Two-dimensional surface charge transport in topological insulators
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 and transport scattering time satisfy , but 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 and explicitly accounts for the absence of backscattering, while screening is included in the random phase approximation. The main contribution to the conductivity is and has different carrier density dependencies for different forms of scattering, while an additional contribution is independent of . 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 . 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.
Topological insulators are a new class of materials that have been intensely researched in recent years. Hasan and Kane (unpublished) They are band insulators in the bulk while conducting along the surfaces, possessing surface states with a spin texture protected by time-reversal symmetry. Within a certain parameter range the surface states are well described by a Dirac cone, allowing for parallels with graphene and relativistic physics, and prohibiting backscattering. Certain physical properties related to the surface states of these materials can be expressed in terms of topologically invariant quantities, and are thus protected against smooth perturbations invariant under time-reversal. For this reason topological insulators are being explored with a view towards applications, as a potential platform for quantum computation,Nayak et al. (2008) and as rich physical systems in their own right.
The concept of a topological insulator dates back to the work of Kane and Mele, who focused on two-dimensional systems. Kane and Mele (2005) These authors demonstrated that, in addition to the well-known TKNN invariant,Thouless et al. (1982) characterizing the time-reversal breaking quantum Hall state, an additional topological invariant exists, which characterizes the surface states of time-reversal invariant insulators. In two dimensions this invariant can be zero or one, the former representing ordinary insulators (equivalent to the vacuum) and the latter topological insulators. The two band structures corresponding to and cannot be deformed into one another. The invariant , which is related to the topology, counts the number of pairs of Dirac cones on the surfaces. Kane and Mele Kane and Mele (2005) proposed realizing a topological insulator in graphene by using the spin-orbit interaction to open a gap, yet the spin-orbit interaction in graphene is rather small. Subsequently Bernevig et al. predicted that a topologically insulating state, the quantum spin Hall insulator, could be realized in HgCdTe quantum wells. Bernevig et al. (2006) This state was observed shortly after its prediction by König et al. König et al. (2007) (for a review of this effect see Ref. König et al., 2008.) The quantum spin Hall state constitutes a topological insulator in two dimensions. Soon afterwards topological insulators were predicted to exist in three dimensions, Fu et al. (2007); Roy (2009); Moore and Balents (2007) in which four topological invariants exist, customarily labeled , , and . The invariants can be calculated easily if the system has inversion symmetry. Fu and Kane (2007) The invariant determines whether a topological insulator is weak () or strong (.) The absence of backscattering gives rise to phenomena such as Klein tunneling and is believed to make strong topological insulators immune to Anderson localization. Hasan and Kane (unpublished)
A significant fraction of theoretical research on these materials to date has focused on their topology. A number of articles have been devoted to the classification of topological insulators. Hatsugai (2005); Roy (2009); Murakami et al. (2004); Moore and Balents (2007); Schnyder et al. (2008) A description of topological insulators in terms of topological field theory has been given in Ref. Qi et al., 2008, and a general theory of topological insulators and superconductors has been formulated in Ref. Hosur et al., 2010. An effective continuous model for the surface states of three dimensional topological insulators has been proposed in Ref. Shan et al., unpublished. Their effective action in the presence of an electromagnetic field has been studied with a focus on the electric-magnetic duality, Karch (2009) and analogies have been made with the topological invariants of the Standard Model of particle physics. Volovik (2010) The surface states of topological insulators in strong magnetic fields have been investigated in Ref. Lee, 2009.
