Landau-Lifshitz theory of the longitudinal spin Seebeck effect

Thermal-bias-induced spin angular momentum transfer between a paramagnetic metal and ferromagnetic insulator is studied theoretically based on the stochastic Landau-Lifshitz-Gilbert (LLG) phenomenology. Magnons in the ferromagnet establish a nonequilibrium steady state by equilibrating with phonons via bulk Gilbert damping and electrons in the paramagnet via spin pumping, according to the fluctuation-dissipation theorem. Subthermal magnons and the associated spin currents are treated classically, while the appropriate quantum crossover is imposed on high-frequency magnetic fluctuations. We identify several length scales in the ferromagnet, which govern qualitative changes in the dependence of the thermally induced spin current on the magnetic film thickness.

Figure 1

Schematic of an N${}_1$$\mid$F$\mid$ N${}_2$ sandwich structure studied in this paper. The normal-metal layer N${}_1$ is treated as a poor spin sink, which blocks spin current, $j_{s1}\approx 0$. The normal-metal layer N${}_2$, on the other hand, is a perfect spin sink, thus establishing a thermal contact between its itinerant electrons and magnons in the ferromagnetic insulator (F), which results in spin current $j_{s2}\equiv j_s$. We assume the phonons in the F layer follow a linear temperature profile from $T_1$ at $x=0$ (N${}_1$$\mid$F interface) to $T_2$ at $x=d$ (F$\mid$ N${}_2$ interface), corresponding to electron temperatures in N${}_1$ and N${}_2$, respectively.

Figure 2

Integrand of the spin Seebeck coefficient, Eq. (27), in arbitrary units, illustrating the first four quasiparticle resonances, $n=0,1,2,3$, according to Eq. (33), for a fixed $q$. We set $\alpha ={10}^{-4}$, ${\alpha }^{\prime }={10}^{-2}$ nm, $q^2+{\xi }^{-2}=1$ nm${}^{-2}$, and $d=10$ nm in the main plot. The inset shows the essentially continuum spectrum when $d=100$ $\mu$m, keeping other parameters unmodified.

Figure 3

Plot of the spin Seebeck coefficient, Eq. (27), as a function of ferromagnet thickness $d$ for $\lambda =1$ nm, $\xi =10$ nm, $l^{\prime }=100$ nm, and $l=10$ $\mu$m. We use Eq. (37) (solid curve on the left) and Eq. (41) (solid line on the right), which are valid when $d\lesssim l$ and $d\gtrsim l$, respectively, when N${}_1$ is a poor spin sink and N${}_2$ a perfect spin sink. To account for the classical-to-quantum crossover, we have inserted factor (45) in the integrands of Eqs. (35) and (36) [with $\omega \to {\omega }_n\left(q\right)$] and Eq. (41). The dotted curve shows the enhanced spin Seebeck coefficient when also N${}_1$ is a perfect spin sink (which increases the effective thermal bias between magnons and electrons at the F$\mid$ N${}_2$ interface).

Figure 4

Plot of the spin Seebeck coefficient using magnetic length scales as in Fig. 3 but calculated for a non-inversion-symmetric N${}_1$$\mid$F$\mid$ N${}_2$ structure with spin-pumping parameters ${\alpha }_1^{\prime }={\alpha }^{\prime }/10$, ${\alpha }_2^{\prime }={\alpha }^{\prime }$, respectively, at the two interfaces (upper trace) and ${\alpha }_1^{\prime }={\alpha }^{\prime }$, ${\alpha }_2^{\prime }={\alpha }^{\prime }/10$ (lower trace). Note that the former case is intermediate between the two curves plotted in Fig. 3 (where the solid curve corresponds to ${\alpha }_1^{\prime }=0$, ${\alpha }_2^{\prime }={\alpha }^{\prime }$ and the dotted curve to ${\alpha }_1^{\prime }={\alpha }_2^{\prime }={\alpha }^{\prime }$).