图 1  磁振子自旋塞贝克效应示意图, 铁磁绝缘体(MI)磁振子流的两个来源是温度梯度$ \nabla T $和化学势梯度$ \nabla \mu $, 磁振子流进入近邻的重金属层(HM), 由HM中逆自旋霍尔效应进行探测

Fig. 1.  Schematic diagram of magnon spin Seebeck effect. The two sources of magnon current in ferromagnetic insulators (MI) are temperature gradient $ \nabla T $ and chemical potential gradient $ \nabla \mu $. Magnon current enters the neighboring heavy metal layer (HM) and is detected by the inverse spin Hall effect.

图 2  非平衡粒子数$ {\delta n}_{{\mathrm{m}}} $随化学势$ {\mu }_{{\mathrm{m}}} $和$ \alpha $的变化规律

Fig. 2.  The variation of nonequilibrium magnon density $ {\delta n}_{{\mathrm{m}}} $with chemical potential $ {\mu }_{{\mathrm{m}}} $ and $ \alpha $.

图 3  非平衡磁振子流在非线性温度梯度$ \nabla T~{10}^{4}{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $下的输运　(a) YIG薄膜上激光辐照光斑附近的温度分布和温度梯度, 数据取自文献[20]; (b)本文的理论结果与位置相关的非平衡磁振子浓度$ \delta {n}_{{\mathrm{m}}} $, 及其与实验数据的对比; (c)本文的理论结果与采用线性近似计算的化学势$ {\mu }_{{\mathrm{m}}} $的对比; (d) MI中的总磁振子流$ {J}_{{\mathrm{m}}} $由两个统计力驱动, 温度梯度$ \nabla T $和化学势$ \nabla {\mu }_{{\mathrm{m}}} $. $ {J}_{{\mathrm{m}}}^{\nabla \mu } $和$ {J}_{{\mathrm{m}}}^{\nabla T} $对总磁振子流$ {J}_{{\mathrm{m}}} $的贡献

Fig. 3.  Out-of-equilibrium magnon transport under a nonlinear temperature gradient $ \nabla T~{10}^{4}{\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $: (a) Distribution of temperature and temperature gradient near the heating laser spot on the YIG film, data taken from^[20]; (b) comparison between our theoretical results and the experimental data for position-dependent nonequilibrium magnon density $ \delta {n}_{{\mathrm{m}}} $; (c) comparison between our theoretical results and the calculated data of linear approximation for position-dependent chemical potential $ {\mu }_{{\mathrm{m}}} $; (d) total magnon current $ {J}_{{\mathrm{m}}} $ in MI driven by two statistical forces: temperature gradient $ \nabla T $ and chemical potential $ \nabla {\mu }_{{\mathrm{m}}}. $ Contribution of $ {J}_{{\mathrm{m}}}^{\nabla \mu } $and $ {J}_{{\mathrm{m}}}^{\nabla T} $ to the total magnon current $ {J}_{{\mathrm{m}}} $.

图 4  MI中恒定温度梯度下的准平衡磁振子输运 (a), (b)不同温度梯度(1—60 $ {\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $)下非平衡磁振子浓度$ \delta {n}_{{\mathrm{m}}} $和化学势$ {\mu }_{{\mathrm{m}}} $的数值结果, 插图化学势$ {\mu }_{{\mathrm{m}}} $空间分布的的示意图; (c), (d)不同的$ \nabla T $值下计算得到的样品中心处 $ \delta {n}_{{\mathrm{m}}}^{0} $ 和 $ {\mu }_{{\mathrm{m}}}^{0} $的值

Fig. 4.  Quasi-equilibrium magnon transport under a constant temperature gradient in MI: (a), (b) Numerical results for the nonequilibrium magnon density $ \delta {n}_{{\mathrm{m}}} $ and chemical potential $ {\mu }_{{\mathrm{m}}} $ at various temperature gradients ranging from 1 to 60 $ {\mathrm{K}}/{\mathrm{m}}{\mathrm{m}} $. Insert is schematic diagram illustrating the spatial-dependent chemical potential $ {\mu }_{{\mathrm{m}}} $, total magnon current $ {J}_{{\mathrm{m}}} $ and its two components $ {J}_{{\mathrm{m}}}^{\nabla \mu } $ and $ {J}_{{\mathrm{m}}}^{\nabla T} $; (c), (d) calculated values of $ \delta {n}_{{\mathrm{m}}} $ and $ {\mu }_{{\mathrm{m}}} $ at the center of the sample for different values of $ \nabla T $.