The interface between a topological insulator and a superconductor was predicted to support Majorana fermions, Fu and Kane (2008) that is, particles that are their own antiparticles. Subsequent articles have focused on probing Majorana fermions, Fu and Kane (2009) including a scheme employing interferometry. Akhmerov et al. (2009) Recently the proximity effect between a superconductor and a topological insulator has been re-examined. Stanescu et al. (unpublished) In addition, other exotic states have also been predicted, including a helical metal, Ran et al. (2009) superconductivity and ferromagnetism induced by the proximity effect, Linder et al. (unpublished) unconventional superconductivity, Linder et al. (2010) a quantized anomalous Hall effect in topological insulators doped with magnetic impurities, Yu et al. (unpublished) and a new critical state attributed to the Coulomb interaction. Ostrovsky et al. (unpublished) Further expected effects include the possibility of Cooper pair injection at the interface between a topological insulator and a superconductor, Sato et al. (unpublished) the inverse spin-galvanic effect at the interface between a topological insulator and a ferromagnet, Garate and Franz (unpublished) giant magnetoresistance, Chu et al. (2009) and a giant Kerr effect. Tse and MacDonald (unpublished)
Variations have also appeared on the theme of topological insulators. Given that superconductors are also gapped, topological superconductors have been predicted to exist. Hasan and Kane (unpublished); Schnyder et al. (2008); Kitaev (2009); Qi et al. (2009) Developments in different directions include the topological Anderson insulator, Groth et al. (2009); Jiang et al. (2009); Li et al. (2009) the topological Mott insulator,Raghu et al. (2008) the topological magnetic insulator, Moore et al. (2008) the fractional topological insulator, Levin and Stern (2009); Lee et al. (2007) the topological Kondo insulator, Dzero et al. (2010) as well as a host of topological phases arising from quadratic, rather than linear, band crossings. Sun et al. (2009) Topological phases in transition-metal based strongly correlated systems have been the subject of Refs. Xu, 2010; Pesin and Balents, unpublished, and a theory relating topological insulators to Mott physics has been proposed in Ref. Rachel and Hur, unpublished.
Following the discovery of the quantum spin-Hall state, several materials were predicted to be topological insulators in three dimensions. The first was the alloy BiSb,Teo et al. (2008); Zhang et al. (2009a) followed by BiSe, BiTe, and SbTe. Zhang et al. (2009b) In particular BiSe and BiTe have long been known from thermoelectric transport as displaying sizable Peltier and Seebeck effects, and their high quality has ensured their place at the forefront of experimental attention. Hasan and Kane (unpublished) Initial predictions of the existence of chiral surface states were confirmed by first principles studies of BiSe, BiTe, and SbTe. Zhang et al. (unpublisheda) On an interesting related note, it has recently been proposed that topologically insulating states can be realized in cold atoms, Stanescu et al. (2009) though these lie beyond the scope of the present work.
After a timid start, experimental progress on the materials above has skyrocketed. Hsieh et al, Hsieh et al. (2008) using angle-resolved photoemission spectroscopy (ARPES), were the first to observe the Dirac cone-like surface states of BiSb, with . Following that, the same group used spin-resolved ARPES to measure the spin polarization of the surface states, demonstrating the correlation between spin and momentum.Hsieh et al. (2009a) Shortly afterwards experiments identified the Dirac cones of BiSe,Xia et al. (2009) BiTe,Chen et al. (2009); Hsieh et al. (2009b) and SbTe.Hsieh et al. (2009b) Recently BiSe nanowires and nanoribbons have also been grown Kong et al. (2010) and experiments have begun to investigate the transition between the three- and two-dimensional phases in this material.Zhang et al. (unpublishedb) BiSe has also been studied in a magnetic field, revealing a magnetoelectric coupling LaForge et al. (unpublished) and an oscillatory angular dependence of the magnetoresistance. Taskin et al. (unpublished) Sample growth dynamics of BiTe by means of MBE were monitored by reflection high-energy electron diffraction (RHEED). Li et al. (unpublished) Doping BiTe with Mn has been shown to result in the onset of ferromagnetism, Hor et al. (unpublished) while doping BiSe with Cu leads to superconductivity.Hor et al. (2010); Wray et al. (unpublished)
Considerable efforts have been devoted to scanning tunneling microscopy (STM) and spectroscopy (STS), which enable the study of quasiparticle scattering. Scattering off surface defects, in which the initial state interferes with the final scattered state, results in a standing-wave interference pattern with a spatial modulation determined by the momentum transfer during scattering. These manifest themselves as oscillations of the local density of states in real space, which have been seen in several materials with topologically protected surface states. Roushan et al. Roushan et al. (2009) have imaged the surface states of BiSb using STM and spin-resolved ARPES demonstrating that, as expected theoretically, in the absence of spin-independent (time-reversal invariant) disorder, scattering between states of opposite momenta is strongly suppressed. Gomes et al. Gomes et al. (unpublished) investigated the (111) surface of Sb, which displays a topological metal phase, and found a similar suppression of backscattering. Zhang et al. Zhang et al. (2009c) and Alpichshev et al.Alpichshev et al. (2010) have observed the suppression of backscattering in BiTe. Theories have recently been formulated to describe the quasiparticle interference seen in STM experiments, accounting for the absence of backscattering. Guo and Franz (2010); Lee et al. (2009); Park et al. (unpublished) Related developments have seen the imaging of a surface bound state using scanning tunneling microscopy. Alpichshev et al. (unpublished) These experimental studies provide evidence of time-reversal symmetry protection of the chiral surface states.
