INTRODUCTION

Spin-orbitronics is the newest branch of spintronics, which is currently being actively developed in the world leading research centers [1–5]. The origin of this term is related to the existence in conducting solids of a special relativistic interaction between the spin moment and the orbital angular momentum of conduction electrons. Spin-orbit interaction (SOI) can produce spin polarization of conduction electrons in a conducting solid even in the absence of any magnetic ordering in it. The phenomenon in which SOI is manifested most clearly is the spin Hall effect [6–13]. The spin Hall effect is observed in nonmagnetic (“normal”) conductive materials, in contrast to the ordinary Hall effect, in the absence of any external magnetic field and is manifested in the fact that the electric current in the conductor, flowing in an arbitrary direction, gives rise to a transverse “pure spin” current [14‒16]. A pure spin current is a spin current that is not accompanied by a macroscopic transfer of electric charge. The spin moment generated by SOI under the conditions of the spin Hall effect and carried by a pure spin current can be transferred to the magnetic subsystem of a magnetically ordered layer adjacent to a normal metal layer [2–5, 13, 17]. The branch of spintronics in which the SOI-caused processes of spin moment transfer in nanostructures consisting of layers with different types of magnetic ordering are studied is called spin-orbitronics.

The transfer of the spin moment induced by SOI in a normal metal can occur both in ferromagnetic layers [18] and in the layers with any other type of magnetic ordering: antiferromagnetic [19–22], ferrimagnetic [23], etc. In this work, we study spin transport in a “normal metal–helimagnet” heterojunction.

Phenomena associated with the spin-moment transfer from a normal metal layer to the magnetic subsystem of a layer with a helical spin structure have been the subject of recent experimental studies [24‒27]. The aim of the present work is to develop a theory that makes it possible to describe the phenomena of spin-orbitronics in terms of the equations of motion and the corresponding boundary conditions for macroscopic quantities: charge densities and spins of conduction electrons, as well as charge and spin currents. On its basis, we will describe the injection of a pure spin current from a normal metal with strong SOI into a helimagnet with helical spin ordering.

1 BASIC EQUATIONS OF SPIN-ORBITRONICS

The equations that describe the electron spin transport in conducting magnets, taking into account the SOI of conduction electrons with scatterers, were formulated within the microscopic approach in [28]. Here, we present these equations for the case where there is no external magnetic field and the motion of electrons is initiated by an electric field E. The interaction of conduction electrons with the magnetic subsystem of localized electrons will be considered within of the s-d(f) exchange model [29]. In the mean-field approximation, the action on the electron spins of a magnetic system of localized electrons with a magnetization M will be described as the action of an effective exchange field \(\Lambda {\mathbf{M}}.\) Here, \(\Lambda \) is a dimensionless parameter characterizing the intensity of the s-d(f)-exchange interaction. Without significant loss of generality, we will assume that the gas of conduction electrons is degenerate. In these approximations, the equations [28] for the conduction electron density N, the electron spin moment density S, the electron flux density I, and the spin current density J take the form

$$\frac{\partial }{{\partial t}}N + \nabla \cdot \mathbf{I} = 0,$$
(1)
$$\frac{\partial }{{\partial t}}\mathbf{S} + \frac{1}{{{{\tau }_{\text{S}}}}}\delta \mathbf{S} + \left[ {\mathbf{S} \times \boldsymbol\Omega } \right] + \nabla \cdot \mathbf{J} = 0,$$
(2)
$$\begin{gathered} \frac{\partial }{\partial t}\mathbf{I}+\frac{1}{{{\tau }_{\text{O}}}}\mathbf{I}+\frac{1}{{{\tau }_{\text{SO}}}}\epsilon\cdot \cdot \mathbf{J}+\frac{v_{\text{F}}^{2}}{3}\nabla \delta N \\ +\,\,\frac{\hbar }{2{{m}_{e}}}\left( \nabla \otimes \boldsymbol\Omega \right)\cdot \mathbf{S}-\frac{e}{{{m}_{e}}}\mathbf{E}N=0, \\ \end{gathered}$$
(3)
$$\begin{gathered} \frac{\partial }{\partial t}\mathbf{J}+\frac{1}{{{\tau }_{\text{O}}}}\mathbf{J}+\left[ \mathbf{J}\times \boldsymbol\Omega \right]+\frac{1}{{{\tau }_{\text{SO}}}}\epsilon\cdot \mathbf{I} \\ +\,\,\frac{v_{\text{F}}^{2}}{3}\nabla \otimes \delta \mathbf{S}+\frac{\hbar }{2{{m}_{e}}}\left( \nabla \otimes \boldsymbol\Omega \right)\delta N-\frac{e}{{{m}_{e}}}\mathbf{E}\otimes \mathbf{S}=0. \\ \end{gathered}$$
(4)

In Eqs. (1)–(4), the quantities S and I are vectors, while J is a second-rank tensor; the symbol \( \epsilon \) denotes the absolutely antisymmetric unit tensor of the 3rd rank. Tensors are denoted here in Italic Bold type, while vectors are denoted in Direct Bold type. The signs “\( \otimes \)”, “\( \cdot \)”, and “\( \cdot \cdot \)” are used to denote the mathematical operations of the tensor, scalar, and double scalar product of vectors and tensors, respectively.

The quantity \(\delta N = N - {{N}_{0}}\) in Eqs. (1)–(4) is the deviation of the electron density N from its local equilibrium value \({{N}_{0}},\) \(\delta \mathbf{S} = \mathbf{S} - {\mathbf{S}_{0}}\) is the deviation of the spin density S from its local equilibrium value \({\mathbf{S}_{0}} = - {{\chi \Lambda {\mathbf{M}}} \mathord{\left/ {\vphantom {{\chi \Lambda {\mathbf{M}}} \mu }} \right. \kern-0em} \mu },\) where \(\chi \) is the Pauli magnetic susceptibility of the electron gas, \(\mu \) is the magnitude of the electron magnetic moment; \(\boldsymbol\Omega = \gamma \Lambda {\mathbf{M}},\) where \(\gamma = {{2\mu } \mathord{\left/ {\vphantom {{2\mu } \hbar }} \right. \kern-0em} \hbar }\) is the gyromagnetic ratio; e, \({{m}_{e}}\), and \({{v}_{\text{F}}}\) are the charge, mass, and Fermi velocity of conduction electrons, respectively; \({{\tau }_{\text{O}}}\) is the momentum relaxation time during the orbital motion of electrons, \({{\tau }_{\text{S}}}\) is the spin relaxation time, and \({{\tau }_{\text{SO}}}\) is the time dimension characterizing the skew spin scattering of electrons. In [28], the relaxation rates \(\tau _{\text{O}}^{{ - 1}},\) \(\tau _{\text{S}}^{{ - 1}}\) and \(\tau _{\text{SO}}^{{ - 1}}\) are expressed in terms of the matrix elements of the scattering amplitude operator, which is determined by the scattering potential and is found as a solution of the Lippmann–Schwinger equation.

