Electronic Structure of Carbon Nanotube Ropes

PHYSICAL REVIEW B VOLUME 61, NUMBER 16 15 APRIL 2000-II

Electronic structure of carbon nanotube ropes

A. A. Maarouf, C. L. Kane, and E. J. Mele Department of Physics, Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104 ͑Received 21 September 1999͒ We present a tight-binding theory to analyze the motion of electrons between carbon nanotubes bundled into a carbon nanotube rope. The theory is developed starting from a description of propagating Bloch waves on ideal tubes, and the effects of intertube motion are treated perturbatively on this basis. Expressions for the interwall tunneling amplitudes between states on neighboring tubes are derived which show the dependence on chiral angles and intratube crystal momenta. We ﬁnd that conservation of crystal momentum along the tube direction suppresses interwall coherence in a carbon nanorope containing tubes with random chiralities. Nu- merical calculations are presented which indicate that electronic states in a rope are localized in the transverse direction, with a coherence length corresponding to a tube diameter.

I. INTRODUCTION Recognizing this, several groups have attempted to estimate the energy scale for similar effects in nanotube ropes.8 A carbon nanotube is a cylindrical tubule formed by Here the situation is much more delicate, since the structure wrapping a graphene sheet. Single-wall carbon nanotubes of a (10,10) nanotube does not permit perfect registry be- ͑SWNT’s͒ can be synthesized in structures 1 nm in diameter tween neighboring tubes when they are packed into a trian- and microns long.1 There has been particular interest in the gular lattice. Nevertheless, it is possible to construct a nano- electronic properties of SWNT’s, which are predicted to ex- tube crystal, a hypothesized ordered structure in which each ist in both conducting and semiconducting forms.2 Remark- ͑10,10͒ tube adopts the same orientation, and to study its ably, it is possible to probe this behavior experimentally by electronic properties with conventional band theoretic contacting individual tubes with lithographically patterned methods.8 Theoretical studies on nanotube ropes show that electrodes or by tunneling spectroscopy on single tubes.3 intertube interactions in a nanotube crystal lead to a mixing However, most methods for synthesizing carbon nanotubes of forward- and backward-propagating electronic states near do not produce isolated tubes; instead the tubes self- the Fermi energy. The level repulsion between these assemble to form a hierarchy of more complex structures. At branches leads to a suppression or ‘‘pseudogap’’ in the elec- the molecular scale tubes pack together to form bundles or tronic density of states, on an energy scale estimated to be a ‘‘ropes’’ which can contain 10–200 tubes. X-ray diffraction few tenths of an electron volt. It should be noted that these reveals that a bundle contains tubes close packed in a trian- effects are qualitatively different from those found in graph- gular lattice,1 and measurements of the lattice constant and ite, where intersurface interactions lead to band overlap and tube form factor led initially to the suggestion that these thus an enhancement of the Fermi-level density of states. ropes contain primarily (10,10) nanotubes, a species pre- It has not yet been possible to extend these ideas to car- dicted to be metallic.2 Subsequent work has demonstrated bon nanotube ropes which contain a mixture of tubes with that the ropes likely contain a distribution of tube diameters various diameters and chiralities. A direct calculation of the and chiralities.4 On larger scales the ropes bend and entangle, electronic structure for such a rope, which we define as so that the macroscopic morphology of a carbon nanotube ‘‘compositionally disordered,’’ is quite complicated since the sample is that of an entangled mat. Carbon nanotubes are system has no translational symmetry either along the rope also formed in various thick multiwalled species which ex- axis or perpendicular to it. hibit their own unique electronic behavior.5 In this paper we develop a tight-binding theory for the The electronic properties of isolated SWNT’s are con- coupling between tubes. In the absence of intertube coupling trolled by the tube’s wrapping vector, curvature, and the electronic states on an isolated tube are essentially the torsion.6 However, in ropes and in multiwalled tubes the in- Bloch waves of the graphene sheet wrapped onto the surface teractions between graphene surfaces is expected to play a of a cylinder and indexed by a two-dimensional crystal mo- major role. This is the case even for crystalline graphite in mentum k. The essence of our theory is to develop the ef- the Bernal structure. Although an isolated graphene sheet is a fects of intertube interactions t(k1 ,k2) perturbatively. We zero gap semiconductor, the small residual interactions be- find that the effects of intertube interactions in a disordered tween neighboring graphene sheets with the ordered A-B rope are quite different from what one obtains for a crystal- stacking sequence lead to a small overlap of bands near the line rope. In fact, we find that compositional disorder intro- Fermi energy and eventually to conducting behavior.7 duces an important energy barrier to intertube hopping Graphite is an ordered three-dimensional crystal for which within a rope, so that intertube coherence is strongly sup- weak intersurface interactions are sufficient to establish pressed. We are led to conclude that eigenstates in a compo- quantum coherence for electronic states on neighboring sitionally disordered rope are strongly localized on indi- sheets. Thus the electrons can delocalize both parallel to the vidual tubes, though they can extend over large distances graphene sheet and perpendicular to it. along the tube direction. Numerical results illustrating this

