafast lasers. These are described in Chapter 3, including the laser systems and associated experimental techniques. Chapter 4 begins by describing the magnetooptics of quantum wells. With the unique facility that exists at the NHMFL, we are able to generate carriers with densities in excess of 1013/cm2 and create a homogeneous 0D system from 2D multiple quantum wells. We find that PAGE 24 24 the high density e h plasma screens the Coulomb interaction, and the introduction of a magnetic field leads to a metal insulator Mott transition. In Chapter 4, we present evidence for this transition and describe the many body effects related with high density e h plasma. Chapter 5 is devoted to investigation of femtosecond transient absorption (TA) dynamics in InxGa1xAs/GaAs multiple quantum wells. Zero field TA shows the relatively slow electron hole recombination process. However, the transmission signal at 17.5T exhibits an abrupt, nonexponential reduction after the initial excitation. The fast decay time along magnetic fields and Landau level index increase is explained as a reduction in confinement effects from the magnetic fields leading to increased scattering and relatively weak Coulomb interaction between electron and holes. We describe time resolved photoluminescence experiments in Chapter 6. These are complementary experiments to the femtosecond TA experiments presented in Chapter 5 and similar dramatic changes are observed in these measurements. We find evidence for multiple bursts of photolumines cence signal at high magnetic fields. Dramatic correlation between time transient absorption and time resolved photoluminescence are discussed. In Chapter 7, we summarize experimental observation of semiconductor quantum well in high magnetic fields and su ggest the further experiments to be carried out in the future. Finally, we include four appendices which contains the definition of basic optical constants (Appendix A and B), a derivation of the D dimension density of states (Appendix C) and fitting metho d for Lorentzian and Gaussian shape of spectrum. PAGE 25 25 Figure 1 1. Calculated band structure of the direct gap semiconductor for GaAs [ 22]. Reprint with permission from J. R. Chelikowsky, M. L. Cohen, PRB 14 556 (1976) ( 1976 American Physical Societ y ) Figure 1 2. Energy band versus lattice constant for a number of semiconductor materials [ 23]. Reprint with permission from 1994 IEEE ( 1994 IEEE) PAGE 26 26 Figure 1 3. Chemical lattice image, a technique based on transmission electron microscopy, of a GaAs/AlGaAs heterostr ucture [ 5 ]. Reprint with permission from A. Ourmazd, D. W. Taylor,et. al., PRL 62 933 (1989) ( 1989 American Physical Society ) A B Figure 1 4. Schematic representation of a A ) type I s emiconductor quantum well and B ) type II structure. PAGE 27 27 CHAPTER 2 MAGNETO OPTICS OF SEMICONDUCTOR QUANTUM WELLS 2.1 Introduction The evolution of modern techniques such as Molecular Beam Epitaxy (MBE) and Meta l Organic Chemi cal Vapor Deposition (MOCVD) make it possible to manufacture ultrathin semiconductor structures of high quality. Ultrathin refers to a thickness comparable to the ex c iton Bohr radius ( 2 2 0e m ar B [24]). The thickness of semiconductors less than 100 A are referred to as quantum confined quasi two dimensional systems and one class of such system is quantum well structures. A single quantum well consists of a thin layer ( A 100 ) of a semiconductor sandwiched be tween two layers of a larger band g ap semiconductor with matched lattic e constant as shown Fig. 2 1. Many investigations have shown that semiconductor quantum well structures exhibit large optical nonlinearities and strong electroabsorption [ 25, 26, an d 27]. Both effects result from strong and well resolved excitonic resonances. In particular, excitons in the presence of magnetic and electrical field are of special interest. The external magnetic field enhances the exciton oscillator strength and results in more excitonic transitions including excited exciton states. Also interaction between excitons can be observed in high magnetic field. From the observation of excited excitonic state s we can deduce the information of the exciton binding energy [ 28, 29, and 30]. In this chapter we introduce the basic background information for the optical properties of quantum well system s including symmetry properties which are related to allowed transitions among excitonic states, magnetic field effects in two or thr eedimensional systems, and finally we introduce the simple two level system in the presence of an external electromagnetic field. PAGE 28 28 2.2 Excitonic Optical effects in Semiconductor Quantum Well Ground state (valence band) electrons of a semiconductor are bound to ions and thus are in an insulating state. However, electrons in the excited state are free to move about the lattice and resemble the conduction band electrons in a metal. By absorbing light, an electron can be excited to the higher energy state. In t he spectral range around the transition energy, the semiconductor exhibits linear and nonlinear optical properties. Here, we present a simple descr iption of a quantum well system in semiconductor and electron hole bound state s i.e. exciton. Basic optical properties of semiconductor s including optical susceptibility ( ), refractive ( n ) and absorption ( ) coefficient are introduced in Appendix A and B for the simple os cillator model and expres sion in the D dimensional density of states are given in Appendix C. In this section we start first symmetry property of 2 dimensional quantum well and then go over energy expression of exciton and we introduce definition of time constant relating with rel axation dynamics of semiconductor system in the next section. 2.2.1 Symmetry Properties and Excitonic effect in Quantum Wells The simplest starting point to describe excitonic effects in quantum well systems is the effective mass approximation (EMA). In EM A, we assume that the diffraction of the particle through the lattices gives the particle its effective mass and other slow varying potentials in the structure are only weak perturbations [ 31]. The most obvious consequence in a quantum well system is that the motions of electrons and holes are quantized in the direction perpendicular to the plane of layers and this layer structure has different symmetry from the bulk material. In particular, the confinement actually changes the masses of the holes for the m otion parallel to the layers [ 31]. New selection rules need to be derived. Lets consider a single quantum well layer with infinitely high potential barriers and assume this system has nondegenerate conduction and valence band. We choose z axis along the PAGE 29 29 normal to the layers. In the plane of the layer the electrons and holes move freely according usual parabolic dispersion law. ) ( 2 ) (2 22 y x e g ck k m E k E 2 1 ) ( 2 ) (2 2 2 y x h vk k m k E 2 2 where em and hm are the effective mass of electron and hole [31]. However, in the direction perpendicular to the layers, potential discontinuities (infinite barriers) confine the particles in the layers. The amplitude of the envelope wave function must vanish at the interfaces and the result can be written as ) sin( )2 ( ) (2 1 z z jL z j L z 2 3 for the electrons and holes and here stands for infinite potential barrier. In Eq. 2 3, zL is the layer thickness and 3 2 1 j is the quantum number of the wave function. These wave functions form a complete orthogonal basis set. Accordingly, the energy eigenvalues are given by 2 2) / ( 2z h e jL j m E 2 4 They depend on the effective mass of the particles. Fig. 2 2 describes 2 D conduction and valance subband. In this kind of infinite quantum barrier, the joint density of states that governs optical transitions for ms a series of step functions [ 24] as expressed in Eq. C 6 in Appendix C. ) ( ) (2 2 jh je DE E E E g 2 5 where is the e h reduced mass 1 1 1 h em m and stands for the Heavyside (step) function. The absorption spectrum is the superposition of steps starting at jh jeE E E with PAGE 30 30 the strong selection rule h ej j j =0 from the orthogonality of the conduction and valance envelope function Eq. 23. Real semiconductor structures have finite potential discontinuities at the interfa ces with different potential discontinuities in the conduction and valance band. The envelope functions now have two parts, a sinusoidal in the layer and exponential tails in the barrier region [ 31]. The energy spectrum in each band consists of a finite number of bound states and a continuum as shown Fig 2 2. Now the energies depend on the effective mass in the well and the barrier. When the discontinuities are the same on either side of the well, the eigenenergies are the solutions of 2 / 1) ( ] ) 2 tan( ) 2 cot( [j B w j j jv m m 2 6 where 1/ E Ej j and 1 ,/ E V vh e are the normalized energies and the potential discontinuities, respectively and wm and Bm are the effective masses in the well and the barrier [9]. The wave functions of the low lying states are well confined similarly to the infinite potential barriers with only a little penetration to the barrier region. As the energy increases, the exponential tails extend further into the barrier reg ion and the bound state envelope functions in conduction and the valance bands are no longer orthogonal even they have the same symmetry properties. The optical transition is still expressed by a step like function, although there are differences compared with infinite quantum well. The transition corresponding ) 2 1 0 ( 1 2 P P j j jh e is still forbidden because of symmetry properties of the system, but in case of ) 0 ( 2 P P j are now allowed because of small overlap even though it is weak. The st ates V Ej are delocalized and form continuum states. PAGE 31 31 2.2.2 Energy Expression near Point and Heavy and Light Hole In the elementary group IV semiconductors, the four electrons in the outer shell of atoms populate the sp3 orbital [ 32]. Th e isoelectronic compound semiconductors of group III V and II VI are also like this. In a cubic symmetry, the valance band state near 0 k is called point and consists of t hree degenerate p like states [ 32]. The conduction band at 0 k is made up of one s like state. If we include the two spin states in this kind of valence band, we find six degenerate states at 0 k 2 / 3 2 / 3 2 / 1 2 / 3 and 2 / 1 ,2 / 1 respecti vely called heavy hole, light hole and split off. A schematic band structure is shown in Fig. 2 3. The general form of Hamiltonian near the point is given by Luttingers phenomenological Hamiltonian for the heavy and light hole for the case of spherica l symmetry at 0 k [33] 2 2 2 3 3 0 2 2 2 0 1) ( 2 3 3 ) ( 9 2 J J J J J J kk k K J K m k m Hij i j j i ij ij j i ij ij ij j i ij 2 7 where 0m is the free electron mass and 2 1, and 3 are Luttinger parameters as shown table 2 1. Based on Eq. 2 7, the energy eigenvalues are [ 32] ) ( 12 2 )] ( [ 2 1 )0 ( ) (2 22 3 2 2 1 2 2 2 2 2 2 2 4 2 2 0 C B A k k k k k k C k B Ak m E kEx z z y y x 2 8 where + gives the heavy hole solution and denotes the light hole solution. From table 2 1, in many case it is true that 3 2 1 2 9 And if we decompose k into zk (perpendicular to the layers) and k (in plane) PAGE 32 32 The upper two ban ds energy can be expressed by [ 24] 2 1 0 2 2 2 1 0 2 22 2 2 m k m k Ez hh for 2 3 ZJ (heavy hole) 2 10 2 1 0 2 2 2 1 0 2 22 2 2 m k m k Ez lh for 2 1 ZJ (light hole) 2 11 2.2.3 Excitons in Two Dimensions Light absorption in semiconductors creates electron hole pairs and Coulomb interaction between them changes the optical properties of the material p articular around the absorption edge. We call this kind of electron hole pair as exciton. To study the Coulomb modifications of the optical semiconductor properties, lets consider a single electron hole pair. Assuming a periodic potential is incorporated into the effective mass of electron em and hole hm we find the Hamiltonian for the system is given by h e h e h h h h e e e e Coulomb confinment h h e er r e z V z y x m z y x m V V m m H 0 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) ( 2 2 2 2 2 12 where subscript e and h denote ele ctron and hole coordinate respectively, t confinemen h eV z V ) (, 0 and 0 is the static dielectric constant of motion. Again the motion is free in y x plane. Putting h em m M and h em m / 1 / 1 / 1 a nd keeping 2 2/ez and 2 2/hz Hamiltonian Eq. 2 12 will be r e z V y x Y X M z m z m Hh e h h e e 0 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) ( 2 2 2 2 2 13 where h er r r and h e h h e em m r m r m R The wave function ) ( r satisfying the above Hamilto nian can be separated z and x y component [1]. PAGE 33 33 ) ( ) ( ) ( ) (h hj e ei xy xy nz z r r 2 14 where ) (xy xy nr is the exciton envelope function in the y x plane and su bscript j i represent the quantum number as shown in Eq. 214. The corresponding energy eigenvalues (including the Coulomb interaction) are [ 24, 22] 2 2 2 2 2 2) 2 / 1 ( 2 j B z r g D nn E L m j E E where 3 2 1 jn and 2 2 0 42 / r Bm e E 2 15 Here the second term is the confinement energy for the electron hole pair with 3 2 1 j and the third term result from the Coulomb interaction. If we compare the 2D Coulomb interaction term with 1 jn with three dimensio nal term, we find: 2 2 2 2) ( 2D B r D Ba m E where 23 2 D B D Ba a and 2 2 0 3e m ar D B 2 16 So Elliotts formula [5] for the absorption leads to cosh ) ( 2/ 1 2 / 1 4 2 2 ) (2 1 2 2 3 2 2 2e E n E E n E mcL n dD g n B D g B r z b cv D 2 17 where B D gE E/ /2 2 2 2 22z r g D BL m E E and cvd and bn is the dipole moment between conduction and valance band and th e background refractive index [ 32]. For the comparison, the D 3 Elliotts formula [5] for the absorption is z e E n E E n E m c n d ez g n B g B r b cv Dsinh ) ( 4 2 ) (1 2 3 2 2 0 2 2 3 2 18 where g BE E z / The first term in the bracket of Eq. 2 18 corresponds to exciton lines at 2 / 1 22 2 2 2 n E L m EB z r g The magnitude of exciton absorption varies with n like 32 / 1 1 n Thus, the absorption line strength decreases more rapidly with n in two dimension than three PAGE 34 34 dimension. The second term gives Coulomb enhanced continuum absorption in a two dimensional system. Schematic absorption spectra are shown at Fig. 2 4 for 3 and 2 dimensional cases. 2.3 Magneto spectroscopy of Semiconductor In semiconductors, a wealth of new phenomena is obtained from the interaction with static external fields, includ ing the quantum hall effect [ 34], and crystallization o f the ele ctron into the Wigner crystal [ 32]. The optical properties are also modified by static fields. In case of optical fields, the properties of system s in magnetic field s are determined by the interaction of the particles with this field. This interac tion modifies the eigenvalues and eigenstates of the original system resulting in a direct response to the magnetic field. Classically, charged particles orbit around the magnetic field axis and in case of free electrons, the frequency of this orbit is given as the cyclotron frequency ( mc eHc/ [35]). The corresponding motion confinement in quantum mechanics changes three dimensional motion in the bulk to one dimension and two dimensional motion to zero dimension for the quantum well, which ca n be realized in quantum wires an d quantum dots, respectively [ 4 ] and hence the behavior of a semiconductor in the present of magnetic field is quite different from its behavior in the absence of a field. In the present section, we discuss the effect of a large magnetic field on the band structure. Consider the motion of the electron with effective mass em under the uniform magnetic field along z direction. The magnetic vector potential A is related to the magnetic field induction H by A H 2 19 PAGE 35 35 We note that A can not be uniquely determined by Eq. 2 19. Since 0 ) ( for arbitrary scalar quantity we can add any gradient of scalar to A In this section, we will work in the Coulomb gauge ( 0 A [36]) and take component of H as 0 y xH H and H Hz and yH Ax The Hamiltonian for the electron is ) ( 2 ) /(0 2 2 2r V m p p c yHe p Hz yx 2 20 where 0 m is the free electron mass and the cr ystal potential respectively. T he Schrodinger equation can be expressed by ) ( ) ( ) ( / 2 12 2 2 0r E r r V z i y i c yHe x i m 2 21 Assuming the perturbation from ) ( r V changes the free electron mass 0m to the effective electron massem, we can remove ) ( r V from Eq. 2 21 [ 35]. By ansatz ) ( )} ( exp{ ) ( y z k x k i z y xz x Eq. 2 21 is expressed as [ 37] ) ( ) ( 2 22 2 2 2 2 2r E r k c yeH k m y mz x e e 2 22 Putting e z e c xm k E E c m eH eH ck y 2 ,2 2 1 0 and 0 1y y y one gets from Eq. 2 22 ) ( ) ( 2 1 21 1 1 2 1 2 2 1 2 2y E y y m y mc e e 2 23 Eq. 2 23 is identical to the one dimensional harmonic o scillator of angular frequency c The corresponding eigenfunctions and eigenvalues are PAGE 36 36 c n z x n c en E z k x k i y y H y y m r 2 1 ) ( exp ) ( ) ( 2 1 exp ) (0 2 0 2 24 where nH is a H ermite polynomial of order n [ 38]. Therefore, the energy of a quasi free electron in a quantizing magnetic field is e z c z nm k n k E 2 2 1 ) (2 2 2 25 The k E relation of a particle is shown in Fig. 2 5 ( A ). The spectra consist of a family of parabolas separated by c along the ener gy axis. The energy levels in the magnetic field are called Landau levels. Correspondingly, the magnetic field restricts the kinetic energy corresponding to x and y motions in the energy range c cn E to c cn E ) 1 ( and assign a value of c cn E ) 2 1 ( This situation is shown in Fig. 2 5 ( B). From Eq. 2 24, the wave function is localized around 0y y ( eH ck yx 0 ) and with the finite sample size, y should be in the range of yL y 0 (where yL is the sample size in y direction. If we apply periodic boundary condition in x direction, interval of xk will be xL / 2 [39]. This means xk is limited to the range 0 to c eH Ly/ ) / ( y c eL m so zk can take up ) / )( 2 ( y c e xL m L different values in the inte rval zdk Including periodicity along the z direction, one finds z c e z z y x c e z y x z zdk m dk L L L m L L L dk k n g 2 2) 2 ( 2 2 2 ) ( 2 26 where factor 2 is included to take into account spin. Rewriting in terms of ) (zk E we find the d ensity of states in Landau level PAGE 37 37 2 / 1 2 / 3 2 2 3) 2 1( 2 ) 2 ( ) ( c e c Dn E m n E g 2 27 where e ccm eH / Therefore, the total density of states is max0) ( ) (n ndE E n g dE E g 2 28 In case of two dimensional confinement, the energy eigenvalues will be the same as Eq. 2 25 except the last zk dependent energy term. ) ) 2 1 ( ( 2 1 ) ( n E n k Ec z n 2 29 The density of states will be c eH m L L m L L n E gc e y x c e yx D 22 ) (2 2 30 As seen from Eq. 2 27 and Eq. 2 30, the three dimensional density of states reduced to a one dimensional density of state as shown in Eq. C 6 in Appendix C and the two dimensional density of states become zero dimensional density of states in the presence of a magnetic field. 2.4 Optical Resonance of the Two Level Dipole System We now turn to examining how light interacts with electrons in the atoms and in atomic like zero dimensional systems. It is almost impossible to describe all the interaction of collections of atoms with light exactly. It is even imposs ible to treat exactly how one atom interacts with light. In this section, we describe a simple two level systems interaction with light. A two level atom is conceptually similar to a spin one half particle in a magnetic field and the basic Hamiltonian i s practically the same as the one appropriate to spins, following the spin vector formulism of Bloch dev eloped for magnetic resonance [ 40, 41]. PAGE 38 38 Below, we first introduce the emission line shape and line width relating with homogeneous and inhomogeneous br oadening and then introduce the Bloch equations for a simple two level system. 2.4.1 Line shape and Line Width In a real dielectric, the dipoles can oscillate at many different natural frequencies, and so every material exhibits many emission lines. Actua lly the polarization density ) ( t P of the medium is due to dipole oscillation of all this lines. In most materials that have optical or near optical emission and absorption lines, the lines are well separated in energy and allow for only t he dipoles connecting with one line need to be considered independently. The emission lines of a typical single dipole are not infinitely sharp but have a width roughly T / 1 because of the finite life time T Becaus e intrinsically all dipoles have the same width, this width is usually called homogeneous line width of the spectrum and denoted by TH1 ~ Since dipole moment decay is exponential, the spectrum shape is Lorentzian [8]. In the more complicated situation, because of Doppler effects, gas atoms having different velocity will have different resonance frequencies even if they are the same kind of atoms. Thus in many cases actual emission lines are the superposition of a large number of Lorentzia n lines shown in Fig. 2 6. Different atomic dipoles oscillating at five distinct frequencies are shown with solid line and the dashed line is the sum of individual line for inhomogeneous line width in Fig. 2 6. In addition to Doppler broadening, homogeneous broadening is due most often to the finite interaction lifetime of emitting or absorbing atoms. Common mechanism s for homogeneous broadening [ 42] are: 1 The spontaneous life time of the excited atoms. PAGE 39 39 2 Collision of an atom in a crystal with phonon. 3 Pressur e broadening of atoms in a gas. The common mechanisms for inhomogeneous broadening are: 1 Random strain 2 Crystal imperfection 3 Doppler shift In a solid state such as the quantum wells that we investigate here, the primary mechanisms for inhomogeneous broadening are crystal imperfections such as lattice defects and dislocation, well width fluctuations, and local strain effects. 2.4.2 Optical Bloch Equation and Two Level System If we are concerned with only electric dipole transitions, we can use the interaction Hamiltonian H given by [ 40] ) ( 0r E d H HA 2 31 where r e d ) (0r E and AH is the atoms dipole moment operator, electric field at atoms position 0r and unperturbed atoms Hamiltonian. For the two level system depicted in Fig. 2 7, we get 0 0 A A A AH H W H W H 2 32 d d d d d d 0 0 2 33 PAGE 40 40 where and stand for ex cited and ground state for two level atomic system. In general dipole matrix elements are complex vector i rd i d d and i rd i d d where rd and id are real vectors and we get 0 0 i r i rd i d d i d d 2 34 With the Pauli metrics, dipole matrix can be expressed as 2 1 i rd d d 2 35 Where 1 0 0 1 0 1 1 0 0 1 1 0 3 2 1 i and commutation relation among Pauli matrix is 3 2 1 2 i With Eq. 2 34 and Eq. 2 35, the total Hamiltonian also can be expressed as 2 1 3 ) ( ) ( ) ( 2 1 ) ( 2 1 E d E d W W I W W Hi r 2 36 where I is the identity matrix. The Heisenberg equation of motion ] [ H O O i ([41], here, represents derivative with respective to time) gives the equations: ) ( )] ( [ 2 ) ( )] ( [ 2 ) ( ) ( )] ( [ 2 ) ( ) ( ) ( )] ( [ 2 ) ( ) ( 1 2 3 3 1 0 2 3 2 0 1t t E d t t E d t t t E d t t t t E d t ti r r i 2 37 where W W0 [40]. In replacing operators by their expectation values i is 3 2 1 i Eq. 2 37 can be expressed as ) ( )] ( [ 2 ) ( )] ( [ 2 ) ( ) ( )] ( [ 2 ) ( ) ( ) ( )] ( [ 2 ) ( ) (1 2 3 3 1 0 2 3 2 0 1t s t E d t s t E d t s t s t E d t s t s t s t E d t s t sr i r i 2 38 PAGE 41 41 If we choose an arbitrary phase for d so that 0 id and putting E E u d E dd r 2 2 where du is the unit vector along rd and d 2 we can simplify Eq. 2 38 to find ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) (2 0 3 3 0 1 0 2 2 0 1t s r t E t s t s r t Et s t s t s t s 2 39 These are the electro dipole equations analogs that govern spin prec ession in magnetic resonance [41]. s is called the pseudospin vector and Eq. 2 39 a re called optical Bloch Equation. Since 12 3 2 2 2 1 s s s [40], the pseudospin vector lives on the unit sphere as shown Fig. 2 8. Introducing ) 0 (0 EF we can simplify to find: ) ( ) ( ) ( t s t t s dt dF 2 40 The above equation is the same equation for the precession of the solid body under the torque ) ( t In the rotating frame with ) 0 ,(0 E ] )[ ( ) (t i t ie e t t E and rotating wave approximation, we find [8] 3 2 11 0 0 0 cos sin 0 sin cos s s s t t t t w v u 2 41 v t w w u t v v t u ) ( ) ( ) ( ) ( ) (0 0 2 42 If we trace back to the basic operator in the Hamiltonian, the physical meaning of w02 1 is the unperturbed energy of atoms and thus w is the single atom population diffe rence between two levels and called population inversion. The inverse transformation of Eq. 2 41 is related to the PAGE 42 42 in phase and in quadrature component of d with the field E From the second equation of Eq. 2 42, v and u are coupled to the optical driving field to produce an energy change. Said another way, v is the absorptive component and u is the dispersive component [ 40]. When only one radiation m ode is excited, the atom and the mode simply trade energy back and f orth (Jaynes Cummings model [ 43]). However, a many modes, one atom problem can also be solved almost exactly in the rotational wave approximation and will be discussed in Chapter 4.2.1. 