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aCentro de Ciencia Naturais e Humanas, Un

SP, BrazilbInstituto de Fısica, Universidade de S~ao PacLaboratorio Nacional de Luz Sıncrotron, CadInstituto de Fısica Teorica, Universidade E

E-mail: reilya@i.unesp.br

† Electronic supplementary informa10.1039/c3nr33185g

Cite this: Nanoscale, 2013, 5, 2798

Received 20th July 2012Accepted 28th January 2013

DOI: 10.1039/c3nr33185g

www.rsc.org/nanoscale

2798 | Nanoscale, 2013, 5, 2798–280

Confinement effects and why carbon nanotube bundlescan work as gas sensors†

Rodrigo G. Amorim,*a A. Fazzio,b Antonio J. R. da Silvabc and Alexandre R. Rochaad

Carbon nanotubes have been at the forefront of nanotechnology, leading not only to a better

understanding of the basic properties of charge transport in one dimensional materials, but also to the

perspective of a variety of possible applications, including highly sensitive sensors. Practical issues,

however, have led to the use of bundles of nanotubes in devices, instead of isolated single nanotubes.

From a theoretical perspective, the understanding of charge transport in such bundles, and how it is

affected by the adsorption of molecules, has been very limited, one of the reasons being the sheer size

of the calculations. A frequent option has been the extrapolation of knowledge gained from single

tubes to the properties of bundles. In the present work we show that such procedure is not correct, and

that there are qualitative differences in the effects caused by molecules on the charge transport in

bundles versus isolated nanotubes. Using a combination of density functional theory and recursive

Green's function techniques we show that the adsorption of molecules randomly distributed onto the

walls of carbon nanotube bundles leads to changes in the charge density and consequently to

significant alterations in the conductance even in pristine tubes. We show that this effect is driven by

confinement which is not present in isolated nanotubes. Furthermore, a low concentration of dopants

randomly adsorbed along a two-hundred nm long bundle drives a change in the transport regime; from

ballistic to diffusive, which can account for the high sensitivity to different molecules.

1 Introduction

Carbon nanotubes (CNTs) are amongst the most promisingnanostructured materials for the development of new nano-scopic electronic devices.1,2 The range of possible applications isunsurmountable3 with transistors,4,5 eld emission sources,6

devices for exible electronics7 and sensors8–12 featuring in thelist. Nevertheless, at the nanoscale even a small amount ofexternal agents such as gas molecules adsorbed on the surfaceof the tubes could add up to signicant changes in the elec-tronic structure and transport properties of such systems. Thus,the effect of defects, vacancies and impurities is an importantaspect that needs to be addressed in CNT-based device engi-neering. On the one hand these molecules can lead to delete-rious effects related to the degradation of the signal inelectronic devices. On the other hand it is the working principleof some types of sensors. In the latter case, changes in theconductance due to small concentrations of gas molecules

iversidade Federal do ABC, Santo Andre,

ulo, S~ao Paulo, SP, Brazil

mpinas, S~ao Paulo, Brazil

stadual Paulista, S~ao Paulo, SP, Brazil.

tion (ESI) available. See DOI:

3

result in high sensitivity. Particularly in CNTs, the high surfacearea to volume ratio makes them stand out as a candidate forpossible gas sensing applications.8–13

Theoretical simulations, however, have demonstrated thatclosed shell gases tend tobeweakly bonded to thewalls of pristinenanotubes.14,15 This is, in principle, good news for electronicdevices and bad news for sensors. In fact, recent experimentalevidence shows that the changes in resistance observed fromexposure to different gases essentially arise due to the moleculesbinding to the metallic electrodes, consequently changing themetal's work function and the Schottky barrier height of themetal–CNT interface.16,17ThusaCNT-based sensor ispossible, butit would not exploit one of the nanotube's main features, namelyits one-dimensional character. Furthermore, from the techno-logical point of view, a single nanotube device is not viable sincethe fabrication process typically leads to nanotubes in bundles,18

forests,19,20 or networks.21 Recent experiments, for instance, haveshown that pristine tubes in a network under constant UV radia-tion can present high sensitivity towards a variety of gases.13 Thesensing mechanism in these cases, however, remains to bedetermined as it is yet unknown whether it would be possible todetect external agents by a direct interactionwith the tubewalls inpristine bundles unlike in isolated CNTs.10

From this perspective, although the electronic andmechanical properties of pristine bundles have been studiedtheoretically,22 there is a very limited number of results dealing

