Abstract
A novel crystalline structure of hybrid monolayer hexagonal boron nitride (BN) and graphene is predicted by means of the firstprinciples calculations. This material can be derived via boron or nitrogen atoms which are substituted by carbon atoms evenly in the graphitic BN with vacancies. The corresponding structure is constructed from a BN hexagonal ring linking an additional carbon atom. The unit cell is composed of seven atoms, three of which are boron atoms, three are nitrogen atoms, and one is a carbon atom. It shows a similar space structure as graphene, which is thus coined as gB_{3}N_{3}C. Two stable topological types associated with the carbon bond formation, i.e., CN or CB bonds, are identified. Interestingly, distinct ground states of each type, depending on CN or CB bonds, and electronic bandgap as well as magnetic properties within this material have been studied systematically. Our work demonstrates a practical and efficient access to electronic properties of twodimensional nanostructures, providing an approach to tackling open fundamental questions in bandgapengineered devices and spintronics.
Keywords:
gB_{3}N_{3}C; Graphene; FirstprinciplesBackground
Twodimensional (2D) nanomaterials, such as graphene and monolayer hexagonal boron nitride (hBN), are expected to play a key role in future nanotechnology as well as to provide potential applications in nextgeneration electronics. Recently, novel hybrid structures consisting of a patchwork of BN and C nanodomains (BNC) were synthesized through the use of a thermal catalytic chemical vapor deposition method [1]. This finding immediately has attracted a great deal of research interest [24], given that it demonstrates a hitherto efficient route to tune the bandgaps of these 2D materials.
It is well known that the perfect hexagonal and planar structure of BNC largely depends on the good matching between BN and C domains. However, it is indeed an outstanding challenge as BN and C phases are naturally immiscible in 2D [1]. This explains why Ci et al. [1] could observe some wrinkles in the atomic force microscopy image. Such mutual contradiction mainly originates from the domain boundary effect and the staggered potentials of B and N atoms in BNC, which doubtlessly affects their continuous tunable electronic energy gaps. It has been confirmed in the related theoretical calculations [59] where the bandgaps show a strong oscillation feature.
The Xray photoelectron spectroscopy of BNC measured in the work of Ci et al. [1] shows two additional types of Cbonding configurations, which correspond to the CB and CN bonds with the bonding energies of around 188.4 and 398.1 eV, respectively. This feature means that two inequivalent Cbonding types, i.e., CB or CN bonds, must be present in the boundaries of the hybridized BN and C domains, which have a significant effect on determining the local structures and subsequently vary the electronic properties in this system. For example, the transport channels show a robust characteristic gap when the topological index changes the sign of the valley Hall effect [3]. In addition, both Raman D band at 1,360 cm^{−1} and D’ band at 1,620 cm^{−1}are also observed in BNC [1], which were attributed to the lattice disorder or the finite crystal size. This lattice disorder effect might directly introduce vacancies to this 2D hexagonal system [1012], which is also true in BNC as shown in Figure two (a) of [1].
For the planar BNC structure, although the larger domains would be preferred to decrease the total domain interfacial energy, the randomly distributed hybrid domains and the immiscible phases, as well as the induced vacancies, must result in various complex structures of BNC [1316]. An immediate consequence is the largely inaccessible synthesis of expected BNC in experiments [17], which next hinders realization of bandgapengineered applications in actual devices. In this endeavour, exploring the structures and electronic properties associated with C bond formation in BNC which contains CN or CB bonds is an interesting topic that must be addressed before widespread synthetic applications. Thus, a simple model of BNC where the C bonds play a crucial role must be considered again. More importantly, a deep theoretical understanding, which originally was concealed behind the complex hybridized structures, is imperative.
