PHYSICAL REVIEW B 82, 155457 ͑2010͒
Two-dimensional surface charge transport in topological insulators
Dimitrie Culcer,1 E. H. Hwang,1 Tudor D. Stanescu,2 and S. Das Sarma1 1Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA 2Department of Physics, West Virginia University, Morgantown, West Virginia 26506, USA ͑Received 1 June 2010; revised manuscript received 19 September 2010; published 29 October 2010͒ We construct a theory of charge transport by the surface states of topological insulators in three dimensions. The focus is on the experimentally relevant case when the Fermi energy F and transport scattering time satisfy F/បӷ1 but F lies below the bottom of the conduction band. Our theory is based on the spin density matrix and takes the quantum Liouville equation as its starting point. The scattering term is determined to linear order in the impurity density ni and explicitly accounts for the absence of backscattering while screening ϰ −1 is included in the random-phase approximation. The main contribution to the conductivity is ni and has different carrier density dependencies for different forms of scattering while an additional contribution is independent of ni. The dominant scattering angles can be inferred by studying the ratio of the transport time to the Bloch lifetime as a function of the Wigner-Seitz radius rs. The current generates a spin polarization that could provide a smoking-gun signature of surface state transport. We also discuss the effect on the surface states of adding metallic contacts.
DOI: 10.1103/PhysRevB.82.155457 PACS number͑s͒: 73.23.Ϫb, 73.20.At, 72.25.Dc, 71.10.Pm
I. INTRODUCTION Ref. 7͒ The quantum spin-Hall state constitutes a topological insulator in two dimensions. Soon afterward topological in- Topological insulators ͑TIs͒ are a new class of materials sulators were predicted to exist in three dimensions,8–10 in 1 that have been intensely researched in recent years. They are which four Z2 topological invariants exist, customarily la- band insulators in the bulk while conducting along the sur- beled 0, 1, 2, and 3. The Z2 invariants can be calculated 11 faces, possessing surface states with a spin texture protected easily if the system has inversion symmetry. The invariant ͑ by time-reversal symmetry. Within a certain parameter range 0 determines whether a topological insulator is weak 0 ͒ ͑ ͒ the surface states are well described by a Dirac cone, allow- =0 or strong 0 =1 . The absence of backscattering gives ing for parallels with graphene and relativistic physics, and rise to phenomena such as Klein tunneling and is believed to make strong topological insulators immune to Anderson prohibiting backscattering. Certain physical properties re- localization.1 lated to the surface states of these materials can be expressed A significant fraction of theoretical research on these ma- in terms of topologically invariant quantities and are thus terials to date has focused on their topology. A number of protected against smooth perturbations invariant under time articles have been devoted to the classification of topological reversal. For this reason topological insulators are being ex- insulators.9,10,12–14 A description of topological insulators in plored with a view toward applications, as a potential plat- terms of topological field theory has been given in Ref. 15, 2 form for quantum computation, and as rich physical systems and a general theory of topological insulators and supercon- in their own right. ductors has been formulated in Ref. 16. An effective continu- The concept of a topological insulator dates back to the ous model for the surface states of three-dimensional ͑3D͒ work of Kane and Mele, who focused on two-dimensional topological insulators has been proposed in Ref. 17. Their ͑2D͒ systems.3 These authors demonstrated that, in addition effective action in the presence of an electromagnetic field to the well-known Thouless-Kohmoto-Nightingale-den Nijs has been studied with a focus on the electric-magnetic ͑TKNN͒ invariant,4 characterizing the time-reversal breaking duality18 and analogies have been made with the topological quantum-Hall state, an additional topological invariant ex- invariants of the standard model of particle physics.19 The ists, which characterizes the surface states of time-reversal surface states of topological insulators in strong magnetic invariant insulators. In two dimensions this invariant can be fields have been investigated in Ref. 20. zero or one, the former representing ordinary insulators The interface between a topological insulator and a super- ͑equivalent to the vacuum͒ and the latter topological insula- conductor was predicted to support Majorana fermions,21 tors. The two band structures corresponding to =0 and that is, particles that are their own antiparticles. Subsequent =1 cannot be deformed into one another. The invariant , articles have focused on probing Majorana fermions,22 in- 23 which is related to the Z2 topology, counts the number of cluding a scheme employing interferometry. Recently the pairs of Dirac cones on the surfaces. Kane and Mele3 pro- proximity effect between a superconductor and a topological posed realizing a topological insulator in graphene by using insulator has been reexamined.24 In addition, other exotic the spin-orbit interaction to open a gap, yet the spin-orbit states have also been predicted, including a helical metal,25 interaction in graphene is rather small. Subsequently superconductivity and ferromagnetism induced by the prox- Bernevig et al.5 predicted that a topologically insulating imity effect,26 unconventional superconductivity,27 a quan- state, the quantum spin-Hall insulator, could be realized in tized anomalous Hall effect in topological insulators doped HgCdTe quantum wells. This state was observed shortly after with magnetic impurities,28 and a new critical state attributed its prediction by König et al.6 ͑for a review of this effect see to the Coulomb interaction.29 Further expected effects in-
1098-0121/2010/82͑15͒/155457͑17͒ 155457-1 ©2010 The American Physical Society CULCER et al. PHYSICAL REVIEW B 82, 155457 ͑2010͒ clude the possibility of Cooper-pair injection at the interface tected surface states. Roushan et al.66 have imaged the sur- 30 between a topological insulator and a superconductor, face states of Bi0.92Sb0.08 using STM and spin-resolved the inverse spin-galvanic effect at the interface between ARPES demonstrating that, as expected theoretically, in the a topological insulator and a ferromagnet,31 giant absence of spin-independent ͑time-reversal invariant͒ disor- magnetoresistance,32 and a giant Kerr effect.33 der, scattering between states of opposite momenta is Variations have also appeared on the theme of topological strongly suppressed. Gomes et al.67 investigated the ͑111͒ insulators. Given that superconductors are also gapped, topo- surface of Sb, which displays a topological metal phase, logical superconductors have been predicted to exist.1,14,34,35 and found a similar suppression of backscattering. Zhang Developments in different directions include the topological et al.68 and Alpichshev et al.69 have observed the suppres- 36–38 39 Anderson insulator, the topological Mott insulator, the sion of backscattering in Bi2Te3. Theories have recently been topological magnetic insulator,40 the fractional topological formulated to describe the quasiparticle interference seen insulator,41,42 the topological Kondo insulator,43 as well as a in STM experiments, accounting for the absence of host of topological phases arising from quadratic, rather than backscattering.70–72 Related developments have seen the im- linear, band crossings.44 Topological phases in transition- aging of a surface bound state using scanning tunneling metal-based strongly correlated systems have been the sub- microscopy.73 These experimental studies provide evidence ject of Refs. 45 and 46, and a theory relating topological of time-reversal symmetry protection of the chiral surface insulators to Mott physics has been proposed in Ref. 47. states. Following the discovery of the quantum spin-Hall state, Despite the success of photoemission and scanning tun- several materials were predicted to be topological insulators neling spectroscopy in identifying chiral surface states, sig- 48,49 in three dimensions. The first was the alloy Bi1−xSbx, natures of the surface Dirac cone have not yet been observed 50 followed by Bi2Se3,Bi2Te3, and Sb2Te3. In particular, in transport. Simply put, the band structure of topological Bi2Se3 and Bi2Te3 have long been known from thermoelec- insulators can be visualized as a band insulator with a Dirac tric transport as displaying sizable Peltier and