Electric and spin current vortices in altermagnets

Abstract

Altermagnets constitute a class of collinear magnets with momentum-dependent spin splitting and vanishing net magnetization. Direct observation of the characteristic altermagnetic spin splitting, however, remains challenging. Indirect signatures can be obtained via transport studies, which so far have only considered homogeneous driving fields. We propose to leverage nonuniform electric fields and spin density gradients to probe the shape and the spin polarization of altermagnetic Fermi surfaces via transport measurements. By using both a semiclassical Boltzmann approach and a lattice Keldysh formalism, we show that altermagnets excite swirling electric and spin currents whose profiles depend on the relative orientation of altermagnetic lobes with respect to the sample boundaries. These currents can be measured via magnetometry techniques. Unlike previous proposals considering the hydrodynamic regime of transport, swirling currents are observed even in the Ohmic regime and rely exclusively on the altermagnetic spin splitting, with no swirls observed in ferromagnets. The electric and spin current vortices predicted here provide a different altermagnetic signature in an experimentally accessible setup.

Figures (4)

• Figure 1: Schematic setup of a magnet with point-like contacts. The streamlines illustrate the flow of electric (spin) currents induced by the applied spin density (electric potential) difference at the contacts. The upper half corresponds to a $d$-wave altermagnet and the lower half represents a ferromagnet; the corresponding Fermi surfaces with spin-up and spin-down polarized bands are shown in red and blue. Despite the system being diffusive and non-interacting, current vortices arise in the altermagnetic case, unlike the ferromagnetic one.
• Figure 2: (a) Electric current $j_{\rm el} = j_{\uparrow}+j_{\downarrow}$ distribution without altermagnetism, $t_1=0$ and $t_2=0$. There are no qualitative changes in the shape of the electric current streamlines at nonzero $t_1$ or $t_2$. Spin current $j_{\rm sp} = j_{\uparrow}-j_{\downarrow}$ distribution in altermagnets for (b) $t_1=t_0/2$ and $t_2=0$, (c) $t_1=0$ and $t_2=t_0/2$, and (d) $t_1=t_2=t_0/(2\sqrt{2})$; the configuration of altermagnetic Fermi surfaces is depicted below each of the panels. In all panels, we fix the electric potentials at the contacts as $e\bar{\phi}_{1,\lambda}/t_0 = -1$ and $e\bar{\phi}_{2,\lambda}/t_0 = 1$. The currents are normalized by $j_{0} = 2e\tau t_0 \, \hbox{max}{\left\{\left|\rho_{\lambda}^{(0)} \left(\bar{\phi}_{2,\lambda}-\bar{\phi}_{1,\lambda}\right) \right|\right\}}/L$, where the maximal value is taken with respect to the spin projections, and we fixed $\rho^{(0)}_{\uparrow}=\rho^{(0)}_{\downarrow}=\rho^{(0)}/2$. Red lines denote the position of the source and drain.
• Figure 3: The spatial dependence of the induced magnetic field $B_z(\mathbf{r},z)$ for $z=0.1\,L$ at (a) $t_1=t_0/2$ and $t_2=0$, (b) $t_1=0$ and $t_2=t_0/2$, and (c) $t_1=t_2=t_0/(2\sqrt{2})$. In all panels, we fixed the potentials at the contacts as $e\bar{\phi}_{1,\lambda}/t_0 = -\lambda$ and $e\bar{\phi}_{2,\lambda}/t_0 = \lambda$, i.e., only the spin imbalance is applied, and we fixed $\rho^{(0)}_{\uparrow}=\rho^{(0)}_{\downarrow}=\rho^{(0)}/2$.
• Figure 4: Spin current distribution obtained via the Keldysh formalism. The contacts of the width $w=10\,a$ are at $x=0,L_x$ surfaces. We use $60\times 60$ lattice with the following parameters: $t_m = 0.3\,t$, $\mu = -2.0\,t$, and the voltage bias $eV = 0.2\,t$.