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with the effects of vacancies or dopants,23 and to the extent ofour knowledge none which addresses the consequences ofimpurities in the electronic transport. In fact, there is usually atendency to transfer our knowledge from isolated tubes tobundles or ropes. This is mostly due to the size of the system,whereby – to fully understand the effects of external agents –

one needs to consider a disordered system with moleculesrandomly distributed along the device.24

The aim of this letter is thus two-fold. Firstly we will show thatinformation about the electronic and transport properties ofpristine bundles of CNTs containing molecules adsorbed on thetube walls cannot be obtained by extrapolating calculations forthe isolated nanotube case. As we will show this is due toconnement effects that are present only when the full system istaken into consideration. Secondly, we will show that in theparticular case of carbon monoxide and ammonia moleculestrapped between nanotube walls in the bundle there is a notice-able change in resistance as a function of molecular concentra-tion even in the case of tubes without any intrinsic defects.

In order to achieve this we study, using a combination of abinitio density functional theory (DFT)25,26 with a van der Waalscorrection, and recursive Green's function techniques,24,27–31 theeffect on the electronic transport properties of CO and NH3

molecules interacting with CNT bundles. Initially we will showthat such molecules are weakly bonded to isolated single wallcarbon nanotubes via van der Waals interactions, and have littleor no effect on their electronic transport properties. On the otherhand they cause noticeable changes when trapped betweennanotubewalls in the case ofbundles.Using the recursiveGreen'sfunction technique extended to treat three-dimensional disor-dered systems we are able to study long nanotubes with varyingconcentration of molecules, and observe signicant changes inthe conductance leading to a change in the transport regime.

2 Methodology

Our electronic structure calculations are based on densityfunctional theory as implemented in the SIESTA package32 withthe inclusion of a parametrized dispersion-corrected atom-centered potential (DCACP),33,34 which accounts for the van derWaals (vdW) interactions. In our DFT simulations we used theGeneralized Gradient Approximation (GGA) within the Perdue–Burke–Ernzerhof (PBE) proposal35 to treat the exchange–corre-lation potential, a double-z polarized basis set (DZP) andstandard norm-conserving pseudopotentials.36 We consideredboth bundles of and isolated (5,5) single wall carbon nanotubes(SWCNTs) with carbon monoxide and ammonia molecules asdopants. The unit cell used for the electronic structure calcu-lations in the bundles is shown in grey in Fig. 1a. The k-spaceintegration was done using a grid of 2 � 2 � 4 k-points, wherethe z-axis lies along the NT growth direction. The congurationswere fully relaxed to equilibrium using a conjugate gradientmethod with residual forces smaller than 0.01 eV A�1.

The electronic structure simulations are combined with anonequilibrium Green's function (NEGF) to calculate the elec-tronic transport properties.37–40 In order to perform thosecalculations for a single dopant, the system is divided into three

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parts, namely a central scattering region corresponding toeither a single tube or the bundle containing the molecule(Fig. 1a, c or d) and two (le- and right-hand side) semi-inniteelectrodes.41 The main quantity that characterizes our system isthen the Green's function which can be written down as

G(E) ¼ [E � SS � HS[r] � SL(E) � SR(E)]�1, (1)

where SS and HS are nite operators for the overlap and theHamiltonian of the central scattering region whereas SL/R arethe so-called self-energies which account for the effect of theelectrodes onto the central region. Here we assume that theHamiltonian is a functional of the charge density r. The self-energies for the electrodes are given by

Sa ¼ VSagaVaS (2)

where VaS ¼ V†Sa (a ¼ {L, R}) are the coupling matrix elementsbetween the electrode and the scattering region and ga is thesurface Green's function for the semi-innite leads. In ourcalculations the electrodes are taken as either pristine isolatednanotubes or pristine bundles. Subsequently, we consider that,at the interface between the electrode and the scattering region,the charge density is identical to the pristine system. Thismeans that any changes to charge rearrangements inside thescattering region will not change the electronic structure of theelectrodes. Thus the Hamiltonian for the electrodes can beobtained using separate DFT calculations, and is then used tocalculate the Green's function of the semi-innite leads byeither a direct39,42 or iterative method.40

Finally, we use DFT to construct the Kohn–Sham Hamilto-nian for the scattering region introducing the effects of theelectrodes via self-energies. The charge density is then self-consistently calculated using the system's Green's function – inthe NEGF approach – in order to solve for the open system.Finally, once convergence has been achieved, one can calculatethe transmission coefficients T(E) in the linear regime,43,44

