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much recent experimental and theoretical work has focused on the electronic transport properties of nanowires , motivated by promises of novel electronic devices and the basic scientific challenges that they present .
one of the key findings in nanowire devices is that the coulomb interaction plays a crucial role in determining device properties .
for example , electrostatics in carbon nanotubes ( nts ) leads to unexpected behavior of intra - tube _
junctions@xcite , of intra - tube schottky junctions@xcite , and of fermi level pinning at contacts between nts and metals@xcite .
the importance of _ intra_-tube coulomb interactions naturally leads to the question of how _
inter_-tube interactions might influence device behavior . answering
this question is gaining increasing importance as experimental devices based on multiple nanotubes are becoming more common@xcite [ an example is shown in fig .
1(a ) ] , and also to address the question of device packing density .    in this paper
, we present self - consistent calculations based on the non - equilibrium green s function technique for planar carbon nanotube transistors containing multiple parallel nts . by analyzing the variations of the transistor characteristics with nt separation , we establish a length scale below which inter - tube interactions become important .
while for small channel lengths this length scale depends on the channel length , for long channel devices it becomes independent of the channel length and is essentially determined by the gate oxide thickness .
importantly , this length scale depends exponentially on the dielectric constant of the medium surrounding the nts , and can be substantially reduced by embedding the nts in a high-@xmath1 dielectric in the channel and using embedded contacts .
figure 1(b - e ) shows the device under consideration : infinitely long single - wall zigzag nts laying on metal electrodes at their two ends and on a dielectric in the channel region . a planar gate 10 nm below the dielectric surface controls the device behavior .
we take a spacing of 0.3 nm between    the nt and the metal or dielectric , and assume that the nt cross - section remains circular , a good approximation for the nts considered heredeformation .
the electronic properties of the nt are described using a tight - binding approach with one @xmath2 orbital per carbon atom , and a coupling @xmath3 ev between nearest - neighbor atoms .
semiconducting tubes are ( 17,0 ) nts ( band gap of 0.55 ev , diameter of 1.33 nm ) while metallic tubes are ( 18,0 ) nts ( diameter of 1.41 nm ) .
we take the nt midgap as the energy reference level , and use a metal workfunction of 5.5 ev , putting the metal fermi level 1 ev below the nt midgap .
the gate insulator dielectric constant is that of sio@xmath4    to apply the non - equilibrium green s function formalismdatta , leonardiccn to this system , we divide the nt in principle layers , with each layer corresponding to a ring of the zigzag nt .
the main quantity of interest is the green s function @xmath5 , from which one can obtain the transmission probability @xmath6 and the zero bias conductance    @xmath7 de\ ] ]    where @xmath8 is the fermi function .
@xmath5 is calculated by solving the matrix equation    @xmath9 ^{-1}\text { , } \label{gr}\ ] ]    where @xmath10 is the tight - binding hamiltonian for the isolated nt and @xmath11 is the electrostatic potential evaluated at the position of each layer . the functions @xmath12 and @xmath13 represent the interaction of the scattering region with the semi - infinite nt leads . in our tight - binding representation ,
the hamiltonian matrix elements for layer @xmath14 are @xmath15 , @xmath16 where @xmath17 is the number of atoms around a nt ring ( @xmath18 or @xmath19 here ) and @xmath20 labels each of the nt bands . in this representation
, we assume that the electrostatic potential on every atom of a ring is the same , a good approximation in this case since @xmath11 varies slowly over the nt diameter .
the electrostatic potential is calculated by solving poisson s equation in three - dimensional coordinates on a variable grid with the source charge on the nt , and with boundary conditions at the source , drain , and gate surfaces , and at the vacuum / dielectric interface .
this is done by combining a finite difference approach in the directions parallel to the nt and perpendicular to the substrate with a fast fourier transform in the direction perpendicular to the nts .
once the three - dimensional electrostatic potential is obtained , the value for @xmath11 on each ring is taken as the average of the electrostatic potentials on the points on the nt nearest and furthest from the substrate , at the position of each ring .    to obtain the charge density , we note that our tight - binding technique provides the total charge on each layer of the nt , which needs to be spatially distributed .
