Symmetry Classification for Alternating Excitons in Two-Dimensional Altermagnets

Abstract

Excitons, bound electron-holes states, often dominate the optical response of two-dimensional (2D) materials and reflect their inherent properties, including spin-orbit coupling, magnetic ordering, or band topology. By focusing on a growing class of collinear antiferromagnets with a nonrelativistic spin splitting, referred to also as altermagnets (AM), we propose a theoretical framework based on the spin space group (SSG) to elucidate their resulting excitons. Our approach is illustrated on 2D AM with spin-polarized valleys, where we classify the combination of conduction and valence bands by the SSG representations into two cases that hosts bright $s$-like and $p$-like excitons, respectively. This analysis is further supported by effective Hamiltonians and the Bethe-Salpeter equation. We identify the excitonic optical selection rules from the calculated absorption spectra and the symmetry of bright excitons from their momentum space envelope functions. Our framework provides optical fingerprints for various cases of AM, while their tunability, such as the strain-induced valley splitting, is also transferred to excitons allowing, additionally, valley-polarized photocurrent generation.

Figures (3)

• Figure 1: (a) Schematic of the band structure of a $d$-wave 2DAM with spin-polarized $X$ (blue) and $Y$ (red) valleys, connected by the symmetry operation $\{U_2||C_{4z}\}$. (b), (c) Absorption spectra of $x$ and $y$ polarized light, with $\Omega_{X,Y}^{1s}$ the excitonic energies in the valleys $X$, $Y$, for the Case 1 (see text). (d), (e) The corresponding momentum-space wave functions, $\mathcal{A}^{1s}_{X,Y,\bm{k}}$, of $1s$-like excitons in the $X$, $Y$ valleys, with $a_0$ the lattice constant and $(k_x,k_y)$ measured from $X$, $Y$.
• Figure 2: Case 2, with the $X$ (blue) and $Y$ (red) valleys. (a) Schematic of the excitonic optical selection rules. (b), (c) Absorption spectra of $x$ and $y$ polarized light, with a single-particle band gap $2\Delta=0.82\;$eV. (d)-(g) The velocity matrix elements $|\mathcal{D}^{x/y}_{\tau,\bm{k}}|$. (h)-(k) The exciton wave function, with $2p_{y/x}$-like exciton in the $X/Y$ valley and $2p_{y/x}$-like exciton in the $Y/X$ valley corresponds to the $x/y$ absorption of the first and second peak as in (b), (c). In (d)-(g), ($k_x$, $k_y$) is measured from the corresponding $X$ or $Y$ valley.
• Figure 3: (a) Schematic of the band edge motion under strain. (b), (c) Evolution of the absorption spectra under strain. Splitting of the peaks with different polarization reflects the broken valley degeneracy. (d)-(f) Anisotropic valley-polarized photocurrent with optical excitation and electric field.