A Defective Graphene Phase Predicted to be a Room Temperature Ferromagnetic Semiconductor

Theoretical calculations, based on hybrid exchange density functional theory, are used to show that in graphene a periodic array of defects generates a ferromagnetic ground state at room temperature for unexpectedly large defect separations. This is demonstrated for defects that consist of a carbon vacancy in which two of the dangling bonds are saturated with H atoms. The magnetic coupling mechanism is analysed and found to be due to an instability in the $\pi$ electron system with respect to a long-range spin polarisation characterised by alternation in the spin direction between adjacent carbon atoms. The disruption of the $\pi$-bonding opens a semiconducting gap at the Fermi edge. The size of the energy gap and the magnetic coupling strength are strong functions of the defect separation and can thus be controlled by varying the defect concentration. The position of the semiconducting energy gap and the electron effective mass are strongly spin-dependent and this is expected to result in a spin asymmetry in the transport properties of the system. A defective graphene sheet is therefore a very promising material with an in-built mechanism for tailoring the properties of the spintronic devices of the future.


I. INTRODUCTION
The miniaturisation of silicon devices has delivered exponentially growing performance for the past fifty years but quantum limits are now being reached as transistor gate lengths approach 5nm. Spintronics, which exploits the electron spin rather than the charge, will operate on a sub 1nm length scale and thus facilitate another generation of devices.
The achievement of many of the promises of spintronics depends on the ready availability of a material that is a semiconductor, ferromagnetic at room temperature, and with a highly tunable band gap and magnetic coupling. The controlled synthesis of a robust room temperature, ferromagnetic semiconductor on a nanoscale has yet to be achieved. Recent reports of high temperature ferromagnetism in metal-free carbon materials have generated a great deal of interest 3 . Proton-bombardment of graphite 4 gives rise to a ferromagnetic phase which x-ray circular dichroism measurements 5 have shown to originate from carbon sp electrons. Pyrolytic treatment of organic compounds 6 also gives rise to a ferromagnetic matetieal. In both cases, the magnetic phase appears to be a minority phase in a non-magnetic host. The detailed composition and structure of the minority phase have not yet been determined but the concentrations of magnetic impurities have been measured sufficiently accurately to suggest that the magnetism is not the result of contamination 7 .
The occurrence of high temperature ferromagnetism in purely sp-bonded materials is a major challenge to current theoretical understanding of magnetic interaction mechanisms. A chemical or structural defect in a carbonbased material leads to the creation of local magnetic moments due to the presence of under-or over-coordinated atoms. The presence of local moments needs to be accompanied by strong long-range coupling between them for the ferromagnetic order to survive thermal fluctuations at room temperature. In extended graphene sheets, local moment formation at point defects is now well established 8 . A significant step forward in the study of the magnetic coupling between local defects has been the demonstration that the presence of spin moments at the edges of graphene ribbons can lead to an instability of the π-electron system, providing a mechanism for long range antiferromagnetic coupling which is robust with respect to competing instabilities involving charge ordering and geometric distortions 9,10,11 . More recently, Yazyev and Helm 12 have studied defective graphene using density functional theory. They find a ferromagnetic ground state which they assume to be of the itinerant type due to the metallicity of their system. Vozmediano et al. 13 studied the interaction between localised moments originating from lattice defects at large distances and concluded that the transition temperature to the ferromagnetic RKKY state is far below room temperature. Dugaev et al. 15 predicted that if the spin-orbit interaction is taken into account, an energy gap opens at the Fermi level and the RKKY model is not applicable. A possibile source of high temperature ferromagnetism in carbon systems has been ascribed by Edwards et al. 16 to a limited screening of the interactions between itinerant electrons in impurity sp-bands. The expected sup-pression of the Stoner critical temperature (typical in the case of transition metal impurity d-bands) is considerably reduced due to a less effective screening of the electron interaction. In the present case we show a completely different scenario, i.e. that high temperature ferromagnetism in a carbon system can be entirely supported by an insulating state of the electronic structure.
The current work demonstrates that local spin moments in defective graphene interact strongly over long distances and may therefore lead to a ferromagnetic semiconducting state at room temperature. The strong, longrange magnetic coupling necessary for magnetic ordering at room temperature in this semiconducting material is provided via a spin alternation mechanism that depends on the presence of partially filled π orbitals in the sp 2 -bonded, bipartite lattice of graphene. Moreover, it is shown that the spin-dependent band gap and magnetic coupling strength can be varied through control of the defect concentration, opening up the possibility of highly tunable graphene-based spintronics nanodevices. This study, therefore, reinforces the case, which has been growing stronger and stronger since the ground-breaking practical realisation of isolated graphene sheets 2 , for considering graphene amongst the most promising materials for building the electronic and spintronic devices of the future.