2 .4.3 Phenomenological Decay Constants When we discuss the classical problems associated with light matter interactions, the role of various dipole decay times are important. The dipole oscillations must damp out in the absence of a driving field. We introduce phenomenological constant 1T and 2T to modify the Bloch equation Eq. 2 42. v T w w t w w T v u t v T u v t ueq 1 2 0 2 0) ( ) ( ) ( ) ( ) ( 2 43 where eqwequilibrium value when 0 Theses are semi ph enomenological equations proposed by Bloch [ 44]. Because interactions such as collisions in a gas and phonon scattering in solids can disturb the dipole oscillation, the population inversion w can decay with a time constant 1T which is different from the dipole moment decay time 2T 1T and 2T are called longitudinal and transverse homogeneous life times as seen Eq. 243. The transverse life time due to inhomogeneous effect such as Doppler broadening is denoted by 2T We define the total transverse life time as PAGE 43 43 2 2 21 11 T T T 2 44 Decays corresponding to 2T are due to the interference among a ll the dipoles with an inhomogeneous frequency distribution. Thus damping from the inhomogeneous broadening can be though as a dephasing process that damps only macroscopic polarization ) ( t P Each dipole oscillates for the time 2T However, the macroscopic polarization may be effectively zero before time 2T because the dipoles can be completely out of phase with one another. PAGE 44 44 Table 2 1. Exa mp les of Luttinger parameter [ 45]. Luttinger parameter 1 2 3 GaAs 6.85 2.1 2.9 InAs 19.67 8.37 9.29 InP 6.35 2.08 2.76 PAGE 45 45 Barrier Well Substrate Figure 2 1. Schematics of a singl e quantum well A B Figure 2 2 S ingle quantum well s and associated ground state and first exc ited state wave functions for A ) infinite and B) finite quantum barrier s. PAGE 46 46 k J=1/2 J=3/2 J=1/2Heavy holeLight holeSplit off Figure 2 3. Schematic band structure in the vicinity of 0 k for a bulk zinc blende semiconductor, such as GaAs PAGE 47 47 n=1 n=3 n=2 With Coulomb interaction No coulomb interactionAbsorption gE E A Absorption E B Figure 2 4. Schematic representati on of the absorption lines in A) 3dimensions and B ) 2 dimensions. The dashed line indicates the spectrum neglecting the Coulomb interaction and the solid line is with the Coulomb interaction. PAGE 48 48 E z k 0 n 1 n 2 n A B Figure 2 5. A ): Energy wave vector relation for e lectrons in a magnetic field. B ): In a magnetic field, the continuous energy levels in the range c cn E to c cn E ) 1 ( evolve into discrete Landau levels, c cn E ) 2 / 1 ( PAGE 49 49 Emission intensityFrequency Figure 2 6 The individ ual Loren t zian emission line s (homogeneous line width) associated with different atomic dipoles oscillating at five distinct frequencies. +> > E Figure 2 7 Energy level of an atomic system, showing two levels connected by near resonance transition. Line b roadening mechanisms are not shown for simplicity. PAGE 50 50 s Figure 2 8. P seudospin vector s traces out orbit on the unit sphere. PAGE 51 51 CHAPTER 3 ULTRAFAST FACILITIES AT NATIONAL HIGH MAGNETIC FIELD LABORATORY 3.1 Introduction In other to investigate the physics of high density magneto plasmas in the strong field limit (>12T) new facilities are required. We have developed an ultrafast spectroscopy laboratory at the National High Magnetic Field Laboratory where DC high ma gnetic fields (31 Tesla Bitter type and 17.5 Tesla superconductor magnets) are available In this chapter we introduce the magnet ic facilities in NHMFL and the operational principle of the light source. Pump probe and streak camera experimental setups for time transient absorption and time resolved photoluminescence measurement defer Chapter 5 and Chapter 6 3.1.1 Magnet system for Magnet optic Experiment The Fast Optics facility consists of two DC magnets, a 31 T NHMFLdesigned Bitter magnet and a 17.5 T Oxford superconducting magnet. Access to both magnets is obtained through free space propagation into the magnet bore. Fig. 3 1 shows the schematic diagram of 31 T Bitter magnet. This resistive magnet consists of a few hundreds of thin copper disks (Bitter disks) with four coils of Bitter disks placed in the magnet housing. A huge current flow ( IDC ~ 37 kA) in the magnet coils is needed to operate at the full magnetic field. Cooling water flows through the holes punched in the Bitter disks to avoid the over heating of the coils from Ohmic heating due to the electric current. The costs of operating the 31 T resistive magnet are appreciable, and as such this magnet is only available for optics experiments on a periodic basis. In addition to the 31 T resistive magnet, a specially modified Oxford 17.5 T superconducting magnet has been commissioned to be able to carry out flexible experiments on a more frequent basis. A new capability around this magnet was developed, including a new PAGE 52 52 magnet probe insert as well a s new ultrafast laser sources. Over the past three years, we have accomplished the development of this facility. The 17.5 T superconducting magnet was configured to allow direct access of laser light to the sample through the magnet center bore at the maxi mum of the magnetic field as shown in Fig. 3 2. A stainless steel center bore surrounding the magnet center separates the sample chamber from the helium reservoir and makes it possible to accomplish direct optical excitation through a transparent window o n the bottom of the magnet. The loading interlock system and the probe designed by us blocks atmospheric leakage into the center bore and allows for easy replacement of samples. Overall experimental scale including magnets is shown in Fig. 3 3. As shown in the Fig. 3 3 (C), the laser installed in Cell3 is introduced to Cell 5 through the hole on the wall between two Cells to excite the carriers in the sample mounted in the inside of magnet. 3.1.2 Cryogenic System for Magnet Optic Experiment Most of magnetooptics experiment for transmission, reflection and photoluminescence usually require low temperature and liquid helium and liquid nitrogen are used to lower the sample temperature to 5K or 77K for each. We specially designed the cryostats and the probe to carry out the experiment for optical spectroscopy at liquid helium temperature with 31 T resistive magnet with bore size around 50mm and 17T super conductor magnet. Fig. 3 4 displays the technical drawing of the cryostat for 31T resistive magnet and 17T s uperconductor magnet installed in Cell 5 and Cell 3 at NHMFL for each. Both cryostat and the probe have long tail, so the sample can be positioned at the center of the magnetic field. For the cryostat f or resistive magnet (Fig. 3 4 (A )), liquid helium is s tored in the center space and separated by vacuum jacket from the liquid nitrogen reservoir. Both spaces for helium and liquid nitrogen are shielded by vacuum jacket to significantly reduce heat into the helium space. PAGE 53 53 Because the probe is inserted in the h elium space, temperature can be lower by heat conduction. Light to excite the sample can be delivered through the optical window mounted on the bottom of the cryostat and signal can be transferred through optical fiber. Cernox temperature sensor and heater are also mounted on the sample mount together with the sample. Fig. 3 4 (B) is the probe for the 17T superconductor magnet. In this case, low pressure helium gas is inserted to cool down the sample temperature by heat exchange. Optical fiber and electrica l wire are connected to the sample position to control and read the temperature in the same way as cryostat for the resistive magnet. The layout of the simple CW magneto optic experimental setup for the superconductor magnet is shown in Fig. 35 and experi mental setup for the resistive magnet is similar to this. Optical probe is positioned from the top of magnet and temperature controller (Cryocon 62) read and set the temperature to create the experimental condition through the sensor and heater located rig ht above to the sample. Input light can delivered by fiber or freely transfer through air into the sample through the optical window depending on the experiment (transmission or reflection). We used the Tungsten or Xenon lamp for the white light transmiss ion measurement and He Ne and Ti: Sapphire to photoluminescence measurement. For transmission experiment, signal is collected to the output fiber located right top of the sample and analyzed 0.75 m single grating spectrometer (McPherson, Model 2075) with a multi channel CCD detector. Apple computer control all of control units including magnet, spectrometer, temperature. All of our measurements were carried out with 31 Tesla Bitter type and 17.5 Tesla superconducting magnets in the Faraday geometry which shown in Fig. 3 6 to make the magnetic field perpendicular to the quantum well plane and for the parallel propagation of the light to the magnetic field. PAGE 54 54 3.1.3 Laser Light Sources For low density characterization of our samples, we use a continuous wave (CW ) helium neon laser (Melles Griot, Model: 25 LHP 632 nm, 30 mW maximum power) and a Tungsten lamp (Oriel Corporation, Model: LPS 220) for low power PL measurements and for white light absorption measurements. The spectrum of the Tungsten lamp is shown in F ig. 3 7 and covers broad range of wavelength. To achieve high excitation densities (in excess of 2 13/ 10 cm ), we developed an optical excitation system based on a femtosecond chirped p ulse amplified system (CPA) [ 6 ] from Coherent (LegendF). As shown in Fig. 3 8, the CPA consists of four basic components: a seed laser, a pulse stretcher, a regenerative amplifier cavity and a pulse compressor. A femtosecond Ti sapphire laser (Coherent Vitesse) is used as the seed laser (88 Mhz, 130 fs, 300mW) a nd is stretched to approximately a 100 ps duration before amplification in a Ti:Sapphire regenerative amplifier cavity to avoid crystal damage. Ultrashort pulse after a stretcher is temporally elongated by wave dependent group delays so that one significantly reduce the brightness (peak power) compared with the original seed pulse to dramatically weaken self focusing in amplification (wave dependent group delayed pulse). This stretched pulse is then amplified in a Ti:Sapphire regenerative amplification stag e. The regenerative amplifier is pumped by a 20 W average power Nd:YLF laser (Coherent Evolution). Two Pockel cells in the cavity pick off one pulse from the pulse train at a user specified repetition rate, typically 1 kHz in our experiments. The pulse ex periences gain via approximately 30 round trips within the cavity through the Ti:Sapphire. Once the amplification saturates, the amplified pulse is switched out of the cavity and propagates through the grating compressor to compress the pulse back to the original pulse width, roughly 150 fs which is PAGE 55 55 nearly transform limited pulse duration. At output, we achieve 1 kHz and 2mJ per pulse from this CPA system. All of this basic principle of CPA is shown in Fig. 3 9. The wavelength from CPA is fixed at 800nm. To allow for more flexible experimentation, and in particular probing through a wider spectrum of wavelengths, an optical parametric amplifier (OPA) [7 ] is used to convert the 800 nm CPA pulses to different wavelengths (300 nm to 20 m ) while preserving the short pulse time width. The term parametric is referred when the initial and the final quantum state are identical [ 7 ]. In this situation only for brief time interval population can be removed from the ground state when it reside in the virtual st ate in a parametric process [ 7 ] Our OPA system is a femtosecond TOPAS Optical Parametric Amplifier from Quantronix Corporation with second harmonic generation (SHG), fourth harmonic generation (FHG) and difference frequency mixing (DFG) units. This system has about 30% conversion efficient and wide spectral coverage (300 nm to 20 m ). TOPAS OPA has one parametric generation and four parametric amplification stages with a single Beta Barium Borate, ( BaB2O4(BBO)) nonlinear crystal to generate high output beam quality and nearly bandwidth limited pulses over the whole tuning range. Wavelength tuning is controlled by computer. Fig. 3 10 shows the schematic di agram of 5 pass Light Conversion TOPAS OPA and Fig. 3 11 displays the schematic diagram of parametric process in case of difference frequency generation. As shown in Fig. 3 11, if optical resonator is used to generate a specified frequency, this frequency,signalh is called signal and the other, unwanted, output frequency is called idler. OPA operation beam path in Fig. 3 10 is as follow. On the first pass, the 800 nm CPA creates a broad white light continuum from a beta barium borate (BBO ) nonlinear crystal, followed by grating selection of the desired output wavelength at the second pass of the beam. PAGE 56 56 The third pass is the seed beam path for the amplification from fourth and fifth pass. The signal and idler pulse are parametrically amplifi ed at the fourth and fifth beam path. The relationship between the signal and idler wavelengths is as follows. idler signal CPA 3 1 At the output of the OPA, a variety of different nonlinear crystals can be placed to frequency double the output pulses via second harmonic generation (SHG), fourth harmonic generation (FHG) and difference frequency mixing (DFG). With all of these nonlinear processes, a wavelength range from 300 nm to 20 m is achievable. PAGE 57 57 Figure 3 1. Schematic technical drawing of the 31 T resistive magnet in cell 5 at NHMFL Reprint with permission National High Magnetic Field Laboratory ( 2009 NHMFL) PAGE 58 58 Figure 3 2. Schematic technical drawing of the 17.5 T superconducting magnet (SCM3) located in Cell 3 at NHMFL [ 46]. PAGE 59 59 Figur e 3 3. Experimental setup in A) Cell 3 and B) Cell 5 in NHMFL. C ): Laser from Cell 3 goes through the hole in the wall between Cell 3 and Cell 5 and is introduced to the mi rror mounted under the 3 1 T resistive magnet in Cell 5[ 46]. PAGE 60 60 Figure 3 4. Technical drawing of c ryostat and optical probe for A ) 31 T r esistive magnet in Cell 3 and B ) 17T superconductor magnet in Cell 3 at NHMFL PAGE 61 61 Figure 3 5. Block diagram of CW spectroscopy of the magneto optical experiment. Main apple computer controls magnet, temperature and spectrometer. PAGE 62 62 Figure 3 6. Schematic diagram of Faraday geometry. Figure 3 7. Tungsten lamp spectrum for CW spectroscopy. PAGE 63 63 Figure 3 8. Sche matic diagram of Chirped pulsed amplifier system (Coherent Legend F). Evolution is pumping source of CPA. Figure 3 9. Principle of chirped pulsed amplification. Input 130fs pulse is stretched to 100ps and then goes to amplifier stage. Final pulse afte r compressor is 1kHz, 150fs and pulse energy is 2mJ PAGE 64 64 Figure 3 10. Schematic diagram of optical parametric amplifier (TOPAS OPA). Figure 3 11. Schematic diagram for Optical Parametric Amplifier system. Solid line denotes the real quantum mechanical state and the dashed is for the virtual state. PAGE 65 65 CHAPTER 4 CONTINUOUS WAVE SPECTROSCOPY OF INGAAS MQWS IN HIGH MAGNETIC FIELD 4.1 Background IIIV compound semiconductors are the basic materials for the various well established commercial technologies and op toelectronic devices such as high electron mobility and heterostructure bipolar transistors, diode lasers, electrooptical modulators, frequency mixing components, photodetectors and light emitting diodes [47]. The characteristics of such devices critical ly depends on the physical properties of the semiconductor material, which are often formulated in quantum heterostructures containing carriers confined to dimensio ns on the order of nanometer [ 47]. Especially, InGaAs ternary alloys are the key components of h igh speed electronic devices [ 1 ], longwavel ength quantum cascade l asers [ 2 ] and infrared lasers [3 ]. Current advance in ultashort pulse laser has greatly accelerated research in the physics of extremely rapid processes. Many linear and nonlinear optical processes occur on the femtosecond time scale, requiring laser pulses with comparable or shorter durations for their observation. Additionally, creating high carrier densities (>10 12 /cm2) or measuring the nonlinear optical responses of the condens ed material typically requires light source s of extremely high intensity. In most cases, the intensities required for observing the nonlinear phenomena and linear response of high carrier densities (>10 12 /cm2) in semiconductors are so high that materials heat, and ultimately damage for pulse durations longer than picoseconds. Presently, high repetition rate amplified mode locked lasers which produce broadband high intensity pulses, allowing for observation of nonlinear optical response and linear response of high carrier densities of semiconductors on a femtosecond time scale. A technique which reveals fascinating new phenomena as well as providing a new powerful tool to probe the properties of excitons is the application of the external magnetic field. PAGE 66 66 St rongly confined electronic states in materials of excellent optical and electronic properties can be obtained by immersing a quantum well in a perpendicular magnetic field. Optically populated electrons and holes are then created to form a so called magnet o plasma (for high carrier densities) or magneto exciton (for low carrier densities). The linear absorption spectrum of these confined states consists of a sequence of isolated spectral peaks, displaying the characteristic behavior of a 0D e h system as de scribed in Chapter 2.3. Moreover, the length scale of the confining potential can be varied over many orders of magnitude by controlling the external magnetic field. The physical properties can be studied continuously from two to zero dimensions, in a single quantum well sample, by changing the magnetic field. In this case, the size distribution of confined electronic states is determined totally by the quality of quantum well layers themselves and so the mature technology for the layer growth determines th e ultimate resolution of the spectral feature. The Optical properties of magneto excitons evolve from that of 2D e h system with strong Coulomb correlations, at low strength external magnetic fields applied perpendicular to the quantum well layer, to that of a noninteracting two level systems in the limit of h igh external magnetic fields [ 8 48]. This occurs in spite of the fact that optically generated e h pairs in a magnetic field are free to move in the plane of two dimensional quantum well. In fact, s cattering interactions of theses magnetoexcitons may be also studied using variety of optical techniques, so physics of magnetoexcitons is richer than that of excitons trapped within a quantum dot. For these reasons, a quantum well in a perpendicular ext ernal magnetic field provides an ideal system in which to study linear and nonlinear optical characteristics of electronic states as they continuously tuned from two to zero dimensions. PAGE 67 67 Numerous studies of the optical and electronic properties of quantum w ells in magnetic fields excited by continuous wave (CW) or pulsed lasers have been reported on in the past 15 20 years [19]. However, most of these investigations were performed at relatively low magnetic fields (<16 T) and using nanosecond pulsed lasers. With Ar+ ion (514.5nm) and Cu vapor pulsed laser (510.5 nm and 20 ns pulse duration), L. V. Butov., et al. [5460] have studied the photoluminescence of inter band Landau level transition of InxGa1 xAs quantum well up to 12 T. In their experiments, the p hoto excited electron hole (e h) gas is considered as a magneto plasma with high excitation powers (carrier density up to 10 12 /cm2 in the quantum well). They observed many body effects and deduced the effective mass of electron from the Landau fan diagra m. Another group, M. Potemski., et. al. [4 9 50] used AlxGa1 xAs to excite the carrier density up to 10 13 /cm2 with 10ns of 20mJ of N2 laser (567nm). With the two band model assuming the parabolic band for the hole and nonparabolic band for the electron [ 49, 50, 51 ], they induced the effective mass for high carrier density and found the many body mass enhancement. L. V. Butov and M. Potemskis experimental works will be introduced in more details in Section 4.5.1. Unlike earlier investigations, 2D quantum well spectroscopy at high magnetic fields (> 16 T) using ultrashort laser pulses (< 1 ps in duration) probes a new regime in which high density magneto plasma are created and interact before substantial recombination dynamics can take place(we will discus s in Chapter 5.2). To discuss our investigation for the magneto optics further, we present first about the basic theory of magnetooptics in the following section and differ the discussion about the relaxation dynamics of Landau levels in Chapter5 and 6. 4.2 Basic Theory of Magneto optics An excited dipole in external electromagnetic radiation decays to the ground state either by spontaneous or stimulated emission. 2 dimensional quantum well in high magnetic field PAGE 68 68 generates multi level dipole systems. To u nderstand relaxation dynamics of 2D system in high magnetic field, we need to categorize the emission process. In this section, we start with discussion of the vacuum fluctuation which is the origin of the spontaneous emission and then introduce amplified spontaneous emission including the cooperative emission process (superflurescence(SF)). The concept of strong field limit which is important in magneto optics and band mixing in high magnetic field also will be discussed. 