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Microcavity effects on the generation, fluorescence, and diffusion of excitons in organic solar cells

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Abstract

We compute the short-circuit diffusion current of excitons in an organic solar cell, with special emphasis on fluorescence losses. The exciton diffusion length is not uniform but varies with its position within the device, even with moderate fluorescence quantum efficiency. With large quantum efficiencies, the rate of fluorescence can be strongly reduced with proper choices of the geometrical and dielectric parameters. Hence, through proper micro-cavity design, the diffusion length can be increased and the device performance significantly improved without recourse to triplet excitonic states.

© 2013 Optical Society of America

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Figures (7)

Fig. 1
Fig. 1 Normalized fluorescent decay rate as a function of distance from a thick Ag electrode in a uniform medium of refractive index n = 1.3. Thin lines: ideal, lossless electrode, nAg = 6.37i. Thick lines: real electrode, nAg = 0.04 + 6.37i[34]. Vaccuum emission wavelength: 900 nm. b||: parallel excitons. b perpendicular excitons.
Fig. 2
Fig. 2 Solar cell geometry and coordinate system. HBL: hole-blocking layer. A: Acceptor. D: Donor; EBL: electron-blocking layer.
Fig. 3
Fig. 3 Decay rates (q = 1), gain functions, and external quantum efficiencies for the parameters given in Table 1. Top: ITO transparent electrode. Bottom: Ag transparent electrode. iso: isotropic case where b = 2 3 b | | + 1 3 b .
Fig. 4
Fig. 4 Absorption efficiency ηA computed for the devices described in Table 1 as a function of sun wavelength. A fixed ITO refractive index nITO = 1.76+0.08i and an absorption length of 70 nm in the active region are assumed over the whole spectral range. Ag refractive index is taken from [34].
Fig. 5
Fig. 5 Gain and decay profiles computed with Table 2 and q = 1. HBL: hole-blocking layer, A: acceptor, D: donor, EBL: electron-blocking layer.
Fig. 6
Fig. 6 b(z) in a structure with refractive indexes nAg/n2/n1/n2/nAg/n3 and thicknesses (nm): 140/29/295/6. n2 = 1.3, n3 = 2.8. Cases a to e: n1 = 2.8, 2.2, 1.7, 1.5, 1.4. Wavelength: 900 nm. Dashed line: numerical. Full line: (n2/n1)5.
Fig. 7
Fig. 7 b(z) and b||(z) computed at λ = 900 nm (thick dashed and thick full lines, respectively), and averaged over the range 860 nm < λ < 940 nm (crosses). Geometrical and dielectric parameters taken in Table 2.

Tables (2)

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Table 1 Two configurations optimized for parallel excitons. A sunlight absorption length of 70 nm is assumed in both photoactive materials.nITO = 1.76 + 0.08i, nAg(900nm) = 0.04 + 6.37i, nAg(750nm) = 0.03 + 5.19i.

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Table 2 Optimized configuration for perpendicular excitons. Absorption length: 70 nm is assumed in both photoactive materials.nAg(900nm) = 0.04 + 6.37i, nAg(750nm) = 0.03 + 5.19i.

Equations (62)

