Generalized quasiclassical theory of the long-range proximity effect and spontaneous currents in superconducting heterostructures with strong ferromagnets

TL;DR

This paper develops a generalized quasiclassical theory for long-range proximity effects and spontaneous currents in superconductor-strong ferromagnet heterostructures, highlighting the role of spin-dependent gauge fields from spin-orbit coupling or magnetic textures.

Contribution

It introduces a unified framework incorporating spin-dependent gauge fields into quasiclassical equations, explaining long-range triplet correlations and spontaneous currents in strong ferromagnetic heterostructures.

Findings

Derivation of Usadel equations for strong ferromagnets

Identification of spin-dependent gauge fields as key to long-range effects

Examples of Josephson systems with spontaneous currents

Abstract

We present the generalized quasiclassical theory of the long-range superconducting proximity effect in heterostructures with strong ferromagnets, where the exchange splitting is of the order of Fermi energy. In the ferromagnet the propagation of spin-triplet Cooper pairs residing on the spin-split Fermi surfaces is shown to be governed by the spin-dependent Abelian gauge field which results either from the spin-orbital coupling or from the magnetic texture. The additional gauge field enters into the quasiclassical equations in superposition with the usual electromagnetic vector potential and results in the generation of spontaneous superconducting currents and phase shifts in various geometries which provide the sources of long-range spin-triplet correlations. We derive the Usadel equations and boundary conditions for the strong ferromagnet and consider several generic examples of the Josephson systems supporting spontaneous currents.

Equations

$$({\check{G}}_0^{-1}+\mu -\check{\Sigma })\check{G}=\check{I}\delta (\mathbf{r}-{\mathbf{r}}^{\prime })$$

$${\check{G}}_0^{-1}=-\hat{\Pi }\frac{1}{2m}\hat{\Pi }-{\hat{\sigma }}_k\{M_{kj},{\hat{p}}_j\}/2+(i\omega +\mathbf{h}\hat{\mathbf{\sigma }}){\hat{\tau }}_3.$$

$$\check{G}_{ES}=\left(\begin{array}[]{cc}G_{0}+G_{z}\hat{\sigma}_{z}&F_{x}\hat{\sigma}_{x}+F_{y}\hat{\sigma}_{y}\\ \tilde{F}_{x}\hat{\sigma}_{x}+\tilde{F}_{y}\hat{\sigma}_{y}&\tilde{G}_{0}+\tilde{G}_{z}\hat{\sigma}_{z}\\ \end{array}\right)$$

$$\check{G}_{MS}=\left(\begin{array}[]{cc}G_{x}\hat{\sigma}_{x}+G_{y}\hat{\sigma}_{y}&F_{0}+F_{z}\hat{\sigma}_{z}\\ \tilde{F}_{0}+\tilde{F}_{z}\hat{\sigma}_{z}&\tilde{G}_{x}\hat{\sigma}_{x}+\tilde{G}_{y}\hat{\sigma}_{y}\\ \end{array}\right).$$

$$i\check{\mathbf{V}}{\hat{\nabla }}_{\mathbf{R}}{\check{G}}_{ES}+[i\omega {\hat{\tau }}_3-M_{zj}{\hat{\sigma }}_z{p}_j-\check{\Sigma },{\check{G}}_{ES}]=0,$$

$${\hat{g}}_{\sigma }=-\oint \frac{d{\xi }_{p\sigma }}{\pi i}{\check{G}}_{\sigma },$$

$$i{\mathbf{v}}_{\sigma }{\hat{\partial }}_{\mathbf{R}}{\hat{g}}_{\sigma }+[i\omega {\hat{\tau }}_3-{\hat{\Sigma }}_{\sigma },{\hat{g}}_{\sigma }]=0,$$

$${\hat{\partial }}_{\mathbf{R}}={\nabla }_{\mathbf{R}}+i\sigma \mathbf{Z}[{\hat{\tau }}_3,\cdot ]-ie\mathbf{A}[{\hat{\tau }}_3,\cdot ]$$

$$\mathbf{Z}=(M_{zx},M_{zy},M_{zz})m_F.$$

$$\mathbf{j}=-\frac{i\pi Te}{2}\sum _{\sigma =\pm }\sum _{\omega }{\nu }_{\sigma }⟨{\mathbf{v}}_{\sigma }\mathrm{Tr}[{\hat{\tau }}_3{\hat{g}}_{\sigma }]⟩,$$

$$2{\tau }_{\sigma }({\mathbf{v}}_{\sigma }{\hat{\partial }}_{\mathbf{R}}){\hat{g}}_{\sigma }=-[⟨{\hat{g}}_{\sigma }⟩,{\hat{g}}_{\sigma }].$$

$${\hat{\mathbf{g}}}_{\sigma }^a=-{\tau }_{\sigma }{\mathbf{v}}_{\sigma }⟨{\hat{g}}_{\sigma }⟩{\hat{\partial }}_{\mathbf{R}}⟨{\hat{g}}_{\sigma }⟩.$$

$$D_{\sigma }{\hat{\partial }}_{\mathbf{R}}({\hat{g}}_{\sigma }{\hat{\partial }}_{\mathbf{R}}{\hat{g}}_{\sigma })-[\omega {\hat{\tau }}_3,{\hat{g}}_{\sigma }]=0,$$

$$\mathbf{j}=\frac{i\pi Te}{2}\sum _{\sigma =\pm }\sum _{\omega }{\nu }_{\sigma }{D}_{\sigma }\mathrm{Tr}[{\hat{\tau }}_3{\hat{g}}_{\sigma }{\hat{\partial }}_{\mathbf{R}}{\hat{g}}_{\sigma }].$$

$$\hat{\tilde{{\cal F}}}=\left(\begin{array}[]{cc}0&{\tilde{{\cal F}}}_{-}\\ {\tilde{{\cal F}}}_{+}&0\\ \end{array}\right).$$

$${\tilde{\mathcal{F}}}_{\sigma }(x,x^{\prime })=e^{-i{n}_x(p_{\sigma n}x-p_{\sigma n}^{\ast }{x}^{\prime })}{S}_{\sigma \perp }{K}_{\sigma },$$

