Theory of Quantum Transport in Carbon Nanotubes

Theory of quantum transport in carbon nanotubes

Stephan Roche Département de Recherche Fondamentale sur la Matière Condensée Commissariat a` l’Energie Atomique DRFMC/SPSMS, 17 avenue des Martyrs, Grenoble (France) [email protected]

Ballistic conduction in SWNTs and MWNTs • – Bandstructure and conducting channels Effects of disorder and doping • – Conduction regimes – Absence of backscattering in undoped nanotubes – Nature of disorder and defects – Elastic mean free path Derivation within the Fermi golden rule ∗ Energy-dependent mean free path ∗ Illustration on vacancies ∗ Illustration on chemical substitutions ∗ Quantum interferences and magnetotransport • Localization regime • Contribution of intershell coupling • – Commensurate nanotubes – Incommensurate nanotubes Role of electrode-nanotube contacts • – landmarks – Role of bonding character (ab-initio results) Annexes : • – Bandstructure and DoS under magnetic field – Derivation of the Fermi golden rule – Standing waves in finite size carbon nanotubes. 1 Ballistic conduction in singlewall and multiwall carbon nanotubes

1.1 Bandstructure and conducting channels

Landauer formula: Tn(E), the transmission amplitude for a given channel n,at energy E :

2e2 (E) = Tn(E) G h n=1X,N ⊥ if the conduction regime = ballistic + contacts are reﬂectionless

Conductance is length independent and QUANTIZED = number N (E) of available quantum channels at a given energy ⊥

2e2 (E) = N (E) G h × ⊥

(5,5) metallic nanotube :

Two quantum channels are available at Fermi energy EF = 0 (with 2 = 2e the quantum resistance) G0 h

(E = 0) = 2 G F G0 (5,5) Armchair metallic nanotube

4 10

Energy G(G0) 10 bands 8 2

6 0 4

−2 2 10 bands

−4 0 −4 −2 0 2 4 −4 −2 0 2 4 wave vector Energy Intrinsic conduction mechanism and conductance scaling are simu- latenously followed through The Kubo formalism (nanotube of length Ltube) :

2e2 (E, Ltube) = lim Tr[δ(E ) ˆ(t)] t τ G Ltube → − H D

δ(E ) = spectral operator (DoS) • − H ˆ(t) = ( ˆ(t) ˆ(0))2/t = the diﬀusion operator • D X − X Contact are fully discarded, and considered as reﬂectionless. •

Ballistic regime = ˆ(t) = ( ˆ(t) ˆ(0))2/t v2 t hD iEF X − X ∼ F with vE = 3accγ0/2¯h, Ltube = vF τ and DoS = N /2πhv¯ F F ⊥

= 2N e2/h G ⊥ 0.8

200

150 Pente : v =3πγ a/h 0.6 /t F 0

〉 100 2 X 〈 50 /atome] 0

γ 0 0 20 40 60 80 0.4 πγ t [h/2 0]

0.2 Filtre Densité [état/

0 −1 −0.5 0 0.5 1 γ Energie [ 0]

Etat filtré Re(ψ) <0 Re(ψ) >0 2 Eﬀects of disorder and doping

2.1 Conduction regimes

Scaling nanotubes : diameter, number of shells within MWNT = ⇒ change of dimensionality !

With disorder (topological, dopants, ...) conduction regime departs from the ballistic regime. diﬀusive regime : •

(t τ) = ` v D ≥ e F

where `e the mean free path is introduced, and τ = `e/vF .

The conductance

2 = 2N e /h(`e/Ltube) G ⊥

Quantum interferences and localization : • In the weak localization regime:

t t 2e2D δ = ∞ dt (r, t)(1 e−τe )e−τφ 0 G −πh¯Ltube Z P −

Ltube ξ N `e (ξ the localization length) ≥ ∼ ⊥

2 (Ltube) = (h/2N e ) exp(ξ/Ltube) G ⊥ 2.2 Absence of backscattering in undoped nanotubes

Symmetry of eigenstates of graphite or metallic nanotubes close to the Fermi level

The eigenvalues at a ~k point in graphite or in carbon nanotubes, write in a general way as

E(~k) = γ0 3 + 2 cos(~k.~a1) + 2 cos(~k.~a2) + 2 cos(~k.(~a1 ~a2)) § r −

At the corners of the Brillouin zones K~ -points one gets E(K~ ) = 0 with two degenerate eigenvectors (graphite or metallic tubes)

iK~ .~` e ~ ΨK~ ,s(~r) = (pz(~r ~rA) + pz(~r ~rB)) bonding state K+ ~` allXcells √2 − − | i eiK~ .~` Ψ ~ (~r) = (pz(~r ~rA) pz(~r ~rB)) antibonding state K~ K,a − X~` √2 − − − | i

At K~ -points, the probability amplitude of a scattering event from a state K~ + to a state K~ as | i | −i

