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Pure Spin Current Injection into a Helimagnet

  • ORDER, DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
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Abstract

The injection of a pure spin current into a conducting helimagnet is investigated. The characteristic decay lengths for the spin current injected into the helimagnet are determined, and their physical meaning is described. It is shown that instead of the spin diffusion length, helimagnets are characterized by the decay length that is always smaller than the spin diffusion length, the difference in these lengths being determined by the ratio of the helimagnet spiral period to the spin diffusion length. We predict the existence of the “effect of the chiral polarization of a pure spin current,” i.e., the emergence of the spin current with longitudinal (transverse) polarization, which depends on the spiral chirality, upon the injection of a pure spin current with the transverse (longitudinal) polarization relative to the spiral axis.

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Funding

This study was supported by the Russian Science Foundation (project no. 22-22-00220).

Author information

Authors and Affiliations

  1. Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, 620137, Yekaterinburg, Russia

    I. A. Yasyulevich, N. G. Bebenin & V. V. Ustinov

  2. Institute of Natural Sciences and Mathematics, Ural Federal University named after the first president of Russia B.N. Yeltsin, 620002, Yekaterinburg, Russia

    V. V. Ustinov

Authors
  1. I. A. Yasyulevich
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  2. N. G. Bebenin
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  3. V. V. Ustinov
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Correspondence to I. A. Yasyulevich or V. V. Ustinov.

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Additional information

Translated by N. Wadhwa

Appendices

APPENDIX A

Using the Cardano formulas, we write the exact solution to characteristic equation (6) in form

$$\begin{gathered} {{\kappa }_{1}} = \sqrt {1 - (2{\text{/}}3){{\eta }^{2}} + U + W} , \\ {{\kappa }_{2}} = (1{\text{/}}\sqrt 2 )(\sqrt {{{{(1 - (2{\text{/}}3){{\eta }^{2}} + \operatorname{Re} V)}}^{2}} + {{{(\operatorname{Im} V)}}^{2}}} \\ + 1 - (2{\text{/}}3){{\eta }^{2}} + \operatorname{Re} V{{)}^{{1/2}}} + i\operatorname{sgn} (\operatorname{Im} V)(1{\text{/}}\sqrt 2 ) \\ \, \times (\sqrt {{{{(1 - (2{\text{/}}3){{\eta }^{2}} + \operatorname{Re} V)}}^{2}} + {{{(\operatorname{Im} V)}}^{2}}} \\ - 1 + (2{\text{/}}3){{\eta }^{2}} - \operatorname{Re} V{{)}^{{1/2}}}, \\ {{\kappa }_{3}} = \kappa _{2}^{*}, \\ \end{gathered} $$
(12)

where

$$\begin{gathered} V = - (U + W){\text{/}}2 + i\sqrt 3 (U - W){\text{/}}2, \\ U = \sqrt[3]{{\frac{{{{\eta }^{2}}}}{6}\left( {\frac{2}{9}{{\eta }^{4}} + 8{{\eta }^{2}} + 5{{\lambda }^{2}}} \right) + \sqrt G }}, \\ W = \sqrt[3]{{\frac{{{{\eta }^{2}}}}{6}\left( {\frac{2}{9}{{\eta }^{4}} + 8{{\eta }^{2}} + 5{{\lambda }^{2}}} \right) - \sqrt G }}, \\ G = \frac{1}{{27}}{{\left( { - \frac{1}{3}{{\eta }^{4}} + 4{{\eta }^{2}} + {{\lambda }^{2}}} \right)}^{3}}\, + \,\frac{{{{\eta }^{4}}}}{{36}}{{\left( {\frac{2}{9}{{\eta }^{4}} + 8{{\eta }^{2}} + 5{{\lambda }^{2}}} \right)}^{2}}. \\ \end{gathered} $$
(13)

APPENDIX B

To find approximate solutions to characteristic equation (6), we can use the following algorithm.

