On the combined use of GW approximation and cumulant expansion in the calculations of quasiparticle spectra: The paradigm of Si valence bands

Since the earliest implementations of the various GW approximations and cumulant expansion in the calculations of quasiparticle propagators and spectra, several attempts have been made to combine the advantageous properties and results of these two theoretical approaches. While the GW-plus-cumulant approach has proven successful in interpreting photoemission spectroscopy data in solids, the formal connection between the two methods has not been investigated in detail. By introducing a general bijective integral representation of the cumulants, we can rigorously identify at which point these two approximations can be connected for the paradigmatic model of quasiparticle interaction with the dielectric response of the system that has been extensively exploited in recent interpretations of the satellite structures in photoelectron spectra. We establish a protocol for consistent practical implementation of the thus established $\mathrm{GW}+\mathrm{cumulant}$ scheme and illustrate it by comprehensive state-of-the-art first-principles calculations of intrinsic angle-resolved photoemission spectra from Si valence bands.

Figure 1

(a) Diagram for the second-order contribution in the expansion of a Green's function (5) of an electron injected with the initial wave vector $\mathbf{k}$ into an empty band. Solid lines denote the unperturbed propagators, wavy lines denote the unscreened Coulomb interaction, and the bubble denotes the electronic response function ${\chi }_{\mathbf{q}}(t_2,t_1)$ of the system. (b) Equivalent diagram in the quasiparticle-boson model outlined in Sec. 2. Solid circles denote the interaction matrix element $v_{\mathbf{q}}$ from Eq. (15) and the dash-dotted line denotes the boson propagator $D_{\mathbf{q}}(t_2-t_1)$ in the intermediate state interval $(t_1,t_2)$. (c) Example of a second-order diagram involving two electrons above and one hole below the Fermi level in the same interval. This term cannot be generated within the single-particle limit of the Hamiltonian (1).

Figure 2

Illustration of the quasiparticle spectrum ${\mathcal{N}}_{\mathbf{K}}\left(\omega \right)$ corresponding to propagation of a hot electron in an initially empty Q2D surface band completely above the Fermi level. The electron is coupled to the electronic charge density excitations in the underlying substrate whose spectrum consists of incoherent $e$-h pairs and surface plasmons of energy $\hslash {\omega }_s=2$ eV. The calculation is performed for intermediate coupling strength and initial 2D electron momentum $K=0.06$ a.u., which produces nearly equal strengths of the threshold and first satellite peak. The satellites appear approximately at multiples of the surface plasmon energy above the threshold energy ${\epsilon }_{\mathbf{K}}={\epsilon }_{\mathbf{K}}^0+{\mathrm{\Delta }}_{\mathbf{K}}$ of the leftmost peak (denoted by the vertical line at $\omega \simeq -2$ eV). Energy zero corresponds to the first moment of the spectrum which measures the energy shift ${\mathrm{\Delta }}_{\mathbf{K}}<0$ of the unperturbed level ${\epsilon }_{\mathbf{K}}^0$.

Figure 3

Spectral function of silicon (in ${\mathrm{eV}}^{-1}$ units) evaluated using (a) the $G_0{W}_0$ approximation, (b) the $G_0{W}_0+\text{cumulant}$ ($G_0{W}_0+\text{C}$) approach, (c) DFT-LDA, and (d) the plasmonic polaron model (PPM). For comparison, we report in panel (c) a replica of the DFT-LDA band structure redshifted by the plasmon energy ${\omega }_{\mathrm{pl}}$ (${\omega }_{pl}\simeq 16.6$ eV for silicon). All energies are relative to the Fermi level.

Figure 4

Spectral function of silicon at the $\mathrm{\Gamma }$ (a) and $W$ (b) high-symmetry points evaluated using the Sternheimer-GW method ($G_0{W}_0$ approximation) and the $G_0{W}_0+\text{cumulant}$ ($\mathrm{GW}+\mathrm{C}$) approach.