Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magneto-electric effects in normal and superconducting systems

TL;DR

This paper derives a generalized Usadel equation for materials with linear spin-orbit coupling, unifying the description of magneto-electric effects in both normal and superconducting states.

Contribution

It introduces a unified theoretical framework linking magneto-electric effects in normal and superconducting systems through an extended Usadel equation.

Findings

Derivation of Usadel equation for gyrotropic materials with spin-orbit coupling

Connection of magneto-electric effects to superconducting counterparts

Application example on bulk spin-galvanic effect

Abstract

The Usadel equation is the standard theoretical tool for the description of superconducting structures in the diffusive limit. Here I derive the Usadel equation for gyrotropic materials with a generic linear in momentum spin-orbit coupling. It accounts for the spin-charge/singlet-triplet coupling and in the normal state reduces to the system of spin-charge diffusion equations describing various magneto-electric effects, such as the spin Hall effect (SHE), the spin-galvanic effect (SGE) and their inverses. Therefore the derived Usadel equation establishes a direct connection of these effects to their superconducting counterparts. The working power of the present formalism is illustrated on the example of the bulk SGE.

Equations

$$\hat{H}=\frac{1}{2m}{\left({\hat{p}}_k-{\mathcal{A}}_k\right)}^2-{\mathcal{A}}_0+V_{dis},$$

$$v_F{n}_k{\tilde{\nabla }}_k\check{g}+[\check{\Omega },\check{g}]-\frac{1}{2m}\{{\mathcal{F}}_{ij},n_i\frac{\partial \check{g}}{\partial n_j}\}=-\frac{1}{2\tau }[⟨\check{g}⟩,\check{g}]$$

$$\check{\Omega }=(\hat{\omega }-i{\mathcal{A}}_0){\tau }_3-i\check{\Delta }$$

$$\check{g}(\mathbf{n})={\check{g}}_0+n_k{\check{g}}_k$$

$$\frac{1}{d}{v}_F{\tilde{\nabla }}_k{\check{g}}_k+[\check{\Omega },{\check{g}}_0]$$

$$\tau v_F{\tilde{\nabla }}_k{\check{g}}_0+\tau [\check{\Omega },{\check{g}}_k]-\frac{\tau }{2m}\{{\mathcal{F}}_{kj},{\check{g}}_j\}$$

$${\check{g}}_k={\check{g}}_k^{(1)}+{\check{g}}_k^{(2)}+{\check{g}}_k^{(3)}$$

$$-\tau v_F{\tilde{\nabla }}_k{\check{g}}_0$$

$$\frac{\tau }{2m}\{{\mathcal{F}}_{kj},{\check{g}}_j^{(1)}\}$$

$$-\tau [\check{\Omega },{\check{g}}_k^{(1)}]$$

$${\check{g}}_0^2=1$$

$$-\tau v_F{\tilde{\nabla }}_k{\check{g}}_0^2=\frac{1}{2}[{\check{g}}_0^2,{\check{g}}_k^{(1)}]$$

$${\check{g}}_k^{(1)}=-\tau v_F{\check{g}}_0{\tilde{\nabla }}_k{\check{g}}_0\equiv \tau v_F({\tilde{\nabla }}_k{\check{g}}_0){\check{g}}_0,$$

$$\frac{v_F{\tau }^2}{2m}\left({\mathcal{F}}_{kj}({\tilde{\nabla }}_j{\check{g}}_0){\check{g}}_0-{\check{g}}_0({\tilde{\nabla }}_j{\check{g}}_0){\mathcal{F}}_{kj}\right)=\frac{1}{2}({\check{g}}_0{\check{g}}_k^{(2)}-{\check{g}}_k^{(2)}{\check{g}}_0)$$

$$\frac{v_F{\tau }^2}{2m}\left({\check{g}}_0{\mathcal{F}}_{kj}({\tilde{\nabla }}_j{\check{g}}_0)-({\tilde{\nabla }}_j{\check{g}}_0){\mathcal{F}}_{kj}{\check{g}}_0\right)=-\frac{1}{2}({\check{g}}_0{\check{g}}_k^{(2)}-{\check{g}}_k^{(2)}{\check{g}}_0)$$

