Extrinsic Spin Hall Effect from Anisotropic Rashba Spin-Orbit Coupling in Graphene

Extrinsic Spin Hall Effect from Anisotropic Rashba Spin-Orbit Coupling in Graphene

H.-Y. Yang Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan    Chunli Huang Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan    H. Ochoa Fundación IMDEA Nanociencia, Cantoblanco 28049, Madrid, Spain.    M. A. Cazalilla Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Department of Physics and National Center for Theoretical Sciences (NCTS), National Tsing Hua University, Hsinchu City, Taiwan 300. Donostia International Physics Center (DIPC), Manuel de Lardizabal, 4. E-20018 San Sebastian, Spain.
July 12, 2019
Abstract

We study the effect of anisotropy of the Rashba coupling on the extrinsic spin Hall effect due to spin-orbit active adatoms on graphene. In addition to the intrinsic spin-orbit coupling, a generalized anisotropic Rashba coupling arising from the breakdown of both mirror and hexagonal symmetries of pristine graphene is considered. We find that Rashba anisotropy can strongly modify the dependence of the spin Hall angle on carrier concentration. Our model provides a simple and general description of the skew scattering mechanism due to the spin-orbit coupling that is induced by proximity to large adatom clusters.

pacs:

I Introduction

The spin Hall effect (SHE)Dyakonov and Perel (1971a, b); Hirsch (1999); Zhang (2000); Nagaosa et al. (2010) has been intensively investigated in the last few decades due to its potential applications in spintronics.Žutić et al. (2004); Nagaosa et al. (2010) Generally speaking, the microscopic mechanisms of SHE can be classified into either intrinsic Sinova et al. (2004); Tanaka et al. (2008) or extrinsic.Tse and Sarma (2006); Ferreira et al. (2014) In both cases, the existence of spin-orbit coupling (SOC) in the material or heterostructure is required. The intrinsic mechanism is a consequence of the band structure of the material whereas the extrinsic mechanism stems from scattering of the charge carriers by impurities that locally induce SOC.

Since SOC is a relativistic effect that is typically strongest in materials containing heavy atoms, the SOC in graphene Novoselov et al. (2005); Castro Neto et al. (2009) is expected to be weak. Huertas-Hernando et al. (2006); Min et al. (2006); Gmitra et al. (2009) Therefore, graphene has been suggested as an ideal material for passive spintronics, for which a long spin diffusion length is required and SOC in the material is a major limiting factor. You et al. (2015)

However, motivated by the search of materials exhibiting the quantum spin Hall effect, L.Kane and Mele (2005) it has been theoretically predicted Weeks et al. (2011); Hu et al. (2012), and experimentally observedBalakrishnan et al. (2014) that SOC can be greatly enhanced in graphene by means of adatom deposition. In the limit of a dilute number of impurities (i.e. adatoms), in which the excellent charge carrier mobility properties of graphene are not strongly modified, heavy adtom clusters have been predicted to induce a sizable SHE. Ferreira et al. (2014) Experimentally, a large spin Hall angle () has been reported by Balakrishnan and coworkers Balakrishnan et al. (2014) in devices made from chemical-vapor-deposited (CVD) graphene. The phenomenon was explained Balakrishnan et al. (2014) by the combination of resonant scattering and the skew scattering off by residual Cu clusters resulting from the CVD process. It was experimentally estimated that the latter can induce a SOC of the order of meV.

Nevertheless, the nature of the SOC induced by the adatoms depends on their arrangement relative to hexagonal unit cell of graphene. The latter can lower the symmetry from the hexagonal symmetry of the carbon monolayer. It also depends on the symmetry of the orbitals that hybridize with the bands of graphene, since this hybridization is ultimately responsible for both the proximity induced SOC and the resonant scattering. This also applies to heavy metal substrates or large clusters, where certain crystalline ordering is possible. In this regard, it is worth mentioning, for instance, the differences between gold intercalation  Marchenko et al. (2012), which leads to a conventional Rashba splitting of graphene bands, and lead, Calleja et al. (2015) which results in a proximity induced Rashba SOC where the two terms of the coupling have different weight. The latter is a consequence of the reduced orthorhombic symmetry of the composite (graphene + substrate) system. Such coupling is therefore an anisotropic generalization of the Rashba SOC, which arises due to the breakdown of both mirror and 6-fold rotation symmetry. Similar features have been reported recently in graphene intercalated with platinum. Klimovskikh et al. (2015)

