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the recent experimental realization of a single layer of carbon atoms arranged in a honey - comb lattice has prompted much excitement and activity in both the experimental and theoretical physics communities @xcite . carriers in graphene ( both electrons and holes ) have a linear bare kinetic energy dispersion spectra around the @xmath1 and @xmath2 points ( the  dirac points " ) of the brillouin zone . the ability of experimentalists to tune the chemical potential to lie above or below the dirac point energy ( by application of voltages to gates in close proximity to the graphene sheets ) allows the carriers to be changed from electrons to holes in the same sample . this sets graphene apart from other two dimensional ( 2d ) carrier systems that have a parabolic dispersion relation , and typically have only one set of carriers , i.e. either electrons or holes . another unique electronic property , the absence of back - scattering , has led to the speculation that carrier mobilities of 2d graphene monolayers ( certainly at room temperature , but also at low temperature ) could be made to be much higher than any other field - effect type device , suggesting great potential both for graphene to be the successor to si - mosfet ( metal - oxide - semicondcutor field effect transistor ) devices and for the discovery of new phenomena that normally accompanies any significant increase in carrier mobility @xcite . it is therefore of considerable fundamental and technological interest to understand the electronic properties of graphene @xcite . graphene samples that are currently being fabricated are far from pure , based on the relatively low electronic mobilities compared to epitaxially grown modulation - doped two - dimensional electron gases ( 2degs ) such as gaas - algaas quantum wells . it is therefore important to understand the effects of disorder on the properties of graphene . disorder manifests itself in the finite lifetimes of electronic eigenstates of the pure system . in the presence of scattering from an impurity potential that is not diagonal in the these eigenstates , the lifetime scattering rate of the eigenstate , @xmath3 , is non - zero and can be measured experimentally by fitting the line - shape of the low - field shubnikov - de haas ( sdh ) oscillations . @xcite the effect of disorder scattering on the sdh line shape is equivalent to increasing the sample temperature and one can therefore measure this dingle temperature ( @xmath4 ) and relate it to the single - particle lifetime through @xmath5 . to avoid potential confusion , we mention that the lifetime damping rate @xmath3 discussed in this paper is _ not _ equal to the _ transport _ scattering rate which governs the electrical conductivity . the lifetime damping rate is the measure of the rate at which particles scatters out of an eigenstate , whereas the transport scattering rate is a measure of the rate of _ current decay _ due to scattering out of an eigenstate . in normal 2degs , the transport scattering time can be much larger than the impurity induced lifetime @xmath6 , particularly in high mobility modulation doped 2d systems where the charged impurities are placed very far from the 2deg@xcite . recently , the issue of transport scattering time versus impurity scattering lifetime in graphene has been discussed.@xcite    the single - particle level broadening due to the impurity potential changes many of the physical properties of the system including the electronic density of states.@xcite the electronic density of states is an important property which directly affects many experimentally measurable quantities such as the electrical conductivity , thermoelectric effects , and differential conductivity in tunneling experiments between graphene and scanning - tunneling microscope tips or other electron gases . changes in the density of states also modify the electron screening,@xcite which is an important factor in the determination of various properties of graphene . it is therefore imperative to take into account the effects of disorder on the density of states , particularly since disorder is quite strong in currently available graphene samples .    in the present work , we present calculations of the average density of states of disordered graphene . this problem has been investigated using various models and techniques , both analytical and numerical.