T(E) ¼ GL(E)GS(E)GR(E)G†S(E), (3)

where the coupling matrices are given by

Ga ¼ i[Sa � S†a]. (4)

For the disordered system, the same procedure could, inprinciple, be employed, nonetheless the number of degrees offreedom turn the direct inversion in eqn (1) into an intractableproblem. Instead, we use a recursive method that allows for thecalculation of the electronic transport properties of longsystems containing a large number of impurities randomlydistributed along the device.24,27–31 This is done by slicing thescattering region into a number of segments. Each segmentconsists of either a piece of a pristine tube (or bundle) or theCNT containing the gas molecule. The Hamiltonian for eachsegment is obtained via a separate ground state DFT calcula-tion. The total Hamiltonian is subsequently reassembled byrandomly arranging the building blocks. In order to ensure thecorrect coupling between each block we assume that the

Nanoscale, 2013, 5, 2798–2803 | 2799

Fig. 1 (a) Unit cell of a (5,5) bundle of carbon nanotubes used in the electronic transport calculations. The irreducible cell is shown in shaded gray and the eight moststable positions for the molecules are presented. (b) Disordered bundle containing molecules randomly dispersed along the walls. Isolated nanotube structures withmolecules adsorbed on the surface for (c) CO and (d) NH3. The tube-molecule distances are also shown.

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interaction between adjacent cells is equal to that of the pristinesystem. This is guaranteed by the inclusion of extra layers of themolecule-free nanotube on either side of the system. As aconsequence, the potential and the charge density at the edgesof the cell should resemble those of the pristine CNTs. Thus, wematch the potential at the edges of the cell to avoid any spuriousscattering. Finally, the inclusion of the effects of disorder in thetransverse direction is obtained by increasing the size of theunit cell perpendicular to transport while making use of peri-odic boundary conditions. This gives rise to eight possiblepositions for the molecules to sit in. The cross-section of thesystem mimicking the bundle is shown in Fig. 1a. The longtubes are then obtained by randomly assembling segments of apristine tube with or without molecules adsorbed along the tubeaxis as can be seen from Fig. 1b. Finally, the recursive procedurethen consists of removing the internal degrees of freedom of thetotal Hamiltonian taking into consideration its sparse form andthe fact that the transmission coefficients only depend on theelements of the Green's function that connects the le and rightelectrodes.31 The concentration of molecules in the tube can bevaried by altering the relative number of building blocks con-taining molecules. For each concentration 50 different cong-urations were taken and an average was calculated.

3 Results

The rst question that we address is how gas molecules such asCO and NH3 can modify the electronic and transport propertiesof an isolated pristine single wall carbon nanotube. We investi-gated the adsorption energy of both molecules onto an isolatednanotube. In the case of the COmolecule we considered it eitherparallel or perpendicular to the tube axis. In the perpendicularcase we also considered the possibility of either atom pointingtowards the CNT, and nally different positions of either C or Owith respect to the hexagonal arrangement forming the nano-tube – in the so-called bridge (B), top (T) or hollow (H) congu-rations –were taken into consideration (more information about

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the different congurations can be found in the ESI†). As shownin Fig. 1b thedistance between themolecule and thenanotube inthe lowest energy conguration – the H conguration with the Cpointing towards the tube – is approximately dt–CO¼ 3.3 A. In thecase of ammonia we performed calculations where the nitrogenatom is eitherpointing awayor towards the carbonnanotube.Wealso considered the different positions of the nitrogen atomwithrespect to the hexagonal arrangement forming the nanotube: T,B or H. The lowest energy conguration was found to be thehollow one followed by the bridge conguration with an energydifference of approximately 0.03 eV. Moreover the relativedistance between nanotube andmolecule was found to be dt–NH3

¼ 3.4 A. In both CO and NH3 cases the distances are consistentwith van der Waals-type interactions.45

From our GGA + vdW simulations of bundles the distancebetween nanotube walls dt–t in the pristine case was calculated tobe 3.3 A. The arrangements containing either CO and NH3

molecules were obtained via quantum molecular dynamics(QMD)46 simulations using a canonical ensemble (NVT) with aNose–Hoover thermostat.47,48 The target temperature used was300 K considering one thousand steps for thermalization. Aerthat the system evolved for 5 ps with a time-step of 0.5 fs. Weobserve that themoleculesmoved freely along the axial direction,butnot along the transversedirection. The transportproperties ofdifferent congurations randomly chosen from snapshots of theQMD simulation were calculated. No signicant differences inthe transport properties between different structures wereobserved (Fig. S5 in ESI†), thus only the lowest energy congu-rations were considered for each molecule. This conguration iswhere the molecule sits in the interstitial position between thenanotubewalls as shown inFig. 1a. In the case ofCO themoleculealigns with the nanotube axis. The tube–molecule distances werefound to be dt–CO ¼ 2.45 A and dt–NH3