In the present work, we do not intend to use these results of the microscopic theory; therefore, we will also regard \({{\tau }_{\text{O}}},\) \({{\tau }_{\text{S}}}\), and \({{\tau }_{\text{SO}}}\) as given phenomenological parameters. The momentum relaxation time \({{\tau }_{\text{O}}}\) determines the conductivity of the free electron gas, \(\sigma = {{{{N}_{0}}{{e}^{2}}{{\tau }_{\text{O}}}} \mathord{\left/ {\vphantom {{{{N}_{0}}{{e}^{2}}{{\tau }_{\text{O}}}} {{{m}_{e}}}}} \right. \kern-0em} {{{m}_{e}}}}\), and the electron diffusion coefficient \(D = {{v_{\text{F}}^{2}{{\tau }_{\text{O}}}} \mathord{\left/ {\vphantom {{v_{\text{F}}^{2}{{\tau }_{O}}} 3}} \right. \kern-0em} 3}.\) The spin relaxation time \({{\tau }_{\text{S}}}\) determines the spin-diffusion length \({{L}_{\text{S}}} = \sqrt {D{{\tau }_{\text{S}}}} .\)

Further consideration will be carried out for the case where the pulling electric field \({\mathbf{E}_{0}}\) is spatially uniform and does not depend on time. We assume that the vector \({\mathbf{E}_{0}}\) lies in the XY plane, so that \({\mathbf{E}_{0}} \cdot {{{\mathbf{e}}}_{z}} = 0,\) where \({{{\mathbf{e}}}_{z}}\) is the unit vector that specifies the direction of the OZ axis. Then, all the quantities \(\delta N,\) \(\delta \mathbf{S}{\text{,}}\) I, and J do not depend on time and their coordinate dependence reduces to the dependence only on the z coordinate. The field \(\mathbf{E} = {\mathbf{E}_{0}} + \delta \mathbf{E}{\text{,}}\) where \(\delta \mathbf{E} = \delta E{{{\mathbf{e}}}_{z}}\) is the SOI-induced nonuniform component of the electric field acting in the metal, directed along the OZ axis.

When writing Eqs. (3) and (4), we omit the penultimate terms on their left-hand side. This means that we neglect the effects due to the action of forces on the electron spin due to the nonuniformity of the effective exchange field \(\Lambda {\mathbf{M}}{\text{.}}\) These effects, which were considered in detail earlier in [30], do not play a fundamental role in the present description of SOI effects because of their smallness.

In writing Eqs. (3) and (4), we linearize them with respect to the electric field E, for which we replace N by \({{N}_{0}}\) and S by \({\mathbf{S}_{0}}\) on the right-hand side of these equations. In addition, we assume to be satisfied the condition \(\Omega {{\tau }_{\text{O}}} \ll 1,\) which allows us to consider the third term on the left-hand side of Eq. (4) small compared to the second one and omit it.

As a result, the system of Eqs. (2)–(4) takes the form:

$$\frac{\partial }{{\partial {\kern 1pt} z}}{{{\mathbf{e}}}_{z}} \cdot \mathbf{I} = 0,$$
(5)
$$\frac{1}{{{{\tau }_{\text{S}}}}}\delta \mathbf{S} + \left[ {\mathbf{S} \times \boldsymbol\Omega } \right] + \frac{\partial }{{\partial {\kern 1pt} z}}{{{\mathbf{e}}}_{z}} \cdot \mathbf{J} = 0,$$
(6)
$$\mathbf{I}=\frac{\sigma }{e}\mathbf{E}-D{{\mathbf{e}}_{z}}\frac{\partial }{\partial {\kern 1pt} z}\delta N-\xi \left(\epsilon \cdot \cdot \mathbf{J} \right),$$
(7)
$$\mathbf{J}=\frac{\sigma }{e{{N}_{0}}}\mathbf{E}\otimes {\mathbf{S}_{0}}-D{{\mathbf{e}}_{z}}\otimes \frac{\partial }{\partial {\kern 1pt} z}\delta \mathbf{S}-\xi \left(\epsilon \cdot \mathbf{I} \right),$$
(8)

where we introduced a parameter \(\xi = {{{{\tau }_{\text{O}}}} \mathord{\left/ {\vphantom {{{{\tau }_{\text{O}}}} {{{\tau }_{\text{SO}}}}}} \right. \kern-0em} {{{\tau }_{\text{SO}}}}}\) that characterizes the relative intensity of the rate of skew scattering of conduction electrons (with respect to the momentum relaxation rate).

Equation (8) clearly describes the phenomenon called the spin Hall effect [6–8]: the conduction electron flux in the last term on the right-hand side of Eq. (8) induces a spin current due to the SOI. Correspondingly, Eq. (7) describes the reverse spin Hall effect: the spin current in the presence of SOI induces an electric current.

In the subsequent transformations of Eqs. (5)–(8), we will use the following relations of tensor algebra, which describe the rules for handling the tensor \(\epsilon\) and hold for arbitrary vectors a and b:

$$\epsilon\,\cdot \,\cdot \,\epsilon=-2,$$
(9)
$$\epsilon\,\cdot \,\cdot \,\,\mathbf{a}\otimes \mathbf{b}=-\left[ \mathbf{a}\times \mathbf{b} \right],$$
(10)
$$\mathbf{a}\cdot \epsilon \cdot \mathbf{b}=-\left[ \mathbf{a\times b} \right].$$
(11)

Substituting expression (7) for I to the right-hand side of Eq. (8) yields

$$\begin{gathered} \mathbf{J}=\frac{{\tilde{\sigma }}}{e{{N}_{0}}}\mathbf{E}\otimes {\mathbf{S}_{0}}-\tilde{D}{{\mathbf{e}}_{z}}\otimes \frac{\partial }{\partial {\kern 1pt} z}\delta \mathbf{S} \\ -\,\,\xi \frac{{\tilde{\sigma }}}{e}\left(\epsilon \cdot \mathbf{E} \right)+\xi \tilde{D}\frac{\partial }{\partial {\kern 1pt} z}\delta N\left(\epsilon \cdot {{\mathbf{e}}_{z}} \right), \\ \end{gathered}$$
(12)

where \(\tilde {\sigma } = {\sigma \mathord{\left/ {\vphantom {\sigma {\left( {1 + 2{{\xi }^{2}}} \right)}}} \right. \kern-0em} {\left( {1 + 2{{\xi }^{2}}} \right)}}\) and \(\tilde {D} = {D \mathord{\left/ {\vphantom {D {\left( {1 + 2{{\xi }^{2}}} \right)}}} \right. \kern-0em} {\left( {1 + 2{{\xi }^{2}}} \right)}}\) are the conductivity and diffusion coefficient renormalized by the spin-orbit interaction. In deriving (12), we neglected the difference between the tensors \(\epsilon\cdot \epsilon \cdot \cdot \mathbf{J}\) and \(\epsilon\cdot \cdot \epsilon \cdot \mathbf{J}\), which is insignificant for the purposes of this work, and used rule (9).

In order to elucidate the spin current J, we use the vector representation of the tensor J, proposed in [28]. Following [28], we introduce into consideration the vectors of polarization \({\mathbf{P}_{i}}\) of the spin currents flowing in directions i, \(i = x,y,z.\) By definition, \({\mathbf{P}_{i}} = {\mathbf{e}_{i}} \cdot \mathbf{J}{\text{.}}\) Specifying three vectors \({\mathbf{P}_{i}}\) is completely equivalent to specifying the tensor J. For the vector \({\mathbf{P}_{z}}\) from formula (12), using relation (11), we obtain

$${\mathbf{P}_{z}} = \frac{{\tilde {\sigma }}}{{e{{N}_{0}}}}\delta E{\mathbf{S}_{0}} - \tilde {D}\frac{\partial }{{\partial {\kern 1pt} z}}\delta \mathbf{S} + \xi \frac{{\tilde {\sigma }}}{e}\left[ {{\mathbf{e}_{z}} \times \mathbf{E}} \right].$$
(13)

It is easy to see that Eq. (6) is written in terms of S and \({\mathbf{P}_{z}}\) as follows:

$$\frac{1}{{{{\tau }_{\text{S}}}}}\delta \mathbf{S} + \left[ {\mathbf{S} \times \boldsymbol\Omega } \right] + \frac{\partial }{{\partial {\kern 1pt} z}}{\mathbf{P}_{z}} = 0.$$
(14)

Substituting expression (8) for \(J\) into the right-hand side of Eq. (7) and using relations (9) and (10), as well as the relation \({\mathbf{S}_{0}} = - {{\chi \Lambda {\mathbf{M}}} \mathord{\left/ {\vphantom {{\chi \Lambda {\mathbf{M}}} \mu }} \right. \kern-0em} \mu },\) we obtain:

$$\begin{gathered} \mathbf{I} = \frac{{\tilde {\sigma }}}{e}\mathbf{E} - \xi \tilde {D}\left[ {{{{\mathbf{e}}}_{z}} \times \frac{\partial }{{\partial {\kern 1pt} z}}\delta \mathbf{S}} \right] \\ + \,\,\xi \frac{{\chi \Lambda \tilde {\sigma }}}{{e\mu {{N}_{0}}}}\left[ {{\mathbf{M}} \times \mathbf{E}} \right] - \tilde {D}\frac{\partial }{{\partial {\kern 1pt} z}}\delta N{{{\mathbf{e}}}_{z}}. \\ \end{gathered} $$
(15)

The four terms on the right-hand side of expression (15) reflect various aspects of the effect of SOI on the electric current in the metal.

