Optical Conductivity from Excitonic States

Xatu can compute the linear optical conductivity using the excitonic states obtained from the BSE. The calculation is performed in the independent-particle approximation (IPA) or the Bethe-Salpeter equation (BSE) formalism, based on the eigenstates previously computed.

General Expression

The optical conductivity tensor is computed from the excitonic eigenstates and their coupling to the electromagnetic field, following the Kubo-Greenwood formalism.

The real part of the conductivity tensor \(\sigma_{ab}(\omega)\) is given by:

\[\sigma_{ab}(\omega) = \frac{\pi e^2 \hbar}{V} \sum_{\bm{k}}^{N_X} \frac{1}{E_{\bm{k}}} \left[ V_{\bm{k}}^a (V_{\bm{k}}^b)^* \right] \delta(\hbar\omega - E_{\bm{k}})\]

where:

  • $ V $ is the system volume

  • \(E_{\mathbf{k}}\) is the exciton energy at momentum \(\mathbf{k}\)

  • \(V_{\mathbf{k}}^a\) velocity matrix elements \(\langle GS| \hat{v}^{a} | X_{\mathbf{k}} \rangle\)

  • $ N_X $ is the number of computed exciton states

  • The delta function is broadened numerically using a specified kernel

This expression is implemented directly in Xatu when the linear response spectrum is requested using the -a flag and a valid kubo_w.in file is provided.

Excitonic Absorption Spectrum

The absorption spectrum is computed by convoluting the excitonic delta functions with a chosen broadening. The user can specify:

  • Broadening type: lorentzian, gaussian, or exponential

  • Broadening width (in eV)

  • Frequency range and resolution

This is controlled by the kubo_w.in input file. The computed spectra include:

  • Independent-particle spectrum (IPA)

  • Excitonic spectrum (BSE)

Each output is saved to separate .dat files for plotting, with names defined in the exciton configuration file.

Reference

For details, see:

Efficient computation of optical excitations in two-dimensional materials with the Xatu code, Computer Physics Communications, 2024 <https://doi.org/10.1016/j.cpc.2023.109001>