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0 = L 2 d 2 ρ d z 2 ρ + g ( z ) , L 2 = D τ ,
0 = L 2 d 2 ρ d z 2 b ( z ) ρ + g ( z ) , L 2 = D τ ,
0 = D i d 2 ρ d z 2 b ( z ) τ i ρ + α i ϕ i N g ( z ) , i = 1 , 1 ,
ρ = α i ϕ i N τ i ρ ,
0 = L i 2 d 2 ρ d z 2 b ( z ) ρ + g ( z ) , L i 2 = D i τ i ,
d ρ d z | z 1 = 0 , ρ ( z 0 ) = 0 , d ρ d z | z 1 = 0 .
I s c = I 1 + I 1 , I 1 = α 1 ϕ 1 N A τ 1 D 1 d ρ d z | z 0 , I 1 = α 1 ϕ 1 N A τ 1 D 1 d ρ d z | z 0 + ,
I i = ± ( A N ) α i ϕ i L i 2 d ρ d z | z 0 ± .
EQE sc = α 1 ϕ 1 L 1 2 d ρ d z | z 0 + α 1 ϕ 1 L 1 2 d ρ d z | z 0
0 L i 2 d 2 ρ d z 2 b ¯ ρ + g ¯ , b ¯ = 1 d A + D b ( z ) d z , g ¯ = 1 d A + D g ( z ) d z ,
EQE sc α g ¯ d × 2 L d b tanh ( d b 2 L ) η A × η D
ε ¯ i = ( ε i , x 0 0 0 ε i , x 0 0 0 ε i , z ) , z i 1 < z < z i .
k e , z , i = ε i , x k 0 2 ε i , x ε i , z k | | 2 ,
k o , z , i = ε i , x k 0 2 k | | 2 .
b ( z ) = 1 + 3 q 2 ε 1 , x 1 / 2 ε 1 , z 2 Re { 1 k 0 3 0 R ^ 0 p + R ^ 1 p + 2 R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p k | | 3 k e , z , 1 d k | | } ,
R ^ 0 p = R 0 p exp [ 2 i k e , z , 1 ( z z 0 ) ] , R ^ 1 p = R 1 p exp [ 2 i k e , z , 1 ( z 1 z ) ] ,
R N p = R N , N + 1 p ,
R j 1 p = R j 1 , j p + R j p exp ( 2 i k e , z d j ) 1 + R j 1 , j p R j p exp ( 2 i k e , z , j d j ) ,
R i j p = k e , z , i ε j , x k e , z , j ε i , x k e , z , i ε j , x + k e , z , j ε i , x .
b | | ( z ) = 1 + 3 q 4 4 ε 1 , x 1 / 2 3 ε 1 , x + ε 1 , z × Re { 1 k 0 0 ( k e , z , 1 ε 1 , x k 0 2 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p + 1 k o , z , 1 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) k | | d k | | } .
R ^ 0 s = R 0 s exp [ 2 i k o , z , 1 ( z z 0 ) ] , R ^ 1 s = R 1 s exp [ 2 i k o , z , 1 ( z 1 z ) ] ,
R N s = R N , N + 1 s , R j 1 s = R j 1 , j s + R j s exp ( 2 i k o , z , j d j ) 1 + R j 1 , j s R j s exp ( 2 i k o , z , j d j ) , R i j s = k o , z , i k o , z , j k o , z , i + k o , z , j .
g ( z ) = Re ( n i , s ) n ext | a i e i n i , s k 0 z + b i e i n i , s k 0 z | 2 .
k z , 1 = n 1 2 k 0 2 k | | 2 , k z , 2 = n 2 2 k 0 2 k | | 2 .
R ^ 0 p R ^ 1 p R = k z , 1 n 2 2 k z , 2 n 1 2 k z , 1 n 2 2 + k z , 2 n 1 2 ,
b ( z ) 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 2 R 1 R k | | 3 k z , 1 d k | | } ,
= 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 k z , 1 n 2 2 k z , 2 n 1 2 k z , 1 k z , 2 n 1 2 k | | 3 d k | | } ,
= 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 ( n 2 2 / n 1 2 k z , 2 1 k z , 1 ) k | | 3 d k | | }
b 1 q + q ( n 2 n 1 ) 5 .
EQE sc = η A ( λ s ) η D ( λ , q ) ,
Γ r ( 1 + γ ( z ) ) ,
Γ n r + Γ r ( 1 + γ ( z ) ) = ( Γ n r + Γ r ) ( 1 + Γ r Γ n r + Γ r γ ( z ) ) .
b ( z ) = 1 + q γ ( z ) .
k 0 2 ε ¯ E = i ω μ J + i ω × B ,
i ω B = × E ,
Z = Z | | + z ^ Z z .
( 2 z 2 + ε x k 0 2 ) E | | = i ω μ J | | + | | E z z i ω z ^ × | | B z ,
( 2 z 2 + ε x k 0 2 ) ( i ω B | | ) = i ω μ z ( z ^ × J | | ) + i ω | | B z z k 0 2 ε x z ^ × | | E z ,
( ε z 2 z 2 + ε x | | 2 + ε x ε z k 0 2 ) E z = i ω μ ε x z ^ ( J + 1 ε x k 0 2 J ) ,
( 2 z 2 + | | 2 + ε x k 0 2 ) B z = μ z ^ ( × J ) .
g e = e i k 0 ε x x 2 + y 2 + ε z z 2 4 π ε x 1 / 2 ε x ( x 2 + y 2 ) + ε z z 2 , g o = e i ε x 1 / 2 k 0 x 2 + y 2 + z 2 4 π x 2 + y 2 + z 2 .
