Hydrodynamic transport and viscosity in two-dimensional conductors

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Universit`

a di Pisa

Corso di Laurea in Fisica

Anno Accademico 2013/2014

Tesi di laurea magistrale

Hydrodynamic transport and

viscosity in two-dimensional

conductors

Candidato:

Iacopo Torre

Relatore:

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Introduction

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 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

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regime, a crossover between “Ohmic” and viscosity-dominated transport occurs as a function of the ratio between the kinematical viscosity ν and W2_{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.

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Chapter 1

Fundamentals of two-dimensional

electron liquids

In this Chapter we introduce the basic concepts used in the study of two-dimensional electron liquids (2DELs).

We start from a brief description of two laboratory systems in which a two-dimensional electron liquid can be realised: GaAs/AlGaAs heterojunctions and graphene. Then we explain the basic theoretical tools used in the description of the homogeneous electron liquid. These are presented referring to the usual, parabolic-band, electron liquid but can be applied as well to graphene away from the neutrality point. Our main purpose in presenting this theoretical material is to give bridges between the microscopic quantum theory of the electron liquid and the hydrodynamic theory developed in the next Chapter.

1.1 Experimental realisations of the two-dimensional electron

liquid

The physics of reduced dimensionality electron systems is one of the most active research fields of modern condensed matter theory. In the last 40 years many models, that where once stud-ied only as academic exercises or as rough approximations of real three dimensional systems, became practically realisable, thanks to advancements in semiconductor technology, and their experimental investigation begun. Most of these system are of great interest both in the study of fundamental physical phenomena and for practical applications.

From the point of view of the study of many-body effects two-dimensional systems offer the great advantage over 3D systems of allowing a simpler tuning of the electron density trough the application of external potentials. This is usually not possible in 3D systems because of the screening effect of electrons that limits the action of an external potential to a region with a thickness of the order of the screening length near the surface of the sample.

Moreover the systems here discussed have both a simple band structure, well described, in the region of interest, by only one parameter (the effective mass in GaAs and the Fermi velocity in graphene). The study of many-body effects in these systems is therefore not complicated by band structure effects that usually occur for example in 3D metals.

To appreciate the importance of two-dimensional electron systems in practical applications it is sufficient to remember that a two-dimensional electron liquid is realised in the channel of Field Effect Transistors (FET) that have substituted the Bipolar Junction Transistor (BJT) of Bardeen, Brattain and Shockley in most modern electronic technologies. On the other side graphene is a recently discovered material and the technological applications of its extraordinary

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properties are only at the beginning.

1.1.1 GaAs/AlGaAs heterostructures

GaAs/AlGaAs heterostructures were developed in late ’70s [1] and since then greatly improved in terms of cleanness and ranges of electronic density that can be reached, as emphasized in Fig. 1.1. They have become almost the standard for the study of the physics of two-dimensional electron in solid state.

Many important discoveries have been made on these systems. One of the most famous among them being the discovery of fractional quantum Hall effect [2] that was allowed by the availability of such clean structures. It must be however pointed out that this effect was discovered using samples with much lower mobilities than those of the samples available nowadays. A lot of new physical phenomena have became accessible to experimental verification with further increase of the samples quality.

On the practical side quantum wells are now routinely employed by the semiconductors industry for example in the production of quantum well-based lasers [3].

These structure are built by growing, through a very precise technique called Molecular

Figure 1.1: Electron mobility as a function of temperature in GaAs heterostructures. The advancements of technology in terms of record mobilities are clearly visible. Taken from Ref. [4].

Beam Epitaxy (MBE)[5, 4, 6], layers of different semiconducting materials one on top of each other with a precision of the single atomic monolayer. Many semiconductors can be grown in this way but the choice of GaAs and AlxGa1−xAs has the great advantage of having very

little lattice mismatch between the two materials. This means that little strain is created at the interface of the two semiconductors, hence reducing also the number of surface defects. Moreover the properties of AlxGa1−xAs can be almost continuously tuned by varying the x

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Gallium arsenide and aluminium gallium arsenide, for sufficiently low aluminium concentration, are both intrinsic, direct-gap, semiconductors with zinc-blende crystal structure. They have a single, non-degenerate, conduction band whose minimum lies at the Γ point of the Brillouin zone. If an electron is added to one of these materials it behaves like a particle with an effective mass determined by the curvature of the conduction band at its minimum. In gallium arsenide this effective mass is m∗ = 0.067mewhere me is the bare electron mass.

Gallium arsenide and aluminium gallium arsenide have however different band gaps. This means that when the two materials are brought together the conduction band minima are offset by an amount of some hundred of meVs. By sandwiching a thin layer of ≈ 10nm of GaAs between barriers of the higher gap AlGaAs one can create a structure in which the spatial profile of the conduction band minimum has the form of a square well in the growth direction of the structure, as shown in Fig. 1.2, while it is invariant under translations in the plane orthogonal to this direction. The conduction band minimum acts as an effective potential for electrons added to the structure that can be approximately described by the single particle Hamiltonian:

Hsp =

p2

2m∗ + Ec(z). (1.1)

The eigenstates of this Hamiltonian are found to be in the form ψnk⊥(r⊥, z) = φn(z)e

ik⊥·r⊥

corresponding to energies Enk⊥ = n~

2_k2

⊥/(2m∗) where φn and n are the eigenfunction and

eigenvalues of the one dimensional problem

− ~ 2 2m∗ d2 dz2 + Ec(z) φ(z) = φ(z). (1.2)

Figure 1.2: Schematic representation of the conduction band minimum and the valence band maximum in the growth direction of a GaAs/AlGaAs quantum well. The two lowest discrete levels of the well are represented with the qualitative behaviour of the corresponding wavefunc-tions.

If the depth of the potential well is large enough the penetration of the electronic wavefunc-tions in the barriers can be neglected, at least for the lowest lying levels. The energy levels can then be interpreted using the simple infinite well result of elementary quantum mechanics n ≈ ~2π2n2/(2m∗L2), where L is the width of the well. By choosing a suitably small L we

can create a structure in which the lowest level is well separated from all the excited levels, the latter being much higher in energy. The electrons trapped in the lowest level of a quantum well

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provide one of the closest realisation of an ideal two-dimensional electron liquid.

This is the structure of single particle energy levels in quantum wells. However, since both GaAs and AlGaAs are intrinsic semiconductor, these levels remain empty unless extra electrons are added to the structure. These electrons must be added by doping. There are essentially two ways to do this: the first is to put donor atoms directly in the well, while the second, called modulation doping, consists in putting an high concentration of donor atoms, for example sili-con, in the barrier at a distance of some tens of nanometers from the well. In this way the extra electrons are able to diffuse to the well populating its energy levels.

Modulation doping has the great advantage of spatially separating the electrons from the donor atoms that, being positively charged, act as a source of scattering for the electrons in the well limiting their mobility.

