Popov V.N., Lambin P. (eds.) Carbon Nanotubes

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155

The main remarkable conductance features for B-doped (10,10) nanotubes

are illustrated in Fig. 5. As the typical length L

dev

increases from the nanometric

scale (~10 nm) to the mesoscopic scale (~ 1 µm), the quantum interference

effects beyond the diffusive regime are correspondingly enhanced. A strong

asymmetric damping of the electronic conductance follows.

Moreover, our results show that the generic conductance G(E,L

dev

) of B-

CNT at Fermi level exhibit a positive derivative with respect to energy. Similar

study on nitrogen doped nanotubes indicates that localized states around the

nitrogen substitution, are responsible for backscattering above the Fermi level

(Latil et al., 2004). Such a difference between N- and B-doped CNTs suggests

that the thermopower measurements

originate from a diffusion mechanism

only (Choi et al., 2003). Gate-dependent studies on individual undoped carbon

nanotubes have recently revealed a quantum connection between conductance

modulations and thermopower (Small et al., 2003). Similar studies on

chemically doped SWNTs would be desirable to confirm these theoretical

predictions.

4. Adsorption of ʌ-conjugated molecules

In this section, the effect of molecular physisorption on the diffusion

mechanisms of metallic CNTs is theoretically investigated. As in Section 3, the

results will be achieved by combining DFT calculations with tight-binding

models. The focus is made on the evaluation of the scattering strength and the

resulting transport length scales produced by a random coverage of ʌ-

conjugated hydrocarbon molecules (benzene (C

) and azulene (C

)).

In order to adjust the parameters of the semi-empirical Hamiltonian, and

since the interaction between the CNT and each molecule is geometry-

dependent, an ab initio study of the adsorption has been carried out prior to the

TB analysis (Tournus et al., 2005). Assuming that the adsorption on a CNT or

on graphene are equivalent, the atomic model is simplified when using a 5 x 5

graphene supercell with one adsorbed benzene or azulene molecule, instead of

the cylindrical structure.

______

The thermopower S gives important information about the sign of charge carriers and the

conductance modulations close to the Fermi level:

3()

kT dG

eG E dE



§·

¨¸

©¹

(Mott formula)

where G(E) is the conductance at Fermi level, k

T the temperature energy scale.

156

Figure 6. The electronic structures of a graphene 5 x 5 supercell with one adsorbed ʌ-conjugated

molecule. Top: the HOMO eigenstate of the isolated benzene C

molecule (shown with a

dashed line in the left hand frame) mixes up with the band structure of the graphene layer, to form

a complex band structure (central frame). Bottom: an azulene C

molecule is physisorbed on

the graphene sheet; the mixing also occurs for HOMO and LUMO eigenstates. Although a

weaker dipersion is observed than for the benzene case, the smaller gap of azulene will allow a

perturbation of the CNT electronic diffusion at the Fermi level (here E

= 0). Note that in both

cases, our TB model gives an excellent description (right frame) compared to the ab-initio

calculation.

157

Figure 7. Schematic view of the graphene-molecule interaction described by equation (14). The

distance d and the angle ĳ are drawn. b) The modified Hückel model for the azulene molecule.

The on-site energies İ are corrected in order to take screening into account and the hopping

integrals are Ȗ = –2.63 eV and Ȗ' = –2.08 eV, respectively.

Here, electronic calculations have been performed with the density

functional theory within its local density approximation to predict the optimal

geometry, and extract its electronic structure.

From the optimized geometries

(displayed in the center of Fig. 6), the LDA band structures are plotted for both

isolated and interacting systems. The interaction acts as a mixing of the

molecular single states with the underlying band structure, resulting in hybrid

eigenstates with low group velocity. However, due to the relatively large gaps

of the isolated molecules (5.217 eV and 2.089 eV for benzene and azulene,

respectively) only weak disturbance is induced around the Fermi energy, and

the linear band dispersion is preserved.

Following the same procedure as in Section 3, the zone folding model is

used for defining the Hamiltonian of the nanotube. The ʌ-conjugated molecules

are treated by the Hückel model (Salem, 1966) and the tight-binding

Hamiltonian of the complete system, containing the interaction between CNT

and N

mol

molecules, reads

CNT

ˆˆ ˆ ˆ

HH H()V()

mol

ªº

 

¬¼

(13)

where

CNT

is the usual zone folding Hamiltonian, equivalent to equation (12)

and

H()

mol

M is the Hückel hamiltonian related to the Mth molecule. The

______

The AIMPRO code (Briddon and Jones, 2000) has been used with standard norm-conserving

pseudopotentials (Bachelet et al., 1982) and a 3 x 3 x 1 k-points sampling for the Brillouin zone

integrations. The basis used is sufficient to obtain correct adsoption curves (C atoms contain five

s- and p-like and three d-like orbitals, and H atoms contain four s- and p-like orbitals).

