Table Of ContentFullerene-based molecular nanobridges: A first-principles study
J. J. Palacios,A. J. P´erez-Jim´enez,and E. Louis
Departamento de F´ısica Aplicada y Unidad Asociada UA-CSIC, Universidad de Alicante, San Vicente del Raspeig, Alicante
03080, Spain
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J. A. Verg´es
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Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, Madrid 28049, Spain.
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(February 1, 2008)
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n Building upon traditional quantum chemistry calculations, we have implemented an ab-initio
a method to study the electrical transport in nanocontacts. We illustrate our technique calculat-
J ing the conductance of C60 molecules connected in various ways to Al electrodes characterized at
3 theatomiclevel. Centraltoacorrectestimateoftheelectrical currentisapreciseknowledgeofthe
2 local chargetransferbetweenmoleculeandmetalwhich,inturn,guaranteesthecorrectpositioning
of the Fermi level with respect to the molecular orbitals. Contrary to our expectations, ballistic
] transport seems to occur in this system.
l
l
a
h
- At the forefront of the carbon-based molecules that trodes with the “Fermi level” of the molecule forces a
s
e are expected to play a key role in nanoelectronic devices certain amount of charge to be transferred one way or
m one can situate carbon nanotubes and fullerenes. From the other. The way this alignment is achieved and how
. Coulomb blockade to Kondo-like phenomena through much charge is transferred constitutes a difficult prob-
t
a ballistic transport,multiple transportregimeshave been lem which turns out to be essential to understand the
m observedinnanotubes(evenforthesamechiralityofthe electrical transport in molecular nanobridges and, more
- nanotube) depending on the way the contact with the generally, in nanocontacts or nanoconstrictions. Curi-
d
electrodes is made [1]. Very little is known, however, ously enough, the prevailing theoretical descriptions in
n
about the true role played by this contact as well as of the great body of work done on various types of metal-
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c the effectofthe substrateonwhichthe carbonnanotube molecule-metal systems [7] as well as on atomic chains
[ has been layed. This is somewhat connected with the andconstrictions[8]arestillbasedonparametrizedtight-
difficulties experiencedinthe interpretationofelectronic bindingorsemi-empiricalmodels. Thetheoreticalframe-
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v transport experiments through C60 molecules. Experi- work to calculate the current is well established [9], but,
9 ments using a scanning tunneling microscope (STM) [2] within these models, Fermi level alignment has to be ei-
5 haveshownthepossibilityof“looking”atindividualC60 ther entirely ignored or imposed with some additional
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molecules when they are adsorbed on a conducting sub- criteria. While local charge neutrality is a sensible cri-
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strate. There is still, however,a lotof controversyin the terion for atomic chains and constrictions made of the
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1 interpretationofthe imagesobtainedbydifferentgroups same material as the electrodes [8], there is not the like
0 [3]. Alternative to STM set-ups, nanoscopic break junc- for metal-molecule-metal systems. This is, in part, the
/ tions also revealedthemselves as powerful tools to study motivation for developing ab-initio methods which can
t
a
transport through individual molecules [4,5]. Recently, provide qualitative and quantitative answers in all pos-
m
for instance, in between the two electrodes of a gold sible scenarios. Some groups have already blazed the
d- break junction, evaporated C60 molecules unexpectedly trail in this direction [10–13], although in their methods
n created “mechanical” bridges for electrical transport be- the problem of the atomic structure of the electrodes is
o tween electrodes [6]. usually put aside. This is not satisfactory when one is
c Archetypal STM and break-junction set-ups above trying to describe STM experiments where the detailed
:
v mentioned can be cataloged under the term molecular atomic structure ofthe tip determines, to a largeextent,
i nanobridge,i.e.,metallicelectrode+molecule+metallic whetherornotthe STMimagescanresolvethe topogra-
X
electrode. The conductance,G,is essentiallydetermined phy or molecular structure of the adsorbate.
