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Disorder, Itinerant Ferromagnetism, and the Anomalous Hall Effect in Two Dimensions


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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 FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Partha Mitra

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To my parents for all their sacrifices to provide me the best education.

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iv 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 hi m, nicely balanced by the much needed guidance and support. I have always (secret ly) admired his unadulterated enthusiasm, willingness to learn and elegant but simp le approach to understanding fundamental physics. I have met some wonderful researcher s 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 th e 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 tha nk 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 in sights regarding my re search during the few discussions we had. I am thankful to the me mbers 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 offi ce 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 th ey made over all these years to provide me

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v the best education. I could not have come this far without their ble ssings. 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 th e 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 wa rmth and affection that I received from her family. I am thankful for having a large a nd wonderful supportive family and appreciate the love and encouragement that I have receiv ed, 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.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION........................................................................................................1 2 EXPERIMENTAL SETUP: SHIVA APPARATUS....................................................7 3 NANO SMOOTHENING DUE TO ION IRRADIATION........................................11 Experimental Details..................................................................................................12 Sample Fabrication..............................................................................................12 Measurement Setup.............................................................................................13 Experimental Results..................................................................................................15 Discussion...................................................................................................................22 4 THE ANOMALOUS HALL EFFECT IN MAGNETIC MATERIALS....................24 Itinerant Ferromagnetism............................................................................................24 Anomalous Hall Effect...............................................................................................26 Anomalous Hall Effect for Itinerant Carriers......................................................27 Skew Scattering Mechanism........................................................................28 Side Jump Mechanism.................................................................................31 Berry Phase Mechanism...............................................................................32 Anomalous Hall Effect in Ferroma gnets with Localized Moments....................33 Superparamagnetism...................................................................................................36 5 QUANTUM CORRECTIONS TO TRANS PORT PROPERTIES IN METALS......38 Weak Localization Effects..........................................................................................39 Magnetoresistance due to Weak Localization.....................................................43 Weak Localization in Presence of Spin-Orbit Interaction...................................44

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vii Weak Localization in Ferromagnetic Films........................................................46 Electron Interaction Effects........................................................................................48 Scattering by Friedel Oscillations.......................................................................48 Magnetoresistance due to Electron Interaction...................................................53 Transport Properties of Granular Metals....................................................................54 Transport in Weakly C oupled Granular Metals..........................................................57 Quantum Corrections to Hall Conductivity................................................................59 6 EXPERIMENTAL RESULTS AND DISCUSSION.................................................64 Experimental Details..................................................................................................64 Sample Fabrication..............................................................................................64 Measurement Setup.............................................................................................66 Weak Disorder: Iron Films.........................................................................................68 Transport Properties at B=0.................................................................................68 Anomalous Hall Effect in Iron............................................................................71 Temperature Dependence of Anomalous Hall Conductivity..............................75 Strong Disorder: Iron/C60 Bilayers.............................................................................84 Experiments on Cobalt Films.....................................................................................94 Discussion of Experimental Results...........................................................................98 Absence of Quantum Correc tions 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 REFERENCES.................................................................................................123 BIOGRAPHICAL SKETCH...........................................................................................127

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viii LIST OF TABLES Table page 6-1 Summary of the results for three different Fe/C60 samples......................................94 6-2 Comparison of some propert ies of bulk iron and cobalt..........................................97

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ix LIST OF FIGURES Figure page 1-1 A cartoon describing the origin of the anomalous Hall effect...................................2 1-2 A cartoon describing the origin of the spin Hall effect..............................................3 2-1 Schematic representation of the SHIVA apparatus....................................................8 3-1 A setup for non-switching van der Pauw technique to measure sheet resistance....14 3-2 Plot of sheet resistance and thickness as a function of time.....................................16 3-3 Check of stability of the reduced re sistance due to ion beam irradiation.................17 3-4 AFM surface topography of two 120 thick Cu films...........................................17 3-5 The two van der Pauw component R1 and R2 plotted separately.............................19 3-6 Plot of the shunting resistance Rs and the time t ....................................................20 3-7 Effect of ion-beam exposure on thicker iron films..................................................21 4-1 The Hall curve for a 20 thick iron film.................................................................27 4-2 Schematic representation of (a) skew scattering and (b) side jump mechanism......30 5-1 Typical wave functions of conduction electrons in presence of disorder................38 5-2 Motion of electrons in presence of impurities .........................................................40 5-3 Schematic diagram of Friedel os cillation due to a single impurity..........................49 5-4 Friedel oscillation due to two impurities..................................................................51 6-1 A d.c. transport measurement setup using Keithly 236............................................66 6-2 An a.c. transport measurement set up using two SR830 lock-in amplifiers ............67 6-3 Typical behaviors for temperature de pendence of resistance for iron films............69 6-4 Plot of numerical prefactor AR for different iron films.............................................71

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x 6-5 The anomalous Hall curves for iron films................................................................72 6-6 The anomalous Hall resistance at T= 5K for different iron films ............................74 6-7 Magnetoresistance as a function of field for an iron film with Ro= 300 ................78 6-8 Relative resistance (RR) scaling behavior at T< 20K for an iron film.....................79 6-9 Plots showing dependence on Ro of the extracted transport coefficients.................80 6-10 An iron film with Ro= 49000 showing deviation from RR scaling.......................82 6-11 The relative changes in AH resistan ces for three different iron films.....................83 6-12 A cartoon of iron/C60 bilayer samples......................................................................85 6-13 Plot of conductivity showing hopping transport in a Fe/C60 sample........................86 6-14 The AH effect in strongly disordered Fe/C60 sample...............................................87 6-15 Magnetoresistance curves at di fferent temperatures for the Fe/C60.........................87 6-16 Plot of as a function of inverse temp erature for three different Fe/C60 samples...90 6-17 Logarithmic temperature dependence of AH resistance of Fe/C60 samples............92 6-18 Temperature dependence of AH conductivity of Fe/C60 samples............................93 6-19 AH resistance for Fe/C60 samples as a function of sheet resistance.........................93 6-20 AH effect in a cobalt film with Ro= 3200 at T= 5K...............................................95 6-21 AH resistance for cobalt films as a function of Ro...................................................97 6-22 The anomalous Hall conductiv ity vs longitudinal conductivity.............................109 6-23 The anomalous Hall conductiv ity vs longitudinal conductivity............................110

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xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 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 addr ess the unsettled issue of how long-range magnetic order in band ferromagnetic metals like iron or coba lt is affected by localization of itinerant conduction electrons due to incr easing disorder. We study a seri es 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 100 to 1,000,000 To protect these ultra-thin air sensitive films fr om oxidation, the experiments were performed in a special homemade high vacuum system, capable of insitu 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 magneto-transport properties for film resistances on the order of k e 1 4 /2. Surprisingly, in the high resistance regime

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xii where the samples were found to systematical ly 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 attr ibute this so called “anomalous Hall insulating” behavior to the granular morphology of the film s, where inter-grain tunneling processes dominate the longitudinal resistan ce, and the anomalous Hall resistance is determined by the intrinsic ferromagnetic nature of the grains. In the insulating phase long-range ferromagnetic order a ppears to be absent in the films and we demonstrate the existence of a new resistance scale much greater than 2/ e where correlation between localized magnetic moments as measured by AH effect, disappears. We also demonstrate how the granular morphology of the films a llows the two different quantum transport mechanisms in metals, namely weak local ization 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 ne w calculations done in collaboration with theorists.

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1 CHAPTER 1 INTRODUCTION Recently there has been cons iderable activity in the fi eld of spintronic devices 1, which rely on the manipulation of spin of conduction electrons in solids and promise to revolutionize microelectronics once spin-polarized electrons can be injected efficiently into semiconductors at room temperatures. Cu rrently, the most widely used spintronic device based on metallic multilayers is the giant magnetoresistive (GMR) spin-valve head for magnetic hard-disk drives. The GMR effect is based on large changes in electrical resistance due to va riations in the relative magnetic orientation of layers on either side of a thin spacer layer. Spintr onic structures are also at the heart of the proposed magnetic random access memory (M RAM), a fast non-volatile new generation of memory. From a more fundamental point spintronics studies involve understanding spin transport and spin intera ction with the solid-state envi ronment 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 syst em, understanding how long the spins are able to reme mber their orientation and finding a way for spins to be detected. Perhaps the simplest way to generate a spin -polarized current is to apply an electric field to the ferromagnetic transition metals, namely iron, cobalt or nickel. The spontaneous magnetization in these band ferro magnets is due to the unequal population

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2 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 a pplied electric field. This spin current generates a transverse voltage due to spin-dependent asymmetric scattering of electrons from impurities or phonons, a phenomenon known as the anomalous Hall (AH) effect 2 as shown schematically in Fig. 1-1 The AH effect (AHE) is observed in all ferromagnetic materials regardless of the nature of the exchange mechanism and also in superparamagnetic systems comprising weak ly-coupled 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 proce ss that couples the volume magnetization in the material to motion of the itinerant car riers via the spin-orbit interaction. The transverse AH voltage relies on itinerant charge carriers and is directly proportional to the volume magnetization. Thus a thorough charac terization of the AH effect is useful for studying magnetic properties in materials especially when direct magnetization measurements via SQUID are not possible. Figure 1-1: A cartoon describing the origin of the anomalous Hall effect in an itinerant ferromagnet with unequal number of spin-up and spin down conduction electrons due to spin dependent scatteri ng of electrons from an impurity X. A transverse electric potential VH is developed due to accumulation of charges at the edge of the sample.

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3 A related phenomenon, which recently has generated considerable interest, is the spin Hall effect (SHE) 3 where an electric field ap plied 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. 1-2. This accumulation of spins results in a gradient of spin population in the transverse direction and is not associated with an electric pot ential because the total number of electrons irrespective of spin orientat ion 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 conduc tivity from any simple scheme analogous to that of the normal or anomalous Hall effect. Figure 1-2: A cartoon de scribing the origin of the spin Hall effect in a paramagnetic conductor due to spin dependent scatteri ng 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 e ffect 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

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4 microscopic scattering mechanisms due to the presence of spin-orbit coupling. Phenomenologically, the two effects are comp lements of each other: in the AHE an electric potential due to the separation of elect ric 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 wa s 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 conductiv ity. Thus the AH effect can provide an important route for making quantitative estimate s 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 material s is known to be due to mobile (itinerant) electrons that are also responsible for el ectrical conduction. Th e balance between the kinetic energy and exchange interaction en ergy of the conduction electrons determines whether a metal with a spin-split band beco mes 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 c onditions are favorable for ferromagnetism with the magnetic properties determined by spin polarized itinerant carriers. We are

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5 investigating whether quench ing the itinerancy of th e conduction electrons with increasing disorder can violate the Stoner criteri on. 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 ai r-sensitive nature of these films rules out the possibility of a magnetization measurem ent using a SQUID magnetometer. As we will show in the following chapters, the AH eff ect is an alternative and effective tool to probe magnetic ordering in conducting materi als. We will show th at with decreasing thickness, long range ferromagnetic order in thicker films gradually disappears and passes over to a system of weakly coupled ma gnetic clusters. Even tually at very high resistances there is an unexpected and pr onounced disappearance of magnetic ordering as measured by the AH effect. We also demons trate that morphology plays an important role in governing the magnetic and electrical pr operties in thin f ilms. 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 fo r this project. In chapter 3 we demonstrate the usefulness of the SHIVA apparatus by describing a short project that studies the eff ect of low energy ion beam i rradiation on the conductivity of metal films. In chapter 4 we describe th e phenomenology of the AH effect and discuss various microscopic mechanisms that are res ponsible 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 experime ntal result and discussions based on the

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6 existing theoretical understanding is provided in chapter 6. Finally, in chapter 7 we summarize and discuss some possible future ex periments related to this dissertation..

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7 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 sensitiv e films. This apparatus, shown schematically in Fig. 2-1,was given a pet na me 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 v acuum system and has mechanical “arms” for transferring samples between th e compartments without brea king 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 cryosta t by gate valves V1 and V2 respectively. This arrangement allows the growth chambe r 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” A1, A2 and A3 c onsist of a strong magnet sliding on the outside of a long hollow stainle ss steel tube attached to the vacuum chamber, and another solid rod inside the tube magnetically coupl ed. Hence by sliding the outside magnet, one can translate and rotate the rod with a thrust er 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 thru ster on the “arms” can be inserted and locked onto the puck by a twisti ng motion. The puck is first mounted on A3 inside the load lock and then pushed inside th e growth chamber. Thruster A1 then comes

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8 Figure 2-1: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: tran sfer arms with thruster mounted at each end; V1,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 A1, is then delivered to a receiver platform, and the design is such that the puck engage s in the receiver and disengages from A1 in one twist, so that A1 can be taken out of the way. Once the sample is grown on the substrate, A1 comes down a nd locks into the puck and disengages it from the receiver. The next step is to bring A3 b ack into the growth chamber lock into the puck, and then disengage A1. The puck is th en brought back into the load lock where thruster A2 engages on it and then A3 is disengaged. Now the sample can be pushed

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9 inside the cryostat and delivered to a receiver platform similar to that in the growth chamber. If necessary, the sample can be tr ansferred back to the growth chamber from the cryostat for further processing, by reversi ng the steps described above. The point to note here is that during the whole procedur e, 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 substrat e. Inside the body of the pucks a set of wires are soldered from inside on to small he mispherical 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 th e substrate before the puck is loaded in the vacuum chamber. Thus the electrodes on th e substrate are electrically connected to different copper heads. The receiver also ha s a set of spring loaded copper heads on the inside, such that when the puck is placed insi de the receiver and enga ged, the two sets of heads press firmly against each other and ma ke good electrical contact. Another set of wires connect each of the copper heads on the receiver, run via vacuum feed-throughs out of the vacuum chamber to a break-out panel. Thus when a puck is engaged properly in the receiver, each of th e electrodes on the substrate is el ectrically 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 ~10-9 Torr and is equipped with a variety of deposition sources namely two AJA magnetron sputter guns, two RADAK thermal evaporation furnaces a nd an ion beam gun. The receiver in the

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10 growth chamber is mounted on a manipulator th at allows us to position the substrate so that it faces the appropriate deposition s ource. The thickness of the films can be monitored with a quartz crystal oscillator firm ly attached on the receiver close to the sample. The growth chamber also has two optic al ports specially desi gned for attaching a Woolam M44 fixed angle ellipsomet er, 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 accu rately 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 Lakeshor e 332 temperature contro ller connected to a second Cernox thermometer and resistiv e heater mounted on the receiver.

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11 CHAPTER 3 NANO SMOOTHENING DUE TO ION IRRADIATION Ion-assisted deposition refers to the tec hnique 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 consequen ces of ion-assisted film growth include densification, modification of nucleation and growth, interf ace mixing, defect generation, and changes in topography and surface roughness 7. In contrast to this general technique of ion-assisted deposition, some investigat ors have reported on the use of a sequential technique in which thin films are first deposit ed and then, after depos ition, exposed to an ion beam. Results of this a pproach include ion bombardmen t induced nanostructuring of Cu(001) surfaces 8, the formation of reproducible ripple structures on Si(001) and Ag(110) and (001) surfaces 9 and the roughening or smoot hing of Si(001) and SiO2 surfaces 10, 11 where the result depends on ion type, en ergy 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 “ion-beam sculpting” 13. Many of these experiments thus provide strong evidence of ion beam induced nanoscale matter transport on solid-state surfaces, a process that promises to be useful in applications requiring nano-textured surfaces and interfaces. We are going to present a systematic study of the effect of low en ergy (~200eV) ion beam exposure on the room temperature resistance of polycrystalline metallic films.

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12 Experimental Details Sample Fabrication Using the SHIVA apparatus described in chap ter 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 ~10-4 Torr and a d.c. bias in the sputter gun ~ 145V. Under these conditions the Iron films grows at a rate of 4/minute. We have also grown a few copper films by thermal evaporation using a RADAK sour ce at a temperature of 1100oC. Film thickness was measured by a calibrated quartz crystal thickne ss monitor placed in close proximity to the sample and varies from 15-45 for the Fe films and 75-130 for the Cu films. The samples were grown on square substrates wi th 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 re-sputtering 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 exposur e steps. The incident noble gas ions are chemically unreactive and simply transfer a fraction of thei r kinetic energy to the atoms in the film, a process which, as discussed above 8-13, can result in a significant

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13 modification of the surface morphology of the film and hence its physical properties. Using atomic force microscopy (AFM) we c onfirm an ion beam induced smoothening of our films and find that there is a reproduc ible correlation of th e 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 th eorem 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 corners (see Fig. 3-1). Apply a current IAB at contact A and take it off at cont act B. Measure the potential difference between C and D and define AB C DI V V R 1 (3-1) Similarly, apply a current ICB and measure the potential difference between D and A define CB D AI V V R 2 (3-2) The sheet resistance R is determined from a mathema tical relation between the above measurements as given by 1 ) exp( ) exp(2 1 R R R R (3-3) The solution of the above equation can be written in a simplified form ) ( 2 2 ln2 1 2 1R R f R R R (3-4)

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14 where f is a numerical factor that depends only on the ratio R1/R2 and is given by the following transcendental equation ) 2 ln exp( 2 1 ) 2 ln 1 / 1 / cosh(2 1 2 1f f R R R R (3-5) which can be easily estimated to any degree of accuracy by numerical methods. Thus the van der Pauw method involves two independent measurements in di fferent electrical configurations. A closer l ook reveal that the two conf igurations for measuring R1 and R2 can be switched from one to other by si mply interchanging any one of the diagonal contacts keeping the other fixed. Usually one uses a mechanical or electronic switching system to change the electrical conf iguration for the two measurements. Figure 3-1: A setup for nonswitching van der Pauw technique to measure sheet resistance of a sample (shaded square ABCD) using two SR830 lock-in amplifiers operating at different frequencies f1 and f2 Different colors are used to represent electrical connections for each lock-in.

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15 We have used a nonswitching technique 15 where the two measurements can be made simultaneously without changing configur ations, by using a.c. excitations instead of d.c. currents as described above. With c ontact 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 lo ckin 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 lock-in amplifiers at frequencies of f1=17Hz and f2=27Hz, and using 1M ballast resistors, we were able to source 1 A of a.c. current into our samples. This arrangement, shown schematically in Fig. 3-1, allows us to simultaneously measure both components, R1 and R2, of the resistance and assess film homogeneity (| R1-R2|) during growth and subsequent ion exposure. We also simulta neously measured the reading of a quartz crystal monitor as a measure of film thickne ss. However, due to lack of calibration of crystal monitor output as a ion beam is incide nt 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.3-2 the resistance of a Cu (top panel) and Fe (bottom panel) film undergoes a pronounced decr ease 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 cons tant rate, as indicated by the linear decrease

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16 of the accumulated thickness of the material de posited 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. Figure 3-2: 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. Th e 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. 3-3). 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. 0 5 10 15 20 50 60 70 80 90 -1000100200 2 4 6 8 10 Rsq(k)time(s) Fe film-1000100200300 0 2 4 6 8 Rsq(k) Cu film thickness(a.u.) thickness(a.u.)

