Playing Pure Spin Current in Helimagnets: Toward Chiral Spin-Orbitronics

Abstract

A quantum theory of electron spin transport in conductive magnets is developed. The theory describes a large number of effects that arise due to spin-orbit scattering of conduction electrons on crystal lattice defects, such as the spin Hall effect, the inverse spin Hall effect, and the anomalous Hall effect. The transport through the contact of two different conductive magnetics is also considered; the phenomenological boundary conditions for the charge and spin flows are formulated, which make it possible to take into account the spin flip at the interface. The developed electron spin transport theory is used to describe the spin-orbitronics of the “helimagnet metal/non-magnetic metal” heterojunction. The spatial distribution of the polarization of the spin current injected into the helimagnet is found, and the characteristic decay lengths of different components of the polarization vector of the spin current are determined.

Appendices

APPENDIX A

Let the states of an electron in an ideal crystal placed in an external magnetic and electric fields be given by the Hamiltonian $\hat {\mathcal{H}}$. Crystal lattice defects (interstitial impurities, vacancies, etc.) lead to electron scattering. If $\hat {\mathcal{V}}$ is the spin-dependent scattering potential created by such defects, then, without significant loss of generality, the full one-electron Hamiltonian ${{\hat {\mathcal{H}}}_{\mathcal{U}}}$, corresponding to a specific configuration of the scattering potential $\hat {\mathcal{V}}$, can be represented as ${{\hat {\mathcal{H}}}_{\mathcal{U}}} = \hat {\mathcal{H}} + \hat {\mathcal{V}}$.

To describe an ensemble of electrons in mixed quantum states, the mathematical apparatus of the density matrix is used. Along with the term “density matrix”, the terms “density operator”, “statistical operator” are also used.

The density operator is a self-adjoint operator operating in the Hilbert space of electronic states, the definition of which allows one to calculate the observed average value of any physical characteristic of an electron defined by the operator $\hat {\mathcal{A}}$ by applying the trace operation to the product of the operator $\hat {\mathcal{A}}$ and the density operator. The statistical operator ${{\hat {\rho }}_{\mathcal{U}}}$ for electrons with a Hamiltonian ${{\hat {\mathcal{H}}}_{\mathcal{U}}}$ obeys equation

$$\frac{\partial }{{\partial t}}{{\hat {\rho }}_{\mathcal{U}}} - \frac{1}{{i\hbar }}\left[ {{{{\hat {\mathcal{H}}}}_{\mathcal{U}}},{{{\hat {\rho }}}_{\mathcal{U}}}} \right] = 0,$$

(65)

which is called the quantum Liouville equation or the von Neumann equation.

The statistical operator ${{\hat {\rho }}_{\mathcal{U}}}$ contains all possible information about the considered quantum system for a specific configuration of scatterers that determines the form of the potential $\hat {\mathcal{V}}$. In practice, we are not interested in the properties of the system for a specific potential configuration $\hat {\mathcal{V}}$, but only in properties averaged over various configurations of the scattering potential. The operation of such configurational averaging we will denote by a bar above the corresponding value. Let us introduce into our consideration the statistical operator $\hat {\rho }$, obtained by averaging the operator ${{\hat {\rho }}_{\mathcal{U}}}$ over all possible scatterer configurations: $\hat {\rho } = \overline {{{{\hat {\rho }}}_{\mathcal{U}}}}$. Representing ${{\hat {\rho }}_{\mathcal{U}}}$ as ${{\hat {\rho }}_{\mathcal{U}}} = \hat {\rho } + \Delta \hat {\rho }$ and substituting this into (65), we can formally solve the resulting equation with respect to $\Delta \hat {\rho }$. Then subjecting the resulting equation to configuration averaging and substituting the previously found formal solution for $\Delta \hat {\rho }$ into the equation obtained after averaging, we obtain a closed equation for $\hat {\rho }$ in the following general form:

$$\frac{\partial }{{\partial t}}\hat {\rho } - \frac{1}{{i\hbar }}\left[ {\hat {\mathcal{H}},\hat {\rho }} \right] + \hat {\mathcal{R}} = 0.$$

(66)

Equation (66) for $\hat {\rho }$, which will be referred to as the quantum kinetic equation, contains a term $\hat {\mathcal{R}}$, which is called collision integral and describes the relaxation of $\hat {\rho }$ to its equilibrium value.

We apply the density matrix formalism to describe a system of electrons that do not interact with each other.

The form of the density matrix depends on the representation in which the matrix elements of the density operator are calculated. Using the eigenvectors $\left| {{\mathbf{p}},\sigma } \right\rangle$ of the Hamiltonian of the electron kinetic energy ${{\hat {\mathcal{H}}}_{0}}$ as a representation basis, the statistical operator $\hat {\rho }$ can be represented by its matrix elements $\left\langle {{\mathbf{p}},\sigma } \right|\hat {\rho }\left| {{\mathbf{p}}{\kern 1pt} ',\sigma {\kern 1pt} '} \right\rangle$. The eigenvectors $\left| {{\mathbf{p}},\sigma } \right\rangle$ are indicated by the spin quantum number $\sigma$ and the quasi-momentum p. In order to bring the form of the quantum description of the electronic system in terms of the density matrix that satisfies the von Neumann equation as close as possible to the classical way of describing in terms of the distribution function that satisfies the classical Boltzmann kinetic equation, we use a special representation of the density matrix, which was introduced by E. Wigner [68].

The density matrix in the Wigner representation, which we call the quantum distribution function, can be defined by relation

$${{f}_{{\sigma \sigma {\kern 1pt} '}}}({\mathbf{r}},{\mathbf{p}},t) = \sum\limits_{\mathbf{Q}} {{{\operatorname{e} }^{{{{i{\mathbf{Qr}}} \mathord{\left/ {\vphantom {{i{\mathbf{Qr}}} \hbar }} \right. \kern-0em} \hbar }}}}\left\langle {{\mathbf{p}} + \frac{{\mathbf{Q}}}{2},\sigma } \right|\hat {\rho }\left| {{\mathbf{p}} - \frac{{\mathbf{Q}}}{2},\sigma {\kern 1pt} '} \right\rangle } .$$

(67)

Four matrix elements ${{f}_{{\sigma \sigma {\kern 1pt} '}}}({\mathbf{r}},{\mathbf{p}},t)$, each of which is a function of coordinate r, quasi-momentum p, and time t, represent an operator $\hat {f}({\mathbf{r}},{\mathbf{p}},t)$ acting only in spin space.

In what follows, when the dependence of the quantum distribution function on the quasi-momentum p plays a significant role, we will use the more compact notation ${{\hat {f}}_{{\mathbf{p}}}}$ for $\hat {f}({\mathbf{r}},{\mathbf{p}},t)$. Similarly, the matrix elements $\left\langle {{\mathbf{p}},\sigma } \right|\hat {\mathcal{R}}\left| {{\mathbf{p}},\sigma {\kern 1pt} '} \right\rangle$ of the collision integral are p-dependent spin operator ${{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$, which is a functional of ${{\hat {f}}_{{\mathbf{p}}}}$.

The quantum kinetic equation for $\hat {f}({\mathbf{r}},{\mathbf{p}},t)$ can be derived from the equation for the one-particle statistical operator $\hat {\rho }$ satisfying Eq. (66). At this stage, it is necessary to specify the form of the Hamiltonian $\hat {\mathcal{H}}$ of an electron in an ideal crystal placed in an arbitrary electromagnetic field. Magnetic induction B and the electric field strength E, depending on the coordinates r and time t, are expressed in terms of the vector potential A and the scalar potential Ф by the relations ${\mathbf{B}} = \left[ {{\partial \mathord{\left/ {\vphantom {\partial {\partial {\mathbf{r}} \times {\mathbf{A}}}}} \right. \kern-0em} {\partial {\mathbf{r}} \times {\mathbf{A}}}}} \right]$ and ${\mathbf{E}} = - {{\partial \Phi } \mathord{\left/ {\vphantom {{\partial \Phi } {\partial {\mathbf{r}}}}} \right. \kern-0em} {\partial {\mathbf{r}}}} - \left( {{1 \mathord{\left/ {\vphantom {1 c}} \right. \kern-0em} c}} \right){{\partial {\mathbf{A}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{A}}} {\partial t}}} \right. \kern-0em} {\partial t}}$. Assuming the electron spectrum to be isotropic and quadratic, we can write $\hat {\mathcal{H}}$ in the form $\hat {\mathcal{H}} = {{\hat {\mathcal{H}}}_{O}} + {{\hat {\mathcal{H}}}_{S}}$, where ${{\hat {\mathcal{H}}}_{O}} = {{{{{\left( {{\mathbf{\hat {p}}} - {{e{\mathbf{A}}} \mathord{\left/ {\vphantom {{e{\mathbf{A}}} c}} \right. \kern-0em} c}} \right)}}^{2}}} \mathord{\left/ {\vphantom {{{{{\left( {{\mathbf{\hat {p}}} - {{e{\mathbf{A}}} \mathord{\left/ {\vphantom {{e{\mathbf{A}}} c}} \right. \kern-0em} c}} \right)}}^{2}}} {2{{m}_{e}}}}} \right. \kern-0em} {2{{m}_{e}}}} + e\Phi$ is the energy operator of a particle with charge e and effective mass ${{m}_{e}}$ moving in an electromagnetic field, characterized by potentials A and Ф; ${{\hat {\mathcal{H}}}_{S}} = \mu {\mathbf{B}} \cdot \hat {\boldsymbol\sigma }$ is the energy operator of the interaction of the spin magnetic moment of an electron of magnitude $\mu$ with a magnetic field B. Substituting $\hat {\mathcal{H}} = {{\hat {\mathcal{H}}}_{O}} + {{\hat {\mathcal{H}}}_{S}}$ into Eq. (66) and using the operator identity

$$\left[ {{\mathbf{B}} \cdot \hat {\boldsymbol\sigma },\hat {\rho }} \right] = \frac{1}{2}\left\{ {\hat {\boldsymbol\sigma },\left[ {{\mathbf{B}},\hat {\rho }} \right]} \right\} + \frac{1}{2}\left\{ {\left[ {\hat {\boldsymbol\sigma },\hat {\rho }} \right],{\mathbf{B}}} \right\},$$

