Uniform Peak Optical Conductivity in Single-Walled Carbon Nanotubes

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by  Jesse M. Kinder

Uniform Peak Optical Conductivity in Single-Walled Carbon Nanotubes
Jesse M. Kinder∗ and Garnet Kin-Lic Chan
Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853
Jiwoong Park
Department of Chemistry and Chemical Biology,
Cornell University, Ithaca, New York 14853 and
Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853,USA.
1 (Dated: January 19, 2011)
1
0 Recent measurements in single-walled carbon nanotubes show that, on resonance, all nanotubes
2 displaythesamepeakopticalconductivityofapproximately8e2/h,independentofradiusorchiral-
ity[Johetal.,NatureNanotechnology 6,51(2011)]. Weshowthatthisuniformpeakconductivityis
n
a consequenceof therelativistic band structure and strength of the Coulomb interaction in carbon
a
J nanotubes. Wefurtherconstructaminimalistmodelofexcitondynamicsthatdescribesthegeneral
phenomenologyandprovidesan accuratepredictionofthenumericalvalueofthepeakoptical con-
8
ductivity. Theworkillustratestheneedforcarefultreatmentofrelaxationmechanismsinmodeling
1
theoptoelectronic properties of carbon nanotubes.

]
l
l
a Usinganewon-chipRayleighscatteringtechnique,Joh modelofacarbonnanotube. Ourmainresultisthatthe
h et al. measured the peak optical conductivity for a vari- peak conductivity will be independent of the nanotube
-
s ety of transitions in semiconducting and metallic nan- radiusRwheneverquasiparticleenergiesareproportional
e otubes [1, 2]. The data reveal a surprising phenomenon: to 1/Rand quasiparticlelifetimes areproportionalto R.
m
on resonance, the optical conductivity is independent of TheCoulombinteractionsatisfiesbothrequirements,and
. the nanotube radius and narrowly distributed around a simple exciton model with interbandCoulombscatter-
t
a σ = 8e2/h. On resonance, all nanotubes respond like ing fits the data quite well. We also consider the effects
m ∗
classicalconducting hollow cylinders with the same con- of phonon and impurity scattering, which may account
- ductivity. The sample included the second, third, and for some of the spread in the data. The analysis sug-
d
n fourth exciton transitions in semiconducting nanotubes geststhatauniformpeakconductivityshouldbeobserv-
o (S22, S33 andS44)andthe firstandsecondexcitontran- able over a wide range of experimental conditions and
c sitions in metallic nanotubes (M11 and M22). The data illustrates the importance of relaxation mechanisms in
[
are shown in Fig. 1. numerical models of nanotube properties.

1 In this Letter we identify the origin of this uniform Our starting point is the low-energy approximationto
v peak conductivity. We analyze the optical conductiv- the tight-binding Hamiltonian for a carbon nanotube,
6
ity within linear response theory for the effective Dirac which is a massless Dirac equation [3]:
4
5
3 =~vF q σ. (1)
H ·
.
1
v is the Fermi velocity and q describes small displace-
0 F
4
1 3 mentsfromthecornersoftheBrillouinzoneofgraphene.
2
1 1" R is the only relevant length scale, and it defines a nat-
v: /%0 ural energy scale:
,
i .-
X +,- E0 =~vF/R. (2)
r *
a () This defines aneffective massas well: E0 =m∗vF2. The
&’ eigenvalues of are E0p(kR)2+∆2, where k is the
H ±
$% wave vector along the nanotube axis and ∆ is the gap
#
" parameter. In metallic nanotubes, ∆ = N, and in semi-
!
conducting nanotubes,∆=N 1/3. The bandindex N
±
is an integer.
5-#6","7%0(64
Thisfreeparticlemodelcanbeextendedtoincludethe
Coulombinteraction[4]. Inasingle-bandmodel, exciton
wave functions are of the form
FIG. 1: Optical conductivity on resonance as a function of
nvaalnuoetuisb8e±dia1m.5eet2e/rh,,asinmdeepaseunrdeedntbyofJtohhe,detiaaml.e[t2e]r..Themean |ni=XAn,k·c†kvk|Ωi. (3)
k
2
c†createsaconductionelectron,v†createsavalenceelec- resonance, w is a function of the other parameters so
tron,and Ω isthegroundstateoffilledvalenceorbitals. that σ = G (λ) e2/h. If x and y are independent
∗ ∗ n n
| i ·
The coefficients A and exciton energies E are ob- ofthe nanotube radius, then so is the peak conductivity.