In this work, we shall investigate the skew scattering mechanism arising from the SOC induced by extrinsic scatterers. Unlike previous studies, Ferreira et al. (2014); Balakrishnan et al. (2014) we shall focus on the understanding of the effects of the Rashba anisotropy on the charge and especially spin transport properties, and in particular, the spin Hall angle. To this end, we shall first solve the scattering problem of an anisotropic SOC-active scatterer. Stauber et al. (2007); Ferreira et al. (2011) From the single-impurity scattering data, we shall derive the relaxation times that parametrize the collision integral of the linearized Boltzmann transport equation (BTE)Ziman (1972), which allows us to compute the spin Hall angle. Finally, we shall also compare the interplay and interference between different scattering potentials.

The rest of the article is organized as follows. We present the details of our theoretical model in Sec. II. First, we discuss the way the symmetries of monolayer graphene decorated with adatoms constrain the form of the SOC in the Hamiltonian of the system. Then, the single scatterer problem is solved. Using the scattering data (i.e. T-matrix) of the single scatterer problem, the linearized Boltzmann transport equation is also solved and the transport properties of the system are obtained. In Sec. III, we discuss the most salient features of our results, namely the change of spin Hall angle and the conductivity of the system as a function of the anisotropy parameter. A summary of the main results of this work is provided in Sec. IV. Finally, let us mention that the Appendices contain the most technical details of the work.

Ii Model

ii.1 Scattering potentials

In this study, we shall consider a dilute ensemble of scatterers that create a (disorder) potential that is smooth in the atomic scale of graphene. As a consequence, we shall neglect scattering between the two valleys at the opposite corners of the hexagonal Brillouin zone (i.e. ). Therefore, most of the discussion below applies to a single valley (i.e. ) unless otherwise stated.

In order to understand the charge and spin transport properties of the system, we shall rely upon the semiclassical Boltzmann transport equation (BTE). The latter applies to doped graphene (i.e. when the Fermi energy measured from the Dirac point ) in the limit where the distance between scatterers is much larger than the Fermi wavelength. Therefore, the results obtained from the BTE should be regarded as providing some sort of interpolation between the hole doped Fermi liquid () and the electron doped Fermi liquid (i.e. ) regime.

In the dilute impurity limit, the collision term of the BTE is determined by the scattering data for a single scatterer. Luttinger and Kohn (1958) Thus, we first analyze the scattering problem of a single scatterer, for which the Hamiltonian describing the electron dynamics in the long-wavelength limit can be generally written as follows:

(1)

where the sign applies to the valley at crystal momentum and () are the Pauli matrices associated with the sublattice degrees of freedom of the wave function. The Pauli matrices acting on electron spin are denoted by . The first two terms in Eq. (1) correspond to the Hamiltonian of the pristine graphene which describes the the electronic bands near the points.

Among the possible time-reversal invariant impurity potentials, we shall focus on the scalar potential (), the intrinsic SOC () and the Rashba potential (). These three are invariant under the point group , which is generated by the 6-fold rotation axis perpendicular graphene intersecting the center of the hexagonal unit cell and 6 reflection planes containing such an axis. The latter group describes the rotation and mirror symmetries of monolayer graphene excluding the mirror reflection about the graphene plane which takes . However, the scalar potential (),

(2)

and intrinsic or Kane-Mele SOC term: L.Kane and Mele (2005)

(3)

are invariant under the larger point group , which includes the mirror reflection for which . On the other hand, Rashba SOC is associated with the lack of the mirror reflection symmetry. Typically, this symmetry is broken and lowered to by the presence of a substrate, adatoms, and/or ripples.

The Rashba SOC is invariant under the group of pristine graphene and takes the following form:

(4)

However, in general, the planar symmetry can be broken, for example, due to the different symmetries of graphene lattice and the substrate, or the arrangement (relative to the hexagonal unit cell of graphene) of the adatoms in a large cluster in the proximity of the carbon layer. As a result, the symmetry can be lowered from hexagonal (i.e. ) to rectangular (i.e. ), for instance. This can be achieved by deposition or intercalation of a metal with either cubic or orthorhombic symmetry.Calleja et al. (2015) In Ref. Calleja et al., 2015, it was shown from symmetry arguments and first principle calculations that the two terms of the Rashba SOC can acquire different weights, which leads to an anisotropic form of the Rashba SOC potential:

(5)

In this work, we shall study the effect of this anisotropy on the skew scattering mechanism and its contribution to the spin Hall effect.

