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carbon nanotubes belong to the most promising new materials for the future molecular electronics , they are believed to potentially replace in the near future the silicon - based conventional electronics . to illustrate the enormous scientific and technological progress that has been made since carbon nanotubes were discovered it is worth to mention new concepts such as : the room temperature single electron transistor,@xcite the ballistic carbon nanotube field - effect transistor@xcite or the non - volatile random access memory for molecular computing.@xcite recently several both experimental@xcite and theoretical@xcite papers have been published on spin - dependent electrical transport in ferromagnetically contacted carbon nanotubes in an attempt to test their ability to operate as spintronic devices . it has been found that carbon nanotubes though intrinsically nonmagnetic reveal quite a considerable gmr effect . however , it should be stated in this context that the experimental results are very diverse , and reflect to a great extent sample - specific features . the poor reproducibility of experiments on carbon nanotubes , or more generally on molecular systems , is due to hardly controllable interface conditions between the molecule and external electrodes . incidentally , judging from the conductance value , this difficulty is sometimes successfully overcome but , to our knowledge , only if nonmagnetic - electrodes are used . the examples proving this statement are numerous as far as single wall carbon nanotubes ( swcnts ) are concerned,@xcite and much more seldom in the case of multiwall carbon nanotubes ( mwcnts ) @xcite . despite intensive studies,@xcite fundamental problems such as ( _ i _ ) the internal structure of nanotubes ( especially multiwall ones ) , ( _ ii _ ) the aforementioned nature of the electrode / nanotube coupling , and ( _ iii _ ) the energy band line - up at the interfaces are still far from being well understood . motivated by these facts , we study a minimal geometrical model of the mwcnt , which enables us to define in a unique way the coupling of the mwcnt to the electrodes . having done that , we can restrict ourselves to the most crucial issues addressed in the present studies , namely : the inter - wall interactions , and the effect of the breakage of the contact between the inner wall and one of the transversally coupled ( burying ) electrodes . so far tangential contact geometries have been mainly studied in connection with stm experiments.@xcite however in experiments with ferromagnetic - electrodes , transition metals are usually deposited by evaporation on top of the carbon nanotube cutting off a segment of the latter , so that the nanotube finds itself buried in the electrode and the contact geometry can be regarded as transversal . ( color online ) view of the ( 2,2)@(6,6 ) carbon nanotube sandwiched between two fcc(111 ) leads and detail of the contact region . what is shown consists of a few ferromagnetic - electrode atoms with the nanotube forming the so - called extended molecule . the other parts of the electrodes ( not shown ) are infinite in all the directions . ] ( color online ) the total conductance for the parallel ( pa ) and antiparallel ( ap ) alignments , as well as the gmr _ vs. _  energy . the outer nanotube length equals that of the inner one ( @xmath0=@xmath1=41 carbon rings ) , the fermi energy is equal to @xmath2 and @xmath3 . left panel : there are no inter - wall interactions ( @xmath4 ) . middle and right panels : the inter - wall interactions are included , @xmath5 and @xmath6 , respectively . ] our system consists of a double - wall carbon nanotube ( dwcnt ) perfectly contacted to semiinfinite electrodes having an fcc(111 ) geometry : every carbon atom at the interface has exactly three nearest neighbors in the electrode ( see fig .  [ fig : fig1 ] ) . such a construction is possible in the case of the ( 6,6 ) armchair swcnt@xcite due to the almost perfect fitting of the armchair nanotube lattice constant ( @xmath7   ) to the interatomic distances at the electrode close packed surfaces ( _ e.g. _  2.51 , 2.49 and 2.55    for co , ni and cu respectively ) . this coincidence makes it possible to put the interface ( contact ) ring of carbon atoms on top of the electrode substrate in such a manner that all carbon atoms sit in the geometrical centers of three adjacent electrode - atoms , and additionally the perimeter of the contact carbon ring equals , within a few percent , the standard value for the ( 6,6 ) swcnt , _ i.e. _  @xmath8 . on geometrical grounds , the only other armchair structure which can be constructed likewise is the ( 2,2 ) one ( see fig .  [ fig : fig1 ] ) . so we will use it in our studies as the inner tube , in spite of the fact that it is known to be energetically unstable.@xcite as a matter of fact , since we focus here on electrical transport properties rather than stability problems , we are convinced that the present model gives a qualitative insight into the role of the inter - wall interaction in real mwcnts , depending on whether or not the inner tubes are fully contacted to the electrodes . in this very context , the present model is advantageous in view of its small number of atoms within the unit cell , ideal geometrical fitting to the electrode surface atoms , and its well - defined inter - wall coordination zones . the last feature enables us to define inter - wall hopping integrals in such a way that each inner - tube carbon atom has just two outer - tube atoms to interact with . thus , again no ambiguity exists concerning the interaction range , and no hidden cutoff parameters are needed . 