Electron-phonon interactions in graphene 

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Two-dimensional materials

Since the experimental isolation of graphene [5, 6, 7], 2D materials have undergone unprecedented scrutiny due to their interesting properties both at the fundamental and applied levels. We first need to state what we mean by « 2D material ». An adequate definition in the context of this work would be « a material constructed by infinitely repeating a pattern of a few atoms in only two dimensions ». This implies that two of the material’s dimensions are much larger than the third one, which is the more general definition of 2D materials. With no periodicity in the third dimension, the thickness is limited to a few atomic layers while in the two other dimensions, the pattern is repeated infinitely. A more precise term for this system would be « 2D-periodic crystal », and this is what we will model. This model will be used to study the properties of more realistic 2D materials in which the periodicity is neither perfect (there can be some defects) nor infinite. In practice, the length and width of 2D materials are at the microscopic scale (∼ m). With a pattern at the atomic scale (∼Å), the number of repetitions is then of the order of ∼ 104 in each direction with periodicity. In the framework of our models and simulations, this number will be considered large enough to neglect edge effects. In other words, we assume that for the physical properties studied here, a 2D material of a few m2 is equivalent to an infinite 2D-periodic crystal. Typical examples of atomically thin monolayer systems are graphene and hexagonal boron nitride, made up of a honeycomb lattice of either carbon, or boron and nitrogen. Since the few atoms included in the pattern are not necessarily in the same plane, the above definition also includes compounds thicker than one atom. Examples of such compounds that are still considered as one layer are the monolayer transition-metal dichalcogenides. A few layers of any of those materials, or a combination thereof are called few-layer materials and are also included in our definition of 2D materials.
In practice, a 2D material can exist in various setups. The purest (yet not the simplest) form is probably when the 2D material is suspended. In that kind of setup, the edges of a sheet of material are attached to a support, while a significant part of the material is suspended in between those points of attach. A more common setup is to deposit the material on a certain supporting material, called substrate. The difference between those setups boils down to the nature of the surroundings of the 2D material in the third direction. Is it vacuum, an insulating dielectric material, a semiconductor? Does the environment interact with the 2D material? Is there some chemical reactions at the interface? Is there an external electric field? A great variety of situations are possible and the physical properties of the 2D material depend on them. In this work, we will focus on the very common field effect transistor (FET) setup [5, 8], as represented in Fig.1.1. In this setup, the 2D material is deposited on an insulating bulk substrate (silicon oxyde for example). On both extremities of the 2D material are contacts, a source and a drain, between which a current can be established. When the source and drain have different electric potential, an in-plane electric field is established between the two, which drives the electrons from one contact to the other. The efficiency of the 2D material in carrying the current from one contact to an other is referred to as its electronic transport properties and will be the main type of experimental measurement studied in the thesis (chapter 6).
On top of the 2D material is a dielectric material, a few tens of nanometers thick. This material is insulating, such that no charge can go through it. Finally, a metallic top gate or grid is deposited on top of the dielectric. The purpose of this gate is to induce additional charges in the 2D material. Following the same principle as a capacitor, applying a voltage difference between the 2D material and the gate results in the accumulation of opposite charges on both sides of the insulating dielectric. This process, called gate-induced doping, is represented in Fig. 1.2. This effect is particularly interesting in 2D materials. Indeed, due to their reduced dimensionality, the accumulated charges translate into large induced densities. Being able to change the density of free carriers implies the ability to tune the electronic properties of 2D materials. For example, in Fig. 1.3, a transition from a semimetallic to a metallic behavior is induced in monolayer graphene by varying the gate voltage [9]. This is of course essential to use graphene as a conductor of charges. Another beautiful example [10, 8] is the case of 2D transition-metal dichalchogenides, in which one can induce sufficiently high doping to obtain superconductivity. Fig. 1.2 is a rather simplistic representation of gate-induced doping. The form of the electric potential in the vicinity of the 2D material will be studied more extensively in this thesis. For now, let us recall some important general features of this setup. The accumulation of charges on either side of the dielectric implies the establishment of an electric field in between, corresponding to a linear variation of the potential. It is important to distinguish this out-of-plane electric field from the in-plane one driving current between source and drain. The accumulation of charges also implies a global dipolar moment for the system. On either side of the system of Fig. 1.2 (on the right of the 2D material and on the left of the region of the gate where the charges accumulate), the electric field is zero. This means that the electric potential is a constant, but this constant is different on either side of the system. Indeed, the finite electric field in the dielectric induces a shift in the potential. This will be particularly important when trying to simulate this system with periodic boundary conditions. The system is quite asymmetric due to this potential shift. In the vicinity of the 2D material, we observe that the electric field is finite on one side while zero on the other.
We now proceed to the description of the electrons and phonons evolving in the 2D material.

