nanocrystal bilayer hybrid solar cells: An analytical quantum efficiency model study

Organic Electronics 47 (2017) 108e116

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Limiting factors of photon-to-current conversion in polymer/ nanocrystal bilayer hybrid solar cells: An analytical quantum efficiency model study Fan Wu a, b, *, Xiaoyi Li a, b, Tiansheng Zhang a, b, Yanhua Tong c a b c

School of Science, Huzhou University, Huzhou, 313000, Zhejiang Province, China Key Lab of Optoelectronic Materials and Devices, Huzhou University, Huzhou, 313000, Zhejiang Province, China School of Engineering, Huzhou University, Huzhou, 313000, Zhejiang Province, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 November 2016 Received in revised form 18 March 2017 Accepted 2 May 2017 Available online 3 May 2017

This paper introduces an analytical external quantum efficiency (EQE) model of planar hybrid solar cells (HSCs) based on photon-to-current conversion processes and uses this to investigate the factors that limit the maximum EQE (EQEm) of devices; i.e., the photon absorption coefficient a, exciton diffusion coefficient Dz, exciton lifetime tz, exciton dissociation rate kdis, electron diffusion coefficient De, electron lifetime te, nanocrystals thickness d, and thickness of the polymer l. Our simulations indicate that relying solely on modifying kdis, De, or te cannot achieve a breakthrough increase in the EQEm of planar HSCs. However, increasing a, Dz, or tz could potentially lead to a large EQEm (30e100%), especially in the context of high kdis values. Moreover, the calculation results indicate that although both Dz and tz contribute to the exciton diffusion length (Lz) via the equation L2z ¼ Dztz, the EQEm has an asymmetric dependence on these variables. With a small kdis (i.e., <104 cm/s), an increase in Dz results in an initial increase and then decrease in EQEm, resulting in a peak value that increases with increasing kdis. When kdis is sufficiently large (~105 cm/s), the EQEm becomes saturated after the initial increase. Thus, although an increase in Dz can adversely affect device performance when the kdis is lower than 104 cm/s, increasing tz always improves device performance, regardless of large kdis becomes. This behavior can be attributed to the detrimental effect of excitons accumulating at the D/A interface, and can be used to optimize the material design and device engineering of planar HSCs and related solar cells for maximum photon-tocurrent conversion performance. In addition, we also demonstrate that the model can fit to the experimental data. © 2017 Elsevier B.V. All rights reserved.

Keywords: Bilayer hybrid solar cells Quantum efficiency model Photon-to-current conversion

1. Introduction Polymer/inorganic nanocrystal hybrid solar cells (HSCs) have garnered significant interest in recent years because of their low cost, mechanical flexibility, and successful integration of organic and inorganic materials [1,2]. Their hybrid fabrication combines the unique advantages of organic and inorganic materials, such as the solution processability and high photosensitivity of polymers, with the high electron mobility and physical/chemical stability of inorganic semiconductors [1e8]. There are currently two architectures commonly used to construct the active layers in HSCs: bilayer

* Corresponding author. School of Science, Huzhou University, Huzhou, 313000, Zhejiang Province, China. E-mail address: [email protected] (F. Wu). http://dx.doi.org/10.1016/j.orgel.2017.05.003 1566-1199/© 2017 Elsevier B.V. All rights reserved.

planar heterojunctions (PHJs) [3e5], which have a simple polymer/ nanocrystal bilayer structure, and bulk heterojunction (BHJs) [6e8], in which the polymer is penetrated by certain nanostructures such as nanoparticles [6], nanorods [7], and nanoarrays [8]. The PHJ architecture has good operational potential in terms of its ease of device fabrication, but an undesirable photon-to-current conversion means that device performance is usually not that impressive. More specifically, the maximum external quantum efficiency (EQEm) that has been reported in the literature is about 0.5e5% [9e13]. The poor EQEm in bilayer HSCs can be explained primarily by the limited interface area; hence, in response to this, many researchers have used a BHJ architecture constructed by embedding nanorod arrays into a polymer. This approach increases the available interface area, thereby dissociating more electrons [14e17]. The incorporation of a ZnO nanorod array into a polymer,