Despite the success of photoemission and scanning tunneling spectroscopy in identifying chiral surface states, signatures of the surface Dirac cone have not yet been observed in transport. Simply put, the band structure of topological insulators can be visualized as a band insulator with a Dirac cone within the bulk gap, and to access this cone one needs to ensure the chemical potential lies below the bottom of the conduction band. Given that the static dielectric constants of materials under investigation are extremely large, approximately 100 in BiSe and 200 in BiTe, gating in order to bring the chemical potential down is challenging. Therefore, in all materials studied to date, residual conduction from the bulk exists due to unintentional doping. One potential way forward was opened by Hor et al., Hor et al. (2009) who demonstrated that Ca doping ( substituting for Bi) brings the Fermi energy into the valence band, followed by the demonstration by Hsieh et al. Hsieh et al. (2009a) that Ca doping can be used to tune the Fermi energy so that it lies in the gap. Checkelsky et al. Checkelsky et al. (2009) performed transport experiments on Ca-doped samples noting an increase in the resistivity, but concluded that surface state conduction alone could not be responsible for the smallness of the resistivities observed at low temperature. Eto et al. Eto et al. (unpublished) reported Shubnikov-de Haas (SdH) oscillations due only to the 3D Fermi surface in BiSe with an electron density of cm, while Analytis et al. Analytis et al. (unpublished) used ARPES and SdH oscillations to probe BiSe. The two methods were found to agree for bulk carriers, but while ARPES identified Dirac cone-like surface states, even at number densities of the order of cm SdH oscillations identified only a bulk Fermi surface, and the surface states were not seen in transport. A series of recent transport experiments claim to have observed separate signatures of bulk and surface conduction. Checkelsky et al. Checkelsky et al. (unpublished) performed weak localization and universal conductance fluctuation measurements, claiming to have observed surface conduction, although the chemical potential still lay in the conduction band. Steinberg et al. Steinberg et al. (unpublished) reported signatures of ambipolar transport analogous to graphene. However, in this work also, two contributions to transport were found, the bulk contribution was subtracted so as to obtain the surface contribution, and the surface effect was observed only for top gate voltage sweeps, not back gate sweeps. Butch et al. Butch et al. (unpublished) carried out SdH, ordinary Hall effect and reflectivity measurements and concluded that the observed signals could be ascribed solely to bulk states. No experimental group has reported SdH oscillations due to the surface states, which are expected at a very different frequency. Butch et al. (unpublished) The above presentation makes it plain that, despite the presence of the Dirac cone, currently no experimental group has produced a true topological insulator. Consequently, all existing topological insulator/metal systems are in practice either heavily bulk-doped materials or thin films of a few monolayersZhang et al. (unpublishedb) (thus, by definition, not 3D.)
In an intriguing twist, two recent works have reported STM imaging of the Landau levels of BiSe. Cheng et al. (unpublished); Hanaguri et al. (unpublished) Cheng et al.Cheng et al. (unpublished) claim to have observed Landau levels analogous to graphene, with energies but the expected spin and valley degeneracy of one, yet puzzling features such as the absence of Landau levels below the Fermi energy and enhanced oscillations near the conduction band edge have not been resolved, and some arbitrariness is involved in extracting the Landau level filling factor . Hanaguri et al.Hanaguri et al. (unpublished) report similar findings in the same material, with analogous features in the Landau level spectrum. Following the pattern set by the imaging of the surface Dirac cone, the Landau levels observed in STM studies have not been seen in transport. These last works illustrate both the enormous potential of the field and the challenges to be overcome experimentally.