@xcite we take into account scattering effects from long - range and short range impurity potentials . we consider both unscreened and screened short - ranged and screened charged impurities , using the born approximation . in addition , for unscreened short - ranged ( usr ) impurities , we go beyond the born approximation and include self - consistent effects . there is another class of disorder in graphene called  off - diagonal " or random gauge potential " disorder , in which the hopping matrix elements of the electrons in the underlying honeycomb lattice are random . in this paper we do not consider in this type of disorder , which can result from height fluctuations ( ripples ) in the graphene sheet and lead to qualitatively different results from the ones presented in this paper.@xcite    the rest of the paper is organized as follows : in sec . [ sec : approx ] , we describe the approximation schemes that we use . secs . [ sec : usr ] and [ sec : screened ] deal with unscreened short range impurities and screened short - range / charged impurities , respectively . in sec . [ sec : comparison ] , we compare our results to those from other workers , and we conclude in sec .  [ sec : conclusion ] . the single - particle density of states for a translationally invariant 2deg is given by@xcite @xmath7 \label{eq : dos}\end{aligned}\ ] ] where @xmath8 is the degeneracy factor ( for graphene @xmath9 due to valley and spin degeneracies ) @xmath10 is the band index , @xmath11 is the retarded green s function , and the @xmath12-integration is over a single valley , which we assume to be a circle of radius @xmath13 . @xmath11 expressed in terms of the retarded self - energy @xmath14 is @xmath15^{-1}.\label{eq : g}\ ] ] where @xmath16 is the bare band energy of the state @xmath17 and @xmath18 is an infinitesimally small positive number . ( in this paper , the green s functions and self - energies are all assumed to be retarded . ) ( [ eq : dos ] ) and ( [ eq : g ] ) show that if @xmath19 and @xmath20 are known , the density of states can be obtained in principle from    @xmath21+\eta}{(e - e_{\bm      k,\lambda } - { \rm re}[\sigma_\lambda(\bm    k , e)])^2+({\rm im}[\sigma_\lambda(\bm k , e)]-\eta)^2}. \label{eq : dos1}\ ] ]    for pure graphene systems , @xmath22 ( excluding electron  electron and electron  phonon interactions , which are not considered here ) , and hence @xmath23 = -\pi \delta(e - e_{\bm    k\lambda})$ ] . close to the dirac points ( which we choose to the the zero of energy ) , the dispersion for graphene is ( we use @xmath24 throughout this paper ) @xmath25 where @xmath26 and @xmath27 for the conduction and valence bands , respectively , @xmath28 is the wavevector with respect to the dirac point , and @xmath29 is the fermi velocity of graphene . performing the @xmath12-integration in eq . ( [ eq : dos ] ) for the pure graphene case gives @xmath30 where @xmath31 is the band energy cut - off . the average density of states for a disordered 2deg can be obtained by averaging the green s function over impurity configurations . the averaging procedure gives a non - zero @xmath32 which , in general , can not be evaluated exactly . various approximation schemes for @xmath32 have therefore been developed , four of which are described below . = 1.7 in    _ born approximation  _ in the born approximation , the self - energy is given by the feynman diagram shown in fig . 1(a)@xcite , and the expression for the self - energy is @xmath33 where @xmath34 is the impurity density , @xmath35 is the fourier transform of the impurity potential , @xmath36 is the bare green s function and @xmath37 is square of the the overlap function between the part of the wavefunctions of @xmath38 and @xmath39 that are periodic with the lattice ( here @xmath40 are band indices ) . for graphene states near the dirac point , @xmath41 where @xmath42 is the angle between @xmath12 and @xmath43 . _ self - consistent born approximation  _ the feynman diagram for this self - energy is the same as fig . 1(a ) , except that the bare green s function is replaced by the full one . consequently , the expression for @xmath44 is the same as in eq . ( [ eq : sigma_b ] ) , except with @xmath36 replaced by @xmath11 . _ @xmath0-matrix approximation  _ the @xmath0-matrix approximation is equivalent to the summation of feynman diagrams shown fig the expression for @xmath45 is the same as in eq . ( [ eq : sigma_b ] ) , except with @xmath46 replaced by @xmath0 , the @xmath0-matrix for an individual impurity .    _ self - consistent @xmath0-matrix approximation  _ the feynman diagram for this approximation is the same as in the @xmath0-matrix approximation , except that the bare green s functions are replaced by full ones . the expression for @xmath47 is the same as in eq . ( [ eq : sigma_b ] ) , except with @xmath46 replaced by @xmath0 , and @xmath36 replaced by @xmath11 .    when the potentials for the impurities are not all identical ( for example , in the case where there is a distribution of distances of charged impurities from the graphene sheet ) , one averages @xmath48 or @xmath49 over the impurities . in the present context , short - ranged impurities are impurities which result from localized structural defects in the honeycomb lattice , which are roughly on the length scale of the lattice constant . in this case , it is acceptable to approximate @xmath50 , a real constant , for intravalley scattering processes . in this paper , we ignore the intervalley processes . ( however , we note that if the matrix element joining intervalley states is constant , inclusion of intervalley scattering in our calculations is not difficult . ) this simplification allows us to obtain some analytic expressions for the self - energies in the approximation schemes mentioned above .      for usr disorder , the four approximation schemes we use give self - energies that are independent of @xmath10 and @xmath51 .      the self - energy for graphene with usr scatterers in the born approximation , using @xmath52 , ( [ eq : dirac_dispersion ] ) and ( [ eq : f ] ) in eq . ( [ eq : sigma_b ] ) , is    @xmath53    as a function of complex @xmath54 , @xmath55 is real and positive ( negative ) along the real axis from @xmath56 to @xmath57 ( @xmath58 to @xmath59 ) . furthermore , it has a branch cut in on the real axis of @xmath54 in between @xmath58 and @xmath56 , so that for @xmath60 , @xmath61 fig . 2 shows @xmath62 .    for real @xmath63 and @xmath64 , @xmath65.\ ] ] the born approximation damping rate for state @xmath66 is @xmath67 = \tilde\gamma_b\,v_f k$ ] . [ see eq . ( [ eq : h_0e ] ) ] , where @xmath68 is the self - energy for usr impurities in the born approximation . ]      the self - energy @xmath69 for unscreened short - ranged scatterers in the self - consistent born approximation is given by the self - consistent equation    @xmath70   \ln\left[1 - \frac{e_c^2 } { [ \zeta-\sigma_{\rm        sb}^{\rm ( usr)}(\zeta ) ] ^2}\right ] \nonumber \\ & = h_0\left [ \zeta-\sigma_{\rm      sb}^{\rm ( usr)}(\zeta)\right ] , \label{eq : sigma_sb}\end{aligned}\ ] ]    this shows that if @xmath71 , then @xmath72 ( except possibly around @xmath73 , which is usually not experimentally relevant ) . in general the impurity averaged @xmath0-matrix for a potential @xmath74 is    @xmath75.\label{eq : tmatrix}\ ] ]    in this approximation , the self energy is @xmath76 if @xmath52 , a constant , the term in the square parentheses in eq . ( [ eq : tmatrix ] ) is @xmath77 . the sum then gives @xmath78      as in self - consistent born approximation , the @xmath79 in eq . ( [ eq : sigma_t ] ) is replaced by the self - consistent @xmath80 , giving    @xmath81    as in the self - consistent born approximation , this shows that if @xmath82 , then @xmath83 ( except possibly around @xmath73 ) . when the self - energy is @xmath12- and @xmath10-independent , the density of states for graphene can be calculated analytically from eq . ( [ eq : dos1 ] ) . the result is    @xmath84\biggr\},\label{eq : dos2}\end{aligned}\ ] ]    where @xmath85 $ ] and @xmath86 $ ] . there has been considerable interest in the minimum dc electrical conductivity of disordered graphene as the fermi energy moves through the dirac point.@xcite there is still no consensus on whether the minimum conductivity is a universal value or not . since the electrical conductivity is directly proportional to the density of states at the fermi energy , it is important to be able to determine the density of states at the dirac point of disordered graphene . because the minimum conductivity is non - zero as the fermi energy passes through the dirac point , the density of states should be nonzero . ( [ eq : dos1 ] ) shows that if @xmath87 when @xmath88 , then the density of states at the dirac point @xmath89 . ( note that we have not included the term that is first order in the impurity potential in our self - energy . since this first - order term merely rigidly shifts the band by an amount @xmath90 , ignoring this term is equivalent to shifting the zero of the energy by @xmath91 , and hence the dirac point is still at @xmath92 . ) a non - zero density of states at the dirac point depends on a non - zero @xmath93 $ ] . since @xmath94 , it is clear from eqs . ( [ eq : sigma_b1 ] ) and ( [ eq : sigma_t ] ) that the born and @xmath0-matrix approximations give zero density of states at the dirac point .    