Reinvestigation of quantum dot symmetry: the symmetric group of the 8-band k⋅p theory Hamiltonian

Abstract

Self-assembled semiconductor quantum dots, usually formed in pyramid or lens shapes, have an intrinsic geometric symmetry. However, the geometric symmetry of a quantum dot is not identical to the symmetry of the associated Hamiltonian. It is a well-accepted conclusion that the symmetric group of the Hamiltonians for both pyramidal and lens-shaped quantum dots is C2v; consequently, the eigenstate of the Hamiltonian is not degenerate because C2v has only one-dimensional irreducible representations. In this paper, we show the above conclusion is wrong. Using the 8-band k · p theory model and considering the action of group elements on both spatial and electron spin parts of the wavefunction, we find the symmetric group of the Hamiltonian is the C2v double group not C2v. C2v is the symmetric group of the spatial part of the conduction band Hamiltonian only when the inter-band coupling is totally ignored. Employing the C2v double group symmetry, we prove that although the C2v double group has both one-dimensional and two-dimensional irreducible representations, the eigenstates of the 8 × 8 Hamiltonian are always two-fold degenerate and that these degenerate states only correspond to the two-dimensional irreducible representation of the C2v double group. The double group symmetry originates from the coupling between spatial potential and electron half spin. This coupling causes a full 2π rotation in the wavefunction space or the Hilbert space not equal to the unity operation. Finally, the connection between the two-fold degeneracy due to the C2v double group symmetry and Kramers’ degeneracy due to the time inversion symmetry is explored.