Phonon modes in InAs quantum dots

Phonon modes in spherical InAs quantum dots (QDs) with up to 11 855 atoms (about 8.5 nm in diameter) are calculated by using a valence force field model, and all the vibration frequencies and vibration amplitudes of the QDs are calculated directly from the lattice-dynamic matrix. The projection operators of the irreducible representations of the group theory are employed to reduce the computational intensity, which further allows us to investigate the quantum confinement effect of phonon modes with different symmetries. It is found that the size effects of phonon modes depend on the symmetry of the modes. For zinc-blende structure, the modes with $A_1$ symmetry has the strongest quantum confinement effect and the $T_1$ mode the weakest. There could be a crossover of symmetries of the highest frequencies from $A_1$ to $T_2$ as the size of the QDs decreases. The behavior of vibration amplitudes and vibration energies of phonon modes in different symmetries are also investigated in detail. These results provide microscopic details of the phonon properties of QDs that are important to the fundamental understanding and potential applications of semiconductor QDs.