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DISORDER, ITINERANT FERROMAGNETISM, AND THE ANOMALOUS HALL EFFECT IN TWO DIMENSIONS By PARTHA MITRA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Partha Mitra To my parents for all their sacrifices to provide me the best education. ACKNOWLEDGMENTS I am grateful to Art Hebard for giving me the wonderful opportunity to be a part of his lab, which for the last five years was my home away from home. I deeply appreciate the freedom I enjoyed while working under him, nicely balanced by the much needed guidance and support. I have always (secretly) admired his unadulterated enthusiasm, willingness to learn and elegant but simple approach to understanding fundamental physics. I have met some wonderful researchers with diverse personalities during my stay at the Hebard lab and I am proud to be able to collaborate and be a part of the team. In particular I am thankful to Steve for the guidance and training he provided during my early days in the lab. I thank Khandelkar Muttalib and Peter Wolfle for their interest in the theoretical aspects of my research. In particular I thank K.M. for the numerous discussions we had in the past couple of months. I am thankful to Dimitrii Maslov for teaching the best solid state course and for his useful theoretical insights regarding my research during the few discussions we had. I am thankful to the members of the departmental cryogenic facility, Greg and John, for the wonderful support in the form of an unlimited supply of liquid He, which was indispensable for my research. I also thank the members of the machine shop, the electronic shop and graduate student office for the excellent standards they have maintained over the years. I am indebted to my parents for their support, encouragement and for always believing in me. I appreciate the sacrifices they made over all these years to provide me the best education. I could not have come this far without their blessings. I appreciate the warmth and affection of my little sister Mala and is thankful for her understandings while I was not there for her as a big brother for the last several years. I thank my loving wife Shweta for being a part of my life and for her unconditional love and affection. I appreciate her support at the home front while I was busy with my work during the past several months. I am also grateful for the warmth and affection that I received from her family. I am thankful for having a large and wonderful supportive family and appreciate the love and encouragement that I have received, especially from my cousin brothers and sisters. In the end I thank all my friends, roommates, tennis buddies etc. who have touched my life in many ways over the years and apologize for not being able to list everybody's name in here. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES .................................................... ........ .. .............. viii LIST OF FIGURES ......... ......................... ...... ........ ............ ix ABSTRACT .............. .......................................... xi CHAPTER 1 IN T R O D U C T IO N ............................................................................. .............. ... 2 EXPERIMENTAL SETUP: SHIVA APPARATUS ................................................ 7 3 NANO SMOOTHENING DUE TO ION IRRADIATION ...................................... 11 E x p erim en tal D details ............................................................................ .............. 12 Sam ple Fabrication .................. ............................ .. ........ .. ........ .... 12 M easurem ent Setup .............................. ........................ .. ........ .... ............13 Experimental Results .................. .......................................... .... ........ 15 D isc u ssio n ............................................................................................................. 2 2 4 THE ANOMALOUS HALL EFFECT IN MAGNETIC MATERIALS ..................24 Itinerant F errom agnetism ................................................................. .....................24 A nom alous H all E effect ................... ........................ ..... .................................. 26 Anomalous Hall Effect for Itinerant Carriers.....................................................27 Skew Scattering M echanism ............................................. ............... 28 Side Jum p M echanism ........................................ ........................... 31 Berry Phase Mechanism.................... ........................................32 Anomalous Hall Effect in Ferromagnets with Localized Moments ..................33 Superparam agnetism ............... ................. .......................... .... ...... 36 5 QUANTUM CORRECTIONS TO TRANSPORT PROPERTIES IN METALS...... 38 W eak L ocalization E effects ................................................. .......................................39 Magnetoresistance due to Weak Localization.................... .................43 Weak Localization in Presence of SpinOrbit Interaction................................44 W eak Localization in Ferromagnetic Film s ................................ ............... 46 Electron Interaction Effects ............................................... ............................. 48 Scattering by Friedel O scillations ............................................ ............... 48 Magnetoresistance due to Electron Interaction ................................................53 Transport Properties of Granular M etals ........................................ ............... 54 Transport in Weakly Coupled Granular Metals .................................. ...............57 Quantum Corrections to Hall Conductivity............... ........... ...............................59 6 EXPERIMENTAL RESULTS AND DISCUSSION.............................................64 E x p erim mental D details ............................................................................ ............... 64 Sam ple Fabrication ................ .............. .............. .... ........ ................ 64 M easurem ent Setup ............................. .................................. ...... ............66 W eak D disorder: Iron F ilm s .............................................................. .....................68 Transport Properties at B =0 ............... ....................................... ... ........... 68 A nom alous H all Effect in Iron ................................ ........... ............... .... 71 Temperature Dependence of Anomalous Hall Conductivity .................................75 Strong Disorder: Iron/C60 Bilayers ............ ..................... ..................... 84 Experim ents on Cobalt Film s .................................. .....................................94 Discussion of Experimental Results ..................... ...........................98 Absence of Quantum Corrections to Hall Conductivity....................... ...98 Finite Quantum Corrections to Hall Conductivity .........................................100 Dependence of Anomalous Hall Conductivity on Disorder............................107 Anomalous Hall Response in Fe/C60 Films ................................................... 113 7 SUMMARY AND FUTURE WORK ........................ ...................116 LIST OF REFEREN CE S ..................................................................... ........... .......... 123 B IO G R A PH IC A L SK ETCH .................................................................. ...............127 LIST OF TABLES Table page 61 Summary of the results for three different Fe/C60 samples. ..............................94 62 Comparison of some properties of bulk iron and cobalt..........................................97 LIST OF FIGURES Figure page 11 A cartoon describing the origin of the anomalous Hall effect ................................ 12 A cartoon describing the origin of the spin Hall effect. ...........................................3 21 Schematic representation of the SHIVA apparatus.................................................8 31 A setup for nonswitching van der Pauw technique to measure sheet resistance. ...14 32 Plot of sheet resistance and thickness as a function of time................................16 33 Check of stability of the reduced resistance due to ion beam irradiation.................17 34 AFM surface topography of two 120 A thick Cu films. ......... .............17 35 The two van der Pauw component R1 and R2 plotted separately. ..........................19 36 Plot of the shunting resistance R, and the time At...........................................20 37 Effect of ionbeam exposure on thicker iron films. ............................................21 41 The Hall curve for a 20A thick iron film. ..................................... ............... 27 42 Schematic representation of (a) skew scattering and (b) side jump mechanism......30 51 Typical wave functions of conduction electrons in presence of disorder ...............38 52 Motion of electrons in presence of impurities ......................................................40 53 Schematic diagram of Friedel oscillation due to a single impurity..........................49 54 Friedel oscillation due to two impurities..................................... ....... .......... 51 61 A d.c. transport measurement setup using Keithly 236.........................................66 62 An a.c. transport measurement setup using two SR830 lockin amplifiers ............67 63 Typical behaviors for temperature dependence of resistance for iron films. ...........69 64 Plot of numerical prefactor AR for different iron films...........................................71 65 The anomalous Hall curves for iron films................. .............. ............... 72 66 The anomalous Hall resistance at T=5K for different iron films ..........................74 67 Magnetoresistance as a function of field for an iron film with Ro=300 ................78 68 Relative resistance (RR) scaling behavior at T<20K for an iron film ...................79 69 Plots showing dependence on Ro of the extracted transport coefficients ...............80 610 An iron film with Ro=49000Q showing deviation from RR scaling....................... 82 611 The relative changes in AH resistances for three different iron films. ....................83 612 A cartoon of iron/C60 bilayer samples.. ........................................ ...............85 613 Plot of conductivity showing hopping transport in a Fe/C60 sample .....................86 614 The AH effect in strongly disordered Fe/C60 sample........................ ...............87 615 Magnetoresistance curves at different temperatures for the Fe/C60 .......................87 616 Plot of y as a function of inverse temperature for three different Fe/C60 samples...90 617 Logarithmic temperature dependence of AH resistance of Fe/C60 samples............92 618 Temperature dependence of AH conductivity of Fe/C60 samples .........................93 619 AH resistance for Fe/C60 samples as a function of sheet resistance......................93 620 AH effect in a cobalt film with Ro 32000 at T=5K. ..........................................95 621 AH resistance for cobalt films as a function of R .. ............................................. 97 622 The anomalous Hall conductivity vs longitudinal conductivity..........................109 623 The anomalous Hall conductivity vs longitudinal conductivity............................110 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 DISORDER, ITINERANT FERROMAGNETISM, AND THE ANOMALOUS HALL EFFECT IN TWO DIMENSIONS By Partha Mitra May 2006 Chair: Arthur F. Hebard Major Department: Physics In this dissertation we address the unsettled issue of how longrange magnetic order in band ferromagnetic metals like iron or cobalt is affected by localization of itinerant conduction electrons due to increasing disorder. We study a series of polycrystalline iron films in the thickness range of 2nm to 10nm. The sheet resistances of the films at T=5K is considered to be a measure of disorder and varies over a wide range from 100Q to 1,000,0000. To protect these ultrathin air sensitive films from oxidation, the experiments were performed in a special homemade high vacuum system, capable of in situ magnetotransport measurement on thin films at low temperatures. To characterize the magnetism in the films, we monitor the anomalous Hall (AH) effect, which refers to the transverse electric potential that develops in magnetic materials in response to an applied charge current, proportional to the volume magnetization. Our experiments reveal a crossover in the magnetotransport properties for film resistances on the order of h/e2 & 4. IkQ. Surprisingly, in the high resistance regime where the samples were found to systematically cross over from a weakly disordered metal to a Mott insulator, the magnitude and temperature dependence of AH resistance does not show a pronounced change. We attribute this so called "anomalous Hall insulating" behavior to the granular morphology of the films, where intergrain tunneling processes dominate the longitudinal resistance, and the anomalous Hall resistance is determined by the intrinsic ferromagnetic nature of the grains. In the insulating phase longrange ferromagnetic order appears to be absent in the films and we demonstrate the existence of a new resistance scale much greater than h/ e2 where correlation between localized magnetic moments as measured by AH effect, disappears. We also demonstrate how the granular morphology of the films allows the two different quantum transport mechanisms in metals, namely weak localization and the Coulomb anomaly, to be distinguished from each other as they have a different effect on the longitudinal and anomalous Hall conductivity. We present a preliminary understanding of the results on the basis of existing theories and some new calculations done in collaboration with theorists. CHAPTER 1 INTRODUCTION Recently there has been considerable activity in the field of spintronic devices , which rely on the manipulation of spin of conduction electrons in solids and promise to revolutionize microelectronics once spinpolarized electrons can be injected efficiently into semiconductors at room temperatures. Currently, the most widely used spintronic device based on metallic multilayers is the giant magnetoresistive (GMR) spinvalve head for magnetic harddisk drives. The GMR effect is based on large changes in electrical resistance due to variations in the relative magnetic orientation of layers on either side of a thin spacer layer. Spintronic structures are also at the heart of the proposed magnetic random access memory (MRAM), a fast nonvolatile new generation of memory. From a more fundamental point, spintronics studies involve understanding spin transport and spin interaction with the solidstate environment under the influence of applied electromagnetic fields and rely heavily on the results obtained in other branches of physics including magnetism, semiconductor physics, optics and mesoscopic physics. However, there are several major challenges in building a useful spintronic device. These challenges include finding an effective way to polarize a spin system, understanding how long the spins are able to remember their orientation and finding a way for spins to be detected. Perhaps the simplest way to generate a spinpolarized current is to apply an electric field to the ferromagnetic transition metals, namely iron, cobalt or nickel. The spontaneous magnetization in these band ferromagnets is due to the unequal population of up (majority) and down (minority) spin electrons in the conduction band. The difference in the spin population results in a spin current coupled with the charge current that flows in response to an applied electric field. This spin current generates a transverse voltage due to spindependent asymmetric scattering of electrons from impurities or phonons, a phenomenon known as the anomalous Hall (AH) effect 2 as shown schematically in Fig. 11 The AH effect (AHE) is observed in all ferromagnetic materials regardless of the nature of the exchange mechanism and also in superparamagnetic systems comprising weaklycoupled localized moments. Unlike the ordinary or normal Hall effect, which depends only on the effective carrier density in the material, the AH effect is a transport process that couples the volume magnetization in the material to motion of the itinerant carriers via the spinorbit interaction. The transverse AH voltage relies on itinerant charge carriers and is directly proportional to the volume magnetization. Thus a thorough characterization of the AH effect is useful for studying magnetic properties in materials especially when direct magnetization measurements via SQUID