Linear magnetoresistance of two-dimensional massless Dirac fermions in the quantum limit

TL;DR

The paper addresses the origin of linear magnetoresistance in 2D massless Dirac fermions under the quantum limit and provides a microscopic framework based on linear-response theory and the self-consistent Born approximation to compute $\rho_{xx}(B)$ for different impurity models. It derives analytic expressions for the quantum-limit conductivity $\sigma_{xx}^{ql}$ in terms of impurity-induced widths and Landau-level structure, and shows how impurity type qualitatively changes the magnetic-field dependence: δ-function impurities give a field-independent resistivity with the minimal conductivity, Gaussian impurities produce linear magnetoresistance when the impurity range is shorter than the magnetic length, and Yukawa impurities yield a $\rho_{xx}\propto 1/B$. The results, including finite-temperature extensions, quantitatively agree with experiments on graphene under ultra-high fields and clarify the impurity-type dependence of magnetotransport in 2D Dirac systems, bridging theory and the Geim et al. observations. The work highlights fundamental differences from 3D Weyl systems and provides a versatile framework for interpreting magnetoresistance in 2D Dirac materials.

Abstract

Linear magnetoresistance is a hallmark of 3D Weyl metals in the quantum limit. Recently, a pronounced linear magnetoresistance has also been observed in 2D graphene [Xin et al., Nature 616, 270 (2023)]. However, a comprehensive theoretical understanding remains elusive. By employing the self-consistent Born approximation, we derive the analytical expressions for the magnetoresistivity of 2D massless Dirac fermions in the quantum limit. Notably, our result recovers the minimum conductivity in the clean limit and reveals a linear dependence of resistivity on the magnetic field for Gaussian impurity potentials, in quantitative agreement with experiments. These findings shed light on the magnetoresistance behavior of 2D Dirac fermions under ultra-high magnetic fields.

Figures (3)

• Figure 1: (a) Calculated magnetoresistivity [Eq. \ref{['Eq: rho_G']}] as a function of the dimensionless parameter $\lambda^2/\ell_B^2$ for various impurity strengths $\gamma$, where $\lambda$ is the decay length of the Gaussian potential, $\ell_B = \sqrt{\hbar/(eB)}$ is the magnetic length, and $\sigma_0=e^2/(2\sqrt{2}\pi h)$ is the minimal conductivity of 2D massless Dirac fermions Ludwig94prbZiegler97prbMudry07prbCserti07prbZiegler07prbDimi22prl. (b) Comparison between linear magnetoresistivity in the quantum limit [Eq. \ref{['Eq: rho_G2']}] and experimental data Geim23nature. (c) Phase diagram for the slope of the magnetoresistivity $d\rho_{xx}^{G,ql}/dB$ as a function of the decay length $\lambda$ and impurity strength $\gamma$. Here, $\gamma$ cannot be zero unless $\lambda$ approaches zero first. Two inset solid curves indicate the ranges of parameters consistent with the experimental slopes (1.2 and 7.3 k$\Omega$/T) shown in panel (b).
• Figure 2: Energy spectrum of massless Dirac fermions [Eq. \ref{['Eq: model']}]. (a) In absence of magnetic field, (b) Landau levels with magnetic field $B=1.6$ T (assuming a constant line-width $\Delta=0.5$ meV), and (c) Landau levels with $B=6.5$ T and $\Delta=2$ meV. The Landau levels are denoted by indices $n = 0,\pm1,\pm2,\cdots$. The dashed line represents the Fermi level $\varepsilon_F$. The model parameter is $v=0.65$ eV$\cdot$nm adopted from Ref. Geim05nature. Feynman diagrams for (d) the self-consistent Born approximation, and (e) the screened length $\kappa$ of the Yukawa potential. In these diagrams, the double solid line represents the dressed Green function, while the single solid line represents the bare Green function, and the dashed line indicates impurity scattering. The solid wavy line represents interaction within the random phase approximation, and the dashed wavy line represents to the bare Coulomb interaction.
• Figure 3: (a) Comparison of calculated magnetoresistivities for different impurity potentials, as given by Eqs. \ref{['Eq: rho_d']}, \ref{['Eq: rho_G']}, and \ref{['Eq: rho_Y']}. The impurity strength is set to $\gamma = 0.1$ for both the $\delta$-function and Gaussian potentials. The decay length is taken $\lambda=4$ nm for the Gaussian potential. For the Yukawa potential, the impurity density is taken $n_i=0.1$ nm$^{-2}$. The other parameters are $\mathcal{C} \simeq 0.2$ and $g \simeq 1.22$ by adopting the experimental values $v=0.65$ eV$\cdot$nm Geim05nature and $\epsilon = 6.9\epsilon_0$Massimo16prb. The inset compares the original magnetoresistivity [Eq. \ref{['Eq: rho_G']}] with the approximated expression [Eq. \ref{['Eq: rho_G2']}] for the Gaussian potential. (b) Finite-temperature magnetoresistivity for the Gaussian potential at $T = 0$ K, 100 K, and 300 K.