Layer Hall effect induced by altermagnetism

TL;DR

This work shows that Bi$_2$Se$_3$, when placed in proximity to $d$-wave altermagnets and subjected to an in-plane layer magnetic field, can realize distinct topological Hall responses via surface Dirac gaps. By controlling Néel-vector alignment (antiparallel vs parallel) on the top and bottom surfaces, the system hosts a layered sequence of phases: a half-quantized layer Hall effect with zero net Hall for antiparallel altermagnets, and a fully quantized anomalous Hall (Chern insulator) state for parallel alignment, with intermediate single-surface half-quantized cases. The Hall response is tunable by the in-plane field orientation and can be made observable with a perpendicular electric field that breaks exact surface cancellation, enabling experimental detection of the layer Hall effect. Collectively, the results establish a versatile route to engineer altermagnet-induced topological phases in ferromagnetic topological insulators and to realize layer-resolved topological transport phenomena.

Abstract

In this work, we propose a scheme to realize the layer Hall effect in the ferromagnetic topological insulator Bi$_2$Se$_3$ via proximity to $d$-wave altermagnets. We show that an altermagnet and an in-plane magnetic field applied near one surface gap the corresponding Dirac cone, yielding an altermagnet-induced half-quantized Hall effect. When altermagnets with antiparallel Néel vectors are placed near the top and bottom surfaces, giving rise to the layer Hall effect with vanishing net Hall conductance, i.e., the altermagnet-induced layer Hall effect. In contrast, altermagnets with parallel Néel vectors lead to a quantized Chern insulating state, i.e., the altermagnet-induced anomalous Hall effect. We further analyze the dependence of the Hall conductance on the orientation of the in-plane magnetic field and demonstrate that the layer Hall effect becomes observable under a perpendicular electric field. Our results establish a route to engineer altermagnet-induced topological phases in ferromagnetic topological insulators.

Figures (5)