$$-\frac{v_F{\tau }^2}{2m}\left({\check{g}}_0\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j{\check{g}}_0\}-\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j{\check{g}}_0\}{\check{g}}_0\right)=({\check{g}}_0{\check{g}}_k^{(2)}-{\check{g}}_k^{(2)}{\check{g}}_0)$$

$${\check{g}}_k^{(2)}=-\frac{v_F{\tau }^2}{2m}\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j{\check{g}}_0\}$$

$$-{\tau }^2{v}_F\left(\check{\Omega }{\check{g}}_0({\tilde{\nabla }}_k{\check{g}}_0)-{\check{g}}_0({\tilde{\nabla }}_k{\check{g}}_0)\check{\Omega }\right)={\check{g}}_0{\check{g}}_k^{(3)}$$

$${\check{g}}_k^{(3)}=-{\tau }^2{v}_F[\check{\Omega },{\tilde{\nabla }}_k{\check{g}}_0]+O({\tau }^3)={\tau }^2{v}_F[{\tilde{\nabla }}_k\check{\Omega },{\check{g}}_0]+O({\tau }^3)$$

$${\check{g}}_k^{(3)}=-i{\tau }^2{v}_F[{\tau }_3{\mathcal{F}}_{k0},{\check{g}}_0]$$

$${\check{g}}_k=-\tau v_F{\check{g}}_0{\tilde{\nabla }}_k{\check{g}}_0-\frac{v_F{\tau }^2}{2m}\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j{\check{g}}_0\}-i{\tau }^2{v}_F[{\tau }_3{\mathcal{F}}_{k0},{\check{g}}_0]$$

$${\tilde{\nabla }}_k{\check{J}}_k+[(\hat{\omega }-i{\mathcal{A}}_0){\tau }_3-i\check{\Delta },\check{g}]=0$$

$${\check{J}}_k=-D\left(\check{g}{\tilde{\nabla }}_k\check{g}+\frac{\tau }{2m}\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j\check{g}\}+i\tau [{\tau }_3{\mathcal{F}}_{k0},\check{g}]\right)$$

$${\tilde{\nabla }}_k\{{\mathcal{F}}_{kj},{\tilde{\nabla }}_j\check{g}\}=\{{\tilde{\nabla }}_k{\mathcal{F}}_{kj},{\tilde{\nabla }}_j\check{g}\}=\{{\tilde{\nabla }}_k{\mathcal{F}}_{kj},{\partial }_j\check{g}\},$$

$$D{\tilde{\nabla }}_k(\check{g}{\tilde{\nabla }}_k\check{g})-[(\hat{\omega }-i{\mathcal{A}}_0){\tau }_3-i\check{\Delta },\check{g}]+\frac{\tau D}{2m}\{{\tilde{\nabla }}_k{\mathcal{F}}_{kj},{\partial }_j\check{g}\}=0$$

$$\delta S^a=-\frac{\pi }{4}{N}_F\mathrm{tr}\{{\sigma }^a{\tau }_3{\check{g}}^K\},$$

$$j_k=\frac{\pi D}{4}{N}_F\mathrm{tr}\{{\tau }_3[\check{g}{\partial }_k\check{g}{]}^K\}-\frac{D\tau }{m}\left[{\mathcal{F}}_{ki}^a{\partial }_i\delta S^a+({\tilde{\nabla }}_i{\mathcal{F}}_{ik}{)}^a\delta S^a\right]$$

$$J_k^a=\frac{\pi D}{4}{N}_F\mathrm{tr}\{{\sigma }^a[\check{g}{\tilde{\nabla }}_k\check{g}{]}^K\}-\frac{D\tau }{m}{\mathcal{F}}_{ki}^a{\partial }_i\delta n$$

$$\frac{1}{2}[\hat{\Gamma }\check{g},\check{g}]-[(\hat{\omega }-i{\mathcal{A}}_0){\tau }_3-i\check{\Delta },\check{g}]=0,$$

$$j_k=-\frac{D\tau }{m}({\tilde{\nabla }}_i{\mathcal{F}}_{ik}{)}^a\delta S^a$$

$$\delta {\chi }^{ab}=\pi N_{F}T\sum _{{\omega }_n}\frac{\Delta }{{\omega }_n^2+{\Delta }^2}{\left[\sqrt{{\omega }_n^2+{\Delta }^2}{\delta }^{ab}+{\Gamma }^{ab}\right]}^{-1}$$

$$j_k=\frac{D\tau }{m}({\tilde{\nabla }}_i{\mathcal{F}}_{ik}{)}^a\delta {\chi }^{ab}{\mathcal{A}}_0^a$$

