Theory of Exciton Energy Transfer in Carbon Nanotube CompositesA. H. Davoody,*,† F. Karimi,† M. S. Arnold,‡ and I. Knezevic*,†

†Department of Electrical and Computer Engineering, University of Wisconsin−Madison, Madison, Wisconsin 53706-1691, UnitedStates‡Department of Materials Science and Engineering, University of Wisconsin−Madison, Madison, Wisconsin 53706-1691, UnitedStates

ABSTRACT: We compute the exciton transfer (ET) rate betweensemiconducting single-wall carbon nanotubes (SWNTs). We showthat the main reasons for the wide range of measured ET ratesreported in the literature are (1) exciton confinement in localquantum wells stemming from disorder in the environment and (2)exciton thermalization between dark and bright states due to intratubescattering. The SWNT excitonic states are calculated by solving theBethe−Salpeter equation using tight-binding basis functions. The ETrates due to intertube Coulomb interaction are computed via Fermi’sgolden rule. In pristine samples, the ET rate between parallel(bundled) SWNTs of similar chirality is very high (∼1014 s−1), whilethe ET rate for dissimilar or nonparallel tubes is considerably lower(∼1012 s−1). Exciton confinement reduces the ET rate between same-chirality parallel SWNTs by 2 orders of magnitude but haslittle effect otherwise. Consequently, the ET rate in most measurements will be on the order of 1012 s−1, regardless of the tuberelative orientation or chirality. Exciton thermalization between bright and dark states further reduces the ET rate to ∼1011 s−1.The ET rate also increases with increasing temperature and decreases with increasing dielectric constant of the surroundingmedium.

1. INTRODUCTION

Carbon nanotubes (CNTs) are quasi-one-dimensional materialswith a unique set of optical and electronic properties.1,2 Today,there is considerable interest in semiconducting CNTs as thelight-absorbing material in organic solar cells, owing to theirtunable band gap, excellent carrier mobility, and chemicalstability.3−9 Improving the efficiency of CNT-based photovoltaicdevices is possible through understanding the dynamics ofexcitons in CNT composites.While the intratube dynamics of excitons in CNTs have been

studied extensively over the past decade,10−18 the intertubeexciton dynamics remain relatively unexplored, owing to thedifficulties in sample preparation and measurements. Thefluorescence from bundled single-wall carbon nanotube(SWNT) samples is quenched and the absorption spectra arebroadened as a result of Coulomb interaction between SWNTswith various chiralities.19 The exciton lifetimes in isolated andbundled SWNTs were measured to be on the order of 100 ps and1 to 2 ps, respectively,15 which underscores the importance ofintertube Coulomb interactions in the dynamics of excitons inSWNT aggregates.There have been a number of measurements of the exciton

transfer (ET) rates in CNTs, but the reported rates differ widely,within 2 orders of magnitude.20 Pump−probe (PP) spectroscopymeasurements have shown a time constant of ∼0.37 ps for theET process from semiconducting SWNTs to metallic SWNTs.21

Time-resolved photoluminescence (PL) spectroscopy found thetime constants of 0.9 and 0.5 ps for ET from semiconducting

SWNTs to metallic and semiconducting SWNTs, respectively.22

In another study, time-resolved PL spectroscopy has been usedto measure ET between semiconducting SWNTs, with a timeconstant τ ≈ 70 ps.23 Qian et al. used spatial high-resolutionoptical spectroscopy to estimate the time constant of ETbetween two nonparallel semiconducting SWNTs as τ≈ 0.5 ps.24

Using PP spectroscopy, the ET time constant between bundledSWNTs was measured to be τ ≈ 10 fs for S11 excitons.

25,26 Thesame study estimated the S22 ET to be very slow due tomomentum mismatch. Another study showed a long-range fastcomponent (τ ≈ 0.3 ps), followed by a short-range slowcomponent (τ ≈ 10 ps) for the ET process in SWNT films.27

Grechko et al. used a diffusion-based model to explain theirmeasurement of ET in bundled semiconducting SWNTsamples.28 They found the time constants of τ ≈ 0.2 to 0.4 psfor ET between bundles of SWNTs and τ ≈ 7 ps for ET withinSWNT bundles. A more recent study by Mehlenbacher et al.revealed ultrafast S22 ET.

29

Many difficulties inherent in experiments can be avoided in atheoretical study of the transfer process; however, to date therehave been only two theoretical studies of ET betweensemiconducting SWNTs.30,31 Wong et al. showed that theideal dipole approximation (known as the Forster theory)overestimates the ET rate by 3 orders of magnitude.30 Postupna

Received: April 21, 2016Revised: June 23, 2016Published: June 24, 2016

Article

pubs.acs.org/JPCC

© 2016 American Chemical Society 16354 DOI: 10.1021/acs.jpcc.6b04050J. Phys. Chem. C 2016, 120, 16354−16366

et al. showed that exciton−phonon coupling could have aprominent effect on the ET process between (6,4) and (8,4)SWNTs;31 however, these studies did not account for someimportant parameters, such as the existence of low-lying opticallydark excitonic states, chirality and diameter of donor andacceptor CNTs, temperature, confinement of excitons, screeningdue to the surrounding medium, and the interaction betweenvarious exciton subbands, all of which have been shown,experimentally and theoretically, to play an important role inexciton dynamics in CNTs.11,12,27,32,33

We present a comprehensive theoretical analysis of Coulomb-mediated intertube exciton dynamics in SWNT composites, inwhich we pay attention to the complex structure of excitonicdispersions, exciton confinement, screening due to surroundingmedia, and temperature dependence of the ET rate. We solve theBethe−Salpeter equation in the GW approximation in the basisof single-particle states obtained from nearest-neighbor tightbinding to calculate the exciton dispersions and wave functions.We calculate the intertube ET rate due to the Coulombinteraction between SWNTs of different chiralities andorientations. For the sake of brevity, in the rest of this paper,we refer to SWNTs simply as carbon nanotubes unless otherwisenoted.We show that momentum conservation plays an important