¼ 2.51 A respectively.The electronic transport properties of an isolated single wall

nanotube presented in Fig. 2a and b show that, although themolecule is bound – albeit weakly – to the nanotube, it does notalter its transmittance. In essence the total transmission at the

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Fig. 2 Total transmission coefficients as a function of energy for, (a) an isolatedSWCNT with CO, (b) an isolated SWCNT with NH3, (c) a CNT bundle with CO, and(d) a CNT bundle with NH3. The insets highlight the total transmission around theFermi level.

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Fermi level (and for a wide range around it) remains unchangedcompared to the pristine case. This is in agreement with resultsthat indicate that, due to the weak nature of the van der Waalsinteraction, the sensitivity of single devices does not arise fromdirect molecular adsorption onto the nanotube walls.45 On theother hand, for the case of a single molecule trapped within thebundle there is a small yetnoticeable change in transmittancepertubearound theFermi level as canbe seen fromthe inset inFig. 2cand d.‡We also performed a calculation with an isolated SWCNT

‡ We note here that, given the weak bonding between the tubes, the transmissionper tube in the molecule-free bundle remains at 2G0.

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with the molecule (CO or NH3) in the same relative distance asobtained in the bundles. In these cases (not shown here) weobserved no variation on the transmittance around the Fermilevel compared to the pristine tube. Thus, proximity to thenanotube alone is not enough to account for changes inconductance.

Thus, in order to understand the origin of this effect we havecalculated the difference in charge density Dr (r) ¼ rt+mol(r) �rt(r), between the cases with (t + mol) and without (t) themolecule for both the isolated CNTs as well as the bundle. Thisway it is possible to determine the effect that the nanotubeshave on the molecule and the consequent feedback onto thetube's electronic structure. The results are shown in Fig. 3. Forthe SWCNT we can note that for both molecules (CO and NH3)the interaction with the tube is weak (Fig. 3a and c), which canbe seen from the lack of distortion in the molecule’s chargedistribution and the invariance in the tube's charge. In partic-ular, the CO molecule retains its cylindrical symmetry. In thecase of the bundle, however, the effect of the molecule on thetube walls is much more pronounced, as can be observed fromFig. 3b and d. This highlights the effect of connement on theelectronic structure of the system. In all cases the nanotube–molecule distance considered was the one in the bundle. Thusone can infer that the effect arises due to an electrostaticinteraction with the molecular orbitals which in turn alters thetube’s properties and inuences the total conductance. In otherwords, connement plays a major role in determining theelectronic transport properties of these systems.

Finally we turn our attention to the calculation of thedisordered bundle containing a large number of molecules. Ournanotubes are �180 nm in length and each building-blockcontains 720 atoms.31 In Fig. 4 the sensitivity, dened as therelative variation in resistance compared to the impurity-freesystem, is presented as a function of the number of moleculesand the relative concentration per mass. As can be seen, thedata can be adjusted to t a straight line which is an indicationthat the system is in the diffusive regime. In that scenario themean free path is proportional to the defect concentration for axed device length.29 Thus one notes that even a very low

Fig. 3 Difference in charge density between a SWCNT (a and c) and bundles(b and d).

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Fig. 4 Molecular sensitivity as a function of the number of molecules for (a)carbon monoxide, and (b) ammonia. The relative concentration of defects permass is also shown on each point of the graphs for comparison.

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concentration of molecules already changes the transportregime from ballistic to diffusive. Furthermore, the sensitivity issignicant, reaching up to �15% in the case of ammonia with aconcentration of 0.2 wt%. For comparison the variation inresistance for the equivalent isolated nanotube with randomlydistributed molecules is shown in the same graph. In this casethere is no noticeable change in resistance compared to thepristine nanotube (RSWCNT � 1/(2G0), where G0 is the quantumof conductance). Consequently this result indicates that even inthe absence of binding sites49 it is possible to detect differentmolecular species in bundles, differently from isolated tubes. Inrealistic samples one should expect a combination of tubes withchiralities presenting both metallic and semiconductingbehavior. Density functional theory calculations by Zhao et al.14

show that the binding energy of the molecules to the tubes issimilar for CNTs with different chirality. At the same time theintroduction of a percentage of semiconducting tubes willmerely drive the overall conductance down (for tubes with andwithout impurities) as they have a gap at the Fermi level andtherefore do not contribute to the transport in the absence of anexternal gate. Thus, these results do provide a possible mech-anism for the detection process in pristine nanotubes wherebytrapped molecules give rise to resistance changes.13

