Quasiparticles and phonon satellites in spectral functions of semiconductors and insulators: Cumulants applied to the full first-principles theory and the Fröhlich polaron

The electron-phonon interaction causes thermal and zero-point motion shifts of electron quasiparticle (QP) energies ${\epsilon }_k\left(T\right)$. Other consequences of interactions, visible in angle-resolved photoemission spectroscopy (ARPES) experiments, are broadening of QP peaks and appearance of sidebands, contained in the electron spectral function $A(k,\omega )=-\Im m{G}_R(k,\omega )/\pi$, where $G_R$ is the retarded Green's function. Electronic structure codes (e.g., using density-functional theory) are now available that compute the shifts and start to address broadening and sidebands. Here we consider MgO and LiF, and determine their nonadiabatic Migdal self-energy. The spectral function obtained from the Dyson equation makes errors in the weight and energy of the QP peak and the position and weight of the phonon-induced sidebands. Only one phonon satellite appears, with an unphysically large energy difference (larger than the highest phonon energy) with respect to the QP peak. By contrast, the spectral function from a cumulant treatment of the same self-energy is physically better, giving a quite accurate QP energy and several satellites approximately spaced by the LO phonon energy. In particular, the positions of the QP peak and first satellite agree closely with those found for the Fröhlich Hamiltonian by Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)] using diagrammatic Monte Carlo. We provide a detailed comparison between the first-principles MgO and LiF results and those of the Fröhlich Hamiltonian. Such an analysis applies widely to materials with infrared(IR)-active phonons.

Figure 1

Quasiparticle energy of the Fröhlich polaron, as a function of the Fröhlich coupling constant, in units of ${\omega }_{\mathrm{LO}}$: accurate results from Ref. [26] (blue squares), from the cumulant approach (which agrees with the lowest-order Rayleigh-Schrödinger theory) (black line), and from Dyson-Migdal spectral function [18, 19] (red circles).

Figure 2

Quasiparticle spectral weight $Z_{k=0}$ of the Fröhlich polaron, as a function of the Fröhlich coupling constant. The blue squares are accurate Monte Carlo results from Ref. [26]. The solid red line comes from the full Dyson-Migdal spectral function, Eq. (11). The dotted red line comes from the linearized approach, Eq. (14). The black line is the cumulant result using Eq. (26) with the retarded Green's function.

Figure 3

Some of the lowest-order Feynman diagrams. (a) Usual Fan self-energy diagram. (b) Diagram that contributes to the self-consistency of the electron propagator. (c) Diagram that contains a vertex correction.

Figure 4

Fröhlich Hamiltonian self-energy. Real part in orange, imaginary part in blue. The functions with a negligible broadening, $\delta =0.001{\omega }_{\mathrm{LO}}$, are represented by a continuous lines, while functions with a broadening $\delta =0.12{\omega }_{\mathrm{LO}}$, similar to the one used in first-principles calculations, are represented by dashed lines.

Figure 5

Fröhlich Hamiltonian spectral function using the cumulant approach $G^{\mathrm{C}}$ (solid, black) and the Dyson-Migdal approach $G^{\mathrm{DM}}$ (dashed, red) for $\alpha =0.34$. The position of the quasiparticle peak slightly differs between the two. The cumulant version deviates from the Fröhlich value $-\alpha {\omega }_{\mathrm{LO}}$ only because a nonzero broadening $\delta \approx 0.12{\omega }_{\mathrm{LO}}$ [85] is used in numerical evaluation of Eq. (29), for consistency with later calculations. In the DM case, the onset of the phonon-emission “satellite” is higher by ${\omega }_{\mathrm{LO}}$ than the bare band energy $\omega ={\varepsilon }_{\mathbf{k}=0,c}^{\left(0\right)}=0$ [18]. By contrast, it is higher by ${\omega }_{\mathrm{LO}}$ than the quasiparticle peak in the cumulant method, corresponding to states that combine the dressed quasiparticle with one LO phonon.