~ ˆ ~ ~ ~ ~ ~ ~ K+ K = d~rdr0 K+ ~r (~r, r0) r0 K h | U | −i Z h | iU h | −i ~ ~ ~ ~ = d~rdr0 K+ ~r (uAδ(~r ~rA) + uBδ(~r ~rB)) r0 K Z h | i − − h | −i 1 A∗ B∗ B A∗ B∗ B = uA (pz + pz )pz + uB (pz + pz )( pz ) 2µ { } { − } 1 = (u u ) 2 A − B CONSEQUENCE: if the disorder potential is long ranged with respect to the unit cell, i.e u u (conservation of pseudospin symmetry) A ' B

K~ + K~ = 0 (full suppression of backscattering). h | U | −i Eigenstates in the vicinity of K~ -points (low energy limit). Around some ~k point, the wavefunction :

~ ~ A ~ ~ B ~ Ψ(k, ~r) = cA(k)p˜z (k, ~r) + cB(k)p˜z (k, ~r) with

A ~ 1 i~k.~` ~ p˜z (k, ~r) = e pz(~r ~rA `) √Ncells X~` − − B ~ 1 i~k.~` ~ p˜z (k, ~r) = e pz(~r ~rB `) √Ncells X~` − − One then has to compute the factors

1 ~ ~ ~ ~ ~ ~ ik.(` `0) A,` A,`0 AA(k) = e − pz pz H N ~X~ h | H | i cells `,`0 1 ~ ~ ~ ~ ~ ~ ik.(` `0) A,` B,`0 AB(k) = e − pz pz H N ~X~ h | H | i cells `,`0 = and so forth

Following the deﬁnitions of Fig.1

A,0 B,0 i~k.~a A,0 B, ~a i~k.~a A,0 B, ~a (~k) = p p + e− 1 p p − 1 + e− 2 p p − 2 HAB h z |H| z i h z |H| z i h z |H| z i = γ α(~k) − 0 y

a cc

a = a 2 = 3 a B 1 cc a 3 a (1/2, 3) a a 1 = cc 2 1 A a 2 = 3 a cc (-1/2, 3) x

Figure 1: Representation of graphite lattice. in other terms

E(~k) γ0α(~k) b1  ~ − ~    = 0  γ0α(k) E(k)  b2      −   

i~k.~a1 i~k.~a2 with α(~k) = 1+e− +e− and ~k = K~ +δ~k (K~ = (4π/(3√3acc), 0)) 2iπ/3 iδkxa/2 2iπ/3 iδkya/2 one gets α(~k) = 1 + e− e− + e e− and the problem is recast to :

~ 3γ0acc E(δk) 2 (δkx + iδky) b1     = 0 3γ0acc ~  (δkx iδky) E(δk)  b2  2     −    CONCLUSION :around the K~ -points linear dispersion relation (~k = K~ + δ~k)

E(δk) = h¯v δ~k § F | | with v = 3γ0acc , and since b2(δk iδk ) = b2(δk + iδk ), the F 2¯h 1 x − y 2 x y corresponding eigenstates can be rewritten as

iθk 1 e− i(κ(n)x+ky) Ψnsk(~r) = ~r n, s,~k =   e √2L C  s  h | i tube h     taking δk + iδk = κ(n) + ik = δk eiθk (θ = Atan(δk /δk )). x y | | k y x

⇓

Amplitude of a backscattering event requires estimate the scattering matrix n, s, k ˆ n, s, +k h − | T | i with

ˆ = ˆ + ˆ 0 ˆ + ˆ 0 ˆ 0 ˆ + . . . T U UG U1 UG UG U 1 1 = ˆ + ˆ ˆ + ˆ ˆ ˆ + . . . U U E U U E U E U − H0 − H0 − H0

Given

1 1 ˆ 1 = 1 U − E ˆ E 0 µ − E 0 ¶ − H0 − U − H − H ˆ ˆ 2 = (1 + U + U + . . .) G0 E E − H0 µ − H0 ¶

Developping n, s, k ˆ n, s, +k on the basis of eigenstates h − | T | i

1 e iθk ˆ iθk − 0 n0, s0, k0 n, s, k = n n0(k k0) e , s   h | U | i √2L C U − − µ ¶ s0 tube h     1 i(θk θ ) k0 = n n0(k k0)(e − + ss0) √2LtubeCh U − −

Backscattering amplitude for states on the same band close to Fermi energy (EF = E(~k = K~ ) = 0),

⇓

i(θ θ ) iπ 0, s, k ˆ 0, s, k = (2k)(e k− k + 1) = (2k)(e + 1) = 0 h | U | − i U0 − U0

Generalization to high order terms, i.e. 0, s, k ˆ (E)p 0, s, +k = h − | T | i

1 0( k kp) 0(kp kp 1) . . . 0(kp k) 1 1 1 . . . U − − U − − U − s [θ k] − [θkp] sp sp 1 [θkp][ − [θk ] sp 1 . . . s1 [θk ] − [θk] s p − − p 1 − 1 (√2LtubeCh) sX1k1 sX1k1 sX1k1 sXpkp (E εsp(kp))(E εsp 1(kp 1)) . . . (E εs1(k1))×h | R R | ih | R R − | i h | R R | i − − − − − where we deﬁne the rotation matrix [θ ] as R kp

