Geometry and edge effects on the energy levels of graphene quantum rings: A comparison between tight-binding and simplified Dirac models

We present a systematic study of the energy spectra of graphene quantum rings having different geometries and edge types in the presence of a perpendicular magnetic field. Results are obtained within the tight-binding (TB) and Dirac models and we discuss which features of the former can be recovered by using the approximations imposed by the latter. Energy levels of graphene quantum rings obtained by diagonalizing the TB Hamiltonian are demonstrated to be strongly dependent on the rings geometry and the microscopical structure of the edges. This makes it difficult to recover those spectra by the existing theories that are based on the continuum (Dirac) model. Nevertheless, our results show that both approaches (i.e., TB and Dirac model) may provide similar results, but only for very specific combinations of ring geometry and edge types. The results obtained by a simplified model describing an infinitely thin circular Dirac ring show good agreement with those obtained for hexagonal and rhombus armchair graphene rings within the TB model. Moreover, we show that the energy levels of a circular quantum ring with an infinite mass boundary condition obtained within the Dirac model agree with those for a ring defined by a ring-shaped staggered potential obtained within the TB model.

Figure 1

Sketch of (a), (c), and (e) armchair and (b), (d), and (f) zigzag rings, with hexagonal, triangular, and rhombus geometries, respectively, as well as (g) and (h) circular rings, considered in this work. The first six geometries are characterized by the number of carbon rings $N_E$ ($N_I$) in their outer (inner) edge. Circular rings are characterized by their width $W$ and average radius $R$. (g) Circular ring defined by cutting the graphene lattice. (h) Circular graphene ring defined by a smooth ring-shaped staggered potential $M_i$, where the color scale goes from $M_i=-M_0$ (red) to $M_i=+M_0$ (blue), and the $M_i=0$ region inside the ring is represented in green. The atoms belonging to sublattices A and B have different colors because of the staggered potential profile.

Figure 2

Energy levels of armchair rhombus quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=17$, $N_I=12$ and (b) $N_E=17$, $N_I=11$. As shown in the insets, the energy spectrum does not have a zero-energy state: States close to $E=0$ are rather similar to the first states above and below this energy, which are composed by branches of two oscillating energy states.

Figure 3

Energy levels of armchair hexagonal quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=15$, $N_I=10$ and (b) $N_E=15$, $N_I=3$. The spectrum is symmetric with respect to $E=0$.

Figure 4

Current density profile for an armchair hexagonal quantum ring corresponding to magnetic flux indicated by (I) and (II) in Fig. 3. The results for the current density are numerically calculated based on the method discussed in Refs. [29, 43] and reproduced in the present paper.

Figure 5

(a) Rhombus armchair quantum ring (blue polygon figure) considered in the TB calculation, with $N_E=17$ and $N_I=12$, along with the one-dimensional $R\approx 32.3$ Å ring (red circle) considered in the simplified model. (b) Energy spectra, obtained from the simplified (solid lines) and TB (dashed lines) models, as a function of the magnetic flux threading the red circle illustrated in (a). Curves with different colors represent different angular momentum index $l$. (c) The results from the TB model with a background mass term $M=0.5E_0$ are also compared to those from the simplified model in this case.

Figure 6

(a) Hexagonal armchair quantum ring (blue polygon figure) considered in the TB calculation, with $N_E=15$ and $N_I=10$, along with the one-dimensional $R\approx 47$ Å ring (red circle) considered in the simplified model. (b) Energy spectra, obtained from the simplified (solid lines) and TB (dashed lines) models, as a function of the magnetic flux threading the red circle illustrated in (a). Curves with different colors represent different angular momentum index $l$. (c) The results from the TB model with a background mass term $M=0.5E_0$ are also compared to those from the simplified model in this case.

Figure 7

Energy levels of armchair hexagonal quantum ring, obtained from the simplified (solid line) and TB (dashed line) models, as function of ring radius $R$, calculated assuming an average radius given by $R={\left[\frac{3\sqrt{3}}{2\pi }{\left(\frac{|L_E+L_I|}{2}\right)}^2\right]}^{1/2}$ and mass term $M=0.1$ eV, for different values of the magnetic flux $\Phi /{\Phi }_0$. Curves with green, red, and blue colors represent angular momentum index $l$ zero, negative, and positive, respectively. The spectrum is symmetric with respect to $E=0$.

Figure 8

Energy levels as a function of the magnetic flux through a single carbon hexagon for (a) the circular graphene ring schematically shown in Fig. 1, and (b) a quantum ring formed by a site-dependent potential given by Eq. (2) and schematically shown in Fig. 1, with smoothness $S=10$ Å and height $M_0=1$ eV. In both cases, the average radius of the ring is $R=80$ Å and the width is 60 Å. The spectrum is symmetric with respect to $E=0$.

Figure 9

Energy spectrum, obtained by the continuum model for $K$ (red solid line) and $K^{\prime }$ (blue dashed line), as a function of the magnetic flux for a graphene quantum ring defined by an infinite mass boundary, with the same average radius and width as the ring in Fig. 8. The spectrum is symmetric with respect to $E=0$.

Figure 10

Energy levels of zigzag hexagonal quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=15$, $N_I=10$ and (b) $N_E=15$, $N_I=9$. In the (former) latter the inner and outer zigzag edges are (anti-)aligned, as sketched in the insets.

Figure 11

Energy levels of zigzag rhombus quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=17$, $N_I=11$ and (b) $N_E=17$, $N_I=9$. For both widths, the energy spectrum does not have a zero-energy state, they are three pairs of oscillating states as shown in the insets.

Figure 12

Energy levels of zigzag triangular quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=17$, $N_I=12$ and (b) $N_E=15$, $N_I=10$. For both widths, the energy spectrum has a zero-energy state.

Figure 13

Energy levels of armchair triangular quantum rings, schematically shown in Fig. 1, as a function of the magnetic flux through a single carbon hexagon for two ring widths: (a) $N_E=15$, $N_I=10$ and (b) $N_E=15$, $N_I=9$. The spectrum is symmetric with respect to $E=0$.