Vacancy-Induced Magnetism in Fluorographene: The Effect of Midgap State

Based on density functional theory, we have systematically investigated the geometric, magnetic, and electronic properties of fluorographene with three types of vacancy defects. With uneven sublattice, the partial defect structures are significantly spin-polarized and present midgap electronic states. The magnetic moment is mainly contributed by the adjacent C atoms of vacancy defects. Furthermore, the strain dependence of the bandgap is analyzed and shows a linear trend with applied strain. This defect-induced tunable narrow bandgap material has great potential in electronic devices and spintronics applications.


Introduction
Graphene has received widespread attention for its unique chemical or physical characteristics and exhibiting great potentials in optics, spintronics, and optoelectronics since its discovery [1][2][3][4][5][6][7]. However, the gapless band structure, which fails to switch current "on" and "off", obstructs its applications in electronics such as field-effect transistors (FET). In order to overcome this obstruction, to date, diverse strategies have been proposed to open the zero bandgap and one of the effective schemes is chemical functionalization [8][9][10][11][12][13]. Functional graphene, including graphene oxide and halogenated graphene, has been proved to possess extraordinary properties as well as an expected tunable bandgap [14][15][16][17][18][19]. For example, by halogen (F, Cl, Br, I) doping, halogenated graphene is suggested to be capable to regulate the bandgap in a wide range and also could enhance the reaction kinetics of the Li-S cathode, leading to a high-performance lithium battery [20,21]. As the F atom possesses a higher electronegativity than other halogen atoms, fluorinated graphene (CF x ), a typical type of halogenated graphene, has been verified to be more stable than other types, and its properties are strongly dependent on the degree of fluorination [22][23][24]. By modulating F/C ratios, graphene, a nonmagnetic semimetal, can be transformed into a nonmagnetic/magnetic semiconductor/insulator [25]. With an F atom attached to each C atom, fluorographene (fully fluorinated graphene) was reported to be a high-quality insulator (resistance >10 GΩ at room-temperature) with a wide optical bandgap (3.8 eV), large negative magnetic resistance (a factor of 40 in 9T field), and remarkable mechanical strength, showing great potential in the electronic applications [26,27].
During the past two decades, many efforts have been made to fabricate fluorographene [27][28][29][30]. In 2010, Cheng, et al. reported the synthesis of graphene fluoride by reacting graphite and fluorine gas. They demonstrated that the band structure and conductivity of CF x were reversible by fluorination or reduction reactions [27]. After that, in 2011, Jeon and his collaborators produced fluorographene with the treatment of graphene with xenon difluoride (XeF 2 ) and pave the way to develop graphene-based semiconductors by the direct chemical fluorination method [28]. Recently, the successful thermal exfoliation of fluorinated graphene at room temperature indicated the feasibility of producing large-scale fluorographene [30]. However, considering the high temperature involved in the fluorination processes, structural defects, especially vacancy defects may appear and deteriorate the performance of structures such as magnetic momentum and conductivity, which has an important impact on the magnetic and electronic applications [27,31]. Many calculations also show that defects will have a great influence on electromagnetic properties of two-dimensional materials [32,33]. Therefore, it is of great meaning to explore the theoretical mechanism of vacancy defects and deeply understand the defects' influence on structural performance. In this article, we studied fluorographene with three types of vacancy defects, including single F atom vacancy (V F -fluorographene), single C-F vacancy (V sCF -fluorographene), and double CF vacancy (V dCF -fluorographene) via first-principle theory. The ab initio molecular dynamics (MD) simulations were performed to estimate the thermodynamical stability. The spin-charge density was analyzed to figure out how magnetic moment induced by vacancy defects. Furthermore, band structures, as well as the density of states, are also investigated. Although fluorographene is an insulator with a large bandgap, it can be transformed into a semiconductor by introducing appropriate vacancy defects and the bandgap can be tuned by the external strains.

