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Transport in three-dimensional topological insulators: theory and experiment

Dimitrie CulcerICQD, Hefei National Laboratory for Physical Sciences at the Microscale,

University of Science and Technology of China, Hefei 230026, Anhui, China

(Dated: August 17, 2011)

This article reviews recent theoretical and experimental work on transport due to the surfacestates of three-dimensional topological insulators. The theoretical focus is on longitudinal trans-

port in the presence of an electric field, including Boltzmann transport, quantum corrections andweak localization, as well as longitudinal and Hall transport in the presence of both electric andmagnetic fields and/or magnetizations. Special attention is paid to transport at finite doping, tothe-Berry phase, which leads to the absence of backscattering, Klein tunneling and half-quantizedHall response. Signatures of surface states in ordinary transport and magnetotransport are clearlyidentified. The review also covers transport experiments of the past years, reviewing the initialobscuring of surface transport by bulk transport, and the way transport due to the surface stateshas increasingly been identified experimentally. Current and likely future experimental challengesare given prominence and the current status of the field is assessed.

I. INTRODUCTION

The conventional band theory picture divides solids

into metals, semiconductors and insulators based on thesize of their band gaps and occupation of the conductionband. Within this picture, an insulator is understoodto be a material with a band gap exceeding room tem-perature by orders of magnitude. In nature, other typesof insulators exist, for example induced by a large dis-order concentration, as in the Anderson insulator, or bystrong Coulomb repulsion between electrons, as in theMott insulator. In this context, the discovery of topolog-ical insulators (TI), first posited in the groundbreakingresearch of Kane and Mele,1 has been a milestone gal-vanizing physics research.14 These novel materials areband insulators in the bulk while conducting along the

surfaces, possessing gapless surface states with a well-defined spin texture protected by time-reversal symme-try. The landmark discovery of topological insulators hasrevolutionized the understanding of insulating behavior.In addition to a variety of exotic effects, TI-based struc-tures may help solve long standing puzzles in physics aswell as contribute to the development of quantum com-putation. Specifically, the interface between a topologi-cal insulator and a superconductor was predicted to sup-port Majorana fermions,5 that is, a species of Bogolyubovquasiparticles represented by real fermions that are theirown antiparticles. Topological insulators are being ex-plored with a view towards applications, as a potentialplatform for quantum computation,6 and as rich physical

systems in their own right.

A. Topological order

On a more profound level, topological insulators are amanifestation of topological order. Unlike other exam-ples of topological order, such as the fractional quan-tum Hall effect or chiral p-wave superconductors, theemergence of topologically insulating behavior is a one-

particle phenomenon, related to the existence of strongspin-orbit coupling. A topologically insulating state in2D, the quantum spin Hall insulator, was predicted in

HgCdTe quantum wells. The topological behavior inthis case is related to the existence of edge states.7 Thekey concept is band inversion consider: two materials,of which one has an inverted band structure. In the casedemonstrated so far this was a CdTe/HgTe quantum well.Each state can be labeled by aZ2 topological invariant(topological quantum number.) As the well width is in-creased the band gap is inverted and the Z2 invariant ofthe ground state changes. From a lay perspective, in thecontext of the quantum spin Hall effect the Z2 invariantcan be thought of as a topological invariant that countsthe number of edge states. The band inversion transi-tion is not characterized by symmetry breaking,2,3 it israther a topological phase transition: the fundamentalsymmetry of the lattice is the same on either side of thetransition, but there is a change in the topological invari-antof the ground state. Kane and Mele1 demonstratedthat in two dimensions the invariantcan be zero or one,the former representing ordinary insulators (equivalent tothe vacuum) and the latter topological insulators. Thetwo band structures corresponding to = 0 and = 1cannot be deformed into one another. The material withan inverted band structure possesses a pair of crossingedge states which realize a quantum spin Hall insulator,that is, they give rise to a quantum spin Hall effect (infact it possesses two pairs, by the fermion doubling theo-rem.) In the quantum spin Hall effect the charge current

is carried by edge states with different spin polarizations.Because of spin-momentum locking due to the spin-orbitinteraction electrons traveling along the edges in differ-ent directions have opposite spin orientations, and thechiral band structure associated with the surface statesprohibits backscattering. The quantum spin Hall insu-lating state was observed experimentally shortly after itsprediction8. For an excellent review of this effect, whichwill not be considered here, the reader is referred to thework of Koenig et al.9.

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Topological insulators were also predicted to exist in3D,1012 where analogous sets of crossing surface statesare present. Three-dimensional topological insulatorswill be the focus of this review. In 3D there arefour Z2 topological invariants characterizing the bandstructure,12 which are related to the time-reversal invari-ant momenta andtime-reversal polarizationat eight spe-cial points in the Brillouin zone. The four Z2 topological

invariants in 3D are customarily labeled 0, 1, 2 and3. They can be calculated easily if the system has inver-sion symmetry.13 TheZ2 topological invariant 0 distin-guishes between strong and weak topological insulators:weak TI have 0 = 0 whereas strong TI have 0 = 1. Ina weak TI the surface states are not protected by timereversal and can be localized by strong disorder. A weakTI can also be interpreted as a 3D stack of 2D quantumspin-Hall insulators. On the other hand, a strong TI isa new class of material and cannot be directly relatedto the quantum spin-Hall insulator. Interestingly, it wasfound that the crossover from a 2D to a 3D TI occursin an oscillatory manner,14 with the material alternatingbetween topologically trivial and nontrivial phases as thelayer thickness is increased.

In an alternative picture of TI that does not rely on Z2invariants, Murakami studied the physics of gap closingin 2D and 3D,15 focusing on whether the gap closes as cer-tain parameters are tuned. Murakami also found that 3Dinversion-asymmetric systems possess a finite crossoverregion of gapless phase between the ordinary insulatorand topological insulator regimes.

The surface states of 3D TI are well described by aRashba spin-orbit Hamiltonian.13,16 A detailed discus-sion of the model Hamiltonian for TI has been given inRef. 17, and a reduced Hamiltonian including surfaceand bulk states has been presented in Ref. 18. Thesestates are chiral, possessing a well-defined spin texture,referred to as spin-momentum locking, and have an en-ergy spectrum akin to a Dirac cone. Chiral Rashba sur-face states have a nontrivial band topology and are char-acterized by a Berry phase, which is associated withKlein tunneling, and provides protection against coher-ent back scattering and therefore weak localization andAnderson localization.

Several materials were predicted to be topological in-sulators in three dimensions. The first was the alloyBi1xSbx,

19,20 followed by the tetradymite semiconduc-tors Bi2Se3, Bi2Te3 and Sb2Te3.

21 These materials havea rhombohedral structure composed of quintuple lay-ers oriented perpendicular to the trigonal c axis. Thecovalent bonding within each quintuple layer is muchstronger than weak van der Waals forces bonding neigh-boring layers. The semiconducting gap is approximately0.3 eV, and the TI states exists along the (111) direc-tion. In particular Bi2Se3 and Bi2Te3 have long beenknown from thermoelectric transport as displaying siz-able Peltier and Seebeck effects, and their high qualityhas ensured their place at the forefront of experimentalattention.2 Initial predictions of the existence of chiral

surface states were confirmed by first principles studiesof Bi2Se3, Bi2Te3, and Sb2Te3.

22 Heusler alloys were re-cently predicted to have topological surface states,23 aswell as chalcopyrites.24

Unlike graphene, the Hamiltonian of topological insu-lators is a function of the real spin, rather than a sub-lattice pseudospin degree of freedom. This implies thatspin dynamics will be qualitatively very different from

graphene. Moreover, the twofold valley degeneracy ofgraphene is not present in topological insulators. Despitethe apparent similarities, the study of topological insu-lators is thus not a simple matter of translating resultsknown from graphene. Due to the dominant spin-orbitinteraction it is also very different from ordinary spin-orbit coupled semiconductors.

Strictly speaking, the first instance of topologically in-sulating behavior discovered in nature is the quantumHall effect (QHE). There is no difference in symmetrybetween the two sides of the quantum Hall transition,involving localized versus extended states. Rather, theLandau levels in the QHE are identified by their Chern

number (TKNN invariant),

25

which is the integral of thegeometrical curvature over the Brillouin zone. From amathematical perspective, the Chern number is a rela-tive of the genus and winding number. An analogy canbe made between the Gaussian curvature and genus onthe one hand and the Berry curvature and Chern num-ber on the other. In the QHE the Chern number can bethought of as the topological invariant that counts thenumber of Landau levels, being therefore equivalent tothe filling factor. Nevertheless, quantum Hall systemsnaturally break time-reversal due to the presence of amagnetic field. Haldane26 devised a model of the QHEwithout an overall magnetic field and therefore withoutLandau levels, in which local magnetic fields were gen-

erated by circulating current loops but averaged to zeroover the whole sample. This model realizes the parityanomaly, but to this day remains a theoretical construct.

In summary, in order to determine whether a certainmaterial is a topological insulator it is necessary to cal-culate a series of topological invariants, for which onerequires the band structure of the material. The exis-tence of chiral surface states is protected by time reversalsymmetry, and is thus robust against smooth perturba-tions invariant under time-reversal, such as non-magneticdisorder and electron-electron interactions. In plain par-lance, in the same manner that a doughnut can be de-formed into a coffee cup, the band structure of a material

cannotbe deformed so as to remove the crossing of a pairof surface states (see the excellent figure in Ref. 2.)

