Refraction Hall Effect

Abstract

We introduce a new mechanism that produces a Hall-like response in time-reversal-invariant materials, driven entirely by geometric effects. Specifically, we demonstrate that a tilted potential interface causes electron wave packets to undergo a refraction-like deflection upon transmission through the barrier, leading to a finite transverse current and a corresponding Hall conductance. Our analytical framework captures the essential features of this refraction Hall effect, and the resulting Hall conductivity profile is corroborated by numerical simulations across different lattice models and device geometries. We further visualize our predicted effect through real-time wave packet dynamics, which reveals its purely geometric origin and the robustness of the transverse response. These findings establish a fundamentally distinct class of Hall-like transport phenomena in mesoscopic systems that preserve time-reversal symmetry.

Figures (4)

• Figure 1: Snell's law for electrons and schematic of the refraction Hall setup. (a) An electron with incident wavevector with magnitude $k$ and angle $\theta$ impinges on a potential step aligned with the $(\bar{X},\bar{Y})$ axes. Upon transmission, its wavevector changes to $k'$ and the refracted trajectory emerges at an angle $\theta'$, satisfying the conservation law $k \sin\theta = k' \sin\theta'$. (b) The device consists of a rectangular slab connected to a source and a drain along the $X-$axis and extending infinitely along $Y$. A potential barrier is engineered at an angle $\theta$ relative to the perpendicular axis ($Y$). The rotated coordinates ($\bar{X},\bar{Y}$) follow the orientation of the barrier. The incident current density from the source, $j_I$, represents a single illustrative angle of incidence. The corresponding reflected and transmitted components are indicated by $j_R$ and $j_T$, respectively, analogous to reflection and refraction in an optical setting.
• Figure 2: Refraction Hall conductivity from analytical calculations. The calculated Hall conductivity, $G_{XY}$, is shown as a function of the Fermi energy $\mu$. The parameters are fixed at a barrier strength $V_{0} = 0.5$ and a tilt angle $\theta = 0.025 ~\pi$. A finite Hall response emerges when the Fermi energy exceeds the barrier height $V_{0}$. The conductivity is expressed in units of $e^{2}/h$.
• Figure 3: Time dynamics illustrating the refraction Hall effect. The time evolution of a Gaussian wave packet initially localized at $(-2,0)$ is shown. The packet is assigned an initial momentum along the $x$-direction, resulting in horizontal propagation prior to interaction with the tilted potential barrier. Upon transmission through the barrier, the wave packet acquires a finite positive momentum component along the $y$-direction, thereby demonstrating the refraction Hall effect of electrons. The dynamics is computed by numerically solving the time-dependent Schrödinger equation. The dashed line indicates the interface of the potential barrier. The potential takes the value $V_{0}$ above the line and vanishes below it. In this case, the barrier is tilted at an angle $\theta = \pi/4$ with respect to the axes, and the barrier strength is $V_{0}=-0.5$.
• Figure 4: Refraction Hall conductivity from lattice simulations. The upper panels (a)-(c) depict the schematic structure of the simulated device geometries, while the corresponding Hall conductivities are shown in the lower panels (f)-(h). Panels (a) and (f) present results for a four-terminal square lattice; panels (b) and (g) correspond to a six-terminal square lattice; and panels (c) and (h) show a four-terminal hexagonal lattice. In all cases, the potential barrier has a tilt angle of $\theta = 0.025~\pi$ and strength $V_{0} = 0.5$. A longitudinal current is applied between the horizontal leads, and the Hall voltage is measured across the vertical leads. The numerical simulations reproduce the analytical trends within the regime of interest. Blue dashed lines denote the moving average of the raw data, while orange curves denote the smoothened profiles. Panels (d) and (e) show the normalized density of states (DOS) with the chemical potential, $\mu$, for the square and hexagonal lattice structures, respectively. All conductivities are expressed in units of $e^{2}/h$. Here the lattice dimensions are $L_X=20, L_Y=200$ for square lattices and $L_X=20, L_Y=400$ for hexagonal lattice geometries.