The value of the first term is determined by the conductivity \(\tilde {\sigma } = {\sigma \mathord{\left/ {\vphantom {\sigma {\left( {1 + 2{{\xi }^{2}}} \right)}}} \right. \kern-0em} {\left( {1 + 2{{\xi }^{2}}} \right)}}.\) From the form of \(\tilde {\sigma }\), it clearly follows that SOI leads to a decrease in the electrical conductivity. This decrease is directly determined by the parameter \(\xi = {{{{\tau }_{\text{O}}}} \mathord{\left/ {\vphantom {{{{\tau }_{\text{O}}}} {{{\tau }_{\text{SO}}}}}} \right. \kern-0em} {{{\tau }_{\text{SO}}}}}.\)

The second term describes the rise of a nonuniform distribution of the electron current density over the sample cross section caused by the SOI. It depends on the nonuniform distribution of the nonequilibrium spin density, which, in turn, is a direct manifestation of the spin Hall effect. Thus, the spin Hall effect manifests itself as the appearance of a nonuniform nonequilibrium electronic magnetization in the sample and a nonuniform distribution of the electric current over the sample cross section.

The third and fourth terms describe the electron fluxes in the direction perpendicular to the pulling electric field E. The third term on the right-hand side of (15) describes the anomalous Hall effect. It is manifested in the fact that the electric current in a system with a magnetization M in the direction of the pulling field \({\mathbf{E}_{0}}\) is accompanied by a current in the transverse direction, \(\left[ {{\mathbf{M}} \times {\mathbf{E}_{0}}} \right].\) The fourth term is the diffusion electron current along the z axis, generated by a nonuniform distribution of nonequilibrium electron density \(\delta N.\)

The appearance of a nonequilibrium electron density \(\delta N\) dependent on the z coordinate is due to the dependence on z of the spin density \({\mathbf{S}_{0}}.\) The equation determining \(\delta N\) is obtained by substituting expression (15) for the current I into Eq. (5) and using the equation \(\operatorname{div} \mathbf{E} = 4\pi e\delta N{\text{:}}\)

$$r_{{\text{D}}}^{2}\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{z}^{2}}}}\delta N - \delta N = \xi \frac{{\chi \Lambda }}{{4\pi e\mu {{N}_{0}}}}\left[ {{\mathbf{E}_{0}} \times {{{\mathbf{e}}}_{z}}} \right] \cdot \frac{\partial }{{\partial {\kern 1pt} z}}{\mathbf{M}},$$
(16)

where \({{r}_{\text{D}}} = \sqrt {{{{{m}_{e}}v_{\text{F}}^{2}} \mathord{\left/ {\vphantom {{{{m}_{e}}v_{\text{F}}^{2}} {12\pi }}} \right. \kern-0em} {12\pi }}{{N}_{0}}{{e}^{2}}} \) is the Debye length of the screening of the electric field by the degenerate electron gas.

The system of Eqs. (13)–(16) will form the basis for describing the phenomena of spin-orbitronics for a “helimagnet–normal metal” heterojunction.

2 SPIN-ORBITRONICS OF A SEMI-BOUNDED NORMAL METAL WITH STRONG SOI

To describe the spin-orbitonics of the “helimagnet–normal metal” heterojunction, we consider a model in which a normal metal with a strong SOI occupies a half-space z ≤ 0, and a helimagnet, in which the SOI is considered negligibly small, is located in the region \(z \leqslant 0.\)

To describe the properties of a normal metal with allowance for SOI, we use Eqs. (13)–(15), in which we should set \(\Lambda = 0.\) All the quantities, both parameters and variables, characterizing a normal metal with strong SOI, will be denoted by a tilde sign. For example, the spin density of electrons in a normal metal will be denoted \(\tilde{\mathbf{S}}{\text{,}}\) the polarization of the spin current \({{{\mathbf{\tilde {P}}}}_{z}},\) and the spin relaxation time \({{\tilde {\tau }}_{\text{S}}}.\) The z index of the quantity \({{{\mathbf{\tilde {P}}}}_{z}}\) will be suppressed if this does not lead to misunderstandings. Equation (14) for \(\tilde{\mathbf{S}}\) and expressions (13) and (15) for \({\mathbf{\tilde {P}}}\) and \({\mathbf{\tilde {I}}}\) in the region \(z \leqslant 0\) take the form

$$\frac{1}{{{{{\tilde {\tau }}}_{\text{S}}}}}\delta \tilde{\mathbf{S}} + \frac{\partial }{{\partial {\kern 1pt} z}}{\mathbf{\tilde {P}}} = 0,$$
(17)
$${\mathbf{\tilde {P}}} = - \tilde {D}\frac{\partial }{{\partial {\kern 1pt} z}}\delta \tilde{\mathbf{S}} + \xi \frac{{\tilde {\sigma }}}{e}\left[ {{\mathbf{e}_{z}} \times {\mathbf{E}_{0}}} \right],$$
(18)
$${\mathbf{\tilde {I}}} = \frac{{\tilde {\sigma }}}{e}{\mathbf{E}_{0}} - \xi \tilde {D}\left[ {{{{\mathbf{e}}}_{z}} \times \frac{\partial }{{\partial {\kern 1pt} z}}\delta \tilde{\mathbf{S}}} \right].$$
(19)

Substituting expression (18) for \({\mathbf{\tilde {P}}}\) into Eq. (17) and taking into account the independence of \({\mathbf{E}_{0}}\) on z, we obtain a closed equation for \(\delta \tilde{\mathbf{S}}{\text{:}}\)

$$\tilde {L}_{\text{S}}^{2}\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{z}^{2}}}}\delta \tilde{\mathbf{S}} - \delta \tilde{\mathbf{S}} = 0,$$
(20)

where \({{\tilde {L}}_{\text{S}}} = \sqrt {\tilde {D}{{{\tilde {\tau }}}_{\text{S}}}} \) is the spin-diffusion length in the metal with allowance for SOI.

The general solution of differential equation (20), which decays at \(z \to - \infty ,\) can be written as

$$\delta \tilde{\mathbf{S}}(z) = \delta \tilde{\mathbf{S}}( - 0){{e}^{{{z \mathord{\left/ {\vphantom {z {{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {L}}}_{\text{S}}}}}}}},$$
(21)

where \(\delta \tilde{\mathbf{S}}( - 0)\) is the quantity to be determined from the boundary conditions. Then, from (18),

$${\mathbf{\tilde {P}}}(z) = - \frac{{\tilde {D}}}{{{{{\tilde {L}}}_{\text{S}}}}}\delta \tilde{\mathbf{S}}( - 0){{e}^{{{z \mathord{\left/ {\vphantom {z {{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {L}}}_{\text{S}}}}}}}} + \xi \left[ {{\mathbf{e}_{z}} \times {{{\tilde{\mathbf{I}}}}_{0}}} \right],$$
(22)

where \({{\tilde{\mathbf{I}}}_{0}} = \left( {{{\tilde {\sigma }} \mathord{\left/ {\vphantom {{\tilde {\sigma }} e}} \right. \kern-0em} e}} \right){\mathbf{E}_{0}}\) is the electron flux density in the depth of the metal (at \(z \to - \infty \)). Substituting (21) into (19), we find the current in the metal \({\mathbf{\tilde {I}}}(z){\text{:}}\)

$${\mathbf{\tilde {I}}}(z) = {{\tilde{\mathbf{I}}}_{0}} + \xi \left\{ {\left[ {{{{\mathbf{e}}}_{z}} \times {\mathbf{\tilde {P}}}( - 0)} \right] + \xi {{{\tilde{\mathbf{I}}}}_{0}}} \right\}{{e}^{{{z \mathord{\left/ {\vphantom {z {{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {L}}}_{\text{S}}}}}}}}.$$
(23)