role in determining the ET rate between parallel CNTs ofdifferent chirality. While the ET rate between similar-chiralitybundled parallel tubes in pristine samples is ∼1014 s−1,34 muchhigher than between misoriented or different-chirality CNTs(∼1012 s−1), exciton confinement due to disorder stronglyreduces the ET rate between parallel tubes of similar chirality buthas little effect on the ET rate otherwise. Consequently, the ETrate dependence on the orientation of donor and acceptor CNTis not as prominent as previously predicted, and the ET rate isinstead expected to be isotropic and 1012 s−1 in mostexperiments.34 Moreover, the ET rate drops by about 1 orderof magnitude if intratube exciton scattering between bright anddark excitonic states is allowed. Our study shows that the transferfrom S22 to S11 excitonic states happens with the same rate as thetransfer process between same sub-band states (S11→ S11 and S22→ S22). The ET rate increases with increasing temperature. Wealso show that the screening of the Coulomb interaction by thesurrounding medium reduces the transfer rate by changing theexciton wave function and energy dispersion as well as byreducing the Coulomb coupling between the donor and acceptorCNTs.The rest of this paper is organized as follows. In Section 2, we

provide a summary of the physics of excitonic states in CNTs.We

review the calculation of exciton wave functions by using thetight-binding states as the basis functions. In Section 3, weintroduce the formulation of excitonic energy transfer in CNTs.Section 4 shows the ET rate between various excitonic states ofCNTs and discusses the effect of the above-mentionedparameters. Section 5 provides a summary of the results.Appendix A shows a detailed derivation of the ET rates betweenvery long CNTs.

2. EXCITONS IN CARBON NANOTUBESIn this section, we provide an overview of excitonic states inCNTs. A detailed derivation of these formulas can be found inpapers of Rohlfing et al.35 and Jiang et al.36

2.1. Single-Particle States. For the purpose of calculatingthe CNT electronic structure, we can consider a CNT as a rolledgraphene sheet.Within this picture, a CNT electronic structure isthe same as for a graphene sheet. Using the tight binding (TB)method, the CNT wave function is37

∑ ∑ψ ϕ= −·r k r RN

C( )1

( )e ( )kk R

au b u

abi

ubub

(1)

where u runs over all of the Nu graphene unit cells, b = A,B runsover all the basis atoms in a graphene unit cell, and a is the bandindex. ϕ(r) is the pz orbital of carbon atom located at the origin.To satisfy the azimuthal symmetry of the wave function, the wavevector is limited to specific values known as cutting lines

μ= + | |k K K Kk /1 2 2 (2)

Here μ is an integer, determining the cutting line. K1 and K2 arethe reciprocal lattice vectors along the circumferential and axialdirections, respectively (Figure 1a). It is noteworthy that thereare always two degenerate cutting lines that pass by the two Diracpoints (K and K′) in the graphene Brillouin zone.

2.2. Excitonic States. According to the Tamm−Dancoffapproximation, the electron−hole excitation (exciton) wavefunction is a linear combination of free electron-free hole wavefunctions38

∑⟩ = ⟩†k k k kn A u v( , ) ( ) ( ) GSk k

n,

c v c v

c v (3)

Here An(kc,kv) is the expansion coefficient, u†(kc) is the creation

operator of an electron in the conduction band with wave vectorkc, and v(kv) is the annihilation operator of an electron from thevalence band with wave vector kv. |GS⟩ is the system groundstate, which corresponds to the valence band full and theconduction band empty of electrons. n represents all of the

Figure 1. (a) Graphene Brillouin zone and the cutting lines of a CNTwith (7,6) chirality. Bold solid lines represent the degenerate cutting lines that passby the K and K′ points in the graphene Brillouin zone. (b) Schematic of excitation type based on the cutting line of electron and hole.

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quantum numbers. We use the nearest-neighbor tight-bindingsingle-particle wave functions as the basis functions. Theexpansion coefficients in the exciton wave function and theexcitonic eigenenergies are calculated by solving the Bethe−Salpeter (BS) equation

∑− + ′ ′ ′ ′

= Ω

′ ′k k k k k k k k k k

k k

E E A A

A

[ ( ) ( )] ( , ) ( , ; , ) ( , )

( , )

k kn n

n n

c c v v c v,

c v c v c v

c v

c v

(4)

where Ec(kc) and Ev(kv) are the quasiparticle energies ofelectrons with wave vectors kc and kv in the conduction andvalence bands, respectively. Ωn is the exciton energy. is theinteraction kernel that describes the particle−particle interaction.Using the GW approximation,39 we can divide the interactionkernel into the direct and exchange terms

δ′ ′ = ′ ′ − ′ ′αk k k k k k k k k k k k( , ; , ) 2 ( , ; , ) ( , ; , )x dc v c v c v c v c v c v

(5a)

where the direct interaction depends on the screened Coulombpotential, w

∫ ψ ψ ω ψ ψ

′ ′ = ′ ′

= ′ * ′ ′ ′ = *′ ′

k k k k k k k k

r r r r r r r r

W

w

( , ; , ) ( , ; , )

d d ( ) ( ) ( , ; 0) ( ) ( )k k k k

dc v c v c c v v

3 3c c v v

(5b)

and the exchange interaction is calculated using the bareCoulomb potential, v

∫ ψ ψ ψ ψ

′ ′ = ′ ′

= ′ * ′ ′ ′ *′ ′

k k k k k k k k

r r r r r r r r

V

v

( , ; , ) ( , ; , )

d d ( ) ( ) ( , ) ( ) ( )k k k k

xc v c v c v c v

3 3c v c v

(5c)

Here ψk is the quasiparticle (electron or hole) wave function. α isthe exciton spin and δα = 1 for singlet excitons (α = 0) and δα = 0for triplet excitons (α = 1).The screened Coulomb interaction can be calculated using the

random phase approximation36,40

κ ω=

ϵ − =πk k k kk k k k

k kW a a a a

V a a a a( , ; , )

( , ; , )( , 0)1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

r 1 2(6)

Here ai represents the conduction or valence band of graphene.κϵr

π is the relative dielectric permittivity of the tube, where κ is thestatic dielectric permittivity that accounts for the effect of coreelectrons and the surrounding environment, while the effect of π-bond electrons is considered in the dynamic relative dielectricfunction ϵr

π(q,ω), which is calculated through the Lindhardformula.40

Using tight-binding single-particle wave functions, theinteraction kernels are

∑

∑

δ

κ

δ

′ ′ = ′ − ′ −× * ′ * ′

− ′ϵ − ′

′ ′ = − ′ − ′× * ′ * ′ −

π

′′ ′

′

′′ ′ ′

k k k k k k k kk k k k

k k

k k

k k k k k k k kk k k k k k

C C C C

v

C C C C v

( , ; , ) ( , )( ) ( ) ( ) ( )

( )

( , 0)

( , ; , ) ( , )( ) ( ) ( ) ( ) ( )

dc v c v

b bb b b b

b b

xc v c v

b bb b b b b b

c c v v

,c c c c v v v v

, c c

r c c

v c v c

,c c v v c c v v , c v

(7)

where vb,b′(q) is the Fourier transform of the overlap matrixbetween two pz orbitals