Before turning our attention to the study of the scattering problem by such anisotropic Rashba potential, it is useful to analyze the symmetries of (1) in the presence of the anisotropic Rashba SOC potential, Eq. (5). For reasons that shall become clear below, it is convenient to write the anisotropic Rashba SOC as the sum of two terms, , where

(6)
(7)

with

(8)

where () sign applies to (), respectively. For we recover the standard Rashba SOC. In the opposite limit, , a SOC to be termed ‘non-standard’ Rashba is obtained. This representation enables us to display more clearly how the anisotropy in the Rashba SOC violates the conservation of the angular momentum projected onto the axis. Let us recall the definition of the -component of angular momentum operator:

(9)

Notice that since are functions of , as we have assumed, the scalar potential, intrinsic and standard Rashba SOC commute with . However, when the Rashba SOC is anisotropic, is no longer conserved and the culprit for this violation is the non-standard Rashba SOC introduced above. Nevertheless, in the special case where (i.e. ), the following quantity:

(10)

is conserved instead of . This will become useful in our investigation of this special limit for a more general type of scatterers than those considered in the following (see Appendix C.1).

Note that the lack of conservation of by the anisotropic Rashba SOC makes it impossible to employ a partial wave expansion to solve the scattering problem as it was done in Ref. Ferreira et al., 2014. Nonetheless, since we are interested in the scattering by clusters of adatoms of characteristic size ( being the interatomic distance in graphene), and for typical experimental parameters in doped graphene (where is the Fermi wave vector), we shall approximate the cluster potentials by Dirac delta functions, i.e.,

(11)
(12)
(13)
(14)

Let us also define , and as the strength of the potentials in units of energy. In passing, we also note that a similar model (with only intrinsic SOC) was successfully employed to account for the giant SHE observed in CVD graphene and attributed to the SOC induced by residual Cu atom clusters. (Balakrishnan et al., 2014) Thus, the potentials in Eq. (1) take the form:

(15)

where is the strength of the potential and are matrices acting upon the sublattice-spin degrees of freedom. Explicitly, the matrices are,

(16)
(17)
(18)

When written in terms of and , the conservation of by and the failure to do so by becomes apparent.

Interestingly, the matrices form a closed group under (matrix) multiplication. This means that the product of two of these matrices can be written as a linear combination of

(19)

The coefficients can be read off from table 1. As a mathematical curiosity, it is worth noting that the group is abelian, as can be expected for a group of order four. Out of the two possible order four groups, this corresponds to the Klein group.

ii.2 Single scatterer problem

In this subsection, the Lippmann-Schwinger (LS) equation for the single impurity problem with the choice of potentials discussed in previous section will be solved. The LS wave equation reads:

0
0
Table 1: The matrices span an order four group under matrix multiplication. Note that is orthogonal to , i.e. the probability amplitude of being scattered consecutively by both standard Rashba SOC and non-standard Rashba SOC is zero.
(20)

where

(21)
(22)

and given by Eq. (15). In the LS equation is the incident wave function from conduction band with momentum and spin state described by the spinor , where . The spin quantization axis is taken to be the axis, which is perpendicular to the graphene plane. The angle of incidence is ; the normalization area of the system is taken to be unity. The scattered wave function is . Note that does not carry a spin index because it is not an eigenstate of in general. The (retarded) Green’s function is a matrix acting both in the sublattice pseudo-spin and electron-spin space (see Appendix A for details).

The Dirac-delta function potential allows us to express the solution to the LS equation in terms of the matrices,

(23)

where . The coefficients are functions of the couplings that appear when we solve for (see Appendix A for details). In particular, it can be seen that for , , and it then follows that the scattered wave function is an eigenstate of because the expression for the scattered wave no longer contains . On the other hand, when then , the scattered wave function becomes an eigenstate of (cf. Eq. 10).