B. Experimental discovery of 3D TI

Thanks to surface-sensitive probes such as scanningtunneling microscopes (STM) and angle-resolved photoe-mission (ARPES), the existence of gapless chiral surfacestates in topological insulators has been established be-

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yond any doubt. Experimental work originally reportedunconventional behavior in Bi2Se3 and Bi2Te3,

27 whichwas later understood to arise from their TI surface states.Several years later, experimental progress on TI skyrock-eted. Hsieh et al,28 were the first to observe the Diraccone-like surface states of Bi1xSbx, with x 0.1, us-ing ARPES. Following that, the same group used spin-resolved ARPES to measure the spin polarization of the

surface states, demonstrating the correlation betweenspin and momentum.29 Shortly afterwards experimentsidentified the Dirac cones in the tetradymite semicon-ductors Bi2Se3,

3032 Bi2Te3,33,34 and Sb2Te3.

34 Reflec-tion high-energy electron diffraction (RHEED) was usedto monitor MBE sample growth dynamics of Bi2Te3,

35

and the electronic structure has been shown to be control-lable in thin films.36 TI growth has been covered in recentreviews.2,3 On a related note, recently Bi2Se3 nanowiresand nanoribbons have also been grown,37 and a topolog-ical phase transition was reported in BiTl(S1dSed)2,

38

which may lead to the observation of charge e/2.

Zhu et al.39 used ARPES to study the surface states

of Bi2Se3. These researchers employed in-situ potassiumdeposition to tackle the instability of electronic proper-ties, which otherwise evolve continuously in time. In theirsamples, as potassium is deposited, new spin-polarizedquantum-well states emerge (with no kzdispersion), orig-inating in the parabolic bulk bands and distinct fromthe Rashba-Dirac cone. These new states also follow aRashba-like dispersion, and their spin splitting is tunableas well as reversible. The potassium-induced potentialgradientV/z enhances the spin splitting. These statesmay affect transport data even from pristine surfaces.

Considerable efforts have been devoted to scanningtunneling microscopy (STM) and spectroscopy (STS),which enable the study of quasiparticle scattering. Scat-tering off surface defects, in which the initial state inter-feres with the final scattered state, results in a standing-wave interference pattern with a spatial modulation de-termined by the momentum transfer during scattering.These manifest themselves as oscillations of the local den-sity of states in real space, which have been seen in severalmaterials with topologically protected surface states.

Roushan et al.40 imaged the surface states ofBi0.92Sb0.08 using STM and spin-resolved ARPESdemonstrating that scattering between states of oppositemomenta is strongly suppressed in the presence of time-reversal invariant disorder. Gomes et al.41 investigatedthe (111) surface of Sb, which displays a topological metalphase, and found a similar suppression of backscattering.Zhang et al.42 and Alpichshev et al.43 observed the sup-pression of backscattering in Bi2Te3, and used STM toimage surface bound states.44 The absence of backscat-tering has been studied theoretically in the context ofquasiparticle interference seen in STM experiments.45,46

Park et al.47 performed ARPES accompanied by theo-retical modelling on Bi2Se3. The observed quasiparticlelifetime of the surface states is attributed to quasiparticledecay predominantly into bulk electronic states through

electron-electron interaction and defect scattering. Stud-ies on aged surfaces show the surface states to be unaf-fected by adsorbed atoms or molecules on the surface,indicating protection against weak perturbations.

Experimental studies have provided evidence of the ex-istence of chiral surface states and of their protectionby time-reversal symmetry. Experimental research hasbegun to investigate the fascinating question concerning

the transition between the three- and two-dimensionalphases in this material.48 Zhang et al.48 used ARPES tostudy a thin 3D Bi2Se3 slab grown by MBE, in whichtunnelling between opposite surfaces opens a small,thickness-dependent gap. The energy gap opening isclearly seen when the thickness is below six quintuplelayers. The gapped surface states also exhibit sizeableRashba spin-orbit splitting due to the potential differ-ence betweenthe two surfaces induced by the substrate.In other words, one of the most important findings ofthis experiment is that thin films need to be thicker thanapproximately six quintuple layers, or 6nm. Bianchi etal.49 showed that the band bending near the Bi2Se3 sur-

face leads to the formation of a two-dimensional electrongas (2DEG), which coexists with the topological surfacestate in Bi2Se3. In the setup used by this experimentalgroup, a topological and a non-topological metallic stateare confined to the same region of space.

Several efforts have focused on the role of magnetic im-purities on the surface states of 3D TI. The work of Horetal. demonstrated that doping Bi2Te3 with Mn results inthe onset of ferromagnetism.50 Chen et al.51 introducedmagnetic dopants into Bi2Se3 to break the time reversalsymmetry and further position the Fermi energy insidethe gaps by simultaneous magnetic and charge doping.The resulting insulating massive Dirac fermion state wasobserved by ARPES. More recently, Wray et al.52 studied

the topological surface states of Bi2Se3 under Coulomband magnetic perturbations. The authors deposited Feon the surface of Bi2Se3, using the fact that Fe has alarge Coulomb charge and sizable magnetic moment tomodify the spin structure of the Bi2Se3 surface states.Interestingly, it was found that this perturbation leadsto the creation of odd multiples of Dirac fermions. Inother words, new pairs of Dirac cones appear as more Feis deposited, also described by Rashba interactions, andthe interaction grows in strength as Fe is added. Numeri-cal simulations indicate that the new Dirac cones are dueto electrons localized deeper inside the material. At thesame time a gap is opened at the original Dirac point, as

one expects.

C. The importance of transport

The key to the eventual success of topological insu-lators in becoming technological materials is inherentlylinked to their transport properties. Potential applica-tions of topological surface states necessarily rely on therealization of an edge metal (semimetal), allowing con-

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tinuous tuning of the Fermi energy through the Diracpoint, the presence of a minimum conductivity (maxi-mum resistivity) at zero carrier density, and ambipolartransport. At this point in time experimental and theo-retical studies of equilibrium TI abound. Transport hasbeen lagging somewhat but is now picking up in boththeory and experiment at a brisk rate.

Despite the success of photoemission and scanning tun-

neling spectroscopy in identifying chiral surface states,signatures of the surface Dirac cone have not yet beenobserved in transport. Simply put, the band structureof topological insulators can be visualized as a band in-sulator with a Dirac cone within the bulk gap, and toaccess this cone one needs to ensure the chemical poten-tial lies below the bottom of the bulk conduction band.Given that the static dielectric constants of materials un-der investigation are extremely large, approximately 100in Bi2Se3 and 200 in Bi2Te3, gating in order to bringthe chemical potential down is challenging. Therefore, inall materials studied to date, residual conduction fromthe bulk exists due to unintentional doping. The above

presentation makes it plain that, despite the presenceof the Dirac cone, currently no experimental group hasproduced a true topological insulator. Consequently, allexisting topological insulator systems are in practice ei-ther heavily bulk-doped materials or thin films of a fewmonolayers (thus, by definition, not 3D.) Following thepattern set by the imaging of the surface Dirac cone, theLandau levels observed in STM studies have not beenseen in transport.53,54 These last works illustrate boththe enormous potential of the field and the challenges tobe overcome experimentally, on which more below.

In this work I will provide a theoretical framework forunderstanding TI transport, review existing TI transport

theories, and survey the impressive experimental break-throughs in transport registered in the past years. Thereview will focus on phenomena in electric fields as wellas electric and magnetic fields, but not magnetic fieldsalone. As it stands, fascinating aspects such as the searchfor Majorana fermions also lie outside the scope of this re-view, since no transport work exists on these topics. Thetransport work covered consists of two main parts, whichin turn can be subdivided into five smaller parts: or-dinary (non-magnetic) transport, comprising Boltzmanntransport and weak localization, and magnetotransport,comprising the quantum Hall, ordinary Hall and anoma-lous Hall effects. Theoretical work in these fields willbe covered first, followed by experiment, and I will closewith a section on experimental challenges, followed bygeneral conclusions and an outline of future directions.

II. EFFECTIVE HAMILTONIAN

The effective Hamiltonian describing the surface statesof non-magnetic topological insulators can be written in

the form of a Rashba spin-orbit interaction16,21

H0k = A k z Ak . (1)

It is understood that k = (kx, ky), and is the tan-gential unit vector corresponding to k. The constant A= 4.1eVA for Bi2Se3.

21 The eigenenergies are denotedby =Ak, so that the spectrum possesses particle-hole symmetry. The Hamiltonian is similar to that ofgraphene, with the exception that the vector of Paulispin matrices represents the true electron spin. Ascalar term Dk2 also exists in principle. The scalar andspin-dependent terms in the Hamiltonian are of the samemagnitude when D kF = A, corresponding to kF 109m1 in Bi2Se3, that is, a density of 10

14cm2, whichis higher than the realistic densities in transport exper-iments. Consequently, in this review scalar terms willnot be considered. I will also not take into account smallanisotropy terms in the scalar part of the Hamiltonian,such as those discussed in Ref. 21. Terms cubic in k inthe spin-orbit interaction are in principle also present, inturn making the energy dispersion anisotropic, but theyare much smaller than the k-linear terms in Eq. (1) andare not expected to contribute significantly to quantitiesdiscussed in this review. All theories described here re-quireF / 1 to be applicable, withthe momentumrelaxation time.