θ e kp/2 0 [θk ] =   p θkp/2 R  0 e−     

1 The product of all amplitude reduces to s [θk] − [θ k] s = θ θ h | R R − | i k− k cos( 2 − ) = 0.

Thus for metallic nanotubes to all orders the backscattering is suppressed in the low energy range around Fermi level 2.3 Nature of disorder and defects

topological defects that yield undoped disordered tubes. Vacan- • cies, heptagon-pentagon pair defects (or Stone-Wales) that might bridge nanotubes with diﬀerent helicities. substitutional impurities. Chemical substitution of carbon atom • by Boron (B), Nitrogen (N) are deﬁned by { }

– Density ni (probability to ﬁnd an impurity inside the carbon- tube = P – Defect strength (resulting from a substitution C N, B) → ∼ the energy diﬀerence between p -orbital ⊥

∆εCN = εp ((Carbon) εp (Nitrogen) = 2.5eV ⊥ − ⊥ − (and ∆εCB = 2.33eV, cf. Harisson)

– Charge transfer = semiconducting tubes may thus become ⇒ either p-doped (B) or n-doped (N)

Anderson-type disorder: less realistic but allow derivation of mean • free path within Fermi golden rule

j i i i = γ0 p p + εi p p H i,j Xn.n. | ⊥ih ⊥| Xi | ⊥ih ⊥|

Site energies εi on carbon atoms are taken randomly within a given interval [ W/2, W/2] (W the disorder strength), hopping integral − is constant γ0 = 2.9eV (disorder with uniform probability 1/W ). Few remarks: Mean free path should roughly scale as ` 1/n . e ∼ c In the TB implementation (dilute alloy model), considering that we have one impurity with probability in the nanotube, the variance of P the uniform distribution :

σε (1 ) ∆εCN = W/2√3 ∼ rP − P | |

2.4 Elastic mean free path 2.5 Derivation within the Fermi golden rule

Weak disorder, disorder eﬀects are treated perturbatively

⇓ Fermi golden rule ` = v τ ⇒ e F 5 1 6 1 v = √3aγ /2¯h 8.10 ms− 10 ms− . F 0 ' −

Total density of states (TDoS) close to Fermi level (kn are deﬁned by E E(k ) = 0) − n ρ(E) = Tr[δ(E )] − H

2 ρ(E) = dkδ(E εn(k)) Ω Xn Z − 2 ∂εn(k) 1 = dkδ(k kn) − Z ¯ ¯ Ω Xn ¯ ∂k ¯ − × ¯ ¯ ¯ ¯ Ω is the volume of k-space per allowed value divided by the spacing ~ 2 2 1 4π h between lines, writes (8π /a √3)/(2π ~ − ), so Ω = |C |. |Ch| √3a2

⇓ The TDoS per carbon atom is thus given by:

2√3acc ρ(EF ) = πγ ~ 0|Ch| Fermi golden rule yields :

1 2π 2 = Ψn1(kF ) ˆ Ψn2( kF ) ρ(EF ) NcNRing 2τ(EF ) h¯ |h |U| − i| × with Nc and NRing are the respective number of pair atoms along the circumference and the total number of rings taken in the unit cell used for diagonalisation. By writting the eigenstates at the Fermi level as

1 imkF Ψn1,n2(kF ) = e αn1,n2(m) | i NRing m=1X,Nring | i r

A ˆ A p (mn) p (m0n0) = εA(random, m, n)δmm0δnn0 h B⊥ |Uˆ| ⊥b i p (mn) p (m0n0) = εB(random, m, n)δmm0δnn0 h ⊥A |U| ⊥A i p (mn) ˆ p (m0n0) = 0 h ⊥ |U| ⊥ i

where εB(random, m, n) and εA(random, m, n), site energies of electron at atoms A and B in position (m, n), are randomly taken in [ W/2, W/2] − 1 2π 1 1 1 = ( ε2 + ε2 )ρ(E ) 2τ(E ) h¯ 4 A B F F NcNRing NcXNRing NcNRing NcXNRing q q

Final estimate of the mean free path:

18a γ2 ` = cc 0 √n2 + m2 + nm (1) e W 2

For a metallic nanotube (NT = 5, NT = 5), with W = 0.2 the mean- free pathesis 560nm which is much more larger than the circumference length. Qualitatively one can recover such result by writting the mean 1 free path as ` T~ Prob.− 2N 500nm (for the (5,5) tube. e ∼| | × × T ∼ 2.5.1 Energy dependent mean free path

1400

1200 500

400 (5,5) 1000 (8,0) (5,5) 300 (15,15) 200 (30,30) 800 100

0 600 0 2 4

Mean−free−path (nm) 400

200

0 −8 −6 −4 −2 0 2 4 6 8 Energy (eV) Figure 2: Energy dependent mean free path (from Triozon et al.[11])

From fig.2 first the scaling law of the mean free path with the • nanotube diameter is confirmed close to the Fermi level of undoped tubes. The strength of disorder used in the calculation is W = 0.2 in γ0 unit. For semiconducting bands, the 1/W 2 is still satisfied, but mean • free pathes are seen to be much smaller. 2.5.2 Illustration on vacancies

Vacancies are generated by removal of one or several carbon-atoms in the tube.

Single lattice vacancy. Reorganized vacancy.

