3 Figure 4: (Color online) (a) Mean free path ( l ) as a function of energy ( E ) for different vacancy concentration ( x ). (b) Mean free path ( l ) as a function of vacancy concentration ( x ) at E $_{F}$. The blue solid line is a fit to the data by l ∝ x − 1 / $^{2}$. Figure 3: (Color online) (a), (b) and (c) Conductivity ( σ $_{xx}$) as a function of energy ( E ) for vacancy concentration x = 0 . 5% , x = 1 . 0% and x = 5 . 0% at the time t$_{1}$ when D reaches its maximum (blue solid line) and t$_{2}$ ≈ 36 . 7 fs (orange dash line). “ E ” indicates the extended state and “ L ” indicates the localized state. (d) The mobility edge ( E $_{c}$) as a function of x − x $_{c}$. (e) The minimal conductivity ( σ $_{min}$) at E c as a function of x . x , but the reason for such trend is unclear. The existence of σ$_{min}$ is possibly due to the quasi -localized nature of electronic states induced by vacancy in graphene. It in- dicates that the transition from the localized states to the extended states at the mobility edge is discontinu- ous as Mott argued [14], suggesting the break-down of one-parameter scaling theory. Also, our simulated σ$_{min}$ varies from 1 to 1 . 3 e $^{2}$/h , different from the universal value of 0 . 628 e $^{2}$/h as expected in conventional 2D lo- calization system [14]. defines the critical energy where the system changes from the localized state to the extended state above the criti- cal vacancy concentration x$_{c}$ . We only need to consider x > x$_{c}$ , because when x < x$_{c}$ the electronic states become extended over the whole energy range. Figures 3(a), (b) and (c) show the energy dependence of conductivity at two different times for three vacancy concentrations of 0 . 5% , 1 . 0% , and 5 . 0% , respectively. One time is chosen at where D reaches its maximum, t$_{1}$ ( D = D$_{max}$ ), and the other is chosen at a much later time ( t$_{2}$ ≈ 36 . 7 fs >> t$_{1}$ ). At the low energies closed to E$_{F}$ , σ$_{xx}$ ( t$_{1}$ ) > σ$_{xx}$ ( t$_{2}$ ) , but at the high energies away from E$_{F}$ , σ$_{xx}$ ( t$_{1}$ ) < σ$_{xx}$ ( t$_{2}$ ) . This is because the low-energy electronic states are lo- calized whose conductivity decreases with time; while the high-energy electronic states are extended whose conduc- tivity increases with time. This is in qualitative agree- ment with the previous analysis of the inverse participa- tion ratio (IPR) of electron wavefunction [6, 8]. Then, the mobility edge E$_{c}$ can be obtained as the energy for σ$_{xx}$ ( t$_{1}$ ) = σ$_{xx}$ ( t$_{2}$ ) . In Figs. 3(a), (b) and (c), we see that the mobility edge E$_{c}$ increases with the increasing vacancy concentration, as the increasing vacancy concen- tration expands the energy range of localization when x > x$_{c}$ . Last, we examine the electron mean free path l , an important length scale characterizing the degree of elec- tron scattering by disorder. In Fig. 4(a), the en- ergy dependence of l is shown, which is obtained by l ( E ) = D$_{max}$ ( E ) /v ( E ) [19, 20]. v ( E ) is the average wave-packet velocity at energy E , extracted from the dif- fusion coefficient in the ballistic regime at short times as D ( E, t ) ≈ v ( E ) $^{2}$t . It is interesting to observe a minimal mean free path at E$_{F}$ . This distinguishes vacancies from other disorders in graphene that give rise to a maximal mean free path at E$_{F}$ [17, 18]. The minimal mean free path means that the lifetime of electrons is the shortest near E$_{F}$ in the localization regime, violating graphene’s ordinary Fermi liquid behavior [21]. Such an unusual be- havior has also been experimentally observed in graphene dosed with a high concentration of H atoms [13]. In Fig. 3(d), we plot E$_{c}$ as a function of x , in a Log- Log scale, which clearly shows a power-law dependence of E$_{c}$ on x . Fitting the simulation data, we obtain the scaling relation of E$_{c}$ ∝ ( x − x$_{c}$ ) $^{β}$, with the scaling expo- nent β ≈ 0 . 81 . Importantly, a non-zero minimum con- ductivity σ$_{min}$ appears at the mobility edge. Figure 3(e) shows that σ$_{min}$ first decreases and then increases with Near E$_{F}$ , the mean free path l is much less than the average distance between vacancies ( d ). This is because the quasi -localized states substantially reduce l in the low-energy region. For example, when x = 0 . 1% , l is found to be 1 . 6 nm, while d is in the order of 5 nm. In