$$\hat{G}=\left({\hat{u}}_1{\hat{u}}_2\right)$$

$$\hat{\tilde{F}}=\left({\hat{v}}_1{\hat{v}}_2\right),$$

$$\left[{\mu }_F-\frac{{\hat{p}}_x^2}{2m_F}-{\sigma }_k\{M_{kj},{\hat{p}}_j\}/2+{\hat{\tau }}_3(i\omega +h{\sigma }_z)\right]\hat{\psi }=0$$

$$\left[{\mu }_S-\frac{{\hat{p}}_x^2}{2m_S}+{\hat{\tau }}_3(i\omega +h{\hat{\sigma }}_z)\right]\hat{\psi }=0,$$

$$\hat{\psi }(x^{\prime }+0)=\hat{\psi }(x^{\prime }-0)$$

$${\nabla }_x{\hat{u}}_1(x^{\prime }+0)-{\nabla }_x{\hat{u}}_1(x^{\prime }-0)=2m_F(1,0{)}^T$$

$${\nabla }_x{\hat{v}}_1(x^{\prime }+0)-{\nabla }_x{\hat{v}}_1(x^{\prime }-0)=0,$$

$$\hat{\psi }(+0)=\hat{\psi }(-0)$$

$$\frac{{\nabla }_x\hat{\psi }(-0)}{m_F}-\frac{{\nabla }_x\hat{\psi }(+0)}{m_S}=(i{M}_{kx}{\hat{\sigma }}_k-2V)\hat{\psi }(0).$$

$$\hat{u}(x>x^{\prime })=e^{i{p}_{+n}x}{\hat{A}}_{+>}+e^{i{p}_{-n}x}{\hat{A}}_{->}$$

$$\hat{u}(x<x^{\prime })=e^{-i{p}_{+n}x}{\hat{A}}_{+<}+e^{-i{p}_{-n}x}{\hat{A}}_{-<},$$

$${\hat{A}}_{+>}=a_{+>}^{(0)}{e}^{-i{p}_{+n}{x}^{\prime }}{\left[1,-(p_{+n}{S}_{\perp +}+p_y{S}_{\parallel +})\right]}^T$$

$${\hat{A}}_{->}=a_{->}{e}^{-i{p}_{+n}{x}^{\prime }}{\left[(p_{-n}{S}_{\perp -}+p_y{S}_{\parallel -}),1\right]}^T$$

$${\hat{A}}_{+<}=a_{+<}^{(0)}{e}^{i{p}_{+n}{x}^{\prime }}{\left[1,(p_{+n}{S}_{\perp +}-p_y{S}_{\parallel +})\right]}^T$$

$${\hat{A}}_{-<}=a_{-<}{e}^{i{p}_{+n}{x}^{\prime }}{\left[(p_y{S}_{\parallel -}-p_{-n}{S}_{\perp -}),1\right]}^T,$$

$$a_{+>}^{(0)}=a_{+<}^{(0)}=-i{m}_F/p_{+n}.$$

$${\hat{u}}_r(x,x^{\prime })=e^{-i{p}_{+n}x}{\hat{D}}_{+}(x^{\prime })+e^{-i{p}_{-n}x}{\hat{D}}_{-}(x^{\prime })$$

$${\hat{v}}_r(x,x^{\prime })=e^{i{p}_{-n}^{\ast }x}{\hat{B}}_{+}(x^{\prime })+e^{i{p}_{+n}^{\ast }x}{\hat{B}}_{-}(x^{\prime }).$$

$${\hat{D}}_{+}(x^{\prime })=d_{+}{e}^{-i{p}_{+n}{x}^{\prime }}{\left[1,(p_{+n}{S}_{\perp +}-p_y{S}_{\parallel +})\right]}^T$$

$${\hat{D}}_{-}(x^{\prime })=d_{-}{e}^{-i{p}_{+n}{x}^{\prime }}{\left[(p_y{S}_{\parallel -}-p_{-n}{S}_{\perp -}),1\right]}^T.$$

Full text

I Introduction

Effective gauge theories has been introduced in many condensed matter systems including spin-triplet superfluid 3He Volovik (2003), cold atom systems Dalibard et al. (2011); Goldman et al. (2014) and magnetic materials Volovik (1987); Nagaosa and Tokura (2013). In spatially inhomogeneous magnetic textures the additional spin-dependent $U(1)$ gauge field of topological origin affects the motion of conduction electrons in the same way as the external electromagnetic fieldVolovik (1987); Aharonov and Stern (1992); Barnes and Maekawa (2007); Hai et al. (2009); Yang et al. (2009). That results in the topological Hall effect and emergent electrodynamics Bruno et al. (2004); Nagaosa and Tokura (2013) observed recently in the chiral magnets with skyrmion lattices Neubauer et al. (2009); Zang et al. (2011); Schulz et al. (2012); Liang et al. (2015).

Geometric flux associated with the spin-dependent gauge field was predicted to generate spontaneous spin and charge currents in mesoscopic rings with spatially-inhomogeneous texture of the Zeeman field Loss et al. (1990); Loss and Goldbart (1992); Tatara and Kohno (2003); Tatara and Garcia (2003). Up to now these effects have not been observed. Experimental detection of persistent currents in normal metals is in general rather challenging Levy et al. (1990); Bleszynski-Jayich et al. (2009) since their magnitude is determined by the single-level contribution which is rather small and highly sensitive to the details of disorder potentialB${\rm\ddot{u}}$ttiker et al. (1984); Cheung et al. (1989).

The situation is completely different in the superconducting state where the locally broken $U(1)$ gauge symmetry leads to the Meissner effect, i.e. the generation of persistent condensate currents in response to the external magnetic field. However effects associated with the geometric spin-dependent flux Loss et al. (1990); Loss and Goldbart (1992); Tatara and Kohno (2003); Tatara and Garcia (2003) have not been identified in usual superconducting systems since the condensate of spin-singlet Cooper pairs is not sensitive to the Zeeman field rotations.