(68)

write equation for $\hat {\rho }$ in the form

$$\begin{gathered} \frac{\partial }{{\partial t}}\hat {\rho } - \frac{1}{{i\hbar }}\left[ {{{{\hat {\mathcal{H}}}}_{O}},\hat {\rho }} \right] - \frac{\mu }{2}\left\{ {\hat {\boldsymbol\sigma },\frac{1}{{i\hbar }}\left[ {{\mathbf{B}},\hat {\rho }} \right]} \right\} \\ - \,\,\frac{\mu }{{2i\hbar }}\left\{ {\left[ {\hat {\boldsymbol\sigma },\hat {\rho }} \right],{\mathbf{B}}} \right\} + \hat {\mathcal{R}} = 0. \\ \end{gathered} $$

(69)

The physical meaning of the representation of the equation for $\hat {\rho }$ in the form (69) is that in Eq. (69) we have explicitly distinguished the terms that are expressed through the so-called quantum Poisson brackets for two pairs of operators: a pair ${{\hat {\mathcal{H}}}_{O}}$ and $\hat {\rho }$, and a pair B and $\hat {\rho }$. The quantum Poisson bracket for the pair of operators ${{\hat {\mathcal{H}}}_{O}}$ and $\hat {\rho }$ is ${{ - {\kern 1pt} \left[ {{{{\hat {\mathcal{H}}}}_{O}},\hat {\rho }} \right]} \mathord{\left/ {\vphantom {{ - {\kern 1pt} \left[ {{{{\hat {\mathcal{H}}}}_{O}},\hat {\rho }} \right]} {i\hbar }}} \right. \kern-0em} {i\hbar }}$, and for the pair $\hat {\rho }$ and B is ${{ - \left[ {{\mathbf{B}},\hat {\rho }} \right]} \mathord{\left/ {\vphantom {{ - \left[ {{\mathbf{B}},\hat {\rho }} \right]} {i\hbar }}} \right. \kern-0em} {i\hbar }}$. A significant simplification of the form of the quantum kinetic equation can be achieved if the semiclassical approximation is used to describe the orbital motion of conduction electrons. The quasi-classical description of the orbital motion of electrons is possible when the de Broglie wavelength of conduction electrons is small compared to all other characteristics of the electron system of dimention of length, e.g., the mean free path of electrons, their spin-diffusion length, the depth of the skin layer, and other quantities that characterize semiclassical picture of the electronic system.

Assuming this condition satisfied, we pass in Eq. (69) to the semiclassical limit. It is well known that such a transition as applied to the equation of motion for the density matrix can be realized by replacing the quantum Poisson brackets with the “classical” Poisson brackets. For a pair of arbitrary operators $\hat {U}$ and $\hat {V}$, corresponding to the classical quantities $U({\mathbf{r}},{\mathbf{p}},t)$ and $V({\mathbf{r}},{\mathbf{p}},t)$, the quantum Poisson bracket ${{ - {\kern 1pt} \left[ {\hat {U},\hat {V}} \right]} \mathord{\left/ {\vphantom {{ - {\kern 1pt} \left[ {\hat {U},\hat {V}} \right]} {i\hbar }}} \right. \kern-0em} {i\hbar }}$ in classical mechanics corresponds to the “classical” Poisson bracket $\left( {{{\partial U} \mathord{\left/ {\vphantom {{\partial U} \partial }} \right. \kern-0em} \partial }{\mathbf{p}} \cdot {{\partial V} \mathord{\left/ {\vphantom {{\partial V} \partial }} \right. \kern-0em} \partial }{\mathbf{r}} - {{\partial U} \mathord{\left/ {\vphantom {{\partial U} \partial }} \right. \kern-0em} \partial }{\mathbf{r}} \cdot {{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial {\mathbf{p}}}}} \right. \kern-0em} {\partial {\mathbf{p}}}}} \right)$. Consequently, after the transition to the semiclassical description of the orbital motion of an electron, the quantum Poisson bracket ${{ - {\kern 1pt} \left[ {{{{\hat {\mathcal{K}}}}_{O}},\hat {\rho }} \right]} \mathord{\left/ {\vphantom {{ - {\kern 1pt} \left[ {{{{\hat {\mathcal{K}}}}_{O}},\hat {\rho }} \right]} {i\hbar }}} \right. \kern-0em} {i\hbar }}$ will take the form ${\mathbf{v}} \cdot {{\partial \hat {f}} \mathord{\left/ {\vphantom {{\partial \hat {f}} \partial }} \right. \kern-0em} \partial }{\mathbf{r}} + {{{\mathbf{F}}}_{O}} \cdot {{\partial \hat {f}} \mathord{\left/ {\vphantom {{\partial \hat {f}} \partial }} \right. \kern-0em} \partial }{\mathbf{p}}$, where v is the electron velocity and ${{{\mathbf{F}}}_{O}} = e\left\{ {{\mathbf{E}} + {{\left[ {{\mathbf{v}} \times {\mathbf{B}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{\mathbf{v}} \times {\mathbf{B}}} \right]} c}} \right. \kern-0em} c}} \right\}$ is the classical Lorentz force acting on a particle with an electric charge e moving with velocity v.

Quantum Poisson bracket ${{ - \left[ {{\mathbf{B}},\hat {\rho }} \right]} \mathord{\left/ {\vphantom {{ - \left[ {{\mathbf{B}},\hat {\rho }} \right]} {i\hbar }}} \right. \kern-0em} {i\hbar }}$ takes the form ${{ - \partial {\mathbf{B}}} \mathord{\left/ {\vphantom {{ - \partial {\mathbf{B}}} {\partial {{r}_{i}}}}} \right. \kern-0em} {\partial {{r}_{i}}}} \cdot {{\partial \hat {f}} \mathord{\left/ {\vphantom {{\partial \hat {f}} \partial }} \right. \kern-0em} \partial }{{p}_{i}}$. Then the third term of the left hand side of Eq. (69) in the semiclassical limit with respect to the variables r and p becomes $\left\{ {{{{{\mathbf{\hat {F}}}}}_{S}},{{\partial \hat {f}} \mathord{\left/ {\vphantom {{\partial \hat {f}} {\partial {\mathbf{p}}}}} \right. \kern-0em} {\partial {\mathbf{p}}}}} \right\}$, where ${{{\mathbf{\hat {F}}}}_{S}} = - {{\mu \partial \left( {\hat {\boldsymbol\sigma } \cdot {\mathbf{B}}} \right)} \mathord{\left/ {\vphantom {{\mu \partial \left( {\hat {\boldsymbol\sigma } \cdot {\mathbf{B}}} \right)} {\partial {\mathbf{r}}}}} \right. \kern-0em} {\partial {\mathbf{r}}}}$ is the operator of the spin-driving force of the inhomogeneous magnetic field. The fourth term of the left hand side of the equation in the classical description of the orbital motion takes the form $i\mu {\mathbf{B}} \cdot {{\left[ {\hat {\boldsymbol\sigma },\hat {\rho }} \right]} \mathord{\left/ {\vphantom {{\left[ {\hat {\boldsymbol\sigma },\hat {\rho }} \right]} \hbar }} \right. \kern-0em} \hbar }$.

As a result, under the assumption of the possibility of a semiclassical description of the orbital motion of electrons, for the quantum distribution function $\hat {f}\left( {{\mathbf{r}},{\mathbf{p}},t} \right)$ we obtain a quantum kinetic equation of the following form

$$\begin{gathered} \frac{\partial }{{\partial t}}\hat {f} + {\mathbf{v}} \cdot \frac{\partial }{{\partial {\kern 1pt} {\mathbf{r}}}}\hat {f} \\ + \,\,\frac{1}{2}\left\{ {e{\mathbf{E}} + \frac{e}{c}\left[ {{\mathbf{v}} \times {\mathbf{B}}} \right] - \mu \frac{\partial }{{\partial {\kern 1pt} {\mathbf{r}}}}\left( {\hat {\boldsymbol\sigma } \cdot {\mathbf{B}}} \right),\frac{\partial }{{\partial {\kern 1pt} {\mathbf{p}}}}\hat {f}} \right\} \\ + \,\,\mu \frac{i}{\hbar }{\mathbf{B}} \cdot \left[ {\hat {\boldsymbol\sigma },\hat {f}} \right] + {{{\hat {\mathcal{R}}}}_{{\mathbf{p}}}} = 0. \\ \end{gathered} $$

(70)

The last term on the left side of Eq. (70), called the collision integral ${{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$, describes the relaxation of the quantum distribution function $\hat {f}$ to its instantaneous local equilibrium value ${{\hat {f}}_{L}}$ and is expressed in terms of the deviation $\delta \hat {f} = \hat {f} - {{\hat {f}}_{L}}$.

APPENDIX B

The density operator makes it possible to calculate the observed average value A of any physical characteristic of an electron defined by the operator $\hat {A}$, according to the rule

$$A = \operatorname{Tr} \hat {A}\hat {\rho }.$$

(71)

In what follows, we will be interested in the average observables of the electron density operator

$$\hat {n}({\mathbf{r}}) = \delta ({\mathbf{\hat {r}}} - {\mathbf{r}}),$$

(72)

of the operator of electron flow

$${\mathbf{\hat {i}}}({\mathbf{r}}) = \frac{1}{2}\left\{ {{\mathbf{\hat {v}}},\delta ({\mathbf{\hat {r}}} - {\mathbf{r}})} \right\},$$

(73)

of spin density

$${\mathbf{\hat {s}}}({\mathbf{r}}) = \delta ({\mathbf{\hat {r}}} - {\mathbf{r}}) \cdot \hat {\boldsymbol\sigma }{\text{,}}$$

(74)

and of spin density flow

$$\hat {j}({\mathbf{r}}) = \frac{1}{2}\left\{ {{\mathbf{\hat {v}}},\delta ({\mathbf{\hat {r}}} - {\mathbf{r}})} \right\} \otimes \hat {\boldsymbol\sigma }{\text{.}}$$

(75)

In (72)–(75), $\delta ({\mathbf{r}})$ is the Dirac delta function, ${\mathbf{\hat {v}}} = {{{\mathbf{\hat {p}}}} \mathord{\left/ {\vphantom {{{\mathbf{\hat {p}}}} {{{m}_{e}}}}} \right. \kern-0em} {{{m}_{e}}}}$ is the electron velocity operator, ${\mathbf{\hat {r}}}$ is the electron coordinate operator.