Here, we report that such a simple model of BNC may be just the graphitic B_{3}N_{3}C (gB_{3}N_{3}C), which is a perfect 2D monolayer graphitelike structure as shown in Figure 1. As mentioned above, the C atom can bond to B and N atoms. Therefore, two topological types of gB_{3}N_{3}C are easily deduced. One is the αgB_{3}N_{3}C which is related to the the higher bonding energy of the CN bond, while the other one is the βgB_{3}N_{3}C which is constructed based on the lower bonding energy of the CB bond. It can be seen that such a material is essentially a Cdoped graphitic BN (gBN) with vacancies [11]. The substitution of the N/B atom with a C atom in gBN with the B/N vacancy will yield the α/βgB_{3}N_{3}C structure. The interactions among the C atoms and/or vacancies as well as the Cbonding types (CB and CN bonds) in gB_{3}N_{3}C significantly alter its electronic properties. To explore this effect, standard density functional theory with different functional (see the following discussion) calculations has been carried out for this predicted material. Remarkably, such material displays two distinct electronic structure properties: αgB_{3}N_{3}C is a semiconductor, while βgB_{3}N_{3}C behaves like a metal and leads to a magnetic ground state.
Figure 1. The total energy per atom as a function of lattice constantaforgB_{3}N_{3}C. Yellow, green, and gray balls represent B, N, and C atoms, respectively. Their respective 2×2 supercells are also given nearby.
This paper is arranged as follows: In the second section, we present the computational method used in this work, followed by the electronic bandgap in αgB_{3}N_{3}C and magnetism in βgB_{3}N_{3}C in the third section. We then conclude this paper in the fourth section.
Methods
For structural optimization, we employed density functional theory with the generalized gradient approximation (GGA) of PerdewBurkeErnzerhof (PBE) [18] for the exchangecorrelation (XC) potential within the projector augmented wave method as implemented in VASP [19,20]. An allelectron description, the projector augmented wave method, is used to describe the electronion interaction. The cutoff energy for plane waves is set to be 500 eV, and the vacuum space is at least 15 Å, which is large enough to avoid the interaction between periodical images. A 7×7×1 MonkhorstPack grid is used for the sampling of the Brillouin zone during geometry optimization. All the atoms in the unit cell were allowed to relax, and the convergence of force is set to 0.01 eV/Å. Additionally, spin polarization is turned on during the relaxation processes. All other calculations of accurate electronic properties were performed using the fullpotential linearized augmented planewave [21] method as implemented in the WIEN2k code [22]. It is well known that different XC potentials can lead, depending on the studied materials and properties, to results which are in very bad agreement with the experiment, e.g., for the bandgap of semiconductors and insulators which is severely underestimated or even absent [23]. For this reason, the modified BeckeJohnson (MBJ) [24,25] potential in the framework of localdensity approximation (LDA) [26,27] is taken to calculate the bandgap of αgB_{3}N_{3}C, while the magnetism in βgB_{3}N_{3}C is described by the PBE (the standard GGA for materials) potential.
Results and discussion
Electronic bandgap in αgB_{3}N_{3}C
Figure 1 plots the total energy per atom against the lattice constant a (the lattice constant c is fixed) for gB_{3}N_{3}C. It can be seen that the total energy of gB_{3}N_{3}C as a function of lattice constant a has a single minimum, meaning that the geometrical structure would be stable. Particularly, the charge population analysis reveals that the electron density around the CN bond in αgB_{3}N_{3}C is much higher than that around the CB bond in βgB_{3}N_{3}C, showing that the CN bond is relatively strong, which is also consistent with the experimental results [1]. This strong interaction between C and N atoms in αgB_{3}N_{3}C directly results in a short CN bond length (see the following paragraph) and can balance the strain of the monolayer graphitelike structure. Thereby, αgB_{3}N_{3}C could be a more thermodynamically stable topological phase against βgB_{3}N_{3}C, which has a lower total energy of around 0.43 eV/atom compared with βgB_{3}N_{3}C.
