### Abstract

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. 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. In the final part of the chapter, we derive the phonon equations-of-motion of thin shells applied to 2D graphene using a differential geometry formulation.

Original language | English |
---|---|

Title of host publication | Physics of quantum rings |

Editors | Vladimir M. Fomin |

Publisher | Springer VS |

Publication date | 1. Jan 2018 |

Edition | 2. udg. |

Pages | 499-533 |

ISBN (Print) | 9783319951584 |

ISBN (Electronic) | 9783319951591 |

DOIs | |

Publication status | Published - 1. Jan 2018 |

Series | NanoScience and Technology |
---|---|

ISSN | 1434-4904 |

### Fingerprint

### Cite this

*Physics of quantum rings*(2. udg. ed., pp. 499-533). Springer VS. NanoScience and Technology https://doi.org/10.1007/978-3-319-95159-1_16

}

*Physics of quantum rings.*2. udg. edn, Springer VS, NanoScience and Technology, pp. 499-533. https://doi.org/10.1007/978-3-319-95159-1_16

**Differential geometry applied to rings and möbius nanostructures.** / Lassen, Benny; Willatzen, Morten; Gravesen, Jens.

Research output: Chapter in Book/Report/Conference proceeding › Book chapter › Research › peer-review

TY - CHAP

T1 - Differential geometry applied to rings and möbius nanostructures

AU - Lassen, Benny

AU - Willatzen, Morten

AU - Gravesen, Jens

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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. 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. In the final part of the chapter, we derive the phonon equations-of-motion of thin shells applied to 2D graphene using a differential geometry formulation.

AB - 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. 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. In the final part of the chapter, we derive the phonon equations-of-motion of thin shells applied to 2D graphene using a differential geometry formulation.

U2 - 10.1007/978-3-319-95159-1_16

DO - 10.1007/978-3-319-95159-1_16

M3 - Book chapter

SN - 9783319951584

SP - 499

EP - 533

BT - Physics of quantum rings

A2 - Fomin, Vladimir M.

PB - Springer VS

ER -