Seebeck ef- cone within the bulk gap, and to access this cone one needs fects, and their high quality has ensured their place at the to ensure the chemical potential lies below the bottom of the forefront of experimental attention.1 Initial predictions of the conduction band. Given that the static dielectric constants of existence of chiral surface states were confirmed by first- materials under investigation are extremely large, approxi- 51 principles studies of Bi2Se3,Bi2Te3, and Sb2Te3. On an mately 100 in Bi2Se3 and 200 in Bi2Te3, gating in order to interesting related note, it has recently been proposed that bring the chemical potential down is challenging. Therefore, topologically insulating states can be realized in cold in all materials studied to date, residual conduction from the atoms,52 though these lie beyond the scope of the present bulk exists due to unintentional doping. One potential way work. forward was opened by Hor et al.,74 who demonstrated that After a timid start, experimental progress on the materials Ca doping ͑Ϸ1% substituting for Bi͒ brings the Fermi en- above has skyrocketed. Hsieh et al.,53 using angle-resolved ergy into the valence band, followed by the demonstration by photoemission spectroscopy ͑ARPES͒, were the first to Hsieh et al.54 that Ca doping can be used to tune the Fermi 75 observe the Dirac conelike surface states of Bi1−xSbx, energy so that it lies in the gap. Checkelsky et al. per- with xϷ0.1. Following that, the same group used spin- formed transport experiments on Ca-doped samples noting resolved ARPES to measure the spin polarization of the sur- an increase in the resistivity but concluded that surface state face states, demonstrating the correlation between spin and conduction alone could not be responsible for the smallness momentum.54 Shortly afterward experiments identified the of the resistivities observed at low temperature. Eto et al.76 55 56,57 57 ͑ ͒ Dirac cones of Bi2Se3, Bi2Te3, and Sb2Te3. Recently reported Shubnikov-de Haas SdH oscillations due only to 58 Bi2Se3 nanowires and nanoribbons have also been grown the 3D Fermi surface in Bi2Se3 with an electron density of and experiments have begun to investigate the transition 5ϫ1018 cm−3 while Analytis et al.77 used ARPES and SdH between the three- and two-dimensional phases in this oscillations to probe Bi2Se3. The two methods were found to 59 material. Bi2Se3 has also been studied in a magnetic field, agree for bulk carriers, but while ARPES identified Dirac revealing a magnetoelectric coupling60 and an oscillatory conelike surface states, even at number densities on the order angular dependence of the magnetoresistance.61 Sample of 1017 cm−3 SdH oscillations identified only a bulk Fermi growth dynamics of Bi2Te3 by means of molecular beam surface, and the surface states were not seen in transport. A epitaxy were monitored by reflection high-energy series of recent transport experiments claim to have observed 62 electron-diffraction. Doping Bi2Te3 with Mn has been separate signatures of bulk and surface conduction. Checkel- shown to result in the onset of ferromagnetism63 while dop- sky et al.78 performed weak localization and universal con- 64,65 ing Bi2Se3 with Cu leads to superconductivity. ductance fluctuation measurements, claiming to have ob- Considerable efforts have been devoted to scanning tun- served surface conduction, although the chemical potential neling microscopy ͑STM͒ and spectroscopy, which enable still lay in the conduction band. Steinberg et al.79 reported the study of quasiparticle scattering. Scattering off surface signatures of ambipolar transport analogous to graphene. defects, in which the initial state interferes with the final However, in this work also, two contributions to transport scattered state, results in a standing-wave interference pattern were found, the bulk contribution was subtracted so as to with a spatial modulation determined by the momentum obtain the surface contribution, and the surface effect was transfer during scattering. These manifest themselves as os- observed only for top gate voltage sweeps, not back gate cillations of the local density of states in real space, which sweeps. Butch et al.80 carried out SdH, ordinary Hall effect have been seen in several materials with topologically pro- and reflectivity measurements and concluded that the ob-
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served signals could be ascribed solely to bulk states. No mechanisms the ratio of the single-particle scattering time to experimental group has reported SdH oscillations due to the the transport scattering time has a qualitatively different de- surface states, which are expected at a very different pendence on the Wigner-Seitz radius rs, which parametrizes frequency.80 The above presentation makes it plain that, de- the relative strength of the kinetic-energy and electron- spite the presence of the Dirac cone, currently no experimen- electron interactions. We study these aspects of the transport tal group has produced a true topological insulator. Conse- problem in detail, covering the possible situations that can be quently, all existing topological insulator/metal systems are encountered experimentally with currently available samples in practice either heavily bulk-doped materials or thin films and technology. Finally, we consider the effect on the topo- 59 ͑ ͒ of a few monolayers thus, by definition, not 3D. logical surface states of adding metal contacts, as would be In an intriguing twist, two recent works have reported the case in experiment. STM imaging of the Landau levels of Bi Se .81,82 Cheng et 2 3 Our work shows that the suppression of backscattering al.81 claim to have observed Landau levels n analogous to seen in experimental work indicates protection against local- graphene, with energies ϰͱnB but the expected spin and val- ization resulting from backscattering, rather than protection ley degeneracy of one, yet puzzling features such as the ab- sence of Landau levels below the Fermi energy and enhanced against resistive scattering in general. The conductivity due oscillations near the conduction band edge have not been to the surface states is determined by scattering of carriers by resolved, and some arbitrariness is involved in extracting the disorder and defects, the density of which is in turn deter- Landau-level filling factor . Hanaguri et al.82 report similar mined by sample quality. Strong scattering will lead to low findings in the same material with analogous features in the mobility and the mobility is not a topologically protected Landau-level spectrum. Following the pattern set by the im- quantity. To see surface transport one therefore requires very aging of the surface Dirac cone, the Landau levels observed clean samples. in STM studies have not been seen in transport. These last Several features set topological insulators apart from works illustrate both the enormous potential of the field and graphene: the twofold valley degeneracy of graphene is not the challenges to be overcome experimentally. present in topological insulators, while the Hamiltonian is a The focus of experimental research has shifted to the function of the real spin, rather than pseudospin. This implies separation of bulk conduction from surface conduction, that spin dynamics will be qualitatively very different from hence it is necessary to know what type of contributions to graphene. Furthermore, the Dirac Hamiltonian is in fact a expect from the surface conduction. The only transport theo- Rashba Hamiltonian expressed in terms of a rotated spin. ries we are aware of have been Ref. 83, which focused on Despite the apparent similarities, the study of topological magnetotransport in the impurity-free limit, and Ref. 84, insulators is thus not a simple matter of translating results which focused on spin-charge coupling. A comprehensive known from graphene. Due to the dominant spin-orbit inter- theory of transport in topological insulators has not been action it is also very different from ordinary spin-orbit semi- formulated to date. This article seeks to remedy the situation. conductors. We begin with the quantum Liouville equation, deriving a The outline of this paper is as follows. In Sec. II we kinetic equation for the density matrix of topological insula- introduce the effective Hamiltonian for the surface states of tors including the full scattering term to linear order in the topological insulators and briefly discuss its structure. Next, impurity density. Peculiar properties of topological insula- in Sec. III we derive a kinetic equation for the spin density tors, such as the absence of backscattering ͑which leads to matrix of