∫

∑

ϕ ϕ

= −

− = ′ − − ′ ′ −

≈| − | +π

′″

· −″ ′

′ ′ ′ ′

ϵ″ ′

″ ′

( )

q R R

R R r r r R r r r R

R R

vN

I

I v

U

U

( )1

e ( ),

( ) d d ( ) ( ) ( )

1

q Rb b

u u

i Ru b b

u b ub ub u b

u b b

,( )

0

3 3 2 2

4

e 0

2

u b b0

02

(8)

The last expression is the Ohno potential,41 where we take theparameterU = 11.3 eV. Note that due to the delta functions in eq7 we can write the interaction kernels in terms of the center-of-mass and relative-motion wave vectors42

=−

=+

Kk k

kk k

2,

2c v

rc v

(9)

Consequently, the center-of-mass wave vector is a good quantumnumber and the quantum number n in the BS equation has twocomponents: n = (s, K). s is the quantum number analogous tothe principal quantum number in a hydrogen atom. The excitedstate in eq 3 is now be written as

∑⟩ = + − ⟩†K K k k K k Ks A u v, ( , ) ( ) ( ) GSk

s r r r

r (10)

Note that both the center-of-mass and relative-motion wavevectors are 2D vectors consisting of two components. (i) Adiscrete circumferential part, which determines the electron andhole subband indices

μ μ μ μ μ= − = +( )/2, ( )/2c v r c v (11)

determines the angular momentum of the exciton center ofmass. (ii) A continuous part, parallel to the axis of the CNT,which determines how fast they are moving along the CNT

= − = +K k k k k k( )/2, ( )/2c v r c v (12)

If the exciton center-of-mass angular momentum is zero( = 0) the exciton is called A type. In an A-type exciton,the electron and the hole belong to the same cutting lines. On thecontrary, if the electron and the hole belong the different cuttinglines, the center-of-mass angular momentum is nonzero( ≠ 0) and the exciton is called E-type (See Figure 1b); E+and E− refer to excitons with positive and negative center-of-massangular momenta, respectively.For an A-type exciton ( = 0), it is easy to show that

′ = − − ′k k K k k K( , ; ) ( , ; )d dr r r r (13a)

′ = − − ′k k K k k K( , ; ) ( , ; )x xr r r r (13b)

′ = − ′k k K k k K( , ; ) ( , ; )x xr r r r (13c)

which results in symmetric (A1) and antisymmetric (A2)excitons43

→ = − −K k K kA A Aexciton ( , ) ( , )s s1 r r (14)

→ =+ −K k K kA A Aexciton ( , ) ( , )s s2 r r (15)

The wave function of an A1(A2) exciton are symmetric(antisymmetric) under C2 rotation around an axis perpendicularto the unwrapped graphene sheet. Also, the E-type excitonscenter-of-mass can rotate clockwise (E+) or anticlockwise (E−)along the circumference of the CNT. Among the excitonsdiscussed here, only the singlet A2 exciton is optically active(bright) and the rest are dark excitons.

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We can calculate the exciton energies for different values ofcenter-of-mass momentum, which yields the exciton dispersioncurves. Figure 2 shows the energy dispersions for singlet andtriplet excitons, with various symmetries and center-of-massmomentum. Here it is assumed that the exciton is a result of anS11 transition (an electron is excited from the highest valenceband to the lowest conduction band).

3. RESONANCE ENERGY TRANSFER: DIRECTINTERACTION

In this section, we calculate the Coulomb interaction matrixelement between excitonic states of two CNTs and the ET ratedue to this interaction (Figure 3).The initial and final states of the system are assumed to be

those of two noninteracting systems. Here we assume that thefirst CNT is the donor CNT, which is initially excited, and as aresult of exciton scattering the excitation is transferred to thesecond, acceptor CNT. Therefore, the exciton initial and finalstates in this two-CNT system can be denoted by |I⟩ = |1*⟩⊗ |2⟩and |F⟩ = |1⟩⊗ |2*⟩, respectively, where the asterisk denotes anexcited state of a given tube. The scattering (transfer) process ispossible through direct and exchange Coulomb interactions

between electrons of donor and acceptor CNTs.44,45 Figure 4shows a schematic of the direct and exchange pathways.The exchange interaction is important for the CNT aggregates

with considerable orbital overlaps between the donor andacceptor systems.44,45 When the separation between the donorand acceptor CNTs is larger than the spatial extent of the pzorbitals (<1 Å), the direct interaction is dominant. Here weassume that the donor and acceptor CNTs are not touching

Figure 2. Energy dispersion of (a) A1 singlet and triplet exciton, (b) A2 singlet, (c) A2 triplet exciton, (d) E-type singlet excitons with positive (blue) andnegative (red) circumferential momentum, and (e) E-type triplet excitons with positive (blue) and negative (red) circumferential momentum for S11transition in (7,5) carbon nanotube. (f) Comparison of lowest-subband exciton dispersions for various exciton types. α denotes the exciton spin, so α = 0refers to singlet and α = 1 to triplet excitons.

Figure 3. Geometry of donor (red) and acceptor (blue) carbonnanotubes.

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(wall-to-wall separation ≈ 2 Å), and hence we only consider thedirect Coulomb interaction. The matrix element of the directinteraction is

∑ ∑= ⟨ ⟩

= * ⟨ | − ′ | ⟩

K K

K k K k k k r r k k

s H s

A A v

M , ; GS GS ; ,

( , ) ( , ) , ( ) ,k k

s s

d 1 1 2 d 1 2 2

1 r 2 r v c c v

r1 r2

1 1 2 2 2 1 2 1

(16)

where we have kc1 = kr1 +K1, kv1 = kr1 −K1, kc2 = kr2 +K2, and kv2 =

kr2 − K2. The Coulomb interaction potential between theelectrons is

π| − ′| =

ϵ| − ′|r r

r rv

e( )

4

2

(17)

Here ϵ = ϵ0κ is the average value of the absolute dielectricpermittivity of the generally nonuniform medium between thetubes; κ is the medium’s average dielectric constant.Now we calculate the overlap integral over the free-particle

tight-binding wave functions

∫

∫

∑ ∑

∑ ∑

ψ ψ ψ ψ

ϕ ϕ

ϕ ϕ

= ⟨ − ′ ⟩

= ′ * | − ′| ′ * ′

= * *

×

× ′ * − − | − ′|

* ′ − ′ −

− · + · + · − ·

()

k k r r k k

r r r r r r r r

k k k k

r r r R r R r r

r R r R

v

v

N NC C C C (

v

, ( ) ,

d d ( ) ( ) ( ) ( ) ( )

