Bandstructure and Contact Resistance of Carbon Nanotubes Deformed by the Metal Contact
BBandstructure and Contact Resistance of Carbon Nanotubes Deformed by the MetalContact
Roohollah Hafizi,
1, 2
Jerry Tersoff, and Vasili Perebeinos ∗ Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 143025, Russia Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran. IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA (Dated: September 30, 2018)Capillary and van der Waals forces cause nanotubes to deform or even collapse under metalcontacts. Using ab-initio bandstructure calculations, we find that these deformations reduce thebandgap by as much as 30%, while fully collapsed nanotubes become metallic. Moreover degeneracylifting, due to the broken axial symmetry and wavefunctions mismatch between the fully collapsedand the round portions of a CNT, leads to a three times higher contact resistance. The latter wedemonstrate by contact resistance calculations within the tight-binding approach.
PACS numbers:
The fascinating mechanical and electrical propertiesof carbon nanotubes (CNTs) have attracted a lot of at-tention for a variety of technologies [1–8]. In partic-ular, semiconducting single-wall CNT field-effect tran-sistors have been considered for sub 10 nm technologynodes [4, 7–9]. CNTs provide high-performance channelsbelow 10 nanometers, but the increase in contact resis-tance with decreasing size dominates the performance ofscaled devices as the channel transport becomes ballistic.Only recently low-resistance end-bonded contacts havebeen demonstrated [7, 9], while most commonly used Pdcontacts establish side contacts [10]. In the latter, anelectron has to overcome two barriers: between the metaland the nanotube under the metal and between the nan-otube under the metal and the nanotube in the channel.While much efforts have been made to describe the for-mer type of barriers [11], less attention has been drawnto the latter.Although CNTs have a very large Young’s modulus inthe axial direction [12], they are rather soft in the ra-dial direction [13, 14], such that they can be deformedby the influence of van der Waals forces on two adjacentCNTs [15]. These deformations are predicted to be muchstronger in CNTs partially covered by a metal contact,due to capillary forces [16]. While much efforts in explor-ing electronic structure in deformed nanotubes have beenmade [17–20] including effects of the external transverseelectric field [21], little is known about how deformationsproduced by a metal modify the electronic structure and,as a result, the contact resistance.In this work, we investigate the effect of such deforma-tions on the electronic structure and contact resistancebetween the metal deformed and the round portions of ananotube. Geometry relaxations are done using the va-lence force model [16, 22, 23], electronic structure calcu-lations using Density Functional Theory (DFT) [24], andcontact resistance calculations within the tight-bindingapproach [25]. In deformed, but still open semiconduct-ing CNTs, we find bandgap reduction by 10%-30% de- pending on a tube diameter. The fully collapsed CNTsare found to be metallic. Most importantly, a new mech-anism for degeneracy lifting, arising from the interactionamong the π orbitals on adjacent sidewalls of a collapsednanotube, leads to qualitatively larger magnitudes of thebands splitting in the fully collapsed CNTs. The bandsplitting is now several hundreds of meV, such that thereis a dramatic impact on transport even at room temper-ature, unlike in previous work.To find atomic positions of carbon atoms in CNTs un-der metal, we employ a semi-atomistic model in whichpositions of carbon atoms are relaxed in the presence ofthe metal and substrate, where both are treated by a con-tinuum model, following Ref. [16]. The CNT interatomicinteractions are described by the valence force model [22],in which stiffness against the misalignment of neighbor-ing π orbitals is adjusted [26] to reproduce the bendingstiffness of D = 1 . eV [16, 27–29]. The substrate isplanar and rigid, and the metal is isotropic. We choosesurface energy γ = 12 . eV /nm as for Pd metal [30, 31],the most-used contact metal.Atomic positions are found by minimizing the totalenergy of the system, described by E = E bending + (cid:88) i,j U cc ( i, j ) + (cid:88) i (cid:90) U sc ( i ) dS s + (cid:88) i (cid:90) U mc ( i ) dS m + γ (cid:90) dS m + (cid:90) (cid:90) U ms dS m dS s (1)where i and j run over carbon atoms in CNT, and U cc , U mc , and U sc are the van der Waals interactions de-scribing CNT self-interaction, the metal-CNT interactionand the substrate-CNT interaction, respectively. Theseinteractions are modelled by the usual 6-12 potentialwith parameters chosen to reproduce the binding energy2 . eV /nm (60 meV/carbon atom) and equilibriumspacing 0.34 nm for a flat graphene sheet [32–34]. Thefifth term is the metal surface energy and the last term isthe interaction between the metal and the substrate. For a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Metal (13,0)