Expressions (22) and (23) directly describe the manifestations of the spin Hall effect near the boundary of a normal metal. The electron flux \(\tilde{\mathbf{I}}\) along the boundary \(z = 0,\) under the action of SOI, generates a spin current with polarization \({\mathbf{\tilde {P}}}(z),\) flowing along the normal to the boundary. Deep in the metal, at distances much greater than the diffusion length \({{\tilde {L}}_{\text{S}}},\) the electron flux is \({{\tilde{\mathbf{I}}}_{0}}.\) The spin current deep in the metal is \({{{{\tilde{\mathbf{P}}}}}_{0}} = \xi \left[ {{\mathbf{e}_{z}} \times {{{\tilde{\mathbf{I}}}}_{0}}} \right].\) With an approach to the boundary, the vector of polarization of the spin current, \({\mathbf{\tilde {P}}}(z)\), changes its length and direction, taking at the boundary a value \({\mathbf{\tilde {P}}}( - 0),\) which depends on the transport properties of the interface. The flux density vector \({\mathbf{\tilde {I}}}(z),\) asymptotically equal to \({{\tilde{\mathbf{I}}}_{0}}\) at \(z \to - \infty ,\) changes at characteristic distances on the order of \({{\tilde {L}}_{\text{S}}}\) and takes at the boundary \(z = 0\) the value

$$\left( {1 + {{\xi }^{2}}} \right){{\tilde{\mathbf{I}}}_{0}} + \xi \left[ {{{{\mathbf{e}}}_{z}} \times {\mathbf{\tilde {P}}}( - 0)} \right].$$

If the interface is impermeable to electrons and there is no spin-flip scattering at the interface, then the boundary condition is the equality \({\mathbf{\tilde {P}}}( - 0) = 0.\) For this simplest case, we obtain

$${\mathbf{\tilde {P}}}(z) = \xi \left( {1 - {{e}^{{{z \mathord{\left/ {\vphantom {z {{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {L}}}_{\text{S}}}}}}}}} \right)\left[ {{\mathbf{e}_{z}} \times {{{\tilde{\mathbf{I}}}}_{0}}} \right],\,\,\,\,{\mathbf{\tilde {I}}}(z) = \left( {1 + {{\xi }^{2}}{{e}^{{{z \mathord{\left/ {\vphantom {z {{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {L}}}_{\text{S}}}}}}}}} \right){{\tilde{\mathbf{I}}}_{0}}.$$

3 SPIN TRANSPORT IN A CONDUCTING HELIMANET

Let us consider the features of spin transport in a semi-bounded helimagnet located in the region \(z \geqslant 0,\) where SOI is assumed to be negligible.

We restrict the consideration to a helimagnet, in which, in the absence of an external magnetic field, a magnetic structure of the “simple helix” type is realized, the axis of which coincides with the axis OZ. The length M of the magnetization vector M of localized electrons will be assumed to be independent of the z coordinate. The direction of the vector M will be specified by a unit vector \(\mathbf{h} = {\mathbf{M} \mathord{\left/ {\vphantom {M M}} \right. \kern-0em} M},\) which changes with increasing z as \(\mathbf{h} = {\mathbf{e}_{x}}\cos Kqz + {\mathbf{e}_{y}}\sin Kqz,\) where q is the wavenumber, \(K = \pm 1\) is the chirality of the magnetization helix, and \({\mathbf{e}_{x}}\) and \({\mathbf{e}_{y}}\) are unit vectors along the axes OX and OY, respectively. Then, the equilibrium spin density of conduction electrons is \({\mathbf{S}_{0}} = - \left( {{{\chi \Lambda M} \mathord{\left/ {\vphantom {{\chi \Lambda M} \mu }} \right. \kern-0em} \mu }} \right)\mathbf{h}{\text{.}}\) In a helimagnet with a wavenumber q, the direction of \({\mathbf{S}_{0}}\) changes in space with a period \({{L}_{\text{H}}} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } q}} \right. \kern-0em} q}.\)

To describe the properties of a helimagnetic metal, we use Eqs. (13)–(15) in which we set \(\xi = 0.\) The relation between the spin current P and the spin density S takes a simple form:

$$\mathbf{P} = - D\frac{\partial }{{\partial {\kern 1pt} z}}\delta \mathbf{S}{\text{.}}$$
(24)

Equation (14) for S, after substituting expression (24) for P, is represented as

$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta \mathbf{S} - \delta \mathbf{S} - \lambda \left[ {\delta \mathbf{S} \times {\mathbf{h}}} \right] = 0.$$
(25)

When writing Eq. (25), we introduced a dimensionless coordinate \(\zeta = {z \mathord{\left/ {\vphantom {z {{{L}_{\text{S}}}}}} \right. \kern-0em} {{{L}_{\text{S}}}}}\) and a dimensionless parameter \(\lambda = {{\tau }_{\text{S}}}\gamma \Lambda M\) characterizing the value of the s-d(f)-exchange interaction.

We will look for a solution of Eq. (25) in the form of a decomposition of \(\delta \mathbf{S}\) in three mutually perpendicular unit vectors \({{{\mathbf{e}}}_{z}},\) h, and \(\left[ {{\mathbf{h}} \times {{{\mathbf{e}}}_{z}}} \right],\) of which the latter two harmonically change direction with an increase in the coordinate z:

$$\delta \mathbf{S} = \delta {{S}_{z}}{{{\mathbf{e}}}_{z}} + \delta {{S}_{\parallel }}{\mathbf{h}} + \delta {{S}_{ \bot }}\left[ {{\mathbf{h}} \times {{{\mathbf{e}}}_{z}}} \right].$$
(26)

From Eq. (25) for the components \(\delta {{S}_{z}},\) \(\delta {{S}_{\parallel }}\), and \(\delta {{S}_{ \bot }}\), we obtain a system of second-order ordinary differential equations with constant coefficients of the form

$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{\parallel }} + 2K\eta \frac{\partial }{{\partial {\kern 1pt} \zeta }}\delta {{S}_{ \bot }} - \left( {1 + {{\eta }^{2}}} \right)\delta {{S}_{\parallel }} = 0,$$
(27)
$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{ \bot }} - 2K\eta \frac{\partial }{{\partial {\kern 1pt} \zeta }}\delta {{S}_{\parallel }} - \left( {1 + {{\eta }^{2}}} \right)\delta {{S}_{ \bot }} + \lambda \delta {{S}_{z}} = 0,$$
(28)
$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{z}} - \delta {{S}_{z}} - \lambda \delta {{S}_{ \bot }} = 0,$$
(29)

where \(\eta = q{{L}_{\text{S}}}.\) Substituting \(\delta {{S}_{\parallel }} = {{C}_{\parallel }}{{e}^{{ - \kappa \zeta }}},\) \(\delta {{S}_{ \bot }} = {{C}_{ \bot }}{{e}^{{ - \kappa \zeta }}}\), and \(\delta {{S}_{z}} = {{C}_{z}}{{e}^{{ - \kappa \zeta }}},\) we obtain a system of equations for the constants \({{C}_{\parallel }},\) \({{C}_{ \bot }}\), and \({{C}_{z}}.\) Equating the determinant of this system to zero, we obtain a characteristic equation for determining \(\kappa {\text{:}}\)

$$\begin{gathered} \left[ {\left( {{{\kappa }^{2}} - 1} \right)\left( {{{\kappa }^{2}} - 1} \right) + {{\lambda }^{2}}} \right]\left( {{{\kappa }^{2}} - {{\eta }^{2}} - 1} \right) \\ + \,\,{{\eta }^{2}}\left( {{{\kappa }^{2}} - 1} \right)\left[ {3{{\kappa }^{2}} + {{\eta }^{2}} + 1} \right] = 0. \\ \end{gathered} $$
(30)