Full text

The influence of spin-orbit coupling (SOC) on transport properties of conducting materials and nanostructures attracts a great deal of attention because of its potential importance for spintronics Zutic et al. (2004); Fabian et al. (2007). The coupling of translational and spin degrees of freedom of charge carriers gives rise to a number magneto-electric effects that allow to control spin dynamics by purely electric means or generate a charge response by acting on the spin subsystem. Probably the most known effect of this sort is the spin Hall effect (SHE) Sinova et al. (2015) which can be observed in practically any material with a sizable SOC. Other examples of magneto-electric effects driven by SOC are the direct and inverse spin-galvanic effects (SGE and ISGE) also known as the inverse and direct Edelstein effects (IEE and EE), see, for example, a recent review Ref.Ganichev et al. (2016) and references therein. The SGE/IEE is the generation of charge current by producing a nonequilibrium spin polarization Ivchenko et al. (1989a, b), while ISGE/EE corresponds to a spin polarization induced by the electric current Aronov and Lyanda-Geller (1989); Edelstein (1990). The SGE is less universal then the SHE as it can exist only in gyrotropic materials/structures (there are 18 gyrotropic crystal classes out of 21 non-centrosymmetric classes) Ivchenko and Ganichev (2008); Ganichev et al. (2016). The gyrotropic symmetry allows for a second rank pseudotensor that is required to convert an axial vector (the spin polarization) to a polar vector (the electric current) and vice versa.

Magneto-electric effects mediated by SOC are also known for non-centrosymmetric superconductors Yip (2014); Smidman et al. (2017). In superconducting materials with the gyrotropic symmetry an external Zeeman field creates a charge supercurrent, and, reversely, a supercurrent flowing in such a system induces a spin polarization Edelstein (1995, 2005); Yip (2002). These effects are the bulk superconducting analogs of the SGE/IEE and ISGE/EE, respectively Konschelle et al. (2015). In the Josephson junction geometry the superconducting SGE manifests itself as an anomalous $\varphi_{0}$-junction that supports a Josephson current at zero phase difference between superconducting leads Buzdin (2008). In superconducting structures SOC leads to an additional channel of the singlet-triplet conversion, which has been recognized as physical mechanism behind the anomalous Josephson effect Bergeret and Tokatly (2015); Konschelle et al. (2015). In the last years the spin-charge and singlet-triplet conversion effects are intensively studied in the context of an emerging field of superconducting spintronics Linder and Robinson (2015); Eschrig (2011).

The magneto-electric spin-charge conversion effects in normal conductors are most naturally described theoretically in terms of a coupled spin-charge diffusion equations Burkov et al. (2004); Shen et al. (2014a); Raimondi et al. (2012). In superconductors the analog of the diffusion equation is the Usadel equation Usadel (1970) for a quasiclassical Green function (GF). Together with the Kupriyanov-Lukichev boundary conditions Kuprianov and Lukichev (1988) the Usadel equation constitute a theoretical basis for the description of transport and spectral properties of most experimentally relevant superconducting structures. Unfortunately the full nonlinear form of the Usadel equation accounting for the spin-charge coupling due to intrinsic SOC is not known. Only its linearized version has been established and used in different contexts Mal’shukov and Chu (2008); Mal’shukov et al. (2010); Bergeret and Tokatly (2015); Konschelle et al. (2015). While the linearized Usadel equation captures some qualitative physics, such as the anomalous Josephson effect Bergeret and Tokatly (2015); Konschelle et al. (2015), it can not fully describe the low temperature regime and does not give an access to the local spectral properties. In the present paper I fill this gap by deriving the full nonliner Usadel equation for gyrotropic materials with intrinsic SOC. In contrast to the linearized theory this nonlinear equation is valid both for superconducting and for normal systems. In the normal state it reduces to the well known spin-charge diffusion equations Burkov et al. (2004); Shen et al. (2014a); Raimondi et al. (2012) thus providing us with a universal theory of magneto-electric effects mediated by SOC.

