2D and Trap-Assisted 2D Langevin Recombination in Polymer:Fullerene Blends

The theory of 2D Langevin recombina-tion is extended to the case with high trap density and demonstrated experi-mentally on the model system poly[2,5-bis(3-tetradecylthiophen-2-yl)thieno[3,2-b]thiophene] (PBTTT):[6,6]-phenyl-C61-bu-tyric acid methyl ester (PCBM) using both transient and steady-state techniques. The recombination capture coeffi cients are derived for trap-assisted and band-to-band recombination, showing that anisotropic charge transport reduces the capture coeffi cients in both cases result-ing in a reduced overall recombination.

M. Nyman ,* O. J. Sandberg , R. Österbacka ................................ X–XX

2D and Trap-Assisted 2D Langevin Recombination in Polymer:Fullerene Blends

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2D and Trap-Assisted 2D Langevin Recombination in Polymer:Fullerene Blends

Mathias Nyman ,* Oskar J. Sandberg , and Ronald Österbacka

M. Nyman, O. J. Sandberg, Prof. R. Österbacka Department of Natural Sciences Physics & Center of Functional Materials Åbo Akademi University Porthansgatan 3 , 20500 , Åbo , Finland E-mail: [email protected]

DOI: 10.1002/aenm.201400890

developed a theory to describe the rate at which anions and cations recombine in an ion gas; the rate depends only on the velocity at which the ions move in their mutual Coulomb fi eld since it is assumed that once the ions are close enough to each other recombination (ion bond formation) is instantaneous. Langevin theory does not describe the reverse process (anion–cation formation); the time reversed pro-cess was described by Onsager and further extended by Braun. [ 2,3 ]

When applying standard Langevin theory to BHJ blends two basic assump-tions are made, namely the charge trans-port in the active material is isotropic and the rate of nongeminate polaron pair formation is much lower than the rate of polaron pair recombination. If these assumptions hold the recombination rate

is simply given by the probability of an electron and a hole to meet in coordinate space, so that R = β L np , and the Langevin recombination coeffi cient β L is:

L

n p

o

eβμ μ

εε=

+

(2)

where e = elementary charge, µ n(p) = the electron (hole) mobility, ε = relative permittivity, and ε 0 = the vacuum permittivity. In general either one of these assumptions—or both, may not be valid. For example, if the charge-carrier mobility is highly aniso-tropic an electron will not necessarily recombine with the hole that is closest; it will recombine with the one that is most easily accessible. This could occur for example in lamellar systems; it has been shown that the mobility in the plane of a polymer lamella can be up to 100 times the out of plane mobility. [ 4 ]

Several BHJ blends have been reported to show a bimolec-ular recombination that is signifi cantly reduced as compared with Langevin, [ 5–7 ] in particular a reduction in the recombina-tion rate of a factor of 10 2 –10 4 has been reported in the BHJ blend poly(3-hexylthiophene) (P3HT) and PCBM. [ 5 ] In these cases a recombination prefactor, defi ned as / Lξ β β= , is typi-cally introduced. Several models have been proposed to clarify the underlying mechanism of reduced recombination, some of which can be directly related to restricted validity of the assump-tions in Langevin theory listed above. Koster et al. [ 8 ] noted that the complex created when an electron and a hole meet in coordinate space (charge encounter complex) can be resplit into a free electron and hole with a certain probability. Murthy et al. [ 7 ] noted that for several BHJ systems exhibiting reduced

The impact of trapping on the recombination dynamics in polymer:fullerene blends is clarifi ed using the highly ordered bulk heterojunction (BHJ) blend poly[2,5-bis(3-tetradecylthiophen-2-yl)thieno[3,2-b]thiophene] (PBTTT) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) at different weight ratios as a model system. The recombination dynamics are determined using both transient charge extraction and steady-state techniques. The results show that both the decay of photogenerated charge and the light ideality factor at a polymer:fullerene weight ratio of 1:4 are fully consistent with 2D Langevin recombination; in the 1:1 case the recombination is seen to be affected by electron trapping. The theory of 2D Langevin recombination is extended to the case with high trap density in agreement with the observations in the 1:1 case. The recombination capture coeffi cients are derived both for trap-assisted and band-to-band recombination and it can be seen that anisotropic charge transport reduces the capture coeffi cients in both cases resulting in a reduced overall recombination.

