Nonlinear spin-Seebeck diode in $f$-wave magnets, third-order spin-Nernst effects in $g$-wave magnets and spin-Nernst effects in $i$-wave altermagnets

Abstract

A prominent feature of $d$-wave altermagnets is that spin current is generated by applying temperature gradient, which is known as the spin-Nernst effect. We show in $f$-wave magnets that spin current is generated proportional to the square of the temperature gradient, which we call the nonlinear spin-Seebeck current. It can be used as a spin current diode. In addition, we show in $g$-wave altermagnets that spin current is generated in the third order of the temperature gradient. We also show in $i$-wave altermagnets that spin current is generated perpendicular to the temperature gradient, which is the spin-Nernst current. We have derived analytic formulas for these spin currents. It is interesting that these phenomena occur in the absence of the spin-orbit interaction. On the other hand, we show in $p$-wave magnets that spin current is not generated by temperature gradient.

Figures (4)

• Figure 1: $d$-wave altermagnet. (a) Fermi surfaces with two nodes. Red oval indicates the Fermi surface with up spin and blue oval indicates down spin. (b) Spin current $j_{\text{spin}}^{\left( x;y\right) }/\partial _{x}T$ in units of $e\tau /\hbar$ as a function of the inverse temperature $\beta =1/k_{\text{B}}T$. Red curve is a numerical result, while blue curve is an analytic result base on the high-temperature expansion. We have set $m=1$, $J=1/2$ and $a=1$.
• Figure 2: $f$-wave magnet. (a) Fermi surfaces with three nodes. (b) Spin current $j_{\text{spin}}^{\left( x^{2};y\right) }/\left( \partial _{x}T\right) ^{2}$ in units of $e\left( \tau /\hbar \right) ^{2}$ as a function of the inverse temperature $\beta =1/k_{\text{B}}T$. See also the caption of Fig.\ref{['FigD']}.
• Figure 3: $g$-wave altermagnet. (a) Fermi surfaces with four nodes. (b) Spin current $j_{\text{spin}}^{\left( x;y\right) }/\partial _{x}T$ in units of $e\tau /\hbar$ as a function of the inverse temperature $\beta =1/k_{\text{B}}T$. See also the caption of Fig.\ref{['FigD']}.
• Figure 4: $i$-wave altermagnet. (a) Fermi surfaces with six nodes. (b) Spin current $j_{\text{spin}}^{\left( x;y\right) }/\left( \partial _{x}T\right) ^{3}$ in units of $e\left( \tau /\hbar \right) ^{3}$ as a function of the inverse temperature $\beta =1/k_{\text{B}}T$. See also the caption of Fig.\ref{['FigD']}.