Theory of quantum transport in carbon nanotubes Stephan Roche D´epartement de Recherche Fondamentale sur la Mati`ere Condens´ee 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
1.1
Ballistic conduction in singlewall and multiwall carbon nanotubes Bandstructure and conducting channels
Landauer formula: Tn(E), the transmission amplitude for a given channel n,at energy E :
2e2 X Tn(E) G(E) = h n=1,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 G(E) = × N⊥(E) h (5,5) metallic nanotube : Two quantum channels are available at Fermi energy EF = 0 (with 2 G0 = 2eh the quantum resistance)
G(EF = 0) = 2G0
(5,5) Armchair metallic nanotube 10
4
Energy
G(G0)
10 bands
8 2
6 0
4 −2
2
10 bands
−4
0 −4
−2
0
wave vector
2
4
−4
−2
0
Energy
2
4
Intrinsic conduction mechanism and conductance scaling are simulatenously followed through The Kubo formalism (nanotube of length Ltube) :
2e2 ˆ G(E, Ltube) = lim Tr[δ(E − H)D(t)] Ltube t→τ • δ(E − H) = spectral operator (DoS)
ˆ = (Xˆ (t) − Xˆ (0))2 /t = the diffusion operator • D(t) • Contact are fully discarded, and considered as reflectionless.
Ballistic regime =
2 2 ˆ ˆ ˆ hD(t)i EF = (X (t) − X (0)) /t ∼ vF t
with vEF = 3accγ0/2¯h, Ltube = vF τ and DoS = N⊥/2π¯hvF
G = 2N⊥e2/h
0.8 200
Pente : vF=3πγ0a/h
0.6
2
〈X 〉/t
Densité [état/γ0/atome]
150 100 50 0
0.4
0
20
40
0.2
0
60
80
t [h/2πγ0]
Filtre
−1
−0.5
0
0.5
Energie [γ0]
Re (ψ) 0
Etat filtré
1
Effects of disorder and doping
2 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 :
D(t ≥ τ ) = `evF where `e the mean free path is introduced, and τ = `e/vF . The conductance
G = 2N⊥e2/h(`e/Ltube) • Quantum interferences and localization : In the weak localization regime: t t − − 2e D Z ∞ τe )e τφ δG = − dtP(r, t)(1 − e π¯hLtube 0 2
Ltube ≥ ξ ∼ N⊥`e (ξ the localization length)
G(Ltube) = (h/2N⊥e2) exp(ξ/Ltube)
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 r
E(~k) = ±γ0 3 + 2 cos(~k.~a1) + 2 cos(~k.~a2) + 2 cos(~k.(~a1 − ~a2)) ~ ~ = 0 with At the corners of the Brillouin zones K-points one gets E(K) two degenerate eigenvectors (graphite or metallic tubes) ~ ~
eiK.` ~ +i √ (pz (~r − ~rA) + pz (~r − ~rB )) bonding state | K ΨK,s r) = ~ (~ 2 ~` all cells X
~ ~
eiK.` ~ −i √ (pz (~r − ~rA) − pz (~r − ~rB )) antibonding state | K ΨK,a r) = ~ (~ 2 ~` X
~ At K-points, the probability amplitude of a scattering event from a ~ +i to a state | K ~ −i as state | K ~ + | Uˆ | K ~ −i = hK =
Z
Z
~ + | ~riU(~r, r~0)hr~0 | K ~ −i d~rdr~0hK ~ + | ~ri(uAδ(~r − ~rA) + uB δ(~r − ~rB ))hr~0 | K ~ −i d~rdr~0hK
1µ ∗ B∗ B A∗ B∗ B = uA{(pA + p )p } + u {(p + p B z z z z z )(−pz )} 2 1 = (uA − uB ) 2
CONSEQUENCE: if the disorder potential is long ranged with respect to the unit cell, i.e uA ' uB (conservation of pseudospin symmetry)
