• 2D materials
• Electronic transport
• Hydrodynamics
• Many-body theory
• Plasmonics

This Thesis is devoted to the study of hydrodynamic transport in two-dimensional (2D)
electron liquids embedded in a solid state matrix.
Chapter 1 presents a brief description of two laboratory systems in which a 2D electron liquid
can be realized: the first is a semiconductor (e.g. GaAs/AlGaAs) quantum well, which hosts
an ordinary parabolic-band electron liquid. The other is graphene, which is an ideal 2D system
being only one-atom thick. The elementary excitations in a graphene sheet have a dispersion
relation that resembles that of ultra-relativistic (i.e. massless) Dirac particles. In this Chapter
we also introduce the basic concepts of the quantum theory of many-electron systems: linear
response theory and the Landau theory of normal Fermi liquids.
A key quantity in the hydrodynamic description of the electron liquid is the electron-electron
scattering time τee . This is evaluated from diagrammatic perturbation theory in the GW ap-
proximation. All these concepts are presented with reference to the 2D parabolic-band electron
liquid, but apply as well to graphene when this is sufficiently far from the neutrality point.
Chapter 2 contains a detailed discussion of the hydrodynamics of the 2D electron liquid. Start-
ing from a brief review of hydrodynamics and the derivation of its main equations from the
Boltzmann transport equation, we present the main peculiarities of the hydrodynamics of an
ideal electron liquid.
The hydrodynamic description is valid in principle only at frequencies much smaller than 1/τee .
However a suitable generalisation of hydrodynamics has been shown to exist also for frequen-
cies higher than 1/τee , in the so-called collisionless regime. The relations between transport
coefficients, such as bulk and shear viscosity, appearing in the Navier-Stokes equation and the
microscopic linear-response functions of the electron liquid have been studied in the literature
both in the collisional and in the collisionless regime. We give a brief review of these relations
and note the existence of two different behaviours of the electron liquid. In the low-frequency
(collision dominated) regime the electron liquid behaves like a normal liquid with a large shear
viscosity and a vanishing shear modulus. Conversely, in the collisionless regime, the electron
liquid behaves like an elastic solid with a finite shear modulus and a small shear viscosity.
We then discuss the regime of parameter space in which the hydrodynamic theory can be ap-
plied to describe transport in a real laboratory system. The main issue here is related to the
breakdown of momentum conservation in a solid state system due to collisions with impurities
and phonons. A precise set of inequalities between different length scales must be satisfied for
a 2D electron liquid to be properly described by hydrodynamics.
The last two Chapters present the main original results of this Thesis, which have been obtained
by applying hydrodynamics to two transport setups. The first problem we tackle is the study of
the impact of the viscosity of the 2D electron liquid on non-local resistance measurements. This
is motivated by the fact that, to the best of our knowledge, experimental measurements of the
1
viscosity of strongly interacting 2D electron liquids are still missing. To this end, we study the
effect of a current flow injected in a rectangular 2D electron liquid from a pair of side contacts.
We first investigate a steady-state (dc) situation and then a dynamical (ac) one. In the dc
regime, a crossover between “Ohmic” and viscosity-dominated transport occurs as a function
of the ratio between the kinematical viscosity ν and W 2 γ, where W is the width of the sample
and γ is the rate at which momentum is damped by electron-impurity and electron-phonon
collisions. In the ac regime, a finite oscillation frequency suppresses the impact of viscosity.
The study of the ac regime leads to the identification of the normal modes of oscillation (plas-
mons) of the 2D electron liquid in a rectangular geometry. These depend strongly on the
screening of the electron-electron interaction by nearby conductors and dielectrics. These elec-
trostatic effects are discussed at a great length, both analytically and numerically. The main
effect of viscosity is to damp these self-sustained oscillations. The excitation of these modes by
an oscillating current of sufficiently high frequency is studied by taking the effect of viscosity
into account.
Due to the non-linearity of the Navier-Stokes equation, the response of the system to a pertur-
bation at a given frequency contains, at second order, also a steady component. For this reason,
the propagation of a plasmon mode generates a steady potential disturbance in the sample.
As a final result, we calculate this steady component of the potential that can in principle pave
the way for an all-electrical detection of plasmons.
The results obtained in the steady and low-frequency regime can be applied to the determination
of the electron liquid viscosity, while the ones relative to frequencies larger than the characteris-
tic frequency of plasma waves can be relevant in the newly emerging field of graphene plasmonics.