The focus of experimental research has shifted to the separation of bulk conduction from surface conduction, hence it is necessary to know what type of contributions to expect from the surface conduction. The only transport theories we are aware of have been Ref. Mondal et al., unpublished, which focused on magnetotransport in the impurity-free limit, and Ref. Burkov and Hawthorn, unpublished, which focused on spin-charge coupling. A comprehensive theory of transport in topological insulators has not been formulated to date. This article seeks to remedy the situation. We begin with the quantum Liouville equation, deriving a kinetic equation for the density matrix of topological insulators including the full scattering term to linear order in the impurity density. Peculiar properties of topological insulators, such as the absence of backscattering (which leads to Klein tunneling and other phenomena), are built into our theory. We determine the polarization function of topological insulators. By solving the kinetic equation we determine all contributions to the conductivity to orders one and zero in the transport scattering time . We find that the scalar part of the Hamiltonian makes an important contribution to the conductivity, which has a different carrier density dependence than the spin-dependent part, and may be a factor in distinguishing the surface conduction from the bulk conduction. We consider charged impurity as well as short-range surface roughness scattering and determine the density-dependence of the conductivity when either of these is dominant. Most importantly, we find that the electrical current generates a spin polarization in the in-plane direction perpendicular to the field, providing a unique signature of surface transport. Moreover, the ratio of the single-particle scattering time to the transport scattering time provides important information on the dominant angles in scattering, which in turn provide information on the dominant scattering mechanisms. At zero temperature, which constitutes the focus of our work, we expect a competition between charged impurity scattering and short-range scattering. For each of these two scattering mechanisms the ratio of the single-particle scattering time to the transport scattering time has a qualitatively different dependence on the Wigner-Seitz radius , which parametrizes the relative strength of the kinetic energy and electron-electron interactions. We study these aspects of the transport problem in detail, covering the possible situations that can be encountered experimentally with currently available samples and technology. Finally, we consider the effect on the topological surface states of adding metal contacts, as would be the case in experiment.
Our work shows that the suppression of backscattering seen in experimental work indicates protection against localization resulting from backscattering, rather than protection against resistive scattering in general. The conductivity due to the surface states is determined by scattering of carriers by disorder and defects, the density of which is in turn determined by sample quality. Strong scattering will lead to low mobility, and the mobility is not a topologically protected quantity. To see surface transport one therefore requires very clean samples.
Several features set topological insulators apart from graphene: the twofold valley degeneracy of graphene is not present in topological insulators, while the Hamiltonian is a function of the real spin, rather than pseudospin. This implies that spin dynamics will be qualitatively very different from graphene. Furthermore, the Dirac Hamiltonian is in fact a Rashba Hamiltonian expressed in terms of a rotated spin. Despite the apparent similarities, the study of topological insulators is thus not a simple matter of translating results known from graphene. Due to the dominant spin-orbit interaction it is also very different from ordinary spin-orbit semiconductors.
The outline of this paper is as follows. In Sec. II we introduce the effective Hamiltonian for the surface states of topological insulators and briefly discuss its structure. Next, in Sec. III we derive a kinetic equation for the spin density matrix of topological insulators, including the full scattering term in the presence of an arbitrary elastic scattering potential to linear order in the impurity density. This equation is solved in Sec. IV, yielding the contributions to orders one and zero in the transport scattering time , and used to determine the charge current. Sec. V discusses our results in the context of experiments, focusing on the electron density and impurity density dependence of the conductivity, ways to determine the dominant angles in impurity scattering, and the effect of the contacts on the surface states. We end with a summary and conclusions.
Ii Effective Hamiltonian for topological insulators
The effective Hamiltonian describing the surface states of topological insulators can be written in the form
It is understood that . Near , when the spectrum is accurately described by a Dirac cone, the scalar term is overwhelmed by the spin-dependent term and can be safely neglected, the remaining Hamiltonian being similar to that of graphene, with the exception that represents the true (rotated) electron spin. On the other hand, at finite doping and at usual electron densities of cm the scalar term is no longer a small perturbation, and should be taken into account. The scalar and spin-dependent terms in the Hamiltonian are of the same magnitude when , corresponding to m in BiSe, that is, a density of cm, which is higher than the realistic densities in transport experiments. Consequently, in this article the terms will be treated as a perturbation. The eigenenergies are denoted by , with . We do not take into account small anisotropy terms in the scalar part of the Hamiltonian, such as those discussed in Ref. Zhang et al., 2009b. Terms cubic in in the spin-orbit interaction are in principle also present, in turn making the energy dispersion anisotropic, but they are much smaller than the -linear and quadratic terms in Eq. (1) and are not expected to contribute significantly to quantities discussed in this work.