are not possible. I I Figure 11: A cartoon describing the origin of the anomalous Hall effect in an itinerant ferromagnet with unequal number of spinup and spin down conduction electrons due to spin dependent scattering of electrons from an impurity X. A transverse electric potential VH is developed due to accumulation of charges at the edge of the sample. A related phenomenon, which recently has generated considerable interest, is the spin Hall effect (SHE) 3 where an electric field applied across a doped semiconductor generates a transverse spin current that in equilibrium leads to an accumulation of opposite spins at the film boundaries 4 as shown schematically in Fig. 12. This accumulation of spins results in a gradient of spin population in the transverse direction and is not associated with an electric potential because the total number of electrons irrespective of spin orientation remains the same along the transverse direction. Since there is no electronic device that can directly detect spin accumulation potential, it is not possible to directly determine the spin conductivity from any simple scheme analogous to that of the normal or anomalous Hall effect. i 4 T I Figure 12: A cartoon describing the origin of the spin Hall effect in a paramagnetic conductor due to spin dependent scattering of electrons from an impurity X. There is a gradient of spin potential in the transverse direction but no net electric potential. The importance of the AH effect and the SH effect lies in the fact that both effects provide a unique way of manipulating electron spins with external electric fields and hence have potential for application in the development of novel spintronic devices. Both the AH effect and the SH effect in itinerant systems result from similar spin dependent microscopic scattering mechanisms due to the presence of spinorbit coupling. Phenomenologically, the two effects are complements of each other: in the AHE an electric potential due to the separation of electric charge develops transverse to a spin polarized current, while in the SHE a spin potential due to spin accumulation develops transverse to a charge current with equal spin populations. Quantitatively, the AH effect will be similar to the inverse spin Hall effect (ISHE) in which a longitudinal spin current carrying no net charge generates a transverse polarization voltage. In other words there is no electric field associated with static spin accumulation but a real electric field associated with spin currents. Recently it was shown theoretically that by redefining the spin current 5, one can demonstrate an Onsager relation that relates the spin Hall conductivity to the anomalous Hall conductivity. Thus the AH effect can provide an important route for making quantitative estimates of spin dependent transport coefficients in magnetic materials. In this dissertation we study the AH effect by addressing a fundamental and unsettled question regarding magnetism in the transition metal elements (iron, cobalt and nickel). Ferromagnetsim in these materials is known to be due to mobile (itinerant) electrons that are also responsible for electrical conduction. The balance between the kinetic energy and exchange interaction energy of the conduction electrons determines whether a metal with a spinsplit band becomes a ferromagnet with long rang order or a paramagnet with enhanced susceptibility. If an inequality known as Stoner criterion 6 is satisfied, then the material is a band ferromagnet. Band structure calculations for the transition metals show that indeed the conditions are favorable for ferromagnetism with the magnetic properties determined by spin polarized itinerant carriers. We are investigating whether quenching the itinerancy of the conduction electrons with increasing disorder can violate the Stoner criterion. In this regard we grow a series of Fe, Co and Fe/C60 films in a very clean environment and systematically reduce the film thickness, thereby increasing resistance. The airsensitive nature of these films rules out the possibility of a magnetization measurement using a SQUID magnetometer. As we will show in the following chapters, the AH effect is an alternative and effective tool to probe magnetic ordering in conducting materials. We will show that with decreasing thickness, long range ferromagnetic order in thicker films gradually disappears and passes over to a system of weakly coupled magnetic clusters. Eventually at very high resistances there is an unexpected and pronounced disappearance of magnetic ordering as measured by the AH effect. We also demonstrate that morphology plays an important role in governing the magnetic and electrical properties in thin films. Our experiments also shed light on another unsettled issue concerning the extent to which weak localization corrections are important in ferromagnetic films. In chapter 2, we describe in detail the SHIVA apparatus capable of in situ magnetotransport measurements, which proved to be indispensable for this project. In chapter 3 we demonstrate the usefulness of the SHIVA apparatus by describing a short project that studies the effect of low energy ion beam irradiation on the conductivity of metal films. In chapter 4 we describe the phenomenology of the AH effect and discuss various microscopic mechanisms that are responsible for this effect. Chapter 5 provides a brief review of the various quantum corrections to the transport properties in disordered metals. A detailed account of our experimental result and discussions based on the 6 existing theoretical understanding is provided in chapter 6. Finally, in chapter 7 we summarize and discuss some possible future experiments related to this dissertation.. CHAPTER 2 EXPERIMENTAL SETUP: SHIVA APPARATUS In this section we describe a unique custom built high vacuum system that was designed for in situ characterization of air sensitive films. This apparatus, shown schematically in Fig. 21,was given a pet name SHIVA, which is the acronym for Sample Handling In VAcuum. SHIVA has a clever design that combines a growth chamber, a cryostat and a load lock into one single vacuum system and has mechanical "arms" for transferring samples between the compartments without breaking vacuum. The load lock LL is a small chamber used to mount and unmount samples from the system and is separated from the growth chamber and cryostat by gate valves VI and V2 respectively. This arrangement allows the growth chamber and the cryostat to be under high vacuum all the time, and the sample is introduced by opening the gate valves only when the load lock is pumped down to a suitable base pressure. The mechanical "arms" Al, A2 and A3 consist of a strong magnet sliding on the outside of a long hollow stainless steel tube attached to the vacuum chamber, and another solid rod inside the tube magnetically coupled. Hence by sliding the outside magnet, one can translate and rotate the rod with a thruster attached to its end, inside the vacuum chamber. Sample holders or pucks are specially designed such that substrates are loaded on one of its faces and slots on other faces where the thruster on the "arms" can be inserted and locked onto the puck by a twisting motion. The puck is first mounted on A3 inside the load lock and then pushed inside the growth chamber. Thruster Al then comes Al A2 growth chamber V1 manipulator <= L A3 thruster Emounted a puck. / $2 Sl lck m ct r; E: ostat ". receiver Figure 21:Schematic representation of the SHIVA apparatus and a blowup (in color) of the receiver platform showing how a puck can be locked in with the help of thruster. LL: Load lock; A1,A2,A3: transfer arms with thruster mounted at each end; V s,V2: gate valves separating the growth chamber from load lock and load lock from cryostat respectively; E: Optical ports for ellipsometry. down and locks into the puck so that A3 can be disconnected from the puck by untwisting and taken out of the chamber. The puck, which is now secured on Al, is then delivered to a receiver platform, and the design is such that the puck engages in the receiver and disengages from Al in one twist, so that Al can be taken out of the way. Once the sample is grown on the substrate, Al comes down and locks into the puck and disengages it from the receiver. The next step is to bring A3 back into the growth chamber, lock into the puck, and then disengage Al. The puck is then brought back into the load lock where thruster A2 engages on it and then A3 is disengaged. Now the sample can be pushed inside the cryostat and delivered to a receiver platform similar to that in the growth chamber. If necessary, the sample can be transferred back to the growth chamber from the cryostat for further processing, by reversing the steps described above. The point to note here is that during the whole procedure, there is no need to break vacuum, the importance of which cannot be overemphasized. Samples are usually grown through shadow masks on a set of electrodes made of thick films of gold pre deposited on the substrate. Inside the body of the pucks a set of wires are soldered from inside on to small hemispherical copper "heads" attached firmly on the outside of the puck. The other ends of the wires emerge from the sample side of the puck and are connected to the electrodes on the substrate before the puck is loaded in the vacuum chamber. Thus the electrodes on the substrate are electrically connected to different copper heads. The receiver also has a set of spring loaded copper heads on the inside, such that when the puck is placed inside the receiver and engaged, the two sets of heads press firmly against each other and make good electrical contact. Another set of wires connect each of the copper heads on the receiver, run via vacuum feedthroughs out of the vacuum chamber to a breakout panel. Thus when a puck is engaged properly in the receiver, each of the electrodes on the substrate is electrically accessible outside the chamber at the break out panel. This allows us to monitor the resistance of the sample during deposition in the growth chamber and also after the puck is delivered into the cryostat and engaged with an identical receiver in the cryostat. The growth chamber can be pumped down to a base pressure 109 Torr and is equipped with a variety of deposition sources, namely two AJA magnetron sputter guns, two RADAK thermal evaporation furnaces and an ion beam gun. The receiver in the growth chamber is mounted on a manipulator that allows us to position the substrate so that it faces the appropriate deposition source. The thickness of the films can be monitored with a quartz crystal oscillator firmly attached on the receiver close to the sample. The growth chamber also has two optical ports specially designed for attaching a Woolam M44 fixed angle ellipsometer, which adds the capability of in situ monitoring of the optical constants and/or thickness of the thin films during deposition. The cryostat is housed inside a CRYOFAB liquid helium dewar with a liquid nitrogen outer jacket. When the dewar is cold the cryostat can reach a base pressure 10.8 Torr. The cryostat has an AMI superconducting magnet, which allows magnetotransport measurement at low temperatures down to 4.5K and magnetic field upto 7T. The temperature of the sample is measured accurately with a Cernox thermometer glued on the puck right underneath the substrate. The temperature of the sample is controlled within a fraction of a Kelvin by a Lakeshore 332 temperature controller connected to a second Cernox thermometer and resistive heater mounted on the receiver. CHAPTER 3 NANO SMOOTHENING DUE TO ION IRRADIATION Ionassisted deposition refers to the technique in which a beam of noble gas ions, usually argon, with energies less than 1 keV, is simultaneously incident on a thin film during growth. The most prominent consequences of ionassisted film growth include densification, modification of nucleation and growth, interface mixing, defect generation, and changes in topography and surface roughness 7. In contrast to this general technique of ionassisted deposition, some investigators have reported on the use of a sequential technique in which thin films are first deposited and then, after deposition, exposed to an ion beam. Results of this approach include ion bombardment induced nanostructuring of Cu(001) surfaces 8, the formation of reproducible ripple structures on Si(001) and Ag(1 10) and (001) surfaces 9 and the roughening or smoothing of Si(001) and SiO2 surfaces 10, 11 where the result depends on ion type, energy and angle of incidence. At the moderately higher energies of a few keV, interlayer exchange coupling in Fe/Cr/Fe trilayers can be controlled with He ion bombardment 12, and microscopic holes can be filled using pulsed Ar ion beams in a process called "ionbeam sculpting" 13. Many of these experiments thus provide strong evidence of ion beam induced nanoscale matter transport on solidstate surfaces, a process that promises to be useful in applications requiring nanotextured surfaces and interfaces. We are going to present a systematic study of the effect of low energy (200eV) ion beam exposure on the room temperature resistance of polycrystalline metallic films. Experimental Details Sample Fabrication Using the SHIVA apparatus described in chapter 2, we have grown a series of iron films on glass substrates at room temperature, using r.f. magnetron sputtering source. We used an r.f. power of 35W and Ar gas flow of 10sccm that resulted in a chamber pressure 104 Torr and a d.c. bias in the sputter gun 145V. Under these conditions the Iron films grows at a rate of 4A/minute. We have also grown a few copper films by thermal evaporation using a RADAK source at a temperature of 1100C. Film thickness was measured by a calibrated quartz crystal thickness monitor placed in close proximity to the sample and varies from 1545A for the Fe films and 75130A for the Cu films. The samples were grown on square substrates with pre deposited gold leads at the four corners and used van der Pauw technique 14 to measure sheet resistances of these films. The use of van der Pauw technique on square shaped samples allowed us to avoid using any shadow mask during growth, thus eliminating the possibility of contamination due to resputtering from the mask during subsequent ion beam exposure. Immediately after growth, the samples were exposed to a beam of Ar+ ion beam generated by 8cm Advanced Energy Kauffman type broad beam ion source. Following are the parameters used to run the ion gun: Ar gas flow 10sccm, beam voltage =200V, beam current =2mA, accelerator voltage= 45V. The experiments are performed in ultra high vacuum (UHV) conditions, and the sheet resistance of the films is monitored in situ without breaking vacuum between the film growth and subsequent ion beam exposure steps. The incident noble gas ions are chemically unreactive and simply transfer a fraction of their kinetic energy to the atoms in the film, a process which, as discussed above s13, can result in a significant modification of the surface morphology of the film and hence its physical properties. Using atomic force microscopy (AFM) we confirm an ion beam induced smoothening of our films and find that there is a reproducible correlation of the reduction in sheet resistance with ion beam parameters (beam voltage and current density) and initial starting sheet resistance. Measurement Setup In a classic paper published in Phillips Technical Review (1958), van der Pauw 14 proposed a novel method based on a mapping theorem for measuring sheet resistance and Hall constant of an arbitrary shaped uniform film. Consider a flat lamellar square film free of holes with four small contacts A, B, C and D at its covers (see Fig. 31). Apply a current IAB at contact A and take it off at contact B. Measure the potential difference between C and D and define V V R, D C (31) IAB Similarly, apply a current ICB and measure the potential difference between D and A define R2 = . (32) ICB The sheet resistance R is determined from a mathematical relation between the above measurements as given by R R, exp( .' )+ exp(; )= 1 (33) R R The solution of the above equation can be written in a simplified form R = R + Rf( (34) In2 2 R2 wherefis a numerical factor that depends only on the ratio R1/R2 and is given by the following transcendental equation RR 1 In2 1 In 2 cosh( /2 ) exp( ) (35) R, /R, +1 f 2 f which can be easily estimated to any degree of accuracy by numerical methods. Thus the van der Pauw method involves two independent measurements in different electrical configurations. A closer look reveal that the two configurations for measuring R1 and R2 can be switched from one to other by simply interchanging any one of the diagonal contacts keeping the other fixed. Usually one uses a mechanical or electronic switching system to change the electrical configuration for the two measurements. Lockin fl Locki 12 Figure 31: A setup for nonswitching van der Pauw technique to measure sheet resistance of a sample (shaded square ABCD) using two SR830 lockin amplifiers operating at different frequenciesfl andf2. Different colors are used to represent electrical connections for each lockin. We have used a nonswitching technique 15 where the two measurements can be made simultaneously without changing configurations, by using a.c. excitations instead of d.c. currents as described above. With contact B of the sample connected to ground, we apply two independent ac currents IAB and ICB with different modulating frequencies. This can be achieved by using lockin amplifiers, which can be used as a constant current source by using a ballast resistor at the voltage source. The same lock in amplifiers can also be used to measure the corresponding voltages VDC and VAD. We have operated two SR830 lockin amplifiers at frequencies offi=17Hz andf2=27Hz, and using 1MQ ballast resistors, we were able to source 1LtA of a.c. current into our samples. This arrangement, shown schematically in Fig. 31, allows us to simultaneously measure both components, R1 and R2, of the resistance and assess film homogeneity (R1R21) during growth and subsequent ion exposure. We also simultaneously measured the reading of a quartz crystal monitor as a measure of film thickness. However, due to lack of calibration of crystal monitor output as a ion beam is incident on it, we quote our thickness in arbitrary units. Experimental Results We report 16 on an additional and unexpected effect of ion milling on the resistance of ultra thin films. As shown in Fig.32 the resistance of a Cu (top panel) and Fe (bottom panel) film undergoes a pronounced decrease which is initiated at time t = 0 when the beam is first directed onto the sample. The resistance reaches a broad minimum and then begins to rise after about 50 s. The initial resistance decreases have been measured to be as large as a factor of 25 for Cu and 2 for Fe. This is an unusual result because the ion beam is expected to erode the film at a constant rate, as indicated by the linear decrease of the accumulated thickness of the material deposited onto the thickness monitor (right hand axes). Apparently, during initial stages of milling, the resistance decreases as material is being removed. As discussed later, we attribute this decrease to an ion beam induced smoothening of the film. 8 90 Cu film i 6 80 n,2 60O S 750 I I I I ' 100 0 100 200 300 10 20 Fe film ( D 8 . / 15 4  05 50 2 .. 