An Analytical Solution for Exciton Generation, Reaction, andDiffusion in Nanotube and Nanowire-Based Solar CellsDarin O. Bellisario,‡ Joel A. Paulson,† Richard D. Braatz,† and Michael S. Strano*,†

†Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States

*S Supporting Information

ABSTRACT: Excitonic solar cells based on aligned or unaligned networks ofnanotubes or nanowires offer advantages with respect of optical absorption, andcontrol of excition and electrical carrier transport; however, there is a lack ofpredictive models of the optimal orientation and packing density of suchdevices to maximize efficiency. Here-in, we develop a concise analyticalframework that describes the orientation and density trade-off on excitoncollection computed from a deterministic model of a carbon nanotube (CNT)photovoltaic device under steady-state operation that incorporates single- andaggregate-nanotube photophysics published earlier (Energy Environ Sci, 2014,7, 3769). We show that the maximal film efficiency is determined by aparameter grouping, α, representing the product of the network density and theeffective exciton diffusion length, reflecting a cooperativity between the rate ofexciton generation and the rate of exciton transport. This allows for a simple,master plot of EQE versus film thickness, parametric in α allowing for optimaldesign. This analysis extends to any excitonic solar cell with anisotropic transport elements, including polymer, nanowire,quantum dot, and nanocarbon photovoltaics.

Nanostructured photovoltaics (PVs), e.g., dye-sensitizedsolar cells (DSSCs), organic PVs, nanoporous metal

oxide, and single-walled carbon nanotube (SWCNT) solar cells,have unique advantages for solar energy with respect to cost,efficiency, morphology, and spectral coverage.1,2 In contrast tobulk-semiconductor solar cells, materials with nanometer scaleheterogeneity remain challenging for mathematical model-ing.3−6 For example, the interaction lengths can be too long forab initio treatment but too short for bulk, averaged properties.This heterogeneity often generates an empirically intractableparameter space. Accordingly, it is not obvious how to improvethe efficiency of recent carbon nanotube photovoltaticdevices.7−9 In recent work, we developed a physics basedapproach to modeling exciton generation, transport, andrecombination in a single walled carbon nanotube photo-voltaic.10 The deterministic model considered a steady-stateplanar-heterojunction device incorporating SWCNT photo-physics. We found that there existed an optimal thicknessbeyond which efficiency decreased rapidly. The system ofnonlinear nonhomogenous integro-differential equations, how-ever, required numerical evaluation. In this work, we perform adetailed parametric analysis of the previous deterministicmodel, and develop an analytical model that describes theoptimal thickness of the photovoltaic as a function of nanotubedensity and photophysical constants. We find that the optimalthickness is approximately the effective exciton diffusion lengthfor a given defect density, nanotube length and intrinsicexcition diffusivity.

Exciton Transport Model. In our previous work, weconsidered a monochiral network of single-walled nanotubes(SWCNT) sandwiched between two type-II exciton dissocia-tion electrode semi-infinite plates separated by a distance T(Figure 1).10 The Cartesian z axis is normal to the plates, withthe incident solar photon flux J0(ω) normally incident at z = 0.A given nanotube in the film has orientation l ≡ (θ, ϕ), where θis the angle with the z axis. When the network density is highenough (above the percolation threshold), the film can then bemeaningfully described by the distribution p(θ, ϕ), with p(θ, ϕ)= δ(θ − θ′)δ(ϕ) for example representing an aligned film atangle θ′.10Exciton diffusion occurs via three channels. Longitudinal

transport is along the nanotube length, with diffusioncoefficient Dl. Exciton energy transfer occurs betweenneighboring tubes either in bundles or at misalignedinterconnects, in both cases being orthogonal to thelongitudinal axis of the originating nanotube. Those effectivediffusion coefficients are DEET,b, dependent on the mean bundlesize, and DEET,I, respectively. Dl is approximately 5 orders ofmagnitude larger than the latter two. The net exciton diffusivityin the z axis is then achieved by integrating these effects overthe film orientation distribution10

