Spin-orbit coupling in fluorinated graphene

We report on theoretical investigations of the spin-orbit coupling effects in fluorinated graphene. First-principles density functional calculations are performed for the dense and dilute adatom coverage limits. The dense limit is represented by the single-side semifluorinated graphene, which is a metal with spin-orbit splittings of about 10 meV. To simulate the effects of a single adatom, we also calculate the electronic structure of a $10\times 10$ supercell, with one fluorine atom in the top position. Since this dilute limit is useful to study spin transport and spin relaxation, we also introduce a realistic effective hopping Hamiltonian, based on symmetry considerations, which describes the supercell bands around the Fermi level. We provide the Hamiltonian parameters which are best fits to the first-principles data. We demonstrate that, unlike for the case of hydrogen adatoms, fluorine's own spin-orbit coupling is the principal cause of the giant induced local spin-orbit coupling in graphene. The $s{p}^3$ hybridization induced transfer of spin-orbit coupling from graphene's $\sigma$ bonds, important for hydrogenated graphene, contributes much less. Furthermore, the magnitude of the induced spin-orbit coupling due to fluorine adatoms is about 1000 times more than that of pristine graphene, and 10 times more than that of hydrogenated graphene. Also unlike hydrogen, the fluorine adatom is not a narrow resonant scatterer at the Dirac point. The resonant peak in the density of states of fluorinated graphene in the dilute limit lies 260 meV below the Dirac point. The peak is rather broad, about 300 meV, making the fluorine adatom only a weakly resonant scatterer.

Figure 1

Unit cells of the dense (50%) and intermediate (2%) fluorination limits. The lattice vectors are shown as black arrows and the unit cell is emphasized by the dashed lines. The inset shows the geometrical structure around a fluorine adatom, with labels for the adatom (F), fluorinated carbon $\left({\mathrm{C}}_{\mathrm{F}}\right)$, the nearest $\left({\mathrm{C}}_{\mathrm{nn}}\right)$, next nearest $\left({\mathrm{C}}_{\mathrm{nnn}}\right)$, and the next-next nearest $\left({\mathrm{C}}_{\mathrm{nnnn}}\right)$ neighbors of the fluorinated carbon.

Figure 2

Calculated electronic properties of single-side semifluorinated graphene ${\mathrm{C}}_2\mathrm{F}$ as obtained from nonmagnetic ab initio calculations. Shown is the electronic band structure (left) and the orbital resolved atomic density of states (right) for fluorine F, fluorinated carbon ${\mathrm{C}}_{\mathrm{F}}$, and its nonfluorinated nearest neighbor ${\mathrm{C}}_{\mathrm{nn}}$. Three relevant bands around the Fermi level are labeled by (a), (b), and (c), respectively. The band (b), which crosses the Fermi level, is formed by the $p_z$ orbitals sitting on ${\mathrm{C}}_{\mathrm{nn}}$.

Figure 3

Calculated band splittings due to SOC for the single-side semifluorinated graphene ${\mathrm{C}}_2\mathrm{F}$. Three figures—(a), (b), and (c)—correspond to SOC splittings of three relevant bands (a), (b), and (c) that are shown in Fig. 2. Dashed lines correspond to the band splittings when turning off SOC on the fluorine adatom.

Figure 4

Calculated electronic properties of fluorinated graphene in the dilute limit represented by a $10\times 10$ supercell with a single fluorine adatom. Band structure (left) along high symmetry lines in first Brillouin zone and orbital resolved atomic density of states (right) are shown. The labels correspond to conduction (a), midgap (b), and valence bands (c). Contribution of different orbitals is indicated by the labeled lines.

Figure 5

Left panel: mutual positions and adopted notation for all relevant atomic sites whose $p_z$ orbitals enter the model tight-binding Hamiltonian including local SOC. Shown are fluorine $\mathrm{F}$, fluorinated carbon $\mathrm{A}={\mathrm{C}}_{\mathrm{F}}$, its three nearest ${\mathrm{B}}_1,{\mathrm{B}}_2,{\mathrm{B}}_3$, and the six next-nearest ${\mathrm{a}}_1,\cdots ,{\mathrm{a}}_6$ neighbors. The principal axis for $C_3$ rotations is defined by the perpendicular $\mathrm{F}-\mathrm{A}$ bond, while ${\sigma }_v$ reflections are given by three $\mathrm{A}-{\mathrm{B}}_i$ lines and the principal axis. Right panel: schematic representation of the dominant orbital and spin-orbital hoppings.

Figure 6

Left panel: electronic band structure of the dilute fluorinated graphene in $10\times 10$ supercell configuration; the first-principles data are presented by black dotted lines and the tight-binding model data by blue solid lines. The model calculation is based on the orbital Hamiltonian ${\mathcal{H}}_0+{\mathcal{H}}^{\prime }$ with the parameters $T=5.5$ eV and ${\varepsilon }_{\mathrm{f}}=-2.2$ eV that are chosen by fitting the conduction (a), midgap (b), and valence band (c) along the M-K-$\Gamma$-M line, respectively. Right panel: DOS per atom and spin corresponding to the tight-binding computed electronic band structure from the left panel.

