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during several years , it was believed that the existence of free - standing two dimensional crystals was impossible , because they would be unstable and ultimately return to a three dimensional object@xcite .
however , it appeared recently that isolated sheets of graphene could be obtained by mechanical exfoliation of a graphite crystal , which therefore proves the predictions of ref . to be inaccurate@xcite .
the experimental accomplishment of synthesizing two - dimensional crystals has lead to the emergence of a truly new physics , since the particular properties of graphene can be considered as a bridge between quantum electrodynamics ( qed ) and condensed matter physics@xcite .
indeed , near the @xmath1 point of the brillouin zone ( bz ) , the one - particle energy dispersion is linear with the momentum , and therefore the corresponding quasiparticles could be described by a dirac - like hamiltonian .
then , qed properties can be studied by investigating the electronic structure of graphene .
later on , it was shown@xcite that the same technique could be used to obtain other compounds , opening the path to the investigation of a large number of two dimensional crystals .
however , it seems that most of the experimental and theoretical efforts in this area are still focusing exclusively on graphene , primarily because of the high quality of samples that could be obtained . in the present paper , we investigate by means of ab - initio calculations the electronic structure of mos@xmath0 and nbse@xmath0 , as they have been synthesized experimentally@xcite , as well as hypothetical two dimensional hexagonal crystals ( with a graphene like structure ) made of si and ge .
to perform the calculations , we have used density functional theory ( dft ) @xcite as implemented in the code vasp ( vienna ab - initio simulation package)@xcite , within the framework of the paw ( projector augmented waves ) method @xcite .
the local density approximation@xcite ( lda ) as well as the perdew burke ernzerhof@xcite variant of the generalized gradient approximation ( gga ) were used for the exchange - correlation potential .
a cut - off of @xmath2 ev was used for the plane - wave expansion of the wave function to converge the relevant quantities . for brillouin zone integrations ,
a mesh of @xmath3 @xmath4-points@xcite was used for mos@xmath0 and nbse@xmath0 , while a mesh of @xmath5 was sufficient to describe two - dimensional silicon and germanium .    concerning the crystal structure mos@xmath0 and nbse@xmath0 we used the experimental bulk values since it was noticed in ref .
that @xmath6d crystals in these cases remains very close to their 3d parents . in this case ,
gga was used .
for the two dimensional hexagonal crystals made of si and ge , we optimized the lattice parameter using either the lda or the gga
. moreover , it appeared that the band structure obtained using either lda or gga were very similar , so we have chosen to show only results corresponding to lda .
for all studies here , the c parameter was taken to be large enough to ensure that no interaction remains between layers , making them effectively isolated @xmath6d objects .
there is a big interest in transition metal dichalcogenides , since due to their layered structure , they have extremely anisotropic properties , and therefore an intercalation process is easy to conduct . in particular , mos@xmath0 is used in the technology of li batteries .
to investigate purely two - dimensional mos@xmath0 , we have used the bulk lattice parameter ( a @xmath7  ) , as reported in ref .
since reduced dimensionality can sometimes leads to magnetic behavior in systems which are not magnetic in bulk , we checked the possibility to have a spin - polarized ground - state , but it was found that two - dimensional mos@xmath0 remains in a non - magnetic state .    .
the fermi level is put at zero ev .
[ fig : dos - mos2 ] , scaledwidth=70.0% ]    our calculated dos is presented in fig [ fig : dos - mos2 ] .
as for three - dimensional mos@xmath0 , the two - dimensional variant of this compound is semiconducting : the bands on each side of the band gap are derived mainly from the mo - d states , which is what a calculation of three dimensional mos@xmath0 also shows@xcite . also , a large gap exists in the occupied states , separating the s - s states ( between -14 and -12 ev ) and the hybridized mo and s - p states ( between -6 and 0 ev ) .     the band structure of two - dimensional mos@xmath0 .
the fermi level is put at zero ev .
, scaledwidth=70.0% ]    in fig .
[ fig : bnds - mos2 ] , we present the energy bands of two - dimensional mos@xmath0 . we see again a lot of similarities with three dimensional mos@xmath0 : the bands around the band - gap are relatively flat , as expected from the d character of the electron states at these energies .
the band gap , which has a value of 1.78 ev , is direct , and occurs at the high - symmetry point @xmath1 , whereas for three dimensional mos@xmath0 the band gap is indirect@xcite .
as for mos2 , the nbse@xmath0 structure is strongly layered , with each nb layer being sandwiched between two se layers , where weak van der waals forces are holding the whole geometry together .
nbse@xmath0 is known to be a prototype to study charge density waves ( cdw ) , and is also a superconductor . here
we focus only on the electronic structure of two - dimensional nbse@xmath0 and a lattice constant of @xmath8 @xcite was used to perform the calculation .    .
the fermi level is put at zero ev .