A topological insulator can doped with magnetic im-purities, which give a net magnetic moment as well asspin-dependent scattering. Magnetic impurities are de-scribed by an additional interaction Hamiltonian

Hmag(r) = I

V(r RI) sI, (2)

where the sum runs over the positions RI of the mag-netic ions with sI. To capture the physics discussedin this review it is sufficient to approximate the poten-tialV(r RI) = J (r RI), with J the exchangeconstant between the localized moments and itinerantcarriers. The magnetic ions are assumed spin-polarizedso that sI = sz. Fourier transforming to the crystalmomentum representation{|k}, the k-diagonal term,Hk=k

mag = nmagJ sz M z, gives the magnetization,with nmag the density of magnetic ions. The k-off-diagonal term causes spin-dependent scattering

Hk=k

mag = Js

V

zI

ei(kk)RI , (3)

with V the volume. The effective band Hamiltonian in-corporating a mean-field magnetization takes the form

Hmk= Ak + M 2 k. (4)

Retaining the same notation as in the non-magnetic case,the eigenenergies are =

A2k2 +M2. The bulk of

the work on magnetic TI has concentrated on out-of-pane

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magnetizations and magnetic fields. An in-plane mag-netization or magnetic field does not alter the Rashba-Dirac cone, though coherence between layers can inducea quantum phase transition.55

Interaction with a static, uniform electric field E isquite generally contained in HE,kk = H

scE,kk+ H

sjE,kk,

the scalar part arising from the ordinary position op-erator HscE,kk = (eE r)kk11, with 11 the identity ma-trix in spin space, and the side-jump part HsjE,kk =e (k E) kk arising from the spin-orbit modifica-tion to the position operator, r r+ k,56 with a material-specific constant.

Elastic scattering off charged impurities, static defectsand magnetic ions, but not phonons or other electrons,is contained in the disorder scattering potential Ukk =Ukk

Je

i(kk)RJ, with RJ the random locations ofthe impurities. The potentialUkk incorporates the mag-netic scattering termHk=k

mag of Eq. (3). The potential of

a single impurity Ukk = (1 i k k)Ukk Js z,withUkk the matrix element of a Coulomb potential be-tween plane waves, and k2F

1. Note that Ukk has

the dimensions of energy times volume.Charged impurities are described by a screened

Coulomb potential. The screening function can be eval-uated in the random phase approximation (RPA.) Inthis approximation the polarization function is obtainedby summing the lowest bubble diagram, and takes theform57 (, = )

(q, ) = 1A

k

f0k f0k+k k+ i

1 + cos

2

,

(5)where f0k f0(k) is the equilibrium Fermi distribu-tion function. The static dielectric function, which is of

relevance to the physics discussed in this review, can bewritten as(q) = 1+v(q) (q), wherev(q) = e2/(20rq),wherer is the relative permittivity. To determine (q),we assume T = 0 and use the Dirac cone approxima-tion as in Ref. 57. This approximation is justified inthe regime T /TF 1, with TF the Fermi tempera-ture. AtT= 0 for charged impurity scattering the long-wavelength limit of the dielectric function is57

(q) = 1 + e2

40rA

kFq

, (6)

yielding the Thomas-Fermi wave vector as kTF =

e

2

kF/(40rA). As a result, in topological insulatorsthe matrix element Ukk of a screened Coulomb poten-tial between plane waves is given by

Ukk = Ze2

20r

1

|k k| +kTF , (7)

whereZis the ionic charge (which I will assume for sim-plicity to be Z= 1) and kTF is the Thomas-Fermi wavevector. The polarization function was also calculated inRef.58. The Wigner-Seitz radius rs, which parametrizesthe relative strength of the kinetic energy and electron-electron interactions, is a constant for the Rashba-DiracHamiltonian, and is given by rs= e

2/(20rA).

The full Hamiltonian Hk = H0k +HEkk +Ukk +

Hk=k

mag . The current operator j has contributions fromthe band Hamiltonian,

j =eA

z, (8)

as well as from the applied electric field jE = 2e2

E, and from the disorder potential jU = 2ie (k

k)Ukk . These latter two cancel on physical grounds, asthey represent the force acting on the system.

Transport experiments on topological insulators seekto distinguish the conduction due to the surface statesfrom that due to the bulk states. As mentioned in the in-troduction, it has proven challenging to lower the chemi-cal potential beneath the bottom of the conduction band,so that the materials studied at present are not strictlyspeaking insulators. It is necessary forFto be below thebulk conduction band, so that one can be certain thatthere is only surface conduction, and at the same timeI recall the requirement that F / 1 for the kineticequation formalism to be applicable. These assumptionswill be made throughout the theoretical presentation out-lined in what follows.

III. TRANSPORT THEORY FORTOPOLOGICAL INSULATORS

As stated above, work on transport in topological in-sulators has begun to expand energetically. The focusof experimental research in particular has shifted to theseparation of bulk conduction from surface conduction,and, ideally, strategies for the elimination of the former.Hence it is necessary to know what contributions to theconductivity are expected from the chiral surface states.

Transport problems are customarily approached withinthe framework of the linear response formalism. The re-sulting Kubo formula is frequently expressed in terms of

the Greens functions of the system. This can be donein the Keldysh formulation, which has been covered incountless textbooks, one of the clearest and most com-prehensive being Ref. 59. Most generally, the finite-frequency conductivity tensor is given by

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() = ine2

m +

e2

2V

Tr

d 0,[vG

R+v(G

A GR) + v(GA GR)vGA]

(9)

In the above formula, n stands for the carrier numberdensity, Tr is the operator trace,

denotes the average

over impurity configurations, 0 is the equilibrium den-sity matrix, v is the velocity operator, and GR,A are theusual retarded and advanced Greens functions. In topo-logical insulators the indices , = x, y. One importantfeature of the Kubo formula is that the conductivity iseffectively given by the impurity average of the prod-uct of two Greens functions, which can be iterated inthe strength of the disorder potential, or alternativelyin powers of the quantity /(F), which as we recallis required to be small for the transport theory outlinedhere to be valid. This iteration is encompassed withinthe Bethe-Salpeter equation, and can be expressed di-agrammatically. Depending on the approximation re-

quired, several sets of diagrams must be summed. In anon-magnetic system, the contribution to leading-orderin /(F) recovers the Boltzmann conductivity, and cor-responds to the sum of ladder diagrams. Since the sumof ladder diagrams can be interpreted classically as rep-resenting diffusion of carriers, this terms also bears thename of diffuson. In the next order, which yields a termindependent of , the weak localization contribution tothe conductivity is obtained as the sum of maximally-crossed diagrams, or the Cooperon (plus some minor cor-rections). Weak localization is usually unimportant forF / 1 unless a magnetic field is applied to measurethe magnetoresistance. However when the Hamiltonian

is spin-dependent/chiral it is possible to have terms in-dependent of even in the weak momentum scatteringlimit. Finally, it must be borne in mind that, in topolog-ical insulators, identifying and explaining all the relevantcontributions to transport necessitates a matrix formula-tion, thus in Eq. (9) the equilibrium density matrix, thevelocity operators and Greens functions are matrices.

The Kubo formula is extremely well known in all itsforms, and there appears to be little motivation for thisreview to focus on conventional linear response. An al-ternative matrix formulation, which contains the samephysics and is potentially more transparent, relies on thequantum Liouville equation to derive a kinetic equation

for the density matrix. This theory was first discussed forgraphene monolayers60 and bilayers,61 and was recentlyextended to topological insulators including the full scat-tering term to linear order in the impurity density.62 Pe-culiarities of topological insulators, such as the absenceof backscattering, which reflects the Berry phase andleads to Klein tunneling, are built into this theory in atransparent fashion.

The derivation of the kinetic equation for a systemdriven by an electric field in the presence of random, un-

correlated impurities begins with the quantum Liouvilleequation for the density operator ,62

d

dt +

i

[H,] = 0, (10)

where the full Hamiltonian H= H0+ HE+ U, the bandHamiltonian H0 is defined in Eq. (1), HE= eE rrepre-sents the interaction with external fields, ris the positionoperator, and U is the impurity potential. We considera set of time-independent states{|ks}, where k indexesthe wave vector and s the spin. The matrix elements of are kk sskk =ks||ks, with the understand-ing that kk is a matrix in spin space. The terms H0and HEare diagonal in wave vector but off-diagonal inspin, while for elastic scattering in the first Born approx-imation Uss

kk = Ukkss . The impurities are assumed

uncorrelated and the average ofks|U|ksks|U|ksover impurity configurations is (ni|Ukk |2ss)/V, whereniis the impurity density,Vthe crystal volume and Ukkthe matrix element of the potential of a single impurity.

The density matrixkk is written as kk =fkkk+gkk , where fk is diagonal in wave vector (i.e. fkkkk) whilegkk is off-diagonal in wave vector (i.e. k =kalways in gkk .) The quantity of interest in determiningthe charge current is fk, since the current operator isdiagonal in wave vector. We therefore derive an effec-tive equation for this quantity by first breaking down thequantum Liouville equation into

dfkdt

+ i

[H0k, fk] = i

[HEk, fk]

i

[U , g]kk,(11a)

dgkk

dt +

i

[ H, g]kk = i

[U , f+ g]kk . (11b)

The term i

[U , g]kk on the RHS of Eq. (11a) will be-

come the scattering term. Solving Eq. (11b) to first order

in the scattering potential U

gkk = i

0

dt eiHt/

U , f(t t)

eiHt/

kk

.

(12)

We are interested in variations which are slow on the scaleof the momentum relaxation time, consequently we donot take into account memory effects and f(tt) f(t).Time integrals such as the one appearing in Eq. (12) are

performed by inserting a regularizing factor et

andletting 0 in the end. It is necessary to carry out anaverage over impurity configurations, keeping the termsto linear order in the impurity density ni.

