Figure 1

Typical quantum scars [red (darker region) indicates higher electron concentration] for a stadium-shaped graphene confinement with zigzag horizontal boundaries and contour plots of energy in the wave vector plane (the band structure). The energies for the patterns in (a–c) are $E/t=0.13252$, 0.4024, and 0.911 88, respectively. The stadium consists of $N=11814$ carbon atoms. Panels (d–f) show the $E-\mathbf{k}$ configuration for an infinite graphene flake for the same lattice orientation and energy values as those for (a–c), respectively. The allowed wave vectors are on the constant energy curves (as indicated by the arrows). The lattice constant is $a=2.46\AA$. Dashed line segments in (a–c) are for eye guidance, and in (d–f) they indicate the first Brillouin zone.Reuse & Permissions

Figure 2

Typical scars in the stadium-shaped graphene confinement as in Fig. 1. The energy values for (a–h) are $E/t=0.25347$, 0.363 58, 0.576 65, 0.606 99, 0.819 56, 0.910 61, 0.977 22, and 0.991 98, respectively. The dashed lines represent classical periodic orbits.Reuse & Permissions

Figure 3

Scars in stadium-shaped graphene confinement with armchair horizontal boundaries. The number of atoms is $N=13694$. The corresponding energy values are $E/t=0.20031$, 0.25 99, 0.3106, 0.549 54, 0.592 38, 0.9168, 0.952 16, and 0.998 01 for (a–h), respectively.Reuse & Permissions

Figure 4

(a) An open stadium-shaped graphene quantum dot with semi-infinite leads on both sides. The shape has a mirror symmetry. (b) Transmission $T$ versus energy $E/t$. (c) Transmission $T$ for the dot after removing the two carbon atoms as indicated by the arrow in (a) so that the mirror symmetry is broken.Reuse & Permissions