Robustness of edge states in graphene quantum dots

We analyze the single-particle states at the edges of disordered graphene quantum dots. We show that generic graphene quantum dots support a number of edge states proportional to circumference of the dot over the lattice constant. Our analytical theory agrees well with numerical simulations. Perturbations breaking electron-hole symmetry such as next-nearest-neighbor hopping or edge impurities shift the edge states away from zero energy but do not change their total amount. We discuss the possibility of detecting the edge states in an antidot array and provide an upper bound on the magnetic moment of a graphene dot.

Figure 1

(Color online) A graphene quantum dot. The excess density of states due to edge states is shown in a color plot (cf. footnote 49) as calculated for a quantum dot with a smooth boundary and no particle-hole symmetry-breaking perturbations (a). In general, edge states are present both near a smooth boundary (b) and a boundary with short-range disorder (c).

Figure 2

(Color online) Electronic states in a graphene quantum dot close to the Dirac point. The graphene quantum dot has the shape of a deformed circle (see Fig. 1 and footnote 57) with $R=160a\approx 40\text{nm}$, and we consider both smooth and rough edges as shown in Figs. 1b, 1c, respectively. The parameters for the edge disorder are $N_{\text{sweep}}=5$ and $p=0.05$ (see the main text for a discussion of the edge disorder model). (a) Number of states per energy interval $\Delta E$ for a quantum dot with smooth (black lines) and rough edges [red lines (gray in print)], with $\Delta E=0.4t/61$. (b) Density of states estimated numerically from Eq. (12) for a quantum dot with smooth (black symbols) and rough edges [red symbols (gray in print)]. For comparison, the blue dashed line shows a $1/E$ dependence.

Figure 3

(Color online) Number of states (black lines) per energy interval $\Delta E$ for a graphene quantum dot with smooth (left panels) and rough (right panels) edges. We show results for situations with broken electron-hole symmetry: (a) finite next-nearest-neighbor hopping and no edge potential ($t^{\prime }=0.1t$ and $U_0=0$) and (b) and (c) finite next-nearest-neighbor hopping including an edge potential [$t^{\prime }=0.1t$, with (b) $p_{\text{edge}}=0.25$ and $U_0=0.2t$, and (c) $p_{\text{edge}}=1$ and $U_0=0.1t$]. The remaining parameters are as in Fig. 2. The blue dashed lines show the number of bulk states $N_{\text{bulk}}$ estimated from the linear density of states of the Dirac dispersion Eq. (13).

Figure 4

(Color online) Magnetic field dependence of the energy levels (black lines) in a desymmetrized quantum dot with $R=100a$ (deformed circle as shown in Fig. 1, cf. footnote 57). The participation ratio $p$ of the states is color encoded, with the most strongly localized states in red. The blue dashed lines indicate the energy of the $n=0,\pm 1$ Landau levels of graphene. The calculations includes finite next-nearest-neighbor hopping $t^{\prime }=0.1t$ and a random edge potential with $p_{\text{edge}}=0.25$ and $U_0=0.2t$.

Figure 5

(Color online) Level-spacing distributions for quantum dots with smooth edges for $t^{\prime }=0$ (solid red curve) and $t^{\prime }=0.1t$ (solid black curve), together with the theoretical predictions for Poisson statistics (dashed line), the Gaussian orthogonal ensemble (dashed-dotted line), and the Gaussian unitary ensemble (dotted line). The inset shows details the integrated level spacing distribution for small level spacings $S$ (same line colors and types as the main plot). The level distribution statistics has been obtained by averaging individual level distributions from 100 quantum dots similar to the type given in footnote 57, with average radius $R=160a$. A state has been identified as an edge state, if its participation ratio $p_i<0.05$ (Ref. 73). For $t^{\prime }=0$ we have also omitted all states with an energy smaller than the numerical precision.

Figure 6

(Color online) Color plot (cf. footnote 49) of wave-function density in a graphene quantum dot (shape as described in footnote 57) for the electron-hole (e-h) symmetric case ($t^{\prime }=0$, left column) and for broken e-h symmetry ($t^{\prime }=0.1t$, right column) on the examples of a mode that is (a) strongly decaying and (b) slowly decaying into the bulk. Note that for presentation purposes we have chosen a rather small dot $\left(R=30a\right)$ but the behavior does not change qualitatively for larger dots.