In the present paper we show that the superconducting condensate in fact can be coupled to the spin-dependent gauge fields emerging in superconductor/ferromagnet (SC/FM) hybrids. In such systems the interplay of superconducting and magnetic orderings results in the generation of the spin-triplet correlations through the proximity effectBergeret et al. (2005) . Of particular interest are the Cooper pairs with spin projections $S_{z}=\pm 1$ in the local basis determined by the exchange field ${\bm{h}}$. They are formed by equal-spin correlations (ESC) between the electrons residing on one and the same spin-up/down Fermi surface [see Fig.(1)a]. Upon the adiabatic rotation of ${\bm{h}}$ such Cooper pairs pick up geometric phases and generate spontaneous superconducting currents.

To understand the behaviour of ESC we develop the gauge theory formalism to treat proximity effect in SC/FM systems with spin-textured strong ferromagnets. So far, proximity and transport calculations in SC/FM hybrids have mostly concentrated on either fully polarized systems, so called half metals Eschrig et al. (2003); Braude and Nazarov (2007); Eschrig and Lofwander (2008); Mironov and Buzdin (2015); Eschrig et al. (2015), or in the opposite limit of weakly polarized systems Bergeret et al. (2005); Buzdin (2005), where the difference between spin subbands is completely neglected. However, most FMs have an intermediate exchange splitting of the energy bands of the order of but less than the Fermi energy. By now the quasiclassical theory for this regime has been formulated for the case of homogeneous magnetization of the strong ferromagnet Grein et al. (2009). To describe the general situation we go beyond those limitations and consider SC/FM structures with arbitrary large and spatially inhomogeneous exchange field.

Our approach relies on the adiabatic approximation for spin transportBliokh and Bliokh (2005) which has been extensively used for studying transport phenomena in spin-textured magnetsVolovik (1987); Nagaosa and Tokura (2013). We derive the quasiclassical equations describing ESC interacting with the spin-dependent $U(1)$ gauge field which can be induced either by the magnetic texture or spin-orbit coupling (SOC). The phases picked up by the $S_{z}=\pm 1$ Cooper pairs in response to this gauge field generate spontaneous superconducting currents through strong FMs.

II Generalized quasiclassical theory

II.1 The Model

We consider the Gor’kov equations in the presence of spin-dependent gradient terms, the exchange field ${\bm{h}}$ and the external vector potential ${\bm{A}}$:

[TABLE]

Here $\hat{\Pi}=\bm{\hat{p}}-e{\bm{A}}\hat{\tau}_{3}$, $\check{G}=\check{G}({\bm{r}},{\bm{r}}^{\prime})$ is the matrix Green function (GF) in spin-Nambu space, $\{,\}$ is the anticommutator added to have the hermitian Hamiltonian in the system with space-dependent field $M_{kj}$, $\mu$ is the chemical potential, $\omega$ is the Matsubara frequency, $m$ is the effective mass which is equal to $m_{F}$ in the ferromagnet and to $m_{S}$ in the superconductor, $e$ is the electron charge, $\hat{\bm{p}}=(\hat{p}_{x},\hat{p}_{y},\hat{p}_{z})$ is the momentum differential operator, $\hat{\sigma}_{k}$ and $\hat{\tau}_{k}$ are the spin and Nambu Pauli matrices. The self-energy term $\check{\Sigma}$ includes the effects related to disorder scattering as well as the non-diagonal superconducting potential. The spin-dependent term $M_{kj}$ can be associated either with the SOC or with the pure gauge SU(2) field $M_{kj}=-i{\rm Tr}\left(\hat{\sigma}_{k}\hat{U}^{\dagger}\nabla_{j}\hat{U}\right)/2m$ where the transformation $\hat{U}({\bm{r}})=e^{i\bm{\hat{\sigma}}\bm{\theta}({\bm{r}})/2}$ rotates spin axes to the local frame where ${\bm{h}}\parallel{\bm{z}}$. It is parametrized by the spin vector $\bm{\theta}=\theta\bm{n}$ defined by the spatial texture of the exchange field distribution $\bm{h}(\bm{r})=\hat{R}(\bm{\theta}(\bm{r}))\bm{h}$, where $\hat{R}$ is the spatially-dependent rotation matrix and we choose ${\bm{h}}=h{\bm{z}}$. Therefore, Eq. (1) is written in the local reference frame, where the quantization axis is aligned with the local exchange field. We assume that the exchange field rotates slowly, on the large scales as compared to the atomic distances. For this reason we neglect second-order spatial derivatives of the exchange field. In the framework of this approach the inhomogeneity of the exchange field enters the equations as the pure gauge SU(2) field.

In the general case of large exchange splitting $|h|\sim\mu$ the spin-dependent Gor’kov Eq. (1) is rather complicated and most importantly one cannot apply here the quasiclassical theory. The quasiclassical approximation is violated by the mixed-spin correlations (MSC) residing in spin-split subbands [Fig. 1b] which are characterized by the spatial length scale of the order of the Fermi wavelength $\lambda_{F}=\sqrt{2m/\mu}$. Therefore MSC yield vanishingly small contribution to the momentum-averaged observables at the distances much larger than the atomic length from the FM/SC interface. Such correlations can be incorporated to the effective boundary conditions as the source terms for the ESC [Fig. 1a]. The ESC survive in the ferromagnet at much larger distances and can be treated within quasiclassics considered separately for each of the spin-up and spin-down Fermi surfaces.

II.2 Adiabatic approximation

To develop the quasiclassical approximation we divide the GF into the parts corresponding ESC of the spin-up/down states (see Fig.1a)

[TABLE]

and the one corresponding to the MSC (see Fig.1b)

[TABLE]

Then from the Gor’kov equation (1) one can see that the amplitude of MSC is in general proportional to $\check{G}_{MS}\propto(M_{ij}/\lambda_{F}h)\check{G}_{ES}$ where $\lambda_{F}$ is the Fermi wavelength. Therefore MSC are small as compared to that of the ESC if the adiabatic criterion is satisfied $|M_{ij}/\lambda_{F}h|\ll 1$.

