Spin orbit interaction induced spin separation in platinum nanostructures

We have demonstrated electrical generation and detection of spin polarization bythe spin Hall effect in platinum.. The spin Hall effect refers to the generation ofa transverse spin current

INDUCED SPIN-SEPARATION

IN PLATINUM NANOSTRUCTURES

KOONG CHEE WENG

(B Eng (Hons.), NUS)

A Thesis Submitted for the Degree of Doctor of Philosophy

NUS Graduate School for

Integrative Sciences and Engineering National University of Singapore

2009

I would like to express my gratitude to all those who contribute in one way oranother in the completion of this thesis I want to thank the A∗STAR GraduateAcademy for the scholarship, and the support of my school, NUS Graduate Schoolfor Integrative Sciences & Engineering.

I am deeply indebted to my supervisors Prof Berthold-Georg Englert fromthe National University of Singapore and Prof Chandrasekhar Natarajan from theInstitute of Materials Research and Engineering I would also like to show myappreciation to the members of the Thesis Advisory Committee, Prof ChristianMiniatura, Prof Ong Chong Kim and Dr Adekunle Adeyeye, whose stimulatingsuggestions and encouragement has helped me in my research and the writing ofthis thesis

My thanks go out especially to Dr Deng Jie, Ms Teo Siew Lang, and Mr.Chum Chan Choy for making the e-beam lithography available for me I wouldalso want to thank Dr Nikolai Yakovlev for doing the SIMS analysis and Ms.Doreen Lai for doing the EDX analysis I would also like to thank Ms Ma HanThu Lwin for helping to coat the silicon wafers with an oxide layer Furthermore Iwould also like to thank countless other colleagues from IMRE and NUS, without

i

their support, this thesis would not have been possible.

Finally, I would like to give my special thanks to my parent and my wife Linda,whose patience and love enabled me to complete this work

1 Introduction 1

1.1 Background 1

1.2 Outline of Thesis 8

2 Theory 12 2.1 Spin-Orbit Interaction 13

2.2 Extrinsic Spin-Orbit Interaction Effect 15

2.2.1 Skew Scattering 15

2.2.2 Side-Jump 18

2.3 Intrinsic Spin-Orbit Interaction Effect 20

2.3.1 Dresselhaus and Rashba Spin-Orbit Interaction 20

2.3.2 Berry Phase 22

2.4 Spin Current 26

2.5 Spin Accumulation 31

2.6 Non-Local Geometry 32

2.7 Spin Relaxation 34

2.8 Spin Hall Effect 37

iii

2.9 Spin Hall Effect in Metals and Semiconductors 38

2.9.1 Platinum 41

2.10 Generation/Detection of Spin 44

2.10.1 Generation of Spin Currents via SHE 45

2.10.2 Detection of Spin Currents via ISHE 46

2.11 Summary 47

3 Sample Design and Fabrication 51 3.1 Design of Experiment (Motivation) 52

3.1.1 Proposed Design 54

3.2 Fabrication 57

3.2.1 Outline of Lithography Procedure 58

3.2.2 Fabrication of Contact Pads with Ultraviolet Lithography 59 3.2.3 E-Beam Lithography 63

3.2.4 Polymethylmethacrylate (PMMA) 63

3.2.5 Bilayer Mask 64

3.2.6 Fabrication of Device with E-Beam Lithography 65

3.3 Measurement Setup 68

3.4 Errors from Electrical Measurements 70

3.5 Electromigration 75

3.6 Transient Protection 76

3.6.1 Shunting For Electrostatic Discharge Protection 77

3.7 Summary 78

4 Experimental Results and Analysis 79

4.1 Background 80

4.2 Spin Transport Equation in a Diffusive Conductor 80

4.2.1 Drift-Diffusion Model 82

4.2.2 Conversion of Spin Current into Hall Voltage 85

4.2.3 Theoretical Model of Device Geometry 88

4.3 The Samples 90

4.4 Characterization of Sample 90

4.4.1 X-Ray Diffraction (XRD) 91

4.4.2 Secondary Ion Mass Spectrometry (SIMS) 92

4.4.3 Energy-Dispersive X-Ray Spectroscopy (EDX) 95

4.5 Sample Testing Procedures 96

4.6 Experimental Data 99

4.6.1 The Spin Hall Resistance RsHand the Misalignment Voltage103 4.7 Discussions 104

4.7.1 RsH/RDversus Temperature 105

4.7.2 Measurement of Spin Diffusion Length λs 106

4.7.3 Intrinsic versus Extrinsic SHE 110

4.7.4 Comparison of RsH with Temperature 111

4.7.5 Skew Scattering due to Fe Impurities 113

4.7.6 Comparison of RsH of Pt with Au and Al 113

4.7.7 RsHin Different Current Ranges 115

4.8 Gold and Platinum — A Comparative Analysis 116

4.9 Device Failure 119

4.10 Shunting For Electrostatic Discharge Protection 1214.11 Summary 123

A Experimental results: VsHagainst Iy 143

B Spin-Orbit Interaction Induced Spin-Separation in Platinum

C Giant spin Hall conductivity in platinum at room temperature 148

We have demonstrated electrical generation and detection of spin polarization bythe spin Hall effect in platinum The spin Hall effect refers to the generation of

