# 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* ·

*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*

**p**_{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.

## Keywords

Quantum dots Symmetry Group theory Double group Modeling and simulation## Notes

### Acknowledgments

The authors thank Dr. Stanko Tomic for directing our attention to his previous work and sending us the paper and its supplementary materials (Tomic and Vukmirovic 2011). We also thank one anonymous reviewer for the valuable comments. Actually, all discussions on Kramers’ degeneracy in this paper result from the suggestions of this reviewer.

### Funding information

This work was partially supported by the Wisconsin Applied Research Grant and WiSys Technology Foundation.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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