Dot-bound and dispersive states in graphene quantum dot superlattices

We consider a square lattice configuration of circular gate-defined quantum dots in an unbiased graphene sheet and calculate the electronic, particularly spectral properties of finite albeit actual sample sized systems by means of a numerically exact kernel polynomial expansion technique. Analyzing the local density of states and the momentum resolved photoemission spectrum we find clear evidence for a series of quasibound states at the dots, which can be probed by optical measurements. We further analyze the interplay of the superlattice structure with dot-localized modes on the electron energy dispersion. Effects of disordered dot lattices are discussed too.

Figure 1

Left: Graphene quantum dot array used in this work. The dots are defined electrostatically, by applying a constant bias $V$. The dot radius is $R$; the square dot superlattice constant is $D$. To ensure the gate potential to be smooth on the scale of the lattice spacing $a$, we adopt a linear interpolation of $V_i$ within a small range $R\pm 0.01R$. The underlying graphene honeycomb lattice structure is shown in the lower left corner; we have zigzag (armchair) edges in $x$ $\left(y\right)$ direction with $N$ $\left(M\right)$ dots. Periodic boundary conditions (PBCs) were used at the edges of the sample. Right: Sketch of Dirac electron scattering at a single quantum dot. For $E<V$, where the dot embodies an $n$-$p$ junction, the incident (${\psi }_i$) and reflected (${\psi }_r$) electron waves reside in the conduction band, while the transmitted (${\psi }_t^{\left(qb\right)}$) wave inside the dot corresponds to a state in the valence band. Owing to the double-cone dispersion (near $K$ and $K^{\prime }$) nonevanescent waves can exist in the dot, i.e., ${\psi }_t^{\left(qb\right)}$ might give rise to a quasibound state.

Figure 2

DOS of the graphene quantum dot superlattice in dependence on $R$, $D$, and $V/t$. Peaks related to quasibound dot modes are designated by $a_m$. Note that the dot parameter $\eta$ is the same in panels (a), (c), and (d). The DOS is calculated by the KPM on a lattice with 3104 (1792) sites in zigzag (armchair) edge direction, using 16 384 Chebyshev moments. To identify effects due to the finiteness of the sample, the PBCs, and the resolution of the KPM—leading to small deviations from the strictly linear increase of the DOS near $E=0$—we included the DOS obtained numerically for the case $V_i=0$ $\forall i$ (dashed lines). The total DOS, of course, fulfills the sum rule ${\int }_{-\infty }^{\infty }\rho \left(E\right)dE=1$.

Figure 3

LDOS intensity plots for the central part of a (larger) square quantum dot superlattice with $8\times 8$ dots (upper panels). The LDOS is given for the energies indicated in the lower panel, showing the mean DOS of the whole panel. To diminish finite-size effects the LDOS was calculated using 4096 Chebyshev moments only; therefore the splitting of the $a_m$ bands is not resolved here (recall that the resolution of the KPM scales with the inverse square root of the number of Chebyshev moments [28]). System parameters are $R=4.775$ nm, $D=19.1$ nm, and $V/t=0.17092$ [as in Fig. 2].

Figure 4

Optical conductivity (in units of ${\sigma }_0=e^2/8\hslash$) for a $20\times 20$ graphene quantum dot superlattice with $V/t=0.17092$ and $R=4.775$ nm, $D=19.1$ nm. We chose $\mu /t=-0.02<0$ so that the Drude peak appears at $\omega =0$ and the transition between the occupied $a_2$ and the unoccupied $a_1$ mode appears at $\omega /t\simeq 0.045$ (note that the transition $a_3$ to $a_2$ does not appear as both modes are occupied).

Figure 5

Single-particle spectral function along the $\overline{\Gamma K}$ direction (horizontal; left-hand panels) and parallel to the $\overline{\Gamma M^{\prime }}$ direction (vertical, right-hand panels) through the Dirac $K$ point, as indicated in the upper central figure (b), showing graphene's Brillouin zone. Panels (a) and (c) give $A(\vec{k},E)$ for a finite sample of pristine graphene ($V/t=0$) with PBC. Below, results for a $20\times 20$ dot superlattice with $R=4.775$ nm and $D=19.1$ nm are shown. In (d) and (e) $V/t=0.10727$ (mode $a_0$ falls on $E=0$), in (f) and (g) $V/t=0.17092$ (mode $a_1$ on $E=0$), and in (h) and (i) $V/t=0.22911$ (mode $a_2$ on $E=0$). The green marker (circle) traces the energy shift of the nodal point for pristine graphene when $V$ is increased. In view of the transfer of spectral weight to other nodal points it should, however, no longer be identified as the “genuine” Dirac point. Note also that for mode $a_1$ at $E=0$ the Dirac point of pristine graphene evolves into the nodal point at $E=0$ when the dot spacing $D$ is reduced [compare panel (e) and Fig. 6 (b)]. Within the KPM 8192 moments were used.

Figure 6

Single-particle spectral function parallel to the $\overline{\Gamma M^{\prime }}$ direction through the Dirac $K$ point for a superlattice of $10\times 10$ dots with $R=4.775$ nm and $D=38.2$ nm. In (a), $V/t=0.10727$ so that the mode $a_0$ falls on $E=0$ [as in Figs. 5 and 5]. In (b), $V/t=0.17092$ so that mode $a_1$ falls on $E=0$ [as in Figs. 5 and 5].

Figure 7

DOS for a $20\times 20$ superlattice sample of graphene quantum dots with random radii. Shown are the results for a single, typical realization, where all the dot radii were drawn from a uniform distribution of radii with mean value $R=4.775$ nm and $R_{(n,m)}/R\in$ [0.975,1.025] (red dashed line), [0.95,1.05] (blue dashed-dotted line), [0.925,1.075] (green double-dot-dashed line), and [0.9,1.1] (violet dotted line). The black line gives the DOS without disorder. Again, we have $D=19.1$ nm and $V/t=0.17092$.

Figure 8

Single-particle spectral function through the Dirac $K$ point for a $20\times 20$ graphene quantum dot superlattice with random $R_{(n,m)}/R\in$ [0.9,1.1]. The parameters $R$, $D$, and $V$ are chosen as in Figs. 5 and 5 so that the mode $a_0$ falls on $E=0$.

Figure 9

Single-particle spectral function through the Dirac $K$ point for a $20\times 20$ graphene quantum dot superlattice where the dots are displaced from their superlattice sites by $r_{(n,m)}\in [0,\Delta r]$ in random direction. In panels (a) and (b) $\Delta r=0.1$; in panels (c) and (d) $\Delta r=0.4$. The parameters $R$, $D$, and $V$ are chosen as in Figs. 5 and 5 so that the mode $a_0$ falls on $E=0$.