a transverse spin current, and the subsequent non-equilibrium spin accumulationnear sample boundaries This occurs when a longitudinal electrical current isapplied to materials with spin-orbit interaction Spin polarization in metals is usu-ally small and requires ferromagnetic metals to create and detect spin polarization.This ferromagnetic-based approach is suitable to be used as a laboratory investi-gation of spin transport phenomenon, but it limits the performance and scalability

of the spintronics devices Therefore, we designed and experimentally strated using a non-local lateral geometry structure to investigate the generationand detection of spin polarization based on the spin Hall effect, without the needfor magnetic materials, external magnetic field, or bulky optical systems

demon-The geometry made use of the spin Hall effect effect to generate spin ization and its reciprocal effect, the inverse spin Hall effect, to detect spin po-larization A large spin Hall effect signal was observed from 10 K up to roomtemperature, which was the largest value reported so far in the literature The mea-surements were also done using gold and aluminum samples Aluminium failed

polar-vii

to demonstrate any signal while gold showed weak signals compared to platinum

in spite of similar atomic number This suggested that the spin Hall effect in inum was unusual The drift-diffusion model was found to be adequate to modelthe spin transport in platinum Based on the experimental results, the spin Hall

plat-effect in platinum was expected to be extrinsic However, the extrinsic tion to the observed effect was not fully understood and further investigation isneeded

contribu-2.1 Theoretical Prediction of Spin Hall Conductivityσxy 422.2 Experimental Measurement of Spin Hall Conductivityσxy 434.1 Measurement of Spin Hall Resistance RsH 1004.2 Measurement of Misalignment Resistance Rmis and the Adjusted

RsH 1034.3 Summary ofλs,σxy, andσyy 1064.4 The effects of Shunt Resistance in Measurements 122

ix

1.1 Schematic of an Electron Spin 2

2.1 Extrinsic Scattering 16

2.2 Dresselhaus and Rashba Spin-Orbit Interaction 22

2.3 Intrinsic Spin-Orbit Interaction 23

2.4 Parallel Transport 25

2.5 Charge Current and Spin Current 28

2.6 Time Transformation of Charge and Spin Current 29

2.7 Spin Valve 33

2.8 Non-Local Measurement Geometry 35

2.9 Anomalous Hall Effect and Spin Hall Effect 39

2.10 Spin Current Injection via Ferromagnetic Resonance 48

2.11 Non-Local Measurement Geometry and Spin Current 49

3.1 Proposed Device 56

3.2 E-Beam Lithography Process 65

3.3 SEM Image of a Typical Device 67

x

3.4 Measurement Setup 69

3.5 LabVIEW Programme 70

3.6 Misalignment Voltage 74

3.7 Electromigration 76

4.1 Schematic of Diffusive Spin Transport 82

4.2 Conversion of Spin and Charge Current 86

4.3 Geometry of the Device Under Test 89

4.4 XRD Analysis 92

4.5 SIMS Analysis 94

4.6 EDX Analysis 95

4.7 SEM Image of Device Under Test and the Contacts Label 97

4.8 Experimental Results for RsH for L= 203 nm 101

4.9 Experimental Results for RsH for L= 323 nm 102

4.10 Ratio of Spin Hall Resistance over Drude Resistance as a Function of Temperature 105

4.11 RsH versus L 107

4.12 Linear Resistance as a Function of L 108

4.13 Log-Log Graph ofσxyagainstσxx 111

4.14 Experimental Results for RsH for L= 115 nm 112

4.15 RsH for Other Metals 114

4.16 Log-Log Graph of Voltage against Current 115

4.17 Comparative Analysis for Pt and Au samples 118

4.18 SEM Image of Effect of Electromigration 120

4.19 Experimental Results for RsH with and without shunting 122A.1 Experimental results spin Hall voltage versus current for variouslength L 144

The electron is one of the most important elementary particles and plays a nent role in all branches of scientific endeavor The electron carries not only a neg-ative electrical charge, but also intrinsic angular momentum which is a relativisticquantum mechanical effect The intrinsic angular momentum of an electron wasonce thought to be associated with self-rotation (e.g a top spinning about its ownaxis), and Niels Bohr used the term “spin” to describe it However, this conve-nient mental picture was subsequently recognized to be incomplete and could notexplain many features of spin The measurement of the spin orientation in anyarbitrary direction results in the projection in either the parallel direction or an-tiparallel directions and could only take values of±~/2 Such a discrete value ofspin has no classical analogue Electrons aligned antiparallel (parallel) with themeasuring magnetic field are traditionally labeled as “spin-up” (“spin-down”)

promi-1

The electron spin angular momentum is antiparallel to the electron magnetic ment µ, so that “spin-up” electrons have their magnetic moment parallel to themagnetic field While it is useful to have a simplistic picture of the spin, it must

mo-be rememmo-bered that spin can only mo-be fully explained in the framework of tic quantum mechanics 1 In Figure 1.1, we illustrate the spin-up and spin-downelectrons The same schematic drawing would be used to denote spin-up andspin-down electrons in subsequent figures

relativis-Figure 1.1: Schematic of an classical picture of spin-up and spin-down electrons.Any measurement of the spin would result in one of the two possible state Thedirection of “up” and “down” is completely arbitrary in the absence of a magneticfield

1 In this thesis, the usage of the label “spin-up” and “spin-down” is interchangeable as no magnetic field or ferromagnet is used to break the symmetry When a reference is made to a fer- romagnet, the spin-up electrons would be the majority electrons, i.e electrons aligned antiparallel with the ferromagnet magnetization.

