Layer-Selective Proximity Symmetry Breaking Enables Anomalous and Nonlinear Hall Responses in 1H-TMD Metals

Abstract

Nonlinear Hall responses are a direct electrical probe of quantum geometry, but they are symmetry-forbidden in many pristine two-dimensional metals. We show that layer-selective magnetic proximity unlocks intrinsic linear and nonlinear Hall effects in metallic $1H-NbX_2$ ($X=\mathrm{S,Se,Te}$), where native $D_{3h}$ symmetry forces both the anomalous Hall conductivity and the Berry-curvature dipole (BCD) to vanish. Fully relativistic density-functional theory combined with Wannier interpolation reveals that an out-of-plane proximity exchange that preserves $C_3$ generates a sizable sheet anomalous Hall conductivity, $σ^{\mathrm{sheet}}_{xy} \sim 10^{-2}(e^2/h)$, while keeping the BCD exactly zero. Breaking $C_3$ by adding an in-plane exchange component (or an orthogonal two-sided exchange texture) produces a strongly tunable BCD and hence a nonlinear Hall conductivity that is odd and approximately linear in the in-plane exchange scale, reaching $|D_y|$ of order $10^{-2}$ angstrom and maximized in NbTe$_2$. These magnitudes imply a readily measurable second-harmonic Hall voltage in micron-scale Hall bars under mA ac drive. We further propose a dual-interface device in which the signs of the first- and second-harmonic Hall voltages provide two-bit readout using the same contacts.

Figures (16)

