Boltzmann-Langevin theory of Coulomb drag

Abstract

We develop a Boltzmann-Langevin description of Coulomb drag effect in clean double-layer systems with large interlayer separation d as compared to the average interelectron distance λF . Coulomb drag arises from density fluctuations with spatial scales of order d. At low temperatures their characteristic frequencies exceed the intralayer equilibration rate of the electron liquid, and Coulomb drag may be treated in the collisionless approximation. As temperature is raised the electron mean free path becomes short due to electron-electron scattering. This leads to local equilibration of electron liquid and consequently drag is determined by hydrodynamic density modes. Our theory applies to both collisionless and hydrodynamic regime and enables us to describe the crossover between them. We find that drag resistivity exhibits nonmonotonic temperature dependence with multiple crossovers at distinct energy scales. At lowest temperatures Coulomb drag is dominated by the particle-hole continuum, whereas at higher-temperatures of the collision-dominated regime it is governed by the plasmon modes. We observe that fast intralayer equilibration mediated by electron-electron collisions ultimately renders stronger drag effect.

Coulomb drag

“…From gauge invariance dr 1 Γ(ω) = dr 2 Γ(ω) = 0. The same result follows from the standard Kubo formula approach within the diagrammatic perturbation theory Kamenev and Oreg, 1995), memory function formalism (Zheng and MacDonald, 1993), and more general Boltzmann-Langevin theory of stochastic kinetic equation (Chen et al, 2015).…”

“…Other developments include mesoscopic fluctuations of Coulomb drag (Narozhny and Aleiner, 2000;Narozhny et al, 2001), frictional drag mediated by virtual photons (Donarini et al, 2003) and plasmons (Badalyan et al, 2007), exciton effects in semiconductors (Laikhtman and Solomon, 2006) and topological insulators (Mink et al, 2012), interlayer Seebeck effect (Lung and Marinescu, 2011) and spin drag (Badalyan and Vignale, 2009;D'Amico and Vignale, 2000;Duine et al, 2011Duine et al, , 2010Duine and Stoof, 2009;Flensberg et al, 2001;Glazov et al, 2011;Pustilnik et al, 2006;Tse and Das Sarma, 2007;Vignale, 2005). Recently, the focus of the theoretical work was shifted towards the drag effect in graphene-based devices (Narozhny, 2007;Narozhny et al, 2015;Song et al, 2013; and strongly interacting high-mobility double-layers with low-density carrier concentration (Apostolov et al, 2014;Chen et al, 2015).…”

“…The frequency integration is now cut off at v F q (or T d ), rather than T , which leads to the linear temperature dependence [first reported in Gramila et al (1991) and Solomon and Laikhtman (1991), see also Jauho and Smith (1993), and recently rediscovered in Chen et al (2015)],…”

“…Definitions of the three crossover scales are given in the main text. [Reproduced from Chen et al (2015).] Hu, 1994); and (ii) the plasmons acquire finite life-time (Hruska and Spivak, 2002;Mishchenko et al, 2004) that regularizes the pole in the interaction propagator.…”

“…In order to accurately describe the plasmon contribution to ρ D , one has to consider intralayer equilibration due to electronelectron collisions (Chen et al, 2015) which gives rise to two important features: (i) the polarization operator acquires nonvanishing spectral weight within the high frequency part of the spectrum at ω > v F q (Flensberg and FIG. 4 (Color online) Schematic illustration for the drag resistivity at high temperatures showing the plasmon peak at T ∼ T h .…”