Equilibrium distances (energies of adsorption) are found to be 3.25 Å (25.1 kJ.mol

-1

) for the

benzene, and 3.28 Å (37.6 kJ. mol

-1

) for the azulene molecules.

158

V(M)

term in equation (13) corresponds to the interaction between the Mth

molecule and the CNT, and is expressed as follow:

CNT

V( ) cos( )exp

mmM

Mnm



§·

¨¸

©¹

¦¦

(14)

Such coupling term was optimized to accurately reproduce the interaction

between shells in a multi-walled CNT (Lambin et al., 2000). In Eq. (14), d

corresponds to the distance between sites, while ĳ is the angle defined in Fig.

4a. The parameters related to the interaction are: į = 3.34 Å, l = 0.45 Å and ȕ =

–0.36 eV.

Figure 8. a) DOS of the (10,10) CNT with a benzene density coverage of 16.3%. The HOMO

molecular state is located by an arrow. b) Same as in a) but for an azulene density coverage of

11.5%. HOMO and LUMO levels are identified by arrows. c) Time-dependent diffusion

coefficient (at Fermi level) for the C

@CNT system, showing quasi-ballistic behavior (solid

line). Ballistic conduction for pristine CNT case is also reported (dashed line). d) Time-dependent

diffusion coefficient for the C

@CNT structure, showing saturation at large times (diffusive

regime).

159

The tight-binding parameters (on-sites energies İ and hopping integrals Ȗ)

are adjusted to reproduce the band structure of graphene sheet and the energy

levels of isolated benzene and azulene molecules, calculated within LDA. A

standard procedure have been used to set the parameters for the graphene or the

CNT: İ

CNT

= 0 eV, Ȗ

CNT

= –2.56 eV, as well as for the benzene molecule: İ

benz

+0.411 eV, Ȗ

benz

= –2.61 eV.

Due to an inhomogeneous charge distribution along the azulene molecule,

the TB model has to be refined to properly account for screening effects. By

adding an electrostatic correction, proportional to the net charge on each carbon

atom (calculated with LDA), renormalized on-site energies for the azulene

molecule have been deduced. The on-sites and the adjusted hopping integrals

are given in Fig. 4b. Within this approach, the TB value of the azulene HOMO-

LUMO gap is 2.198 eV. Finally, the band structure computed with this

modified TB model is compared to the previous ab initio results. As presented

in Fig. 6, this reparametrized semi-empirical model gives an excellent

description of the electronic states for both benzene and azulene adsorption

cases.

The density of states (DOS) of a (10,10) carbon nanotube, with random

coverage of adsorbed molecules is calculated within this framework. Fig. 8

presents the DOS for a nanotube with benzene coverage density of 16.3% (such

density corresponds to the ratio of adsorbed mass of the molecule over CNT

mass) and with azulene coverage density of 11.5%.Since the coupling intensity

is weak, the DOS plotted in Fig. 8 displays the reminiscent discrete molecular

levels, slightly enlarged by the mixing with the underlying continuum of ʌ- or

ʌ*-bands. Peaks arising from molecular HOMO and LUMO levels are labeled.

In the case of the benzene adsorption, the DOS is weakly affected at charge

neutrality point, and the computed diffusion coefficient shows almost no

deviation from the linear scaling in time (ballistic regime). Elastic

backscattering induced by benzene molecules is thus vanishingly low at Fermi

level (also shown in Fig. 8), and does not produce sufficient deviation to

determine an elastic mean free path below ~100 µm.

In contrast, in the case of azulene molecules, the HOMO level is located in

the close vicinity of the last occupied Van Hove singularity of the CNT.

Although the total DOS remains basically unaffected around the Fermi energy,

the azulene adsorption turns out to impact more significantly on the intrinsic

electronic current. Indeed, as shown in Fig. 8, the diffusion coefficients of

propagating electrons at Fermi level depart more strongly from the ballistic

regime.

The non-linear time dependence of D

(t), calculated with equation (7)

allows to extract the elastic relaxation time Ĳ and, knowing the Fermi velocity

, the corresponding electronic mean free path l

Fig. 9 presents the evolution

160

of l

as a function of the azulene coverage density, both at Fermi level (charge

neutrality point) and for an energy closer to the HOMO resonance [E = –0.2Ȗ §

–0.54 eV].