F. Wu et al. / Organic Electronics 47 (2017) 108e116

for example, can increase the interface area by a factor of ~45 when compared to a ZnO film/polymer bilayer structure [9e16]. Note that these comparison is based on nanorods with a density of between 4  102 and 5  102 mm2, and with lengths and diameters of ~400 nm and ~80 nm, respectively. However, despite the increase in surface area, nanorod-array BHJ devices show no significant improvement in EQEm over PHJ devices, with only a 20% increase having been achieved (which represents an improvement factor of 4) [9,10]. The large discrepancy between the significant increase in interface area and moderate increase in EQEm indicates that the interface area is not the only factor limiting the EQEm in bilayer HSCs. As will be discussed later, the limited interface area in bilayer HSCs actually originates from a short exciton diffusion length (Lz) [18]. We believe that this is why investigations focusing on the interface area have not succeeded in isolating the fundamental factors that determine photon-to-current conversion in bilayer HSCs. As shown in Fig. 1, the ideal conversion of sunlight into electricity by an HSC device involves four key steps: the formation of excitons upon photon absorption, diffusion of excitons within the polymer to the D/A interface, dissociation of excitons at the interface to generate free charge carriers, and transport of charge carriers within the inorganic nanocrystal and the polymer to their respective electrodes [1,2,19]. The factors that govern these key optoelectronic responses will influence the final photocurrent generation, and so optimizing each step is obviously fundamental to extracting as much energy as possible from the device [2]. Furthermore, it is the optoelectronic processes that correlate with the photo-induced charge generation, transfer, and collection that should be responsible for the limited photocurrent generation in bilayer HSCs [2,19]. A detailed understanding of the specific factors that limit photon-to-current conversion in bilayer HSCs is therefore required; and to this end, modeling the kinetics of the optoelectronic processes that determine the overall photocurrent should help in understanding the factors that limit quantum efficiency in bilayer HSCs. This, in turn, will be beneficial to material synthesis design and device engineering. There have been several theoretical models dealing with interfacial charge separation and recombination developed in the past few years, but most of these have been based on either Monte Carlo simulations [20e22] or numerical calculations [23]. Only a few

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models have offered analytical expressions for dealing with these processes [24,25,26], but even then the focus has mainly been on understanding the influences of energetic disorder [16], interface morphology [21,22], current-voltage characteristics [25], and charge carrier transport [26]. As a result, there have been few quantitative analyses of the factors limiting photon-to-current conversion in bilayer HSCs. In this paper, an analytical quantum efficiency model involving of all the optoelectronic processes in bilayer HSCs is described, and the factors limiting the EQEm are systematically studied. Note that although bulk HSCs that blend nanowires or nanoparticles into a polymer produce higher efficiency than a bilayer architecture due to their greater D/A interface, we found that the dynamics of charge transport also have an important influence on the EQEm of bilayer HSCs. Based on this, different material design and device engineering approaches for achieving a high quantum efficiency in bilayer HSCs are outlined with the aim of maximizing their photon-to-current conversion.

2. Theory and model The optoelectronic processes involved in bilayer HSCs depicted in Fig. 1 show that the drift rate of electrons in nanocrystals (e.g., TiO2 and ZnO) caused by the built-in electric field is at least an order of magnitude lower than the diffusion rate, and that the electric field is usually not strong enough to induce exciton dissociation [27,28]. The dimensions and high dielectric constant of the nanocrystal also prevent a significant electric field from existing in a nanocrystalline semiconductor [29], and hence, any influences from the electric field can be ignored to simplify the model. In addition, the low mobility and fast recombination means that the contribution of holes to the photocurrent does not need to be considered [2,24,26].

2.1. Exciton transport Under steady-state illumination conditions, the generation, diffusion, and recombination of excitons in conjugated polymers can be described by the continuity equation [18,30,31]:

vZðxÞ v2 ZðxÞ ZðxÞ ¼ aðlÞI0 eaðlÞx þ DZ  ¼ 0: vt tZ vx2

(1)

where Z(x) is the steady-state exciton density, a(l) is the absorption coefficient of the polymer as a function of the photon wavelength l, Dz is the diffusion coefficient of excitons, tz is the exciton lifetime, and I0 is the incident light intensity. This equation can be rewritten as:

v2 ZðxÞ aðlÞI0 aðlÞx 2 ¼ b1 ZðxÞ  e ; DZ vx2

(2a)

in which:

sffiffiffiffiffiffiffiffiffiffiffi 1 : b1 ¼ DZ tZ Fig. 1. Illustration of optoelectronic processes in the planar HSCs under illumination. The coordinates x ¼ -d, 0, and l indicate the electrode/crystalline interface, D/A interface, and polymer/metal electrode interface, respectively. The HOMO and LUMO indicate the highest occupied molecular orbital and lowest unoccupied molecular orbital energy levels of polymer, respectively. CB and VB are the conduction band and valence band of nanocrystalline, respectively. Other physical parameters in picture are defined in the main text afterwards.