The electron-electron interaction in these structures can be written, considering only the lowest subband as: Hee= 1 2 X q6=0 u(q) X kσ,k0_σ0 c†_k−qσc†_k0_+qσ0ck0_σ0c_kσ, (1.3)

where the matrix elements of the Coulomb potential are given by:

u(q) = Z dq⊥ (2π)2 4πe2 q2 ⊥+ q2 |F (q⊥)|2, (1.4)

where F is a form factor related to the z part of the wavefunction by: F (q⊥) =R dz|φ(z)|2e−iq⊥z.

For thin enough wells the interaction is essentially the ideal two-dimensional Coulomb potential v(q) = 2πe2_/q.

The typical range of electron densities that can be reached in these structures is 109_{− 10}11_cm−2

corresponding to values of the Seitz parameter of _{≈ 1 − 18.}

The mobility of the electrons, that is the parameter used to quantify the cleanness of these devices, has been greatly increased thanks to evolution of the fabrication technique reaching record values exceeding 107_cm2_{/V s at low temperatures [7]. The mobility decreases slowly}

when rising the temperature in the regime where the most important factor in limiting the mobility is the scattering against impurities, mainly because of the thermal reduction of the electron screening of impurities and interaction with acoustic phonons. At a temperature of about 35K the electron-phonon scattering becomes dominant because of the interaction with optical phonons and the mobility drops abruptly of several orders of magnitude. While the low temperature mobility could be further improved in the future by using more advanced techniques there is no hope to eliminate the fundamental limit imposed by the GaAs phonons.

1.1.2 Graphene

Graphene has been the first truly two-dimensional material to be discovered [8, 9, 10] and is composed by a single layer of carbon atoms arranged in an honeycomb lattice.

It has been known for a long time that graphite, one of the crystalline forms of carbon, is made up of one atom thick layers superimposed one on top of each other. These planes are kept together by Van der Waals forces that are much weaker than the covalent bonds that binds the atoms in the same layer among them. This fact has a great impact on the macroscopic mechanical and electrical properties of crystalline graphite that are then strongly anisotropic. The basic reason for which graphite can be used in pencils, as in other technological applications where it acts as solid lubricant, is that the carbon layers are loosely bound one to each other. However it was only in 2004 that a single layer of graphene was isolated by Geim and Novoselov from bulk graphite using a technique that still impresses for its simplicity. Acting on a face of an high quality graphite crystal with scotch tape the researchers were able to separate fragments

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of various size. After the tape was dissolved in a chemical solvent the resulting debris were deposited on a silicon substrate covered with a layer of silicon dioxide of the carefully chosen thickness of 300 nm. Under these conditions graphene flakes became visible with an optical microscope and could be spotted and separated from the thicker fragment of graphite.

Nowadays many other techniques have been studied to produce graphene on large scale for technological application but scotch tape exfoliation remains one of the best choices to obtain small samples of very high quality needed in fundamental research experiments.

From the crystallographic point of view, the honeycomb lattice is not a Bravais lattice but is made up of two compenetrating triangular lattices (see Fig. 1.3) shifted one with respect to the other, with carbon atoms occupying the nodes of the two lattices. The symmetry between the two sublattices has an important impact on the band structure properties of this material. The Brillouin zone of graphene is hexagonal and its high symmetry points are the center of the zone Γ, the midpoints of the sides M and the vertices of the hexagon K, K0 (see Fig. 1.4).

t1 t2

d1

d2

Figure 1.3: A portion of graphene lat-tice. Atoms belonging to the two sub-lattices are represented in blue and red. In green it is represented one pos-sible choice for the primitive cell while the vectors t1−2 = (±

√

3/2a, 3/2a) are a possible choice of primitive vec-tors. The vectors d1−2 = (0,±a)

are the basis vectors of the two sub-lattices. The interatomic distance is a = 0.142 nm.

b2 K' K b1

Γ M

Figure 1.4: First Brillouin zone of graphene with the high symmetry points highlighted.

Γ = (0, 0)

M = 2π/(3a)(0, 1) K = 2π/(3a)(1/√3, 1) K0 = 2π/(3a)(−1/√3, 1)

The vectors b1−2 = 2π/(3a)(±

√ 3, 1) are generators of the reciprocal lattice.

Most of the electronic properties of graphene are determined by the two bands that arise, in a tight binding picture, from the pz atomic orbitals of carbon. These bands are the graphene

equivalent of π molecular orbital in benzene.

The main features of these bands are qualitatively well described by the nearest neighbour tight binding approximation [11] that gives the analytical expression for the bands:

E±(k) = ±t|f(k)|

1_{± s|f(k)|}, (1.5)

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Sppπ = 0.129 is the overlap integral, and the modulus of the function f is |f(k)| = v u u t3 + 2 cos( √ 3kya) + 4 cos 3 2kxa cos √ 3 2 kya ! . (1.6)

The most peculiar fact is that these two bands touch one each other at the K and K0 points having a conical dispersion around these points. In undoped graphene the Fermi level fall exactly at the energy where these two bands touch. This makes undoped graphene a zero-gap semiconductor.

ak

_x

2 ₁

0

1

2 ak

y

2

1

0

1

2 E

[

eV

]

5

0

5

10

Figure 1.5: Electron band structure of graphene calculated in the nearest neighbour tight binding approximation (Eq. 1.5). The conduction band is represented in red while the valence band is represented in blue. Energies are calculated from the point where the two bands touch. The black contour marks the boundary of the first Brillouin zone.

The linear dispersion of the bands around the K and K0 points, that is maintained up to energies of the order of 1eV away from the Dirac point, allows a description of the low energy states of electrons in graphene based on an effective Hamiltonian of the type [9, 12]

H = ~vFσ· k, (1.7)

where vF is the Fermi velocity that in graphene is about 108cm/s, k is the wavevector measured

from the K point and σ is a two-dimensional vector of Pauli matrices σxand σy acting on the

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with the true spin of the electron, distinguishes between states localised on the two sublattices. For example the eigenstates of σz with eigenvalue +1 are localised on the sites of sublattice A

while the eigenstates of σz with eigenvalue −1 are localised on the sites of sublattice B.

This Hamiltonian is equivalent to the Dirac Hamiltonian for massless particles in 2+1 dimensions with the speed of light replaced by the Fermi velocity. This is one of the reasons for which graphene has attracted a lot of attention from the theoretical point of view. Many interesting and still not completely clear physical phenomena manifest in graphene near charge neutrality due to the vanishing density of states at the Fermi level.

However in this work we will consider only situations where the Fermi level is shifted away from the Dirac point by doping or electrostatic gating. In this case graphene hosts a liquid of electrons (or holes for p-doping) that displays a behaviour similar to that of the traditional parabolic band 2DEL. Apart from differences introduced by the different single particle part of the Hamiltonian most of the theoretical tools used in the theory of the 2DEL can be applied to graphene away from the neutrality point.

The strength of electron-electron interaction in graphene is not controlled by the Seitz radius but by the value of an effective fine-structure constant given by αee = e2/(~vF) ≈ 2.2/

where is the average dielectric constant of the materials that are on the two sides of the graphene sheet. Here is the only parameter that can be tuned to control the importance of the electron-electron interaction, and for typical values of the dielectric constants of the surrounding materials graphene is in a correlated regime out of the applicability of perturbation theory for the electron-electron interaction.