To explore further the mechanical stability of αgB_{3}N_{3}C, the optimized lattice constant a = 4.67Å is first obtained, as depicted in the left panel of Figure 1. Importantly, the CN bond length converged to 1.31 Å, which is considerably reduced from the typical bond lengths of 1.37 to 1.48 Å in the related materials [28]. This strongly suggests the nature of the higher binding energy of the CN bond [1], which also influences the BN bonding and extends its length. The obtained BN bond length is 1.48 Å, which is slightly bigger than the value of 1.45 Å in hBN. The calculated partial density of states (PDOS) is shown in Figure 2. The valence band is dominated by B p and C p states, while the conduction band is only dominated by B p states. There, one can find the majority of C p states to be semicore, lying 6 to 9 eV below the Fermi level. These states interact with those comprising the valence band with the same symmetry. As a result, there is a small admixture of C p and N p states close to the Fermi level. However, in αgB_{3}N_{3}C, one finds significant admixture of C p and N p states in the semicore energy window with 6 to 9 eV below the Fermi level. This suggests that the C pN p interaction in the semicore region contribution to the CN bonding is significantly more important in αgB_{3}N_{3}C than the C pN p interaction close to the Fermi level.
Figure 2. Total and partial DOS ofαgB_{3}N_{3}C for three XC potentials LDA, PBE, and MBJ. The vertical dotted line denotes the Fermi level and also indicates the end of the fundamental bandgap which starts at E − E_{F} = 0eV. The black, blue, and red lines correspond to the total, p, and s DOS, respectively.
Now, let us look at the band structures of αgB_{3}N_{3}C, as given in Figure 3. It explicitly demonstrates that all three XC potentials give similar band structures. The highest occupied crystalline orbitals are located at the K point of the reciprocal space, while the lowest unoccupied crystalline orbitals appear at the M point. This leads to an indirect bandgap semiconductor. To obtain the bandgap more accurately (see Figure 3c), the MBJ XC functional is used [24,25]. The bandgap of αgB_{3}N_{3}C within MBJ is obtained to be 1.22 eV, which nicely locates the middle region between 0.59 and 1.80 eV for the BNC samples with 12.5% and 50% C contents, respectively [4]. Note that the C content in αgB_{3}N_{3}C is 25%. In the experiments [1], the absorption edges are redshifted as the C concentrations increase, which shows a tunable mechanism of an optical bandgap in actual applications. By comparing with [1] where BNC with around 65% C concentration shows an optical bandgap of 1.62 eV, we infer that such a higher energy absorption edge (take into account the bandgap of 1.22 eV in our case with 25% C concentration) arises from the formation of individual BN and graphene domains. In this way, the even distribution of C in BNC systems might serve as a good guide to find alternative solutions to existing bandgapengineered applications. Future research can test this prediction directly.
Figure 3. Calculated band structures for αgB_{3}N_{3}C model with the three XC potentials.(a) LDA XC potential. (b) PBE XC potential. (c) MBJ XC potential.
In addition, the bandgaps based on the LDA and PBE methods are equal to 0.83 eV (see Figure 3a,b). The relative bandgap correction from MBJ with respect to LDA, with Δ_{MBJ/LDA} being the bandgap, is about 32%. We can see that our calculated for αgB_{3}N_{3}C ( shows an excellent agreement with the 16 sp semiconductors) lies within the range of 16.0% to 100.0% (see Table 1 for details). The correction value is particularly very close to that in BN (25%), GaN (42%), AlP (37%), and AlN (25%). This finding is not surprising because the listed four solids have at least one element close to that of αgB_{3}N_{3}C in the periodic table of elements. It means that the similar chemical circumstances in these materials can be well described by the same XC functionals. This underlyingly confirms our prediction validity and that the αgB_{3}N_{3}C might be carried out experimentally.
Table 1. The bandgaps and the relative bandgap correction of 16 sp semiconductors and the predicted αgB_{3}N_{3}C
From Figure 2, which shows the DOS of αgB_{3}N_{3}C, we can see that the effect of the MBJ potential is to shift up (with respect to LDA/PBE) the unoccupied B 2s and 2p states. Here, three major differences between the LDA/PBE and MBJ methods can be extracted: (a) Another obvious effect of MBJ potentials is to shift down the middle of the valence band at around −4.0 eV. (b) The hybridization of s and p states of dominant B 2s and other atoms’ 2p states at the bottom of the valence band is very strong in MBJ calculations (denoted as down arrows in Figure 2). (c) The correction character of αgB_{3}N_{3}C is much more pronounced with the MBJ than with the LDA/PBE, which narrows the valence band just below the Fermi level. We would like to stress that the MBJ potentials open a bandgap of 0.39 eV in the αgB_{3}N_{3}C model compared to the result of LDA, which is consistent with the orbitaldependent potential principle [24].
