Theory of charge-spin conversion at oxide interfaces:
The inverse spin-galvanic effect
We evaluate the non-equilibrium spin polarization induced by an applied electric field for a tight-binding model of electron states at oxides interfaces in LAO/STO heterostructures. By a combination of analytic and numerical approaches we investigate how the spin texture of the electron eigenstates due to the interplay of spin-orbit coupling and inversion asymmetry determines the sign of the induced spin polarization as a function of the chemical potential or band filling, both in the absence and presence of local disorder. With the latter, we find that the induced spin polarization evolves from a non monotonous behavior at zero temperature to a monotonous one at higher temperature. Our results may provide a sound framework for the interpretation of recent experiments.
Theory of charge-spin conversion at oxide interfaces:
The inverse spin-galvanic effect
Roberto Raimondi: E-mail: firstname.lastname@example.org
Keywords: Spin-orbit coupling, spin-charge conversion, oxides interfaces
It is well known that the breaking of the inversion symmetry leads to the so-called Rashba spin-orbit coupling (SOC)[1, 2, 3], where polar and axial vectors transform similarly. Basically this allows for two major possibilities of charge to spin conversion: The spin Hall (SH)  and the inverse spin galvanic (ISG) effect [6, 7], as well as for their Onsager reciprocal effects. While the SH effect converts an electrical current into a spin imbalance at the sample edges via an induced perpendicular spin current, the ISG effect creates a bulk non-equilibrium spin polarization by a flowing electrical current[8, 6, 9, 10, 11]. The inverse SG effect corresponds then to the production of electrical current via the pumping of spin polarization[12, 13]. Both the SG [12, 13] and the ISG [14, 15, 17, 18, 19, 16, 20] effects have been observed in semiconductors. In the first case an electrical current is measured after pumping spin polarized light (SG) whereas in the second case Faraday and Kerr spectroscopies measure the spin polarization induced by the applied current. The SG effect has also been very effectively measured by spin pumping from an adjacent ferromagnet into a metallic interface, into a topological insulator surface[24, 23] and more recently into the two-dimensional electron gas (2DEG) in oxide LAO/STO heterostructures[26, 25, 28, 27]. These latter materials have emerged[31, 29, 30, 32, 33] as very promising materials for the SG and ISG effect, due to the large values of the Rashba SOC parameter as experimentally observed[34, 35, 36, 37] and also theoretically calculated[38, 39, 40], even though it is likely that, due to their complex band structure, the available theory of the SG/ISG effect developed for the 2DEG in semiconductors may not be able to capture a number of specific features. A first step in this direction has been made recently by a combination of analytical diagrammatic and numerical approaches[42, 43].
The layout of the paper is the following. In the next section we introduce a model for the electron states relevant for describing transport at oxide LAO/STO interfaces. In section 3 we provide the necessary formalism of linear response theory for the SG and ISG effects. In section 4 we introduce an approximate effective model for electron states close to band minima. In section 5 we evaluate analytically the SG response for the effective model, whereas in section 6 we introduce disorder and the necessary formalism to handle it. Finally in section 7 we present a fully numerical approach which includes both cases without and with disorder. We conclude in section 8. A number of technical details are provided in the appendices.
2 The Model
The electronic structure of the 2DEG at LAO/STO interfaces, perpendicular to the crystal direction, is usually described [39, 44, 45] within a tight-binding Hamiltonian for the Ti t orbitals, , supplemented by local atomic spin-orbit interactions with Hamiltonian and an interorbital hopping with Hamiltonian which is induced by the interface asymmetry.
The hopping between the orbitals of two neighbouring cubic cells is mediated via intermediate jumps to orbitals. For instance, the hopping between two orbitals along the x axis occurs via two successive hopping and . In the first hop, the overlap, which is of order , yields a positive sign, whereas the sign is negative in the second one. Hence the effective hopping goes like , with being the energy difference between and orbitals. As a result, in the basis the hopping between similar orbitals reads as
with, setting to unity the lattice spacing,
where the energy difference between the and states is due to the confinement of the 2DEG in the -plane.
The atomic SOC is given by
with denoting the Pauli matrices.