Electrons and phonons in two dimensions

In this section we examine two central objects of condensed matter physics, electrons and phonons. Covering the very large span of physical properties in which they are involved would be a difficult task. Here, we will try to set up the minimal framework necessary to address important topics in this thesis, i.e. electron-phonon interactions, (gate-induced) doping and screening. We will concentrate on the particularities and challenges presented by those topics in a 2D framework.
It is useful to separate each atom of the material in two parts. The first consists of the nucleus plus some core electrons that stay close to the nucleus and do not participate in bonding with neighbors. This subsystem is positively charged. The remaining electrons are valence electrons that participate in the bonding. They can travel in the material. In the following, the term « ion » will refer to the first subsystem (nucleus and core electrons), while the term « electron » will refer to the valence electrons unless specified otherwise. In solids, both ions and electrons can move, generating a gigantic number of degrees of freedom. Fortunately, in the Born-Oppenheimer (BO) approximation [11], we can decouple ionic and electronic degrees of freedom. The BO approximation roughly states that since the electrons are much lighter than the ions, they will react much faster than the ions can move. This means that the positions of the ions can be considered as parameters for the equations ruling the motion of electrons. We can then consider the movements of electrons and ions separately.
We first study an non-interacting electron gas moving in a charge compensating lattice of fixed ions. In this system, we can introduce the concepts of Bloch functions, band structure and charge doping. We then proceed to an interacting electron gas, introducing the concepts of electron-electron interactions and screening. Being aware of the key features of those concepts in 2D is the first step towards their correct treatment in the thesis. We then describe the motion of ions, using phonons. Phonons are quanta of vibrations of the ionic lattice. They have particularly interesting properties in 2D. Finally, we introduce the first order correction to the BO approximation, the electron-phonon interaction. We identify the important issues to consider for their modeling and simulation, which is the central topic of the thesis.

2D non-interacting electron gas in fixed ions

We consider a 2D electron gas moving in a background lattice of fixed ions. The lattice of ions is not included in the system, but its effect on the electrons can be captured by an effective periodic potential U. This potential is a combination of the ion-ion interaction and the interaction between ions and electrons. It is a single-particle potential in the sense that its effect on a given electron does not depend on the other electrons. We start with the Bloch theory for non-interacting electrons moving in such a periodic single-particle potential. The Hamiltonian for the electrons is simply the sum of their kinetic energy and this potential: H = Tel + U
For the infinite and periodic system considered here, a plane-wave basis set is ade-quate to construct the wave functions. The Hamiltonian is solved by the Bloch wave functions:
ψk,s(rp,z) = wk,s(rp,z)eik•rp , (1.2.1)
where rp is a position vector in the plane defined by the 2D material and z is the space variable in the third dimension. The function wk,s(rp,z) has the periodicity of the 2D crystal, and the factor eik•rp represents a plane wave. The quantum numbers k, s represent the in-plane wave vector and band index, respectively.
The particularity of the Bloch wave functions in a 2D material is that they propa-gate only in the plane of the material. This is indicated by the in-plane nature of the variables k and rp in the plane-wave factor. The first quantum number, the momen-tum k, gives the periodicity of the plane wave. The corresponding period is equal or larger than the unit cell of the crystal. The variations of the wave functions on the scale of the unit cell are given by wk,s(rp,z). This function also gives the extension of the wave functions in the third dimension. The electrons are strongly attracted by the positively charged ions in the lattice and cannot go very far from them. This means that if the 2D material is situated around z = 0, wk,s(rp,z) decays rapidly as a function of |z|.
The couple of indices k,s defines an electronic quantum state. The electrons being fermions, the Pauli principle dictates that only two electrons of opposite spin can occupy one of those quantum states. We will not study spin-dependent phenomena and can consider a k,s state as a single state with two degenerate electrons. To each k,s state is associated an energy εk,s. Those allowed energies, as functions of the electron momenta, define the electronic band structure. Several energies might be allowed for a single momentum, thus the need for the quantum number indicating the band s.
In a neutral crystal and at zero temperature, the electrons occupy the lower-energy valence bands. For the purpose of this work, the energy of the highest occupied state at zero temperature will be called Fermi level and noted εF (the norm of the wave vectors for the corresponding states is the Fermi wave vector kF ). Higher in energy are the unoccupied conduction bands. As the temperature increases, some energy is available to promote some electrons to the conduction bands. The occupation of a state of energy εk,s is then given by the Fermi-Dirac distribution:
By adding charges to the system, as can be done via the FET setup mentioned earlier, one can change the Fermi level [12]. This is called doping (or charging) the system. In the case of electron doping, the number of electrons is increased and the Fermi level is shifted higher in energy. In the case of hole doping, the number of electrons is decreased, thus creating vacancies or holes, and the Fermi level is shifted down. At zero temperature, the Fermi level defines a sharp boundary between occupied and unoccupied states. Temperature has the effect of smearing this boundary according to Eq. (1.2.2).
In 2D, an iso-energetic section of the band structure gives a line. When this section is done at the Fermi level, this line is called the Fermi surface (although it should be called the Fermi line). Doping the material changes the form of the Fermi surface. As mentioned earlier, large densities of charges can be added in 2D materials in the FET setup. This corresponds to a wide range of accessible Fermi levels, and thus the opportunity to probe a large portion of the electronic structure. Gate-induced doping is crucial in numerous applications. Although the principle of gate-induced doping seems relatively simple, the details of its effects on the electrons of the 2D materials are not trivial. The presence of an electric field can change the form of the electronic density in the out-of-plane direction. When a few layers of material are present, the added charge will surely not be distributed homogeneously over the layers. In practice, doping the system is more complex than simply shifting the Fermi level. Understanding the side effects of gate-induced doping is a major challenge. Some of them will be treated in chap. 5.
The non-interacting gas in a lattice of fixed ions has provided us with a simple system in which to introduce basic tools. The validity of the Bloch theory extends to a system of electrons moving in any periodic single-particle potential.

