Cumulant Green's function calculations of plasmon satellites in bulk sodium: Influence of screening and the crystal environment

We present ab initio calculations of the photoemission spectra of bulk sodium using different flavors of the cumulant expansion approximation for the Green's function. In particular, we study the dispersion and intensity of the plasmon satellites. We show that the satellite spectrum is much more sensitive to many details than the quasiparticle spectrum, which suggests that the experimental investigation of satellites could yield additional information beyond the usual studies of the band structure. In particular, a comparison to the homogeneous electron gas shows that the satellites are influenced by the crystal environment, although the crystal potential in sodium is weak. Moreover, the temperature dependence of the lattice constant is reflected in the position of the satellites. Details of the screening also play an important role; in particular, the contribution of transitions from $2s$ and $2p$ semicore levels influences the satellites, but not the quasiparticle. Moreover, inclusion of contributions to the screening beyond the random-phase approximation has an effect on the satellites. Finally, we elucidate the importance of the coupling of electrons and holes by comparing the results of the time-ordered and the retarded cumulant expansion approximations. Again, we find small but noticeable differences. Since all the small effects add up, our most advanced calculation yields a satellite position which is improved with respect to previous calculations by almost 1 eV. This stresses the fact that the calculation of satellites is much more delicate than the calculation of a quasiparticle band structure.

Figure 1

$\mathbf{k}$-resolved spectral functions $A(\mathbf{k},\omega )$ for the sodium valence band along $\mathrm{\Gamma }\mathrm{N}$ using TOC (red solid curve) and RC (black dotted curve). The Fermi wave vector is $k_F=0.49$ a.u.

Figure 2

The dispersion of quasiparticles (red) and first (blue) and second (green) satellites in TOC (solid curves) and RC (stars). The stars with double size are satellites from states above the Fermi level $\mu$.

Figure 3

The total $\mathbf{k}$-summed spectra for the sodium valence band in both TOC (in red) and RC (in black), multiplied by the room temperature Fermi function. Inset: Zoom on the first satellite. The two dashed vertical lines mark the positions of the maximum of each satellite; their distance is $0.11\mathrm{eV}$.

Figure 4

The total $\mathbf{k}$-summed spectra of HEG (in red) and sodium (in black), multiplied by the Fermi function. Inset: Zoom on the first satellite. The two dashed vertical lines mark the positions of the maximum of each satellite; their distance is $0.12\mathrm{eV}$.

Figure 5

$\mathbf{k}$-resolved spectral functions $\mathrm{A}(\mathbf{k},\omega )$ of sodium (red curve with circles) and the HEG (black curve) at the $\mathrm{\Gamma }$ point and close to Fermi level ($k_F\sim 0.49$ a.u.).

Figure 6

(a) Dispersion of quasiparticles (in red) and first (in blue) and second (in green) satellite of sodium (solid curves) and of the HEG (stars) along the $\mathrm{N}\mathrm{\Gamma }\mathrm{N}$ direction. (b), (c) Comparison of the quasiparticle (red solid curves), first satellite (blue dashed curve), and second satellite (green dots) dispersions of (b) sodium and (c) the HEG. The satellite energies have been shifted in order to align all energies at the $\mathrm{\Gamma }$ point.

Figure 7

The shifted imaginary part of self-energy of sodium (red solid lines) and HEG (blue dashed lines) at different $\mathbf{k}$ points along $\mathrm{\Gamma }\mathrm{N}$ in the sodium Brillouin zone. Only the removal part $\omega <\mu$ is shown. The Fermi wave vector is at $k_F=0.49$ a.u.

Figure 8

The RPA loss functions of sodium (red curve) and HEG (blue curve) at different momentum transfers $q$ (in a.u.).

Figure 9

RPA loss functions of sodium calculated with lattice parameters corresponding to 5 K (red curve) and 293 K (black curve) at different momentum transfers $q$ (in a.u.).