Received: May 16, 2016Accepted: June 30, 2016Published: June 30, 2016

Letter

pubs.acs.org/JPCL

© 2016 American Chemical Society 2683 DOI: 10.1021/acs.jpclett.6b01053J. Phys. Chem. Lett. 2016, 7, 2683−2688

∫ ∫ θ ϕ θ

θ γ θ ϕ

≡

+ − +

D p D

D b D

( , )[(cos )

(1 cos ) ( )] d d

z2

l

2I EET,I c EET,b (1)

where bc is the bundle fraction in the film (typically near unity)and γI is a sparsity coefficient capturing the density ofinterconnects. The exciton population balance can then beconstructed by accounting for fluorescent emission, non-radiative decay, Auger recombination, and tube-end quenching,

ρ ρ= + − − ′ − −

⟨ ⟩·Γ

⟨ ⟩ ⟨ ⟩

ct

N z Dc

zk c k c

kc

kl

cdd

( )dd

2z

l l

2

2 1EEA 2 end

(2)

where c(z) is the exciton concentration, N(z) is the excitongeneration rate from light absorption, kEEA is the augerrecombination rate constant, kΓ is the radiative decay rateconstant, k1′ is the impurity quenching rate constant propor-tional to the concentration of impurities, kend is the end-quenching rate constant, and ⟨l⟩ is the mean nanotube length.This expression holds only for films of densities above thepercolation threshold and with a uniform distribution ofnanotube end-sites. For a more detailed derivation from single-SWCNT behavior, including broader case analysis, see ref 10.At steady state the exciton transport expression (eq 2) is

therefore described by a nonlinear, nonhomogenous ordinarydifferential equation (ODE) of simplified form,10

− − = −′ ′Dc

zk c k c N z

dd

( )z

2

2 1 22

(3)

Note that all parameters are extracted from individual/aggregate SWCNT photophysics and the distribution ofnanotube orientations in the film. The ODE is subject toRobin boundary conditions describing Type II excitondissociation at each interface, z = 0,T for a film of thickness T,

± = |=

=Dcz

k cddz

z Tz T

0,d 0,

(4)

where kd, the dissociation rate constant, we take to infinity inthe rapid-dissociation limit, checking that the flux converges.Analytic Solution to Photoadsorption of the Solar Spectrum in

SWCNT Films. The first obstacle to analytic evaluation of (3) isN(z), the exciton generation rate, which is the photonabsorption rate. A valid form of N(z) that enables facilesolution to the exciton transport ODE is the usual inhibitor toanalytic or even deterministic solutions to exciton-transport-limited nanomaterial photovoltaic models. Herein we report ananalytic expression with material-specific physical constants thatcaptures that light absorption behavior.N(z) is the product of the absorption cross-section per

nanotube σl(ϵ, l, ω), the number density ρ⟨l⟩, and the solar fluxat depth z Jν(ω, ϵ|z),

ω ρ σ ω ωϵ ϵ ϵ| = · · |ν⟨ ⟩N z l J z( , ) ( , , ) ( , )l l (5)

N(ω, ϵ|z) carries a frequency ω and polarization ϵdependence, the latter of which demands a SWCNTorientation l dependence. N(z) is then a quadruple integralover the nanotube orientation distribution p(θ, ϕ), linear in-plane polarization ϵ (we assume normally incident light), andlight frequency ω,

∫ ∫∫ ∫

ρ ω

σ θ ϕ ω θ ϕ θ ϕ ω

= ϵ|

× ϵ ϵ

π

ν

π π

⟨ ⟩

∞N z J z

p

( ) ( , )

( , [ , ], ) ( , ) d d d d

l

l

0 0

2

0

2

0(6)

where ρ⟨l⟩ is the nanotube density (length of nanotube pervolume of film), Jν is the photon flux, and σl is the single-SWCNT absorption cross-section, per length of nanotube.To proceed, we focus on devices with isotropic in-plane light

absorption and no reflection of light off the back electrode. Thisparametric space is the most relevant to technological interests;in the former case it includes isotropic and vertically alignedfilms, which we previously showed present dominant efficiencyrelative to films with anisotropic in-plane alignment. In thelatter case, a transparent back electrode takes advantage ofSWCNTs near-infrared (nIR) absorption by allowing the solarcell to complement existing visible PVs or coat buildingmaterials. Those approximations make the polarization andnanotube orientation integrals trivial. We then solve thedifferential equation describing the light field attenuation as itpasses through the film,