effect will be presented in this paper. We believe this physics underlies the experimental observation that charge transport at low temperature occurs by hopping conduction in nanotube ropes and mats. In Sec. II we develop the tunneling model for describing tight-binding coupling between neighboring tubes in a rope. In this section we derive an analytic expression giving the tunneling amplitude t(k1 ,k2) between Bloch states on neighboring tubes indexed by momenta k1 and k2. In Sec. III we apply the method to study the electronic structure of a rope crystal, and show that the model reproduces well the results of more complete band theoretic calculations on this ordered FIG. 1. A graphene sheet. The x and y axes are oriented with system. In Sec. IV we then extend the method to study the respect to the armchair and zigzag axes, while the u and v axes low-energy electronic structure in a compositionally disor- point in directions along and around the tube, respectively. The dered rope, and analyze the effects of intertube interactions vectors di are the three nearest-neighbor vectors connecting the A perturbatively. We will also present direct numerical calcu- and B sublattices, and ␳ is the vector pointing to the center of a lations on a compositionally disordered rope which probe the hexagon. ␪ is the chiral angle. transverse localization of the electronic states. A brief dis- cussion of the relation of these results to experimental data is A. Plane-wave basis given in Sec. V. It is useful to express the eigenstates of the individual tubes in a basis of plane-wave states localized on either the A II. TUNNELING MODEL or B sublattice. Let us first focus on a single tube. The eigenstates may be described by considering a two-dimensional In this section we derive an effective tight-binding model graphene sheet with periodic boundary conditions. We will which describes the coupling between the low-energy elec- find it useful to consider two coordinate systems for the two- tronic states on neighboring tubes. This coupling depends on dimensional graphene sheet. As shown in Fig. 1, the x and y the chirality and orientation of the tubes. Our starting point is axes are oriented with respect to the armchair and zigzag a microscopic tight-binding model which describes the cou- axes of the graphene sheet. The u and v axes, on the other pling of the carbon ␲ orbitals both within a tube and between hand, are oriented with respect to the tube, with u pointing tubes: down the tube and v pointing around the circumference. For armchair tubes these axes coincide, and in general the angle HϭH ϩH ͑ ͒ 0 T . 2.1 between the axes is equal to the chiral angle of the tube. Suppressing the tube index, for the moment, we let H 0 is a nearest-neighbor tight-binding model describing uncoupled tubes, 1 ϭ ik ri ͑ ͒ ci ͚ e • c␩(i)k , 2.4 ͱN k H ϭϪ † ͑ ͒ ␩ 0 ͚ ͚ t␲caicaj , 2.2 where specifies the A or B sublattice and N is the number a ͗ij͘ of graphite unit cells on the tube. In this basis, the Hamil- tonian for an isolated tube may be written where the index a labels the tubes and ͗ij͘ is a sum over nearest-neighbor atoms on each tube. Tunneling between H ϭϪ ␥ † ϩ ͑ ͒ tubes is also represented by 0 t␲͚ kcAkcBk H.c., 2.5 k where H ϭ † ϩ ͑ ͒ T ͚ ͚ tai,bjcaicbj H.c. 2.3 ͗ab͘ ij 3 ␥ ϭ ik dj ͑ ͒ k ͚ e • . 2.6 In the following we will assume that t ϭt depends jϭ1 ai,bj rai ,rbj on the positions and relative orientations of the ␲ orbitals on Here dj are the three nearest-neighbor vectors connecting the the i and j atoms. A and B sublattices indicated in Fig. 1. At low energy we The eigenstates of H are plane waves localized in an ϭ␣ ϩ ␣ϭϮ 0 may focus on the points k Kl q, where 1, Kl are individual tube. Due to the translational symmetry of an in- at the corners of the Brillouin zone shown in Fig. 2, and l dividual tube, the eigenstates may be indexed by a tube index ϭϪ 1,0,1. In the u-v system, the Kl vectors can be written a and a two-dimensional momentum k. Of course the peri- ␣ ϭ␣ ␻ ␻ ͒ ͑ ͒ odic boundary conditions imposed by wrapping the graphene Kl K0͑cos l ,sin l , 2.7 sheet into a cylinder will give a constraint on the possible values of k. In the following, we wish to express the Hamil- where tonian in terms of this plane-wave basis. We will focus on ␲ eigenstates with low energy, which have k near one of the 2 ␻ ϭ lϩ␪ ͑2.8͒ corners of the graphite Brillouin zone. l 3 11 158 A. A. MAAROUF, C. L. KANE, AND E. J. MELE PRB 61

1 ␣ ␩ ͉␣ ␩ ͒ϭ ␣ ␳ ϩ␩ ␶ ͒ T͑ a aqa b bqb ͚ exp͓i aKal ͑ a a a ϭϪ a• lalb 1 Ϫi␣ K ͑␳ ϩ␩ ␶ ͔͒ b blb• b b b ϫ ͑ ͒ t␣ K ϩq ,␣ K ϩq . 2.14 a ala a b blb b We shall also ﬁnd it useful to express the tunneling Hamiltonian in a basis in which the bare Hamiltonian describing the tubes is diagonal. This is accomplished by per- forming a rotation in the sublattice index space to make Eq. FIG. 2. Brillouin zone of a graphene sheet. K , lϭϪ1, 0, and 1, ͑2.9͒ diagonal. Speciﬁcally, for a tube with a chiral angle ␪, l ϭ␣ ϩ label the three equivalent Fermi points, and ␪ is the chiral angle of the eigenstates will have momentum k K0 q, with ϭ ␪ ␪ ͑ ͒ the tube formed by wrapping the sheet in the v direction. (qx ,qy) q(cos ,sin ). Equation 2.9 is then H ϭ ␺† ␣ ͑ Ϫi␣␪␴ϩϩ i␣␪␴Ϫ͒␺ ͑ ͒ is the angle that the lth Fermi vector makes with the u axis. 0 v ␣q q e e ␣q . 2.15 We will now focus on the point K0. For small q, ␺ ϭ ␣ ␪ ␺Ј Using the transformation ␣q U( , ) ␣q , with ␥␣ ϩ ϭϪ(ͱ3a/2)(␣q Ϫiq ). Introducing a spinor ␺␩␣ K0 q x y q Ϫ ␣␪␴z Ϫ ␲ ␣␴y ϭc␩␣ ϩ the Hamiltonian may then be written ͑␣ ␪͒ϭ i(1/2) i( /4) ͑ ͒ K0 q U , e e , 2.16

