Valence Electron Photoemission Spectrum of Semiconductors: Ab Initio Description of Multiple Satellites

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.

Figure 1

Experimental XPS spectrum of Si at 800 eV photon energy (blue crosses), compared to the theoretical intrinsic $A(\omega )$ calculated from $G_0{W}_0$ (red dashed line), and from Eq. (4) (green dot-dashed line). On top of the latter the black solid line also includes extrinsic and interference effects. All spectra contain photoabsorption cross sections, a calculated secondary electron background and 0.4 eV Gaussian broadening to account for finite $k$-point sampling and experimental resolution. The Fermi energy is set to 0 eV.

Figure 2

$G_0{W}_0$ spectral function of bulk silicon for the top and bottom valence bands at the $\Gamma$ point (black solid and blue dashed lines, respectively). The corresponding imaginary parts of the self-energy (red empty circles and dashed line, and green full circles and solid-line) and $\omega -{\epsilon }_H-\mathrm{Re}\Sigma$ (red empty squares and dashed line, and green full squares and solid line) are also shown. The Fermi energy is set to 0 eV.