Nonlinear Hall responses in tunable nodal Dirac semimetals

TL;DR

This work addresses how quantum geometry controls nonlinear Hall responses in tunable 2D nodal Dirac semimetals. By analyzing a low-energy model with SN, DN, and NR nodal structures, the authors compute Berry curvature dipole $D_{bd}$ and Berry connection polarizability $G_{ab}$ within a Boltzmann transport framework to obtain SOHE and TOH. The key findings are that SOHE is enhanced in the single-node phase with broken inversion symmetry and vanishes in the nodal-ring phase, while TOH is dramatically amplified in the nodal-ring phase near the band edge due to localized BCP around the nodal ring. These results show how the distribution of quantum geometric quantities in the Brillouin zone governs the magnitude and anisotropy of geometry-driven nonlinear transport, offering a route to engineer nonlinear Hall responses in 2D materials.

Abstract

We investigate the nonlinear Hall responses in tunable two-dimensional Dirac materials. In particular, we study quantum geometry-driven second and third order non-linear responses in a time-reversal symmetric Dirac semimetal that can host single, double and line nodes depending on the model parameters. We find that the second-order Hall response (SOHE), which originates from the Berry curvature dipole, is enhanced in the single-node semimetallic phase as compared to the double node case when inversion symmetry is broken. In contrast, the SOHE vanishes in the nodal line semimetal as the inversion symmetry retains. Notably, the third-order Hall response due to Berry connection polarizabilty becomes much larger in the line-node Dirac semimetal, especially when the Fermi energy lies near the band edge, than in the single- and double-node Dirac semimetals. The reason for this contrasting behavior is attributed to the distinct distribution of the Berry connection polarizability in the Brillouin zone.

Figures (4)

• Figure 1: Energy spectrum of Eq. \ref{['Hamiltonian']} for different values of parameters $\gamma$ and $k_0$, keeping $\lambda$ fixed. (a) SN phase with single nodal point at $\mathbf{k} = 0$ for $k_0=0$ and $\gamma=1.0~\mathrm{eV\!\cdot\!\mathring{A}}$, (b) DN phase with two nodal points located at $\mathbf{k} = (\pm k_0,0)$ for $k_0=1.0~\mathrm{\mathring{A}^{-1}}$ and $\gamma=1~\mathrm{eV\!\cdot\!\mathring{A}}$, and (c) NR phase with a nodal ring of radius $k_0=1.0~\mathrm{\mathring{A}^{-1}}$ and $\gamma=0$. Here we take $\lambda=1.0~\mathrm{eV\!\cdot\!\mathring{A}^2}$ and momentum $k_x$, $k_y$ are in units of $\text{\AA}^{-1}$.
• Figure 2: (a,b) Density plots showing the Berry curvature ($\Omega_z$) multiplied by the velocity component for the single-node and double-node phases, respectively. (c) Plot of the Berry curvature dipole $D_{xz}$ for both these phases. Here, we take gap parameter $\Delta=0.1~\mathrm{eV}$. (d) Density plot of Berry curvature along the energy spectrum in the $k_y=0$ plane for both the SN and DN phases. Here $\Omega_z$ and $D_{xz}$ are in units of $\mathring{A}^2$ and $\mathring{A}^3$ respectively.
• Figure 3: Density plots of the BCP tensor components $G_{xx}$(left column), $G_{yy}$(middle column) and $G_{xy}$(middle column) for all three phases, obtained from equation \ref{['BCP components']}. The top row (a–c), middle row (d–f) and bottom row (g–i) correspond to the SN, DN and NR phases, respectively. The plots are shown only for the conduction band. Here we consider the gap $\Delta=0.1~\mathrm{eV}$.
• Figure 4: The transverse TOH conductivity for different nodal phases. (a-c) shows the variation of the transverse TOH response $\chi_{\perp}$ as a function of the applied electric field direction $\theta$ for different Fermi energies $E_F$ in the SN, DN and NR phase respectively. (d) presents the maximum transverse TOH responses together in different phases near the band edge at $E_F=0.11~\mathrm{eV}$. We take temperature $T$=50K.