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 C_2v; consequently, the eigenstate of the Hamiltonian is not degenerate because C_2v 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 C_2v double group not C_2v. C_2v is the symmetric group of the spatial part of the conduction band Hamiltonian only when the inter-band coupling is totally ignored. Employing the C_2v double group symmetry, we prove that although the C_2v 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 C_2v 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 C_2v double group symmetry and Kramers' degeneracy due to the time inversion symmetry is explored.

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