Electric transport through circular graphene quantum dots: Presence of disorder

The electronic states of an electrostatically confined cylindrical graphene quantum dot and the electric transport through this device are studied theoretically within the continuum Dirac-equation approximation and compared with numerical results obtained from a tight-binding lattice description. A spectral gap, which may originate from strain effects, additional adsorbed atoms, or substrate-induced sublattice-symmetry breaking, allows for bound and scattering states. As long as the diameter of the dot is much larger than the lattice constant, the results of the continuum and the lattice model are in very good agreement. We also investigate the influence of a sloping dot-potential step, of on-site disorder along the sample edges, of uncorrelated short-range disorder potentials in the bulk, and of random magnetic fluxes that mimic ripple disorder. The quantum dot's spectral and transport properties depend crucially on the specific type of disorder. In general, the peaks in the density of bound states are broadened but remain sharp only in the case of edge disorder.

Figure 1

Energy domains for the QD states. I (white), no states; II (black, red), bound states; III (gray, blue), tunneling states; IV (light gray, green), scattering states.

Figure 2

The energies of the bound states in a graphene QD as function of the dot radius (left panel) and potential strength (right panel). We use $\Delta =0.1$ and show the results for $J=1/2$ (solid line), $J=3/2$ (dashed line), and $J=5/2$ (dotted line) only. The intersections of the solutions with the vertical lines yield the bound-states energies for $R=25a$ and $U=0.2$.

Figure 3

Comparison between the eigenenergies of the TB Hamiltonian (circles) of the graphene lattice and the solutions of the Dirac equation (lines) as a function of the potential $U$. The parameters used are $\Delta =0.1$ and $R=25a$.

Figure 4

Potential landscape of a QD with radius $R$ and an additional environment for $r>L$. The vertical dotted arrows indicate $|E-V(r\left)\right|$ in each of the three regions. For $R<r<L$, the wave functions decay. Also, $k^{\prime \prime }$, $k^{\prime }=i{\kappa }^{\prime }$, and $k$ are the wave vectors inside the QD, outside, and in the environment region, respectively.

Figure 5

Transport cross section of a graphene QD ($R/a=30$, $\Delta =\pm 0.1$) with environment (top). The position of the peaks correspond to the energies of the bound states of an isolated graphene QD with dot potential $U=0.2$. They also agree with the results from the corresponding TB model (circles in the bottom panel). The transport cross section (green dashed line) is shown in arbitrary units with the “background” value (red solid line) given by the empty dot $U=0$.

Figure 6

The two-terminal conductance $g$ (top panel) calculated within a TB model compared to the normalized transport cross section (bottom panel) of a continuum Dirac model.

Figure 7

Influence of random magnetic flux disorder on the density of states of a circular QD with radius 30$a$ and dot potential $U/t=0.2$. The sublattice difference is $\Delta /t=\pm 0.1$, as before. With increasing disorder, $f/(h/e)=0.005$ (red line), 0.01 (green line), 0.02 (blue line), and 0.05 (magenta line), the peaks get broadened and start to overlap.

Figure 8

The density of states $\rho (E;W_{\mathrm{ed}})$ within the gap arising from the sublattice difference $\Delta /t=\pm 0.1$ in the presence of edge disorder of strengths $W_{\mathrm{ed}}/t=0.1$ (red line), 0.2 (green line), and 0.5 (blue line). The bound-states energies of the QD (radius 30$a$, potential strength 0.2$t$) remain almost unaffected by the edge disorder.

Figure 9

The density of states $\rho (E;W)$ showing the bound states of the QD (radius 30$a$, potential strength 0.2$t$) within the gap arising from the sublattice difference $\Delta /t=\pm 0.1$ in the presence of on-site disorder of strengths $W/t=0.0$ (red line), 0.05 (green line), 0.1 (blue line), and 0.2 (magenta line). The sharp DOS peaks broaden and finally disappear with increasing disorder strength.