[ fig : dos - nbse2 ] , scaledwidth=70.0% ]    as for mos@xmath0 , we have checked the existence of a magnetic solution , since the non spin - polarized dos ( fig .
[ fig : dos - nbse2 ] ) is very peaked at the fermi level , but we found again a non - magnetic ground state .
however , contrary to mos@xmath0 , two dimensional nbse@xmath0 is metallic , just like the three dimensional version of this compound .
the dos of two - dimensional nbse@xmath0 shows a lot of similarities with the one of mos@xmath0 , with a large hybridization between nb and se states .
in addition , the band state which is pinned at the fermi level is derived primarily from nd d - orbitals , and this band state becomes separated in energy from all other states ( fig .
[ fig : bnds - nbse2 ] ) .    .
the fermi level is put at zero ev .
[ fig : bnds - nbse2 ] , scaledwidth=70.0% ]
with four sp electrons in the valence band , silicon and germanium ( possibly together with bn ) are probably the closest to carbon from a chemical point of view .
therefore , it is of interest to see if in their two - dimensional form they could present some similarities with graphene , and we have investigated this possibility here .    .
lattice parameters of hexagonal si and ge computed with either lda or gga . [ tab : lattice ] [ cols= " < , > , > " , ]      , scaledwidth=70.0% ]     , scaledwidth=70.0% ]    in table [ tab : lattice ] , we present our computed lattice parameters ( see computational section ) for hexagonal si and ge .
as expected , they are larger than the one of graphene ( @xmath9  ) because of the larger radius of si and ge in comparison with c. also , because of the well - known overbinding of lda , gga gives lattice parameters which are larger than the ones given by lda .
, scaledwidth=70.0% ]     , scaledwidth=70.0% ]    we used these lattice parameters for the calculation of the corresponding densities of states and band structures .
as seen in figs [ fig : dos - si ] and [ fig : bnds - si ] , two - dimensional silicon shows a lot of similarities with graphene : in particular the gap is also closing at the k point of the brillouin zone , and the dispersion around this point is linear .
in contrast , two - dimensional germanium is quantitatively different from graphene : a conduction band is partially filled by electrons around the @xmath10 point and therefore the bands at the @xmath1 point are shifted up in energy , so that two - dimensional germanium is not a zero - gap material , but rather a poor metal . for three dimensional germanium , it is known that lda / gga gives a non band - gap behavior , and a more advanced technique like the gw approximation@xcite is required to obtain a band gap .
however , our calculations with dft show that two - dimensional germanium is much more metallic than three - dimensional germanium , and therefore the metallic nature is here much more robust and is likely to be preserved even if a quasiparticle theory ( like the gw approximation ) would be used .
we have studied by means of ab - initio calculations the electronic structure of two - dimensional mos@xmath0 and nbse@xmath0 , as well as hypothetical graphene like structures of si and ge .
we have found that two - dimensional silicon might from an electronic structure point of view be equivalent to graphene , with a linear dispersion of the electronic structure around the k - point .
the possible advantage with si in this regard is that it probably is more easily interfaced with existing electronic devices and technologies .
the obvious disadvantage is that sp@xmath11 bonded si is much less common than for c , and the synthesis of si in a graphene like structure is extremely demanding and is likely to represent a meta - stable material .
the electronic structure of ge in the graphene structure results in a metallic behavior , and the electronic structure of two - dimensional mos@xmath0 and nbse@xmath0 is quite similar to that of the three dimensional counterparts , with mos@xmath0 being a gapped system with a direct gap and nbse@xmath0 a metal .
s.l . acknowledges financial support from anr pnano grant anr- 06-nano-053 - 02 and anr grant anr - blan07 - 1 - 186138 , as well as cines / ccrt for computer time .
o. e. is grateful to snac and vr for support and to a kof initiative at uppsala university .
we report on ab - initio calculations of the two - dimensional systems mos@xmath0 and nbse@xmath0 , which recently were synthesized . 
 we find that two - dimensional mos@xmath0 is a semiconductor with a gap which is rather close to that of the three dimensional analogue , and that nbse@xmath0 is a metal , which is similar to the three dimensional analogue of this compound . 
 we further computed the electronic structure of the two - dimensional hexagonal ( graphene like ) lattices of si and ge , and compare them with the electronic structure of graphene . 
 it is found that the properties related to the dirac cone do not appear in the case of two - dimensional hexagonal germanium , which is metallic , contrary to two - dimensional hexagonal silicon , which has an electronic structure very similar to the one of graphene , making them possibly equivalent .