Dynamical correlation effects in a weakly correlated material: Inelastic x-ray scattering and photoemission spectra of beryllium

Beryllium is a weakly correlated simple metal. Still we find that dynamical correlation effects, beyond the independent-particle picture, are necessary to successfully interpret the electronic spectra measured by inelastic x-ray scattering (IXS) and photoemission spectroscopies (PES). By combining ab initio time-dependent density-functional theory (TDDFT) and many-body Green's function theory in the $GW$ approximation ($GW\mathrm{A}$), we calculate the dynamic structure factor, the quasiparticle (QP) properties and PES spectra of bulk Be. We show that band-structure effects (i.e., due to interaction with the crystal potential) and QP lifetimes (LT) are both needed in order to explain the origin of the measured double-peak features in the IXS spectra. A quantitative agreement with experiment is obtained only when LT are supplemented to the adiabatic local-density approximation (ALDA) of TDDFT. Besides the valence band, PES spectra display a satellite, a signature of dynamical correlation due to the coupling of QPs and plasmons, which we are able to reproduce thanks to the combination of the $GW\mathrm{A}$ for the self-energy with the cumulant expansion of the Green's function.

Figure 1

IXS experimental data of Be from Ref. [30] (grey dots) are compared with dynamic structure factors $S(\mathbf{q},\omega )$ calculated in the RPA (violet curves) and the ALDA (orange curves). Spectra are shown for three different directions in the reciprocal space, from the left to the right, and for increasing momentum transfers going from bottom to top panels. The value of the momentum transfer (in atomic units) is reported in the top-left corner of each panel. Two panels are left blank since the corresponding experimental data are missing in Ref. [30]. Arbitrary units are used everywhere for $S(\mathbf{q},\omega )$.

Figure 2

Comparison between the LDA and the $GW$ band structure along a path in the hexagonal Brillouin zone of Be. Here and in the other figures in this section, the zero of the energy axis is set at the Fermi level.

Figure 3

$GW$ corrections $\mathrm{Re}{E}_{n\mathbf{k}}-{\varepsilon }_{n\mathbf{k}}$ as a function of the LDA eigenvalues ${\varepsilon }_{n\mathbf{k}}$.

Figure 4

Quasiparticle-only (purple curve) and full $GW$ (blue curve) spectral functions compared with experimental photoemission data with photon energy $h\nu =1486$ eV from Ref. [34]. Here and in the following comparisons with the photoemission experiment, the spectra are in arbitrary units.

Figure 5

(a) $GW$ and (b) $GW$+C spectral functions $A_{n\mathbf{k}}\left(\omega \right)$ for the first Be band along the $\mathrm{\Gamma }M$ direction.

Figure 6

$GW$ (blue curve), $GW$+C (red curve), and ${GW}^{\text{ALDA}}+\mathrm{C}$ (green curve) spectral functions compared with experimental photoemission data with photon energy $h\nu =1486$ eV from Ref. [34].

Figure 7

Photoemission spectra simulated from the $GW$+C spectral function $A(\omega$)—see Fig. 6—as a function of photoexcitation cross section ratio ${\sigma }_s/{\sigma }_p$ for $2s$ and $2p$ states, with inclusion of a Shirley background and convoluted with experimental resolution of 0.6 eV, in comparison with experimental photoemission data from Ref. [34].

Figure 8

Absolute value of the imaginary part of the self-energy $|\mathrm{Im}{\mathrm{\Sigma }}^{\mathrm{xc}}|$ calculated in the $GW\mathrm{A}$ as a function of energy (the Fermi level is set at 0). The QP lifetime for holes (negative energies) and electrons (positive energies) is the inverse of $|\mathrm{Im}{\mathrm{\Sigma }}^{\mathrm{xc}}|$.

Figure 9

IXS experimental data of Be from Ref. [30] (grey dots) are compared with dynamic structure factors $S(\mathbf{q},\omega )$ calculated in the RPA+LT (blue curves) and the ALDA+LT (red curves) schemes. See also Fig. 1 where quasiparticle lifetimes (LT) are not considered.