Angle-Resolved Berry Curvature via Nonlinear Hall Effect of Ballistic Electrons

Abstract

Berry curvature fundamentally dictates the topological ground state, anomalous transport and optical properties of quantum materials. However, directly mapping its momentum-space distribution in real materials remains an outstanding experimental challenge. Here, we present an inverse method for reconstructing the abelian Berry curvature of a single band using angle-resolved measurements of the transverse conductance. Our inversion relies on a symmetry-constrained statistical model with two hyperparameters that can be inferred directly from the nonlinear Hall conductance, yielding a parameter-free inversion method. We demonstrate the feasibility of our method using simulated measurements of tight-binding models of WSe$_2$ and $ABC$-stacked trilayer graphene.

Figures (8)

• Figure 1: Part (a) shows the Berry curvature at the $K$ and $K'$ valleys of WSe$_2$. The surplus distribution is at the Fermi surface with positive velocity. Electrons with below the Fermi surface or with negative velocity do not contribute to transport. The 90% contours of the distribution $\delta f = -\frac{\partial n_F}{\partial \varepsilon}\Theta(v_x)$ are shown in red ($k_B T = 0.02$ eV). In our inversion we consider chemical potentials in which the Fermi surface of WSe$_2$ is entirely within these two regions. Part (b) shows the proposed measurement setup. For each angle we have a pair of probes which inject and accept electrons. Orthogonal to these are floating probes which measure the Hall current. The injecting probes are switched to resolve different angles.
• Figure 2: Reconstruction of the Berry curvature (BC) in the $K$ pocket of WSe$_2$. In (a) is the reconstructed Berry curvature, and (b) is the true Berry curvature calculated from the tight-binding model. In (c) is the difference between these two fields. Points not constrained by the data have been masked out of the plot. The hyperparameter tuning chose the regularizing parameter $\hat{\gamma} = 0.0034$. The region of largest error is close to the $K$ point. A detailed analysis of the dependence of the reconstruction and the spectrum on the regularizing parameters is provided in the appendix.
• Figure 3: Reconstruction of the Berry curvature with low noise in the $K$ pocket of ABC-stacked graphene. The $K'$ pocket is determined by time-reversal symmetry, which is explicitly enforced in the inversion procedure. In (a) is the reconstructed Berry curvature and (b) is the true Berry curvature from the tight-binding model. (c) shows the difference between these two fields. The grid search of the log posterior that led to the hyperparameters is shown in the supplementary materials. In (c) examples of the Hall response are shown at different angles used for the inversion. Part (d) shows the MAP estimate and true Berry curvature for the low-noise case of ABC-stacked graphene. Random draws from the marginal posterior are shown to estimate the error of the measurement. For draws to the right of the dashed vertical line, the $k$-point has energy within the chemical potentials measured. The variance is much larger for points outside this range, shown to the left of the dashed line.
• Figure S1: Singular values of the measurement operator with and without regularization for the inversion performed in Figure \ref{['fig:abc']}. The smoothness prior lifts all singular values above the floating point precision cutoff.
• Figure S2: Relative difference of the reconstructed Berry curvature and the actual Berry curvature with respect to the maximum absolute Berry curvature in the domain.