we assume a uniform distribution of the charge in the azimuthal direction , and spatially distribute the total charge in the radial and axial directions with a gaussian smearing function . the three - dimensional charge density
is then given by @xmath21where @xmath22 \\
\left [ -(r - r)^{2}/2\sigma _ { r}^{2}\right ] $ ] with @xmath23 the tube radius , @xmath24 the position of ring @xmath14 , and @xmath25 and @xmath26 the smearing lengths in the axial and radial directions respectively ( this expression for @xmath27 is valid when @xmath28 , and we used values of @xmath29 nm and @xmath30 nm ) .
our overall procedure is to solve the coupled set of nonlinear eqs @xmath31 self - consistently for a given gate - source voltage at zero drain - source bias .
figure 1(e ) shows the unit cells for our calculations to study semiconducting - semiconducting and semiconducting - metallic interactions .
we first discuss interactions between semiconducting nts .
the conductance of the nt transistor as a function of the gate - source voltage is plotted in fig . 2 for different values of the separation between the nts .
clearly , as the nt separation is reduced , the ntfet characteristics degrade ; in the range of gate voltages studied , the current can not be completely turned on for the smallest tube separation , while the behavior in the off state is essentially independent of tube separation .
this behavior arises because of the charge induced on the nts by the gate voltage , and by interactions between nts at the contact . as shown in fig .
2a , the off state corresponds to the fermi level in the middle of the nt bandgap in the channel , with little charge on the nts , so coulomb interactions in the channel between neighboring nts are negligible .
the on state however consists of making the nt _ p_-type in the channel by raising the valence band above the fermi level with the gate voltage ( fig .
2b ) ; this creates a large positive charge on the nt which interacts with the charge on neighboring nts , and lowers the electron energy .
this pushes the bands down , and reduces the conductance .    while interactions in the channel play an important role in determining device behavior
, nt - nt interactions also affects the contacts . because of the mismatch between the metal and nt workfunctions ,
contacts between nts and metals are governed by charge transfer between the metal and the nt , effectively doping the nt _ p_-type at the contact , with the metal fermi level just below the nt valence band edge in the contact@xcite . as the separation between nts decreases , the coulomb interactions between the positive charge at the contacts leads to a lowering of the bands ; at small enough separations , the valence band in the contact can be pushed below the metal fermi level .
thus inter - tube interactions at the contacts can lead to modifications of the nt / metal contacts from ohmic to schottky .
we find that this effect becomes important for inter - tube separations below @xmath32 nm ; the effects can be seen in the bottom curve of fig .
2 , where the on state conductance saturates to a value much smaller than the single - tube limit because of the schottky barrier .
these effects of inter - tube interactions on contact properties can be mitigated by embedding the nts in the metal contact .    to quantify the role of coulomb interactions _ in the channel _ , we calculated the conductance versus gate voltage for devices with different channel lengths ; we plot in fig .
3 the conductance at @xmath33 versus tube separation , for several values of the channel length ( the smallest value of @xmath34 is large enough to avoid contact effects ) . the solid lines in the figure are fits of the form @xmath35 , from which we extract the value of @xmath36 at which the conductance decreases by more than 10% from its @xmath37 value .
this length scale is plotted in fig .
4 as a function of channel length , and delimits regions where inter - tube interactions distort the single - tube behavior . clearly
, very short channel devices can have very high packing densities , while long channel devices are limited to tube separations of 15 nm . a notable aspect of the results of fig .
4 is that the value of @xmath36 is independent of the channel length for large @xmath38 . at first glance
, one would expect that a larger channel length leads to larger total charge on the nts and thus larger interaction energy . however , screening of the coulomb interaction by the planar gate leads to a different behavior , as we now discuss .    to derive approximate analytical expressions for the coulomb interactions between nts , we consider the simplified situation of fig .
5a : a nt of length @xmath38 is at a distance @xmath39 above a metallic substrate held a potential @xmath40 , and carries charge per unit length @xmath41 .
we use an image potential construction to calculate the electrostatic potential @xmath42 at a distance @xmath36 from the charged tube , giving    @xmath43    in the long channel limit @xmath44 , eq .