OPEN (14,0)
OPEN (16,0)
OPEN (17,0)
OPEN (19,0)
OPEN (20,0)
OPEN (22,0)
OPEN (23,0)
COLLAPSED (25,0)
COLLAPSED
Substrate (26,0)
COLLAPSED
FIG. 1. Relaxed atomic positions (purple dots) over curva-ture obtained using a continuum model [16, 23] (pink dash).Grey regions show metal contacts and black lines show theirsurfaces. Axes show scales in nanometers. B a nd G a p ( e V ) Under metalSuspended ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FIG. 2. Bandgap values from DFT of the round and metal-deformed zig-zag CNTs as a function of CNT diameter. R min G a p ( e V ) (13, 0) open(16, 0) open(19, 0) open(22, 0) open(25, 0) open(22, 0) Collapsed(25, 0) Collapsed FIG. 3. Bandgap values as a function of the minimum radiusof curvature from DFT in the open deformed and collapsedzig-zag CNTs. simplicity, we have chosen the same parameters for allvan der Waals interactions, such that the wetting angleof the metal on the substrate is 145 ◦ .CNT geometry corresponding to the energy minimumin Eq. (1) is used for the electronic structure DFT cal-culations, employing the PBE/GGA approximation forthe exchange-correlation energy [24]. The vacuum usedto separate CNTs is 7 ˚A and the k-mesh is 1 × × -0,4-0,200,20,4 E n e r gy ( e V ) K-0,4-0,200,20,4 (b)(c) (d)(a)(19,0) (19,0)(25,0) (25,0)
FIG. 4. DFT band structures of the metal-deformed (19,0)and collapsed (25,0) nanotubes in (a) and (c), correspond-ingly. Tight-binding model calculations [35, 36] for the samegeometries are shown in (b) and (d). Zero of energy cor-responds to the Fermi level position. Note that the doubledegeneracy is lifted due to the broken axial symmetry in bothcases, but by a much greater amount in the fully collapsedgeometry. C ondu c t a n ce ( e / h ) (19,0)(25,0) (a)(b) FIG. 5. Contact conductances as a function of energy be-tween the suspended and metal-deformed CNTs (solid blueline), suspended-suspended CNTs (dashed green line), anddeformed-deformed CNTs (dashed purple line) for (19,0)CNT in (a) and (25,0) CNT in (b). find is (22,0), although metastable collapsed geometriesfor (19,0), (20,0), and (22,0) CNTs are found, which isconsistent with Ref. [16, 23]. We find that atomistic andcontinuum model geometries in Fig. 1 are very similar.Nevertheless, in collapsed CNTs, carbon atoms in the toplayer of the collapsed region prefer not to lie on top ofcarbons in the bottom layer, as shown in Fig. 1. It isenergetically more favorable for carbon atoms to slide by half the C-C bondlength in the curvature direction.Fig. 2 shows DFT bandgaps in suspended, i.e. roundCNTs relaxed in the absence of the substrate and themetal [38], and in the metal-deformed CNTs. After defor-mation, bandgap values are reduced by greater amountsin (3n+1,0) CNTs than in (3n+2,0) CNTs, such thatbandgap reductions fall in between 13% for (17,0) and31% for (22,0) nanotubes. We find all collapsed CNTs tobe metallic. We don’t find drastic differences in the elec-tronic structures as we slide atoms along the curvaturedirection, which is equivalent to rolling the tube alongthe surface obtained from a continuum model.To explore the origins of the electronic structure modi-fications, we produced deformed CNTs structures withinthe continuum model by increasing metal surface energy γ in Eq. (1) and setting van der Waals attraction be-tween the sidewalls to zero, i.e. U cc = 0. The bandgapsgradually reduce with reducing the minimum radius ofcurvature R min and become zero at some critical valueof R min ∝ . − . U cc in the total en-ergy minimization, we stabilize collapsed CNTs in whichthe bandgaps become zero at much larger critical valuesof R min ∝ . − . R min due to the π - π interactions inthe former. Thus we conclude, that π - π interactions be-tween the orbitals on adjacent sidewalls separated at thevan der Waals distance in the collapsed CNTs producemuch stronger electronic structure modifications