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 2D electron gas: electron-electron interactions

Let us now turn on the electron-electron interactions. Those interactions are extremely complex, and the system is not exactly solvable anymore. However, we know that with-out any perturbation, the system stabilizes in its lowest-energy state, or ground state. The electrons then form the so-called Fermi sea. The details of what happens inside the Fermi sea are too complex. However, we can study its reaction to small perturbations. Thus, we start with the lowest-energy state of the system, and consider small exci-tations of the ground state. Within certain approximations (see Fermi liquid theory in Ref. [1]), the low-energy excitations of the Fermi sea can be described as quasi-particles that behave like single electrons weakly interacting with one another. Often, in condensed matter physics and in the following, the term « electron » actually refers to those quasiparticles. The Fermi liquid theory is fundamental because it justifies that we keep using all the previously introduced tools to study an interacting electron gas. We still need to treat the interactions between those electron quasiparticles, what we call electron-electron interactions. This is a very vast topic in condensed matter physics. For now, let us start with the contribution from the Hartree term, included throughout the thesis. The Hartree term accounts for the classical Coulomb repulsion that occurs between similarly charged particles. In our framework, it will be accounted for via the use of an additional term in the Hamiltonian for the electrons: H = Tel + U + VH
where VH is the Hartree potential, which is the electric potential generated by the electronic density. The sum U +VH is still a periodic single particle potential, such that we don’t go out of the Bloch framework. Reducing the electron-electron interactions to the Hartree potential amounts to a mean field approximation, since the sum of the contributions from each individual electron is averaged by the contribution from the electron density.
This « lowest-order » contribution is sufficient to introduce the concept of screening. Screening is the ability of the charges (electrons) to rearrange in order to counteract an external electric field. Any external perturbing potential that induces a varying electric field leads to a movement of the charges. The new configuration of charges will produce a new Hartree potential which tends to counteract the external perturbing potential. A simple example is when a positive external charge is added to an electron gas. Electrons agglomerate around this added charge such that its influence is negated outside of certain region of space around it. This can be generalized to an arbitrary perturbing external potential. The effective potential felt by a test charge is the sum of the perturbing potential and the Hartree potential generated by the electrons (the induced potential). This effective potential is weaker than the original external potential, due to the global reaction of the electron gas. The electrons are said to screen the external potential.
Reduced dimensionality has substantial effects on screening [4]. This can be seen in the Fourier transform of the 2D/3D Coulomb interaction, which rules the interactions between charged particles, and thus the reaction of the electron gas where e is the elementary charge (e > 0). The different power-laws in |q| suggest completely different behaviors. We will see that for the transport applications we have in mind, the small momentum perturbations are most relevant. The divergent behavior of the Coulomb interaction at small |q| must then be carefully treated in the numerical calculations. The importance of dimensionality can also be sensed in the Lindhard function, Fig. 1.4, which gives the electron density response of the non-interacting gas to a static perturbative potential of wave vector q. There we see that dimensionality yields different singularities near twice the Fermi wave vector. In view of the differing singularities of the Coulomb interaction at small momenta and the Lindhard function at twice the Fermi wave vector, we can expect drastically different screening behaviors from the electron gas depending on dimensionality.
In addition to those formal singularities, delicate issues arise on a more practical level. The screening properties of the 2D electron gas are subject to the influence of the environment of the 2D material. Indeed, the driving force of screening is the Coulomb interaction and the electric field generated by charged particles. Electrons in 2D materials are exposed to the influence of other charges from the environment in the out-of-plane direction. Modeling and simulating screening in 2D materials will be one of the major challenges of the following chapters.
