Interaction-controlled transport in a two-dimensional massless-massive Dirac system: Transition from degenerate to nondegenerate regimes

The resistivity of two-dimensional (2D) metals generally exhibits insensitivity to electron-electron scattering. However, it is worth noting that Galilean invariance may not hold true in systems characterized by a spectrum containing multiple electronic branches or in scenarios involving electron-hole plasma. In the context of this paper, we focus on 2D electrons confined within a triple quantum well (TQW) based on HgTe. This system displays a coexistence of energy bands featuring both linear and paraboliclike spectra at low energy and, therefore, lacks the Galilean invariance. This paper employs a combined theoretical and experimental approach to investigate the transport properties of this two-component system across various regimes. By manipulating carrier density and temperature, we tune our system from a fully degenerate regime, where resistance follows a temperature-dependent behavior proportional to $T^2$ to a regime where both types of electrons adhere to Boltzmann statistics. In the nondegenerate regime, electron interactions lead to resistance that is weakly dependent on temperature. Notably, our experimental observations closely align with the theoretical predictions derived in this paper. In this paper, we establish the HgTe-based TQW as a promising platform for exploring different interaction-dominant scenarios for the massless-massive Dirac system.

Figure 1

(a) The conduction and valence band edges of the triple quantum well are schematically shown. The widths $\mathit{d}$ of the HgTe wells and the thickness $d_b$ of the ${\text{Hg}}_{1-x}{\text{Cd}}_x\text{Te}$ barriers ($\mathit{x}$ = 0.3) are indicated. (b) The band structure of the triple HgTe well at $E>0$ is calculated using the tight-binding model described in the text. The black and blue lines represent dispersion curves for massless and massive Dirac fermions. (c) The density of carriers is shown as a function of the chemical potential. $n_e$ and $n_d$ represent the densities of massive and massless electrons, $N_s$ is the total density. An arrow indicates the beginning of the $E^{+}$ subband populations. (d) The resistance of the 6.4 nm sample is plotted as a function of carrier density at T = 4.2 K.

Figure 2

Resistance as a function of the gate voltage at different temperatures for two HgTe triple quantum wells.

Figure 3

Excess resistivity $\mathrm{\Delta }\rho \left(T\right)=\rho \left(T\right)-\rho (T=4.2\text{K})$ as a function of the temperature for various densities for samples (a) A and (b) B. The red lines show $T^2$ dependence. The values of the densities are given in ${10}^{11}{\text{cm}}^{-2}$.

Figure 4

Excess resistivity $\mathrm{\Delta }\rho \left(T\right)=\rho \left(T\right)-\rho (T=4.2\text{K})$ as a function of the temperature for two different densities for sample A. The total density is (a) $N_s=5\times {10}^{11}{\text{cm}}^{-2}$and (b) $N_s=4.1\times {10}^{11}{\text{cm}}^{-2}$. Circles represent the experimental data, and the red lines represent the theory of interparticle scattering between the fully degenerate massless and massive electrons.

Figure 5

Excess resistivity $\mathrm{\Delta }\rho \left(T\right)=\rho \left(T\right)-\rho (T=4.2\text{K})$ as a function of the temperature for two different densities for sample B. The total density is (a) $N_s=4.2\times {10}^{11}{\text{cm}}^{-2}$and (b) $N_s=3.3\times {10}^{11}{\text{cm}}^{-2}$. Circles represents the experimental data, and the red lines represent the theory of interparticle scattering between the fully degenerate massless and massive electrons.

Figure 6

Temperature dependence of the relaxation rate for different density ratios $n_e/n_d$. Solid lines are computed using Eqs. (13) and (15) $[1/{\tau }_{de}^{*}$, dashes are computed using Eqs. (14) and (16) ($1/{\tau }_{ed}^{*})$], dot-dashes represent $1/{\tau }_{\text{int}}^{*}$. Total density $N_s=3\times {10}^{10}{\text{cm}}^{-2}$, (a) ratio $n_e/n_d=0.5$ (black lines) and (b) $n_e/n_d=1.5$ (blue lines).

Figure 7

(a) Resistivity $\rho \left(T\right)$ as a function of temperature for different densities for sample A. The total density is $N_s\approx 0$ (charge neutrality point, blue), $N_s\approx 3\times {10}^{10}{\text{cm}}^{-2}$ (red), $N_s\approx 6.1\times {10}^{10}{\text{cm}}^{-2}$ (black), $N_s\approx 8\times {10}^{10}{\text{cm}}^{-2}$ (cyan), and $N_s\approx 10\times {10}^{10}{\text{cm}}^{-2}$ (dark yellow). (b) Resistivity $\rho \left(T\right)$ as a function of temperature for different densities for sample B. The total density is $N_s\approx 0$ (charge neutrality point, blue), $N_s\approx 3\times {10}^{10}{\text{cm}}^{-2}$ (red), $N_s\approx 6.1\times {10}^{10}{\text{cm}}^{-2}$ (black), and $N_s\approx 8\times {10}^{10}{\text{cm}}^{-2}$ (cyan).

Figure 8

(a) The resistivity ratio$\rho \left(T\right)/\rho (T=4.2\text{K})$ as a function of temperature for different parameters calculated from Eqs. (5), (13), and (14). Total density $N_s=3\times {10}^{10}{\text{cm}}^{-2}$, the ratio ${\tau }_e/{\tau }_d=5,{\tau }_d=0.4\times {10}^{-12}\text{s}$, andthe ratio $n_d/n_e(T=4.2\text{K})=1.5$ (black), 1 (red), and 0.5 (blue). Parameter $\beta =1$.