Table Of ContentScaling of the conductance in gold nanotubes
Miriam del Valle,1,2 Carlos Tejedor,1 and Gianaurelio Cuniberti2
1Departamento de F´ısica Te´orica de la Materia Condensada,
Universidad Aut´onoma de Madrid, Facultad de Ciencias, C-V, E-28049 Madrid, Spain
2Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
6
(Dated: January 17, 2006)
0
0 A new form of gold nanobridges has been recently observed in ultrahigh-vacuum experiments,
2 where the gold atoms rearrange to build helical nanotubes, akin in some respects to carbon nan-
n otubes. Thegoodreproducibilityofthesewiresandtheirunexpectedstabilitywillallowforconduc-
a tancemeasurementsandmakethempromisingcandidatesforfutureapplications. Wepresenthere
J a study of the transport properties of these nanotubes in order to understand the role of chirality
4 and of the different orbitals in quantum transport observables. The conductance per atomic row
2 shows a light decreasing trend as the diameter grows, which is also shown through an analytical
formula based on a one-orbital model.
]
l PACSnumbers:
l
a
h
- Gold is known to be a good conductor for ages but its such as conductance quantization. Experimental evi-
s
use innanoelectronicsdeliversunexpected behaviorsdif- dence points out that a structural transition from the
e
m fering from its bulk properties. Recent experiments1,2,3 thicker,internallycrystallinewiresintothesethinnerreg-
indicate that gold nanowires exhibit fascinating ordered ular but noncrystalline ones takes place at a critical ra-
.
t structures that resemble those of the by now well-known dius below 2 nm. Lattice spacing has been measured to
a
m carbon nanotubes (CNTs).4 bealmost2.88˚A,theneighboringdistanceofgold,forall
these wires with diameters between 0.5 nm and 1.5 nm.
- Highresolutionmicroscopeimagesshowthatthesewires
d consist of coaxial nanotubes of helical atom rows coiled Wethereforetakethisdistanceastheinteratomicspacing
n of the triangular lattice, as we can think of these cylin-
around the wire axis. These wires present therefore dif-
o dricaltubesasrolled-uplatticeplanesoffccAu(111),as
ferent possible helicities as CNTs do, depending on the
c depicted in Fig. 1. Atoms in successive shells appear to
[ angle in which the atom rows rotate around the tube
axis. Experimental evidence so far seems to support the straina bit to maintaincommensurabilitybetweeninner
1 andoutershells. Ab-initiocalculationsshowthoughthat
fact that small chiral angles are preferred by the fab-
v this shear strain should have just a very small effect on
ricated nanotubes. The multi-shell nanotubes keep the
0
difference in the number of atom rows between adjacent
4
5 shells constant, leading to a kind of magic shell-closing
1 number which is always seven. This fact can be under-
0 stood as a new inner shell arises when the radii differ
6 in about a = 2.88 ˚A, the neighboring distance of bulk
0
gold, which implies a perimeter difference (2πa 7a)
/ ∼
t correspondingtosevenatomrowsnearlyparalleltoeach
a
other. Sosingle-wallednanotubesshouldexistwithsixor
m
less atom rows, and actually one single-walledtube with
d- five atom rows has been observed.3 With seven or more
n atomrowsadouble-walledtubeshouldbeseen,whichbe-
o comesatriple-wallednanotubewhenmorethanfourteen
c rows are present, and so on. The good reproducibility
v: of these wires, and their stability,5,6,7,8 which is better
i for those with outer tubes with an odd number of atom
X
rows,makes them promising candidates for future appli-
r
cations. Ballistic conductancemeasurementswill leadto
a
aconfirmationofthe proposedstructuralmodelthrough
its derived electronic structure. Astonishingly enough FIG. 1: Some gold nanotubes are shown as well as the 2D
several works hint at the fact that the conductance per triangular network, from which we can build these models.
The coordinate system used is sketched on the lattice, being
atom row might decrease with the number of circumfer-
ential atoms.9,10 In this Letter, we provide a theoretical a1anda2thebasisvectors,andCthechiralvectorspecifying
the nanotube and corresponding to its circumference. The
understandingofthis phenomenonby meansofboth nu-
highlightedatomsonthelatticearethosedefiningtheregion
merical and analytical calculations. ◦
of inequivalent chiral vectors, which spans an angle of 30
Determining the structure of the nanotubes is funda- as can be easily understood analyzing the symmetry of the
mental in order to understand their physical properties, lattice.
