Alloying effects on the optical properties of ${\text{Ge}}_{1-x}{\text{Si}}_x$ nanocrystals from time-dependent density functional theory and comparison with effective-medium theory

We present the optical spectra of ${\text{Ge}}_{1-x}{\text{Si}}_x$ alloy nanocrystals of a fixed size calculated with time-dependent density functional theory in the adiabatic local-density approximation (TDLDA). The spectra change smoothly as a function of the composition $x$. On the Ge side of the composition range, the lowest excitations at the absorption edge are almost pure Kohn-Sham independent-particle highest occupied molecular orbital–lowest occupied molecular orbital transitions, while for higher Si contents strong mixing of transitions is found. Within TDLDA the first peak is slightly higher in energy than in earlier independent-particle calculations. However, the absorption onset and in particular its composition dependence is similar to independent-particle results. Moreover, classical depolarization effects are responsible for a very strong suppression of the absorption intensity. We show that they can be taken into account in a simpler way using Maxwell-Garnett classical effective-medium theory. Emission spectra are investigated by calculating the absorption of excited nanocrystals at their relaxed geometry. The structural contribution to the Stokes shift is about 0.5 eV. The decomposition of the emission spectra in terms of independent-particle transitions is similar to what is found for absorption. For the emission, very weak transitions are found in Ge-rich clusters well below the strong absorption onset.

Figure 1

(Color online) Example of the studied nanocrystal structures: ${\text{Ge}}_{48}{\text{Si}}_{35}{\text{H}}_{108}$. The colors are: Si (yellow), Ge (blue), and H (white).

Figure 2

(Color online) Absorption spectra as a function of the Si content of the clusters: independent-particle spectra (dashed lines) compared with TDLDA results (solid lines). The green arrows (arrows at lowest energy) mark the HOMO-LUMO gap, while the red arrows (middle arrows) mark the $\Delta \text{SCF}$ excitation energies and the blue arrows (arrows at highest energy) mark the first transitions of Casida’s analysis. The independent-particle curves are divided by a factor of 15. Note the different scale in the two panels.

Figure 3

(Color online) Absorption spectra as a function of the Si content of the clusters: effective-medium theory (dashed lines) using the dielectric function of bulk ${\text{Ge}}_{1-x}{\text{Si}}_x$, compared with TDLDA results (solid lines).

Figure 4

(Color online) Absorption (solid lines) and emission (dashed lines) spectra as a function of the Si content of the clusters. Note the different scales.

Figure 5

(Color online) Absorption (red) and emission (black) spectra of the pure Ge nanocrystal in the independent-particle approximation (dashed) and the TDLDA (solid). Note the different scales.