(2b)

If we assume that the excitons dissociate at the D/A interface at a constant rate of kdis (cm/s) [32,33], then the exciton flow at x ¼ l can be ignored In other words, excitons reaching the polymer/electrode interface (x ¼ l) will be reflected and diffuse back into the inner layer of the polymer [35]. The boundary conditions required for the solution of Eq. (2a) are therefore as follows:

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DZ

F. Wu et al. / Organic Electronics 47 (2017) 108e116

 vZðxÞ ¼ kdis Zð0Þ; vx x¼0

(3a)

 vZðxÞ ¼ 0; vx x¼l

sffiffiffiffiffiffiffiffiffiffi 1 b2 ¼ Dp te

(9)

(3b) 2.3. Electron transport

where kdisZ(0) represents the annihilation flux density of excitons at the D/A interface. Using the above boundary conditions, the solution of Eq. (2a) can be written as:

The transport of electrons in a nanocrystal can be described by the following continuity equation [36]:

ZðxÞ ¼ Aeb1 x þ Beb1 x þ Ceax ;

vnðxÞ v2 nðxÞ nðxÞ ¼ De  ¼0 vt te vx2

(4a)

where:

A¼

aeal ðkdis þ b1 DÞ  b1 eb1 l ðkdis þ aDÞ    ; C  b1 kdis eb1 l þ eb1 l þ Db1 eb1 l  eb1 l

B ¼ C

C¼

aeal ðkdis  b1 DÞ þ b1 eb1 l ðkdis þ aDÞ     ; b1 kdis eb1 l þ eb1 l þ Db1 eb1 l  eb1 l

aI0 : 2 DZ b1  a2 

(4b)

(4c)

(4d)

(10)

where te is the electron lifetime in the nanocrystals and n(x) is the steady-state electron density. The effects of electron trapping and detrapping (Fig. 1) are related to the effective diffusion coefficient of electrons, De, which can be determined by the following expression: De ¼ Dn  (ktrap/ktrapþkdetrap) [37], where D is the diffusion coefficient of electrons in the conduction band. Only De was taken into account in this model without focusing on the trapping effects, and hence, Eq. (5) can be rewritten as:

v2 nðxÞ ¼ b2 nðxÞ vx2

(11a)

where:

sffiffiffiffiffiffiffiffiffiffi 1 b2 ¼ De te

2.2. Hole transport The diffusion equation for the hole concentration is given by

Dp

v2 pðxÞ pðxÞ  ¼0 tp vx2

(5)

where Dp is diffusion coefficient of holes, tp is the hole lifetime in the polymer, p(x) is the steady-state hole density. The hole flow at the polymer/nanocrystalline interface (x ¼ 0) is determined by kdis, and the extraction of holes at nanocrystalline/metal electrode contact (x ¼ l) is assumed to proceed at the constant rate of kp. Therefore, the boundary conditions are

Dp

 vpðxÞ ¼ kdis Zð0Þ vx x¼0

(6a)

Dp

 vpðxÞ ¼ kp pðlÞ vx x¼l

(6b)

pðxÞ ¼ A1 e

þ A2 eb2 x ;

(7)

with the following coefficients:



A1 ¼

A2 ¼

kext Dp b2



kdis Zð0Þ þ A1 Dp b2

where

De

 vnðxÞ ¼ kdis Zð0Þ vx x¼0

(12a)

De

 vnðxÞ ¼ kext nðdÞ vx x¼d

(12b)

Under the above conditions, the solution of Eq. (6a) can be written as:

nðxÞ ¼ A1 eb2 x þ A2 eb2 x

(13)

  kdis Zð0Þ 1  Dkextb ebd f 2 h  i   A1 ¼ De b2 1 þ Dkextb eb2 d þ Dkextb  1 eb2 d e 2