1. Introduction

Bulk heterojunction (BHJ) blends of conjugated polymers and fullerenes show great promise for future solar energy applications. In order to further increase the power conver-sion effi ciencies of these types of solar cells insights about the dominating loss mechanisms are vital. These systems typically exhibit low dielectric constants and charge-carrier mobilities leading to high charge-carrier concentrations which in turn can result in substantial losses due to nongeminate recombination. The decay of photo-generated (free) charges due to bimolecular nongeminate recombination is given (for holes) by:

d

d

p

tG R G npβ= − = −

(1)

where n ( p ) = concentration of electrons (holes), t = time, G = generation rate of free charges, R = recombination rate, and β = the recombination coeffi cient. The bimolecular recombina-tion of free charges in blends of disordered organic materials is typically observed to be of Langevin type. [ 1 ] M. P. Langevin

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recombination the reduction in the recombination seem to be related to the crystallinity of the blend: more crystalline blends tend to have a smaller ξ (i.e., larger reduction in the recom-bination rate). They argue that the reason for this is that the charge encounter complex is energetically close or equivalent to the charge-transfer state thus facilitating a high probability of resplitting. However, the recombination (of free charges) is still of second-order (assuming that β is independent of p ). Integra-tion of Equation ( 1) yields that the decay dynamics should be proportional to t − 1 at long times.

G. Juška et al. [ 9,10 ] have suggested that the charge transport in lamellar systems are mainly restricted to two dimensions, since the mobility in the lamellar plane is signifi cantly larger than the out-of-plane mobility. This fact suggests that in the case of crys-talline blends the standard 3D Langevin formalism should be replaced by a 2D one (2D Langevin). Equation ( 1) should then read, assuming n = p , (derived by Juška et al.: [ 9 ]

d

d2D

5/2p

tG pγ= −

(3)

where 2Dγ = the 2D Langevin recombination coeffi cient, leading to a density decay as

( )1

32

o3/2

2D

2/3

p tp tγ

=+

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟−

(4)

that is proportional to t − 2/3 (at long times) and p 0 = the concen-tration of holes at t = 0. The reduction in the recombination rate (ξ) versus 3D Langevin will be given by (assuming n = p )

3

42D

L

3/2 1/2l pξ ββ

π= =

(5)

where l = the lamellar spacing. Calculated values for P3HT:PCBM are in excellent agreement with measurements. [ 9 ] Reduced recombination has also been suggested to be due to tri-molecular (Auger type) recombination that is a third-order process, in this case the decay dynamics should be proportional to t −1/2 . [ 11,12 ]

If a signifi cant amount of charge carriers are trapped recom-bination will occur not only between free charges; free charges can also recombine with trapped charges. [ 13 ] In the case when recombination between free holes and trapped electrons is dominating R ≈ C p n t p , where n t is the density of trapped electrons and C p is the capture coeffi cient for a hole to fi nd a trapped (immobile) electron. The density of trapped charges is typically described by Shockley–Read–Hall (SRH) statistics. Kuik et al. found the capture coeffi cient (for holes) in the 3D case to be given by

p,L

p

o

C eμεε

= (6)

in accordance with Langevin's theory. [ 14 ] Note that if n t >> n the recombination is essentially monomolecular (since n t is almost constant).