Of the six roots of the characteristic equation (30), we are only interested in three that satisfy the condition \(\operatorname{Re} \kappa > 0,\) which describe solutions that decay as \(\zeta \) → \( + \infty .\) It can be shown that one of these three roots, \({{\kappa }_{1}}\), is real and the other two, \({{\kappa }_{2}}\) and \({{\kappa }_{3}},\) are complex-conjugate. These roots determine the values of two characteristic decay lengths of spin perturbations in a helimagnet, which we define as \({{L}_{\text{D}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {{{\kappa }_{1}}}}} \right. \kern-0em} {{{\kappa }_{1}}}}\) and \({{L}_{\text{P}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {\operatorname{Re} {{\kappa }_{2}}}}} \right. \kern-0em} {\operatorname{Re} {{\kappa }_{2}}}}.\)

Below, we restrict consideration to the case of \(\lambda \gg 1 + {{\eta }^{2}}.\) This condition is satisfied in helimagnets at sufficiently large values of the exchange constant \(\Lambda \) and spin relaxation time \({{\tau }_{\text{S}}}.\) It can be verified that the roots of Eq. (30) can be found by successive approximations in the form of series expansion in a small parameter \({{\left( {1 + {{\eta }^{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {1 + {{\eta }^{2}}} \right)} \lambda }} \right. \kern-0em} \lambda }.\) In the zeroth approximation, Eq. (30) reads

$$\left[ {\left( {{{\kappa }^{2}} - 1} \right)\left( {{{\kappa }^{2}} - 1} \right) + {{\lambda }^{2}}} \right]\left( {{{\kappa }^{2}} - {{\eta }^{2}} - 1} \right) = 0.$$
(31)

Equation (31) is a characteristic equation for a system of differential equations of the form

$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{\parallel }} - \left( {1 + {{\eta }^{2}}} \right)\delta {{S}_{\parallel }} = 0,$$
(32)
$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{ \bot }} - \delta {{S}_{ \bot }} + \lambda \delta {{S}_{z}} = 0,$$
(33)
$$\frac{{{{\partial }^{2}}}}{{\partial {\kern 1pt} {{\zeta }^{2}}}}\delta {{S}_{z}} - \delta {{S}_{z}} - \lambda \delta {{S}_{ \bot }} = 0.$$
(34)

The roots of the characteristic equation (31), which describe the solutions decaying at \(\zeta \to + \infty \), are \({{\kappa }_{1}} = \sqrt {1 + {{\eta }^{2}}} ,\) \({{\kappa }_{2}} = \left( {1 + i} \right)\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \), and \({{\kappa }_{3}} = \kappa _{2}^{*}.\) The characteristic electron magnetization decay lengths in a helimagnet that correspond to these roots can be written as \({{L}_{\text{D}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {\sqrt {1 + {{\eta }^{2}}} }}} \right. \kern-0em} {\sqrt {1 + {{\eta }^{2}}} }}\) and \({{L}_{\text{P}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} }}} \right. \kern-0em} {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} }}.\) The general solution of the system of Eqs. (32)–(34) can be written explicitly:

$$\delta {{S}_{\parallel }}(\zeta ) = \delta {{S}_{\parallel }}( + 0){{\operatorname{e} }^{{ - \sqrt {1 + {{\eta }^{2}}} \zeta }}},$$
(35)
$$\begin{gathered} \delta {{S}_{ \bot }}(\zeta ) = \delta {{S}_{ \bot }}( + 0)\cos \left( {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta } \right){{\operatorname{e} }^{{ - \sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta }}} \\ + \,\,\delta {{S}_{z}}( + 0)\sin \left( {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta } \right){{\operatorname{e} }^{{ - \sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta }}}, \\ \end{gathered} $$
(36)
$$\begin{gathered} \delta {{S}_{z}}(\zeta ) = \delta {{S}_{z}}( + 0)\cos \left( {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta } \right){{\operatorname{e} }^{{ - \sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta }}} \\ - \,\,\delta {{S}_{ \bot }}( + 0)\sin \left( {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta } \right){{\operatorname{e} }^{{ - \sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} \zeta }}}. \\ \end{gathered} $$
(37)

The constants \(\delta {{S}_{\parallel }}( + 0),\) \(\delta {{S}_{ \bot }}( + 0)\), and \(\delta {{S}_{z}}( + 0)\) entering into expressions (35)–(37) are the components of the spin density vector at the boundary:

$$\begin{gathered} \delta S( + 0) = \delta {{S}_{\parallel }}( + 0){{{\mathbf{h}}}_{0}} + \delta {{S}_{ \bot }}( + 0)\left[ {{{{\mathbf{h}}}_{0}} \times {{{\mathbf{e}}}_{z}}} \right] \\ + \,\,\delta {{S}_{z}}( + 0){{{\mathbf{e}}}_{z}}, \\ \end{gathered} $$
(38)

where \({{{\mathbf{h}}}_{0}} = {{\mathbf{M}\left( 0 \right)} \mathord{\left/ {\vphantom {{\mathbf{M}\left( 0 \right)} M}} \right. \kern-0em} M}.\)

It clearly follows from expressions (35)–(37) that the spin density component \(\delta {{S}_{\parallel }}(z)\) decreases exponentially with distance along a length \({{L}_{\text{D}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {\sqrt {1 + {{\eta }^{2}}} }}} \right. \kern-0em} {\sqrt {1 + {{\eta }^{2}}} }}.\) The change in the components \(\delta {{S}_{ \bot }}( + 0)\) and \(\delta {{S}_{z}}( + 0)\) with the distance is described by a combination of a decaying exponential and harmonic functions with a characteristic scale of change \({{L}_{\text{P}}} = {{{{L}_{\text{S}}}} \mathord{\left/ {\vphantom {{{{L}_{\text{S}}}} {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} }}} \right. \kern-0em} {\sqrt {{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}} }}.\) Within the approximations used, \({{L}_{\text{P}}} \ll {{L}_{\text{D}}}.\)

The polarization of the spin current P is found by formula (24) and can be represented in a form similar to (26): \(\mathbf{P} = {{P}_{\parallel }}{\mathbf{h}} + {{P}_{ \bot }}\left[ {{\mathbf{h}} \times {{{\mathbf{e}}}_{z}}} \right] + {{P}_{z}}{{{\mathbf{e}}}_{z}}.\) The exponential decay scale of the component \({{P}_{\parallel }}\) is the length \({{L}_{\text{D}}}\) while \({{P}_{ \bot }}\) and \({{P}_{z}}\) vary over a characteristic distance \({{L}_{\text{P}}}.\)

The spin current polarization vector \(\mathbf{P}( + 0)\) at the boundary can be represented as

$$P( + 0) = {\mathbf{P}_{\parallel }}( + 0){{{\mathbf{h}}}_{0}} + {{P}_{ \bot }}( + 0)\left[ {{{{\mathbf{h}}}_{0}} \times {{{\mathbf{e}}}_{z}}} \right] + {{P}_{z}}( + 0){{{\mathbf{e}}}_{z}},$$
(39)

where

$${{P}_{\parallel }}( + 0) = \frac{D}{{{{L}_{\text{S}}}}}\left[ {\sqrt {1 + {{\eta }^{2}}} \delta {{S}_{\parallel }}(0) - K\eta \delta {{S}_{ \bot }}(0)} \right],$$
(40)
$${{P}_{ \bot }}( + 0) = \frac{D}{{{{L}_{\text{S}}}}}\left\{ {K\eta \delta {{S}_{\parallel }}(0) + \sqrt {\frac{\lambda }{2}} \left[ {\delta {{S}_{ \bot }}(0) - \delta {{S}_{z}}(0)} \right]} \right\},$$
(41)
$${{P}_{z}}( + 0) = \frac{D}{{{{L}_{\text{S}}}}}\sqrt {\frac{\lambda }{2}} \left[ {\delta {{S}_{z}}( + 0) + \delta {{S}_{ \bot }}( + 0)} \right].$$
(42)

The constants \(\delta {{S}_{\parallel }}( + 0),\) \(\delta {{S}_{ \bot }}( + 0)\), and \(\delta {{S}_{z}}( + 0)\) entering into expressions (35)–(37) and (40)–(42) are to be determined from the boundary conditions.