The electron mobility in graphene sheets strongly depends on the method used to produce the graphene flakes and on the characteristics of the substrate over which graphene is deposited. For this reason it is difficult to give a complete review of the values of the mobility that can be reached in various situations.

The mobility of electrons in graphene has a weaker temperature dependence, if compared to that of GaAs, up to room temperature. Even at this temperature the mobility is often impurity-limited. This happens because graphene is a very stiff material, because of the strong bonds between carbon atoms, and the energy scales associated with lattice vibrations are higher than the energy associated to the room temperature 300KkB ≈ 26 meV. This allowed for example

the observation of quantum Hall effect at room temperature [13, 10].

1.2 Fundamentals of the quantum theory of the electron liquid

The ideal two-dimensional electron liquid is described by the Hamiltonian:

H =X k,σ k,σc†_k,σck,σ+ e2 2 X q6=0 vq X k,σ,k0_,σ0 c†_k−q,σc†_k0_+q,σ0ck0_,σ0c_k,σ, (1.8)

where k,σ = ~2k2/(2m) are the single particle energies, while the Fourier transform of the

Coulomb potential is v(q) = 2πe2_{/q. This Hamiltonian is the sum of a kinetic single particle}

term and a two particles interaction terms. For graphene the expression of the single particle energies is different from that of the traditional electron liquid becoming k,σ= vFk. In addition

to this, bands and valley indices must be introduced to completely identify the single particle states. This leads to some complication and from now on in this Chapter we will refer to the parabolic band electron liquid. Almost all the theoretical material we present is discussed in greater detail in Ref. [14] and references quoted therein.

The properties of this model at zero temperature are determined by a single dimensionless parameter called Seitz radius and usually referred as rs. This parameter expresses the radius

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of the circle containing on average one electron in units of the effective Bohr radius: n = (πa∗2

Br2s)−1. The Seitz parameter controls the relative magnitude of the kinetic energy with

respect to the interaction energy. Since the first, when expressed in adimensional units, is proportional to r−2

s while the latter is proportional to rs−1 we can say that for rs 1 the

properties of the electron liquid can be understood in terms of that of a gas of non interacting particles treating the interaction as a small perturbation while in the rs ≈ 1 regime the two

effects are of the same order of magnitude. Finally for rs 1 interaction effects become

dominant with respect to kinetic energy. In a quite counter intuitive way the effect of interaction becomes less and less important when the density increases. This behaviour is fundamentally determined by the Pauli principle that forces the Fermi energy to increase more rapidly than the Coulomb interaction energy when the density is increased.

1.2.1 Phase diagram of the 2D electron liquid

Not much is rigorously known about the phase diagram of the electron liquid as a function of rs and T /TF. From the theoretical point of view the most reliable informations comes from

Quantum Monte Carlo simulations [15, 16], while the experimental investigation is usually lim-ited to moderate values of rs, because at low density disorder effects become dominant, the

characteristics energy scales associated with the unavoidable disorder present in real samples becoming of the same order of the Fermi energy.

However the most widely accepted conjecture about the phase diagram predicts that the ground state of the electron liquid is an unpolarised fluid until a critical value r_s _{Bloch where there} is a phase transition called Bloch transition to a phase with a finite spin polarisation. At an even higher critical value of the Seitz parameter r_s _{Wigner the electron liquid is believed to} crystallise in a hexagonal crystal, known as Wigner crystal [17], in order to minimise the elec-trostatic interaction energy. However most of experimental realisations of the two-dimensional electron liquid squarely fall in the region of the normal liquid behaviour making the search for exotic phases of the electron liquid a very challenging task.

A very important feature of the normal liquid phase at temperatures small with respect to the Fermi temperature, is that its low energy excitation can be understood in term of an adiabatic switching on of the interaction starting from an ideal gas of non-interacting particles even in the regime where the interaction cannot be treated perturbatively. This is the key concept of the Landau theory of the normal Fermi liquid that was originally formulated to explain why the picture of the Sommerfeld model of non interacting electrons qualitatively applies to simple metals whose typical range of Seitz radius is 2 < rs< 6, clearly out of the region where

inter-actions can be treated perturbatively.

1.2.2 Linear-response theory of the 2D electron liquid

One of the most important quantities calculated in quantum many-body theory are the linear response functions. These describe the evolution of an observable of the system as a consequence of an external perturbation that acts on the system coupling to another observable.

The linear response of an homogeneous electron liquid to a general external electromagnetic field is encoded in the current-current response function. For a translationally invariant system this expresses the proportionality between an applied vector potential with a certain wavevector and frequency and the induced current at the same wavevector and frequency. It can be separated into a longitudinal and transverse part:

j_{L(T )}(q, ω) = e

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where the subscript L or T identifies, respectively, the component of a vector in the direction of the wavevector and in the orthogonal direction. The longitudinal response is closely related to the density-density response function that encodes the response to the application of a scalar potential by

χnn(q, ω) =

q2

ω2χL(q, ω), (1.10)

as it can be verified by a gauge transform that eliminates the scalar potential in favour of a longitudinal component of the vector potential. Obviously the exact form of these response functions is not known and we must find some approximations scheme that permits to elucidate the essential physical features of the linear response of the electron liquid.

Non-interacting response functions

The crudest approximation for the response functions consists in completely neglecting electron-electron interactions and treating the electron-electron system as an ideal gas of non interacting fermions. This approximation is reliable only in the regime of extremely low rs, that is not the situation

we are interested in. However, the so calculated response functions, are essential ingredients of more advanced approximations. For a free electron gas the longitudinal and transverse response functions can be evaluated directly from general formulas of linear response theory:

χ(0)_{L(T )}(q, ω, T ) = n m + ~2 m2_L2 X k,σ k_{L(T )}2 nk,σ− nk+q,σ ~ω + k,σ− k+q,σ+ i~η , (1.11)

where kL(T ) is the component of k parallel (perpendicular) to q and nk,σ are the Fermi

occu-pation numbers at k,σ and temperature T. The calculation of the momentum sum leads to the

explicit results in two dimensions and at zero temperature for the non interacting density-density response function also called Lindhard function [18]:

<e [χ(0)nn(q, ω, T = 0)] = −X σ Nσ 1 + 1 ¯ qσ sgn(ν−σ)θ(ν−σ2 − 1) q ν2 −σ− 1 − sgn(ν+σ)θ(ν+σ2 − 1) q ν2 +σ− 1 =m [χ(0)nn(q, ω, T = 0)] =− X σ Nσ 1 ¯ qσ θ(1_{− ν}−σ2 ) q 1_{− ν}2 −σ− θ(1 − ν+σ2 ) q 1_{− ν}2 +σ , (1.12)

where Nσ = m/(2π~2) is the 2D density of states per unit surface at the Fermi energy,

¯

qσ = q/kF σ and ν±σ = [(~ω)/(EFq¯σ)± ¯qσ] /2. At finite temperature a closed expression of

the Lindhard function in terms of known functions does not exists but the Lindhard function can be expressed using a trick due to Maldague [19] as an integral over the Fermi energy of the zero temperature Lindhard function. This one dimensional integral must however be evaluated numerically as we will do in the numerical calculations at the end of this Chapter.