As a specific model I consider a system of conduction electrons with a linear in momentum SOC described by $\hat{H}_{so}=\beta_{k}^{a}p_{k}\sigma^{a}$ in the presence of a spin independent disorder potential $V_{dis}$ and a Zeemann/exchange field, $\hat{H}_{Z}=h^{a}\sigma^{a}$. Here $p_{k}$, $h^{a}$ and $\sigma^{a}$ are the components of the electron momentum, the exchange field and the vector of Pauli matrices, respectively 111Throughout this paper we adopt the summation convention over a pair of repeated indexes. The SOC is parametrized by a second rank pseudotensor $\beta_{k}^{a}$ that is allowed only in gyrotropic materials. In fact, the gyrotropic symmetry is the necessary and sufficient condition for the existence of the linear in ${\bf p}$ SOC in spin-1/2 electronic systems Ivchenko and Ganichev (2008); Winkler (2007) 222An explicit form of the linear SOC for all 18 gyrotropic crystal classes can be found, for example, in Table 2 of Ref.Smidman et al. (2017). Therefore the electron gas with a linear in momentum SOC can be considered as a simple but quite generic model of a gyrotropic material. A very useful feature of this model is that the exchange field and the SOC coefficients can be interpreted as the time and space components of an effective background SU(2) potential ${\cal A}_{\mu}={\cal A}_{\mu}^{a}\sigma^{a}/2$, $\mu=(0,k)$ Mineev and Volovik (1992); Fröhlich and Studer (1993); Jin et al. (2006); Tokatly (2008). Indeed, by defining ${\cal A}_{0}^{a}=2h^{a}$ and ${\cal A}_{k}^{a}=2m\beta_{k}^{a}$ one can represent the Hamiltonian of the model in the following form

[TABLE]

which implies a form invariance of the equations of motion under a local SU(2) rotations of the fermionic fields supplemented with the proper gauge transformation of the effective gauge potentials.

Using the gauge filed representation of SOC one can formulate a physically transparent SU(2) covariant kinetic (Eilenberger) equation for the quasiclassical GF $\check{g}(\bm{n},{\bf r},t,t^{\prime})$ that depends on the direction $\bm{n}={\bf p}/p_{F}$ of the Fermi momentum and describes dynamics of the Fermi surface quasiparticles in the presence of spin-dependent forces generated by the SU(2) gauge fields Gorini et al. (2010); Bergeret and Tokatly (2014, 2015); Konschelle et al. (2015). The covariant Eilenberger equation is the starting point for the derivation of the diffusive Usadel equation presented below.

For superconductors the GF $\check{g}(\bm{n},{\bf r},t,t^{\prime})$ is a $8\times 8$ matrix in the Keldysh-Nambu-spin space and the Eilenberger equation accounting for the spin-charge coupling reads (see e.g. Eq. (8) in Bergeret and Tokatly (2015))

[TABLE]

where $\tilde{\nabla}_{k}\cdot=\partial_{k}\cdot-i[{\cal A}_{k},\cdot]$ is the covariant derivative, ${\cal F}_{ij}=\partial_{i}{\cal A}_{j}-\partial_{j}{\cal A}_{i}-i[{\cal A}_{i},{\cal A}_{j}]$ is the SU(2) magnetic field tensor, $\tau$ is the momentum relaxation time, $\langle\dots\rangle$ stands for the $\bm{n}$-average, and