4 BOUNDARY CONDITIONS

We consider a system of conduction electrons of two metals bordering on the plane \(z = 0.\) In this section, the characteristics of the conductor in the region \(z < 0\) will be indicated by the number 1 and, in the region \(z > 0\), by the number 2. Let \(\varepsilon _{\mathbf{p}}^{{(i)}}\) and \({\mathbf{v}^{{(i)}}} = {{\partial \varepsilon _{\mathbf{p}}^{{(i)}}} \mathord{\left/ {\vphantom {{\partial \varepsilon _{\mathbf{p}}^{{(i)}}} \partial }} \right. \kern-0em} \partial }p\) be the spectrum and velocity of electrons in the metal, \(i = 1,2.\) All electrons in the considered geometry of the system are divided into two groups. The first one comprises the electrons moving from the depth of the conductor toward the interface, and the second one comprises the electrons moving from the interface. In conductor 1, the electrons moving toward the boundary have a positive value of the z component of the velocity, while, for reflected electrons in region 1, this component is negative. In contrast, in conductor 2, the electrons moving toward the boundary have a negative value of the z component of the velocity, while, for the reflected electrons in region 2, these components are positive. The electron flux along the z axis is the sum of the flux \(I_{z}^{ > }\) of electrons with \({{v}_{z}} > 0\) and the flux \(I_{z}^{ < }\) of electrons with \({{v}_{z}} < 0{\text{:}}\) \({{I}_{z}}\left( z \right) = I_{z}^{ > }\left( z \right) + I_{z}^{ < }\left( z \right).\) Similarly, the spin current is \({{{\mathbf{P}}}_{z}}\left( z \right) = {\mathbf{P}}_{z}^{ > }\left( z \right) + {\mathbf{P}}_{z}^{ < }\left( z \right).\)

We will assume that the interface can be passed not by all electrons incident on it. Denote by W the relative fraction of electrons penetrating the boundary. Then, \(1 - W\) is the relative fraction of electrons reflected from the boundary. By definition, \(0 \leqslant W \leqslant 1.\) Below, we consider the simplest case, when the scattering of electrons with spin flip during their interaction with the boundary can be neglected.

The flux of electrons moving from the boundary in conductor 2, \(I_{z}^{ > }\left( { + 0} \right),\) is formed as the sum of two fluxes. The first component is the part of the flux of electrons moving toward the boundary in conductor 1, \(I_{z}^{ > }\left( {z = - 0} \right),\) that penetrated conductor 2, equal to \(WI_{z}^{ > }\left( {z = - 0} \right).\) The second component is the part of the flux of electrons moving toward the boundary in conductor 2, \(I_{z}^{ < }\left( {z = + 0} \right),\) that did not penetrate conductor 1, equal to \( - \left( {1 - W} \right)I_{z}^{ < }\left( { + 0} \right).\) Similar considerations can be given for the flux \(I_{z}^{ < }\left( { - 0} \right).\) The condition for the continuity of the particle flux \({{I}_{z}}\left( z \right)\) can be written as

$$I_{z}^{ > }\left( { + 0} \right) = WI_{z}^{ > }\left( { - 0} \right) - \left( {1 - W} \right)I_{z}^{ < }\left( { + 0} \right),$$
(43)
$$I_{z}^{ < }\left( { - 0} \right) = WI_{z}^{ < }\left( { + 0} \right) - \left( {1 - W} \right)I_{z}^{ > }\left( { - 0} \right),$$
(44)

whence

$${{I}_{z}}\left( { + 0} \right) = {{I}_{z}}\left( { - 0} \right) = W\left[ {I_{z}^{ > }\left( { - 0} \right) + I_{z}^{ < }\left( { + 0} \right)} \right].$$
(45)

The fluxes appearing on the right-hand side of relation (45) can be written in terms of the nonequilibrium part of the electron density distribution functions \(\delta n\left( {z,\mathbf{p}} \right){\text{:}}\)

$$I_{z}^{ > }\left( { - 0} \right) = \sum\limits_{\mathbf{p}, {{v}_{z}} > 0} {v_{z}^{{(1)}}\delta {{n}^{{(1)}}}\left( { - 0,\mathbf{p}} \right)} ,$$
(46)
$$I_{z}^{ < }\left( { + 0} \right) = \sum\limits_{\mathbf{p},{{v}_{z}} < 0} {v_{z}^{{(2)}}\delta {{n}^{{(2)}}}\left( { + 0,\mathbf{p}} \right)} .$$
(47)

In turn, the distribution functions of electrons incident on the boundary, which appear on the right-hand sides of Eqs. (46) and (47), can be represented, as shown in [28], in the form

$$\begin{gathered} \delta {{n}^{{(i)}}}\left( {z,\mathbf{p}} \right) = \delta \left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varepsilon _{\text{F}}^{{(i)}}} \right){{\left[ {\sum\limits_\mathbf{p} \delta \left( {\varepsilon _{\mathbf{p}}^{{(i)}} - \varepsilon _{\text{F}}^{{(i)}}} \right)} \right]}^{{ - 1}}} \\ \times \left[ {\delta {{N}^{{(i)}}}\left( z \right) + {{3{\mathbf{v}^{{(i)}}} \cdot {\mathbf{I}^{{(i)}}}\left( z \right)} \mathord{\left/ {\vphantom {{3{\mathbf{v}^{{(i)}}} \cdot {\mathbf{I}^{{(i)}}}\left( z \right)} {v{{{_{\text{F}}^{{(i)}}}}^{2}}}}} \right. \kern-0em} {v{{{_{\text{F}}^{{(i)}}}}^{2}}}}} \right],\,\,\,\,i = 1,2, \\ \end{gathered} $$
(48)

where \(v_{\text{F}}^{{(i)}}\) and \(\varepsilon _{\text{F}}^{{(i)}}\) are the velocity and Fermi energy of electrons in metal i, respectively, and \(\delta \left( \varepsilon \right)\) is the Dirac delta function (representation (48) is written here for a degenerate electron gas).