n,k n
tainedaseigenvectorsandeigenvaluesofaBetheSalpeter Eq. (5) implies that x and y are independent of the
n n
equation. Thissingle-bandmodelissufficienttodescribe nanotube radius if E and Γ are proportional to 1/R.
n n
directexcitons(S andM )incarbonnanotubes[5],and This proves our central result:
ii ii
the scaling of exciton size and binding energy obtained If quasiparticle energies are inversely proportional to
from this model and ab initio calculations agree to lead- the nanotube radius and quasiparticle lifetimes are pro-
ing order in R [6, 7]. portionaltothenanotuberadius, thentheconductivityon
Atthe leveloflinearresponsetheory,the conductivity resonance is independent of the nanotube radius.
is given by the Kubo formula: Theargumentsofthis sectioncanbeextendedtomul-
~e2 n vˆ Ω 2 2(~ω+i~Γ ) tiplebandsandindirectexcitations. Theresultalsoholds
n
σ(ω)= i2πRLX|h |E| i| E 2 (~ω+i~Γ )2. for unbound electron-hole pairs [18].
n n n
n − The first requirement is satisfied as long as Eq. (1) is
(4)
valid. If the Dirac equation is scaled by E0, then k only
vˆ is the velocity operator projected along the nanotube
enters in the dimensionless combination kR. The scaled
axis. Γ = 1/τ where τ is the lifetime of the excited
n n n Coulomb interaction only depends on an effective fine
state. If |ni were eigenstates of the Hamiltonian in the structure constant, α = e2/2πκ~vF where κ describes
absence of an applied field, Γ could be replaced by an
n staticscreeningfromtheenvironmentandotherbandsof
infinitesimal to enforce causality. In practice, the exact
the nanotube. As a result, the scaled Hamiltonian only
eigenstates are never known. A broadening parameter
depends on the dimensionless variables ξ = kR and α.
1/τ is often introduced to account for relaxation. How-
Thespectrumandeigenvectorsareuniversalfunctionsof
ever, τ is not arbitrary: different scattering mechanisms
these variables, and the energy eigenvalues for both free
leadtoqualitativelydifferent lifetimes, andthe broaden- particles and excitons are proportional to ~v /R. This
F
ing determines the optical conductivity on resonance.
inverse relation between the energy and the radius is a
Eq.(4)givestheconductivityofasingleband. Theto-
consequenceoftherelativisticbandstructureofacarbon
tal conductivity is multiplied a factor of 2 for spin and a
nanotube,andtheconditionsthatleadtoauniformpeak
factorof2fortheKandK′points. Althoughtheconduc-
conductivity in carbon nanotubes are not satisfied in a
tivity is a tensor, scattering in nanotubes is dominated
general quantum wire.
by light polarized along the nanotube axis [8]. Only the
The second requirement is not satisfied in general.
surface current along the axis due to an electric field ap-
Quasiparticle lifetimes arise from interactions not in-
plied along the axis is considered here: σ(ω) σ (ω).
≡ zz cluded in the Dirac equation, and each interaction must
The Kuboformulamaybe simplifiedbyscalingallen-
beanalyzedseparatelytodeterminewhetherthelifetime
ergiesby E0 andrewritingthe expressionintermsofthe
is proportionalto R. The lifetime of a state n due to a
dimensionless parameters | i
potential V can be estimated from Fermi’s golden rule:
w =~ω/E0,
xn =En/E0, (5) Γn =(2π/~)X|Vm,n|2·δ(Em−En). (8)
yn =~Γn/E0. m
All of the dimensional quantities can be collected into
Eq. (4) may be simplified further by introducing a di-
a base scattering rate Γ0. The remaining sum defines
mensionless oscillator strength [4]:
a dimensionless function γ whose analytic form is irrel-
n vˆ Ω 2 evant to the main result. The scattering rate is then
fn =2m∗|h |En| i| . (6) γ · Γ0. If Γ0 ∝ 1/R and γ is independent of R, the
quasiparticle lifetime will satisfy the requirements for a
The oscillator strengths for the solutions of the Dirac
radius-independent peak conductivity.