The solution of Eq. (11b) to first order in U repre-sents the first Born approximation, which is sufficient

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for ordinary Boltzmann transport, with or without mag-netic interactions. In order to recover phenomena suchas skew scattering, elaborated upon below, one needs towork to second order in U. The total scattering termJ(fk) = J

Born(fk) + J3rd(fk), with

JBorn(fk) =

0

dt

2 [U , e

i Ht

[U , f] ei Ht

]

kk

, (13)

while represents averaging over impurity configura-tions. In the absence of scalar terms in the Hamilto-nian the scalar and spin-dependent parts of the densitymatrix, nk and Sk, are decoupled in the first Born ap-proximation. In the second Born approximation, whichis needed in magnetic systems, one obtains the additionalscattering term

J3rd(fk) = i3

0

dt 0

dt[U , ei Ht

[U , ei Ht

[U , f] ei Ht

] ei Ht

]

kk

. (14)

The former can be further broken down into JBorn(fk) =

J0(fk) + JBornss (fk) + JBornsj (fk). The first term, J0(fk),

in which = E = 0 in the time evolution opera-tor, represents elastic, spin-independent, pure momen-tum scattering. In JBornss (fk) we allow to be finite

but E = 0. In JBornsj (fk) both and Eare finite, thusJBornsj (fk) acts on the equilibrium density matrix f0k.Beyond the Born approximation we retain the leadingterm J3rd(fk) J3rdss (fk), with finite but E= 0, andwhich is customarily responsible for skew scattering56.The contribution due to magnetic impurities is also zand is contained in JBornss (fk).

First consider the case in which the magnetizationM = 0. In this case all terms above are zero, andit is safe to take = 0. To first order in E, the diagonalpartfk satisfies

62

dfk

dt

+ i

[H0k, fk] + J(fk) =

Dk. (15)

The driving termDk = i[HEk, f0k], and f0k is theequilibrium density matrix, which is diagonal in k. Thek-diagonal part of the density matrixfkis a 22 Hermi-tian matrix, which is decomposed into a scalar part anda spin-dependent part. One writesfk= nk11 + Sk, whereSk is a 22 Hermitian matrix, which represents thespin-dependent part of the density matrix and is writtenpurely in terms of the Pauli matrices. Every matrixin this problem can be written in terms of a scalar part,labeled by the subscript n, and . Rather than choos-ing Cartesian coordinates to express the latter, it is morenatural to identify two orthogonal directions in recipro-

cal space, denoted by k= andk. One projects ask,= kand k,= k. Note thatk, commuteswithH0k, whilek, does not. One projectsSkonto thetwo directions in k-space, writingSk= Sk,+ Sk,, anddefinesSk, = (1/2) sk,k, and Sk, = (1/2) sk,k,.There is no coupling of the scalar and spin distributionsbecause of the particle-hole symmetry inherent in theRashba-Dirac Hamiltonian.

Introducing projection operators P and P onto thescalar part,k, and k, respectively, equation (15) can

be written as

dSk,dt

+PJ(fk) = Dk,, (16a)dSk,

dt +

i

[Hk, Sk] +PJ(fk) =D

k,. (16b)

The projectorPacts on a matrix M as tr(Mk,), withtr the matrix trace, whileP singles out the part orthog-onal to H0k.

Let represent the angle between k andk. The fol-lowing projections of the scattering term acting on thespin-dependent part of the density matrix are needed

PJ(Sk) =knik

8A

d |Ukk |2 (sk sk)(1 + cos )

PJ(Sk) =knik

8A d |Ukk |2 (sk sk)sin

PJ(Sk) =knik

8A

d |Ukk |2

sk+sk

sin .

(17)Notice the factors of (1 + cos ) and sin which pro-hibit backscattering and give rise to Klein tunneling. Infact, the scattering integrals originally contain energy -functions ( ) = (1/A) (k k), which have al-ready been integrated over above. The -functions of are needed in the scattering term in order to ensureagreement with Boltzmann transport. This fact is al-ready seen for a Dirac cone dispersion in graphene wherethe expression for the conductivity found in Ref. 60usingthe density-matrix formalism agrees with the Boltzmann

transport formula of Ref.63(the definition ofdiffers bya factor of two in these two references.) The necessity ofkeeping the terms is a result of Zitterbewegung: thetwo branchesare mixed, thus scattering of an electronrequires conservation of+ as well as .

For M = 0 the equilibrium density matrix 0k =(1/2) (f0k+ + f0k) 11 (1/2) (f0k+ f0k) . Onedecomposes the driving term into a scalar partDn (notgiven) and a spin-dependent partDs. The latter is fur-ther decomposed into a part parallel to the Hamiltonian

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D and a part perpendicular to itD

D = 12

eE k

(f0k+

k f0k

k ) k=

1

2dkk

D = 12

eE k

(f0k+ f0k) k = 12

dkk.

(18)Equation (16) must be solved perturbatively in the smallparameter /(F) (which is proportional to the impu-rity density ni) as described in Ref. 62. In transportthis expansion starts at order [/(F)]

1 (i.e. n1i ), re-flecting the competition between the driving electric fieldand impurity scattering resulting in a shift of the Fermisurface. The leading terms in [/(F)]

1 appear in nkand Sk,.

62 To leading order in /(F) one finds

SEk, = eE k

4

f0+k

k

1

=

kni4A

d

2|Ukk |2 sin2 .

(19)

The sin2 represents the product (1 + cos )(1 cos ).The first term in this product is characteristic of TI andensures backscattering is suppressed, while the secondterm is characteristic of transport, eliminating the effectof small-angle scattering. The conductivity linear in arises from the parallel part of the density matrix and isgiven by

Boltzxx =e2

h

A kF

4 . (20)

We emphasize again that this result tends to the resultfor graphene60,63 whenD 0 (note that the definition ofin Refs.60,63differs by a factor of 2.) The contribution

to the conductivity independent ofis is zero for finite, which is the case considered here. Thus, the leadingorder term in the conductivity is , and in the limitF / 1 there is no term of order ()0.

The charge current is proportional to the spin operator,as can be seen from Eq. (8). Therefore a nonzero steady-state surface charge current automatically translates intoa nonzero steady-state surface spin density. The spindensity response function can easily be found by simplymultiplying the charge current by 2/(2eA), yielding aspin density of

sy = eEx4

A kF

4 . (21)

For a sample in which the impurities are located on thesurface, with n/ni = 0.5 and Ex = 25000V /m, the spindensity is 5 1014 spins/m2 (where spin ), whichcorresponds to approximately 104 spins per unit cellarea. This number, although small, can be detected ex-perimentally using a surface spin probe such as Kerr ro-tation. It is also a conservative estimate: for very cleansamples, having either a smaller impurity density or im-purities located further away from the surface, this num-ber can reach much higher values. Current-induced spin

polarization is a definitive signature of two-dimensionalsurface transport in topological insulators since there isno spin polarization from the bulk: bulk spin densitiesvanish if the bulk has inversion symmetry, which is thecase in the materials Bi2Se3 and Bi2Te3.

21

I will discuss next implications of these results fortransport experiments.62 Below I will use the terminol-ogy high density as meaning that the density is high

enough that a scalar term in the Hamiltonian, of the formDk2, has a noticeable effect (meaningn 1013cm2 andhigher), withlow densityreserved for situations in whichthis term is negligible. The conductivity is a function oftwo main parameters accessible experimentally: the car-rier number density and the impurity density/scatteringtime. The dependence of the conductivity on the carriernumber density arises through its direct dependence onkFand through its dependence on n through. In termsof the number density the Fermi wave vector is given byk2F = 4n. The number-density dependence ofdependson the dominant form of scattering and whether the num-ber density is high or low as defined above. For charged

impurity scattering the conductivity contains terms nandn3/2 (the latter being due to the scalar term not givenin detail here.) At the same time, in a two-dimensionalsystem surface roughness gives rise to short-range scat-tering as discussed in Ref. 64. For short range scatteringthe two terms in the conductivity are a constant and n1/2

(the latter again due to the scalar term.) These resultsare summarized in Table I,where I have listed only thenumber density dependence explicitly, replacing the con-stants of proportionality by generic constants.

The frequently invoked topological protection of TIsurface transport represents protection only against lo-calization by back scattering, not protection against im-purity or defect scattering, i.e. against resistive scatter-ing in general. The presence of impurities and defectswill certainly lead to scattering of the TI surface carriersand the surface 2D conductivity will be strongly affectedby such scattering. If such scattering is strong (whichwill be a non-universal feature of the sample quality),then the actual surface resistivity will be very high andthe associated mobility very low. There is no guaran-tee or protection against low carrier mobility in the TIsurface states whatsoever, and unless one has very cleansurfaces, there is little hope of studying surface transportin 2D topological states, notwithstanding their observa-tion in beautiful band structure measurements throughARPES or STM experiments. Strong scattering will lead

to low mobility, and the mobility is not a topologicallyprotected quantity. To see surface transport one there-fore requires very clean samples.

At carrier densities such that F /1 the conduc-tivity is given by Boltzxx from Eq. (20). As the numberdensity is tuned continuously through the Dirac point,n 0 and the condition F / 1 is inevitably vio-lated, and a strong renormalization of the conductivityis expected due to charged impurities.6567 Quite gener-ally, charged impurities give rise to an inhomogeneous

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TABLE I: Carrier density dependence of the longitudinalconductivityxx. The numbers a1 a4 represent constants.The table is adapted from Ref.62.

