This chapter provides an overview of quasiparticle calculations in solids, and, in particular, the GW approximation (GWA). A successful approximation for the determination of excited states of solids is based on the quasiparticle concept and the Green function method. The Coulomb repulsion between electrons leads to a depletion of negative charge around a given electron, and the ensemble of this electron and its surrounding positive screening charge forms a quasiparticle. The mathematical description of quasiparticles is based on the single-particle Green function (G), whose exact determination requires complete knowledge of the quasiparticle self-energy ∑. The self-energy ∑ is a non-Hermitian, energy-dependent, nonlocal operator that describes exchange and correlation effects beyond the Hartree approximation. A determination of the self-energy can only be approximate, and a working scheme for the quantitative calculation of excitation energies in metals, semiconductors, and insulators is the so-called dynamically screened interaction or the GWA. The chapter also discusses (1) the physics and extensions of the GWA; (2) numerical aspects of GWA calculations; (3) applications of the GWA to semiconductors, insulators and metals; and (4) the relevance of GWA calculations to optical response. Finally, the chapter presents parallel algorithms both for reciprocal and real-space/imaginary-time GWA calculations and several alternative methods to determine excited states of solids within density functional theory.

Quasiparticle calculations in solids

75 APPENDIX E. 77 APPENDIX F. 79 APPENDIX G. 82 APPENDIX H. Introduction. Generalized Green's functions. Equations of motion for the generalized single-particle Green's function. Bethe-Salpeter equation for the two-particle Green's function. Connection with linear-response theory. Conserving approximations. Gauge invariance, current conservation, and the Ward identities. Optical properties of semiconductors. Single-particle energy levels. Screening of static impurities. Excitonic states. A functional derivative identity. Dyson's equation. Reducible vs. irreducible parts of the correlation functions. Elimination of the spin variables and transformation to the energy-momentum representation. The Thomas-Reiche-Kuhn sum rule. The Clausius-Mossotti relation. The particle-hole Green's function. The Haken potential.

http://bcsbec.df.unicam.it/files/LaRNC11_N12(1988).pdf

Application of the Green's functions method to the study of the optical properties of semiconductors

Reflection anisotropy spectroscopy (RAS) is a non-destructive optical probe of surfaces that is capable of operation within a wide range of environments. In this review we trace the development of RAS from its origins in the 1980s as a probe of semiconductor surfaces and semiconductor growth through to the present where it is emerging as a powerful addition to the wide range of existing ultra-high vacuum (UHV) surface science techniques. The principles, instrumentation and theoretical considerations of RAS are discussed. The recent progress in the application of RAS to investigate phenomena at metal surfaces is reviewed, and applications in fields including electrochemistry, molecular assembly, liquid crystal device fabrication and remote stress sensing are discussed. We show that the experimental study of relatively simple surfaces combined with continuing progress in the theoretical description of surface optics promises to unlock the full potential of RAS. This provides a firm foundation for the application of the technique to the challenging fields of ambient, high pressure and liquid environments. It is in these environments that RAS has a clear advantage over UHV-based probes for investigating surface phenomena, and its surface sensitivity, ability to monitor macroscopic areas and rapidity of response make it an ideal complement to scanning probe techniques which can also operate in such environments.

http://iopscience.iop.org/article/10.1088/0034-4885/68/6/R01/pdf

Reflection anisotropy spectroscopy

For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrodinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrodinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.

/pdf/time-dependent-density-functional-theory-for-extended-3jg2ina8bf.pdf

Time-dependent density-functional theory for extended systems

A brief account of the most important experimental and theoretical methods employed in surface physics is given. The use of optical spectroscopy in the study of clean semiconductor surfaces is described in detail.

Physics of Surfaces

We formulate the theory of the macroscopic dielectric tensor for crystals of lower-than-cubic symmetry, using the local-orbital approach to the dielectric response. In the case of noncubic infinite crystals we give a definition of the macroscopic dielectric tensor that is simpler than any previous formulation and allows us to perform realistic calculations with roughly the same computational effort needed in cubic crystals. In the case of semi-infinite crystals, we give for the first time a definition of the macroscopic dielectric tensor and outline a way of computing it from the two-particle Green's function, which can be computed according to the method of Hanke and Sham, including local-field and excitonic effects. A remarkable result is that in both cases the long-range part of the Coulomb interaction does not affect the macroscopic dielectric tensor. Previous formulations of the optical properties of surfaces, which neglected local-field effects, are shown to be yet valid, provided that the macroscopic dielectric tensor replaces the microscopic one.

E. Fiorino

Papers

Macroscopic dielectric tensor at crystal surfaces

Local-field effects on the optical properties of non-cubic crystals