The equivalent expressions for the longitudinal and transverse non-interacting response func-tions of graphene can be found in [20].

Figure 1.6 shows the imaginary part of the Lindhard function as a function of q and ω. We note immediately that the imaginary part of this response function, that describes the power dissipated by an external field acting on the electron liquid, is zero outside the region delimited by the two parabolas ~ω = EF(¯q2± 2¯q). This is because the only way for an external potential

to feed energy into a free electron gas is to create electron-hole pairs moving an electron from inside the Fermi sphere to outside the Fermi sphere. At zero temperature since all the states under the Fermi energy are occupied and all the states above the Fermi energy are empty the

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Pauli principle imposes constraints on the momentum and energy of the electron-hole pairs that can be created limiting the absorption to the above described region.

0 1 2 3 q/kF 0 1 2 3 ¯hω /E F 0.0 0.5 1.0 1.5 2.0

Figure 1.6: Color plot of_{−=m [χ}(0)nn(q, ω)]/N (0). Thin black lines mark the boundaries of the

electron-hole continuum.

Random phase approximation

A better approximation is to consider the response of electrons as the response of independent particles to an effective field given by the external field plus the electric field generated by the induced charge. This approximation, called for historical reasons Random Phase Approximation (RPA), is the simplest approximation that takes into account the interaction between electrons albeit in a mean field fashion. Using this idea we can write:

n1(q, ω) = χ(0)nn(q, ω)Vsc(q, ω), (1.13)

where the effective potential is given by:

Vsc(q, ω) = Vext(q, ω) + vqn1(q, ω). (1.14)

Solving these equations we can find the RPA density-density response function:

χRPA_nn (q, ω) = χ

(0) nn(q, ω)

1_{− v}qχ(0)nn(q, ω)

. (1.15)

A concept that will be useful in the following is that of proper response function conventionally called ˜χ that encodes the response to the screened potential Vsc instead of Vext. The RPA

consists evidently in setting ˜χnn= χ(0)nn.

The most important new feature of this approximation is the prediction of the existence of col-lective modes of oscillation of the charge density called plasmons. These manifest themselves as delta function singularities in the absorption spectrum due to the vanishing of the denominator of (1.15). The frequency dispersion of plasmons predicted by the RPA in two dimensions is given by: Ωp(q) = r 2πne2_q m v u u t (1 +√q¯ 2rs) 2_{(1 +} q¯3 2√2rs + ¯ q4 8r2 s) 1 + q¯ 2√2rs . (1.16)

The plasmon dispersion lies outside the electron hole continuum but for large enough q it ap-proaches the boundary of this region and the plasmon actually disappears. The RPA does not

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predict a finite linewidth of these modes that at low q is due to exchange and correlation phe-nomena involving the excitation of multiple electron-hole pairs. These effects are not captured by the mean field approximation used in RPA.

We note that the leading order term of the previous expression, that encodes the long wave-length dispersion of plasmons can be obtained using the linearised Euler equation for a classical fluid of charged particles. This reads, in term of current density:

∂tJ1(r, t) = en m∇ Z dr0(−e)n1(r 0_{, t)} |r − r0_| . (1.17)

Making use of the continuity equation and switching to Fourier transformed quantities we obtain: ω2₋nq2vq m n1(q, ω) = 0, (1.18)

that leads to the low q limit of (1.16). Figure 1.7 displays the imaginary part of the RPA density-density response function for zero temperature and rs= 1.

0 1 2 3 q/kF 0 1 2 3 ¯hω /E F 0.0 0.1 0.2 0.3 0.4

Figure 1.7: Color plot of _{−=m [χ}RPA

nn (q, ω)]/N (0) for rs = 1. Thin black lines mark the

boundaries of the electron-hole continuum. The green solid line is the RPA plasmon dispersion while the green dashed lines is the small q approximation of the plasmon disperson.

If we try to apply the same reasoning to the transverse response we obtain no improvement with respect to the non interacting response function because a transverse oscillation of the current does not produce any charge inhomogeneity and hence no electrostatic correction to the external field.

Local field factors

Following the same ideas of RPA we can build more advanced approximations of the interacting response functions always considering the response of the electrons as the response of indepen-dent particles to an effective field. This time however the effective potential contains, beside the external potential and the self consistent Hartree potential considered in RPA, a new correction term that takes into account exchange and correlation effects. These corrections go under the name of local field factors and are the quantum equivalent for electrons of the classical local field corrections introduced by Clausius and Mossotti in their equation that relates the macroscopic dielectric constant to the molecular polarisability of liquids and simple solids. The idea at the basis of local field factors is in fact to take into account the difference between the field felt by

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an electron and the average field considered in RPA. AeffL(q, ω) = AextL(q, ω) + q2 ω2vqJ1L(q, ω)− q2 ω2vqG+L(q, ω)J1L(q, ω) AeffT(q, ω) = AextT(q, ω)− q2 ω2vqG+T(q, ω)J1T(q, ω). (1.19)

Obviously local field factors are not better known than the interacting response functions and must be approximated in some way. Different approximation schemes have been developed and the properties of local field factors can also be studied by Montecarlo techniques, at least for ω = 0. However we are not interested here in the microscopic calculation of these quantities, that we will assume to be known in some way, but on the form that the response functions assume when expressed using the local field factor especially in the long wavelength limit. The proper response functions can be written in this formalism as:

˜ χL(T )(q, ω) = χ(0)_{L(T )}(q, ω) 1 + vqG+L(T )(q, ω)χ (0) L(T )(q, ω) . (1.20)

The long wavelength limit of these response functions can be conveniently expressed introducing the exchange-correlation kernels defined, in terms of the low q limits of the local field factors, by:

fxcL(T )(ω) =− lim

q7→0vqG+L(T )(q, ω). (1.21)

For q_kF the response functions can be approximated by:

˜ χL(T )(q, ω) = n m 1 + αL(T ) EFq2 mω2 + fxcL(T )(ω) nq2 mω2 . (1.22)

where αL= 3/2 and αT = 1/2. From these expression we can show that the local field factors

induce a correction of order q2 _{to the long wavelength limit of the plasmon dispersion and also}

a finite plasmon linewidth. The long wavelength limit of the plasmon frequency as a function of wavevector in this approximation is:

Ω2_p(q) = ω2_p(q) + 3 4v 2 F+ n m<e fxcL(ωp) q2+_O(q4), (1.23) where ωp(q) =p2πe2nq/(2m) is the classical hydrodynamical plasmon dispersion. The linewidth

of the plasma excitations induced by the local field factors is instead found to be:

Γp(q) =

nq2

2mωp(q)=m f

xcL(ωp(q)). (1.24)

We reported here these two expressions because they acquire an interpretation in terms of an enhancement of the plasmon frequency due to the compressibility and the shear modulus of the electron liquid and a damping due to the viscosity as discussed in the next Chapter. The exchange correlation kernels introduced in this section have also a great importance in relation to the theory of time dependent density functional.

