Magneto-conductivity and CME in Dirac semimetals from Keldysh technique in Landau levels basis

TL;DR

The paper develops a kinetic theory for magnetoconductivity in Dirac semimetals using the Keldysh non-equilibrium diagram technique in the Landau-level basis. It questions the primacy of the chiral magnetic effect by deriving transport from a semi-realistic Dirac Hamiltonian that includes disorder and acoustic phonons, yielding a longitudinal conductivity formula that smoothly interpolates between weak and strong magnetic fields and incorporates field- and temperature-dependent relaxation times. A key result is the LL-dominated strong-field limit, where $\sigma_{\text{LLL}} \approx (\rho_{\text{imp}}+\rho'_{\text{ph}} T)^{-1}$, with Landau-level widths $\epsilon_n$ that depend on $B$, $T$, and $\mu$, connecting theory to experimental trends in materials like ZrTe$_5$. The work also introduces a mechanism to suppress observable Landau quantization artifacts via averaging over inhomogeneous magnetic fields, aligning with the absence of clear LL features in experiments. Overall, the study provides a first-principles framework for understanding magnetoconductivity in type-I Dirac/Weyl semimetals and highlights the complex, parameter-dependent role of relaxation processes in real materials.

Abstract

Negative magnetoresistance in Dirac semimetals is conventionally considered as a manifestation of chiral magnetic effect (CME), by means of a postulated Chiral Kinetic equation. In this paper we study magnetoconductivity in large Fermi energy Dirac semimetals, in one of which (ZrTe$_5$) the effect was observed for the first time. Starting with a Hamiltonian for a semi-realistic model of such a Dirac semimetal, we apply the Non-equilibrium Diagram Technique (NDT, or the Keldysh technique) to derive the kinetic equations, to investigate the electrons relaxation due to interaction with phonons and disorder, and, finally, to calculate the DC magnetoconductivity (the longitudinal to magnetic field component of conductivity) as a function of magnetic field strength and temperature. Finally, we compare the obtained temperature dependencies with available to us experimental data.

Figures (4)

• Figure 1: Averaged longitudinal conductivity $\bar{\sigma}_\parallel$\ref{['EqCondG']} and resistivity $\rho_\parallel=\bar{\sigma}_\parallel^{-1}$ as functions of magnetic field. The blue dots are for non-smoothened $\sigma_\parallel$\ref{['EqCondIIExt']}. The red line (stronger oscillating) corresponds to $\delta B/B=0.0{4}$ and the black line (weaker oscillating) --- to $\delta B/B=0.{07}$. The dashed line --- the quadratic magneto-conductivity \ref{['EqCondWF']}. $\nu_B^{-1}=\frac{2|eB|\hbar v_F^2}{\mu^2}$ so that $\lfloor \nu_B\rfloor = N_B$ is the number of the highest occupied LL (LLL's number is zero).
• Figure 2: Large magnetic field resistivity of $ZrTe_5$. Data extracted (points) from Fig.S1 of CMEZrTe5 (see the Methods -- Transport Measurements section in the latest arXiv version, or Supplemental Materials in the journal version) as the average at $B>6$ T and $B<-6$ T, fitted with the obtained here dependence \ref{['EqPlCondLLL']}$\rho_{zz} = \sigma_{zz}^{-1}=\rho_\text{imp}+\rho_\text{ph}'T$; fitting parameters values from the data are $\rho_\text{imp}\sim0.37\pm0.10~m\Omega cm$, $\rho_\text{ph}'\sim0.021~m\Omega cm K^{-1}$, which roughly correspond to values of our model parameters $w\sim {1.3}$ eV, $u_0 \sim {30}$ meV nm$^3$, $n_{imp} \sim 10^{-3}$ nm$^{-3}$. The relatively large error bars result from asymmetry in the data at 'positive' and 'negative' magnetic fields, while for 'positive' ('negative') only data the errors are much smaller. Thus, the slope coefficient variance is small.
• Figure 3: Zero magnetic field resistivity of ZrTe$_5$. Data extracted (points) from Fig.2 of CMEZrTe5 fitted with the obtained here dependence \ref{['EqPlCond0']}, fitting parameters values from the data are $\rho_{0,\text{imp}} \sim 1.1~m\Omega cm \sim 3\rho_\text{imp}$ and $\rho'_{0,\text{ph}} \sim 0.014~m\Omega cm K^{-1} \not\sim 3\rho'_\text{ph}$. We omit error bars since we consider the analysis to be an estimation. The two low-temperature points are ill-defined because of the cusp feature.
• Figure 4: Weak magnetic field magnetoconductivity of $ZrTe_5$. Data extracted (points) from Fig.S1 (c.f. Fig.2) of CMEZrTe5. If all points are included, the fit parameter value may become unphysical $\alpha_\text{imp}<0$. The uncertainty is too large to derive parameters values. We omit error bars since we consider the analysis to be an estimation. The two low-temperature points are ill-defined because of the cusp feature.