4.2.1 Introduction to Emission Pro cess of Simple Two Level Dipole System The well known consequence of the quantization of radiation is the fluctuation at the zero point energy or vacuum fluctuation. The electric and magnetic field o perators can be expressed as [ 52] HC e a e k t r H HC e a e t r Er k i t i k k k k k r k i t i k k k kk k 0 1 ) ( ) ( 4 1 where k k are the wave vector, two polarization basis, unit polarization direction; 2 / 1 02 V ek k and ka and ka are annihilat ion and creation operators of photons with k and HC is the Hermitian conjugate. The vacuum state 0 is defined as 0 0 0 E 4 2 However, the expectation value of intensity operator 2E in an n particle state is given by [77] ) 2 1 ( 22 n n E n 4 3 Therefore, there is a fluctuation in the fields about the zero ensemble average. These vacuum fluctuations form the basis for many interes ting physical phenomena in quantum optics. PAGE 69 69 In an exited dipole twolevel system, the excited atom will spontaneously emit light in a random direction and polarization state (spontaneous emission). In the semi classical theory of light atom interactions, t he atom in the excited state cannot make a transition to the lower level in the absence of a driving field. [ 52]. In the fully quantum mechanical treatment, the explanation of spontaneous decay is more subtle namely, that fluctuations in the vacuum inter act with the atom and stimu late it to emit spontaneously [ 53]. Thus, spontaneous emission results from the interaction between a single atom and all vacuum modes. In the actual vacuum, there are infinitely many vacuum modes, resulting in spontaneous emiss ion in random direction and polarization. We can describe this spontaneous emission as [ 54] 2 2AN dt dNsp 4 4 where 2N is the number of exited states of atom in two level dipoles and A is the spo ntaneous emission probability per unit time, the Einstein coefficient. The quantity Asp/ 1 is called the spontaneous emission life time. A second transition process, related to the first, is amplified spontaneous emission (ASE). In an ens emble of two level atomic systems, when one excited atom emits light spontaneously, the emitted photon can trigger other excited atoms to emit, resulting in a cascade process giving the amplified emission of light. Like spontaneous emission, the emission r esults from the interaction of electromagnetic quanta with an atom. Unlike spontaneous emission, the photon comes from a neighboring atom rather than a vacuum fluctuation. Furthermore, In the case of spontaneous emission, the atom emits an electromagneti c wave which has no definite phase relation with that emitted by another atoms; the wave can be emitted in any direction. In case of amplified spontaneous emission, since the process is forced by the incident light with a well defined phase and polarizatio n, the emission of any atom adds in phase to that of incoming wave PAGE 70 70 and this wave also determines the direction of the emitted light. In this case, also, we can characterize the process by means of the equation [ 54] 2 21 2N W dt dNst 4 5 wher e 21W is called the stimulated transition probability. Unlike A, 21W depends not only on the particular transition but also on the intensity of incident electromagnetic wave as following F W21 21 4 6 where F is the incident photon flux and 21 is the quantity of dimension of area and depends on the characteristic of the given transition. W21 can be computed from the Fermi golden rule. Fig. 4 1 schematically shows the spontaneous and amplified spontaneous emission processes. In the absence of an applied field, each exited atom in a large collection will decay spontaneously to its ground state. However it is misleading to say that a single atom always d ecays monotonically and exponentially to its ground state. Specifically, this does not necessary apply to an ensemble of atoms. When the optical wavelength is compared to the size of atomic system, each exited atom may be substantially influenced during it s decay [ 40]. The interaction between many dipole atoms and vacuum fluctuations was first proposed by Dicke [ 53] and theoretically and experimentally has been extensively studied [ 53 61]. In an N atom ensemble, if the atoms in the exited states are confined within a wavelength, the presence of a spontaneously emitted electromagnetic oscillation can enforce a coherent macroscopic dipole state on the entire ensemble for a short time and as a result, a very short burst of light emission occurs, radiating strong and coherent light. This emission process is called superradiance (SR) or superfluorescence (SF). For SR, the coherent formation of the macroscopic dipole results from an external source. For SF, the exited atoms are initially incoherent and the coherenc e among atoms is built from vacuum fluctuation. PAGE 71 71 Superfluorescence emission is a macroscopic coherence effect of an ensemble of atoms; all atoms radiate photons cooperatively and coherently. Fig. 4 2 shows a cartoon of the SF process. In order to observe th e SF clearly, the following t hree conditions are required [ 62]. (1 ) Because the SF grows from the fluctuation of the electromagnetic field, a population inversion must be established within a short time interval compared to the time delay D of the SF burst. (2 ) c LE/ the time constant required for the electromagnetic wave to escape from an atomic system, must be much shorter than both longitudinal relaxation time 1T and the transverse relaxation time 2T (chapter 2.4.3). Otherwise, the electromagnetic wave feeds back to the atomic system. As a result, a stimulated emission process occurs and suppresses the superfluorescence originating from the spontaneous emission. (3 ) The life time E pulse width R and time delay D must obey the conditions *) ( ,2 2 1T T TD R E 4 7 i.e., since SF is coherent, it must occur before the natural lifetime of the excited state and before the homogeneous (inhomogeneous) dephasing time. (4 ) The key parameter governing the growth rate of SF is the coupling strength between electromagnetic field and optical polarization, expressible as the cooperative frequency SF QW SFL n c N d 2 2 28 4 8 and to observe SF in quantum well system must exceed 2/ 2 T ( or 2 / 1 2 2) /( 2T T if inhomogeneous dephasing time 2 2T T ). Here d is the transition dipole moment; N 2D e h density; overlap of radiation with the QWs; plank constant; n PAGE 72 72 refractive index; wavelength in vacuum; c ,speed of light; QWL width of QW. Therefore the pulse duration is 2 / 1/ 1 NSF R and peak intensity scales as 2 / 3 2 / 3/ B N N IR SF in the magneti c field [ 63, 53 and 61]. 4.2.2 Strong Field Limit Electron hole excitons in high magnetic field experience two main interaction, Coulomb force and magnetic field. Depending on the strength of the external magnetic field, one can take the other interaction term as perturbation. In this section, we define the dimensionless parameter which is the ratio of magnetic and Coulomb energy and conceptually define the weak and strong field regions. In the presence of perpendicular magnetic field H with 0 y xH H and H Hz (cha pter 2.3) and within the effective mass approximation, the relative motion of Schrodinger equation for 2D e h pair with zero center of mass momentum can be written in CGS units ) ( ) ( 2 2 1 2 2 10 2 2 2r E r r e r H c e P m r H c e P mh e 4 9 where r is the e h separat ion and em and hm are electron and hole effective masses. Optical transitions occur at e h pair energies E and the optical transition probability is proportional to the probability of finding electron and hole in the same unit cell, 2) 0 ( r [8 ]. The e h relative motion is determined by an effective potential consisting of two terms 2 2 2 2 0 28 c r H e r e Veff 4 10 where is the reduced mass, 1 1 h em m The first term is the usual Coulomb attraction term and the second is imposed by the external magnetic field and varies by 2r In case of 0 H Eq. 4  PAGE 73 73 10 yields hydrogenlike spectrum which consists of discrete bound states and a continuum. Consequently 0 H states must extrapolate to the bound states solution of the 0 H problem. The strength of the Coulomb interaction can be characterized by the 3 dimensional Rydbe rg energy, 0 4 02 /me E and Bohr radius 2 0 0/ me a [24], while that of magnetic field term is described by the e h cyclotron frequency mc eHc and magnetic length 2 1) / ( eH c lmag[8 48]. The dimensionless para meter 0 2 02 / ) / ( E l ac which is the ratio of magnetic and Coulomb energy is traditionally used to describe the relative importance of these two terms. The region 1 characterizes the weak field limit where Coulomb energy dominat es and the magnetic field can be treated as an external perturbation [ 64]. In strong field limit 1 we need to consider the Landau levels resulting from free particle states by quantization of the motion in the plane perpendicular to m agnetic field, with the Coulomb energy as a perturbation. The intermediate regime 1 is more complicated to describe quantitatively. A further condition to resolve each Landau level can be given as 12 Tc 4 1 1 where 2T is the phase relaxation time (Chapter 2.4.3). In other word, to observe the Landau levels, their broadening must be smaller than the cyclotron energy [ 64]. 4.2.3 Band Mixing In Chapter 2.2, we discussed the multiple subbands due to the quantization of e h motion in z direction. For degenerate bands, we need to modify the band structure of inplane motion of carriers, because quantum confinement generally reduces the original spherical or cubic symmetry around near point (chapter 2.2.2). PAGE 74 74 For the electrons, we assume the effective bulk Hamiltonian (Eq. 213) and for the holes use for the determination of envelope functions. With replacement of zk as z i Pz / the hole band Hamiltonian matrix (Eq. 2 13) for the four degenerate eigenstates with 2 / 3 J states, 2 / 3 ,2 / 1 2 / 1 2 / 3 Jm is given by [ 32] lh hh hh lh m m J JH b c b H c c H b c b H H m H m* '0 0 0 0' 4 12 ) ( 2 ) ( ) ( 2 12 1 0 2 2 2 2 1 2 0 m k k P m Hy x z hh 4 13 ) ( 2 ) ( ) ( 2 12 1 0 2 2 2 2 1 2 0 m k k P m Hy x z lh 4 14 ) ( 32 0 y x zik k P m b 4 15 2 2 0 2) ( 2 3y x zik k P m c 4 16 ) ( ) (2 / 2 / *z z i z dz PC CL L Z 4 17 ) ( ) (2 2 / 2 / 2z z i z dz PC CL L Z 4 18 where 0m is the free electron mass, 2 1, and 3 are Luttinger parameters, and ) ( z is envelope function of confinement quantum well. Now theses four degenerated bands are perturbed by each other and from off diagonal terms b b c and c we can induce the new energy eigenvalues. From Eq. 4 13 and Eq. 4 14, lhH has higher energy than hhH Consequently, the degeneracy at 0 k of bulk semiconductor material is lifted. However, un perturbed bands in the PAGE 75 75 quantum well cross at a finite k With 0 b (in case of a symmetric well, envelope function, ) ( z is odd or even function and hence ZP vanishes because op eration with z i PZ changes the symmetry property in the reverse way), an evaluation of Eq. 4 12 gives the result [ 32] 2 2 2 14 ) ( ) ( 2 1 c H H H H Elh hh lh hh 4 19 The resulting dispersion is shown in Fig. 4 3 for 0 c and 0 c We can see the typical level repulsion and the state mixing where the energies cross for 0 c Therefore when energy levels of exited states are tuned to the same energy, anti crossing behavior is observed depending on cou pling character of the external perturbations. If the coupling matrix element of perturbation terms is nonvanishing, their wave functions are mixed so the crossing behavior is suppressed and replaced by anti crossing behavior. 4.3 Experimental Motivation T o investigate the role of many body interactions in quantum wells, the energy gap between Landau levels( cyclotron frequency, c eHc 2 ) in a magnetic field can be used to determine the elec tron and hole effective mass [ 9 15] and to see the optical interaction of zero dimensional system, applied magnetic fields can be used perpendicular to the sample [ 6566]. The study of magnetoexcitons or magnetoplasma has been extensively studied by Potemski et al. [49, 50] and Butov, et al. [9 15]. F rom high intensity excitation and magneto luminescence measurements, they found new and precise information about many body effects in a quasi twodimensional system by assuming e h system as magneto plasma. Butov., et al. [ 40, 15] carried out experiments at relatively low field strengths (intermediate magnetic field ( 4 ) ) up to carrier densities of 1012/cm2 using a 20 ns Cu vapor pulsed laser source. The major drawback of using nanosecond excitation for their experiment is that the in ter  PAGE 76 76 Landau level transitions (recombination) occurs on shorter (ns) time scales, thus limiting the maximum carrier density and complicating the interpretation. In other to investigate the physics of high density magneto plasmas in the strong field limit 1 new facilities are required. We have developed an ultrafast spectroscopy laboratory at the National High Magnetic Field Laboratory as introduced in the previous chapter where DC high magnetic fields (31 Tesla Bitter type and 17.5 Tes la superconductor magnets) are available. 4.4 Sample Structure and Experimental Methods Our In0.2Ga0.8As samples were grown by molecular beam epitaxy (MBE) on GaAs substrates. These samples consist of a GaAs buffer layer followed by 15 layers of 8 nm In0.2Ga0.8As quantum well separated by 15 nm GaAs barriers and 10 nm GaAs cap layer as shown in Fig. 4 4. All samples were grown at substrate temperatures between 390 and 435 We use the tungsten lamp to measure the white light transmission of the samples. Material parameter is given in Table 4 1. The signal is collected through a 0.6 mm core diameter, 5 m long multimode optical fiber connected to the sample mount. A Princeton Instruments 1024 256 CCD camera attached to 0.75m McPherson spectrometer with a 150lines/800nm grating that disperses and analyzes the time integrated spectrum. We convert the measured transmission signal to an absorption spectra in the following way 0 ) ( 0) ( log 1 ) ( ) ( T T L e T TL L L 4 20 where LT and 0T are the transmission after and before the sample length L and is the wavelength dependent absorption coefficient. The excitation light sources for photoluminescence spectroscopy studies are the CW He Ne laser and CPA for low and high excitation as described in Chapter 3.1.3. Fig. 4 5 shows the schematic diagram for experimental PAGE 77 77 set up. A right angle microprism (area 1 1 mm2) redirects the in plane emission from the edge of the sample to the collection f iber as shown in Fig. 4 6. To align the beam on the center of the sample for maximum collection efficiency, we monitor the maximum power from the center collection fiber with either a power meter or an IR viewer to find the brightest position while adjusti ng the mirror located at right bottom of the magnet. All of our measurements were carried out with 31 Tesla Bitter type and 17.5 Tesla superconducting magnets in the Faraday geometry shown in Fig. 4 6 in which the magnetic field perpendicular to the quantum well plane and for the parallel propagation of the light to the magnetic field. 4.5 Prior Study of Photoluminescence Spectroscopy of 2 D Quantum Well In this section we introduce L. V. Butov., et al and M. Potemski., et. al.s works which are the pioneer ing experiment for 2 dimensional magneto optic experiments and then the first experiment carried out at the new facility at NHMFL will be discussed. 4.5.1 Photoluminescence Spectroscopy at Weak Field Region L. V. Butov., et al. [9 15] studied of InxGa1 xA s quantum well in 12 T superconductor magnet. They used Ar+ ion (514.5nm) and Cu vapor pulsed laser (510.5 nm and 20 ns pulse duration) for low and high excitation. Fig. 4 7 shows the Landau fan diagram with high excitation density (1.7 10 12 /cm2). For the analysis and data fitting, they used simple plasma approximation [ 65]. They assumed the magnetic strength is small enough to apply the plasma approximation and they related the wave vector corresponding to electrons and holes at the n the Landau level according to [ 65] c n eH k / ) 1 2 (2 4 21 PAGE 78 78 and hence determine the quasi particle dispersion relation ) ( k showed in Fig. 4 8. As shown in Fig. 4 8, in case of strained quantum well, reduced mass does not change by wave vector k and hence many body interaction of strained quantum well dose not affect to any strong k dependence of reduce mass In case of unstrained QW, the strong dependent of reduced mass on k was shown in the same figure. Therefore they concluded the band nonparabolicity of unstrained QW is originated from the valence bands complex structure. Summarizing their observation, they found manybody effects such as band gap shri nkage and effective mass renormalization from the fitting method described in Eq. 4 21 and complexity of valance band structure in unstrained quantum well. M. Potemski., et. al. [ 49, 50] also carried out the similar magneto optic experiment for AlxGa1 xAs multiple quantum well. They generate carrier density (10 13 /cm2) higher than L. V. Butov., et al. They assumed the parabolic band for the holes and nonparabolic Landau levels fo r electron in two band model [ 50, 51] to determine the energy structure and the energy expression for this model i s read as follow [ 50] 0 0 0 0 0 2) / )( 2 1 ( ) 2 1 ( 1 2 1 2 1 E m e N m eB N E E E E E E Eh e g g g g N 4 22 Where, N is the Landau level index and em0 and hm0 are the band edge mass of bulk GaAs electron and hole. gE is the 3D bandgap energy. From the Landau fan diagram for absorption spectrum, they decided 0E from the relation of 0 2E E Eg D g Fig. 4 9 shows the fitting result from Eq. 4 22. Their fitting r esult shows 20% reduced mass enhancement for higher carrier density but couldnt see any bandgap renormalization while increasing the carrier density. PAGE 79 79 Electron hole system in 00LL experiences the strong Coulomb interaction and hence, the separate fitting method for 00LL was needed. Both L. V. Butov., et al. and M. Potemski., et. al ignored the this fact and they analyzed all LLs with plasma approximation or simple two level system. One more thing we need to point out in their experiments is that they used nanosecond pulse for excitation of high carrier densities. The drawback of using nanosecond pulse is inter LL carrier relaxation can be taken placed during the excitation pulse. 4.5.2 Prior Study of InxGa1 xAs Photoluminescence Spectroscopy The 3 dimensio nal Bohr radius of In0.2Ga0.8As/GaAs mult iple quantum well is 17.4 nm [14] and magnetic length magl is about 6 nm and 4.6 nm for 17.5 T and 31 T, respectively which correspond to 4 8 and 3 14 Motivat ed by Butov, et al. [ 14, 15] and M. Potemski., et. al.s work at relatively low fields ( 4 ) up to carrier densities of 10 12 /cm2, Jho, et al., [ 63, 67, 68] have previously studied InxGa1xAs samples in the Fast Optics facilities at the National High Magnetic Field Laboratory to understand the baseline physics and magnetooptics of high density magneto plasmas (carrier densities in excess of 2 13/ 10 cm ) and higher factors (up to 3 14 ). At sufficiently high magnetic fields (>12 Tesla) and carrier densities in a multiple quantum well system, Jho, et al. observed a spontaneous macroscopic coherence followed by cooperative recombination, or superfluorescence (SF) [4]. SF had been observ ed in atomic gases [ 69, 7 0 ] and ra refied impurities in crystal [7 1 7 2 ], but the experiments of Jho, et al. were the first to observe this exotic effect in semiconductors. Because of the short decoherence times typical of carrier carrier interactions in semiconductors, the observation of cooperative recombination in semiconductor systems was hampered by the inability to create and maintain sufficiently high carrier densities to allow for the formation of the macroscopic dipole. PAGE 80 80 Jho, et al. [ 63, 67, 68] invest igated light emission processes in an undoped QW system placed in a strongly perpendicular magnetic field. As noted in Ch. 2, the magnetic field fully quantizes the QW system into an atomic like system with a series of Landau levels and thus strongly incre ases the density of states to accommodate highdensity e h plasma. They measured the PL emission as a function of laser fluence ( Flaser) and magnetic field B. Emission characteristics were found to depend linearly on the carrier density N ( Flaser) in the spontaneous and amplified spontaneous emission regimes, but in case of superfluorescence, the emission observed was highly collimated and proportional to N3/2 or B3/2 [63] The successive stages of emission progress from a low densit y conventional PL through ASE to a stochastically determined SF regime. Fig. 4 10 summarized their results. The scaling behavior, the linewidth evolution and directionality of emission for each shot suggests the following physical picture: In the low densi ty limit ( B < 12 T, Flaser< 2/ 5 cm J ), exited e h pair radiate spontaneously through LL interband recombination. The emission is isotropic and inhomogeneously broadened. At a critical fluence ~ 2/ 01 0 cm mJ and B = 12 T, a populatio n inversion is established in the LLs and ASE develops, leading of emission of pulse and stimulated emits any direction in the plane of sample as shown. 4 10 (d). The reduction of linewidth with increasing fluence results from conventional gain narrowing. In the high gain region, the spectral linewidth reduces to 2meV but is still larger than 2/ 2 T required for SF. When density of states and the cooperative frequency SF are sufficiently high and satisfy SF > or 2 / 1 2 2) /( 2T T (from Eq 3 8), the e h plasma establishes a coherent macroscopic dipole after a short delay (<10 ps) and subsequently emits an intense pulse through cooperative recombination. In conclusion, Jho et al. have observed cooperative emission in a highly photoexited and strongly coupled semiconductor system. Using intense ultrafast excitation and strong magnetic PAGE 81 81 fields, the resulting density and energy confinement are sufficient to generate a spontaneous macroscopic polarizat ion that decays through the emission of SF pulse. 4.6 Magneto optical C haracterization of InxGa1 x As Quantum Wells at High Fields and Carrier Densities: Absorption and PL S pectroscopy 4.6.1 S ample C haracteristics Experiments were performed on In0.2Ga0.8As quantum well samples consisting of 15 layers of 8 nm thick QWs separated by 15 nm GaAs buffer layers as shown in Fig. 44. Samples were provided by Glenn Solomon of the National Institutes of Standards and Technology in Gaithersburg, MD. Two samples were used in these studies, designated as M507 and M508. For comparison, we also will occasionally show data on the samples (designated S322 and S324) used in the SF characterization experiments by Jho, et al, described above. All samples have identical structures, but are grown at slightly different sample growing temperatures. To characterize our samples, low power CW absorption and PL were performed. Fig. 4 11 shows the absorption and PL spectra in the absence of a magnetic field at a temperature of 5 K. T he tungsten lamp and He Ne laser (irradiance of 100 mW/cm2) were used for the white light absorption spectra of Fig. 4 11(A ) and l ow excitation PL for Fig. 4 11(B ). The absorption spectra are quite typical of a quantum well system, showing a series of step s corresponding to a 2D density of states (see Eq. C 6 in appendix) punctuated by excitonic absorption peaks at the step e dges. Interestingly Fig. 4 11 (A ) also shows the presence of dark states (defined below) in sample M507, M508 and S324 (displayed by a rrows in the figure). Quantitatively, there are some differences in the energies and absorption strengths. The absorption spectra of M507 and M508 are two times stronger absorption strength than S324 and S322. In addition, the energies of the heavy hole and light hole transitions, e1, h1 and l1 are somewhat shifted with respect to one another, indicating the presence of slightly different amounts of strain in each sample. (e1  PAGE 82 82 h1, e2 h2 and e1 l1 represent transitions corresponding to quantum numbers h ej j and lj as describe d in Chapter 2.2.1). Fig. 4 11(B ) displays the relative PL strengths of the samples. Unlike, the absorption spectra which show all transitions, the PL spectra show emission from only the e1 h1 exciton tr ansition. As shown in the figure, S322 has most efficient PL emission strength. Linewidths of each PL are 7.78meV, 6.74meV, 5.52meV and 8.6meV for S324, S322, M507 and M508 and hence M507 has narrowest band transition. 