Finite-frequency spin conductance of the interface between a ferro- or ferrimagnetic insulator and a normal metal

The interface between a ferro- or ferrimagnetic insulator and a normal metal can support spin currents polarized collinear with and perpendicular to the magnetization direction. The flow of angular momentum perpendicular to the magnetization direction (“transverse” spin current) takes place via spin torque and spin pumping. The flow of angular momentum collinear with the magnetization (“longitudinal” spin current) requires the excitation of magnons. In this article we extend the existing theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal spin conductance nonperturbatively (but in the low-frequency limit) and at finite frequency (but in the limit of low interface transparency). For the paradigmatic spintronic material system $\mathrm{YIG}|\mathrm{Pt}$, we find that nonperturbative effects lead to a longitudinal spin conductance that is ca. 40% smaller than the perturbative limit, whereas finite-frequency corrections are relevant at low temperatures $\lesssim 100\mathrm{K}$ only, when only few magnon modes are thermally occupied.

Figure 1

Illustration of the microscopic mechanisms underlying the transverse (left) and longitudinal (right) components of the spin current through the interface between a ferro- or ferrimagnetic insulator F and a normal metal N. The transverse spin current is mediated by the spin torque and spin pumping involving electrons (red) with spins perpendicular to the magnetization and spin waves (blue) with frequency $\omega$ equal to the frequency at which the spin accumulation ${\mathit{\mu }}_{\mathrm{s}}$ in N is driven. The longitudinal component arises from spin-flip scattering of conduction electrons (red), combined with the creation or absorption of thermal magnons of frequency $\mathrm{\Omega }$ (blue). (The thermal magnon frequency $\mathrm{\Omega }$ is not related to the driving frequency $\omega$.) Alternatively, the longitudinal component can be seen as arising from the spin torque exerted on/spin pumped by the transverse magnetization component induced by thermal fluctuations in F (not shown schematically).

Figure 2

Mode-averaged magnon transmission coefficient $T_{\mathrm{m}}\left(\mathrm{\Omega }\right)$ at a $\mathrm{YIG}|\mathrm{Pt}$-interface. The solid red curve shows the nonperturbative result of Eq. (26) and the blue dashed curve the weak-coupling approximation $T_{\mathrm{m}}^{\left(\mathrm{p}\right)}\left(\mathrm{\Omega }\right)$ of Eq. (41). Both curves are based on the quadratic approximation to the magnon dispersion, which breaks down for magnon frequencies $\mathrm{\Omega }\approx {\mathrm{\Omega }}_{\max }$, which is the frequency of acoustic magnons at the zone boundary. Parameter values are taken from Table 1.

Figure 3

Zero-frequency longitudinal spin conductance per unit area, $g_{\mathrm{s}\parallel }$, at a $\mathrm{YIG}|\mathrm{Pt}$-interface as function of the temperature $T=T_{\mathrm{N}}=T_{\mathrm{F}}$ for $\mu ={\mu }_{\mathrm{s}\parallel }={\mu }_{\mathrm{m}}=0$. The red solid curve shows the nonperturbative result $g_{\mathrm{s}\parallel }/\text{Re}{g}_{\uparrow \downarrow }$ of Eq. (43), the blue dashed curve the perturbative result $g_{\mathrm{s}\parallel }^{\left(\mathrm{p}\right)}/\text{Re}{g}_{\uparrow \downarrow }$ of Eq. (44), and the thin green dot-dashed curve the approximation $g_{\mathrm{s}\parallel }^{\left(\mathrm{p}0\right)}/\text{Re}{g}_{\uparrow \downarrow }$ of Eq. (45). The upper left inset shows the ratios $g_{\mathrm{s}\parallel }/g_{\mathrm{s}\parallel }^{\left(\mathrm{p}0\right)}$ (red solid curve) and $g_{\mathrm{s}\parallel }^{\left(\mathrm{p}\right)}/g_{\mathrm{s}\parallel }^{\left(\mathrm{p}0\right)}$ (blue dashed curve). The lower right inset shows the ratio $g_{\mathrm{s}\parallel }^{\left(\mathrm{pH}\right)}/g_{\mathrm{s}\parallel }^{\left(\mathrm{p}\right)}$, where $g_{\mathrm{s}\parallel }^{\left(\mathrm{pH}\right)}$ is the result of Eq. (44) for the magnon density of states obtained from a Heisenberg model, see Eq. (46), and $g_{\mathrm{s}\parallel }^{\left(\mathrm{p}\right)}$ that of Eq. (44) for the quadratic approximation of the magnon dispersion. Parameter values are taken from Table 1.