Using Eq. (19) and introducing the coefficients

(24)

the solution to the LS equation (20) takes the following compact form:

(25)

Note that the right hand-side of the above expression contains only known quantities. After expanding the Green function asymptotically at large distances, the scattered wave can be written as the sum of an incident and an outgoing wave:

(26)

In the above expression we have introduced the scattering amplitudes given by and . From it, the differential scattering cross-section can be calculated using:

(27)
(28)

where is the scattering angle. We refer the reader to the Appendix A for the detailed form of the scattering amplitude and how it is related with the scattering T-matrix that enters in the collision term of the Boltzmann transport equation.

ii.3 Transport properties

In order to compute the charge and spin transport properties of a dilute random ensemble of identical clusters of areal density , we use the semi-classical Boltzmann transport equation (BTE). Ziman (1972); Ferreira et al. (2014) The details of its solution in the linearized approximation are reviewed in Appendix B. The exact solution of this equation (Ferreira et al., 2014) allows us to obtain the charge, and spin Hall conductivities ( and , respectively):

(29)
(30)

where is the Fermi-Dirac distribution. The different scattering times, are defined in Appendix B and can be derived from the differential scattering cross section. In particular, we would like to point out that

(31)

where and are the areal density of the impurities and the Fermi velocity of pristine graphene, respectively. Here, the notation and . is the single-scatterer transport cross-section which exhibit sharp peaks at the resonance energies of the single scatterer.

The figure of merit determining the efficiency in the charge current to spin current conversion, known as spin Hall “angle”, is defined as the ratio:

(32)

At zero temperature, the spin Hall angle reduces to the ratio (see Appendix B):

(33)
Figure 1: (a) Spin Hall angle versus energy of the carrier () for different values of the strength of scalar potential . We set nm, K, meV, and meV. The location of resonances gradually approaches the Dirac point when v increases, illustrating the general feature of resonant scattering even for the anisotropic case. (b) Corresponding starred transport cross sections of each case. See App. B for the definition of starred transport cross section.
Figure 2: (a) Spin Hall Angle versus the energy of the scattered electron for different anisotropy values of the Rasha SOC anisotropy parameter . We have chosen the strength of the scalar part of the potential, eV and other parameters are the same as Fig. 1. The magnitude of the spin Hall angle is enhanced near the position of the scattering resonance, which is signalled by a peak in the transport scattering cross section (b) Corresponding starred transport cross sections of each case. The curves for the transport cross section and the spin Hall angle are asymmetric about because the Rashba SOC breaks the particle-hole symmetry of the Hamiltonian.
Figure 3: (a)-(d) Temperature dependence of versus plot at meV and four different . (e)-(h) Temperature dependence of versus plot at meV and four different . veV in all the eight plots. Other parameters are the same as in Fig.2. and Fig. 1.

Iii Results and discussion

The following discussion is about the effects of the anisotropy in the Rashba SOC. The degree of anisotropy in Rashba SOC will be phrased in terms of the anisotropy parameter:

(34)

Note that for the standard (“isotropic”) Rashba SOC, . For , the Rashba-like SOC is of the form (with for ), which has been termed non-standard Rashba in Sec, II.1. In the range , is a measure of the degree to which the symmetry is broken by the adatom arrangement within the clusters. Close to the deviation from Rashba and the perfect symmetric situation is small. On the other hand, having requires a strong breaking of the symmetry.

In Fig. 1, we show the dependence of the spin Hall angle and the transport cross section at zero temperature on the carrier energy for different values of strength of the scalar potential (cf. Eq. 11) for a rather anisotropic Rashba-like SOC corresponding to . It can be seen that the enhancement of the spin Hall angle still takes place around the values of for which exhibits a peak, that is, a scattering resonance. This is in agreement with what was already pointed out in Ref. Ferreira et al., 2014 for the isotropic Rashba SOC. Physically, this is also expected, because at resonance the scattering electron or hole spends most time near the scatterer and therefore it can also experience the effect of the locally induced SOC. The enhancement of is suppressed at large values of . To understand this effect qualitatively, let us recall that and are both determined by the T-matrix, which obeys the LS equation:

(35)

where is the retarded Green’s function and , being the scalar potential and the SOC part of the potential. In the limit where , the solution to Eq. (35) can be (loosely) written as:

(36)

where the last expression applies to the large limit. Thus, to leading order, the cross section is determined by the first term of the right hand-side, whereas is determined by the the second term. Hence, is expected to decrease at large , as shown in Fig. 1.

For a given set of , , and , Fig. 2 shows the behaviour of and as a function of the incident electron energy at different values of anisotropy parameter . For the values of close to those corresponding to the non-standard Rashba SOC (i.e. for ), the energy dependence is strongly modified. On the other hand, for the case of a delta function potential, the anisotropy has a less pronounced effect on the cross section .

