Transition to Landau levels in graphene quantum dots

We investigate the electronic eigenstates of graphene quantum dots of realistic size (up to 80 nm diameter) in the presence of a perpendicular magnetic field $B$. Numerical tight-binding calculations and Coulomb-blockade measurements performed near the Dirac point exhibit the transition from the linear density of states at $B=0$ to the Landau-level regime at high fields. Details of this transition sensitively depend on the underlying graphene lattice structure, bulk defects, and localization effects at the edges. Key to the understanding of the parametric evolution of the levels is the strength of the valley-symmetry-breaking $K\text{−}{K}^{\prime }$ scattering. We show that the parametric variation in the level variance provides a quantitative measure for this scattering mechanism. We perform measurements of the parametric motion of Coulomb-blockade peaks as a function of magnetic field and find good agreement. We demonstrate that the magnetic-field dependence of graphene energy levels may serve as a sensitive indicator for the properties of graphene quantum dots and, in further consequence, for the validity of the Dirac picture.

Figure 1

(Color online) Shapes and sizes $\left(50\times 50{\text{nm}}^2\right)$ of graphene quantum dots confined by (a) a smooth valley spin-conserving potential [Eq. (11), the length scale of the confinement is marked by $\Delta d$], (b) atomically sharp armchair and zigzag boundaries. Dots with disorder due to (c) bulk defects or (d) edge roughness.

Figure 2

(Color online) (a) Magnetic-field dependence of the eigenenergies of a graphene quantum dot with smooth confinement which approximately preserves valley symmetry [Eq. (11), see Fig. 1a]. Landau levels for $n=\pm 1,\pm 2$ [dashed lines, see Eq. (2)] are inserted as guide to the eye. The four symbols (▲, ▼, ◼, and ◆) mark parameter values for which eigenstates are shown in Fig. 3. (b) Close-up of the avoided crossing of two eigenstates in (a). Dots represent numerical data, the continuous line is a fit to Eq. (10). (c) Closeup of avoided crossings around the diabatic ridge formed by the first Landau level (see text).

Figure 3

(Color online) Typical eigenstates (plotted is the absolute square of the wave function) of a graphene quantum dot with smooth confinement [see Fig. 1a and Eq. (11)] at high magnetic field $\left(B=25\text{T}\right)$, corresponding to the zeroth [(a) and (b)] and first [(c) and (d)] Landau levels. Symbols (▲, ▼, ◼, and ◆) correspond to those marking the position of these states in the energy level diagram [Fig. 2a].

Figure 4

(Color online) Same as Fig. 2 but for a quantum dot with atomically sharp zigzag and armchair boundaries. The solid line in (b) is a fit to Eq. (14), the dashed line (corresponding to $V_{K{K}^{\prime }}=0$) is inserted as guide to the eye. The evolution of one eigenenergy with magnetic field is drawn with a thick solid line as guide to the eye in (a) and (c). The arrow in (c) marks a kink in the magnetic-field dependence of this state (see text).

Figure 5

(Color online) Same as Fig. 2 for a quantum dot featuring 30 single-lattice vacancies (see inset) out of a total number of 100.000 carbon atoms. Dotted lines mark the evolution of two states localized at one defect. Symbols mark the states for which the corresponding wave function is shown in Fig. 6.

Figure 6

(Color online) Eigenstates of a graphene quantum dot with disorder: (▲ and ▼) 30 single vacancies [one single vacancy shown as inset in Fig. 5a] or (◼ and ◆) edge roughness of $\pm 2\text{nm}$. Positions in the energy level diagram are marked by corresponding symbols in Figs. 5a, 8a.

Figure 7

(Color online) Same as Fig. 2 for a quantum dot featuring 30 double lattice vacancies (see inset) in 100.000 carbon atoms.

Figure 8

(Color online) Same as Fig. 4 for a quantum dot featuring rough edges (with a roughness amplitude $\delta d=\pm 2\text{nm}$). Symbols mark the states for which the corresponding wave function is shown in Fig. 6.

Figure 9

(Color online) (Avoided) crossings for (a) soft edges, (b) hard edges, (c) rough edges, and (d) rough edges plus bulk disorder. The level splitting is (a) 0.1 meV, (b) 2 meV, (c) 3 meV, and (d) 2.5 meV.

Figure 10

(Color online) Variance ${\sigma }_{\epsilon }$ of the mean level spacing [see Eq. (16)] as a function of magnetic field for (a) single-vacancy defects and (b) double-vacancy defects (note the expanded $y$ scale) for three values of disorder concentrations: from top to bottom $n_i=3\times {10}^{-5}$, $6\times {10}^{-5}$, and $2\times {10}^{-4}$.

Figure 11

(Color online) (a) Coulomb-blockade peaks measured as a function of applied plunger gate electrode potential $\left(V_G\right)$ and magnetic field applied perpendicular to the sample (see Ref. 59 for details about the measurement). Lines: fit to Eq. (14) for two measured Coulomb-blockade peak pairs with a level splitting at $B=0$ of $\approx 0.3\text{V}$. (b) Normalized variance of the level spacing ${\sigma }_{\epsilon }$ [see Eq. (16)] as a function of magnetic field. Dots denote an average over 20 consecutive experimental Coulomb-blockade peaks for two values of back-gate voltage $V_{BG}$ [corresponding to the upper $\left(V_{BG}=20\text{V}\right)$ and lower $\left(V_{BG}=38\text{V}\right)$ data points]. Solid lines represent simulations for different edge roughness amplitude $\delta d$ [Fig. 1d] as only fit parameter: $\delta d\approx 0.5$ (0.6) nm, for the upper (lower) curve.