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# Criticality of Vacancy-Induced Metal-Insulator Transition in Graphene 

Shangduan Wu 1 and Feng Liu 2, ∗ 

$^{1}$Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA 

$^{2}$Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA 

(Dated: November 9, 2018) 

The criticality of vacancy-induced metal-insulator transition (MIT) in graphene is investigated by Kubo-Greenwood formula with tight-binding recursion method. The critical vacancy concentration for the MIT is determined to be x c ≈ 0 . 053% . The scaling laws for transport properties near the critical point are examined showing several unconventional 2D localization behaviors. Our theoretical results have shed some new lights to the understanding of recent experiments in H-dosed graphene [Phys. Rev. Lett. 103 , 056404 (2009)] and of 2D disordered systems in general. 

Graphene, a single atomic layer of graphite, has drawn much interest in the last few years [1]. Because of the linear dispersion relation near the Dirac point, many un- conventional transport behaviors have been shown in the pristine 2D graphene [2–4]. On the other hand, defects and impurities are often present in graphene, so their effects on graphene transport properties are of signifi- cant scientific interests and practical implications. Va- cancy, one of the natural defects present in graphene, as observed in experiment [5], has been a subject of intense theoretical study [6–12]. Vacancy is predicted to induce quasi -localized states in the vicinity of Fermi energy [6, 8, 10], leading to a metal-insulator transi- tion (MIT). However, some fundamental properties, es- pecially the criticality of the vacancy-induced MIT in graphene remains unclear. One important question is what is the critical vacancy concentration for MIT. Fur- thermore, what are the scaling laws characterizing the transport properties close to the critical point? The an- swers to these questions will help us not only to explain recent experiment results in graphene [13] but also to better understand, in general, localization and transport behaviors in 2D disordered systems. 

In this Letter, we report a theoretical study of crit- icality of the vacancy-induced MIT in graphene. Most importantly, we determine the critical vacancy concen- tration for MIT to be x$_{c}$ ∼ 0 . 053% , which may be com- pared to the experimentally observed critical H coverage of ∼ 0 . 08% for the H-dosing-induced MIT in graphene [13]. We found that electrical conductivity σ$_{xx}$ scales with the diffusion length L in a power-law, in contrast to the logarithmic scaling predicted for a conventional 2D localization behavior [14, 15]. Above x$_{c}$ , there exists a mobility edge E$_{c}$ defining a transition point of energy from the localized states to the extended states, and the transition is discontinuous as characterized by a non-zero minimum conductivity σ$_{min}$ at E$_{c}$ . σ$_{min}$ is found to vary from 1 to 1 . 3 e $^{2}$/h , different from the universal value of 0 . 628 e $^{2}$/h expected in a conventional 2D localization sys- tem [14]. Our results are consistent with the most salient 

PACS numbers: 73.63.-b, 72.15.Rn, 71.55.-i, 81.05.ue 

features of the H-dosing-induced MIT in graphene ob- served recently [13], suggesting that both vacancy and H belong to short-range disorder in graphene. 

The graphene is described by the π -band, nearest- neighbor tight-binding Hamiltonian as 

where i and j are the neighboring sites on the lattice, α = ( ↑ , ↓ ) is spin index. ε$_{i}$ is the on-site energy set to zero, and γ$_{ij}$ = γ = 2 . 7 eV is the hopping energy. 

To calculate the conductivity, the real-space recursion method [16] in Kubo-Greenwood formula is adopted, which has been proved to be a powerful tool in treat- ing various disordered graphene systems [17, 18]. The DC conductivity σ$_{xx}$ at zero temperature can be calcu- lated as σ$_{xx}$ ( E ) = ( e $^{2}$/S ) ρ ( E ) lim$_{t}$$_{→}$$_{+}$$_{∞}$ D ( E, t ) . Here S is area per atom. ρ ( E ) = T r [ δ ( E − ˆ H )] is average density of states (ADOS). D ( E, t ) is diffusion coefficient, which is defined as D ( E, t ) = 1 t Tr { [ˆ x, ˆ U ( t )] $^{+}$δ ( E − ˆ H )[ˆ x, ˆ U ( t )] } Tr [ δ ( E − ˆ H )] , where ˆ x is the x -component of the position operator, ˆ U ( t ) = exp( − i ˆ Ht ℏ ) is time evolution operator, and t is diffusion time. Periodic boundary conditions are used with a supercell sizes L$_{x}$ ≈ 738 nm and L$_{y}$ ≈ 213 nm. Vacancies are introduced simply by removing the atoms from the lattice [6, 8]. 

Figure 1 shows the time dependence of diffusion coef- ficient D at Fermi energy ( E$_{F}$ = 0 . 0 eV) for different vacancy concentration ( x ) in the range of x ≥ 0 . 1% . The inset shows the behavior of D at large time scale. Three main features are observed: (1) D first increases linearly with time and reaches its maximum D$_{max}$ at a very short time, less than 5 fs. D$_{max}$ is much smaller than D = v 2 $_{F}$t for perfect graphene, where Fermi velocity v$_{F}$ = 3 ta/ 2 ℏ ≈ c/ 300 . This means that the transport leaves the ballistic regime ( d D /dt > 0 ) quickly and en- ters the diffusive regime ( d D /dt = 0 ) due to vacancies. 

However, the diffusive regime is very short, without a 

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