H ϭ ␺† ͑␣ ␴ ϩ ␴ ͒␺ ͑ ͒ the Hamiltonian becomes 0 v ␣q qx x qy y ␣q , 2.9 H ϭ ͑␺Ј† ␺Ј Ϫ␺Ј† ␺Ј ͒ ͑ ͒ 0 vq ␣qR ␣qR ␣qL ␣qL . 2.17 where vϭͱ3t␲a/2, and the ␩ indices are suppressed. The tunneling Hamiltonian may similarly be expressed in In the (R,L) basis, the tunneling matrix has the form this plane-wave basis. Using Eq. ͑2.4͒ the term in Eq. ͑2.3͒ ␣ ␩Ј ͉␣ ␩Ј ͒ϭ † ␣ ␪ ͒ for the bond connecting tubes a and b is TЈ͑ a qa b qb U ͑ a , a ␩Ј␩ a b a a ϫ ͑␣ ␩ ͉␣ ␩ ͒ ͑␣ ␪ ͒ 1 Ϫ T a aqa b bqb U b , b ␩ ␩Ј, i(kb rb ka ra) † ͑ ͒ b b ͚ ͚ tr r e • • c ␩ cb␩ k , 2.10 N a b a aka b b rarb kakb ͑2.18͒ ␩ where a,b label the sublattice of the lattice site ra,b . which may be written as The sums over lattice sites may be evaluated by introducing a Fourier transform of the tunneling matrix element. As 1 Ј͑␣ ␩Ј ͉␣ ␩Ј ͒ϭ ͓ ␣ ␳ T a aqa b bqb ͚ exp i aKal a detailed in the Appendix, we may write ϭϪ a• lalb 1 Ϫ ␣ ␳ ␳ ϩ␩ ␶ Ϫ ␳ ϩ␩ ␶ i bKbl b͔t␣ K ϩq ,␣ K ϩq H ϭ iGa ( a a a) Gb ( b b b) b• a al a b bl b T ͚ ͚ e • • a b ␩ ␩ GaGb a b ϫM ␩Ј␩Ј, ͑2.19͒ † a b ϫt ϩ ϩ c␩ c␩ , ͑2.11͒ ka Gakb Gb k bkb a a where where la lb* la lb* f ␣ f ␣ f ␣ f Ϫ␣ 1 a b a b 1 Mϭ ͫ ͬ , ͑2.20͒ Ϫ ϩ l l l l t ϭ ͵ d2r d2r t͑r ,r ͒e ika•ra ikb•rb, 2 f a f b* f a f b* ka ,kb 2 a b a b Ϫ␣ ␣ Ϫ␣ Ϫ␣ NAcell a b a b ͑2.12͒ l i␾ Ϫi␾ f ␣ϭe lϩ␣e l, ͑2.21͒ and G is a reciprocal-lattice vector. We now specialize to eigenstates in the vicinity of the and Fermi points, kϭ␣K ϩq. We may express the sum over the 0 1 G’s as a sum over equivalent K points which are related to ␾ ϭ ͑2␲Ϫ␻ ͒. ͑2.22͒ l 2 l K0 by a reciprocal-lattice vector. In the following we will see that this sum is dominated by K0 , K1, and KϪ1, which lie in ␶ϭ the ‘‘ﬁrst star’’ in reciprocal space. Since K0• 0, the sum B. Tunneling matrix elements becomes For simplicity, we suppose that the matrix elements tij for tunneling between atoms on different tubes depend only on H ϭ ͑␣ ␩ ͉␣ ␩ ͒␺† ␺ the distance between the atoms, and are of the form T ͚ T a aqa b bqb a␣ ␩ q b␣ ␩ q , ␣ ␩ ␣ ␩ a a a b b b a a b bqaqb Ϫ ͑ ͒ ϭ dij /a0 ͑ ͒ 2.13 tij t0e , 2.23

with where dij is the distance between atoms i and j. PRB 61 ELECTRONIC STRUCTURE OF CARBON NANOTUBE ROPES 11 159

Ϫ Ϫ 2 t ϭt ␦ e (1/4)Ra0(kav kbv) , ͑2.28͒ kakb T kaukbu with ␲ 2 ba0 Ϫ Ϫ 2 ϭ b/a0 (1/2)ba0K ͑ ͒ tT e e 0t0 . 2.29 Acell Using Eq. ͑2.27͒, we then arrive at a ﬁnal expression for the tunneling matrix element relating eigenstates on two tubes:

FIG. 3. Cross section of two parallel tubes of radius R, with a 1 ␣ ␩ ͉␣ ␩ ͒ϭ ␣ ␳ ϩ␩ ␶ ͒ separation b. T͑ a aqa b bqb tT ͚ exp͓i aKal ͑ a a a ϭϪ a• lalb 1 It is useful to introduce two-dimensional coordinates Ϫi␣ K ͑␳ ϩ␩ ␶ ͔͒␦ which are oriented relative to the tube’s axis. Let us define a b blb• b b b kau ,kbu two-dimensional vector rϭ(u,v), where u is the distance Ϫ(1/4)Ra (k Ϫk )2 ϫe 0 av bv ͉ ϭ␣ down the tube axis, and v is the distance around the tube ka(b) a(b) measured from the ‘‘contact line’’ as shown in Fig. 3. ͑ ͒ ϫK ϩq . 2.30 Suppose the two tubes have a separation b as shown in ala(blb) a(b) Fig. 3. Then the distance is given by C. Estimate of t from the band structure of graphite 2 T va vb d͑r ,r ͒2ϭ͑u Ϫu ͒2ϩͩ R sin ϪR sin ͪ The tunneling model described above may be used to de- a b a b R R scribe the coupling between flat graphene sheets. Since the 2 transverse bandwidth of graphite is well known, this allows va vb ϩͩ bϩ2RϪR cos ϪR cos ͪ . us to estimate the prefactor t in the tunneling matrix ele- R R T ment. The coupling between two flat graphene sheets is de- ͑2.24͒ scribed by the R→ϱ limit of the above theory. In this case, Ϫ the Gaussian dependence on kav kbv can be written as a Since the range of the tunneling interaction a0 is of order ͑ ͒ ␦ 0.5 Å, while RϷ7 Å and bϷ3.4 Å, it is useful to expand kronecker function: Eq. ͑2.24͒ for u,vӶb,R: a0 Ϫ Ϫ 2 ͱ (1/4)Ra0(kav kbv) →␦ ͑ ͒ e k k . 2.31 ͉r Ϫr ͉2 v2ϩv2 4␲R av bv ͒ϭ ϩ a b ϩ a b ͑ ͒ d͑ra ,rb b . 2.25 2b 2R We thus obtain