4.6.2 Dark Transition and Anti Crossing Strained In0.2Ga0.8As quantum wells have simple, nearly parabolic hole bands because the strained sample structure induces heavy hole light hole energy splitting, separating heavy hole state from light hole states [ 32]. In an ideal quantum well with inf inite barriers, inter band transition occurs only 0 h ej j j (Chapter 2.2.1). However h ej j (dark) transitions are weakly allowed in real quantum well systems because of finite barrier size, lack of inversion symmetry due t o the quality of quantum well, and perturbations such as strain. Thus, h ej j (dark) transitions can be weakly allowed in our samples (as seen in Fig. 4 11(A )). As a reference, 4 12(a) shows the schematic band structure showing the split ting of the heavy and light hole in the valence band as a result from the confinement effect of small well size of sam ple. Fig. 4 12(B ) displays schematically the forbidden dark state transitions. Fig. 4 13(A ) displays the raw transmission spectrum as a function of magnetic field with white light for sample M508 and. Fig. 413(B ) shows the converted absorption spectrum, obtained from Eq. 3 1. Labels H1, H2 and L1 are representing the transition, e1 h1, e2 h2 and e1 l1. Anti crossing between third Lan dau level (22LL) of H1 and the dark state (forbidden e1 h2 transition) is observed at an energy of ~1.40 eV near 11.5 T. The observation of a normally parity forbidden e1 h2 transition from the broken inversion symmetry arises from the presence of PAGE 83 83 magneti c field; the two different subbands and LLs energies between 22LL and e1 h2 are repulsed and give rise to an anti crossing behavior because of the off diagonal mixing term (explained in Chapter 4.2.3). As reported in Ref. 71 and 88, when a 2D quantum well in a magnetic field experiences a perturbation by applied magnetic field, H this H couples Landau levels of different subbands. The strength of H is proportional to confinement direction z [73] and thus only subbands of opposite parities are coupled by H These off diagonal terms represent the mixing between two different subbands and give a result of anticrossing as shown Fig. 4 13(B ). 4.6.3 Diamagnet ic and Landau Shifts in InGaAs Quantum W ells Fig. 4 14 shows the magnetic fielddependent of the PL emission using low power density excitation (2.1 mW/cm2). Only the 00LL transition is visible at low powers. At magnetic fields above 5T, the 00LL em ission peaks fit linearly with H (Landau shif t) but at low fields, the fit deviates because of a diamagnetic shift proportional to 2H As we show now, the energy dependence is quadratic in H at low fields (the diamagnetic shift) and lin ear in H at high fields (the Landau shift). From Eq. 49, the actual Hamiltonian contains both the Coulomb potential term and a vector potential term from the magnetic field. The Coulomb potential cannot be separated in Eq. 4 9 and so the analytic separati on of wave function into z and x y plane components is not possible. However, if we consider the external magnetic field as a perturbation (which is true if H is sufficiently small), then the perturbed energy shift of ground energy can be expressed as [ 68] 2 3 2 4 24 ) ( H e B Eground 4 23 where is the e h reduced mass, 1 1 1 h em m Eq. 4 23 is the diamagnetic shift of the exciton energy and is only valid for 1 PAGE 84 84 In case of high magnetic fields ( 1 ), the magneto exciton is similar to a free eh pair in magnetic field and thus the Coulomb term can be considered as a small perturbation. In the high magnetic field limit, the energy shift due to magnetic field becomes c groundB E 2 1 ) ( 4 24 and is often called the Landau shift. We defer the discussion of the case of intermediate field strengths to chapter 4.7.2. 4.6.4 Photoluminescence Spectra in High Magnetic Fields To quantify the excitation regime that we can achi eve using ultrafast high intensity excitation, we study the PL spectra obtained from samples (M507 and M508) in high fields. These investigations allow us to i) estimate the carrier densities that can be achieved using femtosecond pulses and ii) the emiss ion regime that we can achieve (section 4.2.1). Fig. 4 15 compares the 0 T luminescence spectra from In0.2Ga0.8As/GaAs multiple quantum well sample (M507) at T =5 K for low intensity (He Ne laser, 2.1W/cm2) excitation and high intensity ultrafast excitatio n (150fs CPA, 1 109 W/cm2). Relative intensities are plotted in the figure and for comparison, the white light absorption spectrum is also shown. Because the energy of the photons greatly exceeds both the In0.2Ga0.8As and the GaAs energy gaps, the pumping excites e h pairs in the wells and the barriers. However, most e h pairs rapidly relax in energy to in the quantum well layers. The radiative recombination appears as a broad asymmetric line with a maximum intensity at near the band gap of In0.2Ga0.8As The line width is increased with excitation power as shown in the Fig. 4 15 (4.9 meV, 70 meV and 95 meV for absorption, 2.1 W/cm2 and 1 109 W/cm2) and for high excitation densities the quantum well is filled by electron and holes and strong excitonic emis sion from GaAs appears as well in the spectrum. PAGE 85 85 Line broadening on the higher energy side is mainly due to the filling of the conduction and valence band states in the quantum well because of the Pauli exclusion principle. However, at sufficiently high excitation densities the emission line of the e h plasma shifts toward lower energies du e to band gap renormalization [ 24, 35, and 32]. This shift arises from the Coulomb interaction among the excited carriers; the energy of electrons and holes in their respect ive bands are reduced. The origin of this reduction is the consequence of the exchange effect for equal spin particles and Coulomb correlation effects for all particles. The exchange term arises from the Pauli exclusion principle the probability of two Fe rmions with the same quantum states are at the same points in real space is zero. For increasing separation between particles, the probability approaches unity. Therefore the Pauli exclusion principle reduces the probability that the equally charged particles come close together and this in turn reduces the repulsive contribution. Fig 4 16 (A ) shows the photoluminescence emitted from the edge of the sample for H = 5.5 T. It is important to note the edge emitted inplane PL experiences ASE while traveling through the sample, resulting in the strong signal intensities [ 63], thus the edge spectra are greatly enhanced in signal relative to the out of plane emission. Comparing with Fig. 4 15, we see a series of discrete LLs up to very high LL indices, indicatin g that femtosecond pumping has produced a sufficiently high enough density of carriers to fill the higher LLs. A series of LLs from the first light hole transiti on is also shown from Fig 4 16(A ) (above 1.40 eV). Because of the increased signal strength, th e step like density of states is evident at approximately 1.40 eV. The spectra shown in Fig. 4 16(A ) was obtained at an excitation dentisy of 1.46 109 W/cm2 and covers almost 15 Landau levels. Therefore we obtain ~1013/cm2 carriers from Landau degeneracy, c eH (Eq. 2 30, here is the filling fa ctor). For 17.5T (Figure 4 16 (B )), the confinement effect of the magnetic field is so strong ( 4 8 ) that the PL emission from in PAGE 86 86 between each Landau levels d rops almost to zero and shows a 0D density of states character. Anti crossings are also evident in the figure at 1.423eV and 1.453eV (associated with the mixing with dark transitions) as shown in Fig. 4 13 for magnetic field dependent of absorption. Like Jho, et al.s data [63 ], we find a low energy Gaussian feature and a high energy Lorentzian feature in the PL emission from 00 LL. For higher LLs, the PL shape is most likely only Lorentzian. Figure 4 17 shows the line shape analysis based on this and deta ils of linewidth analysis refer to Appendix D. The full width half maximum (FWHM) of the narrow peaks shown at each Landau level is ~3 meV for the 00LL and ~2 mev for the 11 and 22 LLs. The line width increasin g region for SF found in Ref. [ 63] is not obse rved at the 00 LL for our samples (M507 and M508). The line width decreases up to 0.07 mJ/cm2 and then saturates at higher excitation fluences. For the 11 LL and 22 LLs, the excitation is not sufficient to observe a signal below 0.07 mJ/cm2 (11 LL) and 0.1 mJ/cm2 (22 LL). A linewidth increase is observed at the PL emission threshold, indicating that for these levels, we have reached the ASE regime. Increasing excitation intensity results in increasing collision probability between carriers and hence the fir st increases in 11 and 22LL may come from damping of one particle decay which increases wit h increasing carrier density [ 13]. The degenerate energy at band maxima (Fig. 2 3: between heavy hole and light hole) can be separated by strain from finite quantum barrier separation. The energy separation between e1 h1 and e1 l1 in Fig 4 11(A ) for each sample is about the same (~76meV) but when compared with the PL measured from sample S322, samples M507 and M508 have more dark states (which is even evident at 0 T (Fig 4 11(A )). This appearance of dark states more defected in sample structure. The correlation of the presence of these dark states and the lack of SF emission from samples M507 and M508 provides strong evidence that defect induced dark states inhibit t he PAGE 87 87 formation of a coherent macroscopic dipole and concomitant SF emission in InGaAs quantum wells. Nonetheless, as we show in chapters 5 and 6, the presence of a high density incoherent magneto plasma leads to complicated, coupled dynamical relaxation and emission processes in this system. 4.7 Energy States in In0.2Ga0.8As/GaAs Quantum Wells at High Magnetic Fields: Excitons, Magneto plasmas, and Mott Transitions 4.7.1 Bandgap Renormalization and Landau Fan Diagram Fig. 4 18(A) and (B ) show the emiss ion spectra from sample M507 for T = 5K as a function of magnetic field. Using He Ne laser excitation (2.1 W/cm2), only the lowest Landau level is occupied and a single emission line is observed. With increased excitation density additional peaks appear, indicating the occupation of higher Landau levels. Spectra at 17.5T are well structured, which is the main advantage of using magnetic field. Fig. 4 18( A ) displays a quadratic diamagnetic shift below 5 T (see Fig. 4 14 ( A )) and a linear dependence at highe r fields. Fig. 4 18( A ) also shows that the LL energies display no significant density dependence, as re ported by M. Potemski et al. [ 49, 50] but in contrast to Butov et al. [ 9 15]. Butov, et al. took special attention to ensure the homogeneity of the e h plasma density in their experiments. First, they avoided the use of multiple quantum well because a distribution of photo carriers varying with the depth in the sample would have given rise to different e h plasma densities in the different quantum wells. M oreover, in the absence of confinement of photocarriers in the quantum well, the density of e h plasma is changed strongly because of the high rate of diffusion of high energy exited e h from the photoexited region. They concluded that the most effective method of confining the propagation of carriers to the quantum well plane was a restriction of the size of these wells to the size of excitation laser beam and used 30 30 m dimension of sample. Therefore our experiment is different f rom that L. V. Butov [ 9 15] carried out because of the PAGE 88 88 relative size of our sample dimension (3 3 2mm and laser spot is 0.5 mm) and the use of multiple quantum well. Fig. 4 18( B) displays the Landau fan diagram measured in white light absorption and via PL emission for low and high excitation densities. A red shift in the PL emission with respect to the white light absorption spectrum is observed at each LL and increases at higher LLs due to the effects of increased screening of the Coulomb interaction. A much smaller red shift is seen for the 00 LL (3 meV) when compared with the 11 and 22 LLs (18 meV), indicative of a stronger excitonic character in the lowest Landau level. If we neglect the excitonic effects at high magnetic field, the LL energies can be expressed as [ 32]. ) 1 ( n c eH E Eg 4 25 i.e, they are linear in field. The observed linear dependence of the 11 and 22 LLs in Fig. 4 18( B) justify that Eq. 4 25 can be used to obtain the reduced e h effective mass from the slope of Eq. 4 25. So far, we have made the assumption that the PL peak change as a function of magnetic field strength follows the diamagnetic shift (Eq.4 23) or the Landau shift (Eq. 4 24). As an example, Butov, et.al sugge sted [ 9 15] that at high densities, the carriers can be assumed to be an e h plasma and the main contribution of the magneto PL arises from the Landau shift. However, in the intermediate magnetic field region, we need to apply both terms (diamagnetic and Landau) and need to find the critical magnetic field to apply each term for a specific region of magnetic strength. As we can see from Fig. 4 14 and Fig. 4 18, the 00 LL does not follow the linear dependence, especially in the small magnetic field strengt h region. Therefore it is obscure PAGE 89 89 whether 00LL has more diamagnetic or Landau shift. We need more careful analysis to discuss in the below. 4.7.2 Coexistence of Exciton and Landau Levels As our above data shows, the energy states in InGaAs MQWs in magnetic fields have a complex character. Discrete LLs are seen, and depending on excitation density, magnetic field strength, and LL index, energy states also can reveal an excitonic character. \ While the concepts of excitons and Landau levels in semiconduct ors are very useful in explaining the observed structure of the fundamental optical absorption edge, the physical interpretation arising from these two concepts is quite different and requires some discussion in relation to our experiments. The exciton is an electron hole bound complex, the discrete level structure of which comes from a long range Coulomb interaction between particles having opposite charge, whereas the Landau levels correspond to purely one particle levels formed in the conduction or valen ce band, their discrete nature resulting from the strong magnetic field. In this section, we will describe the coexistence of these two concepts. To make our problem more specific, consider a simple effective mass Hamiltonian for an e h Coulomb interaction in the presence of an external magnetic field as follows. ) ( ) ( 2 2 1 2 2 10 2 2 2r r r r e r H c e P m r H c e P m Hh e h h h e e e 4 26 For vanishing magnetic field, Eq. 4 26 leads to the simple hydrogenic exciton whose solutions are the well known Coulomb functions of the relative coordinate h er r r On the other hand, if magnetic field is sufficiently strong that the Coulomb interaction term can be neglected in the Hamiltonian, the solution of Eq. 4 26 is a product of the wave functions of the electron and hole PAGE 90 90 free motion in the magne tic field (Landau levels). Thus, the transition from the one limit to the other by increasing the magnetic field evolves from bound exciton to a purly LL. In case of a 2D quantum well having axial symmetry, the angular coordinate of the equation can be se parated, so that the problem reduces to a second order ord inary differential equation [ 74]. ER R r c H e r e R r m dr d r dr d r 2 2 2 2 0 2 2 2 28 1 2 4 27 The separation function that leads to Eq. 4 27 is 2 / 1) 2 ( ) ( ) ( ime r R r 4 28 and energy in Eq. 4 26 is related with E as follows [ 74] m c eH m m m m Eh e h e) 2 ( 4 29 The solution of Eq. 4 27with suitable boundary conditions and determination of the eigenvalues is in general intractable analytically, and we must resort to approximation methods. Akimoto and H asegawa [ 74] solved Eq. 4 27 by using the WKB approximation method [120, 123]. They were able to find a numerical solution and more importantly obtain a simple expression for the two extremes. In the region which in w hich the Coulomb interaction term is sufficiently strong, the energy (in terms of 3D Rydberg co nstant) can be expressed as [ 74] .... ) ( 20 7 ) 2 1 ( ) 2 1 ( 8 5 ) 2 1 ( 12 2 2 2 cn n n E 4 30 where n is the Landau level index and the c cyclotr on frequency ( c eHc 2 ). When 0 n the first term of Eq. 4 30 is seen to be 4 Rydbergs, meaning that the 2D exciton binding energy is 4 PAGE 91 91 times stronger than the dimensional case. The second term is the well known diamagneti c term which is proportional to 2 2Hc In the strong magnetic field region, the energy expression becomes [ 74] 2 / 11 2 3 1 2 n n Ec c 4 31 The first term of Eq. 4 31 is the Landau shift term and the square root dependent term i s in agreement with a perturbation term of Coulomb interaction [ 75, 76]. These two equations (Eq. 4 30 and Eq. 4 31) are a little bit different from Eq. 4 23 and Eq. 4 24. The differences come from including the small perturbation terms in these new equat ions. Fig. 4 19 shows the plots of Eq. 4 30 and Eq. 4 31. As shown in Fig. 4 19 there is a magnetic field at which two solutions intersect. If we define the critical magnetic field ( cH ) as the field at which the two solutions overlap, in the region of cH H we can apply Eq. 4 30 which contains the diamagnetic shift and in the region for cH H we can use Eq. 4 31 expressing Landau shift and the small perturbation term. Akimoto and Has egawa [ 74] did c ompare these two equations with their numerical solution and found the good agreement between them for each region. The calculated critical magnetic field for the case of reduced mass of 005 0 m where 0m is the free ele ctr on mass for each Landau level as shown in table 4.2. 4.7.3 Fitting Method for Landau Levels Eq. 4 30 and Eq. 4 31 are the solutions in case of infinite potential barriers but in real situation, the quantum barrier has finite thickness and height, and el ectron or hole can penetrate into the barrier. In addition, the semiconductor has an intrinsic gap energy Eg separating the valence and conduction bands. We modify Eq. 4 30 and Eq. 4 31 appropriate to the more rea listic situation as follows [ 74] PAGE 92 92 ) ( 20 7 ) 2 1 ( ) 2 1 ( 8 52 2 2 c b gn n E E E 4 32 2 / 11 2 3 1 2 n n E Ec c g 4 33 where gE and bE are 2 dimensional band gap energy and excitonic binding energy. The modification Eq. 4 30 and Eq. 4 31 to Eq. 4 32 and Eq. 4 33 takes into account bandgap energy and 2 dimensional binding energy of finite potential barrier. We can apply eq. 4 32 and 433 to determine how the quantum well energy levels are modified in the presence of a dense electron hole plasma. This will allow us to a ssign a physical character as either Coulomb dominated excitonic (eq. 4 32) or strong fielddominated LL like (eq. 4 33) to each of the energy states in our absorption and PL spectra. Since the effective masses of the electrons and holes and the gap e nergy Eg d epend on the carrier density [ 14], we use as fitting parameters gE and the effective reduced mass ( c eHc 2 ). As shown in Fig. 4 18 and in Table 4 2, with the exception of the lowest 00 LL the higher LLs has relatively small critical magnetic fields (< 2 T). To compare the fits with experimental data, at the 11 and great LLs, we use only PL data beginning at 2 T because broadening of Landau level limits the resolution of the higher LLs PL peak position. Thus, we use only Eq. 4 33 for higher Landau levels (11 LL and above). As shown in Fig. 4 18( B), a clear linear dependent for the higher lying LLs justifies the use of Eq. 4 33 (LL like). For the 00 LL, Eq. 4 32 can be used to fit the absorption and PL data at fields less than 17T because of the large critical magnetic field ( ~21 T). Fig. 4 20 is the spectrum at a few different magnetic fields for absorption and high excitation density. Based on this spectrum, we can draw a Landau fan diagram as shown in Fig. 4 21 presenting a comparison of the (a) experimental white light absorption spectra (low PAGE 93 93 excitation density) and (b) PL spectra obtained from femtosecond laser pumping (1.53mJ/cm2, Ne h > 1013 cm2) with fits based on Eq. 4 32 an d 4 33 using the energy gap gE and the effective reduced mass as fitting parameters. Overall, we find excellent agreement between the fits and the experimental data. In the next three sections, we discuss in mor e detail the results from the fitting. 4.7.4 Binding Energy Physically, the separation between the bandgap energy, Eg, and the extrapolation of the 00 Landau level PL peak position to 0 Tesla is the excitonic binding energy. We find a 1.334 eV bandgap ene rgy based on the absorption data (Fig. 4 21 ( A )), and an 11 meV excitonic binding energy for the first heavy hole. This value is in reasonable agreement with the value obtained by S. Tarucha and H. Okamoto [ 77] in unstrained GaAs/AlGaAs multiple quantum well where the separation of light hole from the heavy hole is not clearly resolved. In our sample (strained InGaAS/GaAs), the strained structure gives sufficient resolution to separate the heavy hole from light hole states in the PL spectra and, as such, we can deduce the excitonic binding energy unambiguously. In contrast to low density excitation, the excitonic binding energy becomes zero at high densities (Fig. 4 21 ( B)), corresponding to an electron hole plasma state. All of the fits for the first hea vy hole converge to one value zero indicating a zero exciton binding energy at 0 T. However, the experimental 00 LL PL data follow the fitting curve of Eq. 4 32 (excitonic character) for increasing magnetic fields. This curve shows a diamagnetic shif t term and binding term as seen in the energy expression Eq. 4 32. Therefore we conclude that as soon as magnetic field is switched on, the plasma stat e present at 0 T undergoes a transition to an excitonic state PAGE 94 94 for the 00LL. For higher Landau levels, al l of the data are best fit with Landau shift term from Eq. 4 33 and hence correspond to a magneto plasma state. 