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17 Figure 3-3: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 an d does not show any significant drift. Figure 3-4:AFM surface topography of two 120 thick Cu films with Ri 2 k The unmilled film (left panel) has a rms average roughness of 30 compared to the 10 roughness of the film (right panel) which was milled to its minimum resistance R = Rm. -30003006009001200 2000 3000 4000 5000 Rsq()time(s) START STOP START STOP

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18 Auger electron spectroscopy of an ion-mille d 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 b eam and thus lower its resistance. The relative decrease in the re sistance of our ultra-thin 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 k the minimum resistance reached during the ion milling process is more than a factor of ten lower for Cu (~200 ) than it is for Fe (~2.5 k ). 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 th an Fe films grown by sputtering. We have also compared the roughness of films not expos ed to an ion beam with films ion milled close to the resistance minimum. As shown in Fig. 3-4, this comp arison for a typical Cu film reveals that the ion-milled film (right panel) has an rms roughness of 10 compared to 30 for an unmilled film (left panel). The smoothening effect is also confirmed in AFM images of Fe films, which, with th eir 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 imme diately after deposition begins, we do not observe an initial resistance decrease. There is also a pronounced incr ease in homogeneity associat ed 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 gr ains, then coalescence of the gr ains and finally formation of a homogeneous system of well-connected grains. At the early stages of growth the local

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19 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 “inhomogeneous” f ilms are exposed to the ion beam, both R1 and R2 individually decrease to approximately the same mini mum. Thus the anisotropy measured by | R1 R2| and the total sheet resistance as measured by the van der Pauw combination of R1 and R2 15 simultaneously decrease. Figure 3-5: The two va n der Pauw component R1 and R2 plotted separately for the copper film shown in Fig. 3-2. 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 conne cting a shunt resistance Rs in parallel with Ri. We use the parallel resist ance formula to calculate Rs as follows: m s iR R R 1 1 1 (3-6) -1000100200300 0 1000 2000 Resistance()time(s) R1 R2

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20 Figure 3-6: Plot of the shunting resistance Rs (panel a) and the time t required for the ion beam to mill the film to its resistance minimum Rm (panel b) as a function of the initial resistance Ri.. The horizontal dashed lines represent the nominally constant values of Rs and t over the indicated ranges of Ri. The crossover to different plateaus near Ri = 1500 represents a change in film morphology in which a smaller ion dose ( t ) gives rise to a larger shunt resistance. In Fig. 3-6(a) we plot the calculated values of Rs versus Ri for twenty-two different Fe films. We note the interesting result that Rs (indicated by the horiz ontal dashed lines) is constant and on the order of 4500 for ultra thin films with Ri > 1500 and constant and on the order of 1400 for thicker films with Ri < 1500 Fig. 3-6(b) shows the dependence on Ri of the ion exposure time t needed to reach the minimum resistance Rm. Since the beam flux is constant for all the experiments, t is proportional to the total number of ions incident on the films or, equivalently, the ion dose. We find that t behaves similarly to Rs. The crossover in both plots near Ri = 1500 corresponding to a thickness of around 25, most likely reflects a change in film morphology in which a 10010002000300040005000 40 80 120 160 200 t(s)Ri() 0 1500 3000 4500 6000 Rs() (a) (b)

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21 smaller ion dose ( t ) gives rise to a larger shunt resi stance. Thus for ultra thin films with Ri > 1500 the shunt resistance increases by a f actor of ~3 and the dose needed to achieve the resistance minimum decreases by a fa ctor of ~2. The relative constancy of the shunt resistance values for a wide range of in itial 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 shun ting resistance that is inde pendent of film thickness. Similar data are found for the Cu films where t ~75 s and the shunting path resistance Rs = 200 is more than an order of magnitude lower than for Fe. If the initial film is very thick, then its conductance dominates and Ri << Rs. Under these conditio ns the resistance decrease, im R R (/)iis R RR due to nano smoothening is negligible as is in fact verified for iron films of thickness greater than 50 ( Ri > 200 ) when the resistance decrease is not observed as shown in Fig. 3-7. Figure 3-7: Effect of ion-beam exposure on thicker iron films (d>50). Note that resistance does not go below its initial value at t =0, when the beam was started. -2000200400 130 135 140 145 150 155 time(s)Rsq()15 20 25 30 35 40 thickness(a.u.) 050010001500 10 15 20 25 time(s)Rsq()0 30 60 90 120 150 180 thickness(a.u.)

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22 The minimum resistance reached during ion milling is not sensitive to small variations in the incident i on 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, t was larger by a factor of 10 and th e resistance minimum was broader. Discussion We surmise that two competing processes are occurring during the ion irradiation process: (1) sputter erosion by the impinging io ns 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) be tween adjacent grains. The resulting nano smoothening process dominates in the initial st ages of the ion exposure, resulting in a decrease in resistance and an increase in hom ogeneity. 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 specu lar 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 give s 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 in cident 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. 3-2, this process eventually do minates over the smoothening process when the resistance starts to rise.

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23 Our results for Fe and Cu films are reminiscent of experiments in which it was found that C60 monolayers deposited onto thin Cu film s give rise to a shunting path with Rs independent of Ri over a similar range 21. In this case the physics is different, since the charge transferred across the Cu/C60 interface begins to fill th e lowest unoccupied band in the C60 monolayer, thus causing the m onolayer 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 a nd 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 surf ace modification. In the initial stages of ion bombardment, in which pronounced resistan ce decreases are obs erved, the lateral transport of material and the associated na no-smoothening dominates over the removal of material. With continued milling the film is uniformly etched away and the resistance increases. While these technique s 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 prepara tion of polycrystalline surfaces prior to the formation of tunnel barriers or the improveme nt of interfaces in me tallic bilayers or superlattices.

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24 CHAPTER 4 THE ANOMALOUS HALL EFFECT IN MAGNETIC MATERIALS Itinerant Ferromagnetism Ferromagnetism in metals arises from unpaired d or f -electrons in the atoms. The experimentally observed values of the magne tic moment per atom for the ferromagnetic transition elements iron, cobalt and nickel ar e 2.22,1.78 and 0.60 respec tively in units of Bohr magneton 22, 23, that is the magnetic moment of one electron. These non-integral values cannot be explained in terms of models where the ma gnetic 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 partic ipate in conduction. The magnetization in this case is due to spont aneously spin-split bands. In 1934, Stoner derived a condition 6 under which a gas of electrons w ith exchange interaction between them becomes a ferromagnet. In a normal metal, in the absence of a ma gnetic field, there are equal numbers of up and down spins in the conduction band. Imagin e a situation when spin-down electrons within an energy range E of the Fermi energy EF are placed in the spin-up band. The number of electrons moved is g(EF)E/2 where g(EF) is the density of states at the Fermi level and the increase in energy is E Hence the change in kinetic energy is 2 .) ( ) ( 2 1 E E g EF E K (4-1)

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25 This increase in energy is compensated if there is an exchange interaction J between electrons that lowers the energy if two spins are parallel. This will lead to a potential energy given by 2 .) ) ( ( 2 1 E E g J EF E P (4-2) Thus the total change in energy is given by )) ( 1 ( ) ( 2 12 F FE Jg E E g E (4-3) and spontaneous ferromagnetism is possible if 0 Ewhich implies : 1 ) ( FE Jg (4-4) This inequality is the Stoner criterion fo r 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 d -bands, which has an imbalance of spins. The calculated values of magnetic moment per atom from band structure are in good agreement with experi mental values. If the Stoner criterion is not satisfied, there will not be any spontan eous long range magnetic order but rather a paramagnet with renormalized susceptibility given by. ) ( 1F PE g J H M (4-5) where P is the Pauli paramagnetic susceptibility Thus there will be a large increase in magnetic susceptibility is known as Stoner e nhancement as is experimentally observed for Pd and Pt which are materials where Jg(EF) is close to but less than unity.

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26 Anomalous Hall Effect The anomalous Hall (AH) effect 2 in magnetic materials refers to the phenomenon when a transverse electric potential proporti onal to the volume magnetization develops in response to an applied charge cu rrent. 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 spin-dependent scattering of the conduction electrons due to spin-orbit coupling with sca ttering 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 ferro magnetic 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 102-103 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 show n in Fig. 4-1. 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 characterist ic field when all the moments are lined up. Thereafter, the Hall resistance ri ses with a much smaller rate that is due to normal Hall effect. This typical behavior can be de scribed using a phenomenological expression 2 B R M Rn s xy (4-6) where the Hall resistivity is described as the sum of the anomalous Hall resistivity proportional to magnetization and the normal Ha ll resistivity proportional to the magnetic

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27 field. Ro is the normal Hall coefficient and depend s only on the effective carrier density in the material. Rs 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 pres ent here a brief account of theoretical understandings, which are relevant to our experimental work. Figure 4-1:The Hall curve for a 20 thick iron film of resistance Ro=2700 at T=5K, showing anomalous behavior. Anomalous Hall Effect for Itinerant Carriers We consider models where a transverse voltage arises due to spin-orbit interaction of the spin polarized current carriers in itinerant ferromagnets with the non-magnetic periodic lattice and or impurities. An electr on in a solid experiences a net electric field say E due to the ionic core in the lattice, other el ectrons or impurities. In the rest frame of the electron there is a magnetic field B as a result of relativist ic transformation as given by mc E p B / (4-7) 024 0 40 80 Rxy()B(T)

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28 where p is the momentum of the electron. This magnetic field interacts with the spin of the electron and favors anti-para llel orientation of orbital an d spin angular momentum of the electron. This describes the intrinsic spin orbit interaction, and results in an additional term in the Hamiltonian given by24: ) ( 4 12 2p V c m HSO (4-8) where ) , (z y x are the Pauli matrices and V E where V is the local potential that the electron experiences. At low temperatures when the dominant scattreing is due to impurities, V is the potential due to a single impurity. The Hamiltonian for such a disordered ferromagnetic metal includi ng spin-orbit interaction as given by: ) ( )] ( 4 ) ( 2 )[ (2 2 3'r V i M r V m r r d Ho z (4-9) where o is effective of spin-orbit scattering st rength and has the dimension of length, ) ( ) ( j jR r v r V is a random potential due to impurities at positions jR, M is a strong exchange interaction in the z-direction and ( ) are spinor fields corresponding to spin-up and spin-down electrons. The Hamilt onian in momentum representation can be written as24: ' '] ). ( 4 1 [ ) 2 (' 2 0 2 k k k k k k k z k kk k i V M m k H (4-10) Skew Scattering Mechanism In 1955 Smit proposed a mechanism 25, 26 for AH effect in ferromagnets which is referred to as Skew Scattering. This mechanis m is based on the fact that the scattering amplitude of an electron wave packet from an impurity due to spin-orbit coupling is

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29 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 0k incident on an impurity site at the origin at time t =0, resulting in a scattered wave given as the sum of nor mal and spin-orbit scattering, k k k so vc (4-11) For a short range impurity potential V(r) it can be shown that scattered wave function from the spin-orbit term far from the origin is given by kr r k kr h q ck k so/ ] ) )( ( [1 1 (4-12) where ] / 1 ) /( [ ) (2 1kr kr i e kr hikr and the scattering wave from the normal potential scattering part is given by ] / ) )( ( ) ( ) [exp(1 1kr r k kr h b kr h b r k i co o k k v (4-13) with kr ie hikr/0 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 “skewness” that arises from spin-orbit interacti on, imagine a plane containing the inci dent direction k and the direction of the spin of the electron assumed to be polarized along a fixed z-direction. It follows from above, the scattered wavefunction for a given spin, the wave func tion 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. 4-2a. Thus for an itinerant ferromagnet with an unequal number of up and down spin electron, a

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30 transverse potential develops as the up/dow n 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 pr oportional to longitudinal conductivity. xx o o o SS xyn n N V ) (2 (4-14) where M is the magnetization, impurity strength ov 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 inve rsion relation between conductivity and resistivity, it follows that the anomalous hall re sistivity due to skew scattering is directly proportional to the longitudinal resistivity. Mxx SS xy ~ (4-15) Figure 4-2:Schematic representation of (a) sk ew scattering and (b) side jump mechanism of AH effect. and represents spin up and spin down electron. 2

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31 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 spinorbit interaction is given by, V m p H r i vo 2] [ (4-16) Thus there is an additional term in the veloc ity operator, transverse to spin polarization which has a sign depending on the spin or ientation. This corresponds to lateral displacement of the center of th e scattered wavepacket with gi ven spin as shown in Fig. 4-2b. The magnitude of the side-jump displacement is proportional to spin-orbit coupling and is expected to be small. We note that for a bare electron in vacuum, the spin-orbit scattering parameter is simply the normalized Compton wavelength i.e mco/ However, Berger has shown that the spin orbit coupling is renormalized by the band structures by factors ~ 104 which results in si de-jump displacements m kF o1110 4 / and is independent of disorder. The characteristic length scale that replaces the mean free path is hence this contribution is small compared to skew scattering, except in the case of short m ean free path i.e. high resistivity. Detail calculation reveals that in this mechanism th e AH conductivity is independent of impurity concentration and depends only on the side jump displacement. ) ( 22 2 n n eo SJ xy (4-17)

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32 This implies that the AH resistance due to side jump scattering is proportional to the square of the longitudinal resistivity as given by Mxx SJ xy2~ (4-18) Berry Phase Mechanism We now discuss a mechanism for AH effect that has drawn cons iderable 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 veloci ty operator in ferromagnetic materials, that can give rise to AH effect. Later, this c ontribution was identified as the effect of Berry phase 29, 30 acquired by Bloch electrons moving in a periodic poten tial of a crystal with spin-orbit interaction with the lattice. The semi-classi cal dynamics of Bloch electrons including the Berry phase may be derived from the Bloch Hamiltonian V k Hn k ) ( (4-19) where ) ( kn are energy bands including the eff ect of spin-orbit interaction and ferromagnetic polarization, and V is the applied external potential such that E e V The semi-classical equations of motion are E e r B r e E e kk n k (4-20) where B is the applied magnetic field and is an effective magnetic field in k space arising from the Berry phase. ) ( ) (k X kk (4-21)

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33 where ) ( ) (*r u i r u r d Xk n cell k n the Berry vector potential and k nuare the Wannier functions for the unit cell of the crystal. The additional term in the velocity operator in equation (4-20) leads to a Hall current given by E n e jH 2 (4-22) implying an AH conductivity given by z AH xyn e 2 (4-23) where ) ( ) (1 k kf k n is the average of Berry magnetic field over all occupied states in k-space. The average is zero unless time reversal symmetry is broken as it is in a ferromagnet where there is spin -orbit coupling between the spin-polarization 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 th e longitudinal resistivity. Mxx B xy2~ (4-24) which is the same dependence found for side jump scattering (equation 4-18). Anomalous Hall Effect in Ferroma gnets with Localized Moments For the sake of completeness we discuss models where the magnetic electrons (either d or f electrons) are not itinerant but rather localized at the ions and the charge carriers (s electrons) are equally distri buted between states of opposite spins. The electron scattering is by thermal disorder in the loca lized spin system th rough direct spin–spin interaction also known as s-d 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

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34 Kondo 32 that the s-d interaction is anisotropi c, but it gives no skew scattering if the orbital ground state of the d or f magnetic electrons is degene rate. On the other hand, the anisotropy disappears when the ground state is non-degenerate. Thus a s-d interaction between the spin of a conduction electron and a spin angular momentum of an incomplete d or f shell cannot give rise to skew scatte ring and hence AH effect. To explain the AH effect in these systems, Kondo introduced an intrinsic spin-o rbit interaction, which is a relativistic effect arising from a magnetic fiel d appearing in the rest frame of the electron as the electron moves past th e nucleus (equation 4-7). This kind of spin-orbit interaction favors anti-parallel alignment of orbital a ngular momentum and the spin. The intrinsic spin-orbit interaction allows odd powers of spin-spin inte raction appropriate to a degenerate ground state to appear in transi tion probabilities and gives rise to skew scattering. Kondo 32 obtained an expression for Hall resistivity in this situation proportional to third moment of magn etization fluctuation as follows: 3) ( ~ M Mxy (4-25) This fluctuation function can be evaluated only under special conditions. At T =0, the spin fluctuations are zero and hence AH resistivit y will be zero. In the paramagnetic region (above Curie temperature), the correla tion function can be evaluated exactly 32 and the Hall resistivity is given by: H J Jxy ) 1 2 2 ( ~2 (4-26) where J is 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 Brillo uin function. Although this theory has had

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35 some success especially for paramagnetic subs tances, it has some major drawbacks. For example gadolinium, a rare earth metal is know n to be in an S state and hence zero orbital angular momentum. Thus in this case the intr insic spin-orbit interact ion will be zero, and hence skew scattering is not expected. However gadolinium is known to exhibit a particularly large AH effect. Another spin-orbit 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 M at the origin of a rectangular co ordinate. It sets up a vector potential at a position vector r given by3/ ) (r r M A The vector potential interacts with a charge carrier with momentum p as given by the term A p mc e p A A p mc e Hso ) . ( 2 (4-27) using 0 A Introducing the angular momentum of the charge carrier about the origin p r L the spin-orbit term can be expressed as L M mc r e Hso .3 (4-28) Clearly, the Hamiltonian changes sign when th e position vector of th e charge carrier is reflected in the plane defined by M and the primary current direc tion, 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 ferromagne tic case obtained the same expression found with intrinsic spin-orbit interaction by K ondo. For the mixed s-orbi t/d-spin interaction,

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36 while evaluating the three spin correlation function in molecular field approximation, the magnetization can be factored out to leav e a two spin functio n giving the following expression for Hall resistivity, Mm xy ~ (4-29) where m is the magnetic spin disorder contribution to conductivity. Superparamagnetism We now consider a unique magnetic beha vior, namely superparamagnetism, which is relevant for high resistance samples near pe rcolation threshold. Th e 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 ) ( sin2 1K E, where K1 is the anisotropy energy and is the angle between the easy axis and the saturation magnetization Ms. Thus if a single domain particle of volume V becomes small enough in size, the energy fluctuation due to finite temperature becomes comparable with V K E1 the energy barrier associated with th e reversal of magnetization. In this situation the magnetization of a particle given by V Ms can be reversed spontaneously even in the absence of an appl ied 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 partic le 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.

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37 If K1=0, so that each particle in the assemb ly can point in any direction and the classical theory of paramagnetism will apply. The magnetization of a superparamagnetic system is thus given by ) / (T k B L n MB (4-30) where n is the number of particles per unit volume, and L is the Langevin function given by x x x L / 1 ) coth( ) ( Thus magnetization curves measured at different temperatures will superimpose when plotted as a fuction of B/T and there will be no hysterisis. We note that superparamagnetism as described above, will disappear and hysterisis will appear when magnetic partic les of certain sizes are cooled below a particular temperature or for a given temperat ure 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 125K T k VB c (4-31) B Bk V K T251 (4-32) TB is known as the blocking temperature, belo w which magnetization will be stable.

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38 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 quant um mechanical effects that cannot be described in terms of classical Bo ltzmann transport theory. To fully understand the nature of disordered c onductors, two new concepts were introduced and have been studied extensively for the last five decad es. 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 deal s with the interacti on among the electrons in the presence of a random potential. These quantum effects become manifest experimentally in the temperature depe ndence of conductivity, magnetoresistance and Hall effect measurements and are more pronounced in low dimensional systems like films and wires. Figure 5-1: Typical wave functions of conduction electr ons in presence of disorder;(a) extended state with mean free path l ; (b) localized state with localization length In the presence of a distribution of random impurity potentials, a conduction electron looses phase coherence at each elas tic scattering event on th e length scale of the

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39 mean free path l but the wave function remains ex tended throughout the sample (Fig.51a). This is the definition of weak disord er. In 1958 Anderson pointed out that if the disorder is sufficiently strong, the wavefuncti on 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.5-1b). An important point to note here is that a linear combinations of infinitely many localized or bitals will not produce an extended state as in the weakly disordered case. Thus enti rely different quantum mechanical processes govern two limits of weak and strong disord er and the understanding of what happens in the intermediate region where the cross over oc curs has lead to one of the most debated and extensively studied topic in condensed matter physics, namely the metal-insulator transition. We present belo w a brief account of how transp ort properties are modified due to quantum properties of conduction electro ns in the presence of varying degree of disorder. Weak Localization Effects A conduction electron in a metal can be treate d as a classical particle only in the limit 1 l kF, where kF is the Fermi momentum and l is the elastic mean free path. At low temperatures when all inelastic pro cesses like phonon scattering are quenched, the conductivity of a metal is dominated elastic scattering of electrons from impurities and is expressed by the Drude expressi on which in 2d is given by, ) (2 2l k h eF d (5-1)

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40 where e2/h is a universal number of the order of 25k and kFl is a dimensionless quantity that determines the degree of disorder. Fo llowing is a simple and elegant reasoning 35 to obtain an order of magnitude estimate of the quantum correct ions to the above classical Drude expression for the conductivity for noninteracting electrons and are known as weak localization corrections. Figure 5-2: Motion of electr ons in presence of impuritie s represented by X (a) Two different paths for an electron to mo ve 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. 5-2a). 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 Ai of individual paths: 2 2 j j i i i i i iA A A A W (5-2)

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41 The first term describes the sum of probabilit ies of traveling each path, and the second term represents interference of various path s. Associated with each path of amplitude Ai there is a phase given by B Al d k 1, which depends on the length of each trajectory. Hence while calculating the total probability W if we consider many distinct trajectories, the net interference term in W will be zero because of the wide distribution of the individual phases. The above argument for neglecting the inte rference term does not hold for certain special trajectories that are se lf-intersecting trajec tories that contain loops (Fig. 5-2b). For each loop there are two amplitudes A1 and A2 corresponding to oppos ite direction of traversal of the loop. However the phase does not depend of the direc tion of traversal of loop and remains same the same, hence for the two amplitudes A1 and A2 are coherent. Thus for a loop trajectory, the quantum mechan ical probability to find the electron at the point O is given by, 2 1 1 2 2 1 2 2 2 1 2 2 14 A A A A A A A A A (5-3) 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 self-i ntersecting 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 self-intersecting 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 Fk/ 1 ~ The mean distance traveled by an

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42 electron diffusing through a configuration of impurities for a certain amount of time t which is much larger than mean collision time from impurities, is given by 2 / 1 2) ( Dt x, where the diffusion constant lv D ~. For a strictly two-dimensional metal film, the area accessible to an electron is Dt. In order for self-intersection to occur during a time d t it is required that the final point of the electron pa th enters the area element t vd. The probability of this event is th e ratio of the two volumes. The total probability of self-intersecting paths is found by integrating over the entire time t The lower limit of the integration is which is the shortest time for the concept of diffusion to apply. To put an upper limit to the inte gral one assumes that there are inelastic processes like electron-phonon and electron-electron interac tion that lead to phase relaxation and hence break down the amplitude c oherence. Lets denote this time scale as phase relaxation time .. The relative change in conductiv ity for the 2d case is given by Dt t vd dd ~2 2 (5-4) The negative sign illustrates that conductivity decreases due to interference. The change in conductivity due to quantum correc tions for a thin film is given by 35, ) ln( ~ ) ln( ~2 2 2l L e ed (5-5) where D L ~. Equation (5-5), although derived for a strictly 2D case, is applicable for thin films of thickness L d The phase-relaxation time has a strong temperature dependence of the form pT~, where p in an integer that depends on th e exact phase rela xation mechanism.