It follows that the tunneling matrix element has a Gaussian t ϭt ␦ , ͑2.32͒ kakb G kakb dependence on ra and rb : with ͉r Ϫr ͉2 v2ϩv2 ͑ ͒ϭ ͩ Ϫ Ϫ 1 ͫ a b ϩ a bͬͪ t ra ,rb t0 exp b/a0 2 . ␲ 2b 2R 2 ba0 Ϫ Ϫ 2 ϭ b/a0 (1/2)ba0K ͑ ͒ ͑2.26͒ tG t0e e 0. 2.33 Acell We may now use Eq. ͑2.26͒ to evaluate the Fourier transform of the matrix elements. The Gaussian form allows this For this calculation, we find it most convenient to use the to be done simply. Using the fact that the total area of the sublattice basis for the electronic eigenstates. Using Eq. ͑ ͒ graphene sheet is given by NA ϭ2␲RL,wefind 2.27 the tunneling Hamiltonian for two graphene sheets is cell then ␲ 2 ba0 Ϫ Ϫ ͉ ͉2ϩ͉ ͉2 t ϭ t e b/a0e (ba0 /4)( ka kb ) kakb 0 H ϭ ͑␣␩ ͉␣␩ ͒␺† ␺ ͑ ͒ Acell T ͚ T aq bq a␣␩ q b␣␩ q , 2.34 ␣␩ ␩ a b q a b a0 Ϫ Ϫ 2 ϫͱ e (Ra0 /4)(kav kbv) ␦ . ͑2.27͒ where 4␲R kaukbu 1 ͑ ͒ ␣ ⌬␳ϩ ␩ Ϫ␩ ␶ We now use Eq. 2.27 to evaluate the low-energy tunneling ͑␣␩ ͉␣␩ ͒ϭ i Kl [ ( a b) ] ͑ ͒ ͉ ͉2 T aq bq tG ͚ e • , 2.35 matrix elements. Due to the exponential dependence on ka lϭϪ1 the sum on K in Eq. ͑2.11͒ will be dominated by three terms a ⌬␳ϭ␳ Ϫ␳ ⌬␳ϭ␶ K in the ‘‘first star’’ in which ͉K͉ϭK ϭ4␲/(3a). Specifi- and a b . For AB stacking of graphite, , so, ai 0 ͚ ␣ ␶ϭ cally, estimating the parameters aϭ2.5 Å, bϭ3.4 Å, and using K exp i K• 0 the only nonzero term is ϭ a0 0.5 Å, the exponent of the last term is approximately ͑␣ ͉␣Ϫ ͒ϭ ͑ ͒ ␲2 2Ϸ ͱ T 1q 1q 3tG . 2.36 8 ba0/9a 2.4. Thus the next star at 3K0 will be suppressed by a factor of exp͓Ϫ2(2.4)͔ϭ0.01, justifying the Thus tunneling only connects the A sublattice on the ‘‘A’’ ϭ␣ ϩ first star approximation. For ka(b) a(b)Kai(bj) qa(b) ,we sheet to the B sublattice on the ‘‘B’’ sheet. This is to be then have expected, since an atom on the B sublattice of the A sheet 11 160 A. A. MAAROUF, C. L. KANE, AND E. J. MELE PRB 61 sits above a hexagon on the B sheet. The Hamiltonian for an AB stacked crystal of graphene planes then has the form

Hϭ ␺† ͑␣ ␴ ϩ ␴ ͒␺ ͚ v s␣q qx x qy y s␣q s

ϩ ␺† ␺ ϩ␺† ␺ 3tG͚ 2s␣Aq 2sϩ1␣Bq 2s␣Aq 2sϪ1␣Bq , s ͑2.37͒ where s indexes the graphene sheets. This can be simpliﬁed FIG. 4. Density of states of a (10,10) crystal. A pseudogap of by introducing a transformation which interchanges the A about 0.09 eV develops at the Fermi level. and B sublattices of the graphene lattice when ␣ϭϪ1, fol- lowed by a transformation which interchanges the A and B Similarly, for a B-B bond, sublattices on the odd (2sϮ1) graphene layers. The Hamil- tonian then has the simpler form 3 TBB͑␣␩ q͉␣␩ q͒ϭ t ͑1Ϫ␴ ͒. ͑3.3͒ a b 2 T z 3 Hϭ͚ ␺† ␴␺ ϩ ͚ ␺†͑ ϩ␣␴ ͒␺ v s q• s tG s 1 z sЈ . For a hexagon-hexagon bond, we have ⌬␳ϭ0, so s ͗ssЈ͘ 2 ͑ ͒ 2.38 AB ␣␩ ͉␣␩ ͒ϭ ␴ ͑ ͒ T ͑ aq bq 3tT x . 3.4 This leads to an energy dispersion The Hamiltonian describing the transverse motion will then ͒ϭ Ϯͱ 2͉ ͉2ϩ ͒2 have the form E͑q,qz 3tG cos bqz v q ͑3tG cos bqz . ͑2.39͒ H ϭ ͑␥ ϩ␥ ͒ϩ␴ ͑␥ Ϫ␥ ͒ϩ ␴ ␥ ͑ ͒ ϭ T ͚ 1 2 z 1 2 2 x 3 , 3.5 The bandwidth for transverse motion is then W 12tG . Ex- q perimentally, the bandwidth of graphite is in the range W ϭ 7 ϭ 1.2–1.6 eV. This leads to an estimate tG 0.1 eV. where Comparing Eqs. ͑2.29͒ and ͑2.33͒ we may relate the tube ␥ ϭ ͑ ͒ tunneling matrix element to that of graphite: i 6tT cos q•ai , 3.6