Figure 9. Electronic mean free path of a C

@CNT nanostructure versus the density of

physisorbed molecules and for two different Fermi level positions. The origin of the error bars is

the use of the random phase states for the expectation values in equation (6).

AKNOWLEDGEMENTS

We acknowledge fruitful discussions and collaborations with Prof. Didier

Mayou, Prof. Jean-Christophe Charlier, Prof. Philippe Lambin, Dr. Xavier

Blase and Prof. Malcolm I. Heggie. S. L. acknowledges financial support from

the Belgian Francqui Fundation for Scientific Research.

A. THE RECURSION TECHNIQUE

As we have established in Section 2, the main issue of our numerical technique

is to compute expectation values, like the local density of states on a given ket

related to the retarded Green's function:

161

(H) lim Im

GM M M



ªº

 

«»



¬¼

(15)

We use the real-space recursion technique, which avoids diagonalization or

inversion of the Hamiltonian and has a numerical cost which grows linearly

with the system size (order N) (Haydock et al., 1972). Starting from a

normalized ket

, an orthonormal basis {

,…} is constructed from

the recurrence relations:

00001

H ab



(16a)

11 1

nnnnn nn

ab b

MM M M

 

 

(16b)

with

. At any step n, a

is given by

, hence it is a real

number. Then b

and



are uniquely determined by the normalization

condition for



and the choice of a real positive b

In this basis, the Hamiltonian is tridiagonal. The Green's function element

mentioned above is simply the first diagonal element of the inverse of the

tridiagonal matrix:

011

Ei a b

bEia b

bEia

 

§·

¨¸



¨¸



¨¸

©¹

It is obtained by a continued fraction expansion:

...





(17)

With N

recurs

recursion steps, an energy resolution of order of W/N

recurs

obtained for the density of states, where W is the energy bandwidth of the

system. In the single-band case, a

and b

converge and the continued fraction is

terminated by the approximation a

N+1

= a



and b

N+1

= b



. A more sophisticated

termination can be used in a multi-band system, as detailed in (Roche, 1999).

162

B. COMPUTATION OF

()

twp

BY CHEBYCHEV POLYNOMIALS

EXPANSION

This section presents the method used to apply the operator

() Z, ()

tut

ªº

¬¼

a state vector

, in view of solving numerically the equation (6), with time

steps ǻt. The concept consists in expanding the evolution operator

()ut'

polynomials of the Hamiltonian operator like:

() ()(H)

ut c tQ

' '

(18)

Practically, sums are limited to N

poly

(hence, the exponential in the evolution

operator is approximated by a polynomial of order N

poly

and Q

are the

Chebychev polynomials, which follow the simple recursive rules





ˆˆ ˆ ˆ

(H) H (H) (H)

nnn

bQ a Q bQ



 

(19a)

(H) 1Q



ˆˆ

(H) H



(19b)

The real numbers a and b are parameters defining a density function on the

mathematical domain [a – 2b, a +2b]

()



§·



¨¸

©¹

(20)

It is then crucial to choose a and b to fit all the spectrum of

within the

domain.

The expansion coefficients c

(ǻt) in equation (18) are then the scalar

product like

() () ()exp( )

ct NEQE EtdE '

(21)

and can be computed. To simplify the equations (18) and (19), we will

introduce the two sets of kets

(H)

(22a)

ˆˆ

Z, (H)

(22b)

that obeys the recursion rules

163





nnn

bab

DDD



 

(23a)





(23b)

and

ˆˆ

Z,H

nnn

bab

EEE



ªº

 

¬¼

(24a)

ˆˆ

Z,H

(24b)

They can be calculated from the starting ket

by applying

and

Once we know

and

, the application of the operators on

simply

poly

() ()

ut c t

' '

(25a)

poly

Z,() ()

ut c t

ªº

' '

¬¼

(25b)

Finally, when the equations (25) are solved for a given time t,

the

following step t + ǻt is calculated by applying the Chebychev expansion on

solution at time t like

ˆˆˆ

() ()()

ut t u tut

' '

(26a)

ˆˆ ˆ

ˆˆˆˆˆ

Z, ( ) Z, ( ) ( ) ( ) Z, ( )ut t u t ut u t ut

ªºªº ªº

' '  '

¬¼¬¼ ¬¼

(26b)

and so on. The evolution of

()At

and

()

can be calculated step by step

from any starting

______

The choice of an orthogonal basis set (within the TB approach) simplifies the action of the

position operator, assuming that

np n

np Z

This means that the two vectors

()ut

and

Z, ( )ut

ªº

¬¼

have already been calculated.

164

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