Dual-space analysis based on symmetries for phonons in AlAs/GaAs shellquantum dots

Abstract

Phonon modes in shell quantum dots (SQDs) composed of a spherical AlAscore embedded in a GaAs shell, with up to about 12 000 atoms (about8.5 nm in outer diameter), are calculated by using a valence force fieldmodel. All the vibration frequencies and amplitudes are evaluated directlyfrom the lattice dynamic matrix by employing the projection operatorsof the group theory. The numerous SQD phonon modes in each of fivesymmetries which suffer more quantum confinement than modes in a usualquantum dot owing to the small scale of the shell’s thickness are classified byusing an analysis method both in real space and in reciprocal space. Itis found that the bulk GaAs-like SQD modes with localization radiuslocated in the interior of the shell have clearly pronounced bulk specific k-pointparentage from a specific part of the Brillouin zone (BZ) (Γderived, X derived etc) and from a definite bulk band (one of sixmodes). In AlAs/GaAs SQDs of all sizes, the bulk GaAs-LO(Γ)-likeSQD modes always have A1 symmetry, while the bulk GaAs-TO(Γ)-like andbulk-GaAs-A(Γ)-likedot modes have T1 symmetry. The bulklike SQD mode of specificsymmetry has a dominant BZ parentage peak around the bulkorigins, so the frequencies of these SQD modes can be approximatelyrelated to a single bulk phonon band at a single wavevector k∗.In addition to the frequencies of bulk GaAs-A(Γ)-like SQDmodes blue-shifting as the shell’s characteristic scale reduces, the bulk-GaAs-like Γ-derivedLO and TO SQD modes red-shift in frequency with decreasing shell characteristicscale. There is almost no LO/TO mixing for bulklike modes. The identificationand classification of SQD modes have fundamental importance in the discussion ofthe Raman spectrum and electron–phonon interaction.