Symmetry-resolved magnetoelastoresistance in multivalley bismuth

TL;DR

We study how magnetic field and uniaxial strain affect charge transport in the multivalley semimetal bismuth by performing symmetry-resolved magnetoelastoresistance (MER) measurements with current, stress, and field along the binary axis. The data are interpreted with a classical transport model that treats three electron valleys with an anisotropic mobility tensor, including an effective field-dependent mobility mu_eff(B) and field-induced valley polarization Delta n_pol, enabling a symmetry-based separation into MER_sym and MER_anti. The main findings are that MER_sym is essentially field-independent while MER_anti shows a non-monotonic field dependence driven by mobility anisotropy and valley polarization, in good qualitative agreement with the model. The work provides a general framework for understanding valley-dependent transport under concurrent magnetic field and strain and demonstrates MER as a sensitive probe of valley states in bismuth.

Abstract

We report a symmetry-resolved study of longitudinal magnetoelastoresistance (MER) in the multivalley material bismuth, with the current, uniaxial stress, and magnetic field all applied along the binary axis. The magnitude of MER exhibits a steep increase at low magnetic fields, reaches a peak, and then gradually decreases at higher fields. By decomposing the strain response into symmetric and antisymmetric symmetry channels, we reveal contrasting magnetic field dependencies. Despite the overall non-monotonic field dependence of the MER, the symmetric component remains nearly constant under magnetic fields, suggesting that the valleys in bismuth preserve a rigid-band nature against strain even in the presence of a magnetic field. In contrast, the antisymmetric component, associated with mobility anisotropy, dominates the MER response in a magnetic field. At low magnetic fields, the applied field effectively modifies the apparent mobility of each valley, leading to an enhancement in the magnitude of the antisymmetric MER. At higher fields, field-induced valley polarization further modifies this mobility anisotropy by altering the contributions from each valley's mobility, accounting for the moderate suppression of the MER. These findings demonstrate that symmetry-resolved MER serves as a powerful probe of valley-dependent electronic states and provides a fundamental platform for understanding the interplay between magnetic field, strain, and charge transport.

Figures (5)

• Figure 1: MER$_{\parallel}$ study of bismuth. (a)(upper inset) Experimental configuration for MER$_{\parallel}$ measurements. (lower inset) Temperature and magnetic field dependent ER/MER$_{\parallel}$. Circle markers represent data points obtained by strain-sweeping at fixed temperatures and fields, and the line curves represent data obtained by field sweeping at three different strain-fixed conditions. (b)Field-induced changes in MER, $\Delta$MER$_{\parallel} = {\rm ER}_{\parallel}(B)-{\rm ER}_{\parallel} (0)$, at each temperature. The data are vertically shifted for clarity. (c) Comparison of quantum oscillations between MER$_{\parallel}$ (left axis) and MR (right axis) at 1.8 K. The orange arrow represents the quantum limit (QL) for e2/e3 valleys.
• Figure 2: Comparison between the low-field behavior of MR and MER at various temperatures. (a),(b) Experimental results of MR (a) and $\Delta {\rm MER}_{\parallel}$ (b). (c),(d) Numerical calculations of the corresponding quantities using the mobility tensor values from Ref. [PhysRevX.5.021022] and valley susceptibility obtained from elasto-quantum oscillation measurements in Ref. [PhysRevResearch.6.033096].
• Figure 3: Field evolution of symmetry-resolved $\Delta$MER at 120 K. Red circles and orange squares represent antisymmetric and symmetric components of $\Delta$MER, respectively.
• Figure 4: Numerical calculations of mobility anisotropy $\gamma$ arising from apparent mobility modification (a) and valley population changes (b). (a) $\gamma$ modified by $\hat{\mu}_{\rm eff}$, evaluated using Eq. (\ref{['eq:sig_B']}) and the mobility tensor from Ref. [PhysRevX.5.021022], with magnetic field applied along the binary axis. The inset shows the corresponding $\Delta {\rm MER_{anti}}$ by using valley susceptibility $\chi_{\rm anti} \sim 85$ determined from zero-field ER measurementsPhysRevResearch.6.033096. (b) $\gamma$ as a function of the deviation in carrier density of e1 valley $\lambda_{\rm e1}$. The carrier densities of e2,e3 valleys are defined by $\lambda_{\rm e2,e3} = k \lambda_{\rm e1}$ for various values of $k$. The red diamonds show $\gamma$ values obtained from the carrier density calculated from Landau quantized energy dispersionPhysRevB.84.115137, which are well described by $k = -3.4$ (dashed curve). The inset displays the corresponding relative changes in $\Delta {\rm MER_{\rm anti}}$ at several magnetic fields. For simplicity, we neglect magnetic field dependence of mobility $\mu$ in this calculation to capture essential effects of valley polarization. The right panel shows the assignment of the electron valleys (e1, e2, e3).
• Figure 5: Numerical calculations of valley populations (a) and mobility anisotropy $\gamma$ (b). (a) Calculated magnetic-field dependence of valley populations, reproduced from Ref. [PhysRevB.84.115137]. The inset shows a schematic picture of field-induced valley polarizations under $B \parallel$ binary. (b) $\gamma$, evaluated using Eq. (\ref{['eq:sig_B']}) and the mobility tensor from Ref. [PhysRevX.5.021022], with magnetic field applied along the binary axis. For $B > 1$ T (vertical dashed line), the effect of valley polarization is incorporated. The inset shows the corresponding $\Delta {\rm MER_{\rm anti}}$ using valley susceptibility $\chi_{\rm anti} = 85$.