Modeling disorder in graphene

We present a study of different models of local disorder in graphene. Our focus is on the main effects that vacancies (random, compensated, and uncompensated), local impurities, and substitutional impurities bring into the electronic structure of graphene. By exploring these types of disorder and their connections, we show that they introduce dramatic changes in the low energy spectrum of graphene, viz., localized zero modes, strong resonances, gap and pseudogap behaviors, and nondispersive midgap zero modes.

Figure 1

(Color online) In (a), ${\vec{a}}_1$ and ${\vec{a}}_2$ are the primitive vectors that define the WS unit cell highlighted as the dashed hexagon. The lattice parameter $a$ is $\simeq 1.4\mathrm{\AA }$. The first BZ of the associated reciprocal lattice is shown in (b), together with the points of high symmetry $\Gamma$ and $M$ and the two nonequivalent $K$ and $K^{\prime }$.

Figure 2

(Color online) (a) Band structure along the symmetry directions in the reciprocal BZ of the honeycomb lattice. (b) Band structure of graphene $(t^{\prime }=0)$ with the two bands touching at the $K$ and $K^{\prime }$ points of the BZ.

Figure 3

(Color online) DOS associated with Eq. (6) for different values of the next-nearest-neighbor hopping $t^{\prime }$.

Figure 4

(Color online) Selected eigenstates in a graphene sheet with ${80}^2$ atoms containing a single impurity at the center (black dot). Only the region near the vacancy is shown. (a) The eigenstate with energy closest, but different, to zero. (b) The eigenstate with $E=0$. (c) and (d) show the presence of two quasilocalized eigenstates even with $t=0.2t$.

Figure 5

(Color online) IPR and LDOS calculated at one site closest to the vacancy. In panel (a), we have results for the IPR with $t^{\prime }=0$ and $t^{\prime }\ne 0$ without any vacancy (top row) and with a single vacancy (bottom) for comparison. In panel (b), we show the dependence of the IPR of the zero mode, $\mathcal{P}(E=0)$, with the system size $N$ (left), and also $⟨\mathcal{P}\left(E\right)⟩$ versus $N$ for the remainder (extended) states (right). Dashed lines are guides for the eye.

Figure 6

(Color online) IPR and DOS for the diluted honeycomb lattice. (a) DOS for selected concentrations $x$ and different values of $t^{\prime }$. (b) IPR for selected values of $t^{\prime }$ using a concentration $x=0.5\%$. For comparison, the corresponding DOS is also plotted in each case. The concentration of vacancies is $x$, and only the vicinity of the Fermi level is shown.

Figure 7

(Color online) Dilution of just one sublattice of the honeycomb. (a) DOS for different dilution strengths, diluting only sublattice $A$. (b) On the left panel, we show a detail of the DOS and the evolution of the gap with vacancy concentrations. On the right panel, we plot the dependence of the missing spectral weight on the band $(=1-w_{\delta })$ with $x$ (circles). The continuous line is the best fit using $f\left(x\right)=ax∕(b-x)$ to the data represented by the circles. The coefficients yielding the best fit are $a=0.997$ and $b=0.998$, thus agreeing with the expected result [Eq. (23)].

Figure 8

(Color online) The gap estimated from the numerical curves in Fig. 7 is plotted against the vacancy concentration $x$. The continuous line is a least squares fit to $f\left(x\right)=a{x}^b$.

Figure 9

(Color online) DOS for the honeycomb lattice using the controlled selective dilution discussed in the text, calculated for different concentrations of vacancies $x$ and several degrees of uncompensation $\eta$. Only the low-energy region close to the Dirac point is shown.

Figure 10

(Color online) Real part of Eq. (30) (top) obtained from the homogeneous DOS (bottom) through the Kramers–Kronig relations.

Figure 11

(Color online) (Top) LDOS at the impurity site $\left[{\rho }_0\left(E\right)\right]$ for different strengths of the scattering potential, as indicated in the legend. (Bottom) The same data divided by the free DOS $\left[{\rho }^0\left(E\right)\right]$.

Figure 12

(Color online) (a) Position of the maximum in the LDOS compared with the roots of Eq. (31). (b) Energy of the bound state.

Figure 13

(Color online) DOS of the honeycomb lattice with a finite density of local impurities. (a) shows the DOS for $U=12t$ and different concentrations of impurities (notice the truncation in the horizontal axis). (b) shows a detail of the low-energy region for different $U$ and $x$ as noted in the different graphs.

Figure 14

(Color online) (a) Variation of the Dirac point energy $E_D$ with impurity concentration and strength. The inset shows $E_D∕EU$ as a function of $U$, in which the curves with $U⩽3$ roughly collapse onto each other. (b) Spectral weight transfer to the impurity band in the presence of local impurities. The values shown correspond to integration of the global DOS beyond $E=4t$ (above the main band edge, cf., Fig. 13).

Figure 15

(Color online) Effect of a single substitutional impurity in the LDOS. In panel (a), we plot the LDOS calculated at the site of the impurity for the four different values of $t_0$ indicated in each frame. In panel (b), the situation is identical but the LDOS is calculated at the nearest-neighboring site of the impurity.

Figure 16

(Color online) Variation of the energy corresponding to the peak in the LDOS with the magnitude of $t_0$. The LDOS in question is the LDOS calculated at the impurity site.

Figure 17

(Color online) The DOS corresponding to the model Hamiltonian (34), with a finite density of impurities. The three panels correspond to different values of the perturbing hopping ($t_0=0.5$, 0.8, 0.9, and 0.95), and within each panel the three curves were obtained at different concentrations ($x=0.01$, 0.05, and 0.1). The inset of the bottom panels is a magnification of the region near $E=0$.