3 FIG. 2: Energetics of the bands in graphane with sp 3 hybridized edges (solid lines) and the defect levels ( π and π ∗ ) induced by H-vacancies (dashed lines): (a) defect-free graphane; (b) graphane containing a single H-vacancy; (c) graphane containing two H-vacancies distributed between the two sides of the graphane plane ( AB -distribution) and separated by a distance of d = 3 a$_{C}$$_{−}$$_{C}$ (antiferromagnetic spin ordering); (d) graphane containing two H-vacancies located on one side of the graphane plane ( AA -distribution) and separated by the distance of d = 4 a$_{C}$$_{−}$$_{C}$ (ferromagnetic spin ordering). The spin density plotted with isovalues of ± 0 . 001 e/ ˚ A 3 is also presented. the AB -distribution the distance between the two defects should be > a$_{C}$$_{−}$$_{C}$ because for d = a$_{C}$$_{−}$$_{C}$ the spins of the two localized states are paired which leads to transfor- mation of a single bond in sp 3 hybridization between the nearest-neighbor carbon atoms to a double bond in sp 2 hybridization ( σ and π bonds) of length 1.36 ˚ A. Such a defect forms the π and π ∗ defect levels which are ener- getically close to the edges of the conduction and valence bands of graphane. Therefore, the size of the band gap defined by the defect levels is 6.38 eV, that is much larger than that for the levels formed by the non-bonded per- pendicular p$_{z}$ orbital (Δ=3.71 eV in Fig. (c)). A significant energy difference between the triplet and the singlet states was found only for d ≤ 2 a$_{C}$$_{−}$$_{C}$ in the AA -distribution for which the relaxation of the graphane lattice stabilizes the state with ferromagnetic spin order- ing, and for d < 2 a$_{C}$$_{−}$$_{C}$ in the AB -distribution. As a result, for the nearest location of the defects, the order- ing of spins is according to Lieb’s theorem [5]. When the defects are spatially separated ( d > 2 a$_{C}$$_{−}$$_{C}$ ) the energy difference between the state with a large spin number and the singlet state diminishes because of decoupling of the magnetic moments of the localized states and the random spin distribution with a minimum number of un- paired spins would be energetically favored. Therefore, for even number of defects the system would prefer to re- main in the singlet state independent of the distribution of defects over the sublattices, while for odd number of defects the triplet state is preferred. We have calculated the fluctuation of the gap size with increasing number of defects for the system in its singlet state when the size of the gap is defined by the energy difference between the induced defect levels, π and π ∗ , formed by the perpendicular p$_{z}$ orbitals. For the AB - distribution ( d > a$_{C}$$_{−}$$_{C}$ ), the size of the gap can fluctu- ate from 1.2 to 3.7 eV depending on the distance between the defects and their locations. The change in the gap size is related to the degree of the broken sublattice sym- TABLE I: Difference in the total energy between the triplet and singlet states for the non-optimized plane of graphane with two defects E 0 ( 2 $_{2}$; 1 $_{2}$) and after its full relaxation E$_{(}$ 2 $_{2}$; 1 $_{$_{2}$)}$. ( ε$_{1}$ − ε$_{2}$ ) is the energy splitting of the π orbitals for the relaxed lattice of graphane with two defects. distance E 0 ( 2 $_{2}$; 1 $_{2}$) (eV) E$_{(}$ 2 $_{2}$; 1 $_{2}$) (eV) ( ε$_{1}$ − ε$_{2}$ ) (eV) AA -distribution (ferromagnetic ordering) d = 2 a$_{C}$$_{−}$$_{C}$ − 1 . 52 − 1 . 23 2.82 × 10 − 1 d = 4 a$_{C}$$_{−}$$_{C}$ − 0 . 26 − 1 . 32 × 10 − 2 1.52 × 10 − 1 d = 6 a$_{C}$$_{−}$$_{C}$ 7.11 × 10 − 5 − 1 . 15 × 10 − 2 1.53 × 10 − 2 d = 8 a$_{C}$$_{−}$$_{C}$ 1.76 × 10 − 5 -7.93 × 10 − 3 6.52 × 10 − 3 AB -distribution (antiferromagnetic ordering) d = a$_{C}$$_{−}$$_{C}$ 1.34 2.98 - d = 3 a$_{C}$$_{−}$$_{C}$ 4.30 × 10 − 3 1.18 × 10 − 2 - d = 5 a$_{C}$$_{−}$$_{C}$ − 1 . 08 × 10 − 4 6.13 × 10 − 3 -