1( ) ( ) ( ) )

e

d d ( ) ( ) ( )

( ) ( )

k k k k

k R k R k R k Ru u b b b b

b b b b

u u u u

i

u b u b

u b u b

TB v c c v

3 3c v c v

, ,c c v v c c v v

, ,

( )

3 3

u b u b u b u b

2 1 2 1

c1 v1 c2 v2

1 2 1 2 3 41 1 2 1 3 2 4 2

1 2 3 4

c1 1 1 v1 2 2 c2 3 3 v2 4 4

1 1 2 2

3 3 4 4 (18)

Now we assume that the last integral is important only when u1b1= u2b2 = ub and u3b3 = u4b4 = u′b′. Then, the overlap integralbecomes

∫

∑

∑

ϕ ϕ

= * * ×

′ − − ′ | ′ − |

′′ ′

′

− · + − ·

′ ′

′ ′

k k k k

r r r R r r r R

N NC C C C

v

1( ) ( ) ( ) ( )

e

d d ( ) ( ) ( )

k k R k k R

u u b bb c b b b

u u

i

ub u b

TB,

c v v c c v v

,

[( ) ( ) ]

3 3 2 2

ub u b

1 21 1 2 2

v1 c1 c2 v2

(19)

Considering the small size of atomic orbitals with respect to theseparation of the atoms in the two system, we can use a transitionmonopole approximation (TMA)30,46

∫ ϕ ϕ

π

′ − − ′ ′ −

≈ϵ| − |

′ ′

′ ′

r r r R r r r R

R R

v

e

d d ( ) ( ) ( )

4

ub u b

ub u b

3 3 2 2

2

(20)

Therefore, we have

∑

∑

π=

ϵ* * ×

| − |

′′ ′

′

− · + ·

′ ′

′ ′

k k k k

R R

eN N

C C C C4

( ) ( ) ( ) ( )

e1K R K R

u u b bb b b b

u u

i

ub u b

TB

2

,c c v v c c v v

,

( 2 2 )ub u b

1 21 1 2 2

1 2

(21)

Taking the wall-to-wall separation between the tubes largeenough that relative positions of basis atoms in donor andacceptor CNTs are not important in calculating the ET rate, wehave

| − |≈

| − |′ ′ ′R R R R1 1

ub u b u u (22)

Therefore, the matrix element is

∑π

=ϵ

*

*′

′

′− · + · ′

K K k k k

k

eN N

J C C C

C

4( , ) ( ) ( ) ( )

( )e K d K d

u u b bb b b

bi

TB

2

1 2,

c c v v c c

v v( 2 2 )b b

1 21 1 2

21 2

(23)

where we have used Rub = Ru + db. J(K1,K2) is the geometric partof the matrix element that contains all of the information aboutrelative orientation and position of donor and acceptor CNTs

∑=| − |′

− · + ·

′

′K KR R

J( , ) e1K R K R

u u

i

u u1 2

,

( 2 2 )u u1 2

(24)

Therefore, the matrix element for direct Coulomb interaction is

∑ ∑

∑

π=

ϵ* ×

* *

×

′′ ′

− · + · ′

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K KK k K k

k k k k

e JN N

A A

C C C C

M( , )

4( , ) ( , )

( ) ( ) ( ) ( )

e

k k

K d K d

u us s

b bb b b b

i

d

21 2

1 r 2 r

,c c v v c c v v

( 2 2 )b b

1 2 r1 r2

1 1 2 2

1 1 2 2

1 2

(25)

Note that based on the normalization of the exciton wave

function we have ∝K kA ( , )s Nr1

u, whereas the number of

terms in the summation over kr increases linearly with Nu.Therefore, we introduce the k-space part of the matrix element,which is independent of the length of donor and acceptor CNTs

∑ ∑

∑π

=ϵ

* ×

* *′

′ ′− · + · ′

K K K k K k

k k k k

QN N

A A

C C C C

( , )e

4( , ) ( , )

( ) ( ) ( ) ( )e

k k

K d K d

u us s

b bb b b b

i

1 2

2

1 r 2 r

,c c v v c c v v

( 2 2 )b b

1 2 r1 r2

1 1 2 2

1 1 2 21 2

(26)

The direct interaction matrix element becomes

Figure 4. Schematic of the exciton transfer processes mediated by direct(top) and exchange (bottom) Coulomb interaction.

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= ×K K K KN N

J QM1

( , ) ( , )u u

d 1 2 1 21 2 (27)

Next, we calculate the exciton scattering rate from donor toacceptor CNTs presented with indices 1 and 2, respectively. Ifthe exciton−phonon and exciton−impurity interaction isstronger than the Coulomb coupling between CNTs, we canassume that the excitons in each CNT have an equilibriumthermal distribution because they are effectively thermalizedwithin each CNT on amuch shorter time scale than the intertubeCoulomb-mediated transfer process. Therefore, the ET rate is anaverage of the exciton scattering rate with the equilibriumthermal distribution. We use Fermi’s golden rule to calculate thescattering rate due to direct interaction between CNTs

∑ ∑π δΓ =ℏ

Ω − Ωβ− Ω

M2 e

( )K Ks s

s s12, ,

d2

s

1 2 1 2

1

1 2(28)

where is the partition function

∑ ∑= = β− − Ωtr{e } eK

k T

s

/ sB

1 1

1

(29)

Assuming the tubes are long enough that we can convert thesummation over K2 into integration, we get

∫∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

π δ

π

π

Γ =ℏ Δ

Ω − Ω

=ℏ Δ Ω

=ℏ Δ

×Ω

β

β

β

− Ω

− Ω

Ω

− Ω

Ω

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟K K K K

KK

KK

N N KJ Q

K

M

M

2 1d

e( )

2 1 e dd

2 1 e( , ) ( , )

dd

K

K

K

s ss s

s s s

u u s s s

M

M

122 ,

2 d2

2 ,d

2 2

2 ,1 2 1 2

2 2

s

s

s

s

s

1 2 1 2

1

1 2

1 2 1 2

1

21

1 2 1 2 1 2

1

21

(30)

To calculate the transfer rate for a limited length CNT, it is betterto write this equations in terms of CNT length; therefore, we usethe following relations

π π π= = Δ =Nr L

AN

r LA

KL

2,

2,

2u

uu

u

1 1 2 22

21 2 (31)

