Figure 1

Formation of a Möbius ring with an inhomogeneous twist. The initial strip [with geometric characteristics shown in panel (a)] is twisted only over the length interval $L_{x2}$ (b) and then rolled up in a Möbius ring (c). A characteristic radius of the Möbius ring is $R\equiv L_x/2\pi$. The relative length of the untwisted part of the Möbius ring is $\eta =L_{x1}/L_x$. The electron states are analyzed for a Möbius ring threaded by a magnetic flux Φ through its opening (shown with arrows).Reuse & Permissions

Figure 2

Squared modulus of the wave functions |Ψ|${}^2$ of an electron confined to a Möbius ring with an inhomogeneous twist (color code: ground state, red; first excited state, purple; second excited state, blue). Three isosurfaces of the squared modulus of the wave function at 0.7, 0.5, and 0.3 of its maximal value are shown with reducing color intensity. The ground-state wave function is expelled from the twisted region. The magnitude of this expulsion increases when increasing the ratio of the length of the untwisted part of the ring to the whole circumference $\eta =L_{x1}/L_x$. The ground state changes from delocalized at η = 0 to effectively localized in the vicinity of the untwisted region at η > 0. The first excited state changes from delocalized at η = 0 to effectively localized in the twisted region at η > 0. The second excited state remains essentially delocalized at η > 0. The localization/delocalization patterns shown for η = 20 are typical also for larger values η > 20. The finite-element calculation is performed for the effective mass $m_e$ = 0.022$m_0$, where $m_0$ is the free-electron mass, and the structural parameters $R=L_x/2\pi =10$ nm, $L_y$ = 8 nm, and $L_z$ = 2 nm.Reuse & Permissions

Figure 3

Eigenenergies of the lowest states of an electron in a Möbius ring as a function of the relative magnetic flux. The calculation is performed for $R=L_x/2\pi =10$ nm, $L_y$ = 8 nm, and $L_z$ = 2 nm. (a) The ground-state energies of an electron in a ring rolled up from the initial strip shown in Fig. 2a are decreased (green arrow down) due to a geometric potential. The ground-state energies of an electron in a Möbius ring are increased (red arrow up) because the increase of the kinetic energy of an electron due to a twist is larger than the energy decrease caused by the geometric potential. (b) When increasing the relative length of the untwisted part η, the expulsion of the electron wave function from the twisted region leads to an overall decrease (except for very small values of η) of the ground-state energy, accompanied by its flattening as a function of the magnetic flux because of an enhanced trend to localization.Reuse & Permissions

Figure 4

“Delocalization-to-localization” transition in a Möbius ring with an inhomogeneous twist. (a) Persistent current as a function of the relative length of the untwisted part. The values of the persistent current are taken at $\Phi /{\Phi }_0=0.25$. The delocalized states (at lower values of η) reveal a slow (quadratic) decay of the persistent current with increasing η. The effectively localized states (at higher values of η) are characterized by a fast (exponential) decay of the persistent current as a function of η. The value ${\eta }_{\mathrm{cr}}\approx 12$ shown with a dashed line indicates a position of a conventional boundary between the delocalized and effectively localized states. The calculation is performed for $R=L_x/2\pi =10$ nm, $L_y$ = 8 nm, and $L_z$ = 2 nm. (b) Phase boundaries for a “delocalized-to-localized” transition obtained from the persistent currents shown in panel (a) plotted as a function of thickness of the Möbius ring. The calculation is performed for $R=L_x/2\pi =10$ nm.Reuse & Permissions

Figure 5

Variational solutions for the amplitude $a$ [Eq. (12)] of the delocalized states (the upper row) and the relative contribution to the electron energy due to twist $\Delta E/E_0$ (the lower row) in a Möbius ring with an inhomogeneous twist. Black lines indicate the level $a$ = 0.9, which serves as a conventional localization boundary. The calculation is performed for $R=L_x/2\pi =10$ nm.Reuse & Permissions