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different carbon - based structures are considered as promising electrode material for field emission ( fe ) cathodes . in particular , the field emission properties of single - walled ( swnt ) and multi - walled ( mwnt ) carbon nanotubes @xcite as well as graphite films @xcite are presently under intensive experimental and theoretical investigations . in experiment , many factors such as inhomogeneities at the cathode surface , surface contamination ( surface adsorbates and oxides ) , local electric fields and barriers , electronic structure of cathode , etc . can drastically change fe results  @xcite . in addition , these factors vary from one experiment to another thus markedly complicating the theoretical description . nevertheless , the electronic characteristics of cathodes should be equally manifested in different experiments . for this reason , the effect of electronic structure on the emission features of cathodes is of definite interest . for swnts this problem was studied numerically in @xcite by using an approach which can be called as a method of _ independent channels_.    in this paper , we present a rather simple modification of this method to study analytically the influence of the 3d band structure on the field emission current ( fec ) of crystalline graphite . as an illustration , we consider the case of swnt where our approach allows us to reproduce the fe results obtained in  @xcite . as is well - known , the electronic structure near the fermi energy of the crystalline graphite markedly depends on the weak interlayer interaction ( see , e.g. , @xcite ) . accordingly , the fec in this case should be sensitive to the specific electronic structure . in order to show this , we consider the simplest possible modification of graphite ( hypothetical @xmath0 stacking ) where the three - dimensional energy spectrum was calculated analytically in  @xcite . two possible orientations of the applied electric field ( along and normal to the graphite layers ) are of our interest . the emitted current density can be written as  @xcite @xmath1 here the field emission is directed along the @xmath2-axis , @xmath3 is the electric charge , @xmath4 the planck constant , @xmath5 the energy , @xmath6 momentum , @xmath7^{-1}$ ] the fermi - dirac distribution function , @xmath8 the transmission probability of an electron through a potential barrier , and @xmath9 the group velocity . the integrals are over the first brillouin zone with account of the positivity of @xmath10 . notice that in most cases the using of infinite limits in integrals is a good approximation . for parabolic bands @xmath11 , and this relation is widely used in deriving the well - known fowler - nordheim equation . in the case of carbon nanotubes , two important differences from the generally accepted consideration should be taken into account . first , an open swnt has a finite small radius which results in quantization of momentum . in this case , the corresponding integrals in ( [ main ] ) transform into sums . for example , choosing the axes like shown in fig . [ figure1 ]     is the circumference of the nanotube , @xmath12 is a thickness of the graphite layer.,width=151 ]    one has @xmath13 where @xmath14 , and @xmath15 is the circumference of the nanotube , @xmath12 the thickness of the graphite layer . for a swnt there exists only a single layer in the @xmath16-direction and , accordingly , there is exactly one term in the sum for @xmath17 . the number of terms for @xmath18 depends on the tube circumference @xmath15 . second , the energy near the fermi level for a single graphite layer ( graphene ) is approximated by @xmath19 , where @xmath20 is the fermi velocity  @xcite . the electrons move in the @xmath21 plane , so that the energy does not depend on @xmath22 . since @xmath23 is quantized the energy @xmath24 turns out to be divided into a set of channels with @xmath25 where @xmath26 takes integer values . therefore , the current density ( [ main ] ) takes the form @xmath27 as is known , the dispersion relation for carbon nanotubes depends on their chirality ( see , e.g. ,  @xcite ) . for a chiral vector @xmath28 it can be written as @xmath29^{1/2}. \label{eheh}\ ] ] generally , there are two symmetric curves with a gap @xmath30 . however , at certain values of @xmath31 , @xmath32 and @xmath26 the gap turns out to be zero and one gets the linear dispersion relation . therefore , at fixed @xmath26 there exist one metallic branch and a set of semiconducting branches for a swnt with a given chirality .    