0 100 0 times) 100 200 Figure 32: Plot of sheet resistance (left hand axis) and thickness (right hand axis) as a function of time for an ultra thin Cu (upper panel) and Fe (lower panel) film. At time t =0 the ion beam is turned on. The resistance and thickness, measured simultaneously, show that as material is being steadily removed the resistance initially decreases to a minimum and then increases. We checked the stability of the minimum resistances achieved due to ion exposure is stable and is not an experimental artifact. We turned off the ion beam when the resistance is near the broad minimum (Fig. 33). The resistance remains stable with no significant drift at the minimum value during the time ion beam was off. The resistance starts to rise as soon as the ion beam exposure is started again. 5000' ''' STOP 4000 C 3000 START S STOP 2000 START 300 0 300 600 900 1200 times) Figure 33:Check of stability of the reduced resistance due to ion beam irradiation. START and STOP refers to ion beam exposure. During the time the ion beam is off the resistance remains stable and does not show any significant drift. Figure 34:AFM surface topography of two 120 A thick Cu films with R, & 2 kM. The unmilled film (left panel) has a rms average roughness of 30 A compared to the 10 A roughness of the film (right panel) which was milled to its minimum resistance R Rm. Auger electron spectroscopy of an ionmilled film compared with that of a pure film does not show the presence of impurity contaminants that might be inadvertently sputtered on the film by the ion beam and thus lower its resistance. The relative decrease in the resistance of our ultrathin films due to ion milling is strongly correlated with their initial surface roughness. Fig. 32 shows that for a Cu and a Fe film with similar initial sheet resistances near 5 kM, the minimum resistance reached during the ion milling process is more than a factor often lower for Cu (200 Q) than it is for Fe (2.5 kM). An important insight into the cause of this difference is provided by our ex situ AFM studies, which show that Cu films grown by thermal sublimation have an rms roughness larger by a factor of 3 than Fe films grown by sputtering. We have also compared the roughness of films not exposed to an ion beam with films ion milled close to the resistance minimum. As shown in Fig. 34, this comparison for a typical Cu film reveals that the ionmilled film (right panel) has an rms roughness of 10 A compared to 30 A for an unmilled film (left panel). The smoothening effect is also confirmed in AFM images ofFe films, which, with their initially smoother topography, exhibit a smaller reduction (factor of 1.2) in rms roughness. For even smoother films such as Gd and Pd, which become conducting almost immediately after deposition begins, we do not observe an initial resistance decrease. There is also a pronounced increase in homogeneity associated with ion milling. If the films are homogeneous, then R1 and R2 should both show the same time dependence and have similar magnitudes. However, a thin film grows through various stages, starting with nucleation of isolated grains, then coalescence of the grains and finally formation of a homogeneous system of wellconnected grains. At the early stages of growth the local resistance is extremely sensitive to local variations in temperature, incident flux and pressure. Any gradients in these quantities can give rise to long length scale anisotropies in the electrical resistance and hence significant differences in R1 and R2. In our experiments, R1 and R2 can differ by factors as large as 3 for Fe and 20 for Cu films. However, when these inhomogeneouss" films are exposed to the ion beam, both R1 and R2 individually decrease to approximately the same minimum. Thus the anisotropy measured by R1 R21 and the total sheet resistance as measured by the van der Pauw combination of R, and R2 15 simultaneously decrease. 2000 , R i 1000  0 100 0 100 200 300 times) Figure 35: The two van der Pauw component R1 and R2 plotted separately for the copper film shown in Fig. 32. We model the ion beam induced decrease in sheet resistance from an initial value R = Ri to a minimum value R = Rm as equivalent to connecting a shunt resistance Rs in parallel with Ri. We use the parallel resistance formula to calculate R, as follows: 1 1 1 + R (36) R, R, R, 6000 A k a) 4500 .Z . '       A.. a . 4500 S3000  200 I I I I 200 160 (b) S120 i 80  40 .....  ^ 40 V 100 1000 2000 3000 4000 5000 Figure 36: Plot of the shunting resistance R, (panel a) and the time At required for the ion beam to mill the film to its resistance minimum Rm (panel b) as a function of the initial resistance R,.. The horizontal dashed lines represent the nominally constant values of R, and At over the indicated ranges of R,. The crossover to different plateaus near R, = 1500 Q represents a change in film morphology in which a smaller ion dose (oc At) gives rise to a larger shunt resistance. In Fig. 36(a) we plot the calculated values of R, versus Ri for twentytwo different Fe films. We note the interesting result that R, (indicated by the horizontal dashed lines) is constant and on the order of 4500 Q for ultra thin films with R, > 1500 Q and constant and on the order of 1400 Q for thicker films with R, < 1500 Q. Fig. 36(b) shows the dependence on R, of the ion exposure time At, needed to reach the minimum resistance Rm. Since the beam flux is constant for all the experiments, At is proportional to the total number of ions incident on the films or, equivalently, the ion dose. We find that At behaves similarly to R. The crossover in both plots near R, = 1500 Q, corresponding to a thickness of around 25A, most likely reflects a change in film morphology in which a smaller ion dose (oc At) gives rise to a larger shunt resistance. Thus for ultra thin films with R, > 1500 Q the shunt resistance increases by a factor of 3 and the dose needed to achieve the resistance minimum decreases by a factor of 2. The relative constancy of the shunt resistance values for a wide range of initial resistances implies that the ion milling is primarily a surface modification effect in which the rearranged surface atoms can be thought of as providing a shunting resistance that is independent of film thickness. Similar data are found for the Cu films where At 75 s and the shunting path resistance R, = 200 Q is more than an order of magnitude lower than for Fe. If the initial film is very thick, then its conductance dominates and R, << R,. Under these conditions the resistance decrease, R, Rm R, (R, / R), due to nano smoothening is negligible as is in fact verified for iron films of thickness greater than 50A (R, > 200 Q) when the resistance decrease is not observed as shown in Fig. 37. 25. 180 155 *' ' 40 S150 150 35 20 120 V 90 \ 140 25X 15 15 3 135 20 30 1 10. ,0 130 0 500 1000 1500 200 0 200 400 times) times) Figure 37: Effect of ionbeam exposure on thicker iron films (d>50A). Note that resistance does not go below its initial value at t=0, when the beam was started. The minimum resistance reached during ion milling is not sensitive to small variations in the incident ion energy. The resistance minima of Fe films, having almost identical initial sheet resistances, separately exposed to beam energies varying from 50 eV to 250 eV do not show any pronounced dependence on beam energy. However at 50 eV, At was larger by a factor of 10 and the resistance minimum was broader. Discussion We surmise that two competing processes are occurring during the ion irradiation process: (1) sputter erosion by the impinging ions preferentially removes atoms from the film at high points where they have less coordination with neighbors and hence less binding energy and (2) the nanoscale transport of material from high points (peaks) near grain centers to low points (valleys) between adjacent grains. The resulting nano smoothening process dominates in the initial stages of the ion exposure, resulting in a decrease in resistance and an increase in homogeneity. From a microscopic point of view, the decrease in resistance can be attributed to a variety of mechanisms including: a transition from the diffuse surface scattering of rough films to specular surface scattering of smooth surfaces 17 18, the removal of foreign surface absorbates 19 and the filling in of high resistivity weak spots at grain boundaries. Equivalently, one can interpret the results in terms of percolation where the ion milling gives rise to a restructuring of the grains and a concomitantly lower critical thickness for the onset of conductivity. Ion beam induced grain growth 20 is probably not relevant because our incident ion energies are too low. At all stages of milling, the erosion of the film at a constant rate is occurring and, as shown in Fig. 32, this process eventually dominates over the smoothening process when the resistance starts to rise. Our results for Fe and Cu films are reminiscent of experiments in which it was found that C60 monolayers deposited onto thin Cu films give rise to a shunting path with R, independent of R, over a similar range 21. In this case the physics is different, since the charge transferred across the Cu/C60 interface begins to fill the lowest unoccupied band in the C60 monolayer, thus causing the monolayer to become conducting. In conclusion, we have shown that post deposition ion milling of ultra thin Fe and Cu films gives rise to a pronounced initial decrease in resistance and a concomitant improvement of electrical homogeneity and film smoothness. The observation of a constant shunt resistance that is independent of the underlying film implies that the initial resistance decrease is due primarily to surface modification. In the initial stages of ion bombardment, in which pronounced resistance decreases are observed, the lateral transport of material and the associated nanosmoothening dominates over the removal of material. With continued milling the film is uniformly etched away and the resistance increases. While these techniques are clearly applicable to fundamental studies of thin films where the resistance can advantageously be externally tuned with an ion beam, they may also have applicability to the preparation of polycrystalline surfaces prior to the formation of tunnel barriers or the improvement of interfaces in metallic bilayers or superlattices. CHAPTER 4 THE ANOMALOUS HALL EFFECT IN MAGNETIC MATERIALS Itinerant Ferromagnetism Ferromagnetism in metals arises from unpaired d orfelectrons in the atoms. The experimentally observed values of the magnetic moment per atom for the ferromagnetic transition elements iron, cobalt and nickel are 2.22,1.78 and 0.60 respectively in units of Bohr magneton 22,23, that is the magnetic moment of one electron. These nonintegral values cannot be explained in terms of models where the magnetic electrons are localized at the core of the atoms forming the lattice. Instead, the magnetic electrons are believed to be itinerant, and are free to move within the crystal and participate in conduction. The magnetization in this case is due to spontaneously spinsplit bands. In 1934, Stoner derived a condition 6 under which a gas of electrons with exchange interaction between them becomes a ferromagnet. In a normal metal, in the absence of a magnetic field, there are equal numbers of up and down spins in the conduction band. Imagine a situation when spindown electrons within an energy range 8E of the Fermi energy EF are placed in the spinup band. The number of electrons moved is g(EF)3E/2, where g(EF) is the density of states at the Fermi level and the increase in energy is bE. Hence the change in kinetic energy is 1 AEK E = g(E, ) (9E)2 (41) 2 This increase in energy is compensated if there is an exchange interaction Jbetween electrons that lowers the energy if two spins are parallel. This will lead to a potential energy given by 1 AEP E = J(g(EF )E)2 (42) 2 Thus the total change in energy is given by AE = g(EF )SE2 (1 Jg(EF)) (43) 2 and spontaneous ferromagnetism is possible if AE < which implies : Jg(E ) > 1 (44) This inequality is the Stoner criterion for ferromagnetic instability that requires the exchange energy to be strong and the density of states at Fermi level to be large. This theory is very successful in explaining the ferromagnetic behavior of the three transition metal elements iron, cobalt and nickel. Band structure calculations for crystals of the transition elements show that the Fermi energy crosses the unoccupied dbands, which has an imbalance of spins. The calculated values of magnetic moment per atom from band structure are in good agreement with experimental values. If the Stoner criterion is not satisfied, there will not be any spontaneous long range magnetic order but rather a paramagnet with renormalized susceptibility given by. My M= I P (45) H 1 J g(E,) where XP is the Pauli paramagnetic susceptibility. Thus there will be a large increase in magnetic susceptibility is known as Stoner enhancement as is experimentally observed for Pd and Pt which are materials where Jg(EF) is close to but less than unity. Anomalous Hall Effect The anomalous Hall (AH) effect 2 in magnetic materials refers to the phenomenon when a transverse electric potential proportional to the volume magnetization develops in response to an applied charge current. This transverse electric potential is different from that due to the normal Hall effect, which is a result of Lorentz force acting on the charge carriers due to an applied magnetic field leading to accumulation of charges at the transverse edges. The AH effect results from spindependent scattering of the conduction electrons due to spinorbit coupling with scattering centers. In principle the AH effect can manifest itself in the absence of any external magnetic field if the sample is a single magnetic domain. However macroscopic ferromagnetic samples in zero magnetic fields consist of randomly oriented multiple magnetic moments with net moment of zero. An external magnetic field has to be applied to line up domains along its direction, in order to observe a finite AH effect. The magnitude of the AH potential due to a certain applied magnetic field is at least 102103 times higher than that of the normal Hall effect, in most ferromagnetic materials, which makes it easier to separate the two effects. A typical Hall curve for a ferromagnetic sample of iron is shown