where ai are the three nearest-neighbor vectors in the trian- a0 t ϭͱ t . ͑2.40͒ gular tube lattice. Then, T 4␲R G E͑q ,q͒ϭ␥ ϩ␥ Ϯͱ͑vq Ϫ2␥ ͒2ϩ͑␥ Ϫ␥ ͒2. ϭ x 1 2 x 3 1 2 For a tube with radius R 7Å,wefind ͑3.7͒ ϭ ͑ ͒ ϭ tT 7.5 meV. 2.41 From the above estimate of tT 7.5 meV, the density of states is plotted in Fig. 4, showing a pseudogap feature, as- sociated with an energy of order 12t Ϸ0.09 eV. The energy III. ELECTRONIC STRUCTURE OF A ROPE CRYSTAL T scale of this pseudogap agrees well with the results of more We now apply the tunneling model described above to the sophisticated electronic structure calculations.8 problem of the electronic structure of nanotube ropes. We begin by considering the simpler problem of an orientation- IV. COMPOSITIONAL DISORDER ally ordered crystal of (10,10) tubes. We then consider a compositionally disordered rope. A compositionally disordered rope contains a random dis- (10,10) tubes can be arranged in a triangular lattice in tribution of chiral tubes. Since tubes with different chiralities which each tube has the same orientation, and the tubes face have different periodicities, Bloch’s theorem is of little use each other via A-A coupling, B-B coupling, and hexagon- for describing the eigenstates of the entire rope. Nonetheless, in the absence of coupling between the tubes, we know that hexagon coupling. For tunneling in the same subband, kav ϭ the eigenstates on each tube are plane waves. Our approach kbv . Again, we find it useful to use the sublattice basis for the tube eigenstates. For each bond, the tunneling between is to describe the coupling between these plane waves per- the pair of tubes is described by Eq. ͑2.27͒, with turbatively. We begin by considering the simpler problem of the electronic structure of two coupled nanotubes of different 1 chirality. ␣ ⌬␳ϩ ␩ ϩ␩ ␶ ␣␩ ͉␣␩ ͒ϭ i Kl [ ( a b) ] ͑ ͒ T͑ aq bq tT ͚ e • . 3.1 ϭϪ l 1 A. Coupling between two tubes of different chirality ␩ ϭ␩ ϭ For an A-A bond it is only nonzero for a b 1. Using a The electronic coupling between two tubes of different matrix notation for the ␩ indices, chirality conserves the momentum along the tube up to reciprocal-lattice vectors in either tube. As we argued in Sec. 3 II, the sum over reciprocal-lattice vectors is dominated by TAA͑␣␩ q͉␣␩ q͒ϭ t ͑1ϩ␴ ͒. ͑3.2͒ a b 2 T z the terms in which kϩG are near the first star of K points. PRB 61 ELECTRONIC STRUCTURE OF CARBON NANOTUBE ROPES 11 161

FIG. 7. A plot showing the minimum energy mismatch ⌬E for different metallic tubes with radii lying between 1.2 and 1.5 nm. The energy offset is taken at a zigzag tube (18,0).

