Electrical contacts to nanotubes and nanowires: why size matters

0.22 MB·English

by  Francois Leonard

Electrical contacts to nanotubes and nanowires: why size matters
∗
Franc¸ois L´eonard and A. A. Talin
Sandia National Laboratories, Livermore, California 94551
(February 6, 2008)
Electrical contacts to semiconductors play a key role in electronics. For nanoscale electronic
6
devices, particularly those employing novel low-dimensionality materials, contacts are expected to
0
play an even more important role. Here we show that for quasi-one-dimensional structures such
0
as nanotubes and nanowires, side contact with the metal only leads to weak band re-alignement,
2
in contrast to bulk metal-semiconductor contacts. Schottky barriers are much reduced compared
n
withthebulklimit,andshouldfacilitatetheformationofgoodcontacts. However,theconventional
a
strategyofheavilydopingthesemiconductortoobtainohmiccontactsbreaksdownasthenanowire
J
diameter is reduced. The issue of Fermi level pinning is also discussed, and it is demonstrated
1
that the unique density of states of quasi-one-dimensional structures make them less sensitive to
3
this effect. Our results agree with recent experimental work, and should apply to a broad range of

quasi-one-dimensional materials.
]
i
c
s TheearlyworkofSchottky,MottandBardeenhaslaid and slowly varying Schottly barriers with NT diameter.
- the course for much of the fundamental understanding In nanowires (NWs), there is a range of diameters with
l
r and improvement in the performance of electrical con- minimized Schottky barriers, providing optimal contact
t
m tacts to bulk semiconductors. For nanoelectronics, con- properties. We also demonstrate that in general, Q1D
. tactsareasignificantfractionofthetotaldevicesizeand structures are much less sensitive to Fermi level pinning
t
a are expected to be crucial for device behavior. However, than their bulk counterparts. Finally, a conventional
m
becauseoftheuniquepropertiesofnanostructures,much strategyformakingohmiccontactsistoheavilydopethe
- of the basic fundamental aspects of contacts need to be semiconductor near the contact; we show that at typical
d
re-examined at the nanoscale. dopingsforSi,thecontactresistanceincreasesrapidlyas
n
o The current understanding of contacts to nanostruc- the nanowire diameter is decreased below 40 nm.
c tures is in its infancy, both from an experimental and
[
theoreticalperspective. Anexampleiscarbonnanotubes (a) Metal (b) (c)

1 (NTs): despitemuchexperimentalwork,itisstillunclear R R
v wether the contacts are Schottky or ohmic, with reports Metal
NT
3 ofSchottkycontactsforTi[1]andohmiccontactsforAu NW
0
[2]andPd[3,4]. However,recentexperimentalwork[5,6]
0
has suggested that the type of contact depends on the metal s emiconductor metal s emiconductor
2
0 NT, with Schottky contacts for small diameter NTs and (d) c (e) c
Bulk system
6 ohmic contacs for large diameter NTs.
mat/0 dsintirmFtarretaoendmdsiitotianhonaatltahnltesahoenermoecsticoticronaunclcdetpuputcertreoossfpreseFn)cedtirsi-mvbienoi,nelffdeiteevdcehtltiavopseimnfbnoeertienaqnglusa(d[7cseri]m-.uocFonionaer--l Energy EFF m -D 0 EF E g EFF m D bEulkg
-
d NTs side-contacted by a metal, modeling has been used W
n to extractSchottky barriersfromexperimentalmeasure- Distance perpendicular to interface Distance perpendicular to interface
o ments of the ON current in NT transistors [5], but have FIG. 1. Panel (a) shows a cross-section of the
c metal-nanostructure contact along the length of the nanos-
not addressed the origin of the Schottky barriers; and
:
v tructure. Panels (b) and (c) show radial cross-sections for
atomistic calculations have provided case-by-case stud-
i metal-nanotube and metal-nanowire contacts. The NT is
X ies [8,9] for planar contacts. However, a more general
separated from the metal by a distance s = 0.3 nm. The
r theoretical understanding for side-contacts to quasi-one- NW is modeled as a solid cylinder with a sharp interface
a dimensional (Q1D) structures is still missing, especially
with the metal. Panel (d) shows the band alignment at a
in light of the recent experimental findings. metal-semiconductorcontactbeforechargetransfer. Inabulk
In this paper, we present a theoretical and modeling contact, panel (e), band bending over a distance W due to
analysisofsidecontactstonanotubesandnanowires. We charge transfer leads to a Schottkybarrier ∆bulk.