as stated above , the condition @xmath33 imposes restrictions on the limits of integral in ( [ off ] ) . in addition , two approximations will be used . first , we consider the zero - temperature limit when the fermi - dirac distribution becomes the step function . second , we suggest that the transmission probability is given by the wkb approximation ( see , e.g. ,  @xcite ) in the form @xmath34\right\ } = b\exp(d\varepsilon)\ ] ] where @xmath35 , @xmath36 , @xmath37 is the electric field , @xmath38 the work function , @xmath39 the dielectric constant , and we denoted @xmath40 and @xmath41 for convenience . the functions @xmath42 and @xmath43 describe a deviation of the barrier from the triangle form due to image effects and can be approximated by @xcite @xmath44 now we are able to calculate the @xmath26th term in the sum ( [ off ] ) . for the metallic branch the integration in ( [ off ] ) spreads from @xmath45 to @xmath46 . one obtains @xmath47 for semiconducting branches the range of integration in ( [ off ] ) is ( @xmath45 , @xmath48 ) and @xmath49 notice that the dispersion relation enters ( [ eee ] ) only through the gap . this agrees with the well - known fact that the group velocity and the density of states are canceled in the one - dimensional case @xcite . the sum over all branches in ( [ off ] ) gives the total fec . [ fig : tube ] shows the calculated emission current density , which is a current divided by the circumference of a nanotube . the parameter set is @xmath50 ev , @xmath51  m / s , @xmath52   , @xmath53 ev@xmath54 m@xmath55.,width=226 ]    for @xmath56 swnt the circumference is defined as @xmath57 with @xmath58 being the lattice constant . in fact , the main contribution to the sum in ( [ off ] ) comes from the first few terms corresponding to branches close to the fermi level . this is due to the exponential dependence of the fec on the gap . for metallic nanotubes , the leading term is @xmath59 , so that @xmath60 . in the case of semiconducting nanotubes , the leading contribution comes from the @xmath26th term with the smaller gap in ( [ eee ] ) and , therefore , @xmath61 $ ] . a similar behavior was found numerically in  @xcite . moreover , comparing our results in fig .  [ fig : tube ] with the exact numerical calculations in  @xcite one can find out a good qualitative agreement . notice that the quantitative difference is also not great and varies from a few to ten percent depending on the diameter of the nanotube . in comparison with  @xcite our points in fig . [ fig : tube ] are situated slightly lower for metallic nanotubes and slightly higher for semiconducting nanotubes . this difference can be explained by at least two reasons . first , we have used the simplified expression for the tunneling probability in ( [ dd ] ) where the image effects were approximated in a standard way ( see , e.g. ,  @xcite ) . second , as distinct from  @xcite we consider the zero - temperature limit . in this section , we study a case of noninteracting graphite layers . the layers are oriented as shown in fig . [ fig : orient ] .        to calculate the fec we will use the method of independent channels described in the previous section . namely , let us consider the 2d graphite lattice with the born - von karman boundary conditions applied in the @xmath62 direction . this gives the natural quantization conditions . for layers of a large ( infinite ) size the sum in ( [ off ] ) can be replaced by the integral @xmath63 , so that finally one obtains @xmath64 where the relation @xmath65 is taken into account . it is interesting to mention that this result is very similar to the fowler - nordheim formula @xmath66 indeed , the exponents are exactly the same and the preexponential factors differ only slightly . what is important , the @xmath67-dependence is equal in both cases . for the