In accordance with (71), the ensemble averages of the electron density $N({\mathbf{r}},t)$, electron flow ${\mathbf{I}}({\mathbf{r}},t)$, spin density ${\mathbf{S}}({\mathbf{r}},t)$, and spin current $\boldsymbol{J}({\mathbf{r}},t)$ for the system of noninteracting electrons are given by

$$N({\mathbf{r}},t) = \operatorname{Tr} \hat {n}({\mathbf{r}})\hat {\rho },$$

(76)

$${\mathbf{I}}({\mathbf{r}},t) = \operatorname{Tr} {\mathbf{\hat {i}}}({\mathbf{r}})\hat {\rho },$$

(77)

$${\mathbf{S}}({\mathbf{r}},t) = \operatorname{Tr} {\mathbf{\hat {s}}}({\mathbf{r}})\hat {\rho },$$

(78)

$$\boldsymbol{J}({\mathbf{r}},t) = \operatorname{Tr} \hat {j}({\mathbf{r}})\hat {\rho }.$$

(79)

The quantum distribution function $\hat {f}({\mathbf{r}},{\mathbf{p}},t)$, represented by the spin matrix ${{f}_{{\sigma \sigma {\kern 1pt} '}}}({\mathbf{r}},{\mathbf{p}},t)$, can be written without any loss of generality as an expansion in Pauli matrices $\hat {\boldsymbol\sigma }$

$$\hat {f}({\mathbf{r}},{\mathbf{p}},t) = \frac{1}{2}n({\mathbf{r}},{\mathbf{p}},t) + \frac{1}{2}{\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t) \cdot \hat {\boldsymbol\sigma }{\text{.}}$$

(80)

The newly introduced functions $n({\mathbf{r}},{\mathbf{p}},t)$ and ${\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)$ are determined by the relations

$$n({\mathbf{r}},{\mathbf{p}},t) = \operatorname{Tr} \hat {f}({\mathbf{r}},{\mathbf{p}},t),$$

(81)

$${\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t) = \operatorname{Tr} \hat {\boldsymbol\sigma }{\text{ }}\hat {f}({\mathbf{r}},{\mathbf{p}},t).$$

(82)

The function $n({\mathbf{r}},{\mathbf{p}},t)$ defined by Eq. (81) has the meaning of the electron density distribution function in momentum space. The function ${\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)$ introduced by (82) by analogy with $n({\mathbf{r}},{\mathbf{p}},t)$ can be called the spin density distribution function. For brevity, when writing the collision integrals for the distribution functions $n({\mathbf{r}},{\mathbf{p}},t)$ and ${\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)$ we will use the compact notation ${{n}_{{\mathbf{p}}}}$ and ${{{\mathbf{s}}}_{{\mathbf{p}}}}$.

Taking the trace from Eq. (70) and perform the same operation after multiplying Eq. (70) by $\hat {\boldsymbol\sigma }$, we obtain a system of coupled kinetic equations for the distribution functions of the electron density and electron spin density

$$\begin{gathered} \frac{\partial }{{\partial t}}n + {\mathbf{v}} \cdot \frac{\partial }{{\partial {\mathbf{r}}}}n + \left\{ {e{\mathbf{E}} + \frac{e}{c}\left[ {{\mathbf{v}} \times {\mathbf{B}}} \right]} \right\} \cdot \frac{\partial }{{\partial {\mathbf{p}}}}n \\ - \,\,\mu \frac{\partial }{{\partial {\mathbf{p}}}}{{s}_{i}} \cdot \frac{\partial }{{\partial {\mathbf{r}}}}{{B}_{i}} + {{R}_{{\mathbf{p}}}} = 0, \\ \end{gathered} $$

(83)

$$\begin{gathered} \frac{\partial }{{\partial t}}{\mathbf{s}} + {\mathbf{v}} \cdot \frac{\partial }{{\partial {\mathbf{r}}}}{\mathbf{s}} + \left\{ {e{\mathbf{E}} + \frac{e}{c}\left[ {{\mathbf{v}} \times {\mathbf{B}}} \right]} \right\} \cdot \frac{\partial }{{\partial {\mathbf{p}}}}{\mathbf{s}} \\ - \,\,\mu \frac{\partial }{{\partial {\mathbf{p}}}}n \cdot \frac{\partial }{{\partial {\mathbf{r}}}}{\mathbf{B}} + \frac{{2\mu }}{\hbar }\left[ {{\mathbf{s}} \times {\mathbf{B}}} \right] + {{{\mathbf{R}}}_{{\mathbf{p}}}} = 0, \\ \end{gathered} $$

(84)

where collision integrals ${{R}_{{\mathbf{p}}}}$ and ${{{\mathbf{R}}}_{{\mathbf{p}}}}$ are defined as ${{R}_{{\mathbf{p}}}} = \operatorname{Tr} {{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$ and ${{{\mathbf{R}}}_{{\mathbf{p}}}} = \operatorname{Tr} \hat {\boldsymbol\sigma }{{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$.

The distribution functions of the electron density $n({\mathbf{r}},{\mathbf{p}},t)$ and spin density ${\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)$, which satisfy Eqs. (83) and (84), allow us to represent the above definitions in a simple form

$$N({\mathbf{r}},t) = \sum\limits_{\mathbf{p}} {n({\mathbf{r}},{\mathbf{p}},t)} ,$$

(85)

$${\mathbf{I}}({\mathbf{r}},t) = \sum\limits_{\mathbf{p}} {{\mathbf{v}}n({\mathbf{r}},{\mathbf{p}},t)} ,$$

(86)

$${\mathbf{S}}({\mathbf{r}},t) = \sum\limits_{\mathbf{p}} {{\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)} ,$$

(87)

$$\boldsymbol{J}({\mathbf{r}},t) = \sum\limits_{\mathbf{p}} {{\mathbf{v}} \otimes {\mathbf{s}}({\mathbf{r}},{\mathbf{p}},t)} .$$

(88)

The summation in (85)–(88) is carried out over all admissible values of electronic quasi-momentums p.

Let us make some remarks about the meaning of the quantities introduced by us. The product $eN({\mathbf{r}},t)$, where e is the electron charge, is the electric charge density of the conduction electrons in the crystal, while $e{\mathbf{I}}({\mathbf{r}},t)$ is the electric current density. The product $\left( {{\hbar \mathord{\left/ {\vphantom {\hbar 2}} \right. \kern-0em} 2}} \right){\mathbf{S}}({\mathbf{r}},t)$ is the vector of the electron spin momentum density, whereas $\left( {{\hbar \mathord{\left/ {\vphantom {\hbar 2}} \right. \kern-0em} 2}} \right)\mathbf{J}({\mathbf{r}},t)$ is the tensor of the flow of electron spin momentum density. Note also that the spin density ${\mathbf{S}}({\mathbf{r}},t)$ has the same dimension as the electron density $N({\mathbf{r}},t)$. Similarly, the electron density flow ${\mathbf{I}}({\mathbf{r}},t)$ and the spin current density $\boldsymbol{J}({\mathbf{r}},t)$ have the same dimension.

APPENDIX C

Further consideration requires specifying the form of the collision integrals ${{R}_{{\mathbf{p}}}} = \operatorname{Tr} {{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$ and ${{{\mathbf{R}}}_{{\mathbf{p}}}} = \operatorname{Tr} \hat {\boldsymbol\sigma }{{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$. It can be shown that in the semiclassical limit, provided that the influence of external fields on scattering processes is neglected, the spin operator ${{\hat {\mathcal{R}}}_{{\mathbf{p}}}}$ of the collision integral can be written in the following form

$$\begin{gathered} {{{\hat {\mathcal{R}}}}_{{\mathbf{p}}}} = \frac{i}{\hbar }\left\{ {\overline {{{{\hat {\mathcal{T}}}}_{{{\mathbf{pp}}}}}\left( {{{\varepsilon }_{{\mathbf{p}}}}} \right)} \delta {{{\hat {f}}}_{{\mathbf{p}}}} - \delta {{{\hat {f}}}_{{\mathbf{p}}}}\overline {\hat {\mathcal{T}}_{{{\mathbf{pp}}}}^{\dag }\left( {{{\varepsilon }_{{\mathbf{p}}}}} \right)} } \right\} \\ - \,\,\frac{{2\pi }}{\hbar }\sum\limits_{{\mathbf{p'}}} {\overline {{{{\hat {\mathcal{T}}}}_{{{\mathbf{pp}}{\kern 1pt} '}}}\left( {{{\varepsilon }_{{\mathbf{p}}}}} \right)\delta {{{\hat {f}}}_{{{\mathbf{p}}{\kern 1pt} '}}}\hat {\mathcal{T}}_{{{\mathbf{pp}}{\kern 1pt} '}}^{\dag }\left( {{{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right)} } \delta \left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right). \\ \end{gathered} $$

(89)

Here $\delta {{\hat {f}}_{{\mathbf{p}}}} \equiv \delta \hat {f}({\mathbf{r}},{\mathbf{p}},t)$, and ${{\hat {\mathcal{T}}}_{{{\mathbf{pp}}{\kern 1pt} '}}}\left( E \right) \equiv \left\langle {\mathbf{p}} \right|\hat {\mathcal{T}}\left| {{\mathbf{p}}{\kern 1pt} '} \right\rangle$ are the matrix elements of the scattering amplitude operator $\hat {\mathcal{T}}$ on the state vectors $\left| {\mathbf{p}} \right\rangle$. The first term in (89) describes the transition of an electron from the state p to all possible states upon scattering by the potential $\hat {\mathcal{U}}$. The second term in (89) describes the arrival of electrons in the state p upon scattering from all other states ${\mathbf{p}}{\kern 1pt} '$. The bar over all different combinations of matrix elements of scattering operators $\hat {\mathcal{T}}$ in (89) means averaging over various configurations of the scattering potential. In what follows, we assume that this operation has been performed and therefore the averaging sign will be omitted.