show that the conventional concepts developed for bulk
metal-semiconductor contacts do not simply carry over We begin by describing the contact geometry consid-
to the nanoscale. In particular, band re-alignement due ered here. Figure 1a shows a sketch of a cross section
to charge transfer is weak due to the limited available of the contact consisting of a Q1D structure embedded
depletion width. In NTs, this leads to relatively small in a metal. For explicit systems, we consider a single-
1
wall NT , as shown in Fig. 1b, or a solid nanowire as in somewherebetween∆0and∆bulk (foranundopedNTor
Fig. 1c. FortheNT,the metalformsacylindricalcavity NW, ∆ = E /2, which would always give relatively
bulk g
of radius R+s where R is the NT radius and s is the high Schottky barriers).
distance between the NT and the metal, while for the Nanotubes are an extreme example of this situation,
NW we consider a solid, continuum cylinder embedded sincethe possible“depletionwidth”isthesizeoftheNT
in a perfect metal, with a sharp interface between the wall; the charge transfer and image charge in the metal
nanowire surface and the metal. create two nested hollow cylinders with opposite charge,
In the simplest picture, the difference between the and an associated electrostatic potential. This electro-
metal Fermi level E and the semiconductor valence staticpotentialinturnshifts the bands,andchangesthe
F
band edge E (the barrier for holes) is simply given by amount of transferred charge. Thus, the charge and po-
v
(Fig. 1d) tentialmustbedeterminedself-consistently. Wecancap-
ture this behavior using analyticalmodels for the charge
1
∆0 =EF Ev = Eg Φm+Φs (1) and potential. The charge per unit area on the NT can
− 2 − be expressed as
whereΦ andΦ arethemetalandsemiconductorwork-
m s
σ =eN D (E)f(E E )dE (3)
functionsrespectively,andE isthesemiconductorband NT F
g −
gap. A positive value for ∆0 indicates a Schottky bar- Z
rier, while a negative value indicates an ohmic contact. where
Because bandgap decreases with increasing diameter for a√3 E+eV
NT
Q1D structures, the value of ∆0 depends on the nanos- DNT(E)= π2Rγ | 2 | 2 (4)
tructure diameter. The behavior of Eq. (1) for undoped (E+eVNT) (Eg/2)
−
NTsisshowninFig. 2foravalueofΦm ΦNT =0.4eV q
− is the NT density of states [11], f(E E ) is the Fermi
(typical of Pd), and using the relation E = 2aγ/d be- F
g function, and N = 4/(3√3a2) is the−atomic areal den-
tween bandgap and NT diameter d (a=0.142 nm is the
sity. We assume a uniform and sharp distribution of the
C-Cbondlengthandγ =2.5eVisthetight-bindingover-
charge on the NT.
lapintegral). Oneproblemwiththis picture(besidesthe
Forthe geometryof Fig. 1, solutionofPoisson’sequa-
fact that the physics is incomplete, as will be discussed
tion gives the potential on the NT as
below) is that Eq. (1) predicts large and negative values
for ∆0, signaling strong ohmic contacts. However, it is eR R+s e2
clear that such strong ohmic contacts are not observed eVNT = σ ln = σ (5)
− ε0 R −C
experimentally.
Ingeneral,becausethemetalFermilevelsofthemetal whereε0 isthepermittivityoffreespaceandC istheca-
andsemiconductorarenotequal,chargetransferbetween pacitance per unit area between the metal and the NT.
the metal and semiconductor occurs, and leads to band Equations (3) and (5) can be solved self-consistently for
re-alignement. At a bulk semiconductor junction (Fig. a given NT. In this model the electrostatic potential in-
1e) this charge transfer leads to the Schottky barrier duced on the NT modifies the barrier to
∆ =E +χ Φ (2) ∆=∆0 eVNT. (6)
bulk g s −
−
Figure2showsresultsofsuchcalculationsforparameters
where χ is the semiconductor electron affinity. This re-
typical of Pd. Clearly, the behavior is different from the
lationship arises because, in the bulk system, a deple-
simpleexpressionsinEqs(1)and(2). Theresultssuggest
tion width perpendicular to the metal-semiconductorin-
that there is a transition between Schottky and ohmic
terface is created in the semiconductor until the band
behaviorat a NT diameter around1.4 nm, in agreement
lineup in Eq. (2) is obtained. However for Q1D struc-
with recently published experimental data for Pd con-
tures, the depletion width depends exponentially on the
tacts [5,6].
