Efficientab initiocalculations of bound and continuum excitons in the absorption spectra of semiconductors and insulators

Abstract

We present calculations of the absorption spectrum of semiconductors and insulators comparing various approaches: (i) the two-particle Bethe-Salpeter equation of Many-Body Perturbation Theory; (ii) time-dependent density-functional theory using a recently developed kernel that was derived from the Bethe-Salpeter equation; (iii) a scheme that we propose in the present work and that allows one to derive different parameter-free approximations to (ii). We show that all methods reproduce the series of bound excitons in the gap of solid argon, as well as continuum excitons in semiconductors. This is even true for the simplest static approximation, which allows us to reformulate the equations in a way such that the scaling of the calculations with number of atoms equals the one of the Random Phase Approximation.PACS numbers: 78.20.Bh, 71.15.Qe Time-dependent density-functional theory (TDDFT) [1] is more and more considered to be a promising approach for the calculation of neutral electronic excitations, even in extended systems [2,3]. In linear response, spectra are described by the Kohn-Sham independentparticle polarizability χ KS 0 and the frequency-dependent exchange-correlation (xc) kernel f xc . The widely used adiabatic local-density approximation [4,5] (TDLDA), with its static and short-ranged kernel, often yields good results in clusters but fails for absorption spectra of solids. Instead, more sophisticated approaches derived from Many-Body Perturbation Theory (MBPT) [6,7,8,9,10] have been able to reproduce, ab initio, the effect of the electron-hole interaction in extended systems, not least thanks to an explicit long-range contribution [6,11,12]. The latter strongly influences spectra like optical absorption or energy loss, especially for relatively small momentum transfer.Here we show that this kernel is even able to reproduce the hydrogen-like excitonic series in the photoemission gap of a rare gas solid. However the kernel has a strong spatial and frequency dependence, and its evaluation requires a significant amount of computer time. We therefore tackle the question of a parameter-free, but quick TDDFT calculation of excitonic effects in solids, which has been so far an unsolved problem, and show that a much more efficient formulation can indeed be achieved. In particular we demonstrate how it is possible for a wide range of materials to obtain good absorption spectra including excitonic effects with a static kernel leading in principle to a Random Phase Approximation (RPA)-like scaling of the calculation with the number of atoms of the system. Atomic units are used throughout the paper. The vectorial character of the quantities r, k, G, q (where k and q are vectors in the Brillouin zone, and G is a reciprocal lattice vector) is implicit. Only transitions of positive frequency (i.e. resonant contributions), which dominate absorption spectra, are considered throughout.Let us first concentrate on the absorption spectrum of solid argon. The low band dispersion, together with the small polarizability ...

Time-dependent stochastic Bethe-Salpeter approach

“…For silicon NCs, the semi-empirical pseudopotential model overemphasizes the lowest excitonic transition in comparison to the plasmonic resonance observed at ∼ 10eV using TDDFT. 35,56,66,101 It also misses the split of the lowest excitonic peak observed experimentally for bulk silicon and reproduced by the BSE approach, [54][55][56][57] but not by the current model ignoring electron-hole correlations. 123 The fact that the current calculation does not capture this split could be a consequence of the approximation used to model the screening.…”

“…The most common flavors are the GW approximation 58 to describe quasiparticle excitations (G indicates the single-particle Green function and W the screened Coulomb interaction) and the Bethe-Salpeter equation (BSE) 59 to describe electron-hole excitations. Both approaches offer a reliable solution to quasiparticle [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] and optical 53,54,56,57,66,[79][80][81][82][83][84][85][86][87][88][89] excitations, even for situations where TDKS often fails, for example in periodic systems [53][54][55][56][57]79 or for charge-transfer excitations in molecules. 86 However, the computational cost of the MBPT methods is considerably more demanding than for TDKS, because conventional techniques require the explicit calculation of a large number of occupied and virtual electronic states and the evaluation of a large number of screened exchange integra...…”

Exciton dispersion from first principles

“…III) represent a prediction for future measurements. Argon is a textbook material that has been studied extensively in the past 2,39,[45][46][47][48][49][50] for its absorption spectrum that shows a hydrogenlike bound-exciton series in which the n = 1 exciton has a strongly localized character and the higher excitons n 2 are more delocalized. 51 The energy-momentum dispersion map represented in Fig.…”

“…8,24 Analogously, the scissor correction is 6.6 eV in Ar. 39 BSE spectra are converged with 20 bands for LiF and 10 bands for Ar. The TDA that is employed for those spectra (see Sec.…”

Estimating Excitonic Effects in the Absorption Spectra of Solids: Problems and Insight from a Guided Iteration Scheme

“…A class of successful kernels has been derived from the BSE [17][18][19][20][21][22]. The nanoquanta kernel [20][21][22][23] gives results close to BSE ones, but with a comparable computational cost, although suggestions for speedups have been made [24]. The long-range corrected (LRC) kernel [23,25] f LRC xc = −α/q 2 with the correct divergence for small wavevectors q is a simple scalar approximation of the nanoquanta kernel.…”