Electric field induced Berry curvature dipole and non-linear anomalous Hall effects in higher wave symmetric unconventional magnets

Abstract

We investigate the second-order anomalous Hall response in two-dimensional higher-wave symmetric magnets, including the recently discovered class of collinear magnets known as altermagnets, when subjected to a symmetry-breaking external electric field. In these systems, the first- and second-order anomalous Hall responses mediated by the first- and second-order multipoles of the Berry curvature over the occupied states vanish by symmetry. However, a symmetry-breaking dc electric field can induce a nonzero Berry curvature dipole by coupling to a non-vanishing quantum metric, also known as the Berry connection polarizability. An applied ac electric field can then generate a finite nonlinear transverse Hall effect characterized by a second harmonic response. In addition, the dc field itself can generate a finite third-order transverse Hall response. We discuss this remarkable effect in a class of higher-order symmetric unconventional magnets (of $p$, $d$, $f$, $g$, $i$ symmetry), including the subclass of altermagnets. We demonstrate that the electric-field-induced anomalous Hall effect in the higher-wave-symmetric magnets can serve not only as a probe of the underlying quantum metric of the occupied states but also as a means to distinguish the even ($d$-,$g$-wave) and odd ($p$-wave) order parameter symmetries defined on the square lattice.

Figures (3)