Within the adiabatic approximation neglecting the MSC in Eq.(1) and substituting the expansion of momentum operator $\hat{p}^{2}=p^{2}-2i{\bm{p}}\bm{\nabla}$, we obtain the quasiclassical equation for ESC part:

[TABLE]

where $\hat{\nabla}_{\bm{R}}=\nabla_{\bm{R}}-ie{\bm{A}}[\hat{\tau}_{3},\cdot]$ and $\bm{\check{V}}=\bm{v}_{+}\check{\gamma}_{+}+\bm{v}_{-}\check{\gamma}_{-}$. Here $v_{\pm}=\sqrt{2(\mu\pm h)/m_{F}}$ are the spin-dependent Fermi velocities determined on each of the spin-split Fermi surfaces labelled by the subscript $\sigma=\pm$. We introduce the projection operators to spin-up and spin-down states $\check{\gamma}_{+}=\hat{\tau}_{\uparrow}\hat{\sigma}_{\uparrow}+\hat{\tau}_{\downarrow}\hat{\sigma}_{\downarrow}$ and $\check{\gamma}_{-}=\hat{\tau}_{\uparrow}\hat{\sigma}_{\downarrow}+\hat{\tau}_{\downarrow}\hat{\sigma}_{\uparrow}$, respectively, where $\hat{\tau}_{\uparrow(\downarrow)}=(\hat{\tau}_{0}+(-)\hat{\tau}_{z})/2$ and $\hat{\sigma}_{\uparrow(\downarrow)}=(\hat{\sigma}_{0}+(-)\hat{\sigma}_{z})/2$.

The paired states on each of the Fermi surfaces are given by the corresponding parts of the equal-spin correlator: $\check{G}_{\pm}=\check{\gamma}_{\pm}\check{G}_{ES}$. This decomposition allows to introduce quasiclassical propagators separately for spin-up and spin-down blocks

[TABLE]

where $\xi_{p\sigma}=p^{2}/2m_{F}+\sigma h-\mu$ and the notation $\oint$ means that the integration takes into account the poles of GF neat the corresponding Fermi surface.

From Eq.(5) we obtain generalized Eilenberger equations fro the spin-less quasiclassical propagators

[TABLE]

where the covariant operator is

[TABLE]

One can see that the Eilenberger-type equations for the spin-up/down correlations contain an additional U(1) gauge field ${\bm{Z}}$ which is added to the usual electromagnetic vector potential ${\bm{A}}$ with the opposite effective charges for spin-up and spin-down Cooper pairs. The U(1) field is obtained by projecting the initial SU(2) field $\hat{\sigma}_{k}M_{ki}$ to the basis of spin-triplet pairing states: ${\bm{Z}}=-i(\hat{U}^{-1}\nabla\hat{U})_{11}$. This reduction means that we neglect spin-flip transitions between the spin-up and spin-down Cooper pairs induced by the SU(2) potential. On a qualitative level it is equivalent to the adiabatic approximation in the single-particle problems that allows to describe the quantum system evolution in terms of the Berry gauge fieldsBliokh and Bliokh (2005).

Finally, the quasiclassical expression for the charge current is given by

[TABLE]

where $\nu_{\sigma}$ are the spin-resolved DOS and $\langle..\rangle$ denotes the averaging over the spin-split Fermi surface.

II.3 Usadel equation for ESC

Let us consider the system with large impurity scattering rate as compared to the superconducting energies determined by the bulk energy gap $\Delta$. In this experimentally relevant diffusive limit it is possible to derive the generalized Usadel theory with the help of the normalization condition $\hat{g}_{\sigma}^{2}=1$ which holds due to the commutator structure of the quasiclassical equations (8).

The impurity self-energy in the Born approximation is given by $\hat{\Sigma}_{\sigma}=\langle\hat{g}_{\sigma}\rangle/2i\tau_{\sigma}$. In the dirty limit we have

[TABLE]

The solution of Eq. (11) can be found as $\hat{g}_{\sigma}=\langle\hat{g}_{\sigma}\rangle+\hat{\bm{g}}_{\sigma}^{a}\frac{\bm{p}_{\sigma}}{p_{\sigma}}$, where the anisotropic part of the solution $\hat{\bm{g}}_{\sigma}^{a}$ is small with respect to $\langle\hat{g}_{\sigma}\rangle$. Making use of the relation $\{\langle\hat{g}_{\sigma}\rangle,\hat{\bm{g}}_{\sigma}^{a}\}=0$, which follows from the normalization condition, one obtains

[TABLE]

Substituting to Eq.(7) and omitting the angle brackets we get the diffusion equation

[TABLE]

where $D_{\sigma}$ are the spin-dependent diffusion coefficients, in the isotropic case given by $D_{\sigma}=\tau_{\sigma}v^{2}_{\sigma}/3$. The current is obtained substituting expansion (12) to the Eq.(10)

[TABLE]

Equations (13,14) together with the boundary conditions derived in the next section provide the framework to study proximity effect-related phenomena in strong ferromagnets with large amount of disorder. This regime describes a different physical situation as compared to the previous works, where other approximations have been used. That concerns papers studying Rashba superconductor with large SOCHouzet and Meyer (2015) and that dealing with weak exchange fields and SOC within SU(2)-covariant formulationBergeret and Tokatly (2013, 2014); Konschelle et al. (2015); Tokatly (2017). The applicability condition of our theory requires that the exchange splitting has to be much larger than all other energy scales except the Fermi energy. In particular, in the bulk superconductor we have omitted the spin-flipping terms produced by various sources like SOC, magnetic texture or magnetic impurity scattering. However the spin-flipping terms are still important in strong ferromagnets/half metals since they provide the source of ESC. Such correlations appear due to the conversion of spin-singlet Cooper pairs leaking from the superconductor electrode into the long-range spin-triplet ones. Below we study this conversion and boundary conditions for ESC using a generic model of SC/FM interface.