~+ | U | K ~ −i = 0 (full suppression of backscattering). hK ~ Eigenstates in the vicinity of K-points (low energy limit). Around some ~k point, the wavefunction :
~ r) ~ r) + cB (~k)˜ pB Ψ(~k, ~r) = cA(~k)˜ pA z (k, ~ z (k, ~ with
1 X i~k.~` e pz (~r − ~rA − ~`) Ncells ~` ~k, ~r) = √ 1 X ei~k.~`pz (~r − ~rB − ~`) p˜B ( z Ncells ~` ~ r) p˜A z (k, ~
= √
One then has to compute the factors
HAA(~k) =
1
X
~ ~ ~0
~
~0
` A,` eik.(`−` )hpA, z | H | pz i
Ncells ~`,`~0 1 X i~k.(~`−`~0) A,~` B,`~0 ~ HAB (k) = e hpz | H | pz i Ncells ~`,`~0 = and so forth Following the definitions of Fig.1 B,0 −i~k.~a1 A,0 B,−~a1 −i~k.~a2 A,0 HAB (~k) = hpA,0 |H|p i + e hp |H|p i + e hpz |H|pzB,−~a2 i z z z z = −γ0α(~k)
y acc B a2
A
a1
a1
=
a2
=
a1 a2
=
3 a cc (1/2, 3)
=
3a
cc
3 acc
(-1/2, 3)
x
Figure 1: Representation of graphite lattice.
in other terms
E(~k) − γ0α(~k) b1 b2 − γ0α(~k) E(~k)
=0
√ ~ ~ ~ ~k (K ~ = (4π/(3 3acc), 0)) with α(~k) = 1+e−ik.~a1 +e−ik.~a2 and ~k = K+δ one gets α(~k) = 1 + e−2iπ/3 e−iδkxa/2 + e2iπ/3 e−iδky a/2 and the problem is recast to :
E(δ~k) 3γ0 acc 2 (δkx − iδky )
3γ0 acc 2 (δkx
b1 + iδky ) b2 E(δ~k)
=0
~ CONCLUSION :around the K-points linear dispersion relation (~k = ~ + δ~k) K
E(δk) = ±¯ hvF | δ~k |
0 acc , and since b2 (δk − iδk ) = b2 (δk + iδk ), the with vF = 3γ2¯ x y x y 2 1 h corresponding eigenstates can be rewritten as
−iθk 1 i(κ(n)x+ky) e ~ Ψnsk (~r) = h~r|n, s, ki = √ e s 2LtubeCh
taking δkx + iδky = κ(n) + ik =| δk | eiθk (θk = Atan(δky /δkx)).
⇓ Amplitude of a backscattering event requires estimate the scattering matrix hn, s, −k | Tˆ | n, s, +ki with ˆ 0 Uˆ + UG ˆ 0 UG ˆ 0 Uˆ + . . . Tˆ = Uˆ + UG 1 1 1 = Uˆ + Uˆ Uˆ + Uˆ Uˆ Uˆ + . . . E − H0 E − H0 E − H0
Given 1 Uˆ ¶−1 1 µ = 1− E − H0 E − H0 E − H0 − Uˆ µ Uˆ ¶2 Uˆ = G0 (1 + + + . . .) E − H0 E − H0
Developping hn, s, −k | Tˆ | n, s, +ki on the basis of eigenstates −iθk0 ¶ µ 1 e 0 iθ 0 0 0 ˆ k Un−n0 (k − k ) e , s 0 hn , s , k | U | n, s, ki = √ s 2Ltube Ch
1 =√ Un−n0 (k − k 0)(ei(θk −θk0 ) + ss0) 2Ltube Ch Backscattering amplitude for states on the same band close to Fermi ~ = 0), energy (EF = E(~k = K) ⇓
h0, s, k | Uˆ | 0, s, −ki = U0(2k)(ei(θk −θ−k ) + 1) = U0(2k)(eiπ + 1) = 0 Generalization to high order terms, i.e. h0, s, −k | Tˆ (E)p | 0, s, +ki =
1 U0(−k − kp)U0(kp − kp−1) . . . U0(kp − k X X X X √ . . . ( 2Ltube Ch)p s1k1 s1k1 s1k1 sp kp (E − εsp (kp ))(E − εsp−1 (kp−1 )) . . . (E − εs1 where we define the rotation matrix R[θkp ] as
R[θkp ] =
e
θkp /2
0
0 e
−θkp /2
The product of all amplitude reduces to hs | R[θk ]R−1[θ−k ] | si = −k ) = 0. cos( θk −θ 2 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. Vacancies, 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 carbontube = 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
H = γ0
X
i,j n.n.
