Kinetic theory of hydrodynamic electron liquids

## Abstract

The purpose of this Medium Compute grant is to numerically describe "hydrodynamic" heat and charge transport in materials with strong mutual interactions between electrons. We will do this using an existing code that evaluates the Boltzmann collision integral for electron interactions, which to date was a considerable challenge in the community. We have recently demonstrated that such calculations are possible using parallelised adaptive Monte Carlo integration methods [published in Phys. Rev. B 108, L121401 (2023) arXiv:2210.16300; and arXiv:2312.09977]. Our results are currently attracting much attention, but we are running our of resources to compute parameter values for different experiments and to include the effect of device constrictions, which requires extended data sets.
For background, hydrodynamic electron transport has recently been discovered in several materials and is marked by electrons behaving like a liquid when moving through the device, which is distinct from a conventional transport picture where they move like billiard balls (i.e., they only interact infrequently with each other and instead collide with impurities or device boundaries). Hydrodynamic transport can have important technological applications: For example, electrons flow with less resistance than in conventional transport, which could create devices with reduced energy losses. Moreover, heat is transported is very efficiently, which is potentially useful for heat removal in miniaturised future electronics. For these reasons, hydrodynamic electron transport is now extensively explored in experiments, which has only recently become possible due to advances in material fabrication.
The theoretical description of hydrodynamic materials, however, is difficult. There are kinetic Boltzmann equations for the electron dynamics, but the challenge is to solve these equations for strong collisions. These collisions are parametrised by so-called collision integrals, which are six-dimensional integrals over phase space with highly unevenly distributed integrands, and the current state-of-the-art are approximate solutions, which may be inaccurate and also miss fundamental physics.
We have recently demonstrated how to solve the collision integrals by developing a basis expansion of the electron distribution functions combined with an efficient high-performance parallel evaluation of the resulting matrix elements of the collision integrals. The numerical calculations rely on the "Cuba" package for adaptive multi-dimensional Monte Carlo integration, which was developed at the Max-Planck Institute for Physics in Munich. In a collaboration with Ulf Gran (Chalmers), we have already published first results, where we compute relaxation rates and the shear viscosity, which is the fundamental transport quantity of hydrodynamic electron gases.
This Medium Compute grant is supposed to consolidate this activity and to evaluate the collision integral for (i) larger basis dimensions (improving precision), (ii) different doping densities in GaAs (corresponding to different electron interaction strengths), (iii) and for higher angular eigenvalues (which is important to describe electron flow in finite constrictions). The aim of this work is to collaborate with experimental groups and describe their viscosity measurements; to provide an accurate description of heat transport in hydrodynamic materials; and to include the effect of finite device dimensions.