Figure 7

Spin-orbit splittings along the M-K-$\Gamma$-M line for the conduction (a), midgap (b), and valence bands (c); refer also to Fig. 6. First-principles data (dotted) are excellently reproduced by the phenomenological tight-binding model (solid) given by the Hamiltonian ${\mathcal{H}}_0+{\mathcal{H}}^{\prime }+{\mathcal{H}}_{\mathrm{SO}}$ from the text. The following orbital and SOC parameters were used in the tight-binding model as the best fits: $T=5.5$ eV, ${\varepsilon }_{\mathrm{f}}=-2.2$ eV, ${\Lambda }_{\mathrm{I}}^{\mathrm{B}}=3.3$ meV, ${\Lambda }_{\mathrm{PIA}}^{\mathrm{B}}=7.3$ meV, and ${\Lambda }_{\mathrm{R}}=11.2$ meV. Turning off the intra-atomic SOC of the fluorine adatom in the first-principles calculations reduces the spin-orbit splitting significantly (dashed line). This residual spin-orbit splitting can be attributed to the structural local $s{p}^3$ distortion caused by fluorination.

Figure 8

Extrapolated band structure and DOS of pristine (dashed) and hydrogenated (solid) graphene for a $40\times 40$ supercell. The left panel shows tight-binding model-calculated band structures along M-K-$\Gamma$-M lines. The right panel shows the corresponding DOS. Tight-binding orbital parameters for hydrogenated graphene, $T_{\mathrm{h}}=7.5$ eV and ${\varepsilon }_{\mathrm{h}}=0.16$ eV, are taken from Ref. [13]. The band structure of hydrogenated graphene shows a narrow resonant peak in the vicinity of the Dirac point.

Figure 9

Extrapolated band structure and DOS for the pristine (dashed) and fluorinated (solid) graphene for a $40\times 40$ supercell configuration. The left panel shows model-calculated band structures along M-K-$\Gamma$-M and the right panel the corresponding DOS. Characteristic changes in DOS for the dilute fluorinated graphene become obvious below $-0.1$ eV. Tight-binding orbital parameters used in the model calculations, $T=5.5$ eV and ${\varepsilon }_{\mathrm{f}}=-2.2$ eV, are best fits to DFT results for $10\times 10$ supercell configuration.

Figure 10

Left panel: change $\Delta \nu \left(E\right)$ in DOS per atom and spin, see Eq. (7), computed for the single-impurity model employing orbital tight-binding parameters $T=5.5$ eV and ${\varepsilon }_{\mathrm{f}}=-2.2$ eV. The pronounced maximum appears at $E\simeq -0.26$ eV with $\mathrm{FWHM}\simeq 0.3$ eV. Right panel: perturbed DOS per atom and spin $\nu \left(E\right)={\nu }_0\left(E\right)+\eta \Delta \nu \left(E\right)$ as functions of energy for the fluorine impurity concentration $\eta =0.5\%$ (solid line). The dashed line shows the unperturbed DOS per atom and spin near the graphene neutrality point.

Figure 11

Calculated electronic band structures of fluorinated (left panel) and hydrogenated (right panel) graphene within $5\times 5$ supercell configurations. Three bands (a), (b), and (c) correspond to the conduction, midgap, and valence band, respectively, as in the main text. Formation of the midgap state around the Fermi level dominates both band structures. However, for hydrogenated graphene this band is hardly dispersive, giving the pronounced resonance at the Dirac point.

Figure 12

Calculated orbital resolved atomic DOS for fluorinated graphene in a $5\times 5$ supercell. The orbital resolved DOS is shown for the fluorine atom $\mathrm{F}$, fluorinated carbon ${\mathrm{C}}_{\mathrm{F}}$, and its nearest, next-nearest, and next-next-nearest neighbors, ${\mathrm{C}}_{\mathrm{nn}},{\mathrm{C}}_{\mathrm{nnn}}$, and ${\mathrm{C}}_{\mathrm{nnnn}}$, respectively.

Figure 13

Top view of the valence charge density plot of fluorinated graphene on a $5\times 5$ supercell. The charge density was obtained by summing the absolute squares of the Kohn-Sham states lying in the energy interval between $-0.2$ eV and $0.6$ eV. The dark (orange) surface is the isovalue of $0.002 {\AA }^{-3}$ and the light (yellow) one corresponds to $0.001 {\AA }^{-3}$. Dashed lines represent the cross-sectional view shown in the bottom right.

Figure 14

SOC splittings for the conduction (a), midgap (b), and valence (c) bands with respect to the Fermi level of the intermediately fluorinated $5\times 5$ system. Dashed lines are representing the SOC splittings without the intra-atomic SOC contributions from the F atom. For comparison the SOC splittings of the hydrogenated $5\times 5$ system are provided by the dashed-dotted lines.