Typeset by REVTEX
2
4 10
total s, p, d orbitals (4,3) (5,2)
d s orbitals 8
s
p
6
2
4
V) h) 2
e 2/
(F 0 2e 0
E - E G ( 8 (6,1) (7,0)
6
-2
4
2
0
-4 -2 -1 0 1 2 -2 -1 0 1 2
0 π/T 0 3 6 9 12
k DOS (arb. u.) G (2e2/h) E - E (eV)
F
FIG. 2: Onedimensional band structure, density of states FIG. 3: Conductance of AuNTs with seven atom rows but
and conductance for the (4,3) gold nanotube. For this tube different chiralities. Red lines indicate results including s or-
the translational vector is T = |T| = 3.03 nm. The energies bitals only, whereas the black ones are showing the calcula-
areshiftedtosettheoriginattheFermienergy,asinthenext tions with all nineouter orbitals.
figures.
structureswithshorterperiods. Thehelicityofthetubes
the electronic structure.11 isobservedexperimentallythroughawavymodulationof
Despite the strong intershell interactions, these struc- the STM images.2,3
tures resemble greatly those of CNTs, where the hon- We study the transport properties of several of these
eycomb network of carbon atoms is replaced by the tri- AuNTs using a multiorbital tight-binding Hamiltonian
angularlatticeofgoldatoms. Theseareactuallycomple- and Green function techniques. In particular we adopt
mentary networks, and the gold nanotube structure can a Slater-Koster-type13 tight-binding approach, with the
beobtainedfromtheoneofCNTsbyputtinggoldatoms parameters taken from the nonorthogonal parametriza-
at the center of honeycomb framework. tion of Papaconstantopoulos and co-workers.14. Our ap-
We use the indices (n,m) to classify the gold nanotubes proach remains of microscopic nature since the symme-
(AuNTs), where n andm areintegersdefining the chiral tries of the atomic s,p, and d valence orbitals are taken
vector, which is wrapped to form the tube. The chiral into account. Ab initio studies including shell effects
vector is then (na +ma ), perpendicular to the tube could be needed for greater quantitative accuracy.
1 2
axis, where the vectors a1 and a2 span the 2D unit cell. As a result we analyze the differences in the transport
The number of atom rows coiling around the tube axis calculations that raise from the inclusion of all outer s,
is then n + m while the nanotube translational vector p and d orbitals instead of just the simplified model of
reads T(n,m) = ((2m+n)/dR)a1 ((2n+m)/dR)a2, s orbitals, which is a tempting approximation for these
−
being dR the greatest common divisor of (2m+n) and kind of systems and has been used in other works.9 In
(2n+m),asTshouldbethesmallestlatticevectorinits order to compare results, we will thus use both a one-
direction. Due to the symmetry of the triangular lattice orbital Hamiltonian describing just the s orbitals, and a
the chiralvectors of all inequivalent tubes are comprised nine-orbital Hamiltonian, containing s,p and d orbitals
ina30◦ angle,thatis,inaone-twelfthirreduciblewedge including all conduction electrons. In doing that we re-
oftheBravaislattice. Wecanthencompromisedanduse strict our calculations to nearest-neighbors interactions.
only indices with n > 0 and 0 < m < n. The tubes of The Hamiltonian describing our systems can be written
the form (n,0) and (n,n) are achiral, presenting thus no as
handedness. Only wires with an even number of strands
may have the structure (n,n).12
For a fixed n, the radius is minimized for higher values = H c† c (1)
H iα,jβ iα jβ
of m (tension decreases with shrinking radius), so as to
hi,Xji,α,β
fulfill simultaneouslytightest externalpackingandmini-
malwire radius.8 The largestm forodd number of atom where the indices i,j run over the atom sites and the
rowsisn 1,leadingalwaystoafinitechiralityandthus indices α,β mark the different orbitals. The transfer in-
−
a helical structure in these cases. The outer tubes with tegralsH arethe on-siteenergiesorhoppingparam-
iα,jβ
an even number of atom rows seem to take non-helical eters, dependent on whether i equals j or not.