|pj⊥ihpi⊥| +
X
i
εi|pi⊥ihpi⊥|
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 `e ∼ 1/nc. In the TB implementation (dilute alloy model), considering that we have one impurity with probability P in the nanotube, the variance of the uniform distribution :
√
r
σε ∼ P(1 − P) | ∆εCN |= W/2 3
2.4
Elastic mean free path
2.5
Derivation within the Fermi golden rule
Weak disorder, disorder effects are treated perturbatively ⇓
Fermi golden rule ⇒ `e √ vF = 3aγ0/2¯h ' 8.105 ms−1 − 106ms−1.
= vF τ
Total density of states (TDoS) close to Fermi level (kn are defined by E − E(kn) = 0) ρ(E) = Tr[δ(E − H)]
2 XZ dkδ(E − εn(k)) ρ(E) = Ωn ¯ ∂ε (k) ¯ 2 XZ ¯ ¯−1 n ¯ = dkδ(k − kn) × ¯¯ ¯ Ωn ∂k
Ω is the volume of k-space per allowed value divided by the spacing √ ~ 2 2 √ Ch | . between lines, writes (8π /a 3)/(2π|C~h |−1), so Ω = 4π| 3a2 ⇓ The TDoS per carbon atom is thus given by:
√ 2 3acc ρ(EF ) = πγ0|C~h| Fermi golden rule yields :
2π 1 ˆ n2(−kF )i|2ρ(EF ) × NcNRing = |hΨn1(kF )|U|Ψ 2τ (EF ) h ¯ 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 X | Ψn1,n2(kF )i = eimkF | αn1,n2(m)i NRing m=1,Nring r
1 NXc 2iπn B (mn)i+ | p | αn1(m)i = √ e Nc (| pA ⊥ (mn)i) ⊥ 2Nc n=1 1 NXc 2iπn B √ | αn2(m)i = e Nc (| pA ⊥ (mn)i− | p⊥ (mn)i) 2Nc n=1
The disorder chosen is a white noise distribution given by
hpA ⊥ (mn) hpB ⊥ (mn) hpA ⊥ (mn)
ˆ pA (m0n0)i = εA(random, m, n)δmm0 δnn0 |U| ⊥ ˆ pb (m0n0)i = εB (random, m, n)δmm0 δnn0 |U| ⊥ ˆ pA (m0n0)i = 0 |U| ⊥
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 = (q 2τ (EF ) h ¯ 4 Nc NRing
X
Nc NRing
ε2A +
q
1 Nc NRing
X
Nc NRing
ε2B )ρ(EF )
Final estimate of the mean free path:
18accγ02 √ 2 n + m2 + nm `e = 2 W
(1)
For a metallic nanotube (NT = 5, NT = 5), with W = 0.2 the meanfree pathesis 560nm which is much more larger than the circumference length. Qualitatively one can recover such result by writting the mean free path as `e ∼| T~ | ×Prob.−1 × 2NT ∼ 500nm (for the (5,5) tube.
2.5.1
Energy dependent mean free path
1400
Mean−free−path (nm)
1200
500 (5,5) (8,0)
400
1000 800
(5,5) (15,15) (30,30)
300 200 100
600
0
0
2
4
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).