As a result, from relations (45), we obtain the boundary conditions for the electron flux in the form

$$\begin{gathered} I_{z}^{{(1)}}\left( { - 0} \right) = I_{z}^{{(2)}}\left( { + 0} \right) \\ = \frac{1}{4}\frac{W}{{1 - W}}\left[ {v_{\text{F}}^{{(1)}}\delta {{N}^{{(1)}}}\left( { - 0} \right) - v_{\text{F}}^{{(2)}}\delta {{N}^{{(2)}}}\left( { + 0} \right)} \right]. \\ \end{gathered} $$
(49)

Similarly, \({\text{we}}\) obtain the boundary conditions for the spin current:

$$\begin{gathered} {\mathbf{P}}_{z}^{{(1)}}\left( { - 0} \right) = {\mathbf{P}}_{z}^{{(2)}}\left( { + 0} \right) \\ = \frac{1}{4}\frac{W}{{1 - W}}\left[ {v_{\text{F}}^{{(1)}}\delta {{{\mathbf{S}}}^{{(1)}}}\left( { - 0} \right) - v_{\text{F}}^{{(2)}}\delta {{{\mathbf{S}}}^{{(2)}}}\left( { + 0} \right)} \right]. \\ \end{gathered} $$
(50)

Phenomenological boundary conditions (49) and (50) are suitable for describing the contact of two metals in which the electron gas can be considered degenerate. Let us apply them to describe the “normal metal–helimagnet” heterojunction. In our case, we neglect the presence of SOI in the helimagnet. We will consider the case where the electron flux \({{I}_{z}} = 0,\) i.e., there is only a pure spin current \({{{\mathbf{P}}}_{z}}\) in the system along the OZ axis. Then, the consideration is noticeably simplified due to the fact that the system remains electrically neutral during the flow of electron and spin currents and \(\delta N\left( z \right) \equiv 0.\)

In the notation adopted earlier, \(\delta {{{\mathbf{S}}}^{{(1)}}}\left( z \right) \equiv \delta {\mathbf{\tilde {S}}}\left( z \right),\) \(\delta {{{\mathbf{S}}}^{{(2)}}}\left( z \right) \equiv \delta {\mathbf{S}}\left( z \right),\) \({\mathbf{P}}_{z}^{{(1)}}\left( z \right) \equiv {\mathbf{\tilde {P}}}\left( z \right),\) \({\mathbf{P}}_{z}^{{(2)}}\left( z \right) \equiv {\mathbf{P}}\left( z \right),\) \(v_{\text{F}}^{{(1)}} \equiv {{\tilde {v}}_{\text{F}}},\) \(v_{\text{F}}^{{(2)}} \equiv {{v}_{\text{F}}}\), and boundary conditions (50) take the form:

$${\mathbf{P}}\left( { + 0} \right) = {\mathbf{\tilde {P}}}\left( { - 0} \right),$$
(51)
$${\mathbf{P}}\left( { + 0} \right) + \frac{1}{4}\frac{W}{{1 - W}}{{v}_{\text{F}}}\delta {\mathbf{S}}\left( { + 0} \right) = \frac{1}{4}\frac{W}{{1 - W}}{{\tilde {v}}_{\text{F}}}\delta {\mathbf{\tilde {S}}}\left( { - 0} \right).$$
(52)

Using relation (22), we eliminate \(\delta {\mathbf{\tilde {S}}}\left( { - 0} \right)\) from Eqs. (51) and (52). As a result, we obtain the desired boundary condition for the helimagnet in the form

$${\mathbf{P}}\left( { + 0} \right) + \tilde {W}\psi {{v}_{\text{F}}}\delta {\mathbf{S}}\left( { + 0} \right) = \tilde {W}\xi \left[ {{\mathbf{e}_{z}} \times {{{\tilde{\mathbf{I}}}}_{0}}} \right],$$
(53)

where we introduced the notation \(\psi = {{\tilde {D}} \mathord{\left/ {\vphantom {{\tilde {D}} {{{{\tilde {v}}}_{\text{F}}}{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {{{{\tilde {v}}}_\text{F}}{{{\tilde {L}}}_{\text{S}}}}}\) and \(\tilde {W} = {W \mathord{\left/ {\vphantom {W {\left[ {W + 4\psi \left( {1 - W} \right)} \right]}}} \right. \kern-0em} {\left[ {W + 4\psi \left( {1 - W} \right)} \right]}}.\)

Writing the diffusion coefficient as \(\tilde {D} = {{{{{\tilde {v}}}_\text{F}}\tilde {\ell }} \mathord{\left/ {\vphantom {{{{{\tilde {v}}}_\text{F}}\tilde {\ell }} 3}} \right. \kern-0em} 3},\) where \(\tilde {\ell }\) is the mean free path of electrons, we obtain a representation of \(\psi \) in the form \(\psi = {{\tilde {\ell }} \mathord{\left/ {\vphantom {{\tilde {\ell }} {3{{{\tilde {L}}}_{\text{S}}}}}} \right. \kern-0em} {3{{{\tilde {L}}}_{\text{S}}}}}.\) Since always \({{\tilde {L}}_{\text{S}}} \gg \tilde {\ell },\) numerically, \(\psi \ll 1.\)

The parameter \(\tilde {W}\) characterizes the properties of the interface. If the probability of electrons passing through the interface W is high, so that \(W \gg \psi ,\) then \(\tilde {W}\) takes its maximum possible value \(\tilde {W} = 1.\) It is natural to call such an interface highly transparent. If \(W \ll \psi ,\) then we have the case of a weakly transparent boundary, when \(\tilde {W} = {W \mathord{\left/ {\vphantom {W {4\psi }}} \right. \kern-0em} {4\psi }} \ll 1.\)

5 POLARIZATION EFFECTS UPON INJECTION OF A PURE SPIN CURRENT INTO A HELIMAGNET

Substituting expressions (38) and (39) and then (40)–(42) into boundary condition (53) we obtain a system of equations for \(\delta {{S}_{\parallel }}( + 0),\) \(\delta {{S}_{ \bot }}( + 0)\), and \(\delta {{S}_{z}}( + 0).\) The solution of this system of equations within the approximation \(1 + {{\eta }^{2}} \ll \lambda \) has the form

$$\begin{gathered} \delta {{S}_{\parallel }}( + 0) \\ = \xi \tilde {W}\frac{{{{L}_{\text{S}}}}}{D}{{{\tilde {I}}}_{0}}\frac{{\left[ {{{{\tilde{\mathbf{i}}}}_{0}} \times {{{\mathbf{h}}}_{0}}} \right] \cdot {\mathbf{e}_{z}} - {{{\tilde{\mathbf{i}}}}_{0}} \cdot {{{\mathbf{h}}}_{0}}{{\eta K} \mathord{\left/ {\vphantom {{\eta K} {\sqrt {2\lambda } }}} \right. \kern-0em} {\sqrt {2\lambda } }}}}{{\sqrt {1 + {{\eta }^{2}}} + a\tilde {W}}}, \\ \end{gathered} $$
(54)
$$\begin{gathered} \delta {{S}_{ \bot }}( + 0) = - \xi \tilde {W}\frac{{{{L}_{\text{S}}}}}{{D\sqrt {2\lambda } }} \\ \times \,\,{{{\tilde {I}}}_{0}}\left\{ {\frac{{K\eta \left[ {{{{\tilde{\mathbf{i}}}}_{0}} \times {{{\mathbf{h}}}_{0}}} \right] \cdot {\mathbf{e}_{z}}}}{{\sqrt {1 + {{\eta }^{2}}} + a\tilde {W}}} + {{{\tilde{\mathbf{i}}}}_{0}} \cdot {{{\mathbf{h}}}_{0}}} \right\}, \\ \end{gathered} $$
(55)
$$\delta {{S}_{z}}( + 0) = - \delta {{S}_{ \bot }}( + 0).$$
(56)

Here, \({{\tilde {I}}_{0}} = \left| {{{{\tilde{\mathbf{I}}}}_{0}}} \right|,\) \({{\tilde{\mathbf{i}}}_{0}} = {{{{\tilde{\mathbf{I}}}}_{0}}} \left/ {{{{\tilde {I}}}_{0}}} \right.\), and \(a = {{{v}_{\text{F}}}\tilde {D}{{L}_{\text{S}}}} \left/ {{{{\tilde {v}}}_{\text{F}}}D{{{\tilde {L}}}_{\text{S}}}} \right..\) The numerical value of the parameter a for a pair of metals with close relaxation times is close to unity.

