Resonance Raman scattering by optical phonons in unstrained germanium quantum dots

The position and shape of the $E_1,E_1+{\Delta }_1$ Raman resonance depending on the size of unstrained Ge quantum dots were studied. The dots were grown in $\mathrm{Ga}\mathrm{As}∕\mathrm{Zn}\mathrm{Se}∕\mathrm{Ge}∕\mathrm{Zn}\mathrm{Se}$ structures on (111)-oriented GaAs substrates using molecular-beam epitaxy. A shift of the $E_1$ and $E_1+{\Delta }_1$ resonance energies by up to $0.3\mathrm{eV}$ was observed. The dependence of the shift on quantum dot size was shown to be well described by a cylindrical model using quantization of the bulk Ge electron-hole states that form excitons at the two-dimensional critical point of the interband density of states. A separate display of the $E_1$ and $E_1+{\Delta }_1$ resonances in the quantum dots related to transformation of the interband density of states into the δ function due to quantization of the energy spectrum was observed.

Figure 1

(a) STM image of an $1800\times 1800\mathrm{\AA }$ surface of a sample with Ge QDs, and (b) the profile measured along the line labeled $a$.

Figure 2

Raman spectra of the $\mathrm{Ga}\mathrm{As}∕\mathrm{Zn}\mathrm{Se}∕\mathrm{Ge}∕\mathrm{Zn}\mathrm{Se}$ structure shown in the inset. The spectra (b) and (c) were obtained from the regions with QDs (the effective thicknesses $d$ of the Ge layer are 15 and $34\mathrm{\AA }$, respectively). The spectrum (d) was taken from the region with a continuous Ge layer $(d=56\mathrm{\AA })$.

Figure 3

Resonance curves of Raman scattering by optical phonons in Ge QDs [(a)–(c)] and in bulk Ge (d). The experimental points are denoted by symbols. The best-fit approximation of the experimental dependencies by Lorenz contours are given by the solid lines. The energy shifts of the ${{E}}_1$ and ${{E}}_1+{\Delta }_1$ resonances are shown by the horizontal arrows.

Figure 4

(a) Energy spectrum of bulk Ge calculated in Ref. 14 for the (111) direction. The horizontal lines ${{E}}_{\mathrm{c}}$ and ${{E}}_{\mathrm{v}}$ indicate the positions of the ZnSe band edges. (b) Dispersion of the electron and hole bands in the ${\mathit{k}}_{x,y}$ plane perpendicular to the (111) direction at the point (A) of the momentum ${\mathit{k}}_Z$.

Figure 5

Dependence of the ${{E}}_1$ and ${{E}}_1+{\Delta }_1$ resonance energy shifts on the radius of cylinder $\mathit{r}$ in the growth plane. The curves shown by the solid lines were calculated using Eq. (1). The observed positions of the resonances in QDs are shown by the symbols.

Figure 6

(a) Resonance curves of Raman scattering by optical phonons calculated for bulk Ge using Eq. (3), for a separate QD (b), and for a QD array with the dispersion of QD sizes taken into account according to Eq. (4).