The scattering amplitude operator $\hat {\mathcal{T}}(E)$ is a solution to the Lippmann–Schwinger equation

$$\hat {\mathcal{T}} = \hat {\mathcal{V}} + \hat {\mathcal{V}}{{\left( {E - {{{\hat {\mathcal{H}}}}_{0}}} \right)}^{{ - 1}}}\hat {\mathcal{T}}{\text{,}}$$

(90)

with ${{\hat {\mathcal{H}}}_{0}} = {{{{{\left( {{\mathbf{\hat {p}}} - {{e{\mathbf{A}}} \mathord{\left/ {\vphantom {{e{\mathbf{A}}} c}} \right. \kern-0em} c}} \right)}}^{2}}} \mathord{\left/ {\vphantom {{{{{\left( {{\mathbf{\hat {p}}} - {{e{\mathbf{A}}} \mathord{\left/ {\vphantom {{e{\mathbf{A}}} c}} \right. \kern-0em} c}} \right)}}^{2}}} {2{{m}_{e}}}}} \right. \kern-0em} {2{{m}_{e}}}}$. Taking into account the spin-orbit interaction we can write the scattering potential operator $\hat {\mathcal{U}}$ as

$$\hat {\mathcal{V}} = V + \frac{\hbar }{{4m_{e}^{2}{{c}^{2}}}}\hat {\boldsymbol\sigma } \cdot \left[ {\frac{\partial }{{\partial {\mathbf{r}}}}V \times {\mathbf{\hat {p}}}} \right],$$

(91)

where V is the scattering potential created by any static defects in the crystal lattice of the conductor under consideration.

Some important information about the structure of $\hat {\mathcal{T}}$ can be obtained by analyzing the symmetry properties $\hat {\mathcal{V}}$, since the scattering amplitude can be represented as a matrix element of the scattering operator, and the symmetry properties of $\hat {\mathcal{T}}$ coincide with the symmetry properties of the Hamiltonian of our system. We restrict ourselves to considering the case when the scatterers, which form the potential $\hat {\mathcal{V}}$ and determine the form of the scattering amplitude operator, do not have their own spin. We assume the Hamiltonian to be invariant under arbitrary rotations and spatial inversion. Consequently, the scattering amplitude operator must also be invariant under these transformations. Based on these fundamental properties of $\hat {\mathcal{T}}$, we can find its general form for spin-orbit scattering.

Consider the properties of matrix elements ${{\hat {\mathcal{T}}}_{{{\mathbf{pp}}{\kern 1pt} '}}}$. Since $\hat {\mathcal{T}}$ is an operator in the space of both orbital and spin functions, the matrix elements can be considered as an operator only in the space of spin variables. This spin operator can always be represented as a linear combination of the Pauli matrices $\hat {\boldsymbol\sigma }$ and the identity operator

$${{\hat {\mathcal{T}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} = \mathcal{A} + \mathcal{B} \cdot \hat {\boldsymbol\sigma }{\text{.}}$$

(92)

In (92), $\mathcal{A}$ and $\mathcal{B}$ are complex-valued functions of the quasi-momenta p and ${\mathbf{p}}{\kern 1pt} '$. The scattering amplitude operator is an arbitrary rotation and inversion invariant only if $\mathcal{A}$ is a scalar and $\mathcal{B}$ is a pseudovector. Let us define three unit vectors

$${\mathbf{b}} = \frac{{{\mathbf{p}} \times {\mathbf{p}}{\kern 1pt} '}}{{\left| {{\mathbf{p}} \times {\mathbf{p}}{\kern 1pt} '} \right|}},\,\,\,\,{{{\mathbf{b}}}_{ + }} = \frac{{{\mathbf{p}} + {\mathbf{p}}{\kern 1pt} '}}{{\left| {{\mathbf{p}} + {\mathbf{p}}{\kern 1pt} '} \right|}},\,\,\,\,{{{\mathbf{b}}}_{ - }} = \frac{{{\mathbf{p}} - {\mathbf{p}}{\kern 1pt} '}}{{\left| {{\mathbf{p}} - {\mathbf{p}}{\kern 1pt} '} \right|}}.$$

(93)

The vectors b, ${{{\mathbf{b}}}_{ + }}$ and ${{{\mathbf{b}}}_{ - }}$ are orthogonal to each other and therefore can be taken as a basis of a three-dimensional space. Then we may write $\mathcal{B} = \mathcal{B}{\mathbf{b}} + {{\mathcal{B}}_{ + }}{{{\mathbf{b}}}_{ + }} + {{\mathcal{B}}_{ - }}{{{\mathbf{b}}}_{ - }}$, where $\mathcal{B}$, ${{\mathcal{B}}_{ + }}$, ${{\mathcal{B}}_{ - }}$ are some scalar or pseudoscalar functions of p and ${\mathbf{p}}{\kern 1pt} '$. Under inversion, p and ${\mathbf{p}}{\kern 1pt} '$ change their sign, and therefore ${{{\mathbf{b}}}_{ + }}$ and ${{{\mathbf{b}}}_{ - }}$ also do so. Therefore, from the invariance of $\hat {\mathcal{T}}$ with respect to spatial inversion it follows that ${{\mathcal{B}}_{ + }} = {{\mathcal{B}}_{ - }} = 0$, and $\mathcal{A}$ and $\mathcal{B}$, which depend on p and ${\mathbf{p}}{\kern 1pt} '$ are scalars. These functions depend on p and ${\mathbf{p}}{\kern 1pt} '$ only through their scalar product ${\mathbf{p}} \cdot {\mathbf{p}}{\kern 1pt} '$. Due to the fulfillment of the condition $\left| {\mathbf{p}} \right| = \left| {{\mathbf{p}}{\kern 1pt} '} \right|$ for elastic scattering, $\mathcal{A}$ and $\mathcal{B}$ are functions of the angle $\theta$ between p and ${\mathbf{p}}{\kern 1pt} '$, which is called the scattering angle below: $\mathcal{A} = \mathcal{A}\left( \theta \right)$, $\mathcal{B} = \mathcal{B}\left( \theta \right)$, and $\mathcal{B}\left( 0 \right) = 0$. The explicit form of $\mathcal{A}\left( \theta \right)$ and $\mathcal{B}\left( \theta \right)$ is determined by the specifics of the potential interaction $V({\mathbf{r}})$ and is not essential for further consideration.

Thus, the scattering amplitude operator has the following general form

$${{\hat {\mathcal{T}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} = \mathcal{A}\left( \theta \right) + \mathcal{B}\left( \theta \right)\left( {{\mathbf{b}} \cdot \hat {\boldsymbol\sigma }} \right),$$

(94)

where b is the unit vector of the normal to the scattering plane, determined by the first of expressions (93).

Substituting into (89) the representations for $\delta \hat {f}$ and $\hat {\mathcal{T}}$ in the form $\delta \hat {f} = {{\left( {\delta n + \delta {\mathbf{s}} \cdot \hat {\boldsymbol\sigma }} \right)} \mathord{\left/ {\vphantom {{\left( {\delta n + \delta {\mathbf{s}} \cdot \hat {\boldsymbol\sigma }} \right)} 2}} \right. \kern-0em} 2}$ and ${{\hat {\mathcal{T}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} = {{\mathcal{A}}_{{{\mathbf{pp}}{\kern 1pt} '}}} + {{\mathcal{B}}_{{{\mathbf{pp}}{\kern 1pt} '}}}{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} \cdot \hat {\boldsymbol\sigma }$, where ${{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} \equiv {\mathbf{b}}$, after performing all operations with spin operators, we obtain the collision integrals in the form

$$\begin{gathered} {{R}_{{\mathbf{p}}}} = {{\nu }_{{\mathbf{p}}}}\delta {{n}_{{\mathbf{p}}}} - \sum\limits_{{\mathbf{p'}}} {\left( {W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{nsf}} \right)}} + W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{sf}} \right)}}} \right)\delta {{n}_{{{\mathbf{p}}{\kern 1pt} '}}}} \\ - \,\,2\sum\limits_{{\mathbf{p}}{\kern 1pt} '} {\operatorname{Re} W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}}\left( {{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} \cdot \delta {{{\mathbf{s}}}_{{{\mathbf{p}}{\kern 1pt} '}}}} \right)} , \\ \end{gathered} $$

(95)

$$\begin{gathered} {{{\mathbf{R}}}_{{\mathbf{p}}}} = {{\nu }_{{\mathbf{p}}}}\delta {{{\mathbf{s}}}_{{\mathbf{p}}}} - \sum\limits_{{\mathbf{p}}{\kern 1pt} '} {\left\{ {\left( {W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{nsf}} \right)}} - W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{sf}} \right)}}} \right)\delta {{{\mathbf{s}}}_{{{\mathbf{p}}{\kern 1pt} '}}}} \right.} \\ + \,\,2W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{sf}} \right)}}{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}}\left( {{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} \cdot \delta {{{\mathbf{s}}}_{{{\mathbf{p}}{\kern 1pt} '}}}} \right) + \left. {2\operatorname{Im} W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}}\left[ {{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}} \times \delta {{{\mathbf{s}}}_{{{\mathbf{p}}{\kern 1pt} '}}}} \right]} \right\} \\ - \,\,\sum\limits_{{\mathbf{p}}{\kern 1pt} '} {2\operatorname{Re} W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}}{{{\mathbf{b}}}_{{{\mathbf{pp}}{\kern 1pt} '}}}\delta {{n}_{{{\mathbf{p}}{\kern 1pt} '}}}} . \\ \end{gathered} $$

(96)

Here we introduce

$${{\nu }_{{\mathbf{p}}}} = - \frac{2}{\hbar }\operatorname{Im} {{\mathcal{A}}_{{{\mathbf{pp}}}}},$$

(97)

$$W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{nsf}} \right)}} = \frac{{2\pi }}{\hbar }{{\left| {{{\mathcal{A}}_{{{\mathbf{pp}}{\kern 1pt} '}}}} \right|}^{2}}\delta \left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right),$$

(98)

$$W_{{\mathbf{pp}{\kern 1pt} '}}^{{\left( {\text{sf}} \right)}} = \frac{{2\pi }}{\hbar }{{\left| {{{\mathcal{B}}_{{{\mathbf{pp}}{\kern 1pt} '}}}} \right|}^{2}}\delta \left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right),$$

(99)

$$W_{{\mathbf{pp}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}} = \frac{{2\pi }}{\hbar }{{\mathcal{A}}_{{{\mathbf{pp}}{\kern 1pt} '}}}\mathcal{B}_{{{\mathbf{pp}}{\kern 1pt} '}}^{*}\delta \left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right).$$

(100)

The quantity ${{\nu }_{{\mathbf{p}}}}$ defined by (97) has the meaning of the frequency of electron scattering processes from a state with a quasi-momentum p to all possible spin states, including both processes without a change in the spin state and scattering processes with spin flip. $W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{nsf}} \right)}}$ is the differential probability of electron scattering from a state with a quasi-momentum ${\mathbf{p}}{\kern 1pt} '$ to a state with a quasi-momentum p per unit time without spin flip (non-spin-flip), while $W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{sf}} \right)}}$ is the differential probability of scattering with spin flip per unit time. $W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}}$ characterizes asymmetric, so-called skew spin scattering, and is a complex function of the electron quasi-momenta before and after scattering, and therefore does not have a simple sense of the probability of any process.