The electron spin has attracted much academic and commercial attention inrecent times due to the convergence of economic forces towards miniaturizationand efficiency in the semiconductor and magnetic-recording industries Aided bythe rapid scientific advancements in condensed matter science, the application ofspin in electronics quickly gives rise to the nascent discipline appropriately labeled

as Spintronics (spin-electronics) The first commercial application of spin is in thearea of magnetoelectronics [1], where the “birth” of the giant magnetoresistance(GMR) and the subsequent commercialization of the highly sensitive GMR read-heads greatly increase the density of magnetic recording media

In recent times, the potential applications of spin have broadened from ing and storing information in the magnetoelectronics industry to the computingand signal processing in the information technology sector Intel co-founder Gor-don E Moore, in a 1965 paper, predicted the number of transistors to double in

sens-a given sens-aresens-a every 12 to 18 months Since then, Moore’s Lsens-aw hsens-as been sens-rate for over half a century However, in 2005 Moore predicted that the scalingprocess will end within 15 years, due to the fundamental limit of miniaturization.Beyond the atomic size limitation of the scaling trends, the impact of Joule heat-ing is more critical than the scaling limitation in determining the continuation ofMoore’s Law The operating temperature of the device and the problem of elec-tromigration constrict the current density, which limits the reduction of the devicedimension [2] Much commercial attention was spent identifying solutions forminimizing the impact of Joule heating, such as using low-k dielectrics materialsand improvement to the cooling mechanism of the integrated circuit (IC) chip In

accu-1989, Datta and Das proposed the spin field-effect transistor (spin FET) [3], and

the event represents a paradigm shift in the use of electrons Spin FET, the “spin”version of the field-effect transistor, uses the spin degrees of freedom to controlthe conductivity of the channel The use of the electron spin for signal process-ing has the potential of operating on a shorter time scale, achieving higher energy

efficiency, and providing further scaling and greater computational capability [4].Beyond making incremental improvements in the performance of contempo-rary electronics devices, research in spintronics also can create new devices withradical functionality [5] For example, the theoretical prediction of the “spin com-puter” based on spin-based logic gate architecture highlights the versatile use ofspin [6] The holy grail for spintronics research is the realization of dissipation-less quantum signal processing and quantum computation with electron spin Thebinary state of the electron spin is a natural candidate for the role as the classi-cal binary bits 0 and 1 As spin interaction with the environment is weaker thanthe Coulomb interaction, the enduring spin orientation makes the spin suitable forprocessing and storing information More significantly, the electron spin exists as

a superposition of the two binary states This quantum bits (qubits) provide a greatcomputational advantages over classical charged-based computer to solve prob-lems that are intractable on conventional machines [7] Although the research isstill in its infancy, its potential to transform all aspects of information technologycannot be understated Together with the rapid progress in other sub-branches ofspintronics, the convergence of the various developments would gradually mergeand mature into a single body of knowledge

The expanding spintronics field can be broadly subdivided into based spintronics, metal-based spintronics and organic-based spintronics The

semiconductor-main commercial and academic research thrust is in semiconductor systems, wherethere exists a substantial infrastructure in the semiconductor industry and accumu-lated technological expertise In particular, much of the investigation is centered

on silicon (Si) which is the dominant material in the semiconductor industry cent breakthrough in the injection of spin-polarized electrons from an iron film [8]and the electronic measurement and control of spin in Si [9] showed that an all-electrical semiconductor spin device is feasible in principle Independent work

Re-by Lou et al also demonstrated an integrated electrical scheme for spin

injec-tion, transport and detection in a single GaAs device [10] Notwithstanding theachievement, the low temperature requirement renders the scheme impractical forcommercial applications Moreover, the conductivity mismatch between ferro-magnet and semiconductor [11] resulted in a small spin polarization in the range

of 10% to 30% at 5 K in Si [8] and 16% at 50 K in GaAs [10] Even the highest ported spin injection polarization of about 70% into GaAs from a CoFe/MgO(100)tunnel spin injector at room temperature [12] is not sufficient to meet the 99% po-larization requirement for a good on-to-off conductance ratio for a spin FET [13].The problem of conductivity mismatch can be addressed through the use of di-lute magnetic semiconductors (DMS) DMS is synthesized using a small amount

re-of transition metals with common semiconductor materials like GaAs [14] DMSemerges as a leading focus of magnetic semiconductor research because of its longrange ferromagnetism and semiconducting properties Although reports showedthat certain DMS has spin polarization of around 90% [15], the low Curie tem-perature (e.g the Curie temperature for the best investigated system, GaMnAs, is