To observe the decay dynamics is a good way of differenti-ating between different recombination mechanisms. However, in some cases transient techniques might not provide all infor-mation about recombination, for example, a buildup of trapped charges might take much longer than the measurement time domain. Furthermore, it is not always easy to experimentally distinguish between different recombination mechanisms due to disorder, dispersion, and the fact that several different mech-anisms may contribute to the overall recombination observed. As a complement to transient spectroscopy steady-state meas-urements can be used, one possibility is to measure the open-circuit voltage ( V OC ) as a function of light intensity. [ 15,16 ]

It has been shown (empirically) that V OC scales with the light intensity I as ( )OCeV mkTln I∝ , where m is the so-called light ideality factor. [ 15,16 ] The light ideality factor is distinctly different for 3D Langevin, 2D Langevin, SRH, and Auger type recombi-nation and thus provides a way of distinguishing between dif-ferent recombination mechanisms, or give hints as to what is the dominant mechanism. Note that the light ideality factor is not necessarily equivalent to the diode ideality factor. [ 16–18 ]

At open-circuit conditions (steady-state d

d0

p

t= ) the photo-gen-

eration rate of free charge carriers is approximately canceled by recombination ( R ) and we have: G R≈ . From the splitting of the quasi-Fermi level under photo-generation it follows that [ 15 ]

lnOC g

DA c veV E kTN N

np= −

⎛⎝⎜

⎞⎠⎟

(7)

where E g DA is the effective donor–acceptor gap and N c and N v is the effective density of states in the acceptor and the donor, respectively. Assuming that the majority of the carriers are photo-generated we set n p≈ . For nongeminate recombination of order α and recombination coeffi cient r we then have that

G R rp≈ = α

(8)

By solving for p and using p = n in Equation ( 7) one fi nds

2ln

2ln( )OC g

DA c v/2

eV E kTr N N

GkT I

α α( )

= −⎛⎝⎜

⎞⎠⎟

∝α

(9)

since, G I∝ . The corresponding light ideality factor is then given by

2m

α= . In 2D Langevin α = 5/2 and hence we expect

0.8m = . For systems displaying 3D Langevin recombination ( α = 2) m = 1 and if the recombination is monomolecular (α = 1) m = 2. [ 13 ] We note that the above discussion is valid for free charges. When the reaction orders for the total charge den-sity (free + trapped) is considered a different situation arises and reaction orders larger than two are commonly observed. [ 19 ] In some cases the extracted charge density might include a con-tribution from charges in shallow traps that would result in an effective increase in the apparent reaction order.

Tress et al. [ 15 ] have shown, using solar cells based on ZnPc and C 60 , that the light ideality factor at low light intensities is close to 2 indicating monomolecular or SRH recombination, whereas at higher intensities m is closer to 1 indicating direct bimolecular recombination. At low light intensities a signifi -cant amount of the charge density is trapped, as the charge



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generation increases the density of free charges also increases resulting in more bimolecular-like recombination.

In the case of charges trapped in an exponential density of tail states the light ideality factor is dependent on the character-istic trap depth E Ch according to: [ 19,20 ]

mkT

EE kT

E

= +⎛⎝⎜

⎞⎠⎟

>

→

⎧

⎨⎪

⎩⎪

−1

2 2, for

1, for 0Ch

1

Ch

Ch

(10)

as shown by Kirchartz et al. [ 19 ] Furthermore, if the amount of free charges is much smaller than the amount of trapped charges the reaction order for free charges (given in this case

by 2

1Chm

kT

Eα = ≈ + ) will decrease for increasing E Ch (deeper

traps) whereas the reaction order for trapped charges (given by E kTα ≈ +1 /t Ch ) will increase. [ 19 ] In P3HT:PCBM devices light ideality factors ranging from 1.2 to at most 1.6 have been reported. [ 13,21–23 ] According to Kirchartz et al. [ 19 ] this can be explained by Equation ( 10) . Wetzelaer et al. [ 24 ] argued that the recombination in P3HT:PCBM devices is a mixture of SRH and Langevin recombination, trap-assisted recombination contrib-utes signifi cantly to the recombination that does occur since the recombination is so strongly reduced (possibly due to 2D Langevin). It should be noted that the transient decay dynamics using both photo-CELIV [ 25 ] and transient absorption [ 26 ] have been shown to follow 2D Langevin, i.e., ( ) 2/3n t t∝ − .