When writing the collision integrals in the form (95), (96), we have taken into account that $\mathcal{A}$ and $\mathcal{B}$ defining the operator $\hat {\mathcal{T}}$ obey the relation, which is known in scattering theory as the “optical theorem”

$$ - \frac{2}{\hbar }\operatorname{Im} {{\mathcal{A}}_{{{\mathbf{pp}}}}} = \frac{{2\pi }}{\hbar }\sum\limits_{{\mathbf{p}}{\kern 1pt} '} {\left( {{{{\left| {{{\mathcal{A}}_{{{\mathbf{pp}}{\kern 1pt} '}}}} \right|}}^{2}} + {{{\left| {{{\mathcal{B}}_{{{\mathbf{pp}}{\kern 1pt} '}}}} \right|}}^{2}}} \right)} \delta \left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{{{\mathbf{p}}{\kern 1pt} '}}}} \right).$$

(101)

Using (101) we get

$${{\nu }_{{\mathbf{p}}}} = \sum\limits_{{\mathbf{p}}{\kern 1pt} '} {\left[ {W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{nsf}} \right)}} + W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}} \right]} .$$

(102)

It is natural to call

$$\nu _{{\mathbf{p}}}^{{\left( {\text{nsf}} \right)}} = \sum\limits_{{\mathbf{p}}{\kern 1pt} '} {W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{nsf}} \right)}}} ,$$

(103)

the scattering frequency without spin flip, and

$$\nu _{{\mathbf{p}}}^{{\left( {\text{sf}} \right)}} = \sum\limits_{{\mathbf{p}}{\kern 1pt} '} {W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}} ,$$

(104)

the frequency of scattering with spin flip. It’s obvious that ${{\nu }_{{\mathbf{p}}}} = \nu _{{\mathbf{p}}}^{{\left( {\text{nsf}} \right)}} + \nu _{{\mathbf{p}}}^{{\left( {\text{sf}} \right)}}$.

Kinetic equations in the form (83), (84) are applicable to describe spin transport when the mean free path of conduction electrons is comparable to or even exceeds the characteristic scale of change in the magnetic and electric fields, as well as the characteristic linear size of the sample. In the other limiting case, when the mean free path of electrons is the smallest parameter of the system of dimension of length, the problem of describing spin kinetics is greatly simplified. In this case, the description of kinetics in the language of distribution functions can be replaced by a much simpler description in terms of densities and flows. Such a description is referred to as reduced. Using the system of Eqs. (83), (84) for distribution functions, below we obtain a system of coupled equations of motion for coarse-grained variables: densities $N\left( {{\mathbf{r}},t} \right)$, ${\mathbf{S}}\left( {{\mathbf{r}},t} \right)$ and flows ${\mathbf{I}}\left( {{\mathbf{r}},t} \right)$, $\boldsymbol{J}\left( {{\mathbf{r}},t} \right)$.

In the absence of external fields, the conduction electron system is in thermodynamic equilibrium and is described by the statistical operator ${{\hat {f}}_{0}} = F\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right)$, where $F\left( \varepsilon \right) = {1 \mathord{\left/ {\vphantom {1 {\left( {\exp \left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {{{k}_{B}}T}}} \right. \kern-0em} {{{k}_{B}}T}}} \right) + 1} \right)}}} \right. \kern-0em} {\left( {\exp \left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {{{k}_{B}}T}}} \right. \kern-0em} {{{k}_{B}}T}}} \right) + 1} \right)}}$ is the Fermi function, ${{\varsigma }_{0}}$ is the chemical potential determined by the condition $\sum\nolimits_{\mathbf{p}} {\text{Tr}F\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right)} = {{N}_{0}}$, in which ${{N}_{0}}$ is the equilibrium density of conduction electrons, which is assume to be independent of coordinates r.

Let us define the instantaneous local equilibrium value of the quantum distribution function for a system placed in a magnetic field B as the operator ${{\hat {f}}_{L}} = F\left( {{{\varepsilon }_{{\mathbf{p}}}} + \mu {\kern 1pt} \hat {\boldsymbol\sigma } \cdot {\mathbf{B}} - \varsigma } \right)$. Here and below we will restrict ourselves to the case when $\left| {\mathbf{B}} \right| = {\text{const}}$. Here, $\varsigma$ is the local chemical potential determined by $\sum\nolimits_{\mathbf{p}} {\text{Tr}F\left( {{{\varepsilon }_{{\mathbf{p}}}} + \mu \hat {\boldsymbol\sigma } \cdot {\mathbf{B}} - \varsigma } \right)} = N$. Assuming that the energy of the spin splitting $\mu B$ and the change in the chemical potential $\left( {\varsigma - {{\varsigma }_{0}}} \right)$ are small compared to ${{\varsigma }_{0}}$, we can write ${{\hat {f}}_{L}}$ in a linear approximation as

$${{\hat {f}}_{L}} = F\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right) + F{\kern 1pt} '\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right)\left[ {\mu \hat {\boldsymbol\sigma } \cdot {\mathbf{B}} - \left( {\varsigma - {{\varsigma }_{0}}} \right)} \right],$$

(105)

where $F{\kern 1pt} '\left( \varepsilon \right) = {{\partial F\left( \varepsilon \right)} \mathord{\left/ {\vphantom {{\partial F\left( \varepsilon \right)} {\partial \varepsilon }}} \right. \kern-0em} {\partial \varepsilon }}$ is the energy derivative of the Fermi function. To simplify the notation, below we will use the notation $F\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right) \equiv {{F}_{0}}$ and $F{\kern 1pt} '\left( {{{\varepsilon }_{{\mathbf{p}}}} - {{\varsigma }_{0}}} \right) \equiv F_{0}^{'}$.

From (105) it follows that the deviation of the local equilibrium distribution function ${{n}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right)$ from the equilibrium value ${{n}_{0}} = 2{{F}_{0}}$ and the local change $\varsigma - {{\varsigma }_{0}}$ in the chemical potential are connected by relation ${{n}_{L}} - {{n}_{0}} = - 2\left( {\varsigma - {{\varsigma }_{0}}} \right)F_{0}^{'}$. Summing this relation over p, we find that the deviation of ${{n}_{L}}$ from ${{n}_{0}}$ and the deviation of the local electron density N from ${{N}_{0}}$ are connected as

$${{n}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right) - {{n}_{0}} = \left( {{{F_{0}^{'}} \mathord{\left/ {\vphantom {{F_{0}^{'}} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right. \kern-0em} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right)\left[ {N\left( {r,t} \right) - {{N}_{0}}} \right].$$

(106)

It also follows from (105) that the local equilibrium spin distribution function ${{{\mathbf{s}}}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right)$ and the magnetic field B obey relation ${{{\mathbf{s}}}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right) = 2\mu F_{0}^{'}{\mathbf{B}}$. Summing ${{{\mathbf{s}}}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right)$ over p, we obtain ${{{\mathbf{S}}}_{L}}\left( {{\mathbf{r}},t} \right) = 2\mu \sum\nolimits_{\mathbf{p}} {F_{0}^{'}} {\mathbf{B}}$. The local equilibrium distribution function ${{{\mathbf{s}}}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right)$ can be expressed in terms of the spin density ${{{\mathbf{S}}}_{L}}\left( {{\mathbf{r}},t} \right)$ as

$${{{\mathbf{s}}}_{L}}\left( {{\mathbf{r}},{\mathbf{p}},t} \right) = \left( {{{F_{0}^{'}} \mathord{\left/ {\vphantom {{F_{0}^{'}} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right. \kern-0em} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right){{{\mathbf{S}}}_{L}}\left( {{\mathbf{r}},t} \right).$$

(107)

Reduced description of the system in terms of densities and flows instead of distribution functions suggests the possibility of directly writing the distribution functions in terms of densities and flows. We postulate that the “reduced” representation ${{\hat {f}}_{R}}$ of $\hat {f}$ has the form ${{\hat {f}}_{R}} = F\left( {{{\varepsilon }_{{{\mathbf{p}} - \hat {\boldsymbol\pi }}}} + \mu \hat {\boldsymbol\sigma } \cdot {\mathbf{B}} - \hat {\varsigma }} \right)$. In this definition, the operator $\hat {\varsigma }$ describes the spin-dependent changes in the chemical potential due to the change in space and time of the densities $N\left( {{\mathbf{r}},t} \right)$ and ${\mathbf{S}}\left( {{\mathbf{r}},t} \right)$, while the vector operator $\hat {\boldsymbol\pi }$ defines the local shift of the quantum distribution function in the momentum space due to the flows ${\mathbf{I}}\left( {{\mathbf{r}},t} \right)$ and $\boldsymbol{J}\left( {{\mathbf{r}},t} \right)$.

