Almost half-quantized planar Hall effects in $X$-wave magnets with $X=p,d,f,g,i$

TL;DR

This work analyzes the planar Hall effect in two-dimensional metals interfaced with higher symmetric X-wave magnets, where the magnetization is encoded by a two-band Hamiltonian with a symmetry function $f_X(\mathbf{k})$ and node count $N_X$. By combining continuum Berry-curvature arguments with tight-binding simulations across p, d, f, g, and i wave families, it shows that the planar Hall conductivity is almost half quantized and given by $\sigma_{xy} \approx \pm \frac{e^2}{2h}\mathrm{sgn}(J\sin N_X\Phi)$ when the Dirac gap is tiny, with the gap scaling as $\Delta \propto B^{N_X}$ and the sign controlled by $J$. The Hall response is periodic in the in-plane field angle $\Phi$ with period $N_X$, enabling identification of the underlying X-wave magnet; the sign of $J$ also provides a potential Ising-like spintronic bit. The study further shows robustness of the half-quantized feature to finite temperature and weak disorder, and discusses tight-binding realizations and material candidates that could realize these effects in experiments. Overall, the work links nodal structure, Berry curvature localization, and planar Hall response to provide a diagnostic and functional framework for X-wave magnets in spintronic contexts.

Abstract

The planar Hall effect is a phenomenon that the Hall conductivity emerges perpendicular to the electric field in the presence of an in-plane magnetic field. We investigate the planar Hall effect in two-dimensional metal coupled with higher symmetric $X$-wave magnets with $X=p,d,f,g,i$,\ where those with $X=d,g,i$ are altermagnets. The $X$-wave magnet is characterized by the number $N_{X}$ of the nodes in the band structure, where $N_{X}=1,2,3,4,6$ corresponding to $X=p,d,f,g,i$. Although the system is metallic, provided the Dirac gap is tiny, we demonstrate that the Hall conductivities are almost half quantized and well approximated by the formula $σ_{xy}=\pm (e^{2}/2h)$ sgn$\left( J\sin N_{X}Φ\right) $, where $J$ is the coefficient of the coupling between the $X$-wave magnet and the electrons, and $Φ$ is the direction of the applied magnetic field. Hence, the Hall conductivity is periodic in $Φ$, and the periodicity is equal to the number $N_{X}$ of the nodes. This property may be used to confirm that the target material is indeed an $X $-wave magnet. Furthermore, the sign of $J$ may be used as a bit for antiferromagnetic spintronics.

Figures (9)

• Figure 1: Illustration for (a) Hall effect and (b) planar Hall effect. In the planar Hall effect, when the electric field is applied along the $x$ axis and the magnetic field $(B\cos\Phi,B\sin\Phi ,0)$ is applied parallel to the system, the Hall current flows along the $y$ axis. The Hall conductivity is predicted to be given by the formula (\ref{['pHall']}).
• Figure 2: $p$-wave magnet. (a) Contour plot of the energy of the bottom band. (b) Berry curvature for the bottom band. (a',b') Color palette for (a,b). We have chosen $\Phi =\pi /2$. We have set $Jak_{0}=\varepsilon _{0}/2$, $\lambda k_{0}=\varepsilon _{0}$ and $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$.
• Figure 3: $p$-wave magnet. (a) Hall conductivity in units of $e^{2}/h$ as a function of chemical potential $\mu$. (b) Energy spectrum for $-\pi <ak_{x}<\pi$ and $k_{y}=0.$ (c) Hall conductivity as a function of angle $\Phi$. In these figures, red curves indicate $B=0.1\varepsilon _{0}$, blue curves indicate $B=0.3\varepsilon _{0}$, and green curves indicate $B=0.5\varepsilon _{0}$. (d) Hall conductivity as a function of magnetic field $B$. A magenta curve indicates $\Phi =\pi /2$ and a cyan curve indicates $\Phi =3\pi /2$. We have chosen $\Phi =\pi /2$ in (a), (b), (d). See also the caption of Fig.\ref{['FigPBerry']}.
• Figure 4: $d$-wave altermagnet. (a) Contour plot of the energy of the bottom band. (b) Berry curvature for the bottom band. (a',b') Color palette for (a,b). We have chosen $\Phi =\pi /4$. We have set $2Ja^{2}k_{0}^{2}=\varepsilon _{0}/2$, $\lambda k_{0}=\varepsilon _{0}$ and $\hbar ^{2}k_{0}^{2}/\left( 2m\right) =\varepsilon _{0}/2$.
• Figure 5: $d$-wave altermagnet. (a) Hall conductivity in units of $e^{2}/h$ as a function of chemical potential $\mu$. (b) Energy spectrum for $\left( k_{x},k_{y}\right) =k\left( \cos \frac{\pi }{4},\sin \frac{\pi }{4}\right)$ with $-\pi <ak<\pi$. (c) Hall conductivity as a function of angle $\Phi$. (d) Hall conductivity as a function of magnetic field $B$. A magenta curve indicates $\Phi =\pi /4$ and a cyan curve indicates $\Phi =5\pi /4$. We have chosen $\phi =\pi /4$ in (a), (b), (d). We have set $2Ja^{2}k_{0}^{2}=\varepsilon _{0}/2$. See also the caption of Fig.\ref{['FigDBerry']}.