To find the polarization of the spin current \({\mathbf{P}}( + 0)\), we use formulas (40)–(42). Within the approximations made, we obtain:

$${{P}_{\parallel }}( + 0) = \frac{{\sqrt {1 + {{\eta }^{2}}} }}{{\sqrt {1 + {{\eta }^{2}}} + \tilde {W}a}}\tilde {W}{{\tilde {P}}_{0}}\sin {{\Phi }_{0}},$$
(57)
$${{P}_{ \bot }}( + 0) = - \tilde {W}{{\tilde {P}}_{0}}\cos {{\Phi }_{0}},$$
(58)
$${{P}_{z}}( + 0) = - \frac{{a{{{\tilde {W}}}^{2}}}}{{\sqrt {2\lambda } }}{{\tilde {P}}_{0}}\left\{ {\cos {{\Phi }_{0}} + K\frac{{\eta \sin {{\Phi }_{0}}}}{{\sqrt {1 + {{\eta }^{2}}} + a\tilde {W}}}} \right\},$$
(59)

where \({{\tilde {P}}_{0}} = \xi {{\tilde {I}}_{0}}\) and \({{\Phi }_{0}}\) is the angle between the vectors \({{\tilde{\mathbf{i}}}_{0}}\) and \({{{\mathbf{h}}}_{0}}.\)

According to (57)–(59), the magnitude of the pure spin current injected from a metal with strong SOI is directly determined by the parameter \(\xi {\text{:}}\) all components of the vector P are directly proportional to \(\xi .\) The transparency of the interface for conduction electrons, described by the parameter \(\tilde {W},\) also significantly affects the injection efficiency. The components \({{P}_{\parallel }}( + 0)\) and \({{P}_{ \bot }}( + 0)\) ∼ \(\tilde {W},\) while \({{P}_{z}}( + 0) \sim {{\tilde {W}}^{2}}.\) This difference is significant for poorly transparent interfaces, when \(\tilde {W} \ll 1.\)

As was shown above, inside a normal metal with strong SOI, the electric current \({{{\mathbf{\tilde {I}}}}_{0}}\) flowing along the OZ axis along the interface \(z = 0\) induces a polarized spin current with a polarization \({{{\mathbf{\tilde {P}}}}_{0}} = \xi \left[ {{\mathbf{e}_{z}} \times {{{\tilde{\mathbf{I}}}}_{0}}} \right]\). The spin current in the depth of a normal metal, flowing along \({\mathbf{e}_{z}},\) is transversely polarized: \({{{\mathbf{\tilde {P}}}}_{0}} \bot {\mathbf{e}_{z}}.\) The vector of the transverse polarization component of the spin current \({{{\mathbf{P}}}_{t}}( + 0)\) injected into the helimagnet turns out to be rotated relative to \({{{\mathbf{\tilde {P}}}}_{0}}\) by an angle \(\Phi \), the value of which, according to (54) and (55), is given by the expression

$$\Phi = - \arctan \frac{{a\tilde {W}\sin {{\Phi }_{0}}\cos {{\Phi }_{0}}}}{{\sqrt {1 + {{\eta }^{2}}} + a\tilde {W}{{{\cos }}^{2}}{{\Phi }_{0}}}}.$$
(60)

Formula (60) describes the effect of rotation of the transverse component in the polarization of the spin current during its injection into a helimagnet. As follows from formula (60), the angle of this rotation is determined mainly by the angle \({{\Phi }_{0}}\) between the vectors \({{\tilde {i}}_{0}}\) and \({{{\mathbf{h}}}_{0}}.\) In short-period helimagnets, for which \(\eta \equiv q{{L}_{\text{S}}} \gg 1,\) the angle \(\Phi \ll 1.\) In the general case, for arbitrary values of the parameters \(\eta ,\) \(a\), and \(\tilde {W}\), the absolute value of the angle, \(\left| \Phi \right|\), does not exceed \({\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-0em} 4}.\)

It should be specially emphasized that at the boundary \(z = 0,\) according to (59), we note the appearance of a nonzero longitudinal (with respect to the OZ axis) spin current polarization component \({{{\mathbf{P}}}_{\ell }} = {{P}_{z}}{\mathbf{e}_{z}}.\) This effect does occur although the spin current deep in a normal metal is transversely polarized and the magnetization of the described helimagnet is a simple transverse helix. The direction of the vector \({{{\mathbf{P}}}_{\ell }},\) according to (59), is directly determined by the chirality of the helimagnet. The magnitude of the longitudinal polarization \({{P}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\ell } }}}\) is small in comparison with \({{P}_{0}},\) the parameter of smallness is the ratio \(a{{{{{\tilde {W}}}^{2}}} \mathord{\left/ {\vphantom {{{{{\tilde {W}}}^{2}}} {\sqrt {2\lambda } }}} \right. \kern-0em} {\sqrt {2\lambda } }}.\) The effect of the appearance in a helimagnet of a longitudinally polarized pure spin current and a longitudinal component of the nonequilibrium electron magnetization, which depend on the chirality of the helix of the helimagnet, upon injection of a transversely polarized spin current from a normal metal with strong SOI, may be called the “effect of chiral polarization of a pure spin current.”

CONCLUSIONS

A theory has been constructed to describe the electron transport phenomena that underlies spin-orbitronics. A system of coupled equations of motion for the charge and spin densities, as well as electric and spin currents, has been formulated, taking into account asymmetric (skew) spin-orbit scattering. The system of equations is supplemented with phenomenological boundary conditions used to describe the spin-orbitronics of the “normal metal–helimagnet” heterojunction. A theory has been developed for the injection of a pure spin current into a helimagnet, which arises in a normal metal as a manifestation of the spin Hall effect.

The theory we developed shows that the injection of a pure spin current into a conducting helimagnet with a simple helix magnetic structure made of a normal metal with strong SOI has a number of significant features associated with the polarization \({\text{of}}\) the spin density of conduction electrons and the spin current. In nonmagnetic materials, the damping of the injected spin current is characterized by one parameter: the spin diffusion length \({{L}_{\text{S}}},\) while in helimagnets, the damping of the spin current injected along the axis of the magnetic helix is described by two characteristic lengths.

The first of these, the length \({{L}_{\text{D}}},\) characterizes the scale of the decrease in the transverse (with respect to the helix axis) component of the nonequilibrium magnetization of conduction electron, which is codirectional with the magnetization of the helimagnet. If the magnetic helix period \({{L}_{\text{H}}}\) is large compared to \({{L}_{\text{S}}},\) then the length \({{L}_{\text{D}}}\) coincides with \({{L}_{\text{S}}}.\) In helimagnets in which the magnetic helix period is small compared to \({{L}_{\text{S}}},\) the length \({{L}_{\text{D}}}\) does not exceed the helix period \({{L}_{\text{H}}}.\)

The second characteristic length, \({{L}_{\text{P}}},\) determines the scale of the decrease in the longitudinal component of the nonequilibrium magnetization of conduction electron and the longitudinal polarization of the spin current. The value of \({{L}_{\text{P}}}\) is determined by the magnitude of the exchange field acting on the magnetization of conduction electrons from localized electrons. The main effect on spin injection on the length scales on the order of \({{L}_{\text{P}}}\) is exerted by the precession of the magnetization of conduction electrons in a nonuniform exchange field created by the magnetic helix of the helimagnet. In short-period helimagnets with \({{L}_{{\text{H}}}} \ll {{L}_{{\text{S}}}}\), significant changes in the polarization of the spin current occur at distances \({{L}_{\text{P}}}\), which are much smaller than the helix period \({{L}_{\text{H}}}.\)

The magnitude of a pure spin current injected from a metal with strong SOI is directly determined by two parameters. The first of these is the ratio of the relaxation time of the momentum of conduction electron to the characteristic time of spin relaxation due to the skew scattering of electrons. The second parameter characterizes the transparency of the normal metal–helimagnet interface for conduction electrons.

If the injection of a spin current into a helimagnet from a normal metal is caused by SOI, which is responsible for the generation of a transversely polarized spin current in a normal metal, which flows along the normal to the interface, then, at the interface, a spin current that has a component with longitudinal polarization is induced. In this case, the direction of the longitudinal polarization vector is directly determined by the chirality of the helimagnet. The predicted effect is called “the effect of chiral polarization of a pure spin current.” It has also been shown that the transverse component of the polarization of conduction electron rotates around the axis of the magnetic helix during injection.