1. Case with λ ≫ 1 + η2. Performing identity transformations, we write characteristic equation (6) in form

$$\begin{gathered} {{\kappa }^{2}} = 1 + {{\eta }^{2}} \\ \, - ({{\kappa }^{2}} - 1)[{{({{\kappa }^{2}} - 1 - {{\eta }^{2}})}^{2}} + 4{{\eta }^{2}}{{\kappa }^{2}}]{\text{/}}{{\lambda }^{2}}. \\ \end{gathered} $$
(14)

In view of smallness of parameter (1 + η2)/λ, this equation can be solved by the method of successive approximation. In the main approximation in small parameter (1 + η2)/λ, we obtain κ = κ1 = \(\sqrt {1 + {{\eta }^{2}}} \). This gives the following expression for LD: LD = LS/\(\sqrt {1 + {{\eta }^{2}}} \).

To obtain root κ2, we write characteristic equation (6) in identical form

$${{\kappa }^{4}} = \frac{{ - {{\lambda }^{2}}}}{{\left( {1 - \frac{1}{{{{\kappa }^{2}}}}} \right)\left[ {\left( {1 - \frac{{(1 + {{\eta }^{2}})}}{{{{\kappa }^{2}}}}} \right) + \frac{{4{{\eta }^{2}}{\text{/}}{{\kappa }^{2}}}}{{1 - (1 + {{\eta }^{2}}){\text{/}}{{\kappa }^{2}}}}} \right]}}$$
(15)

and use the method of iterations again. In the zeroth approximation in (1 + η2)/λ, it follows from expression (15) that κ = κ2 = (1 + i)\(\sqrt {\lambda {\text{/}}2} \). This gives LP = LS/\(\sqrt {\lambda {\text{/}}2} \).

2. Case with λ ≪ 1 + η2. To find the solution to Eq. (6) in the zeroth approximation in small parameter λ/(1 + η2), it is sufficient to solve equation

$$({{\kappa }^{2}} - 1)[{{({{\kappa }^{2}} - 1 - {{\eta }^{2}})}^{2}} + 4{{\eta }^{2}}{{\kappa }^{2}}] = 0,$$
(16)

which has been formally obtained from Eq. (6) for λ = 0. The real-valued solution to this equation is κ = κ1 = 1. Two other roots of Eq. (16) are κ = κ2,3 = 1 ± iη. It follows hence that in the given limiting case, decay lengths LD and LP are identical and coincide with spin diffusion length LS: LD = LP = LS.

APPENDIX C

The coordinate dependence of the nonequilibrium magnetization of conduction electrons in a helimagnet, which has been obtained using exact roots (12) of the characteristic equation, can be written in form

$$\begin{gathered} \delta {{m}_{x}} = {{C}_{1}}{{e}^{{ - {{\kappa }_{1}}\zeta }}}\operatorname{Re} ({{\Psi }_{1}}{{e}^{{iK\eta \zeta }}}) \\ + \operatorname{Re} ({{\Psi }_{2}}{{C}_{2}}{{e}^{{ - {{\kappa }_{2}}\zeta }}}{{e}^{{iK\eta \zeta }}}) + \operatorname{Re} ({{\Psi }_{3}}C_{2}^{*}{{e}^{{ - \kappa _{2}^{*}\zeta }}}{{e}^{{iK\eta \zeta }}}), \\ \delta {{m}_{y}} = {{C}_{1}}{{e}^{{ - {{\kappa }_{1}}\zeta }}}\operatorname{Im} ({{\Psi }_{1}}{{e}^{{iK\eta \zeta }}}) \\ + \operatorname{Im} ({{\Psi }_{2}}{{C}_{2}}{{e}^{{ - {{\kappa }_{2}}\zeta }}}{{e}^{{iK\eta \zeta }}}) + \operatorname{Im} ({{\Psi }_{3}}C_{2}^{*}{{e}^{{ - \kappa _{2}^{*}\zeta }}}{{e}^{{iK\eta \zeta }}}), \\ \delta {{m}_{z}} = {{C}_{1}}{{e}^{{ - {{\kappa }_{1}}\zeta }}} + 2\operatorname{Re} ({{C}_{2}}{{e}^{{ - {{\kappa }_{2}}\zeta }}}), \\ \end{gathered} $$
(17)