4.7.5 Mott Transition and Luminescence from Higher Subbands The findings above are indicative of a metal insulator transition, whereby the Coulomb correlation is changed by density of carriers. Such as transition w as first researched by Mott [ 78, 79] and is known as a Mott transition. Under ordinary conditions (and in the absence of a magnetic field), the insulating phase of a low density excito nic gas becomes unstable at high densities and transforms into a conducting phase of unbound electron hole pairs, an electron hole plasma, as the Coulomb interaction is progressively screened. In a simple direct gap semiconductor, the plasma phase can be g enerated by a sufficient excitation intensity. The gradual change from excitonic gas to electron hole plasma is called the Mott transition [ 24, 78, 79] or exciton ionization. Basically, there are no excitons in an electron hole plasma, and hence, no exciton ic absorption or luminescence features. Therefore, the appearance of Mott transition is accompanied by a drastic change of the optical absorption of the excited volume and in particular, a disappearance of the excitonic absorption lines. A spectral linesha pe corresponding to an e h plasma emission appears as the intensity of excitation is increased, indicating a change of the plasma density as shown in Fig. 4 20 ( B). The appearance of the plasma luminescence is accompanied by the disappearance of the excit on absorption line, as shown in the 0 T spectrum of Fig. 4 20 ( B). The excitonic line bleaches as a result of screening of the Coulomb interaction responsible for binding of the electron and hole and gives the result of zero binding energy as shown in Fig. 4 21 ( B). Fig. 4 21 also shows the fitting results of the second heavy hole and the first light hole emission. Our excitation power generates roughly an electronhole pair density of 1013/cm2 and PAGE 95 95 as shown in Fig. 4 21, this excitation power is sufficient to completely fill the higher subbands. Comparing Fig. 4 21 ( A ) and ( B), the second heavy hole and the first light hole have also zero binding energy for high excitation power. However, the bandgap increases rather than decreases (as in bandgap renormaliza tion). Most of optical properties of semiconductors have been investigated extensively for the first subband, but higher subbands has been well studied yet. This new observed feature of a bandgap increase may be related with many body correlations or ma y be due to state filling and caused by emission from the high energy side of the LL, but it is not completely understood at the present time. 4.7.6 Reduced Mass From our fits, we can also extract the reduced effective mass renormalization due to high carr ier densities. Recall that Butov, et al. [ 9 15] assumed that electron hole pairs under high excitation(~ 1012 cm2 )can be considered as plasma, and that the plasma approximation is valid for very small magnetic fields (when the cyclotron energy is small as compared to the Fermi energy or plasma temperature [ 15]). In this approximation, the wave vector corresponding to electrons and holes at the nth Landau level can be written as: c n eH k / ) 1 2 (2 4 34 From this expression, one can determine the quasiparticle dispersion relation ) ( k E They assume that the plasma approximation describes a dense magnetoplasma at temperatures comparable with cyclotron energy c The approximation should be applicable at moderate magnetic fields when broadening of Landau levels is larger that the excitonic binding energy but on the order of c [15]. The result for the reduced mass is shown in Fig. 4 8. For a strained quantum well, PAGE 96 96 Butov, et al. find that the reduced mass dispersion versus wave number is small and obtained 0.06 reduced mass ratio to free electron mass. Their analysis was performing in the wrong regime, since the 00LL has an excitonic character (Diamagnetic shift) as shown in Fig. 4 1 4( B). The results of our analysis of the reduced effective mass is given in Fig. 4 22, based on Eq. 4 32 and Eq. 4 33. A reduced mass ratio of 00 Landau level is between 0.04 and 0.055. These values are smaller than those of Butovs (~0.06 (refer Fig. 4 8) ). Small discrepancy between these two values comes from exitonic term including Eq. 4 33 which we used for our data fitting. L. V. Butov used Eq. 434 assuming the all LLs follow p lasma approximation mentioned above c n eH k / ) 1 2 (2 4 21 As we can see from the above equation, there is no term related with Coulomb interaction, i.e. they assume most dominant part govern PL emission in magnetic field which is Landau term in Hamiltonian of Eq. 4 26. However as we know the discussion in section 4 73, to fit 00 LL properly, small perturbation terms for higher Landau level should be included as shown in Eq. 4 33. Different slopes between Landau fans for absorption and high excitation power (Fig. 418( B) and Fig. 4 21) originate from different re duced mass because as shown in Eq. 4 31, first term in the equation is proportional to cyclotron frequency c which is inversely proportional to the effective reduced mass ( c eHc 2 ) and this slope difference between them results in 35% enhancement of reduced mass from the many body effect 4.7.7 Summary The magneto optical measurements presented in this chapter have enabled us to observe characterize the 2D e h plasma in strained In0.2Ga0.8As /GaAs quantum wells. A careful analysis PAGE 97 97 for intermediate region of magnetic field was carried out and deduced reduced mass enhancement at high excitation intensity. The bandgap renormalization and anti crossing in strained quantum well nondegenerated valence band were discussed. We found 11meV binding energy of e h exciton with the low carrier excitation densities and observed the clear evidence of insulator metal transition(Mott transition) from zero binding energy for high carrier excitation. Photoluminescence from the first heavy hol e follows the usual Bandgap shrinkage(bandgap renormalization) but PL from higher subband(first light hole and second heavy hole) shows bandgap increase rather then bandgap reduction. This is the new observation for higher subband and need to study future to have complete understanding of many body effects. PAGE 98 98 Table 4 1. Material parameter for InAs and GaAs [68] Parameter InAs GaAs Bandgap (E g eV, T=0) 0.417 1.519 Electron effective mass( 0/ m me T=0) 0.026 0.06 7 Luttinger parameter 1 20.0 6.98 Luttinger parameter 2 8.5 2.06 Luttinger parameter 3 9.2 2.93 Refractive index n r 3.42 3.4 PAGE 99 99 Table 4.2. Critical magnetic field cH Landau level index ) ( T Hc 0 20.79 1 1.54 2 0.37 PAGE 100 100 Figure 4 1. Simplified picture of A) spontaneous emission and B ) amplified spontaneous emission process of a simple two level atomic system. Figure 4 2. The ASE and SF processes. ASE: There is no interaction between atoms on excited states, thus no coherence between the emitted photons. Omnidirectional emission results. SF: Atoms in the excited state interact with vacuum electroma gnetic field fluctuations and establish a coherent ensemble dipole, giving rise to strong emission. A B PAGE 101 101 Figure 4 3. Energy dispersion relations for hole states in a quantum well. The solid lines represent the mixing of heavy and light hole valance bands f rom Eq. 4 19 and the dashed lines shows the energies neglecting band mixing. layer Cap GaAs As Ga In8nm 0.8 0.2 GaAs 15nm layer buffer GaAs 15 layer Cap GaAs As Ga In8nm 0.8 0.2 GaAs 15nm layer buffer GaAs 15 Figure 4 4. Sample structure of In0.2 Ga0.8As/GaAs multiple quantum well. PAGE 102 102 Figure 4 5. Simplified schematic setup of InGaAs multiple quantum well absorption and photolumi nescence spectroscopy. Figure 4 6. Schematic diagram of collection of the PL emission from Faraday geometry. One right angle prism is located at the edge of the sample to collect in plane PL. PAGE 103 103 Figure 4 7. Landau fan diagram for interband transition in 2D electron hole plasma with carrier density 1.7 10 12 /cm2 of InxG a1 xAs multiple quantum well [ 15]. Reprint with permission from L. V. Butov., et al., Phys. Rev. B 46, 15156 (1992).( 1992 American Physical Society) Figure 4 8. The reduced carrier effective mass as a function of square wave vector for different electron hole plasma densities. The dependence is shown for both strained 7.5nm thick In0.28Ga0.72As/GaAs and an unstrained 15nm thick In0.53Ga0.47As/InP single quantum well [ 15]. Reprint with permission from L. V. Butov., et al., Phys. Rev. B 46, 15156 (1992).( 1992 American Physical Society) PAGE 104 104 Figure 4 9. Magnetic field depe ndence of the high excitation luminescence peaks (open circles) and of the absorption peaks (small closed circles) [ 50]. Reprint with permission from M. Potemski, J. C. Maan, K. Ploog and G. Weimann Properties of a dense quasi two dimensional electron ho le gas at high magnetic fields Solid State Communication 75, 185 (1990 ).( 1990 Elsevier ) PAGE 105 105 Figure 4 10. A ) Experimental schematic showing single sh ot excitation and collection. B ) Four representative emission spectra from edge 1(black) and edge 2(red) fibers exited from a single laser pulse and measured simultaneously. Normalized emission strength from 0th LL ve rsus shot number in the C) SF regime and D ) ASE regime [ 63]. Reprint with permission from, Y. D., Wang, X Reitze, D. H. and et. al.. Phys. Rev. Lett. 96, 237401(2006) .( 2006 American Physical Society) PAGE 106 106 A B Figure 4 11. A ) Abso rption Spectra and B ) low excitation PL for four different samples. Arrows in A ) represent emission from the dark states which are not allowed. Each pe ak in the figure is corresponding to e1 h1, e1 l1 and e2 h2 transitions for all samples. The energy separations betw een e1 h1 and e2 h2 in figure A ) are 75.3meV, 77.8meV, 76.8meV and 76.3meV for S324, S322, M507 and M508. A B Figure 4 12. A ) Heav y and light hole transitions as a result of confinement in quantum wells and B) possible allowed electron and hole transitions (solid arrows, 0ehjjj ) and forbidden transitions (dark transition dashed arrows, ehjj ) PAGE 107 107 A B Figure 4 13. A ) White light transmission spectrum for sample M508 and B ) the inferred absorption spectrum obtained using Eq. 3 1. A dark state starts to appear at 11.5T. EGa in A ) indicates the band energy of GaAs PAGE 108 108 A B Figure 4 14. A ) PL emission dependent on magnetic field (He Ne CW laser used with 2.1W/cm2) and B ) peak position for the 00 LL (black squares). Linear fitting (solid red line) of Landau shift from Eq. 4.25 and quadratic fitting (solid blue line) for diamagnetic shift fr om Eq. 4.24 are also shown in B ). The data was taken using sample M508. PAGE 109 109 Figure 4 15. Photoluminescence spectra of M507 at 5K and 0T. He Ne laser and CPA were used for the low (green) and high excitation (red) and for comparison with white light ab sorption, absorption spectrum(blue) from Tungsten lamp is also shown in the figure. All photoluminescence is collected from the center of the samples. PL is shown up to 1.48 eV only for the comparison of InGaAs PL. PL from GaAs (1.49eV) is not shown. A B Figure 4 16. Photolumines cence spectra of M507 at 5.5T A) and 17.5T B ) are taken from the in plane direction. Landau levels from e1 h1 and e1 l1 are also shown in the figure. PAGE 110 110 Figure 4 17. Emission strength (black squares) and linewidth (b lue squares) of the Lorentzian high energy peak for the 00 LL, 11 LL and 22 LL versus laser fluence at 17.5 T for sample M507. Fitting method for Gaussian for lower energy side and Loretzian for higher energy side for 00LL are used and only Lorentzian function is used for 11LL and 22LL. All linewidth analysis is based on Appendix D. PAGE 111 111 A B Fi gure 4 18. 00LL is shown in A ) for white light absorption with Tungsten lamp and PL from low excitation (2.1 w/cm2) with He Ne laser and high excitaion( 3 1010 w/cm2 ) with CPA. Figure B ) display the Landau fan diagrams for white light absorption and high excitation Figure 4 19. Plots of the 2D energies for the Coulomb interaction dominated regime (Eq. 430, dots) and the magnetic field dominated regi me, (Eq. 4 31, solid lines). The energies are plotted in terms the of 3 dimensional Rydberg energy (1 Rydbergy energy is corresponding to 4.23T for In0.2Ga0.8As) PAGE 112 112 Figure 4 20. A ) Absorption spectrum at different magnetic field and B ) time integrated photo luminescence of InGaAs multiple quantum wells. Figure 4 21. Landau fan diagram f or A ) low intensi ty white light absorption and B ) excitation with ultrafast laser (1.53mJ/cm2). Red lines are fitting results based on Eq. 4 32 and Eq. 4 33. Open ci rcles are experimentally obtained data points and filled circles are points for fitting (anti crossing from the dark transition is not used for fitting ( open circles: Section 4.6.2)). Black, blue and yellow circles represent hh1, lh1 and hh2 for each. PAGE 113 113 F igure 4 22. Reduced mass dependence on the Landau level index for A ) the first heavy hole, hh1, B ) the first light hole, lh1 and C ) the second heavy hole, hh2,. Here m0 is the bare free electron mass. Reduced mass for hh1, lh1 and hh2 calculated for 10 LL s, 4LLs and 5LLs for each because as shown Fig.4 21 (B ), hh1, lh1 and hh2 spectrum shows 10 LLs, 4 LLs and 5LLs. PAGE 114 114 CHAPTER 5 TIME RESOLVED ABSORPTION MEASUREMENTS OF INGAAS/GAAS QUANTUM WELLS IN HIGH MAGNETIC FIELDS 5.1 Introduction Ultra fast carrier relaxation dynamics can be best analyzed by photo exciting electrons and holes initially in a non equilibrium distribution using an ultrafast (femtosecond duration) pump pulse and subsequently monitoring their relaxation with a spectrally broa d continuum femtosecond probe pulse. Except for resonance pumping, the usual relaxation dynamics of semiconductor can be measured by introducing the pump laser tuned above the semiconductor band edge and probe laser is the spectral changes around the posit ion which carriers live as shown in Fig. 5 1. The electron and hole population distributions are then monitored by observing absorption changes over the range of the probe spectrum. Many kinds of experimental methods have been employed to study relaxation dynamics and the mechanisms responsible for optical nonlinearities in the absorption spectra in semiconducto rs. Pump probe spectroscopy [ 80], beam dis tortion measurements [ 81], four wave mixing [ 82] and phase conjugation [ 83] are among these techniques. In this chapter we present measurements of relaxation times in In0.2Ga0.8As/GaAs multiple quantum wells using time resolved transient absorption methods. As a basis for interpreting our results, first we introduce some fundamental background on relaxation phase destroying processes and then give an experimental motivation. Experimental result and discussion will be presented in the next. 5.2 Relaxation Process in Semiconductor s The development of femtosecond lasers has allowed the fast dynamics of the opti cal excitations (~ a few tens of femtosecond) in semiconductors to be measured directly in the time domain. Based on the discussion in Chapter 2.4.3, we concentrate again on the relaxation PAGE 115 115 dynamics of electronic excitation in semiconductors and present the basic relaxation mechanism with the help of Fig. 5 2 First, in the collision free or coherent region ( Fig. 5 2( A )), electrons and holes in different k states react independently of each other in response to the exciting light (very fast process PAGE 116 116 still in phase with the exciting laser pulse decay 1 e is called the phase relaxation time 2T (see also Eq. 2 43 and Fig. 53( B )). For the part of polarization wave which is still coherent with the exci ting pulse can be written as [ 64] ) / exp(2 0T t P Pcoh 5 1 In this second stage of relaxation process, the phase is lost before recombination proce ss, which can be written as ([6 4 ] and Chap 2 4 3) 2 2 0 2 1 0) / exp( ) / exp( T t P P T t N N 5 2 where N is the number of de nsity. Eq. 5 2 follows the inequality 1 22 T T 5 3 1 22 T T means there are phase distribution processes other than recombination and the equality is valid if recombination is the only phase destroying process. In t he third stage (intra band relaxation (Fig. 5 2( C )), the main contribution of the scattering process is interaction with phonons. This process increases with increasing lattice temperature (increasing population of phonon modes). Processes involving acoust ic phonon s are shown in the left part of Fig. 5 3 ( A ) in which the initial and the final states are shown to lie in the same level. Process involving the longitudinal optical phonons is shown on the right side of the figure and as shown, the final states l ie in the initial level and also in a different level. The usual intraband relaxation for excess energies above LO ( LO : longitudinal optical phonon frequency) takes place by emission of optical phonons. The rest of the excess energy has to be dissipated by the emission of acoustic phonons. This process is much slower than the emission process of the optical phonon (~sub pico second). PAGE 117 117 Electrons and holes interact first with the lattice via emission of optical phonons and can scatter in k space. This interaction mechanism redistributes the electrons and holes toward their bands extrema and the optical excitation process acts as a local source of new electron hole pairs. Once complete energy relaxation has occurred, the bands are filled up from the band edge to the highest filled k states; the transition is bleached and no more light is absorbed (i.e. the excitation from the probe). This process is sometimes described by time constant 3T (Fig. 5 3 ( B)). When the excess energy of electron hole pair is considerably greater than T kB this process is specially important. After a few scattering process among themselves and with the lattice, the free exciton reaches the thermal distributi on and this state can be described by the exciton temperature eT and by usual Boltzmann statistic If the life time of excitons is long and the coupling to phonons strong, the exictons thermalize with lattice which means lattice tempe rature and exciton temperature in their respective bands become equal. For the excitons created with some excess energy L BT k ( LT : lattice temperature), thermalization means a decrease of the average kinetic energy tow ard a value of L BT k 2 / 3 In contrast if the excitons are generated resonantly of the bottom of the lowest band, the thermalization means increase energy and a spreading in k space. Finally the carriers recombine radiative ly (through emission of a photon) or non radiatively (via emission of acoustic phonons or through two step processes involving an optical phonon and inter band scattering, Fig. 5 2( D )) with a time constant 1T 1T is generally on the nanosecond scale for direct gap semiconductors where radiative recombination is dominant, and PAGE 118 118 can be as long as microseconds or milliseconds for indirect semicond uctors where acoustic phonon [6 4 ]. Fig. 5 4 shows the simplified picture of the relaxation or decay process of electronhole recombination of In0.2 Ga0.8As/GaAs multiple quantum well in a magnetic field. As shown in the figure, phonon, e e and intra Landau level transition occur within a few pico second of time period. Because the applied magnetic fields to 2D semiconductor generates discrete energy levels and the broadening of this each level can be sharp with increasing magnetic field, this is ideal system for studying the dynamics of multi level system With the new facilities a t NHMFL as introduced in Chapter 3, we can observe the Landau level decay process with a resolution of laser pulse (~150fs) 5.3 Motivation The correlated ultrafast electron hole dynamics and lifetimes of a 2D quantum well system in a magnetic field has not been explored to a great extent to date, but it is of significant interest because the quantization effect arising from the external magnetic field in essence creates a quasi 0D system, similar to quantum dots but with an easily variable and controllable degree of confinement via the magnetic field s trength [ 84]. A magnetic field perpendicular to the quantum well layers forces the electrons and holes into the confined orbits; the density of state become a ladder of functions similar to that of quantum dot. Landau and Zeeman splitting lead to externally tunable, nearly equally spaced energy levels (an energy ladder or Landau fan). Electrons and holes in this discrete energy ladder system (0 D system) exhibit a fascinating array of elec tron correlation effects [ 85] and a unique spectroscopic environment which is not achievable in atomic molecular system can be obtained. Various orbital, spin and combined resonances can be spectroscopically studied in Landau quantized systems, especially in two dimensional system, providing insight into low energy quantum dynamics of charge and spin PAGE 119 119 carriers in the frequency domain. However, there hav e been very limited studies [ 80] directly probing carrier dynamics in these ladder systems in the ultrafa st time domain. Recent work has been carried out on the socalled phonon bottleneck which is claimed to inhibit the cooling of carriers in quantum dots when the level separation is no t equal to the phonon energy [ 85 89]. However, as a result of differen t groups using different technique for interband photoluminescence samples, this is controversial and the subject of debates [98 103]. Indeed several mechanisms has been proposed that may bypass the bottleneck, su ch as multiphonon scattering [90] Auger p rocess [ 91], excitonic effects [ 92] a nd defect related processes [93]. In our experiment, we have carried out time transient absorption experiment (TA) at 17.5T corresponding to 46.8meV cyclotron energy. LO phonon energy for InGaAs is 33meV [94] and we tuned the magnetic field high enough (1.8meV higher than LO phonon) to see the decay dynamics from the pure multi level system. 5.4 Experimental Method 5.4.1 Femtosecond Pump Probe Spectroscopy Pump probe spectroscopy is typically employed in order to meas ure the dynamics of exited states of semiconductors and other materials in subpicosecond time domain. The basic idea is illustrated in Fig. 5 5. A relatively strong femtosecond pump pulses is absorbed by the sample and changes its transmittance. A femtos econd probe pulse is used to detect those kinds of changes (Fig. 5 5). We tuned the wavelength corresponding to each Landau level to probe the time transient absorption dynamics from the individual level with Optical parametric amplifier (tunable from 300 nm to 20 m Chapter 3.1.3). The linearly polarized pump and probe pulses are usually orthogonally polarized to make isolation of the probe pulse from the excitation pump PAGE 120 120 pulse. A delay line with variable length optic al path difference between pump and probe beams are then focused to the same spot on the sample. Differential transmission spectra (DTS) yield sensitive and reliable information on the semiconductor transmission spectra. The DTS signal is defined as the di fference between the probe transmission with and without the pump, 2 2 2) ( ) ( ) (off pump off pump on pump P P PE E E DTS 5 4 where ) (PE is the amplitude of the probe after going through the sample. The DST described by Eq. 5 4 depends on the time delay Pt between pump and probe. In the experiment, the DTS is measured at various time delays. In this way, one can obtain transmittance changes with the resolution limited by the probe pulse duration. 