Here Au is the area of the graphene unit cell. r1 and r2 are the radiiof donor and acceptor CNTs, respectively. Therefore, we get

∑ ∑ ∑

πΓ =

ℏ

× Ω

β− Ω

Ω

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟K K K K

LA

r r

J QK

1 e16

e( , ) ( , )

ddK

u

s s s

121

2 2

31 2

2

,1 2 1 2

2 2s

s1 2 1 2

1

21

(32a)

where we have defined the renormalized k-part of the matrixelement as

∑ ∑

∑π

=ϵ

* ×

* *′

′ ′− · + · ′

K K K k K k

k k k k

QL L

A A

C C C C

( , )1

4( , ) ( , )

( ) ( ) ( ) ( )e

k k

K d K d

s s

b bb b b b

i

1 21 2

1 r 2 r

,c c v v c c v v

( 2 2 )b b

r1 r2

1 1 2 2

1 1 2 21 2

(32b)

4. RESULTS AND DISCUSSIONIn this section, we calculate the ET rates between four differenttube chiralities: (7,5), (7,6), (8,6), and (8,7). The energies of thelowest bright excitonic states in these CNTs are shown in Table

1. Figure 5 shows a comparison of the dispersions of opticallyactive S11 and S22 excitonic states in (7,5) and (8,7) CNTs. Wehave taken the separation between the centers of donor andacceptor CNTs to be 1.2 nm, which provides enough wall-to-walldistance (∼2 Å) between the CNTs under consideration that theexchange Coulomb interaction is negligible.We calculate the ET rates across different combinations of

transition subbands (i.e., Sii → Sjj). Moreover, we calculate theET rate between optically bright and optically dark excitonicstates. We report on the dependence of the ET rate on the anglebetween the donor and acceptor tubes. Next, we study the effectof exciton confinement on the ET rate. Furthermore, we showthat the exciton thermalization among both dark and bright statescan reduce the ET rate by an order of magnitude. Also, we studythe exciton-transfer-rate variation with varying electrostaticscreening due to inhomogeneities in the surrounding medium.

4.1. Interband and Intraband Exciton Transfer Rates.First, we calculate the ET rate as a function of the relative anglebetween donor and acceptor CNTs (angle θ in Figure 3).Because of the radiative nature of the direct resonance energytransfer (Figure 4), we expect the ET between bright excitonicstates to be the dominant transfer process. Figure 6 shows thetransfer rate of bright excitons (A2) from donor CNTs with largerband gaps to acceptor CNTs with smaller band gaps (downhilltransfer). The uphill ET process is usually a couple of orders ofmagnitude smaller because in this case the excitonic states in thedonor tube that can resonate with the acceptor-tube states havelow exciton population. In Figure 6, the excitons belong to thesame transition subbands in donor and acceptor CNTs(intraband ET). We assume excitons are confined inside a 10nm long quantum well, similar to the work by Wong et al.30 Weobserve a relatively small dependence of the ET rate on therelative angle between CNTs. This is in contrast with theprediction by Wong et al.,30 who calculated that the transfer ratewould drop to zero when the donor and acceptor tubes areperpendicular. This discrepancy stems from the methodemployed by Wong et al.: They used a formulation that assumesthat the Coulomb interaction matrix element, eq 16, is almostconstant between excitonic states with different energies;however, our detailed derivation shows that one needs tocalculate the interaction matrix element for each pair of states indonor and acceptor CNTs (see eq 26). In addition, ourcalculated rates are in excellent agreement with the study byQianet al., who measured a lifetime of τ ≈ 0.5 ps for ET between twononparallel CNTs.24

Moreover, the ET rate is slightly higher between S11 states thanthe transfer rate between S22 states. This lower transfer rate canbe explained from the point of view that the direct resonanceenergy transfer is a process of simultaneous emission andabsorption of a virtual photon by the donor and acceptorsystems, respectively. The S22 excitonic states that are inresonance between the donor and acceptor tubes on average

Table 1. Energy of Lowest Bright Excitonic States for SelectedSWNTs

chirality S11 energy (eV) S22 energy (eV)

(7,5) 1.136 2.08(7,6) 1.053 1.974(8,6) 1.014 1.849(8,7) 0.9004 1.697

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have a higher center-of-mass momentum, which yields a lowerphoton emission rate.47,48 The lower rate of photon emissionyields a lower ET rate between S22 states.Next, we look at the contribution of dark excitonic states in the

ET process. When the separation between the donor andacceptor molecules is larger than the size of each molecule, thetraditional Forster theory is applied to calculate the ET rate. Inthis case, the transfer process depends on the overlap between

the emission and absorption spectra of the donor and acceptormolecules, respectively. Therefore, the dark excitonic states donot contribute in transfer process; however, in the case of ETbetween neighboring CNTs, the Forster theory fails as the donorand acceptor molecules are relatively large.30 Therefore, somedark excitonic states could contribute to the energy-transferprocess. Figure 7 shows the downhill ET rate to or from dark E-type excitonic states among four CNT types. These transfer rates

Figure 5. Comparison of S11 (a) and S22 (b) excitonic energy dispersions in (7,5) and (8,7) CNTs.

Figure 6. A2 exciton transfer rate versus relative angle between donor and acceptor CNTs. (a) S11 → S11 and (b) S22 →S22.

Figure 7. Transfer rate for processes involving dark E+ excitons versus relative orientation of the donor and acceptor tubes and for different tubechiralities. (a) A2 to E+, (b) E+ to A2, and (c) E+ to E+.