doping [10] and is much longer than the device size for
The results of these calculations can be verified using
non-degenerate doping, leading to slowly varying bands
an atomistic treatment of the NT. For selected zigzag
outside of the contact; and for a three-terminal device
NTs, we apply a tight-binding Green’s function formal-
the band-bending in the channel is governedby the gate
ism, dividing the system into semi-infinite left and right
voltage. In either case, the band alignment is deter-
leadsconnectingacentralscatteringregion(Fig. 1a). In
mined by that in the contact. But for a side-contacted
thecentralregion,wecalculatetheGreen’sfunctionfrom
Q1Dstructure,the semiconductoris onlya few nanome-
ters thick in the direction perpendicular to the metal- GR = (E eV)I H0 ΣR −1, (7)
semiconductor interface; thus only a region of the order − − −
ofthenanostructurecross-sectioncanbedepleted,giving where H0 is the(cid:2)tight-binding Hamilto(cid:3)nian for the iso-
partial band re-alignement. The value of ∆ will then be lated NT and V is the electrostatic potential on each
2
ring of the zigzag NT. The function ΣR represents the σ = eNa√3 EgkTe−βk∆T. (9)
interactionofthe scatteringregionwiththe semi-infinite 2√2βπ3/2Rγ 2
r
NT leads. To obtain the charge density, we assume a
uniform distribution of the charge in the azimuthal di- Combined with Eq. (5) this expression for the charge
rection, and spatially distribute the total chargeon each gives the Schottky barrier
ring in the radial and axial directions with a Gaussion
sdmenesairtiyngisfguinvecntiobny. Thus the three-dimensional charge ∆=∆0+ kTL βα Eg e−βk∆T (10)
β 2kT
r !
e
ρ(r,φ,z)= g(z z ,r R) dEImGR (8)
− − l − π ll where α = e2Na√3 / 2√2βπ3/2RγC and L(x) is
l Z
X
where g(x,y) = 4π2Rσ σ −1exp x2/2σ2 y2/2σ2 Lambert’s W(cid:0)function.(cid:1) A(cid:0) more appealin(cid:1)g formula can
z r − z − r be obtained using asymptotic expansion of L(x) giving
with R the tube radius, z the position of ring l, and σ
l z
(cid:0) (cid:1) (cid:2) (cid:3)
and σ the smearing lengths in the axial and radial di-
rectionrs respectively (this expression for g is valid when kT α 2EkgT
∆= ln . (11)
nRm≫).σr,andweusevaluesofσz =0.14nmandσr =0.06 β lnα 2EkqgT −∆0/kT
 q 
The electrostatic potential is calculated by solving
ThebehaviorofthisfunctionisplottedinFig. 2,showing
Poisson’s equation in cylindrical coordinates on a grid
good agreement with the full calculation. The logarith-
withthegeometryofFig. 1. Oncethe three-dimensional
mic dependence implies relatively slowly varying ∆, at
electrostatic potential is obtained, the value for V on
least compared with Eq. (1). The NT diameter delimit-
each layer is taken as the value of the three-dimensional
ing Schottky from ohmic behavior is [12]
electrostaticpotentialattheatomicpositionofeachring
along the NT. We iterate the calculation of the charge
and the electrostatic potential until self-consistency. kT
d d0 1+α ; (12)
As showninFig. 2, the results ofsuchcalculationsin- ≈ rΦm−ΦNT!
dicate excellent agreement with the analytical approach
introducedabove. This agreementallowsustouseequa- thus the diameter is increased from its bare value by
tions (3) and (5) to gain further understanding of the δd = α Φ −kTΦ d0. Making ohmic contact to a wide
properties of these contacts, as we now discuss. m NT
range oqf NT diameters requires a small δd; this can be

accomplished at low temperature, or with a large metal
0.08 F -F = 0.4 eV workfunction. Because α is inverselyproportionalto the
m NT capacitance,ourresultalsoprovidesone reasonwhyem-
0.06 bedding the NT in the metal and heating can improve
the contact properties: the embedded contact provides
0.04
) Schottky a larger capacitance compared to a planar contact (for
V
e 0.02 typical parameters we find α 2.5 for the embedded
( ≈
contact and up to α 14 for a planar contact), while
Ev 0.00 heating presumably pr≈ovides a smaller average value for
-
F
E s and a smaller capacitance. The large value of α for
-0.02
planar contacts imply that this type of contactis almost
-0.04 Ohmic pathological, since the value of δd should be very large
unless the metal worfunction itself is large. This may
0.8 1.2 1.6 2.0
explain why ab initio calculations find Schottky barriers
Diameter (nm) [8,9] despite the large values of Φ Φ used in the
m NT
−
calculations. It is worth mentioning that changing the
FIG. 2. Schottky barrier at nanotube-metal contacts for
contactgeometryto modify the contactpropertiesisnot
parameters typical of Pd. Dotted line is prediction from Eq.