in Fig. 41. The initial steep increase in the hall resistance at low fields is due to the increasing alignment of magnetic domains along the field direction until some characteristic field when all the moments are lined up. Thereafter, the Hall resistance rises with a much smaller rate that is due to normal Hall effect. This typical behavior can be described using a phenomenological expression 2 p, = R,M + RB (46) where the Hall resistivity is described as the sum of the anomalous Hall resistivity proportional to magnetization and the normal Hall resistivity proportional to the magnetic field. Ro is the normal Hall coefficient and depends only on the effective carrier density in the material. R, is the AH coefficient and depends on the microscopic parameters that describe the scattering mechanism. For years, theoretical studies of the AH effect has generated considerable debate and controversy. We present here a brief account of theoretical understandings, which are relevant to our experimental work. I !I 80 X 40 0 I * 0 2 4 B(T) Figure 41:The Hall curve for a 20A thick iron film of resistance Ro=2700Q at T=5K, showing anomalous behavior. Anomalous Hall Effect for Itinerant Carriers We consider models where a transverse voltage arises due to spinorbit interaction of the spin polarized current carriers in itinerant ferromagnets with the nonmagnetic periodic lattice and or impurities. An electron in a solid experiences a net electric field say E due to the ionic core in the lattice, other electrons or impurities. In the rest frame of the electron there is a magnetic field B as a result of relativistic transformation as given by B = x E / mec (47) where j is the momentum of the electron. This magnetic field interacts with the spin of the electron and favors antiparallel orientation of orbital and spin angular momentum of the electron. This describes the intrinsic spin orbit interaction, and results in an additional term in the Hamiltonian given by24 Hso = 2 (V Vxp) (48) 4m2C2 where a (ox, y, a ) are the Pauli matrices and E = VV, where Vis the local potential that the electron experiences. At low temperatures when the dominant scattreing is due to impurities, Vis the potential due to a single impurity. The Hamiltonian for such a disordered ferromagnetic metal including spinorbit interaction as given by: V2 A2 H= d3 *(r)[ +V(r), M i a (VVxV)]y/(r) (49) 2m 4 where Ao is effective of spinorbit scattering strength and has the dimension of length, V(F) = v(F R ) is a random potential due to impurities at positions R/, M is a strong exchange interaction in the zdirection and f=( f/, y) are spinor fields corresponding to spinup and spindown electrons. The Hamiltonian in momentum representation can be written as24 k2 2 H = cy( 2M )I k +Z kVk[1 k[+ xk').]k (410) k 2m k,k 4 Skew Scattering Mechanism In 1955 Smit proposed a mechanism 25, 26 for AH effect in ferromagnets which is referred to as Skew Scattering. This mechanism is based on the fact that the scattering amplitude of an electron wave packet from an impurity due to spinorbit coupling is asymmetric in the sense that it depends on the relative directions of the scattered and incident waves and of the spin. Consider a Gaussian wave packet with average wave vector k, incident on an impurity site at the origin at time t=, resulting in a scattered wave given as the sum of normal and spinorbit scattering, V = I, + v P = CkV k (411) k For a short range impurity potential V(r) it can be shown that scattered wave function from the spinorbit term far from the origin is given by w,, = c,[qlh(kr)(k xr )' ]/kr (412) k where h, (kr) = e'kr[i/(kr)2 + 1/kr] and the scattering wave from the normal potential scattering part is given by V = ck[exp(ik F) + boho (kr) + b1h, (kr)(k F) /kr] (413) k with h, = ie'kr kr. Thus for the normal case, the scattering amplitude depends on the angle between the incident and scattered direction and does not depend on the spin of the electron. To understand the skewnesss" that arises from spinorbit interaction, imagine a plane containing the incident direction k and the direction of the spin of the electron o assumed to be polarized along a fixed zdirection. It follows from above, the scattered wavefunction for a given spin, the wave function has different sign on both sides of the plane. The signs are reversed when the spin of the electron is reversed. This results in a separation of scattered electrons depending on its spin as depicted in Fig. 42a. Thus for an itinerant ferromagnet with an unequal number of up and down spin electron, a transverse potential develops as the up/down spins are scattered in opposite directions. Detail calculation for a simple parabolic band and short range random impurity potential shows that the AH conductivity is proportional to longitudinal conductivity. =s = VoNoAo (n7 n ) (414) where Mis the magnetization, impurity strength v(k k') = v independent of momentum for, and No is the spin averaged density of states at the Fermi energy. In general it can be shown that for skew scattering, the AH conductivity is directly proportional to conductivity. Using the inversion relation between conductivity and resistivity, it follows that the anomalous hall resistivity due to skew scattering is directly proportional to the longitudinal resistivity. pss pM (415) ,...;(a) @     ^  0 I I i26 (b) Figure 42: Schematic representation of (a) skew scattering and (b) side jump mechanism of AH effect. ) and 0 represents spin up and spin down electron. Side Jump Mechanism In 1970, Berger proposed a new mechanism that can contribute to the AH effect called the Side Jump mechanism 27. It is based on a quantum mechanical effect where the trajectory of the scattered electron is shifted sidewise due to spin orbit scattering from impurities. The quantum mechanical velocity operator for the Hamiltonian with spin orbit interaction is given by, S= i[F, H] = + Ac VV (416) m Thus there is an additional term in the velocity operator, transverse to spin polarization which has a sign depending on the spin orientation. This corresponds to lateral displacement of the center of the scattered wavepacket with given spin as shown in Fig. 42b. The magnitude of the sidejump displacement 5is proportional to spinorbit coupling and is expected to be small. We note that for a bare electron in vacuum, the spinorbit scattering parameter is simply the normalized Compton wavelength i.e Ao = h/mc However, Berger has shown that the spin orbit coupling is renormalized by the band structures by factors a~104 which results in sidejump displacements 3 = aA kF /4 = 10" m and is independent of disorder. The characteristic length scale that replaces the mean free path is 3, hence this contribution is small compared to skew scattering, except in the case of short mean free path i.e. high resistivity. Detail calculation reveals that in this mechanism the AH conductivity is independent of impurity concentration and depends only on the side jump displacement. o = e 2 (n7 n ) (417) 2h This implies that the AH resistance due to side jump scattering is proportional to the square of the longitudinal resistivity as given by Ps prM (418) Berry Phase Mechanism We now discuss a mechanism for AH effect that has drawn considerable attention in the recent times. In a pioneering work in 1958, Karplus and Luttinger 28 pointed out the existence of an additional term in the velocity operator in ferromagnetic materials, that can give rise to AH effect. Later, this contribution was identified as the effect of Berry phase 29,30 acquired by Bloch electrons moving in a periodic potential of a crystal with spinorbit interaction with the lattice. The semiclassical dynamics of Bloch electrons including the Berry phase may be derived from the Bloch Hamiltonian Hk = n (k)+V (419) where (k) are energy bands including the effect of spinorbit interaction and ferromagnetic polarization, and V is the applied external potential such that VV = eE. The semiclassical equations of motion are k =eE +er x B (420) F = Vk nk where B is the applied magnetic field and 0 is an effective magnetic field in k space arising from the Berry phase. Q(k) Vk x X(k) (421) where X = drfu* (r) iVu (F) the Berry vector potential and u are the Wannier cell functions for the unit cell of the crystal. The additional term in the velocity operator in equation (420) leads to a Hall current given by H = e2n()x (422) implying an AH conductivity given by SAH =e 2n(,) (423) where () = n 1,, n,(k) f(FS) is the average of Berry magnetic field over all occupied states in kspace. The average is zero unless time reversal symmetry is broken as it is in a ferromagnet where there is spinorbit coupling between the spinpolarization and orbital motion. The important point is that the AH conductivity due to the Berry phase does not require any impurity scattering and is independent of mean free path and hence conductivity. Thus we have the Berry phase contribution to the AH resistivity proportional to square of the longitudinal resistivity. p ~ PXM (424) which is the same dependence found for side jump scattering (equation 418). Anomalous Hall Effect in Ferromagnets with Localized Moments For the sake of completeness we discuss models where the magnetic electrons (either d orfelectrons) are not itinerant but rather localized at the ions and the charge carriers (s electrons) are equally distributed between states of opposite spins. The electron scattering is by thermal disorder in the localized spin system through direct spinspin interaction also known as sd interaction. Although the sd interaction explains the resistivity of rare earth metals 31, it does not give rise to an AH effect. It was shown by Kondo 32 that the sd interaction is anisotropic, but it gives no skew scattering if the orbital ground state of the d orfmagnetic electrons is degenerate. On the other hand, the anisotropy disappears when the ground state is nondegenerate. Thus a sd interaction between the spin of a conduction electron and a spin angular momentum of an incomplete d orf shell cannot give rise to skew scattering and hence AH effect. To explain the AH effect in these systems, Kondo introduced an intrinsic spinorbit interaction, which is a relativistic effect arising from a magnetic field appearing in the rest frame of the electron as the electron moves past the nucleus (equation 47). This kind of spinorbit interaction favors antiparallel alignment of orbital angular momentum and the spin. The intrinsic spinorbit interaction allows odd powers of spinspin interaction appropriate to a degenerate ground state to appear in transition probabilities and gives rise to skew scattering. Kondo 32 obtained an expression for Hall resistivity in this situation proportional to third moment of magnetization fluctuation as follows: p (M (M))) (425) This fluctuation function can be evaluated only under special conditions. At T=0, the spin fluctuations are zero and hence AH resistivity will be zero. In the paramagnetic region (above Curie temperature), the correlation function can be evaluated exactly 32 and the Hall resistivity is given by: p, (2J2 + 2J + 1)XH (426) where Jis the orbital angular momentum and % is the magnetic susceptibility. Thus the temperature dependence of Hall resistivity comes only from . In the ferromagnetic regime, the spin correlation function in the molecular field approximation was found to vary as the second derivative of the Brillouin function. Although this theory has had some success especially for paramagnetic substances, it has some major drawbacks. For example gadolinium, a rare earth metal is known to be in an S state and hence zero orbital angular momentum. Thus in this case the intrinsic spinorbit interaction will be zero, and hence skew scattering is not expected. However gadolinium is known to exhibit a particularly large AH effect. Another spinorbit interaction discussed in the context of AH effect is the interaction between the magnetic field produced by the localized moments and that due to the itinerant s electrons temporarily localized in the vicinity of the ions. Imagine a localized moment AM at the origin of a rectangular coordinate. It sets up a vector potential at a position vector F given by A = (M x ) / r3. The vector potential interacts with a charge carrier with momentum j as given by the term e e Hs = (.A + A.e) = .A (427) 2mc mc using V.A = 0. Introducing the angular momentum of the charge carrier about the origin L = rx p, the spinorbit term can be expressed as Hso = e M. (428) mcr Clearly, the Hamiltonian changes sign when the position vector of the charge carrier is reflected in the plane defined by AM and the primary current direction, thus giving rise to Skew Scattering. Maranzana 33 has carried out calculation using mixed spin orbit interaction and scattering by thermal disorder and for the ferromagnetic case obtained the same expression found with intrinsic spinorbit interaction by Kondo. For the mixed sorbit/dspin interaction, while evaluating the three spin correlation function in molecular field approximation, the magnetization can be factored out to leave a two spin function giving the following expression for Hall resistivity, pY ~ PmM (429) where m, is the magnetic spin disorder contribution to conductivity. Superparamagnetism We now consider a unique magnetic behavior, namely superparamagnetism, which is relevant for high resistance samples near percolation threshold. The following is a brief account based on a discussion due to Cullity 23. Consider an assembly of uniaxial single domain ferromagnetic particles with an anisotropy energy E = K, sin2 (0), where K1 is the anisotropy energy and Ois the angle between the easy axis and the saturation magnetization Ms. Thus if a single domain particle of volume Vbecomes small enough in size, the energy fluctuation due to finite temperature becomes comparable with AE = KIV, the energy barrier associated with the reversal of magnetization. In this situation the magnetization of a particle given by / = M,V can be reversed spontaneously even in the absence of an applied magnetic field. Thus we have a situation similar to that of paramagnetic material where thermal energy tend to disalign the magnetic ordering and an applied field will tend to align them. However an important distinction is that each ferromagnetic particle can carry an enormous magnetic moment compared to the case of paramagnetism due to atoms or ions, and hence the name superparamagnetsim. This also leads to the saturation of magnetic moments in realistic magnetic fields even at room temperatures, which is impossible in ordinary paramagnetic materials. If K1=O, so that each particle in the assembly can point in any direction and the classical theory of paramagnetism will apply. The magnetization of a superparamagnetic system is thus given by M = nhL(uB/kT) (430) where n is the number of particles per unit volume, and L is the Langevin function given by L(x) = coth(x) 1/ x. Thus magnetization curves measured at different temperatures will superimpose when plotted as a function of B/Tand there will be no hysterisis. We note that superparamagnetism as described above, will disappear and hysterisis will appear when magnetic particles of certain sizes are cooled below a particular temperature or for a given temperature the particle sizes are increases beyond a particular diameter. These critical values of temperature and size are determined by rate at which thermal equilibrium is approached. For uniaxial particles, detailed analysis predicts the critical volume and temperature as given by 25kT Vc = (431) K1 K,1V TB = (432) 25k, TB is known as the blocking temperature, below which magnetization will be stable. CHAPTER 5 QUANTUM CORRECTIONS TO TRANSPORT PROPERTIES IN METALS Transport properties of metals at low temperatures are drastically modified due to presence of disorder 34 35, which leads to nontrivial quantum mechanical effects that cannot be described in terms of classical Boltzmann transport theory. To fully understand the nature of disordered conductors, two new concepts were introduced and have been studied extensively for the last five decades. The first concept is that of Anderson localization, which deals with the nature of a single electron wavefunction in the presence of a random potential. The second concept deals with the interaction among the electrons in the presence of a random potential. These quantum effects become manifest experimentally in the temperature dependence of conductivity, magnetoresistance and Hall effect measurements and are more pronounced in low dimensional systems like films and wires. V VV (a) (b) Figure 51: Typical wave functions of conduction electrons in presence of disorder;(a) extended state with mean free path 1; (b) localized state with localization length E. In the presence of a distribution of random impurity potentials, a conduction electron looses phase coherence