Valence Electron Photoemission Spectrum of Semiconductors:Ab InitioDescription of Multiple Satellites

Abstract

The experimental valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approximation for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare experimental high energy photoemission spectra with GW calculations and analyze the origin of the GW failure. We then propose an approximation to the functional differential equation that determines the exact one-body Green's function, whose solution has an exponential form. This yields a calculated spectrum, including cross sections, secondary electrons, and an estimate for extrinsic and interference effects, in excellent agreement with experiment. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.Photoemission is a prominent tool to access information about electronic structure and excitations in materials. Modern synchrotron sources can provide detailed insight, thanks to their high intensity and broad photon energy range. But the interpretation of the experimental data is far from obvious, and theory is an essential complementary tool. However, ab initio calculations typically focus on bulk bandstructure [1, 2]; thus surface effects are ignored, and satellites are not included. The latter are a pure many-body effect due to coupling to excitations of the material. Such many-body effects are contained in approaches developed for correlated materials [3,4] however, these are usually based on models with short-range interactions, whereas satellites such as plasmons involve long-range effects. Plasmon satellites have been extensively studied in core-level experiments [5]. There they can be described by a theoretical model where a single dispersionless fermion couples to bosons. The resulting exact Green's function has an exponential form given by the so-called cumulant expansion (CE). A Taylor expansion of the exponential leads to a well defined quasi-particle (QP) peak followed by a decaying series of plasmon satellites at energy differences given by the plasmon energy, consistent with experimental observations [6][7][8][9][10]. In the valence region, plasmon satellites are much less studied, though ab initio approaches can provide a good starting point. At high photoelectron energies the photoemission spectrum is approximately proportional to the intrinsic spectral function A(ω) = −(1/π)Im G(ω), where G is the one-particle Green's function. The latter is typically calculated using the widely used GW approximation (GWA) [7,11,12]. In principle, the GWA contains correlations effects beyond the quasiparticle approximation. However, these additional features are rarely calculated due to computational complexity and, more importantly, the serious discrepancies between GWA and experiment (see e.g. [13][14][15][16]). The CE has also been used for the homogeneous electron gas [17] and simple metals [14,15], yielding an improved description of satellites over GW. Silicon [16] and graphite [18] ...

Dispersion and line shape of plasmon satellites in one, two, and three dimensions

“…This is achieved by the GW plus cumulant (GW+C) approach [13,14], where the cumulant expansion of the electron Green's function G is truncated at second order in the screened Coulomb interaction W . GW+C calculations yielded good agreement with experimental photoemission and tunneling spectra in a wide range of physical systems [6][7][8][15][16][17] and also with highly accurate coupled-cluster Green's function calculations [18]. While Green's function methods, such as the GW+C approach, often produce highly accurate results, gaining intuition and insights into the underlying manybody processes can be difficult.…”

“…Such plasmon satellites have long been known in core-electron photoemission spectra [4,5]. In recent years, valence band plasmon satellites, which were observed experimentally in three-dimensional metals and semiconductors [6][7][8][9], but also in two-dimensional systems, such as doped graphene and semiconductor quantum-well electron gases [10][11][12], received much attention.…”

“…While the first solution corresponds to a standard quasiparticle excitation, he assigned the second solution to a novel particle, the plasmaron, a strongly coupled, coherent hole-plasmon state. Despite several reports claiming the observation of the plasmaron [10,11], it has become clear recently that no such excitation exists in known materials and that its spurious prediction signals the inability of the GW method (and, equivalently, the Brillouin-Wigner perturbation theory) to describe plasmon satellites [6,7,15,16].…”

“…Eq. (6) shows that the satellite contribution to the spectral function closely related to the coupling-strength weighted electron-plasmon jointdensity of states…”

“…Previous models of plasmon satellites in threedimensional systems [6,32,33] assumed that plasmon dispersion is a minor effect and approximated the satellite simply as a shifted, broadened copy of the quasiparticle peak. Such approaches fail to describe the asymmetric lineshape of the satellite and also cannot be applied straightforwardly to lower-dimensional systems, which we discuss below.…”