Polar unidirectional magnetotransport in $p-$type tellurene from quantum geometry

TL;DR

The paper demonstrates polar eMChA in the valence bands of 2D tellurene, showing that a finite quantum metric dipole arises only after coupling to remote Weyl-containing bands via Löwdin downfolding, with the lone-pair polarization providing the intrinsic field ${\cal E}_0$ that activates the effect. Using a ${\bf k}\cdot{\bf p}$ model and semiclassical Boltzmann transport, the authors derive the eMChA tensor $G_{ijk\ell}$ and its scaling with chemical potential, identifying a crossover controlled by the valence-band splitting $\Delta_1$ and predicting a $G/\sigma \sim \varepsilon^{-3/2}$ behavior deep in the valence regime and a gap-regularized rise near the band edge. Numerical calculations reveal divergences of $G$ near Weyl-node energies and discontinuities tied to Fermi-surface topology changes, while experimental second-harmonic measurements under gate-tuning and angular-field scans validate the multiband geometric mechanism, showing concurrent chiral and polar contributions. The results unify conduction- and valence-band eMChA in tellurene, establishing a platform for quantum-geometric rectification in multiband noncentrosymmetric systems, with potential for gate-tunable, intrinsic nanoscale rectification.

Abstract

Unidirectional magnetoresistance, or electric magnetochiral anisotropy (eMChA), is a nonlinear magnetotransport phenomenon that arises in noncentrosymmetric conductors , where changes in resistance $R(B)$ are: (i) chiral, $ΔR(B)/R(0)=2\,χ\, {\bf I}\cdot{\bf B}$, or (ii) polar, $ΔR(B)/R(0)=2\,γ\, {\bf I}\cdot({\bf P}\times{\bf B})$, with eMChA coefficients $χ$ and $γ$. In [Phys. Rev. Lett. 135, 106602 (2025)], we showed that the eMChA in the conduction band of tellurene is polar ($χ=0$, $γ\neq 0$) and emerges from the quantum metric dipole due to its Weyl node and from the lone pair polarization ${\bf P}$. Here, we extend our work to the valence band of tellurene, where the eMChA is usually said to be chiral ($χ\neq 0, γ= 0$). We show that also a polar coefficient $γ\neq 0$ emerges naturally through a downfolding procedure, in which remote Weyl-node containing bands induce momentum-space gradients of the quantum metric in the low-energy levels, activating finite metric dipoles. Combining semiclassical Boltzmann transport with a ${\bf k}\cdot{\bf p}$ description of tellurene, our numerical calculations agree quantitatively with doping ($μ$) dependent second-harmonic measurements of the longitudinal voltage $V^{2ω}_\parallel(μ)$ in perpendicular field. The combined chiral and polar characters ($χ\neq0, γ\neq 0)$ of the eMChA in tellurene also explains the shift in the angular ($φ$) dependence of $V^{2ω}_\parallel(φ)$ for in plane fields. Our results demonstrate that the polar eMChA can arise in topologically trivial bands through multiband effects and establishes tellurene as a platform for quantum-geometric rectification in both electron and hole regimes.

Figures (10)

• Figure 1: Extrinsic vs intrinsic rectification.$(\hbox{a})$ In a conventional $p$–$n$ junction diode, rectification originates from the built-in electric field ${\bf E}_{{built\text{-}in}}$ created at the interface between $p$-type and $n$-type regions. $(\hbox{b})$ In tellurene, instead, the intrinsic lone pair polarization electric field $\boldsymbol{{\cal E}}_{0}\parallel\hat{x}$ arising at the surfaces of the material plays the role of the built-in field. When an external magnetic field ${\bf B}\parallel\hat{y}$ is applied, carriers experience a nonreciprocal response along the direction $\boldsymbol{{\cal E}}_{0}\times{\bf B}\parallel\hat{z}$, contributing to the longitudinal eMChA.
• Figure 2: Structure of tellurene. (a) Crystal structure of $\alpha$-Te, highlighting the helical arrangement of Te atoms and the crystallographic axes. The red hexagon indicates the projection of the unit cell onto the basal plane. (b) First Brillouin zone with high-symmetry points and reciprocal lattice directions, showing the orientation of the $k_x$, $k_y$, and $k_z$ axes used throughout this work. (c) Representative band structure along the $k_z$ direction, illustrating the conduction and valence sectors relevant for nonlinear transport. (d) Calculated polarization components $(P_x,P_y,P_z)$ as a function of an applied electric field, evidencing the dominant response of the lone-pair polarization along the $x$ direction for a tellurene thin film or flake. (e) Real-space visualization of the lone-pair electronic density in a two-layer tellurene film, showing how the intrinsic dipoles align and generate a polarization $\mathbf{P}$.
• Figure 3: Valence band structure and device geometry. Valence band structure of Te along the line $k_x=k_y=0$, calculated using the Hamiltonian in Eq. \ref{['Eq: Hamiltonian-Te']} with the parameters listed in Table \ref{['tab:band_params']}. The energy scales $\Delta_1$ and $\Delta_2$ represent the gaps between the $H_{4,5}$ and $H_{6,6'}$ band blocks, as indicated by the purple and gray vertical arrows, respectively. Inset: Pictorial representation of the four-terminal device used for nonreciprocal transport measurements within the $x-z$ plane.
• Figure 4: Geometric origin of the nonreciprocal response. (a) Fermi surfaces of the two valence bands in the $(k_x,k_z)$ plane overlaid on a heatmap of the energy--normalized quantum metric $\tilde{g}(\mathbf k)$, which is even under $k_z\rightarrow -k_z$, calculated using Eq. \ref{['eq: gzz_plot']}. (b)-(c) The same Fermi surfaces overlaid on a heatmap of the $\Gamma_{zzx}$ and $\Gamma_{xzx}$ Christoffel-symbols, respectively, which is odd in $k_z$ and encodes the momentum gradient of the metric. While the metric-induced Stark energy shift is symmetric in momentum space, its derivative produces asymmetric velocity corrections; in the presence of a magnetic field, this asymmetry leads to nonreciprocal transport.
• Figure 5: Scaling behavior of the nonlinear magnetochiral response in the valence band of tellurene. Log--log plot of the theoretical scaling of the nonlinear magnetochiral conductivity $G$ normalized by $\sigma$ as a function of the energy depth $\varepsilon=|\mu|$ measured from the top of the valence band. Three distinct regimes are identified: (i) a gap-regularized regime near the band edge ($\varepsilon\ll\varepsilon_\Delta$) where $G(\varepsilon)\sim\varepsilon^{3/2}$, (ii) a crossover regime ($\varepsilon\simeq\varepsilon_\Delta$) where $G(\varepsilon)\sim G_0^{max}$, and (iii) a gapless high-doping regime ($\varepsilon\gg\varepsilon_\Delta$) where $G(\varepsilon)\sim\varepsilon^{-1/2}$.