ωω ω ω ρ σ ω

|= − | ⇒ | = · − · ·ν

ν ⟨ ⟩

dJ z

dzN z J z J z

( )( ) ( ) ( ) exp[ ( ) ]l l0

(7)

where J0(ω) is the incident solar flux. That produces the finalintegral over frequency space

∫ρ ω σ ω ρ σ ω ω= − · ·⟨ ⟩

∞

⟨ ⟩N z J z( ) ( ) ( ) exp[ ( ) ] dl l l l0 0 (8)

Figure 1. (A) Cartoon of a SWCNT solar cell of thickness T, withdiameters exaggerated. (B) Numerical solution to the steady-statebehavior of the vertically aligned film, with the external quantumefficiency (EQE) plotted (colormap) versus the film thickness T andthe density ϕ as a fraction of the close-packed density, at two differentmean nanotube lengths ⟨l⟩. Longer tubes exhibit less end-quenching,increasing EQE and allowing the film to be thicker to capture morelight while still collecting excitons at the electrodes.

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To solve that integral, we recognize several simplifications.First we can very closely approximate the nanotube absorptionspectrum as a series of Gaussian functions for each absorptionpeak, with magnitude Sa,

∑σ ωω μ

σ= · −

−⎡⎣⎢⎢

⎤⎦⎥⎥S( ) exp

( )

2la

aa

a

2

2(9)

where μa is the energy of the transition, and σa is the peak width( σ=FWHM 2 2 ln 2 a). Next, we find that because theSWCNT absorption peaks lie on only the red side of thesolar spectrum peak, the spectrum can be sufficientlyapproximated as a Gaussian as well,

ωω μ

σ≅ · −

−⎡⎣⎢⎢

⎤⎦⎥⎥J J( ) exp

( )

2J

J0 0

2

2(10)

Those simplifications make N(z) an insoluble convolution ofa Gaussian and an exponential decay of a Gaussian (see SI). Wecan observe, however, that while the Exponential-Gaussian termis in general not well captured by the first-order Taylorexpansion, its product with the Gaussian term is because eitherthe function extrema are close or the Gaussian-Exponentialreduces to unity. That observation simplifies the integral,allowing us to solve it as

∑ρ γ π= · ρ⟨ ⟩

=

− ⟨ ⟩N z J S e( ) 2la

a aS z

01

2l a

(11)

where γa can be understood as the area of overlap of the solarflux and absorption peak a,

γ σ σσ σ

σ σ≡ ≡

+μ μ σ σ− − +e ,a Ja Ja

J a

J a

( ) /2( )2 2

2 2J a J a

2 2 2

(12)

The sum must include two peaks for each SWCNT chirality,corresponding to the E1u and E2u transitions. For a solid film of(6,5) SWCNT,7 the empirical constants are S1 = 1.64 × 10−10

m2/m, γ1 = 293 nm, S2 = 0.13 × 10−10 m2/m, and γ2 = 74 nm.This biexponential analytic expression closely matches the

numerical solution (Figure 2A). The curve can be understoodas light absorption on two length-scales, one for each portion ofthe frequency spectrum absorbed. Because there are twoabsorption peaks with different absorption cross sections, near-IR absorption being stronger, light at nIR frequencies is, relativeto initial intensity, attenuated rapidly while visible light isabsorbed more gradually, providing a long tail to our generationrate. That effect is emphasized by the greater incident solarintensity at the visible vs nIR frequencies. We can also seeevidenced that our first-order Taylor expansion of theGaussian-exponential is sufficient, with essentially no benefitto a third-order expansion.Analytical Solution to the Exciton Transport Model. With an

analytic expression for N(z) we are able to nondimensionalizethe ODE (eq 3). The problem symmetry yields thecharacteristic exciton concentration and Cartesian length,