The observations made above remain largely unchanged when the effect of finite temperature is taken into account, see Fig 3. As shown there, thermal fluctuations and the associated smearing of the Fermi distribution, smooth out the sharper features of the (Fermi) energy dependence of found at and suppress the magnitude of . This can be seen in the left panel in Fig. 3 for case of a pure (i.e. ) anisotropic Rashba SOC and on the right for . The plots on the left panel also illustrate that, (i.e. ) the spin current as well as the spin Hall angle vanish (cf. second plot from the bottom on the left). This is because the quantization axis for the spin current is aligned along the axis, whereas for , commutes with the Hamiltonian. As pointed out above for , the energy dependence (relative to the isotropic case), is most strongly affected as approaches (see plot for ). However, the effect of the anisotropy is less pronounced for . This conclusion still holds true when the scatterer also induces intrinsic SOC on the graphene layer (i.e. for ), as it is shown in the right panel of Fig. 3.

Finally, it is also worth mentioning that the observation of a very different energy dependence as is independent of the assumption of a Dirac delta potential. This is investigated in detail in Appendix C, where circular (i.e. ‘pill-box’ shaped) scatterer is assumed and the scattering properties in the the case of standard and non-standard Rashba are obtained. The results for the energy dependence of and are displayed in Fig. 4. The more complicated internal structure of the finite-radius circular scatter, whose wave functions are distorted in different ways by the standard and non-standard Rashba SOC, shows up in a very different resonant peak structure exhibited by the transport cross section and the spin Hall angle .

Figure 4: (a) Comparison between the spin Hall angle () of a standard Rashba (SR) and non-standard Rashba (NS) SOC for a circular (i.e. ‘pill-box’) scattering potential. (b) Transport cross section. We have assumed the radius of the scatterer nm, eV and meV. The same asymmetric resonant property as that discussed in the main context recurs in the finite-disk modeling.

Iv Summary and Conclusions

We have analyzed a simple model to understand the effects of the anisotropy of the proximity-induced Rashba spin-orbit coupling (SOC) on the spin Hall effect. The anisotropy arises as a consequence of the arrangement of adatoms in the clusters decorating a single layer of graphene and takes the form:

(37)

On symmetry grounds, such a SOC is effectively generated when the arrangement lowers the symmetry of the system from the hexagonal symmetry (i.e. the group) of graphene.

From our analysis we conclude that the anisotropy in the Rashba SOC does not modify the observation that the spin Hall angle in graphene is enhanced by the scattering resonances Ferreira et al. (2014) that appear near the Dirac point. In addition, the dependence on the carrier concentration (or equivalently the Fermi energy) of the spin Hall angle is also not strongly modified for weak anisotropy. However, when the parameter and especially when approaches , we have found the Fermi energy dependence to strongly deviate from the one observed in the isotropic case (corresponding to or ). This conclusion is robust against finite temperature effects, which somewhat smoothes out the Fermi energy dependence and suppress the value of . It is also not modified by relaxing our assumption of a zero-range (i.e. Dirac-delta) potentials.

In our study, we assumed a single type of single scatter model. In a realistic experiment (like the one envisaged in Ref. Balakrishnan et al., 2014), several kinds of scatterers may be present, some of which do not induce SOC. However, we expect that the above qualitative features will remain unchanged. Experimentally, it would be interesting to study the differences in the Fermi energy (i.e. doping) dependence of the spin Hall angle for clusters of different atomic species, which can lead to different anisotropic Rashba couplings. Indeed, experimental evidence for intercalated Pb islands obtained in Ref. Calleja et al., 2015 seems to indicate that this metal can induce a rather anisotropic Rashba coupling with . Similar deviations from the standard Rashba splitting of graphene bands have been recently reported in platinum intercalated devices.Klimovskikh et al. (2015) Our study identifies the signatures of such deviations in the carriers’ skew scattering properties, providing a way to probe different spin textures in transport.

Acknowledgements.
C.H’s work was supported in part by the Singapore National Research Foundation grant No. NRFF2012-02. HYY and MAC acknowledge support from the Ministry of Science and Technology in Taiwan. HO acknowledges support from the European Union’s Seventh Framework Programme (FP7/2007-2013) through the ERC Advanced Grant NOVGRAPHENE (GA No. 290846).