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are about 2 orders of magnitude smaller than the transfer ratesbetween bright excitonic states (Figure 6), owing to the largeangular momentum of E-type excitons. However, Postupna etal.31 suggested that the phonon-assisted processes could facilitateefficient ET from/to dark excitonic states.31 Postupna et al. usedtime-dependent density functional theory (TDDFT) inconjunction with molecular dynamics (MD) to study phonon-assisted exciton hopping. Because of numerical reasons, theirstudy was limited to the ET process between two arrays of CNTswith a relative angle of 90°.We should note that the dark A1 excitonic states still do not

contribute to the ET process due to symmetry considerationsthat hold beyond the dipole approximation.36

Next, we look at the ET process from the S22 excitonic states inthe donor tube to the S11 excitonic states in the acceptor tube(interband ET). As we can see in Figure 8, the interband energy-

transfer process occurs almost as fast as the intraband ET processshown in Figure 6. To better understand the role of differentexcitonic states in the excitation energy transfer process, wecalculated the intraband (S11 → S11 and S22 →S22) and interband(S22 →S11) ET rates considering only the ET process betweentightly bound excitonic states below the continuum level49

(white dashed line in the inset to Figure 8a). The calculatedintraband transfer rates did not change significantly from the casethat included both tightly bound excitonic states and thecontinuum states. (In other words, they remain similar to thosedepicted in Figure 6.) However, the interband ET ratesdecreased by about 2 orders of magnitude when the continuumstates were eliminated (compare Figure 8a to Figure 8b). Weconclude that although there are many transition states in thecontinuum region that resonate between the donor and acceptorstates, they do not contribute to the intraband ET process. Incontrast, most of the interband ET processes occur from tightlybound excitonic states to these continuum states.4.2. Exciton Confinement Effect. Owing to various forms

of disorder (e.g., the nonuniformity in the dielectric properties ofthe surrounding environment and the presence of chargedimpurities in the CNT samples), an exciton can be confined inquantumwells along the CNT. In this section, we study the effectof quantum-well size on the ET rate. For relatively wide quantumwells, which yield excitonic states with small energy spacing, thematrix element of the Coulomb interaction and the calculated ET

rate are affected only by the change in the geometric part of thematrix element (eq 24). Therefore, we calculate the ET ratethrough eq 32a, where the spatial extent of the quantum well isused in calculating the geometric part of the matrix element (eq24).Figure 9a (Figure 9b) shows the ET rate between bright

excitonic states when the donor and acceptor SWNTs have

similar (different) chiralities. It is assumed that the sizes ofquantum wells in donor and acceptor SWNTs are the same.When the CNTs are not parallel, the ET rate drops with

increasing size of the quantum wells because the average spacingbetween the donor and acceptor systems increases. We can seethis length dependence in eq 32a; however, in a CNT samplewith a constant density of tubes, the number of available acceptortubes is proportional to the length of the donor CNT. Therefore,we introduce the ET rate per unit length of the donor tube(Figure 10a). The ET rate per unit length changes up to a factor

of 3 due to the variation of the geometric part of matrix element(eq 32a). Nevertheless, the transfer rate stays relatively constantas we go to the limit of free-ET rates. The derivation of ananalytical expression for the matrix element and the ET rate inthe case of a free exciton in infinitely long donor and acceptortubes is shown in Appendix A.We observe a different behavior when the donor and acceptor

tubes are parallel. This case is particularly important because inmany samples the CNTs stick together and form CNT bundles.Unlike the case of nonparallel CNTs, we do not observe a drop in

Figure 8. (a) Transfer rate of bright S22 excitonic states to bright S11excitonic states as a function of tube orientation when considering allexcitonic states, both tightly bound and continuum. (Inset) Dispersionsof bound and continuum excitonic states, separated by a dashed whiteline that denotes the lowest continuum level. (b) Transfer rate of brightS22 excitonic states to bright S11 excitonic states as a function of tubeorientation when considering only bound excitonic states.

Figure 9. Exciton transfer rate from bright excitonic states of a donor(7,5) SWNT to the bright excitonic states of (7,5) (a) and (8,7) (b)acceptor SWNTs. Different colors show the transfer rates of excitonswith various quantum well sizes.

Figure 10. (a) Exciton transfer rate per unit length betweenperpendicular donor and acceptor SWNTs as a function of excitonconfinement length. (b) Exciton transfer rate between parallel donorand acceptor SWNTs as a function of exciton confinement length.

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the ET rate with increasing size of quantum wells, as the averagedistance between the donor and acceptor states is almostindependent of this size. We can predict this behavior based onthe analytical expressions derived in Appendix A.Furthermore, based on the chirality of the donor and acceptor

CNTs, the ET rates follow different trends as the size of quantumwells increases. For small confinement lengths, the excitoncenter-of-mass momenta in initial and final states are notimportant factors in determining the strength of Coulombcoupling; therefore, there are many states in the acceptor tubethat can resonate with the donor-tube excitonic states; however,as the confinement length increases, the contribution from theexcitonic states that do not conserve momentum drops. In thelimit of free-ET, only the states that conserve both the center-of-mass momentum and energy can transfer between the CNTs.Therefore, when the excitonic energy dispersions in the donorand acceptor tubes are dissimilar (e.g., due to differentchiralities), a limited number of states contribute to the ETprocess. This results in a decrease in the ET rate. On the contrary,if the donor and acceptor CNTs have similar dispersion curves,there are many excitonic states that conserve both momentumand energy in donor and acceptor CNTs, which increases the ETrate by ∼2 orders of magnitude (Figure 10).These findings are in excellent agreements with measure-

ments. Luer et al. have measured ET rates between S11 stateswithin bundles of CNTs that exceed 1014 s−1.26 They alsoreported limited ET rate between S22 states, which is due to thesame momentum matching considerations that we havediscussed here. Recently, Mehlenbacher et al. studied ET insamples where CNTs are wrapped in polymers and in sampleswith no residual surfactant.27,29,34 They found that, in thesamples with no polymer wrapping, the ET rate between parallelCNTs is extremely fast, with <60 fs time scales. Two-dimensionalanisotropy measurements showed much slower ET ratesbetween nonparallel CNTs in these samples. On the contrary,Mehlenbacher et al. found picosecond time scales for ET rates insamples that CNTs are wrapped in polymers. In these samples,ET shows no preference between parallel and nonparallel relativeorientation of donor and acceptor CNTs. Our calculations agree

very well with these experimental findings: In pristine samples(thus no exciton confinement), we expect high transfer ratesbetween same-chirality parallel tubes (1014 s−1) and much lowerrates when the tubes are misoriented. In samples with polymerresidue, excitons exhibit confinement, which drastically reducesthe rate of transfer between parallel tubes and results in isotropicET rates of ∼1012 s−1.