Figure 6

The lower part shows the DM self-energy (in units of ${\omega }_{\mathrm{LO}}$) for a Fröhlich electron with $\alpha =1.62$ (typical of the conduction-band minimum of MgO), using Eq. (30) except broadened with $\delta =0.12{\omega }_{\mathrm{LO}}$ in Eq. (28). The position of the DM QP peak is at the crossing between the real part and the line $\Re e\mathrm{\Sigma }=\omega$. The upper part shows both the resulting DM spectral function and the cumulant version from $G^{\mathrm{C}}$. The satellite setting in at $\omega /{\omega }_{\mathrm{LO}}=1$ in the DM case is barely visible in this picture.

Figure 7

Fröhlich Hamiltonian spectral function using the cumulant $G^{\mathrm{C}}$ (in black) and the Dyson-Migdal approach $G^{\mathrm{DM}}$ (dashed, in red) for $\alpha =4.01$ (typical of the conduction-band minimum of LiF), from the Migdal self-energy broadened by $\delta =0.12{\omega }_{\mathrm{LO}}$.

Figure 8

Fröhlich Hamiltonian spectral function using the cumulant $G^{\mathrm{C}}$ (in black) and the Dyson-Migdal approach $G^{\mathrm{DM}}$ (dashed, in red) for $\alpha =8$, from the Migdal self-energy broadened by $\delta =0.12{\omega }_{\mathrm{LO}}$.

Figure 9

The MgO conduction-band minimum retarded self-energy with $\delta =0.01$ eV and a ${96}^3 q$ grid. Full black line: imaginary part; dotted red line: real part. The reference energy ${\varepsilon }^0$ is the unrenormalized conduction-band energy minimum. This figure is surprisingly similar to Fig. 6, considering that it includes all phonon and interband effects rather than just the analytic Fröhlich result.

Figure 10

Retarded self-energy for the bottom of the conduction band of MgO in a wider range of energy than in Fig. 9: imaginary part in black, real part in dashed red. The electronic DOS is also shown (dotted blue) for comparison.

Figure 11

Retarded self-energy for the top of the valence band of MgO: imaginary part in black, real part in dashed red. The electronic DOS is also shown (dotted blue), for comparison. The VBM eigenenergy is the reference energy ${\varepsilon }^0$, which explains the horizontal shift of the DOS with respect to Fig. 10.

Figure 12

Cumulant (black) and Dyson-Migdal (dashed red) spectral functions for the conduction-band minimum of MgO. The Fröhlich spectral function (dotted blue) with $\alpha =1.62$ is also shown for comparison. The good correspondance between the Fröhlich and Dyson-Migdal peaks is fortuitous, see text.

Figure 13

Cumulant (black) and Dyson-Migdal (dashed red) spectral functions for the conduction-band minimum of LiF. The Fröhlich spectral function (dotted blue) with $\alpha =4.01$ is also shown for comparison.

Figure 14

Cumulant (black) and Dyson-Migdal (dashed red) spectral functions for the valence-band maximum of MgO.

Figure 15

Cumulant (black) and Dyson-Migdal (dashed red) spectral functions for the valence-band maximum of LiF.

Figure 16

Spectral function (Dyson-Migdal) of the MgO CBM self-energy with a ${20}^3 q$ grid and decreasing $\delta =0.1$, 0.05, and 0.02 eV. The separation between the quasiparticle peak and the satellite is still not complete with the smallest $\delta$ value.

Figure 17

The imaginary part of the MgO CBM self-energy with a ${20}^3 q$ grid and decreasing $\delta =0.1$, 0.05, and 0.02 eV.

Figure 18

The MgO CBM self-energy with $\delta =0.01$ eV and a ${20}^3 q$ grid. Full black line: imaginary part; dotted red line: real part.

Figure 19

Cumulant spectral function of the MgO VBM with $\delta =0.01$ eV and ${20}^3, {32}^3, {48}^3$, and ${64}^3 q$ grids.