Results and Discussion
To avoid the interactions between defects, 5 × 3 supercell of fluorographene is established. We chose three typical vacancy structures (single F, single C-F, and double C-F vacancy) by referring to the vacancy defect structures in other functional graphene [14,15]. Figure 1 displays the structures of V F -, V CF -and V dCF -fluorographene, respectively.
Molecules 2021, 26, x FOR PEER REVIEW 2 of xenon difluoride (XeF2) and pave the way to develop graphene-based semiconductors b the direct chemical fluorination method [28]. Recently, the successful thermal exfoliatio of fluorinated graphene at room temperature indicated the feasibility of producing large scale fluorographene [30]. However, considering the high temperature involved in th fluorination processes, structural defects, especially vacancy defects may appear and de teriorate the performance of structures such as magnetic momentum and conductivity which has an important impact on the magnetic and electronic applications [27,31]. Man calculations also show that defects will have a great influence on electromagnetic proper ties of two-dimensional materials [32,33]. Therefore, it is of great meaning to explore th theoretical mechanism of vacancy defects and deeply understand the defects' influenc on structural performance. In this article, we studied fluorographene with three types o vacancy defects, including single F atom vacancy (VF-fluorographene), single C-F vacanc (VsCF-fluorographene), and double CF vacancy (VdCF-fluorographene) via first-principl theory. The ab initio molecular dynamics (MD) simulations were performed to estimat the thermodynamical stability. The spin-charge density was analyzed to figure out how magnetic moment induced by vacancy defects. Furthermore, band structures, as well a the density of states, are also investigated. Although fluorographene is an insulator wit a large bandgap, it can be transformed into a semiconductor by introducing appropriat vacancy defects and the bandgap can be tuned by the external strains.

Results and Discussion
To avoid the interactions between defects, 5 × 3 supercell of fluorographene is estab lished. We chose three typical vacancy structures (single F, single C-F, and double Cvacancy) by referring to the vacancy defect structures in other functional graphene [14,15 Figure 1 displays the structures of VF-, VCF-and VdCF-fluorographene, respectively.  The C-F bond length (1.384 Å) and angle between adjacent C-C bonds (110.8 • ) are in good agreement with the previous calculation (1.37 Å/111 • ) [34]. With an F absence, the C 0 atom in V F -fluorographene connects to three nearest neighbor C 1 atoms, and there is a slight distortion in the lattice, especially in the red rectangular where the lattice is greatly perturbed by the vacancy defect. The buckling height of the C 0 atom decreases remarkably due to the enhancement of the C 0 -C 1 bonding strength. Similarly, for the V sCF -and V dCF -fluorographene, the bond strength of C 1 -C enhanced since C 1 atoms move close to their adjacent C atoms. The vacancy defects break the original symmetry of fluorographene. V F -and V sCF -fluorographene fluorographene shows mirror symmetry, while V dCF -fluorographene shows central symmetry.
The structural data of fluorographene and fluorographene with vacancy defects are summarized in Table 1. The decrease of the average bond length d C 1 −C in all these structures confirms the enhancement of C 1 -C bonding strength. With the F defect, the average C 1 -F bond length d C 1 −F in V F -fluorographene show an unexpected increase, which is quite different from V sCF -and V dCF -fluorographene. This shows that compared to fluorographene, the interaction between C 1 and F atoms of V F -fluorographene is weaker, while that of V sCF -and V dCF -fluorographene are stronger. The deviation of lattice angle θ in V dCF -fluorographene demonstrates that the deformation caused by the double C-F vacancy is stronger than others. Table 1. The structural data including lattice information, average bond length (d C 1 −C and d C 1 −F ), average angle between adjacent C 1 -C (ϕ C−C 1 −C ) and formation energy of fluorographene, V F -, V sCF -, and V dCF -fluorographene. C 1 is the nearest neighbor C atoms of vacancy defects.