In linear approximation

$${{\hat {f}}_{R}} = {{F}_{0}} + F_{0}^{'}\left( {\mu \hat {\boldsymbol\sigma } \cdot {\mathbf{B}} - \left( {\hat {\varsigma } - {{\varsigma }_{0}}} \right) - {\mathbf{v}} \cdot \hat {\boldsymbol\pi }} \right).$$

(108)

$\hat {\varsigma }$ and $\hat {\boldsymbol\pi }$ are defined by

$$\begin{gathered} N = \sum\limits_{\mathbf{p}} {\operatorname{Tr} {\kern 1pt} {{{\hat {f}}}_{R}}} ,\,\,\,\,{\mathbf{S}} = \sum\limits_{\mathbf{p}} {\operatorname{Tr} \hat {\boldsymbol\sigma }{{{\hat {f}}}_{R}}} , \\ {\mathbf{I}} = \sum\limits_{\mathbf{p}} {\operatorname{Tr} {\mathbf{v}}{{{\hat {f}}}_{R}}} ,\,\,\,\,\boldsymbol{J} = \sum\limits_{\mathbf{p}} {\operatorname{Tr} {\mathbf{v}} \otimes \hat {\boldsymbol\sigma }{{{\hat {f}}}_{R}}} , \\ \end{gathered} $$

(109)

therefore we obtain

$${{\hat {f}}_{R}} = \frac{1}{2}{{n}_{R}} + \frac{1}{2}{{s}_{R}} \cdot \hat {\boldsymbol\sigma }{\text{,}}$$

(110)

$${{n}_{R}} = 2{{F}_{0}} + \left( {{{F_{0}^{'}} \mathord{\left/ {\vphantom {{F_{0}^{'}} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right. \kern-0em} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right)\left( {N - {{N}_{0}} + {{{\mathbf{v}} \cdot {\mathbf{I}}} \mathord{\left/ {\vphantom {{{\mathbf{v}} \cdot {\mathbf{I}}} {v_{0}^{2}}}} \right. \kern-0em} {v_{0}^{2}}}} \right),$$

(111)

$${{s}_{R}} = \left( {{{F_{0}^{'}} \mathord{\left/ {\vphantom {{F_{0}^{'}} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right. \kern-0em} {\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right)\left( {{\mathbf{S}} + {{{\mathbf{v}} \cdot \boldsymbol{J}} \mathord{\left/ {\vphantom {{{\mathbf{v}} \cdot J} {v_{0}^{2}}}} \right. \kern-0em} {v_{0}^{2}}}} \right),$$

(112)

where

$${{v}_{0}} = {{\left( {\frac{1}{3}\frac{{\sum\limits_{\mathbf{p}} {{{{\mathbf{v}}}^{2}}F_{0}^{'}} }}{{\sum\limits_{\mathbf{p}} {F_{0}^{'}} }}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}.$$

(113)

${{v}_{0}}$ is the quasi-momentum-mean-square value of any of the electron velocity components near the isoenergetic surface ${{\varepsilon }_{{\mathbf{p}}}} = {{\varsigma }_{0}}$. For degenerate electron gas $F_{0}^{'} = - \delta ({{\varepsilon }_{{\mathbf{p}}}} - {{\varepsilon }_{\operatorname{F} }})$, where ${{\varepsilon }_{\operatorname{F} }}$ is the Fermi energy, and ${{v}_{0}} = {{{{v}_{\operatorname{F} }}} \mathord{\left/ {\vphantom {{{{v}_{\operatorname{F} }}} {\sqrt 3 }}} \right. \kern-0em} {\sqrt 3 }}$, where ${{v}_{\operatorname{F} }}$ is the Fermi velocity.

Let us proceed to the derivation of equations for densities and currents from equations for distribution functions. To do this, we sum Eqs. (83), (84) over p, and then perform the same operation, preliminarily multiplying each of Eqs. (83), (84) by velocity v. When performing the above operations, neglecting the difference between the distribution functions n and s from ${{n}_{R}}$ and ${{s}_{R}}$, respectively, we obtain a system of coupled equations for N, S, I, and J:

$$\frac{\partial }{{\partial t}}N + \frac{\partial }{{\partial {\mathbf{r}}}} \cdot {\mathbf{I}} = 0,$$

(114)

$$\frac{\partial }{{\partial t}}{\mathbf{S}} + \frac{\partial }{{\partial {\mathbf{r}}}} \cdot \boldsymbol{J} + \frac{{2\mu }}{\hbar }\left[ {{\mathbf{S}} \times {\mathbf{B}}} \right] + \frac{1}{{{{\tau }_{S}}}}\delta {\mathbf{S}} = 0,$$

(115)

$$\begin{gathered} \frac{\partial }{\partial t}\mathbf{I}+v_{0}^{2}\frac{\partial }{\partial {\kern 1pt} \mathbf{r}}\delta N-\frac{e}{{{m}_{e}}}\mathbf{E}N \\ +\,\,\frac{e}{{{m}_{e}}c}\left[ \mathbf{B}\times \mathbf{I} \right]+\frac{\mu }{{{m}_{e}}}\left( \frac{\partial }{\partial {\kern 1pt} \mathbf{r}}\otimes \mathbf{B} \right)\cdot \mathbf{S} \\ +\,\,\frac{1}{{{\tau }_{O}}}\mathbf{I}+\frac{1}{{{\tau }_{\operatorname{SO}}}}\epsilon \cdot \cdot \,\boldsymbol{J}=0, \\ \end{gathered}$$

(116)

$$\begin{gathered} \frac{\partial }{\partial t}\boldsymbol{J}+v_{0}^{2}\frac{\partial }{\partial \mathbf{r}}\otimes \delta \mathbf{S}-\frac{e}{{{m}_{e}}}\mathbf{E}\otimes \mathbf{S} \\ +\,\,\frac{e}{{{m}_{e}}c}\left[ \mathbf{B}\times \boldsymbol{J} \right]+\frac{2\mu }{\hbar }\left[ \boldsymbol{J}\times \mathbf{B} \right]+\frac{\mu }{{{m}_{e}}}\left( \frac{\partial }{\partial \mathbf{r}}\otimes \mathbf{B} \right)\delta N \\ +\,\,\frac{1}{{{\tau }_{O}}}\boldsymbol{J}+\frac{1}{{{\tau }_{\operatorname{SO}}}}\epsilon \cdot \mathbf{I}=0. \\ \end{gathered}$$

(117)

In (114)–(117) there appear: the deviation $\delta N = N - {{N}_{0}}$ of the electron density N from the equilibrium value ${{N}_{0}}$ and the deviation $\delta {\mathbf{S}} = {\mathbf{S}} - {{{\mathbf{S}}}_{L}}$ of the spin density S from its local equilibrium value ${{{\mathbf{S}}}_{L}}$, which determines the local equilibrium value of the conduction electron magnetization ${{{\mathbf{m}}}_{L}} = - \mu {{{\mathbf{S}}}_{L}}$, with $\chi = - 2{{\mu }^{2}}\sum\nolimits_{\mathbf{p}} {F_{0}^{'}}$ being the Pauli magnetic susceptibility of the electron gas.

To describe spin-dependent scattering, the following quantities are introduced in above equations, which characterize the rate of various relaxation processes:

${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{S}}}}} \right. \kern-0em} {{{\tau }_{S}}}}$ is the spin relaxation rate (or the frequency of spin flip scattering):

$$\frac{1}{{{{\tau }_{S}}}} = \frac{4}{3}\frac{{\sum\limits_{{\mathbf{pp}}{\kern 1pt} '} {W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}F_{0}^{'}} }}{{\sum\limits_{\mathbf{p}} {F_{0}^{'}} }};$$

(118)

momentum relaxation rate ${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{O}}}}} \right. \kern-0em} {{{\tau }_{O}}}}$ (or the frequency of scattering processes with a change in the orbital motion of electrons):

$$\frac{1}{{{{\tau }_{O}}}} = \frac{{\sum\limits_{{\mathbf{pp}}{\kern 1pt} '} {{{{\left[ {W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{nsf}} \right)}} + W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}} \right]}}_{0}}\left( {{{v}^{2}} - {\mathbf{v}} \cdot {\mathbf{v}}{\kern 1pt} '} \right)F{\kern 1pt} '} }}{{\sum\limits_{\mathbf{p}} {{{v}^{2}}F{\kern 1pt} '} }};$$

(119)

rate of skew scattering of electrons ${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{{\operatorname{SO} }}}}}} \right. \kern-0em} {{{\tau }_{{\operatorname{SO} }}}}}$ (characteristic of asymmetric spin-orbit scattering):

$$\frac{1}{{{{\tau }_{{\operatorname{SO} }}}}} = \frac{{\sum\limits_{{\mathbf{pp}}{\kern 1pt} '} {\operatorname{Re} W_{{{\mathbf{pp}}{\kern 1pt} '}}^{{\left( {\text{as}} \right)}}\left| {\left[ {{\mathbf{v}} \times {\mathbf{v}}{\kern 1pt} '} \right]} \right|F{\kern 1pt} '} }}{{\sum\limits_{\mathbf{p}} {{{v}^{2}}F{\kern 1pt} '} }}.$$

(120)

It is easy to see that Eq. (114) is nothing but the continuity equation for the electron flow. The presence of two terms on the left hand side of the equation reflects the fulfillment of the law of conservation of the number of particles: the rate of change in the density of particles at a given point is equal with the opposite sign to the divergence of the density vector of the particle flow at the same point.

Equation (115) is the well-known equation of motion for the spin density. It can be viewed as a continuity equation for the spin current. The second term on the left hand side of the equation is responsible for the local change in the spin density due to the transfer of spin during the flow of the spin current J from one region of space to another region with a different spin density in magnitude or direction. The third term describes the precession of the spin density of electrons in a magnetic field B. The fourth term describes the spin relaxation of electrons. As one can see from (118), the rate of spin relaxation of conduction electrons ${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{S}}}}} \right. \kern-0em} {{{\tau }_{S}}}}$ and, accordingly, the spin relaxation time ${{\tau }_{S}}$ are determined by the differential probability of spin-flip scattering $W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}$.

Equation (116) is used to find the electron flow density I for given E and B. The second term on its left hand side describes the diffusion motion of electrons that occurs in the presence of an electron density gradient. A characteristic of electron diffusion is the diffusion coefficient $D = v_{0}^{2}{{\tau }_{O}}$. The third term describes the conduction current induced by the field E, the value of which is determined by the electrical conductivity of a metal $\sigma = {{{{N}_{0}}{{e}^{2}}{{\tau }_{O}}} \mathord{\left/ {\vphantom {{{{N}_{0}}{{e}^{2}}{{\tau }_{O}}} {{{m}_{e}}}}} \right. \kern-0em} {{{m}_{e}}}}$. The fourth describes the change in the density of the electric current due to the action of the Lorentz force, forcing the electrons to move in cyclotron orbits and leading to the appearance of the Hall effect. The fifth term takes into account the change in electronic conductivity in the presence of a dependence of the magnetic field acting in the metal on coordinates. It is this contribution that determines the change in the conductivity of inhomogeneously magnetized conductors. The sixth term describes the rate of change in the orbital state of electrons, which is determined by the transport momentum relaxation time ${{\tau }_{O}}$. The momentum relaxation rate ${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{O}}}}} \right. \kern-0em} {{{\tau }_{O}}}}$, as can be seen from definition (119), is given by the sum of the differential scattering probabilities without spin flip and with spin flip $W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{nsf}} \right)}} + W_{{{\mathbf{p}}{\kern 1pt} '{\kern 1pt} {\mathbf{p}}}}^{{\left( {\text{sf}} \right)}}$. Finally, the last term on the left hand side of (116) is due to the asymmetry of electron spin scattering, the intensity of which is given by the relaxation time ${{\tau }_{{\operatorname{SO} }}}$. This term describes the specific features of a physical phenomenon called the “inverse spin Hall effect”. The essence of the latter lies in the fact that the presence of a spin current causes the appearance of an electric current.

