Hidden anisotropy in the Drude conductivity of charge carriers with Dirac-Schrödinger dynamics

We show that the conductivity of a two-dimensional electron gas can be intrinsically anisotropic despite isotropic Fermi surface, energy dispersion, and disorder configuration. In the model we study, the anisotropy stems from the interplay between Dirac and Schrödinger features combined in a special two-band Hamiltonian describing the quasiparticles similar to the low-energy excitations in phosphorene. As a result, even scalar isotropic disorder scattering alters the nature of the carriers and results in anisotropic transport. Solving the Boltzmann equation exactly for such carriers with pointlike random impurities, we find a hidden knob to control the anisotropy just by tuning either the Fermi energy or temperature. Our results are expected to be generally applicable beyond the model studied here, and should stimulate further search for the alternative ways to control electron transport in advanced materials.

Figure 1

(a) The band structure given by the eigenvalues of (1) is parabolic and isotropic. The energy $E$ is measured from the middle of the band gap and characterized by the dimensionless parameter $\epsilon$ changing between 0 (band edge) and 1 (theoretical limit $\left|E\right|\gg \mathrm{\Delta }/2$). The in-plane pseudospin texture for conduction (b) and valence (c) bands suggests anisotropic scattering even for isotropic delta-correlated disorder. At $\epsilon \ll 1$, it emulates the pseudospin texture for carriers in phosphorene.

Figure 2

Conductivity $\sigma$ in units of the conventional Drude conductivity ${\sigma }_0=e^2{\tau }_{0}n/m$ as a function of the current direction in the metallic (a) and semiconducting (b) regimes. The Fermi energy is counted from the middle of the band gap, i.e., it intersects the bottom of the conduction band at $E_F=\mathrm{\Delta }/2$. The limit $E_F\to \infty$ is nonphysical and shown for the sake of completeness.

Figure 3

Pseudospin texture in true phosphorene: (a) conduction band, (b) valence band. The region in $k$ space shown has dimensions $1{\mathring{\mathrm{A}}}^{-1}\times 1{\mathring{\mathrm{A}}}^{-1}$. Our Hamiltonian (1) emulates this texture near the band edges ($\epsilon \ll 1$).

Figure 4

Scattering matrix element squared (B3) in units of $u_0^2$ as a function of scattering angle $\varphi$ for different incident angles shown by thick arrows: (a) ${\varphi }^{\prime }=0$, (b) ${\varphi }^{\prime }=\pi /2$. Different colors encode different $\epsilon$.

Figure 5

The anomaly parameter $a_{\epsilon }$ (red solid curve) and its low-energy approximation (dashed line).

Figure 6

Conductivity ratio ${\sigma }_{xx}/{\sigma }_{yy}$ (red solid curve) and its linear approximation (dashed line) in the metallic regime ($E_F>\mathrm{\Delta }/2$, $T=0$). Note the nonmonotonic dependence.