The role of the exchange-Coulomb potential in two-dimensional electron transport

TL;DR

The study develops a quantum kinetic theory for 2D electron gases that includes exchange at the Hartree–Fock level by deriving a Hartree–Fock–Wigner equation for the Wigner function $f_{oldsymbol{k}}(oldsymbol{r},t)$ with a self-consistent, momentum-dependent Fock potential $ ext{Φ}^ ext{F}(oldsymbol{r},oldsymbol{k},t)$. A semiclassical fluid closure introduces exchange-c corrected pressure and force, yielding a modified degenerate plasmon dispersion and, in bilayer geometries, exchange-driven instabilities and acoustic–optical coupling. Numerical simulations of the full kinetic equation reveal nonlinear exchange effects on screening, bilayer charge-imbalance patterns, and a substantial enhancement of Coulomb drag at low density that quantitatively matches experiments in dilute GaAs bilayers. The results demonstrate that exchange is a dynamic, nonlocal field that can dominate in degenerate 2D systems, offering a framework applicable to plasmonics, hydrodynamics, and spin-transport in low-dimensional materials.

Abstract

We develop a quantum kinetic theory of two-dimensional electron gases in which exchange is treated self-consistently at the Hartree-Fock level and enters as a nonlocal, momentum-dependent field in phase space. By starting from the Coulomb Hamiltonian, we derive a Hartree-Fock-Wigner equation for the electronic Wigner function and obtain a closed fluid model with exchange-corrected pressure, force, and current. For a single layer, we show that exchange renormalizes the Fermi velocity and can drive a long-wavelength plasmonic instability at low densities. In coupled layers, the same framework predicts acoustic-optical mode coupling, and an instability forming long-lived charge-imbalance patterns that are not predicted by classical Vlasov and Boltzmann models. Finally, we apply the kinetic model to the Coulomb drag problem and show how exchange substantially enhances the drag resistivity in dilute GaAs double wells, quantitatively matching experimental observations.

Figures (14)

• Figure 1: Normalized Fock potential of Eq. \ref{['ExchH']}.
• Figure 2: Real (a) and imaginary (b) parts of the plasmon dispersion including the effect of exchange (solid curves) and neglecting exchange (dashed curve) for varying densities $n_0 = 10^{10}\, \text{cm}^{-2}$ (orange) and $n_0 = 10^{11} \,\text{cm}^{-2}$ (blue) with equilibrium velocity $u_0 = 5\times 10^{4} \,\text{m}\text{s}^{-1}$. (c): real part of the plasmon dispersion for growing values of the density. The solid curve corresponds to the critical density ($n_c \sim 2\times 10^{14}\, \text{cm}^{-2}$) above which the dispersion is real. ($m = 0.067 m_e$)
• Figure 3: Acoustic and optical plasmons (a) if exchange is neglected and (b) if exchange is considered, the latter displaying acoustic-optical mode coupling ($n_{0,2}/n_{0,1} = 0.1$, $u_{0,2}/u_{0,1} = 0.5$, $d=1\, \textup{\AA}$).
• Figure 4: Comparison between the gains of (a) lower acoustic and (b) lower optical modes excluding and including exchange effects. The simulation parameters are those of Fig. \ref{['plasmons_CD']}.
• Figure 5: Mean ratio between the effective Fock and Hartree forces, $R = \langle |F^{F}| / |F^{H}| \rangle_{x}$, as a function of the dimensionless modulation wavelength $k_{F}\lambda$ and reduced temperature $T/T_{F}$. Exchange is negligible in the upper left blue region ($k_{F}\lambda \gg 1$ or $T \gg T_{F}$), and becomes comparable to or larger than the Hartree force in the red region where $k_{F}\lambda \lesssim 1$ and $T \lesssim T_{F}$.