Electron-electron interactions and topology in the electronic properties of gated graphene nanoribbon rings in Möbius and cylindrical configurations

We present a theory of the electronic properties of gated graphene nanoribbon rings with zigzag edges in Möbius and cylindrical configurations. The finite width opens a gap and nontrivial topology of the Möbius ring leads to a single edge with edge states with an induced, effective gauge field, in analogy to topological insulators. The single zigzag edge leads to a shell of degenerate states at the Fermi level and a ferromagnetic (FM) ground state at half-filling, i.e., at charge neutrality, due to electron-electron interactions. For fractional fillings, both the magnetic moment and the energy gap are found to oscillate as a function of the shell filling. In cylindrical rings, the two edges lead to AF ground state at half-filling but FM ground state at fractional fillings.

Figure 1

(a) Heilbronner ring with $M$ $p_z$ orbitals. Different gray scale (blue and red colors online) represent sign of the wave function. The Möbius condition requires a sign inversion between the 0th and $M-1$th sites. (b) For a Heilbronner ring with odd $M$, $p_z$ orbitals can be chosen to be alternating their sign from site to site.

Figure 2

Polyacene ring with (a) cyclic and (b) Möbius periodicity. In the Möbius polyacene ring, $p_z$ orbitals are chosen to be alternating their sign from site to site.

Figure 3

Graphene nanoribbon with three polyacetylene chains. $p_z$ orbitals are chosen to be alternating their sign from site to site, shown by different gray scales (red and blue colors online). In any given unit cell, the edge site that has only two neighbors is assigned to be the first site of the unit cell.

Figure 4

Single particle energy spectrum of polyacene ring with $N=2$, $M=26$ (total of $N_a=52$ atoms), in cyclic (solid line) and Möbius (circles) configurations. Cyclic and Möbius configurations share the same valence band edge state energy levels.

Figure 5

Single particle energy spectrum of nanoribbon ring with $N=14$, $M=26$ (total of $N_a=364$ atoms), in cyclic (solid line) and Möbius (circles) configurations. The Möbius configuration (circles) has nine degenerate edge states occupied by eight electrons.

Figure 6

Single particle energy spectrum of a nanoribbon ring with $N=13$, $M=25$. Only Möbius configuration is allowed. Eight degenerate edge states are occupied by nine electrons.

Figure 7

Total spin $S$ as a function of number of electrons occupying the edge states for a cyclic ribbon with length $M=26$ for different widths, $N=2$, 8, and 14.

Figure 8

Total spin $S$ as a function of number of electrons occupying the edge states for a Möbius ribbon with length $M=26$ for different widths, $N=2$ and 14.

Figure 9

Total spin $S$ as a function of number of electrons occupying the edge states for a Möbius ribbon with length $M=25$ for different widths, $N=3$ and 13.