5.4.2 Experimental Setup In0.2Ga0.8As samples are used for time resolved absorption measurement The size of the sample is 3 3mm 2 and orientation of crystal is c plane (c axis perpendicular to the plane). In0.2Ga0.8As multiple quantum well were grown by MBE as refer to Figure 4.4. For the excitatio n of carriers, we used 150fs CPA laser and we restricted the maximum energy of pumping CPA laser as 7mJ/cm 2 to avoid sample damage. The pump pulse from CPA was sent through a computer controlled optical delay line (Maximum delay: 1ns, Newport) and focus ed into the sample at normal incidence In all cases, In0.2Ga0.8As samples were exited 800nm CPA laser. The schematic diagram of pump probe experiment is shown at Fig. 5 1. To probe each Landau level we varied the OPA center of wavelength from 850nm to 950n m which correspond s to 1.46eV and 1.31eV for each and covers at least three Landau levels. The beam was focused to a spot size of 0.5mm on the sample through a 1m focal length PAGE 121 1 21 of lens and two beam paths from pump CPA probe OPA beam are collinear and we separated the OPA signal from the pump laser with spectrometer or monochrometor. We monitored the OPA fluctuation directly by using beam splitter in the OPA beam path and used it for the probe fluctuation normalization for our final result. Spatial overlappin g which is crucial for our pump probe experiment is checked by introducing the pump and probe beam at long range distance and for the temporal overlapping to find time zero, we cross correlated with BBO crystal. In any pump probe experiment, it is very important to ensure that special walk off between the pump and the probe beams did not occur during the measurement. Both sources of walk off in measurement were corrected by careful alignment and the signal monitoring procedure. In order to ensure parallel b eams into and out of the stepper stage, the outgoing beam was projected onto the iris approximately 10m apart from the stage. As stepper stage translated, small angles between the incoming and outgoing beams is realigned until no spot motion occurred. As s hown in Fig. 5 6, we used monochrometor (Optometrics) to select the probe signal from the pump (800nm) and this signal is transferred to SR560 Stanford Research preamplifier for band pass filtering. To increase the signal noise ratio, we chopped the pump pulse with 271Hz and used this frequency as external triggering for 7265 DSP Signal Recovery Lockin amplifier with 1s time constant. With the same time constant and band pass filtering, another Lock in amplifier and preamplifier are used to collect the OPA reference signal for signal fluctuation normalization of our final result. Fig. 5 7 shows the different way of the signal collection method. The two signals transferred through fibers for the reference beam of OPA fluctuation and the signal through out o f the sample are connected together to the specially made fiber coupler with 2mm separation which is attached to 0.75m McPHERSON spectrometer. The two different light signals through PAGE 122 122 this fiber coupler propagate onto the different location of Princeton ins truments 1024 256 2 dimensional CCD array. We used the upper part of 2D array as the OPA fluctuation reference and the bottom half as the absorption signal. In the same as using the Lock in amplifier and preamplifier, we normalized our final result with th e OPA reference. With spectrometer, we could measure the wide range of spectrum of time relaxation dynamic simultaneously such as between Landau levels. 5.5 Experimental Results and Discussion The application of a strong magnetic field perpendicular to the plane of a semiconductor quantum well generates a series of Landau levels (LL) in which the zero field step like2 dimensional density of states transform into like 0 dimensional density of states as shown the spectra in the previous chapter (Fig. 4 16) as magnetic field increased. Intense ultrafast laser excitation results in a highly quantized dense (Ne h~1013cm2) electron hole plasma tightly confined in energy. The presence of many Landau levels results in complicated relaxation d ynamics which can occur through intra or inter tran sitions. Earlier experiments [ 84, 951 05] have probed the ultrafast dynamics of magnetoexcitons in LLs at much lower magnetic fields and most of experiments have been carried out for doped semiconductor and probe only first Landau level. Our measurement is for multi level systems and probed each Landau level by tuning the probe wavelength to cover at least 3 Landau levels. In this section, first we introduce the result and the discussion of time transien t absorption measurement at 0T and then go over TA results at 17.5 Tesla. 5.5.1 Time Transient Absorption Measurement at 0 Tesla We carried out the time transient absorption measurement at 0T for the reference to compare with TA results at 17.5T. To probe the TA at photoluminescence peak, we tuned the PAGE 123 123 probe wave length as shown in Fig. 5 8 ( A ). As shown in the figure, PL spectrum shifts to the lower energy side relative to the transmission peak with Tungsten lamp because of bandgap renormalization as discus sed in chapter 4.6.4. Red shift of the PL energy relative to the transmission increases along Landau level index increase as shown in Fig. 58 ( B ) (D ). We tuned the wavelength of the probe with OPA at the Landau PL peak positions for 17.5T for TA of LL at 17.5T (next section). The first transient absorption measurement is shown in Fig. 5 9 displaying absorption dynamics proving at the lowest heavy hole (e1 h1) level in the absence of magnetic field with 0.5mJ/cm2 excitation power. Probe pulse intensity is 0.08mW (0.045mJ/cm2) which is much less than the carrier excitation density to reduce the effect of carrier excitation from the probe pulse itself. The sharp increase of signal starts at 0 ps time delay and if we defined the decay time as the half of the m aximum peak, it is about 360 ps. We pumped the carriers more than 150meV above the bandgap minimum (Fig. 5 1), so this data shows that the hot carriers relaxed down to the lowest band and decay with 360ps time scale. A relatively long life time of e1 h1 tr ansition can be explained with the e h plasma as discussed in Chapter 4.7.5. With less binding energy (effectively zero binding energy), decay rate of e1 h1 transition will be reduced because of less Coulomb interaction to create recombination of e1 h1. Be cause the probe wavelength was tuned to the first heavy hole transition energy, Fig. 5 9 is directly related with inter band (e1 h1) transition of the first heavy hole. Maximum PL doesnt mean that carriers stay at that energy level. Complicate relaxation and interaction dynamics can shift the PL peaks from the original energy which the carriers sit on. We scanned the probe pulse to find the position of maximum pump probe signal. Fig 5 10 shows this 2D span of pump probe measurement based on the experiment al setup of Fig. 5 7. We PAGE 124 124 used the spectrometer to span the all the range of probe pulse signal change and the long decay time at PL maximum position also shown with the same time constant with Fig. 5 9. Based on this 2D pump probe, time transient absorpti on signal at zero time delay is shown in Fig. 5 11. High excitation PL (1.326eV, red) is 1.4meV shifted from the white light absorption (1.327eV, green) by bandgap renormalization. Interestingly, the maximum TA (1.351eV, blue) is 23.4meV higher than the ab sorption peak and shows very broad range of signal (28.2meV). This result presents that the position of carriers with high excitation density is not the same as the both of PL maximum and the assumption peak because of filling of carriers from the band min imum for high excitation power. The very broad range of signal in TA represents hot carriers around the maximum TA signal. From the Boltzmann distribution function, T k EBAe f/ (where A and Bk represen t amplitude and Boltzmann constant), we can induce the temperature of hot carriers (Fig. 5 12) and results in 30.4meV which is corresponding to 350K of hot carrier temperature (at room temperature, energy is ~26meV (T=300K)) with the smallest excitation power we used, ~0.047mJ/cm2 Therefore the use of femtosecond laser having high peak intensity generates very hot carriers (>350K) even with small power. If we assume the carriers at the first heavy hole lost their energy with phonon interaction and radiate with less energy, the exchange energy result in the creation of phonons The separation between PL signal and TA signal in Fig. 5 11 is about 25meV and this is comparable to LO phonon energy of InAs (28meV, [ 1 06] and for GaAs, 33.4meV [ 106]). Fig 5 13 show s the transient absorption with various time delays. As shown in the figure, at 0 time delay we can see the largest signal. One more peak around at 1.375 appears until 100ps and this signal would come from dark states from the forbidden transition and afte r 200ps this second signal disappears. This signal from dark states is shown in Fig. 5 10 also. As increasing PAGE 125 125 time delay signal, the maximum TA peak moves to lower energy side representing the reduction of carrier filling from the inter level transition.( i.e., relaxation starts from the hot carriers(higher energy side of TA) and relax down to lower energy). 5.5.2 Time Transient Absorption Measurement at 17.5 Tesla Actual our main concern is carrier relaxation dynamics of multi level system. To create multi level system, we increased the magnetic field up to 17.5T and create 0 dimensional multi level system. First we tuned OPA wavelength at the Landau level of low excitation density (white light absorption peak with Tungsten lamp) as shown in Fig. 5 8 ( B) (D ). Fig. 5 14 ( A ) (C) display the dynamics for B=17.5T and 2.2mJ/cm2 excitation power with probe tuned to the low excitation density (white light) absorption peak for 00, 11 and 22LLs. As noted in the previous section, at high excitation densities, band ga p renormalization red shifts the energy levels with respect to the low density case, thus the probe does not directly interrogate the carrier occupancy. When the field is applied, larger transmission increases are observed and increasingly fast relaxation times occurs at progressively higher LLs. In addition, the residual long lived transmission increase is dismissed relative to the peak transmission at small time delay. Similar data is shown in Fig. 5 15, but probe was tuned to the energy where the maximum PL emission is observed. The data is qualifiedly similar, with a few striking differences, however. At high magnetic fields, the transmission of signal exhibits an abrupt reduction and an abrupt reduction occurs at earlier times for higher LLs (90ps, 00LL ; 75Ps, 11LL; 50Ps, 22LL). In addition the transmission time from high to low transmission becomes increasingly more rapid as the LL index increases. After the maximum rise time of transient absorption, the signal doesnt decay for 80ps and 70ps for 22LL a nd 33LL (plateau in Fig. 515 (b) and (c)) and decay rate increase fast right after the plateau. This non exponential relaxation type is very unique result for PAGE 126 126 our pump probe experiment and should be carefully analyzed with suitable models including the de crease of the decay time for higher LLs. G. Wang, et. al [107] carried out time resolved PL measurement for InGaAs/GaAs quantum well and quantum dot and they observed the decay time increase by reducing dimensionality. However, as shown in Fig. 5 16, we ha ve the opposite data. Fig. 5 16 displays the decay time for each Landau level at 17.5T and the decay time of the first heavy hole at 0T also is shown in the same plot for comparison. As discussed in Chapter 4.7.4, the finite magnetic field can mediate elec tron hole interaction and transform plasma state into the excitonic bound state in finite magnetic fields. Therefore the decrease of decay time between 0T and 00LL at 17.5T can be interpreted as the transition from the plasma to excitonic state mediated by finite magnetic fields and increase recombination rate of electron and hole excitons. If we consider quantum well with large separation, the decrease of the well separation results in the large energy difference between adjacent energy levels, i.e decrea se of density of states. In quantum well system with applying magnetic fields perpendicular to the well direction, motions of electron and hole is confined in all three direction. However, the function of magnetic fields increases the density of states pro portional to the amount of magnetic fields also (density of state~ c eH ). Therefore the magnetic field reduces the dimensionality of 2D well system and increase the density of states around at the specific discrete level as well as shown in Fig. 5 17. Therefore finite magnetic fields mediate electron and hole interaction and increase scattering rate by increasing density of state. These facts can lead the shorter decay time for 00LL than the first heavy hole time transient absorption and g ive a different result from G. Wang, et. al [154]. To discuss about the decay time of higher LLs, we need to remind of Eq. 4 32 which was used to fit the Landau fan diagram for higher LLs (Fig. 4 21) PAGE 127 127 2 / 11 2 3 1 2 n n Ec c 4 31 The second terms in Eq. 4 31 represents small perturbation from the Coulomb interaction and c eHc2 As shown Fig. 4 22, reduced mass increases along increasing Landau level index and so the second terms of Eq. 4 31 (perturbation from the coulomb interaction ) is decreased along increase of LL index. Hence, increasing LL index gives less interaction from Coulomb force and so results in relatively less inter LL transition rate or longer life time along increasing LL index However, as shown in Fig. 5 16 decay time is shorter in increasing Landau level index. In high magnetic field, the contribution of the second term of Eq. 4 31 is relatively smaller than the first Landau term and so we can not simply related our decay data for higher LL with only the second te rm of the equation. If we focus the first Landau term in Eq. 4 31, we can notice energy increase for higher LL, in other word, confinement effect by magnetic fiels for higher LL is smaller than the lower LLs. This decrease of confinement effect of electron and hole gives a result of the increase of scattering rate making more inter and intra landau level transition. Therefore we can say the origin of the decrease of LL decay time for higher LL is from the decrease of the confinement effect by the magnetic f ields. In summarizing, reduced dimensionality and increase of density of state by external magnetic fields induces the decrease of decay time for higher LL and give a shorter decay time for LL system compared with usual quantum dot system. For the spectral ly resolved time transient absorption measurement, we carried out pump probe measurement by scanning the probe pulse as shown in Fig. 5 18 and cover 3 LLs at 17.5T. Fig. 5 19 displays T/T at 0 ps time delay and white light absorption and high excitation P L emission for the comparison. As shown in figure, transmission signals for each LL are placed PAGE 128 128 between white light absorption and PL. If we compare the signal width at 0 ps time delay, we have 9meV, 10meV and 17.3meV for 00LL, 11LL and 22LL at 17.5T and th is value is smaller than that of 28.2meV bandwidth of 0T (Fig. 5 11). By increasing magnetic field, we can increase number of state of each LL by factor of c eH and this is corresponding to carrier density of 8.45 1011/cm2 at 17.5T. Com paring with the step like energy band (continuous energy band) of 0T, magnetic field generate like 0 dimensional density of states at high magnetic fields. This dense energy states at the give level creates a relatively sharp signal bandwidth. One more interesting feature in TA spectrum is that the second maximum appears at the lower energy side of the first maximum of each LL. Some of groups observed this kind of satellite at around multiple integer of phonon energy below the main PL spectrum [ 106]. However in our sample phonon ener gy of InGaAs is about 33meV [ 94] and the second signal appear 13meV (for 00LL and 11LL) and 20meV (for 22LL) below the main peaks. Therefore we couldnt simply connect with carrier phonon interaction and ne ed a different mechanism to explain. Fig. 5 20 display the maximum peak of high excitation PL(red), white light absorption(green), TA maxima(blue) and the position of the second maxima(black star). Except 0T, TA maxima are located between PL peaks and the white light absorption peaks. Again position of TA (higher than the white light absorption) at 0T shows the filling of the plasma state and presence of hot carriers at the initial time. However for LLs at 17.5T, TA are located at lower energy than that of white light absorption from the band gap renormalization. This plot clearly shows the position of carrier (TA signal) is not the same as PL or white light absorption peaks. Figure 5 20 and Figure 521 shows the TA spectrum change at various time delays for sample M507 and M508. As shown in Fig. 520,the maximum peak of 00LL move toward to the PAGE 129 129 lower energy side but for 11LL and 22LL, TA signal is rather constant motion or has small deviation to the higher energy side. Increasing time delay gives a results of less filling of carriers in 00LL because the number of carriers is used up for the inter and intra LL transition but for the higher LL, the decrease of carrier density results in less band gap renormalization and the TA signal move to the peak of low exci tation peaks(white light absorption peaks). To understand scattering mechanism of GaAs, Y. S. Sun and C. J. Stanton [ 109] numerically solved the time dependent coupled Boltzmann equation with the method of computational mote Carlo approach. To describe th e complicate decay dynamics of quantum well in high magnetic fields, we need to develop the physical model of multiple N level system and related TA signal to time resolved photoluminescence. With the use of streak camera, we carried out the experiment to see the time evolution of photoluminescence and will be discussed in the following chapter. 5.6 Summary We discussed the time transient absorption dynamics at 0T and 17.5T. By comparing the first heavy hole decay dynamics at 0T with 00LL at 17.5T, we desc ribe the effect of magnetic fields. Applying magnetic fields reduce the dimensionality continuously from 2dimensional quantum well to 0 D quantum dot system and increase density of state proportional to applying magnetic fields. Shorter decay at higher LL can be explained with less confinement effect to increase scattering rate. Time evolution of the band filling at 00LL and band gap renormalization for higher LL is discussed in this chapter. The abrupt TA increase at initial time delay and pleatue at highe r LL should be explained with the proper physical model describing multi level system. To correlate with emission dynamics, we will discuss time resolved photoluminescence in the next chapter. PAGE 130 130 Figure 5 1. Schematic diagram of pump probe experiment of 2D quantum well in magnetic fields. A series of LLs in conduction and valance band are shown in the figure. Pump laser is tuned 150meV above the band edge and probe light is tuned at the emission wavelengths of allowed transition. 800nm central wavelength of pump pulse excites carriers in quantum well system and probe pulse(OPA) is tuned at the selective wavelengths which are carrier states Figure 5 2. Relaxation processes and time constants for each process. PAGE 131 131 A B Figure 5 3. A ) Scattering of exciton by phonons. Acoustic (left) and LO (right) phonon scattering. B ) Schematic drawing of the dispersion relation of excitons and of the main time constants Figure 5 4. Simplified picture of relaxation process in 2D semic onductors in magnetic fields: A ) laser excitation with fem tosecond (very fast process), B ) phonon and electron el ectron scattering (~1ps) C ) intra Landau level transition (~5 ps) and D ) inter Landau level transition(~100ps) PAGE 132 132 Figure 5 5. Schematic of femtosec ond spectroscopy. Pump pulse and probe pulse are delayed by tP with respect to each other. The transmission of the probe pulse is measured with and without the presence of the pump to obtain DTS. Figure 5 6. Schematic diagram of pump probe experiment with Lock in. The chopping frequency of the pump is 271Hz and the dT/T is fluctuation normalized by the OPA reference signal. PAGE 133 133 Figure 5 7. Schematic diagram of the pump probe experiment with McPHERSON spectrometer, Two inset show the fiber coupler and 2 dimensional image on CCD camera. Figure 5 8. Spectra of transmission with Tungsten lamp(Black), PL with C PA (Red) and probe (blue) for A) 0T and B,C and D ) 17.5T. Probe pulses are tuned at transmission spectra minimum assuming carriers live in transmission peaks (or absorption peak). PAGE 134 134 Figure 5 9 Normalized transient absorption at 0T. Probe was tuned at the peak of Tungsten lamp white light transmission minimum (Sample M507). Figure 5 10. Two dimensional pump probe measurement at 0T (refer t o the setting, Fig. 5 7). Exictation power is 0.8mJ/cm2 and M507 was used for the measurement PAGE 135 135 Figure 5 11. T/T(blue) at 0 time delay with high excitation PL spectrum(red) and white light absorption (green) based on Fig. 5 10. Figure 5 12. P hotoluminescence with 0.047mJ/ cm2 excitation power at 0T and exponential fitting. Fitting function follows the Boltzmann distribution function, T k EBAe f/ and energy, T kB is about 30.4meV( corresponding to 350K) from the fitting curve. PAGE 136 136 Figure 5 13. Transient absorption signals at various time delays. PAGE 137 137 Figure 5 14. Normalized transient absorption at 17.5T. Probe was tuned at white light absorption peak of each LL (Sample M507 and excitation power=2.2mJ/cm2). PAGE 138 138 Figure 5 15. Normalized transient absorption at 17.5T. Probe was tuned at PL peaks of each LL. (Sample M507 and excitation power=2.2mJ/cm2). Figure 5 16. Decay time for each Landau levels. We used TA data for PL maximum. 0T represent first heavy hole at 0T and 00LL, 11LL and 22LL measured at corresponding LL at 17.5T (Sample M507) PAGE 139 139 A B Figure 5 17. The function of reduced dimensionality A ): By reducing the separation of confinement, separation between adjacent energy level increases and result in re duction of density of state. B ): Function of applying magnetic files in quantum well reduces dimensionality and increase the density of state. Figure 5 18. Two dimensional pump probe measurement at 17.5T (refer to the setting, Fig. 5 7). Excitation po wer is 4.6mJ/cm2 and M507 was used for the measurement PAGE 140 140 Figure 5 19. T/T(blue) at 0 time delay with high excitation PL spectrum(4.6mJ/cm2 red) and white light absorption (green) based on Fig. 5 18. There are the second maximum around the first maximum of T/T and the arrows represent position of this second maximum Figure 5 20. Maximum PL peaks(red), white light absorption peaks(green) and time transient absorption peak(blue).0T represent first heavy hole at 0T and 00LL, 11LL and 22LL measured a t corresponding LL at 17.5T (Sample M507). Black star is the position of the second peaks around LL as shown in Fig. 5 19. PAGE 141 141 Figure 5 21. Transient absorption signals at various time delays. Red arrow displays for guiding of main peaks (from Fig. 518). PAGE 142 142 Figure 5 22. Transient absorption signals for M508 at various time delays (excitation power=4.8mJ/cm2. Red arrow displays for guiding of main peaks and red dot and green dot for guiding of LL peaks for PL and white light absorption. Black arrows repre sent the position of the second peak. High excitation PL(solid red) and white light absorption(solid green) are plotted for the comparison. PAGE 143 143 CHAPTER 6 TIME RESOLVED CHARACTERIZATION OF ULTRASHORT PHOTOLUMINESCENCE BURSTS: DIRECT OBSERVATION OF INTER LANDAU LEVEL RECOMBINATION 6.1 Introduction In Chapter 5, we presented femtosecond transient absorption (TA) measurements of Landau levels created in two dimensional In0.2Ga0.8As/GaAs multiple quantum well in high magnetic fields. To understand time dynamics of each Landau level, the experiments related with the radiative mechanism (or emission mechanism) should be carried out and need to compare and correlate this to TA measurements. The name photoluminescence is emission process taking place in carriers in excess of thermal equilibrium values which is created by optical absorption. In this chapter, we investigate high density Landau level dynamics in a complementary way, by spectrally and temporally characterizing the emitted photoluminescence. F or better understanding of multilevel system we present basic background of photoluminescence of 2 level system first and then continue to talk about the investigation of the radiative recombination process in a semiconductor quantum well in high magnetic fields, the rates of processes and also the nature of the emission spectra from the experimental observation. Finally, we correlate time resolved photoluminescence with transient absorption measurement (TA) in Chapter 5. In semiconductors, absorption of vi sible or near infrared photons generates high energy carriers in well in excess of their thermal equilibrium values. The opposite process of absorption is emission. As noted in Chapter 5, semiconductors excited electrons ultimately return to their equili brium state through the emission of a photon upon recombination with a hole. In direct gap semiconconductors, this is a direct recombination whereas in indirect semiconductors, phonon assisted recombination occurs. Fig. 6 1 illustrates different kind of em ission processes. PAGE 144 144 In the k space of direct gap semiconductors, the optical transition corresponding to electron hole recombination is vertical, and the photon momentum involved in the transition is negligible. For indirect gap semiconductors, phonons or ot her momentum conserving agents are required during the process. The simplest process involving the recombination for direct and indirect gap semiconductors are shown in Fig 6 1 ( A ) and ( B). A conceptual view of the various kinds of recombination processes via different channels is given in Fig. 6 1 ( C). Fig. 6 1 simplifies the physical picture somewhat, as carriers in the bands are depicted as free and there is no correlation between electron and hole. In real systems, the Coulomb force mediates the elect ron and hole interactions giving rise to excitons. When the excitation density is increased, more pairing of carriers can be generated an d such as excitonic molecules [ 24], Bose co ndensed excitons [ 1 08 110, 11 3 144] and electron hole droplet [ 24] give rise to charact eristic emission spectra. There is a fundamental difference between absorption and emission in semiconductors. In absorption, all the states in the bands are related while in emission only a very narrow band of states about a few T kB fr om the band edges containing the thermalized (e xcited) carriers are involved [ 35]. The emission spectra in semiconductors are therefore very narrow. In the following section, we introduce the basic background of photoluminescence of two level system for the better understanding of the rest of this chapter. 