r
a by the electronic structure of the isolated molecule since What we present here is an ab-initio alternative to
the number of available channels for conduction deep in parametrized tight-binding or semi-empirical models in
the electrodesis alwaysmuchlargerthanthose provided order to address electrical transport through generic
by the molecule. However, as already mentioned, this nanocontacts. Our method automatically accounts for
conductance is strongly influenced by the chemistry of Fermi level alignment and charge transfer whenever the
the electrode-molecule contacts. Behind this contact lies latter is expected to occur. This is the case in the
a key problem: Charge transfers between electrode and system we study here: Al∞-C60-Al∞. Isolated C60
molecule when they come in close proximity. The ne- molecules present a large gap (≈ 2 eV.) and behave like
cessity of aligning the Fermi level of the metallic elec- intrinsic semiconductors. At low temperatures, metal-
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semiconductor-metal systems do not conduct when the ber of atoms describing the metallic electrodes the en-
semiconductor is intrinsic regardless of the sign and the ergy of the highest occupied molecular orbital (HOMO)
amount of charge transferred at the interfaces. In the of the whole structure sets the Fermi level. In addition,
metal-fullerene-metal system, however, a tunnel current the correct charge transfer (to or from the molecule) is
can always be expected due to the nanoscopic size of guaranteed. Next, we perform a Lo¨wdin orthogonaliza-
the fullerene [14]. What is more surprising is the fact tionoftheoriginalnon-orthogonalgaussianbasissetinto
that C60 molecules can conduct ballistically (G > e2/h) anorthogonalonewhich,havingbasiselementsmoreex-
[13]. Some of the lowest unoccupied molecular orbitals tendedthantheoriginalone,stillpreservesthesymmetry
(LUMO’s) of the isolated fullerene still retain its ex- of the orbitals and the atomic character. The hamilto-
tended characterwhen contact is made to the electrodes nian of the electrode+molecule+electrode system, Hˆ, as
and they become partially or fully occupied. In solid- it stands, is finite and, according to the usual theoreti-
state language: The Fermi level lies above the bottom cal transport schemes [9], its Green’s functions are un-
of the conduction band all across the (nanoscopic) semi- suitable for any current determination since they simply
conductor. Noticethatbiggermoleculesexhibitingagap havepoles. Thefollowing(andcrucial)stepinourproce-
likesomefamiliesofundopednanotubesarenotexpected dure is to transform this finite system into an effectively
to conduct at zero bias [1]. Whether or not this occurs infinite one. In order to do this we first remove from the
depends on the size of the molecule and the details of hamiltonian all but the N atoms forming the relevant
the charge transfer process which, in turn, relies heavily atomic structure of each electrode close to the molecule
on the relaxationof the atomic structure of the fullerene [seeFig. 1(b)]. TheretardedGreen’sfunctionassociated
when in contact with the metal. with this reduced hamiltonian Hˆ is now transformed
r
into the retarded Green’s function of an infinite system:
Gr(ǫ)=(ǫIˆ−Hˆ +iδ)−1 →[ǫIˆ−Hˆ −Σˆ(ǫ)]−1. (1)
r r
In this expression Σˆ = Σˆ +Σˆ where Σˆ (Σˆ ) denotes
R L R L
a self-energy matrix that accounts for the right(left) in-
finite electrode, part of which has been included in the
DFcalculation[seeFig. 1(a)]. Theaddedself-energycan
onlybeexplicitlycalculatedinidealsituations,which,in
principle, might seem to limit the desired applicability
of the above step. However,as pointed out by Landauer
in his scattering description of electrical transport [9],
the bulk details of a metallic electrode are not neces-
sarily relevant for the electrical transport properties of
the region with the smallest number of channels, i.e.,
the contacted molecule. With this crucial observation
in mind we choose to describe the bulk electrode with
a Bethe lattice tight-binding model [17] with the appro-
priate coordination and appropriate parameters. More
FIG.1. Schematicviewofthetwo-stepprocessforcalculat-
specifically: We require our Bethe lattice model to re-
ingtheGreen’sfunctionusedinthecalculationoftheconduc-
produce the electrode bulk density of states and to have
tance: (a) A density functional calculation of a C60 molecule
the same Fermi energy as that of the system on which
contactedbytwoelectrodes(hereintheform ofpyramids)is
the DF calculation was initially performed [18]. The ad-
performed. (b) Atomsthat are irrelevant for electrical trans-
vantage of choosing a Bethe lattice resides in that Σˆ can
port are removed and substituted by an effective selfenergy
“attached” to the contact left and right atoms remaining in be easily calculated through a well-known iteration pro-
theelectrodes (see text). cedure [17] which we do not detail here. It suffices to
saythat,foreachatominthereducedelectrodethathas
Standard ab-initio or quantum chemistry calculations been stripped from, at least, one nearest-neighbor atom,
[15,16] are only possible for finite or periodic systems, acontributionfromaninfiniteBethelatticecalculatedin
whereasthetransportproblemonetypicallywantstoad- thedirectionoftheremovedatomisaddedtoΣˆ. Assum-