Hopping between different orbitals may occur if inversion symmetry is broken, see Fig. 1. Consider, for instance, the two hops along the y direction, and While the first hop is as the first hop of the effective hopping between two orbitals discussed above, the second hop will be forbidden in the presence of inversion symmetry. To see this consider that
where is the lattice site of the orbital. In the same way
In both cases, is the full Hamiltonian. If is invariant with respect to the inversion , then necessarily , because is even, while is odd. Clearly if has terms which are not invariant for , then . As a result the interface asymmetry hopping reads
In the following we use the parameters, eV, eV, eV, eV, eV, eV, which have been derived in Ref.  from projecting DFT on the Wannier states. Note that for the splitting we take a value intermediate between the theoretical ( eV) and the experimental one ( eV). The left panel of Fig. 2 shows the band dispersions along the x axis for these values of the parameters. The bands come naturally in three pairs, which are split by the combined effect of the spin-orbit coupling and the inversion symmetry breaking. For our analysis we have selected three different values of the chemical potential for corresponding filling regimes. For eV, only the lowest pair of bands (1,2) is occupied. The chemical potential eV is close to the Lifshitz point, where the spin-orbit splitting is large and the pairs of bands (3,4) and (5,6) start to be filled. Finally, the chemical potential eV is in the regime, where all pairs of bands (1,2), (3,4) and (5,6) are occupied.
We now analyze the chirality for each eigenstate band by computing the spin at each momentum point of the Fermi surface (FS) according to
where are the eigenfunctions of the system at momentum . The indices and label the orbital and its spin. Then the chirality of the -th band can be obtained from
Fig. 2 shows the chiralities for each pair of bands at selected chemical potentials and the corresponding FSs. For the lowest pair of bands (1,2) the momentum dependent spin pattern displays a vortex-type structure with the core centered at . Thus, even when the FS changes from electron- to hole-like between and , the corresponding chiralities are always confined to without any sign change in . For the middle pair of bands (3,4) the spin structure is composed of two vortex patterns (with the same vorticity) centered at and . As a consequence, the spin texture vanishes along the diagonals and a Rashba-type description along this direction fails. In section 4 we will come back to this point. However, for small and all other momenta the chirality also starts at but then on average becomes smaller with increasing chemical potential and eventually changes sign for . An analogous situation occurs for the uppermost pair of bands where the ’spin-vortex core’ is centered at . In this case the chiralities also change sign upon increasing the chemical potential while at small one again recovers .
3 Linear Response Theory
In this paper we aim at evaluating the spin polarization induced by an externally applied electric field. To be definite we take the electric field along the x axis and the spin polarization along the y axis. To linear order in the applied field we write the spin polarization as
where , the “conductivity” for the ISG effect, can be obtained by the zero-momentum limit of the Fourier transform of the response function (henceforth the symbols in capital letters indicate the operators for spin density and charge current) defined as
where the brackets stand for the quantum-statistical average and is the Heaviside step function. The frequency-dependent ISG conductivity reads
where the first term will be referred to as the Drude singular term and the second as the regular term, in analogy with the terminology used in the case of the optical conductivity. Because under time reversal both the charge current and the spin polarization are odd, according to the Onsager relation, the SG and ISG conductivities are equal. For this reason we will use the term SG conductivity (SGC) for both direct and inverse effects. The calligraphic symbol stands for the principal part. The real and the imaginary parts of the response function are related by the Kramers-Kronig relation (KKR)
By integration over the frequency, thanks to the KKR, the SGC satisfies the following sum rule
due to the fact that for the SGC there is no ’diamagnetic’ contribution as opposed to the optical conductivity.
In the following we are going to apply the above formulae to the model introduced in section 2. To this end, it is instructive to consider first the case of the Rashba SOC for a 2DEG with quadratic dispersion relation in the effective mass approximation. The insight gained in this simpler case will guide us also in the analysis of the model with a complex band structure. We consider then the Rashba-Bychkov Hamiltonian
where is the effective mass and the SOC. The 2DEG is confined to the xy plane and and are the momentum operators along the two coordinate axes. Clearly there are two eigenvalues with the corresponding eigenstates of (9) being plane waves whose spin quantization axis is fixed by the momentum direction
where The ISG response function at finite frequency and momentum reads
where is the Fermi distribution function at temperature . Depending on the values of the spin indices, one has intraband () and interband () contributions. In the dynamic limit, when the momentum goes to zero at finite frequency, the intraband contribution vanishes. For the model of Eq. (9) the interband matrix elements for spin density and charge current read
and the zero-momentum response function becomes
At zero temperature, there are two FSs corresponding to the two spin helicity bands with Fermi momenta . The evaluation of the imaginary part of the zero-momentum response function leads to ()
showing an antisymmetric behavior with respect to the frequency . The spectral weight, at positive frequency, is confined in the range The two frequencies delimiting the interval are nothing but the spin-orbit splitting at the two Fermi surfaces. We note, and this will turn out useful when discussing the numerical calculations, that at finite , the imaginary part remains finite and acquires a linear-in-frequency behavior around the origin, whose slope vanishes as . The Drude weight, according to Eq. (6) can be easily obtained by the KKR relation (7) to read
where is the single-particle density of states of the 2DEG. For the sake of simplicity we have chosen units such . There are two features worth noticing. The first is that the Drude weight is controlled by the sign of the SOC. The second is that the Drude weight arises from the interband transitions between the spin-orbit split bands. This must be compared with the case of optical conductivity for the electron gas, where the Drude weight arises from the diamagnetic contribution to the current. In the present case, due to the sum rule (8), the Drude low-frequency peak yields information about the spectral weight of interband transitions at finite frequency. To the best of our knowledge this feature has not been noticed before.