2. Results and Discussion

The effect of charge trapping on the decay dynamics in the case of 2D Langevin has previously not been clarifi ed. If, for example, there is signifi cant electron trapping occurring there will be a buildup of trapped electrons at open-circuit conditions (steady-state).

To generalize Equation ( 3) to the case with n p≠ , we con-sider the recombination mechanism of a hole, recombining with an electron, in a 2D lamellar system. In general, the bimo-lecular recombination rate can be written by RR v npσ= ⟨ ⟩ , where v is the magnitude of the relative velocity between the electron and the hole and σ R is the bimolecular recombina-tion cross section. In Langevin’s original theory, the carriers drift toward each other in their mutual Coulomb attraction so

that: 4

n po

2veμ μ

πεε ρ( )= + . Taking a spherical cross section

4R,3D2σ πρ= one fi nds 3D L

on pv

eσ βεε

μ μ( )⟨ ⟩ = = + . In the 2D

case, however, a cylindrical cross section has to be assumed. In this case, 2R,2D lσ πρ= ( l = interlamellar distance) and

1

2

1R 2D 2D

on pv

elσ β

εεμ μ

ρ( )⟨ ⟩ = = + ⟨ ⟩ , where l ρ<< . Averaging

over a volume with radius ρ R , one fi nds 1 2

31 R

ρρ

⟨ ⟩ =− , where

ρ R is the average distance between an electron and a hole. This is the corresponding distance an electron and a hole have to drift in their mutual electric fi eld before they meet and recom-bine, and is related to the carrier densities as R

2 1l npπρ ( )=

−.

Consequently,

3

42D

3 1/2

Ll npβ π β( )= (11)

Noting that 3

42D

3/2Llγ π β= , the trap-free case then reads

d

d( )2D 2D

5/4p

tG np G npβ γ= − = −

(12)

When n p≈ Equation ( 12) reduces to Equation ( 3) . Con-versely, Equation ( 12) can be reobtained by setting 2p np→ in Equation ( 3) .

A similar derivation for the capture coeffi cient C p between free holes and trapped electrons, noting that the trapped elec-trons are immobile ( µ n = 0), reveals that

3

4p,2D

3t

1/2

p,LC l n p Cπ ( )=

(13)

where C p,L is given by Equation ( 6) . If the majority of elec-trons are trapped in deep trap states, such that ,tn n p>> , trap-assisted recombination between free holes and trapped elec-trons become dominating and the recombination term of Equa-tion ( 12) is instead given by

p,2D t 2D

p

p nt

54 5/4R C n p n p pγ

μμ μ

( )= =+

⎛⎝⎜

⎞⎠⎟

∝

(14)

Now, since α = 5/4 we obtain a light ideality factor m = 8/5 = 1.6. In a system where the nongeminate recombination is gov-erned by 2D Langevin light ideality factors between 0.8 and 1.6 are thus expected. Furthermore, anisotropic charge transport will reduce the capture coeffi cients both for Langevin and SRH recombination.

To illustrate the effect of charge trapping on the recombi-nation in a lamellar system we measure the recombination dynamics in the BHJ blend of PBTTT and PCBM using photo-CELIV and correlate the observed recombination rate to the light ideality factor. This particular blend was chosen due to the possibility of controlling the electron transport by varying the PBTTT: PCBM weight ratio. In addition this system displays a high level of order; in a 1:1 weight ratio PBTTT:PCBM forms a bi-crystal with all the PCBM fully intercalated between the poly mer side chains. [ 27 ] Since, all the fullerenes in this blend are intercalated between the side-chains of the polymer the electron transport should be very limited—the photo-generated holes can delocalize along the polymer chain but the electrons will be highly localized. In a 1:4 weight ratio the excess PCBM forms conducting pathways outside of the polymer lamellae. [ 27 ] Due to the high level of order and lamellar formation 2D Lan-gevin recombination is expected in these systems.

