Wiedemann-Franz law for magnon transport

One of the main goals of spintronics is to improve transport of information carriers and to achieve new functionalities with ultra-low dissipation. A most promising strategy for this holy grail is to use pure magnon currents created and transported in insulating magnets, in the complete absence of any conducting metallic elements. Here we propose a realistic solution to this fundamental challenge by analyzing magnon and heat transport in insulating ferromagnetic junctions. We calculate all transport coefficients for magnon transport and establish Onsager relations between them. We theoretically discover that magnon transport in junctions has a universal behavior, i.e., is independent of material parameters, and establish a magnon analog of the celebrated Wiedemann-Franz law, which governs charge transport at low temperatures. We calculate the Seebeck and Peltier coefficients, which are crucial quantities for spin caloritronics, and demonstrate that they assume universal values in the low-temperature limit. Finally, we show that our predictions are within experimental reach with current device and measurement technologies.

Figure 1

Schematic representation of the ferromagnetic insulating junction. Magnons, i.e., the quanta of spin-waves, carry magnetic momentum ${\mu }_{\mathrm{B}}$ and heat $k_{\mathrm{B}}$. Those thermomagnetic transport properties are described by the WF law [Eq. (7)]. The exchange interaction between the nearest-neighbor spins in each FI (whose width is $L$) is $J$ and the magnetic field in the left (right) FI along the $z$ axis is denoted by $B_{l\left(r\right)}$. The time reversal symmetry is broken by the ferromagnetic order and the magnetic field. The boundary spins ${\mathbf{S}}_{{\mathrm{\Gamma }}_l}$ in the left FI and ${\mathbf{S}}_{{\mathrm{\Gamma }}_r}$ in the right FI are relevant to the transport of such magnons and they are weakly exchange-coupled with the strength $J_{\mathrm{ex}} \left(\ll J\right)$.

Figure 2

Plot of the ratio ${(g{\mu }_{\mathrm{B}}/k_{\mathrm{B}})}^2[K/\left(GT\right)]$ as function of $b$ where $\epsilon ={10}^{-10}$. At low temperatures $b=\mathcal{O}\left(10\right)$, the ratio reaches the constant “1” and the WF law for magnon transport [Eq. (7)] is realized.

Figure 3

Plot of magnon Seebeck coefficient $\mathcal{S}$ as function of $b$ where $\epsilon ={10}^{-10}$. At low temperatures $b=\mathcal{O}\left({10}^2\right)$, the rescaled coefficient $g{\mu }_{\mathrm{B}}\mathcal{S}/\left(k_{\mathrm{B}}b\right)$ approaches unity asymptotically and realizes the universal relation [Eq. (10)].