topological insulators, including the full scattering Klein tunneling and other phenomena͒, are built into our term in the presence of an arbitrary elastic scattering poten- theory. We determine the polarization function of topological tial to linear order in the impurity density. This equation is insulators. By solving the kinetic equation we determine all solved in Sec. IV, yielding the contributions to orders one contributions to the conductivity to orders one and zero in and zero in the transport scattering time , and used to de- the transport scattering time . We find that the scalar part of termine the charge current. Section V discusses our results in the Hamiltonian makes an important contribution to the con- the context of experiments, focusing on the electron density ductivity, which has a different carrier density dependence and impurity density dependence of the conductivity, ways to than the spin-dependent part, and may be a factor in distin- determine the dominant angles in impurity scattering, and the guishing the surface conduction from the bulk conduction. effect of the contacts on the surface states. We end with a We consider charged impurity as well as short-range surface summary and conclusions. roughness scattering and determine the density dependence of the conductivity when either of these is dominant. Most II. EFFECTIVE HAMILTONIAN FOR TOPOLOGICAL importantly, we find that the electrical current generates a INSULATORS spin polarization in the in-plane direction perpendicular to the field, providing a unique signature of surface transport. The effective Hamiltonian describing the surface states of Moreover, the ratio of the single-particle scattering time to topological insulators can be written in the form the transport scattering time provides important information H = Dk2 + A · k. ͑1͒ on the dominant angles in scattering, which in turn provide 0k ͑ ͒ information on the dominant scattering mechanisms. At zero It is understood that k= kx ,ky . Near k=0, when the spec- temperature, which constitutes the focus of our work, we trum is accurately described by a Dirac cone, the scalar term expect a competition between charged impurity scattering is overwhelmed by the spin-dependent term and can be and short-range scattering. For each of these two scattering safely neglected, the remaining Hamiltonian being similar to
155457-3 CULCER et al. PHYSICAL REVIEW B 82, 155457 ͑2010͒
͗ ͉͉ ͘ that of graphene, with the exception that represents the = ks ˆ kЈsЈ with the understanding that kkЈ is a matrix in ͑ ͒ true rotated electron spin. On the other hand, at finite dop- spin space. The terms Hˆ and Hˆ are diagonal in wave vector 11 12 −2 0 E ing and at usual electron densities of 10 –10 cm the but off-diagonal in spin while for elastic scattering in scalar term is no longer a small perturbation and should be Ј the first Born approximation Uss =U ␦ . The impuri- taken into account. The scalar- and spin-dependent terms in kkЈ kkЈ ssЈ ties are assumed uncorrelated and the average of the Hamiltonian are of the same magnitude when DkF =A, Ϸ 9 −1 ͗ ͉ ˆ ͉ Ј Ј͗͘ Ј Ј͉ ˆ ͉ ͘ corresponding to kF 10 m in Bi2Se3, that is, a density of ks U k s k s U ks over impurity configurations is 14 −2 ͑ ͉¯ ͉2␦ ͒/ 10 cm , which is higher than the realistic densities in ni UkkЈ ssЈ V, where ni is the impurity density, V the crys- transport experiments. Consequently, in this article the terms ¯ tal volume, and UkkЈ the matrix element of the potential of a ϰD will be treated as a perturbation. The eigenenergies are 2 single impurity. denoted by ⑀ =Dk +Ak with =Ϯ. We do not take into ␦ The density matrix kkЈ is written as kkЈ= fk kkЈ+gkkЈ, account small anisotropy terms in the scalar part of the ͑ ϰ␦ ͒ where fk is diagonal in wave vector i.e. fkkЈ kkЈ while Hamiltonian, such as those discussed in Ref. 50. Terms cubic g is off-diagonal in wave vector ͑i.e., kkЈ always in in k in the spin-orbit interaction are, in principle, also kkЈ g ͒. The quantity of interest in determining the charge cur- present, in turn making the energy dispersion anisotropic, but kkЈ rent is f since the current operator is diagonal in wave vec- they are much smaller than the k-linear and quadratic terms k tor. We therefore derive an effective equation for this quan- in Eq. ͑1͒ and are not expected to contribute significantly to tity by first breaking down the quantum Liouville equation quantities discussed in this work. into There are contributions to the charge current from the scalar and spin parts of the density matrix. The current op- df i i i k + ͓H , f ͔ =− ͓HE, f ͔ − ͓Uˆ ,gˆ͔ , ͑4a͒ erator is given by dt ប 0k k ប k k ប kk e j =− ͑2Dk + A͒. ͑2͒ dg i i ប kkЈ ͓ ˆ ͔ ͓ ˆ ˆ ͔ ͑ ͒ + H,gˆ kkЈ =− U, f + gˆ kkЈ, 4b dt ប ប Unlike graphene, where the effective Dirac Hamiltonian is valid even at large electron densities and the current operator and solving Eq. ͑4b͒ to first order in Uˆ , is independent of k for all ranges of wave vector relevant to ϱ i ˆ ˆ transport, in topological insulators the scalar term in the g =− ͵ dtЈe−iHtЈ/ប͓Uˆ ,ˆf͑t − tЈ͔͒eiHtЈ/ប͉ . ͑5͒ kkЈ ប kkЈ Hamiltonian quadratic in k also contributes to the current, 0 and will yield a different number-density dependence. Transport experiments on topological insulators seek to The details of this derivation have been included in the Ap- distinguish the conduction due to the surface states from that pendix. We are interested in variations which are slow on the due to the bulk states. As mentioned in Sec. I, it has proven scale of the momentum relaxation time, consequently we do challenging to lower the chemical potential beneath the bot- not take into account memory effects and ˆf͑t−tЈ͒Ϸˆf͑t͒,so tom of the conduction band so that the materials studied at that fk obeys present are not strictly speaking insulators. It is necessary for df i i k ͓ ͔ ˆ͑ ͒ ͓ E ͔ ͑ ͒ F to be below the bulk conduction band, so that one can be + H0k, fk + J fk =− Hk , fk , 6a certain that there is only surface conduction, and at the same dt ប ប /បӷ time the requirement that F 1 is necessary for the ki- ϱ netic equation formalism to be applicable. These assump- 1 ˆ /ប ˆ /ប Jˆ͑f ͒ = ͵ dtЈ͓Uˆ ,e−iHtЈ ͓Uˆ ,ˆf͑t͔͒eiHtЈ ͔ . ͑6b͒ k ប2 kk tions will be made in our work. 0 The integral in Eq. ͑6b͒, which represents the scattering III. KINETIC EQUATION term, is performed by inserting a regularizing factor e−tЈ and → ͉ ͉ϰ We wish to derive a kinetic equation for the system driven letting 0 in the end. We consider a potential UkkЈ 1, by an electric field in the presence of random, uncorrelated which does not have spin dependence, and carry out an av- impurities. To this end we begin with the quantum Liouville erage over impurity configurations, keeping the terms to lin- equation for the density operator ˆ, ear order in the impurity density ni. The scattering term has the form dˆ i + ͓Hˆ ,ˆ͔ =0, ͑3͒ n d2kЈ ប ˆ͑ ͒ i ͵ ͉¯ ͉2 dt J fk = 2 lim 2 UkkЈ ប →0 ͑2͒ ˆ ˆ ˆ ˆ where the full Hamiltonian H=H0 +HE +U, the band Hamil- ϱ Ј Ј/ប Ј/ប ˆ ͑ ͒ ˆ ϫ͵ Ј − t ͕ −iHkЈt ͑ Ј͒ iHkt ͖ tonian H0 is defined in Eq. 1 , HE =eE·rˆ represents the in- dt e e fk − fk e + H.c. . teraction with external fields, rˆ is the position operator, and 0 Uˆ is the impurity potential. We consider a set of time- ͑7͒ ͕͉ ͖͘ independent states ks , where k indexes the wave vector For this term to be cast in a useful form one needs to com- ϵssЈ and s the spin. The matrix elements of ˆ are kkЈ kkЈ plete the time integral and analyze the spin structure of the