g e = i 4 π ε z 0 J 0 ( k | | ρ ) k e , z e i k e , z | z | k | | d k | | , g o = i 4 π 0 J 0 ( k | | ρ ) k o , z e i k o , z | z | k | | d k | | ,
E z = i ω μ ε x ( z ^ j 0 + 1 ε x k 0 2 z j 0 ) g e + E , B z = i μ z ^ ( j 0 ) g o + B ,
E z = ω μ ε x j 0 4 π ε z 2 k 0 2 0 k | | 3 k e , z J 0 ( k | | ρ ) ( e i k e , z | z | + C e i k e , z z + D e i k e , z z ) d k | | ,
D = R 1 p e 2 i k e , z 1 ( 1 + C ) R ^ 1 p ( 1 + C ) .
C = R 0 p e 2 i k e , z 0 ( 1 + D ) R ^ 0 p ( 1 + D ) .
C = R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p , D = R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p ,
E z = ω μ ε x j 0 4 π ε z 2 k 0 2 0 k | | 3 k e , z J 0 ( k | | ρ ) ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | .
( 2 z 2 + ε x k 0 2 ) E | | = ω μ ε x j 0 4 π ε z 2 k 0 2 | | z 0 k | | 3 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | .
E | | = ω μ j 0 4 π ε z k 0 2 | | z 0 k | | J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z k e , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ,
2 Re { E j 0 * } h ¯ ω = μ ε x | j 0 | 2 2 π h ¯ ε z 2 k 0 2 Re { 0 k | | 3 k e , z ( 1 + 2 R ^ 0 p R ^ 1 p + R ^ 0 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p ) d k | | . }
Γ r = μ ε x | j 0 | 2 2 π h ¯ ε z 2 k 0 2 0 ε z 1 / 2 k 0 k | | 3 k e , z d k | | = μ | j 0 | 2 k 0 3 π h ¯ ε x 1 / 2 .
Γ r ( 1 + 3 ε x 1 / 2 2 ε z 2 k 0 3 Re { 0 k | | 3 k e , z 2 R ^ 0 p R ^ 1 p + R ^ 0 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p d k | | } ) ,
E z = ω μ j 0 4 π ε z k 0 2 2 x z 0 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) k | | d k | | ,
B z = i μ j 0 4 π y 0 J 0 ( k | | ρ ) k o , z ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) k | | d k | | .
δ ( x ) = δ ( z ) 2 π 0 J 0 ( k | | ρ ) k | | d k | | ,
( 2 z 2 + ε x k 0 2 ) E | | = i ω μ j 0 2 π x ^ δ ( z ) 0 J 0 ( k | | ρ ) k | | d k | | ω μ j 0 4 π ε z k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) k | | d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) k | | d k | | .
E | | = ω μ j 0 4 π ε x 1 / 2 k 0 x ^ δ ( z ) 0 J 0 ( k | | ρ ) e i ε x 1 / 2 k 0 | z | k | | d k | | ω μ j 0 4 π ε x k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z k | | ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z k e , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z k | | ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z k o , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ,
E | | = ω μ j 0 4 π ε x k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z k | | ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z k | | ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) d k | | .
2 Re { E x j 0 * } h ¯ ω = μ | j 0 | 2 4 π h ¯ Re { 0 [ k e , z ε x k 0 2 ( 1 + 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p ) + 1 k o , z ( 1 + 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) ] k | | d k | | } .
Γ r = μ | j 0 | 2 4 π h ¯ ( 1 ε x k 0 2 0 ε z 1 / 2 k 0 k e , z k | | d k | | + 0 ε x 1 / 2 k 0 1 k o , z k | | d k | | ) = μ | j 0 | 2 k 0 3 π h ¯ ( 3 ε x + ε z 4 ε x 1 / 2 ) .
Γ r ( 1 + 3 4 k 0 4 ε x 1 / 2 3 ε x + ε z Re { 0 ( k e , z ε x k 0 2 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p + 1 k o , z 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) k | | d k | | } ) ,
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