A peak of localized state at the Fermi level is found for for undoped • tubes (EF = 0). At such resonance the diﬀusion is completely screen and electrons remain in the neighbourhood of the vacancy (see Figure below).

0.2 Time [femtoseconds] 0 0.05 0.1 0.15 0.2 10 9 8

] 7 Localized states 2 6

0.1 .nm 0

γ 5

D(t) [ 3

Density of states [electron/site] 1

0 0 -2 -1 0 1 2 0 200 400 600 800 1000 γ -1 Energy [eV] Time [units of 0 ] DoS for a metallic tube with single lattice vacancy. Diﬀusion coeﬃcient (Latil et al.[?])

Differently if Fermi energy is slightly shifted away from E = 0, • F the diffusion is much affected by defects, and in the time scale considered in Fig.2.5.2, a saturation of diffusity (diffusive regime), with a further slight decrease (due to quantum interferences) are obtained. A single vacancy on an infinite long tube yields stepwise reduction • of conductance at a resonant energies corresponding to quasibound states (see Right-charge density countour plots) in the (10,10) tube.

2.5.3 Relation to chemical substitutions

Work of Liu et al[3], experiments on boron-doped nanotubes

The probability density of boron atom with respect to carbon atom • is evaluated to be 1% ' Diameters of tubes are in between [17nm, 27nm] • Fits of weak localization yield mean free pathes in the order of • ` = 220 250nm e − Applying equation (1), one finds a theoretical estimate of ` 274nm e ' for the tube with diameter 17nm For small nanotubes contacted in between metallic leads, the effect of boron or nitrogen substitution can be investigated through ab-initio methods: Effect of a Boron (Nitrogen) impurity (Quasibound states and scattering)[7] A single impurity per nanotube unit cell yield stepwize recution of quantum condutance (1 quantum channel suppressed). Realistic situation sould addressed random distribution of subtituted impurities. (a) 6 (b) 6 5 5

/h) 4 /h) 4 2 2 3 3 2 2 G (2e G (2e 1 1 0 0 -0.5 0 0.5 -0.5 0 0.5 E (eV) E (eV) boron.002.density_eigen.fixed.ps input file: 002.density_eigen.dat E = -0.69860 eV G = 0.99961 state: 2 th eigenchannel k mixed v = 0.00261 c T = 0.00088 6 max = 2.20710 Angstrom^(-2) 5

/h) 4 2 3 perfect tube 2 1 with boron 0

0 odd component of channel 1 π even component of channel 2 0 phase shift LDOS G (2e -0.5 0 0.5 E (eV) 0.000 0.025 0.050 0.100 0.200 0.400 nitrogen.002.density_eigen.fixed.ps input file: 002.density_eigen.dat E = 0.53140 eV G = 1.00056 state: 2 th eigenchannel k mixed v = 0.00250 c T = 0.00125 6 max = 5.17620 Angstrom^(-2) 5

/h) 4 2 3 perfect tube 2 1 with nitrogen 0

π even component of channel 1 0 odd component of channel 2 phase shift LDOS G (2e -0.5 0 0.5 E (eV) 0.000 0.025 0.050 0.100 0.200 0.400 3 Quantum interference eﬀects and magnetotransport

The magnetoresistance depends on the probability for an electronic P wavepacket to go from one site P to another Q , which can be | i | i written as

2 i(αi αj) P Q = i + i je − P → Xi | A | iX=j A A 6 with eiαi the probability amplitude to from from P to Q via the Ai i-path. Switching on a magnetic ﬁeld removes time-reversal symmetry of these pathes

⇓ Increase of conductance or decrease of resistance(negative magne- • toresistance). Modulation of the ﬁeld-dependent resistance that become Φ /2- • 0 periodic. Indeed (A~ the potential vector)

e 2π α = A.d~ ~r = A.d~ ~r § §h¯c I §Φ0 I

2 i(α+ α ) 2 yielding 1 + e − − so modulate by a cos(2πΦ/Φ ) | A | | | 0 factor. Results on simulation on metallic tubes (ﬁeld-dependent studies of diffusion coeﬃcients) CASE ` < < L(τ ) e C φ

Diﬀusivity increases at low ﬁeld (negative magnetoresistance) • Oscillations are dominated by a Φ /2-periodic Aharonov-Bohm • 0 period, that is

(τ , Φ + Φ /2) = (τ , Φ) D φ 0 D φ 40 3

35 2 ) φ 60 40 τ

( Ballistic ψ (b)

D 40 le=3 nm 30 35 1 20 (a) le=0.5 nm 0 30 0 400 0 0.5 1 t φ/φ0 25 0 0 0.25 0.5 0.75 1 φ/φ0

2 Figure 3: Main frame : (τφ, Φ/Φ0) (in A˚ γ0/h¯ unit) for a metallic SWNT (9, 0) evaluated D at time τφ τe, for two disorder strengths,W/γ0 = 3 and 1, such that the mean free path À (è 0.5 and 3 nm, respectively) is either shorter (dashed line) or larger (solid line) than the nanotub∼ e circumference ( 2.3nm). The right y-axis is associated to the dashed line and the C ∼ left y-axis to the solid line. inset : (τφ, Φ/Φ0) for è = 3 nm and L(τφ) < 2è (from [?]) D agrees with weak localization theory’s predictions CASE ` > C, L(τ < 2` ) = e φ e ⇒ Positive magnetoresistance •

(τ , Φ + Φ ) = (τ , Φ) D φ 0 D φ 4 Localization regime

Low dimensional systems mean free path and localization length are related through the Thouless relation [8]

ξ = 2`e (1D)

ξ N `e (quasi-1D) ∼ ⊥

Metallic carbon nanotubes at EF

36a γ2 ξ = cc 0 √n2 + m2 + nm (1) e W 2 that also scales linearly with the tube diameter for low disorder.