155457-4 TWO-DIMENSIONAL SURFACE CHARGE TRANSPORT IN… PHYSICAL REVIEW B 82, 155457 ͑2010͒ scattering operator Jˆ͑f ͒, which is done below. n d2kЈ k ˆ͑ ͒ i ͵ ͉¯ ͉2͑ ͒ ͓͑ ˆ ˆЈ͒ J S = UkkЈ Sk − SkЈ · 1−k · k 4ប ͑2͒2 A. Scattering term + ͑kˆ · ͒kˆЈ + kˆ͑kˆЈ · ͔͓͒␦͑⑀Ј − ⑀ ͒ + ␦͑⑀Ј − ⑀ ͔͒ ϫ + + − − The k-diagonal part of the density matrix fk isa2 2 n d2kЈ Hermitian matrix, which is decomposed into a scalar part i ͵ ͉¯ ͉2͑ ͒ ͑ˆ ˆЈ͓͒␦͑⑀Ј ⑀ ͒ + 2 UkkЈ Sk − SkЈ · k + k + − + and a spin-dependent part. We write fk =nk1+Sk, where nk 4ប ͑2͒ represents the scalar part and 1 is the identity matrix in two ␦͑⑀Ј ⑀ ͔͒ ͑ ͒ ϫ − − − − . 10 dimensions. The component Sk is itself a 2 2 Hermitian matrix, which represents the spin-dependent part of the den- sity matrix and is written purely in terms of the Pauli The details of this calculation are contained in the Appendix. The energy ␦ functions ␦͑⑀ϮЈ −⑀Ϯ͒=͓1/͑AϮ2Dk͔͒␦͑kЈ−k͒. matrices. Thus, the matrix Sk can be expressed as Sk 1 1 The ␦ functions of ⑀ are needed in the scattering term in = S ·ϵ S . With this notation, the scattering term is in − 2 k 2 ki i order to ensure agreement with Boltzmann transport. This turn decomposed into scalar- and spin-dependent parts, fact is already seen for a Dirac cone dispersion in graphene 2 ϱ where the expression for the conductivity found in Ref. 85 ni d kЈ Ј Ј/ប Ј/ប Jˆ͑f͒ = ͵ ͉U¯ ͉2͑n − n ͒͵ dtЈe− t e−iHkЈt eiHkt ប2 ͑ ͒2 kkЈ k kЈ using the density-matrix formalism agrees with the Boltz- 2 0 mann transport formula of Ref. 86 ͑the definition of differs ͒ n d2kЈ by a factor of 2 in these two references . The necessity of + H.c. + i ͵ ͉U¯ ͉2͑S − S ͒ · ⑀ ប2 ͑ ͒2 kkЈ k kЈ keeping the − terms is a result of Zitterbewegung: the two 2 2 branches = Ϯ1 are mixed, thus scattering of an electron ϱ requires conservation of ⑀ as well as ⑀ . In the equivalent −tЈ −iH tЈ/ប iH tЈ/ប + − ϫ͵ dtЈe e kЈ e k + H.c. ͑8͒ kinetic equation for graphene there is no coupling of the 0 scalar and spin distributions because of the particle-hole symmetry inherent in the Dirac Hamiltonian. In topological We assume electron doping, which implies that the chemical 2 potential is in the conduction band ⑀ . For the scalar part n insulators the Dk term breaks particle-hole symmetry and + k gives rise to a small coupling of the scalar and spin distribu- and spin-dependent part Sk of the density matrix the above results in tions. Later below we quantify this coupling by an effective →ϱ → time D, which as D 0. iHˆ tЈ/ប iHˆ tЈ/ប We find a series of scattering terms coupling the scalar e− kЈ e k + H.c. and spin distributions. Let ␥ represent the angle between kˆ ប → ͑1+kˆ · kˆЈ͓͒␦͑⑀Ј − ⑀ ͒ + ␦͑⑀Ј − ⑀ ͔͒ and kˆЈ, and the unit vector ˆ in polar coordinates represent 2 + + − − the direction perpendicular to kˆ, that is the tangential direc- ប ˆ ˆЈ ˆ ˆ Ј + · ͑kˆ + kˆЈ͓͒␦͑⑀Ј − ⑀ ͒ − ␦͑⑀Ј − ⑀ ͔͒, tion. The unit vectors k, k , , and are related by the 2 + + − − transformations,
ˆ Ј/ប ˆ Ј/ប ˆЈ ˆ ␥ ˆ ␥ e−iHkЈt eiHkt + H.c. k = k cos + sin , ប → ͓͑1−kˆ · kˆЈ͒ + ͑kˆ · ͒kˆЈ + kˆ͑kˆЈ · ͔͓͒␦͑⑀Ј − ⑀ ͒ 2 + + ˆ Ј =−kˆ sin ␥ + ˆ cos ␥, ប ␦͑⑀Ј ⑀ ͔͒ ͑ˆ ˆЈ͓͒␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ + − − − + k + k + − + − − − − , 2 kˆ = kˆЈ cos ␥ − ˆ Јsin ␥. ͑11͒ ͑9͒ We decompose the matrix Sk =Skʈ +SkЌ and write those two ˆ ͑ / ͒ ͑ / ͒ where k is a unit vector in the direction of k. We emphasize parts in turn as Skʈ = 1 2 skʈ kʈ and SkЌ = 1 2 skЌ kЌ. The ˆ that there is no backscattering so the physics of topological small skʈ and skЌ are scalars and are given by skʈ =Sk ·k and insulators is built into the kinetic equation from the begin- ˆ ˆ ˆ s Ќ =S ·, and similarly ʈ =·k and Ќ =·. We intro- ning. Collecting all terms, we obtain the scattering term as k k k k duce projection operators Pn, Pʈ, and PЌ onto the scalar part, ˆ͑ ͒ ˆ͑ ͒ ˆ͑ ͒ J f =J n +J S with kʈ and kЌ, respectively. We will need the following pro- jections of the scattering term acting on the spin-dependent n d2kЈ ˆ͑ ͒ i ͵ ͉¯ ͉2͑ ͒͑ ˆ ˆЈ͒ part of the density matrix: J n = UkkЈ nk − nkЈ 1+k · k 2ប ͑2͒2
2 Ј kni 1 1 ni d k ˆ͑ ͒ ͫ ͬ ϫ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ͵ ͉¯ ͉2 PʈJ Skʈ = + − + + − − + UkkЈ ប ͑ ͒ ͑ ͒ + − 2ប ͑2͒2 16 A +2Dk A −2Dk ϫ͑ ͒ ͑ˆ ˆЈ͓͒␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ϫ͵ Ј͉¯ ͉2͑ ͒͑ ␥͒ nk − nkЈ · k + k + − + − − − − , d UkkЈ skʈ − skЈʈ 1 + cos kʈ,
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kn 1 1 As a result, in topological insulators the matrix element ˆ͑ ͒ i ͫ ͬ PЌJ Skʈ = + ¯ 16ប ͑A +2Dk͒ ͑A −2Dk͒ UkkЈ of a screened Coulomb potential between plane waves is given by ϫ Ј͉¯ ͉2͑ ͒ ␥ ͵ d UkkЈ skʈ − skЈʈ sin kЌ, Ze2 1 U¯ = , kkЈ ⑀ ⑀ ͉ ͉ 2 0 r k − kЈ + kTF kn 1 1 ˆ͑ ͒ i ͫ ͬ PʈJ SkЌ = + Z2e4 1 2 W 16ប ͑A +2Dk͒ ͑A −2Dk͒ ͉¯ ͉2 ͩ ͪ ϵ UkkЈ = , 4⑀2⑀2 ͉k − kЈ͉ + k ␥ k 2 0 r TF ͩsin + TF ͪ ϫ͵ Ј͉¯ ͉2͑ ͒ ␥ ͑ ͒ d UkkЈ skЌ + skЈЌ sin kʈ, 12 2 2kF ͑16͒ and PЌJˆ͑S Ќ͒ will be deferred to a later time. Furthermore, k where Z is the ionic charge ͑which we will assume to be Z the scattering terms coupling the scalar and spin distributions =1 below in this work͒ and k is the Thomas-Fermi wave are TF vector. The parallel projection of the scattering term becomes n d2kЈ ˆ ͑ ͒ i ͵ ͉¯ ͉2͑ ͓͒ ͑ ␥͒ kn W 1 1 J → n = U n − n ʈ 1 + cos i n S k 2 kkЈ k kЈ k ˆ͑ ͒ ͫ ͬ 2ប ͑2͒ PʈJ Sʈ = + 16ប ͑A +2Dk͒ ͑A −2Dk͒ + Ќ sin ␥͔͓␦͑⑀Ј − ⑀ ͒ − ␦͑⑀Ј − ⑀ ͔͒, k + + − − 1 + cos ␥ ϫ͵ Ј ͑ Ј͒ d sʈ − sʈ ʈ ␥ k 2 k n d2kЈ ͩ TF ͪ ˆ ͑ ͒ i ͵ ͉¯ ͉2͓͑ ͒͑ ␥͒ sin + JS→n Sk = UkkЈ skʈ − skЈʈ 1 + cos 2 2k 4ប ͑2͒2 F kni 1 1 + ͑s Ќ + s Ќ͒sin ␥͔͓␦͑⑀Ј − ⑀ ͒ − ␦͑⑀Ј − ⑀ ͔͒. ͫ ͬ k kЈ + − = + + − 8ប ͑A +2Dk͒ ͑A −2Dk͒ ͑13͒ ͑ ␥͒ ϫ͵ Ј͑␥͒͑ Ј͒ Notice the factors of 1+cos which prohibit backscatter- d sʈ − sʈ kʈ, ing ͑and lead to Klein tunneling͒. ␥ B. Polarization function and effective scattering potential cos2 2 The above form of the scattering term is valid for any ͑␥͒ = . ͑17͒ ␥ k 2 scattering potential, as long as scattering is elastic. To make ͩsin + TF ͪ our analysis more specific we consider a screened Coulomb 2 2kF potential and evaluate the screening function in the random- We expand ͑␥͒ and s ʈ͑͒ in Fourier harmonics phase approximation. In this approximation the polarization k function is obtained by summing the lowest bubble diagram ͑␥͒ = ͚ eim␥, 87 m and takes the form m
1 f0k − f0kЈЈ 1+Ј cos ␥ ⌸͑ ͒ ͩ ͪ ͑͒ im q, =− ͚ , skʈ = ͚ sʈme , A + − + i 2 kЈ k kЈ Ј m ͑14͒ kn 1 1 ϵ ͑ ͒ ˆ͑ ͒ i ͫ ͬ PʈJ Sʈ = + where f0k f0 k is the equilibrium Fermi distribution ប ͑ ͒ ͑ ͒ function. The static dielectric function, of interest to us in 4 A +2Dk A −2Dk this work, can be written as ⑀͑q͒=1+v͑q͒⌸͑q͒, where v͑q͒ −im ϫ͚ ͑ − ͒sʈ e ʈ, ͑18͒ 2 /͑ ⑀ ⑀ ͒ ⌸͑ ͒ 0 m m k =e 2 0 rq . To determine q , we assume T=0 and use m the Dirac cone approximation as in Ref. 87. This approxima- ͑Ј͒ →Ј tion is justified by the fact that we are working in a regime in and to obtain the expansion of skʈ replace . The which T/T Ӷ1 with T the Fermi temperature. The results absence of backscattering, characteristic of topological insu- F F ͑␥͒ ͑ ͒ obtained are in fact correct to linear order in D.AtT=0 for lators, is contained in the function . Equation 17 shows ͑ ␥͒ charged impurity scattering the long-wavelength limit of the that this function contains a factor of 1+cos which sup- ␥ dielectric function is87 presses scattering for = . e2 k ⑀͑q͒ =1+ ͩ F ͪ, ͑15͒ ⑀ ⑀ IV. SOLUTION OF THE KINETIC EQUATION 4 0 rA q The Hamiltonian Hˆ describing the interaction with the yielding the Thomas-Fermi wave vector as kTF E 2 /͑ ⑀ ⑀ ͒ ˆ =e kF 4 0 rA . electric field is given by HE =eE·rˆ with rˆ the position opera-
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tor. It then follows straightforwardly that the kinetic equation nk and skʈ, the coupling to skЌ can be neglected. We are then takes the form left with two coupled equations for nk and skʈ, which are written as df i k ͓ ͔ ˆ͑ ͒ D ͑ ͒ + Hk, fk + J fk = . 19 dn dt ប k ˆ ͑ ͒ ˆ ͑ ͒ D + Jn→n nk + JS→n Sk = n, dtץ D eE 0k k , whereץ · The driving term in the kinetic equation is = ប 0k represents the equilibrium density matrix, given by 0k 1 ds ʈ 1 k ˆ ͑ ͒ ˆ ͑ ͒ ͑ ͒ =͑f + f ͒ +͑1/2͒͑f − f ͒·kˆ. The first term repre- kʈ + PʈJS→S Sʈ + PʈJn→S nk = dkʈ kʈ. 23 0k+ 0k− 1 0k+ 0k− 2 dt 2 sents the charge density while the second represents the spin D density. We decompose the driving term into a scalar part n We must carry out the projections in order to determine the D and a spin-dependent part s,as scattering terms above. In the equations involving Pʈ we have eliminated kʈ since the equations for Sk have been cast ץ ץ ˆ eE · k f0k+ f0k− in terms of equations for s . To first order in D, the required D = ͩ + ͪ1, k k scattering terms can be written asץ kץ n ប
kniW ,͒ Jˆ͑n͒ → ͵ dЈ͑␥͒͑n − n ץ ץ ˆ 1 eE · k f0k+ f0k− ប k kЈ D = · kˆ ͩ − ͪ 4 A kץ kץ s 2 ប kn WDk 1 eE · ˆ ˆ ͑ ͒ → i ͵ Ј͑␥͒͑ ͒ ˆ ͑ ͒ ͑ ͒ JS→n Sk − 2 d skʈ − skЈʈ , + · f0k+ − f0k− . 20 4បA 2 បk