4 Conclusion

In conclusion we have performed ab initio DFT-based electronicstructure and transport calculations on isolated and bundledCNTs containing adsorbed molecules. We have shown that inthe isolated case there are no signicant changes in the elec-tronic transport properties for both the single- and many-impurity case. In the bundle, connement effects lead tonoticeable changes in the transport properties, showing thatone cannot extrapolate single-tube results to bundles andforests of CNTs. Furthermore, in the many-impurity case, oneobserves a change in the transport regime – from ballistic todiffusive – and variations in resistance up to �15% for smallconcentrations of molecules. On the one hand it shows that adiffuse transport regime is predominant in these systems – as

2802 | Nanoscale, 2013, 5, 2798–2803

impurities are almost inevitably present – and must be takeninto consideration in realistic electronic devices. At the sametime it provides a possible pathway for sensor design usingpristine bundles, since they give rise to noticeable changes inresistance even in the absence of tube functionalization.

Acknowledgements

We acknowledge support from the Brazilian Agencies CNPq andFAPESP, and LCCA and CENAPAD-SP for computational time.

References

1 S. Iijima, Nature, 1991, 354, 56.2 S. Iijima and T. Ichihashi, Nature, 1993, 363, 603.3 R. H. Baughman, A. A. Zakhidov and W. A. de Heer, Science,2002, 297, 787–792.

4 S. J. Tans, A. R. M. Verschueren and C. Dekker, Nature, 1998,393, 49.

5 A. D. Franklin, M. Luisier, S.-J. Han, G. Tulevski,C. M. Breslin, L. Gignac, M. S. Lundstrom andW. Haensch, Nano Lett., 2012, 12, 758–762.

6 W. A. de Heer, A. Chatelain and D. Ugarte, Science, 1995, 270,1179–1180.

7 D.-m. Sun, M. Y. Timmermans, Y. Tian, A. G. Nasibulin,E. I. Kauppinen, S. Kishimoto, T. Mizutani and Y. Ohno,Nat. Nanotechnol., 2011, 6, 156–161.

8 P. G. Collins, K. Bradley, M. Ishigami and A. Zettl, Science,2000, 287, 1801–1804.

9 J. Kong, N. R. Franklin, C. Zhou, M. G. Chapline, S. Peng,K. Cho and H. Dai, Science, 2000, 287, 622–625.

10 P. Bondavalli, P. Legagneux and D. Pribat, Sens. Actuators, B,2009, 140, 304–318.

11 A. Goldoni, L. Petaccia, S. Lizzit and R. Larciprete, J. Phys.:Condens. Matter, 2010, 22, 013001.

12 S. Ammu, V. Dua, S. R. Agnihotra, S. P. Surwade,A. Phulgirkar, S. Patel and S. K. Manohar, J. Am. Chem.Soc., 2012, 134, 4553–4556.

13 G. Chen, T. M. Paronyan, E. M. Pigos and A. R. Harutyunyan,Sci. Rep., 2012, 2, 343.

14 J. Zhao, A. Buldum, J. Han and J. P. Lu,Nanotechnology, 2002,13, 195.

15 S. Peng, K. Cho, P. Qi and H. Dai, Chem. Phys. Lett., 2004,387, 271.

16 S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller andP. Avouris, Phys. Rev. Lett., 2002, 89, 106801.

17 S. Heinze, J. Tersoff and P. Avouris, Appl. Phys. Lett., 2003, 83,5038.

18 A. Kis, G. Csanyi, J. P. Salvetat, T.-N. Lee, E. Couteau,A. J. Kulik, W. Benoit, J. Brugger and L. Forro, Nat. Mater.,2004, 3, 153–157.

19 A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu,Y. H. Lee, S. G. Kim, A. G. Rinzler, D. T. Colbert,G. E. Scuseria, D. Tomanek, J. E. Fischer and R. E. Smalley,Science, 1996, 273, 483–487.