1 corresponds to an odd number of carbon rings ( inverse - symmetric system ) . the case of even numbers of rings makes no serious complication@xcite , but the drain electrode has to be rotated by @xmath9 with respect to the source one if the perfect geometrical matching at the drain interface is also required . ( color online ) unlike in the previous figures , here the inner tube is out of contact to the right electrode ( see the upper right diagram ) . a drastic drop in the conductance ( at @xmath10 ) , accompanied by a positive gmr , takes place . no inter - wall interactions ( left panel ) , and @xmath11 and @xmath6 ( middle and right panels , respectively ) . ] the device under consideration is described by means of a single - band tight - binding hamiltonian , assuming @xmath12 and @xmath13-electrons in the double - wall carbon nanotube and ferromagnetic - electrodes , respectively . the total hamiltonian can be compactly written as @xmath14 with vanishing initial on - site potentials , @xmath15 , in the dwcnt , whereas in the electrodes , they are spin - dependent and chosen so as to give the required magnetization . to solve the green function problem , we use the partitioning technique,@xcite treating the whole device as a left - electrode  dwcnt right - electrode system ( shorthand notation l - c - r ) . here c stands for the central carbon - based part , which incorporates not only the dwcnt but also the first two closest atomic layers ( with @xmath16 atoms ) from both the l- and r - electrode , implementing the extended molecule concept ( as shown in fig . [ fig : fig1 ] ) . the green function of the extended molecule reads @xmath17 whereas the density matrix and the conductance ( per spin ) are given by :     ( color online ) as fig.2 but for 40 carbon rings in each nanotube . ] @xmath18    @xmath19    with @xmath20 and @xmath21 being the fermi - dirac distribution function ( @xmath22  l , r ) . the @xmath23-th electrode surface green function , @xmath24 , is calculated as described in ref .  . while back fourier transforming to the real space , the integration over the two - dimensional brillouin zone has been performed by the _ special - k - points method_.@xcite the giant magnetoresistance is defined as gmr@xmath25 , where the arrows denote the aligned and antialigned magnetic configuration . to parameterize the hamiltonian , we have put the polarization of the electrodes equal to 50% ( with the number of electrons per atom being 0.75 and 0.25 for majority and minority bands , respectively ) . the energy scale is given by the hopping integral @xmath26  ev and the nanotube lattice constant is @xmath7   , fixing the energy and length units , respectively . in order to minimize the number of free parameters the nearest neighbor hopping integral is kept constant in the entire device . as regards the inter - tube interaction we treat the hopping integral as a free parameter of the order of @xmath27 ( as for the interlayer distance in graphite@xcite ) , although in general @xmath28 depends exponentially on distance and also on angles between the @xmath12 orbitals of the involved atoms.@xcite it should be also admitted that another complexity , not captured by our simple structural model ( two armchair tubes ) , is a possible additional off - diagonal disorder due to incommensurability between shells ( cf . ref . ) . +     ( color online ) the conductance in the case of paramagnetic - electrodes . the breakage of the inner - tube contact to one of the electrodes results in a drastic drop of the conductance ( the lower curves ) . the inter - wall hopping integrals , @xmath29 , are equal to @xmath30 and @xmath6 ( solid , dashed and dotted curves , respectively ) . ] our energy band line - up procedure goes as follows : first , @xmath10 is fixed at 0 , next the on - site potentials in the electrodes are set so as to yield the required number of electrons per atom ( @xmath31 ) and magnetic moment ( 0.5 and 0 in the ferromagnetic and paramagnetic cases , respectively ) . the so - determined fermi energy is kept fixed at the charge neutrality point and no doping effects are taken into account . finally , all the on - site potentials within the extended molecule are self - consistently computed from the global charge neutrality condition and from eq . ( [ eq : rho ] ) . rather than restrict ourselves to the fermi energy , we present our results in an energy window , accessible e.g. by applying a gate potential ( see e.g ref .  ) . we have first considered a perfectly contacted case , when both walls are in touch with the source and drain electrodes . the results , presented in fig .  [ fig : fig2 ] , show that for the mwcnt length equal to 20.5_a _ ( 41 carbon rings ) the gmr is negative regardless of whether inter - wall interactions are included or not . next , the inner tube has been allowed to be out of contact to , say , the drain electrode , see fig . [ fig : fig3 ] . it turns out that then , not only the conductance of the whole system decreases roughly twice , but also the gmr becomes positive . the effect of the inter - wall interactions is moderate ( at least up to @xmath32 ) , they happen to increase the conductance , always keeping the gmr positive . the inverse gmr effect in the case of full end - contacted nanotubes , and @xmath29 not exceeding a length - dependent threshold value , has been also found for the nanotube lengths of 40- and 39-rings , and is depicted in fig . [ fig : fig4 ] for @xmath33=40 . as it is seen , the gmr is now far less robust than for @xmath33=41 . judging from the conductance behavior at @xmath10 for 3 consecutive nanotube lengths , one can anticipate what happens for an arbitrary length , because the conductance is a quasi - periodic function with a period equal to 3-rings spacing . thus , for dwcnts which are @xmath34 rings long we predict a quite pronounced negative gmr effect ( @xmath33 is an integer ) . the @xmath34 length rule ( in a/2 units ) was numerically found@xcite , with these lengths an ideal nanotube has got maximum conductance (  on - resonance device  ) . disorder at the interfaces makes the maximum of the conductance shift beyond @xmath10 , but the quasi - periodicity is still preserved . the aforementioned quasi - periodicity originates from the fact that ideally the swcnt armchair energy spectrum crosses the [ @xmath35-point for particular lengths only , and additionally conductance is a measure of the squared electron wave function .    