where

$${{\Psi }_{n}} = \frac{{i\lambda }}{{\kappa _{n}^{2} - 1 - {{\eta }^{2}} - 2iK\eta {{\kappa }_{n}}}},$$
$${{C}_{1}} = {{\Delta }_{{{{C}_{1}}}}}{\text{/}}\Delta ,$$
$${{C}_{2}} = ({{\Delta }_{{\operatorname{Re} {{C}_{2}}}}} + i{{\Delta }_{{\operatorname{Im} {{C}_{2}}}}}){\text{/}}\Delta ,$$
$$\begin{gathered} {{\Delta }_{{{{C}_{1}}}}} = - 2({{\Phi }_{5}}\operatorname{Im} {{\kappa }_{2}} + {{\Phi }_{6}}\operatorname{Re} {{\kappa }_{2}}){{{\tilde {P}}}_{x}} \\ \, + 2({{\Phi }_{2}}\operatorname{Im} {{\kappa }_{2}} + {{\Phi }_{3}}\operatorname{Re} {{\kappa }_{2}}){{{\tilde {P}}}_{y}} + ({{\Phi }_{2}}{{\Phi }_{6}} - {{\Phi }_{3}}{{\Phi }_{5}}){{{\tilde {P}}}_{z}}, \\ \end{gathered} $$
$$\begin{gathered} {{\Delta }_{{\operatorname{Re} {{C}_{2}}}}} = (2{{\Phi }_{4}}\operatorname{Im} {{\kappa }_{2}} + {{\kappa }_{1}}{{\Phi }_{6}}){{{\tilde {P}}}_{x}} \\ \, - (2{{\Phi }_{1}}\operatorname{Im} {{\kappa }_{2}} + {{\kappa }_{1}}{{\Phi }_{3}}){{{\tilde {P}}}_{y}} + ({{\Phi }_{3}}{{\Phi }_{4}} - {{\Phi }_{1}}{{\Phi }_{6}}){{{\tilde {P}}}_{z}}, \\ \end{gathered} $$
$$\begin{gathered} {{\Delta }_{{\operatorname{Im} {{C}_{2}}}}} = - ({{\kappa }_{1}}{{\Phi }_{5}} + 2{{\Phi }_{4}}\operatorname{Re} {{\kappa }_{2}}){{{\tilde {P}}}_{x}} \\ \, - (2{{\Phi }_{1}}\operatorname{Re} {{\kappa }_{2}} + {{\kappa }_{1}}{{\Phi }_{2}}){{{\tilde {P}}}_{y}} + ({{\Phi }_{1}}{{\Phi }_{5}} - {{\Phi }_{2}}{{\Phi }_{4}}){{{\tilde {P}}}_{z}}, \\ \end{gathered} $$
$$\begin{gathered} \Delta = - 2{{\Phi }_{1}}({{\Phi }_{5}}\operatorname{Im} {{\kappa }_{2}} + {{\Phi }_{6}}\operatorname{Re} {{\kappa }_{2}}) \\ \, + {{\Phi }_{2}}(2{{\Phi }_{4}}\operatorname{Im} {{\kappa }_{2}} + {{\kappa }_{1}}{{\Phi }_{6}}) - {{\Phi }_{3}}({{\kappa }_{1}}{{\Phi }_{5}} - 2{{\Phi }_{4}}\operatorname{Re} {{\kappa }_{2}}), \\ \end{gathered} $$
$${{\tilde {P}}_{x}} = ({{\tau }_{S}}{\text{/}}{{L}_{S}})({{{\mathbf{P}}}_{0}} \cdot {{{\mathbf{e}}}_{x}}),$$
$${{\tilde {P}}_{y}} = ({{\tau }_{S}}{\text{/}}{{L}_{S}})({{{\mathbf{P}}}_{0}} \cdot {{{\mathbf{e}}}_{y}}),$$
$${{\tilde {P}}_{z}} = ({{\tau }_{S}}{\text{/}}{{L}_{S}})({{{\mathbf{P}}}_{0}} \cdot {{{\mathbf{e}}}_{z}}),$$