170 K) limits its prospect as a practical spin source Co-doped ZnO DMS system

has Curie temperature which is above room temperature, and is considered to be

a promising DMS material for spintronics application [16]

Organic materials are the latest emerging trend in spintronics research, withthe driving factor being their long spin relaxation time and length [17] The firstorganic spintronics devices - organic spin valve - was demonstrated in low tem-perature (refer to Reference [18] for a review of the development of organic spinvalve) The main drawbacks of the organic conductors are their low electricalconductivity and the difficulties of interfacing the ferromagnet with the organiclayer without damaging the film [19] The poor interconnection prevents effectivetransfer of spin-polarized electrons from the ferromagnet contacts to the organicmaterial, and hinders the adoption of organic in complex spintronics system

In contrast to the semiconductors and organic materials, room temperaturemetal-based spintronics is well established Currently, metal-based spintronicsproducts, such as GMR read-head and magnetoresistive random access memory(MRAM), already played a pivotal role in sensing, storing and recording of bits.The earliest spin-based experiment can be traced back to the spin accumulationexperiment in a hybrid aluminium/permalloy structures by Johnson and Silsbee

in 1985 [20] (refer to [21] for a review of metal-based magnetoelectronics) Thehybrid normal metal/ferromagnet structures do not suffer from conductivity mis-match affecting semiconductors and organic materials

Beyond acting as a passive device, a growing number of schemes are proposedfor utilizing spin in active devices such as spin logic devices [6], spin-flip andspin-torque transistors [21] Johnson [22, 23] was the first to propose an all-metalspin transistor, in which the device can perform both logic and memory function

The incorporation of logic and memory function on a single device would createthe necessary integration technologies for future all-spin computation architec-ture Such an approach avoids the shortcoming of minimization of charge-baseddevices, such as Joule heating and stray magnetic fields from charge current [24].Recent demonstrations of ferromagnetic switching using a pure spin current [25],and the generation and detection of spin at room temperature [26] highlight thepotential of metal-based spintronics for creating the first fully functional pure spindevice at room temperature.

The key ingredients of spintronics devices can be categorized into three ponents: (1) spin generation: generate and control the polarization and direction

com-of the spin current; (2) spin manipulation: control the spin current as it travels

in the device; (3) spin detection: read the spin state at the detector accurately.There are different physical mechanisms to generate, manipulate and detect spincurrents in devices The earliest work used mainly magnetic materials (or exter-nal magnetic fields) or optical techniques based on the selection rules However,there is a more promising scheme to generate and detect spin polarization throughthe spin Hall effect (SHE) mediated by spin-orbit interaction Spin-orbit inter-action in solids was first studied by Dresselhaus [27] and Rashba [28] in 1955and 1960 respectively Interest in the subject was rejuvenated when a later the-

oretical study by Murakami et al [29] showed a dissipationless intrinsic spin current that is independent of impurity scattering Soon after, Sinova el al also

predicted a dissipationless spin currents in two-dimensional electron system withRashba spin-orbit coupling [30] Interest in the intrinsic spin Hall effect extends tothe impurities dependent (extrinsic) spin Hall effect, which D’yakonov and Perel’

[31] and Hirsch [32] predicted earlier in 1971 and 1999 respectively SHE ated a strong interest within the community, motivated by its potential as a spininjection and detection tool in all-electrical spin devices Of particular interest

gener-is the SHE in metals, where room temperature operation has been demonstrated[26, 33–36] The studies in this relatively new unexplored field have turned out aplethora of discoveries on new phenomena, and have opened a new chapter in thedevelopment of spintronics

The objective of the research is to study the spin Hall effect due to its potential as aspin source and detector in the next generation of spintronics devices Inspired bythe theoretical prediction of dissipationless spin current in semiconductor system[29], we aim to conduct experiments to verify the existence of the spin Hall effect.Subsequent literature reviews revealed that SHE was predicted earlier in param-agnetic materials [31, 32], but there was no experimental verification at the time

of commencing the studies Also, numerical calculation showed that the spin Hall

effect is stronger and more robust in metals than in semiconductors These bination of factors add a further incentive for the investigation in metallic SHE[37]

com-At the beginning of the research, there was no experimental work in the area

of metallic SHE Thus our early works focused on the investigation the spin Hall

effect in metal with different magnitude of spin-orbit interaction We chose Al,

Au and Pt as targets of our initial investigation Al was chosen as the spin

trans-port was demonstrated in 1985 on a aluminium bar at temperatures below 77 K[20] Au and Pt were chosen because of their large spin-orbit scattering strengths,which scale with the fourth power of the atomic number [38] This was expected

to translate into a large SHE signal Our experiments found that Al had no tectable spin current, while Au had a faint signature at 100 K This observationwas consistent with our assumption that the difference in the atomic number be-tween Al and Au is large Surprisingly, the effect in Pt was an order of magnitudelarger than in Au, even though the atomic numbers were similar This unexpectedobservation meant that the rule of thumb was insufficient to explain the extraordi-nary nature of the SHE in Pt Subsequently, the research was based solely on the

