Nonequilibrium steady-state transport properties of magnons in ferromagnetic insulators
Authors: Yang Dong-Chao , Yi Li-Zhi , Ding Lin-Jie , Liu Min , Zhu Li-Ya , Xu Yun-Li , He Xiong , Shen Shun-Qing , Pan Li-Qing , John Q. Xiao ,   Hubei Engineering Research Center of Weak Magnetic-field Detection, College of Science, China Three Gorges University, Yichang 443002, China,   Department of Physics, Chongqing Three Gorges University, Chongqing 404100, China,   Department of Physics, University of Hong Kong, Hong Kong 999077, China,   Department of Physics and Astronomy, University of Delaware, Newark 19716, USA
Published: May 2024

Understanding nonequilibrium transport phenomena in bosonic systems is highly challenging. Magnons, as bosons, exhibit different transport behavior from fermionic electron spins. This study focuses on the key factors influencing the nonequilibrium transport of magnons in steady states within magnetic insulators by taking Y<sub>3</sub>Fe<sub>5</sub>O<sub>12</sub> (YIG) for example. By incorporating the Bose-Einstein distribution function with a non-zero chemical potential <inline-formula><tex-math id="M15">\begin{document}$ {\mu }_{m} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M15.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M15.png"/></alternatives></inline-formula> into the Boltzmann transport equation, analytical expressions for transport parameters in power of <inline-formula><tex-math id="M16">\begin{document}$ \alpha $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M16.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M16.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M17">\begin{document}$ =-{\mu }_{{\mathrm{m}}}/({k}_{{\mathrm{B}}}T) $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M17.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M17.png"/></alternatives></inline-formula>) are obtained under the condition <i>α</i><1. It is the biggest different from previous researches that our theory establishes a nonlinear relationship between the chemical potential and the nonequilibrium particle density <inline-formula><tex-math id="M18">\begin{document}$ \delta {n}_{{\mathrm{m}}}\propto -{\alpha }^{1/2}\propto $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M18.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M18.png"/></alternatives></inline-formula><inline-formula><tex-math id="M18-1">\begin{document}$ -{(-{\mu }_{{\mathrm{m}}})}^{1/2} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M18-1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M18-1.png"/></alternatives></inline-formula> for magnons under <i>α</i><inline-formula><tex-math id="Z-20240629142100">\begin{document}$\ll 1 $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_Z-20240629142100.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_Z-20240629142100.png"/></alternatives></inline-formula>. For a large chemical potential, higher-order terms of <i>α</i> must be taken into account. Owing to this nonlinear relationship, the magnon diffusion equation markedly differs from that governing electron spin,which evolves into more complex nonlinear differential equation. We specifically focus on the ferrimagnetic insulator YIG by making a comparison of the spatial distribution of the nonequilibrium magnon density <inline-formula><tex-math id="M19">\begin{document}$ \delta {n}_{m} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M19.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M19.png"/></alternatives></inline-formula> and chemical potential <inline-formula><tex-math id="M20">\begin{document}$ {\mu }_{m} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M20.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M20.png"/></alternatives></inline-formula> between two extreme temperature gradients, namely, <inline-formula><tex-math id="M21">\begin{document}$ \nabla T \sim 1\;{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M21.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M21.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M22">\begin{document}$ {10}^{4}\;{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}}, $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M22.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M22.png"/></alternatives></inline-formula> which correspond to <inline-formula><tex-math id="M23">\begin{document}$ {\mu }_{{\mathrm{m}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M23.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M23.png"/></alternatives></inline-formula> values on the order of <inline-formula><tex-math id="M24">\begin{document}$ -0.1\;{\text{μ}}{\mathrm{e}}{\mathrm{V}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M24.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M24.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M25">\begin{document}$ -6.2\;{\mathrm{m}}{\mathrm{e}}{\mathrm{V}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M25.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M25.png"/></alternatives></inline-formula>, respectively, while still satisfying the prerequisite <i>α</i> < 1. Given the known temperature gradient distribution, the nonequilibrium magnon density <inline-formula><tex-math id="M26">\begin{document}$ \delta {n}_{{\mathrm{m}}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M26.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20240498_M26.png"/></alternatives></inline-formula> calculated based on our theory is in good agreement with the experimental result. Our theoretical and numerical findings greatly contribute to a profound understanding of the nonequilibrium magnon transport characteristics in magnetic insulators.