Hamiltoniansatisfyasumrule[17]: f =L/πR. For
Pk k TheCoulombinteractionlargelydeterminesthephoto-
anexciton,f =φ L/πR,wherethefractionaloscillator
n n
· physicsofcarbonnanotubes. Theabsorptionofaphoton
strength φ is independent of R.
n
producesaparticle-holepair,andthesechargedparticles
After these substitutions, Eq. (4) becomes
interact strongly through their mutual Coulomb attrac-
e2 1 w+iy tion. The interaction between particles and holes in the
n
σ(ω)= h iπ Xφn· x 2 (w+iy )2. (7) same band leads to strong exciton binding. Interband
n n
n −
scatteringgivesthe excitonafinite lifetime[9]. Thebase
This expression implies σ(ω) = G(w;λ) e2/h, where rate is
G is a dimensionless function and λ is a s·et of dimen-
sionless parameters derived from xn and yn . On ~Γ0 =α2E0, (9)
{ } { }
3
and γ is independent of R. Thus, interband Coulomb orientation of the bonds with respect to the axis of a
scattering leads to a peak conductivity that is indepen- nanotubedependsonitschirality,andphononscattering
dent of the radius. rates depend on the chiral angle θ .
c
In fact, an exciton model with dissociation due to in- There are four acoustic modes in carbon nanotubes.
terband Coulomb scattering accounts for all of the qual- The longitudinal acoustic (LA) modes compress and ex-
itative features of the data: the peak conductivity is in- pandthelatticealongthenanotubeaxis. Thetransverse
dependentoftheradius,doesnotdependstronglyonthe acoustic(TA)ortwistmodesrotatethelatticeaboutthe
transition responsible for the resonance, and is approxi- nanotubeaxis. Bothofthesemodeshavealineardisper-
mately equal in semiconducting and metallic nanotubes. sion relation: ω = cq where c is the sound velocity.
q
Dissociationratesshowlittle variationbetweenbands. These modes do not carry any angular momentum and
The lifetime ofanexcitonis approximatelyequaltothat lead to small-momentum scattering within a band. The
of a particle-hole pair at the band edge [9]. For the S33, two flexure modes bend the nanotube in the plane of its
S44, M11, and M22 excitons, γ falls between 0.7 and 1.2. axis. These modes have a quadratic dispersion relation:
TheestimatedscatteringrateoftheS22 excitonissignif- ωq = cq2R where c has units of velocity. These modes
icantly larger because of the small overlap of wave func- carry one quantum of angular momentum and mediate
tions in the first and second bands [19]. interband transitions.
The bands in metallic nanotubes are twofold degen- Aradialbreathingmode(RBM)isaperiodicvariation
erate, which suggests the peak conductivity of metallic intheradiusofananotubealongitsaxis. Itsfrequencyis
nanotubes could be more than twice that of semicon- inversely proportional to the nanotube radius. At small
ducting nanotubes. However, screening in metallic nan- wave vector, ω c/R, where c has units of velocity. In
q
≈
otubes reduces the exciton binding energy and oscillator nanotubewitha1nmdiameter,the energyofthelowest
strength. Trigonal warping lifts the degeneracy and fur- RBM is about 27 meV [16].
therreducesthe conductivity inchiralnanotubes. These The scatteringrates ofthe RBM, LA, TA, and flexure
factors lead to similar peak conductivities for semicon- modes all have the same form. At zero temperature,
ducting and metallic nanotubes.
g2
accDoiussnotcsiafotriotnhdeumeatgoniinttuedrebaonfdthCeoupleoamkbcosncadtutcetriivnigtya.lsIof ~Γ0 = ρ0cvFR2. (11)
alltheoscillatorstrengthofabandislocalizedinasingle In the high-temperature limit,
transition and γ 1, the peak conductivity is
≈ g2k T
σ = e2 2κ2 , (10) ~Γ0 = ~ρ0c2BvFR. (12)
∗ h · πα02
g characterizesthe electron-phononinteractionstrength,
where α0 = e2/2π~vF 0.42, and κ is the dielectric T is the temperature, and ρ0 is the mass density of the
≈
constantoftheenvironment. Theexperimentalmeasure- graphenelattice. Although a single parameterg appears
ments shown in Fig. 1 were taken on a quartz substrate above,thedeformationpotentialisanorderofmagnitude
2
in glycerol (n=1.46). Setting κ= n gives a maximum larger than the gauge potential [15]. As a result, the
conductivity of σ 16e2/h. A peak conductivity of LA and RB modes, which contribute to the deformation
∗
8e2/h follows if the≈exciton transition accounts for half potential, generally have largerscattering rates than the
thetotaloscillatorstrength,afractionconsistentwithab othermodes. γ depends onc/v andθ ,but notR. The
F c
initio calculations [6]. contribution of these modes to the peak conductivity is
Other interactions also contribute to the exciton life- independent of the radius in the high-temperature limit.