at each elastic scattering event on the length scale of the mean free path I but the wave function remains extended throughout the sample (Fig.5 la). This is the definition of weak disorder. In 1958 Anderson pointed out that if the disorder is sufficiently strong, the wavefunction of the electron may be localized and can be described as a bound state arising because of deep fluctuations in the random potential. This is the strong disorder limit when the envelope of the electronic wavefunction decays exponentially from some point in space on a lengthscale (the localization length) (Fig.51b). An important point to note here is that a linear combinations of infinitely many localized orbitals will not produce an extended state as in the weakly disordered case. Thus entirely different quantum mechanical processes govern two limits of weak and strong disorder and the understanding of what happens in the intermediate region where the cross over occurs has lead to one of the most debated and extensively studied topic in condensed matter physics, namely the metalinsulator transition. We present below a brief account of how transport properties are modified due to quantum properties of conduction electrons in the presence of varying degree of disorder. Weak Localization Effects A conduction electron in a metal can be treated as a classical particle only in the limit kFl >> 1, where kF is the Fermi momentum and / is the elastic mean free path. At low temperatures when all inelastic processes like phonon scattering are quenched, the conductivity of a metal is dominated elastic scattering of electrons from impurities and is expressed by the Drude expression which in 2d is given by, 2 C2d (kFl) (51) h where e2/h is a universal number of the order of 25kf and kFI is a dimensionless quantity that determines the degree of disorder. Following is a simple and elegant reasoning 35 to obtain an order of magnitude estimate of the quantum corrections to the above classical Drude expression for the conductivity for noninteracting electrons and are known as weak localization corrections. A.. ,B (a) AX A oB /\ (b) x Figure 52: Motion of electrons in presence of impurities represented by X (a) Two different paths for an electron to move from point A to B (b) A self intersecting path with two possibilities of traversing the loop. Consider an electron moving from point A to point B along various paths while being scattered from impurities (Fig. 52a). Quantum mechanically the total probability for the particle to reach from one point to another is the square of the modulus for the sum of the amplitudes A, of individual paths: 2 W = ZA =IA2 + AA (52) I I J The first term describes the sum of probabilities of traveling each path, and the second term represents interference of various paths. Associated with each path of amplitude A, B there is a phase given by A(p = h Jk dl, which depends on the length of each A trajectory. Hence while calculating the total probability W, if we consider many distinct trajectories, the net interference term in Wwill be zero because of the wide distribution of the individual phases. The above argument for neglecting the interference term does not hold for certain special trajectories that are selfintersecting trajectories that contain loops (Fig. 52b). For each loop there are two amplitudes A1 and A2 corresponding to opposite direction of traversal of the loop. However the phase does not depend of the direction of traversal of loop and remains same the same, hence for the two amplitudes A] and A2 are coherent. Thus for a loop trajectory, the quantum mechanical probability to find the electron at the point O is given by, A1 + A2 = A1 A2 + A1A2 + A2A1 = 4 A2 (53) which is twice as large if interference is neglected (classical situation). This simple example demonstrates that due to quantum mechanical nature of electron paths, interference effect for selfintersecting paths increases. Thus the probability of an electron leaving point A and reaching point B decreases, which leads to an increase in resistivity. The relative magnitude of this correction due to interference effect is determined by the probability of selfintersecting trajectories, which can be estimated as follows. The quantum mechanical path of an electron can be visualized as a tube of diameter of the order of its wavelength A 1/kF. The mean distance traveled by an electron diffusing through a configuration of impurities for a certain amount of time t, which is much larger than mean collision time r from impurities, is given by V_7 = (Dt)1"2, where the diffusion constant D ~ Iv For a strictly twodimensional metal film, the area accessible to an electron is Dt. In order for selfintersection to occur during a time dt it is required that the final point of the electron path enters the area element vdtA. The probability of this event is the ratio of the two volumes. The total probability of selfintersecting paths is found by integrating over the entire time t. The lower limit of the integration is r, which is the shortest time for the concept of diffusion to apply. To put an upper limit to the integral one assumes that there are inelastic processes like electronphonon and electronelectron interaction that lead to phase relaxation and hence break down the amplitude coherence. Lets denote this time scale as phase relaxation time rT,.. The relative change in conductivity for the 2d case is given by Ao2d vdt (54) (54) 2d I Dt The negative sign illustrates that conductivity decreases due to interference. The change in conductivity due to quantum corrections for a thin film is given by 35 22d eln() 2 L A 2d In n(' (55) hi T h I where L, ~ VT. Equation (55), although derived for a strictly 2D case, is applicable for thin films of thickness d < L(. The phaserelaxation time r, has a strong temperature dependence of the form Tr ~ TP, where in an integer that depends on the exact phase relaxation mechanism. For example p 1 for electronelectron interaction with small energy transfer 36, which is the dominant phase relaxation mechanism in low dimensions. Thus in general for a metallic film, the conductivity will have logarithmic temperature dependence due to interference corrections and is given by, 2 2d, + p ln(T) + const. (56) h Magnetoresistance due to Weak Localization The above correction is drastically modified if one places the sample in a magnetic field. For a vector potential A describing the magnetic field, one should replace the momentum 3 by the canonical momentum p This results in a phase difference c in the amplitudes A1 and A2, for traversing loops (Figure 52b) in opposite directions as given by 2e A(H = Adl = 2 (57) ch OQ where Ois the magnetic flux enclosed by the loop and o=hc/2e is the flux quantum. The appearance of a phase difference results in the destruction of the interference and hence a decrease in resistivity. To estimate this negative magnetoresistance due to an applied magnetic field, we introduce a new time scale H Since the average diffusion length is (Dt)12 and using it as a characteristic size of loops, the magnetic flux through such loops is D HDt. We define rH so that A(H ~ 2i that gives TH (58) The characteristic magnetic fields are determined by the condition so that The characteristic magnetic fields are determined by the condition r. z, So that H (59) Dr Substituting D ~ Iv ~ ET / m, and using o)B = eH / mc, we find that oB) << 1. This means that one needs small magnetic fields. The asymptotic formula for quantum corrections in presence of magnetic field Ac2d (H), for H is much larger than characteristic field given in equation (59), is obtained by replacing r, by TH in equation (55). Thus for the 2d case the magneto conductance is given by 35, e2 eHDr o(H) o(0) = AC2d (H) A2d (H = 0) ln( ') (510) h he In the above formula H refers to the component perpendicular to the film. The component of applied field parallel to the film does not affect the weak localization correction in a 2d sample as the magnetic flux does not penetrate through any closed electron paths. Weak Localization in Presence of SpinOrbit Interaction Quantum interference depends significantly on electron spin if there exists a scattering mechanism leading to a flip of electron spin. The following is a brief description of an estimate of the interference correction in the presence of spinorbit scattering. Since the spin is not conserved and can flip while the electron moves from one point to another, one needs to consider all possibilities of initial and final spin while calculating the interference term of the selfintersecting loops (Figure 52b). If the initial and final states are given by the wave functions (p and (p respectively, the interference term is given by C = AA = (1)(2)P*(1)*( (511) C=AA2 = P vv (511) 2 r where the sum is taken over the final spin / and the average is taken over the initial spin state a. To simplify the expression, instead of assuming two possible trajectories along the loop (Figure 52b) for a single particle, we assume two particles moving simultaneously in opposite directions. The interference term can be written in terms of eigenfunctions of total spin denoted by af0 for total spin zero and 1,, for spin and projections m=1,2,3 as given by V0 1 (1)9 @(2) (1) (2)) (1)2)(512) 1 (1) (2) 1 ( ( 1)9(2) (+ (1) (2)) Thus the interference term in equation 511 can be expressed as follows, C = ( 2 2) (513) In the presence of spinorbit scattering with characteristic times To << r,, the states yli that carry spin information are damped with a characteristic time ro, while V0 is damped with time r,. Therefore for the 2d case it was shown 37 AO2d r 2vdt 3 et/l, 1) (514) O2d Dt 2 2 Thus depending on the spinorbit scattering time rT, relative to r, we have the following two cases for 2d samples 37 e2 Z Ao2d n(Y) for ro >> (515) h r e2 3 1 Ac2d (ln( ~)+ ln( )) for r,, << r (516) h 2 r 2 r thus it follows from equation (516) that the spinorbit correction reverses the sign of the temperature dependence of conductivity due to r, In weak magnetic fields and under strong spinorbit, the magnetoresistance becomes positive. For sufficiently strong magnetic field such that rH < r, the magnetoresistance changes sign and becomes negative. The combined effect of positive quantum correction to conductivity and positive magnetoresistance is known as 'weak antilocalization'. If however, scattering occurs from paramagnetic impurities, then both the singlet and the triplet wavefunctions in equation (513) decay with time of the order of r,, so that for r, << r, the corrections to conductivity are no longer temperature dependent. Weak Localization in Ferromagnetic Films We now review the weak localization corrections in ferromagnetic materials with strong spinorbit scattering based on a theoretical treatment by Dugaev et. al 38. The relevant Hamiltonian considered in this case is given by H = d3F r* [ M + V(F) ( x VV(F)) V]/(F) (517) 2m 4 where ./ ( ,,V T) is a spinor field ( = (ox, cy, a, )are the Pauli matrices, Mis the magnetization assumed to be along zdirection and Vis the random impurity potential, and Ao is the spinorbit scattering strength. The exchange term Mo acts only on the spins and has no direct effect on the orbital motion. In the language of manybody theory, weak localization corrections arise from the particleparticle channel with two propagators describing electrons with vanishing total momentum and with very close energy (Cooper channel). Physically, the socalled "Cooperon" propagator represents two electrons traversing a self intersecting loop (Figure 52b) in opposite directions. There are two possible situations depending on the relative spin orientation of the two electrons. In a ferromagnet the exchange energy is strong so that M >> rT r1, where rz and r, are momentum relaxation times for spin up and spin down conduction electrons. Physically, this condition implies that spinflip processes are suppressed due to the exchange field M, and it was shown that for a bulk 3d ferromagnetic sample, the contribution to the "Cooperon" from the singlet pairs i.e. electrons with opposite spins is small compared to that of triplet pairs of electrons with parallel spins by the factor 1/Mr(,). The exclusion of the singlet channel is crucial and leads to the absence of weak antilocalization in ferromagnets and the weak localization correction to the conductivity is found to be a direct generalization of the non magnetic case with two bands of electrons of opposite spin polarization. We are interested in the two dimensional ferromagnetic samples and quote the final result in two dimensions 38 as follows, e 1 1 1 1 Ac2d ({ln[, + )] +ln[r (+ )]} (518) 4;r rh, ,, so where r()is the temperature dependent phase breaking time. r,) is the momentum relaxation time and Trso(, is the effective spinorbit scattering time for spin up (down) electrons and depends on the relative orientation of the magnetization M with respect to the plane of the film. The important point is that for r << rzs, r, the above correction to conductivity is negative just like the nonmagnetic case. Electron Interaction Effects In this section we discuss the corrections to conductivity due to interactions between conduction electrons. We note that within the Boltzmann transport equation, electronelectron collisions cannot affect the conductivity in the case of a single band structure and in the absence of Umklapp processes. This is because electronelectron collisions conserve the total momentum. Inclusion of the Fermi liquid corrections that takes into account a finite inter electron interaction potential, renormalizes the residual conductivity but does not introduce any essential temperature dependence. However, taking into account the interference of elastic scattering by impurities with electron electron interaction produces nontrivial temperature dependence of the conductivity and the one particle density of states. The following is brief account of some simple physical ideas that illustrates the origin of quantum corrections to transport properties due to electronelectron interaction. Scattering by Friedel Oscillations We now discuss a very important concept of the Friedel oscillations that arise due to standing waves formed as a result of interference between incoming and backscattered electron waves. To illustrate this we consider a simple situation in ID with an infinitely high barrier at x=0. For each wave vector k, the wavefunction is a superposition of the incoming plane wave exp(ikx)/ V and a reflected wave exp(ikx) / and is given by / = 2i sin(kx) / Accordingly, the probability density is given by y]2 = 4 sin2 (kx) / L and oscillates in space. The probability that a state with momentum k is occupied is given by the Fermi functionfk. The electron density is described by n(dk 2 dk sin(2kFx) n(x)= 2 fk Vf =8j sin 2(kx)=n, (519) k 2;r 2;r Rx where no = 2kF /zI is the density of the homogeneous electron gas, is oscillatory and damps from the origin as x1. C B  A Figure 53: Schematic diagram of Friedel oscillation due to a single impurity due to backscattering described by path C. Interference between two paths A and B contributes mostly to backscattering. A single impurity at the origin with a general potential U(F) also induces a modulation of electron density close to the impurity. The oscillating part of the modulation in 2D(shown schematically in Fig. 53) is given by 39, vA 3p(F) = 2 sin(2kFr) (520) where r is the distance from the impurity, which has a potential treated in the Born approximation A = JU(F)dF, and v = m/2 is the density of states in 2D. Taking into account the electronelectron interaction Vo (r r,) one finds additional scattering due to the Friedel oscillation. This potential can be presentation as a sum of the direct (Hartree) and exchange (Fock) terms 39 V(rF,r = F VH(rF)8(r rFV(rFrI2) VH () = drV,( F,)3p(F) (521) 1(  VF(F7r) =I V,(r7 rFsn(r7, r2 where p(F) is the diagonal element of the one electron density matrix n given by, n(F,F2)= I T (1 )k (F,) (522) k The factor 12 indicates that only electrons with the same spin participate in exchange interaction. As a function of the distance from the impurity, the HartreeFock energy oscillates similarly to the Friedel oscillation. The leading correction to conductivity is a result of interference between two semiclassical paths as shown in Fig.53. If an electron follows path A, it scatters off the Friedel oscillation created by the impurity and path B corresponds to scattering by the impurity itself. Interference is most important for scattering angles close to 7t backscatteringg), since the extra phase factor on path A is cancelled by the phase of Friedel oscillation exp(i2kFR), so that the amplitudes corresponding to the two paths are coherent and interfere constructively. This interference persists to large distances R and is limited only by temperature, R 1/ lk kF < hvF kT At finite temperature the amplitude