ρ γ≡

· ·⟨ ⟩cJ S

k nl

00 1 1

1 1 (13)

≡ =zD

k nLz

D01 1 (14)

Figure 2. Comparison of analytic and numerical solutions. (A) Solution to the photon absorption rate, comparing the numerical result (black) to theanalytic solutions with one or two absorption peaks and first- or third-order Gaussian-exponential Taylor expansion. Solutions for 20% and 80%close-packed density are shown, with insets zoomed for clarity. While it is necessary to include absorption from both SWCNT electronic transitions,we can see that a third-order Taylor expansion of the frequency integrand is unnecessary. (B) Nondimensional exciton concentration profilescomparing the numerical and analytic solutions at different densities for a film of thickness z = 4. (C) Film EQE as a function of film thickness atdifferent densities.

The Journal of Physical Chemistry Letters Letter

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where peak 1 (implied in S1, γ1) is the dominant absorptionmode, and k1n1 is the product of the single-nanotube first-orderquenching rate constant and the concentration of impuritiesincluding nanotube ends. Multiple impurities with different rateconstants can be treated with a weighted sum. The character-istic variables c0 and z0 reflect the physical trade-offs of thesystem. The nondimensional concentration is a ratio of theexciton generation rate to the dominant exciton quenching rate,which compete to increase and decrease, respectively, theexciton concentration. The nondimensional Cartesian coor-dinate normalizes the film thickness as η ≡ T/LD, whichbalances the film thickness against the effective exciton diffusionlength in the film. A thicker film or lower diffusion lengthreduces the number of excitons that reach the electrodes.Substituting nondimensional concentration c ≡ c/c0 and

Cartesian dimension z ≡ z/z0 into the exciton balance, we areleft with

∑

ρ γ

πγγ

πγγ

− + −

= − ·

= − + ·

ρ

ρ ρ

Γ ⟨ ⟩

=

−

− −

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

cz

kk n

ck J S

k nc

S

Se

eSS

e

dd

1( )

( )

2

2

l

a

a a S D k n z

S L z S L z

2

21 1

2 0 1 11

1 12

2

1

2

1 1

/

2 2

1 1

l a z

l D l D

1 1

1 2

(15)

We then compare the magnitude of terms, finding that thosemuch less than unity are negligible to yield the far simplifiedexciton balance

πγγ

− = − + ·τ α τ α− − ⎛⎝⎜⎜

⎞⎠⎟⎟c

zc e

SS

edd

2 z z2

22 2

1 1

1 2

(16)

κ κ η

=

= − η= =

cz

ccz

cdd

(0),dd

( )z z0 (17)

α ϕ ϕρ

ρη κ

τ ρ

≡ · ≡ ≡ ≡

≡

⟨ ⟩

⟨ ⟩

⟨ ⟩

LTL

kD k n

S

, , , ,Dl

lCP

D

d

z

a a lCP

1 1

(18)

Solving with the method of undetermined coefficients andtaking the rapid-dissociation limit, we find the concentrationprofile is a balance of four competing exponentials,

β βτ α

πγγ

τ α

= + −−

−

−

τ α

τ α

− −

−

c z e eSS

e

( ) e1

( ) 12

1( ) 1

z z z

z

1 21

22 2

1 1

22

1

2

(19)

βτ α

πγγ τ α

= −−

−−

+ −−η η

η τ αη η τ αη

−

− − − −⎛⎝⎜⎜

⎞⎠⎟⎟e e

e e SS

e e1( ) 1

2( ) 11

12

2 2

1 1 22

1 2

(20)

βτ α

πγγ τ α

= +−

−−

+ −−η η

η τ αη η τ αη

−

− −⎛⎝⎜⎜

⎞⎠⎟⎟e e

e e SS

e e1( ) 1

2( ) 12

12

2 2

1 1 22

1 2

(21)

The result we again check against the numerical solution(Figure 2B), finding excellent agreement to validate ourapproximations. Our biexponential light field decay due to aweak and strong absorption peak manifests in a pronounced tailin exciton concentration toward the back electrode; nIR light(the stronger peak, τ1) is rapidly collected, while visible light(the weaker peak, τ2) is more weakly absorbed to generateexcitons deeper in the film.