III Boundary conditions

The quasiclassical Eq. (7) deals with the long-range ESC transformed adiabatically in space under the action of spin gauge fields. In FM/SC systems with usual spin-singlet superconductors such correlations can appear only in result of the non-adiabatic spin flip which converts MSC into the ESC Houzet and Buzdin (2007); Eschrig and Lofwander (2008); Eschrig et al. (2015). This process occurs within the thin layer near FM/SC interface and can be described by the effective boundary conditions. To derive them we consider the simple microscopic model of the boundary with the non-magnetic potential barrier. The FM/SC interface is located at $x=0$, the normal ${\bm{n}}=n_{x}{\bm{x}}$ directed from SC to FM and $n_{x}=\pm 1$. The momentum components ${\bm{p}}_{\parallel}$ in $yz$ plane parallel to the boundary are conserved. The effective masses $m_{S}$ and $m_{F}$ in the superconducting and ferromagnet regions are assumed to be different and the FM/SC boundary is characterized by the interfacial potential barrier of the strength $V\delta(x)$ added to the Hamiltonian $\hat{G}_{0}^{-1}$ in Eq.(1).

Let us outline the general strategy to deriving the boundary conditions for quasiclassical ESC propagators. We need to solve the exact Gor’kov equations (1) near the boundary with accuracy up to the first order in SU(2) terms, which provide the conversion of MSC to ESC. Therefore we will use an expansion by the small parameter $|M_{ij}p_{j}/h|\ll 1$.

In result we will find the slow component of the anomalous function $\hat{\tilde{F}}(x,x^{\prime})=(\check{G})_{21}$, where the index corresponds to the Nambu space. This component does not contain fast oscillations as a function of the center of mass coordinate $X=(x+x^{\prime})/2$. Let us denote such components as

$\hat{\tilde{{\cal F}}}(x,x^{\prime})$ . Below we will show that these correlations have the form

[TABLE]

The spin-up ${\tilde{{\cal F}}}_{+}$ and spin-down ${\tilde{{\cal F}}}_{-}$ pairing amplitudes are given by

[TABLE]

where $p_{\sigma n}=\sqrt{2m_{F}(\mu+\sigma h)-p_{||}^{2}}$, and $S_{\sigma\perp}=(M_{xi}+i\sigma M_{yi})n_{i}/h$ is the combination of SU(2) field components that generate ESC in the ferromagnet near the superconducting interface. Here the coefficient $K_{\sigma}$ incorporates the dependence of the pairing amplitude on the interface barrier strength, order parameter and effective masses.

By writing Eq.(16) we assume the averaging over the directions of the in-plane momentum. This is enough in the dirty limit although in the clean case additional important effects resulting from the in-plane gradients of the exchange field ${\bm{h}}$ can be obtained beyond this approximation. Eq.(16) is valid for $\omega>0$ and for $\omega<0$ the amplitude can be obtained using symmetry relations, as discussed below.

We need to find the GF near the FM/SC interface determined by the 1D Gorkov’ equation (1) along $x$ coordinate. Since each GF is the 2x2 matrix in spin space they can be represented in the form of two spinors

[TABLE]

where the spinor elements are $\hat{u}_{k}=(u_{k1},u_{k2})^{T}$ and $\hat{v}_{k}=(v_{k1},v_{k2})^{T}$. Let us consider the components $\hat{u}_{1}$, $\hat{v}_{1}$ in detail. The other pair $\hat{u}_{2}$, $\hat{v}_{2}$ is given by $\hat{u}_{2}=\sigma_{x}\hat{u}_{1}$ and $\hat{v}_{2}=\sigma_{x}\hat{v}_{1}$ and changing the sign of the fields $h$ and $M_{yi}$.

For the Nambu spinor $\hat{\psi}=(\hat{u}_{1},\hat{v}_{1})^{T}$ from (1) we obtain the equation in the ferromagnet

[TABLE]

and in the superconductor

[TABLE]

where $\mu_{S,F}=\mu-p_{\parallel}^{2}/2m_{S,F}$ . These equations look similar to the Bogolubov-de Gennes equation but taken at the imaginary frequency. The boundary condition are obtained by integrating Eq.(1) near the singularity points $x=x^{\prime}$ and $x=0$. In this way we obtain the boundary conditions at $x=x^{\prime}$:

[TABLE]

ant at $x=0$

[TABLE]

Here we neglect the impurity scattering and the inverse proximity effect in the superconducting region. The impurity self energy can be included into the consideration in case of the tunnelling limit when the surface barrier is strong enough to suppress the inverse proximity effect in the superconductor. This consideration demonstrates that the impurity scattering does not change boundary conditions for quasiclassical propagators. In the opposite case of weak barriers one should strictly speaking take into account how the impurity self energy in the superconductor is modified by the inverse proximity effect, which in principle can affect the boundary conditions.

Let’s assume that ${n_{x}<0}$. Then the solution for electron wave in bulk normal ferromagnet can be written as $\hat{v}=0$ and

[TABLE]

where the labels $>(<)$ denote right- and left-going waves, $p_{\sigma n}=\sqrt{2m_{F}(\mu_{F}+\sigma h+i\omega)}$ and we neglect SO corrections to the momenta. Expressions (26) are valid for the half-metal when $h>\mu_{F}$ as well. In this case one should take $p_{-n}=i\sqrt{2m_{F}(h-\mu_{F})}$.

To the first order by the small parameter $|M_{ij}p_{j}/h|\ll 1$ we get expressions for the amplitudes in Eq.(26)

[TABLE]

where $S_{\sigma\parallel}=(M_{xy}+i\sigma M_{yy})/h$ and $S_{\sigma\perp}$ is defined above. The zero - order amplitudes are given by

[TABLE]

We neglect the second term in Eq.(26) $\hat{A}_{->}$ since its amplitude is much smaller than $a^{(0)}_{+>}$ due to the prefactors $p_{\sigma n}S_{\sigma\perp}$ and $p_{y}S_{\sigma\parallel}$. Therefore up to the first order in these small parameters we should take into account the reflected holes generated by $\hat{A}_{->}$ without spin flip which have the same wave vector as the incident wave and therefore does not contribute to the slow-varying correlation (15).

The solution (26) can be considered as the incident electronic wave at the FM/SC interface. The reflected wave consists of electronic $\hat{u}_{r}$ and hole $\hat{v}_{r}$ components having the form

[TABLE]

Here the structure of $\hat{D}_{\sigma}$, is similar to that of $\hat{A}_{\sigma<}$

[TABLE]

The reflected hole-like wave is given by (34) with the amplitudes

[TABLE]

We are interested in the wave $\hat{B}_{-}$ because the corresponding contribution to the reflected hole amplitude $\hat{v}_{r}$ given by the second term in Eq.(34) does not contain fast oscillations as the function of $X=(x+x^{\prime})/2$. Hence it provides the source of the long-range superconducting correlations. On a qualitative level the wave $\hat{B}_{-}$ determines the spin-flip Andreev reflection leading to the generation of spin-triplet Cooper pairs.