of the Friedel oscillation assumes temperature dependence. Explicit calculations for the scattering amplitude as function of scattering angle exhibit a sharp peak for back scattering with a width and height proportional to JV The correction to conductivity with respect to the classical Drude conductivity is given by 39, S= v[V (0) 2V (2kF)] (523) O SF where the first term is the exchange correction and the second term is the Hartree correction. The above is the conductivity correction in the ballistic limit defined by T << 1. We note some important points in the above expression. Firstly, the sign of the Hartree and exchange corrections are opposite. Secondly, the leading temperature correction comes from the Fourier component at q=0 and q=2kF for the exchange and Hartree term respectively. The sign of the total correction is not universal and depends on the details of the electronelectron scattering. B A Figure 54: Friedel oscillation due to two impurities created by the selfintersecting path C. Scattering at all angles are affected by interference. So far we have considered the effect of single impurity. For the case of multiple impurities, the Friedel oscillations can occur from selfintersecting paths of electrons. In Fig. 54 we show scattering process that involves two impurities and the resulting Friedel oscillation due to path C shown in dashed lines. In this case the scattering amplitude at all angles and not just the back scattering are affected. Scattering by multiple Friedel oscillations have been calculated in the framework of many body theory and is known as the AltshulerAronov 35 correction. Although interaction between two electrons is independent of spin, summation of terms in the perturbation theory depends on the spin state of the two electrons involved. The total number of channels is 4. These channels are classified by the total spin of the two electrons; one state with total spin zero (singlet channel) and three states with total spin 1(triplet channel) differing by the projection of the spin. For long range interaction the perturbation theory for the Hartree corrections singlet and triplet channels is different. The singlet channel contribution combined with exchange corrections as a renormalization of coupling constant and the final result is still universal. The triplet channel contribution depends on the Fermiliquid constant F The total conductivity correction in 2D is given by 35, 39 e2 ln(1 + F) = In( )[1+ 3(1)] (524) 272 h T Fo The above equation was derived for the socalled "Diffusive regime" characterized by Tr << 1. The sign and magnitude of the correction is nonuniversal and depends on the competition between the universal and positive exchange term and the coupling dependent and negative Hartree contribution. For shortrange electronelectron interactions due to screened Coulomb interactions in good metals, the Hartree term is neglected and one has a universal correction to conductivity due to the exchange term. The above result is obtained by treating interaction in lowest order perturbation theory and in weak impurity scattering regime. Now we quote some results of scaling theory of interaction problem that goes beyond the perturbation theory. For a two dimensional conductor with spinsplit bands like in ferromagnet and long range Coulomb interaction, the scaling theory predicts a universal correction 40 given by 2 Ac = (2 21n 2)ln(Tr) (525) The above results were obtained for the case when weak localization corrections are completely suppressed. For short range electron interaction the logarithmic coefficient is found to be nonuniversal 40 Moreover, for 2d conductors with strong spinorbit or spinflip scattering, with long range Coulomb interaction, it was argued 41 that the triplet channel is suppressed and only the exchange term survives, predicting a universal correction given by, A = In(Tr) (526) 27r2h Magnetoresistance due to Electron Interaction We have discussed earlier how even a weak magnetic field can suppress the localization effects resulting in a negative magnetoresistance. In the language of many body theory, weak localization effects arise from the particleparticle channel and are supposed to be sensitive to magnetic flux. The electron interaction effects arise from the particlehole diffusion channel and do not have similar sensitivity to magnetic field. The dominant effect of magnetic field in this case is the splitting of the spin up and spin down bands 34. This physical idea is most simply illustrated for the selfenergy correction, where the singular correction is due to the correlation between the wavefunction of the added electron and the wavefunctions of the occupied electrons that are nearby in energy. In the presence of a magnetic field, the triplet term is divided into an Sz=0 and two S, = 1 terms. The exchange (singlet) and the Sz=0 triplet terms involve correlation with electrons with the same spin and are unaffected by the spin splitting. For the S, = +1 terms, the spin splitting produces a gap guH between the lowest unoccupied spinup and the highest occupied spindown electron. The singularity of that term is cut off for gBH greater than kBT. In a magnetic field, the correction to the conductivity can be written as a sum of two terms, So(H, T)= oSc (T)+ &s (H, T) (527) The first term is the field independent "charge channel" contribution which is the sum of exchange and Sz=0 Hartree contribution, is same as equation (521). The second term is the S I = 1 triplet contribution or the "spin channel", with a field dependence for the 2d case given by 4, 2 In(l+Fc) Inh h>>1 s(HT)9s(o0,T) =e (1  )In h 1 (528) 27r2 F0o h2 h <<1 where h = gJBH / kT The quantity in the parenthesis is the Hartree contribution to conductivity and is nonuniversal in both sign and magnitude and depends on the details of the potential describing electronelectron interaction. Transport Properties of Granular Metals So far we have discussed quantum transport properties of homogeneous systems with a uniform distribution of impurities that determine the mean free path. In this section we discuss the transport properties of granular metals with Coulomb interaction between electrons. The motion of electrons inside each grain is diffusive and they can tunnel from one grain to another. In principle the grains can be clean so that electron scattering is mainly from surfaces. In this limit the tunnel conductance is smaller than the grain conductance and intergranular transport can be distinguished from intragranular transport. The process of electron tunneling from grain to grain that governs the transport properties are accompanied by charging of grains. This may lead to Coulomb blockade especially in the limit of weak coupling between grains. A step towards formulating a theory on transport in granular medium was due to Beloborodov et. al.42. It was shown that depending on the dimensionless tunneling conductance gr one observes exponential (at gr<<1) or logarithmic (at gT>>l) temperature dependence of conductivity. This theoretical approach was based on an earlier paper by Ambegaokar 43, which however was applicable only at temperatures T > g,3, where dis the mean energy level spacing in a single grain. In this regime the electron coherence does not extend beyond the grain size. The low temperature regime T < g,3, where the electron moves coherently over distances exceeding the single grain size, was discussed in a later paper by Beloborodov et.al42 for large tunneling conductance gr. The following Hamiltonian describes a system of weakly coupled metallic grains, H = Ho + HC + t [P (r,)(r)+ (r) (r,)] (529) where t,, is the tunneling matrix element between ith andjth grain, Ho is the Hamiltonian for noninteracting isolated grains, and He describes the Coulomb interaction inside (i=j) and between grains (isj) as described by 2 H, = e ,Clhj (530) 2 where C, is the capacitance matrix and i, is the operator of electron number in the ith grain. Beloborodov et. al. 42 show that in the low temperature regime, properties of the granular metal depend on dimensionality and corrections to conductivity and density of states due to Coulomb interaction are similar to those obtained for homogeneous metals. The critical grain size in 3D where a metal to insulator transition occurs is estimated to be, 1 Ec g = ln( ) (531) 67r 3 where Ec is the charging energy of an isolated grain. The conductivity of a granular metal is given by o = o + SO,1 + 2 (532) The classical Drude conductivity for a granular metal in a general dimension d, with grain size a is given by 2 o, = 2ga2d (533) h The correction to conductivity due to large energy scales E > g,8 is given by 90 1 gEC ln[ E (534) ao 2igd max(T, g 3) We note that the dimensionality in this case appears only as a coefficient but the logarithmic temperature dependence remains same for all dimensions. This means tunneling of electrons with energies E > g,3 can be considered as incoherent. On the other hand corrections from low energy scale E < g,8 arises from coherent electron motion on the scales larger than grain size, and is given by a d=3 12a T TT 12i2g1 g13 1 gd = I=d In( ) =d 2 (535) S 4i 2g T 4.V Tg, where ac and 3 are numerical constants. We summarize the results for a 2D granular system for T < g,8 as follows, 0 [4.'"2gT + iln() +ln(  )] (536) 22"2h Ec gr8 and for T > g, 2 T S= [42 2gT + ln( )] (537) 2 2h g ,E For samples with weak inter grain coupling g, <<1 at low temperatures T << E, the conductivity was shown to be 42 a = 20, exp(Ec /T) (538) where Ec is the charging energy. However this behavior is usually not observed experimentally because of the distribution of grain sizes in real samples, as discussed in the next section. Transport in Weakly Coupled Granular Metals A theory of transport in granular metallic films was developed by Sheng et.al 44 in the limit of weak coupling between grains. They proposed a picture of granular metal represented by a conductance network in which the metal grains are interconnected by conductances of the form: a, exp(2s E' /2kT) (539) where s is the tunnelbarrier thickness and Z = (2m(p/h2 )1/2 for a barrier height of p . The calculations are simplified by assuming the grains to be spherical with a distribution of grain diameters d and the charging energy E 1/ d, such that the product sEc is constant for a given film: sEc = C, where C and x are constants that depend only on the volume fraction of the metal. The second assumption was to include only tunneling between nearest neighbors, which are nearly equal in size. This implies that for a given temperature there is an optimum tunnelbarrier thickness given by s, = (C / k T)/2 / 2 for which the inter grain conductivity is maximum. The final assumption is that the temperature dependence of the conductivity network is given by that of the maximum intergrain conductivity so that, o(T) exp[2(C/ kT)'/2] (540) The constant C is proportional to the charging energy Ec, which is inversely related to the mean grain diameter temperatures is due to tunneling between small grains (large Ec) separated by thin tunnel barriers, while at low temperatures the dominant contribution is due to large grains separated by thick tunnel barriers. Now we consider the case for ferromagnetic metallic grains so that in addition to the charging energy Ec there is a magnetic exchange energy EM associated with a tunneling event. The exchange energy arises when the magnetic moments of the participating grains are not parallel and electron energy is conserved during tunneling. The intergrain conductance in this case is given by 45 1 1 o, exp(2xs){ (1+ P)exp[(E + E,)/2kT]+ (1P)exp[(E E)/2kT]} 2 2 (541) Here P is the polarization of tunneling electrons, so that the coefficients (1+P)/2 and (1 P)/2 are the probabilities that an electron tunneling from one grain to another has its spin parallel and antiparallel, respectively, to that of the initial grain. Using the same assumptions for the nonmagnetic grains discussed earlier, the magnetoconductivity is 45 given by 45, o(H, T) = o(0, T)[cosh(E, / k T) P sinh(E, / kT)] (542) The magnetic exchange energy can be expressed in terms of spin correlations of two neighboring grains and is given by 45 E = J[ (S,S,)/S2] (543) An important point to note is that by applying a magnetic field strong enough to align all the moments, the exchange energy is zero, and the temperature dependence of conductivity reduces to that of the nonmagnetic case shown in equation (540). Quantum Corrections to Hall Conductivity The normal Hall coefficient defined as R, = E / JH, is another quantity in addition to magnetoresistance that behaves differently for weak localization and electron interaction effects. The quantum correction to the Hall conductivity due to weak localization effects was first calculated by Fukuyama 46 and it was shown that the Hall resistance given by R" = R,B remains unchanged so that R" / R" =0 XY XY (544) Thus in a disordered conductor subject solely to weak localization effects, the normal Hall resistance at given magnetic field will remain constant as temperature is varied. The longitudinal resistance R, will have the usual logarithmic temperature dependence due to interference effects. The normal Hall conductivity has logarithmic temperature dependence and has a slope twice that of longitudinal conductivity. This is easily deduced using the fact that o," R" /R' and taking logarithmic derivative we have n=_ R 2 'R (545) ay R' R Thus it follows from equation (544) that the normal Hall conductivity has logarithmic temperature dependence with a slope twice that of longitudinal conductivity as shown by the relation, S= 2 = 2 (546) Oxy R o For the case of only electron interaction effect (no weak localization) in the diffusion channel it was shown by Altshuler et. al.35 that the normal Hall conductivity has zero quantum correction and hence remains independent of temperature so that, 5r //oI = 0 (547) Thus it follows from equation (545) the effect of only electron interaction implies that =2 (548) R" R This means that both R and R will exhibit logarithmic temperature dependence due to This means that both R, and Rn will exhibit logarithmic temperature dependence due to xy electron interaction, and the slope of R" will be twice that ofR,. Quantum corrections to the anomalous Hall conductivity have not been studied as extensively as the normal Hall conductivity. The effect of shortrange electron interactions to the AH conductivity within the framework of the skew scattering mechanism were first studied by Langenfeld et. al. 47 who showed that there is no finite correction due to the exchange(Fock) part of interaction, so that 3]?AH R oAH = 0 =2 (549) Y RAH R Thus the AH effect was found to have a behavior similar to that of the normal Hall effect. We note that the above calculation does not include the contributions from the Hartree terms to the interactions and also assumes the absence of weak localization corrections. The above theoretical prediction (equation 549) was found to be in good agreement with experimental results of Bergmann and Ye 48 where the AH conductivity in thin amorphous films of iron was found to independent of temperature. Weak localization contributions to the AH conductivity were studied by Dugaev et. al. 24 for the Hamiltonian given by equation (517). For the case of side jump (SJ) mechanism in a 2d ferromagnetic sample it was predicted 24 that 3oAH(SI) xy (550) o.YHI (kFl)3 The corresponding weak localization correction to longitudinal conductivity is given by ,x / a, c~ (kFl) 1. We note that the weak localization calculations are valid in the metallic regime that corresponds to kFl >> 1. Thus it was concluded that the weak localization correction to AH conductivity due to side jump mechanism is negligible 24 3goAH(SJ) 4H << (551) XH(SJ) The above theoretical prediction provided an alternative explanation24 to the experimental results ofBergmann and Ye 48 For the case of skew scattering(SS) mechanism, the weak localization corrections were shown 24 to give rise a finite correction as given by AH(SS) A vok t ln[r ( +1 )]k, vIn[r( + )]} d0AH FTss FS F 36zh r rsoZ Tso (552) The above expression was derived for the case of a random shortrange impurity potential of the form V(F) = vo 3(F R ) is assumed. Within this model, the contribution to the AH conductivity solely due to impurity scattering, in the absence of weak localization or electron interactions was shown 24 to be, C.AH(SS) 2 vk v 2 k2 2 (553) 18 v {kFV Vr (553) Using the fact that the residual longitudinal conductivity is given by n^.2 z,, 2T a = 2 (554) m m and assuming a parabolic band, one can simplify the expression for weak localization correction to the AH conductivity for each of the two bands with opposite spin polarization and is given by the following expression, SAH(SS) 1 e2 1 1 1 AH 2 22 ln[rj,)( + )] (555) A H(SS) 2 2 x xt() ,,t( ) 'so(1) 63 Comparing