Photogalvanic Effects in Surface States of Topological Insulators under Perpendicular Magnetic Fields

TL;DR

The paper addresses the nonlinear magneto-optical shift response of Bi$_2$Se$_3$ surface states in a perpendicular magnetic field by deriving a microscopic second-order shift conductivity $σ^{(2);\alphaetaeta}(-ω,ω)$ for circularly polarized light. It shows that $C_3$ symmetry restricts nonzero components to $σ^{(2);+--}$ and $σ^{(2);-++}$ with $σ^{(2);-++}=[σ^{(2);+--}]^*$, and that the response arises from inter- and intraband Landau-level transitions governed by selection rules $|s_1|-|s_2|=±1,±2$. In the clean limit the spectra consist of discrete Lorentzian lines centered at Landau transition energies, while damping broadens these features; a nonzero shift current requires non-circular polarization, and Pauli blocking via the chemical potential $μ$ tunes which transitions contribute. The results reveal strong tunability of the shift current with $μ$ and $B$, and indicate potential for tunable, strong nonlinear magneto-optical devices based on TI surface states, with magnitudes potentially reaching hundreds of μA/V$^2$ under feasible conditions.

Abstract

We present a theoretical study of the nonlinear magneto-optical shift conductivity in the surface states of the prototypical topological insulator Bi$_2$Se$_3$ under a perpendicular quantizing magnetic field. By describing the electronic states as Landau levels and using a perturbative approach, we derive the microscopic expression for the shift conductivity $σ^{(2);αβγ}(-ω,ω)$, where $α,β,γ=\pm$ stand for the circular polarization of light and $ω$ is the light frequency; the spectra are further decomposed into contributions from the interband and intraband optical transitions, for which the selection rules are identified. Considering that the system possesses $C_3$ point group of symmetry, the nonzero components of the conductivity tensor are $σ^{(2);-++}=[σ^{(2);+--}]^\ast$. Therefore, a pure circularly polarized light generates zero shift current. In the clean limit, the conductivities are nonzero only for discrete photon energies because of the discrete Landau levels and energy conservation, and they become Lorentzian lineshapes with the inclusion of damping, which relaxes the condition of energy conservation. The dependence of the spectra on the damping parameters, the magnetic fields, and the chemical potentials is investigated in detail. Our results reveal that the shift current is highly tunable by the chemical potential and the magnetic field. These results underscore the potential of topological insulators for tunable, strong nonlinear magneto-optical applications.

Figures (5)

• Figure 1: (a) Illustration of the selection rules for Berry connection $\xi^+_{s_1 s_2}$ with $|s_1|-|s_2|=3l+1$, where thick (thin) arrows correspond $l=0$ ($l=\pm1$) with large (small) values. See details in Ref. PhysRevB.111.205408. (b) Illustration of the selection rules for coefficient $S_{s_1s_2}$ in shift conductivity with $|s_1|-|s_2|=1, -2$, which is defined in Eq.\ref{['eq:S']}. Note, both in (a) and (b), the blue and red arrows represent interband and intraband transitions, respectively. (c) All nonzero $S_{s_1 s_2}$ versus their respective discrete transition energies $|\hbar\omega_{s_1s_2}|$, where the Landau energies are calculated at $B=5$ T. $\sigma_0 = {e^2}/({4 \hbar})$.
• Figure 2: (a) The spectra of $\sigma_r(\omega)$ and the values of ${\cal A}_{s_1s_2}$ with $|s_1|-|s_2|=\pm1$ (star symbols) and with $|s_1|-|s_2|=\pm2$ (square symbols). (b) The spectra of $\sigma_i(\omega)$ and the values of ${\cal B}_{s_1s_2}$ with $|s_1|-|s_2|=\pm1$ (star symbols) and with $|s_1|-|s_2|=\pm2$ (square symbols).
• Figure 3: Spectra of (a) $\sigma_{r}(\omega)$ and (b) $\sigma_{i}(\omega)$ for different relaxation parameters $\Gamma$.
• Figure 4: Shift conductivities with different $\mu_s$. (a) displays the spectra of $\sigma_{r}(\omega)$ for $\mu_s=0, \pm1, \pm2$. (b, c) show the spectra of $\sigma_{r}(\omega)$ and $\sigma_{i}(\omega)$, respectively, where the black curve denotes $\mu_0$, and other curves above (below) it correspond to increasing (decreasing) values of $\mu_s$. Illustration of the selection rules for the coefficient $S_{s_1s_2}$ when the highest occupied Landau level is (d) $0$ and (e) $s$ with $s > 0$, corresponding to the filling factors $\mu_0$ and $\mu_s$, respectively. Arrows of the same color denote a pair of degenerate transitions (see Table \ref{['tab:1']}), while crosses mark forbidden transitions.
• Figure 5: The spectra of (a) $\sigma_r(\omega)$ and (b) $\sigma_i(\omega)$ as functions of $\hbar\omega/\hbar\omega_c$ for $\hbar\omega<300$ meV at $\mu=0$ eV and different magnetic field $B=0.5$, $2$, and $5$ T. The spectra of (c) $B^{-1}\sigma_{r}(\omega)$ and (d) $\sigma_{i}(\omega)$ with different chemical potentials $\mu=0$, $20$, and $40$ meV for $B = 0.05$ T.