Valence Band X-Ray Emission Spectra of Compressed Germanium

We report measurements of the valence band width in compressed Ge determined from x-ray emission spectra below the Ge $K$ edge. The width of the valence band does not show any pressure dependence in the semiconducting diamond-type structure of Ge below 10 GPa. On the other hand, in the metallic $\beta$-Sn phase above 10 GPa the valence band width increases under compression. Density-functional calculations show an increasing valence band width under compression both in the semiconducting phase (contrary to experiment) and in the metallic $\beta$-Sn phase of Ge (in agreement with observed pressure-induced broadening). The pressure-independent valence band width in the semiconducting phase of Ge appears to require theoretical advances beyond the density-functional theory or the $GW$ approximation.

Figure 1

Overview of valence-band emission spectra in Ge at selected pressures. We show the schematics of the transition from $4p$ to $1s$ states on the left. The $4p$ valence-band structure is adopted from 3, and the full bandwidth is indicated by the arrow (red online) in the inset. The details of the measurements and setup were reported in 31. Spectra are normalized and shifted on the energy scale to match at the high-energy side of the peak. Emission spectra at 1, 5, and 9 GPa were collected from the semiconducting diamond-type phase and have nearly identical line shape, while the spectra at 13, 21, and 31 GPa from the metallic $\beta$-Sn phase demonstrate the linewidth which is increasing under pressure.

Figure 2

Theoretically calculated valence-band region of the x-ray emission spectra. Black solid line—ambient pressure; black dashed line—diamond-type Ge at 9 GPa; red dash-dotted line—$\beta$-Sn Ge at 10 GPa. (a) XES calculated using a supercell method; (b) XES calculated from single unit cell; (c) theoretical spectra from Fig. 2b convoluted with Lorentzian (1.96 eV FWHM) and Gaussian (1.5 eV FWHM) to simulate effects of the core-hole lifetime and experimental resolution.

Figure 3

Changes in the linewidth of the valence band in Ge under pressure. Circles with error bars—experiment; squares (red)—theory, single unit cell calculations. Solid line is an estimate for the half-width of the valence band in the free-electron approximation ($E_F/2$ for the free-electron gas); dashed line represents the free-electron calculation corrected according to the GWA results from Fig. 1 in Ref. 25.

Figure 4

Changes in the $p$-PDOS of the valence band of Ge under pressure. The width of the main peak (extending from 0 to $-5\mathrm{eV}$) is increasing under compression. See text for detailed discussion.