20 Z. J. Han and K. K. Ostrikov, J. Am. Chem. Soc., 2012, 134,6018.

This journal is ª The Royal Society of Chemistry 2013

Paper Nanoscale

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nloa

ded

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cMas

ter

Uni

vers

ity o

n 15

Mar

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013

Publ

ishe

d on

31

Janu

ary

2013

on

http

://pu

bs.r

sc.o

rg |

doi:1

0.10

39/C

3NR

3318

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21 M. Musameh, M. R. Notivoli, M. Hickey, I. L. Kyratzis,Y. Gao, C. Huynh and S. C. Hawkins, Adv. Mater., 2011, 23,906–910.

22 T. Filleter, S. Yockel, M. Naraghi, J. T. Paci, O. C. Compton,M. L. Mayes, S. T. Nguyen, G. C. Schatz andH. D. Espinosa, Nano Lett., 2012, 12, 732–742.

23 A. J. R. da Silva, A. Fazzio and A. Antonelli, Nano Lett., 2005,5, 1045–1049.

24 A. R. Rocha, M. Rossi, A. J. R. da Silva and A. Fazzio, Phys.Rev. Lett., 2008, 100, 176803.

25 P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136,B864.

26 W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133.27 C. Gomez-Navarro, P. J. de Pablo, J. Gomez-Herrero, B. Biel,

F. J. Garcia-Vidal, A. Rubio and F. Flores, Nat. Mater., 2005, 4,534.

28 R. Avriller, S. Roche, F. Triozon, X. Blase and S. Latil, Mod.Phys. Lett. B, 2007, 21, 1955.

29 T. Markussen, R. Rurali, M. Brandbyge and A.-P. Jauho, Phys.Rev. B: Condens. Matter Mater. Phys., 2006, 74,245313.

30 C. G. Rocha, A. Wall, A. R. Rocha and M. S. Ferreira, J. Phys.:Condens. Matter, 2007, 19, 346201.

31 A. R. Rocha, M. Rossi, A. Fazzio and A. J. R. da Silva, J. Phys.D: Appl. Phys., 2010, 43, 374002.

32 E. Artacho, D. Sanchez-Portal, P. Ordejon, A. Garcıa andJ. M. Soler, Phys. Status Solidi B, 1999, 215, 809.

33 O. A. von Lilienfeld, I. Tavernelli, U. Rothlisberger andD. Sebastiani, Phys. Rev. Lett., 2004, 93, 153004.

34 M. P. Lima, A. Fazzio and A. J. R. da Silva, Phys. Rev. B:Condens. Matter Mater. Phys., 2009, 79, 153401.

This journal is ª The Royal Society of Chemistry 2013

35 J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.,1996, 77, 3865–3868.

36 N. Troullier and J. L. Martins, Phys. Rev. B: Condens. MatterMater. Phys., 1991, 43, 1993–2006.

37 S. Datta, Electronic Transport in Mesoscopic Systems,Cambridge University Press, Cambridge, 1st edn, 1995,pp. 57–67, 86–88.

38 M. Brandbyge, J.-L. Mozos, P. Ordejn, J. Taylor andK. Stokbro, Phys. Rev. B: Condens. Matter Mater. Phys.,2002, 65, 165401.

39 A. R. Rocha, V. M. Garcıa-Suarez, S. Bailey, C. Lambert,J. Ferrer and S. Sanvito, Phys. Rev. B: Condens. MatterMater. Phys., 2006, 73, 085414.

40 F. D. Novais, A. J. R. da Silva and A. Fazzio, Braz. J. Phys.,2006, 36, 799.

41 C. Caroli, R. Combescot, P. Nozieres and D. Saint-James,J. Phys. C: Solid State Phys., 1971, 4, 916.

42 S. Sanvito, C. J. Lambert, J. H. Jefferson and A. M. Bratkovsky,Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 11936–11948.

43 M. Buttiker, Y. Imry, R. Landauer and S. Pinhas, Phys. Rev. B:Condens. Matter Mater. Phys., 1985, 31, 6207.

44 D. S. Fisher and P. A. Lee, Phys. Rev. B:Condens. Matter Mater.Phys., 1981, 23, 6851.

45 P. Giannozzi, R. Car and G. Scoles, J. Chem. Phys., 2003, 118,1003.

46 R. Car and M. Parrinello, Phys. Rev. Lett., 1985, 55, 2471.47 S. Nose, J. Chem. Phys., 1984, 81, 511.48 W. Hoover, Phys. Rev. A: At., Mol., Opt. Phys., 1985, 31, 1695.49 B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane,

G. A. Weiss and P. G. Collins, Science, 2007, 315, 77–81.

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