1.2.3 Landau Theory of normal Fermi liquids

Since the first observations of the electronic properties of metals it was noted that electrons seemed to behave like independent particles, like in the Sommerfeld model of conduction, also at

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densities where the interaction energy and the kinetic energy are of the same order of magnitude. The origin of this surprisingly ineffectiveness of the electron-electron interaction remained a mystery for decades until it was clarified by the introduction of Landau’s theory of the normal Fermi liquid. This theory describes the excitations of an interacting Fermi system (not only the electron liquid here considered but also liquid He3 _{and nuclear matter) at energies much lower}

than the Fermi energy in situations where rsis not too large and the temperature is much lower

than the Fermi temperature.

The basic idea of this theory is that the ground state and the lowest lying excited states of an interacting electron liquid can be obtained from the ground state and the excited states of a non interacting gas by an adiabatic switching on of the interaction. Starting from a state of the non interacting system described by the occupation numbers _Nk,σ and turning on the

interaction we obtain an approximate eigenstate of the system that can still be classified by the numbers_Nk,σ. These are not however the physical occupation numbers nk,σ=hc

†

k,σck,σi of the

interacting system but only approximate quantum numbers that classify the quasi-eigenstates of the system. For example the ground state occupation numbers of the interacting electron liquid has still a discontinuity at kF like in the non interacting situation but the value of this

discontinuity is less than one.

Since the above described states are not true energy eigenstates but only approximated ones they decay on a finite lifetime τk,σ. A fundamental point for the validity of the Landau theory

is that, at least for weakly excited states, the lifetime is very long with respect to the oscillation frequency associated to the energy of the excited state, that is τk,σk,σ/~ 1. The possibility

for these lifetimes to be very long depends fundamentally on two physical reason that are Pauli blocking and phase space limitations. When a quasi-particle tries to decay into a lower energy state almost all the possible final states compatible with momentum and energy conservation are already occupied. The decay is then prevented by the Pauli principle that operates irrespectively of how the interaction may be strong. Dimensionality plays an important role in limiting the phase space available for quasi-particles decay, forcing τk,σ to be proportional to (k− kF)−2 at

zero temperature in three dimensions and to (k− kF)−2(ln|k − kF|)−1 in two dimensions. This

ensures that τ_k,σ−1 goes to zero more rapidly than the energy k,σ≈ ~vF(k− kF) approaching the

Fermi surface.

In one dimension the situation is completely different because τk,σ behaves only like (k− kF)−1,

in this case the Fermi-Landau theory is not valid and the one dimensional electron liquid must be described under the different paradigm of the Luttinger liquid that has a completely different spectrum of excitations.

The Landau’s idea was based only on physical intuition and phase space arguments rather than on a rigorous derivation from the microscopic many-body theory. Later theoretical works however developed strong connections between the macroscopic Landau theory and the many-body theory based on the diagrammatic approach.

Here however we review only some aspect of the macroscopic theory that can be relevant to the hydrodynamic description of the electron liquid. In particular we will show that the long wavelength behaviour of the Landau-Fermi liquid can be described by a kinetic equation for quasi-particles from which hydrodynamics can be derived as from the Boltzmann equation. In the Landau theory of Fermi liquids the total energy of the fluid is expressed as a functional of the occupation numbers _{Nk,σ} of the non interacting system from which the system has

evolved, approximated to second order in the deviations of these numbers from their ground state values N0 k,σ= θ(kF− k), E[{Nk,σ}] = E0+ X k,σ k,σδNk,σ+ 1 2 X kσ,k0_σ0 fkσ,k0_σ0δN_k,σδN_k0_,σ0, (1.25)

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where k,σ = δE δNk,σ δNk,σ=Nk,σ0

are the quasi-particle energies while the function f , called

Lan-dau interaction function, and formally defined by fkσ,k0_σ0 =

δ2_E δNk,σδNk0,σ0 δNk,σ=Nk,σ0 describe the interaction between the quasi-particles. This function is obviously unknown but many im-portant quantities can be expressed in terms of only few parameters, called Landau parameters that describe the weight of the various harmonic components of the Landau interaction function evaluated at the Fermi surface. The Landau parameters are defined as follows:

F_ls,a= L

2_N∗₍₀₎

2 Z

dθ

2π[f↑↑(cos(θ))± f↑↓(cos(θ))] cos(lθ), (1.26) where the interaction function is evaluated for_{|k| = |k}0_{| = k}

F and θ is the angle between k and

k0.

These parameters are helpful to establish relations between various properties of the Fermi liquid and can be approximatively calculated using microscopic theories or by numerical Montecarlo techniques.

One of the most important results of the Landau theory is the derivation of a kinetic equation for the quasi-particles that describes the response of the liquid to the application of slowly varying, on the scales of k_F−1 and ~/EF, external fields.

The quantum mechanical uncertainties are negligible on the scales of variation of the external fields and the quasi-particles can be described by the classical Hamiltonian:

Hcl(r, ~k, σ) = kσ− eφσ(r, t) +

X

k0_σ0

fkσ,k0_σ0δN_k0_σ0(r, t). (1.27)

This Hamiltonian is self consistent since it takes into account the interaction between the quasi-particles in a mean-field approximation. Applying the Liouville theorem to this Hamiltonian leads to: ∂Nkσ(r, t) ∂t + 1 ~ ∂Hcl ∂k · ∂Nkσ(r, t) ∂r − 1 ~ ∂Hcl ∂r · ∂Nkσ(r, t) ∂k = ∂Nkσ(r, t) ∂t coll , (1.28)

where the collision term takes into account the collision processes that are not taken into account by the classical Hamiltonian being effects of correlations between the motion of particles that goes beyond the mean field approximation. When linearised with respect to deviations from the equilibrium distribution function the equation becomes:

∂δ_Nkσ(r, t) ∂t + vkσ· ∂δ_Nkσ(r, t) ∂r + vkσ· Fσ(r, t)δ(kσ− µ) = ∂δNkσ(r, t) ∂t coll , (1.29)

where the force acting on the quasi-particles is given by

Fσ(r, t) =−∇r " −eφσ(r, t) + X k0_σ0 fkσ,k0_σ0δN_k0_σ0(r, t) # . (1.30)

Using this equation it is possible to derive approximate expressions of the long wavelength limit of the response functions of the electron liquid in terms of Landau parameters in the collision-less regime χL(q, ω) = n m 1 +qvF ω 23 + 2Fs 0 + F2s 4(1 + Fs 1) χT(q, ω) = n m 1 +qvF ω 2 1 + Fs 2 4(1 + Fs 1) . (1.31)

Also these expression are reported because they are useful to establish connections between the results of the hydrodynamics and the Landau theory of the electron liquid.