with the minimum momentum mismatch, for the uncoupled system ͓Fig. 6͑a͔͒, and the coupled one ͓Fig. 6͑b͔͒. To lowest FIG. 5. Brillouin zones of two tubes of different chiralities. The order, the effect of the coupling is only important near points zones are rotated such that the ua and ub axes coincide. In the lower of degeneracy, i.e., where we have band crossing. This oc- part of the figure we show the low-energy band structure of the two curs in two cases. The first, which we refer to as a ‘‘back- metallic tubes. Solid bands belong to one tube, and dotted bands scattering gap,’’ occurs when the right- and left-moving belong to the other tube. The bands are replicated three times, re- bands on each tube cross. The second, which we refer to as a flecting the three equivalent K points. ‘‘tunneling gap’’ occurs when the left-moving band on one tube crosses the right-moving band on the other tube. When the coupling between the tubes is weak, it is useful to The effect of the coupling depends on two crucial energy view this as momentum-conserving coupling between states scales: ͑1͒ the tunneling matrix element t, which we esti- located near the three equivalent K points. It must be kept in mated in Eq. ͑2.27͒ to be less than 7.5 meV, and ͑2͒ the ϩ ϩ energy mismatch mind that two states K0 q and K1 q in the vicinity of different K’s are actually the same state. ⌬Eϭv͓␣ ͑K ͒ Ϫ␣ ͑K ͒ ͔, ͑4.1͒ Figure 5 shows the Brillouin zones of two nanotubes with a ala u b blb u chiral angles ␪ and ␪ , oriented so that the u and u axes a b a b which determines the energy at which the right- and left- coincide. Since the tunneling Hamiltonian conserves k ,itis u moving bands on tubes a and b cross. In general, this energy convenient to view the band structure of the pair of tubes as mismatch depends on the chiral angles ␪ and ␪ of the two a function of k . Consider first the band structure in the a b u tubes as well as the Fermi point indices i and j.InFig.7we absence of coupling. The solid bands describe the low- show the variation of the energy mismatch ⌬E with the tube energy states on one tube, while the dotted bands show the chirality for all the metallic tubes with diameters between 1.2 states on the other tube. Each set of bands is replicated three and 1.5 nm. Tubes which are mirror images of one another times, reflecting the three equivalent K points. In Fig. 6 we have the same diameter and energy difference. The offset of show the band structure in the vicinity of the Fermi points the u momentum is taken at a zigzag tube; in that case we have an (18,0) tube. As we see the typical energy mismatch is a few hundred meV. The fact that tӶ⌬E simplifies the problem considerably and justifies our perturbative approach. Of course it breaks down in special cases when ⌬E is zero or very small, which occurs, for instance, when the two tubes are mirror images of each other. The tunneling Hamiltonian couples left and right movers in first order, and therefore the tunneling gap is linear in the tunneling strength t. The gap opens at ⌬E/2, and since t is much smaller than ⌬E, one concludes that tube-tube coupling has a small effect near the Fermi energy. FIG. 6. ͑a͒ Band structure of the two ͑uncoupled͒ tubes in the Backscattering gaps form through second-order coupling vicinity of the Fermi point with the minimum momentum mismatch between left and right movers on the same tube, and hence ⌬k, which defines the important energy scale ⌬Eϭv⌬k. ͑b͒ Band the gap is second order in the tunneling strength; Eg 2 structure of the coupled tubes near the Fermi energy. Backscattering ϳt /⌬E, which is of order 1 meV. Furthermore, because the gaps open quadratically with the tunneling strength t, whereas tun- tunneling matrix elements are not invariant under the inter- neling gaps are linear in t. In general, backscattering gaps open at change of the two tubes, the backscattering gap of each tube different energies. opens at a different energy. This means that the effect on the 11 162 A. A. MAAROUF, C. L. KANE, AND E. J. MELE PRB 61 density of states near the Fermi energy is weakened by the nential is due to the fact that the tubes face each other from wiggling of the gaps. In what follows, we present quantita- the outside. As we see from Fig. 7, the average energy sepa- tive arguments justifying our expectations. ⌬ ϳ T ration E 300 meV, whereas the tunneling gap Eg We focus first on the tunneling gaps. The movers on each Ͻ30 meV. This means that in a rope with a random distri- tube couple in first order. In general, crossing occurs at three bution of chiralities, the opening of such gaps will have a different energies, as dictated by the momentum mismatch negligible effect near the Fermi level. ͑ ͒ between the Fermi points of the two tubes see Fig. 5 . Since Now we focus on the backscattering gaps which form we are ultimately interested in the effect of tube interactions near the Fermi level. Since left- and right-moving states on on the states nearest to the Fermi level, we only consider the same tube do not couple in first order, we use second- points with the lowest-lying crossing. We denote these by order degenerate perturbation theory. We are interested in ␣ K and ␣ K . To find the magnitude of the gaps and a ai b bj calculating the size of the resulting gap as well as the offset their offsets, we need to diagonalize the first-order perturba- of the gap. In order to calculate these, we need to diagonalize tion matrix. We thus consider the Hamiltonian which couples the right-moving states on tube a with the left-moving states the matrix which arises from the second-order coupling be- on tube b, and we diagonalize the matrix tween the states. In general, we have tunneling between all Fermi points on each tube. Since the magnitude of the back- v͑qϪ␣ K ͒ TЈ͑␣ Rq ͉␣ Lq ͒ scattering gaps varies quadratically with the tunneling H Tϭͫ a ai u a a b b ͬ a ␣ ͉␣ ͒ Ϫ Ϫ␣ ͒ , strength and inversely with the energy difference ⌬E, the TЈ*͑ aRqa bLqb v͑q bKbj u ͑4.2͒ most effective contributions are those with the highest tunneling strength and lowest ⌬E. This argument makes us only where q is measured from the Fermi point of tube a. For include the set of nearest K points. Therefore, we diagonal- ␣ ␣ ϭϩ a , b 1, the magnitude of the tunneling gap is ize 1 ETϭ2͉TЈ ͉ϭ4t expͩ Ϫ Ra K2͑sin ␻ ϩsin ␻ ͒2 ͪ g RL T 4 0 0 ai bj Ϫ␣ ͒ ϩ a a v͑q aKa0 u E E H aϭͫ RR RL ͬ ␻ ␻ a a , ai bj E Ϫv͑qϪ␣ K ͒ ϩE ϫͯcos sin ͯ. ͑4.3͒ LR a a0 u LL 2 2 ͑4.4͒ Tϭ⌬ It is centered about an energy Eo E/2 above the Fermi energy. Notice that the ϩ sign in the argument of the expo- where

␣ ␭ ͉␣ ͒ ␣ ␭ ͉␣ ͒Ϫ ␣ ␭ ͉␣ ͒ ␣ ␭ ͉␣ ͒ ͓TЈ͑ a qa bLqb TЈ*͑ a Јqa bLqb TЈ͑ a qa bRqb TЈ*͑ a Јqa bRqb ͔ Ea ϭ ͚ ͯ , ͑4.5͒ ␭␭Ј ͗l l ͘ v͑␣ K Ϫ␣ K ͒ a b a ala b blb u ϭ qa 0 and ␭,␭ЈϭR,L. Diagonalizing, we obtain

a ϩ a a Ϫ a 2 ELL ERR ELL ERR Ea ͑q͒ϭ Ϯͱͩ v͑qϪ␣ K ͒ Ϫ ͪ ϩ͉Ea ͉2. ͑4.6͒ Ϯ 2 a a0 u 2 RL

aϭ ͉ a ͉ aϭ a ϩ a The backscattering gap is Eg 2 ERL , and it opens around an energy offset Eo (ERR ELL)/2. We thus ﬁnd

