Differential Geometry Applied to Rings and Möbius Nanostructures

  • Benny Lassen
  • Morten WillatzenEmail author
  • Jens Gravesen
Part of the NanoScience and Technology book series (NANO)


Nanostructure shape effects have become a topic of increasing interest due to advancements in fabrication technology. In order to pursue novel physics and better devices by tailoring the shape and size of nanostructures, effective analytical and computational tools are indispensable. In this chapter, we present analytical and computational differential geometry methods to examine particle quantum eigenstates and eigenenergies in curved and strained nanostructures. Example studies are carried out for a set of ring structures with different radii and it is shown that eigenstate and eigenenergy changes due to curvature are most significant for the groundstate eventually leading to qualitative and quantitative changes in physical properties. In particular, the groundstate in-plane symmetry characteristics are broken by curvature effects, however, curvature contributions can be discarded at bending radii above 50 nm. In the second part of the chapter, a more complicated topological structure, the Möbius nanostructure, is analyzed and geometry effects for eigenstate properties are discussed including dependencies on the Möbius nanostructure width, length, thickness, and strain.


Separable Solution Strain Term Nanowire Structure Local Cross Section Graphene Strip 
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  1. 1.
    S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, N. Hatekenaka, Nature (London) 417, 397 (2002) CrossRefGoogle Scholar
  2. 2.
    M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi, S.C. Zhang, Science 318, 766 (2008) CrossRefGoogle Scholar
  3. 3.
    Y. Ran, Y. Zhang, A. Vishwanath, Nat. Phys. 5, 298 (2009) CrossRefGoogle Scholar
  4. 4.
    Z.L. Guo, Z.R. Gong, H. Dong, C.P. Sun, Phys. Rev. B 80, 195310 (2009) ADSCrossRefGoogle Scholar
  5. 5.
    J. Gravesen, M. Willatzen, Phys. Rev. A 72, 032108 (2005) ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    E.L. Starostin, G.H.M. van der Heijden, Nat. Mater. 6, 563 (2007) CrossRefGoogle Scholar
  7. 7.
    E.L. Starostin, G.H.M. van der Heijden, Phys. Rev. B 79, 066602 (2009) CrossRefGoogle Scholar
  8. 8.
    D.J. Ballon, H.U. Voss, Phys. Rev. Lett. 101, 247701 (2008) ADSCrossRefGoogle Scholar
  9. 9.
    M. Yoneya, K. Kuboki, M. Hayashi, Phys. Rev. B 78, 064419 (2008) ADSCrossRefGoogle Scholar
  10. 10.
    C. Rockstuhl, C. Menzel, T. Paul, F. Lederer, Phys. Rev. B 79, 035321 (2009) ADSCrossRefGoogle Scholar
  11. 11.
    N. Zhao, H. Dong, S. Yang, C.P. Sun, Phys. Rev. B 79, 125440 (2009) ADSCrossRefGoogle Scholar
  12. 12.
    Z. Li, L.R. Ram-Mohan, Phys. Rev. B 85, 195438 (2012) ADSCrossRefGoogle Scholar
  13. 13.
    V.M. Fomin, S. Kiravittaya, O.G. Schmidt, Phys. Rev. B 86, 195421 (2012) ADSCrossRefGoogle Scholar
  14. 14.
    B. Lassen, M. Willatzen, J. Gravesen, J. Nanoelectron. Optoelectron. 6, 68 (2011) CrossRefGoogle Scholar
  15. 15.
    J. Gravesen, M. Willatzen, Physica B 371, 112–119 (2006) ADSCrossRefGoogle Scholar
  16. 16.
    J. Gravesen, M. Willatzen, L.C. Lew Yan Voon, J. Math. Phys. 46, 012107 (2005) ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    L.D. Landau, E.M. Lifshitz, Theory of Elasticity, 3rd edn. Course of Theoretical Physics, vol. 7 (Butterworth Heinemann, Oxford, 1999) Google Scholar
  18. 18.
    M. Sadowski, in Verh. 3. Kongr. Techn. Mechanik, vol. II (1930), pp. 444–451 Google Scholar
  19. 19.
    E.O. Kane, J. Phys. Chem. Solids 1, 249 (1957) ADSCrossRefGoogle Scholar
  20. 20.
    P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, 4th edn. (Springer, Berlin, 2010) CrossRefGoogle Scholar
  21. 21.
    T. Randrup, P. Røgen, Arch. Math. 66, 511–521 (1996) CrossRefzbMATHGoogle Scholar
  22. 22.
    G. Schwarz, Pac. J. Math. 143, 195 (1990) CrossRefzbMATHGoogle Scholar
  23. 23.
    The MathWorks Inc., MATLAB Version 7.8.0 (The MathWorks Inc., Natick, 2009) Google Scholar
  24. 24.
    L.C. Lew Yan Voon, M. Willatzen, The kp Method. Springer Series in Solid State Physics (Springer, Berlin, 2009) Google Scholar
  25. 25.
    I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Benny Lassen
    • 1
  • Morten Willatzen
    • 1
    • 2
    Email author
  • Jens Gravesen
    • 3
  1. 1.Mads Clausen InstituteUniversity of Southern DenmarkSønderborgDenmark
  2. 2.Department of Photonics EngineeringTechnical University of DenmarkKgs. LyngbyDenmark
  3. 3.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark

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