Screening occurs as soon as there are mobile charges due to the classical Coulomb interaction between charged particles. However, electron-electron interactions beyond the Hartree term can of course play a role. Those interactions, of a more quantum nature, will be called electron correlations. Let us now quickly discuss their treatment. In general, they affect a wide variety of phenomena, basically everything that can be described as a reaction of the electrons to a certain perturbation. In strongly correlated systems, electron-electron interactions are very strong, and the formalism of the non-interacting gas (Bloch wave-functions, band structure, …) may not make sense anymore. In weakly correlated systems, as studied in this thesis, the formalism of the non-interacting gas can be kept. Electron correlations then bring corrections to the band structure, finite lifetimes to the electrons, and they can renormalize the response of the electrons to a perturbation. Estimating those corrections is an important and non-trivial task. In principle, one can use an additional potential, similar to the Hartree potential, to account for further electron-electron interactions. However, the exact form of this potential is not known, such that electron correlations are always treated within a certain approximation.
By discussing the 2D electron gas, we were able to introduce most of the peripheral topics treated in the thesis. Understanding the side effects of gate-induced doping, simulating the peculiarities of screening in 2D and estimating corrections for electron correlations are among the objectives of the next chapters. We now proceed to the description of the second central object of the thesis.

2D phonons

When thermal energy is available, the ions of the crystal vibrate around their equi-librium positions. Those vibrations are usually (in the scope of this work at least) well described by harmonic oscillators. They can be decomposed in a set of conve-nient quanta of vibrations called phonons. It does not concern an individual particle, but a collective movement of a very large (infinite) number of atoms. In the Born-Oppenheimer approximation used here, electrons adapt instantaneously to the changes in the positions of the ions. They simply act as the restoring force of the harmonic oscillators, always bringing the ions back to their equilibrium positions.
Phonon excitations are described by wave functions defining the displacements uq,ν (ra) of each atom a with respect to its equilibrium position ra. Those functions bear resemblance with the wave functions of the electrons. The function uq,ν is also a wave, and is similarly characterized by two quantum numbers q, ν. The momentum q gives the periodicity of the propagating wave of displacements and is an in-plane quantity in the case of a 2D material. The other index indicates a phonon branch, or mode. It specifies how the ions move. For example, for a small momentum, one may distinguish three kind of modes: (i) longitudinal modes in which the ions move in the same direction as the momentum; (ii) transverse modes in which the atoms move in the plane of the 2D material, in the direction perpendicular to the momentum; (iii) out-of-plane modes in which the ions move in the out-of-plane direction. If there are several atoms in a unit cell, the modes also distinguish whether those atoms move in phase (acoustic phonons) or out of phase (optical phonons), see Fig. 1.5. To a given phonon uq,ν is associated an energy ωq,ν , where is Planck’s constant and ω is a frequency. Plotting those energies as functions of momenta gives the phonon dispersion, Fig. 1.5. Electrons and phonons are described in a quite similar way, using momentum, energy and bands or branches. A fundamental difference, however, is that phonons are bosons. This means that unlike fermions, there is no limit to the population of a given state. More precisely, the population of a given phonon state q, ν of energy ωq,ν is given by the Bose-Einstein distribution
where the use of the indices q,ν in nq,ν distinguishes the Bose-Einstein distribution from the electronic density.
In 2D materials there is a clear distinction between in-plane and-out-plane phonon modes. The term in-plane phonons is used when the ions move only in the plane. We will see that having a good model to describe the in-plane motions of the ions is essential for modeling the transport properties of graphene. Out-of-plane phonons, and in particular flexural acoustic phonons represented in Fig. 1.6, are a peculiarity of 2D materials. In single-layer materials, they present a quadratic dispersion at small momenta ωq,ν ∝ |q|2, see Fig. 1.5. As a consequence, flexural phonons dominate low-energy, small-momentum structural properties of free-standing single-layer materials. At finite temperature, the quadratic dispersion makes the number of phonons divergent in the thermodynamic limit of an infinite system, prohibiting long-range structural order. This has generated surprise and debate concerning the existence of free-standing graphene. More in the scope of this thesis are the properties of out-of-plane phonons in the FET setup. The zero-momentum limit of the acoustic out-of-plane phonon is equivalent to a displacement of the whole 2D material in the out-of-plane direction. When the 2D material is enclosed between a substrate and a dielectric, the energy required to achieve such a displacement is finite. Contrary to a free standing material, the dispersion should then tend to a constant in the small-momentum limit. This is one of the side-effects of the FET setup that would be interesting to simulate.
Phonons are involved in many properties of the material. Their vibrational nature and the fact that their abundance is ruled by temperature makes them central for the description of the thermodynamical and mechanical properties of crystals. They are involved in many other physical properties via their coupling to the electrons.