Screened charges Short rangeLow density a1n a3High density a1n+a2n

3/2 a3+a4n1/2

0 0.1 0.2 0.3 0.4 0.5

n/n

0

50

100

150

200

250

(

e

/h)

i

2

d=10A

d=5A

d=0

FIG. 1: Calculated conductivity limited by screened chargedimpurities as a function of the surface carrier density for dif-ferent impurity locations d = 0, 5, 10A. In this figure weuse the following parameters: an impurity densityni = 10

13

cm2, A = 4.1 eVA, which corresponds to the Fermi veloc-ity vF = 6.2 10

7 cm/s, and the static dielectric constantr = 100. The figure is adapted from Ref. 62.

Coulomb potential, which is screened by both electronsand holes. The net effects of this potential are an in-homogeneity in the carrier density itself and a shift inthe Dirac point as a function of position. At high den-sities the spatial fluctuations in the carrier density areof secondary importance, and do not modify the lineardependence of the conductivity onn, yet as the chemicalpotential approaches the Dirac point, where the averagecarrier densityn = 0, these fluctuations play the dom-inant role in conduction.65 At low densities, the carrierscluster into puddles of electrons andholes, and a residualdensity of carriers is always present, making it impossibleto reach the Dirac point experimentally.65 Consequentlythe renormalization due to the presence of electron andhole puddles displays a strong sample dependence.63,66

In addition, exchange and correlation effects make a sig-nificant contribution to this minimum conductivity.67 Anaccurate determination of this enhancement for topolog-ical insulators requires detailed knowledge of the impu-rity density distribution, which can only be determinedexperimentally or modeled numerically by means of e.g.

an effective medium theory.67 However, a self consistenttransport theory provides a physically transparent wayto identify the approximate minimum conductivity. Themagnitude of voltage fluctuations can be calculated inthe random phase approximation, and the result used todetermine the residual density and the critical numberdensity at which the transition occurs to the regime ofelectron and hole puddles, where the carrier density will

be highly inhomogeneous. Both the residual density andthe critical density are proportional to ni by a factor oforder unity,65 and as a first approximation one may takeboth of them as ni. In view of these observations,the minimum conductivity plateau will be seen approx-imately at Boltzxx , in which with the carrier densityn isreplaced by ni. Using the expressions found above forBoltzxx and , this yields for topological insulators

minxx e2

h

8

Itc

, (22)

where the dimensionless Itc is determined by rs.62 With

rs

much smaller than in graphene due to the large dielec-tric constant, the minimum conductivity may be substan-tially larger.67 (Note that for a Rashba-Dirac cone thedefinition ofrs contains some arbitrariness.

63)Transport measurements require the addition of metal-

lic contacts on the surface of the topological insulator.Since the properties of the surface states depend cru-cially on the boundary conditions, the natural questionis how these properties are affected by the metallic con-tacts. To address this question numerical calculationsfor the minimal tight-binding model studied in Ref. 68were performed, focusing on the case of the case of largearea contacts.62 When the TI surface is in contact with ametal, the surface states penetrate inside the metal and

hybridize with the metallic states. A typical hybridizedstate is shown in Fig. 2a. The amplitude of this statenear the boundary of the topological insulator is reduced

by a factor 1/L(m)z with respect to the amplitude of a

pure surface state with the same energy, where L(m)z is

the width of the metal in the direction perpendicular tothe interface. However, the local density of states nearthe boundary is not reduced, as the number of hybridized

states also scales withL(m)z . The spectrum of a topologi-

cal insulator in contact with a metal is shown in Fig. 2b.For each state the total amplitude within a thin layer oftopological insulator in the vicinity of the boundary wascalculated, shown in Fig. 2c. Instead of the sharply de-fined Dirac cone that characterizes the free surface, onehas a diffuse distribution of states with boundary contri-butions. The properties of the interface amplitude distri-bution shown in Fig. 2c, i.e., its width and the dispersionof its maximum, depend on the strength of the couplingbetween the metal and the topological insulator. If thedistribution is sharp enough and the dispersion does notdeviate significantly from the Dirac cone, the transportanalysis presented above should be applied using bare pa-rameters (i.e. parameters characterizing the spectrum of

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Mkx ky

E(k)

E(k)

TI Metal

Distance (z/a)Amplitude

b

c

a

FIG. 2: (a) Amplitude of a metallic state hybridized with atopological insulator surface state as function of distance fromthe interface (in units of interlayer spacing). (b) Spectrumof a topological insulator in contact with a metal (black).The states of a topological insulator with a free surface arealso shown (purple - bulk states, red - surface states). (c)Density plot of the total amplitude within a thin region oftopological insulator in the vicinity of the interface (red/darkgrey area in panel a). Note that, instead of a well-definedsurface mode, one has a diffuse distribution of states withboundary contributions. The dispersion of the maxima ofthis distribution is represented by green points in panel (b).The figure is adapted from Ref.62.

a free surface.) Nonetheless, significant deviations fromthe free surface dispersion are possible. For example, thelocation of the Dirac point in the model calculation con-sidered here is fixed by symmetry. However, in real sys-tems the energy of the Dirac point can be easily modifiedby changing the boundary conditions. Consequently, theeffective parameters (e.g. A Aeff) entering transportcoefficients may differ significantly from the correspond-ing parameters extracted from ARPES measurements.

FIG. 3: Contributions to the Hall conductivity of a TI thinfilm due to the orbital magnetic moment plotted as a functionof Fermi level. The two plots are for film thicknesses of (a)20Aand (b) 32A respectively. This effect can be understoodas a spin-Hall effect in much the same manner as the 2D quan-tum spin-Hall insulator, and is accompanied by a topologicalphase transition as the film thickness is changed. This figurehas been adapted from Ref. 69.

In topological insulator thin films tunneling may bepossible between the top and bottom surfaces. Evidentlyinclusion of this physics requires one to start with a 4 4Hamiltonian, rather than the 2 2 model discussed sofar. Interlayer hybridization induces a gap in the surfacestates of thin films, as was discussed in Ref. 69. Oncethe gap has been taken into account, the theory can bereduced to an effective 2 2 Hamiltonian for each sur-face, each of which contains a z term due to interlayertunneling. This mass term turns the original Dirac conesof the surface states into two massive parabolas. TwofoldKramers degeneracy is preserved since time-reversal sym-metry is not broken. Consequently, the parabolas are in-dexed by z = 1, describing two sets of Dirac fermions(rather than two independent surfaces). As a result of

the mass term the Berry curvature of the surface statesis finite, as is the orbital magnetic moment arising fromthe self-rotation of Bloch electron wave packets. Themass term oscillates as a function of film thickness andchanges sign at critical thicknesses, where a topologicalphase transition occurs due to a discontinuous change inChern number, in the same fashion as in the 2D quantumspin-Hall insulator.

Li et al.70 studied transport in topological insulatorsin the neighborhood of a helical spin density wave. Thenontrivial magnetic structure results in a spatially local-ized breakdown of time reversal symmetry on the TI sur-face. The helical spin density wave has the same effect as

an external one-dimensional periodic magnetic potential(of period L) for spins on the surface of the TI, with anadditional term in the Hamiltonian

U(x) = Uyycos

2x

L

+Uzzsin

2x

L

. (23)

As expected the Dirac cone of the TI surfaces becomeshighly anisotropic, and, as a result, transport due to thetopological surface states displays a strong anisotropy.The helical spin density wave on the TI surface leads to

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FIG. 4: Spatial distribution of local spin density Sz for asingle Dirac fermion with a Fermi velocity of 5105ms1. (a)nnjunction,V2 = 40 meV, and the incoming electron energyis 60 meV. The angle of incidence = 63 deg [(blue) solid line]and = 18 deg [(red) dashed line]. (b)npjunction,V2 = 60meV and the incoming electron energy is 35 meV. The angleof incidence = 63 deg [(blue) solid line] and = 36 deg [(red)dashed line]. This figure has been adapted from Ref. 71.

striking anisotropy of the Dirac cones and group veloc-ity of the surface states. The decrease in group velocityalong the direction defined by the one-dimensional pe-riodic magnetic potential is twice as large as that per-pendicular to this direction. More importantly, the re-

searchers found that at the Brillouin zone boundaries, theperiodic magnetic structure gives rise to new semi-Diracpoints that have a linear dispersion along the directiondefined by the magnetic potential but a quadratic disper-sion perpendicular to this direction. The group velocityof electrons at these new semi-Dirac points is predictablyalso highly anisotropic. The authors suggest that sucha structure could be realized in a topological insulatorgrown on a multiferroic substrate. It could be the directmanifestation of the elusive 2D surface TI transport evenin the presence of considerable bulk conduction, since thebulk transport is isotropic and presumably unaffected bythe presence of the helical spin density wave.

A fascinating publication recently predicted a giantpolarization due to the topological surface states ofBi2Se3.

71 The structure proposed by the authors envis-ages a TI thin film to which a step-function potentialis applied by means of a back gate, given by V(x) =V2(x). A voltageVy across the surface drives a chargecurrent. The charge current in turn generates a spin-Hall current along thex-direction, which strikes the step-potential boundary atx = 0. Near the potential bound-ary atx = 0, a spin-zpolarization (20%) exists along thex-direction, induced by an electric current along the y-

direction in the ballistic (disorder-free) transport regimeat zero temperature. Thanks to the Klein paradox, elec-trons with energies less than V2 are not confined. How-ever, the incident electrons are no longer in eigenstateson thex >0 side of the potential boundary. The spin po-larization oscillates across the potential boundary with aperiod given by 1/kF, reflecting the interference of elec-tron and hole states. This is shown in Fig. 4. Note that

the plot represents the spin polarization of a singleelec-tron. The local spin density has a strong dependence onthe angle of incidence of the electrons. The induced spinpolarization is found to be insensitive to the surface F,because the spin polarization is approximately inverselyproportional to the Fermi velocity, which is a constantfor Dirac electrons. The authors expect the effect to beobservable in a spin resolved STM.