for the self - consistent born approximation , eq . ( [ eq : sigma_sb ] ) can be rewritten as @xmath95^{-1},\end{aligned}\ ] ] which shows that @xmath96 , and therefore the self - consistent born approximation also gives @xmath89 .    in the case of the self - consistent @xmath0-matrix approximation , re - writing eq  ( [ eq : sigma_st ] ) as @xmath97 setting @xmath92 , using eq . ( [ eq : h_0 ] ) and taking the imaginary parts of both sides of this equation gives @xmath98 \equiv\gamma(0 ) & = { \rm i m } \left[\frac{n_i u_0}{\displaystyle 1-i\frac{u_0\gamma(0)}{2        v_f^2}\ln\left(1+\frac{e_c^2}{\gamma(0)^2}\right)}\right ] . \label{eq : gamma0}\end{aligned}\ ] ]    in the weak scattering limit , when @xmath99 , the imaginary term in the denominator of eq . ( [ eq : gamma0 ] ) is much less than one ( we check for self - consistency later ) , and this gives @xmath100 = 2\tilde\gamma_b\gamma_w(0 ) \ln\left(\frac{e_c}{\gamma_{w}(0)}\right),\end{aligned}\ ] ] which implies that @xmath101 note that the result is non - analytic in @xmath102 . inserting eq . ( [ eq : gamma0result ] ) into the imaginary part of the denominator of eq . ( [ eq : gamma0 ] ) ( which we had assumed to be much smaller than 1 in magnitude ) gives the self - consistent criterion @xmath103 for the validity of eq . ( [ eq : gamma0result ] ) . substituting this into eq . ( [ eq : dos2 ] ) gives an average density of states at the dirac point for weak scattering of approximately @xmath104 similar results to eq . ( [ eq : dosweak ] ) have been reported@xcite in studies of disordered systems of fermions with linear dispersions using other methods . we mention that our calculation of the graphene density of states at the dirac point should only be considered as demonstrative since electron - electron interaction effects are crucial @xcite at the dirac point , and the undoped graphene system is not a simple fermi liquid at the dirac point . free carriers will move to screen a bare impurity potential @xmath105 , resulting in a screened interaction @xmath106 , where @xmath107 is the static dielectric function . the @xmath107 results in an @xmath108-dependent effective electron - impurity potential @xmath46 , even in the case of short - ranged ( @xmath108-independent ) bare impurity potentials . this makes the calculations much more involved than in the usr case . therefore , in this paper , we limit our investigation of @xmath108-dependent screened potentials to the level of the born approximation .    for the dielectric function @xmath107 , we use the random phase approximation ( rpa ) for dielectric function appropriate for graphene , given by @xmath109 where @xmath110 is the two - dimensional fourier transform of the coulomb potential ( @xmath111 is the dielectric constant of the surrounding material ) , and @xmath112 is the static irreducible rpa polarizability for graphene.@xcite we use @xmath113 for charged impurities , and @xmath114 , a constant , for short - range point defect scatterers . we first look at the density of states at the fermi surface ; _ i.e. _ , at energy @xmath115 . to obtain this , we calculate the single - particle lifetime damping rate @xmath3 in the born approximation , which is given by @xmath116 \nonumber \\ & = & \frac{n_i}{2\pi } \frac{k}{v_f}\int_0^{\pi}d\theta\ \frac { \langle\left |{v_{ei}(q)}\right |^2\rangle}{\varepsilon(q)^2 } ( 1+\cos\theta ) , \label{scat_s}\end{aligned}\ ] ] where @xmath117 . then , assuming that @xmath118 $ ] is relatively constant for @xmath51 close to @xmath119 , we substitute @xmath120 for @xmath121 $ ] into eq . ( [ eq : dos1 ] ) , which gives eq . ( [ eq : dos2 ] ) with @xmath122 . we assume that the charged or neutral impurities are distributed completely at random on the surface of the insulating substrate on which the graphene layer lies , the areal density for the charged and neutral impurities is @xmath123 and @xmath124 , respectively , and the density of carriers in the graphene layer is @xmath125 , where @xmath119 is the fermi wavevector relative to the dirac