Efficient dielectric matrix calculations using the Lanczos algorithm for fast many-body G0W0G_0W_0 implementations

We present a $G_0W_0$ implementation that assesses the two major bottlenecks of traditional plane-waves implementations, the summations over conduction states and the inversion of the dielectric matrix, without introducing new approximations in the formalism. The first bottleneck is circumvented by converting the summations into Sternheimer equations. Then, the novel avenue of expressing the dielectric matrix in a Lanczos basis is developed, which reduces the matrix size by orders of magnitude while being computationally efficient. We also develop a model dielectric operator that allows us to further reduce the size of the dielectric matrix without accuracy loss. Furthermore, we develop a scheme that reduces the numerical cost of the contour deformation technique to the level of the lightest plasmon pole model. Finally, the use of the simplified quasi-minimal residual scheme in replacement of the conjugate gradients algorithm allows a direct evaluation of the $G_0W_0$ corrections at the desired real frequencies, without need for analytical continuation. The performance of the resulting $G_0W_0$ implementation is demonstrated by comparison with a traditional plane-waves implementation, which reveals a 500-fold speedup for the silane molecule. Finally, the accuracy of our $G_0W_0$ implementation is demonstrated by comparison with other $G_0W_0$ calculations and experimental results.Comment: 19 pages, 2 figure

Méthode de calcul à N-corps basée sur la G0W0 : étude du couplage électron-phonon dans le C60 et développement d’une approche accélérée pour matériaux organiques

La présente thèse porte sur les limites de la théorie de la fonctionnelle de la densité et les moyens de surmonter celles-ci. Ces limites sont explorées dans le contexte d'une implémentation traditionnelle utilisant une base d'ondes planes. Dans un premier temps, les limites dans la taille des systèmes pouvant être simulés sont observées. Des méthodes de pointe pour surmonter ces dernières sont ensuite utilisées pour simuler des systèmes de taille nanométrique. En particulier, le greffage de molécules de bromophényle sur les nanotubes de carbone est étudié avec ces méthodes, étant donné l'impact substantiel que pourrait avoir une meilleure compréhension de ce procédé sur l'industrie de l'électronique. Dans un deuxième temps, les limites de précision de la théorie de la fonctionnelle de la densité sont explorées. Tout d'abord, une étude quantitative de l'incertitude de cette méthode pour le couplage électron-phonon est effectuée et révèle que celle-ci est substantiellement plus élevée que celle présumée dans la littérature. L'incertitude sur le couplage électron-phonon est ensuite explorée dans le cadre de la méthode G0W0 et cette dernière se révèle être une alternative substantiellement plus précise. Cette méthode présentant toutefois de sévères limitations dans la taille des systèmes traitables, différents moyens théoriques pour surmonter ces dernières sont développés et présentés dans cette thèse. La performance et la précision accrues de l'implémentation résultante laissent présager de nouvelles possibilités dans l'étude et la conception de certaines catégories de matériaux, dont les supraconducteurs, les polymères utiles en photovoltaïque organique, les semi-conducteurs, etc.This thesis studies the limitations of density functional theory. These limits are explored in the context of a traditional implementation using a plane waves basis set. First, we investigate the limit of the size of the systems that can be treated. Cutting edge methods that assess these limitations are then used to simulate nanoscale systems. More specifically, the grafting of bromophenyl molecules on the sidewall of carbon nanotubes is studied with these methods, as a better understanding of this procedure could have substantial impact on the electronic industry. Second, the limitations of the precision of density functional theory are explored. We begin with a quantitative study of the uncertainty of this method for the case of electron-phonon coupling calculations and find it to be substantially higher than what is widely presumed in the literature. The uncertainty on electron-phonon coupling calculations is then explored within the G0W0 method, which is found to be a substantially more precise alternative. However, this method has the drawback of being severely limitated in the size of systems that can be computed. In the following, theoretical solutions to overcome these limitations are developed and presented. The increased performance and precision of the resulting implementation opens new possibilities for the study and design of materials, such as superconductors, polymers for organic photovoltaics and semiconductors

