Electron-phonon interactions in graphene 

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EPC parameters at finite phonon momentum from DFPT

In this section we perform direct ab initio calculations of acoustic EPC matrix elements by using DFPT [20] in the linear response. The parameters O, K for optical phonons have already been evaluated using this method [14, 27] and compared to experimental Raman measurements. Their numerical values are reported in Table 3.1. We will mainly focus here on the acoustic phonon EPC parameters. DFPT calculations of EPC can be performed at chosen k and q vectors. This enables the verification of the angular and |q| dependencies of the matrix elements. EPC matrix elements can also be calculated for a chosen set of phonon modes. This involves minor modifications of the QE code to print out the EPC matrix in the basis of atomic displacements along the Cartesian directions, and the development of a small post-processing program to project the matrix on a chosen set of phonon modes. This allows for the calculation of both the canonical (˜A) and effective (A) gauge field parameters to verify the consistency of our model.
By choosing the phonon momentum q along the high symmetry directions 􀀀 ! K and 􀀀 ! M, we have q = 0 and /6, respectively. If initial and scattered states are taken on a circular iso-energetic line, i.e. if |k| = |k + q|, then k+q+k 2 = q± 2 . From Eqs. (2.2.11) and (2.2.12) we obtain: |H (􀀀!K) q,TA (k)| = |q|A.

EPC in the tight-binding model

In this section we compare the results of DFPT with other results obtained within the tight binding (TB) model. In particular, we compare with the TB model of EPC used in the collaborative work of Ref. [28].
The TB model provides a computationally cheap way to calculate EPC to a first approximation. In the two-nearest-neighbors TB model of EPC, App. B, the gauge field parameters are all proportional to the derivative 1 of the first-neighbor hopping integral with respect to bond length. We also have a link between the bare deformation potential bare and the derivative 2 of the second-neighbor hopping integral. Such relationships are obviously very specific to the TB model and are not enforced in the symmetry-based model used here. The hopping to the first neighbors is linked to the Fermi velocity, while hopping to the second neighbors is linked to the work function. By performing DFT calculations of the band structure of graphene for varying bond lengths, one can calculate 1 and 2. Those DFT calculations of 1 and 2 result in the numerical values of EPC parameters reported in Table 3.1, column « TB-DFT ». It is important to underline that within this TB model, the canonical phonon modes are used to calculate the perturbation to the TB Hamiltonian. Thus, we obtain the canonical EPC parameters fA, fO. There is no further modeling of the phonon modes, and therefore, no values for the effective parameters A and O. This is not really relevant for the optical phonons, because we have seen that the phonon model has DFPT EPC: direct DFPT LDA calculation of EPC, Sec. 3.1. This method does not give access to unscreened bare. (ii) |q| = 0: from zero-momentum model, see Sec. 3.3. Acoustic parameters are obtained by calculating the magnitude of strained-induced scalar and vector potentials. Optical parameters are obtained with the frozen phonon method from Ref. [34]. (iii)TB-DFT: results obtained in Ref. [28] using a TB model (App. B) and DFT to calculate the derivative of the hopping parameter with respect to bond length (see Sec. 3.2). (iv) GW calculations of EPC parameters. For phonons at 􀀀, the renormalization is 20%, as the Fermi velocity. For the A01 mode, results are taken from Ref. [35]. (v) Exp: obtained by fitting our numerical solution of Boltzmann transport equation to experiment. ˜A and bare are not used in the simulations. The values of K in the last line are doping dependent, see chapter 6, Sec. 6.3 for plots (Fig. 6.3) and discussion.

EPC at zero momentum from static strain method

In order to calculate the electron-phonon matrix elements in the GW approximation, we calculate the GW band structure for suitably chosen deformation patterns. If the displacement pattern is chosen to reproduce the zero-momentum limit of a given phonon, the matrix elements of the resulting perturbation Hamiltonian can be linked to the EPC parameters. Following this approach, the frozen phonons method [34] was used to calculate the electron-electron renormalization of the coupling to optical modes (LO, TO, A01) within GW. In order to perform GW calculations and to check the consistency of the small momentum EPC model, we also seek an interpretation of the acoustic EPC parameters at momentum exactly zero. This is achieved by linking acoustic EPC parameters to the perturbation potentials induced by a mechanical strain. This link is then verified numerically at the DFT level. Finally, we present the results of GW calculations for acoustic EPC parameters using this method, and summarize the already existing results on the optical EPC parameters.

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Acoustic EPC and strain-induced potentials

For acoustic phonons at 􀀀, a static phonon displacement in the zero momentum limit is equivalent to a strain deformation. We consider the long wavelength limit of an acoustic phonon and the corresponding perturbation occurring on a portion of the graphene sheet of scale d << 2 such zones distant to each other, the phonon perturbation can be seen locally as a simple mechanical strain of the sheet. Author of Ref. [16] derived the |q| ! 0 limit of the electron-phonon interaction (Eqs. (2.2.7) and (2.2.8)) for the canonical acoustic modes presented in Sec. 2.2 (Eq. (2.1.5)). Since screening was ignored in this EPC model, the magnitude of the bare deformation potential bare was used. Without screening, there is no long-range interaction and strain can be considered to be the exactly |q| = 0 equivalent of the |q| ! 0 limit of an acoustic phonon. We will discuss the consequences of screening on the interpretation of the deformation potential in the zero-momentum limit in paragraph 3.3.2. We first review the model of strain introduced in Ref. [16]. This model will be called canonical. The strained unit cell is defined with the lattice vectors b01, b02 such that: b0 i = (I + U) bi.

2D materials periodically repeated in the third dimension

In ab initio calculations, in addition to the non-zero thickness of the simulated electronic density, another issue arises. Current DFT packages such as QE rely on the use of 3D plane waves, requiring the presence of periodic images of the 2D system in the out-of-plane direction, separated by a distance c (interlayer distance). For many quantities, imposing a large distance between periodic images is sufficient to obtain relevant results for the 2D system. However, simulating the electronic screening of 2D systems correctly is computationally challenging due to the long-range character of the Coulomb interaction. As illustrated in Eq. (4.1.13), the Hartree potential induced by a 2D electronic density perturbed at wave vector qp goes to zero in the out-of-plane direction on a length scale 1/|qp|. For the layers (or periodic images) to be effectively isolated, they would have to be separated by a distance much greater than 1/|qp|. The computational cost of calculations increasing linearly with interlayer distance, fulfilling this condition is extremely challenging for the wave vectors considered in the following.
In order to isolate the layers from one another, the long-range Coulomb interaction is cut off between layers, as previously proposed in such context [74, 40, 75]. We use the following definition of the Coulomb interaction in real space: ¯vc(rp, z) = e2(lz − |z|) q |rp|2 + z2 .

Table of contents :

Notations
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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