point . ( this relationship between @xmath126 and @xmath119 takes into account the spin and valley degeneracy @xmath127 and @xmath128 . ) we use the rpa screening function at @xmath129,@xcite to obtain the effective impurity potential . the key dimensionless parameter that quantifies the screening strength is @xmath130 , which is corresponding to the interaction strength parameter of a normal 2d system ( _ i.e. _ the ratio of potential energy to kinetic energy ) . the born approximation lifetime damping rates for screened charged - impurities @xmath131 and @xmath132-correlated neutral impurities @xmath133 at @xmath119 are    @xmath134    [ eq : gamma ]    in these equations ,    @xmath135    where @xmath136                    & \mbox{for   $ x < 1 $ } ; \\                     1                    & \mbox{for $ x = 1 $ } ; \\ \frac{1}{\sqrt{x^2 - 1}}\cos^{-1}\frac{1}{x }                    & \mbox{for $ x > 1$}. \end{array } \right . \label{fx}\end{aligned}\ ] ]    in fig . [ damping ] we show the calculated damping rates scaled by @xmath137 as a function the interaction parameter @xmath138 . for @xmath139 we have @xmath140 and @xmath141 . for @xmath142 we have @xmath143 and @xmath144 . thus , for small ( large ) @xmath138 the damping rate due to the short - ranged impurity dominates over that due to the long - ranged charged impurity . on the other hand , since @xmath145 and @xmath146 [ eq .  ( [ eq : gamma ] ) ] , in the low ( high ) carrier density limit the lifetime damping of single particle states at the fermi surface is dominated by charged impurity ( short - ranged impurity ) scattering . the crossover takes place around a density @xmath147     as a function of @xmath138 . graphene on a sio@xmath148 ( air ) substrate has an @xmath149 ( 2 ) . solid lines indicate damping rates ( @xmath131 ) due to charged impurities with an impurity density @xmath150 for different electron densities @xmath151 , 10 , 50@xmath152 @xmath153 ( from top to bottom ) , respectively . dashed line indicate the damping rate ( @xmath154 ) due to short - ranged impurity with impurity density @xmath155 @xmath153 and potential strength @xmath1561 kev @xmath157 , which correspond to @xmath158 . note @xmath159 is independent on the electron density . ] using @xmath160 in eq . ( [ eq : dos2 ] ) gives ( assuming @xmath161 , @xmath162 ) @xmath163.\end{aligned}\ ] ] for @xmath164 , this gives @xmath165 \right\ } , \label{eq : dosdelta}\ ] ] and for @xmath166 @xmath167 + \frac{g_sg_v}{2\pi^2 } \frac{\gamma } { v_f^2}\ln\left(\frac{e_c}{\gamma}\right ) . \label{eq : doscharge}\ ] ]    we can apply eq . ( [ eq : dosdelta ] ) for short - ranged impurity scattering and for charged impurity scattering in high carrier density limits , and eq . ( [ eq : doscharge ] ) for charged impurity scattering in low density limits . taking the limit @xmath168 in eq . ( [ eq : doscharge ] ) , it appears that for the case of screened charged impurities , we obtain a finite density of states at the dirac point with the born approximation ( since @xmath169 ) . however , recall that in deriving eqs . ( [ eq : dosdelta ] ) and ( [ eq : doscharge ] ) , we have assumed that @xmath170 is constant with respect to @xmath51 , which is not necessarily the case at the dirac point .     for screened coulomb scattering potential ( solid lines ) and for screened neutral short - ranged scattering potential ( dashed lines).,title="fig : " ]   for screened coulomb scattering potential ( solid lines ) and for screened neutral short - ranged scattering potential ( dashed lines).,title="fig : " ]    in general , the damping rate ( or the imaginary part of the self energy ) is a function of energy and wave vector rather than a constant . from eq . ( [ eq : sigma_b ] ) , we calculated the self - energy of disordered graphene . [ self ] we show the self - energy of a conduction band electron ( @xmath171 ) for both screened coulomb scattering potential and screened neutral short - ranged potential . for coulomb scatterers we use the impurity density @xmath172 @xmath153 , and for neutral short - ranged scatterers the impurity density @xmath155 @xmath153 and potential strength @xmath1561 kev @xmath157 . the self - energies in the valence band ( @xmath173 ) are related to the self - energy in the conduction band by @xmath174 and @xmath175 . as @xmath176 , @xmath177 , and @xmath178 for both scattering potentials . however , for large value of @xmath179 the asymptotic behaviors are different , that is , as @xmath180 @xmath181 for coulomb scattering potential and @xmath182 for short - ranged potential . note that by using only the ( non - self - consistent ) born approximation in this section , we assume weak scattering and ignoring multiple - scattering events in calculated @xmath183 . therefore , the results are unreliable in the strong disorder limit ( _ i.e. _ , when @xmath183 is modified significantly from its lowest - order form ) .    , 1 , @xmath184 @xmath153 ( from bottom to top ) , and in ( b ) the short - ranged impurity density @xmath185 , 0.5 , @xmath186 @xmath153 ( from bottom to top ) and potential strength @xmath1871 kev @xmath157 . , title="fig : " ] , 1 , @xmath184 @xmath153 ( from bottom to top ) , and in ( b ) the short - ranged impurity density @xmath185 , 0.5 , @xmath186 @xmath153 ( from bottom to top ) and potential strength @xmath1871 kev @xmath157 . , title="fig : " ]    in fig . [ dos_f ] the density of states in the presence of impurity is shown for different impurity densities . in fig . [ dos_f](a ) we show the density of states for coulomb impurity potential with impurity densities , @xmath188 , @xmath189 @xmath153 , @xmath184 @xmath153 ( from bottom to top ) , and in fig . [ dos_f](b ) we show the density of states for neutral short - ranged impurity potential with densities , @xmath185 , @xmath190 @xmath153 , @xmath189 @xmath153 ( from bottom to top ) and potential strength @xmath1871 kev @xmath157 . the calculated density of states is normalized by @xmath191 . the density of states is enhanced near dirac point ( @xmath92 ) , but as @xmath192 it goes zero as @xmath193 for both types of screened impurity scattering . based on the results of section iii , we expect that this result is an artifact of the born approximation , and that the density of states should in fact be non - zero . the enhancement of the density of states can be explained as follows . in normal 2d system with finite disorder the band edge @xmath194 of the pure conduction ( valence ) band is shifted to @xmath195 ( @xmath196 ) and a band tail forms below ( above ) the band edge of a pure system . thus , the density of states in the presence of impurities is reduced for @xmath197 ( @xmath198 ) because the states have been shifted by the impurity potential into the band tail . however , for graphene since the conduction band and the valence band meet at the dirac point the band tail ( or shift of band edge ) can not be formed , which gives rise to enhancement of density of states near dirac point .    before concluding we point out that our perturbative calculation of the graphene density of states assumes that the system remains homogeneous in the presence of impurities . it is , however , believed @xcite that graphene carriers develop strong density inhomogeneous ( i.e. electron - hole puddle ) at low enough carrier densities in the presence of charged impurities due to the breakdown of linear scattering . in such an inhomogeneous low - density regime close to the dirac point , our homogeneous perturbative calculation does not apply . in this section , we compare and contrast our model of disorder and results to other works in the field . et al._@xcite studied the effect of disorder in graphene by considering the effect of vacancies on the honeycomb lattice . for a finite density of vacancies , they found that the density of states at the dirac point is zero for the  full born approximation " ( equivalent to our @xmath0-matrix approximation ) and non - zero for the  full self - consistent born approximation " ( equivalent to our self - consistent @xmath0-matrix approximation ) . our results are consistent with theirs , even though the regimes that are studied are different . vacancies correspond to the limit where the impurity potential @xmath199 , whereas this work is more concerned with the weak impurity - scattering limit .    pereira _ at al._@xcite considered , among several different models of disorder , both vacancies and randomness in the on - site energy of the honeycomb lattice . they numerically calculated the density of states for these models of disorder . for compensated vacancies ( same density of vacancies in both sub - lattices of the honeycomb structure ) they found that the density of states increased around the dirac point . ( ref .   