[TABLE]

with $\check{\Delta}$ being the anomalous self energy describing the superconducting ordering, and $\hat{\omega}_{t,t^{\prime}}=\partial_{t}\delta(t-t^{\prime})$. In the equilibrium Matsubara formalism $\hat{\omega}=\omega_{n}=\pi(2n+1)$ and $\check{g}(\bm{n},{\bf r},\omega_{n})$ becomes a $4\times 4$ matrix in the Nambu-spin space. The commutator part of the covariant derivative in Eq.(1) describes the inhomogeneous spin precession due to SOC, which can generate a long range proximity effect Bergeret and Tokatly (2013, 2014) accompanied with a highly nontrivial modifications of the local density of states in Josephson junctions Jacobsen and Linder (2015); Jacobsen et al. (2015); Arjoranta and Heikkilä (2016). The last term in the left hand side in Eq.(1) corresponds to the spin dependent Lorentz force produced by the SU(2) magnetic field ${\cal B}_{k}=\frac{1}{2}\varepsilon_{kij}{\cal F}_{ij}$. An accurate treatment of this term is our main concern as it is responsible for the charge-spin/singlet-triplet coupling and eventually for the SHE, SGE, and the anomalous Josephson effect. The main technical difficulty is related to the anticommutator structure of the Lorentz force term, which leads to the violation of the normalization condition for the quasiclassical GF $\check{g}(\bm{n})$. I will show that the normalization condition reappears in the diffusive limit for the isotropic part of the GF.

The diffusive limit of Eq.(1) is formally obtained as an asymptotic expansion in $\tau$ at $\tau\to 0$. I will perform this expansion to the order $\tau^{2}$ that is necessary to capture the effects of the SU(2) Lorentz force. With the accuracy of $O(\tau\text{\texttwosuperior})$ it is sufficient to represent the GF $\check{g}(\bm{n})$ by its 0th and 1st moments 333One can show that corrections from higher moments are at least of the order of $\tau^{3}$.

[TABLE]

where $\check{g}_{0}=\langle\check{g}(\bm{n})\rangle$ and $\check{g}_{k}=\langle n_{k}\check{g}(\bm{n})\rangle/d$ (here $d$ is the dimensionality of space). By taking the 0th and the 1st moment of Eq.(1) and using the representation of $\check{g}(\bm{n})$ given by Eq.(3) we obtain the following system of equations for $\check{g}_{0}$ and $\check{g}_{k}$, which should be then solved perturbatively in $\tau$

[TABLE]

Equation (5) relates the 1st moment $\check{g}_{k}$ of the GF to the 0th moment $\check{g}_{0}$ (the isotropic part of GF). Importantly, this equation is consistent only when solved up to the second order in $\tau$. The contribution $\sim\tau$ comes from the last two terms in the left hand side of Eq.(5). To make the perturbative structure explicit I represent $\check{g}_{k}$ as follows

[TABLE]

where $\check{g}_{k}^{(1)}\sim\tau$ and $\check{g}_{k}^{(2)},\check{g}_{k}^{(3)}\sim\tau^{2}$ are determined from the following equations

[TABLE]

In principle Eqs.(6)-(9) and (4) fully determine the diffusive dynamics up to the required order in $\tau$. It is however possible to solve Eqs.(7)-(9) explicitly and obtain a closed equation of motion for $\check{g}_{0}$

• the Usadel equation.

The first observation is that the commutator structure of the right hand side in Eq.(7) implies the normalization condition for $\check{g}_{0}$,

[TABLE]

Indeed from Eq.(7) we find the following equation for $\check{g}_{0}^{2}$

[TABLE]

which has a unique solution given by Eq.(10) provided it is fulfilled at the spatial infinity. Using the normalization condition we find $\check{g}_{k}^{(1)}$ from Eq.(7)

[TABLE]

which is the standard expression for the 1st moment of GF in the leading $\sim\tau$ order of the diffusive approximation. The $\tau^{2}$ corrections are obtained from Eqs.(8) and (9). By inserting Eq.(11) into Eq.(8) I rewrite it as follows

[TABLE]

Multiplying this equation with $\check{g}_{0}$ from both sides, such as $\check{g}_{0}(\dots)\check{g}_{0}$, and using Eq.(10) I obtain an alternative representation of Eq.(8)

[TABLE]

The subtraction of the last two equations from each other yields yet another form of Eq.(8)

[TABLE]

from which the solution is read out immediately as

[TABLE]

Finally I find $\check{g}_{k}^{(3)}$ from Eq.(9) that reads explicitly as

[TABLE]