The optical absorption spectrum for αgB_{3}N_{3}C within MBJ potentials (see Figure 4c) shows a big absorption packet with two adjacent peaks in the range of 2.5 to 5.0 eV, which originates from the band structure as shown in Figure 3c. As an indirect bandgap material, the transition from the K point to Γpoint is very weak as the momentum conservation rule is not satisfied here, and thus, the corresponding characteristic photoluminescence in the optical absorption spectrum can be negligible. This is also true in our case for αgB_{3}N_{3}C. The first peak corresponds to the direct bandgap transition at the M point. The second peak comes from the larger direct gap from higher energy states located at the Γ point. More importantly, these two peaks are not separated distinctly. This feature can be attributed to the even distribution of C atoms in αgB_{3}N_{3}C as mentioned above, which shows a different formation mechanism compared with the hybrid BNC in the experiment [1].
Figure 4. The optical absorption expressed by the imaginary part of the dielectric tensorε_{2}forαgB_{3}N_{3}C. The imaginary part of the dielectric tensor is averaged for four different directions. Three XC potentials including (a) LDA, (b) PBE, and (c) MBJ are shown in the calculations.
Correction effects were taken into account by adding a LDA correction potential in MBJ [25]. This important physical effect opens an additional bandgap by mimicking very well the behavior of orbitaldependent potentials and causes a rigid blueshift of the absorption spectrum compared with the LDA/PBE curves as shown in Figure 4. This explains the excellent qualitative agreement of the hybrid exchangepotential optical absorption spectrum seen in Figure 4, due to a compensation of significant errors within the standard DFT methods.
Magnetism in βgB_{3}N_{3}C
The predicted structure of βgB_{3}N_{3}C is shown in the right panel of Figure 1. The lattice constant of βgB_{3}N_{3}C is obtained to be 5.26 Å as shown in Figure 1. The results show the equilibrium value d_{BC} = 1.52 Å, which is close to the value of graphitelike BC_{3}, which is 1.55 Å [29]. It is to be noticed that all the equilibrium values, d_{BN}, in αgB_{3}N_{3}C are equal to 1.42 Å, which is slightly less than the value of 1.45 Å in the pristine BN sheet. This implies the stronger BN bonds formed in βgB_{3}N_{3}C. Our calculations show that the βgB_{3}N_{3}C leads to a ground state with a magnetic moment of 0.68 μ_{B}. The nonmagnetic state is 0.07 eV higher than this ground state.
From the band structures, we see that although both the pristine BN sheet and graphene are nonmagnetic, the βgB_{3}N_{3}C model can be spinpolarized. It is necessary to discuss magnetism in more detail from its electronic structures. Figure 5a,b,c presents the band structure and spinresolved total density of state (TDOS). Remarkably, two bands cross the Fermi level (black and red circles in Figure 5a) and make the Fermi energy level occupied completely along the entire highsymmetry lines. In contrast, the spindown one does not possess such a strongly localized feature, just one band (black circles) crosses the Fermi level monolithically as shown in Figure 5c. The calculated magnetic moment in βgB_{3}N_{3}C should originate from this asymmetric spindependent localization. The corresponding strong spin splitting can be further confirmed from TDOS as shown in Figure 5b. In addition, close examination of the top valence bands (see Figure 6a) indicates that the strong localization mainly comes from the 2p atomic orbitals of C and N atoms. Here, we also show the Fermi surfaces of βgB_{3}N_{3}C in the first Brillouin zone. The unique feature of the Fermi surface which is almost parallel to the highsymmetry lines (see Figure 5d,e,f) is the direct manifestation of the bands near the Fermi level in Figure 5a,c.
Figure 5. Band structures, spinresolved TDOS, and Fermi surfaces forβgB_{3}N_{3}C.(a, b, c)Band structures and spinresolved TDOS for βgB_{3}N_{3}C. The dotted line indicates the Fermi level. The arrow denotes the spin polarization direction: up for spin up and down for spin down. (d, e, f) Fermi surfaces drawn in the first Brillouin zone and the corresponding highsymmetry points; (d) and (e) for spin up and (f) for spin down.
Figure 6. PDOS and spin density forβgB_{3}N_{3}C.(a) PDOS on the interstitial region and on the 2s and 2p orbitals of C and N atoms in βgB_{3}N_{3}C. (b) The 3D isosurface plot of spin density for the 2×2 supercell at the value of 0.04 e/Å^{3}. (c) A side view of spin density corresponding to (b).
It should be noticed that such magnetism is induced without transition metals and without external perturbations so that βgB_{3}N_{3}C behaves as the first, theoretically predicted, metalfree magnetic material in the hybrid BNC system. Apparently, our finding points out a new direction for further related experimental investigations in spintronics.