6.2 Photoluminescence of Simple 2 Level System The optical spontaneous emission can not be predicted when the electromagnetic field is described classically. However, one can relate its intensity to that o f absorption by means of the Einstein coefficients (Chap. 4.2.1). In presenting the theoretical considerations on the lineshape of recombination processes in semiconductor, the information one can extract from the PAGE 145 145 photoluminescence concerning the energy le vels of a given heterostructure is often very scarce and depends to a large extend on the prior knowl edge of that heterostructure [ 73]. Actual 2 dimensional quantum well can be described as 2 level system consisting of conduction and valence band as a simp lest picture. Because Landau level splitting is enhanced by applied magnetic fields, we could approximate multilevels of Landau energy ladder to simple two level system if the Landau splitting is large enough. Therefore consideration of simple 2 level syst em gives us some basic intuition for multi level system. For absorption, a photon is absorbed an electron from the level 1 to the level 2 with 1 2E E On the other hand, the emission involves the transition 2 1 with 2 1E E and photon with energy 2 1E E is emitted during the process. Such a process is possible when the system is initially exited at state 1 Thus it is not e quilibrium and as a result, optical emission processes occur. In addition to the radiative channel, the nonradiative relaxation channel is in competition for the energy relaxation dynamics describe in the previous chapter. Therefore, emission is different from absorption because instead of being 100% efficient (in absorption one photon creates one electron hole pair), emission is only one possible mechanism for the energy relaxation. To consider this nonradiative channel, lets think about 2 level system as shown in Fig. 6 2. System is excited from level 1 to the level above 2 as shown in the figure and we consider photoluminescence signal of level 2 to 1 The parameters 21 and 21T denote the time constants for nonradiative channel and radiative channel from level 2 to 1 respectively. Ignoring the effect of the final state population (above level 2 ) and considering only level 2 and 1 we can write the rate equation as 21 2 21 2 0 2T n n n dt dnP 6 1 PAGE 146 146 In the steady state, 02 dt dn 21 21 21 0 2/ 1 T T n nP 6 2 The photoluminescence signal at energies 1 2 21E E is proportional to 2n From Eq. 6 1 and Eq.62, if 21 21T almost all population of level 2 is used to decay radiatively t o the level 1 and thus does contribute to the emission which occur at 21 Lets now consider the transient phenomena (time resolved photoluminescence). Assuming the pump rate of level 2 is d escribed by a function ) ( t g ( ) ( t g vanishes at 0 t ), we obtain tdt T t t g T t t n' 2 2 2) / exp( ) ( ) / exp( ) ( 6 3 where 1 21 1 21 1 2 T T In the case of fast pumping, g(t) can be represented by a delta function, ) (0t n and Eq. 6 3 can be written ) / exp( ) ( ) (2 0 2T t t n t n 6 4 where ) ( t is the Heaviside step function. The population ) (2t n increases first with time and then decreases as shown at Fig. 6 3. The decay time, 2T between state 2 and 1 involves both radiative and nonradiative rates. Therefore, any transient photoluminescence experiment should be cautiously interpreted. 6.3 Motivatio n Two dimensional quantum well systems in high magnetic fields mimic zero dimensional systems because applied magnetic fields squeeze in plane motion of electrons and holes. LLs provide an evenly spaced energy ladder in which the spacing between the states can be increased PAGE 147 147 or decreased by increasing or decreasing the applied magnetic field. However, research on two dimensional quantum well in high magnetic fields is also very limited [1 11]. IIIV compound semiconductors possess many interesting properties and have attracted much attention. Much research has been devoted on exciton dy namics and many body e ffects [1247] especially on zero dimensional systems. However, experiments on the zero dimensional system relaxation dynamics are hard to achieve because inhomogeneous of different size of quantum dots make obscure signal and hard to analyze. The progress made in recent years at the National High Magnetic Field Laboratory makes it possible to carry out experiments on recombination dynamics of carriers in high magnetic field (up to 31 T). In complementary fashion to the time resolved transient absorption measurements presented in the previous chapters, probing the time resolved photoluminescence allows for a better understanding of the relaxation dynamics of Landau Levels In the rest of this chapter, we present the results of time resolved photoluminescence experiments of In0.2Ga0.8As/GaAs QWs in high magnetic fields. We examine the emission time delay and time structure for each Landau level transition. Th e excess carriers in each level are characterized by a lifetime. The relationship between the lifetime and recombination rate will be developed. This relationship will be used to predict the emission spectra. 6.4 Experimental Methods 6.4.1 Installation of Streak Camera at NHMFL To measure the photoluminescence from In0.2Ga0.8As/GaAs semiconductor quantum wells, we have developed a facility for measuring time resolved photoluminescence in high magnetic fields using a Hamamatsu streak camera (Fig. 6 4). Fig. 6 4 shows the operational principle of the streak tube at the heart of the streak camera. The light collected from the sample passes through a slit located at the entrance of the streak camera and propagates to the photocathode PAGE 148 148 through imaging optics. The incident light on the photocathode is converted into the electrons proportional to the incident light intensity. The time structure of the light pulse is linearly mapped onto the time structure of the electron bunch. The electrons then pass through a pair of sweep electrodes, where they are accelerated and impinge against a phosphor screen. The sweep electrodes convert the time structure of the electron bunch to a spatial structure through the application of an ultrafast linear voltage ramp applied to the sweep electrodes. To obtain the optimal time resolution in our experiments, we had to overcome three main problems: 1 Increase the photoluminescence signal: Minimum sensitivity of streak camera is about 0.72nW/cm2 for 150fs and 1Khz repetition rate of laser. In case of photoluminescence measurement, time for electron and hole recombination is relatively broader (~a few hundred of picosecond) than laser pulse itself (150fs). As shown in Fig. 6 4, this broadening of PL signal spreads out the photo excited elect rons in phosphor screen which result in big signal noise ratio. 2 Reduce and compensate the dispersion of fiber: PL signal is transferred by multimode fiber and it has broad bandwidth (77meV at 0T (Fig. 4 11)). Group velocity dispersion from this kind of bro ad bandwidth through the fiber should be compensated after measurement with correct time relation. 3 Reduce the magnetic field effect which strongly affects the time sweep electrode system in The Streak Camera: PAGE 149 149 Motion of electrons from the photocathode can be changed by external magnetic fields. Especially for the use of streak camera near 17T super conducting magnet and 31T resistive magnet, the special magnetic field shielding material should be prepared. To maximize the photoluminescence signal from the s ample in the center of the magnet bore, we installed a 0.6 mm core diameter, 5m long graded index multimode fiber (obtained from Mitsubishi) which allows for simultaneous propagation of many different wavelengths with relatively low dispersion. The fiber w as carefully inserted in the cryostat and butted up against the sapphire substrate that serves as the sample mount as shown in Fig. 6 5. The minimum achievable time resolution of the streak camera is 2ps as shown in Fig. 6 6 and for this measurement, we us ed 150fs of laser through free space. In real experimental situation for time resolved PL measurement, we collect signal through multimode fiber as shown in Fig. 6 5 and hence need to consider dispersion from the multimode fiber. To determine the fiber dis persion, we tuned OPA to get different group velocity dispersion from different wavelength. We specially overlapped two laser pulses at the same time with BBO crystal to decide zero timing before 7m long multimode fiber(one from CPA(800nm and 150fs) and t he other from OPA(800nm~1200nm and 150fs)). These two pulses have no time difference before going into the fiber but after transferring through fiber, fiber dispersion generates different group velocities at different wavelengths of pulses and this relativ e dispersion is depicted in Fig. 6 7. Fig. 6 7 was used to compensate effect of fiber dispersion after time resolved photoluminescence measurement. To minimize the influence of the magnetic field, mu metal shielding was used. The magnetic strength on the top of the magnet is approximately 50 80 Gauss. The sweep electrodes PAGE 150 150 are strongly affected by magnetic field. The magnetic field strength is large enough to provide an addition deflection of the electron bunch via the Lorentz force which distorts the ele ctron bunch and results in a potentially significantly distorted measurement. The left panel of Fig. 6 8 shows the streak camera image tilting with 50G of magnetic field. To minimize this coupling, we designed and ordered a special mu metal shield for the streak camera (Mu Shield Co.) to prevent the magnetic flux from penetrating to the streak tube. Inside of the mu metal box, the magnetic field was measured to be less than 1 Gauss and we found no image distortion or tilting. The right panel of Fig. 6 8 shows the image when the mu metal cover is present; the tilt is essentially removed. To minimize the fiber dispersion we located the Streak Camera system close the magnet to reduce the fiber length but it increased the magnetic field effect and mu metal was saturated. We found 5m long fiber is the best length in considering the above including the physical size of the magnet. 6.4.2 Experimental Methods Similar to the transient absorption experiments described earlier, we excite the In0.2Ga0.8As/GaAs quantum wells using the CPA laser operating at a1 kHz repetition rate with 150 fs pulse durations and 800 nm center wavelength. Fig. 69 shows the schematic diagram of our streak camera measurement setup. The laser beam is directed on the sample through the optical window attached on the bottom of 17.5T superconducing magnet. To control the pump power, we mounted a neutral density filter wheel to a stepper motor. PL from the sample was collected from the center and the edge of the sample with the graded index multi mode fiber described above and shown in Fig. 65. Both center and edge collections geometry of PL are shown in Fig. 6 5. The graded index multimode fiber was attached to monochrometer ( Acton SpectraPro 2300i), with entrance slit and two exit slits and PAGE 151 151 thi s two slits width are fixed for all measurement for consistent bandwidth resolution (5nm). A nitrogen cooled CCD camera (Roper Scientific) was mounted to the side exit of the spectrometer and served as a monitor for the peak wavelength for every streak cam era measurement. The back exit slit was used as the beam line to the streak camera. For the proper beam focusing at the input slit of spectrometer, we mounted 7.5cm and 20cm focal length of lens system which can collimates and refocus the beam from the fib er to the input slit of the spectrometer. The diameter of these two lenses was selected to be 2 to match the f number of the spectrometer and to maximize the light collection from the fiber. We used a 600 lines/mm grating set to a 750 nm center wavelength to collect the peak PL emission to the streak camera. The light passed through 7.5 cm and 10 cm focal length lenses after the exit slit of spectrometer; these output optics collimated and refocused the beam into the entrance slit of streak camera. The spa tial size of the electron beam bunch emitted from the photocathode defines the time resolution. We selected a streak camera slit width of typically 40 m to maximumize the PL signal and achieve the minimum time resolution. Including the effects of dispers ion from the 5m long fiber, the spectrometer and the imaging, we achieved a time resolution of 2527ps. With the streak camera set to the highest temporal resolution, a small drift of the signal to earlier times appears, especially when the streak camera was first started. To ensure that the drift did not lead to broadening, we warmed up streak camera for more than one hour. After one hour, the image drift was measured to be less than 5 ps per 10 minutes. Since integration times in our measurements are ty pically 50 second, this drift was not a problem. For every measurement, we also measured the 800 nm excitation laser to determine the zero time delay and correct timing based on fiber dispersion shown in Fig. 6 7. PAGE 152 152 The streak camera is sensitive enough to detect stray light from the background illumination. To minimize this, we covered the beam path, spectrometer and the streak camera with a black, absorbing cloth and recorded the background for the every measurement. Background subtraction was used to obt ain the final measurement. To get a clean signal, we repeated the measurement more than 5 times with a 10 s exposure time. 6.5 Experimental Results and Discussion In the measurements presented below, the PL emissions were obtained both as a function of tim e and wavelength (to probe specific Landau levels). Fig. 6 10 shows the normalized PL emission spectra of the InGaAs/GaAs multiple quantum well M508 sample at 0T and 17.5T when pumped by 7.13mJ/cm2 excitation intensity. In this spectra, the signal is inte grated in time form Princeton instruments 1024 256 2 dimensional CCD array mounted on 0.75m McPHERSON spectrometer. The linewidth of the peaks are 11meV for 0T, and 3meV for the 00, 11 and 22LLs at 17.5T (obtained from the Gaussian and Lorentzian fitting m ethod discussed in Appendix D). The 3 meV linewidths at high magnetic field correspond to the amplified spontaneous emission region (as shown in Fig. 4 17). The diamagnetic shift toward higher energies side is evident as the magnetic field is increased and signal at 17.5 Tesla is 10 times stronger than that of 0 Tesla because of increase of density of state at each Landau level while increase of magnetic fields (~ c eH ) and signal amplification. Fig. 6 11 shows the measured time resolved photoluminescence for sample M508 at 0Tesla and a temperature of 5 K at 1.317 eV corresponding to the e1h1 transition. For the measurement, we used the signal from monochrometer ( Acton SpectraPro 2300i) with streak camera as shown in Fig. 6 9 PAGE 153 153 As discussed in Chapter 6 2, in simple two level system, number of carriers at upper level decay exponentially with simple delta function like pumping excitation. In real system such as InGaAs semiconductor, we need to consider multi energy levels. In simplest assu mption and just to focus on one specific level, we can assume that the carriers in the specific energy level decay exponentially and are exponentially populated from upper levels. To extract the time constants, we use an exponential fitting for the rise an d fall times, we used two time constants for decay and rise time as following ) ( ) () / () / (r t dbe ae t It 6 5 where a b d and r are fitting parameters( a and b are amplitude of each decay and d and r are decay and rise time for each). Figure 6 11 displays the fitting results of the time resolved photoluminescence at three different exc itation intensities of sample 508 at 0 Tesla and e1 h1 transition. As shown in the figure, relatively similar rise (232ps) and decay time (233ps) are observed for all three powers and relative change of a and b i n Eq. 6 5 doesnt change the value of d and r within 3ps. If we compare the signal strength in Fig. 6 11, intensity of (a) is three times smaller than (b) and (c) ( (b) and (c) have similar signal intensity). Wh en we measured TR, we fixed the emission wavelength at 1.317eV which is corresponding to e1 h1 transition and hence even we increase the excitation power from (b) 1.95mJ/cm2 to (c) 6.27mJ/cm2, number of carriers in energy level of e1 and h1 (or at 1.317 eV ) is not changed much because carriers are already saturated and as a result just increase the carriers in higher energy side of e1h1 peaks from the band filling. Figure 6 11 ( D ) shows the photoluminescence from Sample M507 and has similar time constant ( r =232ps) and d =233ps) as M508. The origin of the PL signal at 0 Tesla can be understood as follows. After the initial pump pulse excites carriers high into the InGaAs and GaAs bands, the electrons and holes rel ax PAGE 154 154 quickly (within a few picoseconds) to the fill up the well states e1, h1, as e2, and h2. This is corroborated by the TR TA data in Fig. 6 12, which shows a rapid rise at t = 0. Initial, no PL is observed from the e1 h1 transition until a sufficient den sity of electrons and holes has relaxed into the e1 and h1 levels. As shown in the figure, TA signal slowly decays after 230ps and this value is corresponding to rise time (232ps) from time resolved photoluminescence. This consistence and correlation betwe en these two values suggest the recombination process of the lowest subband (e1 h1) transition is mostly radiatively recombined. Fig. 6 13 and Fig. 6 14 present in plane (edge emission) time resolved photoluminescence signal at different magnetic fields f rom two different samples. Photoluminescence signals are qualitatively similar for both samples as shown in the figures. These two figures show that at high magnetic fields, the emission characteristic is qualitatively different multiple bursts of PL sep arately in time appear from each LL. The high field emissions are much brighter than the zero field emissions, suggestive of a stimulated emission process enhanced by the magnetic field. As shown in the figure (( B),( C) and ( D ) of Fig. 6 13 and Fig. 6 14), maximum peaks at around zero time delay move toward to the earlier time along magnetic field increase. This kind of observation appears even in higher LLs suggesting fast rise time along magnetic field increase. As shown in Fig. 6 14 ( C), if we define time duration of the signal as fullwidth half maximum, the data shows that the duration is decreased from 33ps to 26ps between 12Tesla and 17.5Tesla which is comparable the resolution of streak camera(~28ps). This short duration comparable to streak camera mi nimum time resolution also indirectly suggest that we are in the amplified spontaneous emission region. As presented in Fig. 6 15 ( A ) and ( B), depending on magnetic fields, PL intensity increases linearly along magnetic field, H because of linear dependenc y of density of state (~ c eH ). Fig. 6 15( B) shows this linear PAGE 155 155 dependency of PL emission. This is the consistent with our spectrum analysis as shown in Fig. 4 17 investigation the time integrated emission, which showed that in plane well emission undergoes a transition from low intensity spontaneous PL emission to a bright amplified spontaneous emission process at a critical magnetic field or laser fluence. As discussed in chapter 4.2.1(Eq. 4 5), this linear dependency of PL signal and rel atively short time duration comparable minimum streak camera resolution are the evidences of amplified spontaneous emission. We measured time resolved photoluminescence with different powers at 17.5 Tesla. As shown in Fig. 6 16, multi burst of TR signals change the position in the time depending on excitation power for all Landau levels showing the complicate radiation dynamics. Figure 6 17 presents the peak position of multi burst at each Landau level at 17.5 Tesla. If we focus on 6.2mJ/cm2 excitation power in Fig. 6 17, four burst of signals in 00LL appear and the separation between two adjacent peaks of burst are 46, 90 and 97ps.