as a test illustrating the quality of the present approach , we have completed our studies by additional computations for dwcnts sandwiched between _ paramagnetic _ electrodes with the same number of electrons per lattice site ( @xmath36 ) as in the ferromagnetic calculations . our attention has been again focused both on the effect of the inner nanotube contact to one of the electrodes , and the importance of the inter - wall interactions . the latter are now allowed to take values up to @xmath37 . [ fig : fig5 ] shows that the conductance in the vicinity of @xmath10 does not reach the maximum theoretical value of @xmath38 for a dwcnt ( @xmath39 contribution for every wall obtained for infinite mwcnts ) . the conductance suppression in mwcnts below the expected value for the infinite homogeneous tube has been experimentally well documented.@xcite in our model system charge - transfer induced changes in the band structure of the dwcnt lead to a reduction ( roughly to @xmath40 ) of the conductance -as one can see in the upper curves in fig . [ fig : fig5 ] . this finding is in line with the well - known scenario , that the reduction of the conductance is due to the interface - induced suppression of one of the two transport channels.@xcite obviously , if a finite armchair nanotube is contacted to external electrodes it can no longer be viewed as a periodic one with the repeat unit consisting of two carbon rings , so the band degeneracy is lifted and the two channels couple very differently to their counterparts in the leads . it is noteworthy that recent _ ab initio _ results on single wall carbon nanotubes end - contacted to paramagnetic electrodes also show that the interface mismatch may result in roughly halving the conductance.@xcite    the lower part of fig . [ fig : fig5 ] shows the effect of the inner - tube contact breakage . the conductance gets reduced by a factor close to two , very much like in the ferromagnetic computations . more importantly however , this behavior is in qualitative agreement with experimental results@xcite if we associate the lower and upper bunch of curves in fig . [ fig : fig5 ] with the major plateaux of the conductance of mwcnt immersed into hg . incidentally , the peculiar peak in fig . [ fig : fig5 ] more than 0.15 @xmath41 ( ca . 0.4 ev ) away from the charge neutrality point , is due to model - specific features of no universal nature .    both the gmr and the conductance are oscillatory , non - monotonic functions of the inter - wall coupling , and reveal a quasi - periodic behaviour with the dwcnt length . it is also noteworthy that the conductances in the ferromagnetic case are always clearly smaller ( regardless of the electrode magnetization alignment ) than their paramagnetic counterparts . we attribute this behavior to the increased energy band mismatch at the interfaces in the ferromagnetic case . in conclusion , we have investigated the effect of inter - wall interactions on the giant magnetoresistance of double wall carbon nanotubes , depending on whether or not the inner wall is fully contacted to the electrodes . we have found that the gmr of perfectly contacted dwcnts is very sensitive to the inter - wall coupling strength , and show a tendency to become negative . on the contrary , dwcnts with inner tubes being out of contact with one of the electrodes reveal ( close to the fermi energy ) a robust positive gmr only weakly dependent on the inter - wall coupling . these observations may be related to the experimental results showing either positive or negative ( sample - dependent ) gmr of comparable value ( @xmath42),@xcite as well as recently reported negative gmr values in mwcnts strongly contacted to permalloy electrodes.@xcite moreover , our complementary calculations for the case with paramagnetic electrodes show that the conductance of the perfectly contacted dwcnt is suppressed by a factor two with respect to the case of a homogeneous infinite dwcnt ( that is @xmath40 instead of @xmath43 ) . consequently , for the case in which the inner wall gets disconnected , we observe conductance values close to @xmath44 instead of @xmath40 . this is related to the charge transfer induced energy band rearrangements in the extended molecule . the present approach may be regarded as a reference for further generalizations . the diameters and the number of walls forming the mwcnt may be easily increased and the respective stable positions of the interface atoms can be readily determined by a simple relaxation procedure involving just a few variational parameters ( displacements and rotations ) . we are indebted to gerrit e. w. bauer , uwe krey and nitesh ranjan for fruitful discussions . sk acknowledges the support from the daad foundation and the kbn research project pbz - kbn-044/p03 - 2001 . this work was partially funded by the volkswagen stiftung and the eu centre of excellence contract g5ma - ct-2002 - 04049 .
we report on the giant magnetoresistance ( gmr ) of multiwall carbon nanotubes with ultra small diameters . 
 in particular , we consider the effect of the inter - wall interactions and the lead / nanotube coupling . 
 comparative studies have been performed to show that in the case when all walls are well coupled to the electrodes , the so - called inverse gmr can appear . 
 the tendency towards a negative gmr depends on the inter - wall interaction and on the nanotube length . 
 if , however , the inner nanotubes are out of contact with one of the electrodes , the gmr remains positive even for relatively strong inter - wall interactions regardless of the outer nanotube length . 
 these results shed additional light on recently reported experimental data , where an inverse gmr was found in some multiwall carbon nanotube samples .