Figure 10

$\mathbf{k}$-resolved spectral functions of sodium along the $\mathrm{\Gamma }\mathrm{N}$ direction using lattice parameters corresponding to 5 K (black curve with dots) and 293 K (red solid curve) at the $\mathrm{\Gamma }$ point and close to Fermi level ($k_F=0.49\mathrm{a}.\mathrm{u}.$).

Figure 11

(a) The QP and plasmon satellite dispersions along $\mathrm{\Gamma }\mathrm{N}$. The QP and first satellite energies of Na at 5 K are represented in red and blue solid curves, respectively. For Na at 293 K QP and satellite energies are stars. (d) The total spectra summed over $\mathbf{k}$ points and two bands, using lattice parameters corresponding to 5 K (black curve with dots) and 293 K (red solid curve).

Figure 12

Comparison between Na with lattice parameter at 5 K ($a_0=4.23$ Å) and Na with artificially expanded lattice parameter ($a_0=4.44$ Å): $\mathbf{k}$-resolved spectral functions (a) at the $\mathrm{\Gamma }$ point and (b) close to the Fermi level; (c) band and satellite dispersions; (d) $\mathbf{k}$-integrated spectral function.

Figure 13

The loss functions $-\mathrm{Im}{\epsilon }^{-1}$ including different transitions. The yellow filled curve contains all transitions from $2s, 2p, 3s$ states. The red dashed and black solid curves contain transitions from $2p+3s$ and $3s$ only, respectively. The black diamonds are calculated using a pseudopotential containing only $3s$ electrons as valence states.

Figure 14

(a) The loss functions $-\mathrm{Im}{\epsilon }^{-1}$ containing the core transitions (red dashed curve) and transitions of valence states only (black solid curve), together with their real (${\epsilon }_1$) and imaginary (${\epsilon }_2$) parts at $q=0.11$ a.u. (b) Zoom around the plasmon energy for ${\epsilon }_1$. (c) Zoom around the core-level contributions for ${\epsilon }_2$. Note that the fast wiggles in the blue curve in panel (c) are due to the finite $\mathbf{k}$-point sampling.

Figure 15

Same as Fig. 14, but for $q=0.45$ a.u.

Figure 16

$\mathbf{k}$-resolved spectral functions $\mathrm{A}(\mathbf{k},\omega )$ of sodium taking into account the core polarization (black curve with dots) and without core polarization (red curve) (a) at the $\mathrm{\Gamma }$ point and (b) close to Fermi level ($k_F=0.49$ a.u.).

Figure 17

(a) Band and satellite dispersions along $\mathrm{\Gamma }\mathrm{N}$ and (b) $\mathbf{k}$-integrated spectral function, calculated with or without the core polarization contribution.

Figure 18

The loss functions $-\mathrm{Im}{\epsilon }^{-1}$ of Na calculated in RPA (black solid lines) and ALDA (red dashed lines) at different momentum transfers $\mathbf{q}$ in a.u.

Figure 19

$\mathbf{k}$-resolved spectral functions $A(\mathbf{k},\omega )$ of sodium using RPA (black dotted curve) and ALDA screening (red curve) (a) at the $\mathrm{\Gamma }$ point and (b) close to Fermi level ($k_F=0.49$ a.u.).

Figure 20

(a) Band and satellite dispersions and (b) $\mathbf{k}$-integrated spectral functions of sodium using RPA and ALDA screening.

Figure 21

The black and red dashed curves are intrinsic TOC from Ref. [16] (using the 5 K lattice constant, transitions from valence only, without intraband transitions, and RPA screening) and RC spectral functions (using room temperature lattice constant, including transitions from semicore and intraband transitions, and ALDA screening), respectively. The TOC spectral function (using the 5 K lattice constant, transitions from valence only, without intraband transitions, and RPA screening) with extrinsic and interference effects, together with secondary electron background (magenta solid curve), is compared with experimental data from Ref. [37] (green dots). The black solid curve is obtained by adding the extrinsic and interference effect on the black dashed curve. All curves have been aligned on the low-binding energy side of the QP.