interlayer distance @xmath68 one can estimate @xmath69 . generally , there are known three possible configurations of crystalline graphite : @xmath70 stacking sequence of hexagonal layers ( bernal structure ) , rhombohedral @xmath71 stacking , and @xmath72 stacking when layers of carbon atoms are located directly on top of each other  @xcite . the @xmath0 stacking is called hypothetical because it has not been observed yet in crystalline graphite . however , this configuration is expected in disordered or pregraphitic carbon  @xcite . in this paper , we consider the model of @xmath0 stacking which is the simplest one and allows us to study the effect of interlayer interaction on the emission properties analytically . let us consider the interacting graphite layers oriented parallelly to the electric field ( see fig . [ fig : orient ] ) . in the framework of the @xmath0 model the interaction modifies the 3d band structure near the fermi level which can be written in a simple analytical form @xcite @xmath73 here the @xmath74 parameter represents the interaction between first - neighboring atoms in a layer , @xmath75 is related to the interaction between two atoms of the same projection on the @xmath21 plane , from two neighboring layers , and @xmath76 is the interlayer spacing . the influence of other parameters @xmath77 and @xmath78 introduced in @xcite is suggested to be negligible and only linear terms in @xmath79 perturbation expansion are taken into account . actually , the analysis in  @xcite shows that the maximum effect of the next - to - leading term ( breaking the cylindrical symmetry ) is of the order of five percent . the upper sign in ( [ 3ds ] ) corresponds to the conduction band , and the lower sign corresponds to the valence band , @xmath80 .    in compliance with ( [ 3ds ] ) [ fig : fermi1 ] represents the fermi surface of the @xmath0 graphite . the fermi surface is composed of a hole pocket ( the valence band , sigh `` minus '' in ( [ 3ds ] ) ) and two half pockets of electrons ( sign `` plus '' in ( [ 3ds ] ) ) . in our case , the emission occurs along the z - axis . generally , the possible values of the momentum of emitting electrons with respect to the fermi surface can be collected into five different groups . we call them _ independent channels_.        solid lines in fig . [ fig : fermi1 ] indicate five possible types of independent channels : ( 1 ) for an intermediate electron - hole region , ( 2,3 ) for holes , and ( 4,5 ) for electrons . for finite - size layers , quantization of momentum in the @xmath81 plane occurs . in this case , the spectrum is written as @xmath82 where @xmath83 are integer , and @xmath84 lies in the region @xmath85 . the total fec is a sum of all channels . let us consider these contributions separately . the hole region is defined as @xmath86 . as is shown in fig . [ fig : fermi1 ] there are two types of channels for the hole region and the channel 1 can be considered as an intermediate case . [ holechannel ] shows all possible one - dimensional dispersion relations for this case . the channel 3 crosses the fermi surface at two points while the channel 2 does not cross the fermi surface . for each channel @xmath87 . the integration here is over the occupied states , and @xmath88 is the distance from the fermi level to the extremum point of the branch ( see fig . [ holechannel ] ) . for the channel 1 @xmath89 , and the current density is found to be equal to @xmath59 . notice that there are only two channels of this type . analytically , the channel 2 is defined as @xmath90 , where @xmath91 . for this channel one obtains @xmath92 here @xmath93 takes the form @xmath94 , which can be easily found from ( [ 3dsij ] ) . replacing the sum @xmath95 by the integral one gets @xmath96 d p_y = \frac{2eb}{h^2d^2 c\upsilon_f}.