In the following of the paper we will consider the effect of disorder, but it is instructive to make here an heuristic discussion. In the presence of spin-independent disorder, due to the form (10) of the eigenstates, the electron spin acquires a finite relaxation rate . This mechanism, which is known as the Dyakonov-Perel relaxation, arises because, at each scattering event, the change in momentum also affects the spin eigenstate. As a result, in the diffusive approximation, , the spin density obeys a Bloch equation
where represents the steady-state nonequilibrium spin polarization induced by an applied electric field along the x axis and is the momentum relaxation scattering time (not to be confused with the Pauli matrices ). According to Ref. the Dyakonov-Perel relaxation rate reads
By Fourier transforming (15) to frequency , one obtains the SGC in the form
which has a Lorentzian lineshape and evolves to a singular contribution in the weak scattering limit . More precisely by integrating over frequency one obtains
which reproduces the Drude weight of Eq. (14). Notice that in the last step we made use of the fact that the spin relaxation time becomes twice the momentum relaxation time in the weak scattering limit according to Eq. (16). Eq. (18) seems to violate the sum rule (8), but this is not the case. The form (17) for the SGC has been derived in the diffusive approximation, which is valid for frequencies well below the region of the interband spectral weight. Hence, the form (17) captures only the low frequency spectral weight, which evolves in the singular Drude weight in the limit of vanishing disorder. The effect of disorder is then to eliminate the Drude singular contribution and to yield a finite SGC at zero frequency, which is the result of a finite slope of the imaginary part of the response function. The microscopic approach in the presence of disorder is discussed in section 6 and details about the frequency dependence are developed in the appendix B.
4 Effective Models
Around the point the non-interacting part of the Hamiltonian (1) reads
where . The atomic SOC [, cf. Eq. (2)] lifts the degeneracy between and but leaves the spin degeneracy, cf. Fig. 3. One obtains the new , corresponding to the pairs of bands (3,4) and (5,6) respectively,
In the basis the asymmetry hopping is given by
and the residual coupling of with the -level reads
In the following we restrict to the region close to the point, where , and neglect therefore the two latter terms in resulting in the effective coupling structure depicted in panel (b) of Fig. 3. We can now calculate the effective interactions between levels , in 2nd order perturbation theory
and , either corresponds to the or to the levels. For the states one finds
and similarly for . Inserting the matrix elements yields an effective Rashba SOC
with a negative coupling constant with .
From Fig. 3 one can see that the same matrix elements also mediate the 2nd order interaction between the and states. Since in this case the denominator in Eq. (26) is negative we obtain a positive coupling for the states
The effective interactions between and can be again obtained from 2nd order perturbation theory in the limit which now involves the matrix elements represented by the dashed lines in Fig. 3. The resulting effective coupling reads
and therefore corresponds to a linear Rashba SOC but with a coupling constant .
5 The Clean Limit
In this section we evaluate the Drude weight for the effective models discussed in section 4.
In this case the eigenvalues and eigenvectors corresponding to the Hamiltonian (27) read
with and . As for the Rashba model (9), the spin operator is simply the Pauli matrix and the charge current is similar to the 2DEG case The interband matrix elements read
and the response function (in the zero-temperature limit) gives
which leads to the Drude weight
with an opposite sign as compared to the 2DEG case of Eq. (14).
