In this thesis, we have dealt with the description of optical properties of materials, with a particular interest in bulk materials. The state-of-the-art of first principle calculations of optical absorption of solids has been represented, up to recently, by the Bethe-Salpeter equation (BSE) approach. The results one can achieve within BSE are in good agreement with experiments and, moreover, the range of applicability of BSE goes well beyond the solids. Absorption spectra of atoms, molecules, clusters or surfaces are usually well described by the Bethe-Salpeter approach. The heaviness of the calculations involved, however, prevents a large-scale application to more complex systems, that are the systems of great interest in material science like, e.g., quantum dots, multi-wall nanotubes, biological molecules or defects in solids. The most prominent alternative to the Bethe-Salpeter approach is the Time Dependent Density Functional Theory, which is density-based and, therefore, could in principle lead to simpler calculations. A widely used approximation to the TDDFT is the adiabatic local density approximation (ALDA). Despite its (partial) success in finite systems the ALDA has led to poor results in the description of optical absorption of solids. The search for an efficient description of optical spectra of solids in TDDFT has become, hence, one of the major problems related to the optical response of a material. In this thesis new inroads have been made in the comprehension of the optical response of a material. Starting from a joint analysis of the BSE approach and the TDDFT framework, it has been possible to calculate the optical spectra of solids, semiconductors and insulators, with the inclusion of the excitonic effects, without solving the BSE but obtaining results of the same precision. To do so, we have derived, or re-derived in a more general way, the most relevant formulas, and implemented them in existing computer codes.7 In order to increase the efficiency and the stability of the calculations, the codes have also been optimized, and, in particular, a linear solver technique has been implemented to avoid all matrix inversions. A large number of calculations for a detailed analysis of the models used and for all hypotheses made, had then to be performed. Finally we have illustrated our findings for bulk silicon, silicon carbide, and solid argon. More in detail, after the theoretical framework, presented in chapters 1-4, • In chapter 5 we have recalled the fact that TDDFT and BSE have the same mathematical structure, and that the response function in both approaches can be written as S = S0 + S0KS whose solution gives the searched absorption or energy loss spectrum. The link between the response of an independent particle (or quasi-particle) system S0 and the full response S is given by the kernel K in this Dyson-like screening equation. The kernel K contains the Coulomb interaction v, which is common to both TDDFT and BSE, and the exchange-correlation (in TDDFT) or electron-hole (BSE) contribution, which is instead different in the two approaches. • The role of the Coulomb potential v has been elucidated in chapter 6. We have discussed its long-range component v0, which is responsible for the difference between the absorption and the electron energy loss spectra in solids. To further illustrate this result, we have shown, analytically and numerically that, in the limit of an isolated system, where the long-range component of the Coulomb interaction is negligible, the absorption and the electron energy-loss (at vanishing momentum transfer) spectra are the same. The microscopic components of the Coulomb interaction, instead, are responsible for the local field effects. • We have then addressed the second term of the kernel K, namely the exchangecorrelation kernel fxc (in TDDFT). The study made in chapter 7 has allowed us to show that simple static approximations for the exchange-correlation kernel fxc can yield spectra of semiconductors and insulators in qualitative agreement with the experiments. • In chapter 8 we have generalized a previously proposed, but never tested, expression for an exchange-correlation kernel fxc, within the time dependent density functional theory framework, which is fully ab initio and parameter-free. We have also investigated the different contributions to the kernel and found out how and when the generalized kernel 0fxc0 works in principle. • In chapter 9 the kernel here developed is applied and tested for two semiconductors, namely bulk silicon and silicon carbide, while • in chapter 10 the kernel is tested for an insulator, taking solid argon as an example. In both semiconductors and insulators the results of the TDDFT (using the kernel developed in chapter 8) and those of BSE are almost indistinguishable. We have hence contributed to the solution of the long-standing problem of how to calculate the absorption spectra of solids, in the framework of TDDFT, without solving the BSE. To have dealt with this problem, has allowed us to address, answer or simply illustrate several questions: ! classical Coulomb (Hartree) contributions play an important role in electronic spectra, and it is worthwhile to discuss their effects before addressing the problems of exchange and correlation. Their short-range and long-range parts are crucial for understanding the link between absorption and energy loss, and the transition between finite and infinite systems. ! satisfactory optical spectra of semiconductors and insulators can be obtained by using very simple models for the electron-hole interaction (in BSE) or for the exchangecorrelation kernel (in TDDFT); these models, of course, depend on parameters that one has to adjust to fit the experiment. However the computational complexity of these calculations is that of the RPA, and one might try to find ways to determine these parameters from first principles; ! to obtain an exchange-correlation kernel from the BSE is not straightforward. However, apart from some counter examples, discussed in chapter 8, in most cases an approximate mapping from one theory (the BSE) to the other (the TDDFT) is possible. A first consequence of the existing difficulties is that the resulting fxc kernel can suffer of convergence problem (requiring a lot of G-vectors or presenting strong fluctuations in frequency). There can also be several possible approximate mappings: an example is reported in chapter 7 where two exchange-correlations kernels, a long-range and an ultra short-range one, give similar (good) results. These two kernels are derived starting from the same model (the contact exciton model) within the BSE; ! an important finding, following the discussions of chapter 8, is that the key quantity of the theory is not the kernel fxc, but fxc multiplied by a response function; ! finally, we are now able to obtain very good results for the optical spectra of solids, well describing both the continuum and bound excitons, within a parameter-free TDDFT framework. Future Developments The thesis focused on the description of optical spectra of solids, in the appealing framework of TDDFT. However the concepts and approaches here developed have to be ex-tended in order to improve their applicability and “workability”. Concerning the latter, we will have to search for the most convenient algorithm to numerically evaluate the proposed expressions. We also plan to take advantage of the fact that, in our formulation, k-points are simply summed over, instead of being the indices of matrices that have to be inverted or diagonalized. Concerning the applications, a short term perspective is to apply the method to the description of the electron energy loss spectra, for which the resonant and anti-resonant part of the response function have to be taken into account. We also plan to study finite systems. Moreover the introduction of the spin degree of freedom could allow to enlarge the target of our study, towards polarised systems, both finite and infinite, also subject to magnetic fields.
| Identifer | oai:union.ndltd.org:CCSD/oai:pastel.archives-ouvertes.fr:pastel-00000739 |
| Date | 29 September 2003 |
| Creators | Sottile, Francesco |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | French |
| Detected Language | English |
| Type | PhD thesis |
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