1 ͩ Ϫ 2͑␣ ␻ ϩ␣ ␻ ͒2 ͪ 2 exp Ra K sin sin 4t 0 0 a ala b blb a T ͯ 2 ͯ ϭ ␻ ␻ ͑ ͒ Eg ͚ sin al cos bl , 4.7 vK ͗l l ͘ ␣ cos ␻ Ϫ␣ cos ␻ a b 0 a b a ala b blb and

1 ͩ Ϫ 2͑␣ ␻ ϩ␣ ␻ ͒2 ͪ 2 exp Ra K sin sin 2t 2 0 0 a ala b blb aϭ T ␻ ͑ ͒ Eo ͚ cos bl . 4.8 vK ͗l l ͘ ␣ cos ␻ Ϫ␣ cos ␻ b 0 a b a ala b blb PRB 61 ELECTRONIC STRUCTURE OF CARBON NANOTUBE ROPES 11 163

Equations ͑4.7͒ and ͑4.8͒ show that, in general, the gap offset Eo is greater than the gap Eg . In addition, one expects that the offsets and gaps of both tubes will generally be different. This means that the gaps ‘‘wiggle’’ around the Fermi energy as the chiral angle is changed, leading to the conclusion that in a rope formed of a random collection of chiralities, the effect on the density of states around EF is very small. Let us now have a closer look at the contributions of different tunneling points. In general, only one set of Fermi points will dominate, unless the tubes are mirror images of each other. We now argue that in a case when the tubes have different chiralities, it is a certain set of Fermi points that is FIG. 8. ͑a͒ A compositionally disordered metallic rope. Differ- actually important. ent gray scales indicate different tube chiralities. ͑b͒ A composition- We want to understand which tubes signiﬁcantly couple ally disordered rope with one/third of its tubes metallic. The vacant to each other, and for those tubes, the Fermi points at which circles denote semiconducting tubes. It is clear that the semicon- the coupling is most effective. To do this, we study the quan- ducting tubes percolate along the rope. tity B. Compositionally disordered rope In this section we study the electronic structure of a nano- 1 expͩ Ϫ Ra K2͑␣ sin ␻ ϩ␣ sin ␻ ͒2 ͪ tube rope composed of tubes with a random distribution of 0 0 a ala b blb t tT 4 diameters and chiralities. We expect that the momentum mis- ϭ . ⌬E vK ␣ cos ␻ Ϫ␣ cos ␻ match between the Fermi points of neighboring tubes will 0 a ala b blb ͑4.9͒ suppress the tunneling and lead to localization. In real ropes, we expect two-thirds of the tubes to be semiconducting. As indicated in Fig. 8, this will make the localization effects This quantity will be dominated by the exponential factor, even stronger. To emphasize our point, we consider a com- and in cases where different sets of points have nearly equal positionally disordered rope with only metallic tubes. exponential contribution, the denominator will dominate. To solve the problem, we employ the ‘‘dominant Fermi ␻ For nearly armchair tubes, the maximum is at a(b)l(lЈ) point approximation’’ introduced in Sec. IV A. In this ap- ϳ ͒ ϭ ϩ 0, i.e., around the K0 points (K0-K0 tunneling . In that proximation, the momentum k K0u q is conserved by the case, the exponential factor is approximately 1, and the de- tunneling Hamiltonian. We may thus write nominator takes its minimum value (ϳ7 meV) as the two ␻’s are closest to zero. As the tubes shift from being in the Hϭ H ϩ H ͑ ͒ ͚ i ͚ ij , 4.10 armchair region, the denominator increases as the Fermi i ͗ij͘ points rotate away from the u axis, thereby making the tun- with neling less effective. For tubes which are nearly mirror images of each other, H ϭ ͑ Ϫ␣ i ͒͑␺† ␺ Ϫ␺† ␺ ͒ the dominant set is also K -K . While moving away from i ͚ v k K0u i␣kR i␣kR i␣kL i␣kL , 0 0 ␣ the armchair region does not signiﬁcantly change the expo- k ͑4.11͒ nential contribution, it makes the denominator larger hence making the coupling less important. and For tubes which are nearly zigzag ͑but are not mirror ͒ images of each other , tunneling is the least effective, as both H ϭ ˜ Ј͑␣␩Ј͉␣␩Ј͒␺† ␺ ͑ ͒ ij ͚ T i j i␣k␩Ј j␣k␩Ј, 4.12 the exponential argument and the denominator are larger. In ␣␩Ј␩Ј i j i j k some of these cases, the K0-K0 tunneling may not be the dominant one. where ˜TЈ is the tunneling matrix given by We thus conclude that tunneling is most effective between

tubes which are nearly in the armchair region, and that the 1 2 1 ˜ Јϭ ͩ Ϫ ͑ ␻ ϩ ␻ ͒2 ͪ T 2tT exp Ra0K0 sin i0 sin j0 K0-K0 tunneling is the most important one, and hence ERR 4 ͑and similar sums͒ are dominated by one term. In other ␻ ␻ ␻ ␻ words, the sum over reciprocal-lattice vectors in the tunnel- i0 j0 i0 j0 cos cos i cos sin ing Hamiltonian is dominated by a single term. If we ignore 2 2 2 2 the other terms, then there is no reciprocal-lattice vector sum, ϫͫ ͬ . ␻ ␻ ␻ ␻ i0 j0 i0 j0 and the system effectively has translational invariance in the Ϫi sin cos sin sin direction parallel to the tubes. This ‘‘dominant Fermi point 2 2 2 2 approximation’’ simpliﬁes our problem considerably, since it ͑4.13͒ allows us to assign a conserved momentum to each state. This will allow us to compute the band structure for an entire The Hamiltonian may now be diagonalized for each k by rope in Sec. IV B. diagonalizing a 4Nϫ4N matrix, where N is the number of 11 164 A. A. MAAROUF, C. L. KANE, AND E. J. MELE PRB 61