Still on the topic of spin transport, Dora and Moess-ner studied the spin-Hall effect of massive Dirac fermionsin topological insulators instrongelectric fields, near thethreshold of electrical breakdown.72 It is well known thata single Dirac cone with a mass term gives rise to a quan-

tized spin-Hall effect, as discussed above in the case ofRef. 69. Dora and Moessner consider switching on anelectric field, following which a steady-state Hall currentis reached via damped oscillations of the transient com-ponents. They find that the spin-Hall conductivity re-mains quantized as long as the electric field does not in-duce Landau-Zener transitions. However, as the Landau-Zener threshold is approached. the spin-Hall conduc-tance quantization breaks down, and the conductivitydecreases as 1/

E. In other words, electrical breakdown

affects not only longitudinal transport, but also trans-verse transport.

The topological order present in TI is a result of one-

particle physics. Interaction effects have been stud-ied in TI with a focus on phenomena in equilibriumand in the quantum Hall regime. A fundamental ques-tion is whether basic TI phenomenology survives in-teractions out of equilibrium (i.e. in an electric field),since in transport topology only protects against back-scattering. A second fundamental question concerns thenon-equilibrium spin polarization due to spin-momentumlocking in an electric field. In interacting TI, the mean-field Hamiltonian is spin-dependent, and it is naturalto ask whether a Stoner criterion exists for the non-equilibrium spin polarization. A systematic understand-ing of electron-electron interactions in non-equilibriumTI remains to be constructed. It was demonstratedrecently73 that many-body interactions enhance thecharge conductivity and non-equilibrium spin polariza-tion of TI, and that the non-equilibrium enhancementis intimately linked to Zitterbewegung and not diver-gent under any circumstances. Although disorder renor-malizations give a nontrivial doubling of the enhance-ment, the equivalent of the Stoner criterion is neverfulfilled for TI. These findings can be understood bymeans of the following physical picture, illustrated inFig.5. The enhancement of the charge conductivity and

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FIG. 5: Spin polarization in interacting TI. At each k, redarrows represent the effective spin-orbit field k and greenarrows the spin. (a) Equilibrium non-interacting case. Thespin follows the local k and the spin polarization averagesto zero. Out of equilibrium (not shown) the spin at k also

follows k, resulting in a net spin polarization. (b) Non-equilibrium interacting case. Interactions tilt the spin awayfromk (here the tilt y) and enhance the non-equilibriumspin polarization. Scattering from k= k1 to k = kx furtherincreases the local(i.e. k-dependent) spin polarization, whilescattering fromk = k1to k = kxdecreases it. In an electricfield the asymmetry of the Fermi surface means that the neteffect of scattering is to increase the spin polarization. Thisfigure is adapted from Ref. 73.

non-equilibrium spin polarization reflects the interplay ofspin-momentum locking and many-body correlations. A

spin at k feels the effect of two competing interactions.The Coulomb interaction between Bloch electrons withk and k tends to align a spin at k with the spin at k,equivalent to a z-rotation. The total mean-field inter-action tends to align the spin at k with the sum of allspins at all k, and the effective (Hartree-Fock) Hamil-tonian describing electron-electron interactions encapsu-lates the amount by which the spin at k is tilted as aresult of the mean-field interaction with all other spinson the Fermi surface. The effective field k tends toalign the spin with itself. As a result of this latter fact,out of equilibrium, an electrically-induced spin polariza-tion is already found in the non-interacting system,62 asdiscussed extensively above. Interactions tend to alignelectron spins with the existing polarization. The effec-tive z-rotation explains the counterintuitive observationthat the enhancement is related to Zitterbewegung, orig-inating as it does in Sk. Many-body interactions givean effectivek-dependent magnetic field z, such that forE x the spins sy andsy are rotated in opposite di-rections, reinforcing each other. Due to spin-momentumlocking, a tilt in the spin becomes a tilt in the wave vec-tor, increasing the conductivity: spin dynamics create afeedback effect on charge transport. This feedback effect

is even clearer in the fact that scattering doubles the en-hancement, a feature that was demonstrated rigorouslyusing the density matrix formalism. This doubling isvalid for any elastic scattering. Takek = k1 in Fig. 5.Scattering from k = k1 to k = kx increases the spinpolarization at that point on the Fermi surface, whilescattering from k = k1 to k =kx decreases the spinpolarization at that point. Because the Fermi surface is

shifted by the electric field, the net effect of this mecha-nism is to increase the spin polarization.

This concludes the discussion of the theory of non-magnetic surface transport in topological insulators.Aside from the predictions of some fascinating effects,the central message of this section is that threeprimarysignatures of surface transport may be identified: (a)a non-universal minimum conductivity as in graphene,due to the formation of electron and hole puddles as theDirac point is approached (b) a current-induced spin po-larization and (c) a thickness independent conductancewould be a likely indication of TI surface transport. Al-though not elaborated upon, the latter fact is obvious,

since the number of surface carriers does not change withfilm thickness, wheres the number of bulk carriers does.

IV. MAGNETOTRANSPORT THEORY

Both in theory and in experiment, considerably moreattention has been devoted to magnetotransport than tonon-magnetic transport. The primary reason for this fo-cus is that specific signatures of surface transport canbe extracted without having to tune through the Diracpoint or measure the spin polarization of a current. InShubnikov-de Haas oscillations and in the anomalous

Hall effect, both discussed below, qualitatively differentbehavior is expected from the bulk and surface states.Broadly speaking, magnetotransport encompasses two

categories: transport in magnetic fields and transport inthe presence of a magnetization. A magnetic field couplesboth to the orbital degree of freedom, giving rise to theLorentz force and cyclotron orbits, and to the spin degreeof freedom via the Zeeman interaction. A magnetizationcouples only to the spin degree of freedom. The electricalresponse in the presence of a magnetic field and that inthe presence of a magnetization do share common terms,yet they ultimately contain different contributions andare best treated separately. I will concentrate first ontransport in a magnetic field, then on transport in the

presence of a magnetization. Well-known aspects thathave been covered extensively before, primarily the topo-logical magnetoelectric effect,24 will not be covered here.

A. Magnetic fields

In a magnetic field B the energy spectra of both thebulk and the surface consist of Landau levels. The cy-clotron frequency c = eB/m, with m the effective

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mass, is conventionally used to differentiate weak andstrong magnetic fields. In a weak magnetic field c 1,while in a strong magnetic field c 1, withthe mo-mentum relaxation time as in the previous section. Inthe bulk case, withB z, the diagonal conductivity xxcontains a classical contribution as well as an oscillatoryterm periodic in 1/B, referred to as Shubnikov-de Haas(SdH) oscillations. Unlike non-magnetic transport, the

classical term represents scattering-assisted transport,59

which can be interpreted as the hopping of electrons be-tween the centers of the classical cyclotron orbits. Inboth weak and strong magnetic fields this contributionis 1/. For details of its form in different regimesthe reader is referred to the comprehensive discussion inRef.59. The oscillatory contribution to the diagonal con-ductivity reflects the fact that consecutive Landau levelspass through the Fermi energy as the magnetic field isswept. The oscillatory contribution is present even atsmall magnetic fields, that is, in the ordinary Hall effect.The corresponding oscillations in the magnetization aretermed the de Haas-van Alphen (dHvA) effect. In thebulk in a magnetic field only the ordinary Hall effect willoccur, and experimentally one expects SdH oscillationsfor all orientations of the magnetic field.

In a strong perpendicular magnetic field in 2D systemsthe quantum Hall effect is observed. The diagonal con-ductivityxx consists of a series of peaks, while the Hallconductivity xy consists of a number of plateux. TheLandau levels corresponding to the topological surfacestates are effectively the same as in graphene and thetheory will not be developed here, primarily because itis either obvious or it has been covered elsewhere.2,3 Ne-glecting the Zeeman splitting, the Landau level energiesindexed by n are given by the formula

n= 0+ sgn(n)A

2eB|n|, (24)

with 0 the energy at the Dirac point, which is inde-pendent of the magnetic field for massless Rashba-Diracfermions. The Landau levels show the

|n|Bdependence

on the magnetic field and Landau level index character-istic of the linear Rashba-Dirac cone dispersion. In two-dimensional systems the considerations above for xxre-main valid, while in a large perpendicular magnetic fieldthe quantization of xy becomes apparent. The quan-tized Hall conductivity is given by

xy =

n+1

2e2

h. (25)

The half-quantization is due to the Berry phase, whichis also present in the scattering term in non-magnetictransport. The half-quantization of the QHE can also beunderstood in terms of a term in the bulk Lagrangian E B. Shubnikov-de Haas oscillations in xx are de-scribed by the same formalism as in graphene. Only thelevels at the Fermi energy at one particular time con-tribute to the diagonal conductivity xx, yet the Hall

conductivityxy has contributions from all levels belowthe Fermi energy.