Hence we obtain the component of the ESC in the form ${{\tilde{{\cal F}}}_{+}(x,x^{\prime})=b_{-}e^{i(p_{+n}x-p_{+n}^{*}x^{\prime})}}$ . The other component is obtained from the other pair of spinors $(\hat{u}_{2},\hat{v}_{2})$ and has the form ${{\tilde{{\cal F}}}_{-}(x,x^{\prime})=b_{+}e^{i(p_{-n}x-p_{-n}^{*}x^{\prime})}}$. Averaging over in-plane momentum directions we get ${\langle b_{\bar{\sigma}}\rangle=S_{\sigma\perp}K_{\sigma}}$ in accordance with the Eq.(16) where we put ${n_{x}=-1}$.

General expressions for the amplitudes $K_{\sigma}$ are rather involved (see Appendix A). However in the tunnelling limit they read

[TABLE]

where $F_{bcs}=\Delta/\sqrt{\omega^{2}+|\Delta|^{2}}$ and $v_{Sn}=\sqrt{2\mu_{S}/m_{S}}$. For the half-metal Eq.(39) is valid with the imaginary momentum $p_{-n}=i\sqrt{2m_{F}(h-\mu)+p_{||}^{2}}$. The other Nambu component of the anomalous function $\hat{F}(x,x^{\prime})=(\check{G})_{12}$ can be obtained from the general particle-hole symmetry

[TABLE]

Now having in hand the expression for the slowly-varying amplitude (16) we can derive the boundary conditions for the components of quasiclassical propagator. Following Ref. Zaitsev, 1984 we write them in the form

[TABLE]

where $\hat{\tau}_{\pm}=(\hat{\tau}_{x}\pm i\hat{\tau}_{y})/2$. The quasiclassical propagators can be obtained by taking the Fourier transform of the slow-varying exact GF components (16) and then using the definition (6). In this way we obtain the propagators as functions of the momentum direction ${\bm{\hat{p}_{\sigma}}}$, which determines the quasiclassical trajectories in each of the spin subbands. We will use the notations $\tilde{f}_{\sigma,in(out)}$ for the ”incoming” (”outgoing”) trajectories with $\bm{\hat{p}_{\sigma}}\bm{n}<(>)0$ .

Then Eq. (16) yields the quasiclassical ESC propagators at the interface $x=0$:

[TABLE]

where $v_{\sigma n}=p_{\sigma n}/m_{F}$ is the spin-dependent Fermi-velocity. At the same time $\tilde{f}_{\sigma,out}(x=0,\omega>0)=0$.

The other anomalous Green’s functions can be obtained from $\tilde{f}_{\sigma}(x=0,\omega>0,\bm{\hat{p}_{\sigma}})$ according to the following symmetry relations Serene and Rainer (1983): $\tilde{f}_{\sigma}(x=0,\omega<0,\bm{\hat{p}_{\sigma}})=-\tilde{f}_{\sigma}(x=0,\omega>0,-\bm{\hat{p}_{\sigma}})$ and $f_{\sigma}(x=0,\omega,\bm{\hat{p}_{\sigma}})=\tilde{f}_{\sigma}^{*}(x=0,-\omega,\bm{\hat{p}_{\sigma}})$. The normal part of the GF is given by $g_{\sigma,in(out)}=1$ due to the normalization condition. Thus Eq. (42) gives the value of the ESC, generated by the magnetic inhomogeneity/SOC at the FM/SC interface. To find this equal-spin GF in the ferromagnetic region we solve in general the transport equation (7) with the boundary condition (42).

The advantage of the boundary conditions (42) is that they give an explicit value of the Green’s function at the ferromagnetic side of the interface. The price, which we have paid for it, is that, strictly speaking, they are only valid for an isolated interface, because the asymptotic conditions at the infinity were essentially used in the derivation. However, they can be safely applied to the dirty systems with more than one interface if the distances between the interfaces are large as compared to the mean free path. Below we are only interested in the dirty case.

The boundary conditions to the Usadel equations can be obtained from Eq. (42) in a straightforward way. As usual, one can show Kupriyanov and Lukichev (1988) that in the isotropization region near the interface the matrix current is

[TABLE]

where $\langle...\rangle_{-(+)}$ means the averaging over the part of the ferromagnet FS corresponding to $\bm{p_{\sigma}}\bm{n}<(>)0$ and real values of $p_{\sigma n}$ and $p_{sn}$, $l_{\sigma}=v_{\sigma}\tau_{\sigma}$ is the mean free path and $\hat{\partial}_{\bm{n}}=\bm{n}\hat{\partial}_{\bm{R}}$. The boundary condition to the Usadel equation is obtained using (43), taken at $x=0$ with $\hat{g}_{\sigma,in(out)}$ from Eq.  (42) and the symmetry relations discussed above:

[TABLE]

where $\kappa_{\sigma}=-i\langle K_{\sigma}v_{\sigma n}^{2}/v_{\sigma}\rangle_{-}$.

From the boundary condition (44) one can see that the generation of ESC is determined by the non-adiabatic spin-flipping terms near the boundary. In case if these terms are of the SOC origin, e.g. having the Rashba form the magnitude of ESC correlations is given by $S_{\sigma\perp}\sim\alpha/h$, where $\alpha$ is the SOC constant. Otherwise if it comes from the magnetic texture with the characteristic scale $\xi_{\theta}$ the estimation is $S_{\sigma\perp}\sim 1/(m_{F}\xi_{\theta}h)$. However as shown below, that smallness affects only the overall amplitude of the critical current in the generic SC/FM/SC Josephson systems , but not the spontaneous phase shift of the current-phase relation. Indeed the emergent gauge field $\bm{Z}$ which drives the spontaneous supercurrents through strong ferromagnets does not contain any small parameter. Therefore the anomalous current at zero phase difference across the junction can be of the order of the critical one.