Figure 3. (A) Photovoltaic external quantum efficiency (EQE) as a function of film thickness η and parameter grouping α. Note that a nonlinearcolor bar was used for visual clarity. At a given α there is an optimal thickness η* that maximizes the EQE, indicated by the black line overlay. (B) theEQE-maximizing thickness T* = η*LD versus density ϕ and exciton diffusion length LD. (C) Maximum achievable EQE versus density ϕ and excitondiffusion length LD, showing the monotonic increase in both. In both panels B and C, α is the product of the axes. (D) Master curve (black) ofoptimal device thickness, nondimensionalized with the exciton diffusion length η* ≡ T*/LD, versus α, the product of diffusion length and normalizeddensity and the sole system parameter. Measuring a SWCNT film’s α dictates the optimal device thickness for that material. Blue dashed curvesindicate the bounds of higher and lower thickness that yield an EQE above 99.5% of the maximum. The broad EQE tolerance suggests that in lieu ofmeasuring α, choosing η between 0.6 and 1.4, i.e., T* ≅ LD will provide essentially optimal performance.

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In the exciton-transport-limited approximation, we canestimate the EQE from the boundary solutions

η γτ

α

∝ − =

− η

= =

= =

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

Dcz

cz

J

cz

cz

EQE( )dd

dd

dd

dd

zz z T

z z

00 1 1

0 (22)

We can optimize eq 22, setting its derivative equal to zero tofind the maximum as (Figure 2C),

η=

= − +

+ − + −

+ + + −

+ + −

+ − + +

+ + − −

η τ α η τ α

η τ α η τ α

η τ α η τ α

η τ α η τ α

η τ α η τ α

η η η

− * − * −

− * − − * −

− * − − *

− * − − * −

− * − − * −

* * *

F p e p e

m p e m e

m e F p e

p e m p e

m e m e

m p F m p F e e e

dEQEd

0

((1 ) (2 )

(2 2) ( 2 )

(1 ) ) ((1 )

(2 ) (2 2)

( 2 ) (1 ) )

(2 2 )(2 )

1 1( )

1( 1)

1 1( 2)

1( 3)

1( 4)

2 2( )

2( 1)

2 2( 2)

2( 3)

2( 4)

1 1 1 2 2 22 3

1 1

1 1

1 2

2 2

2 2

(23)

τ α τ απ

γγ

≡−

≡+

≡m p FS

S1

1,

11

, 2aa

aa

aa a

1 1 (24)

Crucially, there is only one film parameter in this expression,α, the product of film density and the effective exciton diffusionlength in the film. The optimal thickness therefore depends ononly this single variable, and all possible films collapse to asingle η−α space, as in Figure 3A.Discussion. Examining the efficiency variation with thickness

and α (Figure 3A), we can observe that EQE monotonicallyincreases with α; higher α materials, which have higherdensities and exciton diffusion lengths, are unambiguouslybetter (Figure 3C). In particular, there is a sharp drop inefficiency for α below 10 nm, indicating that a mean excitondiffusion length of over 100 nm for a 10% close-packed film or10 nm for a 100% close-packed film is a crucial materialproperty that must be achieved to advance SWCNT Solar Celltechnology. Additionally, at a given α there is an optimal η*,indicated by the black line in Figure 3A. The optimumrepresents a balance between a thicker film collecting morelight but increasing the distance that excitons must travel toreach the electrodes. At higher densities (ϕ) more light iscollected per unit thickness, increasing the maximum EQE andreducing the optimal thickness required to achieve it. At longerdiffusion lengths (LD) excitons are collected more efficiently,allowing the film to be thicker to achieve the same efficiency orto increase the EQE at the same thickness. These cooperativeeffects are captured in the product α, with the only differencebeing that LD’s thickness-increasing attribute is balanced by η’snormalization to LD, which is clear in Figure 3B; as LD increasesand the density decreases, the optimal thickness rises. From thisexamination we can finally address the practically achievablemorphology trade-off between increasing density and increasingdiffusion length, such as encountered when choosing betweenisotropic and vertically aligned films using current techniques:because the product of ϕ and LD is the driver of EQE, theyshould match. This is demonstrated in the symmetry of Figure