Time [femtoseconds] 0 10
0.05
0.1
0.15
0.2
9 8 2
D(t) [γ0.nm ]
Density of states [electron/site]
0.2
Localized states 0.1
7 6 5 4 3 2 1
0
-2
-1
0
1
2
Energy [eV] DoS for a metallic tube with single lattice vacancy.
0 0
200
400
600
800
1000
-1
Time [units of γ0 ]
Diffusion coefficient (Latil et al.[?])
• Differently if Fermi energy is slightly shifted away from EF = 0, 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 `e = 220 − 250nm Applying equation (1), one finds a theoretical estimate of ` e ' 274nm 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.
5
5
4
2
2
G (2e /h)
(b) 6
G (2e /h)
(a) 6
3 2 1
4 3 2 1
0
0 -0.5
0
E (eV)
0.5
-0.5
0
E (eV)
0.5
G (2e2/h)
perfect tube with boron
LDOS
6 5 4 3 2 1 0
bor inp E= G= sta k v= T= ma
phase shift
0 π 0
odd component of channel 1 even component of channel 2
-0.5
0
E (eV)
0.5 0.000 0.025 0.050 0.100 0.200 0.400
6 5 4 3 2 1 0
perfect tube with nitrogen
LDOS
G (2e2/h)
nitr inp E= G= sta k v= T= ma
phase shift
0 π 0
even component of channel 1 odd component of channel 2 -0.5
0
E (eV)
0.5 0.000 0.025 0.050 0.100 0.200 0.400
3
Quantum interference effects and magnetotransport
The magnetoresistance depends on the probability P for an electronic wavepacket to go from one site | P i to another | Qi, which can be written as
PP →Q =
X
i
| A i |2 +
X
i6=j
AiAj ei(αi−αj )
with Aieiαi the probability amplitude to from from P to Q via the i-path. Switching on a magnetic field removes time-reversal symmetry of these pathes ⇓ • Increase of conductance or decrease of resistance(negative magnetoresistance). • Modulation of the field-dependent resistance that become Φ0/2~ the potential vector) periodic. Indeed (A
e I ~ 2π I ~ α± = ± A.d~r = ± A.d~r h ¯c Φ0 yielding | A |2| 1 + ei(α+−α−) |2 so modulate by a cos(2πΦ/Φ0 ) factor.
Results on simulation on metallic tubes (field-dependent studies of diffusion coefficients) CASE `e < C < L(τφ) • Diffusivity increases at low field (negative magnetoresistance) • Oscillations are dominated by a Φ0/2-periodic Aharonov-Bohm period, that is
D(τφ, Φ + Φ0/2) = D(τφ, Φ) 3
35
2
Dψ(τφ)
40
60
40
Ballistic
(b)
le=3 nm
40
1
35
30 20
(a)
le=0.5 nm 0
0 25
0
t
30
400
0.25
0.5
φ/φ0
0
0.5
φ/φ0
1
0.75
1
0
˚2 γ0 /¯h unit) for a metallic SWNT (9, 0) evaluated Figure 3: Main frame : D(τφ , Φ/Φ0 ) (in A at time τφ À τe , for two disorder strengths,W/γ0 = 3 and 1, such that the mean free path (`e ∼ 0.5 and 3 nm, respectively) is either shorter (dashed line) or larger (solid line) than the nanotube circumference (C ∼ 2.3nm). The right y-axis is associated to the dashed line and the left y-axis to the solid line. inset : D(τφ , Φ/Φ0 ) for `e = 3 nm and L(τφ ) < 2`e (from [?])
agrees with weak localization theory’s predictions CASE `e > C, L(τφ < 2`e) =⇒ • Positive magnetoresistance
D(τφ, Φ + Φ0) = D(τφ, 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
36accγ02 √ 2 n + m2 + nm ξe = 2 W
(1)
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).