time andmayaccountfor someofthe spreadinthe data Incontrast,opticalmodesdonot contributetoauniform
of Fig. 1. Intrinsic sources of scattering include phonons peak conductivity.
and lattice defects. External perturbations such as the Analyzing variations in the peak conductivity with
substrate, applied fields, or atoms adsorbed on the sur- temperature would provide a way to extract the phonon
face of the nanotube also affect the conductivity. Here contribution. The contributionof the RBM should show
we consider phonon and impurity scattering to illustrate a crossover to radius-dependent scaling as the temper-
howdifferentscatteringmechanismsleadtoqualitatively ature is reduced, and the contribution of the acoustic
different quasiparticle lifetimes. modes should scale linearly with temperature.
Electron-phonon interactions are a significant source Impurities in the nanotube or its environmentprovide
of scatteringin carbonnanotubes [10–14]. Lattice defor- another source of scattering. A short-range potential lo-
mations introduce an effective potential to Eq. (1). A calizedonthesurfaceofananotubecanbeapproximated
general deformation has two effects [3, 15]. First, varia- by a point-like impurity potential: V(r) V0a2δ(r),
≈
tions in the lattice charge density can produce a scalar where a . R is the range of the potential. This might
deformation potential. Second, bending and stretching represent a topological defect in the lattice, a substitu-
of bonds canintroduce an effective gaugepotential. The tion impurity at a lattice site, or an atom adsorbed on
4
the surface of the nanotube [20]. Incoherentelastic scat- areproportionalto the radius. The Coulombinteraction
tering from identical impurities gives satisfiesboth requirementsandexplains boththe unifor-
mity and mean value of the conductivity data in Fig. 1.
~Γ0 = ρ~av4VR02, (13) Phonon, impurity scattering, and trigonal warping may
F account for small variations between nanotubes.
where ρ is the surface defect density. Our analysis suggests the peak conductivity will be
A long-range potential that is nearly uniform around uniform overa wide rangeofexperimentalconditions: it
the circumference of a nanotube but localized along its will be the same for all nanotubes in a sample, indepen-
axismaybeapproximatedbyaone-dimensionalimpurity dent of diameter or chirality. The peak conductivity is
potential: V(r) aV0δ(z), where V0 is the average po- not universal, however, and may depend on factors such
≈
tentialaroundthecircumference. Thismightrepresenta as the dielectric environment, temperature, or doping.
chargedefectinthesubstrateoralocalgatingpotential. A uniform peak conductivity could be useful in opti-
Theresultingscatteringrateisproportionaltothelinear cal devices that utilize carbon nanotubes. Many proper-
defect density and independent of R. ties of a nanotube depend strongly on its radius or chi-
Elastic scattering from short-rangeimpurities leads to rality. Applications designed to exploit these properties
apeakconductivitythatisindependentoftheradius;in- are faced with the difficult task of separating nanotubes
elastic scattering and long-range impurity scattering do based on their geometry. In applications that only de-
not. The high mobilities observedin DC transport mea- pend on the resonant conductivity, nanotubes would be
surementssuggestthatimpurityscatteringofanytypeis interchangeable: on resonance, all nanotubes behave as
insignificant compared with phonon scattering [10, 11]. classical wires with the same conductivity.
The lifetime due to phonon and impurity scattering Our analysis also illustrates the importance of the
in metallic nanotubes is twice as large as that in semi- broadening term in numerical studies: a scattering rate
conducting nanotubes. Phonon and impurity scattering inversely proportional to the nanotube radius is essen-
alsoleadto strongerdependence onthe bandindex than tial to reproduce the uniform conductivity observed by
Coulomb scattering. No general scaling arguments re- Joh et al. In calculations of nanotube properties, it is
quire the peak conductivity to be independent of the common to introduce a phenomenological parameter to
band index. Although ∆ E0 defines a natural energy account for scattering mechanisms not included in the
·
scale for a band, the peak conductivity depends on the model. If the same parameter is used for different nan-
gap parameter even if En and Γn are proportional to otubes, calculations will yield incorrectscaling relations.
