Thermodynamic transport theory of spin waves in ferromagnetic insulators

We use the Boltzmann transport theory in the relaxation time approximation to describe the thermal transport of spin waves in a ferromagnet. By treating spin waves as magnon excitations we are able to compute analytically and numerically the coefficients of the constitutive thermomagnetic transport equations. As a main result, we find that the absolute thermomagnetic power coefficient ${\epsilon }_M$, relating the gradient of the potential of the magnetization current and the gradient of the temperature, in the limit of low temperature and low field, is a constant ${\epsilon }_M=-0.6419k_B/{\mu }_B$. The theory correctly describes the low-temperature and magnetic-field dependencies of spin Seebeck experiments. Furthermore, the theory predicts that in the limit of very low temperatures the spin Peltier coefficient ${\mathrm{\Pi }}_M$, relating the heat and the magnetization currents, tends to a finite value which depends on the amplitude of the magnetic field. This indicates the possibility to exploit the spin Peltier effect as an efficient cooling mechanism in cryogenics.

Figure 1

Thermomagnetic coefficients computed from the magnon transport theory (low-temperature approximation). The coefficients are normalized to ${\sigma }_{M_0}={\tau }_{c}m{v}_m^2/{\mu }_0,{\epsilon }_{M_0}=-k_B/\left(2{\mu }_B\right),{\kappa }_0=n{k}_B{\tau }_c{v}_m^2,{\tau }_{M_0}={\tau }_r/m,l_{M_0}={\left({\tau }_c{\tau }_r\right)}^{1/2}{v}_m,v_{M_0}={({\tau }_c/{\tau }_r)}^{1/2}m{v}_m$. For all graphs, the values of $h$ are ${10}^{-5},{10}^{-4}$, and ${10}^{-3}$ (red, green, and blue, respectively).

Figure 2

Spin Seebeck effect for the magnon transport theory (low-temperature approximation). $j_{MS}$ is the magnetization current source normalized to $j_{M{S}_0}=-{\epsilon }_{M_0}{\sigma }_{M_0}\mathbf{\nabla }T$. The main plot shows the temperature dependence at different fields. The inset shows the field dependence at different temperatures.

Figure 3

Thermomagnetic coefficients computed from the magnon transport theory (high-temperature approximation). The coefficients are normalized to the same values reported in the caption of Fig. 1. For all graphs, the values of $h$ are ${10}^{-5},{10}^{-4}$, and ${10}^{-3}$ (red, green, and blue, respectively).

Figure 4

Spin Seebeck effect for the magnon transport theory (high-temperature approximation). $j_{MS}$ is the magnetization current source normalized as in Fig. 2. Main plot and inset have the same meaning as in Fig. 2.

Figure 5

Magnon dispersion relations. The dashed line is the low-$q$ approximation of Eq. (6). The full line is the high-$q$ isotropic approximation of Eq. (A16).