• Figure 1: BCD induced by a dc electric field for a $p$-wave magnet (Eq. (\ref{['eq:hamiltonian']}) with form factor Eq. (\ref{['eq:p-wave']})) with model parameters $t=1$ eV, $\lambda=0.1\,t$, $\Delta_{p}=0.5\,t$, $E^{\rm{dc}}=3$ kV/m, scattering time $\tau=10^{-12}~s$, and lattice constant $a= 4~\text{\AA}$. (a) The first-order correction to Berry curvature $\Omega^{(1)}$ in the presence of $\bm{E}^{\rm{dc}}$ along the $x$-direction, i.e. $\phi=0$, where the induced BCD is perpendicular to the dc field. The dipolar nature of the distribution is evident with mirrored positive and negative regions. BCD distributions (b) $\partial_{y}\Omega^{(1)}$ with $\bm{E}^{\rm{dc}}$ along $\phi = 0$, and (c) $\partial_{x}\Omega^{(1)}$ with $\bm{E}^{\rm{dc}}$ along $\phi = \pi/2$, illustrating that the two cases are not equivalent. This accounts for the anisotropy in the magnitudes of $\tilde{\mathcal{D}}_{x}$ and $\tilde{\mathcal{D}}_{y}$. (d) Unequal amplitudes of $\boldsymbol{\mathcal{D}}^{(1)}$ (see Eq. (\ref{['eq:BCDamp']})) along $x$ and $y$ directions (in the log scale) as a function of the chemical potential varied near the band-touching point at $\mu=0$. (e) Polar plot of the field-induced BCD $\boldsymbol{\mathcal{D}}^{(1)}$ showing directional anisotropy. The direction of the symmetry-reducing electric field ${\bm{E}^{\rm{dc}}}$ is indicated by orange arrows, and the induced $\boldsymbol{\mathcal{D}}^{(1)}$ is understood to be the vector starting at the origin and ending at the base of the corresponding arrow. For the $p$-wave system under consideration, $\boldsymbol{\mathcal{D}}^{(1)}$ is only perpendicular to $\bm{E}^{\rm{dc}}$ when the applied dc field is along the $x$- or $y$-directions. As ${\bm{E}^{\rm{dc}}}$ rotates clockwise, $\boldsymbol{\mathcal{D}}^{(1)}$ rotates in the same sense for the chemical potential chosen. (f) The second-order Hall conductivity $\chi^{2\omega}$ (Eq. (\ref{['eq:secondOrderAC']})) versus the angle $\theta$ (w.r.t. the $x$-direction) of the probing field $\bm{E}^{\omega}$ for various $\bm{E}^{\rm{dc}}$ orientations $\phi$, with the chemical potential set to $\mu=0.5$ meV. The vertical dashed lines mark the expected angular positions of the maximal values of $\chi^{(2\omega)}$ for each $\phi$ (denoted by the corresponding color) in an isotropic system. The directional anisotropy of $\boldsymbol{\mathcal{D}}^{(1)}$ is manifest in $\chi^{(2\omega)}$. The units for $\Omega^{(1)}$ are $\text{\AA}^2$, those of $\mathcal{D}^{(1)}$ are $nm$, $\chi^{2\omega}$ has units of $V^{-1}Snm$.
• Figure 2: Third-order dc anomalous Hall conductivity $\chi^{(\rm{dc})}$ (see Eq. (\ref{['eq:thirdOrderDC']})) originating from field-induced BCD for a Hamiltonian (Eq. (\ref{['eq:hamiltonian']}), showing a $\sin{2\phi}$ dependence for a $p$-wave form factor (Eq. (\ref{['eq:p-wave']})), and vanishing for a $d$-wave form factor (Eq. (\ref{['eq:d-wave']})). Units for $\chi^{(3)}$ are $V^{-2}Sµm^{2}$. The model parameters are the same as the ones mentioned in the caption of Fig. \ref{['fig:p_wave']}.
• Figure 3: BCD induced by dc electric field for a $d$-wave altermagnet for which the form factor is Eq. (\ref{['eq:d-wave']}) with model parameters $t=1$ eV, $\lambda=0.1\,t$, $\Delta_{p}=0.5\,t$, $E^{\rm{dc}}=3$ kV/m, scattering time $\tau=10^{-12}~s$ and lattice constant $a= 4~\text{\AA}$. (a) The first-order correction to Berry curvature $\Omega^{(1)}$ in the presence of $\bm{E}^{\rm{dc}}$ along the $x$-direction, i.e., $\phi=0$, where the induced BCD is perpendicular to the dc field. The dipolar nature of the distribution is evident with mirrored positive and negative regions. BCD distributions (b) $\partial_{y}\Omega^{(1)}$ with $\bm{E}^{\rm{dc}}$ along $\phi = 0$, and (c) $\partial_{x}\Omega^{(1)}$ with $\bm{E}^{\rm{dc}}$ along $\phi = \pi/2$, illustrating that the two cases are equivalent. This explains the isotropy in the amplitudes of of $\mathcal{D}^{(1)}_{x}$ and $\mathcal{D}^{(1)}_{y}$. (d) Equal amplitudes of $\boldsymbol{\mathcal{D}}^{(1)}$ (see Eq. \ref{['eq:BCDamp']}) along $x$ and $y$ directions (in the log scale) as a function of the chemical potential varied near the band-touching point at $\mu=0$. (e) Polar plot of the field-induced BCD $\boldsymbol{\mathcal{D}}^{(1)}$ showing directional isotropy. The direction of the symmetry-reducing electric field ${\bm{E}^{\rm{dc}}}$ is indicated by orange arrows, and the induced $\boldsymbol{\mathcal{D}}^{(1)}$ is understood to be the vector starting at the origin and ending at the base of the corresponding arrow. For the $d$-wave system under consideration, $\boldsymbol{\mathcal{D}}^{(1)}$ remains perpendicular to $\bm{E}^{\rm{dc}}$ as the field is rotated in the plane. As ${\bm{E}^{\rm{dc}}}$ rotates clockwise, $\boldsymbol{\mathcal{D}}^{(1)}$ rotates in the same sense for any chemical potential. (f) The second-order Hall conductivity $\chi^{2\omega}$ (Eq. (\ref{['eq:secondOrderAC']})) versus the angle $\theta$ (w.r.t. the $x$-direction) of the probing field $\bm{E}^{\omega}$ for various $\bm{E}^{\rm{dc}}$ orientations $\phi$, with the chemical potential set to $\mu=0.5$ meV. The vertical dashed lines mark the expected angular positions of the maximum value of $\chi^{(2\omega)}$ for each $\phi$ (denoted by the corresponding color) in an isotropic system. The directional isotropy of the BCD is manifest in $\chi^{(2\omega)}$. The units for $\Omega^{(1)}$ are $\text{\AA}^2$, those of $\mathcal{D}^{(1)}$ are $nm$, $\chi^{2\omega}$ has units of $V^{-1}Snm$.