Model of Raman scattering in self-assembled $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots

Multiphonon resonant Raman scattering in self-assembled quantum disks is investigated using a nonadiabatic approach. The optical phonons and the electron-phonon interaction are considered within the multimode dielectric continuum model. The model exploits both electrostatic and mechanical boundary conditions for the relative ionic displacement vector, as well as the phonon spatial dispersion in bulk. The confined phonon modes in a quantum dot are hybrids of bulklike and interface vibrations. It is shown that nonadiabatic effects substantially increase the Raman scattering probabilities and the relative multiphonon integral intensities with respect to the one-phonon intensity. The calculated ratio of the two- and one-phonon integral intensities is close to the experimental value for self-organized $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots.

Figure 1

Schematic picture of a disk-shaped quantum dot.

Figure 2

(Color online) Frequencies of the (a) even and (b) odd hybrid phonon modes in an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ disk-shaped quantum dot with the parameters $h=8\mathrm{nm}$ and $\hslash {\Omega }_0=20\mathrm{meV}$ as a function of the modulus of the in-plane wave vector ${\mathbf{q}}_{\parallel }$. The dotted curves in panel (a) in the frequency ranges $0.9{\omega }_{1,\mathrm{LO}}⪅\omega ⪅0.95{\omega }_{1,\mathrm{LO}}$ and $1.15{\omega }_{1,\mathrm{LO}}⪅\omega ⪅1.21{\omega }_{1,\mathrm{LO}}$ correspond, respectively, to the even interfacelike InAs and GaAs modes, which have a considerable dispersion. For the odd modes, a strong coupling of the bulklike and interface phonons occurs in the frequency range $0.95{\omega }_{1,\mathrm{LO}}⪅\omega ⪅0.98{\omega }_{1,\mathrm{LO}}$. The interfacelike odd GaAs phonon mode lies in the frequency range $1.11{\omega }_{1,\mathrm{LO}}⪅\omega ⪅1.15{\omega }_{1,\mathrm{LO}}$. Frequencies of other hybrid modes in the figure are close to those for the bulklike phonons and possess a relatively weak dispersion.

Figure 3

(Color online) Frequencies of the (a) even and (b) odd hybrid phonon modes in an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ disk-shaped quantum dot as a function of the quantum-dot height at the modulus of the in-plane wave vector $q_{\parallel }=2$ [in units ${(m_{1,e}{\omega }_{1,\mathrm{LO}}∕\hslash )}^{1∕2}$]. Solid black curves are hybrid InAs phonon modes. Solid and dashed red curves are interfacelike and half-space GaAs modes, respectively. Dot-dashed blue curves are interface InAs modes given by DCM.

Figure 4

(Color online) [(a), (c), and (e)] One-phonon and [(b), (d), and (f)] two-phonon Raman spectra for disk-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots with different values of the height and of the in-plane confinement frequency parameter. Solid curves show results obtained taking into account effects of nonadiabaticity. Dashed curves show results of the adiabatic approximation. The units for the Raman intensity are the same in all panels.

Figure 5

(Color online) (a) The parameter $S$ determined as $S\equiv 2I_2∕I_1$, where $I_1$ and $I_2$ are the integral intensities of the one-phonon and the two-phonon Raman bands, respectively, as a function of the confinement energy $\hslash {\Omega }_0$. In the adiabatic approximation, $S$ is the Huang-Rhys parameter. The solid curves are obtained taking into account nonadiabatic effects. The dashed curves correspond to the adiabatic approximation. (b) The parameter $S$ as a function of the quantum-dot height of the disk-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot with $h∕R_0=0.4$.

Figure 6

(Color online) Absolute and (inset) relative contributions of all hybrid InAs and the interfacelike GaAs phonon modes into the one-phonon Raman scattering spectra as a function of the height of the disk-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot with $h∕R_0=0.4$.