Magnetic field induced formation of a stationary charge density wave in a conducting Möbius stripe

Magnetic field induced edge currents propagate in opposite directions at opposite edges of a Hall bar. We consider a system that possesses just one edge: a Möbius stripe. We show that if the carrier mean free path is less than the width of the stripe, magnetic field induced edge currents vanish. Instead, a stationary charge density wave is formed. Its profile is governed by the interplay between the magnetic field dependence of the chemical potential and the screened Coulomb repulsion of charge carriers. This effect offers an experimental tool for studies of the electron density of states in two-dimensional crystal films.

Figure 1

(a) A conducting Möbius stripe subjected to a magnetic field. The red line along the edge corresponds to the area where the normal to the plane projection of the magnetic field is positive, the green line shows where it is negative. (b) The trajectories of cyclotron orbits of charge carriers for a section of the Möbius stripe near the points 0 and $2\pi$.

Figure 2

Variation of the electron density at the edge of a metallic Möbius stripe as a function of the angular coordinate calculated at different values of the applied magnetic field: (a) $H_0=15\mathrm{T}$, (b) $H_0=10\mathrm{T}$, (c) $H_0=5\mathrm{T}$. Blue, green, and red lines correspond to the values of the Dingle temperature: $T_D=2\mathrm{K}$, $T_D=10\mathrm{K}$, and $T_D=50\mathrm{K}$, respectively. The insets show on a larger scale the distribution of the electron density in the vicinity of the $3\pi$ point (${\theta }_1=3\pi -\frac{\pi }{32};{\theta }_2=3\pi +\frac{\pi }{32}$).

Figure 3

Variation of the electron density at the edge of the graphene Möbius stripe as a function of the angular coordinate calculated at different values of the applied magnetic field: (a) $H_0=15\mathrm{T}$, (b) $H_0=10\mathrm{T}$, (c) $H_0=5\mathrm{T}$. Blue, green, and red lines correspond to the Dingle temperatures $T_D=2\mathrm{K}$, $T_D=10\mathrm{K}$, and $T_D=50\mathrm{K}$, respectively.

Figure 4

The electronic chemical potential as a function of the normal-to-plane magnetic field calculated for a graphene Möbius stripe in the presence (red line), and in the absence (blue line) of the electron density redistribution caused by the CDW effect. Labeled lines show the energies of the corresponding Landau levels as functions of the magnetic field. $T_D=10\mathrm{K}$ is assumed.

Figure 5

The ballistic limit: clockwise and anticlockwise edge currents in a Möbius stripe are shown by green and red lines, respectively. No CDW is formed. We have used the parameters of a metallic foil $n=0.2\times {10}^{16}{\mathrm{cm}}^{-2},R=5\mu \mathrm{m}$.