Tight-binding and density-functional study of the Raman tensor in two-dimensional massive Dirac fermion systems

TL;DR

The paper addresses how the Raman response of out-of-plane phonons in 2D massive Dirac systems encodes the sign of the Dirac mass and valley physics. It extends previous continuum-model predictions by performing tight-binding calculations on a honeycomb lattice with Semenoff and Haldane masses and by carrying out first-principles DFT calculations on monolayer 2H-RuCl$_2$, both of which corroborate the predicted Raman-tensor features near the Dirac gap. Key findings include the quantized phase difference $\phi_{xy}-\phi_{xx}=\pm\frac{\pi}{2}$ and a circular-polarization selection rule for extinctions in out-of-plane phonons, with no analogous rule for in-plane phonons; DFT results also reveal valley Zeeman splitting consistent with a Semenoff-insulating regime. The work provides a robust framework for interpreting Raman experiments in 2D Dirac materials and motivates tunable-frequency measurements near electronic gaps to probe valley and mass-sign effects.

Abstract

Recently, two unusual features were theoretically predicted for the Raman response of out-of-plane phonons in magnetic two-dimensional materials hosting massive Dirac fermions. First, the phase difference between certain Raman tensor elements was found to be quantized to $\pm π/2$, sensitive only to the sign of the Dirac fermion mass. Second, a selection rule was identified in the Raman intensity under circularly polarized light, which generalizes the well-known optical valley selection rule. These predictions were based on a low-energy effective model in the continuum approximation. Here, we test the robustness of those results for more realistic theoretical approaches. First, we calculate the Raman tensor for an electronic tight-binding model on a honeycomb lattice with broken time-reversal and inversion symmetries. Second, we compute the Raman tensor from density functional theory (DFT) for a monolayer of ferromagnetic 2H-RuCl$_2$. Both calculations corroborate the analytical results found in the continuum model, thereby theoretically confirming the peculiar behaviour of the Raman tensor for two dimensional massive Dirac fermion systems.

Figures (9)

• Figure 1: Electronic bands of the honeycomb lattice, with nearest-neighbor electronic hopping parameter $t_1$, Semenoff mass $M=0.3 t_1$ and second-neighbor hopping amplitude $t_2=0.01 t_1$. The Fermi level sits at the zero of energy. The inset shows the $K$ and $K^\prime$ valleys in the Brillouin zone.
• Figure 2: Raman scattering intensity for an out-of-plane phonon as a function of the incident light frequency, in a tight-binding model of a honeycomb lattice with broken inversion and time-reversal symmetries. For all panels $t_1=1$ eV, $\omega_0=0.05$ eV and $\eta=0.01$ eV. The red and blue curves correspond to right- and left-circularly polarized light, respectively (same polarization for incident and detected light, denoted as $LL$ and $RR$ configuration). The vertical dashed lines are the frequencies corresponding to the two lowest electronic interband transition energies across the energy gap of the insulator, one at valley $K$ and the other at valley $K'$ a) Semenoff insulator regime, with $t_2=0.01$ eV, $M=0.3$ eV and a valley Zeeman splitting of $\approx 200$ meV. b) Haldane/Chern insulator regime, with $t_2=0.1$ eV, $M=0.05$ eV and a valley Zeeman splitting of $\approx 200$ meV. Inset shows the zoom of the figure to signify a nonzero but small signal in the $RR$ configuration. c) Semenoff insulator with time-reversal symmetry ($t_2=0$).
• Figure 3: Phase difference between Raman tensor elements $R_{xx}^{\rm op}$ and $R_{xy}^{\rm op}$ for the out-of-plane phonon in a two-dimensional honeycomb lattice with a broken inversion and time-reversal symmetry, modelled with a tight-binding electronic structure. Different curves represent different electronic decay rate values. The vertical dashed lines are the frequencies corresponding to the two lowest electronic interband transition energies across the energy gap of the insulator, one at valley $K$ and the other at valley $K'$. Horizontal lines are guides to the eye. The phonon frequency is taken as $\omega_0=0.05$ eV. a) $t_2=0.01$ eV and $M=0.3$ eV, corresponding to the Semenoff insulator regime. b) $t_2=0.1$ eV and $M=0.3$ eV, corresponding to the Haldane/Chern insulator regime.
• Figure 4: Raman scattering intensity for the $E_2$ in-plane phonons, as a function of the incident light frequency, in a tight-binding model of a honeycomb lattice with broken inversion and time-reversal symmetries. For all panels $t_1=1$ eV, $\omega_0=0.05$ eV, $\eta=0.01$ eV and $M=0.3$ eV. The red and blue curves correspond to right- and left-circularly polarized light, respectively (opposite polarizations for incident and detected light, denoted as $LR$ and $RL$ configurations). The vertical dashed lines are the frequencies corresponding to the two lowest electronic interband transition energies across the energy gap of the insulator, one at valley $K$ and the other at valley $K'$. a) Semenoff insulator regime, with $t_2=0.01$ eV and a valley Zeeman splitting of $\approx 200$ meV. b) Haldane/Chern insulator regime, with $t_2=0.1$ eV and a valley Zeeman splitting of $\approx 1.20$ eV. c) Semenoff insulator with time-reversal symmetry ($t_2=0$).
• Figure 5: The ratio of the Raman tensor elements $R_{xx}$ and $R_{xy}$ for the in-plane phonon mode $E_2$ (red), and the out-of-plane phonon $A_1'$ (blue) in a two-dimensional honeycomb lattice with a broken inversion and time-reversal symmetry, modelled with a tight-binding electronic structure. The vertical dashed lines are the frequencies corresponding to the two lowest interband transition energies across the energy gap of the insulator, one at valley $K$ and the other at valley $K'$. Horizontal dashed lines are guides to the eye for $|R_{xx}/R_{xy}|=1$. In both panels, $\omega_0=0.05$ eV, $t_1=1$ eV, and $\eta=0.01$ eV. a) $t_2=0.01$ eV and $M=0.3$ eV, corresponding to the Semenoff insulator regime. b) $t_2=0.1$ eV and $M=0.3$ eV, corresponding to the Haldane/Chern insulator regime.