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43 For example p= 1 for electron-electron 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 l ogarithmic temperature dependence due to interference corrections and is given by, ) ln( ~2 2const T p ed (5-6) Magnetoresistance due to Weak Localization The above correction is drastically modi fied if one places the sample in a magnetic field. For a vector potential A describing the magnetic field, one should replace the momentum p by the canonical momentum A c e p This results in a phase difference in the amplitudes A1 and A2 for traversing loops (Figure 52b) in opposite directions as given by 02 2 l d A c eH (57) where is the magnetic flux enclosed by the loop and 0=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 magneto-resistance due to an applied magnetic field, we introduce a new time scale H Since the average diffusion length is (Dt)1/2 and using it as a characteristic size of loops, the magnetic flux through such loops is HDt ~ We define H so that 2 ~H that gives HDH0~ (5-8) The characteristic magnetic fields are determined by the condition ~H, so that

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44 D H0~ (59) Substituting m E lv DF/ ~ ~ and using mc eHB/ we find that 1 B. This means that one needs small magnetic fields. The asymptotic formula for quantum corr ections in presence of magnetic field ) (2Hd for H is much larger than characteristic field given in equation (5-9), is obtained by replacing by H in equation (5-5). Thus for the 2d case the magnetoconductance is given by 35, ) ln( ) 0 ( ) ( ) 0 ( ) (2 2 2c eHD e H H Hd d (5-10) In the above formula H refers to the component perpendi cular to the film. The component of applied field parallel to the film does not affect the weak locali zation correction in a 2d sample as the magnetic flux does not penetr ate through any closed electron paths. Weak Localization in Presence of Spin-Orbit 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 spin-orbit scattering. Since the spin is not conserved a nd can flip while the el ectron moves from one point to another, one needs to consider all possibilities of initial and final spin while calculating the interferen ce term of the self-int ersecting loops (Figure 5-2b). If the initial and final states are given by the wave functions and respectively, the interference term is given by ') 2 ( ) 1 ( ) 2 ( ) 1 ( 2 12 1 A A C (5-11)

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45 where the sum is taken over the final spin and the average is taken over the initial spin state To simplify the expression, instead of assuming two possibl e trajectories along the loop (Figure 5-2b) for a single part icle, we assume two particles moving simultaneously in opposite directions. The inte rference term can be written in terms of eigenfunctions of total spin denoted by 0 for total spin zero and m, 1 for spin1 and projections m =1,2,3 as given by ) ( 2 1 ) ( 2 1) 2 ( ) 1 ( ) 2 ( ) 1 ( 0 1 ) 2 ( ) 1 ( 1 1 ) 2 ( ) 1 ( 1 1 ) 2 ( ) 1 ( ) 2 ( ) 1 ( 0 (5-12) Thus the interference term in equatio n 5-11 can be expressed as follows, ) ( 2 12 0 2 1 m mC (5-13) In the presence of spin-orbit scat tering with characteristic times so, the states m, 1 that carry spin information are da mped with a characteristic time so while 0 is damped with time Therefore for the 2d case it was shown 37, ) 2 1 2 3 ( ~/ 2 2 sot d de Dt vdt (5-14) Thus depending on the sp in-orbit scattering time so relative to we have the following two cases for 2d samples 37, ) ln( ~2 2 ed for so (5-15)

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46 )) ln( 2 1 ) ln( 2 3 ( ~2 2 so de for so (5-16) thus it follows from equation (5-16) that the spin-orbit correction reverses the sign of the temperature dependence of conductivity due to In weak magnetic fields and under strong spin-orbit, the magnetoresistance b ecomes positive. For sufficiently strong magnetic field such that so H the magneto-resistance changes sign and becomes negative. The combined effect of positive quantum correction to conductivity and positive magnetoresistance is known as ‘weak anti-localization’. If however, scattering occurs from paramagnetic impurities, then both the singlet and the triplet wavefunctions in equation (5-13) decay with time of the order of s so that for s 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 spin-orbit scattering based on a theoretical treatment by Dugaev et. al 38. The relevant Hamiltonian considered in this case is given by ) ( ] )) ( ( 4 ) ( 2 [2 0 2 3r r V i r V M m r d Hz (5-17) where ) ( is a spinor field ) , (z y x are the Pauli matrices, M is the magnetization assumed to be along z-direction and V is the random impurity potential, and 0 is the spin-orbit scattering strength. The exchange term Mz acts only on the spins and has no direct effect on the orbital motion. In the language of many-body th eory, weak localization co rrections arise from the particle-particle channel with two propagators describing el ectrons with vanishing total

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47 momentum and with very close energy (C ooper channel). Physic ally, the so-called “Cooperon” propagator represents two elec trons traversing a self intersecting loop (Figure 5-2b) in opposite dire ctions. There are two possibl e situations depending on the relative spin orientation of the two electrons In a ferromagnet the exchange energy is strong so that 1 1, M, where and are momentum relaxation times for spin up and spin down conduction electrons. Physically this condition imp lies that spin-flip 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 pa irs of electrons with parallel spins by the factor ) (/ 1 M. The exclusion of the singl et 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 gene ralization of the non magnetic case with two bands of electrons of opposite spin polarization. We are interested in the two dimensional ferromagnetic samples and quo te the final result in two dimensions 38 as follows, )]} ~ 1 1 ( ln[ )] ~ 1 1 ( {ln[ 42 2 2 SO SO de (5-18) where ) ( is the temperature depende nt phase breaking time. ) ( is the momentum relaxation time and ) ( ~ SO is the effective spin-orbit 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 im portant point is that for ~ SO the above correction to conductivity is nega tive just like the nonmagnetic case.

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48 Electron Interaction Effects In this section we discuss the correcti ons to conductivity due to interactions between conduction electrons. We note that within the Boltzmann transport equation, electron-electron collisions ca nnot affect the conductivity in the case of a single band structure and in the absence of Umklapp pr ocesses. This is because electron-electron collisions conserve the total momentum. Incl usion of the Fermi liquid corrections that takes into account a finite inter electron inter action potential, renor malizes the residual conductivity but does not intr oduce any essential temperat ure dependence. However, taking into account the interference of el astic scattering by impurities with electronelectron interaction pr oduces nontrivial temperature de pendence of the conductivity and the one particle density of states. The follo wing is brief account of some simple physical ideas that illustrates the origin of quantum corrections to transport properties due to electron-electron interaction. Scattering by Friedel Oscillations We now discuss a very importa nt concept of the Friedel os cillations that arise due to standing waves formed as a result of in terference between incoming and backscattered electron waves. To illustrate this we consider a simple situation in 1D with an infinitely high barrier at x=0. For each wave vector k, the wavefunction is a superposition of the incoming plane wave L ikx/ ) exp(and a reflected wave L ikx/ ) exp( and is given by L kx i/ ) sin( 2 Accordingly, the probabili ty density is given by L kx/ ) ( sin 42 2 and oscillates in space. The probabil ity that a state with momentum k is occupied is given by the Fermi function fk. The electron density is described by

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49 x x k n kx dk f dk x nF o k k kF ) 2 sin( ) ( sin 2 8 2 2 ) (2 0 2 (5-19) where / 2F ok n is the density of the homogeneous electron gas, is oscillatory and damps from the origin as x-1. Figure 5-3: Schematic diagram of Friedel oscillation due to a single impurity due to backscattering described by path C. In terference between two paths A and B contributes mostly to backscattering. A single impurity at the orig in with a general potential ) ( r U also induces a modulation of electron density close to the impurity. Th e oscillating part of the modulation in 2D(shown schematically in Fig. 5-3) is given by 39, ) 2 sin( 2 ) (2r k r rF (5-20) where r is the distance from the impurity, whic h has a potential treated in the Born approximationr d r U ) (, and 2/ m is the density of states in 2D. Taking into account the electron-electron interaction ) (2 1r r Vo one finds additional scattering due to

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50 the Friedel oscillation. This potential can be presentation as a sum of the direct (Hartree) and exchange (Fock) terms 39: ) ( ) ( 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2 1 2 1 2 1 3 3 1 3 1 2 1 2 1 1 2 1r r n r r V r r V r r r V r d r V r r V r r r V r r Vo F o H F H (5-21) where ) ( r is the diagonal element of th e one electron density matrix n given by, ) ( ) ( ) (2 1 2 1r r r r nk k k (5-22) The factor indicates that onl y electrons with the same spin participate in exchange interaction. As a function of the distance from the impurity, the Hartree-Fock 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.5-3. If an electron follows path A, it scatters off the Friedel oscillation created by the impurity a nd path B corresponds to scattering by the impurity itself. Interference is most important for scattering angles close to (backscattering), since the ex tra phase factor on path A is cancelled by the phase of Friedel oscillation ) 2 exp( R k iF, so that the amplitudes co rresponding to the two paths are coherent and interfere constructively. Th is interference persists to large distances R and is limited only by temperature, T k v k k RB F F/ / 1 At finite temperature the amplitude of the Friedel oscillation a ssumes 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 T. The correction to conductivity with respect to the clas sical Drude conductivity is given by 39,

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51 F F o oT k V V )] 2 ( 2 ) 0 ( [ (5-23) where the first term is the exchange corre ction and the second term is the Hartree correction. The above is the conductivity correction in the ballistic limit defined by 1 T We note some important points in the a bove expression. Firstly, the sign of the Hartree and exchange corre ctions are opposite. Secondly, the leading temperature correction comes from th e Fourier component at q=0 and q=2kF for the exchange and Hartree term respectively. The sign of the to tal correction is not uni versal and depends on the details of the electron-electron scattering. Figure 5-4: Friedel oscillation due to two im purities created by the self-intersecting 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 fr om self-intersecting paths of electrons. In Fig. 5-4 we show scattering process that i nvolves 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

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52 oscillations have been calc ulated in the framework of many body theory and is known as the Altshuler-Aronov 35 correction. Although interacti on between two electrons is independent of spin, summation of terms in th e perturbation theory depends on the spin state of the two electrons invol ved. The total number of channe ls 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) differi ng by the projection of the spin. For long range intera ction the perturbation theory for the Hartree corrections singlet and triplet channels is different. The singlet channel contribution combin ed with exchange corrections as a renormalization of coupling cons tant and the final result is still universal. The triplet channel contribution depends on the Fermi-liquid constant 0F. The total conductivity correction in 2D is given by 35, 39 : )] ) 1 ln( 1 ( 3 1 )[ ln( 20 0 2 2 F F T e (5-24) The above equation was derived for the so-c alled “Diffusive regime” characterized by 1 T. The sign and magnitude of the correcti on is non-universal and depends on the competition between the universal and positive exchange term and the coupling dependent and negative Hart ree contribution. For shor t-range electron-electron interactions due to screened Coulomb interactions in good metals, the Hartree term is neglected and one has a univers al correction to c onductivity due to th e exchange term. The above result is obtained by treating in teraction 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

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53 dimensional conductor with spin-split bands li ke in ferromagnet and long range Coulomb interaction, the scaling theory predicts a universal correction 40 given by ) ln( ) 2 ln 2 2 ( 22 2 T e (5-25) The above results were obtained for the cas e when weak localization corrections are completely suppressed. For shor t range electron inte raction the logarithmic coefficient is found to be non-universal 40. Moreover, for 2d conductors with strong sp in-orbit or spin-fli p scattering, with long range Coulomb intera ction, it was argued 41 that the triplet cha nnel is suppressed and only the exchange term survives, pred icting a universal co rrection given by, ) ln( 22 2 T e (5-26) 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 manybody theory, weak localization effects arise fr om the particle-particle channel and are supposed to be sensitive to magnetic flux. Th e electron interaction e ffects arise from the particle-hole diffusion channel and do not have similar sensitivity to magnetic field. The dominant effect of magnetic fi eld in this case is the splitti ng of the spin up and spin down bands 34. This physical idea is most simply illu strated for the self-energy correction, where the singular correction is due to the correlation betw een the wavefunction of the added electron and the wavefunctions of the oc cupied electrons that are nearby in energy. In the presence of a magnetic field, th e triplet term is divided into an Sz=0 and two 1 zSterms. The exchange (singlet) and the Sz=0 triplet terms invol ve correlation with

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54 electrons with the same spin and are unaffected by the spin splitting. For the 1 zS terms, the spin splitting produces a gap H gB between the lowest unoccupied spin-up and the highest occupied spindown electron. The singularity of that term is cut off for H gB greater than kBT. In a magnetic field, the correction to the conductivity can be written as a sum of two terms, ) ( ) ( ) (T H T T HS C (5-27) 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 (5-21). The second term is the 1 zStriplet contribution or th e “spin channel”, with a field dependence for the 2d case given by 34, 1 1 ln ) ) 1 ln( 1 ( 2 ) 0 ( ) (2 0 0 2 2h h h h F F e T T HS S (5-28) where T k H g hB B/ The quantity in the parenthesis is the Hartree contribution to conductivity and is non-universal in both sign and magnitude and de pends on the details of the potential describing electron-electron interaction. Transport Properties of Granular Metals So far we have discussed quantum transpor t properties of homoge neous systems with a uniform distribution of impurities that determin e 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 diffusi ve and they can tunnel from one grain to another. In prin ciple the grains can be clean so that electron scattering is mainly from surfaces. In this limit the t unnel conductance is smaller than the grain

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55 conductance and inter-granular transport can be distinguis hed from intra-granular transport. The process of elect ron tunneling from grain to grai n that governs the transport properties are accompanied by charging of grai ns. 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 gT one observes exponential (at gT<<1) or logarithmic (at gT>>1) temperature dependence of conductivity. This theoretical approach was based on an earlier paper by Ambegaokar 43, which however was applicable only at temperatures Tg T where is the mean energy level spacing in a single grain. In this regime the electron coheren ce does not extend beyond the grain size. The low temperature regime Tg T where the electron moves coherently over distances exceeding the single gr ain size, was discussed in a later paper by Beloborodov et.al 42 for large tunne ling conductance gT. The following Hamiltonian describes a system of weakly coupled metallic grains, )] ( ) ( ) ( ) ( [* i j j i ij ij C or r r r t H H H (5-29) where tij is the tunneling matr ix element between i-th and j-th grain, Ho is the Hamiltonian for non-interacting is olated grains, and Hc describes the Coulomb in teraction inside (i=j) and between grains (i j) as described by j ij ij i cn C n e Hˆ ˆ 21 2 (5-30) where Cij is the capacitance matrix and inˆ is the operator of electron number in the i-th grain.

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56 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 intera ction are similar to those obt ained for homogeneous metals. The critical grain size in 3D where a metal to insulator transition occurs is estimated to be, ) ln( 6 1 C c TE g (5-31) where EC is the charging energy of an isolated grain. The conductivity of a granular metal is given by 2 1 o (5-32) The classical Drude conductivity for a gr anular metal in a general dimension d, with grain size a is given by d T oa g e2 22 (5-33) The correction to c onductivity due to la rge energy scales Tg is given by ] ) max( ln[ 2 11 T C T T og T E g d g (5-34) We note that the dimensionality in this case appears only as a coefficient but the logarithmic temperature dependence remain s same for all dimensions. This means tunneling of electrons with energies Tg can be considered as incoherent. On the other hand corrections from low energy scale Tg arises from coherent electron motion on the scales larger than grain size, and is given by

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57 1 4 2 ) ln( 4 1 3 122 2 2d Tg d T g g d g T gT T T T T o (5-35) where and are numerical constants. We summari ze the results for a 2D granular system for Tg T as follows, )] ln( ) ln( 4 [ 22 2 2 T C Tg T E g e (5-36) and for Tg T )] ln( 4 [ 22 2 2C T TE g T g e (5-37) For samples with weak inter grain coupling 1 Tg at low temperatures CE T the conductivity was shown to be 42, ) / exp( 2T EC o (5-38) where EC is the charging energy. However this behavior is usua lly not observed experimentally because of the distribution of gr ain 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 be tween grains. They proposed a picture of granular metal represented by a conductance network in whic h the metal grains are interconnected by conductances of the form:

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58 ) 2 / 2 exp( ~T k E sB o c s (5-39) where s is the tunnel-barrier thickness and 2 / 1 2) / 2 ( m for a barrier height of The calculations are simplified by assuming the gr ains to be spherica l with a distribution of grain diameters d and the charging energy d EC/ 1 ~, such that the product sEc is constant for a given film: C sEC where C and are constants that depend only on the volume fraction of the metal. The sec ond 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 tunnel-barrier thickness given by2 / ) / (2 / 1T k C sB m for which the inter grain c onductivity is maximum. The final assumption is that the temperature dependence of the conductivity ne twork is given by that of the maximum inter-grain conduc tivity so that, ] ) / ( 2 exp[ ~ ) (2 / 1T k C TB (5-40) The constant C is proportional to the charging energy EC, which is inversely related to the mean grain diameter . Thus, the dominant contri bution to conductivity at high temperatures is due to tunneli ng between small grains (large Ec) separated by thin tunnel barriers, while at low temperatures the dom inant 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,

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59 ]} 2 / ) ( exp[ ) 1 ( 2 1 ] 2 / ) ( exp[ ) 1 ( 2 1 ){ 2 exp( ~T k E E P T k E E P sB M o c B M o c s (5-41) Here P is the polarization of tunneling electrons, so that the coefficients (1+P)/2 and (1P)/2 are the probabilities that an electron tunneling from one gr ain to another has its spin parallel and anti-parallel, respectively, to that of the initial grain. Using the same assumptions for the non-magnetic grains di scussed earlier, the magneto-conductivity is given by 45, )] / sinh( ) / )[cosh( 0 ( ) ( T k E P T k E T T HB M B M (5-42) The magnetic exchange energy can be expresse d in terms of spin correlations of two neighboring grains and is given by 45, ] / 1 [ 2 12 2 1S S S J EM (5-43) An important point to note is that by applying a magnetic field strong enough to align all the moments, the exchange energy is zer o, and the temperature dependence of conductivity reduces to that of the non-magnetic case shown in equation (5-40). Quantum Corrections to Hall Conductivity The normal Hall coefficient defined asH J E Rx y n/ is another quantity in addition to magnetoresistance that behaves di fferently for weak lo calization and electron interaction effects. The quantum correc tion to the Hall conductivity due to weak localization effects was fi rst calculated by Fukuyama 46 and it was shown that the Hall resistance given byB R Rn n xyremains unchanged so that 0 / n xy n xyR R (5-44)

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60 Thus in a disordered conducto r subject solely to weak lo calization effects, the normal Hall resistance at given magnetic field will remain constant as temperature is varied. The longitudinal resistance Rxx will have the usual logarithmic temperature dependence due to interference effects. The normal Hall c onductivity has logar ithmic temperature dependence and has a slope twi ce that of longitudi nal conductivity. This is easily deduced using the fact that 2/xx n xy n xyR R and taking logarithmic derivative we have xx xx n xy n xy n xy n xyR R R R 2 (5-45) Thus it follows from equation (5-44) that the normal Hall conductivity has logarithmic temperature dependence with a slope twice that of longitudinal conductivity as shown by the relation, xx xx xx xx n xy n xyR R 2 2 (5-46) 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, 0 / n xy n xy (5-47) Thus it follows from equation (5-45) the effect of only electron interaction implies that xx xx n xy n xyR R R R 2 (5-48) This means that both Rxx and n xyR will exhibit logarithmic temperature dependence due to electron interaction, and the slope of n xyRwill be twice that of Rxx.