The latter is further decomposed into a part parallel to the kn WDk ˆ ͑ ͒ → i ͵ Ј͑␥͒͑ ͒ Hamiltonian Dʈ and a part perpendicular to it DЌ, PʈJn→S nk − d nk − nkЈ kʈ, 2បA2 f 1 ץ f ץ ˆeE · k 1 D ͩ 0k+ 0k− ͪ ʈ = − kʈ = dkʈ kʈ, kn W ͒ ͑ k 2 ˆ͑ ͒ → i ͵ Ј͑␥͒͑ ͒ץ kץ ប 2 PʈJ Skʈ d skʈ − skЈʈ kʈ. 24 8បA 1 eE · ˆ 1 The scattering terms have exactly the same structure. We DЌ = ͑f − f ͒ Ќ = d Ќ Ќ. ͑21͒ 2 បk 0k+ 0k− k 2 k k proceed to solve the equations for nk and skʈ. We note that the right-hand side ͑RHS͒ on both equations in this case has Evidently, since we consider electron doping, f0k− =0. Nev- only Fourier component 1, so we can make the Ansatz nk −i i −i i ertheless we retain f0k− for completeness since the term =n1e +n−1e and skʈ =sʈ1e +sʈ−1e . Observing in addi- ͑ ͒ D f0k+ − f0k− will be shown to give rise to a contribution to the tion that n =dkʈ allows us to write the coupled equations in conductivity singular at the origin. the steady state as We solve the kinetic equation. The first step is to divide it 2n s ʈ into three equations: one for the scalar part, one for the part k − k = D , n parallel to the Hamiltonian and one for the part orthogonal to D the Hamiltonian, D dn 2nk skʈ n k ˆ͑ ͒ D ͑ ͒ − + = , ͑25͒ + PnJ fk = n, 22a dt D 2 where , the transport relaxation time, is given by the expres- dSkʈ ˆ sion + PʈJ͑f ͒ = Dʈ, ͑22b͒ dt k 1 kn W = i ͑ − ͒͑26͒ 2បA 0 1 dSkЌ i + ͓H ,S Ќ͔ + PЌJˆ͑f ͒ = DЌ. ͑22c͒ dt ប k k k and we have also introduced We search for the solution as an expansion in the small pa- 1 kn WDk i ͑ ͒ ͑ ͒ ប/͑ ͒ ϰ −1 = 0 − 1 , 27 rameter F , where ni , to be defined below, repre- បA2 sents the transport relaxation time. We label the orders in the D ͑m͒ expansion by superscripts with, e.g., Sk representing Sk as a measure of the coupling of the scalar and spin distribu- evaluated to order m. From these equations it is evident that tions by scattering. The equations are trivial to solve and we /ប the expansions of nk and skʈ begin at order F , in other obtain in the steady state, ͓ប/͑ ͔͒͑−1͒ words F , whereas skЌ begins at order D ͓ប/͑ ͔͒͑0͒ n F , in other words the leading term in it is indepen- n = ͑1+/2 ͒, /ប k D dent of F . Therefore, in determining the leading terms in 2
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D e2 Ak 2 Dk2 n ͑ / ͒ ͑ ͒ Boltz ͫ F ͩ ͪ F ͬ ͑ ͒ ʈ = 1+ + . 35 sk = 1+2 D . 28 xx ប ប 2 h 4 D 4 / The equation is valid to first order in D. Note that, as D We emphasize again that this result tends to the result for → →ϱ 85,86 → ͑ 0, D and we recover the results for the bare Dirac graphene when D 0 note that the definition of in cone. Refs. 85 and 86 differs by a factor of 2͒. The contribution to ͑0͒ Next we examine the term SkЌ. The kinetic equation for the conductivity independent of is S͑0͒ kЌ is 2 0Ќ e 1 1 ͑ ͒ = ͩ ͪ limͫ − ͬ. dS 0 xx ͑+ប/2͒ ͑−ប/2͒ kЌ i ͑0͒ ˆ ͑−1͒ ˆ ͑−1͒ 8h →0 1+e 1+e + ͓H ,S ͔ = dЌ − PЌJ͓n ʈ ͔ − PЌJ͓S ʈ ͔. ប k kЌ k k dt ͑36͒ ͑29͒ The latter term however is zero for finite , which is the case The projections of the scattering terms do not contribute to we are considering. Thus, the leading order term in the con- ϰ /បӷ the conductivity. The bare source term dЌ is ductivity is and in the limit F 1 there is no term of order 0. eE · ˆ dЌ = ͑f − f ͒. ͑30͒ បk 0k+ 0k− V. IMPLICATIONS FOR TRANSPORT EXPERIMENTS The result for S͑0͒ is kЌ In this section, as in the calculation of screening above, it ϱ eE · ˆ is assumed the system has a low enough density that the ͑0͒ ͑ ͒͵ −t SkЌ = f0k+ − f0k− dte Dirac-cone approximation is applicable, which for Bi2Se3 2បk Ͻ 8 −1 0 implies kF 10 m . Below we use the terminology high 2Akt 2Akt density as meaning that the density is high enough that the ϫͩ ˆ ϫ ˆ ͪ ͑ ͒ 2 ͑ Ќ cos + · k sin . 31 Dk k ប ប term in the Hamiltonian has a noticeable effect meaning nϷ1013 cm−2 and higher͒, with low density reserved for ϰ ˆ ϫˆ situations in which this term is negligible. Nevertheless we The part ·k averages to zero over directions in mo- mentum space and is not considered any further. Integrating retain the scattering time D, which allows us to examine the the cosine over time gives correction to the low-density expressions when these terms become noticeable. ˆ ͑0͒ eE · S Ќ = ͑f − f ͒ Ќ. ͑32͒ k 2បk 0k+ 0k− 4A2k2 k 2 A. Density dependence 2 + ប The conductivity is a function of two main parameters accessible experimentally: the carrier number density and the Current operator and conductivity impurity density/scattering time. The dependence of the con- The current-density operator is decomposed into a parallel ductivity on the carrier number density arises through its and a perpendicular part, the electric field is assumed ʈxˆ and direct dependence on kF and through its dependence on n through . In terms of the number density the Fermi wave we are looking for the diagonal conductivity 2 vector is given by kF =4 n. The transport time is given by 1 1 Eq. ͑26͒ and W is defined in Eq. ͑16͒. The number-density ͑ ͒ ͑ ͒ ji k = ji,n + ji,ʈ ʈ + ji,Ќ Ќ. 33 2 2 dependence of depends on the dominant form of scattering and whether the number density is high or low as defined The three contributions to the current operator, representing above. For charged impurity scattering Wϰk−2 so that ϰk the scalar part, the part parallel to the Hamiltonian and the and the two terms in the conductivity are ϰn and n3/2. At the part orthogonal to the Hamiltonian, are given explicitly by same time, in a two-dimensional system surface roughness 2eDk gives rise to short-range scattering as discussed in Ref. 88. j =− cos , For short-range scattering of this type W is a constant and x,n ប ϰk with the two terms in the conductivity being a constant and n1/2. These results are summarized in Table I. At high 2eA ϰ density the term Dk becomes noticeable through the scat- jx,ʈ =− cos , ប tering time D. Since D has an extra factor of kF, retaining terms to first order in D gives us an additional term ϰDn1/2, resulting in a further contribution to the conductivity of the 2eA / j Ќ = sin . ͑34͒ ϰ 3 2 ͑ ͒ x, ប form Dn , as is manifest in Eq. 35 . Taking this fact into account, in order to keep the expressions simple, in Table I Using this form for the current operator it is trivial to show we have listed only the number density dependence explic- that the conductivity linear in , keeping only terms linear in itly, replacing the constants of proportionality by generic D, is given by constants.
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TABLE I. Carrier-density dependence of the conductivity. High- 250 density and low-density regimes for scattering by screened charged impurities and for short-range impurities. The numbers a1 −a4 rep- 200 resent constants. d=10A 150 Screened charges Short range 2
e /h) d=5A Low density a1na3 100 3/2 1/2 σ( High density a1n+a2n a3 +a4n 50 d=0
B. Scattering times 0 0 0.1 0.2 0.3 0.4 0.5 Two scattering times commonly encountered in momen- n/ni tum relaxation: the transport relaxation time and the quan- tum lifetime Q. The former represents momentum relaxation FIG. 1. ͑Color online͒ Calculated conductivity limited by in transport and is weighted by a factor of ͑1−cos ␥͒ to screened charged impurities as a function of the surface carrier reflect the fact that forward scattering does not alter the density for different impurity locations d=0, 5, 10 Å. In this figure charge current. This is the time measured in transport experi- we use the following parameters: an impurity density ni ments. The latter represents the time taken to scatter from =1013 cm−2, A=4.1 eV Å, which corresponds to the Fermi veloc- 7 one Bloch state into another, thus all scattering amplitudes ity vF =6.2ϫ10 cm/s, and the static dielectric constant ⑀r =100. are weighted equally. This time determines observables such as the Dingle temperature and the broadening of 2 1 k n U¯ ͑q͒ Shubnikov-de Haas oscillations. As a result of the different = F i ͵ d␥ͫ ͬ ͑1 + cos ␥͒. ͑38͒ ប ⑀͑ ͒ weighting of the scattering amplitudes in and Q, the ratio Q 4 A 0 q / Q characterizes the scattering by providing information 89 on which angles are dominant. We determine the ratio of We recall that for elastic scattering of carriers on the Fermi these two scattering times in topological insulators. First, we ͉ Ј͉ϵ → ␥/ surface k−k q 2kF sin 2. We consider first charged provide expressions for and Q in the presence of charged impurity scattering, denoted by the subscript c. The two scat- impurity scattering. To begin with, we write the quantum tering times can be expressed in terms of a prefactor with the lifetime Q and the transport lifetime in a slightly different units of frequency multiplying a dimensionless angular inte- way from the way they were written in Sec. IV ͑with D=0 / ͑ 2 / ͒ / ͑ 2 / ͒ ͒ gral, thus 1 Qc= rs 0c IQc and 1 c = rs 0c Itc, where the and d=0 , 2 /͑ ⑀ ⑀ ͒ Wigner-Seitz radius rs =e 2 0 rA , the quantity
2 1 kFniW = ͵ d␥͑␥͒͑1 − cos ␥͒, 1 Ani ប = ͱ ͑39͒ 4 A 0 ប 0c 32 n
2 and the angular integrals I and I , 1 k n W t Q = F i ͵ d␥͑␥͒, ͑37͒ ប Q 4 A 0 60 where Q is discussed at length in Ref. 89. d=10A
We can also express them in terms of the original scatter- Qc ¯ ϵ ¯ ͑ ͒ ing potential UkkЈ U q as 40
tc d=5A
2 τ/τ 1 k n U¯ ͑q͒ = F i ͵ d␥ͫ ͬ ͑1 − cos2 ␥͒, 20 ប ⑀͑ ͒ d=0 4 A 0 q
TABLE II. Ratio of the transport and Bloch scattering times 0 / Ӷ ӷ 012345 Q for rs 4, rs 4 and different scattering mechanisms. b1 12 -2 −b8 are constants. n (10 cm ) ͑ ͒ Screened charges Short range FIG. 2. Color online The ratio of the transport relaxation time to the quantum lifetime for screened charged impurities as a func- Ӷ −1 rs 4 b1rs −b2 ln rs b5 +b6rs ln rs tion of the carrier density for different impurity locations. We use ӷ −1 the same parameters as Fig. 1. The ratio for the short-range impurity rs 4 b3 +b4rs b7 +b8rs is independent of carrier density and we find t␦ /Q␦ =1.46.