Equation (117) is an equation for finding the spin current tensor J. The second term on its left hand side describes the diffusion component of the spin flow, which is determined by the spin diffusion coefficient D. The third term describes the effects of spin density drift under the action of an electric field. The fourth term, which contains the vector product of the vector B and the spin current tensor J, describes the effect of the Lorentz force, similarly to the fourth term in (116). The fifth term, the vector product of the spin current tensor J and vector B, describes the precession of electrons. The sixth term describes the effect of magnetic field inhomogeneities on the spin current, the seventh term takes into account the damping of the spin current with the momentum relaxation rate ${1 \mathord{\left/ {\vphantom {1 {{{\tau }_{O}}}}} \right. \kern-0em} {{{\tau }_{O}}}}$. The last term, due to the asymmetry of the spin-orbit scattering of electrons, is responsible for the spin Hall effect.

APPENDIX D

In this Appendix we give a definition of the mathematical operations of the scalar, double scalar, vector and tensor product of vectors and tensors used in the work. A reader can find more detailed information in books on tensor analysis [69, 70].

The definition of the scalar and vector products of vectors are well-known: if $\mathbf{A} = {{A}_{i}}{\mathbf{e}_{i}}$, $\mathbf{C} = {{C}_{i}}{\mathbf{e}_{i}}$, where ${\mathbf{e}_{i}}$ is unit vector in ith direction ($i = x,y,z$) of our Cartesian coordinate system, then the scalar product is $\left( {\mathbf{A} \cdot \mathbf{C}} \right) = {{A}_{i}}{{C}_{i}}$ and vector product is vector ${{\left[ \mathbf{A}\times \mathbf{C} \right]}_{i}}={{\epsilon}_{ijk}}{{A}_{j}}{{C}_{k}}$, where $\epsilon$ is the completely antisymmetric unit tensor of third rank.

Scalar product of vector A and second rank tensor F is vector: ${{\left( {\mathbf{A} \cdot \mathbf{F}} \right)}_{j}} = {{A}_{i}}{{F}_{{ij}}}$, ${{\left( {\mathbf{F} \cdot \mathbf{A}} \right)}_{i}} = {{F}_{{ij}}}{{A}_{j}}$.

Double scalar product of rank 2 tensor F and rank 3 tensor G is vector: ${{\left( {\mathbf{F} \cdot \cdot \,\mathbf{G}} \right)}_{k}} = {{F}_{{ij}}}{{G}_{{jik}}}$, ${{\left( {\mathbf{G} \cdot \cdot \,\mathbf{F}} \right)}_{i}} = {{G}_{{ijk}}}{{F}_{{kj}}}$.

Tensor product of vectors A and C is a tensor of the 2nd rank: ${{\left( {\mathbf{A} \otimes \mathbf{C}} \right)}_{{ij}}} = {{A}_{i}}{{C}_{j}}$.

Vector product of vector A and rank 2 tensor F is a rank 2 tensor: ${{\left[ \mathbf{A}\times \mathbf{F} \right]}_{mj}}={{\epsilon}_{mki}}{{A}_{k}}{{F}_{ij}}$, ${{\left[ \mathbf{F}\times \mathbf{A} \right]}_{im}}={{\epsilon}_{mjk}}{{F}_{ij}}{{A}_{k}}$.

Scalar product of vector A and rank 3 tensor G is rank 2 tensor: ${{\left( {\mathbf{A} \cdot \mathbf{G}} \right)}_{{jk}}} = {{A}_{i}}{{G}_{{ijk}}}$, ${{\left( {\mathbf{G} \cdot \mathbf{A}} \right)}_{{ij}}} = {{G}_{{ijk}}}{{A}_{k}}$.

Double scalar product of 3rd rank tensors F and G is tensor of the 2nd rank: ${{\left( {\mathbf{F} \cdot \cdot \,\mathbf{G}} \right)}_{{im}}} = {{F}_{{ijk}}}{{G}_{{kjm}}}$.

APPENDIX E

Consider a system of conduction electrons of two metals bordering on a plane $z = 0$. In this section, the characteristics of the conductor in the region $z < 0$ are labeled as 1, and in the region $z > 0$ as 2. Let $\varepsilon _{\mathbf{p}}^{{(i)}}$ and ${\mathbf{v}^{{(i)}}} = {{\partial \varepsilon _{\mathbf{p}}^{{(i)}}} \mathord{\left/ {\vphantom {{\partial \varepsilon _{p}^{{(i)}}} \partial }} \right. \kern-0em} \partial }\mathbf{p}$ be the spectrum and velocity of electrons in ith metal. All electrons are divided into two groups. The first one is the electrons that moving from the depth of the conductor towards the interface; the second one is the electrons that moving from the interface. For conductor 1, the z-component of the velocity of the electrons moving towards the boundary is positive, while for the reflected electrons in region 1, this component is negative. On the contrary, for conductor 2, electrons moving towards the boundary have negative z‑component of velocity, while for reflected electrons in region 2, this component is positive. The flow of electrons along z axis is the sum of the flow $I_{z}^{ > }$ of electrons with ${{v}_{z}} > 0$ and the flow $I_{z}^{ < }$ of electrons with ${{v}_{z}} < 0$: ${{I}_{z}}\left( z \right) = I_{z}^{ > }\left( z \right) + I_{z}^{ < }\left( z \right)$. Similarly, the spin current is ${{{\mathbf{P}}}_{z}}\left( z \right) = {\mathbf{P}}_{z}^{ > }\left( z \right) + {\mathbf{P}}_{z}^{ < }\left( z \right)$.

Generally, not all electrons incident on the interface can pass through it. Let W denote the relative fraction of electrons penetrating the boundary; then $R = 1 - W$ is the relative fraction of electrons reflected from the boundary. By definition, $0 \leqslant W \leqslant 1$, $0 \leqslant R \leqslant 1$, and $W + R = 1$.

The flow $I_{z}^{ > }\left( { + 0} \right)$ of electrons moving from the boundary in conductor 2, is the sum of two flows. The first is a part of the flow $I_{z}^{ > }\left( {z = - 0} \right)$ of electrons moving towards the boundary in conductor 1, which penetrated into conductor 2; it equals to $WI_{z}^{ > }\left( {z = - 0} \right)$. The second is the part of the flow of electrons moving towards the boundary in conductor 2, which did not penetrate into conductor 1; this component equals to $- RI_{z}^{ < }\left( { + 0} \right)$. Similar considerations can be made for $I_{z}^{ < }\left( { - 0} \right)$. The flow continuity condition can then be written as:

$$I_{z}^{ > }\left( { + 0} \right) = WI_{z}^{ > }\left( { - 0} \right) - RI_{z}^{ < }\left( { + 0} \right),$$

(121)

$$I_{z}^{ < }\left( { - 0} \right) = WI_{z}^{ < }\left( { + 0} \right) - RI_{z}^{ > }\left( { - 0} \right).$$

(122)

Taking into account $I_{z}^{{(1)}}\left( { - 0} \right) = I_{z}^{ < }\left( { - 0} \right) + I_{z}^{ > }\left( { - 0} \right)$ and $I_{z}^{{(2)}}\left( { + 0} \right) = I_{z}^{ > }\left( { + 0} \right) + I_{z}^{ < }\left( { + 0} \right)$ we see that from (121) and (122) the equality of flows to the left and right of the boundary follows:

$$I_{z}^{{(1)}}\left( { - 0} \right) = I_{z}^{{(2)}}\left( { + 0} \right) = W\left[ {I_{z}^{ > }\left( { - 0} \right) + I_{z}^{ < }\left( { + 0} \right)} \right].$$

(123)

In other words, the boundary conditions (121) and (122) automatically ensure the continuity of the electron flow at the interface: ${{I}_{z}}\left( 0 \right) = I_{z}^{{(1)}}\left( { - 0} \right) = I_{z}^{{(2)}}\left( { + 0} \right)$.

The incident on the boundary flows $I_{z}^{ > }\left( { - 0} \right)$ and $I_{z}^{ < }\left( { + 0} \right)$, appearing on the right hand side of (123), can be written in terms of the non-equilibrium part of the electron density distribution functions $\delta n\left( {z,\mathbf{p}} \right)$

$$I_{z}^{ > }\left( { - 0} \right) = \sum\limits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} > 0} {v_{z}^{{(1)}}\delta {{n}^{{(1)}}}\left( { - 0,\mathbf{p}} \right)} ,$$

(124)

$$I_{z}^{ < }\left( { + 0} \right) = \sum\limits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} < 0} {v_{z}^{{(2)}}\delta {{n}^{{(2)}}}\left( { + 0,\mathbf{p}} \right)} .$$

(125)

In turn, the distribution functions of electrons incident on the boundary appearing on the right-hand sides of Eqs. (124) and (125) can be represented in the form

$$\begin{gathered} \delta {{n}^{{(i)}}}\left( {z,\mathbf{p}} \right) = \frac{{F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)}}{{\sum\limits_{\mathbf{p}} {F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} }} \\ \times \,\,\left[ {\delta {{N}^{{(i)}}}\left( z \right) + {{{\mathbf{v}^{{(i)}}} \cdot {\mathbf{I}^{{(i)}}}\left( z \right)} \mathord{\left/ {\vphantom {{{{v}^{{(i)}}} \cdot {{I}^{{(i)}}}\left( z \right)} {v{{{_{0}^{{(i)}}}}^{2}}}}} \right. \kern-0em} {v{{{_{0}^{{(i)}}}}^{2}}}}} \right], \\ \end{gathered} $$

(126)

where ${{\varsigma }^{{(i)}}}$ is the chemical potential in ith metal, determined from the normalization condition $\sum\nolimits_{\mathbf{p}} {2F\left( {{{\varepsilon }_{\mathbf{p}}} - \varsigma _{0}^{{(i)}}} \right)} =$ N₀; $v{{_{0}^{{(i)}}}^{2}} = \sum\nolimits_{\mathbf{p}} {{{{\mathbf{v}}}^{{(i)2}}}F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right){{{\left[ {3F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} \right]}}^{{ - 1}}}}$ is the root-mean-square value of any of the components of the electron velocity near the isoenergetic surface $\varepsilon _{\mathbf{p}}^{{(i)}} = \varsigma _{0}^{{(i)}}$, for a degenerate electron gas $v_{0}^{{(i)}} = {{v_{\operatorname{F} }^{{(i)}}} \mathord{\left/ {\vphantom {{v_{\operatorname{F} }^{{(i)}}} {\sqrt 3 }}} \right. \kern-0em} {\sqrt 3 }}$.