IV Spontaneous Josephson current through strong ferromagnets

Having in hand the machinery of the generalized quasiclassical theory described above we can calculate Josephson current-phase relations for different systems with spin-dependent fields. Example of such a system with magnetic helix texture is shown in Fig. 1c .

Here we work in the dirty limit using the linearized (with respect to the anomalous Green’s function) version of the spin-less Usadel equations (13) and boundary conditions (44). This simplification is adequate if the proximity effect at the SC/FM interface is weak, for example when the interface is low-transparent. The absolute value of the order parameter is assumed to be the same in the superconducting leads, while there is the phase difference $\chi$ between them. The electric current in the ferromagnetic interlayer can be calculated according to Eq. (14), which in the linearised form is reduced to

[TABLE]

The anomalous GF $f_{\sigma}$ and $\tilde{f}_{\sigma}$ should be calculated from the linearized version of the Usadel equation (13):

[TABLE]

The solution of these equations takes the form:

[TABLE]

The coefficients $C_{\sigma,\pm}$ and $\tilde{C}_{\sigma,\pm}$ are to be found from the boundary conditions (44) taken at the S/F interfaces $x=\mp d/2$. The resulting expressions take the form:

[TABLE]

and

[TABLE]

where $\lambda_{\sigma}=\sqrt{2|\omega|/D_{\sigma}}$. Substituting the anomalous GF from (47) with the coefficients $C_{\sigma,\pm}$ and $\tilde{C}_{\sigma,\pm}$ from (48) and (49) into Eq. (45) we obtain the general current-phase relation (CPR):

[TABLE]

where $\chi$ is the Josephson phase difference $R_{\sigma}=1/(e^{2}\nu_{\sigma}D_{\sigma})$ is the spin-resolved resistivity and $\lambda_{\sigma}=\sqrt{2|\omega|/D_{\sigma}}$.

The spin gauge field $Z_{x}\neq 0$ and finite spin splitting $D_{+}\neq D_{-}$ in the CPR (50) lead to the spontaneous current at zero phase difference known as the anomalous Josephson effect. The ground state phase difference $\chi_{0}$ can be found from the zero-current condition $I(\chi=\chi_{0})=0$

[TABLE]

The spontaneous phase shift of Josephson current has been obtained in several FM/SC systems Braude and Nazarov (2007); Buzdin (2008); Reynoso et al. (2008); Grein et al. (2009); Zazunov et al. (2009); Liu and Chan (2010); Malshukov et al. (2010); Mironov and Buzdin (2015); Brunetti et al. (2013); Yokoyama et al. (2014); Kulagina and Linder (2014); Nesterov et al. (2016); Konschelle et al. (2015); Moor et al. (2015a, b); Bobkova et al. (2016); Szombati et al. (2016); Silaev et al. (2017). Here we demonstrate that this effect is essential only for the case of strong ferromagnets. When the ferromagnet is weak and treated within the usual quasiclassical approximation, the difference between $I_{-}$ and $I_{+}$ is neglected and the anomalous Josephson effect disappears.

The spin gauge field $\bm{Z}$ is generated by the spin helix shown schematically in Fig.  1. Recently the proximity effect in helical magnets has been observed experimentallyDi Bernardo et al. (2015); Satchell et al. (2017). In this case the magnetization texture is described by $\bm{h}=h(\cos\alpha,\sin\alpha\cos\theta,\sin\alpha\sin\theta)$, where we assume that the angle $\alpha$ is spatially independent and $\theta=\theta(x)$. The spin rotation is given by $\hat{U}=e^{-i\hat{\sigma}_{x}\theta/2}e^{-i\hat{\sigma}_{z}\alpha/2}e^{-i\hat{\sigma}_{y}\pi/4}$ yielding $Z_{x}=-\cos\alpha\partial_{x}\theta/2m_{F}$. The surface ESC-generating term in Eq.(16) is provided by $S_{\sigma\perp}=i\sigma n_{x}\sin\alpha\partial_{x}\theta/(2m_{F}h)$ .

The general theory developed above describes the proximity effect in homogeneous ferromagnet with a linear in momentum SOC Bergeret and Tokatly (2013, 2014). For example let us consider the SC/FM/SC junction through the quasi-2D ferromagnet in $xz$ plane, interfaces in $yz$ planes and the exchange field in the plane of the ferromagnet ${\bm{h}}\parallel{\bm{z}}$ . In case of the Rashba SOC in the ferromagnetic region this system is characterized by the spin-dependent fields $M_{zx}=-M_{xz}=-\alpha/2$ which leads to

[TABLE]

while the Dresselhaus SOC yields $M_{zz}=-M_{xx}=\beta/2$ and therefore

[TABLE]

In each of these cases the Josephson CPR can be found substituting the fields into the general Eqs.(50,51). Since the ground state phase shift $\chi_{0}$ is determined by the component $Z_{x}$ parallel to the Josephson current, for $\bm{h}\parallel{\bm{z}}$ we have $\chi_{0}\neq 0,\pi$ only for the Rashba SOC, but not for the Dresselhaus one. This is natural, because in general case of a magneto-electric effect the spontaneous current and the magnetization are perpendicular to each other for the Rashba SOC Buzdin (2008); Malshukov and Chu (2008); Bobkova and Bobkov (2017), but they are parallel for the Dresselhaus SOC. Therefore, in order to get the anomalous Josephson current for the Dresselhaus SOC, ${\bm{h}}$ is to have a component parallel to the current.