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2 / 1 − cos ␥ that the ratio Q will have a qualitatively different behavior I ͑x͒ = ͵ d␥ , tc ␥ 2 as a function of rs for charged impurity scattering and for 0 ͩsin + xͪ short-range scattering. It must be emphasized also that, de- 2 spite the superficial similarity to graphene, the behavior of this ratio as a function of r for real materials is also quite s 1 + cos ␥ different, due to the absence of the spin and valley degen- I ͑x͒ = ͵ d␥ . ͑40͒ Qc ␥ 2 eracy, and the fact that r for a topological insulator differs 0 s ͩsin + xͪ from its value in graphene ͑for the same parameters͒ by an 2 overall factor of 2. Even before evaluating the integrals explicitly it will be ob- Substituting for the screening function we can determine served that the transport time is not sensitive to either small the expressions for these times as a function of rs in the or large angle scattering while the quantum lifetime Q is limits of small and large rs. The results are qualitatively the insensitive to large angle scattering. The qualitatively differ- same as in Ref. 89 and are summarized in Table II. For ent dependence on the scattering angles makes plain the fact charged impurity scattering, we find for the integral IQc,
1−x ͑2−x2͒arccothͱ1−x2 − ͑1−x2͒arccothͩ ͪ 4 1+x x Ͻ 1, −2 +8x x ͑1−x2͒3/2 I ͑x͒ = ͑41͒ Qc 8−2 x =1, 4 8x x +1 −2 + − ͩarccotͱx2 − 1 − arctanͱ ͪ x Ͼ 1 x ͱx2 −1 x −1
while Itc can be written as
16x͑3x2 −2͒ 1+x 2͑ +12x −6x2͒ + ͩarccothͱ1−x2 − arctanhͱ ͪ x Ͻ 1, ͱ1−x2 1−x ͑ ͒ ͑ ͒ Itc x = 32 − 10 x =1, 42 16x͑3x2 −2͒ x +1 2͑ +12x −6x2͒ − ͩarccotͱx2 − 1 − arctanͱ ͪ x Ͼ 1. ͱx2 −1 x −1
/ 2ͱ Setting x=rs 4 we find that the ratio of the transport and 1 n v n = i 0 ͑45͒ quantum lifetimes for charged impurity scattering in the limit ͱ Ӷ 0␦ 2A ប rs 4 can be expressed as and the integrals appearing in Eq. ͑44͒ are 8 16 ln r ͩ ͪ = − s + . ͑43͒ 2 ¯ ␥ 2 Q c rs sin ␦͑ ͒ 2 For short-range scatterers characterized by a potential v0 r , ͑ ͒ ͵ ␥ ͑ 2 ␥͒ I ␦ x = d 1 − cos , t ␥ the transport and quantum scattering times are given by the 0 sin + x following expressions: 2 1 1 r ͩ s ͪ 2 = I ␦ , ␥ t ␦ 0␦ 4 sin 2 ͑ ͒ ͵ ␥ ͑ ␥͒ ͑ ͒ I ␦ x = d 1 + cos . 46 Q ␥ 1 1 rs 0 ͩ ͪ ͑ ͒ = I ␦ , 44 sin + x Q Q␦ 0␦ 4 2 where the quantity 0␦ is defined by Evaluating the integrals explicitly yields
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1−x 8x͑3x2 −2͒ͩarccothͱ1−x2 − arccothͱ ͪ 1+x x Ͻ 1, +12x −6x2 + ͱ1−x2 ͑ ͒ ͑ ͒ IQ␦ x = 16 − 5 x =1, 47 x −1 8x͑3x2 −2͒ͩarccothͱx2 − 1 − arccothͱ ͪ x +1 x Ͼ 1, +12x −6x2 − ͱ1−x2 and
1−x 16x3͑5x2 −4͒ͩarccothͱ1−x2 − arccothͱ ͪ 16x 1+x x Ͻ 1, − +6x2 +40x3 −20x4 + 2 3 ͱ1−x2 1 I ␦͑x͒ = ͑256 − 81͒ x =1, ͑48͒ t 6 x −1 16x3͑5x2 −4͒ͩarccotͱx2 − 1 − arccotͱ ͪ 16x x +1 x Ͼ 1. − +6x2 +40x3 −20x4 − 2 3 ͱx2 −1
/ Setting again x=rs 4 the ratio of the transport and quantum We note that the analytical results determined in the pre- Ӷ lifetimes in the limit rs 4 for short-range impurities takes vious sections for d=0 agree with the numerical results the form shown in Figs. 1 and 2. Nevertheless, for d0, it is not possible to compare directly the numerical results with the ͑ ͒ 4rs ln rs 2 17−18ln2rs analytical results, the latter being valid for d=0, i.e., the ͩ ͪ =2+ + + . ͑49͒ ¯ Q ␦ 3 impurities located on the surface. Reference 80 measured a mobility on the order of In the range covered here this ratio does not exceed 2 for 1m2 /͑Vs͒ at a number density on the order of short-range scatterers. We would also like to point out that Ϸ1017 cm−3. For example, for one sample the surface carrier ϰ / 12 −2 rs 1 A for the surface of Bi2Te3 is approximately twice that density is estimated at Ϸ7ϫ10 cm , while the effective Ϸ Ӷ 90 in graphene, i.e., rs 0.14, which is 4. scattering rate is 0.8 ps, which is obtained from transport. /បϷ For such a sample F 35, placing it within the range of C. Numerical results applicability of our theory. However the experiment is not sensing surface state transport so a direct comparison is not We conclude our discussion with numerical results on the feasible at present. conductivity and the ratio of the two scattering times. From Our theory has focused on scattering due to charged im- the transport scattering time calculated in the previous sec- purities and interface roughness, whereas in a realistic tion we can calculate the conductivity using Eq. ͑35͒ for d sample other scattering mechanisms may exist, primarily =0. In Fig. 1 we have plotted the conductivity versus the phonons. Phonons can be included easily in our theory as an / ratio n ni of the carrier number density to the impurity num- additional contribution to the scattering term. However in ber density for several values of d, the distance of the impu- this work we have considered zero temperature, therefore rities from the surface of the topological insulator. For d=0, there is no phonon absorption and there is only phonon emis- when the impurities are located right on the surface, the con- sion, which usually has a negligible contribution to the con- / ductivity is a linear function of n ni.Ifd is large, meaning ductivity. Ͼ kFd 1 then large angle scattering is strongly suppressed. We note that our theoretical results would be useful in / This leads to a large increase in Q, since the Fourier trans- discerning transport by the surface topological 2D states only form of the scattering potential is suppressed by e−qd. after a real topological insulator material, i.e., a system In Fig. 2 we have plotted the ratio of the two scattering which is a bulk insulator, becomes available. All currently times as a function of the carrier number density for charged existing TI materials are effectively bulk metals because of impurity scattering for different values of the distance d of their large unintentional doping. Such systems are, by defi- the impurities from the surface. For d=0 this ratio is not a nition, unsuitable for studying surface transport properties function of n. It should also be noted that, for short-range since bulk transport completely overwhelms any surface / scatterers, the ratio ␦ Q␦ is a constant as a function of n, transport signature. Discussing TI surface transport in such which we find to be 1.46. bulk-doped TI materials is not particularly meaningful since
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it must necessarily involve complex data fitting and assump- the chemical potential approaches the Dirac point, where the tions in order to distinguish bulk versus surface transport average carrier density ͗n͘=0, these fluctuations play the contributions. Although such analyses are being carried out dominant role in conduction.91 At low densities, the carriers in bulk doped TI materials, in order to understand the trans- cluster into puddles of electrons and holes, and a residual port behavior ͑and our theoretical results should certainly density of carriers is always present, making it impossible to help such analyses͒, we believe that any real progress will reach the Dirac point experimentally.91 Consequently, al- come only when surface TI transport can be carried out un- though Eq. ͑36͒ appears to give a universal value for the ambiguously, without any complications arising from the minimum conductivity, the renormalization due to the pres- ͑more dominant͒ bulk transport channel. We believe that our ence of electron and hole puddles displays a strong sample 0Ќ theoretical results presented in this paper would become im- dependence: in graphene it results in an enhancement of xx portant in the context of such surface TI studies in the labo- by a factor of 2–20.86,92 In addition, exchange and correlation ratory. effects make a significant contribution to the minimum What we have shown in this paper is that the so-called conductivity.93 topological protection of TI surface transport is a protection An accurate determination of this enhancement for topo- only against localization by backscattering, not a protection logical insulators requires detailed knowledge of the impu- against impurity or defect scattering. The presence of impu- rity density distribution, which can only be determined ex- rities and defects will certainly lead to scattering of the TI perimentally or modeled numerically by means of, e.g., an surface carriers and the surface 2D conductivity will be effective medium theory.93 However, a self consistent trans- strongly affected by such scattering. If such scattering is port theory