Generalized Two-Temperature Model for Coupled Phonon-Magnon Diffusion

We generalize the two-temperature model [Sanders and Walton, Phys. Rev. B 15, 1489 (1977)] for coupled phonon-magnon diffusion to include the effect of the concurrent magnetization flow, with a particular emphasis on the thermal consequence of the magnon flow driven by a nonuniform magnetic field. Working within the framework of the Boltzmann transport equation, we derive the constitutive equations for coupled phonon-magnon transport driven by gradients of both temperature and external magnetic fields, and the corresponding conservation laws. Our equations reduce to the original Sanders-Walton two-temperature model under a uniform external field, but predict a new magnon cooling effect driven by a nonuniform magnetic field in a homogeneous single-domain ferromagnet. We estimate the magnitude of the cooling effect in an yttrium iron garnet, and show it is within current experimental reach. With properly optimized materials, the predicted cooling effect can potentially supplement the conventional magnetocaloric effect in cryogenic applications in the future.

Figure 1

(a) The steplike magnetic field, smeared out as a Fermi-Dirac function. $B_0$ is fixed to be 0.5 T in the following calculation. (b) The temperature distribution of phonons and magnons when $B_1=1.5\mathrm{T}$. One end of the sample ($x=0$) is thermally connected to a reservoir at 20 K, and the other end is isolated. (c) The dependence of the phonon temperature difference between the two ends on the difference of the magnetic field when $B_0$ is set to 0.5 T.

Figure 2

(a) The temperature profiles of phonons and magnons when $T_h=20\mathrm{K}$, $T_h-T_c=30\mathrm{mK}$ and $(B_0,B_1)=(0.5\mathrm{T},1.5\mathrm{T})$. (b) COP versus the change of magnetic field when the temperature difference is fixed at 30 mK. The hot-side temperature is fixed at 20 K. (c) COP versus the temperature difference when the hot-side temperature is fixed at 20 K and the magnetic field is fixed at $(B_0,B_1)=(0.5\mathrm{T},1.5\mathrm{T})$.