Figure 4

Real part $\text{Re}{g}_{\mathrm{s}\parallel }$ of the finite-frequency longitudinal spin conductance of an $\mathrm{F}|\mathrm{N}$ interface in the perturbative regime of weak coupling as a function of the driving frequency $\omega /2\pi$ for various temperatures $T=T_{\mathrm{N}}=T_{\mathrm{F}}$ (solid colored lines). Material parameters are taken for an $\mathrm{F}|\mathrm{N}$ interface with F=YIG and N an arbitrary normal metal, see Table 1. The black dashed curve shows the result of the Rayleigh-Jeans approximation, see Eq. (47). The inset displays the same curves. The time-independent background magnon chemical potential and spin accumulation have been set to zero, ${\overline{\mu }}_{\mathrm{m}}={\overline{\mu }}_{\mathrm{s}\parallel }=0$.

Figure 5

Same as Fig. 4, but for the imaginary part $\text{Im}{g}_{\mathrm{s}\parallel }$ of the finite-frequency spin conductance. The black dashed lines show the limiting behavior for small and large $\omega$ according to Eqs. (51) and (53) for $T_{\mathrm{N}}\to 0$.

Figure 6

Real part $\text{Re}{g}_{\mathrm{s}\parallel }$ of the finite-frequency spin conductance of an $\mathrm{F}|\mathrm{N}$ interface in the weak-coupling regime as function of the driving frequency for two values of the temperature $T=T_{\mathrm{N}}=T_{\mathrm{F}}$. Material parameters are taken for an $\mathrm{F}|\mathrm{N}$ interface with F=YIG and N is an arbitrary normal metal, see Table 1. Curves are shown for three combinations of the time-independent background potentials ${\overline{\mu }}_{\mathrm{m}}$ and ${\overline{\mu }}_{\mathrm{s}\parallel }$: The solid colored curves correspond to ${\overline{\mu }}_{\mathrm{m}}={\overline{\mu }}_{\mathrm{s}\parallel }=0$; the dashed curves correspond to ${\overline{\mu }}_{\mathrm{m}}={\overline{\mu }}_{\mathrm{s}\parallel }=0.5\hslash {\omega }_0$, and the dot-dashed ones to ${\overline{\mu }}_{\mathrm{m}}={\overline{\mu }}_{\mathrm{s}\parallel }=-0.5\hslash {\omega }_0$.

Figure 7

Spin-Seebeck coefficient $L_{\mathrm{SSE}}$ at a $\mathrm{YIG}|\mathrm{Pt}$-interface in the linear-response regime as function of the temperature $T={\overline{T}}_{\mathrm{N}}={\overline{T}}_{\mathrm{F}}$. All three curves in the main panel are obtained using the parabolic approximation of the magnon dispersion. The red solid curve shows $L_{\mathrm{SSE}}$ according to the nonperturbative theory, see Eq. (56), the blue dashed curve the perturbative result $L_{\mathrm{SSE}}^{\left(\mathrm{p}\right)}$ of Eq. (57), and the thin green dot-dashed curve includes the approximation $L_{\mathrm{SSE}}^{\left(\mathrm{p}0\right)}$ of Eq. (58). The upper left inset shows the ratios $L_{\mathrm{SSE}}/L_{\mathrm{SSE}}^{\left(\mathrm{p}0\right)}$ (red solid curve) and $L_{\mathrm{SSE}}^{\left(\mathrm{p}\right)}/L_{\mathrm{SSE}}^{\left(\mathrm{p}0\right)}$ (blue dashed curve). The lower right inset shows the ratio $L_{\mathrm{SSE}}^{\left(\mathrm{pH}\right)}/L_{\mathrm{SSE}}^{\left(\mathrm{p}\right)}$, where $L_{\mathrm{SSE}}^{\left(\mathrm{pH}\right)}$ is the result of Eq. (57) for the magnon density of states obtained from a Heisenberg model, see Eq. (46), and $L_{\mathrm{SSE}}^{\left(\mathrm{p}\right)}$ that of Eq. (57) for the quadratic approximation of the magnon dispersion. Parameter values are taken from Table 1.