Electronic properties of disordered graphene antidot lattices

Regular nanoscale perforations in graphene (graphene antidot lattices, GALs) are known to lead to a gap in the energy spectrum, thereby paving a possible way towards many applications. This theoretical prediction relies on a perfect placement of identical perforations, a situation not likely to occur in the laboratory. Here, we present a systematic study of the effects of disorder in GALs. We consider both geometric and chemical disorder, and evaluate the density of states as well as the optical conductivity of disordered GALs. The theoretical method is based on an efficient algorithm for solving the time-dependent Schrödinger equation in a tight-binding representation of the graphene sheet [Yuan et al., Phys. Rev. B 82, 115448 (2010)], which allows us to consider GALs consisting of 6400 $\times$ 6400 carbon atoms. The central conclusion for all kinds of disorder is that the gaps found for pristine GALs do survive at a considerable amount of disorder, but disappear for very strong disorder. Geometric disorder is more detrimental to gap formation than chemical disorder. The optical conductivity shows a low-energy tail below the pristine GAL band gap due to disorder-introduced transitions.

Figure 1

Sketch of the different kinds of disorder considered. (a) The center of the holes is shifted randomly with respect to the original position in the perfect periodic array $(x,y)$ to a new position in the range $(x\pm l_C,y\pm l_C)$ ($l_C=2a$). (Notice the different relative distance between the holes.) (b) The radius of the holes is randomly shrunk or enlarged within the range $[R-r_R,R+r_R]$ ($r_R=a$). (Notice the different relative size of the holes.) (c) GAL with additional randomly distributed vacancies, signaled by the missing carbon atoms ($n_x=1\%$). (d) GAL with randomly distributed hydrogen adatoms, signaled by the red dots ($n_i=1.75\%$). Notice that two other kinds of disorder are considered in the text, namely noncorrelated and correlated long-range (Gaussian) changes in the on-site potentials, which are not sketched in this figure.

Figure 2

DOS (top panels) and optical conductivity (bottom panels) for a $\{10,6\}$ GAL with geometrical disorder. In the first column we show (a) DOS and (c) $\sigma \left(\omega \right)$ for a disordered GAL in which the center of the holes is shifted randomly with respect to the original position in the perfect periodic array within the range $(x\pm l_C,y\pm l_C)$, as sketched in Fig. 1a. The different colors correspond to different values of $l_C$ (in units of $a$), as denoted in the inset of the figures. In the second column we show (b) DOS and (d) $\sigma \left(\omega \right)$ for a GAL where the radius of the holes is randomly shrunk or enlarged within the range $[R-r_R,R+r_R]$, as sketched in Fig. 1b. Different colors correspond to different values of $r_R$ (in units of $a$).

Figure 3

Distribution of quasieigenstates of a $\{10,6\}$ GAL with geometrical disorder, where the radius of the holes is randomly shrunk or enlarged within the range $[5.75,6.25]$. The holes with unchanged radius ($R=6$) are indicated by the red arrows. The quasieigenstates are calculated at energy $E=0.056t$ and $E=0.12t$, corresponding to the states at the low-energy peaks marked by the black and red arrows, respectively, in the DOS of Fig. 2b ($r_R=0.25$). The highest contribution to the quasieigensate at $E=0.056t$ (a) is concentrated around holes with radius $R>6$, for which the first ring of atoms, as compared to perfect GAL with $R=6$, has been removed. The quasieigenstate at $E=0.12t$ with highest amplitude (b) are localized around the holes with $R=6$ (marked by the red arrows), as in perfect GAL.

Figure 4

DOS (top panels) and optical conductivity (bottom panels) for a $\{10,6\}$ GAL with resonant impurities. In the first column we show (a) DOS and (c) $\sigma \left(\omega \right)$ for a GAL with a random distribution of vacancies, as sketched in Fig. 1c. The different colors correspond to different amounts of missing dangling bonds, as denoted in the inset of the figures. In the second column we show (b) DOS and (d) $\sigma \left(\omega \right)$ for a GAL with hydrogen adatoms, as sketched in Fig. 1d. Different colors correspond to different percentage of adatoms in the sample.

Figure 5

DOS (top panels) and optical conductivity (bottom panels) for a $\{10,6\}$ GAL with on-site potential disorder. In the first column we show (a) DOS and (c) $\sigma \left(\omega \right)$ for a GAL with a noncorrelated random distribution of short-range potential, which can take the values within the range $[-v_r,v_r]$. The different colors correspond to different concentrations of disorder, as denoted in the inset of the figures. In the second column we show (b) DOS and (d) $\sigma \left(\omega \right)$ for a GAL with a long-range Gaussian potential disorder. The potential is given by Eq. (2), where $V_k$ is uniformly random in the range between $-V_0$ and $V_0$, and $d$ is the effective potential radius (see text).