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61 Quantum corrections to the anomalous Hall conductivity have not been studied as extensively as the normal Hall conductivity. The effect of short-range electron interactions to the AH conduc tivity within the framewor k 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 xx xx AH xy AH xy AH xyR R R R 2 0 (5-49) 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 contribu tions to the AH conductivity were studied by Dugaev et. al 24 for the Hamiltonian given by equation (5 -17). For the case of side jump (SJ) mechanism in a 2d ferromagnetic sample it was predicted 24 that 3 ) ( ) () ( 1 ~ l kF SJ AH xy SJ AH xy (5-50) The corresponding weak localiza tion correction to longitudina l conductivity is given by 1) ( ~ /l kF xx xx We note that the weak localizati on calculations are valid in the metallic regime that corresponds to 1 l kF. Thus it was concluded that the weak localization correction to AH conductivity due to side jump mechanism is negligible 24,

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62 xx xx SJ AH xy SJ AH xy ) ( ) ( (5-51) The above theoretical prediction prov ided an alternative explanation24 to the experimental results of Bergmann and Ye 48. For the case of skew scattering(SS) mechan ism, the weak localization corrections were shown 24 to give rise a finite correction as given by )]} ~ 1 1 ( ln[ )] ~ 1 1 ( ln[ { 362 2 0 2 0 2 ) ( SO F SO F SS AH xyk k v e (5-52) The above expression was derived for the case of a random short-range impurity potential of the form i i oR r v r V ) ( ) ( 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, } { 182 2 2 2 2 2 0 2 0 2 ) ( F F F F SS AH xyv k v k v e (5-53) Using the fact that the residual l ongitudinal conductivity is given by m e n m e nxx 2 2 (5-54) and assuming a parabolic band, one can simp lify 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, )] ~ 1 1 ( ln[ 1 2 2 1) ( ) ( ) ( ) ( 2 2 ) ( ) ( ) ( ) ( SO xx SS AH xy SS AH xye (5-55)

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63 Comparing the above equation wi th equation (5-18) it follows that both the longitudinal and the AH conductivity have finite logarithmi c corrections due to weak localization.

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64 CHAPTER 6 EXPERIMENTAL RESULTS AND DISCUSSION Experimental study of magnetism in ultr athin 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 fe w monolayers thick, which were quenchcondensed on antimony substrates at liqui d helium temperature. These experiments revealed an important finding that the Anomal ous Hall effect behaves similar to that of normal Hall effect in non-magnetic material s and has no quantum correction at low temperatures due to electron-electron interactio ns or weak localizations effects. However, one might argue that the presence of a “pol arizable” substrate, namely antimony, might dope the few atomic layers of iron on top a nd 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 micr oscope slides. Using the SHIVA apparatus described in chapter 2, we were able to investigate the magnetic properties of iron and cobalt films with polycrystal line 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 spu ttering techniques under identical growth conditions. We used r.f. power of 35W with an argon flow of 10

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65 sccm, which developed a DC bias of around -145 V with respect to target. The pressure in the chamber is of the order of 10-4 Torr. The samples were grown in the Hall bar geometry through a shadow mask onto glass s ubstrates at room te mperatures. Under the conditions described above, films are known to grow through various stages of different morphology 49 rather than gradual layer by layer gr owth. Initially, film growth proceeds via nucleation of isolated grains of meta ls. With more arriving adatoms the grains continue to grow in size and at some critic al thickness, the grains coalesce into several discrete and continuous percolating channels. As the substrates ar e exposed further the film eventually becomes homogeneous with we ll-connected microscopi c grains, such that the film resistivity scales with thickness 17, 18. Such a film behaves like a good metal with a low temperature residual conductivity de termined by impurities and imperfections. Even in the case of metallic grains not physic ally touching each other, there could be electrical conduction du e to tunneling of electrons betw een grains. The polycrystalline films in our experiments were thicker than the quench-condensed f ilms in the previous investigation 48 by Bergmann and Ye, but are outsid e the homogeneous regime where film resistance scales with thickness. Thus resistivity is not a we ll-defined quantity, and we use sheet resistance, to characterize our f ilms. The sheet resistance is defined as the resistance of a square film and is independe nt of the lateral dime nsions of a film and depends only on thickness and morphology. In the Hall-bar geometry, where the sample is rectangular shaped, where the current I is uniformly distri buted along the width W and where voltage V is 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) / /( W L R Rsq

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66 Measurement Setup We have investigated films over a wide range of sheet resistances from 50 to 1000000 All our samples are in the Hall bar ge ometry with six terminals as shown in Figs. 6-1and 6-2. We performed standard four terminal techniques us ing 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 meas ure longitudinal and transverse resistances simultaneously, depending on the magnitude of the two terminal resistances of the leads. Figure 6-1: A d.c. transport measurement se tup using Keithly 236 for sourcing a constant current and measure longitudinal voltage and Keithly 182 nano-voltmeter to measure transverse voltage. The sample in Hall-bar shape is shown as a shaded. Figure 6-1 shows the circuit diagram fo r d.c. measurement using a Keithley 236 Source-Measure Unit and a Keithley 182 Nanovoltmeter. We programmed the 236 unit to

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67 source a constant current through the sample and measure the voltage developed across the longitudinal leads. Simultaneously the Keithley182 is used to measure the voltage developed across the Hall leads. This setup is particularly usef ul for measuring high resistance samples as the Keithley 236 is e quipped 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 6-2:An a.c. transport measurement setup using two SR830 lock-in amplifiers operating at same frequency to measure longitudinal and transverse resistance, used in samples with low contact resist ances. A constant current is generated from the voltage source of the upper SR830 and by placing a ballast resistor Rb of 1M in series with the sample. The sample in Hall-bar geometry is shown as shaded

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68 Figure 6-2 shows the circuit diagram fo r 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 1nV. The SR 830 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 meas ure longitudinal and transverse voltages simultaneously. The input resistance for the SR830 is only 1M hence only samples with two terminal contact resistances less than 10k are measured using this setup. Weak Disorder: Iron Films Transport Properties at B=0 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 magneto-transport 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 indi vidual grains is much smaller than the intergrain tunneling resistance and hence the late r determines the low temperature residual resistance Ro. We grew a series of iron f ilms on glass substrates with Ro varying over a range of 50 to 50000 We observed a crossover in the temperature dependence of sheet resistance Rxx(T) as Ro is systematically increased Figure 6-3a shows Rxx(T) for an iron film of thickness d= 100 and Ro= 70 This represents typical behavior for all films with Ro<1000 with resistance decreasing linearl y with temperature and reaching a

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69 minimum at some temperature Tmin 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 lo w temperatures. For films with Ro<1000 the grains are well connected and Ro is determined by impurities and lattice imperfections in the grains. For T1000 the resistance increases monotonically with decreasing temperature and with no minimum. Figure 6-3b shows Rxx(T) for an iron film with d= 20 and Ro= 8400 At high temperatures, the increase in resistance with decreasing temperature is due to decreased ph onon assisted “hopping” processes of conduction electr ons over the tunnel barrier s between grains. At low 5100200300 70 80 90 100 5101520 69.68 69.72 R()T(K) (a)5100200300 6500 7500 8500 5101520 7650 8000 8350 R()T(K)(b)

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70 temperatures the residual resistance should be due to temperature independent inter-grain tunneling processes. However at low temperat ures (T<15K), the resistance is found to increase logarithmically as shown in the inse t of Fig. 6-3b, which is a manifestation of quantum corrections in the presence of tunneli ng processes. The crossover in the sign of temperature coefficient of resistance dR/dT in 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 cm. Thus the iron films used in our experiment with Ro in the range 50 to 50000 exhibit a logarithmic increase at low temper atures and are considered to be weakly disordered in the context of quantum theory of transport. To compare the experimental data with existing theo ries 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): ) ln( 1 const T L A R Loo R xx xx (6-1) where 1 2 2) 81 ( 2 / k e Loo,the quantum of conductance and AR is a numerical prefactor that depends on the microscopic scattering parameters that determine the quantum corrections. Figure 6-4 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 3000 For low resistance samples, the prefactor is constant with AR = 0.95 0.03 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= 49000 the prefactor AR= 0.326 0.001. We note that the red square da ta points refer to films that had

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71 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 re duction in the re sistance, as discussed in detail in chapter 3. However as seen from our data, the ion-milled 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. Figure 6-4:Plot of numerical prefactor AR for logarithmic temperature dependence of longitudinal conductance (equation 6-1) fo r different iron films, as a function of Ro, sheet resistance at T=5K Red square points corresponds to ion-milled 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 el ectric field, even in the absence of an 1021031041050.0 0.5 1.0 ARRo()

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72 external magnetic field. However, an extern al magnetic field is required to align the magnetic domains thereby maximizing the magn itude 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 pot ential between the Hall leads ( Vxy) will be due to the longitudi nal 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 2 / )) ( ) ( ( B V B V Vxy xy H Figure 6-5: The anomalous Hall cu rves for iron films with (a) Ro= 300 and (b) Ro= 2700 Note decrease in Bs and simultaneous increase of Rxy as Ro increases. Typical AH curves for two different iron films are shown in Figs 6-5(a) and 65(b) corresponding to Ro= 300 and Ro= 2700 respectively. Both Hall curves exhibit anomalous behavior with a st eep rise in Hall resistance w ith increasing magnetic field B, due to moments lining up along the field until the saturation value at B=Bs (shown by the vertical arrow), followed by a much slower increase due to the normal Hall effect. An -4-2024 -8 -4 0 4 8 Rxy()B(T)(a) Bs-4-2024 -80 -40 0 40 80 Rxy()B(T)(b) Bs

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73 important point to note is that for Ro= 300 the AH curve saturates at an applied field of Bs~1.7T to a value Rxy~ 8 while for the sample with Ro= 2700 the saturation field Bs~1.2T and high field value Rxy~80 This points out an impor tant trend seen in our samples: as Ro increases, Bs decreases while the high fi eld saturation value of Hall resistance increases as Ro increases. However, as we will show subsequently, this monotonic dependence on sheet resistance breaks down above Ro~ 2000 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 depende nce of the high field saturation value of AH resistance AH xyR on Ro, into the very high resistance regime ~1M which to the best of our knowledge has not been studi ed experimentally. We use the following scheme to analyze our data and extract the saturation value of AH resistance, which allows us to systematically compare AH xyRfor 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 Rxy 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 : B R M R Rn s s o xy (6-2) where Rs and Rn are the anomalous and normal Hall coefficients respectively in two dimensions. Thus the intercept of such a fitte d straight line is the contribution to Hall

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74 resistance at zero applied field B and arises only from the spontaneous magnetization in the material. We identify the intercept as the AH resistance s s AH xyM R R0, which is the contribution at zero applied field and so lely due to magnetization of the film. Figure 6-6:The anomalous Hall resistance at T= 5K for different iron films as a function of Ro on a log-log scale. Red square points are ion-milled films. The dashed line represents the average value of the AH resistance at 80 Figure 6-6 shows the dependence of AH xyR on iron samples with different Ro using a log-log scale. For iron samples with sheet resistances as high as Ro= 49000 we observe the anomalous behavior in Hall resistan ce indicating the presence of a local finite magnetic moment in the films. However we observe a distinct crossover in the dependence of AH xyR with increasing Ro. For Ro>2000 the monotonic increase in AH xyR seen at lower resistances, ceases to hold. Instead the AH resistance attains a constant value 10 80AH xyR, independent of Ro. We point out that the crossover is observed around values of Ro where the zero field coefficien t of logarithmic temperature dependence AR, starts to deviate from unity. 102103104105100101102 RAH xy()Ro()

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75 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, Rxx(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 ) ( T Rxxand ) ( T RAH xy are simultaneously measured while temperature is slow ly increased. We will show that AH xyRalso has logarithmic temperature dependence for T <20K for all the iron samples, and discuss the relative resistance (RR) scaling of ) ( T Rxxand AH xyR for each sample with different Ro. In most samples there is always a misalignm ent in the Hall leads as result of which a fraction of the transverse potential has c ontribution from the l ongitudinal 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 Vxx(T) and transverse potential Vxy(T) for each field. Then we extract the symmetric response as l ongitudinal resistance I B T V B T V B T Rxx xx xx2 / )) ( ) ( ( ) ( and antisymmetric response as the AH resistanceI B T V B T V B T Rxx xx AH xy2 / )) ( ) ( ( ) ( 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 ) ( KN that we call the “normalized relative change” in a transport quantity, say K with respect to some reference temperature T0 evaluated for temperatures T >To:

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76 ) ( 1 ) ( ) ( ) ( ) (0 00 0T R L T K T K T K Kxx o N (6-3) where 00L= (81k )-1 is the quantum of conductance and Rxx(T0) is the sheet resistance at T0. We note that the “rela tive change” is defined as: ) ( ) ( ) (0 0T K T K T K K K (64) Thus the normalized relative change ) ( KN is the relative change in the quantity K K / divided by the factor L00Rxx(T0) a dimensionless quantity that is a measure of the effective disorder in a two dimensional syst em. Using the above notation and keeping in mind the fact that the low te mperature behavior of both xxR and AH xyR is logarithmic, we employ the following equations to fit our data: ) ln( ) ( ) ( ) ( ) (2 00o R o xx o xx xx xx NT T A T R L T R T R R (6-5) ) ln( ) ( ) ( ) ( ) ( ) (00o AH o xx o AH xy o AH xy AH xy AH xy NT T A T R T R L T R T R R (6-6) where AR and AAH are coefficients of the logarithmic temperature dependence of xxR and AH xyR respectively. The coefficients depend on the parameters describing the quantum corrections to longitudinal and AH resistance respectively. Using the approximation that ) ( ) ( ) ( T R T R T Rxx o xx xx the longitudinal c onductivity is calculated from the raw data as xx xxR L / 1 so that ) / ( /xx xx xx xxR R L L Thus it follows from equations (6-5) that ) ln( ) ( ) ( ) ( ) ( 1 ) ( ) (2 00 0o R o xx oo o xx xx o xx xx xx xx NT T A T R L T R T R L T R T L L L (6-7)

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77 The AH conductivity has contributions from both xxR and AH xyR, and is calculated from raw data as ) ) /((2 2xx AH xy AH xy AH xyR R R L Using the approximation) ( ) ( T R T Rxx AH xy, which is true for most materials, we have2/xx AH xy AH xyR R L. It follows from equations (6-5) and (6-6), ) ln( ) 2 ( ) ( 1 ) 2 ( ) ( 1 ) (00 00 0o AH R o xx xx xx AH xy AH xy xx AH xy AH xy AH xy NT T A A L T R R R R R L T R L L L (6-8) Moreover, it follows directly from equations (6-5) and (6-6), R AH xx N AH xy NA A R R ) ( ) ( (6-9) Thus the coefficient of logarithmic te mperature dependence of conductivity is AR and that of the AH conductivity is 2AR-AAH. Also it follows from equations (6-8) and (6-9) that AAH / AR =2 implies that0 ) ( AH xy NL. Any deviation of this ratio from 2 implies a nonzero logarithmic temperature dependence ofAH xyL. We note that the logarithmic prefactor AR, as defined in equation (6-1) and equation (6-5), 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 Lxx. We have already discussed how AR varies with changing Ro when B= 0(Figure 6-4). We found that, even in the presence of B= 4T, the magnitude of AR did not show any significant change. The reason is that the iron films under c onsideration have very small ma gnetoresistance(MR). In Fig. 6-7 we show the MR for a film with Ro= 300 is of the order of 0.15%. The magnetoresistance curves for all the iron samp les are found to be predominantly negative with small hysteresis and showed a pronounced saturation at applied fields that also

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78 corresponds to saturation of AH resistance (compare Figs. 6-5(a) and 6-7). The hysteretic behavior progressively d ecreases 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. Figure 6-7: Magnetoresistance as a func tion of field for an iron film with Ro= 300 To show the relative scaling behavior of resistance and AH resistance, we have plotted in Fig. 6-8 on a logarithmic temper ature scale the normalized relative changes ) ( T Rxx N ) ( T RAH xy N and ) ( T LAH xy N, with T0=5K as reference temperature and at an applied magnetic field of 4T, for a film with R0=2733 We observe a distinct feature that the curves for ) (AH xy NR and ) ( T Rxx N exactly overlap each other while obeying logarithmic temperature dependen ce at low temperatures up to T~ 20K. At higher temperatures, we observe that ) (AH xy NRdeviates from logarithmic behavior and decreases at a faster rate than ) ( T Rxx N The importance of this behavior becomes -4-2024 -0.0015 -0.0010 -0.0005 0.0000 (R(B)-R(0))/R(0)B(T)

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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 AH xyR was found to be a factor of two higher than that of xxR .The deviation of ) (AH xy NR from ) (xx NR at high temperatures(Figure 6-8) is also intere sting and is a possible indication that the dominant quantum corrections to xxR and AH xyRare due to different mechanisms. We note that the quench-condensed amorphous films 48 in Bergmann’s experiments could not be heated above 20K without incu rring irreversible morphologica l changes, hence we could not compare our high temperature data. Figure 6-8: Relative resistance (RR) scaling behavior at T< 20K for an iron film with Ro= 2700 The uppermost (blue) curve e xhibits finite logarithmic temperature dependence of ) (AH xy NL. 510100200 -6 -5 -4 -3 -2 -1 0 1 2 Normalized relative change T(K)N (Rxy AH)N (Rxx)N(Lxy AH)

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80 Figure 6-9: Plots s howing dependence on Ro of the extracted transport coefficients (a) AR and (b) AAH/AR showing a crossover in value (shown by arrow) near Ro~ /e2= 4100 Values close to unity in both cases correspond to RR scaling. (c) 2AR-AAH 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. 0.0 0.5 1.0 0.0 0.5 1.0 1021031041050.0 0.5 1.0 AR AAH/AR (a)(b) R0() 2AR-AAH(c)

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81 Fitting the data in Fig. 6-8 for low temperature ( T =5-20K) part to equations (6-5) and (6-6), we find for this particular film that AR= 0.897 0.001, and AAH= 0.908 0.005. The AH conductivity AH xyLcalculated from the raw data also exhibits a logarithmic temperature dependence below 20K with a pos itive slope as shown by the blue data points in Figure 6-8 a nd the prefactor for ) (xx NL is found to be 0.908 0.005 close to the value of 2AR-AAH =0.89 0.01 in accordance with equation (6-7). For simplicity and future reference we call the low temperatur e scaling behavior for samples where the AAH/AR=1 (Figure 6-8) so that the relative changes in ) ( T Rxxand ) ( T RAH xyare equal, the relative resistance (RR) scaling. The RR sca ling behavior is obser ved in all of our samples with Ro<3000 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 toAH xyL In Fig. 6-9(a) we have re-plott ed for reference the coefficient AR as a function of Ro. The plot is essentially the same as Fig. 6-4 and shows a deviation from AR~1 for Ro>3000 There is also a crossover in the rela tive scaling behavior of AH resistance and resistance, which is measured quant itatively by the ratio of coefficients AAH/AR as shown in Fig.6-9(b). All samples with Ro in the range 300-3000 were found to exhibit RR scaling behavior (Fi gure 6-8) with the average value for the ratio AAH/AR=1.07 0.1. However as Ro increases beyond 3000 the ratio AAH/AR systematically decreases from unity, which according to equation (6-9) im plies that the relative change in xxR is larger than the relative change in AH xyR. For a sample with R0= 50000 for example, which is

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82 the last data point in Figure 6-9b, we found that AAH/AR=0.122 0.002. In Fig. 6-10 we have plotted ) (xx NR and ) (AH xy NRfor this sample, which clearly shows that the logarithmic slope for ) (xx NR is much greater than ) (AH xy NR. We emphasize that even if the ratio AAH/AR decreases from unity in the high resistance regime, there remains a finite and positive quantum correction to the AH c onductivity as shown by the finite value of 2AR-AAH =0.6 (Figure 6-9c). Figure 6-10:An iron film with Ro= 49000 showing deviation from RR scaling. The ratio AAH/AR =0.12 as shown in the final point in Fig. 6-9c. An important observation related to the hi gh resistance regime is shown inFig. 611, where we have plotted the relative changes in the AH resistances AH xy AH xyR R /as defined in equation (6-4), for three samples with Ro varying over a wide range from 2400 to 49000 In the low temperature range of T =5-20K, 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 th e increasing longitudinal resistance. Thus for 58121620 -0.4 -0.3 -0.2 -0.1 0.0 Normalized Relative changeN (Rxx)N (Rxy AH) T(K)

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83 Ro>3000 in addition to the fact that the magnitude of AH xyR at T =5K attains a constant value ~80 as shown in Fig. 6-6, the relative change in AH xyR over the temperature range of 5-20K also remains constant. Figure 6-11:The relative changes in AH resist ances 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 AH xyLis observed. Interestingly, the coefficient of AH conductivity 2AR-AAH as defined in equation (6-8), does not show any pronounced dependence on Ro and is scattered around an average value of 0.86 0.2(Figure 6-9c). 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 conductiv ity that showed a crossover behavior around 2/ ~ e Ro. This is a strong indication that the dominant 5101520 -0.04 -0.02 0.00 R0= 2.7k Ro=26.1k Ro=48.8k RAH xy(T)/RAH xy(5K)T(K)

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84 mechanisms that are responsible for the te mperature dependence of the longitudinal (xxL ) and AH ) (AH xyLconductivity 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 50-50000 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 ca rriers 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 inter-grain barrier separati on and/or barrier height is la rger 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 subs trate and then iron films were grown top. C60 in solid form is known to be an insu lator with complete ly filled bands. However, C60 molecules have a high electron affin ity; 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 othe rwise isolated grains, as s hown schematically in Fig. 6-

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85 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. Usin g these techniques, we were able to grow high resistance stable ferromagnetic films. Figure 6-12: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 Ro= 1600 and 4100 which are well within the range of iron films grown on bare glass and compared their transport properties with iron films having simila r resistances. The AH resistance at T= 5K was found to beAH xyR=64 and 81 respectively, which is wh at 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 ) ( T Rxx, and ) ( T RAH xy obeyed a logarithmic temperature dependence at low temperat ures, with the logarithmic prefactors AR AAH 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 resist ance when grown on C60 rather than bare glass, because of the shunting path provided by the C60 monolayer.

