### Abstract

In this paper we analyze a long standing problem of the appearance of spurious, non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures. The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem. We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coefficients to small k components would lead to the appearance of non-physical solutions. We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution. This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures. Finally, based on the above requirement of small k, we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.

Original language | English |
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Journal | Communications in Computational Physics |

Volume | 6 |

Issue number | 4 |

Pages (from-to) | 699-729 |

ISSN | 1815-2406 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Keywords

- Effective envelope theory
- Quantum confinement
- Abrupt interfaces
- Multiband models
- k space
- Fourier coefficients
- Highly oscillatory integrals
- Variational formulation
- Coupled systems of PDEs
- Multiple scales
- Continuum and atomistic models
- Eigenvalue problem
- Interface boundary conditions
- Band gap
- Spurious solutions
- Low dimensional semiconductor nanostructures

## Cite this

Lassen, B., Melnik, R. V. N., & Willatzen, M. (2009). Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures.

*Communications in Computational Physics*,*6*(4), 699-729. https://doi.org/10.4208/cicp.2009.v6.p699