( 101ps and 104 ps for 11LL and 22LL). Both the magnitude and the temporal separation of the bursts depend on the laser fluenc e, indicating that the emission is driven by the underlying magnetoplasma density and dynamics rather than sample geometry (feedback from facet reflection). Figure 6 18 present the separation between first two pulses along Landau level index and shows the separation increase along the index. Our sample size is about 3 3mm2 and from Table 4.1 the refractive index of InGaAs is 3.4. Based on simple calculation, round trip time in the sample is 68 ps which is corresponding none of values in adjacent signal separations (Fig. 6 18). Therefore the origin of multi burst of signal changing along the Landau index and input excitation power is rather related with intrinsic properties of multi level system and should not be related with the sample geometry simply. PAGE 156 156 From a physical standpoint, the presence of multiple bursts from the LLs suggests a complicated relaxation process mediated by the magnetic field whereby the photoexcited exciton initially cascades down and relaxes into the LLs, followed by subsequence e h recombination and emission. Because the densities are so high, a relaxation bottleneck occurs until the emission takes place, and LLs can subsequently reload from higher occupied LL states. These results should be discussed in the framework of a N level system. Fig. 6 19 schematically shows the procedure of multiple burst of emission signal. Based on this picture, increase of interval between signals along Landau index as shown in Fig. 6 18 comes from the decrease number of levels above the specific level and result on less chance to reload the carriers on that level. Fig. 6 20 displays the overlapped transient absorption data (shown in Chapter 5.5.2) and the time resolved PL signal for sample M507. Fig. 6 20 red line is time resolved PL for F=4.9 mJ/cm2 for (a) B=0T, heavy hole emission and (b) (d) for 00, 11 and 22LL at 17.5T.As shown in the figure, correlation between two different measurements is observed in all the Landau level. When the transient absorption signal decreases, radiative PL signal starts to increase meaning carriers start to recombine and expanse energy with radiative field. As discussed in Chapter 5, there is sharp increase of signal at time 0 for the transient absorption and signal last until it decay abruptly. After carriers decay abru ptly, the maximum emission occurs and increasing Landau level indexes, the maximum peak of emission appear at earlie r time. This is depicted in Fig. 6 21. Full width half maximum for 0T and the time to start abrupt decay in TA for 17.5T are used to define decay time. In t he figure As presented in Fig. 6 21, 22LL at 17.5T shows the photoluminescence emission right after TA signal decays. At 0 T, carriers are populated to continuum energy states PAGE 157 157 which mean there will be more probability to reload carriers at the vacancy of electrons in conduction band and this procedure will be occurred with relatively continuously. However, in finite magnetic fields, all the continuous energy levels will be discretized and each energy level has dense energy states which can r esult in the bottlenecks as shown in PL emission and the pleatue in TA signal in Fig. 6 20. From the comparison with time resolved PL, at the edge of the plateau of the transient absorption signal, it relaxes radiatively. This kind of correlation is observ ed in all LLs and all magnetic fields even there is multiple burst of the peaks in time resolved PL in finite magnetic fields. For more strict analysis, we need theoretical background of multi level system. 6.6 Summary We have reported time resolved photol uminescence dynamics of multiple quantum well in high magnetic fields. We carried out experiments for different excitation powers and different magnetic fields. We found 230ps decay and rise time for 0 Tesla with two exponential fitting functions. We obser ved that multiple sharp signals from TR measurement have non exponential decays at high magnetic fields. This non exponential sharp emission is related with amplified spontaneous emission discussed in Chapter 4 and existence of multiple emissions is rather intrinsic phenomena in quantum well system rather than the effect from sample geometry. Increasing Landau level index, radiative decay occurs in earlier time and this radiative decay follow abrupt decay of time transient absorption signal. Correlation be tween time transient absorption and time resolved photoluminescence are discussed. This correlation occurs in all LLs and magnetic fields. We need theoretical framework of N level system to discuss the physical origin of the signal in the context of intra LL relaxation and inter level recombination process. PAGE 158 158 Figure 6 1. Differe nt recombination processes in A) direct and B indirect gap mater ials in k space and C) real space. Five differe nt recombination processes in C are shown (1) donor to accepter, (2) conduction band to valance band, (3) conduction band to acceptor, (4) donor to valance band and (5)conduction band to donor. Figur 6 2. Schematic diagram of a simple two level system, showing radiative and nonradiative recombination channels. The pa rameters 21 and 21T denote the time constants for nonradiative channel and radiative channel from level 2 to 1 PAGE 159 159 Figure 6 3. Calculated time evolution of the 2n population in a two level system. Level 2 has been assumed to be populated by a () t pulse at t = 0. Figure 6 4. Operating principle of the streak tube. Light pulse at t1, t2 an d t3 is converted to three electrons by photocathode. These three electrons deflected by time varying voltage bias and impinge against a phosphor screen at different positions. PAGE 160 160 Figure 6 5. Sample mount for time resolved PL collections. Sample is attac hed to sapphire substrate and multimode fibers butted up against it. Micro prism redirects PL from the edge of the sample and transfers the signal to the multimode fiber. PAGE 161 161 Figure 6 6. Top panel: raw 2 dimensional streak camera image of the CPA laser pu lse. For this measurement, the laser was attenuated and sent directly into the streak camera. The measured duration was 2 ps, corresponding to the minimum achievable resolution of the streak camera. PAGE 162 162 A B Figure 6 7. (a) Measurement of the dispersion (relative time delay) as a function of wavelength after propagation through a 7m long graded index multi mode fiber (GIMF). (b) Measurement of the dispersion (relative time delay) including the Acton spectrometer in the beamline. The OPA was used to vary the wavelength center wavelength. A B Figure 6 8. A) Ra w 2 dimensional streak image of 150fs and 800 CPA laser pulse without mu metal shielding (50 Gauss magnetic f ield at the streak camera an d B) 2 dimensional streak image with mu metal covered PAGE 163 163 Figure 6 9. Schematic diagram of the experimental setup for the time resolved PL measurement. The laser beam is injected into the 17.5 T superconducting magnet (in Cell 3 at the NHMFL). A graded i ndex multimode fiber (GIMF) collects the PL from the sample and directs it to the streak camera injection optics. The injection optics, Acton spectrometer, and Hamamatsu streak camera are on the roof of Cell 3 to reduce the magnetic field effect and minim ize the length of the GIMF. PAGE 164 164 Figure 6 10. Timeintegrated photoluminescence spectra of In0.2Ga0.8As/GaAs multiple quantum wells (sample M508) at 0T (dash) and 17.5T (solid) with 7.13mJ/cm2 excitation intensity. Signal at 17.5 Tesla is 10 times stronger than that of 0 Tesla (for comparison normalization of signals are used) PAGE 165 165 Figure 6 11. The time resolved photoluminescence signal for sample M508 at 0 T for pump fluences with three di fferent excitation powers for A),B ) and C ) and for samples M508 with and M507 ( A ) 0.4456 mJ/cm2 ,B ) 1.9502 mJ/cm2 C ) 6.2652 mJ/cm2 and D ) 1.4413 mJ/cm2). The signal is collected from the edge. Exponential fitting is used to determine the rise time, 232ps, and decay time, 233 ps of the PL for A), B), C) and D ) with 3ps. Data is normalized for visuali zation. Signal intensities of B) and C ) are almost same and intensity of A) is three times weaker than B) and C ). All the data is taken at 0 Tesla and at 1.317eV for A), B ) and C) and 1.326ev for D ) corresponding to e1 h1 tran sition of sample M508 and M507. PAGE 166 166 Figure 6 12. TA( time transient absorption, red) and TR(time resolved photoluminescence, blue) of M507 with 4 mJ/cm2 excitation power at 0 Tesla. Figure 6 13. Time resolved photolu minescence for sample M507 at A) 0 T, B) 00LL C) 11LL and D ) 22LL from different magnetic fields with 4.9042 mJ/cm2 excitation fluence. PL was collected from the in plane (edge collection). PAGE 167 167 Figure 6 14. Timeresolved photoluminescence for sample M508 at A) 0 T, B) 00LL C) 11LL and D ) 22LL from different magnetic fields with 6.265 mJ/cm2 excitation fluence. PL was collected from the in plane (edge collection). Figure 6 15. A ) M507 spectrum at different magnetic fields from the edge collection with 4.9042 mJ/cm2 excitation power a nd B ) magnetic field dependent of 00LL peak intensity ( black circle) and linear fit(solid red). PAGE 168 168 Figure 6 16. M508 time resolved photoluminescence from the edge collection at different excitation power and Landau level. All measurements are carried at 17.5 Tesla Figure 6 17. Time position of multiple bursts along excitation power at 17.5T (based on Figure 6 16) PAGE 169 169 Figure 6 18. Time separation between first two adjacent burst of signals with 3.7679 mJ/cm2 excitation intensity based on Fig. 6 18. Figure 6 19. Schematic diagram of multi level system. Because of high carrier density, the relaxation bottleneck occurs. Carriers are reloaded after T1 and T2 As a result multiple bursts in emission appear with time interval T1 and T2. To simpli fy the picture, higher energy levels are represented by dot line. PAGE 170 170 Figure 6 20. Correlation between time transient absorption (blue) and time resolved photoluminescence (red) for sample M507. This plot compares the transient absorption signal probed at the center sample in transmission with the edge emission for time resolved PL. Excitation intensity is 4.1 mJ/cm2 and 4.9 mJ/cm2 for TA and TR PAGE 171 171 Figure 6 21. Decay time (blue circle) at each LL and position of maximum peak of time resolved photoluminesce nce (red cross). For the comparison, 0 Tesla data from e1 h1 transition is plotted. 22LL at 17.5 Tesla shows the emission of photoluminescence right after TA signal decrease. (Data is taken from Fig. 6 20) PAGE 172 172 CHAPTER 7 CONCLUSION AND FUTUR E WORKS In other to investigate the physics of semiconductor quantum well at high magnetic fields (>12T) and high carrier density w e have developed an ultrafast spectroscopy laboratory at the National High Magnetic Field Laboratory where DC high magnetic fields (31 Tesla Bitter type and 17.5 Tesla superconductor magnets) are available We specially designed the cryostats and the probe to carry out the experiment for optical spectroscopy at liquid helium temperature. To achieve high excitation densities (in excess of 2 13/ 10 cm ), we developed an optical excitation system based on a femtosecond chirped pulse amplified system (CPA) including optical parametric amplifier (OPA). Pump probe experimental setup and installation of Streak camera have bee n carried out for time transient absorption and time resolved photoluminescence. A detailed description of magneto optic facilities at NHMFL was introduced in Chapter 3. To understand the optical spectrum of semiconductors with high and low excitation inte nsities, the detailed theoretical background for electronic states of semiconductor quantum well and radiation process for simple two level systems including many body physics was discussed. The sharp emission in the high energy side in spectrum of each La ndau level and the superliner dependence of emission strength depending on excitation fluencies and magnetic fields presents cooperative emission process of dense electron hole plasma in high magnetic fields. With white light absorption spectrum (low excit ation regime), we observed the clearly resolved Landau level transitions following the selection. The s pectrum along magnetic fields shows dark exciton transition (forbidden, e1 h2) including anticrossing behavior between e1 h2 and e1 h1. By comparing the spectrum of In0.2Ga0.8As /GaAs quantum wells in magnetic fields for high and low powers, we found 11meV binding energy of e h exciton with the low carrier excitation densities and observed the clear evidence of insulator metal transition (Mott transition) PAGE 173 173 from zero binding energy for high carrier excitation. With a careful analysis for intermediate region of magnetic field, we observed magnetic field mediate electron hole interaction at 00 Landau level. The new observation of bandgap increase for higher subbands was reported including bandgap reduction of the e1h1 transition. This new observation for higher subband is needed to study future to have complete understanding of many body effects and need theoretical models for many body physics in higher subban ds. We discussed the time transient absorption dynamics (TA) at 0T and 17.5T in Chapter 5. Relatively long life time s (~ 360 ps) in e1h1 transition at zero Tesla were found and this can be explained with less binding energy (effectively zero binding energy). Zero binding energy means that Coulomb interaction mediating e h interaction disappears and as a result probability of e h recombination is decreased. At high magnetic fields, we observe a dramatic and qualitative change in the transient absorption dyna mics with respect to zero field dynamics all LLs exhibit an initial fast increase in transmission however the transmission signal exhibit an abrupt, non exponential reduction after the initial excitation. Similar dramatic changes are observed in time re solved photoluminescence measurements. This fast decay time along magnetic fields and Landau level index was explained with less confinement effect from the magnetic fields to increase scattering rate. Time evolution of the band filling at 00LL and band ga p renormalization for higher LL was also discussed. We have reported on the characterization of emitted photoluminescence in complementary to transient absorption measurement. We measured photoluminescence for different excitation powers and different magn etic fields. We found 230ps decay and rise time for 0 Tesla with two exponential fitting functions. We observed nonexponential decay of multiple bursts of photoluminescence signal at high magnetic fields. This non exponential sharp emission was PAGE 174 174 related wi th amplified spontaneous emission discussed in Chapter 4 and time interval between these multiple burst of photoluminescence show that the origin of multi burst is from the intrinsic properties of quantum well sample in high magnetic fields rather than sim ple geometry. Dramatic correlation between time transient absorption and time resolved photoluminescence are discussed. This correlation occurs in all LLs and magnetic fields. We need theoretical framework of N level system to understand the physical orig in of the signal in the context of intra LL relaxation and inter level recombination process such as numerical approach with Monte Carlo method ([1 09]) For the future, we proposed the following experiments Time integrated photoluminescence depending on te mperature. 1 Time integrated photoluminescence with circularly polarized pumping This can show Zeeman splitting and give us information for g factor. 2 Transient absorption measurement and time resolved photoluminescence for superflurescence regime depending on temperature and magnetic fields. This provides us the building time of cooperative emission and will be the direct evidence of SF. 3 Transient absorption measurement and time integrated photoluminescence at magnetic fields corresponding to phonon energy Optical phonon scattering process can be shown in this measurement. 4 Transient absorption measurement and time integrated photoluminescence with the excitation power corresponding to Landau filling factor 1. Spectrum depending on Landau filling factor give us symmetry property of the system as reported in ref [1 12] PAGE 175 175 APPENDIX A HARMONIC OSCILLATOR AND OPTICAL SUSCEPTIBILITY Electric field polarized in the x direction causes a displacement x of e charge of electron from its equilibrium position. The resulting polarization can be defined as dipole moment per unit volume. d n ex n L P0 0 3 P A 1 where V L 3 is the volume, ex d is the electric dipole moment and 0n is electron density per unit volume. Under the influence of electric field ) ( t E as a damped driven oscillator, we can write Newtons equation as ) ( 22 0 0 0 2 2 0t eE x m dt dx m dt x d m A 2 where is the damping constant, 0m is th e free electron mass and 0 is resonanc e frequency of the oscillator [ 32, 36]. Often it is convenient to consider the complex electric field t ie E t E ) ( ) ( A 3 and take the real part of it as the final result. With the an satz t ie x t x ) ( ) ( A 4 we get from Eq.A 2 ) ( ) ( ) 2 (2 0 2 0 eE x i m A 5 and from Eq.A 1 ) ( 2 12 0 2 0 2 E i m e 0n ) P( A 6 From Eq.A 6, we define the optical susceptibility PAGE 176 176 i i m e n' 0 0 0 0 2 01 1 2 ) ( A 7 where 2 2 0 is the shifted resonance frequency of the damped harmonic oscillator and in general is tensor. From Eq. A 7, ) ( has singularities at 0 i A 8 If we consi der complex frequencies, i and from Eq. A 7, ) ( has poles in the lower half of complex plane, i.e. for 0" but in the whole upper half plane, it is analytic. This prospe rity can be related with causality [ 113] in other word, the polarization ) ( tP at time t can only be influenced by fields ) ( t E acting at early time ( 0 ). From this fact, the real and imaginary parts are Hil bert transforms of each other [ 113]. ) ( ) (" 'd P A 9 ) ( ) (' "d P A 10 where ') ( ) ( ) ( i Splitting the integral of Eq. A 9 into two parts ) ( ) ( ) (" 0 0 'd P d P A 11 an d using ) ( ) (" (from real function of ) ( t ), we get 2 2 0 '2 ) ( ) ( d P A 12 This is the Kramer Kroning relation, which can compute the real part of ) ( if we know the imaginary part of ) ( for the all positive frequencies. PAGE 177 177 APPENDIX B ABSORPTION AND REFRACTION We can establish some relation between ) ( and the other physical constant from the oscillator model we get in the appendix A. The displaceme nt field ) (D can be expressed with the polarization and the electric field ) ( ) ( ) ( )] ( 4 1 [ ) ( 4 ) ( ) ( E E E D P B 1 where the optical dielectric function ) ( is obtained from Eq. A 7 as i ipl 0 0 0 21 1 2 1 ) (4 1 ) ( B 2 Here pl is de fined as the plasma frequency [ 36] 0 2 04 m e npl B 3 The physical meaning of plasma frequency is it is eigenfrequency of electron plasma density oscillation around the ions, i.e., charge dens ity oscillations with the eigenfrequency pl around equilibrium density 0n Returning the optical dielectric function, Eq. B 1, we see that ) ( has poles at 0 i describing the resonance and nonresonant part in each like optical susceptibility ) ( If our interest is in the spectral region around 0 and if 0 is large enough to ignore the nonresonant part, we can simpli fy Eq. B 2 in case of 0 so 0 0 ipl 0 0 21 2 1 ) ( B 4 2 2 0 0 0 2 ') ( 2 1 1 ) ( pl B 5 PAGE 178 178 2 2 0 0 2 ") ( 2 4 ) ( pl B 6 where ) ( ) ( ) (" i Example of the spectral dis persion in case of 1 0 /0 is shown in Fig.B 1. As described in Eq. B 6, spectral shape of the imaginary part of dielectric constant determined by Lorenzian shape function 2 2 0) ( 2 here 0/ is line broaden ing for the case of Lorenz ian [40]. In order to understand the physical information in ) (' and ) (" we need to think about how to the light propagates in the dielectric medium. From Maxwells equations [ 36] ) ( 1 ) ( t r D t c t r B B 7 ) ( 1 ) ( t r H t c t r E B 8 with H B ) ( 1 ) ( ) (2 2 2t r D t c t r H t t r E B 9 Using ) ( we can get the following with 0 ) ( t r E 0 ) ( 1 ) (2 2 2 2 t r D t c t r E B 10 Fourier transformation of Eq. B 10 and from Eq. B 1 yields 0 ) ( ) ( ) ( ) ( ) (" 2 2 2 2 2 r E c i r E c r E B 11 For a plane wave propagating with wave number ) (k and extinction coefficient ) ( in the z direction like PAGE 179 179 z i k ie E r E)] ( ) ( [ 0) ( ) ( B 12 we can get the relation like the following )] ( ) ( [ )] ( ) ( [" 2 2 2 i c i k B 13 By arranging the above to the real and imaginary part, this equation result in ) ( ) ( ) (' 2 2 2 2 c k B 14 ) ( ) ( ) ( 2" 2 2 c k B 15 Next lets define the index of refraction ) (n as the ration between the wave number ) (k in the medium and the vacuum c k ) (0 c n k ) ( ) ( B 16 And the absorption coefficient ) ( as ) ( 2 ) ( B 17 From Eq. B 14 to Eq. B 17, we obtain the relations )] ( ) ( ) ( [ 2 1 ) (2 2 n B 18 ) ( ) ( ) (" c n B 19 Hence Eq. B 6 and Eq. B 19 yield a Lorenzian absorption line and Eq. B 5 and Eq. B 18 describes the frequency dependent refractive index. In case of ) ( ) (' (u sually true in semiconductors [ 32] ) refractive index Eq. B 18 can be expressed by ) ( ) (' n B 20 PAGE 180 180 Furthermore, if ) (n is almost constant over the wide range of frequency, one can approximate Eq. B 19 as ) ( 4 ) ( ) (" c n c nb b B 21 where bn is the background refractive index. In summary, we have discussed the most important optical constants, their relation and analytic properties and explicit forms in the classical oscillator model. PAGE 181 181 0.0 0.5 1.0 1.5 2.0 0.2 0.0 0.2 0.4 0.6 0.8 1.0 max max Figure B 1. Dispersion of the real and imaginary part of the dielectric function of Eq. B 5 and Eq. B 6, respectively where 1 0 /0 and 0 2 max "2 / pl PAGE 182 182 APPENDIX C DENSITY OF STATES FOR D DIMENSIONAL SYSTEM The solution of electron wave function in a periodic lattice in a 3D volume z y xL L L V is given by ) ( exp 1 1 ) ( z k y k x k i V e V rz y x r k i C1 From the periodic boundary condition ( i i iL n k2 z y x i and ... 2 1 0 in ), we find e nergy eigenvalues in the 3 dimensional confinement 2 2 2 2 2 2 0 2 2 0 2 2 32 2zz y y x x DL n L nL n m m k E C 2 where 0mis the free electron mass. In the cases of lower dimensionality 2D confinement along the z direction, 1D confinement with motion free along direction, and 0D confinement, the electron energies can be expressed as 2 0 2 2 0 2 2 0 2 0 2 2 0 0 2 2 2 0 2 2 0 2 1 2 0 2 0 2 2 22 2 2 2 2 2 2 2 2 z z y y x x D y z z x x D z DL j m L j m L j m m k E m k L j m L j m E L j m m k E C 3 where k is wave vector in plane for 2 dimensional system and y xj j j and zj run from 0 to 2 1 For the case of a 0D system such as a quantum dot, the energy states are discrete and separated in reciprocal lattice space by V k3) 2 ( If we d efine a probability distribution function ) ( s k f ( s is spin state) as the probability that the state } { s k is occupied, from k d V k V k kk k k 3 32 2 1 we get that the electron density n is given by: PAGE 183 183 ) () 2 ( 1 23k f dk V N n C4 where N is the total number of electrons in the system and the first factor 2 for the two spin state for the electrons. By changing the integration over k to energy, we find the d ensity of state ) ( E g of the system dk E f E dEg n3) 2 ( 1 2 ) ( ) ( C5 Eq. C 5 allows for the evaluation of ) ( E g using Eq. C 4. From the energies given i n Eq. C 2 and Eq. C 3, we get [ 24] ) ( 2 ) ( 1 2 1 ) ( ) () ( 2 2 1 ) (0 2 1 2 0 1 2 0 2 2 1 2 3 2 0 2 3 j j D j j j j D j j D DE E E g E E E m E g E E m E g E mE gz x z x C6 where z xj jE E is quantum confinement energy along x and z direction. Figure C 1 shows the density of states for each dimension. PAGE 184 184 Density of statesE Figure C 1. Schematic representation of the den sity of state for each dimension PAGE 185 185 APPENDIX D FITTING METHOD OF PHOTOLUMINESENCE Figure D 1 shows the first Landau level spectrum of In0.2Ga0.8As/GaAs at 17.5T. The figure shows inhomogeneous broadened photoluminescence (~9meV) from l ower energy side and a narrow peak (~2meV) at higher energy side. For the fitting method of this spectrum, we apply the Gaussian function for the inhomogeneous line from the lower energy side of spectrum as follows 2 2/ ) ( 22 /G Gw x x G Ge w A y D 1 where Gw A and Gx are amplitude, width and the center of Gaussian function. The higher energy side of homogeneous spectrum can be read as Lorentzian function 2 24 1 2L L L Lw x x w B y D 2 where Lw B and Lx are amplitude, width and the center of Lorentzian function. We add these two function and used as fitting function such as 2 2 / ) ( 24 1 2 2 /2 2L L L w x x Gw x x w B e w A C yG G D 3 where C is the offset of the function and Gw A Gx Lw B Lx and C are fitting parameters. Fitting result is shown in Figure D 1. PAGE 186 186 Figure D 1. 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He worked on several projects in Gainesville involving pulse shaping and high power femtosecond laser systems before moving to the National High Magnetic Field Laboratory (NHMFL) in January 2006 to perform magnetospectroscopic studies of quantum wells with femtosecond lasers. Th e work he performed at the NHMFL is the topic of this dissertation. 