Effects of $5d$ electrons and spin-orbit interaction on the characteristics of bulk plasmons in lead

An ab initio study of the dynamical dielectric response of bulk lead is presented. The influence of the $5d$ semicore states on the characteristics of the bulk plasmon is analyzed by means of first-principles pseudopotential calculations. The effects of spin-orbit interactions and local-fields are also studied in detail. The inclusion of the $5d$ semicore states in the valence configuration completely changes the high-energy-transfer dielectric properties of bulk Pb. In particular, it lowers the computed bulk plasmon energy by about 3.5 eV, bringing its frequency to good agreement with experimental data. In general, the high-energy-transfer dielectric response of bulk Pb is found to be shaped mostly by the interplay between the interband transitions involving the semicore $5d$ states and the spin-orbit coupling interaction. Local-field effects are found to affect the relative spectral weight of the high-energy excitations, while leaving their dispersion mostly unaffected.

Figure 1

Schematic representation of the binding energy of the semicore $5d$ electrons of lead in the atom, bulk, and surface. The colored rectangles represent the $6s$- and $6p$-like energy bands in bulk Pb, with a finite bandwidth. For simplicity the bulk and surface Fermi levels ($E_F$) are considered to be equal.

Figure 2

Electronic band structure of bulk lead, calculated with (red solid lines) and without (black dashed lines) spin-orbit interaction included. The $5d$ semicore electrons are considered in the valence configuration, their corresponding energy bands are located at energies below −15 eV. The horizontal thin dashed line represents the Fermi level set to zero.

Figure 3

Comparison of the present results at $q\to 0$ with the data reported in Ref. [63]. Top panel: real part of the dielectric function. Middle panel: imaginary part of the dielectric function. Bottom panel: energy-loss function. The black solid (green dashed-dotted-dotted) curves represent the results obtained in the present work with the semicore electrons included in the valence (core) configuration. The red dashed and blue dashed-dotted curves stand for the experimental and theoretical results, respectively, of Ref. [63]. The insets are zooms into relevant $\omega$ ranges (see text). The vertical orange line in the bottom panel represents the free-electron-gas (FEG) plasmon energy of ${\omega }_p^{\mathrm{FEG}}\simeq 13.5$ eV.

Figure 4

Energy-loss function evaluated (a) without and (b) with inclusion of the semicore $5d$ states for momentum transfers along the $\Gamma \text{−}X$ symmetry direction. Calculations were performed taking into account the SO coupling, LFEs, and the TDLDA XC kernel.

Figure 5

Imaginary part of the noninteracting density-response function, $\text{Im}\left[{\chi }^0(\mathbf{q},\omega )\right]$, for momentum transfers along the $\Gamma \text{−}X$ direction with the SO included and (a) excluding and (b) including the semicore $5d$ electrons in the valence configuration. Vertical arrows show the position of $q_c$ where the plasmon dispersion curve enters the region with increased number of electron-hole pairs. Horizontal dotted lines show the energy thresholds for interband transitions involving the energy bands originated from the $5d_{5/2}$ and $5d_{3/2}$ states. Vertical arrows point to regions where the plasmon peak enters the electron-hole pair continuum.

Figure 6

Energy-loss function surfaces calculated with the semicore $5d$ states included in the valence configuration. The upper, middle, and bottom rows correspond to the results with q along $\Gamma \text{−}X$, $\Gamma \text{−}K$, and $\Gamma \text{−}L$ symmetry directions, respectively. High-symmetry points are marked by thin white vertical lines. The left column stands for $\text{Im}\left[{\epsilon }_{0,0}^{-1}(\mathbf{q},\omega )\right]$ obtained with the SO coupling and LFE included, whereas SO coupling (LFE) was excluded from the calculations corresponding to the right (middle) column. See text for the meaning of the symbols in panels (g) and (i).

Figure 7

Calculated energy-loss function at three different values of $q$ along $\Gamma -L$. Black solid, red dashed and green dashed-dotted curves stand for the results obtained with SO and LFE included, SO coupling excluded, and LFE ignored, respectively.