3
15
r = 0.24 nm (3,3) r = 0.28 nm (4,3)
10
5
0
r = 0.36 nm (5,4) r = 0.44 nm (6,5)
h) 15
2/
e 10
2
(
G 5
0
r = 0.52 nm (7,6) r = 0.55 nm (7,7)
15
10
5
0
-3 -2 -1 0 1 2 3 -2 -1 0 1 2 3
FIG. 5: Schematic illustration of the reciprocal lattice of
E - E (eV)
F the 2D hexagonal Bravais lattice of Au (111) where the first
Brillouin zone is highlighted through a darker background.
FIG. 4: Conductance of AuNTs with different radii r. Red Thek-vectors lying at theFermi surface giverise toa nearly
lines indicate results including s orbitals only, whereas the perfect circle in this 2D structure. The black cutting lines
black ones are showing the calculations with all nine outer represent the Brillouin zone of a nanotube in an extended
orbitals. zone scheme which allows the use of the dispersion relation
calculated for the 2D gold layer to get the bands of a folded
nanotube. In this example we show the wave vectors giving
Theconductanceiscalculatedfromthetransmissionval- the bands in a (6,0) AuNT, a good candidate for this visu-
alization as it has a small number of bands. These lines are
ues applying the Landauer formula, using Green func-
tion techniques.15 In particular, we derive the elas- plottedassolid bandsinsidethefirstBrillouin zone,and col-
ored when folded back into it through the reciprocal lattice
tic linear response conductance via the Fisher-Lee for-
vectors of the gold layer. In the inset the conductance per
mula for the quantum mechanical transmission16: G =
atom row at the Fermi energy based on theanalytical model
2e2Tr Γ Γ † ,whereΓ =i Σ Σ† be- accountingonlyfors-orbitalsdescribedinthisLetterisshown
h LG RG L/R L/R− L/R for all possible (n,m) NTs, reproducing exactly the values of
ing ΣL(cid:8)/R the self e(cid:9)nergy of the left o(cid:16)r right lead re(cid:17)spec- thenumerical calculations.
tively, and where is the Green function of the central
G
region dressed by the electrodes.
1
We consider semi-infinite single-walled nanotubes voltages the current will show the difference in energies
(SWNTs) as leads. Considering the leads as bulk gold at which new channels start taking part in transport.
will yield an increased imaginary part of the lead self In Fig. 4 the conductance is plotted for nanotubes of
energies, which will slightly decrease the conductance different radii, chosen as to have the smallest chirality
values. For multi-walled nanotubes (MWNTs) it has possible for a given radius. The number of atom rows
been suggested that the conductance G is mainly aroundthe nanotubeaxisineachofthemis6,7,9,11,13
determined by the outer shell which is also responsible and14respectively. Weseethat,thoughtheconductance
for the structural stability.17 A slight reduction of the increases for thicker nanotubes, this is not the general
conductance is expected from the finite size effects. trend if divided by the number of atom rows. We can
The calculateddensity ofstates andthe conductance for observe again how the expansion of the orbital basis to
thegoldnanotube(4,3)areshowninFig.2. Thistubeis include all conduction electrons changes the energies at
one of the simplest possible candidates with seven atom whichnewchannelsopen,beingcriticaltoobtaincorrect
rows around its circumference (Fig. 1). The colored ar- current values.