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1.2.4 Electron-electron scattering time

The most important microscopic quantity of the electron liquid in relation to hydrodynamics is the electron-electron scattering time τee. This quantity relates to hydrodynamics in at least

three ways: it determines the electron-electron mean free path and hence the length scale where hydrodynamics can be applied, it fixes the time scale over which the behaviour of the electron liquid switches from a viscous to an elastic one and controls the value of viscosity in the collision dominated regime.

A good knowledge of this quantity is therefore of paramount importance for every discussion on the hydrodynamics of electrons.

The electron-electron scattering time can be expressed in terms of the electron self energy, a quantity that can be calculated by diagrammatic perturbation theory. Roughly speaking the self-energy represents the correction introduced by many-body effects to the energy of plane wave states with respect to the non-interacting energies k. The real part of this quantity is a

correction of the dispersion relation and in particular of the Fermi velocity, while the imaginary part introduces a damping rate to the plane wave states that are no more exact eigenstates of the system. For this reason it is natural to relate the electron-electron scattering time to the imaginary part of the self-energy calculated at the Fermi wavevector and energy as [14, 21]:

Γee= ~ τee = 2ZkF =m h Σ(R)kF, kF ~ i ≈ −2=m h Σ(R)kF, kF ~ i , (1.32)

where the normalization constant, ZkF has been neglected since it is of the order of 1 if rs is

not too high.

Like all the microscopic quantities of the electron liquid the self-energy is not exactly known and must be approximated in some way. Here we will use the simplest approximation that gives a non zero estimate for the imaginary part of the self-energy. This is the G0WRPA approximation

[22] that is popularly used to calculate quasi-particle properties from first principles.

An exact expression of the self-energy would require the knowledge of the interacting Green function, of the screened interaction and of the vertex correction. Our approximation con-sists in neglecting the vertex correction, approximating the interacting Green function with the non-interacting one and assuming as screened interaction the RPA screened interaction WRPA(q, ω, T ) = vq/RPA(q, ω, T ). We note that the previous order approximation that

ne-glects the screening of the interaction is simply the Hartree-Fock approximation that gives a purely real self-energy. On the other side trying to go beyond the G0WRPA, for example

intro-ducing vertex corrections, is a very difficult task that is outside the scopes of this work. The expression for the imaginary part of the G0W self-energy, as evaluated by Matsubara finite

temperature perturbation theory reads [14]:

=m hΣ(R)(k, ω)i = Z dq (2π)2v 2 q Z ∞ −∞ dω0_=m χnn(q, ω0) (nF(−˜k−q) + nB(˜k− ˜k−q)) δ ω_{− ω}0₋µ ~− ˜ k ~ , (1.33)

where ˜k= k−µ are the energies of the eigenstates of the non-interacting electron gas measured

from the chemical potential, χnn is the density-density response function and nF (B)(E) =

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using the random phase approximation for χnn this formula can be transformed into: =m hΣ(R)k,k ~ i = ~ (2π)2 Z ∞ −∞ dω 1− nF(˜k− ~ω) 1_{− exp(−β~ω)} − nF(˜k− ~ω) 1_{− exp(β~ω)} Z ∞ 0 dqq vq RPA(q, ω, T ) 2 =m hχ(0)(q, ω, T )iA(k, q, ω), (1.34)

where χ(0)_{(q, ω, T ) is the Lindhard function,}

RPA(q, ω, T ) is the RPA dielectric constant and

the functionA resulting from the angular integration is given by:

A(k, q, ω) = Z 2π 0 dθk,qδ(k− k−q− ~ω) = 2Θ1₋ q2+2mω ~ 2kq ~2kq m r 1₋ q2₊2mω ~ 2kq . (1.35)

Formula (1.34) is worth some comment: the two terms in the square bracket can be interpreted as the contributions to the lifetime of a state with wavevector k of the corresponding quasi-electron and quasi-hole states. For low temperatures the first term is non negligible only for positive frequencies due to the exponential in the denominator, at the same time the numerator is different from zero only if ˜k > ~ω. As a consequence at low temperature the first term

is important only for state above the Fermi surface. Following the same reasoning one can show that at low temperature the second term is non negligible only for states under the Fermi surface. If k = kF and T TF the two contributions are equal.

We observe moreover that the same expression for the lifetime of an electron could be obtained applying the Fermi golden rule with an effective interaction given by the RPA screened Coulomb interaction.

From this expression, the leading order contribution to τeein the limit T TF can be extracted.

This reads [23, 14]: ~ τee =₋π 4EFξ2(rs) T TF 2 ln T 4TF , (1.36) where ξ2(rs) = 1 + rs/(rs+ √ 2)2

/2 is a small, rs dependent, correction that comes from the

region of integration where q_{≈ 2k}F.

We note that there is some confusion in the literature on the numerical prefactor of formula (1.36). Some of this confusion is due to the correction factor ξ2 that is set to one in almost all

calculations with the exception of Ref. [24]. Even neglecting this correction the above presented result is different from those obtained in Refs. [21, 25] while becomes equivalent to those found in Refs. [26, 27]. A critical discussion of these different results can be found in Ref. [23]. When the temperature is lowered the electron-electron scattering time increases going to infinity at T = 0 because of the Pauli blocking effect. This is a fundamental requirement for the validity of the Fermi liquid theory in two dimensions.

As we will see, hydrodynamics is valid in a regime of parameters where the Fermi energy, the thermal energy and the interaction energy are of the same order of magnitude. In this regime the leading order approximation (1.36) cannot be trusted. For this reason we evaluated numerically the exact formula for the G0WRPA self-energy (1.34) up to temperatures of some TF for typical

values of the Wigner-Seitz parameter. Our results are shown in Fig. 1.8.

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10−3 10−2 10−1 100 ₁₀1 T /TF 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 Γee [Ryd ∗ ] rs=1.0 rs=2.0 rs=5.0

Figure 1.8: Electron-electron damping rate as a function of reduced temperature for different values of rs, in increasing order from the higher curve to the lower. Dots represent numerical

data, the full curves are the leading order contribution (1.36), the dashed curves are the empirical estimate τee= n−1/2/vF.

It turns out that the asymptotic formula (1.36) is in good agreement with numerical data up to temperatures close to TF.

Some authors [28] based their discussion on the validity of hydrodynamics on an empirical approximation for τee at temperatures near the Fermi temperature, where Pauli blocking is

suppressed: τee = n−1/2/vF. A more refined analysis based on the numerical evaluation of

the G0WRPA self energy does not justify in any way the choice of a characteristic value of τee

because this is a monotonically and rapidly decreasing function of temperature also near TF.

However this simple estimate gives the correct value of τee at a temperature of about 0.8TF

independently of the values of rs.

Very similar calculations can be done for graphene, the main differences being the introduction of an inter-band contribution and angle-dependent chirality factors. A derivation of the corre-sponding formulas for graphene can be found in Refs. [29, 30].

The quasiparticle lifetime in GaAs/AlGaAs quantum wells has been measured using momentum-conserving tunneling between coupled quantum wells [31]. The obtained results show a signif-icant discrepancy between the measured and calculated τee, whose origin has not yet been

completely clarified.