Electron-phonon coupling in the C60 fullerene within the many-body GW approach

We study the electron-phonon coupling in the C60 fullerene within the first-principles GW approach, focusing on the lowest unoccupied t1u three-fold electronic state which is relevant for the superconducting transition in electron doped fullerides. It is shown that the strength of the coupling is significantly enhanced as compared to standard density functional theory calculations with (semi)local functionals, with a 48% increase of the electron-phonon potential Vep. The calculated GW value for the contribution from the Hg modes of 93 meV comes within 4% of the most recent experimental values. The present results call for a reinvestigation of previous density functional based calculations of electron-phonon coupling in covalent systems in general.Comment: 4 pages, 0 figur

Bromophenyl functionalization of carbon nanotubes : an ab initio study

We study the thermodynamics of bromophenyl functionalization of carbon nanotubes with respect to diameter and metallic/insulating character using density-functional theory (DFT). On one hand, we show that the activation energy for the grafting of a bromophenyl molecule onto a semiconducting zigzag nanotube ranges from 0.73 eV to 0.76 eV without any clear trend with respect to diameter within numerical accuracy. On the other hand, the binding energy of a single bromophenyl molecule shows a clear diameter dependence and ranges from 1.51 eV for a (8,0) zigzag nanotube to 0.83 eV for a (20,0) zigzag nanotube. This is in part explained by the transition from sp2 to sp3 bonding occurring to a carbon atom of a nanotube when a phenyl is grafted to it and the fact that smaller nanotubes are closer to a sp3 hybridization than larger ones due to increased curvature. Since a second bromophenyl unit can attach without energy barrier next to an isolated grafted unit, they are assumed to exist in pairs. The para configuration is found to be favored for the pairs and their binding energy decreases with increasing diameter, ranging from 4.34 eV for a (7,0) nanotube to 2.27 eV for a (29,0) nanotube. An analytic form for this radius dependence is derived using a tight binding hamiltonian and first order perturbation theory. The 1/R^2 dependance obtained (where R is the nanotube radius) is verified by our DFT results within numerical accuracy. Finally, metallic nanotubes are found to be more reactive than semiconducting nanotubes, a feature that can be explained by a non-zero density of states at the Fermi level for metallic nanotubes.Comment: 7 pages, 5 figures and 3 table

Electron-phonon coupling in C60 using exact-exchange functional

The superconductivity in C60 doped crystals is now well understood as a phonon mediated interaction. The strength of the electron-phonon coupling can be deduced by Raman and PES measurments which can then be used to assess the density-functional theory results. Although experimental and computed electron-phonon coupling agree on the total magnitude of the coupling, they do not on the contributions of the individual vibrational modes. Density-functional theory calculations indicate that high frequency modes are responsible for most of the coupling whereas experiments suggest that low frequency modes are the dominating contribution. Up to now, only calculations using the local density approximation (LDA) were performed. In this study, we investigate the effect of exact-exchange functionals, such as B3LYP, on the computed electron-phonon coupling of the different vibrational modes

The temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation

The renormalization of electronic eigenenergies due to electron-phonon interactions (temperature dependence and zero-point motion effect) is important in many materials. We address it in the adiabatic harmonic approximation, based on first principles (e.g. Density-Functional Theory), from different points of view: directly from atomic position fluctuations or, alternatively, from Janak's theorem generalized to the case where the Helmholtz free energy, including the vibrational entropy, is used. We prove their equivalence, based on the usual form of Janak's theorem and on the dynamical equation. We then also place the Allen-Heine-Cardona (AHC) theory of the renormalization in a first-principle context. The AHC theory relies on the rigid-ion approximation, and naturally leads to a self-energy (Fan) contribution and a Debye-Waller contribution. Such a splitting can also be done for the complete harmonic adiabatic expression, in which the rigid-ion approximation is not required. A numerical study within the Density-Functional Perturbation theory framework allows us to compare the AHC theory with frozen-phonon calculations, with or without the rigid-ion terms. For the two different numerical approaches without rigid-ion terms, the agreement is better than 7 $\mu$eV in the case of diamond, which represent an agreement to 5 significant digits. The magnitude of the non rigid-ion terms in this case is also presented, distinguishing specific phonon modes contributions to different electronic eigenenergies

