On the Geometry of Nanowires and the Role of Torsion

Jens Gravesen, Morten Willatzen*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

A detailed analysis of the Schrödinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2π equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.
Original languageEnglish
Article number1800357
JournalPHYSICA STATUS SOLIDI (RRL) - RAPID RESEARCH LETTERS
Number of pages11
ISSN1862-6254
DOIs
Publication statusPublished - 2018

Keywords

  • Acoustics
  • Frame orientation
  • Quantum mechanics
  • Torsion

Cite this

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title = "On the Geometry of Nanowires and the Role of Torsion",
abstract = "A detailed analysis of the Schr{\"o}dinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2π equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.",
keywords = "Acoustics, Frame orientation, Quantum mechanics, Torsion",
author = "Jens Gravesen and Morten Willatzen",
year = "2018",
doi = "10.1002/pssr.201800357",
language = "English",
journal = "Physica Status Solidi. Rapid Research Letters",
issn = "1862-6254",
publisher = "Wiley - V C H Verlag GmbH & Co. KGaA",

}

On the Geometry of Nanowires and the Role of Torsion. / Gravesen, Jens; Willatzen, Morten.

In: PHYSICA STATUS SOLIDI (RRL) - RAPID RESEARCH LETTERS, 2018.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - On the Geometry of Nanowires and the Role of Torsion

AU - Gravesen, Jens

AU - Willatzen, Morten

PY - 2018

Y1 - 2018

N2 - A detailed analysis of the Schrödinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2π equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.

AB - A detailed analysis of the Schrödinger equation in curved coordinates, exact to all orders in the cross sectional dimension is presented, and we discuss the implications of the frame rotation for energies of both open and closed structures. For a circular cross-section, the energy spectrum is independent of the frame orientation for an open structure. For a closed curve, the energies depend on the holonomy angle of a minimal rotating frame (MR) which is equal to the area enclosed by the tangent image on the unit sphere. In the case of a curve with a well-defined torsion at all points this is up to a multiple of 2π equal to the total torsion, a result first found in 1992 by Takagi and Tanzawa. In both cases we find that the effect on the eigenstates is a phase shift. We validate our findings by accurate numerical solution of both the exact 3D equations and the approximate 1D equations for a helix structure and find that the error is proportional to the square of the diameter of the cross section. We discuss Dirichlet versus Neumann boundary conditions and show that care has to be taken in the latter case.

KW - Acoustics

KW - Frame orientation

KW - Quantum mechanics

KW - Torsion

U2 - 10.1002/pssr.201800357

DO - 10.1002/pssr.201800357

M3 - Journal article

JO - Physica Status Solidi. Rapid Research Letters

JF - Physica Status Solidi. Rapid Research Letters

SN - 1862-6254

M1 - 1800357

ER -