Electron Hydrodynamics with X-momentum Conservation

Hicks, Andrew

Undergraduate Honors Thesis

Electron Hydrodynamics with X-momentum Conservation Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/jh343t989

Abstract

The flow of electrons in most materials is nearly Ohmic – that is the current density is uniform

and proportional to the applied voltage. In these materials the contributions from electron-electron

collisions are negligible when compared to electron-ion collisions and even moreso when compared

to the contributions from electron-impurity collisions [1]. Historically, the approach in solid-state

physics has been to treat these contributions collectively neglecting the nuance between momentum

relaxation and conservation – embodied in the Drude model under a single collision time τ [2].

However, as far back as the 1960s it has been suggested that hydrodynamic flow characterized

by viscous effects may be observed in ultra-pure, low temperature metals when electron-electron

interactions again become significant [3]. A schematic depicting these two types of flows is shown in

Figure 1, in the hydrodynamic flow the velocity is maximal at the center and suppressed along the

edges. This phenomena went mostly ignored until the early 21st century when rapid advancement

in the fabrication of ultra-pure materials enabled experimental detection of these effects. Since

then signatures of electron hydrodynamics has been detected in a variety of correlated electron

materials such as graphene [4], W T e2 [5], P dCoO2 [6], W P2 [7]. The purity of these materials may

be observed by the fact that the resistance of the materials is directly proportional to their scattering

rates. Experimental evidence for this flow is shown in Figure 2 . This technological advancement

and the commercial success of materials such as graphene has created renewed interest in this

field and necessitated the development of theories which accommodate the effects of hydrodynamic

effects. In this paper we will develop one such theory – but first we will provide a primer on modern

hydrodynamics.

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