Time-dependent density-functional theory for ultrafast interband excitations

We formulate a time-dependent density-functional theory (TDDFT) in terms of the density matrix to study ultrafast phenomena in semiconductor structures. A system of equations for the density-matrix components, which is equivalent to the time-dependent Kohn-Sham equation, is derived. From this, we obtain a TDDFT version of the semiconductor Bloch equations, where the electronic many-body effects are taken into account, in principle, exactly. As an example, we study the optical response of a three-dimensional two-band insulator to an external short-time pulsed laser field. We show that the optical absorption spectrum acquires excitonic features when the exchange-correlation potential contains a $1∕q^2$ Coulomb singularity. A qualitative comparison of the TDDFT optical absorption spectra with the corresponding results obtained within the Hartree-Fock approximation is made.

Figure 1

Hartree-Fock optical absorption spectra for parabolic bands at different values of the valence bandwidth $W^v$ (divided by ${\pi }^2$). We consider external pulses of the form of Eq. (28), with pulse duration $\tau =0.05$ and electric field amplitude $E_0=2.0$. The conduction bandwidth and the band gap are $W^c=0.4{\pi }^2$ and ${\omega }_g=5$. Frequencies are measured with respect to ${\omega }_g$.

Figure 2

Absorption spectra in ALDA at different values of the electric field amplitude and the free electron bands $W^v=0.15,W^c=0.06$. The other parameters are ${\omega }_g=0.375,a_0=20,\tau =10,\Gamma =0.0025$, and $\alpha =7.5$.

Figure 3

Optical absorption spectra in the Slater approximation at different values of the valence bandwidth (divided by ${\pi }^2$). The other parameters are $W^c=0.025{\pi }^2,{\omega }_g=0.2,a_0=10,\tau =4,E=0.0166,\Gamma =0.012$, and $\alpha =7.5$.

Figure 4

Optical absorption spectra in the KLI approximation at different values of the electric field. The other parameters are $W^v=0.0416{\pi }^2,W^c=0.0208{\pi }^2,{\omega }_g=0.166,a_0=10,\tau =4.8,\Gamma =0.012$, and $\alpha =7.5$.

Figure 5

Optical absorption spectra in different approximations at the same values of the model parameters. The valence bandwidth is equal to $0.05{\pi }^2$ and the other parameters are given in the caption to Fig. 3. The Hartree-Fock result is obtained using a Bloch function approximation for the xc potential matrix elements (see the Appendix).

Figure 6

The three-dimensional density of states [Eq. (A18)] as a function of energy.