(1),solidlineiscomputedfromEqs(3)and(5),dashedlineis possible in bulk contacts, andis thus a unique feature of
Eq. (11) and circles are calculated from atomistic approach. nanostructures.
We now consider side-contacts to nanowires. The dif-
To proceed further we focus on the small and posi- ference betweenNWs and NTs is two-fold: first, in NWs
tive∆regime,andapproximatethefermidistributionas the possible depletion width increases with diameter,
1
f(E E ) exp[ β(E E )/kT] where β = 0.7. while in a NT it stays constant; second, while the band
− F ≈ 2 − − F
Asymptotic expansion of the integral in Eq. 3 leads to gapalso decreaseswith diameter for NWs [13], the large
the charge diameter limit can lead to a finite band gap, while for
3
NTs it leads to a zero band gap. It is not clear a priori age of E E (r); the results plotted in Fig. 3a in-
F v
−
how these effects influence the contact behavior. To ad- dicate a remarkable behavior: the contact is always of
dressthisissue,we consideramodelNWwithdensityof Schottky character, with the barrier minimized at a di-
states ameter of about 4 nm. Thus, while in NTs the barrier
heightdecreasesmonotonicallywithdiameter,thebehav-
D (E)= √2m∗ (E E /2)−1/2 (13) iorinotherQ1Dstructuresmaybenon-monotonic,with
NW g
π¯h −
a range of diameters providing optimal contact proper-
∗
where m is the effective mass. For silicon NWs, it has ties.
been shown experimentally [13] that the band gap de- We now discuss the issue of Fermi level pinning. As
pends on diameter as Eg = E0+C/d2 where E0 = 1.12 showninFig. 4a,inabulkmetal-semiconductorcontact,
eV and C = 4.33 eVnm2. We consider the situation metal-induced gap states (MIGS) lead to band-bending
Φm ΦNW = 0.7 eV typical of contacts to Si. Fig. 3a overadistance l W,andmodificationofthe Schottky
show−s the expected Schottky barrier heights from Eq. barrier height to≪∆ . The question is to what extent
pin
(1), which predicts ohmic contacts to NWs with diam- this mechanisminfluences the propertiesofside contacts
eters larger than 4 nm. To study the effects of charge to NTs and NWs. To model this effect, we consider a
transfer, we perform a self-consistent calculation of the radial pinning charge [7]
chargeandpotential,usingEq. (13)toobtainthecharge
andsolvingPoisson’sequationnumericallyintheNWto
σpin(r)=D0NA[EF EN(r)]h(r) (14)
obtain the potential (we use an atomic volume density −
N =5 1028 atoms/m3).
v
× wheretheneutralitylevelE isatmidgap[i.e. E (r)=
N N
eV(r)], h(r) = e−r/l for a NW and h(r) = δ for
0.3 −a NT, and N = N for a NT and N = N2r/,R3 for
A A v
(a) F -F = 0.7 eV a NW. We choose l = 0.3 nm, a typical value for
m s
0.2 metal-semiconductor interfaces [14]. We add this pin-
ning charge to Eq. (3) or to the charge calculated from
)
V
e 0.1 Eq. (13) and repeat our self-consistent calculations.