II. COMPUTATIONAL DETAILS
Spin localisation at defects and long range spin coupling are studied here using all electron, twodimensionally periodic, hybrid exchange density functional theory (B3LYP 17,18,19 ) which significantly extends the reliability of the widely used local and gradient corrected approximations to density functional theory in strongly interacting systems 20,21,22 . In the CRYSTAL package, used for this study, the crystalline wavefunctions are expanded as a linear combination of atom centred Gaussian orbitals (LCAO) with s, p, d, or f symmetry. The calculations reported here are all-electron, i.e., with no shape approximation to the ionic potential or electron charge density. Basis sets of double valence quality (6-21G * for C and 6-31G * for H) are used. A reciprocal space sampling on a Monkhorst-Pack grid of shrinking factor equal to 6 is adopted after finding it to be sufficient to converge the total energy to within 10 −4 eV per unit cell.
In the present work we consider three types of structure, characterised by different arrangements of defects. The first structure contains defects arranged in pairs within a rectangular unit cell that is 25Å wide and of length 2L, which is convenient for studying inter-defect coupling. The system under study is therefore infinitely periodic in the 2 dimensions parallel to the graphene plane and non-periodic in the direction perpendicular to the plane. In Fig. 3 the unit cell containing a pair of defects is shown. The defects are therefore arranged along parallel chains running along the horizontal direction and the chains are 25Å apart. The second structure is an hexagonal cell used to quantify inter-chain effects. The third structure is a triangular superlattice used for representing the band structure in a form compatible with that of graphene (Fig. 4). All structures considered were initially relaxed using a force-field model (GULP 23 ) in which the carbon-carbon and carbon-hydrogen interactions are described via the Brenner bond order potential 23 . In the case of the calculation of the stabilisation energy ∆E (see below and Fig. 2), full quantum mechanical relaxation was carried out for the structure with L=7.5Å (see below) and resulted in a small increase in ∆E (see Fig. 2) of 25 meV. As the additional relaxation is confined to the region of the defect, the energy shift is expected to be depend weakly on the defect separation especially at large distances and thus full quantum mechanical relaxation was not required for larger structures. The magnetic stablisation energy reported in Fig.  2, and the transition temperature based on it, is therefore slightly underestimated. In the analysis of the electronic band structure shown in Fig. 4, where defects are displayed in a triangular lattice and are 20Å apart, we performed a full quantum-mechanical optimisation in order to quantify the magnitude and decay law of the semiconducting gap.

