Light-induced half-quantized Hall effect and axion insulator

TL;DR

This work addresses realizing half-quantized Hall, axion insulator, and Chern insulator phases in 3D topological insulators by combining surface-selective TR breaking through circularly polarized Floquet driving with bottom-layer magnetic doping. Using a high-frequency Magnus expansion, the authors derive an effective Floquet Hamiltonian for a realistic TI heterostructure and show that the gap openings on the top and/or bottom surfaces yield half-quantized contributions $\pm\frac{e^2}{4\pi\hbar}$ per Dirac cone, which can be added coherently to produce zero (axion), one (Chern), or half-integer (semi-Floquet) Hall plateaus depending on polarization and penetration depth. They substantiate these phases with band-structure and Berry-curvature calculations, including real-space surface-state localization and $k$-space curvature distributions, demonstrating tunable phase control via light polarization and/or magnetic doping. The work also outlines an all-optical route that drives both surfaces, enabling axion/Chern phases without doping, and discusses Floquet quench protocols that switch between phases, highlighting potential polarization-controlled topological transistors and other devices with practical experimental feasibility.

Abstract

Motivated by the recent experimental realization of the half-quantized Hall effect phase in a three-dimensional (3D) semi-magnetic topological insulator [M. Mogi et al., Nature Physics 18, 390 (2022)], we propose a scheme for realizing the half-quantized Hall effect and axion insulator in experimentally mature 3D topological insulator heterostructures. Our approach involves optically pumping and/or magnetically doping the topological insulator surface, such as to break time reversal and gap out the Dirac cones. By toggling between left and right circularly polarized optical pumping, the sign of the half-integer Hall conductance from each of the surface Dirac cones can be controlled, such as to yield half-quantized ($0+1/2$), axion ($-1/2+1/2=0$), and Chern ($1/2+1/2=1$) insulator phases. We substantiate our results based on detailed band structure and Berry curvature numerics on the Floquet Hamiltonian in the high-frequency limit. Our paper showcases how topological phases can be obtained through mature experimental approaches such as magnetic layer doping and circularly polarized laser pumping and opens up potential device applications such as a polarization chirality-controlled topological transistor.

Figures (11)