Comparing Eqs.(51) and (53,54) one can see that in the considered geometry the Dresselhaus SOC produce the long-range ESC even in the case of the homogeneous magnet (while in general the both Rashba and Dresselhause SOC can produce ESC Bergeret and Tokatly (2014)). However their amplitudes are determined by the SOC constants which in general are rather small in metals. Although the anomalous phase shift is also determined by SOC, it can become rather large for sufficiently long junction, i.e. when $|Z_{x}d|\geq 1$. Therefore even the weak SOC leads to the significant phase shifts of the CPR although the overall critical current amplitude is rather small . In reality however the long-range ESC can be generated by the magnetic inhomogeneity near the interface Eschrig and Lofwander (2008). Let us consider the following model :

[TABLE]

where $\theta=\theta(x)$ changes linearly in the region $\xi_{\theta}\ll d$ near the interfaces and $\theta=0$ in the bulk FM. Neglect the effect of SOC we get $S_{\sigma\perp}=-i\sigma n_{x}\partial_{x}\theta/(2m_{F}h)$ in the CPR Eq. (50), where the spin gauge field $Z_{x}$ is determined by the SOC in the bulk FM. For small exchange fields $h\ll\mu$ the anomalous phase shift $\chi_{0}$ is determined by the prefactor $(I_{+}-I_{-})/(I_{+}+I_{-})\sim h/\mu$. If we assume additionally (in general it can be not the case) that $Z_{x}d\ll 1$, then $\chi_{0}\sim m_{F}(hd/\mu)\alpha$, in agreement with Ref. Buzdin, 2008.

V Conclusion

To conclude, we have developed the generalized quasiclassical formalism to calculate the behaviour of long-range ESC in the ferromagnet. These correlations can be generated at the ferromagnetic/superconductor interface in the presence of either magnetization inhomogeneity or SOC. The general conditions for ESC generation are derived in terms of the SU(2) gauge fields. In the ferromagnetic material the behaviour of ESC is shown to be governed by the adiabatic spin gauge field which generates spontaneous superconducting currents through strong FMs with magnetic texture or SOC. These results demonstrate that spontaneous superconducting currents exist as a robust and experimentally observable phenomenon in many superconducting/ferromagnetic systems studied in connection to the superconducting spintronics Linder and Robinson (2015); Eschrig (2015).

VI Acknowledgements

We thank A.S. Mel’nikov for stimulating discussions. This work was supported by the Academy of Finland. I.V.B. and A.M. B. acknowledge the support by the Program of Russian Academy of Sciences “Electron spin resonance, spin-dependent electron effects and spin technologies” and Russian-Greek Project N 2017-14-588-0007-011 “Experimental and theoretical studies of physical properties of low-dimensional quantum nanoelectronic systems”.

Appendix A Spin-flip Andreev reflection coefficient

Here we find the reflection coefficients of the electronic and hole-like waves in Eqs.(33,34). For this purpose we use Eq.(26) as the incident wave coming from the ferromagnet to the FM/SC interface. To apply the boundary conditions (24, 25) we write solution in the superconductor, as the superposition of two terms decaying at $x\to\infty$ :

[TABLE]

where $q_{1}=\sqrt{2m_{S}(\mu_{S}+i\Omega)}$ and $q_{2}=-\sqrt{2m_{S}(\mu_{S}-i\Omega)}$ and $\Omega=\sqrt{\omega^{2}+|\Delta|^{2}}$. Below we will neglect the imaginary part of $q_{1,2}$ and use $q_{1}\approx/q_{2}\approx p_{Sn}$ where $p_{Sn}=\sqrt{2m_{S}\mu_{S}}$.

First, let us find reflection coefficients without spin-flip, in zero order by small parameter $M^{+}_{x,y}p_{x,y}/h$. For this purpose we obtain the following system of equations :

[TABLE]

where

[TABLE]

The solutions are

[TABLE]

The spin-flip reflection amplitude $b_{-}$ can be found taking into account first-order corrections in $M^{+}_{x,y}p_{x,y}/h$ when matching the electron and hole waves at FM/SC boundary. In this way we get the linear system

[TABLE]

where the coefficients $\hat{\alpha}=\alpha_{0}-M_{kx}\hat{\sigma}_{k}$ and $\tilde{\hat{\alpha}}=\alpha_{0}^{*}+M_{kx}\hat{\sigma}_{k}$ take into account the correction to the boundary condition from the effective spin-orbital term. The spinors $\hat{A}^{(0)}_{+}$, $\hat{B}^{(0)}_{+}$, $\hat{D}^{(0)}_{+}$ are given by Eqs.(28,35,37) with the amplitudes (64,65,66) and without the spin-flip terms

[TABLE]

The solution of this system reads

[TABLE]

where

[TABLE]

where $v_{\Sigma}=v_{+n}+v_{-n}$ and $v_{d}=v_{+n}-v_{-n}$.

In this way we obtain the spin-flip Andreev reflection amplitude in the form ${b_{-}=K_{x+}S_{+\perp}+K_{y+}S_{+\parallel}}$ with

[TABLE]

(ii) Let us now consider the other pair of spinors in Eq.(17) $\hat{u}_{2}$, $\hat{v}_{2}$ which determine the correlation function ${\tilde{\cal F}_{-}}$ . They can be obtained from $\hat{u}_{1}$, $\hat{v}_{1}$ by transforming the Hamiltonian $\check{H}\to\sigma_{x}\check{H}\sigma_{x}$ which flips the spin index $\sigma$. The reflected hole-like states are given by the same Eq.(37), but this time we are interested in the wave $\hat{B}_{+}(x^{\prime})$. This wave has the form (37) with the amplitude ${b_{+}=K_{x-}S_{-\perp}+K_{y-}S_{-\parallel}}$, where

[TABLE]

where the amplitudes $d_{-}^{(0)}$, $b_{-}^{(0)}$, $a_{-}^{(0)}$ and coefficients $Z_{-}$, $Y_{-}$ are obtained from (64,65,66,74,75, 76,77) with $+$ changed by the $-$ and vice versa.

**(iii) Limiting cases: ** Consider large barrier $V\gg v_{S},v_{\sigma},v_{\sigma},h/p_{\sigma}$ so that $\alpha\approx iV$ and $Z_{0}\approx Y\approx\Omega V^{2}$. Then from Eqs. we get $d_{+}^{(0)}\approx-a_{+}^{(0)}$ and $b_{+}^{(0)}\approx a_{+}^{(0)}i\Delta p_{Sn}p_{+n}/(2\Omega V^{2})$.

Substituting to the Eq.(78) and taking into account (64) we get

[TABLE]

These expressions are valid for the half-metal when $h>\mu_{F}$ as well. In this case one should take $p_{-n}=i\sqrt{2m_{F}(h-\mu_{F})}$.