Symmetry Classes in Graphene Quantum Dots: Universal Spectral Statistics, Weak Localization, and Conductance Fluctuations

We study the symmetry classes of graphene quantum dots, both open and closed, through the conductance and energy level statistics. For abrupt termination of the lattice, these properties are well described by the standard orthogonal and unitary ensembles. However, for smooth mass confinement, special time-reversal symmetries associated with the sublattice and valley degrees of freedom are critical: they lead to block diagonal Hamiltonians and scattering matrices with blocks belonging to the unitary symmetry class even at zero magnetic field. While the effect of this structure is clearly seen in the conductance of open dots, it is suppressed in the spectral statistics of closed dots, because the intervalley scattering time is shorter than the time required to resolve a level spacing in the closed systems but longer than the escape time of the open systems.

Figure 1

Systems studied numerically (schematic). (a),(b) Africa billiard. (c),(d) Half-stadium with two identical leads; left-right symmetry is broken by cutting out circular segments at the top left and bottom right. The graphene lattice is terminated abruptly in (a) and (c), while smooth mass confinement is used in (b) and (d).

Figure 2

Level-spacing distribution $P(S)$ for an Africa flake consisting of 68 169 carbon atoms using about 3000 energy levels in the range [$-0.5t$, $0.5t$]. (a),(b) Abrupt lattice termination. (c),(d) Smooth mass confinement with $W=4.5\sqrt{3}a$, $\omega =0.15\sqrt{t}/a$. (e),(f) Smooth mass confinement with $W=16.5\sqrt{3}a$, $\omega =0.041\sqrt{t}/a$. $a\approx 0.25\mathrm{nm}$ is the graphene lattice constant. The left panels are for $\Phi =0$, while the right panels are for $\Phi =0.7{\Phi }_0$. Insets in each panel present the integrated distributions $C(S)={\int }_0^{S}P(S^{\prime })d{S}^{\prime }$. Numerical results are shown with solid thick black lines, whereas the thin lines are for Poisson (green solid), GOE (red dotted), and GUE (blue dashed) statistics.

Figure 3

Average conductance: weak localization. (a) Transmission as a function of energy for an abruptly terminated billiard [Fig. 1c] with zigzag leads (solid line). The dashed line shows the number of open channels in the leads, $M$. (b) Average transmission as a function of $M$ for the same system with $\Phi =0$ (solid black line, open triangles) and $\Phi =1.6{\Phi }_0$ (dashed red line, full triangles). (c) Change in the average transmission as a function of the magnetic flux. Circles: Abrupt termination with armchair leads (1–7 open channels). The fit (solid line) yields $A_0=1.5A_B$ and $\mathcal{R}=0.19$. Triangles: Abrupt termination with zigzag leads (3–7 open channels). The fit (dashed line) yields $A_0=1.0A_B$ and $\mathcal{R}=0.18$. Diamonds: Smooth mass confinement [Fig. 1d] (2–8 open channels). The fit (dotted line) yields $A_0=0.54A_B$ and $\mathcal{R}=0.057$ (parameters of the billiards given in 29).

Figure 4

Conductance fluctuations: Variance of the transmission as a function of the number of open channels in the leads (same cavities as in Fig. 3). (a) Abruptly terminated boundary. $B=0$ (black open symbols) and $B\ne 0$ (red full symbols) results are in good agreement with the corresponding RMT values, orthogonal (COE, black solid line) and unitary (CUE, red dashed line). The unitary data uses several values for the magnetic field in the range $\Phi \in [0.8,2.4]{\Phi }_0$; both armchair leads (triangles) and zigzag leads (circles) are used. (b) Smooth mass confinement. Zero field (black open symbols) and $\Phi =2.0{\Phi }_0$ (red full symbols) results are compared to 1, 2, and 4 times the CUE values (black dotted, red dashed, and black solid lines).