also studied the case of uncompensated vacancies , but that has no analogue in our model of disorder . ) for the case of random on - site impurity potential , they find that `` there is a marked increase in the dos ( density of states ) at @xmath200 ( the dirac point ) '' and  the dos becomes finite at @xmath200 with increasing concentration " of impurities . our self - consistent @xmath0-matrix approximation result is consistent with their numerical results , although it should be mentioned again that it strictly does not apply to the case of vacancies . et al._@xcite numerically investigated the average density of states of graphene for the case of on - site disorder . they found that for weak disorder , the density of states at the dirac point increased with both increasing density of impurities and strength of the disorder . ( in their work , they did not absorb the shift in the band due to the impurity potential in their definition of the energy , so the dirac point had a shift @xmath201 where @xmath202 is the concentration of impurities and @xmath203 is the on - site impurity energy . ) their numerical results for the minimum in the average density of states ( fig . 3(b ) in ref .  ) seem to indicate a non - linear dependence of the value of the minimum as a function of the strength of the disorder potential , and is at least not inconsistent with eq . ( [ eq : dosweak ] ) . the issue of the effect of screening of the impurity interactions on the density of states discussed in section iv , to the best of our knowledge has not yet been treated in the literature . qualitatively , the effect of screened impurities away from the dirac point is the increase the density of states , and is consistent with numerical results for random on - site disorder.@xcite ( since our treatment of screened impurities is at the level of the born approximation , we do not obtain either a non - zero density of states at the dirac point nor resonances in the density of states.@xcite ) we have calculated the density of states for disordered graphene . in the case of unscreened short - ranged impurities , we utilized the non - self - consistent and self - consistent born and @xmath0-matrix approximations to calculate the self - energy . among these , only the self - consistent @xmath0-matrix approximation gave a non - zero density of states at the dirac point , and the density of states is a non - analytic function of the impurity potential . we investigated the density of states in the case of screened short - ranged and charged impurity potentials at the level of the born approximation . we find that , unlike the case of parabolic band 2degs , in graphene near the band - edge ( _ i.e. _ , the dirac point ) the density of states is enhanced by impurities instead of being suppressed . at very low carrier densities , however , graphene develops strong carrier density inhomogeneity in the presence of charged impurities , an effect not captured by the homogeneous many - body theory in our description . morozov , k.s . novoselov , m.i . katsnelson , f. schedin , d.c . elias , j.a . jaszczak , a.k . geim , phys . . lett . * 100 * , 016602 ( 2008 ) ; j. h. chen , c. jang , s. xiao , m. ishigami , m. s. fuhrer , nature nanotechnology * 3 * , 206(2008 ) ; e. h. hwang and s. das sarma , phys . b * 77 * , 115449 ( 2008 ) . victor m. periera , f. guinea , j. m. b. lopes dos santos , n. m. r. peres , and a. h. castro neto , phys . . lett . * 96 * , 036801 ( 2006 ) ; victor m. periera , j. m. b. lopes dos santos , and a. h. castro neto , phys . b * 77 * , 115109 ( 2008 ) .                see _ e.g. _ , g.  rickaysen , _ green s functions and condensed matter _ ( academic press , new york , 1980 ) ; s.  doniach and e.  h.  sondheimer , _ green s functions for solid state physicists _ ( imperial college , london , 1998 ) ; g. d. mahan , _ many particle physics _ , 3rd ed . ( plenum , new york , 2000 ) .
we calculate the average single particle density of states in graphene with disorder due to impurity potentials . for unscreened short - ranged impurities , 
 we use the non - self - consistent and self - consistent born and @xmath0-matrix approximations to obtain the self - energy . among these 
 , only the self - consistent @xmath0-matrix approximation gives a non - zero density of states at the dirac point . 
 the density of states at the dirac point is non - analytic in the impurity potential . for screened short - ranged and charged long - range impurity potentials , 
 the density of states near the dirac point typically increases in the presence of impurities , compared to that of the pure system .