In a magnetic field of several Tesla, if the magneto-transport signal can be ascribed to the topological sur-face states, one thus expects to observe the quantum Halleffect, and expects to see SdH oscillations only for a mag-netic field normal to the TI surface. An in-plane mag-netic field would not yield SdH oscillations. In addition,

due to the different effective mass, the frequency of SdHoscillations for the bulk is considerably different fromthat for the surface. Specifically, the cyclotron frequencyc = eB/m

is a constant for bulk transport. For sur-face transport one may define a cyclotron effective massm = 2kF/A, which depends on the Fermi wave vector,given by kF =

4n. In the case of coexisting bulk and

surface transport one therefore expects two sets of SdHoscillations with different frequencies, as well as angulardependence of the SdH oscillations, both of which shouldbe clearly distinguishable experimentally.

Shubnikov-de Haas and de Haas-van Alphen oscilla-tions in Bi2Se3 were studied theoretically in Ref.74. The

authors determine the band structure from first princi-ples, then calculate the magnetization induced by an ex-ternal magnetic field. Depending on whether the zero-mode Landau level is occupied or empty, the locationof center of the dHvA oscillations in the magnetizationchanges. Interestingly, the magnetic oscillation patternsfor the cases of filled and half-filled n = 0 Landau levelare out phase.

Chuet al.75 studied numerically the transport proper-ties of the topological surface states in a Zeeman field.The main argument of their work is that a chiral edgestate forms on the surface, but is split into two spatially-separated halves. Each half carries one half of the con-

ductance quantum. They propose a four-terminal setupin which this theory can be tested. Numerical simula-tions reveal that the difference between the clockwise andcounterclockwise transmission coefficients of two neigh-boring terminals is 1/2. This suggests a half-quantizedordinary Hall conductance could be measured in a four-terminal experiment.

Zhanget al.76 set out to explain why the half-quantizedQHE has not been observed in experiments to date. Theauthors investigated a 3DTI in a uniform magnetic fieldusing tight binding model.They point out that all surfacestates of a finite 3DTI are connected to each other. Ina finite 3D TI, the two surfaces cannot simply be sepa-rated by the bulk as in usual 2D quantum Hall systems.As a result, surface states may move from one side ofthe sample to another. Hall voltage measurements for3DTI are therefore necessarily ambiguous. The Hall cur-rent is due to 2D surface states, rather than 1D edgestates. The structures of these side surface states aremore complicated than the edge states in 2DEG. Quan-tum Hall plateaux of (2n+ 1) e2/h can only be definedfrom transverse current instead of the Hall voltage. Thetight-binding calculations are in agreement with earlierpredictions based on an effective theory of Dirac fermion

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in curved 2D spaces,77 which showed that, based on theargument above that the current is carried by 2D ratherthan 1D states, the robustness of the QH conductanceagainst impurity scattering is determined by the oddnessand evenness of the Dirac cone number.

Tse and MacDonald78 developed a theory of themagneto-optical and magneto-electric response of a topo-logical insulator thin film in a strong (quantizing) mag-

netic field. Their work was driven by the realization thatthe quantum Hall effect due to the topological surfacestates could be observed optically, even when bulk carri-ers are present in the system and contribute to conduc-tion. Interestingly, these authors showed that the low-frequency magneto-optical properties depend only on thesum of the Dirac-cone filling factors of the top and bot-tom surfaces. In contrast, the low-frequency magneto-electric response is determined by the difference in thesefilling factors. Furthermore, the Faraday rotation angleis quantized, being given by the sum of the filling factorsof the top and bottom surfaces multiplied by the finestructure constant. The Faraday rotation angle exhibitssharp cyclotron resonance peaks and changes sign neareach allowed transition frequency, as shown in Fig. 6.At half-odd-integer filling factors a single dipole-allowedintraband transition exists. At other filling factors tworesonances are allows at different frequencies associatedwith transitions into and out of the partially filled Lan-dau level. This behavior is completely different fromusual 2DEGs, in which all dipole-allowed transitions havethe same energy. The Kerr rotation angle is /2, asfound by the same authors in the case of magnetically-doped TI (see below.) It is strongly enhanced at thefrequencies at which cyclotron resonance occurs. Unlikethe low-frequency giant Kerr effect, the resonant peakscorresponding to absorptive Landau level transitions are

non-universal.Tkachov and Hankiewicz79 studied the quantum Hall

effect due to the topological surface states in a TI filmof thicknessd in the situation in which bulk carriers alsocontribute via the ordinary Hall effect. The authors tookinto account the contributions of the bulk as well as thetop and bottom surfaces, showing that the contributionto the conductivity due the bulk carriers can be sup-pressed by an external magnetic field and an ac-electricfield. They work in the regime , c , where isthe frequency of the applied electric field, and is theZeeman-energy gap in the Dirac cone. Under these con-ditions, the cyclotron resonance frequency is shifted by a

thickness-dependent term,

c c+ dbulkxx

|sfcxy|, (26)

where bulkxx is the bulk dc conductivity at zero mag-netic field, andsfcxy is the surface Hall conductivity. Thenonzero surface conductivity alters the cyclotron drift inthe direction of the current, and induces a linear nega-tive magnetoresistivity. At the same time, the Hall angledepends on the magnetic field quadratically

FIG. 6: (a) Faraday rotation angle versus frequency /(c)at equal filling factors on both surfaces for a 30nm-thickBi2Se3 film. The densities on both surfaces are 510

11cm2.The bulk band gap is 0.35 eV, the Fermi velocity is vF =5 105ms1 and the dielectric constant r = 29. (b). Kerrrotation angle versus frequency for the same parameters. Thisfigure is adapted from Ref. 78.

Zyuzin et al.80 investigated a thin TI film in a strongperpendicular magnetic field, allowing for hybridizationbetween the top and bottom surfaces. The authors deter-mined the Landau-level spectrum of the film as a functionof the applied magnetic field and the magnitude of thehybridization matrix element, taking into account boththe orbital and the Zeeman spin splitting effects of thefield. Most interestingly, they identified a quantum phasetransition between a state with a zero Hall conductivityand a state with a quantized Hall conductivity of e2/has a function of the magnitude of the applied field. The

transition is driven by the competition between the Zee-man and the hybridization energies.

B. Magnetizations

Doping with magnetic impurities, whose momentsalign beyond a certain doping concentration, makes topo-logical insulators magnetic and opens a gap. Topologicalinsulators in the presence of a strong out-of-plane magne-tization exhibit rather different physics than their coun-terparts placed in magnetic fields.

The interplay of magnetism, strong spin-orbit cou-pling, disorder scattering and driving electric fields leadsto the anomalous Hall effect in magnetic TIs. Theanomalous Hall effect has been a mainstay of condensedmatter physics, studied extensively over the years.81

A recent groundbreaking paper by Yu et al.82 studiedtetradymite semiconductors doped with Cr and Fe usingfirst principles calculations, determining a critical tem-perature as high as 70K. Yu et al. further demonstratedthat in a thin TI film near a band inversion, in which thechemical potential lies in the magnetization-induced gap

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between the surface valence and conduction bands, edgestates give rise to an integer quantized anomalous Halleffect.82 The quantized anomalous Hall effect requires aband inversion, provided by spin-orbit coupling, and fer-romagnetic order, provided by doping. The 4 4 Hamil-tonian describing the surface states can be broken up intotwo blocks, and a sufficiently large exchange field givesrise to a difference in Chern number of one between the

two blocks, resulting in an anomalous Hall conductivityofe2/h. Remarkably, this is true regardless of whetherthe system is in the topologically trivial phase or in thetopologically insulating phase. The quantized anoma-lous Hall effect is to be distinguished from the anoma-lous quantum Hall effect, which has also been predictedto occur in topological insulators.2

In contrast to the 2D case, in 3D magnetic TIs withthe chemical potential in the band gap the anomalousHall conductivity due to one surface is half-quantizede2/2h,83,84 rather than the integer quantization seen for2D edge states. The value of e2/2h found for F ingap was demonstrated to be robust against disorder.84

Ref.85studied the anomalous Hall effect due to the sur-face states of 3D topological insulators in whichFlies inthe surface conduction band, showing that the topologi-cal band structure contribution (e2/h) found in previousstudies, with a substantial renormalization due to dis-order, yields the dominant term, overwhelming all con-tributions due to skew scattering and side jump. Theend result is of the order of the conductivity quantumand independent of the magnetization. These results areexplained in what follows.

Theoretically, the already challenging problem of theanomalous Hall effect in 3D TI is complicated by the ne-cessity of a quantum mechanical starting point and ofa matrix formulation describing strong band structure

spin-orbit coupling, spin-dependent scattering includingrelativistic corrections, Klein tunneling and Zitterbewe-gung on the same footing. The density matrix-Liouvilleequation formalism has a quantum mechanical founda-tion and accounts for the three mechanisms known tobe important in the anomalous Hall effect. The inter-play of band structure spin-orbit interaction, adiabaticchange ink in the external electric field, and out-of-planemagnetization leads to a sideways displacement of carri-ers even in the absence of scattering (intrinsic).86 Thespin dependence of the impurity potentials via exchangeand extrinsic spin-orbit coupling causes asymmetric scat-tering of up and down spins (skew scattering),87 and a

sideways displacement during scattering (side jump).88

In magnetically doped TI the magnetization M= 0and the kinetic equation can be written in the form85

dfkdt

+ i

[Hmk, fk] + J(fk) = Dk, (27)

withHmkgiven by Eq.(4). All extrinsic spin-orbit terms must be taken into account, including the secondBorn approximation scattering term J3rd(fk) of Eq.14,the spin-dependent correction to the position operator

and the corresponding modification of the interac-tion with the electric field. Consequently, the drivingtermDk contains a host of contributions not found inthe non-magnetic case. It is given in detail in Ref. 85.At zero temperature the splitting due to M is resolved,and the Fermi energyF |M| as usual lies in the sur-face conduction band. Three orthogonal directions areidentified in reciprocal space,

k= ak +bk z

keff =k

zeff =ak z+bk ,

(28)

where ak = 2Ak/k and bk = 2M/k, with k de-fined in Eq. (4), and aF andbF atk = kF, with bF 1at usual transport densities. Considering the negligiblesize ofbk at k = kF, the condition yielding the suppres-sion of backscattering is effectively unmodified from thenon-magnetic case. The vector of Pauli matrices is

projected onto the three directions in reciprocal space inan analogous fashion to the non-magnetic case, and thedetails will not be reproduced here.85 LikewiseSk is alsoprojected onto these three directions.

