Microscopic theory of magnon-drag electron flow in ferromagnetic metals

A temperature gradient applied to a ferromagnetic metal induces not only independent flows of electrons and magnons but also drag currents because of their mutual interaction. In this paper, we present a microscopic study of the electron flow induced by the drag due to magnons. The analysis is based on the s-d model, which describes conduction electrons and magnons coupled via the s-d exchange interaction. Magnetic impurities are introduced in the electron subsystem as a source of spin relaxation. The obtained magnon-drag electron current is proportional to the entropy of magnons and to $\alpha -\beta$ (more precisely, to $1-\beta /\alpha$), where $\alpha$ is the Gilbert damping constant and $\beta$ is the dissipative spin-transfer torque parameter. This result almost coincides with the previous phenomenological result based on the magnonic spin-motive forces, and consists of spin-transfer and momentum-transfer contributions, but with a slight disagreement in the former. The result is interpreted in terms of the nonequilibrium spin chemical potential generated by nonequilibrium magnons.

Figure 1

(a) Self-energy of electrons, $\mathrm{\Sigma }$. (b) Self-energy of magnons, $\mathrm{\Pi }$. (c) Spin vertex ${\mathrm{\Lambda }}^{\alpha }$ renormalized by impurity-ladder corrections. (d) Four-point vertex ${\mathrm{\Gamma }}_{\sigma {\sigma }^{\prime }}$, which we call the diffusion-type vertex correction, or simply, the diffusion propagator. The solid (wavy) lines represent electron (magnon) propagators, and the dashed line with a cross represents impurity scattering.

Figure 2

Feynman diagrams for $K_{ij}\left(i{\omega }_{\lambda }\right)$ [Eq. (26)], which describe the magnon-drag processes. The solid (wavy) lines represent the electron (magnon) Green's functions. (a) Processes for $\mathit{Q}=\mathit{0}$. The gray triangles are defined in Fig. 1. (b) Additional processes that contribute when $\mathit{Q}\ne \mathit{0}$. The gray rectangle (or trapezoid) represents the diffusion propagator ${\mathrm{\Gamma }}_{\sigma {\sigma }^{\prime }}$ defined in Fig. 1. The diagrams in (b) vanish for $\mathit{Q}=\mathit{0}$, but contribute for finite $\mathit{Q}$ and lead to Eq. (57).

Figure 3

Feynman diagrams for the electric current induced by magnetization dynamics. Arrows in the electron lines (solid lines) are suppressed for simplicity. (a) Nonlinear response to the (classical) magnetization dynamics. The wavy lines represent the perturbations due to (classical) magnetization. Because of causality (retarded response), the incoming Matsubara frequencies should satisfy the conditions ${\nu }_l+{\omega }_{\lambda }>0$ and $-{\nu }_l>0$ [25]. (b) Part of the diagram of the present magnon-drag process (Fig. 2). The wavy lines represent (quantum) magnon propagators. Note that the flow of the Matsubara frequency in the lower magnon line is reversed compared to (a). The same causality relation as (a) leads to the analytic continuation $D(i{\nu }_l+i{\omega }_{\lambda })D\left(i{\nu }_l\right)\to D^{\mathrm{R}}(\nu +\omega )D^{\mathrm{A}}\left(\nu \right)$ for the pair of magnon propagators, and this is associated with ${\mathcal{E}}_i^{\left(2\right)}$ given by Eq. (31).

Figure 4

Feynman diagrams for the electron density $\delta n_{\mathrm{el}}$ [Eq. (51)] and the current density $j_{\mathrm{el},i}$ [Eq. (52)] induced by $\mathbf{\nabla }T$. $h_{\mathrm{el}}$ and $j_{\mathrm{el},i}^Q$ are the Hamiltonian density and heat current density, respectively, of the conduction electrons. The shaded rectangle represents the diffusion-type ladder vertex correction due to impurities [Fig. 1], which describes diffusive motion of the electrons.