Appendix A Single scatterer problem

In this Appendix, we shall provide the details of the solution of the Lippmann-Schwinger (LS) equation. To this end, we first recall the form of the retarded Green’s function in the continuum limit of single-layer graphene, , where is the identity matrix in spin space. The function is given by

(38)

Hence,

(39)

The integral in the above expression is the Green’s function for the two-dimensional Helmholtz equation, which reads:

(40)

where is the Hankel function of the first kind. Inserting this result in the expression for and using yields:

(41)

where

(42)

Due to translational invariance, the Green function is only a function of the difference in position. Here , with , is the angle between the vector and the x-axis. Note that we have chosen the retarded Green’s function for both electrons (i.e. ) and holes (i.e. ). Setting , we arrive at

(43)

To simplify the discussion, in what follows, we shall limit ourselves to the study of the scattering of electrons within the conduction band (i.e. for ), although in the main text both the valence () and conduction () bands have been considered.

As described in the Sec. II.2, in order to obtain the asymptotic wave function describing the outgoing scattered wave, we need to consider the limit , in which the Green’s function becomes:

(44)
(45)

where is the momentum of the scattered wave and is the angle subtended between the scattered momentum with axis.

Accounting for the spin degree of freedom, the asymptotic form of Green’s function reads:,

(46)

Thus, we are equiped to solve the Lippmann-Schwinger (LS) equation of the scattering problem. Assuming an incident electron from the conduction band with momentum and means that (we work assuming a normalization area equal to unity):

(47)

Thus, the LS equation ( Eq. 20 ) becomes

(48)

where we have introduced the T-matrix, which can be defined by the equation . Note that the scattered wave does not carry a spin index because it is not an eigenstate of . Indeed, for a non-spin conserving potential like Rashba, the scattered wave is a combination of the incident wave with momentum and spin and an scattered radial spin up (spin down) wave with amplitude given by ().

From the above result, the scattering amplitude can be related to the T-matrix by the following expression:

(49)

For elastic scattering (i.e. ), the scattering amplitude is a function of the scattering angle , i.e.

(50)

Recalling that the impurity potential is given by Eq. (15), the T-matrix can be written as follows:

(51)
(52)
(53)

where we have used (see below) . Writing where the coefficient can be read off from group multiplication table (cf. 1). In addition, let us define:

(54)

The T-matrix can be obtained as follow,

(55)

In the above equations, it is understood that the indices run over the set . Upon setting in Eq. (20),

(56)

Hence,

(57)
(58)

The coefficients are obtained by inverting the matrix and (exactly) projecting onto the basis of matrices, which yields:

(59)
(60)
(61)
(62)

In the above expressions is a scalar and is the matrix Green’s function at the origin. The is obtained by the imposing a cut-off at high momenta. Setting in Eq. (A), we have

(63)

The integral is cut-off at momenta , where is of the order of the actual spatial range of the scatterer potential.

Appendix B Solution of the Boltzmann Equation

In this appendix, in order to make the article self-contained, we review the solution of the linearized Boltzmann equation (BTE) obtained in Ref. Ferreira et al., 2014. For an external DC electric field, the linearized BTE takes the form:Ziman (1972)

(64)

where is the applied electric field, the electron charge, is the equilibrium Fermi-Dirac distribution, and

(65)

is the carrier velocity in graphene with the angle (not to be confused with with the free Hamiltonian eigenstate ); is the band index ( for electrons, for holes) and is the distribution function for electrons with spin projection and Bloch wave-vector . The spin quantization axis is chosen to be the axis perpendicular to the graphene plane, which we take to be the z-axis. The term denotes the collision integral,Ferreira et al. (2014)

(66)

is the scattering rateLuttinger and Kohn (1958) from state to due to the presence of impurities:

Here is the T-matrix that has been explicitly obtained in Appendix A and is the density of impurities. The linearized BTE can be solved exactly by the following ansatz for :

(68)

Introducing this ansatz in (64) and setting and for longitudinal and transverse response respectively, we obtain the following system of algebraic equations for and :

(69)
(70)

where and the coefficients are defined as

(71)
(72)
(73)

where is the scattering angle.

Note that time-reversal symmetry imposes several constraints on Eq. (69) and Eq. (70). In particular, it requires that , , , , and , where we further denote as the opposite spin of to make the notation more compact. These relations are used to simplify the above system of equations. Using these coefficients