Dual Space Analyzing Based on Symmetry Properties for Phonons fo Si Quantum Dot

Phonon modes in spherical Si quantum dots (QDs) with up to 7.9 nm in diameter are calculated by using the projection operators of the group theory into valence force field model. The phonons of dot modes in each of five irreducible representations (symmetries) are classified by using a dual space analysis method. It is found that the bulk-like modes with localization radius much smaller than the dot's radius have clearly pronounced bulk specific-k point parentage, T_λ(n, k_BZ, K_cut), from specific part of the Brillouin zone (BZ) (Γ-derived, X-derived etc.) and from definite bulk band (one in six modes). In Si dots of all sizes, each specific bulk-like dot mode has specific symmetry. The bulk TO(Γ)-like and the bulk LO(Γ)-like dot modes always have T₂ and A₁ symmetries, respectively. Except the bulk-A(Γ)-like dot modes of which the frequencies blue-shift as the dot size reducing, the bulk-like Γ-derived LO and TO dot modes and bulk-like X-derived TA and LA dot modes red-shift in frequency with decreasing dot size. There is almost not LO/TO mixing for bulk-like modes. As for the surface-like modes localized at the periphery of the dot, their eigenmodes have not a dominant bulk specific-k point parentage or a dominant BZ parentage around some special point. They are a superposition of many bulk bands with k from all over the bulk BZ. They have much significant mode mixing than the bulk-like phonons. The classification of dot modes based on the symmetry of group theory will bring advantageous to the discussion of Ramam spectrum, electron-phonon interaction and other phonon-assisted effects in QDs.