Intertwined lattice deformation and magnetism in monovacancy graphene

Using density functional calculations we have investigated the local spin moment formation and lattice deformation in graphene when an isolated vacancy is created. We predict two competing equilibrium structures: a ground-state planar configuration with a saturated local moment of 1.5 ${\mu }_B$ and a metastable nonplanar configuration with a vanishing magnetic moment, at a modest energy expense of 50 meV. Though nonplanarity relieves the lattice of vacancy-induced strain, the planar state is energetically favored due to maximally localized defect states $(v\sigma$, $v\pi )$. In the planar configuration, charge transfer from itinerant (Dirac) states weakens the spin polarization of $v\pi$ yielding a fractional moment, which is aligned parallel to the unpaired $v\sigma$ electron through Hund's coupling. As a by-product, the Dirac states $\left(d\pi \right)$ of the two sublattices undergo a minor spin polarization and couple antiferromagnetically. In the nonplanar configuration, the absence of orthogonal symmetry allows interaction between $v\sigma$ and local $d\pi$ states, to form a hybridized $v{\sigma }^{\prime }$ state. The nonorthogonality also destabilizes the Hund's coupling, and an antiparallel alignment between $v\sigma$ and $v\pi$ lowers the energy. The gradual spin reversal of $v\pi$ with increasing nonplanarity opens up the possibility of an intermediate structure with a balanced $v\pi$ spin population. If such a structure is realized under external perturbations, diluted vacancy concentration may lead to $v\sigma$-based spin-$\frac{1}{2}$ paramagnetism. Carrier doping, electron or hole, does not alter the structural stability. However, the doping proportionately changes the occupancy of $v\pi$ state and hence the net magnetic moment.

Figure 1

Equilibrium lattice structures for (a) the ground state (planar) and (b) the metastable state (nonplanar). The displacements of the atoms along the $z$ direction are indicated in the figure. The side view of the nonplanar structure showing local rippling induced by the vacancy is shown in panel (c). Here, out-of-plane displacements are exaggerated by a factor of 5 for clarity.

Figure 2

Red (blue) represents spin-up (spin-down) channel here as well as in other relevant figures. (a) Band structure near the Fermi level showing the dangling $v\sigma$, the quasilocalized $v\pi$, and the itinerant Dirac ($d\pi$) states of the ground-state planar configuration. The shaded portions indicate the amount of charge transfer from the $d\pi$ to $v\pi$ states. (b) The spin-polarized density of states. (c) The itinerant spin density showing the antiferromagnetic coupling between the two sublattices. (d) The $\pi$ spin density ($v\pi +d\pi$) and the total spin density ($v\pi +d\pi +v\sigma$) are shown in panels (d) and (e), respectively. The isovalues for the spin density plots have not been mentioned since different, but appropriate energy ranges have been chosen to isolate charge densities of different states.

Figure 3

(a) The band structure near the Fermi level of the metastable nonplanar configuration. Hybridization between $v\sigma$ and $d\pi$ leads to band splitting around $\mathrm{\Gamma }$. The dotted lines at the band splitting are artificially constructed to act as a guide for the eyes to distinguish the hybridized $v{\sigma }^{\prime }$ and $d{\pi }^{\prime }$ states. (b) The spin-polarized density of states showing the antiparallel coupling between $v\pi$ and $v{\sigma }^{\prime }$. The $v{\sigma }^{\prime }$ spin density, $v\pi$ spin density, and total spin density ($v{\sigma }^{\prime }+v\pi +d{\pi }^{\prime }$) are shown in panels (c), (d), and (e), respectively. The isovalues for the spin density plots have not been mentioned since different, but appropriate energy ranges have been chosen to isolate charge densities of different states.

Figure 4

The relative total energy plotted as a function of the magnetic moment for the planar and nonplanar equilibrium configurations. The results were obtained using fixed spin moment calculations.

Figure 5

(a) Density of states plots showing the evolution of the electronic structure as the system moves from the ground-state planar lattice to the metastable-state nonplanar lattice. The results are obtained by carrying out a virtual experiment where the apical carbon atom (C1) is gradually displaced in the out-of-plane ($z$) direction from the planar case of 0 Å to a maximum of 0.6 Å. The magnitude of displacement ($\mathrm{\Delta }z$) is indicated in the respective figures. The gradual change in the spin alignment of $v{\sigma }^{\prime }$ and $v\pi$ moments from parallel to antiparallel is schematically shown on the right. (b) The net magnetic moment with respect to $\mathrm{\Delta }z$.

Figure 6

The top and bottom panels show the spin-polarized DOS for one-electron and one-hole doped systems, respectively. The left panels [panels (a) and (c)] represent the planar ground state, and the right panels [panels (b) and (d)] represent the nonplanar metastable state. In the case of electron doping, the $v\pi$ state is completely occupied to give 1 ${\mu }_B$. In the case of hole doping, the $v\pi$ state is partially occupied in one spin channel only. Other features of the electronic structure shown in Figs. 2 and 3 are largely retained.

Figure 7

(a) Total energy with respect to the fixed spin moment for the metastable state, with $U=3$ eV. The minimum energy state is achieved at 0.4 ${\mu }_B$. (b) The DOS corresponding to the minimum energy state is shown on the right.

Figure 8

The various energy contributions with respect to out-of-plane deformation using the model discussed in the text is shown in panel (a), with the total energy magnified in the inset for clarity. (b) Histogram of bond lengths in the planar and nonplanar equilibrium configurations.