where E C and E F are the energy of C and F atom obtained from diamond and F 2 , respectively. The m and n represent the numbers of the C and F atoms. The formation energies of the fluorographene with vacancy defects are shown in Table 1. Even though the energies of vacancy configurations are slightly larger than that of fluorographene, the small deviations (less than 40 meV/atom) suggest that V F -, V sCF -, and V dCF -fluorographene could be stabilized at nonequilibrium conditions. We performed the ab initio molecular dynamics (AIMD) simulation to verify the thermodynamic stability of fluorographene with vacancy defects at room temperature (300 K). The results are shown in Figure 2. As the variations in the total energies are within 0.15 eV/atom and the atomic structures maintain well during AIMD simulation for 10 ps, V F -, V sCF -, and V dCF -fluorographene are predicted to be thermodynamically stable at room temperature.
By absorbing the F atom, the depletion of the local π bond causes charge transfer in fluorographene. Since F atom possesses a higher electronegativity than C atom, electrons transfer from the C atom to its connected F atom, indicating that C-F is a polar covalent bond. The calculated charge of the C atom and F atom, obtained by the Hirshfeld-based method [9], are +0.48, −0.48, respectively. Sounding F atom vacancy, C 0 atom remains 4 by maintaining its unpair electron instead of reducing its electron passes to F atom, while the charge sharing of C 1 -F bonds adjacent to the vacancy has some little deviations. The charge sharing of the C-F bond is closely related to the third-order nonlinear optical response [35]. The Bader charge of nearest C-F bonds in V sCF -and V dCF -fluorographene also has been investigated and it is convinced that the charge transfer scheme will be affected by inducing vacancy. This result is in agreement with the change of bond strength shown in Table 1. By absorbing the F atom, the depletion of the local π bond causes charge transfer in fluorographene. Since F atom possesses a higher electronegativity than C atom, electrons transfer from the C atom to its connected F atom, indicating that C-F is a polar covalent bond. The calculated charge of the C atom and F atom, obtained by the Hirshfeld-based method [9], are +0.48, −0.48, respectively. Sounding F atom vacancy, C0 atom remains ~4 by maintaining its unpair electron instead of reducing its electron passes to F atom, while the charge sharing of C1-F bonds adjacent to the vacancy has some little deviations. The charge sharing of the C-F bond is closely related to the third-order nonlinear optical response [35]. The Bader charge of nearest C-F bonds in VsCF-and VdCF-fluorographene also has been investigated and it is convinced that the charge transfer scheme will be affected by inducing vacancy. This result is in agreement with the change of bond strength shown in Table 1.
To figure out whether vacancy can induce magnetic moment in fluorographene, we analyzed the spin density of defects structures and found that both VFand VsCF-fluorographene are magnetic and hold 1μB magnetic moment, whereas VdCF-fluorographene is nonmagnetic. This is consistent with the reports that only fluorinated graphene with uneven F atoms in the double sides is magnetic and can be explained by Lieb's theorem [36]. Figure 3 demonstrates the spin densities of VFand VsCF-fluorographene. It is obvious that the magnetic moment of VF-fluorographene mostly comes from the C0 atom (0.72 μB) nearest to F vacancy, while that of VsCF-fluorographene is mainly provided by C1 atoms (0.48 μB, 0.49 μB, and −0.29 μB) next to the C-F vacancy. The discrepancy can be explained by the asymmetry structure as a consequence of vacancy defects. Furthermore, the three F atoms adjacent to C1 atoms have nonnegligible contributions: 0.06 μB each F aligned ferromagnetically for VF-fluorographene, (0.09 μB, 0.09 μB, −0.06 μB) for VsCF-fluorographene. To figure out whether vacancy can induce magnetic moment in fluorographene, we analyzed the spin density of defects structures and found that both V F -and V sCFfluorographene are magnetic and hold 1µ B magnetic moment, whereas V dCF -fluorographene is nonmagnetic. This is consistent with the reports that only fluorinated