\ ] ] for the channel 3 one has @xmath97 and @xmath98 . like before , one obtains @xmath99      the electron region is defined as @xmath100 and @xmath101 or , taken into account a periodicity of the brillouin zone , @xmath102 . there are two kinds of channels in the electron region with spectra shown in fig . [ electronch ] . the channel 4 does not cross the fermi surface and , therefore , the current density turns out to be zero . for the channel 5 one has @xmath103 . this channel contains occupied states below the fermi level and , hence , there is nonzero contribution to the fec . one gets @xmath104,\ ] ] where @xmath105\}d p_y \nonumber\\   = \frac{2 eb[4 d \alpha_1/\pi + \bigl(\textbf{i}_0(2 d\alpha_1 ) - \textbf{l}_0(2 d \alpha_1 ) - 1\bigr ) ] } { d^2 c \upsilon_f h^2},&\end{aligned}\ ] ] where @xmath106 is the modified struve function , and @xmath107 is the modified bessel function . the total current density is found to be @xmath108}{h^2d^2 c \upsilon_f } .\ ] ] notice that @xmath109 reduces to ( [ g - j ] ) for @xmath110 . let us consider interacting graphite layers placed normally to the electric field ( see fig . [ fig : orient ] ) . this situation differs markedly from the previous case . let us denote @xmath111 in the dispersion relation in ( [ 3ds ] ) . quantization of momentum results in the replacement @xmath112 . there are only two types of channels in this case ( see fig . [ channelsnormal ] ) . -direction.,width=226 ]    as before , let us consider them separately . the hole region is defined by @xmath113 . there are two kinds of channels in the hole region with spectra shown in fig . [ fig : holech ] . the channel 1 is defined by @xmath114 . the current density reads @xmath115,\ ] ] where @xmath116 . one obtains @xmath117\ } 2 \pi \rho d \rho \\ \nonumber = \frac{4 \pi e b}{(d h)^3 \upsilon_f^2}\left(\frac{(2\alpha_1 d)^2}{2 } - 1 + \exp(- 2\alpha_1 d)(2\alpha_1 d + 1 ) \right).\end{aligned}\ ] ] the channel 2 is defined by @xmath118 , and @xmath119 finally , @xmath120 2\pi\rho d\rho \\ \nonumber = \frac{4 \pi e b(2\alpha_1 d + 1)}{(d h)^3 \upsilon_f^2 } \exp(-2\alpha_1 d)(\exp(2\alpha_1 d ) - 1).\end{aligned}\ ] ]      in the electron region @xmath121 . there are also two kinds of channels in this region with spectra shown in fig . [ fig : electronch ] . the channel 3 is equivalent to the channel 5 in the previous section . it is defined by @xmath114 . the current is @xmath104.\ ] ] here @xmath122 ( see fig .  [ fig : electronch ] and ( [ 3dsij ] ) . in this case , @xmath123\ } 2 \pi \rho d \rho\\   = \frac{4 \pi e b}{(d h)^3 \upsilon_f^2}\left(\frac{(2\alpha_1 d)^2}{2 } - 2\alpha_1 d + \exp(2 \alpha_1 d ) - 1 \right).\end{aligned}\ ] ] the channel 4 is defined by @xmath118 . since it does not cross the fermi surface the current density is equal to zero . the total current density is the sum of all channels @xmath124.\ ] ] [ fig : jf ] shows the calculated current densities as functions of the applied electric field . for comparison , the fowler - nordheim curve is drawn .    ) and normally ( @xmath125 ) to the electric field with @xmath126 ev , @xmath127 ev . for comparison the fawler - nordheim curve ( @xmath128 ) is shown.,width=226 ]    the most important difference comes from preexponential factors . as is known , the fowler - nordheim theory gives @xmath129 at all @xmath37 . in accordance with ( [ result ] ) the preexponential factor has a different field - dependence . at small @xmath37 one obtains @xmath130 . when @xmath37 increases ( which means @xmath131 ) the current density comes to @xmath132 and , finally , @xmath133 as was shown in section [ nonint ] . indeed , at large @xmath16 the difference @xmath134 tends to @xmath135 while at small @xmath16 it tends to @xmath136  @xcite . this is clearly seen in fig . [ fig : all ] where the comparative curves are demonstrated . for @xmath125 we have a similar behavior . according to ( [ result2 ] ) , @xmath137 at small @xmath37 , and @xmath138 at large @xmath37 , so that @xmath139 tends to a constant with increasing @xmath37 . one can conclude that the bigger is the electric field the lesser is the role of the interlayer interaction . the anisotropy of the emission from the 3d graphite is also shown in fig . [ fig : all ] . as is seen , @xmath140 , that is almost a constant in the considered interval of @xmath37 . therefore , we obtain three times increase in fec when graphite layers are oriented in parallel with the electric field .     