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86 Figure 6-13:Plot of conductivity sh owing hopping transport in a Fe/C60 sample as given by equation (6-10). We now focus our attention on iron/C60 films that are in the strongly disordered regime. Figure 6-13 shows Rxx(T) at B =0 for such a film with resistance at T =5K, Ro 20M We observe that this film does not show the logarithmic temperature dependence of conductivity. Instead the longi tudinal conductance f its to the following functional form as predicted 44 by theories of hopping conduc tion in granular metallic films: ) ) ( exp( ) (2 / 1 T T L T LO xx xx (6-10) where 0 xxL and T are characteristic resistance and energy scales in a two dimensional system with localized electronic states. The fit in Fig. 6-13 yields T=266K and OO O xxL L 754 1 Table 6-1 summarizes the fitting resu lts for all the strongly disordered films under consideratio n. We note that the T is directly related to the Coulomb 0.10.20.30.40.5 10-210-1100 Lxx/LOOT-1/2(K-1/2) 4 100 25 11T(K)

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87 energy44 d e EC/ ~2 associated with the charging of grains when an electron hops from one grain to another, being the average grain size. Our data as shown in table 6-1, reveal that T increases with increasing Ro indicating that for hi gh resistance samples the average grain size is smaller. Figure 6-14: 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 so lid lines represent fits to the Langevin function. Figure 6-15:Magnetoresistance curves at different temperatures for the Fe/C60 sample shown in Figs. 6-13 and 6-14. The curv es do not show shar p saturation at a particular field as seen in iron films (Figure 6-7). 0246 0 50 100 Rxy()B(T)124K Rxx=70k79K Rxx=95k25K Rxx=309k -4-2024 -0.03 -0.02 -0.01 0.00 MRB(T) T=25K T=79K T=124K

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88 The resistance of the particular sample s hown in Fig. 6-13 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. 6-14. The film still exhibits anomalous behavior in the Hall curves, indicating the presence of fi nite local magnetic moments at such high resistances. However, the AH curves do not e xhibit a sharp “knee” like saturation at any definite applied field similar to that seen in iron films (Figure 6-5); instead there is a smooth and gradual cross over to a high fiel d saturated value of Hall resistance. The sample also exhibits negative magnetoresistance (Figure 6-15), but in this case also we do not observe a sharp saturation at any characteristic field as seen for thicker iron samples (Figure 6-7). These are possib le indications of the absence of long range ferromagnetic coupling between the grains comprising the f ilms. However, an important observation is that for samples exhibiting hopping trans port behavior, the re sistance increases dramatically with decreasing temperatur e, but the corresponding AH curves do not change significantly. As deduced from the legend of Fig. 6-14, for the resistance increases by 440% while the AH re sistance 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 x x x L / 1 ) coth( ) ( (6-11) which describes the magnetization of a parama gnetic system consisting of non interacting particles or clusters of magnetic moment as follows ) / ( ) ( T k B L M T B MB s (6-12)

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89 where Ms is the saturation magnetization and is the magnetic moment of an individual magnetic entity. The motivation behind fitting the AH curves to Langevin functions is based on the expectation that the magnetizati on response of a film consisting of weakly coupled isolated iron grains will be similar to that of superparamagnetic system. Since AH resistance is proportional to the magnetization (equation 62) of the film, we fit our data to the following functional form: ) ). ( 1 ) ) ( )(coth( ( ) ( ) ( ) (0B T B T T R B T M T R B T Rxy s AH xy (6-13) where and o xyR are fitting parameters and are functions of temperature. The solid lines in Fig. 6-14 show Langevin fits to the AH curves at different temperatures. For the ideal case of non-i nteracting magnetic grains one expects a paramagnetic response so that T k n TB B/ ) ( and ) ( ) ( T R n T Rs B o xywhere n is the average number of Bohr magnetons B in each grain. Ideally, in a system with paramagnetic response, the magnetization curves at differe nt temperatures collapse onto a single curve when plotted as a function of B/T Accordingly, since AH re sistance is proportional to magnetization, one might expect the B/T scaling to hold for the AH curves also. However we failed to observe the B/T scaling behavior in any of our Fe/C60 samples. Thus the assumption that the ferromagnetic grains ar e non-interacting and act independently of each other does not hold in our samples. We already have evidence of electrons tunneling between grains in the temperature depende nce of resistance and expect the magnetic response to be modified by these transport processes. However, to the best of our knowledge we are not aware of any theoretical treatment of AH response to an applied field in the strongly disordered hopping regime.

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90 Figure 6-16:Plot of (see equation 6-13) as a function of inverse temperature for three different Fe/C60 samples. Solid lines represents linear fits given by equation 614. In Fig. 6-16 we have plotted the extracted fitting parameter used in equation (613), as function of corresponding inverse temperatures for three different Fe/C60 samples. We note that for the case of an ideal n on-interacting parama gnetic response with B/T scaling behavior one would expect the data for as a function on T-1, to fit a straight line with positive slope goin g through the origin. Surprisingly our data show a negative slope and finite positive intercept as shown by the solid lines in Fig. 6-16. This result leads us to propose the following phenomenological expression: ) 1 1 ( ) ( T T k n TJ B B (6-14) 0.00.10.2 0 4 8 Ro=70k Ro=300k Ro=10M (T-1)1/T (K-1)5 1020T(K)

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91 where TJ and n are fitting parameters. Table 6-1 su mmarizes the results of the fit to equation (6-14) for data on the three sample s shown in the Fig. 6-16. The observed values of TJ indicate a new and much weaker en ergy scale compared to the charging energy of grains as measured by T listed in Table 6-1, or the long range ferromagnetic exchange as measured by the Curie temperature Tc listed in Table 6-2. The important point to note is that as T approaches TJ, approaches zero. Thus, as JT T from above, one has to apply increasingly strong enough magnetic fields B so that the AH resistance reaches its saturation value o xyR. We note here that for the ferromagnetic samples the AH resistance for B>Bs, is independent of the applied magne tic field. This is not true for a paramagnetic sample, where the magnetizati on and hence AH resistance depend on the applied field. At a particul ar temperature the AH resist ance reaches its saturation valueo xyRfor the argument B>>1 However, according to the experimentally determined dependence of equation (6-14), ex trapolation to a temperature T=TJ with corresponding resistance Rxx=RJ, the parameter =0 which in turn implies that the AH resistance 0AH xyR irrespective of the applied fi eld. Thus, the dependence of as a function of temperature points to a phase transition at T=TJ, whre the AH effect disappears. This behavior is similar to that observed for the case of AH resistance with localized moment 32, 33, when AH xyR approaches zero as temperature is reduced because of decreasing spin disorder at lower temperature. The fitting parameter o xyRin equation (6-13) repr esents the saturation AH resistance at a given temperature and is considered as AH xyR. Surprisingly, for these strongly disordered iron/C60 samples, AH xyR showed a logarithmic decrease with

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92 increasing temperatures (Figure 6-17), which is a very weak ly varying function compared to the exponential rise of longitudinal resistan ce. Thus one would e xpect the temperature dependence of the AH conductivity to be same as that of the long itudinalconductivity as expressed by the equation: ) ) ( exp(2 1 0 2T T L R R Lxy xx xy xy (6-15) Figure 6-18 shows the temperature depe ndence of AH conductivity for two Fe/C60 samples with Ro = 70k and 300k fitted to the functional form as predicted by equation (6-15). The result of the fits of conductiv ity to equation (6-10) and AH conductivity to equation (6-15) are summarized in Table 6-1. Figure 6-17: Logarithmic temperature dependence of AH resistance of Fe/C60 samples in the strongly disordered regime. 10100 -0.6 -0.4 -0.2 0.0 Ro=70k Ro=300k Rxy AH/Rxy AHT(K)

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93 Figure 6-18:Temperature dependence of AH conductivity of Fe/C60 samples. Figure 6-19: AH resistance for Fe/C60 samples (green square points) as a function of sheet resistance, superimposed on the data for iron samples shown in previously in Fig. 6-6. The dashed line represents the average value of AH resistance of 80 in the high resistance limit. 0.10.20.30.40.5 10-410-310-2 Ro=300k Ro=70k Lxy/L00T-1/2(K-1/2) 4 100 25 11T(K)102103104105106100101102 Ro xy()Ro xx()

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94 Table 6-1:Summary of the results fo r three different Fe/C60 samples. Ro Lxx 0 T Lxy 0 T 1 TJ RJ n 70k 5 .03L00 11K 0.025 L00 36.7K 0.34K 4.76M 4 300k 5.15 L00 47K 0.019 L00 133.9K3.7K 0.56M 40 10M 5.78 L00 266K 0.014 L00 652.4K4.6K 28M 39 We now discuss how the AH resistance for the strongly disordered Fe/C60 films compare with that of the weakly disordered iron films discussed earlier. In Figure 6-19 we have superimposed the AH xyR for several Fe/C60 films on the data on pure iron films, already shown in Figure 6-6. We observe th at not only the dependence of AH resistance for the Fe/C60 samples on Ro follows the same trend as that of pure iron, but also the magnitudes of AH resistances AH xyR are within the experimentally observed scatter, determined for the iron films. Thus these re sults consolidate the argument that in the Fe/C60 bilayers, the underlying monolayer of C60 does not affect the magnetism due to the iron film as measured by AH resistance. The central and significant result of our investigation is that the AH resistance for iron and iron/C60 films attains a constant magnitude of the order of 80 and is unaffected by increasing resistance as seen experimentally up to 1M At higher resistances, ex trapolations show that AH xyR collapses to zero. Experiments on Cobalt Films To understand whether the results obtained for iron samples are typical of other itinerant ferromagnetic materials grown under similar conditions, we decided to investigate the behavior of AH conductivity in thin films of cobalt. Figure 6-20 shows an AH curve for a cobalt film with Ro= 3200 at T =5K, which has two important distinction from that of an iron film of similar resi stance (Figure 6-5b). The high field saturation

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95 value of AH resistance for this particular cobalt film is close to 6 which is one order of magnitude less than that of an iron film of similar resistance that had an AH resistance around 80 Moreover, the slope of the Hall curve in the high field region is negative unlike iron, which had a positive slope. These findings are in agreement with existing data on bulk material 22, where iron and cobalt are know n to exhibit hole-like and electron-like carriers respectively. Th e magnitude of the AH coefficient Rs 22 in bulk iron is about one order of magnitude higher than that of cobalt as shown in Table 6-2. Figure 6-20:AH effect in a cobalt film with Ro= 3200 at T= 5K. Inset is a magnification of the high field region to emphasize the negative slope for the normal Hall effect. Next we investigate th e scaling behavior of xxR and AH xyR for a particular film with varying temperature. For a cobalt film with Ro= 756 we found RR scaling behavior with logarithmic temperature dependence of Rxx and AH xyR with slopes 1 AH RA A. For another cobalt film with Ro= 3185 we observed deviation from RR scaling with 69 0RAand 391 0AHA, so that the logarithmic coefficient of AH conductivity -4-2024 -6 -4 -2 0 2 4 6 24 5.6 5.8 6.0 Rxy AH()B(T)

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96 985 0 2 AH RA A. We note that the above behavi or is in agreement with our observations of the high resistance iron samples, where the prefactor AR of the longitudinal conductivity de viates from unity as Ro increases beyond 3k but the prefactor ( 2AR-AAH) for AH conductivity does not change appreciably. We measured the AH resistance of several cobalt films at T=5K to find the dependence on Ro. Figure 6-21 shows a monotonic increase in AH xyR for Ro<1000, followed by a saturation at constant average value of AH xyR~5.8 for Ro in the range of 1000-15000 This leads to the important result that for both iron and cobalt polycrystalline films, the AH resistance ceases to increase and attains a constant value as the sheet resistance increases beyond some characteristic value of the order of /e2 4100 However, we found that the valu e of the AH resistance in the high resistance regime depends on the material itself. Table 6-2 quotes some important prope rties of bulk samples of iron and cobalt 22. The anisotropy energy K1 which is related to spin-orbit coupling strength in the material is about one order of magnitude higher fo r cobalt and than iron. Interestingly, the observed values of saturation of AH xyR in our experiments in the high resistance (ultrathin) regime, is higher for iron(~80 ) by about one order of magnitude than that of cobalt(~6 ). The reported values of anomalous Hall coefficients Rs for bulk samples are also in agreement with the trend mentione d above. We note that the moments per atom 22 in bulk iron and cobalt are very close. These data em phasizes that although the AH resistance is proportional to ma gnetization, it is a magnetotran sport property that depends

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97 strongly on the spin-orbit coupling which in turn depends on band structure and other material dependent parameters. Figure 6-21:AH resistance for cobalt films as a function of Ro. The dashed line represents the average value for the AH resistance in the limit of high resistances. Table 6-2:Comparison of some properties of bulk iron and cobalt. The last row quotes our results of AH resistance in ultra-thin films Iron Cobalt Moments per atom, M 2.12B 1.72B Curie temperature, Tc 1043K 1404K Anisotropy energy, K1 4.8X105erg/cm3 41.2X105 erg/cm3 Anomalous Hall coefficient, Rs 7.22X10-12 cm/G 0.24X10-12 cm/G Anomalous Hall resistance,AH xyR80 6 1021031042 3 4 5 6 7 Rxy AH()Ro()

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98 Discussion of Experimental Results Absence of Quantum Corrections to Hall Conductivity We begin with the discussion of the previous experiment 48 by Bergmann and Ye, which focused on the low temperature quantum corrections to AH conductivity. The samples in this case are believed to have a different morphology compared to those grown for our experiments. Freshly grown antimony films around 10 atomic layers thick kept at liquid He temperatur e, were used as substrates and iron films were then condensed on top by e-beam evaporation. Th e arriving adatoms on the cold substrate do not have enough energy to diffuse and form cr ystalline grains. They simply stick to the surface where they land. Hence, these quench-condensed films were believed to be amorphous, homogeneous and highly disordered. These films could not be heated above 20K because of irreversible transformations into crystalline form. The films reported in the experiments were in the thickness range of 2 to 10 atomic layers with corresponding sheet resistances Ro varying from 1900 to 500 The magnitude of AH resistance was found to be higher than what we find for our pol ycrystalline films of similar resistance. For example, a film with Ro= 1910 the AH resistance was reporte d to be of the order of 110 almost double the value for our pol ycrystalline films with the same Ro. The longitudinal conductivity Lxx of the quench condensed films showed a logarithmic decrease with temperat ure with the normalized prefactor AR close to unity, in agreement with the results of our expe riments on polycrystalline films with Ro<3000 However, the important finding of Bergma nn and Ye’s investigation was that the logarithmic slope for the AH resistance ) ( T RAH xy was twice the slope of the longitudinal resistance ) ( T Rxx. In terms of our notation this implies AAH/AR=2. This in turn implies

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99 that the AH conductivity AH xyL, does not have a logarith mic temperature dependence because the prefactor ( 2AR-AAH)=0. This is a consequence of inverse relationship between conductivity and resistivity tensor and is illustrated as follows: xx xx AH xy AH xy AH xy AH xy xx AH xy AH xy xx AH xy AH xyR R R R L L R R L R R L 2 ) ln( 2 ) ln( ) ln(2 (6-16) The above is a general relation, which is true under all circumstances. Thus experimental observation for the quench-conde nsed films that th e slope of relative changes in ) ( T RAH xy is twice the slope of the relative changes in Rxx(T) directly implies the absence of temperature dependence in AH xyL as shown below, 0 2 AH xy xx xx AH xy AH xyL R R R R (6-17) It is known that in non-magnetic metals the normal Hall c onductivity does not have any finite quantum corrections due to electron in teractions alone 36. Thus, having ruled out the weak localization effects (expl ained below) it was concluded that the quantum corrections to th e longitudinal conductivity Lxx arise due to electron interactions but the corresponding quan tum corrections to AH xyL are zero. This is a surprising result that both the anomalous and the normal Hall effect behave similarly although they originate from different physical mechanisms. Motivated by Bergmann and Ye’s experimental results, theoretical calculations 47 were performed to include the e ffect of quantum corrections to AH xyL. The effect of weak

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100 localization that arises from interference of coherent electronic wavefunctions was neglected in these systems because of the pr esence of strong spin-orbit scattering that destroys the phase coherence of electrons. The logarithmic temperature dependence of conductivity was attributed to the Coulomb anomaly that aris es from interaction between conduction electrons in the pr esence of disorder. Many body calculations were carried out to include the exchange corrections to AH conductivity due to the short range electron-electron interaction, a nd the sum of all contributions, turned out to be zero, in agreement with the experimental findings. The corresponding Hartree terms were not included in the calculations as they are im portant only for the long range interaction. Thus the scaling behavior obser ved in quench-condensed films 48 was explained within the framework of short range electron interactions 47. Finite Quantum Correctio ns to Hall Conductivity For our polycrystalline films with Ro<3000 which is the range of resistance for the quench-condensed films, we observed the relative resistance (RR) scaling behavior (Figure 6-8) for T <20 K where the relative changes in Rxx(T) and ) ( T RAH xyare exactly equal and using equation (6-16) it follows, xx xx xx xx AH xy AH xy xx xx AH xy AH xyL L R R L L R R R R (6-18) Thus we have a finite logarithmic temperature dependence of AH conductivity) ( T LAH xy and the relative changes in) ( T LAH xyequal in magnitude and sign to that of longitudinal conductivity Lxx(T) According to present understandings of quantum theory of transport,