Quantum interferences and localization can be achieved by following the participation ration of electronic eigenstates

This quantity give at a given energy, how many sites in a tight binding basis contribute to weight the eigenstate (+scaling analysis).

2 2 ( ψi(E) ) Xi | | α P R(ψ(E)) = 4 N ψi(E) ' Xi | |

+ periodic boundary conditions and a unit cell with N atoms. With disorder If α = 1 = states are purely extended, • ⇒ α 0 the PR give the localization length. • → Intermediate scaling provide informations about the increasing • contributions of quantum interferences in the diﬀusive regime (al- though states remain extended).

t t 2e2D −τφ 1 δσIQ = ∞ dt (r, t)(1 e−τe )e P R− − πh¯ Z0 P − ∼ −

Eigenstates characterized by a linear scaling in N are uniformly extended and associated with a vanishing contribution of QIE, i.e. (σBB is the Bloch-Boltzman result)

δσ/σ 0 BB →

Localized states are related to strong contributions of QIE, i.e.

δσ/σ 1 BB ' Scaling laws P R(N) = N α, with 0 < α < 1, indicate the relative strength of QIE. The conductance thus change from

G = e2(v2τ δσ)/L e − tube to an exponentional decrease as soon as L ξ tube ≥

PR 4000 4000 3000 (6,6) 2000

1000 3000 0 0 1000 2000 3000 4000

2000 (7,5)

1000

0 0 1000 2000 3000 4000 Number of atoms

Figure 4: The participation ratio for a metallic or semiconducting tubes. Fermi energies are given by the charge neutrality point for (6, 6) and is chosen to be EF 0.333γ0 for the (7, 5) tube (to mimick ∼ doping). Values of disorder strength =0.054eV (open circles), 0.136eV (filled circles), 0.98eV (filled diamond). From the results, one sees that whereas the metallic tube remain nearly insesnitive to disorder (as manifested by linear scaling), the PR for the (7, 5) semiconducting tube is much more affected, with a PR tending to a constant value for W 0.98eV . By taking the limit of PR, one finds ' a mean free path of è = ξ/2 20nm. ∼ 4.1 Contribution of intershell coupling

Multiwalled nanotubes are made of a few to thenths of shells with random helicities and weakly coupled through Van der waals intershell interaction.

Description of intershell coupling (one p -orbital per carbon atom is ⊥ kept, with zero onsite energies, whereas constant nearest-neighbor hopping on each layer n (n.n.), and hopping between neighboring layers (n.l.))

d a j i ij− j i = γ0 p p β cos(θij)e− δ p p H ·i,j Xn.n. | ⊥ih ⊥|¸ − ·i,jXn.l. | ⊥ih ⊥|¸ ∈

Incommensurate MWNT (6, 4)@(10, 10)@(17, 13) Commensurate MWNT (6, 4)@(12, 8)@(18, 12).

i j θij is the angle between the p⊥ and p⊥ orbitals, and dij denotes their relative distance. The parameters used here are : γ0 = 2.9eV , a = 3.34A˚, δ = 0.45A˚. Ab-initio estimate gives β γ0/8 ' THERE EXIST TWO CLASSES OF MWNTs (for given metallic/semiconducting characters of inner shells)

Periodic objects as (6, 4)@(12, 8)@(18, 12) case (Fig.4.1-right). with • a common unit cell for all shells, that is deﬁned by an unique translational vector T~ 18.79A˚. | |' Incommensurate objects such as (6, 4)@(10, 10)@(17, 13)tube • (Fig.4.1-left) The translational vector along each shell are respectively T~ = | (6,4) | 3√19a , T~ ) = √3a , T~ = 3√1679a , (ratio cc | (10,10) | cc | (17,13) | cc of lengthes of individual shell translational vectors are irrational numbers).

4.1.1 Commensurate multiwall nanotubes

(5,5) @ (10,10) @ (15,15) 78 G(E) 72 66 60 54 48 42 36 30 24 18 12 6 0 −1.5 −1 −0.5 0 0.5 1 1.5 Energy(eV)

Figure 5: Conductance pattern of the (5, 5)@(10, 10)@(15, 15) MWNT.

PROPERTIES OF COMMENSURATE OBJECTS (full metallic jacket) Commensurate MWNTs are periodic objects with a well deﬁned • unit cell. Bloch theorem applies and the bandstructure can be computed. Depending on SYMETRIES

– Full translational + rotational symetries – In the ballistic regime, the conductance spectrum of the MWNT would be given at ﬁrst order by the sum of total conducting channels for a given energy(Fig.5).