4.3. Effect of Static Screening by Surrounding Media.In this section, we study another effect of the CNT surroundingenvironment on the ET process. Like most quasi-1Dnanostructures, the electronic and optical properties of CNTsare influenced by their surrounding media. One of theseenvironmental effects is the screening of electrostatic elec-tron−electron and electron−hole interactions inside a CNT. Aswe discussed before, the relative dielectric permittivity κ in eq 6accounts for the screening due to the surrounding medium andthe core electrons in CNTs. As κ increases, the self-energy due torepulsive electron−electron interaction and the binding energydue to the attractive electron−hole interaction decreases.50 Asshown in Figure 11a, the net effect is a decrease in exciton energywith increasing permittivity. In the limit of infinite permittivity,we retrieve the noninteracting electron results. Figure 11 showsthe energy dispersions for bright excitonic states assumingvarious values of κ. As expected, the binding energy and thenumber of tightly bound excitonic states decrease with increasingpermittivity.Screening of the Coulomb interactions via the surrounding

medium affects the ET rates in two ways: by changing the excitonwave function and dispersion energy (intratube screening effect)and by changing the electron−electron interaction between thedonor and acceptor CNTs (intertube screening effect). The firsteffect stems from the change in the permittivity of theenvironment in the small area around the donor/acceptorCNT (we denote this permittivity by κ), whereas the secondeffect happens through changes in the average permittivity of theenvironment over long distances in the sample (we capture thiseffect through κ). While κ and κ are generally not the same, forsimplicity here we assume that they are. Figure 12a shows the ETrate between bright excitonic states as a function of the

Figure 11. (a) Lowest transition energies in (7,5) and (8,7) SWNTs as a function of relative permittivity. (b−d) Exciton dispersions with various relativedielectric permittivities of the environment.

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environment relative permittivity when only the first effect istaken into account. The drop in the ET rate with increasingrelative permittivity can be explained by looking at the photonabsorption and emission rates. For higher permittivities, theexcitonic states are more like free-electron and free-hole statesthan like bound excitons and thus have lower photon absorptionand emission rates.48 Consequently, the rate of ET, which issimultaneous photon emission and photon absorption by thetwo tubes, decreases. Figure 12b shows the ET rate betweenbright excitonic states as a function of Coulomb screening whenaccounting for both effects of screening. In this case, the drop inthe ET rate with increasing permittivity is much more significantthan in the previous case. This major drop is a result of a smallerperturbing Hamiltonian, which has a dependence of ∝

κHd

1 .

4.4. Interband Exciton Thermalization. As we previouslydiscussed, the excitonic states in CNTs are classified as bright anddark states. The former exciton type is created via opticalstimulation of ground-state electrons and the latter is usuallypopulated through some second-order processes, such as Ramanscattering or the scattering of bright excitons into dark excitonsby phonons and impurities. So far, we have studied the intrinsicET rate from either bright or dark excitonic states; however, if theexciton scattering between bright and dark states is fast enough(compared with the ET process), the excitons are thermalized

among both bright and dark states and we have to consider bothin the transfer process.Figure 13 shows the ET rate as a function of temperature for

the full exciton thermalization among bright and dark excitonicstates (Figure 13a) and among bright states only (Figure 13b). Inthe presence of exciton thermalization process between brightand dark excitonic states, we observe a 20-fold decrease in the ETrate. This is due to the presence of low-lying triplet states and thesymmetric singlet states that do not transfer through the directCoulomb interaction. As the temperature decreases to T = 0 K,the excitons only populate these low-lying states and the transferrate goes to zero. In addition, we note that the intrinsic transferrate between bright excitonic decreases when temperature drops.This trend is contrary to the behavior of radiative exciton decayrate predicted by Perebeinos et al.11

5. CONCLUSIONS

We calculated the ET rates between semiconducting CNTs ofdifferent chiralities and relative orientations. The ET rate isweakly dependent on the orientation of the tubes. This finding isin contradiction with previous theoretical studies but in goodagreement with experiments. The ET rate between brightexcitonic states is ∼2 × 1012 s−1. The transfer rates betweenbright and dark states are at least 2 orders of magnitude smallerthat the transfer rates between bright excitonic states. We alsolooked at the ET from S22 to S11 transition energies. We foundthat this type of ET process is as fast as transfer from S11 to S11and from S22 to S22 states. This process is facilitated by couplingof tightly bound excitonic states in S22 to the continuum levelstates (equivalent of free-electron/free-hole states) in S11.Furthermore, we studied the environmental effects on the ET

rates. We calculated the ET rate for excitons confined toquantum wells with various sizes. When the quantum-well sizeincreases, we observed a decrease in the transfer rate betweennonparallel tubes. This is due to the increase in the averagedistance between donor and acceptor systems. By introducing amore relevant quantity (i.e., transfer rate per donor tube length),we showed that the ET rate is almost independent of the excitonconfinement length; however, ET between parallel tubes followsa different trend: Transfer between same-chirality donor andacceptor tubes is extremely sensitive to confinement and free

Figure 12. Exciton transfer rate between (7,5) and (8,7) CNTs as afunction of environment relative permittivity when (a) only theintratube screening effect is taken into account and (b) both intertubeand intratube screening effects are taken into account.

Figure 13. Exciton transfer rate of excitons between CNTs (a) when the excitons are thermalized between both bright and dark excitonic states and (b)when only the bright excitonic states are populated. The donor and acceptor CNTs are parallel and the excitons are assumed to be confined in a 10 nmwide quantum well.

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excitons have the highest transfer rate (>1014 s−1). The transferrates between different-chirality tubes are not as sensitive toconfinement and only slightly increase with decreasing confine-ment length. These findings are a result of momentumconservation rules.Moreover, we looked at the effect of Coulomb screening due

to the surrounding media. The ET rate decreases with increasingscreening. We also showed that the ET rate increases withincreasing temperature. This behavior is the opposite of what onewould expect based on the emission and absorption spectra ofdonor and acceptor systems. Also, we showed that the ET ratedrops by about 1 order of magnitude if excitons are thermalizedbetween bright and dark excitonic states via extrinsic scatteringsources (e.g., impurities and phonons).We conclude that the wide range of ET-rate measurements,

spanning 2 orders of magnitude, stems from variations in samplepreparation and thus the degree of environmental disorder andhomogeneity, as the ET rates in pristine samples and in thesamples in which environmental disorder results in excitonconfinement differ by 2 orders of magnitude.