• Figure 1: Schematic of topological phases in a three-dimensional topological insulator Bi$_2$Se$_3$ in the presence of $d$-wave altermagnetic order on the top or bottom layers. The unit vector $\hat{\bf n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ denotes the orientation of the Néel vector shao2021spinli2024creation, where $\theta$ and $\phi$ are the polar and azimuthal angles, respectively. For simplicity, $\theta\!=\!0$ or $\pi$. (a) Altermagnet penetrating only the top layers (yellow) gaps the top surface Dirac cone via time-reversal-symmetry breaking, giving rise to an altermagnet-induced half-quantized Hall effect. (b) Altermagnet penetrating only the bottom layers (green) gaps the bottom surface Dirac cone, leading to a half-quantized Hall effect. (c) Antiparallel Néel vectors on the top and bottom surfaces gap both Dirac cones with opposite Hall contributions, resulting in an altermagnet-induced layer Hall effect with zero net Hall conductance. (d) Parallel Néel vectors on the two surfaces gap both Dirac cones with identical Hall contributions, yielding an altermagnet-induced anomalous Hall effect and a fully quantized Chern insulating phase.
• Figure 2: Band structures and Hall conductances for the tight-binding Hamiltonian \ref{['eq:Hz_TB_r']} of Bi$_2$Se$_3$ in proximity to $d$-wave altermagnetic layers under an external in-plane magnetic field. [(a1)-(d1)] Energy spectra. Black curves show numerical results obtained under OBCs along $z$ direction and PBCs along $x$ and $y$ directions. Red (blue) circles and asterisks denote the analytical surface-state spectra for the top (bottom) surface, derived from Eqs. \ref{['eq:Ek_top']} and \ref{['eq:Ek_bot']}. Subscripts label the surface bands: $t+$ (top conduction), $t-$ (top valence), $b+$ (bottom conduction), and $b-$ (bottom valence). (a1) Top-surface altermagnetic layer with thickness $d_{t}\!=\!2$ nm, which gaps the top Dirac cone. Here, $\theta\!=\!0$, $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm and $G(z)\!=\!F(z)\!=\!0$ elsewhere. (b1) Bottom-surface inverse altermagnetic layer with thickness $d_{b}\!=\!2$ nm, which gaps the bottom Dirac cone. Here, $\theta\!=\!\pi$, $G(z)\cos\pi\!=\!-F(z)\!=\!-1$ for $8\!\leqslant\!z\!\leqslant\!10$ nm and $G(z)\!=\!F(z)\!=\!0$ elsewhere. (c1) Combined top altermagnetic layer ($d_{t}\!=\!2$ nm) and bottom inverse altermagnetic layer ($d_{b}\!=\!2$ nm), which gap both surface Dirac cones. Here, $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm, $G(z)\cos\pi\!=\!-F(z)\!=\!-1$ for $8\!\leqslant\!z\!\leqslant\!10$ nm, and $G(z)\!=\!F(z)\!=\!0$ elsewhere. (d1) Top and bottom altermagnetic layers with parallel Néel vectors, $d_{t}\!=\!d_{b}\!=\!2$ nm and $\theta\!=\!0$, simultaneously gap both surfaces. Here, $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm and $8\!\leqslant\!z\!\leqslant\!10$ nm, and $G(z)\!=\!F(z)\!=\!0$ elsewhere. Yellow shaded regions indicate the bandwidths of the corresponding surface states. [(a2)-(d2)] Hall conductance as a function of the Fermi energy $E_{F}$ for the configurations in (a1)-(d1), exhibiting positive half-quantized, negative half-quantized, vanishing, and integer-quantized values in the energy gaps, respectively. In (c2), red (blue) dots represent the summed layer Hall conductance of the top (bottom) two layers. Other parameters: $M\!=\!0.28$ eV, $A_{1}\!=\!0.22$ eV$\cdot$nm, $A_{2}\!=\!0.41$ eV$\cdot$nm, $B_{1}\!=\!0.1$ eV$\cdot$nm$^2$, $B_{2}\!=\!0.566$ eV$\cdot$nm$^2$, $\Delta\!=\!0.1$ eV, $J_{d}\!=\!B_{2}$, $\varphi\!=\!0$, $a_{z}\!=\!a_{||}\!=\!1$ nm, and sample thickness $L_{z}\!=\!10$ nm along the $z$ direction.
• Figure 3: Upper row: [(a1), (b1)] Hall conductances of the top and bottom surfaces as functions of the angle $\varphi$, calculated from the analytical expressions in Eqs. \ref{['eq:Hallconductance_top_1']} and \ref{['eq:Hallconductance_bot_1']}, respectively. (a1) Top-surface Hall conductance with $\theta\!=\!0$ and $G(z)\cos0\!=\!F(z)\!=\!1$. (b1) Bottom-surface Hall conductance with $\theta\!=\!\pi$ and $G(z)\cos\pi\!=\!-F(z)\!=\!-1$. Here, $\varphi$ specifies the orientation of the external in-plane magnetic field. The integration range is $k_{x},k_{y}\in[-\pi,\pi]$ nm$^{-1}$, and the Fermi energy $E_{F}$ is set within the surface-state gap. Lower row: [(a2), (b2)] Summed layer Hall conductance of the top (bottom) two layers as a function of $\varphi$, for a system with both a top altermagnetic layer ($d_{t}\!=\!2$ nm) and a bottom inverse altermagnetic layer ($d_{b}\!=\!2$ nm). The spatial profiles are given by $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm, $G(z)\cos\pi\!=\!-F(z)\!=\!-1$ for $8\!\leqslant\!z\!\leqslant\!10$ nm, and $G(z)\!=\!F(z)\!=\!0$ elsewhere. (a2) Summed Hall conductance of the top two layers. (b2) Summed Hall conductance of the bottom two layers. Here, $E_{F}\!=\!0$, and all other parameters are the same as in Fig. \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}.
• Figure 4: Band structures and Hall conductances for the altermagnet-induced layer Hall effect under different strengths $V_{0}$, which could be induced by applying the perpendicular electric field. Upper row: [(a1)-(d1)] show the energy spectra for (a1) $V_{0}\!=\!0$, (b1) $V_{0}\!=\!3$ meV, (c1) $V_{0}\!=\!10$ meV, and (d1) $V_{0}\!=\!20$ meV. Black curves show numerical results under OBCs along $z$ direction and PBCs along $x$ and $y$ directions. Red (blue) circles and asterisks represent analytical surface-state spectra for the top (bottom) surface obtained from Eqs. \ref{['eq:Ek_top_Ez']} and \ref{['eq:Ek_bot_Ez']}. In (a1) and (b1), the yellow shaded region marks the bandwidth of the surface states, while in (c1) and (d1), the cyan (green) shaded regions indicate the bandwidths of the top (bottom) surface states. Lower row: [(a2)-(d2)] display the corresponding Hall conductance as a function of the Fermi energy $E_{F}$. Magenta lines show the total Hall conductance, $\sigma_{xy}\!=\!\sum_{j_z}\sigma_{xy}(j_z)$. In (a2), red (blue) dots denote the summed layer Hall conductance of the top (bottom) two layers. The system consists of a top altermagnetic layer ($d_{t}\!=\!2$ nm) and a bottom inverse altermagnetic layer ($d_{b}\!=\!2$ nm), with $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm, $G(z)\cos\pi\!=\!-F(z)\!=\!-1$ for $8\!\leqslant\!z\!\leqslant\!10$ nm, and $G(z)\!=\!F(z)\!=\!0$ elsewhere. These parameters are identical to those used in Figs. \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}(c1) and \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}(c2). All other parameters are the same as those used in Fig. \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}.
• Figure 5: Total Hall conductance as a function of $\varphi$ and $V_{0}$ at a fixed Fermi energy $E_{F}\!=\!10$ meV. The system consists of a top altermagnetic layer ($d_{t}\!=\!2$ nm) and a bottom inverse altermagnetic layer ($d_{b}\!=\!2$ nm). The spatial profiles are specified as $G(z)\cos0\!=\!F(z)\!=\!1$ for $0\!\leqslant\!z\!\leqslant\!2$ nm, $G(z)\cos\pi\!=\!-F(z)\!=\!-1$ for $8\!\leqslant\!z\!\leqslant\!10$ nm, and $G(z)\!=\!F(z)\!=\!0$ elsewhere. These parameters are identical to those used in Figs. \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}(c1) and \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}(c2). All other parameters are the same as those used in Fig. \ref{['Fig:E_C_JB2_D01_Lz10_n2_inverse_together']}.