In this case the eigenvalues and eigenvectors corresponding to the Hamiltonian (29) read
where with and . In this case the spin operator reads , while the charge current has a more complicated structure as compared to the 2DEG case . The interband matrix elements read
In the response function
the factors disappear and the Drude weight reads
In this case the eigenvalues and eigenvectors corresponding to the Hamiltonian (28) read
with and . In this case the spin operator reads , while the charge current is . The interband matrix elements are
The response function is
and the Drude weight is
6 The Disordered Limit
It is well known that in the presence of disorder, the Drude weight in the formula for the optical conductivity is suppressed and the spectral weight goes into the regular part. In the Drude model, the regular part, as function of the frequency, has a Lorentzian shape whose width is controlled by the scattering rate (not to be confused with the Pauli matrices). Such a transfer of spectral weight from the singular to the regular part occurs also in the case of the SGC. To this end we need to introduce disorder in our model. This will be done in the numerical computation of the next section, whereas in this section we introduce disorder within the effective models derived in section 4 by using the standard diagrammatic impurity technique. This technique has been applied to the Rashba model for the evaluation of the ISG effect, anisotropy magnetoresistance[47, 50] and spin Hall effect. We review here the basic aspects by focusing on the case of the xy-bands, which is equivalent to the Bychkov-Rashba model in the 2DEG. By following the standard procedure, disorder is introduced as a random potential , with zero average and white-noise correlations , with being the impurity concentration. By Fermi golden rule, one associates a scattering rate , where is the single-particle density of state previously introduced in Eq. (14). We will consider the weak-disorder limit which is controlled by the small parameter , with the Fermi energy. In the diagrammatic impurity technique, the first step is the introduction of the irreducible self-energy in the self-consistent Born approximation for the electron Green function.
6.1 The case of the bands
The Green function, due to the SOC of the lowest pair of bands of Eq. (27), can be expanded in Pauli matrices as and explictly reads
and the self-energy has the form
the minus and plus signs applying to the retarded (R) and advanced (A) sectors, respectively. The scattering time entering Eq. (38) is exactly the one required by the Fermi golden rule. It is worth noticing that the self-energy is proportional to the identity matrix in the spin space. Once the Green function is known, we may compute the SGC by means of the Kubo formula
which can be obtained from the expression (5), after averaging over the disorder configurations, represented as . In the above the symbol involves all degrees of freedom, i.e. spin and space coordinates. The disorder average in Eq. (39) enters in two ways. The first is to use the disorder-averaged Green function given in Eq. (36). The second is the introduction, to lowest order in the expansion parameter , of the so-called ladder diagrams, which lead to vertex corrections. The vertex corrections procedure can be performed either for the spin or charge vertex of Eq. (39). Here we consider the vertex correction for the charge current vertex. The dressed vertex obeys the Bethe–Salpeter equation
which results from the infinite summation of ladder diagrams, as shown in Fig. 4. In terms of the dressed vertex the SGC reads
where now the lower case trace symbol involves the spin degrees of freedom only. The problem is then reduced to the solution of the Bethe–Salpeter equation (40) and to the evaluation of the bubble (41). In general the Bethe–Salpeter equation is an integral equation. However, in the present case of white-noise disorder, the Bethe–Salpeter equation becomes an algebraic one, even though still having a spin structure. In the appendix A we provide the details of the solution of Eq. (40), which leads to
which shows that the vertex corrections exactly cancel the interband matrix elements of the charge current vertex. As a result, the evaluation of Eq. (41) leads to
which must be compared with the Drude weight evaluated in Eq. (31).
6.2 The case of the bands
According to the analysis of appendix A, the dressed charge current vertex reads
The evaluation then of Eq. (41) leads to
which has a sign opposite to that of the bands. In Eq. (44) the combination plays the role of an effective SOC, whereas is the spin dressing factor accounting for the interactions in the original model.
6.3 The case of the bands
According to the analysis of appendix A, the dressed charge current vertex reads
The evaluation then of Eq. (41) leads to
which shows again a change of sign with respect to that of the bands. Also here the combination is the spin dressing factor accounting for the interactions in the original model.
7 The Numerical Approach
In this section we present our numerical results. The starting point is the response function defined in Eq. (5), which may be expressed as follows
where and are quantum numbers labelling the eigenstates of the Hamiltonian. For instance, in the absence of disorder, the index includes the crystal momentum, the orbital and spin degrees of freedom. The symbol stands for the Fermi function evaluated at the energy of the eigenstate . In Eq. (46) is the number of lattice sites. The numerical evaluation is performed on a finite system and then it is convenient to separate from the outset the Drude singular weight from the regular part as follows