(
E v Schottky Figure 4b shows the Schottky barrier calculated for
-F 0.0 several NTs as a function of the density of gap states
E
Ohmic (∆ = E /2). Clearly, there is a rapid onset of pin-
pin g
-0.1 ning at D0 ∼ 0.1 states/(atom·eV); this value of D0
is rather larger considering the van der Waals bonding
of NTs to metals. Furthermore, atomistic calculations
-0.2
0 2 4 6 8 10 [8] have obtained values at least an order of magnitude
Diameter (nm) smaller. Thus,asinend-bondedcontacts,weexpectthat
Fermilevelpinningwillplayaminorroleinside-contacts

1.8 (b) 0.6 (c) toNTs. TheratherlargevalueofD0 requiredtoseepin-
1.2 0.3 ning effect is a generic property of Q1D structures as we
V) 0.6
ergy (e -00..60 -00 ..03 noFwigduisrceus4sc. shows the effects of Fermi level pinning
En -1.2 -0.6 on the barrier height in SiNWs. The results also in-

-1.8 -0.9
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 dicate a value of D0 0.1 states/(atomeV) required
r/R r/R to see pinning effects.∼ For comparison,·the inset in
this figure shows the same calculation for a bulk metal-
FIG.3. Panel(a): Schottkybarrieratnanowire-metalcon-
semiconductor interface with the same parameters, in-
tacts for parameters typical of SiNWs. Dotted line is predic-
tion from Eq. (1) and solid line is self-consistent calculation. dicating that only 0.002 states/(atomeV) are needed to
·
Panels (b) and (c): Band-bending across nanowires with di- reach the onset of pinning. Thus, the Q1D system re-
ameters of 2 nm and 10 nm, respectively; dotted lines are quires almost two orders of magnitude larger density of
Fermi level. pinning states compared with the bulk interface.
Theoriginofthisbehaviorcanbetracedtotheunique
Fig. 3b,c shows the self-consistent band-bending for
density of states of Q1D systems. Indeed, for Si, we
NWs of 2 and 10 nm radius. Clearly, the nanoscale
can repeat the analysis leading to Eq. (9) using the
dimension of the NWs prevents the bands from reach-
density of states for the NW and for the bulk system
ing their asymptotic value; instead, there is only a rel- [D (E) = √2(m∗)3/2 π2¯h3 −1 E E /2]. This
atively weak band-bending present. To quantify the bulk − g
Schottky barrier height, we calculate the spatial aver- leads to the ratio σNW/σb(cid:0)ulk =(cid:1)2πpNv1/3β /(m∗kT).
(cid:16) (cid:17)
4
metal semi conductor
(a) c V) 0.5 0.5
e
(
y 0.0 0.0
F W g
m er
n -0.5 -0.5
E E
g
E -1.0(a) -1.0 (b)
F D -20 -10 0 10 20 -4 -2 0 2 4
pin D r (nm) r (nm)
bulk
l << W
100
e
Distance perpendicular to interface c
n
a
st
1.0 (b) pinning esi
0.8 d = 1 nm d r 10
d = 1.4 nm e
z
0.6 d = 1.8 nm ali
d = 2.2 nm m
pin 0.4 or (c)
D/ N 1
D 0.2 5 10 15 20 25 30 35 40
Diameter (nm)
0.0
FIG. 5. Band-bending across Si NWs with doping of
-0.2 1019e/cm−3 for diameters of 40 nm (a) and 10 nm (b). The
10-3 10-2 10-1 100 arrow indicates tunneling of electrons throught the Schottky
D (states/eV/atom) barrier. Thenormalized resistance is shown in figure(c) as a
0 function of NW diameter.

1.2
(c) Our discussion has so far focused on the situation of
pinning
1.0 lowdoping,wherethestrategyformakingohmiccontacts
1.0
0.8 is byselectionofametalwith appropriateworkfunction.
0.8 0.6 In traditional metal-semiconductor contacts, an alterna-

0.4 tive approach is to heavily dope the semiconductor, and
pin 0.6 0.2 relyontunneling throughthe Schottky barrierto reduce
D/ 0.10E-5 1E-4 1E-3 0.01 0.1 1 thecontactresistanceandobtainohmic-likecontatcs. To
0.4
D address the feasibility of this approach for contacts to
0.2 NWs, we repeat our self-consistent calculations for the
d = 2.5 nm
Si NW, focusing on the situation where the metal Fermi
d = 4 nm
0.0 d = 6 nm level is in the middle of the NW bandgap at the inter-
face,andaddingauniformdopingchargeof1019e/cm−3.