According to Eq.(4) the commutator $[\check{\Omega},\check{g}_{0}]\sim\tau.$ Therefore, to the required accuracy of $\tau^{2}$, in first term in the above equation one can safely interchange the order of $\check{\Omega}$ and $\check{g}_{0}$ to find the following result

[TABLE]

From the explicit form of $\check{\Omega}$ in Eq.(2) I identify $\tilde{\nabla}_{k}\check{\Omega}=-i\tau_{3}\tilde{\nabla}_{k}{\cal A}_{0}=-i\tau_{3}{\cal F}_{k0}$ as the SU(2) electric field. Thus $\check{g}_{k}^{(3)}$ can be represented as follows

[TABLE]

Equations (6), (11)-(12), and (9) define the anisotropic part of the quasiclassical GF to the order of $\tau^{2}$ in the diffusive limit

[TABLE]

The Usadel equation is obtained by inserting this result into Eq.(4).

It is convenient to introduce the matrix current $\check{J}_{k}=\frac{1}{d}v_{F}\check{g}_{k}$ and rewrite the Usadel equation as follows

[TABLE]

with the matrix current defined as

[TABLE]

where $D$ is the diffusion coefficient and index 0 of the GF is suppressed for brevity. This matrix current $\check{J}_{k}$ enters the boundary conditions at the interfaces Kuprianov and Lukichev (1988); Bergeret and Tokatly (2016), while its Keldysh component determines the physical charge $j_{k}=-\pi N_{F}{\rm tr}\{\tau_{3}\check{J}_{k}^{K}(t,t)\}/4$ and spin $J_{k}^{a}=-\pi N_{F}{\rm tr}\{\sigma^{a}\check{J}_{k}^{K}(t,t)\}/4$ currents. It is important to emphasize that Eqs.(15)-(16) should be supplemented with the normalization condition $\check{g}^{2}=1$.

A very appealing property of Eq.(16) is that the intrinsic spin Hall contribution (the second term) has exactly the same structure as the corresponding contribution in the case of the extrinsic SHE (see Eq.(4) in Ref.Bergeret and Tokatly (2016)) with the extrinsic spin Hall angle being replaced by the intrinsic one $\theta_{ij}^{a}=\frac{\tau}{m}{\cal F}_{ij}^{a}$. It is therefore natural to expect that, like in the normal case, the Hall angles will add up if both intrinsic and extrinsic SOC is present.

One can argue that the last, proportional to ${\cal F}_{k0}$ term in Eq.(16) gives a small correction to the effect of the Zeeman/exchange field and therefore can be ignored. Indeed, the contribution of this term to the Usadel equation (15) has the same global (in the Nambu-spin subspace) structure as the Zeeman term. Therefore it simply renormalizes the Zeeman field by a negligible amount $\sim(\Delta_{so}\tau)^{2}\ll 1$ where $\Delta_{so}\sim\beta v_{F}$ is the SOC-induced spin splitting (the inverse spin precession rate).

The Usadel equation simplifies further if SO fields ${\cal A}_{k}$ are spatially uniform. In this case the covariant divergence of the spin Hall contribution to Eq.(16) simplifies as

[TABLE]

and the Usadel equation (15) takes the following form

[TABLE]

The physical observables are calculated from the Keldysh component of Eq.(16) (without the last term) or the GF. In particular, we have the induced spin density $\delta S^{a}=S^{a}-\chi_{P}{\cal A}_{0}^{a}$ (its deviation from the Pauli response value $\chi_{P}{\cal A}_{0}$, where $\chi_{P}$ is the Pauli susceptibility)

[TABLE]

the charge current

[TABLE]

and the spin current

[TABLE]

where $\delta n=-\pi N_{F}{\rm tr}\{\check{g}^{K}\}/4$ is the induced charge density. It is worth noting that there is no Hall contribution to the spin current in equilibrium because of the vanishing induced charge density.