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101 finite quantum corrections to normal Hall conductivity 36 can arise only from weak localization effects while the co rrections due to electron inte ractions are zero. The weak localization theory predicts th at the normal Hall coefficient Rn and hence the Hall resistance at a given field B R Rn xy should be temperature i ndependent. Hence, relative change in Hall conductivity s hould be twice that of the re lative changes in conductivity itself, as explained earlier in chapter 5. The finite logarithmic slope for AH xyL in our samples raises the possibility of existence of weak localization corrections in AH effect. We discussed earlier the case of shor t range interactions between conduction electrons in the presence of disorder, which gives rise to a logarithmic temperature dependence of conductivity for weak disorder and does not lead to any corrections to anomalous Hall conductivity 47. We revisited the issue of eff ect of electron interactions to AH conductivity in the light of the new results ob tained in our experiments, with the help of our theoretical collaborators 52. We realized that in the earlier calculations 47 both impurity scattering and electron-electron intera ction strength were considered only as first order perturbations. In good metals like iron, impurity scatterings are considered to be weak due to screening effects by other el ectrons, and first order perturbation theory turned out to be a good approximation. However, in the polycrystalline samples used in our experiments, scattering from grain bounda ries and inter-grain tunneling processes dominates over the ordinary impurity scatte ring. The AH conductivity was recalculated 53 by treating the impurity scattering in all or ders (strong scattering) with short range electron interactions. These calculations reve al that in the strong scattering limit, the coupling constants that determ ine the strength of potential and spin-orbit scattering are renormalized thus modifying the expression of AH conductivity for both skew scattering

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102 and side jump mechanisms. However the surprising result52 was that by including the effect of a short range electr on interactions in the presence of strong impurity scattering to calculate the temperature dependent quantum corrections to the residual values of AH conductance, the sum of all contributions wa s found to be zero, just like the weak scattering scenario. Although some details still have to be worked out, the calculations suggest the existence of a profound gauge symmet ry associated with electron interactions that leads to zero corrections to diagonal conductivity under all conditions. The Hartree terms are not considered as they are important only for long range interactions that are ruled out in good metals like iron. This is supported by the fa ct that the experimentally observed numerical prefactor for longitudinal conductivity AR~1 over a wide range of resistances from 50-2000 which is thought to be the unive rsal exchange correction that does not depend on the details of interaction stre ngth or sign, as discussed in chapter 5. The quench-condensed films of Bergmann and Ye 48 were amorphous in nature and considered to be highly disordered with mean free path of the order of few angstroms. The phase relaxation time due to spin-flips may be comparable to elastic scattering time ; thus completely quenching the weak lo calization effects. However having ruled out electron interactions as a source of the finite quantum corrections to AH conductivity we revisited the possibility of weak lo calization effects in our samples. Assuming that the grains comprising our samples are good metals with crystalline structure and the overall resistance is dete rmined by the tunnel barriers between grains. Even in the ideal limit of no impurities in side the grains, the electrons can undergo a diffusive motion due to specular scattering fr om grain boundaries. It was shown that in granular media 42 at low temperatures, Tg T electrons can move coherently over

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103 several grains where is the mean energy level spacing in the grains and gT is a dimensionless inter-grain tunnel conductance. Thus it is possible for weak localization effects to be important in granular films. However, our films are ferromagnetic in nature and hence there exists a str ong internal magnetic induction Bint~104T that is responsible for the polarization of conduction electrons giving rise to ferromagnetism. This Bint is a fictitious field that aligns spins but does not affect the orbital moti on of the electrons and therefore should not affect the weak localization effects. In fact it was shown for the case of homogeneous itinerant ferromagnetic syst ems that the presence of strong internal magnetic field nullifies the effect 38 of strong spin-orbit scat tering responsible for spin flip processes that lead to the so called “weak an ti-localization”. The results of these theoretical calculations 38 lead to weak localization corre ctions to conductivity similar to that in non-magnetic systems but modified to include two bands of up and down spin electrons. With the above picture in mind, the inte rference corrections are calculated for strong impurity scattering with spin-orb it coupling and a finite correction to AH conductivity is obtained 52 with the numerical prefactor of the logarithmic temperature dependence close to unity The weak lo calization correction to AH conductivity is further supported by the hi gh temperature behavior ( T >20K) as shown in Fig. 6-8, where the AH conductivity rapidly deviates from l ogarithmic behavior, probably due to phase relaxation of electrons from inelastic pro cesses, like phonon scattering, that destroy interference effects. However, the resistance Rxx(T) continues to exhibit a logarithmic behavior over a longer temperature range (F igure 6-8) raising the possibility that the

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104 mechanism for quantum corrections to longitudi nal conductivity is different from that of AH conductivity. Theoretically it is believed that the e ffect of weak local ization and electron interactions are additive 35 and since both effects pred ict logarithmic temperature dependence, the numerical prefactor is simply the sum of individual prefactors. However, this fact has not been established with certa inty in any experimental investigation. For example, in a previous magneto-tra nsport study on palladium thin films 54, the normal Hall resistance was found to be independent of temperature, which according to current understanding indicates dominan ce of localization effects over interaction effects. This observation is made over the same temperat ure range where the resistance varies logarithmically with temperature. Howe ver such a conclusion was found to be inconsistent with the apparent need to include interaction effects to explain the zero field temperature dependence of resist ance as in palladium spin-o rbit scattering is known to be strong. We note that the films used in this in vestigation were grown on glass substrates at room temperatures by e-beam evaporation. We believe that these palladium films have a granular morphology similar to those in our experiments and arguments regarding the separation of quantum corrections for l ongitudinal and Hall conductivity is also applicable in this case. We propose a novel mechanism to explain co nsistently the temperature dependence of both the longitudinal c onductivity and the AH conductivity We emphasize that the important feature of granular films is that the conductivity of each grain is much larger compared to tunneling conductivity between gr ains and hence the later dominates the longitudinal transport properties. However the AH conductivity is a pr operty that arises

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105 from the ferromagnetic nature of the grains a nd is affected by scatte ring processes within the grains and not the inter gr ain tunneling process. Thus if we track an electron in a granular film, the AH voltage is developed due to electr on motion within the magnetic grains, while the longitudinal voltage drop that determines the resistance is predominantly across the tunnel barriers betw een the grains. The transverse Hall voltage in equilibrium (no current in transverse direction), is due to accumulation of static charges at the sample boundary and is unaffected by granular morphology. Thus we propose that for the samples that exhibit RR scaling beha vior, so that the relative change in longitudinal resistance Rxx and the AH resistance AH xyRare equal over the temperature range T =5-20K, the underlying mechanism responsible for the quantum corrections are different. The quantum correct ions for longitudinal conductivity are due to electron interaction in granular metallic system as discussed in chapter 5, which is cutoff by temperature above a charac teristic value on the order of gT. This temperature could be large for low resistance samples. The AH conductivity assumes quantum corrections that are due to weak localization effects associated with the diffusive motion of electrons within each gr ain and are destroyed either by inelastic processes like phonons at higher temperatures or by the in ter-grain tunneling pr ocess depending on the tunnel resistance for that sample. For 2/ e Ro and at low temperatures when phonon scatterings are quenched, conduction electrons can move coherently over several grains and loose their phase information due to elect ron interaction processe s that determine the quantum corrections to longitudinal, thus resulting in RR scaling behavior. As temperature is raised, phonons inside the crystalline grai ns might destroy the phase coherence of electrons within the grain. He nce the weak localization corrections in AH

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106 resistance are cutoff at characteristic te mperatures due to phonons. The correction to longitudinal conductivity depends on the coherence of severa l tunneling processes, which are cutoff at characteristic temperatures that depend on the average grain sizes and tunnel barrier height given by the product gT. The above explanation is also consistent with the data on high resistance samples with Ro>3000 As shown in Fig. 6-11the relative change in AH resistance is unaffected by increasing resistance. In this situation the coherent motion of electrons that contribute to weak localization effects is confined to individual grains. Since the intrinsic nature of the iron grains does not change significantly with increasing Ro, the relative change in AH resistance is unaffected. However, with in creasing resistance the average grain size and the tunneling conductance becomes smalle r. Thus the effect associated with charging of grains d e EC/ ~2becomes increasingly important for the thinner films. We speculate that the charging effect of gr ains might introduce an unscreened long range Coulomb force in the system and thus the Ha rtree term in the expression for conductivity 35 given by ) / ) 1 ln( 1 (0 0 F F becomes important, where 0F is the interaction parameter. A finite negative value of 0F gives rise to a negative contribution from the Hartree terms and reduces the value of the co efficient from unity due to the exchange term 35. This is a possible explan ation of the deviation of AR from unity at high resistances shown in Fig. 6-9a. We note that the effect of Coulomb forces due to charging of grains becomes prominent in the strongly disordered Fe/C60 samples where the conductivity rapidly diverges as given by e quation (6-10) as predicted by theoretical calculations 44 that explicitly takes into account th e effect of charging energy as an electron hops from one grain to another. Th e onset of the so called Coulomb blockade

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107 happens for resistances of the order of 2/ eand results in the decrease in the value of AR with increasing Ro and eventually reaches the hopping regime. However, the surprising observation is that even in the strongl y disordered samples the AH resistance AH xyR, continues to vary logarithmically with temper ature (Figure 6-17) and further consolidates our argument for a separation of inter a nd intra grain mechanisms affecting the longitudinal and AH resistances respectively. We note here the results of a previous experiment on quench-condensed copper films 55 with resistance varying from 100 to 40000 where the prefactor AR was found to be close to unity over the entire range These films are homogeneous and amorphous and exhibit the universal exchange corre ction to conductivity. The deviation of AR from unity for our polycrystalline films at high resistances is probably due to granular morphology of the films where the Hartree term s could be important, as explained above. We also note here th at the deviation of AAH from unity (Figure 6-9b) for the high resistance iron samples is due to the defin ition of normalized relative change in equation (6-6), where one divides the relative change in AH resistance by the low temperature resistance. Thus even though the relative change in the AH resistance does not change (Figure 6-11), AAH decreases with increasing Ro. Dependence of Anomalous Hall Conductivity on Disorder So far we have discussed the sc aling of longitudi nal conductivity Lxx and AH conductivity AH xyL with respect to varying temper atures. Next we discuss another surprising result shown in Fig. 6-19, which e xhibits the dependence of the magnitude of the low temperature ( T =5K) value of AH resistance in our polycrystalline films on Ro. We observe an important crossover behavior near Ro~ 2000 In the low resistance

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108 regime, we observed a monotonic increase in the AH resistance with increasing Ro. However in the high resistance regime with Ro varying over a wide range from 2000 to 1M the magnitudes of the corresponding AH resistances attain a constant value 10 80AH xyR. We emphasize that Fig. 6-19 is on a log-log scale and spans a considerably wide range of resistances starting from Ro~2000 where the samples are weakly disordered iron films with l ogarithmic temperature dependence to Ro~1M where the Fe/C60 samples are strongly disordered exhibiting hopping transport behavior. A similar behavior was also observed in c obalt Fig. 6-20, where the AH resistance was found to saturate at 1 6AH xyR as Ro varied from 1000 to 10000 Our observation that AH xyR attains a constant value while Ro increases can be expressed in terms of conductiv ities as shown in Fig. 6-22 for iron and Fig. 6-23 for cobalt. In both figures we have plo tted on log-log scale the AH conductivity AH xyL as a function of longit udinal conductivity Lxx in units of the quantum of conductance L00. We found that for both iron and cobalt films, as Lxx of the films decreases (resistance increases), the corresponding AH xyL also decreases and the data fit to a straight line as shown by dashed lines. The corresponding sl opes (shown in inset) for both iron and cobalt are found to be close to 2. Thus we conclude that, in the limit of decreasing conductance of the sample i.e. 0 xxL the AH conductance also decreases 0AH xyL, but the approach to zero happens at different rates so that .2 2const L L R L Lxx AH xy AH xy xx AH xy (6-19)

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109 The behavior described in the equation (6 -19) has been discussed theoretically 56 and verified experimentally in two dimensional electron gas 57 in the context of quantum Hall effect and is known as a “Ha ll Insulator”. We note that in the measurement of the quantum Hall effect one is measuring the norma l Hall effect in very high magnetic fields. Our AH measurements on disordered itinerant ferromagnets reflect distinctly different physics. We denote the behavior described by equation (6-19) as the “anomalous Hall insulator”. Figure 6-22:The anomalous Hall conductivity vs longitudinal conduc tivity in units of quantum of conductance L00 for iron films on a log-log plot. Triangular data points represents ion-milled iron films. Green square points represents Fe/C60 samples. The red dashed line represents the linear fit for the high resistance samples with slope =1.92 0.03 and intercept ~80 10 slope=1.91 0.03 10-110010110210310-610-410-2100102 Lxy AH/L00Lxx/L00

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110 Figure 6-23: The anomalous Hall conductivity vs longitudinal conduc tivity in units of quantum of conductance L00 for cobalt films on a log-log plot. The red dashed line represents the linear fit for the high resistance samples with slope =2.0 0.03 and intercept =6 1 The various microscopic mechanisms for AH effect discussed in chapter 4 assume that at low temperatures, when impurity scattering is the dominant mechanism, each impurity is responsible for both potential sca ttering and skew scattering due to spin-orbit scattering. These theories predict a monotonic dependence of the AH resistance on the sheet resistance of the formM R Rxx AH xy~, where 2 1 However, as we have discussed in the previous section, granular films provide a unique situation where the dominant scattering mechanism determining resistance a nd AH resistance are different especially at high resistances. The fact that AH xyRassumes a value independent of the resistance is a manifestation of the fact that with increasing Ro, the average inter-grain coupling gets weaker, but the intrinsic ferromagnetic nature of the grains compri sing the films does not change significantly. 10110210310-210-1100101 Lxy/LooLxx/Loo slope=2.0 0.03

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111 A refined theoretical model for the residual AH conductivity 53 that might be relevant in granular samples would be to c onsider two sets of s cattering centers in the system: one is the normal potential scatteri ng centers that do not give rise to skew scattering with concentration Nn and the other is the set of scattering centers that gives rise to skew scattering with concentration Nm. In the limit that the skew scattering is weak and the normal scattering is strong and hen ce determines the mean free path, it was shown by Muttallib and Wolfle 53 that the AH conductivity for skew scattering mechanism is given by 2 ) (~n m SS AH xyN N (6-20) and for side jump mechanism n m SJ AH xyN N ~) ( (6-21) The longitudinal conductivity in this model is not affected by the weak skew scattering processes and has the usual Drude kind of dependence due to th e potential scattering centers and is given by n xxN / 1 ~ (6-22) The above dependences for AH conductivity on the impurity concentrations are different from those obtained earlier in th e framework of skew scattering 25 and side jump mechanism 27. In our model of granular ferromagnetic films, we assume that the properties of individual grain do not change w ith increasing resistance. This is equivalent to saying that Nm remains unchanged as resistance increases. Thus it follows from equations (6-20) and (6-21) that for skew scattering mechanism, the scaling behavior 2~xx AH xy appears naturally implying that AH xyR is independent of Rxx.

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112 Now we briefly discuss a pr evious experiment by Lee et.al 58 on metallic ferromagnetic copper-chromiu m selenide spinel CuCr2Se4 that addresses an important issue of whether the AH current due to the Berry phase mechanism28 is dissipationless, i.e. independent of impurity concentration. The magnetic moments in this spinel compound are localized on the chromium atom s and the coupling between the moments is due to superexchange along Cr-Se-Cr bonds. A series of these spinel compounds were grown by chemical vapor deposition and Se at oms were systematically substituted by Br atoms to form CuCr2Se4-xBrx, without significantly changing the saturation magnetization. However, as x varies from 0 to 0.85, the low temperature resistivity increases by factor of 270, which was attr ibuted predominantly due to increase in impurity scattering rate / 1 and partly due to a decrease in the hole density. It was shown that by varying the resistivity xx of the samples over several decades, the AH resistivity AH xy at T =5K for each of the samples normalized by the hole density nH obeys the relation 2~ /xx H AH xyn implying that the normalized AH conductivity H AH xyn / at T =5K is independent of impurities and probabl y has an origin of a topological nature28. The above result is different from our results where we find the AH conductivity tends to zero with increasing resistance. We note the important distinctions between Lee et. al experiment and ours. The spinel compoun ds are not itinerant ferromagnets unlike our iron films. The resistance in the spinel samples was varied by changing the impurity concentration, whereas in our experiment s we are changing the inter grain tunnel resistance without significantly changing th e intrinsic grain pr operties like impurity concentration. Moreover we note that the resis tivity of the spinel samples were less than

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113 200 cm, which is close to the Mooij limit 50, unlike our samples where the resistivity is inferred to be much higher than 100 cm. Anomalous Hall Response in Fe/C60 Films So far we have discussed experimental evidence for a crossover in the magnetotransport properties in granular ferromagnetic films around a sheet resistance of the order of 2/ e. The crossover is a manifestation of granular morphology of our films that results in a separation of in tra-grain mechanism that aff ect the AH resistance from the inter grain processes that affect the longitudi nal resistance. This leads to the anomalous Hall insulating behavior, where the AH resistance AH xyR remains constant while the sheet resistance of the samples varies over a wide range from the weakly disordered regime to the strongly disordered hopping regime Although the anomalous Hall insulating behavior is found to hold fo r films resistances up to 1M the question remains open whether this behavior holds for arbitrarily high resistan ces or breaks down at some characteristic resistance much larger than 2/ e. This question is related to our initial motivation to understand the nature of magne tism in itinerant ferromagnets in the limit of zero conductivity. We wanted to fi nd out if the Stoner criterion 6 for the occurrence of band ferromagnetism breaks down in the limit of high resistance when the conduction electrons are increasingly localized as in a strongly disordered samples and conduction is via hoping between localized states. The existence of a superparamagnetic response in thin ferromagnetic systems consisting of weakly interacting magnetic cl usters has been esta blished in several investigations 59-61, using SQUID measurements. The magnetization curves at a given temperature as a function of applied magnetic field B exhibit a Langevin dependence,

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114 obey the characteristic B/T scaling behavior and show no hys teresis below a characteristic blocking temperature TB related to the volume of individual magnetic grains. The conventional view of a paramagnetic response is that lowering temperature or increasing the magnetic field should increase the magnetic order in the system. This is the basis of B/T scaling behavior in paramagnetic systems where the magnetization curves at different temperatures collapse onto one curve when plotted as function of B/T Our high resistance (>50k ) Fe/C60 films consist of isolated weakly coupled ferromagnetic grains as supported by data on transport and magnetoresistance properties. In our experiments we could not measure di rectly the magnetization of the films using SQUID or any equivalent method, which are thermodynamic measurements of volume magnetization. Instead we have performed AH measurements, which are essentially a transport measurement where, by introducing a charge current th rough the sample and relying on the spin-orbit coupling between the itinerant electrons and volume magnetization M we measure a transverse vo ltage that is proportional to M The AH curves for our strongly disordered Fe/C60 samples, as a function of applied field fit to a high degree of accuracy to Langevin function (equation 6-12) as shown by the solid lines in Fig. 6-14, further supporting the picture of isolated magnetic particles and absence of long range ferromagnetic order. However in a strongly disordered film, as temperature is lowered, in addition to increasing magnetic order between isolated magnetic grains, there is an opposing pro cess of the quenching of itinerancy of conduction electrons which leads to a rapid increase in the resistance. This manifests itself in the systematic analysis of the Langevin coefficient defined in equation (6-13), which is found to decrease linearly with inve rse temperature as de scribed by equation (6-

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115 14). As temperature is lowered and appr oaches a characteristic temperature TJ, the coefficient also decreases, which implies that on e needs to apply a stronger magnetic field to saturate the AH response. However at T=TJ, the AH response will always be zero irrespective of the applied magne tic field, thus signaling a po ssible phase transition. We note that to demonstrate the anomalous Hall insulating behavior in the Fe/C60 samples we have used the AH resistance as the phenomenological fitting parameter o xyR, which is independent of applied field. We emphasize th at this is a valid procedure as long as T>TJ. Thus our results suggest that the anomal ous Hall insulating behavior holds only for resistances lower than RJ determined by extrapolating equation (6-10) to TJ. RJ is a new resistance scale and as shown in Tabl e 6-1 has values much larger than 4100 /2e For higher resistances R>RJ there will be no AH effect. We note that the trend of decreasing with decreasing temperature starts at relatively high temperatures of the order of 100K, thus allowing us to estimate TJ even from high temperature measurement, where the Hall signal can be meas ured with greater accuracy.