graphene with uneven F atoms in the double sides is magnetic and can be explained by Lieb's theorem [36]. Figure 3 demonstrates the spin densities of V F -and V sCF -fluorographene. It is obvious that the magnetic moment of V F -fluorographene mostly comes from the C 0 atom (0.72 µ B ) nearest to F vacancy, while that of V sCF -fluorographene is mainly provided by C 1 atoms (0.48 µ B , 0.49 µ B , and −0.29 µ B ) next to the C-F vacancy. The discrepancy can be explained by the asymmetry structure as a consequence of vacancy defects. Furthermore, the three F atoms adjacent to C 1 Figure 4. It is revealed that fluorographene is an insulator with a band 3.09 eV, which is in good agreement with early reports (3.10 eV) [23], and VdCF-fluo phene has a 3.21 eV bandgap. Both fluorographene and VdCF-fluorographene have n splitting (see Figure S1 in Supplementary Materials). With uneven F atoms in the d sides induced by a single F vacancy defect, midgap states appear and VF-fluorogra  Figure 4. It is revealed that fluorographene is an insulator with a bandgap of 3.09 eV, which is in good agreement with early reports (3.10 eV) [23], and V dCF -fluorographene has a 3.21 eV bandgap. Both fluorographene and V dCF -fluorographene have no spin splitting (see Figure S1 in Supplementary Materials). With uneven F atoms in the double sides induced by a single F vacancy defect, midgap states appear and V F -fluorographene is a semiconductor with a direct bandgap of 1.48 eV. The bandgap is tuned by the arise of the spin splitting. The flatness of the midgap band means that electrons are strongly localized and the localizations mainly come from p z orbital of both C and F atoms, conforming to the characteristics of defects states. This defect level is caused by spin-down states only. Different from V F -fluorographene, in V sCF -fluorographene, the valence band maximum (VBM) and conduction band minimum (CBM) are mainly contributed by p x and p y orbitals of spin-up states. The bandgap decrease to 0.61 eV and V sCF -fluorographene is a semiconductor with a direct bandgap. For both V F and V sCF -fluorographene, because of the exchange splitting of the defect states, the p orbital of C and F atoms is hybridized and produces exchange split bonding and antibonding states which are the origin of the induced magnetic moment near vacancy.  Considering strain is an inevitable factor during fabrication, we further examined the strain dependence of bandgap and exchange-splitting of VsCF-fluorographene. The results are shown in Figure 5. By applying strain from −0.02 to 0.02 in zigzag direction, the bandgap of VsCF-fluorographene shows a linear increase from 0.51 eV to 0.78 eV. It should be noted that the position of the VBM changes infinitesimally, while the CBM changes greatly. The strain dependences of exchange-splitting are shown in Figure 5b. It is worth mentioning that the defect states related to ∆ , which is mostly contributed from py orbitals, has a significant increase with applied strain, while that of ∆ and ∆ change slightly. The changes of exchange-splitting eventually tune the bandgap of VsCF-fluorographene. In contrast to the bandgap, the magnetic moment of VsCF-fluorographene remain 1μB magnetic moment and show no obvious change with the uniaxial strain applied in the For the magnetic structures, the PDOS is shown; the black line represents the spin-up and the red line represents the spin-down; in PDOS, the black, red, blue, and green represent the s, p x , p y , p z orbital of C and F, respectively. The shadow represents the total density of states and the light blue dot represents the fermi-level. ∆ε = ε ↑ − ε ↓ , defined as the difference between spin-up and spin-down, is the exchange splitting energy.
Considering strain is an inevitable factor during fabrication, we further examined the strain dependence of bandgap and exchange-splitting of V sCF -fluorographene. The results are shown in Figure 5. By applying strain from −0.02 to 0.02 in zigzag direction, the bandgap of V sCF -fluorographene shows a linear increase from 0.51 eV to 0.78 eV. It should be noted that the position of the VBM changes infinitesimally, while the CBM changes greatly. The strain dependences of exchange-splitting are shown in Figure 5b.