We now address the possibility of the origin of magnetism in βgB_{3}N_{3}C. Based on the orbitalresolved density of state as shown in Figure 6a, the magnetic moment is mainly ascribed to the 2p orbitals of C and three N atoms. The spin polarization of C atom offers a magnetic moment of 0.18 μ_{B}. A total magnetic moment of 0.18 μ_{B} is shared equally by the 2p orbitals of three N atoms. The remaining magnetic moments distribute evenly in the interstitial region among N atoms. This is conceivable because βgB_{3}N_{3}C, if compared with other BNC systems, has large interspaces between N atoms (see the 2×2 supercell in Figure 1 or Figure 6b). The remarkable feature is that the equilibrium surface density of βgB_{3}N_{3}C is around 1.27 times larger than that of αgB_{3}N_{3}C. Owing to the large interspaces between atoms, the hydrogen storage in βgB_{3}N_{3}C may be expected [30]. The detailed description of hydrogen storage related to βgB_{3}N_{3}C is beyond the scope of this work.
Concerning the special carbon atom in βgB_{3}N_{3}C, the counting of four valence electrons is as follows: Three electrons participate in the sp^{2} hybrid orbital, which forms a planar structure. The remaining one electron is then redistributed in the whole unit cell due to the enhanced BN covalent bond (with shorter bond length compared with the value in pristine hBN), which makes the magnetic properties more complicated. From the fourth electron, only 18% of the electron still fills the Πorbital of the C atom and contributes a magnetic moment of 0.18 μ_{B}. This is in fairly good agreement with theoretical description of the Πorbital state to the local magnetic moment of approximately 0.3 μ_{B}in the graphenelike systems [31,32]. Around 32% of the fourth electron mainly resides at the interstitial region, which plays a crucial role in βgB_{3}N_{3}C as follows: (a) enhances the BN covalent bond, (b) provides the main interstitial magnetic moment of 0.32 μ_{B}, and (c) promotes the 2p_{z}of the N atom spinpolarized slightly with a magnetic moment of 0.06 μ_{B}per N atom. The remaining percentage of the fourth electron acts as the conduction electrons and makes the system metallic, which dominates the mechanism of ferromagnetic ordering in βgB_{3}N_{3}C. In the case of βgB_{3}N_{3}C, we can see that the electron spin at the localized Πorbital state of C and N atoms as well as the interstitial region compels two energy bands localized strictly along the entire highsymmetry lines, i.e., the ΓMKΓline (see Figure 5a). Thus, the RKKY interaction [10,33] among the magnetic sites through the residual conduction electrons forms a spin ordering in these orbitals, which is the physical origin of ferromagnetism in βgB_{3}N_{3}C. Figure 6b,c respectively plot the top and side views of the 3D isosurfaces for net magnetic charge density in the xy plane. This finding is insightful, and three major points deserve comment: (a) The C site is more spinpolarized as compared with each N site. (b) The dumbbelllike magnetic moment distribution along the z direction implies that the 2p_{z} orbital becomes partially filled with one spinup electron. (c) The induced moments are ferromagnetic coupled between the N and C sites based on the RKKY exchange interaction model as mentioned above.
Conclusions
In summary, we have predicted a novel crystalline material gB_{3}N_{3}C, which displays two distinct electronic properties where the selective bonding type of the C atom is a key parameter for future industrial processes. αgB_{3}N_{3}C is a semiconductor, while βgB_{3}N_{3}C behaves like a metal and holds a magnetic moment of 0.68 μ_{B}. Importantly, compared with the hybrid BNC, gB_{3}N_{3}C is proposed to have a simple structure, which can be applied in various fields due to its unique properties.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MSS conceived, designed, and optimized the structure of gB_{3}N_{3}C. JYL performed the calculations, analyzed the data, and drew the figures. DQG, NXN, and DSX discussed the results. MSS wrote the paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Basic Research Program of China under No. 2012CB933101. This work was also supported by the National Science Foundation of China (NSFC) under No. 10804038, 11034004, and 50925103 and the Fundamental Research Fund for the Central Universities and Physics and Mathematics of Lanzhou University. We acknowledge that part of the work was as done at the National Supercomputing Center in Shenzhen.
References