the above equation with equation (518) it follows that both the longitudinal and the AH conductivity have finite logarithmic corrections due to weak localization. CHAPTER 6 EXPERIMENTAL RESULTS AND DISCUSSION Experimental study of magnetism in ultrathin films of ferromagnetic transition elements like iron, cobalt and nickel pose a serious challenge because of their air sensitive nature. Previously, Bergmann and Ye 48 have reported, in situ transport measurements on pure amorphous iron films few monolayers thick, which were quench condensed on antimony substrates at liquid helium temperature. These experiments revealed an important finding that the Anomalous Hall effect behaves similar to that of normal Hall effect in nonmagnetic materials and has no quantum correction at low temperatures due to electronelectron interactions or weak localizations effects. However, one might argue that the presence of a "polarizable" substrate, namely antimony, might dope the few atomic layers of iron on top and these might affect its properties. We undertook a study of magnetic properties of thin films of iron and cobalt grown on inert glass substrates, ordinarily used as microscope slides. Using the SHIVA apparatus described in chapter 2, we were able to investigate the magnetic properties of iron and cobalt films with polycrystalline morphology, while protecting them from oxidation, using the Anomalous Hall 2 measurement. The following is a detailed account of our experimental findings. Experimental Details Sample Fabrication Iron and cobalt samples were grown by r.f magnetron sputtering techniques under identical growth conditions. We used r.f. power of 35W with an argon flow of 10 seem, which developed a DC bias of around 145V with respect to target. The pressure in the chamber is of the order of 104 Torr. The samples were grown in the Hall bar geometry through a shadow mask onto glass substrates at room temperatures. Under the conditions described above, films are known to grow through various stages of different morphology 49 rather than gradual layer by layer growth. Initially, film growth proceeds via nucleation of isolated grains of metals. With more arriving adatoms the grains continue to grow in size and at some critical thickness, the grains coalesce into several discrete and continuous percolating channels. As the substrates are exposed further the film eventually becomes homogeneous with wellconnected microscopic grains, such that the film resistivity scales with thickness 17 Such a film behaves like a good metal with a low temperature residual conductivity determined by impurities and imperfections. Even in the case of metallic grains not physically touching each other, there could be electrical conduction due to tunneling of electrons between grains. The polycrystalline films in our experiments were thicker than the quenchcondensed films in the previous investigation 48 by Bergmann and Ye, but are outside the homogeneous regime where film resistance scales with thickness. Thus resistivity is not a welldefined quantity, and we use sheet resistance, to characterize our films. The sheet resistance is defined as the resistance of a square film and is independent of the lateral dimensions of a film and depends only on thickness and morphology. In the Hallbar geometry, where the sample is rectangular shaped, where the current I is uniformly distributed along the width W and where voltage Vis measured between leads separated by L. The number of squares involved in such a measurement is L/W. The sheet resistance is given by dividing the measured resistance R=V/I by the total number of squares so that R, = R/(L / W). Measurement Setup We have investigated films over a wide range of sheet resistances from 50Q to 1000000Q. All our samples are in the Hall bar geometry with six terminals as shown in Figs. 6land 62. We performed standard four terminal techniques using separate pair of leads for sourcing current in the sample and measuring transverse and longitudinal voltages, to eliminate the effect of contact resistances. We have used two different experimental setups to measure longitudinal and transverse resistances simultaneously, depending on the magnitude of the two terminal resistances of the leads. Keitliley 236 Keithley 182 Figure 61: A d.c. transport measurement setup using Keithly 236 for sourcing a constant current and measure longitudinal voltage and Keithly 182 nanovoltmeter to measure transverse voltage. The sample in Hallbar shape is shown as a shaded. Figure 61 shows the circuit diagram for d.c. measurement using a Keithley 236 SourceMeasure Unit and a Keithley 182 Nanovoltmeter. We programmed the 236 unit to source a constant current through the sample and measure the voltage developed across the longitudinal leads. Simultaneously the Keithleyl82 is used to measure the voltage developed across the Hall leads. This setup is particularly useful for measuring high resistance samples as the Keithley 236 is equipped with guarding buffers which increase the input resistances of source/sense leads to greater than1014 and also reduce cable capacitance, thus leading to more accurate high resistance measurements with faster settling times. However, since the sensitivity of the Keithley 236 is only 1[ V, it is not suitable for measuring small changes in resistance. Figure 62:An a.c. transport measurement setup using two SR830 lockin amplifiers operating at same frequency to measure longitudinal and transverse resistance, used in samples with low contact resistances. A constant current is generated from the voltage source of the upper SR830 and by placing a ballast resistor Rb of 1MQ in series with the sample. The sample in Hallbar geometry is shown as shaded Figure 62 shows the circuit diagram for an a.c. measurement setup using two Stanford Research SR830 lockin amplifiers. Lockin amplifiers use phase sensitive detectors to make low noise measurement of a.c. signals of a given reference frequency with sensitivity of InV. The SR830 also has a voltage output that can be used to source constant current through a ballast resistor in series with the sample. Both SR830's are programmed to measure signals at the same frequency and phase as the current through the sample, and make it possible to measure longitudinal and transverse voltages simultaneously. The input resistance for the SR830 is only 1MQ, hence only samples with two terminal contact resistances less than 10kQ are measured using this setup. Weak Disorder: Iron Films Transport Properties at B=O We monitor the resistances of all samples during growth, which allowed us to grow samples with specific sheet resistances and transfer them to the cryostat for magnetotransport measurements. As a measure of the disorder characteristic of each film, we use the sheet resistance at T=5K, which we denote by Ro. We note that in our polycrystalline films, the resistance of individual grains is much smaller than the inter grain tunneling resistance and hence the later determines the low temperature residual resistance Ro. We grew a series of iron films on glass substrates with Ro varying over a range of 500 to 50000Q. We observed a crossover in the temperature dependence of sheet resistance R,(T) as Ro is systematically increased Figure 63a shows R,(T) for an iron film of thickness d=100A and Ro=70Q. This represents typical behavior for all films with Ro<1000Q, with resistance decreasing linearly with temperature and reaching a minimum at some temperature Tm,, that shifts towards higher temperature as Ro increases. This linear decrease of resistance is typical of homogeneous metallic samples and is due to decreased phonon scattering at low temperatures. For films with Ro<10000, the grains are well connected and Ro is determined by impurities and lattice imperfections in the grains. For T the inset of Fig. 63a, which is a manifestation of low temperature quantum corrections discussed in chapter 5. 10o. eroi i ig ,1,, 100 8500 69.,"i 90 a 69.66 S 5 10 15 20/ C: 80 L , 1 5 70. (a). 69(b) 5 100 T( K 200 300 5 100 T( 200 300 Figure 63:Typical behaviors for temperature dependence of resistance for iron films in the absence of magnetic field for (a) Ro=70f showing a good metallic behavior with decreasing resistance with temperature and (b) R= 8400Q showing a monotonic increase in resistance with decreasing temperature. Inset of both graphs is a blow up of low temperature behavior showing logarithmic divergence of resistance in both cases. For samples with higher sheet resistances, Ro>1000Q, the resistance increases monotonically with decreasing temperature and with no minimum. Figure 63b shows R,(T) for an iron film with d=20A and Ro 8400Q. At high temperatures, the increase in resistance with decreasing temperature is due to decreased phonon assisted "hopping" processes of conduction electrons over the tunnel barriers between grains. At low temperatures the residual resistance should be due to temperature independent intergrain tunneling processes. However at low temperatures (T<15K), the resistance is found to increase logarithmically as shown in the inset of Fig. 63b, which is a manifestation of quantum corrections in the presence of tunneling processes. The crossover in the sign of temperature coefficient of resistance dR/dTin the high temperature regime, is consistent with the usual observation for thin metallic films described by the so called Mooij limit 50, 51 for film resistivity around 100[tQcm. Thus the iron films used in our experiment with Ro in the range 500 to 50000Q, exhibit a logarithmic increase at low temperatures and are considered to be weakly disordered in the context of quantum theory of transport. To compare the experimental data with existing theories on quantum corrections to conductivity in 2D metals, we use the following functional form to fit the data at low temperature (T=4.5 15K): 1 L, ARLoo In(T) + const. (61) R, where Loo = e2 / 2.2h = (8 lk) ',the quantum of conductance and AR is a numerical prefactor that depends on the microscopic scattering parameters that determine the quantum corrections. Figure 64 shows a plot of AR as a function of Ro. We note that there is a distinct crossover in the value of AR as Ro is systematically increased beyond 3000Q. For low resistance samples, the prefactor is constant with AR 0.950.03 and did not show any pronounced dependence on Ro. However, for samples with higher sheet resistances, AR systematically decreases as Ro increases. For example, for a sample with Ro=49000Q, the prefactor AR=0.3260.001. We note that the red square data points refer to films that had been exposed to an Ar ion beam prior to the transport measurement, a process that smoothens the film surfaces 16 and results in a pronounced reduction in the resistance, as discussed in detail in chapter 3. However as seen from our data, the ionmilled films follow similar trends to those of the pristine films as far as low temperature transport property as measured by the value of AR is concerned. These data further justify the use of sheet resistance is a measure of disorder. 1 o  k A    A W' A 0.a 102 103 104 105 Figure 64:Plot of numerical prefactor AR for logarithmic temperature dependence of longitudinal conductance (equation 61) for different iron films, as a function of Ro, sheet resistance at T=5K. Red square points corresponds to ionmilled films. Anomalous Hall Effect in Iron In this section we present anomalous Hall (AH) measurements on our films at T=5K, and discuss their dependence on Ro which is a measure of effective disorder in the film. In principle, ferromagnetic samples with uniform magnetization should exhibit a transverse Hall potential due to an applied electric field, even in the absence of an external magnetic field. However, an external magnetic field is required to align the magnetic domains thereby maximizing the magnitude of magnetization and hence the AH resistance. In most samples, there may be a small unavoidable misalignment of the Hall leads, as a result of which a fraction of the measured potential between the Hall leads (Vxy) will be due to the longitudinal potential drop along the sample. Hence, to properly calculate the Hall potential, we scan the magnetic field in both positive and negative directions to measure the transverse potential Vxy(B), keeping the sample at a fixed temperature. The Hall potential is extracted from the raw data as the antisymmetric part given by VH = (V(B) V (B))/2. 4 1 180  S(a) (b) 440 B B 0. s % > 0 s 4440 4 2 0 2 4 4 2 0 2 4 B(T) B(T) Figure 65: The anomalous Hall curves for iron films with (a) Ro=300Q and (b) Ro 2700Q. Note decrease in B, and simultaneous increase of R, as Ro increases. Typical AH curves for two different iron films are shown in Figs 65(a) and 6 5(b) corresponding to Ro=300Q and Ro=2700Q respectively. Both Hall curves exhibit anomalous behavior with a steep rise in Hall resistance with increasing magnetic field B, due to moments lining up along the field until the saturation value at B=B, (shown by the vertical arrow), followed by a much slower increase due to the normal Hall effect. An important point to note is that for Ro=300Q, the AH curve saturates at an applied field of B,1.7T to a value Rx,8Q, while for the sample with Ro=2700Q, the saturation field B,1.2T and high field value Rxy80Q. This points out an important trend seen in our samples: as Ro increases, B, decreases while the high field saturation value of Hall resistance increases as Ro increases. However, as we will show subsequently, this monotonic dependence on sheet resistance breaks down above Ro20000. We also note that the sign of the slopes of both anomalous and normal part of the Hall curve are positive, which is in agreement with experiments on bulk iron samples 22 We have undertaken a study of the dependence of the high field saturation value of AH resistance RH on Ro, into the very high resistance regime ~1MQ, which to the best of our knowledge has not been studied experimentally. We use the following scheme to analyze our data and extract the saturation value of AH resistance, which allows us to systematically compare RH for different Ro at T=5K. As discussed earlier, the Hall resistance in a ferromagnet is the sum of the anomalous contribution proportional to M and the normal Hall effect. When the applied magnetic field exceeds Bs, the net magnetization along the field remains constant at Ms and the observed slow increase in Ry with increasing field is due to the normal Hall effect plus any background susceptibility effects. So the high field part of the AH curve can be fit to the following phenomenological expression 2 linear in magnetic field B: R, = poRMs + RnB (62) where Rs and Rn are the anomalous and normal Hall coefficients respectively in two dimensions. Thus the intercept of such a fitted straight line is the contribution to Hall resistance at zero applied field B and arises only from the spontaneous magnetization in the material. We identify the intercept as the AH resistance R, = /,R,M,, which is the contribution at zero applied field and solely due to magnetization of the film. 102 A 10'  AAA A 100 102 10 R ) 104 105 Figure 66:The anomalous Hall resistance at T=5K for different iron films as a function of R on a loglog scale. Red square points are ionmilled films. The dashed line represents the average value of the AH resistance at 800. Figure 66 shows the dependence of RH on iron samples with different Ro using a loglog scale. For iron samples with sheet resistances as high as Ro=49000Q, we observe the anomalous behavior in Hall resistance indicating the presence of a local finite magnetic moment in the films. However we observe a distinct crossover in the dependence of RH with increasing Ro. For Ro>2000Q, the monotonic increase in RH seen at lower resistances, ceases to hold. Instead the AH resistance attains a constant value RH = 80 100, independent of Ro. We point out that the crossover is observed around values ofRo where the zero field coefficient of logarithmic temperature dependence AR, starts to deviate from unity. Temperature Dependence of Anomalous Hall Conductivity We have shown in the previous section that at low temperatures (4.5 to 15K), for all iron samples, Rx(T) exhibits a logarithmic dependence at B=0. To find the quantum corrections to the AH conductivity a magnetic field of 4T, which is well above the saturation field Bs, is applied to each sample and R, (T) and R"H (T) are simultaneously measured while temperature is slowly increased. We will show that RH also has logarithmic temperature dependence for T<20K for all the iron samples, and discuss the relative resistance (RR) scaling of R, (T) and RH for each sample with different Ro. In most samples there is always a misalignment in the Hall leads as result of which a fraction of the transverse potential has contribution from the longitudinal potential drop along the sample. Similarly, there may be a