de-Pt system in order to investigate its spintronics potential

We have developed a model for the spin Hall effect, which allows us to vestigate both spin Hall conductivity and spin diffusion length at the same time.This is novel, since current models could only determine the spin Hall conductiv-ity We have investigate multiple samples to study the effect of temperature andincreasing spatial separation of adjacent channels on the SHE Our efforts haveresulted in the experimental demonstration of electrical spin injection, transport,and detection in Pt up to room temperature The gist of the thesis was summa-rized and submitted for publication The preprint of the manuscript is reproduced

in-in Appendix C

The results of the research are presented in the following chapters

Chapter 2: In this chapter, the fundamentals of spin physics are introduced tofamiliarize readers with the framework of spin physics, transport and dy-

namics The review would start off with the key concept of sporbit teraction, which lies at the foundation of spintronics Spin-orbit interaction

in-in the solid state leads to many notable phenomena, among them the spin-inHall effect Another fundamental concept is the notion of a spin current.Spin current is the transport of spin without the net transfer of charge Animportant feature of the spin current is that it cannot be directly measuredwith conventional electrical technique However, the spin current can beindirectly determined with a non-local geometry setup through the spin ac-cumulation Another important area is the electron spin relaxation processesbecause they limit the length-scale on which coherent spin transport can beobserved The review would introduce the main spin relaxation mechanism

in metal which is the Elliot-Yafet mechanism SHE is closely related tothe anomalous Hall effect (AHE) in ferromagnetic metal The bulk of theSHE theoretical framework originates from the investigation of the AHE Inthis review, we would introduce both the extrinsic and intrinsic mechanismscommon for both SHE and AHE Next, we would review some theoreticalpredictions and experimental measurements of the spin Hall conductivity inmetals and semiconductors Thereafter, there would be a discussion on thegeneration of spin via the SHE and the generation of charge via its recip-rocal effect, the inverse SHE Lastly we would introduce a modified drift-

diffusion equation to model the spin transport in a diffusive medium Thedrift-diffusion transport model, although limited in scope, provides a goodqualitative description of the main characteristics of the spin transport in a

diffusive medium The review is not meant to be a comprehensive

examina-tion of the whole field of spintronics but focuses on the selected area of spintransport in metals, generated by the spin Hall effect The review would fo-cus on metallic systems, but necessary references to semiconductor systemswould be made to elucidate the physics behind the metallic spin Hall effect.Chapter 3: The main focus of this chapter is to introduce the device geometry,the fabrication and the measurement processes First, the design process

of the sample geometry would be described It would be shown that ourproposed non-local geometry allows an all-electrical measurement of theelectron spin Next, the device fabrication procedures would be sequen-tially detailed It would be followed by a description of the measurementsetup used to determine the spin Hall effect Finally, there would be a briefdiscussion on the damaging effect of transient current on the device duringthe measurement and the protection mechanism against transient current.Chapter 4: In this chapter, the experimental results are presented and analyzed.First, we would introduce the theoretical model for our sample geometry,and describe the measurement procedure The results presented show a clearspin Hall signal for platinum, and a weaker signal for gold and aluminum.The experimental data are shown to be in good agreement with the drift-

diffusion model The measurements support the view that the SHE in Ptwas extrinsic

Chapter 5: The conclusion and outlook will be given in this last chapter

The basic ingredients for a spintronics device are the mastery of generation, portation, and detection of spin In order to understand the mechanism behindthe phenomenon, one needs to start off with some basic theoretical framework.First, we begin with a discussion on the spin-orbit interaction and how it can lead

trans-to the harnessing of spin current Later on, we present and discuss about thecontroversies regarding the precise definition of the spin current Next, the spinaccumulation on the boundary due to the spin current is described Finally, weconsider in detail the spin Hall effect and, its reciprocal effects, the inverse spinHall effect The chapter would focus on spin Hall effect in metallic system, and

on how it can be used to generate and detect spin current

12

from the Dirac equation for a free electron of mass m with charge −e The

expres-sion for the SOI in vacuum is

Hso= e~

4m2c2σ · [∇V(r) × p] (2.1)

where V is the Coulomb potential, p is the electron momentum andσ is the vector

of the Pauli spin matrices For a detailed explanation of the origin of SOI startingfrom the relativistic Dirac equation, refer to the Chapter 2 of Reference [39].One intuitive way to understand the SOI is to consider the situation of anelectron in orbit around a positively charged nucleus In the rest frame of theelectron, the electric field of the nucleus E = ∇V(r) transform to an effective

magnetic field B = −v× E

c2 The magnetic moment µ of the electron interactswith the magnetic field B to produce a torque on the electron, aligning the spin inthe magnetic field direction The interaction energy after taking into account theThomas precession is

m2c2r3S · L (2.2)

where S is the spin angular momentum vector and L is the orbital angular mentum vector of a particle The S · L is the coupling term between the spin and

