Fourth-order and six-order nonlinear spin current diode in $h$-wave and $j$-wave odd-parity magnets

Abstract

Higher-order symmetric $X$-wave magnets consist of two groups. One includes $d$-wave, $g$-wave and $i$-wave altermagnets, while the other includes $p$-wave and $f$-wave odd-parity magnets. Recently, the possibility of $h$-wave magnets has been discussed. Motivated by this development, we systematically construct an $X$-wave magnet with $\left( N_{X}+1\right) $ nodes in three dimensions from an $X$-wave magnet with $N_{X}$ nodes in two dimensions by means of a dimensional extension, where $N_X=1,2,3,4,6$ for $X=p,d,f,g,i$, respectively. Based on this method, we predict $j$-wave magnets in three dimensions. Then, we argue how to identify each of these $X$-wave magnets experimentally. We show that the $X$-wave magnet is completely identified by measuring the nonlinear spin currents. In particular, we predict that there are no spin currents other than the fourth-order ones such as $σ_{\text{spin}}^{x^{3}y;z}$ in $h$-wave odd-parity magnets in three dimensions and the sixth-order ones such as $σ_{\text{spin}}^{x^{5}y;z}$ in $j$-wave odd-parity magnets in three dimensions. They function as spin-current diodes because the spin current exhibits unidirectional flow independent of the applied electric field.

Figures (4)

• Figure 1: Illustration of the Fermi surface. (a1) $h$-wave magnets in 2D. (b1) $j$-wave magnets in 2D. (a2) $h$-wave magnets in 3D. (b2) $j$-wave magnets in 3D. (a3) $g$-wave altermagnet in 3D. (b3) $i$-wave altermagnet in 2D. There is no lattice realization of $h$-wave ($j$-wave) magnets in 2D due to the five-fold (seven-fold) rotational symmetry. The $h$-wave ($j$-wave) magnet in 3D is constructed as a dimensional extension of the $g$-wave ($i$-wave) altermagnet in 2D.
• Figure 2: Fermi volume as a function of $J$. (a) $h$-wave magnets in 2D. (b) $j$-wave magnets in 3D. The vertical axis is the Fermi volume in units of $8\sqrt{2}\pi (m\mu )^{3/2}/(3\hbar ^{3})$. The horizontal axis is $J$ in units of $\varepsilon _{0}$. Cyan curves represent numerical results based on the tight-binding model. Red lines represent the analytic reult $V_{0}$ based on the continuum model. Purple curves represent analytic results based on the continuum model up to the second order in $J.$We have set $\hbar ^{2}/\left( ma^{2}\right) =\varepsilon _{0}/4$ and $\mu =0.1\varepsilon _{0}$.
• Figure 3: Spin conductivity as a function of $J$. (a) $h$-wave magnets in 2D. We have set $\mu =0.05\varepsilon _{0}$. (b) $j$-wave magnets in 3D. The vertical axis is $\sigma _{\text{spin}}$ in units of $e^{N_{X}}\tau ^{N_{X}-1}\varepsilon _{0}/\left( \hbar ^{N_{X}}k_{0}\right)$. The horizontal axis is $J$ in units of $\varepsilon _{0}$. We have set $\hbar ^{2}/\left( ma^{2}\right) =\varepsilon _{0}/4$ and $\mu =0.01\varepsilon _{0}.$
• Figure 4: Spin conductivity as a function of $\mu$. (a) $h$-wave magnets in 2D. (b) $j$-wave magnets in 3D. The vertical axis is $\sigma _{\text{spin}}$ in units of $e^{N_{X}}\tau ^{N_{X}-1}\varepsilon _{0}/\left( \hbar ^{N_{X}}k_{0}\right)$. The horizontal axis is $\mu$ in units of $\varepsilon _{0}$. We have set $\hbar ^{2}/\left( ma^{2}\right) =\varepsilon _{0}/4$ and $J=0.1\varepsilon _{0}$.