V. CONCLUSION In this paper we have shown that the constraints of energy and crystal momentum conservation severely restrict the electronic coupling between carbon nanotubes. The electronic coupling between two nanotubes is only effective when the eigenstates near the Fermi energy have the same momentum, which requires that the graphene sheets of the two nanotubes are oriented parallel to one another. This only occurs when the two tubes are mirror images of one another. Thus, in contrast to a crystalline rope of armchair nanotubes, in which eigenstates are extended throughout the rope, we find that the electronic eigenstates of a compositionally dis- ͑ ͒ ordered rope are strongly localized on individual nanotubes. FIG. 9. a Low-energy band structure of a compositionally dis- This conclusion has important consequences for the trans- ordered metallic rope, showing tunneling and backscattering gaps. port properties of nanotube ropes. In particular, it provides a The latter wiggle around the Fermi energy, leading to a negligible natural explanation of the nonlocal effects observed by effect on the density of states, which is shown in ͑b͒. Bockrath et al.9 in their multiterminal conductance measurements. These effects can arise when different electrical leads tubes in the rope. For each k, the mth eigenstate may be make contact to different tubes within a rope, allowing the ␨m described by a ‘‘wave function’’ ␣␩(i), which is the ampli- current in a tube to ‘‘bypass’’ an electrical lead which it does tude for the particle to be in state ␣, ␩ on tube i. not contact. A portion of the band structure of the metallic rope is In the absence of impurities, the eigenstates will be local- shown in Fig. 9͑a͒. It is clear that there is no significant ized on a single tube, but extend across the entire length of a change in the vicinity of the Fermi energy. As we argued tube. Scattering, due either to impurities or tube ends, will above, the backscattering gaps wiggle around the Fermi en- tend to localize the states in the tube direction. Paradoxically, ergy, thereby negligibly changing the density of states, by relaxing the constraint of momentum conservation such which is shown in Fig. 9͑b͒. Therefore, no pseudo gap de- scattering will increase the coupling between tubes. None- velops. theless, we are led to a picture of highly anisotropic local- The extent of localization of the eigenstates may be quan- ization. titatively measured by computing the correlation function This picture may help to explain some apparently para- doxical transport data on nanotube mats. At low tempera- tures, nanotube mats are observed to obey the three- ͑ ͒ϭ ͉␨m ͑ ͉͒2͉␨m ͑ ͉͒2 C rЌ ,E ͚ ␣␩ i ␣Ј␩Ј j dimensional Mott variable-range hopping law, R ␣␩␣ ␩ ϭ 1/4 10 ijm Ј Ј R0 exp(T0 /T) , with T0 of order 100 K. If one uses the standard formula for isotropic variable range hopping and ϫ␦͉͑R ϪR ͉ϪrЌ͒␦͑E ϪE͒, ͑4.14͒ i j m knowledge of the nanotube’s density of states, one extracts a localization length of order 200 Å. In contrast, Fuhrer 11 at the Fermi energy, where Ri is the position of tube i in the et al. analyzed the scaling of the hopping conductivity with rope. As shown in Fig. 10, the correlation function decays electric field and temperature R(E,T)ϭ f (␰E/T), and argued ϰ Ϫ2rЌ /␰Ќ exponentially with distance, C(rЌ ,EF) e , indicating that the localization length is much longer, of order 6000 Å. that the eigenstates are localized perpendicular to the tube Our picture of anisotropic localization offers a possible reso- axes with a localization length ␰Ќϳ10 Å. Thus the eigen- lution to this discrepancy. In the simplest model of aniso- states are predominantly on a single tube. tropic variable range hopping, T0 depends on the geometric 2 1/3 mean of the localization lengths, (␰ʈ␰Ќ) , while the scaling with electric field depends on the longest localization length, ␰ʈ . In this paper, we have developed a general framework for describing the electronic coupling between graphene-based structures. This approach should prove useful for other problems, including the coupling between neighboring shells of multiwalled tubes as well as the coupling between crossed single walled tubes. Analysis of these problems will be left for future work.

APPENDIX: LATTICE FOURIER TRANSFORMS In this appendix, we explicitly work out the Fourier trans-

FIG. 10. The correlation function C(rЌ ,EF) deﬁned by Eq. form of Sec. II A. As shown in Fig. 1 the positions of the ϭ ϩ␳ϩ␩␶ ͑4.14͒, which gives a quantitative measure of the localization of the lattice may be written as ri R , which may be states on single tubes. speciﬁed by a lattice vector R and a sublattice index ␩ϭ PRB 61 ELECTRONIC STRUCTURE OF CARBON NANOTUBE ROPES 11 165

Ϯ1. The position of the center of the hexagon is given by ␳. Deﬁning the Fourier transform Consider ﬁrst a sum over lattice sites of the form 1 1 Ϫ 1 2 Ϫik r ͒ ik riϭ ͒ ik r͉ ϭ ͵ ͑ ͒ • ͑ ͒ ͚ f ͑ri e • ͚ f ͑r e • rϭRϩ␳ϩ␩␶ . f k d rt r e , A3 ͱ ͱ ␩ ͱ N i N R Acell N ͑A1͒ The sum over R may be performed by introducing we may then write reciprocal-lattice vectors G:

1 2 ϪiG (rϪ␳Ϫ␩␶) Ϫik r 1 Ϫ ϭ ͵ • ͑ ͒ • ͑ ͒ ͒ ik riϭ iG d␩ ͑ ͒ ͚ d re f r e . A2 ͚ f ͑ri e • ͚ e • f Gϩk . A4 ͱ G␩ ͱ i G␩ Acell N N

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