3C. Roughly, EQE is maximal when ϕ is about 1/1000th of LDin nm.In hindsight the reduction of the system to α is intuitive: the

rate of light absorption (exciton generation) is dictated by thefilm density, while the rate of exciton collection is dictated bythe diffusion length (and film thickness). Higher density andhigher LD increase the rate of charge collection cooperatively.Having reduced the system to this single parameter, we canconstruct the master curve Figure 3D. To employ this curve, anexperimentalist only needs to measure her SWCNT film’s α,ideally having increased it as much as possible (Figure 3C).Above very low α (10 nm), The EQE has a plateau-like cross-section (Figure 2C) that makes it drop off suddenly far fromη*, but fairly invariant close to η*. As a result, there is a highdegree of EQE-tolerance in the region around η*, exemplifiedby the blue bounds in Figure 3D within which the EQE is≥99.5% of the maximum possible EQE. That tolerance meansthat even if experimenters do not measure their α, they cancomfortably approximate 0.6 ≤ η* ≤ 1.4 or, equivalently, T* ≅LD. In other words, near-maximal performance can be achievedby setting the device thickness equal to the exciton diffusionlength in the charge-collecting axis.Under the simplifications we have made, that result has quite

broad applicability to any excitonic photovoltaic material withbell-shaped absorption modes. The rule of thumb breaks downat two extremes: very high and very low α. The latter case isphysically realistic; it is sparse films, or materials with very lowoptical absorptivity. In those cases, the optimal thicknessbecomes many times the diffusion length due to the poor lightcollection. The other limit of very high α can be imagined asextremely high light absorption, where the exciton diffusivitybecomes irrelevant and the film should be as thin as possible,i.e., η* ≪ 1, T* ≪ LD. This case is not physically realistic withexistent materials however; SWCNT already have very highoptical absorptivity compared to most photovoltaic materials,and density is constrained to a maximum of close-packeddensity. As a result, our maximum α considered in Figure 3D isunlikely to be exceeded by any other material.In summary, we find that the characteristic spatial length

scale describing SWCNT PVs is the exciton diffusion length,=L D k n/D z 1 1 , where Dz is the exciton diffusivity in the

charge-collecting axis and k1n1 is the first-order quenching rateconstant, i.e., the balance of exciton reaction and diffusion.Nondimensionalizing device thickness as η ≡ T/LD, theefficiency-maximizing thickness η* depends on a singlegrouping of parameters, α = ϕ·LD, where ϕ is the SWCNTnumber density normalized to close-packed. That generates amaster η*(α) curve (Figure 3D) that any device can be placedon, empowering device-makers to know their optimal thicknesssimply by measuring their material’s α. We further find thatclose to the optimum, external quantum efficiency (EQE) isonly weakly variant with η, yielding the rule of thumb 0.6 ≤ η*≤ 1.4, or equivalently T* ≅ LD, the thickness equals thediffusion length, which provides an EQE within 0.5% of themaximum. Finally, our solution is enabled by a new method ofapproximating absorption of the solar flux that is applicable toany film with bell-shaped, isotropic absorption peaks. The lightabsorption (carrier generation) gradient is generally the mostdifficult component of photovoltaic performance to solve dueto the convolution of nonlinear incident intensity andabsorption spectrum over frequency and polarization. Ourtreatment as a whole applies to any exciton-transport-limited

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film with those absorption properties, with an analytic vsnumerical error less than 1% of the resulting EQE.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpclett.6b01053.

Step-by-step derivation of our solution, as well asadditional figures examining model results (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: strano@mit.edu.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSM.S.S. acknowledges a grant from Eni S.p.A. in the frame of theeni-MIT Solar Frontiers Center. D.O.B. thanks the UnitedStates Department of Defense for the National Defense Scienceand Engineering Graduate Fellowship.

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