( [ v ] ) becomes@xmath45the important point here is that in this limit the potential shift is independent of the channel length @xmath38 .
taking a potential change of @xmath46 volts as a criterion for the importance of inter - tube effects , we obtain a separation@xmath47below which inter - tube effects become important .
thus the gate oxide thickness @xmath39 sets the length scale @xmath48 . in the above expression
, @xmath1 is the effective dielectric constant for the device geometry . for a simple comparison with our simulations
we take @xmath1 as the dielectric constant of sio@xmath49 , and use the computed value of @xmath50 e / c atom in the on state .
this gives @xmath51 nm in good agreement with our numerical results .
a key point is that the length scale @xmath48 depends _ exponentially _ on the gate insulator dielectric constant ; thus , replacing sio@xmath49 with high-@xmath1 dielectric materials ( ntfets have been fabricated with hfo@xmath49@xcite , sro@xmath49@xcite and srtio@xmath52@xcite ) allows a reduction of @xmath53 to extremely small values , thus permitting high device densities .
the general behavior of interactions between semiconducting nts can be understood from further analysis of eq . @xmath54 . near threshold ,
the charge on the nt can be written as @xmath55 where @xmath56 is the threshold voltage .
substitution in eq .
gives@xmath57hence , the gate voltage is rescaled and shifted by the interaction between nts .
this behavior is indicated in the inset of fig .
2c , showing good collapse of the data after appropriate rescaling and shifting of @xmath58 .    the results discussed above focused on interactions between semiconducting nts
. however , in multi - nt devices , there is often a mixture of semiconducting and metallic nanotubes .
a question therefore is how the presence of nearby metallic nts affects the properties of the semiconducting nts . to address this issue
, we performed calculations for the geometry depicted in fig .
there , a semiconducting nt is separated from two metallic nts by a distance @xmath36 .
figure 6 shows the calculated conductance vs gate voltage dependence for a 10 nm channel device .
( the conductance plotted here is that of the semiconducting nt only . for single electrodes making contact to all the nts
, the conductance of the metallic tubes would have to be included to obtain the total device conductance . ) the behavior in fig .
6 is similar to that observed for interactions between semiconducting nts , i.e. a degradation of the characteristics with decreasing separation between the nts .
however , because the charge on the metallic tubes is not strongly modulated by the gate voltage , the physics is somewhat different ; indeed , the metallic nts essentially consist of lines of constant potential . to their effect on the semiconducting nts we consider the situation of fig .
5b , where a line held at a constant potential @xmath59 is above a metallic plane held at potential @xmath58 .
solution of the laplace equation for this geometry gives the potential on the nearby nt as@xmath60thus the presence of the metallic tubes shifts and rescales the gate voltage seen by the semiconducting tube .
for small separations between the nts , the potential on the semiconducting nt approaches that of the metallic nt , which is larger than that of an isolated semiconducting nt .
thus , at the contact , the fermi level is pushed above the valence band edge , leading to a schottky barrier and saturation of the conductance to lower values .
the behavior expressed in eq .
( [ scaling ] ) can be verified by looking at the off state behavior , and scaling and shifting the gate voltage appropriately ; the inset in fig .
6 shows the resulting collapse of the data .
in summary , we have shown that coulomb interactions between carbon nanotubes can have a strong influence on device behavior .
below a characteristic length scale , interactions between nts can significantly degrade device behavior ; this can be understood in terms of a rescaling and shift of the applied gate voltage . for long channel devices ,
the tube separation above which interactions between nts is negligible becomes independent of the channel length , and is set by the gate oxide thickness .
this length scale can be substantially reduced by using high-@xmath1 dielectrics , due to an exponential dependence on dielectric constant . while this paper focused on the static device properties
, we expect that interactions during current flow ( e.g. coulomb drag ) would also have intriguing properties .
we hope that this work will stimulate controlled experiments to further explore interactions between nanotubes .
sandia is a multiprogram laboratory operated by sandia corporation , a lockheed martin company , for the united states department of energy under contract de - ac01 - 94-al85000 .
we thank k. mcdonald , f. jones , and a. talin for providing the sem image in fig . 1 .
at reduced dimensionality , coulomb interactions play a crucial role in determining device properties . 
 while such interactions within the same carbon nanotube have been shown to have unexpected properties , device integration and multi - nanotube devices require the consideration of inter - nanotube interactions . 
 we present calculations of the characteristics of planar carbon nanotube transistors including interactions between semiconducting nanotubes and between semiconducting and metallic nanotubes . 
 the results indicate that inter - tube interactions affect both the channel behavior and the contacts . for long channel devices , 
 a separation of the order of the gate oxide thickness is necessary to eliminate inter - nanotube effects . because of an exponential dependence of this length scale on dielectric constant , very high device densities are possible by using high-@xmath0 dielectrics and embedded contacts .    2