mo-the orbital angular momentum

While the atomic picture of the SOI illustrates the concept easily, most cations of SOI happen in solid state systems In the case where the whole lattice isconsidered, the Hamiltonian for the extended system becomes complicated Phe-nomenologically, we can assume that the spin-orbit interaction Hamiltonian insolid takes the form

appli-Hso∝ σ · B(k) (2.3)where B is an effective magnetic field, which contains all the various contribu-tions such as the external field, the crystal structure and any impurities scattering.There are two distinct physical mechanisms that determines B The first mecha-

nism is the impurities-driven extrinsic effect, where B is induced by the impuritiespotential Within the extrinsic mechanism, it is further divided into two uniquecontributions: (i) skew-scattering and (ii) side-jump The second mechanism isrelated to the spin-dependent band structure of the material This effect is inde-

pendent of the impurities and thus is considered to be an intrinsic effect The twomain theoretical models for the intrinsic effects are the Dresselhaus model and theRashba model

In solids without inversion symmetry, the Hsoleads to a spin-splitting for trons in the absence of a magnetic field This inversion asymmetry comes fromthe bulk inversion asymmetry (BIA) or the structural inversion asymmetry (SIA)

elec-The BIA originated from the lack of an inversion center (non-centrosymmetric)

in certain crystal lattice such as zinc blende structure (e.g GaAs) For crystalstructure with an inversion center such as diamond structure (e.g Si), the symme-

try can be broken by inducing an internal electric field in the device structure by

means of artificial heterostructures or quantum wells

2.2 Extrinsic Spin-Orbit Interaction E ffect

The extrinsic spin-orbit interaction refers to the scattering of Fermi electrons by

local potentials caused by impurities or defects in the material, i.e external to the

host material There are two distinct scattering mechanisms in the SHE literature,namely the skew scattering (SS) mechanism and the side jump (SJ) mechanism,which are illustrated in Figure 4.17 Here, we will discuss both mechanisms inmore details

In 1929, Nevill Mott carried out a mathematical investigation on the problem ofelectrons scattering off nuclei by a double scattering method [40] The investi-gation centered on the fundamental question of the existence of spin; is spin anintrinsic properties of electrons or just another quantum state for a bound electron?Starting off from the spin-orbit interaction term discovered by Dirac’s theory twoyears previously, Mott showed that the scattering of an unpolarized electron beamwas asymmetrical and thus provided evidence for the existence of the free electron

(a) Skew Scattering (b) Side-Jump

Figure 2.1: The semiclassical representations of the skew scattering and the jump mechanisms (a) The trajectory of the scattered spin-up (spin-down) elec-trons is deflected to the left (right) side of the scattering center by an angleδ (b)The trajectory of the scattered spin-up (spin-down) electrons is side shifted to the

side-left (right) of the scattering center by distance d In a collision, the side-jump

deflection is superimposed on the skew scattering

spin Mott called this effect skew scattering 1

The skew scattering can be understood using a simple qualitative picture ofthe electron-impurity collision process When an electron moves with velocity v

through the electric potential V(r) (due to the charged impurities), it experiences

an electric field E= ∇V(r) In the rest frame of the electron, the “moving charged

center” creates a magnetic field B = −v× E

c2 in its vicinity The non-uniformmagnetic field is perpendicular to the electron trajectory and points in the oppositedirection on either side of the charge center The magnetic field then exerts a spin-dependent force (proportional to the gradient of the Zeeman energy 2µBB· S) onthe electron such that the force at one end of the dipole will be slightly greaterthan the opposing force at the other end of the dipole The effect is that electronswith opposite spin states would be deflected by the inhomogeneous magnetic field

in opposite directions, leading to spin separation along the transverse direction.Not surprisingly, this skew scattering is sensitive to the type and range of theimpurities Likewise the rate of spin separation depends on the rate of collisionbetween the electrons and the impurities Thus the skew scattering resistivityρSS

scales linearly with the Drude resistivityρxx, i.e ρSS ∝ ρxx As shown in Figure(2.1(a)), skew scattering is characterized by a constant spontaneous angle, δ, atwhich the scattered electrons are deflected from their original trajectories Thespontaneous angleδ is also referred to as the spin Hall angle αsH and it is defined

as the inverse tangent of the ratio between the transverse spin Hall current andthe longitudinal electric current (or equivalent to the ratio between the spin Hall

1 A historical review of the early controversies surrounding the essence of the spin in the late 1920s can be found in Ref [41].

conductivity and the Drude conductivity, i.e tanαsH = σxy

σxx

) In other words, spinHall angle measures the efficiency of converting charge current to spin current.This skew scattering effect [40, 42] is the basis of Smit’s postulation in 1954[43] that the skew-scattering of electrons is the cause of the AHE found in ferro-magnets Similarly as in the case of paramagnetic materials, the electron beam in

a non-polarized charge current would be spin separated in the transverse direction.But the skew scattering would not lead to a net charge separation in the transversedirection because of the equal population for both the spin-up and spin-down elec-trons This accumulation of spin, and no charge, at the samples’ boundaries givesrise to the SHE