It is worth mentioning that the defect states related to ∆ε 1 , which is mostly contributed from p y orbitals, has a significant increase with applied strain, while that of ∆ε 2 and ∆ε 3 change slightly. The changes of exchange-splitting eventually tune the bandgap of V sCFfluorographene. In contrast to the bandgap, the magnetic moment of V sCF -fluorographene remain 1µ B magnetic moment and show no obvious change with the uniaxial strain applied in the zigzag direction (see Table S1).

Method
Density functional theory (DFT) calculations are completed using the Vienna ab initio simulation package (VASP). The projector augmented wave (PAW) method and generalized gradient approximation (GGA) are performed to describe the core valence interaction and exchange-correlation [37][38][39]. A grid of 5 × 5 × 1 kpoints generated by Monkhorst-Packscheme method [40] is used for the defect fluorographene and the cutoff energy is set as 500 eV to verify the accuracy of energy convergence. All structures are relaxed to maximum atomic forces allowance of 1 × 10 −2 eV/Å and total threshold energy of 1 × 10 −7 eV. Spin polarization is considered in all of the calculations by setting ISPIN = 2, and the noncollinear version of VASP is used to complete the magnetic calculation. The thickness of the vacuum layer between the monolayers is set as 15 Å to avoid the spurious interlayer interaction in the out-of-plane direction.

Conclusions
In summary, we studied the magnetic and electronic properties of fluorographene with three types of vacancy defects by using first-principle calculations. Our results indicate that all the three structures: VF-, VsCF-, VdCF-fluorographene are stable at room temperature. Due to the uneven F atoms in the double sides caused by defects, VF-, VsCF-fluorographene has been proved to be magnetic and possesses 1μB magnetic moments. The magnetic moment is mainly contributed by the adjacent C atoms of vacancy defects. We also investigated the strain dependence of VdCF-fluorographene, and it is found that the bandgap, as well as exchange-splitting energy, can be tuned by applied strain, especially the position of the valence band. The study of fluorographene paves the way for fabricating and analyzing fluorographene-based devices.
Supplementary Materials: The following are available online, Figure S1: The band structures and DOS of fluorographene and VdCF-fluorographene, Table S1: Magnetic moment of VsCF-fluorographene with the uniaxial strain [23].

Method
Density functional theory (DFT) calculations are completed using the Vienna ab initio simulation package (VASP). The projector augmented wave (PAW) method and generalized gradient approximation (GGA) are performed to describe the core valence interaction and exchange-correlation [37][38][39]. A grid of 5 × 5 × 1 kpoints generated by Monkhorst-Packscheme method [40] is used for the defect fluorographene and the cutoff energy is set as 500 eV to verify the accuracy of energy convergence. All structures are relaxed to maximum atomic forces allowance of 1 × 10 −2 eV/Å and total threshold energy of 1 × 10 −7 eV. Spin polarization is considered in all of the calculations by setting ISPIN = 2, and the non-collinear version of VASP is used to complete the magnetic calculation. The thickness of the vacuum layer between the monolayers is set as 15 Å to avoid the spurious interlayer interaction in the out-of-plane direction.

Conclusions
In summary, we studied the magnetic and electronic properties of fluorographene with three types of vacancy defects by using first-principle calculations. Our results indicate that all the three structures: V F -, V sCF -, V dCF -fluorographene are stable at room temperature. Due to the uneven F atoms in the double sides caused by defects, V F -, V sCFfluorographene has been proved to be magnetic and possesses 1µ B magnetic moments. The magnetic moment is mainly contributed by the adjacent C atoms of vacancy defects. We also investigated the strain dependence of V dCF -fluorographene, and it is found that the bandgap, as well as exchange-splitting energy, can be tuned by applied strain, especially the position of the valence band. The study of fluorographene paves the way for fabricating and analyzing fluorographene-based devices.