First-principles approach to excitons in time-resolved and angle-resolved photoemission spectra

In this work we put forward a first-principles approach and propose an accurate diagrammatic approximation to calculate the time-resolved (TR) and angle-resolved photoemission spectrum of systems with excitons. We also derive an alternative formula to the TR photocurrent which involves a single time-integral of the lesser Green's function. The diagrammatic approximation applies to the relaxed regime characterized by the presence of quasistationary excitons and vanishing polarization. The nonequilibrium self-energy diagrams are evaluated using excited Green's functions; since this is not standard, the analytic derivation is presented in detail. The final result is an expression for the lesser Green's function in terms of quantities that can all be calculated in a first-principles manner. The validity of the proposed theory is illustrated in a one-dimensional model system with a direct gap. We discuss possible scenarios and highlight some universal features of the exciton peaks. Our results indicate that the exciton dispersion can be observed in TR and angle-resolved photoemission.

Figure 1

Schematic description of a TR and angle-resolved PE experiment.

Figure 2

(a) Diagram for the self-energy. (b) Diagram for $L$. Wiggly lines denote the bare interaction $v$ and doubly wiggly lines denote the statically screened interaction $W$.

Figure 3

Self-energy diagrams for the model Hamiltonian of Eq. (44).

Figure 4

Log plot of $L^{q,<}$ (in arbitrary units) at $q=0$ according to Eq. (37) (dashed black line) and Eq. (34) (solid red line). The inset shows the difference between the two curves.

Figure 5

(Left) Lesser Green's function $-i{G}_{cc,0}^{<}\left(\omega \right)$ (in arbitrary units) for different densities of the conduction electrons $n_c$. (Right) Dependence of the exciton weight $Z_X$ on $n_c$.

Figure 6

Lesser Green's function $-i{G}_{cc,0}^{<}\left(\omega \right)$ (in arbitrary units) for different interaction strength $U$ (top) and temperatures $T$ (bottom).

Figure 7

Lesser Green's function $-i{G}_{cc,k}^{<}\left(\omega \right)$ (in arbitrary units) for different momenta $k$ of the conduction electron. The (red) curve in the background is the integrated quantity $-i\int dk{G}_{cc,k}^{<}\left(\omega \right)$.

Figure 8

Momentum-resolved and energy-resolved excited spectral function $A_k\left(\omega \right)$ in arbitrary units. The dashed line corresponds to the exciton dispersion of the system in equilibrium.