Berger pointed out a different extrinsic mechanism for AHE called the jump” mechanism in 1970 [44] Berger’s side-jump model originated from theanomalous velocity operator in systems with spin-orbit interaction He describedthe side-jump mechanism as a discontinuous and finite sideways displacement

“side-d = ∆rsoof the center of mass of a wave-packet during collisions in the presence

of electron-impurity potential The wave-packet then continued to move in theoriginal direction

The relativistic part vsoof the velocity operator v = (i/~)[H, ˆr] in the presence

of SOI takes the form [45],

vso= 2e~λ(σ × E) (2.4)

where λ is the coupling constant, E is a homogeneous external field and σ isthe Pauli spin matrices We can determine the side-jump displacement rsoof theelectron by integrating vsowith respect to time Substituting ˙k= eE/~ in Equation

(2.4), we have

∆rso =

∫

vsodt = 2λ(σ × ∆k) (2.5)The direction of the displacement is spin-dependent, and it is directed along

σ × ∆k Similar to the skew scattering case, if the average spin polarization

⟨σ⟩ = 0, then the side-jump leads to a pure spin current in the transverse direction

to the driving field E= ∆k/e and σ The side-jump contribution is independent of

scattering time and depends solely on the external E field Unlike the skew tering, the side-jump contribution scales quadratically with the Drude resistivity

scat-ρSJ ∝ ρ2

xx 2

The different resistivity dependence of the skew scattering and side-jump anism allows us to differentiate the dominating contribution in various parameterregimes Skew scattering dominates when ρxx is small (at low temperature orlow impurities concentration) and the side-jump contribution dominates when

mech-ρxx is large (at high temperature or in a dirty sample) The experimental servation is expected to be a linear combination of the two contributions with

2 Interestingly, the side-jump mechanism was found to be related to the intrinsic Berry curvature

in the momentum space for electrons in GaAs type semiconductors [45].

2.3 Intrinsic Spin-Orbit Interaction E ffect

The intrinsic effect was first proposed by Karplus and Luttinger in 1954 [46] toprovide an explanation for the experimental observation of AHE Unlike the ex-trinsic effect, the intrinsic effect is related to the spin-dependent band structure

of the materials, and is not directly related to electron-impurity collision To put

in another way, the intrinsic effect is due to the spin-orbit coupling (SOC) in thewhole Fermi sea and not merely due to the scattering of electrons at the Fermilevel

We start by considering the SOI-induced zero-field spin splitting in inversionasymmetry systems In a periodic crystal, the spin degeneracy is dependent on the

spatio-temporal symmetry In spatial tranformation, i.e r → −r, the wavevector

k is transformed to -k Whereas in temporal tranformation, i.e t → −t, both k

and spin s change sign If both symmetries are present, such as diamond structure

of Si, we have E+(k)= E−(−k) in accordance with Kramers theorem However insolids without crystal inversion symmetry, such as zinc blende structure of GaAs,the spin degeneracy is removed for k , 0, i.e E+(k) , E−(−k) Thus we obtain

two branches of the energy dispersion E+(k) and, E−(−k)

There are two main kinds of inversion asymmetry: The inversion asymmetry inates from bulk (Dresselhaus) or structure (Rashba) inversion asymmetry Simple

orig-expressions for the spin-orbit term in the Hamiltonion for the linear Dresselhausand the Rashba coupling in a 2D system can be obtained.

H D= ~2k2

2m + β(k xσx − k yσy), (2.6)

H R = ~2k2

2m + α(k xσy − k yσx) (2.7)

Figure 2.2 illustrates the two branches of the energy dispersion E+(k) and

E−(k) in the presence of Dresselhaus and Rashba coupling The zero-field ting of two branches due to Dresselhaus and Rashba coupling are∆E = 2βk and

split-∆E = 2αk respectively It is noted that the effective magnetic field direction is pendicular to k for the Rashba term, while it has a more complicated geometrical

per-dependence for the Dresselhaus term

We can use a simple qualitative example in a 2D system to appreciate the trinsic mechanism As mentioned, the intrinsic effect can be related to an effectivek-dependent B(k) field We consider the case in a 2D system where the electronspin lies in the in-plane direction When a weak external electric field Ex is ap-

in-plied, the whole Fermi sphere is displaced in the negative x-direction in k-space.

As the Fermi sphere shifted, the B(k) → B(k+∆ k) which causes the electronspins to go out of alignment with the magnetic field B(k+∆ k)) then exerts an

effective torque which tilts the spins back into alignment For two electrons onthe opposite side on the Fermi sphere, the B(k+∆ k)) points in the opposite di-rection Therefore, if we consider electrons with±k y, the intrinsic effect creates a

spin current in the y-direction with polarization in the z-direction.