The important point for our subsequent discussions is that the measured lifetime, even after the subtraction of a disorder originated contribution, is shorter than the calculated one. This can be seen in the comparison between analytical formulas and the experimental data of Ref. [31] presented in Ref. [23]. We can therefore use the numerical data or the asymptotic formula (1.36) as a conservative upper bound for τee.

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Chapter 2

Hydrodynamic theory of a

two-dimensional electron liquid

In this Chapter we discuss the hydrodynamic description of transport in a two-dimensional electron liquid. We start from a brief introduction to hydrodynamics showing its fundamental equations and their derivation from the Boltzmann transport equation in the simple case of a liquid of classical particles interacting via short range forces. The basics steps for the calculation of the corresponding transport coefficients are also sketched.

After that, we analyse some specific feature of the hydrodynamics of the ideal electron liquid trying to establish contacts between hydrodynamics and the microscopic theories presented in the first Chapter. The hydrodynamics for two-dimensional Massless Dirac Fermions (MDFs) in graphene is also discussed and it is shown to give qualitatively similar results to the ordinary hydrodynamics with an appropriate substitution of parameters.

Electrons in solid state systems are however not ideal particles moving in vacuum but are sub-ject to the effects of the host lattice. These effects can limit the applicability of hydrodynamics to electrons, breaking momentum conservation. The inequalities that a system must fulfil in order to be properly described by hydrodynamics are discussed and the relevant length scales estimated. A phenomenological way to take into account the smooth disorder effects is also introduced.

At the end of the Chapter we write down the hydrodynamic equations for the two-dimensional electron liquid, which we will use in the original calculations presented in the subsequent Chap-ters.

2.1 Introduction to hydrodynamics

Hydrodynamics is a macroscopic theory that is useful to describe the behaviour of many par-ticles systems, either classical or quantum, when these are viewed at length scales much larger than the mean free path between interparticle collisions and on time scales much longer than the average time between collisions.

The basic idea underlying hydrodynamics is to consider, instead of the enormous amount of information contained in a complete description of the state of the system, that is in the coor-dinate and momenta of the particles for a classical system or in the many-body wavefunction in quantum mechanics, only few collective and slowly varying variables that still encode the important physics of the system.

Hydrodynamics has been known for a long time and used to describe the motion of classical fluids. Its equations are the continuity, Navier-Stokes and heat transport equations. These are

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expressed as [32] ∂tn(r, t) +∇ · (n(r, t)v(r, t)) = 0 mn(r, t) [∂tv(r, t) + (v(r, t)· ∇) v(r, t)] + ∇P (r, t) − ∇ · σ0(r, t) = n(r, t)F(r, t) ∂t 1 2mn(r, t)v 2_{(r, t) + n(r, t)u} +∇ · n(r, t)v(r, t) 1 2mv 2_{(r, t) + h} − v(r, t) · σ0(r, t) + q0 = n(r, t)F(r, t)_{· v(r, t) + Q(r, t),} (2.1) where n is the particle density, v is the local velocity of the fluid, P , u and h are the pressure, energy density and hentalpy density of the fluid, σ0 _{is the viscous stress tensor, q}0 _{is the heat}

flux, while F and Q are the external forces and heat sources.

Since the fluid is considered on length scales much longer than the interparticle mean free path the thermodynamic quantities appearing in the equations can be identified with the ones calculated for the fluid in equilibrium at the local density and temperature. Moreover σ0 and q0 can be considered as depending only on first derivatives of velocity and temperature respectively. The isotropy of the fluid imposes that the most general form for the viscous stress tensor is

σ_ij0 = η(n, T ) ∂ivj+ ∂jvi− 2 dδij∂kvk + ζ(n, T )δij∂kvk, (2.2)

where η and ζ are, respectively, the shear and bulk viscosity, d is the dimensionality of the sys-tem, while the heat flux is simply proportional to the temperature gradient q0 ₌_{−k(n, T )∇T .}

The first equation in (2.1) simply states that the number of particles in the fluid is conserved, the second expresses the fact that the momentum of a fluid element can be varied by the forces acting on the fluid elements that are the external forces F, the force due to the pressure gradient and the forces due to the viscosity of the fluid. The third equation is the expression of energy conservation: the time derivative of the energy density is the sum of the energy fed into the fluid element by external forces or heat sources minus the energy that is driven away from the fluid element, expressed by the divergence of a vector on the left hand side. This vector has three contributions that corresponds respectively to the transport of energy due to the flow of particles, the energy exchanged by viscous forces and the heat conducted without average current of particles due to the thermal conductivity of the fluid.

Setting the viscosities and the thermal conductivity to zero in the above set of equations leads to the equations for an ideal fluid, that are the Euler equation instead of the more general Navier-Stokes equation and the equation of entropy conservation that expresses the fact that no energy is dissipated in an ideal fluids.

The hydrodynamic equations can be derived as macroscopic equations relying only on macro-scopic laws of conservation of particle number, momentum and energy and on the assumption that the momentum and heat flows are respectively proportional to the velocity and tempera-ture gradients, neglecting the contributions from higher order derivatives of these quantities. In this framework the coefficients appearing in the equations, as the the viscosities or the thermal conductivity, are treated as phenomenological parameter that describe the behaviour of the various fluid.

With the birth of kinetic physics, whose cornerstone is the Boltzmann’s transport equation [33], the investigation of relations between the macroscopic transport coefficients appearing in the Navier-Stokes equations and the microscopic properties of the fluids at molecular level became possible at least for the simplest fluids like diluted gases. As we will show the equations of hydrodynamics can be derived from the moments of the Boltzmann equation.

After the discovery of quantum mechanics our knowledge of the behaviour of matter at the mi-croscopic level was improved allowing to consider appropriate generalisations of the Boltzmann

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transport equation to the quantum case.

Nevertheless the hydrodynamic model remains useful to describe the macroscopic behaviour of fluids also in situations where the microscopic behaviour is largely determined by quantum mechanics, that is in the case of quantum fluids [34]. This is true in particular when dealing with transport phenomena in spatially inhomogeneous systems for which the application of mi-croscopic many-body theory leads rapidly to cumbersome calculations.

Hydrodynamics is currently used to describe a variety of quantum fluids [35] ranging from the oldest known quantum fluid, that is liquid Helium, to the quark-gluon plasmas generated in heavy ions collision experiments and including the quantum fluids of cold atoms of both bosonic and fermionic type realised by trapping with optical lattices.

In this work we focus on the application of the hydrodynamic equations to the two-dimensional electron liquid that can be realised in solid state systems. The existence of situations where hydrodynamics can provide a useful description of the behaviour of two-dimensional electron liquids has already been noted in the literature for both GaAs/AlGaAs heterojunctions [28, 36] and graphene [37, 38].

Hydrodynamic electron flows have indeed been generated in real experiments and different hy-drodynamic effects have been measured. These include, for example, the transition between Poiseuille and Knudsen flow in semiconductor nanowires [39], the effect of the injection of hot electrons beams in a 2DEL [40, 41], and the Fermi liquid analogous of the Venturi effect [42]. Many theoretical works have also used hydrodynamics to describe the collective behaviour of Quantum Hall phases of the electron liquid [43, 44].