– Very small intrinsic resistance at EF for a metallic MWNT.

Broken translational and rotational symetries • – Splitting of degeneracy occurs and pseudogaps are formed[4, 6]. An example is shown on Fig.6 where apart from the presence of pseudogaps, the intershell interaction also results in aﬀecting the local density of states of the outer shell, eﬀect that can be investigated with STM experiments[6].

Figure 6: Electronic bandstructure of (5, 5)@(10, 10) (right) and LDoS on sites labeled 1-8 in the external layer[6]. – The opening of pseudogaps have direct consequences on the total number of conducting channels available at a given energy = STEPWIZE reduction of quantized conductance ⇒

0.03

0.02

0.01 (a) (b) DOS [states/eV/atom] 0.00

0 4 G/G 2 (c) (d) 0 -0.2 0.0 0.2 -0.2 0.0 0.2 E [eV] E [eV]

Figure 7: Electronic density of states and conductance for the (10, 10)@(15, 15) [(a) and (c)] and (5, 5)@(10, 10)@(15, 15) [(b) and (d)]. LANDMARKS

In commensurate systems, a relaxation with a typical time scale of •

τ hγ¯ /β2 ll ∼ 0

in good agreement with the Fermi Golden Rule(see Fig.8). By increasing the amplitude of β in the range [γ0/8, γ0], the expected scaling form of τll is checked.

1.0

0.9 (6,4)@(12,8)@(18,12) Ψ 2 (i,j) 0.8 β γ /10 = 0 0.7 (18,12) β=γ /3 0.6 0

0.5 β=γ0 0.4

0.3 (12,8)

0.2 (6,4) 0.1

0.0 0 100 200 300 400 500 600

Diffusion time (h/γ0)

Figure 8: Wavepacket spreading as a funtion of intershell coupling strength.

Two electrodes separated by 1µm and assuming ballistic transport • 6 1 with a Fermi velocity of 10 ms− ,

⇓

t 4500¯h/γ 102τ ∼ 0 ∼ Important contribution of interwall coupling in experiments. 4.1.2 Incommensurate multiwall nanotubes

Incommensurate shells = no Bloch theorem

Redistribution of the wavepacket amongst inner shell induced by • intershell coupling. (slower with a higher weight lost from the outer shell).Intermediate objects between periodic and disordered systems.

1.0 1.0 (6,4)@(12,8)@(18,12) (6,4)@(10,10)@(17,13) 0.9 Ψ 2 (i,j) 0.8

0.7 (17,13) 0.6 0.78 (18,12)

0.5 0.73

0.4 0.68 0 100 200 300 (10,10) 0.3

0.2 (6,4) (12,8) 0.1 (6,4) 0.0 0.0 0 500 1000 0 500 1000 1500 2000 γ ) Diffusion time (h/ 0 Diffusion time (h/γ0) Figure 9: WP spreading for commensurate and incommensurate triple-wall(from[?]).

Wavepacket redistribution in incommensurate MWNT indicates randomization of quantum phase, and homogeneous spreading. Diffusion coefficient (energy averaged) DEPARTURE from ballis- • tic motion because of multiple scattering effects in the non-periodic systems. Conduction is said to be non-ballistic, with

= (t) t tη L rD × ∼ A

4000 (9,0)@(18,0)

(9,0)@(10,10) 3000 ) φ τ 2000 D(

1500

1000 1000

500 (6,4)@(10,10)@(17,13)

0 0 200 400 600 800 0 0 200 400 600 800 1000 τφ

Figure 10: Main Frame: Averaged diffusion coefficient (arb. unit) for (9, 0)@(18, 0) and (9, 0)@(10, 10) with β = γ0/3. Inset: Avergared diffusion coefficient for the incommensurate MWNT (6, 4)@(10, 10)@(17, 13).

The coeﬃcient η is found to decrease from 1 to 1/2 by in- ∼ ∼ creasing the number of coupled incommensurate shells 5 Role of electrode-nanotube contacts

6 Some general landmarks

End-contact configuration Side-contact configuration

Nanotube Nanotube

Figure 11: Diﬀerent contact conﬁguration between nanotube and electrodes.

Two diﬀerent kinf of metal/nanotube junctions can be deﬁned: metal- metallic nanotube-metal junction •

ikmp ikF p If km = e ϕm (resp. kF = e ϕNT ) the propagating states with km (kF ) | i p | i | i p | i the wavectorP in the metal (resp. nanotubPe). We take ϕNT the localized basis vectors, that | i will have nonzero overlap with ϕm only for a few unit cells (p) deﬁning the contact area. The | i scattering rate between the metal and the nanotube can be written following the Fermi golden

rule and will be related to ( contact the coupling operator between the tube and electrodes) H

i(km kF )p km contact kF γ0 ϕNT ϕm e − h | H | i ∼ h | i Xp

Various physical aspects can be outlined:

– γ0 is related to the chemical nature of interface bonding (covalent, ionic,...). In the most favourable case of a covalent 2 coupling, lowest contact resistance is given by Rc = h/2e , whereas ionic bonding would mostly favoured tunneling con- 2 2 tact resistance R h/¯ (2πe γ0 ). c ∼ | | – ϕ ϕ is related with the geometry and conﬁguration of h NT | mi contact between nanotube and electrodes: end or side contacts, length of the contact area,... – The last term is maximized whenever wavevector conservation ( δ(k k )) is best satisﬁed. For instance in case of a ∼ m − F metallic armchair tubes, larger coupling will be achieved for k 2π/3√3a . Much smaller metallic wavector will yield m ' cc small coupling rate. – The tunneling rate from the metal to the nanotube is given by

1 2π k k 2 ρ (E )ρ (E ) τ ∼ h¯ | h m | Hcontact | F i | NT F m F

with ρNT(EF ) (resp. ρm(EF )) the density of states of the nanotube (metal) at Fermi level.

metal-semiconducting nanotube-metal junction subject to • Schotkky barriers that forbid electronic transmission at zero or low bias voltage

Schottky barrier heights are related to the energy difference between the metallic work functions and the semiconducting electronic affinity. Sensitivity to the tube-diameter which range, according to ab-initio calculations([12]), from 5.4eV down to the value for graphene (for large tube) that is 4.91eV (Fig.12). Met- ∼ als such as Au and Ni with work functions 5.1eV will behave ' differently from Al or Ti with respective work functions of 4.28eV and 4.33eV . 5.5

) 5.4 (7,0) V

e 5.3 (

n 5.2 (10,0) o i

t 5.1

c (13,0) n 5.0 (17,0) u f

(19,0)

k 4.9 r (12,12) o 4.8 (10,10) (8,8) (6,6) W (15,0) (5,5) 4.7 (12,0) (4,4) 4.6 0.05 0.10 0.15 0.20 o 1/D(A-1)

Figure 12: Workfunctions calculated for several semiconducting and metallic nanotube 7 Atomic scale interface and bonding character

Ab-initio study demontsrate that electrodes in Titanium are more suitable to achieve high transmission at the contact (when compared to Al, Au). Two contact conﬁgurations were considered: the side contact and the end contact as shown on Fig.11

In Fig.13, the energy dependent conductance for a N=15 (5,5) open tube en-contacted to Al(111) surfaces is reported in (a)-Fermi level is set to zero, whereas the inset shows a Schematic spectrum showing the four states responsible for the resonances. Transmission spectrum for the highest conducting states is reported in (b).

Shown on Fig.13 are the conductance and transmission patterns of a Al-nanotube-Al junction. Instead of the 2 0 plateau, only individual resonances persist (whose number increases with system length). G Nonetheless from such calculation it ﬁrst appears that π states are less backscattered when compared ∗ twith π states. Fig.?? shows the diﬀerence with similar junctions but with Au and Ti instaed of Al.

Results point towards better matching between Ti-orbitals with carbon π, π orbitals than do Al or ∗ Au-based interfaces. 2.0 * k k2 π •1 •

1.5 * E (a.u.) * ∗ • k1 k π * • 2 k /h) 1 k2 Enter text here 2 k k2 1 1.0 k G (2e 0.5 Enter text here (a) 0.0 π 0.8

0.6

0.4

0.2 π∗ (b) Transmission eigenvalues 0.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Energy (eV)

Figure 13: Al-(5,5)-Al end-contacted(from [13]) 2.0 2.0

1.5 1.5 /h) /h)

2 Enter text here Ti Enter text here 2 1.0 1.0 G (2e G (2e 0.5 0.5 Enter text here Enter text here Au (a) (a) 0.0 0.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 (b) (b) Transmission eigenvalues

Transmission eigenvalues 0.0 0.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Energy (eV) Energy (eV)

Same for N=10 (5,5) end-contacted to Au(111). Same for a Ti(111) surface.

8 ANNEXES

8.1 The Fermi golden rule

To investigate quantum conduction in carbon nanotubes one needs to solve the time dependent Schr¨odinger equation

∂ ih¯ Ψ(~r, t) = ( ˆ + ˆ)Ψ(~r, t) ∂t H0 U where

2 2 h¯ 2 h¯ 2 ˆ0 = − + VˆXtallin = − H 2m ∇ 2m∗ ∇ (under the effective mass approximation). The operator ˆ describes U the effect of superimposed disorder due to e.g. substitutional or ad- sorbed impurities, topological defects, ... When disorder in the systems is low, its effect is to limitate the lifetime of eigenstates of ˆ inducing H0 transition between different allowed vectors at a given energy (elastic scattering). If one sets ˆ k = ε k H0| i k| i the eigenstates of the unperturbated Hamiltonian, then in presence of ˆ, one can assume a trial wavefunction Ψ(~r, t) = k ~r k k Ψ U P h | ih | i which is taken as k ck(t)ψk(~r). Introducing this form in the general equation one gets: P

ˆ d iεk0 k0 k ck0(t) + ck0(t) = h |U| ick(t) dt h¯ Xk ih¯ If at time t = 0 the electron is in state k then at ﬁrst order its | i evolution is driven by

ˆ d iεk k0 k iε t/h¯ c (t) + 0 c (t) = h |U| ie− k dt k0 h¯ k0 ih¯ and the general solution writes