Ci L, Song L, Jin C, Jariwala D, Wu D, Li Y, Srivastava A, Wang ZF, Storr K, Balicas L, Liu F, Ajayan PM: Atomic layers of hybridized boron nitride and graphene domains.
Nat Mater 2010, 9:430435. PubMed Abstract  Publisher Full Text

Rubio A: Hybridized graphene: nanoscale patchworks.
Nat Mater 2010, 9:379380. PubMed Abstract  Publisher Full Text

Jung J, Qiao Z, Niu Q, MacDonald AH: Transport properties of graphene nanoroads in boron nitride sheets.
Nano Lett 2012, 12:29362940. PubMed Abstract  Publisher Full Text

Bernardi M, Palummo M, Grossman JC: Optoelectronic properties in monolayers of hybridized graphene and hexagonal boron nitride.
Phys Rev Lett 2012, 108:226805226809. PubMed Abstract  Publisher Full Text

Li J, Shenoy VB: Graphene quantum dots embedded in hexagonal boron nitride sheets.
Appl Phys Lett 2011, 98:013105013107. Publisher Full Text

Lam K, Lu Y, Feng YP, Liang G: Stability and electronic structure of two dimensional Cx(BN)y,compound.
Appl Phys Lett 2011, 98:022101022103. Publisher Full Text

Seol G, Guo J: Bandgap opening in boron nitride confined armchair graphene nanoribbon.
Appl Phys Lett 2011, 98:143107143109. Publisher Full Text

da Rocha Martins J, Chacham H: Disorder and segregation in BCN graphenetype layers and nanotubes: tuning the band gap.
ACS Nano 2010, 5:385393. PubMed Abstract  Publisher Full Text

Manna AK, Pati SK: Tunable electronic and magnetic properties in BxNyCz nanohybrids: effect of domain segregation.
J Phys Chem C 2011, 115:1084210850. Publisher Full Text

Lehtinen PO, Foster AS, Ayuela A, Krasheninnikov A, Nordlund K, Nieminen RM: Magnetic properties and diffusion of adatoms on a graphene sheet.
Phys Rev Lett 2003, 91:017202017205. PubMed Abstract  Publisher Full Text

Si MS, Xue DS: Magnetic properties of vacancies in a graphitic boron nitride sheet by firstprinciples pseudopotential calculations.