PBTTT:PCBM (from Merck and Solenne, respectively) were spin-cast from ≈50 mg mL −1 solution in dichlorobenzene on ITO-covered glass (Präzisions Glas & Optik GmbH), the resulting thicknesses were ≈200 nm for the 1:4 devices and 470 nm for the 1:1 devices. Subsequently, a 5 nm layer of LiF followed by 60 nm of Al was thermally evaporated. For the light ideality measurements the LiF layer was 1 nm, in addition an electron blocking layer of 1 nm MoO 3 was evaporated on the



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ITO prior to spin casting of the active material. The mobility was measured by photo-CELIV to ≈10 −3 cm 2 (Vs) −1 in the 1:4 case and 6·10 −5 cm 2 (Vs) −1 in the 1:1 case. The resulting 3D Langevin recombination rates are β L ≈ 10 −9 cm 3 s −1 and 7·10 −11 cm 3 s −1 , respectively (with ε = 3).

Figure 1 shows photoinduced charge extraction by a lin-early increasing voltage (photo-CELIV) transients with varying laser intensity of a 1:1 and a 1:4 PBTTT:PCBM device. In the 1:4 case, a clear saturation can be seen when the intensity is high. According to Juška and co-workers [ 28 ] the recombination prefactor ξ can be determined as j ( 0 )/Δ j sat that in the 1:4 case is roughly 0.15. In the 1:1 case, the saturation is not as clear as in the 1:4 case due to the low extraction effi ciency; however, j ( 0 )/Δ j sat is signifi cantly larger than 1, which would indicate typical 3D Langevin recombination. However, especially in thin fi lms this is a problematic approach to determine the recom-bination prefactor since a large amount of charge will be col-lected in the outer circuit before the CELIV pulse is applied (in our setup the shortest possible delay between laser pulse and voltage pulse is 2 μs). To obtain the transient decay we perform measurements with increasing delay times ( t del ) between laser pulse and voltage pulse. The concentration of extracted charge n ext ( t del ) is obtained by integrating the current transient and subtracting the dark response.

Figure 2 shows n ext as a function of t del + t max of a 1:4 PBTTT:PCBM device as measured by photo-CELIV ( t max is the time at that the transient reaches its highest value). In this case, the majority of the charge carriers are photo-generated and n p≈ and Equation ( 12) reduces to Equation ( 3) . The red line is a fi t to the 2D Langevin decay (Equation ( 4) ) and the blue line represent the corresponding 3D Langevin case (with β L = 10 −9 cm 3 s −1 ). The experimental data are seen to fi t well with the 2D Langevin model, due to the high charge-carrier mobility of this blend it is not possible to have this large carrier concentrations at such long times if the recombination would be 3D Langevin with β = β L . The reduction in the recombina-tion rate as compared with 3D Langevin ( ξ ) can be calculated by multiplying γ 2D with n 0 1/2 and then divide by β L to obtain β / β L . In this case, the resulting β / β L is roughly 0.03. One should note, however, that a good fi t can also be made with a 3D Langevin model, provided that the 3D recombination rate is reduced compared with Langevin (with a factor of ≈0.03). Hence it is not possible from these data alone to distinguish between 2D and

reduced 3D Langevin. However, in the 2D Langevin model the reduction factor is not ad hoc since it has a physical meaning.

Figure 3 shows β / β L as a function of n 0 for a PBTTT:PCBM 1:4 (blue) and 1:1 (red) device at different temperatures. The black line represents the reduction factor as calculated by Equa-tion ( 5) using 3 nm as the interlamellar distance. [ 27 ] In the 1:4 case the measured recombination prefactors correspond well to the 2D Langevin model with only slight variations in tem-perature. In the 1:1 case, however, the recombination rate is reduced but much less so than in the 1:4 case. We note that the low mobility in the 1:1 case makes the measurements and analysis challenging, especially at lower temperatures.