eas show the contribution of the different orbitals to the By analyzing the Brillouin zone of the two-dimensional
total density of states, indicating a dominance of s or- gold lattice taking into account only s orbitals, one
bitalsaroundtheFermilevel. Consideringonlysorbitals caneasilydemonstrate afterstudying its dispersionrela-
around the Fermi energy is thus a good approximation, tion that the Fermi surface is with great accuracy ap-
but when applying a bias it can be critical to include all proximated by a circle with a radius close to k =
F
s,p,d outer orbitals, as they play an important role in 2√π/(√43a),wherea=2.88˚A. Thissimplemodelallows
the opening of new conduction channels. usto haveananalyticalapproachtothe conductance,as
A comparison of the conductance of nanotubes of the the number of energy bands crossing the Fermi surface,
samediameterbutdifferentchiralities,asshowninFig.3, since the Brillouin zone of the gold nanotubes consist of
suggeststhatnosignificantdifferenceisexpectedintheir parallellinesegmentsduetothequantizationofthewave
conductancevaluesattheFermienergy. Forappliedbias vectoralongthe circumferentialdirection. Forsimplicity
4
8 Goldnanotubeshavebeen observedasshort(L 4 nm)
∼
conductors between bulk gold electrodes. This calls for
6
) an extension of our theory to account for finite size ef-
h
2/ fects. We didthis byparametricallyintroducingintothe
e 4
2 nanotube two tunneling barriers at a distance L. Fig. 6
(
G infinite system shows the conductance between two homologous semi-
2
with artificial barriers
infinite electrodes which are taken into account through
with wide band leads
modified self energies. This modification is introduced
0
-1.5 -0.75 0 0.75 1.5 by a multiplicative factor α, which simulates the bar-
E - E (eV)
riers. The finite size effects can be equally made ap-
F
parent by considering wide band leads, i.e. Σ (E) =
WB ∼
FIG.6: Conductanceofthe(4,3)-AuNTforaperfectinfinite iImΣ(E =EF). As we cansee inFig.6, we obtainthe
−
NT(blackline),forafinitetubewithbarriersintheconnect- same kind of oscillating behavior with both approxima-
ingregionstotheelectrodes(wheretheconstantαintroduced tions. This shows that the results of our model provide
in the text equals 0.6) and for a finite tube like in the previ- an upper limit for the conductance.
ous case but in the wide band limit approximation taken at
Inconclusion,weinvestigatedtheelectronicpropertiesof
the Fermi energy. The colored lines show oscillations due to
severalgoldnanotubes. WebuilttheHamiltoniansofthe
scatteringattheendsofthefinitesizetube. Thethickerlines
correspond to room temperatureconductance.19 systems by applying a tight-binding model for s orbitals
as well as for all outer orbitals. All AuNTs are equally
metallic independently of their chirality or diameter in
contrasttotheresultsobtainedfortheircarboncounter-
we can restrict this counting to the first Brillouin zone
parts,20 butexpectedfromgoldelectronicnature. Inthe
by folding back to this area the segments lying outside
case of goldthe quantizationof electronwaves along the
it, which will elongate the lines inside it, as pictured in
Fig. 5.18 The conductance can then be written as circumferenceofthetuberesultsinaBrillouinzonecom-
posedof line segments that alwayscut the Fermi surface
oftheAu[111]layerduetothecontinuityofthissurface.
2e2 n2+m2+nm Thes-orbitalcalculationsdemonstratethataone-orbital
G= 2Int +1 (2) approach to the problem of gold nanotubes is not a bad
h " s π√3 ! #
approximation, but should be avoided when applying a
finite bias voltage to the system.
and is only dependent of the nanotube indices and not
Ananalyticalformulafortheconductanceofthes-orbital
on hopping parameters or on-site energies. This mod-
calculations, matching perfectly the numerical results,
els yields a perfect matching with the numerical calcula-
shows that there is a slight decrease of the conductance
tions described before. The result of this approximation
per atom row as the radius increases.
of the conductance at the Fermi level is plotted in the
insetofFig.5fornanotubesofdifferentradiiandchirali-
ties,presentingtheconductanceperatomrow. Thesolid
Acknowledgments
linesinthisfigureputanaccentonthe resultsofAuNTs
with the smallest helicity as characterized in Fig. 4. We
can observe how the envelope of all these points slowly ThisworkwaspartiallyfundedbytheVolkswagenFoun-
decreases with increasing number of coiling atom rows, dation(Germany)undergrantNo.I/78340,bytheMEC
or likewise with increasing diameter. The conductance (Spain) under contracts MAT2005-01388, NAN2004-
values will nevertheless be reduced by considering more 09109-CO4-04, and by the CAM under contract No. S-
realistic gold leads. This trend has been also found in a 0505/ESP-0200. MDacknowledgesthe supportfromthe
first-principles one-orbital calculation.9 FPI Programof the Comunidad Aut´onoma de Madrid.
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