than thecurvature induced σ - π interactions.It is expected that changes in the electronic structurecaused by the deformations would introduce contact re-sistance at the interface between the round and the de-formed portions of a nanotube. To quantify the magni-tude of resistance we solve a scattering problem withina tight-binding formulation [25]. Following Ref. [39], wedescribe nanotube bandstructure by a four-orbital tight-binding model to account for curvature-induced mixingof π and σ orbitals [35]. In the case of collapsed CNTs,an additional long-range interaction between carbons be-longing to different layers is added [36]. The resultingband structure agrees fairly well with the DFT results,as shown in Fig. 4.The contact resistances of the round-deformed inter-faces in (19,0) and (25,0) nanotubes are shown in Fig. 5aand 5b, correspondingly. As expected, in both cases theconductance is smaller than for a uniform tube, whetherround or uniformly deformed, as shown by the dashedcurves in Fig. 5. For the interface between the de-formed and the round portions of (19,0) nanotube, wesee very little scattering due to wavefunction mismatch,such that the contact conductance is around 95% ofthe ideal conductance, at a typical metal-induced dop-ing level of about 0.1-0.2 electrons per nm. However, amuch stronger effect on the resistance is found for theround-collapsed contact interface. Lifting the double de-generacy alone in the fully collapsed (25,0) CNT resultsin a two times higher contact resistance as compared tothe ideal resistance of R = h/ e = 6 . W = 0 .
35 eV [40].The metal-CNT capacitance C M = 2 πε /ln (1 + 2 d /d )depends on CNT diameter d and electrostatic distance d = 2 . ρ = C M eφ = − (cid:90) ∞ E NP DOS ( E − E NP ) f ( E − E F ) dE + (cid:90) E NP −∞ DOS ( E − E NP )(1 − f ( E − E F )) dE (2)where charge neutrality point is E NP = ∆ W − eφ , DOS ( E ) is the density of states from the DFT calcu-lations, and f is the Fermi-Dirac function. Dopingdetermines the number of conduction modes M at theFermi energy E F and the maximum on-state conductance G on [41–43] (in the absence of the tunneling current con-tribution [40]): G on = 1 R on = 2 e h (cid:90) ∞−∞ T ( E ) M ( E ) (cid:18) − ∂f ∂E (cid:19) dE (3)where T ( E ) is the transmission coefficient and the spindegeneracy is included. Using DOS for a round CNTin Eq. (2), the self consistent doping level would be E V − E F = 8 meV, where E V is the top of the valenceband, and an ideal on-state resistance of R on = 11 . T ( E ) = 1 and T = 300 K.In the deformed CNT, the self consistent doping level islower E V − E F = 20 meV due to the smaller bandgapand the ideal on-state resistance is R on = 9 . T ( E ) as a function ofdoping level in the round portion of the nanotube, whichis controlled by the backgate voltage and which we modelby introducing a rigid shift of the bandstructure in theround CNT with respect to that in CNT under the metal,we find resistance according to Eq. (3), shown in Fig. 6.We estimate a typical carrier density in the on-state usinga wrap around SiO gate with dielectric constant ε = 3 . R on = 10 . T ( E ) = 1.In the case of fully collapsed (25,0) CNT, the self-consistent Fermi energy of a round CNT under the metal FIG. 6. Contact resistance in (19,0) (blue) and (25,0) (red)CNTs at T = 300 K as a function of carrier density in theround portions of CNT. The Fermi level in CNT under themetal is assumed to be fixed, as explained in the text. Thevertical dashed lines correspond to the estimated carrier den-sities in the on-state (see text). The inset shows a schematicof a CNT field-effect transistor contact [44]. would be E V − E F = 36 meV and the correspondingon-state resistance R on = 8 . DOS of thefully collapsed CNT, self consistent doping is found to be E V − E F = 208 meV and the ideal contact resistance 6 . T ( E ) = 1and the doubly degenerate bands.In conclusion, we identified the major effect of metal-induced nanotube deformations on the electronic struc-ture and the electrical contact resistance at the inter-face between the deformed and the round portions of ananotube. While the bandgap reduction in the deformedCNTs increases metal-induced doping of a nanotube, andthus reduces the resistance by increasing the number ofthe conduction channels, wavefunction mismatch intro-duces an additional scattering at the contact, hence, par-tially compensating the effect of doping. 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