$${{\Phi }_{1}} = {{\kappa }_{1}}\operatorname{Re} {{\Psi }_{1}} + K\eta \operatorname{Im} {{\Psi }_{1}},$$
$$\begin{gathered} {{\Phi }_{2}} = K\eta \operatorname{Im} {{\Psi }_{2}} + K\eta \operatorname{Im} {{\Psi }_{3}} \\ + \operatorname{Re} ({{\Psi }_{2}})\operatorname{Re} ({{\kappa }_{2}}) - \operatorname{Im} ({{\Psi }_{2}})\operatorname{Im} ({{\kappa }_{2}}) \\ + \operatorname{Re} ({{\Psi }_{3}})\operatorname{Re} ({{\kappa }_{2}}) + \operatorname{Im} ({{\Psi }_{3}})\operatorname{Im} ({{\kappa }_{2}}), \\ \end{gathered} $$
$$\begin{gathered} {{\Phi }_{3}} = K\eta \operatorname{Re} {{\Psi }_{2}} - K\eta \operatorname{Re} {{\Psi }_{3}} \\ - \operatorname{Re} ({{\Psi }_{2}})\operatorname{Im} ({{\kappa }_{2}}) - \operatorname{Im} ({{\Psi }_{2}})\operatorname{Re} ({{\kappa }_{2}}) \\ - \operatorname{Re} ({{\Psi }_{3}})\operatorname{Im} ({{\kappa }_{2}}) + \operatorname{Im} ({{\Psi }_{3}})\operatorname{Re} ({{\kappa }_{2}}), \\ \end{gathered} $$
$${{\Phi }_{4}} = {{\kappa }_{1}}\operatorname{Im} {{\Psi }_{1}} - K\eta \operatorname{Re} {{\Psi }_{1}},$$
$$\begin{gathered} {{\Phi }_{5}} = - K\eta \operatorname{Re} {{\Psi }_{2}} - K\eta \operatorname{Re} {{\Psi }_{3}} \\ + \operatorname{Re} ({{\Psi }_{2}})\operatorname{Im} ({{\kappa }_{2}}) + \operatorname{Im} ({{\Psi }_{2}})\operatorname{Re} ({{\kappa }_{2}}) \\ - \operatorname{Re} ({{\Psi }_{3}})\operatorname{Im} ({{\kappa }_{2}}) + \operatorname{Im} ({{\Psi }_{3}})\operatorname{Re} ({{\kappa }_{2}}), \\ \end{gathered} $$
$$\begin{gathered} {{\Phi }_{6}} = K\eta \operatorname{Im} {{\Psi }_{2}} - K\eta \operatorname{Im} {{\Psi }_{3}} \\ + \operatorname{Re} ({{\Psi }_{2}})\operatorname{Re} ({{\kappa }_{2}}) - \operatorname{Im} ({{\Psi }_{2}})\operatorname{Im} ({{\kappa }_{2}}) \\ - \operatorname{Re} ({{\Psi }_{3}})\operatorname{Re} ({{\kappa }_{2}}) - \operatorname{Im} ({{\Psi }_{3}})\operatorname{Im} ({{\kappa }_{2}}). \\ \end{gathered} $$

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Yasyulevich, I.A., Bebenin, N.G. & Ustinov, V.V. Pure Spin Current Injection into a Helimagnet. J. Exp. Theor. Phys. 136, 509–518 (2023). https://doi.org/10.1134/S1063776123040143

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