P R(ψ(E)) =
( | ψi(E) |2)2 X
i
X
i
| ψi(E) |4
' Nα
+ 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 (although states remain extended).
t t − − 2e D Z ∞ τe )e τφ ∼ −P R−1 =− dtP(r, t)(1 − e π¯h 0 2
δσIQ
Eigenstates characterized by a linear scaling in N are uniformly extended and associated with a vanishing contribution of QIE, i.e. (σ BB is the Bloch-Boltzman result)
δσ/σBB → 0 Localized states are related to strong contributions of QIE, i.e.
δσ/σBB ' 1
Scaling laws P R(N ) = N α , with 0 < α < 1, indicate the relative strength of QIE. The conductance thus change from
G = e2(v 2τe − δσ)/Ltube to an exponentional decrease as soon as Ltube ≥ ξ PR
4000
4000
(6,6)
3000 2000 1000
3000
0
0
1000
2000
3000
4000
2000
(7,5)
1000
0
0
1000
2000 3000 Number of atoms
4000
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 `e = 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.))
H = γ0
·
X
i,j n.n.
¸ j i |p⊥ihp⊥|
−β
·
X
i,j∈n.l.
Incommensurate MWNT (6, 4)@(10, 10)@(17, 13)
cos(θij )e
d −a − ijδ
¸ j i |p⊥ihp⊥|
Commensurate MWNT (6, 4)@(12, 8)@(18, 12).
θij is the angle between the pi⊥ and pj⊥ orbitals, and dij denotes their relative distance. The parameters used here are : γ0 = 2.9eV , a = 3.34˚ A, δ = 0.45˚ A. 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~ |' 18.79˚ A. • 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 19acc, | T(10,10) ) |= 3acc, | T(17,13) |= 3 1679acc, (ratio of lengthes of individual shell translational vectors are irrational numbers). 4.1.1
Commensurate multiwall nanotubes (5,5) @ (10,10) @ (15,15) G(E)
78 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
DOS [states/eV/atom]
0.03
0.02
0.01
(a)
(b)
(c)
(d)
0.00
G/G0
6 4 2 0 -0.2
0.0 E [eV]
0.2
-0.2
0.0 E [eV]
0.2
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
τll ∼ h ¯ γ0/β 2 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
(6,4)@(12,8)@(18,12)
0.9
Ψ(i,j)
2 0.8
β=γ0/10
0.7
(18,12)
β=γ0/3
0.6 0.5
β=γ0
0.4
(12,8)
0.3 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 with a Fermi velocity of 106ms−1, ⇓ t ∼ 4500¯h/γ0 ∼ 102 τ 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
Ψ(i,j)
1.0
(6,4)@(10,10)@(17,13)
(6,4)@(12,8)@(18,12)
0.9 2 0.8 0.7 0.6
0.78
0.5
0.73
(17,13) (18,12)
0.4 0.68 0.3
0
100
200
(10,10)
300
0.2
(6,4)
(12,8) 0.1
(6,4) 0.0
0.0
0
500 1000 Diffusion time (h/γ0)
0
500
1000
1500
2000
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 ballistic motion because of multiple scattering effects in the non-periodic systems. Conduction is said to be non-ballistic, with r
L = D(t) × t ∼ Atη (9,0)@(18,0)
4000
(9,0)@(10,10)
D(τφ)
3000
2000
1500 1000
1000
500 0 0
0
200
400
(6,4)@(10,10)@(17,13) 0
200
τφ
600
400
600
800
800
1000
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 increasing the number of coupled incommensurate shells
5
Role of electrode-nanotube contacts
6
Some general landmarks Side-contact configuration
End-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 If | km i =
P
pe
ikm p
| ϕm i (resp. | kF i =
P
pe
ikF p
| ϕNT i) the propagating states with km (kF )
the wavector in the metal (resp. nanotube). We take | ϕNT i the localized basis vectors, that will have nonzero overlap with | ϕm i only for a few unit cells (p) defining the contact area. The scattering rate between the metal and the nanotube can be written following the Fermi golden rule and will be related to (Hcontact the coupling operator between the tube and electrodes)
hkm | Hcontact | kF i ∼ γ 0hϕNT | ϕmi
X
p
ei(km−kF )p
Various physical aspects can be outlined: – γ 0 is related to the chemical nature of interface bonding (covalent, ionic,...). In the most favourable case of a covalent coupling, lowest contact resistance is given by Rc = h/2e2 , whereas ionic bonding would mostly favoured tunneling contact resistance Rc ∼ h ¯ /(2πe2 | γ 0 |2).