Current–voltage characteristics were measured using a Keithley 2636 source meter, as light source an argon ion laser (Innova 90 °C series, Coherent) at 514 nm was used. The light ideality factors measured at light intensities relevant to operational conditions of a 1:4 and a 1:1 PBTTT:PCBM device are shown in Figure 4 a). The light ideality factor in the 1:4 case is ≈0.87 that is very close to what is expected for 2D Langevin (0.8). In the 1:1 case, however, the light ideality factor is ≈1.15, slightly over what is expected from 2D- and 3D Langevin recombination. At lower light intensities the light



-200 -100 0 100 200 300 400 500 600 700-40

-20

0

20

40

60

80

A = 2.5V/500µsV

OFF = -0.5V

delay = 2 µs

j ext /(

µA

/cm

2 )

time /µs

dark 0.2 µJ 2 µJ 20 µJ 200 µJ 500 µJ

a)

-100 0 100 200 300-1

0

1

2

j ext /(

mA

/cm

2 )

time /µs

dark 2 µJ 20 µJ 200 µJ

A = 2V/200µsV

OFF = -0.4V

delay = 2 µs

b)

Figure 1. Light intensity dependent photo-CELIV transients of a) a 1:1 PBTTT:PCBM device and b) a 1:4 PBTTT:PCBM device.

1E-6 1E-5 1E-4 1E-3

∼

∼

1E13

1E14

1E15

1E16

n ext /c

m-3

t-3/2

t-1

N0 = 1.2E162D Langevin = 4.8E-19γ3D Langevin = L ~ 1E-9β β

Photo-CELIV 2D Langevin fit 3D Langevin fit

(tdel + tmax) /s

Figure 2. The concentration of extraction charge as a function of t del + t max for a 1:4 PBTTT:PCBM device (open squares), the red and blue lines rep-resent a fi tting to the 2D- and 3D Langevin models, respectively.

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ideality factors increase for both systems indicating a crossover to a trap-assisted recombination regime similar to what was reported in ref. [15]. Kirchartz et al. [ 18 ] showed that at very high light intensities the V OC will saturate to the built-in voltage due to increased surface recombination. However, this cannot be the case here; the built-in voltage was determined to 0.9 V from temperature-dependent V OC, which is much larger than the voltages in Figure 4 a). Figure 4 b) shows simulated light ideality factors for the trap-free case with different types of bimolecular recombination coeffi cients β using a drift-diffusion model pre-sented elsewhere, [ 29 ] the relevant input parameters are shown in Table 1 . It can be seen that a device with no recombination at all ( ξ = 0) would have a light ideality factor of roughly one. Furthermore, a reduction of the recombination coeffi cient ( ξ = 0.02) in a 3D Langevin case ( α = 2) will only result in ideality factors very close to one. Hence, the only possibility of having a light ideality factor signifi cantly lower than one is if α is larger than two (assuming Ohmic contacts and device thicknesses > 100 nm). However, in this case we can rule out tri-molecular and quadrimolecular recombination since this is not con-sistent with the transient case, leaving 2D Langevin as the only

possibility. We note that the V OC is fairly similar in the 2D and reduced 3D cases but the slope is distinctly different.

In the 1:4 case, the recombination is consistent to what is expected from 2D Langevin—both in the transient and the steady-state cases, which has previously not been reported for any other BHJ blend. We conclude that this is due to the high level of order and balanced mobilities exhibited in the 1:4 case. [ 30 ] However, it appears that the recombination in the 1:1 case is heavily infl uenced by trapping due to the limited elec-tron transport resulting in an ideality factor considerably larger than 0.8. Figure 5 shows simulated light ideality factors for a 2D Langevin system with both direct and trap-assisted recom-bination with capture coeffi cients given by Equations ( 11) and (13), respectively. The traps, having a concentration N t and residing at a trap depth E trap , are assumed to be acceptor-type and described with SRH statistics. [ 14 ] It can be seen that an increasing trap depth results in a higher light ideality factor as expected, but never larger than 1.6. For a fi xed trap depth an increase in the trap density will also result in an increase in the light ideality factor (not shown).

