Weak localization and antilocalization corrections to nonlinear transport: a semiclassical Boltzmann treatment

TL;DR

This work shows that weak localization and weak antilocalization corrections can strongly influence nonlinear (second-order) transport in inversion-broken 2D systems within a semiclassical Boltzmann framework. By solving the Boltzmann equation with a time-nonlocal collision kernel and applying it to a minimal Dirac-like dispersion with trigonal warping, it demonstrates that the nonlinear conductivity $\tilde{\sigma}$ can change sign as a function of the WL/WAL parameter $\alpha(0)$ and carrier density, including a sign change prior to any unphysical divergences. The authors connect their results to graphene-based heterostructures, showing qualitative agreement with experimental data where the nonlinear signal $V^{2\omega}_{xx}/\rho_{xx}$ exhibits sign reversal as gate voltage (density) is tuned. The findings highlight that quantum interference effects can manifest in nonlinear transport and offer density- and disorder-tunable signatures relevant for interpreting nonlinear responses in 2D materials.

Abstract

The nonlinear transport regime is manifested in the nonlinear current-voltage characteristic of the system. An example of such a nonlinear regime is a setup in which current is injected into the sample and the measured voltage drop is quadratic in the injected current. Such a quadratic nonlinear regime requires inversion symmetry to be broken. This is the same symmetry condition as one needs to observe weak antilocalization, which can be prominent in two-dimensional systems. Here, we study the effects of weak (anti)localization on second-order nonlinear transport in two-dimensional systems using the semiclassical Boltzmann approach. We solve for quasiparticle distribution function up to the second order in the applied external electric field and calculate linear and nonlinear conductivity tensors for a toy model. We find that localization effects could lead to a sign change of the nonlinear conductivity tensor -- a phenomenon observed in single-layer graphene devices.

Figures (2)

• Figure 1: Results for the minimal model and comparison with experiment: a) Fermi surface contours for the minimal model for $v_D = 1, \mu = 0.1$. For the blue curve $c=0.03$, whereas for the red curve $c=0.3$. The momentum is measured in units of inverse lattice spacing. We set the lattice constant $a=1$. b) Dependence of linear conductivity $\sigma$ Eq. \ref{['lincondres']} on $\alpha = \alpha(0)$ calculated for $\tau=10$. Two gray grid lines indicate $\alpha = \pm \tau$. Note that linear conductivity is not well defined for $\alpha < -0.1$ here as it becomes negative. c) Nonlinear conductivity $\tilde{\sigma} (\alpha)$ (see Eq. \ref{['nonlincondres']}) for the same value of $\tau$. Note the two types of discontinuities: for $\alpha=-0.1$ both linear and nonlinear conductivities diverge, while for $\alpha = 0.1$ only nonlinear conductivity is divergent. The discontinuity of $\tilde{\sigma} (\alpha)$ for positive $\alpha$ (weak localization case) has a similar shape to the Hall number discontinuity across the Van Hove singularity doping. Importantly, in the weak antilocalization regime $\alpha<0$, nonlinear conductivity changes sign for intermediate values of $\alpha$. In both plots we used $v_D = 1, \mu=0.1, c = 3.3 \times 10^{-2}$. d) Linear conductivity dependence on chemical potential for $\alpha>0$ (localization) and $\alpha<0$ (antilocalization) for $\ln \frac{l_{\phi}}{l} = 0.3$. The unphysical discontinuity in the antilocalization case at small $\mu$ is due to $\alpha \simeq 1/\tau$. For such values of $\mu$, our simplistic description clearly breaks down. e) Plot of ratio $\tilde{\sigma}/\sigma^2$ as a function of chemical potential for the same value of $\ln \frac{l_{\phi}}{l}$. Note the sign change for $\mu \sim 0.2$, which is away from the unphysical value regime of divergent linear conductivity. f) Experimental data extracted from Ref. He2022: nonlinear voltage drop $V^{2 \omega}_{xx}$ divided by linear resistivity $\rho_{xx}$ as a function of gate voltage $V_g$ which controls electrochemical potential in the system. As we show in the SI, $V^{2 \omega}_{xx}/\rho_{xx} \propto \tilde{\sigma}/\sigma^2$. Note the sign change of the signal and the overall similarity of the curve shape to the orange curve in panel e).
• Figure S1: Data analysis details: a) Extracted $\rho_{xx}$ vs $V_g$ data from Supplementary Fig. 5d of Ref. He2022 (scatter points) and interpolation of the data (continuous line); b) Extracted $\tilde{\sigma}_{xxx}$ vs $V_g$ data from Supplementary Fig. 5e of Ref. He2022; c) Calculated $- \frac{V_{x}^{2 \omega}}{\rho_{xx} }$ using raw data from panel b) and values of interpolation function from panel a). Panel c) is presented in the main text.