There are contributions to the charge current from the scalar and spin parts of the density matrix. The current operator is given by
Unlike graphene, where the effective Dirac Hamiltonian is valid even at large electron densities and the current operator is independent of for all ranges of wave vector relevant to transport, in topological insulators the scalar term in the Hamiltonian quadratic in also contributes to the current, and will yield a different number-density dependence.
Transport experiments on topological insulators seek to distinguish the conduction due to the surface states from that due to the bulk states. As mentioned in the introduction, it has proven challenging to lower the chemical potential beneath the bottom of the conduction band, so that the materials studied at present are not strictly speaking insulators. It is necessary for to be below the bulk conduction band, so that one can be certain that there is only surface conduction, and at the same time the requirement that is necessary for the kinetic equation formalism to be applicable. These assumptions will be made in our work.
Iii Kinetic equation
We wish to derive a kinetic equation for the system driven by an electric field in the presence of random, uncorrelated impurities. To this end we begin with the quantum Liouville equation for the density operator ,
where the full Hamiltonian , the band Hamiltonian is defined in Eq. (1), represents the interaction with external fields, is the position operator, and is the impurity potential. We consider a set of time-independent states , where indexes the wave vector and the spin. The matrix elements of are , with the understanding that is a matrix in spin space. The terms and are diagonal in wave vector but off-diagonal in spin, while for elastic scattering in the first Born approximation . The impurities are assumed uncorrelated and the average of over impurity configurations is , where is the impurity density, the crystal volume and the matrix element of the potential of a single impurity.
The density matrix is written as , where is diagonal in wave vector (i.e. ) while is off-diagonal in wave vector (i.e. always in .) The quantity of interest in determining the charge current is , since the current operator is diagonal in wave vector. We therefore derive an effective equation for this quantity by first breaking down the quantum Liouville equation into
and solving Eq. (4b) to first order in
The details of this derivation have been included in Appendix A. We are interested in variations which are slow on the scale of the momentum relaxation time, consequently we do not take into account memory effects and , so that obeys
The integral in Eq. (6b), which represents the scattering term, is performed by inserting a regularizing factor and letting in the end. We consider a potential , which does not have spin dependence, and carry out an average over impurity configurations, keeping the terms to linear order in the impurity density . The scattering term has the form
For this term to be cast in a useful form one needs to complete the time integral and analyze the spin structure of the scattering operator , which is done below.
iii.1 Scattering term
The -diagonal part of the density matrix is a Hermitian matrix, which is decomposed into a scalar part and a spin-dependent part. We write , where represents the scalar part and is the identity matrix in two dimensions. The component is itself a Hermitian matrix, which represents the spin-dependent part of the density matrix and is written purely in terms of the Pauli matrices. Thus, the matrix can be expressed as . With this notation, the scattering term is in turn decomposed into scalar and spin-dependent parts
We assume electron doping, which implies that the chemical potential is in the conduction band . For the scalar part and spin-dependent part of the density matrix the above results in
where is a unit vector in the direction of . We emphasize that there is no backscattering, so the physics of topological insulators is built into the kinetic equation from the beginning. Collecting all terms, we obtain the scattering term as , with
The details of this calculation are contained in Appendix A. The energy -functions . The -functions of are needed in the scattering term in order to ensure agreement with Boltzmann transport. This fact is already seen for a Dirac cone dispersion in graphene where the expression for the conductivity found in Ref. Culcer and Winkler, 2008 using the density-matrix formalism agrees with the Boltzmann transport formula of Ref. Das Sarma et al., unpublished (the definition of differs by a factor of two in these two references.) The necessity of keeping the terms is a result of Zitterbewegung: the two branches are mixed, thus scattering of an electron requires conservation of as well as . In the equivalent kinetic equation for graphene there is no coupling of the scalar and spin distributions because of the particle-hole symmetry inherent in the Dirac Hamiltonian. In topological insulators the term breaks particle-hole symmetry and gives rise to a small coupling of the scalar and spin distributions. Later below we quantify this coupling by an effective time , which as .