In a series of seminal papers [28, 45, 46, 47] Dyakonov and Shur (DS) used the hydrodynamic equations to predict a number of interesting non linear effects that can take place in the channel of a Field Effect Transistor (FET). Of particular interest is the possibility to use the hydro-dynamic nonlinearity to generate and detect radiation in the TeraHertz region of spectrum. This led to a substantial amount of experimental work both in traditional semiconductor FETs [48, 49] and in graphene-based FETs [50].

In my calculations of Chapter 3 and 4 I will give particular attention to the impact of viscosity on the electron flow. This is motivated essentially by two reasons.

The first is the lacking of experimental measurements of the electron liquid viscosity. As re-cently pointed out by Tomadin et al. [51], despite the theoretical interest of this quantity no viscometry experiments have been performed on the 2DEL. These authors proposed a method to measure the electron liquid viscosity that has not yet been implemented. In Chapter 3 I will analyse a different geometrical configuration in which an effect of the viscosity could be measured.

The second reason is the evaluation of the impact that the viscosity of the electron liquid can have on the functionality of devices proposed by DS and on similar devices.

To introduce the reader to the basic relations between hydrodynamics and the microscopic the-ory we present a brief derivation of the hydrodynamic equations from the Boltzmann equation following the treatment in reference [33]. For sake of clarity this is presented in the case of a classical gas of neutral particles interacting by short range forces. We defer the new features introduced by the long range Coulomb interaction and the quantum nature of electrons, in particular by the Fermi-Dirac statistics, as well as the discussion of the effects due to medium hosting the electron liquid to later sections.

2.1.1 Derivation of the hydrodynamics equations from the Boltzmann equa-tion

We start from the simplest case of a gas of classical particles interacting via short range two body forces and subjected to an external force field F. We assume that all the interesting

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statistical information about the state of the fluid is contained in the one particle distribution function f (r, p, t). This function expresses the average number of particles present in a small volume of phase space around the coordinates (r, p, t). The function f satisfies the Boltzmann transport equation

∂tf (r, p, t) +

pα

m∂rαf (r, p, t) + Fα(r, t)∂pαf (r, p, t) = I{f}(r, p, t), (2.3)

where I_{{f} is a functional of f that takes into account the effect of collisions between particles} and whose specific form is not necessary at this stage. Of crucial importance in the derivation are instead the following properties of I that reflect the conservation laws of particle number, momentum and energy during a collision between two particles:

Z dp I_{{f}(r, p, t) = 0} Z dp p I_{{f}(r, p, t) = 0} Z dp (p) I{f}(r, p, t) = 0. (2.4)

by multiplying the transport equation by respectively 1, pβ and (p) and integrating on the

momentum we obtain the three equations:

∂tn(r, t) + ∂rα(n(r, t)vα(r, t)) = 0

∂t(mn(r, t)vβ(r, t)) + ∂rαΠαβ(r, t)− Fβ(r, t)n(r, t) = 0

∂t(n(r, t)¯(r, t)) + ∂rαqα(r, t)− Fα(r, t)n(r, t)vα(r, t) = 0,

(2.5)

where the various quantities appearing in the equations are defined in terms of integrals of the distribution function as:

Density of particles n(r, t) = Z dp f (r, p, t) Local velocity vα(r, t) = 1 n(r, t) Z dp pα mf (r, p, t)

Momentum flux density Παβ(r, t) =

1 m Z dp pαpβf1(r, p, t) Energy density ¯(r, t) = 1 n(r, t) Z dp (p) f (r, p, t)

Energy flux density qα(r, t) =

Z

dp pα

m(p) f (r, p, t).

(2.6)

The above expression for momentum flux, energy density and energy flux may be simplified by switching to a frame of reference that moves with the same local velocity of the fluid. The momentum p0 measured in this reference frame is related to the momentum in the rest reference frame by a Galileian transformation p = p0_{+ mv. This allow us to write:}

Παβ(r, t) = 1 m Z dp0 (mvα(r, t) + p0α)(mvβ(r, t) + p0β)f (r, p 0 + mv, t) = mn(r, t)vα(r, t)vβ(r, t) + δαβ md Z dp0 p02f0(r, p0, t) + 1 m Z dp0 p0_αp0_β−δαβ d p 02 f0(r, p0, t) = mn(r, t)vα(r, t)vβ(r, t) + P (r, t)δαβ + σαβ0 (r, t). (2.7)

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where f0 _{is the distribution function of the fluid at rest. In a similar way we can relate the}

energy of a particle in the rest frame to that calculated in the co-moving frame as (p) = (p0)+p0·v(r, t)+1/2mv2_{(r, t), and substituting in the energy density and energy flux expression}

we obtain: ¯ (r, t) = 1 2mv 2_{(r, t) +} 1 n(r, t) Z dp0(p0) f0(r, p0, t) = 1 2mv 2_{(r, t) + u(r, t)} qα(r, t) = vα(r, t)n(r, t) ¯ 0(r, t) + 1 2mv 2_{(r, t) +} 1 dn(r, t) Z dp0p 02 mf 0 (r, p0, t) + Z dp0 1 m (p0)p0_α+ vβ(r, t)(p0αp0β− δαβp02 d ) f0(r, p0, t). (2.8)

If the distribution function is assumed to be everywhere the equilibrium distribution function, the above formulas for the momentum flux tensor and energy flux reduce to the results for an ideal fluid: Παβ(r, t) = mn(r, t)vα(r, t)vβ(r, t) + P (r, t)δαβ qα(r, t) = vα(r, t)n(r, t) 1 2mv 2_{(r, t) + h(r, t)} . (2.9)

leading to the Euler equation for momentum and to the equation of energy transport without viscosity and thermal conduction.

To calculate the next order correction to these equations that include the effects of viscosities and thermal conductivity we must find a solution of the Boltzmann equation that is slightly different from the equilibrium distribution.

We therefore seek a solution of the Boltzmann equation in absence of external forces in the form:

f (r, p, t) = feq((p), t) + δf (r, p, t), (2.10)

where the deviation from equilibrium is usually written as

δf (r, p, t) =

−∂f_∂eq

χ(r, p, t), (2.11)

the left hand side of the transport equation can be approximated by

∂t+ pα∂α m feq((p), t), (2.12)

while the right hand side requires a linearised form of the collision integral that can be calculated from the knowledge of the interaction law. Equating the two expressions leads to an integral equation for χ that can be solved for example by the methods described in [33]. The obtained expression for the distribution function can then be used to calculate the additional terms in the momentum and energy flux densities and therefore the transport coefficients.

2.2 Hydrodynamics for the electron liquid

When compared to the hydrodynamics of classical neutral particles the hydrodynamic treatment of electrons in solids presents some new features. These arise from the long range character of Coulomb interaction, from the quantum mechanical nature of electrons, and from the effect of the medium that hosts the electron liquid. The first two kinds of effect may be theoretically interpreted within the framework of the ideal electron liquid while the effects of the host medium