10-3 10-2 10-1 100
D (states/eV/atom) Figure 5 shows the band-bending in the presence of this
0
doping charge for NWs of 40 and 10 nm diameters. We
FIG. 4. Panel (a) shows a sketch of the band-bending
calculate the contact conductance from
in the presence of metal-induced gap states at a bulk
∞
metal-semiconductor interface. Panels (b) and (c) show the ∂f
G T(E) dE (15)
cstaalctueslaftoerdseSvcehroatltNkyTbsaarnriderNaWsas,fruenscpteicotnivoeflyt.heTdheenisnisteytoifng(acp) ∼ZEcmin (cid:18)−∂E(cid:19)
showsthebehaviorforaplanarmetal-semiconductorcontact. where the tunneling probability T(E) is obtained from
the WKB approximation. The normalized contact resis-
tanceisthenG∞/GwhereG∞ istheconductanceinthe
limit of large diameters. The behavior of the normalized
TheappearanceofthekT factorinthedenominatoris resistanceasa function ofNW diameter is showninFig.
entirely due to the Q1Ddensity of states of the NW and 5c, indicating a rapid increase of the resistance with de-
the presence of a van Hove singularity at the band edge. crease in diameter. The origin of this behavior is due
At room temperature, we find that σ /σ > 100; to the increasedtunneling distance andreducedrangeof
NW bulk
thus the MIGS are competing with a much largercharge tunneling energies because of the poor band-bending in
density in the Q1D system. the NW (the increase in bandgap also plays a role for
5
the smaller diameters). One implication of this result is [1] Heinze, S.; Tersoff, J.; Martel, R.; Derycke, V.; Appen-
thatdifferentdiameterNWswillrequiredifferentdoping zeller,J.;Avouris,Ph.Phys.Rev.Lett.2002,89,106801.
levels to achieve the same contact quality. [2] Yaish, Y.; Park, J.-Y.; Rosenblatt, S.; Sazonova, V.;
Insummary,wefindthattheconceptsdevelopedtode- Brink, M.; McEuen, P. L. Phys. Rev. Lett. 2004, 92,
046401.
scribe traditional metal-semiconductor interfaces fail to
[3] Javey,A.;Guo,J.; Wang,Q.;Lundstrom,M.; Dai,H.J.
properly account for the properties of contacts to quasi-
Nature (London) 2003, 424, 654.
one-dimensional structures such as nanowires and nan-
[4] Javey, A.; Guo, J.; Former, D. B.; Wang, Q.; Wang, D.
otubes. The nanometer cross-section of nanowires and
W.;Gordon,R.G.;Lundstrom,M.;Dai,H.J.NanoLett.
nanotubes prevents the formation of an appropriate de-
2004, 4, 447.
pletion width, leading to only weak band re-alignement; [5] Chen, Z.; Appenzeller, J.; Knoch, J.; Lin, Y.-M.;
and because of the diverging density of states at the Avouris, Ph. Nano Lett. 2005, 5, 1497.
band edge, these nanostructures are much less sensitive [6] Kim, W.; Javey, A.; Tu, R.; Cao, J.; Wang, Q.; Dai, H.
to Fermilevelpinning thantheir bulk counterparts. Op- Appl. Phys. Lett. 2005, 87, 173101.
timizing device performance will not only require select- [7] L´eonard, F.; Tersoff, J. Phys. Rev. Lett. 2000, 84, 4693.
ingQ1Dstructuresfortheirbehaviorinthechannel,but [8] Xue, Y.; Datta, S.Phys. Rev. Lett. 1999, 83, 4844.
[9] Shan, B.; Cho, K. Phys. Rev. B2004, 70, 233405.
also for their contact properties. We expect that our
[10] L´eonard, F.; Tersoff, J. Phys. Rev. Lett. 1999, 83, 5174.
results will be applicable to a broadrange of Q1Dstruc-
[11] Mintmire, J. W.; White, C. T. Phys. Rev. Lett. 1998,
tures.
81, 2506.
∗ [12] To derive this relation we assumed a constant capaci-
email:fl[email protected]
tanceovertherangeofNTdiametersofinterest.Amore
Acknowledgement. Sandia is a multiprogram labo-
detailed treatment would not changethe keyphysics.
ratoryoperatedbySandiaCorporation,aLockheedMar-
[13] Ma, D.D.D.;Lee,C.S.;Au,F.C.K.;Tong,S.Y.;Lee,
tin Company, for the United States Department of En- S. T. Science 2003, 299, 1874.
ergy under contract DE-AC01-94-AL85000. [14] Louie,S.G.;Cohen,M.L.Phys. Rev. B 1976,13,2461.
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