III. RESULTS AND DISCUSSION
A number of point defects are possible. Here we choose to study a simple carbon vacancy in which two of the three dangling carbon bonds are saturated with H-atoms. This defect has been proposed previously, studied extensively 8 , and is displayed in Fig. 1. Within a valence bond picture there are two unpaired electrons in the defect as the breaking of three π-bonds generates 1/3 of an unpaired electron on each of the C-atoms neighbouring the vacancy and the breaking of the σ-bond also generates an unpaired electron. Hund's rule coupling within the uncoordinated C-atom and the spin-alternation rule operating between the π-electrons ensures that these spins are aligned and thus the overall magnetic moment on the defect is 2µ B (µ B is the Bohr magneton). 34 In the current study this defect is used as an example of the many possible structures at which electron localisation might occur in a graphene sheet. The conclusions drawn below depend on the arrangement of defects and on their ability to generate a local magnetic moment through an instability of the electronic system, but are not expected to depend on the structural and chemical details of the defects.
The interaction of the localised moments is quantified by computing the quantity ∆E (stabilization energy) which is defined as the difference between the total electronic energy of the antialigned and aligned arrangements of the defects' spins. The decay of the stablisation energy with L is approximately ∆E = 4208L −1.43 meV and it is shown in Fig. 2. The calculated coupling energy can be related to the Curie temperature (T c ), at which the ferromagnetic order is lost due to thermal fluctuations, through an effective Heisenberg model with an interaction J = ∆E/2 per pair of spins. According to the Mermin-Wagner theorem 24 , a two dimensional isotropic Heisenberg ferromagnet is unstable with respect to thermal fluctuations at any finite temperature and therefore does not support long range order. However such effects are seen to reduce significantly when a small coupling along the third direction is introduced. Recent Quantum Monte Carlo (QMC) studies on quasi two dimensional Heisenberg ferromagnets 25,26 have shown that the introduction of an interlayer coupling 0.1% of that in-plane is enough to restore a finite value of T c . In particular, for a square lattice of spin-1 T c = JS(S + 1)0.44/K B (see Fig. 3 in Ref. 26 ) which, compared to a mean-field expression T c = JS(S + 1)z/3K B (z is the coordinaton number), differs by a factor of about 3. On the right-hand side of the graph in Fig. 2, the transition temperature T c for a square lattice arrangement of defects within the QMC formula above is shown for different interdefects distances. For this arrangement of the defects, the ferromagnetic state is lower in energy over the whole range of distances considered and is predicted to be stable at room temperature up to very large defect separations (Fig. 2).
The present system should be considered as a model system for the study of more complex and realistic cases such as defective graphite 4 or multilayer defective graphene or defective graphene on a substrate where a small but non zero exchange interaction in the third direction is usually found. The ferromagnetic long range order at room temperature is in agreement with recent experiments on bombarded graphite where a room temperature ferromagnetic ground state is found and the average inter-defect distance was determined to be about 1 nm 27 .
Effects due to interchain interaction expected at distances L close to the interchain fixed distance of 25Å are seen not to change the picture qualitatively. To confirm this, we independently considered a series of calculation of ∆E for a strictly 2-dimensional arrangement of defects (a hexagonal superlattice of defects) for which the qualitative nature of the long ranged interaction is unchanged and the power of the decay law is found to be 1.7.
The bipartite nature of the lattice and the presence of the partially filled π system of electrons in the sp 2 network of atoms are essential for establishing the strong, long-range coupling mechanism that gives magnetic ordering at high temperature. This mechanism is the result of local exchange repulsion, which ensures that the partially filled π orbital of each carbon atom is coupled in an anti-parallel fashion to that on its nearest neighbour. The operation of this mechanism to produce a long range spin polarisation within the sheet is evident in the spin density displayed in Fig. 3 and has been observed previously in π bonded carbon networks 28 . It is notable that this mechanism extends the validity of Lieb's theorem beyond a nearest neighbour model 29 . The predicted ferromagnetic nature of the overall interaction is also entirely consistent with previous studies of interactions between extended defects in graphene ribbons 9,10 . It is clear from the spin alternation mechanism that the ferromagnetic coupling between the localised spin moments depends on the defects being arranged on the same sublattice of the graphene sheet. A regular arrangement on opposite sublattices would lead to antiferromagnetic coupling, as seen by Yazyev et al. at short separation 12 , and a random arrangement of defects would be expected to lead to a ground state of smaller spontaneous magnetisation, while retaining the strong local coupling. This prediction agrees with the current experimental observation by Esquinazi et. al. 4,27 (i.e. high transition temperature and small total magnetisation), where the defect density is controlled, to an extent, via the defect implantation technique but the defects are most probably distributed randomly. The production of a graphene sheet and control of its defect density is not unrealistic using current  To analyse the semiconducting properties of the electronic structure of the system considered in the present study, the electronic band structure is show in Fig. 4 for the ferromagnetic state of a triangular array of defects at a separation of 20Å . This choice of array facilitates the comparison with that electronic structure of perfect graphene. The previously analysed structures (parallel chains and hexagonal arrangement of defects) show the same qualitative features as for the triangular array.
The presence of the defects and the magnetic ordering break the symmetry of the graphene π system, opening band gaps of 0.51 and 0.55 eV in the majority and minority spin band structures, respectively. The energy gap of the majority states is shifted upwards by about 0.20 eV with respect to the energy gap of the minority states, and the shape of the bands around the Fermi energy is markedly spin dependent. A strong spin asymmetry in the conductivity of the sheet is therefore to be expected. The size of the band gap is reduced to 0.1 eV when adopting the PBE generalised gradient approximation 33 .
Calculations at a variety of defect separations establish that the band gap scales as L −2 . This decay law is consistent with that recently calculated for graphene ribbons where the edges are line defects at which the spin moments localise and the band gap scales as L −1 with the ribbon width.
Recently Yazyev et al. 12 reported GGA calculations that produced ferromagnetism in graphene induced by the presence of defects but the magnetic coupling mechanism proposed differs strongly from that described in the current work. In Ref. 12 a metallic state is obtained and the ferromagnetic order is therefore assumed to be supported by itinerant electrons. In the present study the ground state is semiconducting and the magnetic cou- pling is clearly not due to itinerant electrons. For this reason we adopt an effective Heisenberg model in order to assess the temperature related properties of the system rather than the Stoner model of Ref. 12 . The explanation for this different behaviour does not reside in the morphology or in the concentration of defects nor in the treatment of electronic exchange and correlation (the band gap is still present when using the gradient corrected functional PBE as adopted in Ref. 12 ) but is due to the arrangement of defects. In Ref. 30 , the authors study the same H-chemisorption defect as in Ref. 12 , but consider a different arrangement of defects (specifically, the unit cell in Ref. 30 has zig-zag borders whereas armchair borders are used in Ref. 12 ). This results in a semiconducting ground state as found here. In Ref. 30 the magnetic coupling between defects was not analysed and this is the aim of the present study.
The opening of a gap obtained in this study is an effect of much larger magnitude than the opening due to spinorbit coupling described in Ref. 15 . The effect of the spinorbit coupling is not included in our study since it is expected to be negligible in carbon systems 32 .

IV. CONCLUSIONS
In conclusion the ground state of a two dimensional graphene sheet containing a periodic array of point defects has been shown to be both ferromagnetic at room temperature and semiconducting for defect separations up to 20Å and thus for defect concentrations as low as 10 13 /cm 2 . The energy gap and magnetic coupling de-pend strongly on defect concentration. We conclude that a doped or defective graphene sheet is a very promising material with an in built mechanism for tailoring properties for a variety of spintronics applications.