Si MS, Li JY, Shi HG, Niu XN, Xue DS: Divacancies in graphitic boron nitride sheets.
Europhys Lett 2009, 86:4600246007. Publisher Full Text

Pruneda JM: Native defects in hybrid C/BN nanostructures by density functional theory calculations.

Mazzoni MSC, Nunes RW, Azevedo S, Chacham H: Electronic structure and energetics of BxCyNz layered structures.

Enyashin A, Makurin Y, Ivanovskii A: Quantum chemical study of the electronic structure of new nanotubular systems: αgraphynelike carbon, boronnitrogen and boroncarbonnitrogen nanotubes.
Carbon 2004, 42:20812089. Publisher Full Text

Dutta S, Pati SK: Halfmetallicity in undoped and boron doped graphene nanoribbons in the presence of semilocal exchangecorrelation interactions.
J Phys Chem B 2008, 112:13331335. PubMed Abstract  Publisher Full Text

Han W, Wu L, Zhu Y, Watanabe K, Taniguchi T: Structure of chemically derived mono and fewatomiclayer boron nitride sheets.
Appl Phys Lett 2008, 93:223103223105. Publisher Full Text

Perdew JP, Burke K, Ernzerhof M: Generalized gradient approximation made simple.
Phys Rev Lett 1996, 77:38653868. PubMed Abstract  Publisher Full Text

Kresse G, Furthmüller J: Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set.
Comp Mater Sci 1996, 6:1550. Publisher Full Text

Kresse G, Furthmüller J: Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set.
Phys Rev B 1996, 54:1116911186. Publisher Full Text

Madsen GKH, Blaha P, Schwarz K, Sjöstedt E, Nordström L: Efficient linearization of the augmented planewave method.

Blaha P, Schwarz K, Madsen GKH, Kvasnicka D, Luitz J: WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties. Techn. Universität Wien, Austria; 2001.

Heyd J, Peralta JE, Scuseria GE, Martin RL: Energy band gaps and lattice parameters evaluated with the HeydScuseriaErnzerhof screened hybrid functional.
J Chem Phys 2005, 123:174101174108. PubMed Abstract  Publisher Full Text

Becke AD, Johnson ER: A simple effective potential for exchange.
J Chem Phys 2006, 124:221101221104. PubMed Abstract  Publisher Full Text

Tran F, Blaha P: Accurate band gaps of semiconductors and insulators with a semilocal exchangecorrelation potential.
Phys Rev Lett 2009, 102:226401226404. PubMed Abstract  Publisher Full Text

Kohn W, Sham LJ: Selfconsistent equations including exchange and correlation effects.
Phys Rev 1965, 140:A11331138. Publisher Full Text

Perdew JP, Wang Y: Accurate and simple analytic representation of the electrongas correlation energy.
Phys Rev B 1992, 45:1324413249. Publisher Full Text

Heyrovska R: Structures of the molecular components in DNA and RNA with bond lengths interpreted as sums of atomic covalent radii.
Open Struct Biol J 2008, 2:17. Publisher Full Text

Tomanek D, Wentzcovitch RM, Louie SG, Cohen ML: Calculation of electronic and structural properties of BC3.
Phys Rev B 1988, 37:31343136. Publisher Full Text

Mapasha RE, Ukpong AM, Chetty N: Ab initio studies of hydrogen adatoms on bilayer graphene.

Tada K, Haruyama J, Yang HX, Chshiev M, Matsui T, Fukuyama H: Ferromagnetism in hydrogenated graphene nanopore arrays.
Phys Rev Lett 2011, 107:217203217207. PubMed Abstract  Publisher Full Text

Lee H, Son Y, Park N, Han S, Yu J: Magnetic ordering at the edges of graphitic fragments: magnetic tail interactions between the edgelocalized states.

Bruno P, Chappert C: RudermanKittel theory of oscillatory interlayer exchange coupling.
Phys Rev B 1992, 46:261270. Publisher Full Text