∆ E0. A simple model illustrates this. The peak con- Tofitadataset,aseparateparametercouldbe adjusted
·
ductivity can be calculated analyticallyin a free particle for each nanotube. The experiments of Joh, et al. and
modelwithaconstantscatteringrate. IfΓ0 =c∆/R,the the analysis above suggest a different approach: for a
peak conductivity is independent of the radius but pro- given set of experimental conditions, a single parameter
portional to 1/∆. For phonon and impurity scattering, can describe all nanotubes in a sample if the lifetime is
the lifetime is independent of ∆, and the peak conduc- proportional to the nanotube radius: τ =τ0 (R/R0).
tivity is proportional to 1/√∆. ·
J.M.K.wouldliketothankDanielJohandLihongHer-
Phononandimpurityscatteringmayaccountforsome
mann for many useful discussions of the experiment.
of the spread in the data of Fig. 1. Experimental error
Thisworkwasfundedbythe CornellCenterforMate-
accounts for some variation. Acoustic phonon scattering
rials Research, the Center for Molecular Interfacing, the
and short-rangeimpurity scattering would introduce de-
NSF CAREER grant, the Air Force Office of Scientific
pendenceonthechiralangle;opticalphonons,long-range
Research (NE and IO), the Camille and Henry Dreyfus
impurities, and inelastic scattering would introduce de-
Foundation, the David and Lucile Packard Foundation,
pendence on the radius. Other factors such as doping
and the Alfred P. Sloan Foundation.
or fluctuations in the local dielectric environment might
also play a role.
Trigonal warping may also account for some variation
betweennanotubes. TheDiracHamiltonianisagoodap-
proximation for a nanotube whose radius is much larger ∗ Electronic address: [email protected]
thanthelatticespacing. Whentheradiusissmall,trigo- [1] D. Y. Joh, L. H. Herman, S.-Y. Ju, J. Kinder,
nal warping could lead to variations in the peak conduc- M. A. Segal, J. N. Johnson, G. K.-L. Chan,
tivity throughits effect on scatteringrates, analogousto and J. Park, Nano Lett. 11, 1–7 (2011),
http://pubs.acs.org/doi/pdf/10.1021/nl1012568, URL
the family behavior of exciton binding energies [5].
http://pubs.acs.org/doi/abs/10.1021/nl1012568.
To summarize, linear response theory predicts a peak
[2] D. Y. Joh, J. Kinder, L. H. Herman, S.-Y. Ju,
conductivity that is independent of the nanotube ra-
M. A. Segal, J. N. Johnson, G. K.-L. Chan, and
diuswheneverquasiparticleenergiesareinverselypropor- J. Park, Nature Nanotech. 6, 51 (2011), URL
tional to the nanotube radius and quasiparticle lifetimes http://dx.doi.org/10.1038/nnano.2010.248.
5
[3] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 78, 1932 [14] Y. Miyauchi and S. Maruyama, Phys. Rev. B 74, 35415
(1997). (2006).
[4] T. Ando,J. of thePhys. Soc. of Jpn.66, 1066 (1997). [15] H. Suzuura and T. Ando, Phys. Rev. B 65, 235412
[5] M. Dresselhaus, G. Dresselhaus, R. Saito, and A. Jorio, (2002).
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[6] V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev. properties of carbon nanotubes (Imperial college press
Lett.92, 257402 (2004). London, 1999).
[7] R. Capaz, C. Spataru, S. Ismail-Beigi, and S. Louie, [17] ThisfollowsfromEq.(6)whenthesumoverkisapprox-
Phys.Rev.B 74, 121401 (2006). imated by an integral.
[8] M.Islam,D.Milkie, C.Kane,A.Yodh,andJ.Kikkawa, [18] Although the fractional oscillator strength is no longer
Phys.Rev.Lett. 93, 37404 (2004). proportional to L/R, transforming the sum in Eq. (4)
[9] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 90, 207401 into an integral over k leads to the same dependence on
(2003). w and λ.
[10] J. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, [19] The single S22 transition in Fig. 1 has a peak conduc-
H. U¨stu¨nel, S. Braig, T. Arias, P. Brouwer, and tivity of 10.8e2/h. This is smaller than predicted bythe
P.McEuen, NanoLett. 4, 517 (2004). Dirac model, but corrections will be significant because
[11] X.Zhou,J.Park,S.Huang,J.Liu,andP.McEuen,Phys. the nanotuberadius is so small (0.35 nm).
Rev.Lett. 95, 146805 (2005). [20] Apotentialthataffectsthetwosublatticesdifferentlycan
[12] M. Dresselhaus, G. Dresselhaus, R. Saito, and A. Jorio, be modeled by replacing V0 with V0nˆ·σ. This does not
Phys.Rep.409, 47 (2005). affect the base scattering rate Γ0.
[13] V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev.
Lett.94, 27402 (2005).