dress requires infinitely large leads with no symmetry at ingthattheinitialsystemwaslargeenoughandthatthe
all. DrawingonRefs.[12,13],the mainideahereconsists reduced system still contains the relevant atomic details
ofperformingadensityfunctional(DF)calculationofthe of the electrode close to the contact with the molecule,
molecule including part of the leads with the desired ge- to replace a piece of the electrode by our effective self-
ometry [seeFig. 1(a)]. This canbe efficiently done using energyshouldintroducenospuriouseffects,andit makes
the Gaussian-98 code [15]. For a sufficiently large num- the system effectively infinite. The conductance can now
be simply calculated through the expression [9]
2
2
G= 2e Tr[Γ GrΓ Ga], (2) covalent chemical bond with carbon molecules which, in
L R
h principle, makes it idealfor the future fabricationof sta-
ble nanostructures. Forthe DFcalculationwe haveused
where Tr denotes the trace over all the orbitals of the
the Becke’s three-parameter hybrid functional using the
reduced hamiltonian and the matrices Γ and Γ are
R L Lee, Yang and Parr correlation functional with a mini-
given by i(Σr −Σa) and i(Σr −Σa), respectively. As
R R L L malbasisset[19]. Theorientationofthefullereneandthe
in the self-energy matrices, the matrix elements of Γ
R separationandatomicstructureoftheelectrodesconsid-
and Γ are only different from zero for the atoms in the
L ered in Fig. 2 (see insets) are reasonable, but arbitrary
reduced electrodes that have been stripped from at least
[20]. In Fig. 2(a) we plot the conductance around the
one near-neighbor atom.
Fermienergy(heresettozero)ofthe fullerenecontacted
by two opposite electrodes in the form of pyramids [see
inset or Fig.1(a)]. These are composed of 30 atoms each
andtheir apex atoms areseparatedby 10.1˚A . We have
orientated the fullerene so that two opposite C atoms
are as close as possible to the apex atoms of the pyra-
mids(i.e.,1.4˚AbetweentheCatomofthefullereneand
the Al apex atom). In order to illustrate the reliability
of the method, we present the results for three differ-
ent approximation degrees: N = 1 (dotted line), N = 5
(dashed line), and N = 14 (solid line), corresponding to
an increasingly detailed contact structure. Notice how,
as expected, all the curves are very similar since the rel-
evant atomic structure close to the molecule is simply
representedby the atom atthe pyramidapex. Logically,
the results using 5atoms (twoatomic planes)arealmost
identical to those using 14 atoms (three atomic planes).
The gap of the isolated fullerene reflects itself in a van-
ishing Gbelow the Fermienergyandthe LUMO’s ofthe
isolated molecule are now partially occupied. Note that
they retain their extended character since the conduc-
tance around the Fermi level goes up to 2e2/h at cer-
tain energies. Nonetheless, the three-fold degeneracy of
the f1u and f1g LUMO’s has been entirely removed by
the interaction with the electrodes (this interaction is
alsoresponsibleforthe broadeningofthe peaks). Inthis
situation Coulomb blockade physics must dominate and
spin-splitting will occur in the fullerene [21].
FIG.2. (a)ConductancearoundtheFermienergy(hereset
InFig. 2(b)wepresenttheconductanceofthefullerene
tozero) ofaC60 molecule contactedbytwooppositeAlelec-
orientedasinthe previouscase,butwherethe twopyra-
trodes in the form of pyramids composed of 30 atoms each
(see inset). We present three different cases: N = 1 (dot- midshavebeenreplacedbyAl(001)parallelsurfaces(see
ted line), N = 5 (dashed line), and N = 14 contact atoms inset). The different types of lines correspond to an in-
(solid line), corresponding to an increasingly detailed elec- creasingdefinitionofthesurfaceplane(1,5,and9surface
trodestructureinthereducedsystem(seetext). (b)Sameas atoms). Now one atom is not enough to define the con-
in(a),butwith thefullereneinbetweenAl(001)parallel sur- tactsincethereisanimportantcouplingbetweenseveral
faces, each one composed by two atomic planes with a total Alsurfaceatomsandthefullerene. Noteven5atomssuf-
of33atoms(seeinset). Wepresentresultsfor N =1(dotted fice to define the metallic contact region quite precisely.
line), N = 5 (dashed line), and N = 9 contact atoms (solid Now the value of the conductance rises up to ≈ 2.5e2/h
line), corresponding to an increasing surface plane.
right at the Fermi energy. This was somehow expected
since,inthissituation,thehybridizationwiththesurface
We applyhere the technique just describedto the sys-
isstrongerandtherearemorechannelsfortheelectronto
tem Al∞-C60-Al∞ with different geometries. Although travelbetweenelectrodeswhiletheLUMO’sappeartobe
they are not commonly used in experiments, we have
also extended. The gap still reflects itself in G, but has
consideredAlelectrodesfortwoindependentreasons: (i)
been partially closed by the strong hybridization. The
The number of electrons per atom is small enough to excesschargein the fullerene is ≈2.6whichagreesfairly
avoidpseudo-potentialsintheDFcalculationsand,(ii)in
well with a related and recent calculation [13].
contrasttonoblemetals,Alpresentsastrong,essentially
3
We have chosen to study a system of recent interest: A
fullerene contacted by Al electrodes. Our main goal was
to describe correctly the contact charge transfer so that
realistic predictions on the electrical conduction proper-
ties of these molecules can be made.