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116 CHAPTER 7 SUMMARY AND FUTURE WORK In this dissertation we address an unsettl ed issue regarding itinerant ferromagnetic exchange known to exist in the transition metal elements (iron, cobalt, nickel) and investigate the nature of magnetic ordering in ultrathin films of these materials as the conduction electrons are progressively localiz ed due to increasing disorder. We have presented a systematic in situ magnetotransport study on a series of Fe, Co and Fe/C60 films with sheet resistances va rying over a wide range of 100 to 1M These films are believed to consist of polycrystalline metallic grains of very low resistance and the sheet resistance in these films is a measure of average inter-grain tunneling resistance. We investigate the magnetic properties of these films by monitoring the anomalous Hall (AH) effect, which is the transverse potential proportional to the volume magnetization in response to a longitudinal charge current. Our results provide strong evidence that due to the granular morphology of the films, the micr oscopic scattering mech anisms that affect the electrical conduction and band ferromagne tism are different and lead to unusual properties that may seem c ounterintuitive in the light of existing theories on homogeneous systems. The temperature dependence of longitudina l electrical conduct ance in iron films exhibits a crossover in behavior arou nd a sheet resistance of the order of 4100 /2e. For resistances below the crossover value, th e films show weakly disordered behavior with logarithmic temperature dependence and a universal prefactor of unity arising from the exchange interaction between conducti on electrons. For resistances above the

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117 crossover, the logarithmic pref actor was found to systematica lly decrease from unity. We propose that as the inter-grai n coupling becomes progressive ly weaker with increasing resistance of the sample, the el ectrostatic energy associated wi th the charging of grains as an electron hops from one grain to another as sumes an unscreened long-range character. This leads to a finite contribution from the Hartree term in the expression for conductivity which is opposite in sign to that of the excha nge correction, thus e xplaining the decrease in the value of logarithmic prefactor from un ity due to the universal exchange correction. The high resistance iron samples (>50k ) were grown on a monolayer of C60 to increase film stability and exhibited Coulomb blockade with an exponential of 2 / 1 T increase in resistance. The granular morphology of the films mani fests itself in the relative scaling behavior of the longitudinal resistance (xxR ) and AH resistance (AH xyR) as temperature is varied in the range 5-20K. For resistances less than 2/ e we find a unique scaling behavior in which the relative changes in xxR and AH xyR are equal with logarithmic prefactors close to unity. Howeve r for temperatures above 20K, AH xyR deviates from logarithmic behavior and decreases at a faster rate compared to xxR which continues to exhibit logarithmic behavior to higher temp eratures. We propose a model in which the AH resistance derives from quantum corrections arising from weak localization effects associated with the diffusive motion of elect rons inside the grains. These contributions are suppressed at higher temperatures by inel astic process such as phonon scattering. The longitudinal resistance, on the other hand, is dominated by Coulomb interaction effects

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118 associated with charging of grains due to inter-grain tunneling pro cesses, which are cut off at relatively higher temperatures that depend on grain size and tunnel resistance42. For resistances higher than 2/ e, the scaling behavior described above breaks down. The striking observation in this regime is that the temperature dependent relative changes in AH xyR for different samples do not show any significant differences from each other while the residual resistance Rxx at T =5K varies over a wide range of 3k to 50k and the corresponding relative changes in xxR systematically increase. We argue that the increasing residual resistance is due to the weakening of coupling between grains. While the grain sizes may be smaller for a thinner film with higher resistance, the intrinsic ferromagnetic properties of individual grains that determine the AH resistance do not change significantly. This argument is also c onsistent with our data on the dependence of the AH resistance on the residual re sistance as discussed below. The AH resistance at T =5K for iron samples was found to increase monotonically with increasing resistance until around 2k when the AH resistance ceases to increase any further. In the high resistance regime 2/ e Ro including the strongly disordered Fe/C60 films, the AH resistance is found be c onstant with an average value near 80 while the longitudinal residual resistance is varied over several decades from 3k to 1M Similar observations have been made in cobalt films where the AH resistance remains constant around 6 while the longitudinal re sistance varies from 1k to 10k This is the “anomalous Hall insulating” behavior where the Hall resistance remains finite in the limit of conductivity appr oaching zero. The fact that the iron and cobalt samples saturate at different values of the AH resist ance reveals an important aspect of our AH

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119 measurements, that is, although the AH effect is proportional to volume magnetization, the proportionality constant is a transport property depending on the strength of the spinorbit coupling in each material. The magnitude s of the magnetization per atom of iron and cobalt as measured in SQUID magnetome ters are found to be very close to each other. However the anisotropy energy K1 for cobalt is one order of magnitude higher than that of iron. Both K1 and the AH effect are related to spin orbit coupling in the material which explains the differe nce in the magnitude of AH xyR for iron and cobalt. Although the Hall insulating behavior is found to hold for strongly disordered samples with resistances as high as 1M the question remains open whether these trends will continue to be obeyed for films with arbitrarily low conductance where they become indistinguishable from insulators. To shed so me light on this possibi lity, we performed a detailed analysis of the dependence on field and temperature of the AH curves for strongly disordered Fe/C60 films. At each temperature, th e AH curves as a function of magnetic applied field B, were found to obey the classical paramagnetic response described by the Langevin function ) / 1 ) (coth(0B B Rxy indicating the presence of isolated magnetic clusters and the absen ce of ferromagnetic coupling between them. However the argument of the Langevin function extracted from the fits showed a linear decrease with inverse temper ature and points toward a char acteristic low temperature TJ where the coefficient will be zero. This tr end is opposite to that expected in the magnetization response of a paramagnetic or superparamagnetic system, where the coefficient of the Langevin function is known to increase linearly with inverse temperature. The extrapolation of the coefficient to zero at some characteristic low temperature T = TJ implies that the AH resistance will be identically zero irrespective of

PAGE 132

120 the applied magnetic field. Thus for a samp le characterized by given grain sizes and tunneling resistance, there is a certain temperature and hence resistance when the magnetic order as measured by AH resistan ce is destroyed. The anomalous Hall insulating behavior will only hold for resistances less than an extrapolated resistance RJ determined by the temperature TJ. An important point is that the trend towards vanishing begins at relatively high temperature and one can ther efore make an estimate of TJ and RJ from high temperature measurements. The existence of a sample-dependent resistance RJ that is orders of magnitude higher than /e2 and which delineates a pronounced disappearance of ferromagnetism as measured by the AH effect is unexpected and to our knowledge has not been anticipated for hi ghly disordered itinerant ferromagnets. We have presented new results on the magnetotransport properties of thin ferromagnetic films and provided preliminary explanation of several apparent unusual behaviors by taking into account the granular nature of the samples. Transport properties of granular media have drawn considerable attention in r ecent times, although a complete and well-established theoretical understanding is yet to be achieved. A simple way to test the validity of our arguments based on granular ity of the sample is to perform a similar experiment on films that are homogeneous and reproduce the results pr eviously reported by Bergmann and Ye 48. This can likely be achieved by choosing suitable substrates and/or different growth conditions. A popular method of growing homogeneous films is to maintain the substrate at very low te mperatures during growth, during which the arriving atoms simply condense onto the s ubstrate without having enough energy to diffuse and form crystalline grains. In our expe rimental set up it is difficult to cool down the sample receiver efficiently for growi ng samples via quench-condensation. However

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121 one can also choose a different material that can be grown in both granular and homogeneous forms with similar properties For example, depending on the substrate temperature, manganite materials 62 with ferromagnetic properti es can be grown in both granular and homogeneous forms. The mangan ites have the added advantage of not being air sensitive like iron or cobalt ; hence they can be easily measured and differences in the properties, if any, in the two cases can be attributed to morphology. In this dissertation we tried to understa nd how band ferromagnetism is affected by increasing disorder by reducing the film thic kness. Another important route to address this question is by changing the density of char ge carriers in the films by the field effect and monitor the changes in AH resistances. The field effect experiments are performed in a capacitor geometry, where conduc ting “gate” electrode is sepa rated from the film under observation by a layer of dielectr ic. By applying a potential across the gate and film one can change the carrier density in the films, the induced charge density being limited by the breakdown voltage of the dielectric. Fiel d effect experiments are prominent only in low conductivity systems where carrier dens ity can be modulated significantly with experimentally achievable induced charges. As a possible low carri er density material, we propose the use of iron/C60 bilayers that exhibit prominent magnetic ordering as measured by the AH effect. Using degenerately doped silicon as th e gate electrode, the native silicon oxide as the dielectric, and the Fe/C60 bilayer as the top layer, a simple field effect device suitable for experiments can be realized. It will be quite interesting to investigate how the AH effect varies as the ca rriers are systematically depleted from the film.

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122 Another interesting experime nt would be to systematically study the AH effect in ferromagnetic metals with localized moments, e.g., gadolinium, and look for a similar high resistance scale where the AH resistance disappears. The exchange mechanisms for band ferromagnets like iron and Heisenberg ferromagnets like gadolinium are quite different and might yield additional results th at will shed more light on the microscopic origin of magnetism in metals.

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123 LIST OF REFERENCES 1 G.A. Prinz, Science 282, 1660 (1998). 2 C.M. Hurd, The Hall effect in metals and alloys (Plenum Press, New York, 1972). 3 J.E. Hirsch, Physical Review Letters 83, 1834 (1999). 4 Y.K. Kato, R.C. Myers, A.C. Gossard, and D. D. Awschalom, Science Express, 1105514 (2004). 5 P. Zhang, J. Shi, D. Xiao, and Q. Niu, condmat 0503505 (2005). 6 E.C. Stoner, Magnetism and matter (Methuen, London, 1934). 7 A.R. Gonzalez-Elipe, F. Yubero, and J.M. Sanz, Low energy ion assisted film growth (Imperial College Press, di stributed by World Scientific Publishing Co, 2003). 8 M. Ritter, M. Stindtmann, M. Farle, and K. Baberschke, Surface Science 348, 243 (1996). 9 S. Rusponi, G. Costantini, F.B. de Mongeot, C. Boragno, and U.V.Valbusa, Applied Physics Letters 75, 3318 (1999). 10 T.M. Mayer, E. Chason, and A.J. Howard, Journal of Applied Physics 76, 1633 (1994). 11 J. Erlebacher, M.J. Aziz, E. Chason, M.B. Sinclair, and J.A. Floro, Physical Review Letters 84, 5800 (2000). 12 S.O. Demokritov, C. Bayer, S. Poppe, M. Rickart, J. Fassbender, B. Hillebrands, D.I. Kholin, N.M. Kreines, and O.M. Liedke, Physical Review Letters 90, 097201/1 (2003). 13 D. Stein, Li Jiali, and J.A. Go lovchenko, Physical Review Letters 89, 276106/1 (2002). 14 L.J. van der Pauw, Phillips Technical Review 26, 220 (1958). 15 G.T.Kim, J.G. Park, Y.M.Park, C. Muller-Schwanneke, M. Wagenhals, and S. Roth, Review of Scientific Instruments 70, 2177 (1999).

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124 16 P.Mitra and A.F.Hebard, Applied Physics Letters 86, 063108 (2005). 17 K. Fuchs, Proceedings Cambridge Philosophical Society 34, 100 (1938). 18 E.H. Sondheimer, Advances Physics 1, 1 (1952). 19 B.N.J. Persson, Physical Review B. 44, 3277 (1991). 20 S.G.Mayr and R.S. Averback, Physi cal Review B (Condensed Matter and Materials Physics) 68, 75419 (2003). 21 A. F. Hebard, R. R. Ruel, and C. B. Eom, Physical Review B 54, 14052 (1996). 22 E.P. Wohlfarth, Ferromagnetic materials (North-Holland, Amsterdam, 1980). 23 B.D. Cullity, Introduction to magnetic materials (Addision-Wesley, New York, 1972). 24 V.K. Dugaev, A. Crepieux, and P. Bruno, Physical Review B, 104411 (2001). 25 J. Smit, Physica (Amsterdam) 21, 877 (1955). 26 J. Smit, Physical Review B 8, 2349 (1973). 27 L. Berger, Physical Review B 2, 4559 (1970). 28 R. Karplus and J.M. Luttinger, Physical Review 95, 1154 (1954). 29 M. Onada and N. Nagaosa, Journal of Physical Society of Japan 71, 19 (2002). 30 G. Sundaram and Q. Niu, Physical Review B 59, 14915 (1999). 31 T. Kasua, Progress in Theoretical Physics(Japan) 16, 45 (1956). 32 J. Kondo, Progress in Theo retical Physics(Japan) 27, 772 (1962). 33 F.E. Maranzana, Physical Review 160, 421 (1967). 34 P.A. Lee and T.V. Ramakrishnan, Reviews of Modern Physics 57, 287 (1985). 35 B.L. Altshuler and A. G. Aronov, edited by A. L. Efros and M. Pollok (North-Holland, Amsterdam,1985).

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125 36 B.L. Altshuler, D. Khmel'nitzkii, A.I. Larkin, and P.A. Lee, Physical Review B 20, 5142 (1980). 37 S.Hikami, A.I. Larkin, and y.Nagaoka Progress in Theoretical Physics Letters 63, 707 (1980). 38 V.K. Dugaev, P. Bruno, and J. Barnas, Physical Review B 64, 14423 (2001). 39 G. Zala, B.N. Narozhny, and I.L. Aleiner, Physical Review B 64, 214204 (2001). 40 C. Castellani, C. DiCastro, P.A. L ee, and M. Ma, Physical Review B 30, 527 (1984). 41 B.L. Altshuler and A.G. Aronov, Solid State Communication 46, 429 (1983). 42 I.S. Beloborodov, K.B. Efetov, A.V. Lopatin, and V.M. Vinokur, Physical Review Letters 91, 246801 (2003). 43 V. Ambegaokar, U. Eckern, and G. Schon, Physical Review Letters 48, 1745 (1982). 44 P. Sheng, B. Abeles, and Y. Arie, Physical Review Letters 31, 44 (1973). 45 J.S. Helman and B. Abeles, Physical Review Letters 22, 1429 (1976). 46 H. Fukuyama, Journal of Physical Society of Japan 49, 64 (1980). 47 A. Langenfeld and P. Wolfle, Physical Review Letters 67, 739 (1991). 48 G. Bergmann and F. Ye, Physical Review Letters 67, 735 (1991). 49 L.I. Maissel and R. Glang, Handbook of thin film technology (McGraw Hill Book Company, New York, 1970). 50 J.H. Mooij, Phys. Status Solidi A 17, 521 (1973). 51 C.C. Tsuei, Physical Review Letters 57, 1943 (1986). 52 K. Muttalib, Private communication (2005). 53 P. Wolfle and K. A. Muttalib, condmat 0510481 (2005). 54 W.C. McGinnis and P.M. Chai kin, Physical Review B 32, 6319 (1985). 55 H. White and G. Bergma nn, Physical Review B 40, 11594 (1989).

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126 56 S. Kivelson, D.H. Lee, and S.C. Zhang, Physical Review B 46, 2223 (1992). 57 M. Hike, D. Shahar, S.H. Song, D.C. Tsui, Y.H. Xie, and D. Monroe, Nature 395, 675 (1998). 58 W. Lee, S. Watauchi, V.L. Miller, R.J. Cava, and N. P. Ong, Science 303, 1647 (2004). 59 A. Frydman, T.L. Kirk, and R.C. Dynes, Solid State Communication 114, 481 (2000). 60 C. Binns and M.J. Maher, New Journal of Physics 4, 85.1 (2002). 61 Y. Park, S. Adenwalla, G.P. Felcher, and S.D. Bader, Physical Review B 52, 12779 (1995). 62 A. Biswas, Physical Review B 61, 9665 (2000).

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127 BIOGRAPHICAL SKETCH The author is the first child of Mr. Pr abhat Kumar Mitra and Mrs. Ratna Mitra, born on 8th May 1976 in the great city of Calcutta, lo cated in the eastern part of India. He grew up in his 150 years old ancestral hous e in a small town called Halisahar, around 40km north of Calcutta. Just like most small town kids the author had his own share of climbing trees, fishing, swimming, biking a nd several other cherished memories. At a very young age his father introduced him to the wonderful world of books during a visit to the Calcutta book fair. By the time the author was in high school he was the proud owner of a small library of sc ience books, many of which were from Russia. He went to school in the neighboring town of Kalyani wh ere he used to bike everyday and made many friends. His father was the only tutor he had on all subjects, till his tenth standard. He got interested in physics in particular among other science subjects during his high school days and where he found a wonderful physics teacher, Mr. Misra. He joined Jadavpur University in Calcut ta in 1993 for undergraduate studies and graduated with a Bachelor of Science degree with honors in physics in the summer of 1996. The author showed keen interest in the laboratory cl asses and was well known among his friends for his ability to troubleshoot. During his final y ear at J.U., he took a course on advanced condensed matter physics, where he was expos ed to various topi cs in contemporary physics like superconductivity, superfluidity, NM R, etc. and decided to pursue advanced studies in this field. He then joined the P hysics Department of th e Indian Institute of Science for post–graduate studies after being selected for the Integrated PhD program.

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128 He received the Master of Physical Sciences degree in 1999 with a thesis on experimental soft condensed matter under Prof. Ajay Sood. The education and rese arch experience at IISc were extremely fruitful and gave the author confidence to explore new and better career opportunities outside of I ndia. He joined the graduate program at the University of Florida in the Fall of 2000 for pursuing a PhD degree in physics. During a casual “meet the faculty session” followed by a lab tour, the author came in touch with Prof. Art Hebard and found a kindhearted gentleman as s upervisor and one of the most resourceful labs and a very enthusiastic group in the department. During the summer of 2004 fate brought him to close to a wonderful and frie ndly soul, Mahasweta, and eventually they got married in the winter of same year. The author is considered to be very quiet and taciturn person and believes in using his ears more than his voi ce. He is also particularly interested in cooking and watching movies.


Permanent Link: http://ufdc.ufl.edu/UFE0013425/00001

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Title: Disorder, Itinerant Ferromagnetism, and the Anomalous Hall Effect in Two Dimensions
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Copyright Date: 2008

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Permanent Link: http://ufdc.ufl.edu/UFE0013425/00001

Material Information

Title: Disorder, Itinerant Ferromagnetism, and the Anomalous Hall Effect in Two Dimensions
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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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 Spin-Orbit 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

6-1 Summary of the results for three different Fe/C60 samples. ..............................94

6-2 Comparison of some properties of bulk iron and cobalt..........................................97
















LIST OF FIGURES


Figure page

1-1 A cartoon describing the origin of the anomalous Hall effect ................................