• Figure 1: (Color online) Symmetry route to Hall responses in monolayer $1H$-Nb$X_2$. (a) Crystal structure (top/side views) with horizontal mirror $\sigma_h$ through the Nb plane and threefold axis $C_3$; chalcogen sublayers $X_{\mathrm{top}}$/$X_{\mathrm{bot}}$ host layer-selective exchange. (b) Schematic proximity configurations: one-sided $m_z$ [$R_{\mathrm{int}}$ allowed; AHE allowed, BCD forbidden by $C_3$], two-sided Hall valve [parallel alignment ON, antiparallel alignment OFF], and orthogonal two-sided [AHE and BCD both allowed]. Here $R_{\mathrm{int}}$ denotes the interface-induced Rashba term linear in the in-plane crystal momentum, $H_R=\alpha(k_y s_x-k_x s_y)$. (c) Minimal-model Berry curvature $\Omega^z(\mathbf{k})$ from the interface-induced $k$-linear Rashba--Zeeman Hamiltonian, Eq. (\ref{['eq:RZ']}). For $R_{\mathrm{int}}$ ON with $\Delta_z>0$ and $\boldsymbol{\Delta}_\parallel=\mathbf{0}$, the curvature is an isotropic hot spot and the BCD vanishes. For $R_{\mathrm{int}}$ OFF ($\alpha=0$, $\sigma_h$ preserved), $\Omega^z=0$within the interface-induced $k$-linear Rashba--Zeeman minimal model; in the full material, however, a finite AHE can still arise in the $\sigma_h$-preserving two-sided $m_z$ geometry from pre-existing valley-textured Berry curvature (Supplemental Material Sec. S3 supplemental_material). In the orthogonal case ($\Delta_z\neq 0$ and $\boldsymbol{\Delta}_\parallel\neq\mathbf{0}$), the term $\hat{\mathbf z}\!\cdot(\mathbf{k}\times\boldsymbol{\Delta}_\parallel)$ shifts the avoided crossing in momentum space, creating a dipolar curvature asymmetry and hence a finite $\mathbf D\parallel \hat{\mathbf z}\times\boldsymbol{\Delta}_\parallel$.
• Figure 2: (Color online) First-principles anomalous Hall conductivity under layer-selective proximity exchange in $1H$-Nb$X_2$. (a) One-sided (layer-odd) proximity exchange on a single chalcogen sublayer of NbSe$_2$: an out-of-plane exchange field $\boldsymbol{\Delta}_{\rm ex}\parallel\hat{z}$ generates a sizable intrinsic AHC with sign changes near SOC-induced avoided crossings, whereas a purely in-plane field $\boldsymbol{\Delta}_{\rm ex}\parallel\hat{x}$ yields $\sigma_{xy}\approx 0$; a small canting $\boldsymbol{\Delta}_{\rm ex}(\theta=10^\circ)$ with $\theta$ the canting angle between $\boldsymbol{\Delta}_{\rm ex}$ and $\hat{z}$ (tilt from $\hat{z}$ toward $\hat{x}$ in the $xz$ plane) interpolates between the two limits. (b) Two-sided proximity exchange in a magnet/NbSe$_2$/magnet stack, characterized by the relative orientation of the exchange fields at the top and bottom interfaces, $\boldsymbol{\Delta}^{\rm top}_{\rm ex}$ and $\boldsymbol{\Delta}^{\rm bot}_{\rm ex}$: parallel alignment ($\boldsymbol{\Delta}^{\rm top}_{\rm ex}\parallel \boldsymbol{\Delta}^{\rm bot}_{\rm ex}$) yields a large AHC (Hall valve ON), antiparallel alignment ($\boldsymbol{\Delta}^{\rm top}_{\rm ex}=-\boldsymbol{\Delta}^{\rm bot}_{\rm ex}$) nearly cancels the net exchange (OFF), and the orthogonal configuration ($\boldsymbol{\Delta}^{\rm top}_{\rm ex}\perp \boldsymbol{\Delta}^{\rm bot}_{\rm ex}$) remains AHC-active. (c) Chalcogen dependence of $\sigma_{xy}$ in the orthogonal configuration for NbS$_2$, NbSe$_2$, and NbTe$_2$ (curves rescaled as indicated), showing a pronounced enhancement from S$\rightarrow$Se$\rightarrow$Te. Energies are referenced to the DFT Fermi level; unless stated otherwise, $|\boldsymbol{\Delta}_{\rm ex}|=30~\mathrm{meV}$ is applied to the proximitized chalcogen $p$ orbitals.
• Figure 3: (Color online) Berry-curvature dipole under layer-selective proximity exchange in $1H$-Nb$X_2$. The leading intrinsic second-order Hall response is governed by the BCD component $D_y \equiv D_{yz}=\sum_n\!\int_{\rm BZ}\frac{d^2k}{(2\pi)^2}\, f_{n\mathbf{k}}\;\partial_{k_y}\Omega^z_n(\mathbf{k})$ (in units of Å), which yields $\chi_{yxx}=(e^3\tau/2\hbar^2)D_{yz}$ for $\omega\tau\ll 1$. (a) $D_y$ of NbSe$_2$ for representative exchange textures: one-sided $\boldsymbol{\Delta}_{\rm ex}\parallel\hat{z}$, one-sided canted $\boldsymbol{\Delta}_{\rm ex}(\theta=10^\circ)$ (tilted from $\hat{z}$ toward $\hat{x}$ in the $xz$ plane), two-sided parallel $\boldsymbol{\Delta}^{\rm top}_{\rm ex}\parallel\boldsymbol{\Delta}^{\rm bot}_{\rm ex}$, and two-sided orthogonal $\boldsymbol{\Delta}^{\rm top}_{\rm ex}\perp\boldsymbol{\Delta}^{\rm bot}_{\rm ex}$. (b) Tunability of $D_y$ in the orthogonal geometry by independently varying the magnitudes of the top-interface in-plane exchange and the bottom-interface out-of-plane exchange (here $\boldsymbol{\Delta}^{\rm top}_{\rm ex}\parallel\hat{x}$ and $\boldsymbol{\Delta}^{\rm bot}_{\rm ex}\parallel\hat{z}$); parameter sets are indicated in the legend. (c) Chalcogen dependence of $D_y$ in the orthogonal configuration for NbS$_2$, NbSe$_2$, and NbTe$_2$ (curves rescaled as indicated), showing a pronounced enhancement from S$\rightarrow$Se$\rightarrow$Te. Energies are referenced to the DFT Fermi level; unless stated otherwise, $|\boldsymbol{\Delta}_{\rm ex}|=30~\mathrm{meV}$ is applied to the proximitized chalcogen $p$ orbitals.
• Figure 4: (Color online) Orthogonal two-sided magnetic proximity enables independent first- and second-harmonic Hall readout in monolayer NbSe$_2$. (a) Dual-interface heterostructure: the bottom ferromagnetic insulator (FMI) provides predominantly out-of-plane proximity exchange, the top FMI predominantly in-plane; gating tunes the chemical potential $\mu$. (b) Hall-bar geometry driven by an ac current $I_x(t)=I_0\cos\omega t$, with transverse voltages recorded at the first and second harmonics. (c) Intrinsic AHC $\sigma_{xy}(\mu)$ for four magnetic configurations, showing sign control primarily by reversal of the out-of-plane component. (d) Intrinsic nonlinear Hall coefficient $\chi_{yxx}(\mu)$ (BCD contribution) for the same configurations; its sign is controlled mainly by the in-plane component. The vertical dashed line marks an operating point $\mu^*$ used to define the four-state harmonic Hall readout discussed in the text.
• Figure S1: Quality of Wannier interpolation for monolayer $1H$--Nb$X_2$. Comparison of fully relativistic DFT bands (gray) and Wannier-interpolated bands from wannier90 (blue) along the high-symmetry path $\Gamma$–M–K–M–K$'$–$\Gamma$ for (a) NbS$_2$, (b) NbSe$_2$, and (c) NbTe$_2$. The horizontal red dashed line marks a representative chemical potential used in the Hall response analysis in the main text. All Wannier Hamiltonians are constructed from $50$ Bloch bands and $22$ spinor MLWFs (5 Nb $d$-like and 6 chalcogen $p$-like orbitals per spin) using the energy windows described in Sec. S4.2. Over the energy range shown, and in particular near the Fermi level that controls the Hall responses discussed in the main text, the Wannier bands track the DFT dispersions to within $\mathcal{O}(10)$ meV, faithfully capturing the spin–orbit-induced splittings and valley structure.