provides a physically transparent way to identify strong ͑which will be a nonuniversal feature of the sample the approximate minimum conductivity. The magnitude of quality͒, then the actual surface resistivity will be very high voltage fluctuations can be calculated in the random-phase and the associated mobility very low. There is no guarantee approximation, and the result used to determine the residual or protection against low carrier mobility in the TI surface density and the critical number density at which the transi- states whatsoever, and unless one has very clean surfaces, tion occurs to the regime of electron and hole puddles, where there is little hope of studying surface transport in 2D topo- the carrier density will be highly inhomogeneous. Both the logical states, notwithstanding their observation in beautiful residual density and the critical density are proportional to ni band-structure measurements through ARPES or STM ex- by a factor of order unity,91 and as a first approximation one Ϸ periments. may take both of them as ni. In view of these observations, the minimum conductivity plateau will be seen approxi- Boltz mately at xx , in which with the carrier density n is re- D. Minimum conductivity at the Dirac point Boltz placed by ni. Using the expressions found above for xx Thus far we have concentrated on the conductivity xx at and , this yields for topological insulators, carrier densities such that /បӷ1. This is the limit that is F e2 8 immediately relevant to experiment given the high uninten- min Ϸ ͩ ͪ ͑ ͒ xx . 50 tional doping in topological insulators. At such densities the h Itc Boltz ͑ ͒ conductivity is given by xx from Eq. 35 while the term At carrier densities approaching zero the D-dependent term 0Ќ ͑ ͒ xx from Eq. 36 does not contribute. At present it is not in the Hamiltonian will not contribute. Therefore, when ex- possible to tune the number density continuously through the periment will be able to tune the carrier density to zero we Dirac point as is done in graphene. Nevertheless, given the expect a minimum conductivity of the order of min to be similarity of the Hamiltonians of graphene and topological xx observed. We note that, with rs much smaller than in insulators, we expect a similar behavior at the Dirac point. graphene due to the large dielectric constant, the minimum We predict that, when such tuning does become possible, a conductivity may be substantially larger.93 minimum conductivity plateau around the Dirac point will be observed, analogously to graphene. 0Ќ The term xx is expected to provide one contribution to E. Current-induced spin polarization the minimum conductivity, which can also be obtained using The charge current is proportional to the spin operator, as a field-theoretical method in the absence of disorder.86 This can be seen from Eq. ͑2͒. Therefore a nonzero steady-state method can in fact yield several values, depending on the surface charge current automatically translates into a nonzero choice of regularization procedure, which are close in mag- → /បӷ steady-state surface spin density. The spin density can easily nitude. However, as n 0 the condition F 1 is inevi- 2 Ќ be found by simply multiplying the charge current by ប /͑ tably violated and a strong renormalization of 0 is ex- xx −2eA͒, yielding pected due to charged impurities.91–93 Quite generally, charged impurities give rise to an inhomogeneous Coulomb eE Ak 2 Dk2 x ͫ F ͩ ͪ F ͬ ͑ ͒ both sx =− 1+ + . 51 potential, which is screened by electrons and holes. The 4 4ប 4ប net effects of this potential are an inhomogeneity in the car- D rier density itself and a shift in the Dirac point as a function Since in the Hamiltonian represents the rotated spin, this of position. At high densities the spatial fluctuations in the corresponds to a net density of sy. The effective Hamiltonian carrier density are of secondary importance, and do not of Eq. ͑1͒ is obtained from the original Rashba Hamiltonian → → modify the linear dependence of the conductivity on n, yet as by replacing y x and x − y. For a sample in which
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/ the impurities are located on the surface, with n ni =0.5 and / Ϸ ϫ 14 / 2 Ex =25000 V m, the spin density is 5 10 spins m ͑where spinϵប͒, which corresponds to approximately 10−4 spins/unit-cell area. This number, although small, can be detected experimentally using a surface spin probe such as Kerr rotation. It is also a conservative estimate: for very clean samples, having either a smaller impurity density or impurities located further away from the surface, this number can reach much higher values. Current-induced spin polar- ization is a signature of two-dimensional transport in topo- logical insulators since there is no spin polarization from the bulk.
VI. EFFECT OF METALLIC CONTACTS A question of major practical importance concerns the stability of the surface states in the presence of metallic con- tacts. Transport measurements require the addition of metal- lic contacts on the surface of the topological insulator. Since the properties of the surface states depend crucially on the boundary conditions, the natural question is how are these properties affected by the metallic contacts. To address this question, we consider the case of large area contacts and perform numerical calculations for the minimal tight-binding model studied in Ref. 24. A topological insulator in contact with a metal along a planar interface is described by the Hamiltonian FIG. 3. ͑Color online͒͑a͒ Amplitude of a metallic state hybrid- ized with a topological insulator surface state as function of dis- ͑ ͒ Htot = HTI + HM + Ht. 52 tance from the interface ͑in units of interlayer spacing͒. ͑b͒ Spec- trum of a topological insulator in contact with a metal ͑black͒. The The topological insulator term H is given by the tight- TI states of a topological insulator with a free surface are also shown binding model on a diamond lattice with spin-orbit ͑ ͒ ͑ ͒ 8 purple—bulk states, red—surface states . c Density plot of the interactions total amplitude within a thin region of topological insulator in the ͓ ͑ ͔͒ ͚ † ͚ ͑ 1 ϫ 2 ͒ † vicinity of the interface red/dark gray area in panel a . Note that, HTI = t cicj + i SO S,Ј · dij dij cicjЈ, instead of a well-defined surface mode, one has a diffuse distribu- ͗ij͘, ͗ij͘ tion of states with boundary contributions. The dispersion of the ͑53͒ maxima of this distribution is represented by green points in panel ͑ ͒ where the first term is the nearest-neighbor hopping on the b . diamond lattice and the second term connects second-order surface states.24 These surface states are exponentially local- neighbors with a spin- and direction-dependent amplitude. ized near the boundary with a characteristic length scale of a d1,2 The direction dependence is given by the bond vectors ij few lattice spacings. When the TI surface is in contact with a i j ͑ ͒ traversed between sites and . The metallic term in Eq. 52 metal, the surface states penetrate inside the metal and hy- is given by bridize with the metallic states. A typical hybridized state is Ј † ͑ ͒ shown in Fig. 3͑a͒. Note that the amplitude of this state near HM = ͚ tijaiaj. 54 ͗ij͘, the boundary of the topological insulator is reduced by a / ͑m͒ factor 1 Lz with respect to the amplitude of a pure surface The model is defined on a simple hexagonal lattice with a ͑m͒ state with the same energy, where Lz is the width of the lattice constant for the triangular basis that ensures simple metal in the direction perpendicular to the interface. How- interface matching conditions between the metal and the ever, the local density of states near the boundary is not ͑ ͒ ͑ ͒ 1,1,1 surface of the TI. The tunneling term in Eq. 52 is reduced, as the number of hybridized states also scales with ͑m͒ † † Lz . The spectrum of a topological insulator in contact with H = ͚˜t͑a c + c a ͒, ͑55͒ t i j j i a metal is shown in Fig. 3͑b͒. Also, we calculated for each ͗ij͘ state the total amplitude within a thin layer of topological were ˜t is the tunneling matrix element that characterizes the insulator in the vicinity of the boundary. The result is shown transparency of the interface between the topological insula- in Fig. 3͑c͒. Instead of the sharply defined Dirac cone that tor and the metal. characterizes the free surface, one has a diffuse distribution For a free surface, the topological insulator spectrum is of states with boundary contributions. characterized by a bulk gap occupied in the vicinity of the M The properties of the interface amplitude distribution point by an anisotropic Dirac cone associated with gapless shown in Fig. 3͑c͒, i.e., its width and the dispersion of its