ˆ i(εk ε )t iε T/h¯ k0 k T − − k0 ck (t = T ) = e− k0 h |U| i e ¯h dt 0 ih¯ Z0 The Fermi golden rule is thus derived by computing the probability per unit time to obtain a transition from the state k to any other 2 | i allowed state k0 ( c ), that is | i | k0 |

1 2π k0 Uˆ k 2 = h | | i δ(εk εk) ¯ ¯ 0 τ h¯ Xk ¯ ih¯ ¯ − 0 ¯ ¯ ¯ ¯ Let’s deﬁne (t, t ) as the probability amplitude of a transition be- S 0 tween two states induced by the disorder potential acting during an inﬁnite period from t = t t = + , then 0 −∞ ∞

1 2 + ˆ ∞ iωkk (t2 t1) (t, t0) = 2 k0 k dt1dt2e 0 − ¯ ¯ Z S h¯ ¯h |U| i¯ −∞ ¯ ¯ 1 ¯ ¯2 + ∞ = 2 k0 k 2πhδ¯ (εk εk ) dt ¯ ¯ 0 Z h¯ ¯h |U| i¯ × − × −∞ ¯ ¯ ¯ ¯ Then the transition probability per unit time is given by

1 2π 2 = k0 k δ(εk εk ) ¯ ¯ 0 τ h¯ Xk ¯h |U| i¯ − 0 ¯ ¯ which is known as the Fermi Golden¯ Rule¯ (FGR)

8.2 Bandstructure and DoS with magnetic ﬁeld

To investigate Aharonov–Bohm phenomena, we start from the Hamil- tonian for electrons moving on a nanotube under the influence of Hkk0 a magnetic field: ie 1 i(k.R k0.R0) ∆ϕ − − − h¯ R,R0 kk0 = e H N RX,R0 (1) p2 ψ(r R) ¯ + V ¯ ψ(r R0) * ¯ ¯ + × − ¯2m ¯ − ¯ ¯ ¯ ¯ ¯ ¯ where the phase factors are given ¯by ¯ 1 ∆ϕR,R = (R0 R) ( (R + λ[R0 R]))dλ, (2) 0 Z0 − · A − and ψ(r R0) is the localized atomic orbital, and p and V are, | − i respectively, the momentum and disorder operators. As shown here- after, different physics is found according to the orientation of the magnetic field with respect to the nanotube axis. In the former case, the vector potential is simply expressed as = (φ/ , 0) in the two- A |Ch| dimensional ~ / ~ , T~/ T~ coordinate system, and the phase factors Ch |Ch| | | become ∆ϕ = i( 0)φ/ for R = ( , ). This yields new R,R0 X − X |Ch| X Y magnetic field-dependent dispersion relations ε(δk, φ/φ0). Close to the Fermi energy, this energy dispersion relation is affected according to k k + 2πφ/(φ0 h ) which leads to a φ0-periodic variation of ⊥ → ⊥ |C | the energy gap ∆g. In the semiconducting case, the oscillations in the DOS correspond 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 1 0.1 0.9 0 0 −3 −1 1 3 −3 −1 1 3 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 0.2 0.8 0.5 0 0 −3 −1 1 3 −3 −1 1 3 to the following variations of the gap widths [?]:

φ φ0  ∆0 ¯1 3 ¯ if 0 φ ¯ ¯  ¯ − φ0 ¯ ≤ ≤ 2 ∆ =  ¯ ¯ (3) g  ¯ ¯  ¯ φ ¯ φ0 ∆0 ¯2 3 ¯ if φ φ0  ¯ ¯  ¯ − φ0 ¯ 2 ≤ ≤  ¯ ¯  ¯ ¯  ¯ ¯ where ∆0 = 2πaC Cγ0/ ~h is a characteristic energy associated with − |C | the nanotube. It turns out that at φ values of φ0/3 and 2/3φ0, in accordance with the values of ν = 1, there is a local gap-closing § in the vicinity of either the K or K 0 points in the Brillouin zone. This can be seen simply by considering the coeﬃcients of the general wavefunction in the vicinity of the K and K 0 points, which can be written as Ψ ~ ~ (~r + ~h). Since periodic boundary conditions apply K+δkK C in the ~h direction, one can write exp[i(K~ + δ~kK) ~h] = 1. For C ~ ~ ~ · C ~ ν = +1, we write δkK = (2π/ h )(q 1/3)k +δk k whereas δkK = |C | − ⊥ k k 0 (2π/ h )(q + 1/3)k~ + δk k~ . When K K0, then 1/3 1/3 |C | ⊥ k k → § → ∓ in the above expressions, as ν goes from +1 1, which makes → − the situation between K and K0 symmetrical. Similarly for metallic nanotubes, the gap-width ∆g is expressed (see ﬁgure for illustration) by

φ φ0  3∆0 if 0 φ  φ0 ≤ ≤ 2 ∆ =  (4) g   φ φ0 3∆0 ¯1 ¯ if φ φ0 ¯ ¯  ¯ − φ0 ¯ 2 ≤ ≤  ¯ ¯  ¯ ¯  ¯ ¯ References

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