As a result, from relations (123) we obtain the boundary conditions for the electron flow in the form:

$$\begin{gathered} I_{z}^{{(1)}}\left( { - 0} \right) = I_{z}^{{(2)}}\left( { + 0} \right) \\ = \frac{1}{2}\frac{W}{R}\left[ {{{v}^{{(1)}}}\delta {{N}^{{(1)}}}\left( { - 0} \right) - {{v}^{{(2)}}}\delta {{N}^{{(2)}}}\left( { + 0} \right)} \right], \\ \end{gathered} $$

(127)

where ${{v}^{{(i)}}} = {{2\sum\nolimits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} > 0} {v_{z}^{{(i)}}F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} } \mathord{\left/ {\vphantom {{2\sum\nolimits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} > 0} {v_{z}^{{(i)}}F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} } {\sum\nolimits_{\mathbf{p}} {F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} }}} \right. \kern-0em} {\sum\nolimits_{\mathbf{p}} {F{\kern 1pt} '\left( {\varepsilon _{p}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} }}$, for a degenerate electron gas ${{v}^{{(i)}}} = {{v_{\operatorname{F} }^{{(i)}}} \mathord{\left/ {\vphantom {{v_{\operatorname{F} }^{{(i)}}} 2}} \right. \kern-0em} 2}$.

Equation (127) is nothing more than a phenomenological boundary condition that allows one to stitch electron flows when two conductors come into contact.

When describing spin transport through the boundary between two conductors, it is necessary to take into account the fact that the processes of electron passage through the boundary and reflection from it can occur both without spin flip and with spin flip [71]. Let ${{W}^{{(\text{sf})}}}$ and ${{W}^{{(\text{nsf})}}}$ be the probabilities of electron penetration through the boundary with and without spin flip, ${{R}^{{(\text{sf})}}}$ and ${{R}^{{(\text{nsf})}}}$ be the probabilities of electron reflection from the boundary with spin flip and without, respectively. By definition, $W = {{W}^{{(\text{nsf})}}} + {{W}^{{(\text{sf})}}}$ and $R = {{R}^{{(\text{nsf})}}} + {{R}^{{(\text{sf})}}}$.

The spin flow of electrons moving from the boundary in the conductor 2, ${\mathbf{P}}_{z}^{ > }\left( { + 0} \right)$, is the sum of four flows. The first is a part of the flow of electrons moving to the boundary in conductor 1, ${\mathbf{P}}_{z}^{ > }\left( { - 0} \right)$, which penetrate into conductor 2 without spin flip, equal to ${{W}^{{(\text{nsf})}}}{\mathbf{P}}_{z}^{ > }\left( { - 0} \right)$. The second component is a part of the flow of electrons moving to the boundary in conductor 1, ${\mathbf{P}}_{z}^{ > }\left( { - 0} \right)$, which penetrate into conductor 2 as a result of scattering with a spin flip, equal to $- {{W}^{{(\text{sf})}}}{\mathbf{P}}_{z}^{ > }\left( { - 0} \right)$. The third one is a part of the flow of electrons moving to the boundary in conductor 2, ${\mathbf{P}}_{z}^{ < }\left( { + 0} \right)$, which does not penetrate into conductor 1 and is reflected without spin flip, equal to $- {{R}^{{(\text{nsf})}}}{\mathbf{P}}_{z}^{ < }\left( { + 0} \right)$. The fourth component is a part of the flow of electrons moving to the boundary in conductor 2, which does not penetrate into conductor 1 and is reflected as a result of the spin flip scattering, equal to ${{R}^{{(\text{sf})}}}{\mathbf{P}}_{z}^{ < }\left( { + 0} \right)$. Similar considerations can be given with respect to flow ${\mathbf{P}}_{z}^{ < }\left( { - 0} \right)$.

The condition of the balance of spin flow can be written as

$$\begin{gathered} {\mathbf{P}}_{z}^{ > }\left( { + 0} \right) = \left[ {{{W}^{{(\text{nsf})}}} - {{W}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ > }\left( { - 0} \right) \\ - \,\,\left[ {{{R}^{{(\text{nsf})}}} - {{R}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ < }\left( { + 0} \right), \\ \end{gathered} $$

(128)

$$\begin{gathered} {\mathbf{P}}_{z}^{ < }\left( { - 0} \right) = \left[ {{{W}^{{(\text{nsf})}}} - {{W}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ < }\left( { + 0} \right) \\ - \,\,\left[ {{{R}^{{(\text{nsf})}}} - {{R}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ > }\left( { - 0} \right). \\ \end{gathered} $$

(129)

As the spin flow in conductor 1 is $\mathbf{P}_{z}^{{(1)}}\left( { - 0} \right) = {\mathbf{P}}_{z}^{ > }\left( { - 0} \right) + {\mathbf{P}}_{z}^{ < }\left( { - 0} \right)$, and in conductor 2 is $\mathbf{P}_{z}^{{(2)}}\left( { + 0} \right) = {\mathbf{P}}_{z}^{ > }\left( { + 0} \right) + {\mathbf{P}}_{z}^{ < }\left( { + 0} \right)$ it follows from (128) and (129)

$$\begin{gathered} \mathbf{P}_{z}^{{(2)}}\left( { + 0} \right) = \left[ {W - 2{{W}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ > }\left( { - 0} \right) \\ + \,\,\left[ {W + 2{{R}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ < }\left( { + 0} \right), \\ \end{gathered} $$

(130)

$$\begin{gathered} \mathbf{P}_{z}^{{(1)}}\left( { - 0} \right) = \left[ {W - 2{{W}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ < }\left( { + 0} \right) \\ + \,\,\left[ {W + 2{{R}^{{(\text{sf})}}}} \right]{\mathbf{P}}_{z}^{ > }\left( { - 0} \right). \\ \end{gathered} $$

(131)

${\mathbf{P}}_{z}^{ > }\left( { - 0} \right)$ and ${\mathbf{P}}_{z}^{ < }\left( { + 0} \right)$ can be expressed in terms of the non-equilibrium part of the electron spin density distribution function $\delta s\left( {z,p} \right)$:

$${\mathbf{P}}_{z}^{ > }\left( { - 0} \right) = \sum\limits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} > 0} {v_{z}^{{(1)}}\delta {\mathbf{s}^{{(1)}}}\left( { - 0,\mathbf{p}} \right)} ,$$

(132)

$${\mathbf{P}}_{z}^{ < }\left( { + 0} \right) = \sum\limits_{{{p}_{x}},{{p}_{y}},{{p}_{z}} < 0} {v_{z}^{{(2)}}\delta {\mathbf{s}^{{(2)}}}\left( { + 0,\mathbf{p}} \right)} .$$

(133)

Spin density distribution functions $\delta {\mathbf{s}^{{(i)}}}\left( {z,\mathbf{p}} \right)$ of electrons incident on the boundary from the depth of ith metal for a degenerate electron gas can be written as

$$\begin{gathered} \delta {\mathbf{s}^{{(i)}}}\left( {z,\mathbf{p}} \right) = \frac{{F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)}}{{\sum\limits_{\mathbf{p}} {F{\kern 1pt} '\left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varsigma _{0}^{{(i)}}} \right)} }} \\ \times \,\,\left[ {\delta {\mathbf{S}^{{(i)}}}\left( z \right) + {{v_{z}^{{(i)}}\mathbf{P}_{z}^{{(i)}}\left( z \right)} \mathord{\left/ {\vphantom {{v_{z}^{{(i)}}P_{z}^{{(i)}}\left( z \right)} {v{{{_{0}^{{(i)}}}}^{2}}}}} \right. \kern-0em} {v{{{_{0}^{{(i)}}}}^{2}}}}} \right]. \\ \end{gathered} $$

(134)

Taking into account (134) we can rewrite (130) and (131) in the form

$$\begin{gathered} \mathbf{P}_{z}^{{(1)}}\left( { - 0} \right) - \mathbf{P}_{z}^{{(2)}}\left( { + 0} \right) \\ = \frac{\varepsilon }{{1 - \varepsilon }}\left[ {{{v}^{{(1)}}}\delta {\mathbf{S}^{{(1)}}}\left( { - 0} \right) + {{v}^{{(2)}}}\delta {\mathbf{S}^{{(2)}}}\left( { + 0} \right)} \right], \\ \end{gathered} $$

(135)

$$\begin{gathered} \mathbf{P}_{z}^{{(1)}}\left( { - 0} \right) + \mathbf{P}_{z}^{{(2)}}\left( { + 0} \right) \\ = \frac{\beta }{{1 - \beta }}\left[ {{{v}^{{(1)}}}\delta {\mathbf{S}^{{(1)}}}\left( { - 0} \right) - {{v}^{{(2)}}}\delta {\mathbf{S}^{{(2)}}}\left( { + 0} \right)} \right]. \\ \end{gathered} $$

(136)

Here $\varepsilon = {{W}^{{(\text{sf})}}} + {{R}^{{(\text{sf})}}}$ is the probability of an electron spin flip when interacting with the interface, $\beta = {{W}^{{(\text{nsf})}}} + {{R}^{{(\text{sf})}}}$ is the probability that an electron passes the interface without changing its spin state, or be reflected with a change in its spin state.