Our geometry consists of a finite circular graphene quantum dot with 1,011 carbon atoms. For electronic transport, the quantum dot is connected to two semi-infinite leads. In Figure

1, we show the quantum dot and, partially, the semi-infinite leads. We employ a tight-binding model that only takes into account one

*π*-orbital per atom. The overlap energy between nearest neighbours is taken as

*t*=2.66 eV, where second-neighbour interactions are neglected. The advantage of using a single-band

*π*-orbital model resides in its simplicity, being the general features of electronic transport in very good agreement with those obtained by more sophisticated approaches. The hamiltonian can then be written as

$\u0124=-t\sum _{<\mathit{\text{ij}}>\sigma}({c}_{i}^{\u2021}{c}_{j}+\mathit{\text{hc}}),$

(1)

where${c}_{i}^{\u2021}/{c}_{i}$ are the creation/annihilation operators of an electron in site *i*. We expand the wave function in terms of the site base.$|{\Psi}^{k}\u3009=\sum _{i}{a}_{i}^{k}|i\u3009$, where${a}_{i}^{k}$ is the amplitude probability that the electron is to be in site *i* for the eigenstate *k*. We need to solve$\u0124|{\Psi}^{k}\u3009={E}^{k}|{\Psi}^{k}\u3009$. Four quantities are calculated to characterize the nature of the electronic and transport properties on two-circled structures, with PD and defect-free (ND) structures: the total density of states *N*(*E*), the local density of states *ρ*(*i*,*E*), the participation number *P*(*E*) and the transmission function *T*(*E*).

### Electronic properties for the closed system

The density of states is determined from the energy spectrum as

$N\left(E\right)=\frac{1}{D}\sum _{k=1}^{D}\delta (E-{E}_{k}).$

(2)

Another useful property is the local density of states:

$\rho (i,E)=\frac{1}{D}\sum _{k=1}^{D}|\u3008i|{\Psi}_{k}\u3009{|}^{2}\delta (E-{E}_{k}),$

(3)

which measures how each site

*i* contributes to the complete spectrum. For a fixed

*E*, it characterizes the spatial nature of the state: it is localized when only few sites contribute to that energy, or extended when more sites participate. Finally, the participation number is defined as[

16]

$P\left(E\right)=\frac{1}{\sum _{k=1}^{D}|\u3008{\Psi}^{k}|{\Psi}^{k}\u3009{|}^{4}}.$

(4)

It assesses the wave function spreading so it can help to find out the localized or extended nature of an electronic state. For a completely localized wave function *Ψ*^{
k
}(*i*) is approximately *δ*_{
k
i
}→*P*≈1 while for a typical delocalized wave function on *D* atoms, *Ψ*^{
k
}(*i*) is approximately$\frac{1}{\sqrt{D}}$, and then *P*≈*D*.

### Electronic transport properties: Green’s function method

We calculate transport properties for our graphene structures such as the density of states and the transmission function using Green’s function method. In order to obtain Green’s function, we use the following expression[

17]:

$\u011c\left(E\right)={\left(\mathrm{E\xce}-\u0124-{\widehat{\Sigma}}_{\mathrm{L}}-{\widehat{\Sigma}}_{\mathrm{R}}\right)}^{-1},$

(5)

where${\widehat{\sum}}_{\mathrm{L}}$ and${\widehat{\sum}}_{\mathrm{R}}$ are the self-energy terms of left and right leads, respectively, and$\u0124$ is the Hamiltonian of the conductor, i.e., in our case, the circular graphene sheet plus a few unit cells of the leads. In our approach, the contact leads at opposite sides of the circular graphene sheet is the graphene sheet itself extended to make the leads semi-infinite. This is equivalent to have reflectionless contacts in macroscopic conductors. Self-energy terms are calculated using the prescription${\widehat{\Sigma}}_{\mathrm{R}/L}(i,j)={t}^{2}{\u011c}_{\mathrm{R}/L}(k,l)$, where${\u011c}_{\mathrm{R}/L}(k,l)$ is Green’s function of the semi-infinite lead (right or left) evaluated on sites k and l, which are in contact with sites i and j in the circular graphene sheet.

We only need to calculate${\u011c}_{\mathrm{R}/L}$ in the sites in contact with the conductor. To do that, we use the formalism developed by López Sancho et al.[18]. This method has the advantage that the number of iterations close to singularities is very low compared to other transfer matrix methods, so it converges very fast and has been applied to graphene layers by other authors (see e.g.[19]). In this scheme, Green’s function is${\u011c}_{\mathrm{R}/L}={\left(\mathrm{E\xce}-{\u0124}_{00}-{\u0124}_{01}\widehat{T}\right)}^{-1}$, where${\u0124}_{00}$ is the Hamiltonian of one isolated graphene cell in the lead, and${\u0124}_{01}$ is the matrix that takes into account the interaction between two consecutive cells. For the calculation of *T*, we use the iterative method described in[18].

From Green’s function of the graphene structure, we calculate the transmission function and the density of states as[

17]

$T\left(E\right)=\text{Tr}\left[{\widehat{\Gamma}}_{\mathrm{L}/R}{\u011c}^{\mathrm{R}}{\widehat{\Gamma}}_{\mathrm{R}/L}{\u011c}^{\mathrm{A}}\right]$

(6)

$\mathrm{N}\left(E\right)=-\frac{1}{\pi}\text{Tr}\left(\text{Im}\u011c\right).$

(7)

In Equation 6, *G*^{R/A} are the retarded and advanced Green’s functions, respectively, and${\widehat{\Gamma}}_{\mathrm{L}/R}=i[{\widehat{\sum}}_{\mathrm{L}/R}^{\mathrm{R}}-{\widehat{\sum}}_{\mathrm{L}/R}^{\mathrm{A}}]$. We denote the trace of the matrix considered by “Tr”, which is extended over the whole matrix.