155457-13 CULCER et al. PHYSICAL REVIEW B 82, 155457 ͑2010͒ maximum, depend on the strength of the coupling between APPENDIX: DERIVATIONS OF CERTAIN EXPRESSIONS the metal and the topological insulator. If the distribution is APPEARING IN THE SCATTERING TERM sharp enough and the dispersion does not deviate signifi- Firstly, Eq. ͑5͒ is derived as follows: cantly from the Dirac cone, our transport analysis should be applied using bare parameters, i.e., parameters characterizing dgkkЈ i i + ͓Hˆ ,gˆ͔ =− ͓Uˆ ,ˆf + gˆ͔ . ͑A1͒ the spectrum of a free surface. Nonetheless, significant de- ប kkЈ ប kkЈ viations from the free surface dispersion are, in principle, dt possible. For example, the location of the Dirac point in our To first order in Uˆ we do not retain the gˆ-dependent term on model calculation is fixed by symmetry. However, in real the RHS. We also drop the k and kЈ indices for simplicity, systems the energy of the Dirac point can be easily modified writing the equation as an operator equation by changing the boundary conditions. Consequently, the ef- fective parameters Deff and Aeff entering transport coeffi- dg i i + ͓Hˆ ,gˆ͔ =− ͓Uˆ ,ˆf͔. ͑A2͒ cients may differ significantly from the corresponding pa- dt ប ប rameters extracted from ARPES measurements. To clarify these issues, as well as to determine the dependence of these −iHˆ t/ប iHˆ t/ប Now let g=e gIe , a transformation equivalent to effects on the size of the contacts, further studies are neces- switching to the interaction picture sary. dg i ˆ ˆ I =− eiHt/ប͓Uˆ ,ˆf͔e−iHt/ប, ប VII. SUMMARY dt
We have determined the charge conductivity of the sur- t i ˆ ˆ g =− ͵ dtЈeiHtЈ/ប͓Uˆ ,ˆf͑tЈ͔͒e−iHtЈ/ប, face states of three-dimensional topological insulators basing I ប our theory on the quantum Liouville equation and accounting −ϱ fully for the absence of backscattering. We identify two con- t tributions to the conductivity with different dependencies on i ˆ ˆ −iH͑t−tЈ͒/ប͓ ˆ ˆ͑ ͔͒ iH͑t−tЈ͒/ប the carrier number density. At low densities the conductivity g =− ͵ dtЈe U, f tЈ e ប ϱ is ϰn if charged impurity scattering is dominant and is inde- − ϱ pendent of n if short-range scatterers such as surface rough- i ˆ ˆ =− ͵ dtЉe−iHtЉ/ប͓Uˆ ,ˆf͑t − tЉ͔͒eiHtЉ/ប. ͑A3͒ ness are dominant. As the density increases an additional ប term is present in the conductivity, which is ϰn3/2 for 0 / charged impurity scattering and ϰn1 2 for short-range scatter- Projecting onto k and kЈ we obtain Eq. ͑5͒. ing. The ratio of the transport and quantum lifetimes in the We consider in more detail the derivation of the contribu- Ӷ region rs 4, which we believe to be the most realistic ex- tion ϰ͑k+kЈ͒ in the scattering term. The scattering term has −1 perimentally, is of the form b1rs −b2 ln rs for charged impu- the general form ͑with summation implied over the index i͒, rity scattering and of the form b5 +b6rs ln rs for surface 2 ϱ roughness scattering. The analysis we have presented should ni d kЈ Ј Ј Ј Jˆ͑f͒ = ͵ ͉U ͉2͑n − n ͒͵ dtЈe− t e−iHkЈt eiHkt ប2 ͑ ͒2 kkЈ k kЈ enable one to obtain a complete characterization of all the 2 0 features of topological insulator samples accessible in trans- /បӷ n d2kЈ port experiments, provided F 1 and the chemical po- i ͵ ͉ ͉2͑ ͒ + H.c. + UkkЈ Ski − SkЈi tential is below the bottom of the conduction band. 2ប2 ͑2͒2 The current form of the theory presented in this paper ϱ Ј Ј Ј gives results equivalent to those of Boltzmann transport ϫ͵ Ј − t −iHkЈt iHkt ͑ ͒ dt e e ie + H.c. A4 theory. Nevertheless, this theory accounts explicitly for inter- 0 band coherence effects such as Zitterbewegung, which should become important near the Dirac point in ballistic In the time evolution operator one substitutes the general 2 ͑ / ͒ ⍀ samples. We expect the advantages of this formulation to expression for the band Hamiltonian H0k =Dk + 1 2 · k, ⍀ become manifest in the presence of a magnetic field. A mag- where k =2Ak. In what follows we shall shorten the time netic field induces additional coherence between the two evolution operators by omitting the factors of ប. These will bands, influences the spin dynamics, and gives rise to a be restored in the final expressions. The time evolution op- qualitatively different scattering term. The flexibility of the erator can be written as theory outlined in this work enables us to study a variety of ⍀ ⍀ 2 kt kt multiband effects beyond Boltzmann transport, which we re- −iHkt −iDk tͩ ⍀ˆ ͪ ͑ ͒ e = e cos − i · k sin . A5 serve for a future publication. 2 2 We recall the identity ACKNOWLEDGMENTS ͑ ⍀ˆ ͒͑ ⍀ˆ ͒ ⍀ˆ ⍀ˆ ⍀ˆ ϫ ⍀ˆ ͑ ͒ · k · kЈ = k · kЈ + i · k kЈ. A6 We are grateful to H. D. Drew, N. P. Butch, G. S. Jenkins, J. Paglione, A. B. Sushkov, Roman Lutchyn, and Jay Sau for This allows us to express the products of two time evolution discussions. This work was supported by LPS-NSA-CMTC. operators as
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⍀ ⍀ For the spin-dependent part of the density matrix, ͑ Ј2 2͒ kЈt kt e−iHkteiHkЈt = ei Dk −Dk tͫcos cos 2 2 ប −iHkt iHkЈt → ͓ ͑ ˆ͒͑ ˆЈ͔͓͒␦͑⑀Ј ⑀ ͒ ⍀ e e + · k · k + − + kЈt ⍀ t 4 − i · ⍀ˆ cos sin k k 2 2 ប + ␦͑⑀Ј − ⑀ ͔͒ + ͓͑ · kˆ͒ + ͑ · kˆЈ͔͒ ⍀ − − kЈt ⍀ t 4 + i · ⍀ˆ sin cos k kЈ ϫ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ 2 2 + − + − − − − , ⍀ kЈt ⍀ t ͑⍀ˆ ⍀ˆ ⍀ˆ ϫ ⍀ˆ ͒ k ͬ ˆ ˆ + kЈ · k + i · k kЈ sin sin , −iH ЈtЈ iH tЈ −iH tЈ iH ЈtЈ 2 2 e k e k + e k e k ប ⍀ ⍀ → ͓ͭ2 + ͑ · kˆ͒͑ · kˆЈ͒ + ͑ · kˆЈ͒͑ · kˆ͔͒ ͑ Ј2 2͒ kЈt kt e−iHkteiHkЈt = ei Dk −Dk tͫ cos cos 4 2 2 ប ϫ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ͓͑ ˆ͒ ͑ ˆЈ͒ ⍀ ⍀ + − + + − − − + · k + · k kЈt kt 4 − i͑ · ⍀ˆ ͒ cos sin k 2 2 + ͑ · kˆЈ͒ + ͑ · kˆ͔͓͒␦͑⑀Ј − ⑀ ͒ − ␦͑⑀Ј − ⑀ ͔͒ͮ. ⍀ t ⍀ t + + − − ͑ ⍀ˆ ͒ kЈ k + i · kЈ sin cos 2 2 ͑A10͒ ⍀ t ⍀ t ͑ ⍀ˆ ͒͑ ⍀ˆ ͒ kЈ k ͬ The product of three sigma matrices simplifies as follows: + · k · kЈ sin sin . 2 2 ͑ ˆ͒͑ ˆЈ͒ ͑ ˆЈ͒͑ ˆ͒ ͑A7͒ · k · k + · k · k ˆ ˆ ˆ ˆ ˆ ˆ The time integrals of these products of exponentials and =2͑k · ͒kЈ −2͑k · kЈ͒ +2k͑kЈ · ͒, ͑A11͒ trigonometric functions give yielding finally 2 2 ⍀ t ⍀Јt ប i͑DkЈ −Dk ͒t k k → ͓␦͑⑀ ⑀Ј͒ ␦͑⑀ ⑀Ј͒ e cos cos + − + − − −iHkЈtЈ iHktЈ −iHktЈ iHkЈtЈ 2 2 4 + − e e + e e ␦͑⑀ ⑀Ј͒ ␦͑⑀ ⑀Ј͔͒ ប + + − − + − − + , → ͓͑1−kˆ · kˆЈ͒ + ͑kˆ · ͒kˆЈ + kˆ͑kˆЈ · ͔͓͒␦͑⑀Ј − ⑀ ͒ 2 + + ⍀ 2 2 kЈt ⍀ t ប ប i͑DkЈ −Dk ͒t k → ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀Ј͒ e cos sin + − − + − − − + ␦͑⑀Ј − ⑀ ͔͒ + ͑kˆ + kˆЈ͓͒␦͑⑀Ј − ⑀ ͒ − ␦͑⑀Ј − ⑀ ͔͒. 2 2 4i − − 2 + + − − ␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ͑ ͒ − + − + − − − + , A12
⍀ The scattering term takes the form Jˆ͑f͒=Jˆ͑n͒+Jˆ͑S͒, 2 2 kЈt ⍀ t ប i͑DkЈ −Dk ͒t k → ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͒ e sin cos + − + + + − − 2 2 4i n d2kЈ Jˆ͑n͒ = i ͵ ͉U¯ ͉2͑n − n ͒͑1+kˆ · kˆЈ͓͒␦͑⑀Ј − ⑀ ͒ ␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ប ͑ ͒2 kkЈ k kЈ + + − − − − − − − + , 2 2 2 Ј ni d k 2 2 2 ⍀ t ⍀Јt ប ␦͑⑀Ј ⑀ ͔͒ ͵ ͉¯ ͉ ͑ ͒ ͑ Ј ͒ k k + − − + UkkЈ nk − nkЈ ei Dk −Dk t sin sin → − ͓␦͑⑀ − ⑀Ј͒ + ␦͑⑀ − ⑀Ј͒ − 2ប ͑2͒2 2 2 4 + − − + ϫ ͑ˆ ˆЈ͓͒␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ ␦͑⑀ ⑀Ј͒ ␦͑⑀ ⑀Ј͔͒ · k + k + − + − − − − , − − − − − + − + . ͑A8͒ n d2kЈ ˆ͑ ͒ i ͵ ͉¯ ͉2͑ ͒ ͓͑ ˆ ˆЈ͒ The Hermitian conjugates are trivial. For the scalar part of J S = UkkЈ Sk − SkЈ · 1−k · k 4ប ͑2͒2 ⍀ˆ ˆ the density matrix, letting k =k the above results in ͑ˆ ͒ˆ ˆ͑ˆ ͔͓͒␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ + k · kЈ + k kЈ · − + + − − ˆ Ј ˆ Ј ˆ Ј ˆ Ј ប + − e−iHkЈt eiHkt + e−iHkt eiHkЈt → ͑1+kˆ · kˆЈ͓͒␦͑⑀Ј − ⑀ ͒ + + n d2kЈ 2 i ͵ ͉¯ ͉2͑ ͒ ͑ˆ ˆЈ͓͒␦͑⑀Ј ⑀ ͒ + 2 UkkЈ Sk − SkЈ · k + k + − + ប 4ប ͑2͒ ␦͑⑀Ј ⑀ ͔͒ ͑ˆ ˆЈ͒ + − − − + · k + k ␦͑⑀Ј ⑀ ͔͒ ͑ ͒ 2 − − − − . A13 ϫ͓␦͑⑀Ј ⑀ ͒ ␦͑⑀Ј ⑀ ͔͒ + − + − − − − . Note the factor of 2 difference between Jˆ͑n͒ and Jˆ͑S͒, which 2 ͑ / ͒ ⍀ ͑A9͒ arises solely from our definition H0k =Dk + 1 2 · k.
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͕ ͖ ␦ ͑ ͒ Note also that there are no second-order terms in the Pauli i, j =2 ij. A14 matrices i. From the algebra of these matrices we know that ␦ ⑀ i j = ij + ijk k, In other words the second-order terms can give zero, the ͓ ϰ͑ ͒ ͑ˆ ˆ ͔͒ identity matrix leading to the terms Sk −SkЈ · k+kЈ or ͓ ͔ ⑀ i, j =2 ijk k, one Pauli matrix.
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