contribution of transverse Hall voltage between misaligned longitudinal leads. Thus, for each sample we perform two sets of experiments, at B=+4T and B=4T, and measure simultaneously the longitudinal potential Vx,(T) and transverse potential Vxy(T) for each field. Then we extract the symmetric response as longitudinal resistance R, (T, B) = (V, (T, B) + V, (T, B)) / 21 and antisymmetric response as the AH resistance RH (T, B) = (V, (T, B) V, (T, B)) / 21, with B=4T. The contribution from the normal Hall effect is found to be negligible. To quantify our experimental results and facilitate comparison between different samples, we define a function AN (K) that we call the "normalized relative change" in a transport quantity, say K, with respect to some reference temperature To evaluated for temperatures T> To: SK(T) K(T) 1 K(To) LooR, (To) where Loo = (81kQ)1 is the quantum of conductance and Rx(To) is the sheet resistance at To. We note that the "relative change" is defined as: 9K K(T) K(T) (64) K K(To) Thus the normalized relative change AN (K) is the relative change in the quantity K / K, divided by the factor LooRxI(To), a dimensionless quantity that is a measure of the effective disorder in a two dimensional system. Using the above notation and keeping in mind the fact that the low temperature behavior of both R, and R"H is logarithmic, we employ the following equations to fit our data: RAH (T) RAH (To) T A (RAH)= = A, In() (66) LooRHy To)RA(T o) (T where AR and AAH are coefficients of the logarithmic temperature dependence of R, and RAH respectively. The coefficients depend on the parameters describing the quantum corrections to longitudinal and AH resistance respectively. Using the approximation that Rxx (T) Rx (To) << Rxx (T), the longitudinal conductivity is calculated from the raw data as L, = 1/Rx so that dL, / L, = (R, / R,) Thus it follows from equations (65) that L L, 1 R, (T) R, (TO) T A (L,)= 2 ( =T AR In() (67) L (TO) R,(To)Loo LR (T,) T, The AH conductivity has contributions from both Rand RH and is calculated from raw data as L = RAH /((RAH )2 + R) Using the approximation RAH (T) << RX (T), which is true for most materials, we have LAH RH / R. It follows from equations (65) and (66), NA AH 1 RAH R 1 T A (LA)= y =2 ( 2" = (2A A )ln() LAH R(TO)LOO R AH R R (To)LoO T (68) Moreover, it follows directly from equations (65) and (66), S(R AH) AH (R AAH (69) A"(R,) AR Thus the coefficient of logarithmic temperature dependence of conductivity is AR and that of the AH conductivity is 2ARAAH. Also it follows from equations (68) and (69) that AAHIAR =2 implies thatA(LAH) = 0. Any deviation of this ratio from 2 implies a non zero logarithmic temperature dependence ofL AH We note that the logarithmic prefactor AR, as defined in equation (61) and equation (65), are self consistent and always give the same value for a given data set as they refer to the slope of the temperature dependence of L,. We have already discussed how AR varies with changing Ro when B=0(Figure 64). We found that, even in the presence ofB=4T, the magnitude of AR did not show any significant change. The reason is that the iron films under consideration have very small magnetoresistance(MR). In Fig. 67 we show the MR for a film with Ro=3000 is of the order of 0.15%. The magnetoresistance curves for all the iron samples are found to be predominantly negative with small hysteresis and showed a pronounced saturation at applied fields that also corresponds to saturation of AH resistance (compare Figs. 65(a) and 67). The hysteretic behavior progressively decreases with increasing Ro, which we will argue later, is due to weakening of ferromagnetic coupling between grains, and for very high resistance samples the hysteresis disappears, as it should for a paramagnetic response. 0.0000 A I\ S 0.0005 I I I o.oo*o I 0.0010 1 0.0015 I I I I I 4 2 0 2 4 B(T) Figure 67: Magnetoresistance as a function of field for an iron film with Ro=3002. To show the relative scaling behavior of resistance and AH resistance, we have plotted in Fig. 68 on a logarithmic temperature scale the normalized relative changes ANR (T), ANRAH (T) and ANLAH (T) with T=5K as reference temperature and at an applied magnetic field of 4T, for a film with Ro=2733Q. We observe a distinct feature that the curves for AN (RH ) and ANR, (T) exactly overlap each other while obeying logarithmic temperature dependence at low temperatures up to T20K. At higher temperatures, we observe that AN (R'H) deviates from logarithmic behavior and decreases at a faster rate than ANR, (T). The importance of this behavior becomes 79 apparent by comparing with the results of a previous similar experiment by Bergmann and Ye on ultrathin amorphous iron 48, where the logarithmic slope of RH was found to be a factor of two higher than that of R, .The deviation of AN (RH ) from AN (R,) at high temperatures(Figure 68) is also interesting and is a possible indication that the dominant quantum corrections to R, and RH are due to different mechanisms. We note that the quenchcondensed amorphous films 48 in Bergmann's experiments could not be heated above 20K without incurring irreversible morphological changes, hence we could not compare our high temperature data. 2 (V 1 (LAH C, 0)Figure 68 Relative resistance (RR) scaling behavior at T20K for an iron film with 1 A (R 2 "o3 E 4 A 0 A (R ) z 5 xy 6 1 1 1 o I I I I I III 5 10 T(K) 100 200 Figure 68: Relative resistance (RR) scaling behavior at T<20K for an iron film with Ro=2700Q. The uppermost (blue) curve exhibits finite logarithmic temperature dependence of AN (LXH ). A A... A.................. I AA AA AA m(a) I I I I I mill a . I I I I 1 I AI I I I I I I I I I I I, I I I I I ll A A AA   i   4 A (b) A AA AA I I ,11111 I I I ,IIIII I I 1111, A A SAt (c) , ,I ml i I m ,,,Ii m m m mi 10 R() 104 Figure 69: Plots showing dependence on Ro of the extracted transport coefficients (a) AR and (b) AAH/AR showing a crossover in value (shown by arrow) near Rohle2 41000. Values close to unity in both cases correspond to RR scaling. (c) 2ARAAH is the numerical prefactor for AH conductivity showing no pronounced dependence on Ro. The average value of 0.8 is shown by the dotted line. 1.0 z 0.5 I I z a < CM1 0.0 1.0 0.5 0.0 1.0 0.5 0.0 102 105 Fitting the data in Fig. 68 for low temperature (T=520K) part to equations (65) and (66), we find for this particular film that AR 0.897+0.001, and AH= 0.908+0.005. The AH conductivity LA calculated from the raw data also exhibits a logarithmic temperature dependence below 20K with a positive slope as shown by the blue data points in Figure 68 and the prefactor for AN (L,) is found to be 0.9080.005 close to the value of 2ARAAH =0.890.01 in accordance with equation (67). For simplicity and future reference we call the low temperature scaling behavior for samples where the AAH/AR=I (Figure 68) so that the relative changes in R, (T) and R H (T) are equal, the relative resistance (RR) scaling. The RR scaling behavior is observed in all of our samples with Ro<3000Q, and implies a finite temperature dependence of anomalous Hall conductivity. This behavior is significantly different from that reported in previous experiments by Bergmann and Ye on amorphous iron films 48 where AAH/AR=2 implying that there are no temperature dependent quantum corrections to L . In Fig. 69(a) we have replotted for reference the coefficient AR as a function of Ro. The plot is essentially the same as Fig. 64 and shows a deviation from ARI for Ro>30000. There is also a crossover in the relative scaling behavior of AH resistance and resistance, which is measured quantitatively by the ratio of coefficients AAH/AR as shown in Fig.69(b). All samples with R, in the range 3003000Q were found to exhibit RR scaling behavior (Figure 68) with the average value for the ratio AAH/AR=1.070.1. However as Ro increases beyond 3000Q, the ratio AAH/AR systematically decreases from unity, which according to equation (69) implies that the relative change in R= is larger than the relative change in RH For a sample with Ro=50000Q for example, which is XY the last data point in Figure 69b, we found that AAH/AR=0.122+0.002. In Fig. 610 we have plotted AN (R,) and AN (R H ) for this sample, which clearly shows that the logarithmic slope for AN (R) is much greater than AN (R H). We emphasize that even if the ratio A1A/AR decreases from unity in the high resistance regime, there remains a finite and positive quantum correction to the AH conductivity as shown by the finite value of 2ARAAH =0.6 (Figure 69c). 00.0 *:. * *** I I M 0.1 (R ) xy 5 8 12 16 20 T(K) Figure 610:An iron film with Ro=4900002 showing deviation from RR scaling. The ratio AA/AR =0.12 as shown in the final point in Fig. 69c. An important observation related to the high resistance regime is shown inFig. 6 11, where we have plotted the relative changes in the AH resistances SRAH /RIR as defined in equation (64), for three samples with Ro varying over a wide range from 24000 to 490000. In the low temperature range of T=520K, the curves for all three samples are shown to overlap each other, indicating that the AH resistance in these polycrystalline films is not affected by the increasing longitudinal resistance. Thus for 83 Ro>3000Q, in addition to the fact that the magnitude of R, at T=5K attains a constant value 80Q as shown in Fig. 66, the relative change in RH over the temperature range of 520K also remains constant. 0.00 A4 AA 0.02 A S* Ro= 2.7k S R =26.1 kQ 0.04 R =48.8k I P . I . ..I........I, 5 10 15 20 T(K) Figure 611 :The relative changes in AH resistances for three different iron films. The curves overlap each other even though the corresponding Ro as shown in the legend varies over a wide range. Thus for all iron films a finite positive logarithmic correction to LIH is observed. Interestingly, the coefficient of AH conductivity 2ARAAH as defined in equation (68), does not show any pronounced dependence on Ro and is scattered around an average value of 0.86+0.2(Figure 69c). Thus, there is a "universal" behavior in the low temperature logarithmic dependence of AH conductivity over the whole range of Ro, in comparison with the magnitude of variation of longitudinal conductivity that showed a crossover behavior around Ro, h/e2. This is a strong indication that the dominant mechanisms that are responsible for the temperature dependence of the longitudinal (L ) and AH (L" ) conductivity are different. Strong Disorder: Iron/C60 Bilayers So far we have presented data on iron films directly grown on glass substrates with Ro varying from 5050000Q. The transport properties of all the films showed logarithmic temperature dependence, which in the context of quantum theory of transport should be considered as 'weakly disordered' metals. This means that despite the granular morphology of the films, at low temperatures the phase coherence length extends over several grains 42, and the motion of the carriers can be considered as diffusive. To make the samples insulating where the dominant conduction process is hopping from one localized state to another, one needs to grow even higher sheet resistance samples where the intergrain barrier separation and/or barrier height is larger compared to iron films. However, it is an experimental challenge to grow a sample with arbitrarily high sheet resistance, mainly because of resistance drifts up at room temperature even at very high vacuum. These drifts, which result from slow oxidation and/or thermally activated annealing, decrease as temperature is lowered. To grow films with even higher sheet resistances we used a novel technique where a monolayer of C60 is grown first on the glass substrate and then iron films were grown top. C60 in solid form is known to be an insulator with completely filled bands. However, C60 molecules have a high electron affinity; hence when in contact with a metal, electrons are transferred from the metal to the C60 and can move freely within the monolayer 21. Thus, the underlying C60 monolayer provides an extra shunting path for conduction electrons between otherwise isolated grains, as shown schematically in Fig. 6 12. The critical thickness at which a metal film starts to conduct is much less when grown on C60 than when grown directly on glass. Using these techniques, we were able to grow high resistance stable ferromagnetic films. Fe film SC60 monolayer Glass Substrate Figure 612:A cartoon of iron/C60 bilayer samples. Big circles represent C60 molecules and the small brown circles represent iron atoms. The small open circles represent electrons transferred from the iron film to C60 ,which can move freely in the monolayer. We grew two Fe/C60 films with sheet resistances R= 1600Q and 41000, which are well within the range of iron films grown on bare glass and compared their transport properties with iron films having similar resistances. The AH resistance at T=5K was found to be RH =640 and 81 respectively, which is what one would expect from monolayer of iron on glass. These films also showed RR scaling behavior similar to that seen in low resistance iron films where both R (T), and RH (T) obeyed a logarithmic temperature dependence at low temperatures, with the logarithmic prefactors A .4 1. These observations demonstrate that the underlying C60 layer does not alter the transport properties of iron. However, an important point to note is that it takes less iron to grow a film of certain resistance when grown on C60 rather than bare glass, because of the shunting path provided by the C60 monolayer. T(K) 100 25 11 4 S101 1 10 r 102 N 0.1 0.2 0.3 0.4 0.5 T1/2(K1/2) Figure 613:Plot of conductivity showing hopping transport in a Fe/C60 sample as given by equation (610). We now focus our attention on iron/C60 films that are in the strongly disordered regime. Figure 613 shows Rxx(T) at B=0 for such a film with resistance at T=5K, Ro,20Mf. We observe that this film does not show the logarithmic temperature dependence of conductivity. Instead the longitudinal conductance fits to the following functional form as predicted 44 by theories of hopping conduction in granular metallic films: T 1/2 L~ (T) = L exp(() ) (610) T where LO and T are characteristic resistance and energy scales in a two dimensional system with localized electronic states. The fit in Fig. 613 yields T=266K and L = 1.754Loo. Table 61 summarizes the fitting results for all the strongly disordered films under consideration. We note that the T is directly related to the Coulomb energy44Ec e2 (d) associated with the charging of grains when an electron hops from one grain to another, reveal that T increases with increasing Ro indicating that for high resistance samples the average grain size is smaller. 100 5 S50 2 4 B(T) Figure 614: The AH effect in strongly disordered Fe/C60 sample at three different temperatures. Legend quotes resistances at given temperatures corresponding to the hall curves of same color. The solid lines represent fits to the Langevin function. 0.00 0.01 0.02 0.03 1 1 1 1 1 1* 4 2 0 2 4 B(T) Figure 615 :Magnetoresistance curves at different temperatures for the Fe/C60 sample shown in Figs. 613 and 614. The curves do not show sharp saturation at a particular field as seen in iron films (Figure 67). The resistance of the particular sample shown in Fig. 613 is high so that the Hall signal was much less than the longitudinal potential due to the misalignment of transverse leads. Thus for this sample we could extract Hall curve accurately only for temperatures at and above T=25K as shown in Fig. 614. The film still exhibits anomalous behavior in the Hall curves, indicating the presence of finite local magnetic moments at such high resistances. However, the AH curves do not exhibit a sharp "knee" like saturation at any definite applied field similar to that seen in iron films (Figure 65); instead there is a smooth and gradual cross over to a high field saturated value of Hall resistance. The sample also exhibits negative magnetoresistance (Figure 615), but in this case also we do not observe a sharp saturation at any characteristic field as seen for thicker iron samples (Figure 67). These are possible indications of the absence of long range ferromagnetic coupling between the grains comprising the films. However, an important observation is that for samples exhibiting hopping transport behavior, the resistance increases dramatically with decreasing temperature, but the corresponding AH curves do not change significantly. As deduced from the legend of Fig. 614, for the resistance increases by 440% while the AH resistance increases only by 33%. An important feature of the high resistance Fe/C60 samples is that the AH curves could be fitted with very high accuracy to the Langevin function L(x)= coth(x)1/x (611) which describes the magnetization of a paramagnetic system consisting of non interacting particles or clusters of magnetic moment /, as follows M(B, T) = M,L(,uB/ k T) (612) 