Theory of exciton-phonon coupling

The effects of the electron-phonon interaction on optical excitations can be understood in terms of exciton-phonon coupling and require a careful treatment in low-dimensional materials with strongly bound excitons or strong electron-hole interaction in general. Through phonon absorption and emission processes, the optically accessible excitons are scattered into otherwise optically dark finite-momentum exciton states. We derive a practical expression for the phonon-induced term of the exciton self-energy (denoted as the exciton-phonon self-energy) that gives the temperature dependence of the optical transition energies and their lifetime broadening resulting from the exciton's interaction with the phonons. We illustrate this theory on a two-dimensional model and show that our expression for the exciton-phonon self-energy differs qualitatively from previous expressions found in the literature that neglect the exciton binding or electron-hole correlations.

Figure 1

Diagramatic representation of the Bethe-Salpeter equation. The BSE Kernel is expressed as the sum of the bare exchange Coulomb repulsion (single line) and the screened Coulomb attraction (double line) between the electron and the hole.

Figure 2

Dyson equation for the one-particle propagator and the electron-phonon self-energy, expressed as the sum of the Fan-Migdal term and the Debye-Waller term.

Figure 3

The Dyson equation for the IEHPP ${\mathrm{\Lambda }}^0$ involving the IEHPP self-energy ${\mathrm{\Xi }}^0$ (without electron-hole interactions), and the Dyson equation for the interacting exciton propagator $\mathrm{\Lambda }$ involving the exciton-phonon self-energy $\mathrm{\Xi }$.

Figure 4

The different contributions to the IEHPP self-energy ${\mathrm{\Xi }}^0$, corresponding to a free electron-hole pair interacting with phonons. Contributions on the right-hand side of the equation are the Fan-Migdal terms (first line), the phonon exchange terms (second line), and the Debye-Waller terms (third line).

Figure 5

Some of the diagrams contributing to the exciton-phonon self-energy $\left(\mathrm{\Xi }\right)$ in the quasiparticle basis. This series contains all the diagrams with one or many Coulomb interaction lines (wavy lines) between the phonon vertices, in addition to the diagrams of the IEHPP self-energy $\left({\mathrm{\Xi }}^0\right)$. Diagrams consisting of a phonon exchange line and Coulomb interaction lines are present in this series, but are not depicted.

Figure 6

(a) Two-band model electronic structure for the different spin states $|\uparrow \rangle$ (orange) and $|\downarrow \rangle$ (blue). (b) Exciton band structure for the different spin states $|\uparrow \downarrow \rangle$ (orange) and $|\downarrow \uparrow \rangle$ (blue). The filled regions represent the continuum of excited states. [(c)–(f)] Real-space wave function $|A^S{(\mathbf{R},\mathbf{Q})|}^2$ of the first four optically accessible excitons $\left(S=1-4,\mathbf{Q}=\mathrm{\Gamma }\right)$.

Figure 7

[(a) and (b)] Convergence of the imaginary part of the self-energy with respect to the number of $\mathbf{q}$-points for the lowest two excitons $\left(S=1,2\right)$. $\mathbf{q}$ grids are increasing (right to left) from $6\times 6$ to $96\times 96$, for different broadening parameter $\eta$. (c) Temperature dependence of the optical excitation energies (solid line) and their line broadening (filled color). The dash lines indicate the bare exciton energies, without electron-phonon coupling.

Figure 8

Real (solid lines) and imaginary part (dashed lines) of the exciton-phonon self-energy with (black) and without (blue) electron-hole interaction. The height of the filled region is the contribution of the completion term to the interacting self-energy.

Figure 9

Comparison between the exciton-phonon self-energy (black discs) and an approximate expression (blue circles) for the first four bright excitons. (a) Energy shift. (b) Inverse lifetime.

Figure 10

Spectral function (imaginary part of the propagator) for the first bright exciton at $T=0$ computed with the model hamiltonian at different levels of theory (in all cases, the imaginary part of the self-energy of the particle due to electron-electron interaction is neglected): noninteracting (gray), electron-hole interaction only (orange), electron-phonon interaction only (blue), and both interactions (black).