■ APPENDIX A: DERIVATION OF THE EXCITONTRANSFER RATE FOR FREE EXCITONS

In this Appendix, we derive the transfer rate of free excitonsbetween two very long CNTs of radii r1 and r2. We assume thatthe CNTs are far enough that, from the point of view of eachCNT, the other one looks like a continuous medium, so we canconvert the last sum in eq 24 to an integral

∫

∑| − |

≈ × ′ ×| − ′|

′

− · + ·

′ ′

− · · ′

′ ′

R R

r rr rA

e1

1d d

e e

K R K R

K r K r

u u

i

ub u b

u

i i

,

( 2 2 )

22 2

2 2

ub u b1 2

1 2

(33)

Assuming that the CNTs are shifted by a center-to-centerdistance D along the z axis and are misoriented by angle θ in thexy plane, the position vectors are

ϕ ϕ

θ ϕ θθ ϕ θ

ϕ

= + +

′ = ′ − ′ + ′ + ′ + + ′

r x y z

r xy

z

x r r

x rx rD r

cos sin

( cos cos sin )( sin cos cos )( sin )

1 1

2

2

2 (34)

Therefore, the geometric part of the matrix element becomes

∫ ϕ ϕ

θ ϕ θ

θ ϕ θ ϕ

ϕ ϕ

= ′ ′ ′ ′

× ′ − ′ −

+ ′ + ′ −

+ + ′ −

ϕ ϕ− + + +

−

⎡⎣

⎤⎦

K KJr rA

x x

x r x

x r r

D r r

( , ) d d d d e

( cos cos sin )

( sin cos cos cos )

( sin sin )

u

i K x i K x1 2

1 22

2 ( ) 2 ( )

22

2 12

2 11/2

1 1 2 2

(35)

If the CNTs are infinitely long, the integrals over x and x′ in thegeometric part of matrix element can be calculated analytically51

∫πϕ ϕ ϕ ϕ

ϕ ϕ θ ϕ

ϕ θ θ

θ

θ

= ′ ′ −

× ′ − + −

′

×− + −

+ −

ϕ ϕθ

| + ′ − |( )

K KJr r

Ai

i K r r K r r

K K K K

K K K K

( , ) d d exp(2 ( ))

exp(2 ( ( cos cos cos ) ( cos

cos cos ))/(sin ))

exp 2 2 cos

2 cos

u

D r r

1 21 2

2 2 1

1 2 1 2 1 2

sin sinsin 2

212

1 2

22

12

1 2

2 1

(36)

Using this relation in eq 32a, we can calculate the ET rate for freeexcitons; however, the transfer rate between two CNTs goes tozero due to infinitely long donor CNT; however, we can considerthe transition rate from a donor CNT to an array of CNTs thatare misoriented by angle θ. This is effectively equivalent to havinga network of CNTs. The number of acceptor CNTs that theexciton can be transferred to from the initial donor CNT withlength L1 is N = L1sin θ/W, where W is the center-to-centerdistance in the array of acceptor CNTs. Therefore, the totaltransfer rate is

∑ ∑ ∑

θπ

Γ =ℏ

×

′ × Ω

β− Ω

Ω

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟K K K K

WA

r r

J QK

sin4

e( , ) ( , )

ddK

u

s s s

12

2

21 2

2

,1 2 1 2

2 2s

s1 2 1 2

1

21

(37)

This equation turns out to be divergent at θ→ 0. This is simplydue to the fact that the transfer rate between completely parallelCNTs is length-independent and we expect the transfer rate fromthe initial CNT to an infinite number of final CNTs placed at acertain distance to become infinite.52

A.1. Special Case of Free-Exciton Transfer between ParallelTubesWhen the CNTs are very long and parallel to each other, thegeometric part of the matrix element yields a Kronecker deltafunction

δ= × ×K KJ L K K C K( , ) ( , ) ( , ; )1 2 1 2 1 2 1 (38a)

∫ ϕ ϕ

ϕ ϕ ϕ ϕ

= ′ ′

× − ′ + + − ′

ϕ ϕ−C Kr r

A

K r r D r r

( , ; )2

d d e

( 2 ( sin sin ) ( cos cos ) )

u

i1 2 1

1 22

2 ( )

0 1 1 22

1 22

2 1

(38b)

L1 = L2 = L→∞ is the length of CNTs. 0 is the modified Besselfunction of the second kind.Therefore, the matrix element of direct Coulomb interaction

becomes

πδ= × K K

Ar r

K K C K QM4

( , ) ( , ; ) ( , )ud

2

21 2

1 2 1 2 1 1 2

(39)

Using the conservation of the continuous components of wavevectors K1 and K2 in the matrix element d in eq 28, we get

∫

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

ππ

δ

ππ π

δ

π

Γ =ℏ

× ′

Ω − Ω′

=ℏ

′

Ω − Ω′

=ℏ

| ″ × ″ ′ |Ω

β

β

β

− Ω

− Ω

″

− Ω

Ω″

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

K K

K K

K K

Ar r

C K Q

L Ar r

K C K Q

L Ar r

C K QK

24

e( , ; ) ( , )

( )

22 4

de

( , ; ) ( , )

( )

4e

( , ; ) ( , )d

d

K

K

u

s s

s

s s

u

s s

s

s s

u

s s

s

ss

12

2

21 2

2

1, 2 1 2

11 2 1 1 2

2

1 2

2

21 2

2

1, 2 1 21

11 2 1 1 2

2

1 2

2

21 2

2

1, 2 1 2

11 2 1 1 2

2 1

11

(40)

The primed quantities in the last relation show the excitonicstates in the acceptor CNT that conserve the continuouscomponent of the center-of-mass wave vector, that is, K2 = K1.The double-primed quantities represent the excitonic states onthe donor CNT that conserve both the energy and thecontinuous part of the center-of-mass momentum in the transfer

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process, that is, Ω = ΩK K( , ) ( , )s s1 1 2 11 2. We should note

that the final transfer rate is independent of the CNT length asthe partition function, , is linearly dependent on the length ofthe donor CNT

∫

∑ ∑

∑ ∑π

=

=

β

β

− Ω

− ΩLK

e

2d e

K

K

K

s

s

( )

1( )

s

s

1 1

1 1

1 1

1 1

(41)

which cancels the parameter L in the eq 40, so we obtain

∫∑ ∑

∑ ∑ ∑

ππ

Γ =ℏ

× ″ × ″ ′Ω

β

β

− Ω

−

″

− Ω

Ω″

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡

⎣⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥⎥

K K

Ar r

K

C K QK

24

d e

e ( , ; ) ( , )dd

K

K

u

s

s s s

12

2

21 2

2

1( )

1

,1 2 1 1 2

2 1

s

s

s

1 1

1 1

1 2 1 2

1

11

(42)

■ AUTHOR INFORMATIONCorresponding Authors*A.H.D.: E-mail: davoody@wisc.edu.*I.K.: E-mail: iknezevic@wisc.edu. Phone: (608) 890-3383.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was primarily funded by the U.S. Department ofEnergy (DOE), Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering, Physical Behavior ofMaterials Program, under award no. DE-SC0008712. Prelimi-nary efforts (prior to the start of DOE funding) were supportedas part of the University of Wisconsin MRSEC, IRG2 (NSFGrant No. 1121288).

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