Exact wave functions for an electron on a graphene triangular quantum dot

We generalize the known solution of the Schrödinger equation, describing a particle confined to a triangular area, for a triangular graphene quantum dot with armchair-type boundaries. The quantization conditions, wave functions, and the eigenenergies are determined analytically. As an application, we calculate the corrections to the quantum dot’s energy levels due to distortions of the carbon-carbon bonds at the edges of the quantum dot.

Figure 1

Sextet of plane waves which compose the wave function $\psi$ in Eq. (14). The dashed lines are $k_y=\pm \sqrt{3}{k}_x$.

Figure 2

(Color online) The absolute value $\left|\psi \left(x,y\right)\right|$ of the wave function $\psi \left(x,y\right)$, Eq. (14), for arbitrary $\mathbf{k}$. The wave function vanishes at $y=0$ and at $y=\pm \sqrt{3}x$.

Figure 3

The hexagon shown is the Brillouin zone of graphene. The white polygon is where the allowed wave vectors for the triangular quantum dot are located (see Figs. 8, 9).

Figure 4

Triangular graphene quantum dot with armchair edges (here, $N_0=3$ and $N=4$). The solid lines represent covalent bonds between neighboring carbon atoms (black circles). The thick lines at the edges represent deformed bonds, whose effect on the spectrum is studied in Sec. 6. The dashed lines represent fictitious bonds connecting real carbon atoms and auxiliary atoms. The latter are represented by hatched circles. Dotted lines correspond to the effective boundaries of the triangular dot.

Figure 5

A triangular graphene dot can be split into $\left(N_0+1\right)N_0/2$ aromatic rings, where the integer $N_0$ is proportional to $L_0$, the dot size: $L_0=3a_0{N}_0$. For the dot shown on the figure, $N_0=3$. Thus, the triangular dot consists of six rings.

Figure 6

Sequence of steps to rotate the triangular quantum dot about its center. First, the dot is rotated around the origin (black arrow). After this transformation, the original white TQD becomes the gray TQD. Point A is transformed into ${\text{A}}^{\prime }$ and point B is mapped on ${\text{B}}^{\prime }$. Afterwards, the dot is shifted by $L$ (white arrow) to restore the original position.

Figure 7

(Color online) The absolute value of the wave function ${\Psi }^{n,m}$ on a large graphene lattice. The wave function vanishes on the black sites (blue sites when the figure is in color), which are the auxiliary atoms. The lines of the auxiliary atoms split the whole lattice in $n_{\text{dot}}$ triangular dots. Note that a given auxiliary atom is shared by two dots. The bar on the right shows the correspondence between the dot color and the wave function value.

Figure 8

The allowed values of ${\mathbf{k}}^{n,m}$ occupy the sector $\sqrt{3}\left|k_x\right|<k_y$ of the graphene Brillouin zone (this sector is drawn in white in Fig. 3). Every filled circle represents a state. Points A and B (open circles near the top) correspond to the same state. This is also true for C and D, see Sec. 5D. The thick solid line at the top of the figure is the Brillouin-zone boundary.

Figure 9

Counting the quantum states of a TQD. The thick solid V-shaped lines show how we group our states to form an arithmetic progression. Numbers from 1 to 4 enumerate terms of the progression. Note that each pair of open circles (points A and B; C and D) counts as one state, see Sec. 5D.

Figure 10

(Color online) Probability density for the state with $n=39$ and $m=41$ for a triangular graphene dot. The dot’s effective size, $L=3N{a}_0$, is fixed by the value of the constant $N=41$. The total number of atoms in such a dot is $N_a=4920$. The probability density for the same state is presented in Fig. 4(a) of Ref. 14.

Figure 11

(Color online) Probability density for the state with $n=38$ and $m=41$ for a triangular graphene dot. The dot’s effective size, $L=3N{a}_0$, is fixed by the value of constant $N=41$. The total number of atoms in such a dot is $N_a=4920$. The probability density for the same state is presented in Fig. 4(c) of Ref. 14.