Equation (17) and expressions for the currents Eqs.(19), (20) are the main results of the present paper. In the normal state $\check{g}(t,t^{\prime})$ becomes diagonal in the Nambu space and $g^{R,A}(t,t^{\prime})=\pm\delta(t-t^{\prime})$. In this case one can set $t^{\prime}=t$ directly in the Keldysh component of Eq.(17) to obtain a closed system of equations for $\delta S^{a}({\bf r},t)$ and $\delta n({\bf r},t)$. These equations together with the expressions for the currents recover the known system of coupled spin-charge diffusion equations in normal conductors with intrinsic SOC Burkov et al. (2004); Shen et al. (2014a); Raimondi et al. (2012). Hence the general Eqs.(17)-(20) provide us with unified theoretical tool for addressing magneto-electric phenomena both in superconducting and in normal systems. To illustrate the working power of these equations I will consider the SGE in bulk materials.

For a homogeneous bulk system the Usadel equation (17) reduces the following form

[TABLE]

where $\hat{\Gamma}$ is the Dyakonov-Perel (DP) spin relaxation kernel that is defined as $\hat{\Gamma}\check{g}=D[{\cal A}_{k},[{\cal A}_{k},\check{g}]]$. The kernel $\hat{\Gamma}$ acts only on the spin part of the GF and has the following explicit form $\Gamma^{ab}=D({\cal A}_{k}^{c}{\cal A}_{k}^{c}\delta^{ab}-{\cal A}_{k}^{a}{\cal A}_{k}^{b})$ Bergeret and Tokatly (2014). In the expression of Eq.(19) for the charge current only the last term survives, so that we have

[TABLE]

This equation presents a very interesting result. The spin-galvanic relation between the charge current and the induced spin is valid universally both for the normal and for the superconducting state.

In superconductors the Zeeman field induces a nonzero $\delta S^{a}$ even in equilibrium because of the Knight shift - the Cooper paring reduces the paramagnetic susceptibility so that it becomes smaller than $\chi_{P}$. By solving the equilibrium Matsubara version of Eq.(21) one readily finds the spin $\delta S^{a}$ induced by the Zeeman field in the presence of the DP relaxation, $\delta S^{a}=-\delta\chi^{ab}{\cal A}_{0}^{b}$, where

[TABLE]

is the deviation of the paramagnetic susceptibility from $\chi_{P}$. This equation generalizes the result of Ref.Gor’kov and Rashba (2001) to the case of diffusive superconductors and arbitrary SOC. By inserting the induced spin into Eq.22 we obtain the anomalous supercurrent generated by the Zeeman field

[TABLE]

For the special case of Rashba SOC and $T$ close to the critical temperature Eqs.(23), (24) reduce to the Edelstein result Edelstein (2005). The present formalism clearly demonstrates that the equilibrium SGE in gyrotropic superconductors is directly related to the Knight shift.

In the normal state $\delta S^{a}$ entering Eq.(22) can be nonzero only away from equilibrium. In semiconductors the SGE can occur due to optically generated spin polarization Ivchenko et al. (1989a, b). Below I consider the spin generation by a time dependent Zeeman field ${\cal A}_{0}(t)$. In this case from the Keldysh component of Eq.(21) we obtain the following equation of motion for the induced spin $\delta S=\delta S^{a}\sigma^{a}/2$,

[TABLE]

The linear response solution of this equation reads

[TABLE]

Equations (25) and (22) determine the charge current generated via the SGE in a normal conductor. At $\omega$ much smaller than the DP relaxation rate the expression for the current takes the form identity

[TABLE]

which is the expected results for IEE in the presence of intrinsic SOC Shen et al. (2014b).

In conclusion, I derived the full nonlinear Usadel equation and the expressions for the charge and spin currents for gyrotropic materials with a generic linear in momentum intrinsic SOC. This equation takes into account the spin-charge/singlet-triplet coupling and provides a unified description of magneto-electric effects in diffusive systems. In particular it makes a direct connection of the usual SHE (ISHE) and SGE (ISGE) to their superconducting phase-coherent counterparts. As a simple illustrative example I considered the description of SGE in bulk systems. However the most useful applications of this formalism are expected for inhomogeneous systems, hybrid metallic nanostructures, and Josephson junctions. The presented Usadel equation is perfectly suited for studying various magneto-electric transport phenomena and accompanying them modifications of the local density of states in such systems.