• Figure 1: (Color online) Schematic of various topological phases that result when a 3D topological insulator (Bi,Sb)$_{2}$Te$_{3}$ or Bi$_2$Se$_3$ is optically driven from above, and/or doped with magnetic Cr in its bottom layers. (a) Ordinary 3D $\mathbb{Z}_2$ topological insulator without Cr doping and optical pumping, with two opposite gapless surface Dirac cones. (b) When irradiated with light that penetrates only the top layer (orange), the top surface Dirac cone becomes gapped due to time-reversal breaking, resulting in a semi-Floquet topological insulator (Floquet-induced half-quantized Hall effect). (c) Floquet-induced Chern phase, where the radiation penetrates both layers (orange) and gaps out both Dirac cones. (d) When optical driving is absent, but Cr doping (blue) breaks time reversal in the bottom layers, gapping out its Dirac cone, a semi-magnetic topological insulator (magnetic doping-induced half-quantized Hall effect) results. [(e),(f)] When there is optical driving upon one surface and Cr-doping in the other, both surfaces have Dirac cones gapped out, but their chiralities can be controlled. Depending on whether left-handed (e) or right-handed (f) circularly polarized light (purple circular arrow) is used, we obtain either (e) an axion insulator with zero net Chern number or (f) an effective Chern phase with nonzero Hall conductivity.
• Figure 2: (Color online) Floquet band structures and their corresponding Hall conductivity for the Floquet optical-driven Hamiltonian \ref{['eq:H_F']} of Bi$_2$Se$_3$ without Cr doping. [(a1),(b1)] Floquet band structures under open boundary conditions along the $z$ direction and periodic boundary conditions along the $x$ and $y$ directions. In (a1), $A_{0}=0.8$ nm$^{-1}$ and the driving radiation only gaps out the top surface Dirac cone (red). In (b1), $A_{0}=2.5$ nm$^{-1}$ and the driving radiation penetrates both surfaces, gapping both the top and bottom surface Dirac cones (red and blue). Here, $E_{t\pm}$ and $E_{b\pm}$ are the energies of the top and bottom surface bands; the subscripts "$\pm$" respectively denote the lowest conduction band or highest valence band. The green and yellow shaded intervals indicate the band widths of the top and bottom surfaces, respectively. [(a2),(b2)] Hall conductance as a function of the Fermi energy $E_{F}$, corresponding to the light intensities in (a1) $A_{0}=0.8$ nm$^{-1}$ and (b1) $A_{0}=2.5$ nm$^{-1}$. They exhibit half-integer (semi-Floquet) and integer-quantized (Chern-Floquet) Hall conductivity in the gap, respectively. The other parameters are $k_y=0$, sample thickness $L_z=30$ nm, $a_{z}=0.5$ nm, $a=1$ nm, $t_{z}=0.40$ eV, $t_{||}=0.566$ eV, $\lambda_{z}=0.44$ eV, $\lambda_{||}=0.41$ eV, $m_{0}=0.28$ eV, $\delta=16.3$ nm, $\hbar\omega=3.82$ eV, and $\varphi=-\pi/2$.
• Figure 3: (Color online) Band and Floquet band structures and their corresponding Hall conductivity for the Hamiltonian of Bi$_2$Se$_3$ with Cr doping. [(a1)-(c1)] Energy bands for the tight-binding Hamiltonian of the Bi$_2$Se$_3$ under open boundary conditions along the $z$ direction and periodic boundary conditions along the $x$ and $y$ directions. In (a1), $V_z=0.1$ eV, $A_{0}=0$ nm$^{-1}$, and the magnetic doping gaps out the bottom surface Dirac cone (blue). In (b1), $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, $\varphi=-\pi/2$, and the driving radiation penetrates the top surface, gapping the top surface Dirac cone (red). The magnetic doping gaps out the bottom surface Dirac cone (blue). In (c1), $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, $\varphi=\pi/2$, and the driving radiation penetrates the top surface, gapping the top surface Dirac cone (red). The magnetic doping gaps out the bottom surface Dirac cone (blue). Here, $E_{t\pm}$ and $E_{b\pm}$ are the energies of the top and bottom surface bands, subscripts "$\pm$" respectively denote the lowest conduction band or highest valence band. The green and yellow shaded intervals indicate the band widths of the top and bottom surfaces, respectively. [(a2)-(c2)] Hall conductance as a function of the Fermi energy $E_{F}$, corresponding to the parameters in (a1) $V_z=0.1$ eV, $A_{0}=0$ nm$^{-1}$; (b1) $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, $\varphi=-\pi/2$; and (c1) $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, $\varphi=\pi/2$. They exhibit half-integer (semi-magnetic), zero (axion), and integer-quantized (Chern) Hall conductivity in the gap, respectively. The thickness of the Cr-doped layer is $d=1$ nm from the bottom surface, and the laser penetration depth is $\delta=16.3$ nm under $\hbar\omega=3.82$ eV humlivcek2014raman accompanying the light incoming from the top surface. The black dashed line in (b2) is the combination of the Hall conductances of Fig. \ref{['Fig:E_C_Vz0_A_Lz30_delta163_together']}(a2) and Fig. \ref{['Fig:E_C_Lz30_delta163_inverse_together']}(a2). The other parameters are the same as those in Fig. \ref{['Fig:E_C_Vz0_A_Lz30_delta163_together']}.
• Figure 4: (Color online) Spatial state distribution $|\psi(z)|^{2}$ along the vertical direction $z$ for the lowest conduction (subscript "+'') and highest valence (subscript "-'') bands of the top surface state ($|\psi_{t\pm}|^{2}$ -- red lines) and bottom surface state ($|\psi_{b\pm}|^{2}$ -- blue lines) at $k_{x}=k_{y}=0$. Indeed, the supposed top and bottom surface states are concentrated near the top ($z=0.5$ nm) and bottom $z=30$ nm boundaries. (a) Semi-Floquet topological insulator with $V_z=0$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=-\pi/2$. (b) Semi-magnetic topological insulator with $V_z=0.1$ eV and $A_{0}=0$ nm$^{-1}$. (c) Axion insulator with $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=-\pi/2$. (d) Chern topological insulator with $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=\pi/2$. The other parameters are the same as those in Fig. \ref{['Fig:E_C_Lz30_delta163_inverse_together']}.
• Figure 5: (Color online) Berry curvature distributions for the lowest conduction ("+'') and highest valence bands ("-'') of the top (t) and bottom (b) surfaces, for the four scenarios from Fig. \ref{['Fig:P_Lz30_delta163_together']}. [(a1)-(d1)] Semi-Floquet topological insulator with $V_z=0$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=-\pi/2$. [(a2)-(d2)] Semi-magnetic topological insulator with $V_z=0.1$ eV and $A_{0}=0$ nm$^{-1}$. [(a3)-(d3)] Axion insulator with $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=-\pi/2$. [(a4)-(d4)] Chern topological insulator with $V_z=0.1$ eV, $A_{0}=0.8$ nm$^{-1}$, and $\varphi=\pi/2$, which is equal but opposite to that of (c) due to the reversed polarization of the optical driving. We see Berry curvature peaks around $k_x=k_y=0$ when gapped Dirac cones exist, and another ring of peaks at larger $k_x,k_y$ when the bands merge into the bulk. The other parameters are the same as those in Fig. \ref{['Fig:E_C_Lz30_delta163_inverse_together']}.