Eigenstates of Möbius nanostructures including curvature effects

Möbius-shell structures and their physical properties have recently received considerable attention experimentally and theoretically. In this work, eigenstates and associated eigenenergies are determined for a quantum-mechanical particle bounded to a Möbius shell including curvature contributions to the kinetic-energy operator. This is done using a parametrization of the Möbius shell–found by minimizing the elastic energy of the full structure–and employing differential-geometry methods. It is shown that inclusion of curvature contributions to the kinetic energy leads to splitting of the otherwise doubly degenerate groundstate and significantly alters the form of the groundstate and excited-state wavefunctions. Hence, we anticipate qualitative changes in the physical properties of Möbius-shell structures due to surface confinement and curvature effects.

Figure 1

(Color online) The Möbius structure color coded with the value of $M^2$ (in ${\mathrm{\AA }}^{-2}$), the square of the mean curvature. The units on the axes are in ångstroms.

Figure 2

(Color online) Plot of the $M^2$ contribution to the kinetic energy versus coordinates $u^1$ and $u^2$ in the domain $(u^1,u^2)=[0;L]\times [-w;w]$. Parameter values for $L$ and $w$ are as in Table . The maximum value of the $M^2$ contribution is approximately $5.6\times {10}^{-3}{\mathrm{\AA }}^{-2}$ (dark red) corresponding to [in energy, i.e., $({\hslash }^2∕2m)M^2$] a maximum value of approximately 21 meV.

Figure 3

(Color online) Contour plots of the first three eigenstates for the flat-Möbius structure case (upper left, middle left, and lower left plots) and the Möbius-structure case including the $M^2$ contributions (upper right, middle right, and lower right plots). Eigenstates are plotted versus coordinates $u^1$ and $u^2$ in the domain $(u^1,u^2)=[0;L]\times [-w;w]$. Parameter values for $L$ and $w$ are as in Table .