2.3.2 Berry Phase

The treatment of theoretical models for Dresselhaus and Rashba spin-orbit pling largely ignored the details of the whole band structure There have beenseveral works attempting to understand the effects of the SOI by performing first-principles relativistic band calculations [37, 47, 48] It is observed that the in-trinsic effect is closely related to the geometrical phase of conduction electrons

cou-in the momentum space [49] This geometrical phase, or Berry’s phase, plays avital role in the dynamics of electrons in a crystal Berry [50] discovered in 1984that a quantum system that is transported adiabatically in the time interval [0, T]

round a closed path C (i.e H(T) = H(0)) would acquire a geometrical phase or

Figure 2.2: (a) The Dresselhaus term and (b) the Rashba term which result fromthe breaking of inversion symmetry The effective magnetic field direction is per-

pendicular to k for the Rashba term, while it has a more complicated geometrical

dependence for the Dresselhaus term The arrows indicate the direction on the

effective magnetic field

Figure 2.3: The change in the effective magnetic field in the presence of an electricfield The displacement of the Fermi sphere causes the magnetic field to change,i.e B(k)→ B(k+∆ k) The field B(k+∆ k) then exerts an effective torque whichtilts the spins back into alignment For electron with ±k y , the magnetic field z-

components points in the opposite directions, which creates a spin current in the

y-direction with polarization in the z-direction.

Berry’s phase, in addition to the familiar dynamical phase in its Hamiltonian Ifthe initial state is in an eigenstate ofH, then | ψ(T)⟩ = e −iβ n e −iϕ n | ψ(0)⟩ where βn

is the Berry’s phase andϕn is the dynamical phase βn is entirely determined bythe geometry of the parameter space and the trajectory of the closed path, and isindependent of the temporal evolution of the Hamiltonian, whileϕnis sensitive toboth the path and the rate at which the Hamiltonian is evolving

In Berry’s paper, he approached the situation by considering the parallel port of eigenstates, which were represented by unit vectors, in Hilbert space Ku-

trans-gler et al [51] explained that Berry’s phase can also be derived classically by

considering locally inertial coordinate frame undergoing parallel transport abatically A lucid example which helps to introduce the Berry’s phase is theparallel transport of a vector along a close path on a sphere as shown in Figure2.4

adi-Figure 2.4: The parallel transport of tangent vector v(r) (denoted by the red row) on the spherical surface path marked by the dotted lines v(r) is restricted

ar-to remain orthogonal ar-to the normal vecar-tor e3(r) at every point along the path, and does not rotate about e3(r), i.e v(r) always remains in tangent plane The parallel

transport of spin eigenvector in Hilbert space causes the eigenstate to acquire a ometrical phase or Berry’s phase The Berry’s phase is the cause of the anomalousvelocity contribution, and is the source of the intrinsic SHE

ge-In this example, a tangent vector v(r) (denoted by the red arrow) is held

con-stant in a local inertial coordinate frame, and travels in a closed path On a flatsurface, such as a plane or a cylinder, the direction of the tangent vector doesnot change during the course However on a curved surface, like on a surface

of a sphere, the v(r) gains a phase angle related to the integral of the curvature

on the surface upon returning to its initial state We first define an orthonormalbasis {v, w, e3}, where e3(r) is the normal vector and w = e3 × v We move v(r) adiabatically along the sphere, such that v(r) remains on the tangent plane of the surface at any given point, i.e e3(r) × dv(r) = 0 The rate of change of v(r) is,

˙v(r) = −[v(r)· ˙e3(r)]e3(r) Relative to the local reference frame {e1, e2, e3}, the slow

moving v(r) acquires a small geometric angle, d α = e1(r) · de2(r) The integral of

the whole enclosed path would give the total geometric angleα,3

˙r= ∂ϵn (k)

∂k − ˙k × Ω(k)

˙k = −eE − e˙r × B

(2.9)

where ϵn is the energy of the nth band, Ω is the Berry’s curvature of Bloch

elec-trons, k is the crystal momentum, E and B are the perturbing electrical and

mag-netic fields The second term in the velocity equation,−˙k×Ω(k), is the anomalous

velocity contribution to the electron dynamics This is the source of the intrinsicSHE

stood in a classical framework Unlike charges, spin is a quantum mechanicalphenomenon and it cannot be interpreted classically Like charge, spin can betransported to process and store information.

There are two different kinds of spin transport; spin-polarized charge currentand spin current Spin-polarized current transport refers to the transfer of electronswith one dominant spin type Such transport mechanism is endemic in ferromag-netic materials In contrast, spin current can be thought of as a transport of the spin

degree of freedom along a conductor, without the net transportation of charge.

A spin current consists of electrons with different spin states, each of equal

numbers, moving in opposite directions, i.e j↑ = − j↓ where j↑(↓) = n↑(↓)v is the

product of the carrier density for spin-up (spin-down) electrons and the drift

ve-locity The most natural way to define a spin current is Js= ~

The prediction of a dissipationless quantum spin current at room temperature[29] sparked a widespread interest to understand and utilize this revolutionary dis-covery The dissipationless quantum spin current is induced when a charge current

flows in p-doped semiconductors with spin-orbit coupling This theoretical

pre-diction highlights the potential of the electron spin to transfer information Thecentral argument for the purely topological spin current is that it is invariant un-der the time reversal In a time reversal operation, both the velocity and the spinangular momentum of the electron are odd in time, while the electron charge is