– hϕNT | ϕmi is related with the geometry and configuration of contact between nanotube and electrodes: end or side contacts, length of the contact area,... – The last term is maximized whenever wavevector conservation (∼ δ(km − kF )) is best satisfied. For instance in case of a metallic armchair tubes, larger coupling will be achieved for √ km ' 2π/3 3acc. Much smaller metallic wavector will yield small coupling rate. – The tunneling rate from the metal to the nanotube is given by 1 2π ∼ | hkm | Hcontact | kF i |2 ρNT (EF )ρm(EF ) τ h ¯ 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). Metals 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 .
Work function (e
5.4
(7,0)
5.3 5.2
(10,0)
5.1
(13,0) (17,0)
5.0
(19,0) 4.9 4.8
(12,12) (10,10)
4.7 4.6 0.05
(8,8)
(15,0)
(6,6)
(5,5)
(12,0) 0.10
(4,4) o
0.15
0.20
-1
1/D (A )
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 2G0 plateau, only individual resonances persist (whose number increases with system length). 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.
1.5
E (a.u.)
2.0
G (2e /h)
k1* k1
2
k2
1.0
* k k1 2
• • * ∗ • k2 π k k• π
* 1
2
k
0.5
Transmission eigenvalues
(a) 0.0
π
0.8 0.6 0.4
π∗
0.2 0.0 -3.0
-2.0
-1.0
(b) 0.0
1.0
Energy (eV)
2.0
3.0
2.0
1.5
1.5
G (2e2/h)
G (2e2/h)
2.0
1.0
0.5
Au
(a)
0.0 0.8 0.6 0.4 0.2
(b)
0.0 -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Enter text here
Transmission eigenvalues
0.5
Transmission eigenvalues
Ti
Enter text here
1.0
(a)
0.0 0.8 0.6 0.4 0.2 0.0 -3.0
(b) -2.0
-1.0
0.0
1.0
2.0
Energy (eV)
Energy (eV)
Same for N=10 (5,5) end-contacted to Au(111).
Same for a Ti(111) surface.
8
3.0
ANNEXES
8.1
The Fermi golden rule
To investigate quantum conduction in carbon nanotubes one needs to solve the time dependent Schr¨odinger equation i¯h
∂ ˆ 0 + U)Ψ(~ ˆ r, t) Ψ(~r, t) = (H ∂t
where 2 2 −¯ h −¯ h 2 2 ˆ0 = ∇ + VˆXtallin = H ∇ 2m 2m∗ (under the effective mass approximation). The operator Uˆ describes the effect of superimposed disorder due to e.g. substitutional or adsorbed impurities, topological defects, ... When disorder in the systems ˆ 0 inducing is low, its effect is to limitate the lifetime of eigenstates of H transition between different allowed vectors at a given energy (elastic scattering). If one sets
ˆ 0|ki = εk |ki H the eigenstates of the unperturbated Hamiltonian, then in presence of ˆ one can assume a trial wavefunction Ψ(~r, t) = Pk h~r | kihk | Ψi U, P which is taken as k ck (t)ψk (~r). Introducing this form in the general equation one gets: 0 ˆ d iεk0 X hk |U|ki ck0 (t) + ck0 (t) = ck (t) dt h ¯ i¯h k
If at time t = 0 the electron is in state | ki then at first order its evolution is driven by ˆ d iεk0 hk 0|U|ki 0 0 ck (t) + ck (t) = e−iεk t/¯h dt h ¯ i¯h and the general solution writes ck0 (t = T ) = e
