Transport in Two-Dimensional Disordered Semimetals

We theoretically study transport in two-dimensional semimetals. Typically, electron and hole puddles emerge in the transport layer of these systems due to smooth fluctuations in the potential. We calculate the electric response of the electron-hole liquid subject to zero and finite perpendicular magnetic fields using an effective medium approximation and a complementary mapping on resistor networks. In the presence of smooth disorder and in the limit of a weak electron-hole recombination rate, we find for small but finite overlap of the electron and hole bands an abrupt upturn in resistivity when lowering the temperature but no divergence at zero temperature. We discuss how this behavior is relevant for several experimental realizations and introduce a simple physical explanation for this effect.

Figure 1

(a) A two-dimensional electron and hole mixture, red region, is driven by a bias voltage imprinted from the potential difference in the left and the right lead (side view). (b) Definitions of the energy scales: $\mathrm{\Delta }$ is the distance between the conduction and the valance band edge, which is taken to be negative when they overlap. ${\mu }^{\mathrm{ec}}$ is the electrochemical potential, which at equilibrium is identical for electrons and holes. The electrical potential $U(r)$ and thus also the chemical potentials of the electrons ${\tilde{\mu }}_n(r)$ and holes ${\tilde{\mu }}_p(r)$ vary smoothly in space due to the disorder. (c) The long-range fluctuations in the potential create regions where only electrons (red) or only holes (blue) are occupied, and regions where both carriers coexist (purple) (top view).

Figure 2

Resistivity $\rho$, Eq. (8), as a function of the gate voltage $V_g$ for different temperatures $T$, characteristic scale of the potential fluctuations $W=1$, vanishing electron-hole recombination rate $\gamma =0$, and (a) band overlap $\mathrm{\Delta }=-0.5$ and (b) $\mathrm{\Delta }=-0.025$. Solid curves are evaluated for disordered systems using the effective medium approximation, while the dashed line is evaluated for a clean system. For a small band overlap (b) the random local potential leads to an increase of resistivity.

Figure 3

Resistivity $\rho$ as a function of temperature $T$ for $V_g=-0.02$, $\mathrm{\Delta }=-0.025$, $W=1$, and $\gamma =0$ evaluated with the EMA, solid line, and resistor networks, squares. These data are compared with the resistivity $\rho$ in the presence of a weak electron hole recombination rate $\gamma =0.0001$, dotted line, and of the clean system $W=0$, dashed line. At low temperatures $\rho$ is considerably enhanced by disorder.

Figure 4

(a) Longitudinal ${\rho }_{xx}$ and (b) transverse ${\rho }_{xy}$ resistivity as a function of the magnetic field $B$ applied perpendicular to an electron-hole mixture for the same temperatures $T$ as in Fig. 2. The curves are taken at gate voltage $V_g=-0.16$, band overlap $\mathrm{\Delta }=-0.025$, disorder strength $W=1$, and relaxation rate $\gamma =0$.