Table of contents :

Introduction 
1 Electrons and phonons in 2D materials 
1.1 Two-dimensional materials
1.2 Electrons and phonons in two dimensions
1.2.1 2D non-interacting electron gas in fixed ions
1.2.2 2D electron gas: electron-electron interactions
1.2.3 2D phonons
1.2.4 Electron-phonon interactions
1.3 Simulation challenges
2 Electron-phonon interactions in graphene 
2.1 Electrons and phonons models
2.1.1 Dirac Hamiltonian for electrons
2.1.2 Phonons
2.2 Electron-phonon coupling matrix elements
2.2.1 Coupling to canonical phonon modes at 􀀀
2.2.2 Coupling to DFT phonon modes at 􀀀
2.2.3 Coupling to inter-valley A01 mode at K
3 Ab initio calculations of EPC in graphene 
3.1 EPC parameters at finite phonon momentum from DFPT
3.2 EPC in the tight-binding model
3.3 EPC at zero momentum from static strain method
3.3.1 Acoustic EPC and strain-induced potentials
3.3.2 Calculation of strain-induced potentials at the DFT level
3.3.3 EPC parameters at the GW level
4 Static screening in 2D 
4.1 Static dielectric function
4.1.1 Three-dimensional materials
4.1.2 2D materials
4.1.3 2D-periodic materials with finite thickness
4.1.4 2D materials periodically repeated in the third dimension
4.2 Static screening properties of graphene
4.2.1 Analytical and semi-numerical solutions
4.2.2 DFPT LDA solution
4.3 Results in graphene
4.3.1 Importance of cutting off the Coulomb interactions
4.3.2 Comparison of analytical and LDA methods: band structure effects
4.3.3 Estimation of the screened deformation potential
5 DFT/DFPT for 2D materials in the FET setup 
5.1 Description of a 2D material doped in the FET setup
5.2 Treatment of the periodic images
5.2.1 Inadequacy of 3D PBC
5.2.2 Isolate the layers with 2D Coulomb cutoff
5.3 Implementation
5.3.1 KS Potential
5.3.2 Total Energy
5.3.3 Forces
5.3.4 Phonons and EPC
5.4 Results in graphene
5.4.1 Finite frequency for ZA phonons at 􀀀
5.4.2 Screening
5.4.3 Finite coupling to out-of-plane phonons
6 Phonon-limited resistivity of graphene 
6.1 Boltzmann transport theory
6.2 EPC included in the transport model
6.3 Results
6.4 Approximated solutions
6.4.1 Semi-analytical approximated solution
6.4.2 Additivity of resistivities
Conclusion 
Appendices
A Computational Methods
B EPC in TB model
C Analytical susceptibility
D 2D Coulomb cutoff at G = 0
E Boltzmann transport equation
Résumé en français
Bibliography

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