−iεk0 T /¯h hk
0
ˆ Z T −i(εk −εk0 )t |U|ki e ¯h dt 0 i¯h
The Fermi golden rule is thus derived by computing the probability per unit time to obtain a transition from the state | ki to any other allowed state | k 0i (| ck0 |2), that is 1 2π X¯¯¯ hk 0|Uˆ |ki ¯¯¯2 = ¯ ¯ δ(εk 0 − εk ) τ h ¯ k0 i¯h Let’s define S(t, t0 ) as the probability amplitude of a transition between two states induced by the disorder potential acting during an infinite period from t0 = −∞ t t = +∞, then 1 ¯¯¯ 0 ˆ ¯¯¯2 Z +∞ S(t, t0 ) = 2 ¯hk |U|ki¯ −∞ dt1dt2eiωkk0 (t2−t1) h ¯ ¯2 Z +∞ 1 ¯¯¯ 0 ¯ ¯ = 2 ¯hk |U|ki¯ × 2π¯hδ(εk − εk0 ) × −∞ dt h ¯
Then the transition probability per unit time is given by ¯2 1 2π X¯¯¯ 0 ¯ ¯ = ¯hk |U|ki¯ δ(εk − εk 0 ) τ h ¯ k0 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 Hamiltonian Hkk0 for electrons moving on a nanotube under the influence of a magnetic field: ie 1 X −i(k.R−k0.R0)− ∆ϕR,R0 h ¯ e Hkk0 = N R,R0 ¯ ¯ ¯ 2 ¯ * + ¯p ¯ ¯ ¯ 0 × ψ(r − R) ¯¯ + V ¯¯ ψ(r − R ) ¯ 2m ¯
(1)
where the phase factors are given by ∆ϕ
R,R0
=
Z 1
0
(R0 − R) · (A(R + λ[R0 − R]))dλ,
(2)
and |ψ(r − R0)i is the localized atomic orbital, and p and V are, respectively, the momentum and disorder operators. As shown hereafter, 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 A = (φ/|Ch |, 0) in the twodimensional C~h/|C~h|, T~ /|T~ | coordinate system, and the phase factors become ∆ϕR,R0 = i(X − X 0)φ/|Ch | for R = (X , Y). This yields new 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 |Ch|) which leads to a φ0-periodic variation of 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 −3 0.8
0
0.1
1 −1
1
0
3
−3 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2 0
−3
−1
0.8 1
3
0
0.9 −1
1
3
1
3
0.5 −3
−1
to the following variations of the gap widths [?]: ¯ ¯ ¯ ¯ φ0 φ ¯ ¯ ¯ if 0 ≤ φ ≤ ¯ ∆ 1 − 3 ¯ 0 ¯¯ ¯ φ 2 0 ¯ ¯ ∆g = (3) ¯ ¯ φ φ ¯ ¯ 0 ¯ ¯ if ≤ φ ≤ φ0 ¯ ∆0 ¯¯2 − 3 φ0 ¯ 2 where ∆0 = 2πaC−C γ0/|C~h| is a characteristic energy associated with 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 ΨK+δ r + C~h). Since periodic boundary conditions apply ~ ~kK (~ ~ + δ~kK ) · C~h] = 1. For in the C~h direction, one can write exp[i(K ν = +1, we write δ~kK = (2π/|Ch |)(q −1/3)k~⊥ +δkkk~k whereas δ~kK 0 = (2π/|Ch |)(q + 1/3)k~⊥ + δkkk~k. When K → K 0, then ±1/3 → ∓1/3 in the above expressions, as ν goes from +1 → −1, which makes the situation between K and K 0 symmetrical. Similarly for metallic nanotubes, the gap-width ∆g is expressed (see figure for illustration)
by
∆g =
3∆0
φ φ0
if
0≤φ≤
φ0 2
φ0 φ ¯¯¯ if 1 − 3∆ ≤ φ ≤ φ0 ¯ φ0 ¯ 2 ¯ ¯ ¯ ¯ 0 ¯¯
¯
(4)
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