Asymptotic spectral analysis in semiconductor nanowire heterostructures

Abstract

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between material composition and geometric parameters like curvature and torsion. First, the macroscopic behaviour of a nanowire made of composite fibres with microscopic periodic texture is analysed, which amounts to determining the asymptotic behaviour of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross-section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm–Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d–1d dimension reduction for the two-scale problem in a variational framework by means of Γ-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross-section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.

Keywords:

• spectral analysis
• dimension reduction
• homogenization
• Γ-convergence
• quantum waveguides

AMS Subject Classifications:

• 49R05 (primary)
• 49J45
• 35P20
• 78A50
• 81Q15

Acknowledgements

The authors would like to express their thanks to Luís Trabucho for several stimulating conversations and to Christoph Kreisbeck for his helpful comments on the physics background of the problem. C.K. was supported by the Fundação para a Ciência e a Tecnologia through the ICTI CMU-Portugal Program in Applied Mathematics and UTA-CMU/MAT/0005/2009. Part of this research was carried out while C.K. was at Universität Regensburg. Travelling funds for C.K. provided by CMA are thankfully acknowledged. L.M. was supported by the Fundação para a Ciência e a Tecnologia, through PEst-OE/MAT/UI0297/2011, PTDC/MAT109973/2009, and UTA-CMU/MAT/0005/2009.