1-2 A cartoon describing the origin of the spin Hall effect. ...........................................3

2-1 Schematic representation of the SHIVA apparatus.................................................8

3-1 A setup for non-switching van der Pauw technique to measure sheet resistance. ...14

3-2 Plot of sheet resistance and thickness as a function of time................................16

3-3 Check of stability of the reduced resistance due to ion beam irradiation.................17

3-4 AFM surface topography of two 120 A thick Cu films. ......... .............17

3-5 The two van der Pauw component R1 and R2 plotted separately. ..........................19

3-6 Plot of the shunting resistance R, and the time At...........................................20

3-7 Effect of ion-beam exposure on thicker iron films. ............................................21

4-1 The Hall curve for a 20A thick iron film. ..................................... ............... 27

4-2 Schematic representation of (a) skew scattering and (b) side jump mechanism......30

5-1 Typical wave functions of conduction electrons in presence of disorder ...............38

5-2 Motion of electrons in presence of impurities ......................................................40

5-3 Schematic diagram of Friedel oscillation due to a single impurity..........................49

5-4 Friedel oscillation due to two impurities..................................... ....... .......... 51

6-1 A d.c. transport measurement setup using Keithly 236.........................................66

6-2 An a.c. transport measurement setup using two SR830 lock-in amplifiers ............67

6-3 Typical behaviors for temperature dependence of resistance for iron films. ...........69

6-4 Plot of numerical prefactor AR for different iron films...........................................71









6-5 The anomalous Hall curves for iron films................. .............. ............... 72

6-6 The anomalous Hall resistance at T=5K for different iron films ..........................74

6-7 Magnetoresistance as a function of field for an iron film with Ro=300 ................78

6-8 Relative resistance (RR) scaling behavior at T<20K for an iron film ...................79

6-9 Plots showing dependence on Ro of the extracted transport coefficients ...............80

6-10 An iron film with Ro=49000Q showing deviation from RR scaling....................... 82

6-11 The relative changes in AH resistances for three different iron films. ....................83

6-12 A cartoon of iron/C60 bilayer samples.. ........................................ ...............85

6-13 Plot of conductivity showing hopping transport in a Fe/C60 sample .....................86

6-14 The AH effect in strongly disordered Fe/C60 sample........................ ...............87

6-15 Magnetoresistance curves at different temperatures for the Fe/C60 .......................87

6-16 Plot of y as a function of inverse temperature for three different Fe/C60 samples...90

6-17 Logarithmic temperature dependence of AH resistance of Fe/C60 samples............92

6-18 Temperature dependence of AH conductivity of Fe/C60 samples .........................93

6-19 AH resistance for Fe/C60 samples as a function of sheet resistance......................93

6-20 AH effect in a cobalt film with Ro 32000 at T=5K. ..........................................95

6-21 AH resistance for cobalt films as a function of R .. ............................................. 97

6-22 The anomalous Hall conductivity vs longitudinal conductivity..........................109

6-23 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 long-range 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 ultra-thin 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 magneto-transport 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 inter-grain 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

long-range 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 spin-polarized 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) spin-valve

head for magnetic hard-disk 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 non-volatile new generation

of memory. From a more fundamental point, spintronics studies involve understanding

spin transport and spin interaction with the solid-state 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 spin-polarized 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 spin-dependent asymmetric scattering of electrons from impurities or

phonons, a phenomenon known as the anomalous Hall (AH) effect 2 as shown

schematically in Fig. 1-1 The AH effect (AHE) is observed in all ferromagnetic

materials regardless of the nature of the exchange mechanism and also in

superparamagnetic systems comprising weakly-coupled 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 spin-orbit 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 1-1: A cartoon describing the origin of the anomalous Hall effect in an itinerant
ferromagnet with unequal number of spin-up 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. 1-2. 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 1-2: 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 spin-orbit 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 spin-split 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 air-sensitive 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. 2-1,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 2-1: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 feed-throughs out

of the vacuum chamber to a break-out 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 -10-9 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

Ion-assisted 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 ion-assisted 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 ion-assisted 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 "ion-beam sculpting" 13. Many of

these experiments thus provide strong evidence of ion beam induced nanoscale matter

transport on solid-state surfaces, a process that promises to be useful in applications

requiring nano-textured 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

-10-4 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 15-45A for the Fe films and 75-130A 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

re-sputtering 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 s-13, 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. 3-1). 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 (3-1)
IAB

Similarly, apply a current ICB and measure the potential difference between D and A

define


R2 = .-- (3-2)
ICB

The sheet resistance R is determined from a mathematical relation between the above

measurements as given by

R R,
exp(- .' )+ exp(-; -)= 1 (3-3)
R R

The solution of the above equation can be written in a simplified form


R = R + Rf( (3-4)
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( ) (3-5)
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.


Lock-in fl


Lock-i 12


Figure 3-1: A setup for non-switching van der Pauw technique to measure sheet
resistance of a sample (shaded square ABCD) using two SR830 lock-in
amplifiers operating at different frequenciesfl andf2. Different colors are
used to represent electrical connections for each lock-in.









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 lock-in 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. 3-1, allows us to simultaneously measure both components,

R1 and R2, of the resistance and assess film homogeneity (|R1-R21) 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.3-2 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 3-2: 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. 3-3). 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 3-3: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 3-4: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 ion-milled 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 ultra-thin films due to ion milling is

strongly correlated with their initial surface roughness. Fig. 3-2 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. 3-4, this comparison for a typical Cu

film reveals that the ion-milled 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 well-connected 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 3-5: The two van der Pauw component R1 and R2 plotted separately for the copper
film shown in Fig. 3-2.

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 (3-6)
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 3-6: 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. 3-6(a) we plot the calculated values of R, versus Ri for twenty-two 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. 3-6(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. 3-7.

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 3-7: Effect of ion-beam 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. 3-2, 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 nano-smoothening 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- orf-electrons 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 non-integral

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 spin-split 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 spin-down electrons

within an energy range 8E of the Fermi energy EF are placed in the spin-up 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 (4-1)
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 (4-2)
2

Thus the total change in energy is given by


AE = g(EF )SE2 (1- Jg(EF)) (4-3)
2

and spontaneous ferromagnetism is possible if AE < which implies :

Jg(E ) > 1 (4-4)

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 d-bands, 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 (4-5)
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 spin-dependent scattering of the conduction

electrons due to spin-orbit 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 102-103 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. 4-1. 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 (4-6)

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 4-1: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 spin-orbit interaction

of the spin polarized current carriers in itinerant ferromagnets with the non-magnetic

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


(4-7)









where j is the momentum of the electron. This magnetic field interacts with the spin of

the electron and favors anti-parallel 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) (4-8)
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 spin-orbit interaction as given by:

V2 A2
H= d3 *(r)[ -+V(r), -M -i a- (VVxV)]y/(r) (4-9)
2m 4

where Ao is effective of spin-orbit 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 z-direction and f-=( f/, y) are spinor fields corresponding to

spin-up and spin-down electrons. The Hamiltonian in momentum representation can be

written as24

k2 2
H = cy( 2-M )I k +Z kVk-[1 k[+ xk').]k (4-10)
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 spin-orbit 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 spin-orbit scattering,

V = I, + v P = -CkV k (4-11)
k

For a short range impurity potential V(r) it can be shown that scattered wave function

from the spin-orbit term far from the origin is given by

w,, = c,[-qlh(kr)(k xr )-' ]/kr (4-12)
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] (4-13)
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 spin-orbit 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 z-direction. 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. 4-2a. 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 ) (4-14)

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 (4-15)






,-.--..;(a)






@ ---- -- --- | ^ ------------------
0 I I i26 (b)




Figure 4-2: 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 (4-16)
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.

4-2b. The magnitude of the side-jump displacement 5is proportional to spin-orbit

coupling and is expected to be small. We note that for a bare electron in vacuum, the

spin-orbit 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 side-jump 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 ) (4-17)
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 (4-18)

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

spin-orbit interaction with the lattice. The semi-classical dynamics of Bloch electrons

including the Berry phase may be derived from the Bloch Hamiltonian

Hk = n (k)+V (4-19)

where (k) are energy bands including the effect of spin-orbit interaction and

ferromagnetic polarization, and V is the applied external potential such that VV = -eE.

The semi-classical equations of motion are

k =eE +er x B
(4-20)
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)


(4-21)









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 (4-20) leads to a Hall current given by

H = -e2n()x (4-22)

implying an AH conductivity given by

SAH =e 2n(,) (4-23)

where () = n 1,, n,(k) f(FS) is the average of Berry magnetic field over all

occupied states in k-space. The average is zero unless time reversal symmetry is broken

as it is in a ferromagnet where there is spin-orbit coupling between the spin-polarization

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 (4-24)

which is the same dependence found for side jump scattering (equation 4-18).

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 spin-spin

interaction also known as s-d interaction. Although the s-d 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 s-d 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 non-degenerate. Thus a s-d 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 spin-orbit 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 4-7). This kind of spin-orbit interaction

favors anti-parallel alignment of orbital angular momentum and the spin. The intrinsic

spin-orbit interaction allows odd powers of spin-spin 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))) (4-25)

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 (4-26)

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 spin-orbit interaction will be zero, and

hence skew scattering is not expected. However gadolinium is known to exhibit a

particularly large AH effect.

Another spin-orbit 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 (4-27)
2mc mc

using V.A = 0. Introducing the angular momentum of the charge carrier about the origin

L = rx p, the spin-orbit term can be expressed as


Hso = -e M. (4-28)
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 spin-orbit interaction by Kondo. For the mixed s-orbit/d-spin 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 (4-29)

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) (4-30)

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 =- (4-31)
K1

K,1V
TB = (4-32)
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 5-1: 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.5-1b). 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 metal-insulator

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) (5-1)
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 non-interacting electrons and are known as

weak localization corrections.

A.. ,B (a)


AX






A oB
/\ (b)

x






Figure 5-2: 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. 5-2a). 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 (5-2)
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 self-intersecting trajectories that contain loops (Fig. 5-2b). 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 (5-3)

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 self-intersecting 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 self-intersecting 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 two-dimensional metal

film, the area accessible to an electron is Dt. In order for self-intersection 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 self-intersecting 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 electron-phonon and electron-electron 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

Ao-2d vdt (54)
(5-4)
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(' (5-5)
hi T h I

where L, ~ VT. Equation (5-5), although derived for a strictly 2D case, is applicable

for thin films of thickness d < L(.

The phase-relaxation 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 electron-electron 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. (5-6)
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 5-2b) in opposite directions as

given by

2e
A(H = A-dl = 2- (5-7)
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 magneto-resistance 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 (5-8)
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 -(5-9)
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 (5-9), is

obtained by replacing r, by TH in equation (5-5). Thus for the 2d case the magneto-

conductance is given by 35,

e2 eHDr
o(H)- o(0) = AC2d (H) A2d (H = 0) -ln( -') (5-10)
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 Spin-Orbit 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 spin-orbit

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 self-intersecting loops (Figure 5-2b). 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)*( (5-11)
C=AA2 = P vv (5-11)
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 5-2b) 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)(5-12)
1 (1) (2)


1 ( ( 1)9(2) (+ (1) (2))


Thus the interference term in equation 5-11 can be expressed as follows,


C = ( 2 2) (5-13)


In the presence of spin-orbit 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) (5-14)
O2d Dt 2 2

Thus depending on the spin-orbit scattering time rT, relative to r, we have the following

two cases for 2d samples 37

e2 Z
Ao2d --n(Y) for ro >> (5-15)
h r









e2 3 1
Ac2d (-ln( -~)+ ln(- )) for r,, << r (5-16)
h 2 r 2 r

thus it follows from equation (5-16) that the spin-orbit correction reverses the sign of the

temperature dependence of conductivity due to r, In weak magnetic fields and under

strong spin-orbit, the magnetoresistance becomes positive. For sufficiently strong

magnetic field such that rH < r, the magneto-resistance changes sign and becomes

negative. The combined effect of positive quantum correction to conductivity and

positive magnetoresistance is known as 'weak anti-localization'. If however, scattering

occurs from paramagnetic impurities, then both the singlet and the triplet wavefunctions

in equation (5-13) 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 spin-orbit 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) (5-17)
2m 4

where ./ ( ,,V T) is a spinor field ( = (ox, cy, a, )are the Pauli matrices, Mis the

magnetization assumed to be along z-direction and Vis the random impurity potential,

and Ao is the spin-orbit scattering strength. The exchange term Mo- acts only on the spins

and has no direct effect on the orbital motion.

In the language of many-body theory, weak localization corrections arise from the

particle-particle channel with two propagators describing electrons with vanishing total









momentum and with very close energy (Cooper channel). Physically, the so-called

"Cooperon" propagator represents two electrons traversing a self intersecting loop

(Figure 5-2b) 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 spin-flip

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 (-+-- )]} (5-18)
4;r rh, ,, so

where r()is the temperature dependent phase breaking time. r,) is the momentum

relaxation time and Trso(, is the effective spin-orbit 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,

electron-electron collisions cannot affect the conductivity in the case of a single band

structure and in the absence of Umklapp processes. This is because electron-electron

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

electron-electron 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, (5-19)
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 5-3: 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. 5-3) is given by 39,

vA
3p(F) = 2 sin(2kFr) (5-20)

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 electron-electron 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 -rF-V(rFrI2)
VH () = drV,( F,)3p(F) (5-21)
1( -
VF(F7-r) =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,) (5-22)
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 Hartree-Fock 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.5-3. 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)]- (5-23)
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 electron-electron scattering.






B A












Figure 5-4: Friedel oscillation due to two impurities created by the self-intersecting 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 self-intersecting paths of electrons. In

Fig. 5-4 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 Altshuler-Aronov 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 Fermi-liquid constant F The total conductivity

correction in 2D is given by 35, 39

e2 ln(1 + F)
= In( )[1+ 3(1-)] (5-24)
272 h T Fo

The above equation was derived for the so-called "Diffusive regime" characterized by

Tr << 1. The sign and magnitude of the correction is non-universal and depends on the

competition between the universal and positive exchange term and the coupling

dependent and negative Hartree contribution. For short-range electron-electron

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 spin-split 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) (5-25)


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 non-universal 40

Moreover, for 2d conductors with strong spin-orbit or spin-flip 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) (5-26)
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 particle-particle channel and are

supposed to be sensitive to magnetic flux. The electron interaction effects arise from the

particle-hole 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 self-energy 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 spin-up

and the highest occupied spin-down 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) (5-27)

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 (5-21). 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 (5-28)
27r2 F0o h2 h <<1

where h = gJBH / kT The quantity in the parenthesis is the Hartree contribution to

conductivity and is non-universal in both sign and magnitude and depends on the details

of the potential describing electron-electron 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 inter-granular transport can be distinguished from intra-granular

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,)] (5-29)


where t,, is the tunneling matrix element between i-th andj-th grain, Ho is the Hamiltonian

for non-interacting isolated grains, and He describes the Coulomb interaction inside (i=j)

and between grains (isj) as described by

2
H, = e- ,Clhj (5-30)
2

where C, is the capacitance matrix and i, is the operator of electron number in the i-th

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( ) (5-31)
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 (5-32)

The classical Drude conductivity for a granular metal in a general dimension d, with

grain size a is given by

2
o, = 2-ga2-d (5-33)
h

The correction to conductivity due to large energy scales E > g,8 is given by

90- 1 gEC
ln[ E (5-34)
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
12i-2g1 g13
1 gd =
I=d In( ) =d 2 (5-35)
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( -- )] (5-36)
22"2h Ec gr8

and for T > g,

2 T
S= [42 2gT + ln( )] (5-37)
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) (5-38)

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) (5-39)

where s is the tunnel-barrier 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 tunnel-barrier 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

inter-grain conductivity so that,

o(T)- exp[-2(C/ kT)'/2] (5-40)

The constant C is proportional to the charging energy Ec, which is inversely related to the

mean grain diameter . Thus, the dominant contribution to conductivity at high

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]+ (1-P)exp[-(E E)/2kT]}
2 2

(5-41)

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 anti-parallel, respectively, to that of the initial grain. Using the same

assumptions for the non-magnetic grains discussed earlier, the magneto-conductivity is

45
given by 45,

o(H, T) = o(0, T)[cosh(E, / k T) -P sinh(E, / kT)] (5-42)

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] (5-43)


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 non-magnetic case shown in equation (5-40).



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


(5-44)









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 (5-45)
ay R' R

Thus it follows from equation (5-44) 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 (5-46)
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 (5-47)

Thus it follows from equation (5-45) the effect of only electron interaction implies that


=2 (5-48)
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 short-range 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 (5-49)
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 5-49) 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 (5-17). For the case of side jump (SJ)

mechanism in a 2d ferromagnetic sample it was predicted 24 that

3o-AH(SI)
xy (5-50)
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 <<- (5-51)
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

(5-52)

The above expression was derived for the case of a random short-range 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 (5-53)
18 v {kFV Vr (5-53)

Using the fact that the residual longitudinal conductivity is given by

n^.2 z,, 2T
a = 2 (5-54)
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,)( + )] (5-55)
A H(SS) 2 2 x xt() ,,t( ) 'so(1)






63


Comparing the above equation with equation (5-18) 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 non-magnetic materials and has no quantum correction at low

temperatures due to electron-electron 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 10-4 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 well-connected 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 quench-condensed 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 well-defined 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 Hall-bar 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. 6-land 6-2. 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 6-1: A d.c. transport measurement setup using Keithly 236 for sourcing a constant
current and measure longitudinal voltage and Keithly 182 nano-voltmeter to
measure transverse voltage. The sample in Hall-bar shape is shown as a
shaded.



Figure 6-1 shows the circuit diagram for d.c. measurement using a Keithley 236

Source-Measure 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 6-2:An a.c. transport measurement setup using two SR830 lock-in 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 Hall-bar geometry is
shown as shaded









Figure 6-2 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

magneto-transport 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 6-3a 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. 6-3a, which is a manifestation of low temperature quantum corrections

discussed in chapter 5.
10o. ero-----i 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 6-3: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 6-3b 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 inter-grain

tunneling processes. However at low temperatures (T<15K), the resistance is found to

increase logarithmically as shown in the inset of Fig. 6-3b, 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. (6-1)
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 6-4 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 ion-milled 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 6-4:Plot of numerical prefactor AR for logarithmic temperature dependence of
longitudinal conductance (equation 6-1) for different iron films, as a function
of Ro, sheet resistance at T=5K. Red square points corresponds to ion-milled
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

4-4-40



-4 -2 0 2 4 -4 -2 0 2 4
B(T) B(T)
Figure 6-5: 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 6-5(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 Ro-20000. 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 (6-2)


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 6-6:The anomalous Hall resistance at T=5K for different iron films as a function
of R on a log-log scale. Red square points are ion-milled films. The dashed
line represents the average value of the AH resistance at 800.

Figure 6-6 shows the dependence of RH on iron samples with different Ro using

a log-log 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) (6-4)
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(--) (6-6)
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 (6-5)

that

L L, 1 R, (T) R, (TO) T
A (L,)= 2 ( =T AR In(-) (6-7)
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 (6-5)

and (6-6),

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

(6-8)

Moreover, it follows directly from equations (6-5) and (6-6),
S(R AH) AH
(R AAH (6-9)
A"(R,) AR

Thus the coefficient of logarithmic temperature dependence of conductivity is AR and that

of the AH conductivity is 2AR-AAH. Also it follows from equations (6-8) and (6-9) 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 (6-1) and

equation (6-5), 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 6-4). 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.

6-7 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. 6-5(a) and 6-7). 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 6-7: 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. 6-8 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 T-20K. 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 6-8) 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 quench-condensed 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 6-8 Relative resistance (RR) scaling behavior at T20K for an iron film with
-1 A (R
-2

"o-3

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 6-8: 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 A-A ---- --- 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 6-9: 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
Ro-hle2 41000. Values close to unity in both cases correspond to RR scaling.
(c) 2AR-AAH 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. 6-8 for low temperature (T=5-20K) part to equations (6-5)

and (6-6), 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 6-8 and the prefactor for AN (L,) is found to be 0.9080.005 close to the

value of 2AR-AAH =0.890.01 in accordance with equation (6-7). For simplicity and

future reference we call the low temperature scaling behavior for samples where the

AAH/AR=I (Figure 6-8) 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. 6-9(a) we have re-plotted for reference the coefficient AR as a function of

Ro. The plot is essentially the same as Fig. 6-4 and shows a deviation from AR-I 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.6-9(b). All samples with R, in the range 300-3000Q were found to exhibit

RR scaling behavior (Figure 6-8) 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 (6-9) 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 6-9b, we found that AAH/AR=0.122+0.002. In Fig. 6-10 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

2AR-AAH =0.6 (Figure 6-9c).

00.0 *:. * *** I I


M -0.1 (R )
xy










5 8 12 16 20
T(K)

Figure 6-10: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. 6-9c.

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 (6-4), for three samples with Ro varying over a wide range from

24000 to 490000. In the low temperature range of T=5-20K, 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. 6-6, the relative change in RH over the temperature range

of 5-20K 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 6-11 :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 2AR-AAH as defined in equation (6-8),

does not show any pronounced dependence on Ro and is scattered around an average

value of 0.86+0.2(Figure 6-9c). 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 50-50000Q. 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 inter-grain 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 6-12: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
10-2 N



0.1 0.2 0.3 0.4 0.5
T1/2(K-1/2)

Figure 6-13:Plot of conductivity showing hopping transport in a Fe/C60 sample as given
by equation (6-10).

We now focus our attention on iron/C60 films that are in the strongly disordered

regime. Figure 6-13 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(-(-) ) (6-10)
T

where LO and T are characteristic resistance and energy scales in a two dimensional

system with localized electronic states. The fit in Fig. 6-13 yields T=266K and

L = 1.754Loo. Table 6-1 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, being the average grain size. Our data as shown in table 6-1,

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 6-14: 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 6-15 :Magnetoresistance curves at different temperatures for the Fe/C60 sample
shown in Figs. 6-13 and 6-14. The curves do not show sharp saturation at a
particular field as seen in iron films (Figure 6-7).









The resistance of the particular sample shown in Fig. 6-13 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. 6-14. 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 6-5); 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 6-15), but in this case also we do

not observe a sharp saturation at any characteristic field as seen for thicker iron samples

(Figure 6-7). 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. 6-14, 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 (6-11)

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) (6-12)