Disentangling contributions to longitudinal magnetoconductivity for Kramers-Weyl nodes

TL;DR

The paper analyzes longitudinal magnetoconductivity of an isolated isotropic Kramers-Weyl node, where a dominant quadratic dispersion yields two concentric Fermi surfaces for $\mu>0$. Using an exact semiclassical Boltzmann framework that includes Berry curvature, orbital magnetic moment, and spin magnetic moment, the authors derive the full $B$-dependence beyond relaxation-time approximations. A key finding is that the spin-magnetic moment induces a linear-in-$B$ contribution to the conductivity (satisfying Onsager reciprocity), while the Berry-curvature and orbital-moment terms give $B^2$-like corrections, with the interband scattering strength $\beta_{\text inter}$ critically shaping the response and potentially flipping its sign. The work clarifies how KWNs in chiral crystals produce distinctive magnetotransport signatures, guiding experiments and motivating extensions to noncollinear fields and finite-temperature transport.

Abstract

We set out to compute the longitudinal magnetoconductivity for an isolated and isotropic Kramers-Weyl node (KWN), existing in chiral crystals, which forms an exotic cousin of the conventional Weyl nodes resulting from band-inversions. The peculiarities of KWNs are many, the principal one being the presence of two concentric Fermi surfaces at any positive chemical potential ($μ$) with respect to the nodal point. This is caused by a dominant quadratic-in-momentum dispersion, with the linear-in-momentum Dirac- or Weyl-like terms relegated to a secondary status. In a KWN, the chirally-conjugate node typically serves as a mere doppelgänger, being significantly separated in energy. Hence, when $μ$ is set near such a node, the signatures of a lone node are probed in the transport-measurements. The intrinsic topological quantities in the forms of Berry curvature and orbital magnetic moment contribute to the linear response, which we determine by exactly solving the semiclassical Bolzmann equations. Another crucial feature is that the two bands at the same KWN node carry actual spin-quantum numbers, thus providing an additional coupling to an external magnetic field ($\boldsymbol B$), and affecting the conductivity. We take this into account as well, and demonstrate that it causes a linear-in-$B$ dependence, on top of the usual $B^2$-dependence.

Figures (4)

• Figure 1: Energy bands for $s=1$ (light red) and $s=-1$ (light blue): (a) Bare dispersion ($E$) of the two bands at an isotropic Kramers-Weyl node against the $k_x k_y$-plane (or, equivalently, $k_y k_z$- and $k_z k_x$-planes). The yellow plane depicts a positive chemical potential ($\mu$) cutting the bands, giving rise to two concentric Fermi surfaces. (b) Schematics of the Fermi-surface projections in the the $k_z k_x$-plane, when a nonzero $\boldsymbol B$ (red arrow) is applied along the $z$-axis. To provide an eye-estimate, the dotted curves show the unperturbed projections in the absence of an external magnetic field.
• Figure 2: $\delta \sigma_{zz} (s)$ from each of the bands considering no interband interactions, with $\beta_{\rm intra}$ set to unity: While subfigure (a) depicts the variation of the full conductivity with $B$ (in eV$^2$) when OMM is taken into account appropriately, subfigure (b) represents the conductivity versus $B$ characteristics when OMM is neglected. The number in each plot-legend indicates the index of the band (i.e., $1$ or $-1$). The three frames in each subfigure represent three distinct sets of parameter values, as indicated in the plot-labels, with $c$ in eV$^{-1}$, $v_0$ being unitless, and $\mu$ in eV.
• Figure 3: $\delta \sigma_{zz} (s)$ from each of the bands considering both intraband and interband interactions, with $\beta_{\rm intra}$ set to unity: While subfigure (a) depicts the variation of the full conductivity with $B$ (in eV$^2$) when OMM is taken into account appropriately, subfigure (b) represents the conductivity versus $B$ profiles when OMM is neglected. The two numbers in each plot-legend indicate the index of the band (i.e., $1$ or $-1$) and the value of $\beta_{\rm inter} / \beta_{\rm intra}$, respectively. The three frames in each subfigure represent three distinct sets of parameter values, as indicated in the plot-labels, with $c$ in eV$^{-1}$, $v_0$ being unitless, and $\mu$ in eV.
• Figure 4: Longiudinal conductivity from the relaxation-time approximation [cf. Eqs. \ref{['eqrelax']} and \ref{['eqrelax2']}] for $s=1$ (left frame) and $s=-1$ (right frame). The parameter values, as indicated in the plot-labels, with $c$ in eV$^{-1}$, $v_0$ being unitless, and $\mu$ in eV.