PartofthisworkwassupportedbytheSpanishCICYT
under Grants Nos. 1FD97-1358 and by the Generalitat
Valenciana under Grant No. GV00-151-01. Discussions
with E. San Fabia´n, J. C. Sancho, L. Pastor-Abia, and
J. M. P´erez-Jorda´are acknowledged.
FIG. 3. Conductance around the Fermi energy (here set
to zero) of a C60 molecule contacted by two parallel Al(111)
surfaces separated by 10.1˚A and composed of 31 atoms each [1] For a nice review on nanotubes see C. Dekker, Phys.
Today 52, No. 5, 22 (1999).
(see inset). The dashed line is the result for the non-relaxed
[2] C. Joachim et al., Phys. Rev. Lett. 74, 2102 (1995); D.
fullereneandthesolidoneistheresultwhenthefullerenehas
Porath et al., Phys. Rev. B 56, 9829 (1997); J. G. Hou
been allowed to relax.
et al., Phys. Rev. Lett. 83, 3001 (1999); J. I. Pascual et
al., Chem. Phys.Lett. 321, 78 (2000).
Figure 3 shows the conductance when the fullerene is
[3] J. I. Pascual et al., Phys. Rev. Lett. 85, 2653 (2000); J.
placedinbetweentwoparallelAl(111)surfaces(31atoms G. Hou et al.,Phys. Rev.Lett. 85, 2654 (2000).
each, 7 of which have been used in the calculation of G) [4] M. A.Reed et al.,Science 278, 252 (1997).
with parallel Al surface and fullerene hexagons. This [5] C. Kergueris et al., Phys.Rev.B 59, 12505 (1999).
orientation has been chosen on the basis of several ex- [6] H. Park et al.,Nature407, 57 (2000).
perimental observations [22]. In one case (dashed line) [7] For references see C. Joachim, J. K. Gimzewski, and A.
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(solid line) an ab-initio relaxation has been performed. [8] J. C. Cuevas et al., Phys.Rev.Lett. 80, 1066 (1998).
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by H. Ahmed, M. Pepper, and A. Broers (Cambridge
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University Press, Cambridge, 1995).
surface and fullerene atoms intervene. While the de-
[10] N. D. Lang and Ph. Avouris, Phys. Rev. Lett. 81, 3515
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(1998).
the excess charge is ≈ 2.1. Surprisingly enough, when [11] G. Taraschi et al.,Phys.Rev. B 58, 13138 (1998).
the fullerene is allowed to relax (stretch in this case), [12] S. N. Yarilaki et al.,J. Chem. Phys.111, 6997 (1999).
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[17] L. Mart´ın-Moreno and J. A. Verg´es, Phys. Rev. B 42,
where bonding takes place. (The distance between sur-
7193 (1990).
faces was chosen to be slightly larger than the diameter
[18] The nearest-neighbor hopping elements of the tight-
of the fullerene so the structural changes of the fullerene
binding model for the Bethe lattice can be obtained in
do not play any role). The small value of the conduc-
multipleways.Wehavechosenheretoextractthemfrom
tance below the Fermi energy cannot be attributed to
aDFcalculation ofadimeratthebulknearest-neighbor
the gap of the fullerene (which has entirely disappeared distance of the considered electrode. The diagonal ele-
from the density of states), but to the localization pro- ments are shifted at will to reproduce the desired Fermi
cess whichdoes notreflect inthe globaldensity of states level.
ofthe fullerene. Thislastresultbridgesthe gapwiththe [19] A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
intuitive picture for transportin a metal-semiconductor- [20] A structural relaxation is expected to occur, but this is
metal system and shows that the chemical details of the not relevant to show the reliability of themethod.
[21] Work in progress.
metal-molecule contact are determinant in the transport
[22] R. Fasel et al.,Phys.Rev. Lett.76, 4733 (1996).
properties of molecular bridges.
In summary,we have developeda methodology to cal-
culateab-initiotransportpropertiesofnanoconstrictions.
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