We find a series of scattering terms coupling the scalar and spin distributions. Let represent the angle between and , and the unit vector in polar coordinates represent the direction perpendicular to , that is the tangential direction. The unit vectors , , and are related by the transformations
We decompose the matrix and write those two parts in turn as and . The small and are scalars and are given by and , and similarly and . We introduce projection operators , and onto the scalar part, and respectively. We will need the following projections of the scattering term acting on the spin-dependent part of the density matrix
and will be deferred to a later time. Furthermore, the scattering terms coupling the scalar and spin distributions are
Notice the factors of which prohibit backscattering (and lead to Klein tunneling.)
iii.2 Polarization function and effective scattering potential
The above form of the scattering term is valid for any scattering potential, as long as scattering is elastic. To make our analysis more specific we consider a screened Coulomb potential, and evaluate the screening function in the random phase approximation. In this approximation the polarization function is obtained by summing the lowest bubble diagram, and takes the form Hwang and Das Sarma (2007)
where is the equilibrium Fermi distribution function. The static dielectric function, of interest to us in this work, can be written as , where . To determine , we assume and use the Dirac cone approximation as in Ref. Hwang and Das Sarma, 2007. This approximation is justified by the fact that we are working in a regime in which , with the Fermi temperature. The results obtained are in fact correct to linear order in . At for charged impurity scattering the long-wavelength limit of the dielectric function is Hwang and Das Sarma (2007)
yielding the Thomas-Fermi wave vector as .
As a result, in topological insulators the matrix element of a screened Coulomb potential between plane waves is given by
where is the ionic charge (which we will assume to be below in this work) and is the Thomas-Fermi wave vector. The parallel projection of the scattering term becomes
We expand and in Fourier harmonics
and to obtain the expansion of replace . The absence of backscattering, characteristic of topological insulators, is contained in the function . Equation (17) shows that this function contains a factor of which suppresses scattering for .
Iv Solution of the kinetic equation
The Hamiltonian describing the interaction with the electric field is given by , with the position operator. It then follows straightforwardly that the kinetic equation takes the form
The driving term in the kinetic equation is , where represents the equilibrium density matrix, given by . The first term represents the charge density, while the second represents the spin density. We decompose the driving term into a scalar part and a spin-dependent part , as
The latter is further decomposed into a part parallel to the Hamiltonian and a part perpendicular to it
Evidently, since we consider electron doping, . Nevertheless we retain for completeness, since the term will be shown to give rise to a contribution to the conductivity singular at the origin.
We solve the kinetic equation. The first step is to divide it into three equations: one for the scalar part, one for the part parallel to the Hamiltonian and one for the part orthogonal to the Hamiltonian.
We search for the solution as an expansion in the small parameter , where , to be defined below, represents the transport relaxation time. We label the orders in the expansion by superscripts with e.g. representing evaluated to order . From these equations it is evident that the expansions of and begin at order , in other words , whereas begins at order , in other words the leading term in it is independent of . Therefore, in determining the leading terms in and , the coupling to can be neglected. We are then left with two coupled equations for and , which are written as
We must carry out the projections in order to determine the scattering terms above. In the equations involving we have eliminated , since the equations for have been cast in terms of equations for . To first order in , the required scattering terms can be written as
The scattering terms have exactly the same structure. We proceed to solve the equations for and . We note that the RHS on both equations in this case has only Fourier component 1, so we can make the Ansatz and . Observing in addition that allows us to write the coupled equations in the steady state as
where , the transport relaxation time, is given by the expression
and we have also introduced
as a measure of the coupling of the scalar and spin distributions by scattering. The equations are trivial to solve and we obtain in the steady state
The equation is valid to first-order in . Note that, as , and we recover the results for the bare Dirac cone.
Next we examine the term . The kinetic equation for is
The projections of the scattering terms do not contribute to the conductivity. The bare source term is
The result for is
The part averages to zero over directions in momentum space and is not considered any further. Integrating the cosine over time gives
iv.1 Current operator and conductivity
The current density operator is decomposed into a parallel and a perpendicular part, the electric field is assumed and we are looking for the diagonal conductivity
The three contributions to the current operator, representing the scalar part, the part parallel to the Hamiltonian and the part orthogonal to the Hamiltonian, are given explicitly by