The only other BHJ blend to date that has been proposed to exhibit 2D Langevin recombination is P3HT:PCBM, however, the discrepancy between the recombination dynamics in steady-state and in the transient case has previously not been under-stood. If the recombination in P3HT:PCBM is governed by 2D Langevin our results suggest that both the capture coeffi cients for direct and trap-assisted recombination are reduced as com-pared with ordinary 3D Langevin. Furthermore, trap-assisted 2D Langevin should have light ideality factors m ≤ 1.6 instead of m ≤ 2, which is in good agreement with the reported values of 1.2–1.6. [ 13,21–23 ] The fact that the light ideality factor is seen to depend on the trap concentration might explain the fairly large range of the reported m values for P3HT, since the trap concen-tration can be highly sensitive to varying production methods. The discrepancy between the transient and steady-state cases seems to indicate that there is a slow buildup of trapped charge.

3. Conclusion

The PBTTT:PCBM system is one of the most ordered BHJ blends, equally, or even more ordered than for example P3HT:PCBM as evidenced by the narrow Gaussian density of



1015 1016 101710-3

10-2

10-1

100

10

β = β

ξ

1

297K 280K 270K 260K 250K

n0 /cm-3

2D-Langevin (l = 3 nm)

L

Figure 3. The recombination reduction factor ξ for a 1:1 PBTTT:PCBM device (red) and a 1:4 PBTTT:PCBM (blue) as a function of generated charge at varying temperatures. The black line is calculated from Equa-tion ( 5) .

Figure 4. Light ideality factors a) experimentally determined b) simulated with R = βnp (where β = ξ β L ) using a drift-diffusion model. For 2D Langevin β = β 2D as per Equation ( 11) was used in the simulations.

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states and low transport activation energy. [ 30 ] However, contrary to the conclusion by Murthy et al. that the reduction in the recombination should be larger for more ordered systems, the reduction is not larger in PBTTT:PCBM than in P3HT:PCBM. This indicates that the issue of reduced recombination is more complex and cannot be directly related to crystallinity alone, which was previously pointed out also by Clarke et al. [ 31 ] The reduction factor in the 1:4 system is, however, in agreement with what is expected from 2D Langevin, providing an under-lying physical reason for the reduction. Murthy et al. further note that the recombination is reduced by a factor of ≈20 in a BHJ consisting of a thiophene and quinoxaline copolymer blended with a fullerene despite the fact that this system does not exhibit any long range order. This shows that in general reduced recombination can be due to both a reduced prob-ability for creating a charge encounter complex as well as a reduced probability of the charge encounter complex to recom-bine. Trapping and energetic and structural disorder makes the experimentally observed total nongeminate recombination more complex, and different mechanisms dominate under different experimental settings and for varying charge car-rier densities. However, as we have shown, anisotropy in the

charge transport will affect the capture coeffi cients for both trap-assisted and free-to-free recombination. We argue that in ordered systems the effect of anisotropic charge transport on recombination cannot be neglected.

Acknowledgements The Academy of Finland is acknowledged for funding through projects #135262 and #141115. M.N. acknowledges funding from the Magnus Ehrnrooth Foundation, O.J.S from the National Graduate School of Nanosciences (Nano-NGS), and R.Ö. a personal grant from The Swedish Cultural Foundation in Finland.

Received: May 28, 2014 Revised: October 13, 2014

Published online:

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Table 1. Input parameters for device modeling.

Temperature 300 K

Dielectric constant 3.0

Bandgap 1.0 eV

Semiconductor thickness 250 nm

Electron mobility 5 × 10 −8 m 2 V −1 s −1

Hole mobility 5 × 10 −8 m 2 V −1 s −1

Effective density of states 10 26 m −3

Electron and hole injection barrier 0.05 eV

Charge generation rate at 1 mW cm −2 10 26 m −3 s −1

Figure 5. Simulated light ideality factors m for a 2D Langevin system with varying trap depths ( E trap ) for N t = 10 17 cm −3 , and C p = C p,2D as per Equation ( 13) . For simplicity the SRH capture coeffi cients for electrons and holes were assumed to be equal in the simulations.

FULL P

APER

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (7 of 7) 1400890wileyonlinelibrary.comAdv. Energy Mater. 2014, 1400890


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2D and Trap-Assisted 2D Langevin Recombination in Polymer:Fullerene Blends

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