Quantum metric-based optical selection rules

Abstract

The optical selection rules dictate symmetry-allowed/forbidden transitions, playing a decisive role in engineering exciton quantum states and designing optoelectronic devices. While both the real (quantum metric) and imaginary (Berry curvature) parts of quantum geometry contribute to optical transitions, the conventional theory of optical selection rules in solids incorporates only Berry curvature. Here, we propose quantum metric-based optical selection rules. We unveil a universal quantum metric-oscillator strength correspondence for linear polarization of light and establish valley-contrasted optical selection rules that lock orthogonal linear polarizations to distinct valleys. Tight-binding and first-principles calculations confirm our theory in two models (altermagnet and Kane-Mele) and monolayer $d$-wave altermagnet $\mathrm{V_2SeSO}$. This work provides a quantum metric paradigm for valley-based spintronic and optoelectronic applications.

Figures (3)

• Figure 1: Schematics of the quantum metric-based optical selection rules for linearly polarized light (a) and the Berry curvature-based optical selection rules for circularly polarized light (b). (a) The two valleys are located at mirror-invariant lines or rotation axes of in-plane two-fold rotational symmetry and related by $M$ symmetry (e.g., mirror symmetry or four-fold rotational symmetry) with non-zero quantum metric $g^{xy}(\boldsymbol{k})=-g^{xy}(M\boldsymbol{k})$ and vanishing Berry curvature. For two orthogonal linearly polarized lights along specific directions, one (another) light exclusively excites electrons at one (the other) valley. (b) The two valleys are located at $n$-fold rotation axes ($n\geq3$) and related by $T$ symmetry (e.g., time-reversal symmetry or mirror symmetry) with non-zero Berry curvature $\Omega^{xy}(\boldsymbol{k})=-\Omega^{xy}(T\boldsymbol{k})$ and vanishing quantum metric. The left (right) circularly polarized light exclusively excites electrons at one (the other) valley.
• Figure 2: The quantum metric-based optical selection rule for linearly polarized light in the altermagnet model and Kane-Mele model. (a) Schematic of the altermagnet model (a1) and Kane-Mele model (a2). The red dots and blue dots denote the spin-up sublattice and spin-down sublattice, respectively. (b) Relevant mirror symmetries in the Brillouin zone for the altermagnet model (b1) and Kane-Mele model (b2). (c) The corresponding energy bands. The insert in (c1) shows the magnified view of the two valleys. (d) The corresponding $k$-resolved oscillator strengths and degree of linear polarization $\eta(\boldsymbol{k})$. Parameters: The photon energies are $0.4t$ for the altermagnet model and $2t$ for the Kane-Mele model, while the smearing parameters are $0.1t$ for both. $t_1=-t_2=0.1t$, $\lambda_{\mathrm{Z}}=0.6t$, and $t_{\mathrm{SOC}}=0.6t$.
• Figure 3: Valley-contrasted quantum metric-based optical selection rules and fully spin-polarized currents induced by linearly polarized light in altermagnet $\mathrm{V_2SeSO}$. (a) The crystal structure. (b) The mirror symmetries. (c) The spin-splitting band structures. (d) The $k$-resolved degree of linear polarization $\eta(\boldsymbol{k})$ for photon energy $\hbar\omega$ (shown in (c)) with a smearing parameter of 0.05 eV. The insets are $\eta(\boldsymbol{k})$ around X and Y, with the white dashed lines the mirror invariant lines. (e) Schematic of the generation of the fully spin-polarized current based on valley-contrasted quantum metric-based complete optical selection. Here only the electrons are drawn, while the holes carry the different spin and move in the opposite direction.