The kinetic equation is solved perturbatively in/(F), characterizing the disorder strength, as well asin , which quantifies the strength of the extrinsic spin-orbit coupling. When a magnetization is present a num-ber of terms independent ofappear in the conductivity,representing transvers (Hall) transport. The dominantterm in the anomalous Hall conductivity is85

yx = e2

2h(1 ), (29)

where is a disorder renormalization, equivalent to avertex correction in the diagrammatic formalism.85 The

bare contribution to yx is e2

2h, which can also be ex-

pressed in terms of the Berry (geometrical) curvature89,and is thus a topological quantity. It is similar to the re-sult of Ref.83in the vicinity of a ferromagnetic layer, andcan be identified with a monopole in reciprocal space. Itarises from the integral kF0

AkM

(A2k2 +M2)3/2 =

1

A

1 M

A2k2F+M

2

. (30)

If we ignore terms of order b2F it can be approximatedas simply 1/A, corresponding to the limit kF . Inthis regime the integral is therefore a constant, and the

bare conductivityyx = e2

2h, independent of the magne-

tization, number density, or Rashba spin-orbit couplingstrengthA. Contributions turn out to be negligiblein comparison with this term.85

Ref. 85 showed that there are no terms from extrin-sic spin-orbit scattering that commute with the band

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Hamiltonian, and thus, following the argument above,no terms in the density matrix due to extrinsicspin-orbit mechanisms. Neither extrinsic spin-orbit cou-pling nor magnetic impurity scattering give a drivingterm in the kinetic equation parallel to H0k (that is, k), thus the anomalous Hall response does not con-tain a term of order [/(F)]

1 due to extrinsic spin-orbit scattering or magnetic impurity scattering. Such a

term would have overwhelmed the renormalized topolog-ical term yx = e

2

2h(1 ) in the ballistic limit.

In TI the spin and charge degrees of freedom are in-herently coupled and the anomalous Hall current canalso be viewed a steady-state in-plane spin polarizationin a direction parallel to the electric field. The (bare)topological term has two equal contributions, which arepart of the correction to the density matrix orthogonalto the effective Zeeman field, therefore they represent anelectric-field induced displacement of the spin in a direc-tion transverse to its original direction. They are also ob-tained in the Heisenberg equation of motion if one takesinto account the fact that k is changing adiabatically.Physically, thex-component of the effective Zeeman fieldk is changing adiabatically, and the out-of-plane spincomponent undergoes a small rotation about this new ef-fective field. Consequently, each spin acquires a steadystate component parallel to E, which in turn causes k toacquire a small component in the direction perpendicularto E. Elastic, pure momentum scattering [contained in

J0(fk)] reduces this spin polarization because, in scatter-ing from one point on the Fermi surface to another, thespin has to line up with a different effective field k. Theextra spin component of each electron is M, howeverthe final result is independent ofM, as the integrand con-tains a monopole located at the origin ink-space.83,89 AsM 0 the effect disappears, since the correction to theorthogonal part of the density matrix vanishes. Finally,though the form of this term would hint that it is observ-able for infinitesimally small M, it was tacitly assumedthatM exceeds the disorder and thermal broadening.

The appearance of terms to order zero in /(F),not related to weak localization, is standard in spin-dependent transport. The expansion of the density ma-trix starts at order /(F)

(1), i.e. the leading-orderterm is , so the fact that the next term is indepen-dent of /(F)

(1) is to be expected. Of course thisterm is not independent of the angular characteristics ofthe scattering potential (above depends on the scatter-

ing potential), which is why it is not enough to consideronly the band structure contribution, but also its disor-der renormalization. Disorder renormalizations are cru-cial in spin-related transport, where they can go as far asto cancel band structure contributions, as they do in thespin-Hall effect in spin-orbit coupled systems describedby the Rashba Hamiltonian.90

Experimentally, for charged impurity scattering thefigures depend on the Wigner-Seitz radius rs, represent-ing the ratio of the Coulomb interaction energy and the

kinetic energy, with r the relative permittivity. ForBi2Se3 with rs = 0.14

62, the dominant term by far isyx 0.53 (e2/2h) e2/4h. We can consider alsothe (artificial) limit rs 0, which implies r , arti-ficially turning off the Coulomb interaction. As rs 0,the prefactor ofe2/2h tends to 0.61, while at rs = 4,the limit of RPA in this case, it is 0.12. Interestingly,for short-range scattering, the anomalous Hall current

changes sign, with yx 0.18 (e2

/2h). The expressionabove represents the contribution from the conductionband, and in principle an extra e2/2h needs to be addedto the total result to obtain the signal expected in ex-periment. Since the conduction band is expected to con-tribute e2/4h, this does not make a difference in ab-solute terms. However, if one bears in mind that the barecontribution from the conduction band ise2/2h, theinescapable conclusion is that the remaining, observablepart of the conductivity, is exactly the disorder renormal-ization. The same issue as in the quantum Hall effectis present with regard to the detection of the anomalousHall effect, since there is always more than one surfaceand the surfaces are connected. Unlike the quantum Halleffect, the surfaces give opposite contributions to the half-quantized anomalous Hall effect.84 But since the disorderrenormalizations should be different on the two surfacesa finite total signal should exist.

In a related context, Tse and MacDonald91 studied aTI thin film weakly coupled to a ferromagnet, but fo-cused on the magneto-optical Kerr and Faraday effects,since these are less likely to be affected by unintentionalbulk carriers. Using linear-response theory, these scien-tists discovered that at low frequencies the Faraday rota-tion angle has a universal value, which is determined bythe vacuum fine structure constant, when F lies in theDirac gap for both surfaces. The Kerr rotation angle inthis regime, is again universal and equal to /2, hencethe nomenclature of a giant Kerr rotation. The physicsoverlaps significantly with the anomalous Hall effect de-scribed above leading to the half-quantized anomalousHall effect, except in this case the effect explicitly involvesboth surfaces. The effect reflects the interplay betweenthe chiral nature of the topological surface states andinterference between waves reflected off the top surfaceand waves scattered off the bottom surface. The surfaceHall response creates a splitting between the reflectedleft-handed and right-handed circularly polarized fieldsalong the transverse direction. The presence of the bot-tom topological surface is important since the first-order

partial wave reflected from the top surface undergoes onlya small rotation. The higher-order partial waves fromsubsequent scattering with the bottom surface yield acontribution that strongly suppresses the first-order par-tial wave along the incident polarization plane. As aresult, the left-handed and right-handed circularly polar-ized fields each acquire a phase of approximately /2 inopposite directions, leading to a/2 Kerr rotation.

This concludes the discussion of the anomalous Halleffect. The topic of spin transfer in magnetic topological

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insulators has also been studied recently. Nomura andNagaosa92 investigated magnetic textures in a ferromag-netic thin film deposited on a 3D TI, showing that theycan be electrically charged thanks to the proximity effectwith the Dirac surface states. They consider an insulat-ing magnet on the TI surface, with a magnetization Mpointing along. Due to the interplay of the chiral surfacestates with the proximity-induced magnetization, electric

charge and current densities are induced in the TI,

n=

MHevF

M

j =

MHevF

M

t ,

(31)

whereHis the quantized Hall conductivity. The electri-cal current, which we have established corresponds to asteady-state spin density, is the analog of the spin trans-fer torques. Spin transfer in systems with strong spin-orbit interactions was discussed in the related context offerromagnetic semiconductors in Ref. 93. In topological

insulators, precisely because of the equivalence of chargecurrents and spin densities, this electric charging of mag-netic textures can be used for electrical manipulation ofdomain walls and vortices. In the case of vortices, anelectric field breaks the vortex-antivortex symmetry onthe TI surface. In the case of domain walls, the authorsshow that an electric field can displace a domain wall ina topological insulator, much like a magnetic field canaccomplish this in an ordinary magnet.

The action of an electrical current on a magnetizationat the interface between a topological insulator and a fer-romagnet was also studied by Garate and Franz.94 Whiletheir results were similar, these authors focused on the

magnetization dynamics induced by the steady-state spinpolarization, inlcuding the possibility of current-inducedmagnetization reversal. In their language, the Hall cur-rent changes the effective anisotropy field, which in turnaffects the magnetization dynamics in the ferromagnet.Interestingly, their work shows that the magnetizationmay be flipped by the Hall charge current in tandemwith a small magnetic field of approximately 20mT.

The same authors subsequently focused on the magne-toelectric response of a disorder-free time-reversal invari-ant TI.95 By determining the spin-charge response func-tion, these researchers confirmed the observation thata uniform static in-plane magnetic field does not pro-duce a spin polarization in topological insulators. Theequilibrium RKKY interaction between localized spinsseparated by R decays as 1/R3 for kF = 0, while forkFR 1 it oscillates and decays as 1/R2. Unlike ordi-nary two-dimensional electron systems, the RKKY coef-ficients depend on the ca