Faster G0W0 calculations using Lanczos algorithm and Sternheimer equation

G0W0 corrections to DFT band structures are a popular way to go beyond the accuracy DFT is able to provide. However, the calculation of such corrections with the ABINIT code is currently prohibitive for systems with more than a few hundreds of electrons. What limits the calculations to this system size is the need in the current implementation to invert the dielectric matrix and to carry out some summation over conduction bands. This poster presents a strategy to avoid both of these limitations for the screened-exchange contribution to the self-energy. In ABINIT’s implementation, the dielectric matrix is expressed in a plane wave basis, which needs to be relatively big to properly describe the matrix. This poster explains how a Lanczos basis can be generated to substantially reduce the size of the matrix. Also, the number of conduction bands needed to reach convergence in the summation is usually an order of magnitude bigger than the number of valence bands. Here, the calculation of all the conduction states is avoided by reformulating the summation into a linear equation problem (Sternheimer equation), which also substantially reduces the computation time

Faster G0W0 implementation for more accurate material design

Density-functional theory (DFT) is currently the ab initio method most widely used to predict electronic energy levels of new materials. However, approxima- tions intrinsic to the theory limit the accuracy of calculated energy levels to about 0.5 eV. The G0W0 approach is an alternate ab initio method that provides an enhanced precision (about 0.05 eV). However, its computational cost is currently prohibitive for systems with more than a few tens of electrons, thus limiting its use in the simulation and design of technologically relevant materials. This limitation of current G0W0 implementations can be traced to two bottle- necks : the need to invert a large matrix (the dielectric matrix) and the need to carry out summations over a large number of electronic states (conduction states). The first bottleneck is caused by the choice of the basis in which the dielectric matrix is represented : traditional G0W0 implementations use a plane wave basis, which needs to be relatively large to properly describe the matrix. This talk will explain how a Lanczos basis can be generated to substantially reduce the size of the matrix. Also, the number of conduction bands needed to reach convergence in the summations is usually an order of magnitude larger than the number of valence bands. Here, the calculation of the conduction states is avoided by refor- mulating the summations into linear equation problems (Sternheimer equations), which also substantially reduces the computation time. Finally, we will discuss how the introduction of a model dielectric operator allows further speedup in the calculation of the dielectric matrix without introducing further approximation in the method. Preliminary calculations on silane, thiophene and benzene will be presented

Electronic properties of double-walled carbon nanotubes (DWNT) and the effect of functionalization

Although promising for many electronic applications, further understanding of carbon nanotubes systems are required for practical designs. A difficulty currently hindering further development in this field is the considerable degradation of transport properties in a single-wall carbon nanotube (SWCNT) when it is subjected to ambient conditions or functionalized. Double-wall carbon nanotubes (DWCNT) could solve this problem, by allowing the outer tube to be functionalized while the inner tube would retain a pristine structure and it's promising electronic properties. However, our understanding of interactions between the tubes and their consequences on the system's electronic properties is still incomplete. In this presentation, we investigate those interactions using density-functional theory (DFT) calculations. In particular, we investigate separately the effects of structural deformations, Fermi energy realignment and electronic orbital overlap on the band structure of DWCNT. The effects of functionalization will also be addressed

G0W0 implementation using Lanczos algorithm and Sternheimer equation

The G0W0 approach is an accurate method to give a physical meaning to the eigenvalues obtained in adensity-functional theory (DFT) calculation.However, the calculation of such corrections with plane wave codes is currently prohibitive for systems with more than a few hundreds of electrons. What limits calculations to this system size is the need in current implementations to invert the dielectric matrix and the need to carry out summations over conduction bands. This talk presents a strategy to avoid both of these bottlenecks. In traditional plane wave implementations of G0W0, the dielectric matrix is expressed in a plane wave basis, which needs to be relatively big to properly describe the matrix. Here, we will explain how a Lanczos basis can be generated to substantially reduce the size of the matrix. Also, the number of conduction bands needed to reach convergence in the summations is usually an order of magnitude larger than the number of valence bands. Here, the calculation of the conduction states is avoided by reformulating the summations into linear equation problems (Sternheimer equations), which also substantially reduces the computation time. Preliminary results will be presented