is almost a constant in the considered interval of @xmath37 . , width=6 ]    it is interesting to discuss the dependence of fec from the parameter @xmath75 which characterizes the interlayer interaction . it was found in @xcite that this parameter is very sensitive to the interlayer distance . based on their results one can approximate @xmath141 where @xmath142 and @xmath75 is measured in ev . , @xmath143 for two field orientations . @xmath125 is markedly more sensitive to @xmath16 than @xmath144.,width=226 ]    as is seen from fig . [ interl ] , there is a strong dependence of the fec from the interlayer distance . the less is this distance the more is the emission current . this is valid for both orientations . it would be interesting to check this finding in experiments with graphite crystals under pressure . notice that this result follows from the fact that the dos at the fermi level ( which is of the most importance in the emission process ) is determined by @xmath75 ( see @xcite ) . as is seen from fig . , [ interl ] , @xmath125 is more sensitive to @xmath16 than @xmath144 . moreover , for @xmath145 one has @xmath146 , which follows from the fact that the movement of electrons between layers is suppressed in the absence of the interlayer interaction . in conclusion , we have found that the band structure of the 3d graphite has a marked impact on the field emission current . experimentally , the field emission from carbon materials was studied in @xcite . unfortunately , the polycrystalline carbon films used in experiment can not be properly described in the framework of our approach because for this purpose we have to consider a mixture between different crystalline structures . in fact , the fermi surface of the @xmath147 structure of graphite is found to be much more complex and , in particular , it does not possess the cylindrical symmetry ( see , e.g. ,  @xcite ) . in this case , our approach should be markedly modified . besides , many additional factors like the presence of a diamond - like phase on the surface of samples and the absence of any information about the local electric fields do not allow us to clarify the role of the band structure in this case . therefore , specific emission experiments with graphite single crystals at different orientations of the electric field would be of evident interest .    this work has been supported by the russian foundation for basic research under grant no . 05 - 02 - 17721 jonge b n and bonard j 2004 _ phil . r. soc . lond . a _ * 362 * 2239 obraztsov a n , volkov a p and pavlovskii i yu 1998 _ jetp letters _ * 68 * 59 modinos a 1984 _ field - , thermionic - and secondary electron emission spectroscopy _ ( new york : plenum press ) shi - dong l and xu n s 2003 _ app . lett . _ * 83 * 1213 shi - dong l , huang , n y , deng s z and xu n s 2004 _ app . lett . _ * 85 * 813 xinghui l , changchun z and yukui l 2003 _ physica b _ * 344 * 243 wallace p r 1947 _ phys . rev . _ , * 71 * 622 slonczewski j c 1958 _ phys . rev . _ * 109 * 272 mcclure j w 1964 _ ibm . _ * 8 * 255 charlier j c , michenaud j p , gonze x and vigneron j p 1991 _ phys . b _ * 44 * 13237 saito r , fujita m , dresselhaus g , and dresselhaus m s _ app . lett . _ * 60 * 2204 gadzuk j w and plummer e w 1973 _ rev . * 45 * 487 edgcombe c j 2002 _ phil . mag . b _ * 82 * 1009 hawkes p w and kasper e 1989 _ principles of electron optic vol . 2 _ ( london : academic press ) abramowitz m , stegun i 1984 _ handbook of mathematical function _ ( new york : dover publication ) * 12.2.6*. charlier j c , michenaud j p and gonze x 1992 _ phys . rev . b _ * 46 * 4531
the field emission of crystalline @xmath0 graphite is studied within a simple analytical approach with account of the exact dispersion relation near the fermi level . 
 the emission current is calculated for two crystal orientations with respect to the applied electric field . 
 it is found that the exponent of the fowler - nordheim equation remains the same while the preexponential factor is markedly modified . for both field directions 
 , the linear field dependence is found in weak fields and the standard quadratic fowler - nordheim behavior takes place in strong fields . a strong dependence of the emission current from the interlayer distance is observed . as an illustration of the method the known case of a single - walled carbon nanotube 
 is considered .