A2 ¼ ebl

kdis Zð0Þ 1  h  i   Dp b2 1 þ Dkextb eb2 l þ Dkextb  1 eb2 l p 2

The electron flow at the polymer/nanocrystalline interface (x ¼ 0) is determined by kdis; and so at the nanocrystalline/ITO electrode contact, the electrons are efficiently drawn off as a photocurrent. Thus, if the extraction of electrons at x ¼ d is assumed to proceed at a constant rate of kext [26,37], then the boundary conditions required for the solution of Eq. (6a) are:

with the following coefficients:

Under the above conditions, the solution of Eq. (5) can be written as: b2 x

(11b)

(8a)

e 2

kdis Zð0Þ þ A1 De b2

(14b)

At this stage, the photocurrent density from electron and hole contributions is given by Ref. [36e38]:

p 2

j ¼ qDe (8b)

(14a)

  vnðxÞ vpðxÞ þ qD p vx x¼d vx x¼l

(15)

and the quantum efficiency (incident photon-to-electron) can be written as:

F. Wu et al. / Organic Electronics 47 (2017) 108e116

EQEðlÞ ¼ j=qI0

(16)

In this paper, we use the maximum absorption coefficient of the conjugated polymer to obtain the EQEm value by Eq. (16), and then study the factors that limit the optoelectronic processes in bilayer HSCs.

EQEm ¼ j=qI0 jaðlÞ¼maxa

(17)

3. Results and discussion The dependencies of EQEm on different kinetic parameters, as calculated using Eq. (12), are shown in Figs. 2e6. These calculations were based on P3HT as the polymer and ZnO nanocrystals, as these are typically used in HSCs [1,2,4,6,7,10,11]. The thicknesses of the polymer and nanocrystals were assumed to be l ¼ 200 nm and d ¼ 40 nm, respectively, based on previously reported experimental results [5,39e41]. The theoretical effects of different l and d values on EQEm were also studied and are presented in the supplementary material (Fig. S1). This revealed that l and d are not the primary limiting factors of EQEm in bilayer HSCs. 3.1. Effects of exciton dissociation rate kdis Polymer-metal oxide hybrid photovoltaic devices do not currently achieve power conversion efficiencies comparable to those of organic solar cells [42,43]; however, previous studies have not considered optimization of the electron transfer process at the D/A interface in HSCs [1e3]. As shown in Fig. 2, a higher kdis does indeed produce a higher calculated EQEm for the device, which indicates that exciton dissociation at the D/A interface is crucial to device performance. An improvement in exciton dissociation (i.e., a larger kdis) can be achieved by modifying the interface between the polymer and inorganic nanocrystals [2,3,39,44,45], but the poor compatibility between inorganic and polymer phases, as well as the serious charge recombination at the polymer/inorganic interface, present major hurdles to creating more efficient devices. The incompatibility between hydrophilic nanocrystals and hydrophobic polymers frequently causes phase separation and poor interfacial contact between D/A components, thereby limiting charge transfer from the polymer to the nanocrystals. Modification of the D/A

Fig. 2. Dependence of the calculated EQEm response on kdis, with d ¼ 80 nm, l ¼ 200 nm, De ¼ 2.5  105 cm2/s, Dp ¼ 1.0  105 cm/s, tp ¼ 1  106 s, Dz ¼ 3  103 cm2/s, tz ¼ 300  1012 s, te ¼ 1 ms, a ¼ 105 cm1, kp ¼ ∞, and kext ¼ ∞.

111

interface has been attempted using dyes [3,44], organic acids [39], inorganic materials [4] and self-assembled monolayers (SAMs) [45] to improve the quality of the interface, enhance the charge separation efficiency and reduce charge recombination [2,3,39,44,45]. These interfacial modifiers assist with exciton dissociation, which accounts for the enhanced short-circuit current Jsc. These experimental results agree well with the results in Fig. 2, in that a higher kdis will generate a higher EQEm, and so prove the validity of our model. When kdis is higher than 105 cm/s, which is about one order of magnitude lower than the exciton quenching velocity of ~106 cm/s at the interface between perylene bis(phenethylimide) and the substrate [46], the EQEm reaches a saturation value of ~8.5% (Fig. 2). This phenomenon indicates that a D/A interface with a kdis value higher than 105 approaches perfect dissociation, allowing excitons at the D/A interface to split rapidly. Although the theoretical saturation value (~8.5%) is much higher than typical experimental values without modification (i.e., 0.5e5%), the theoretical saturated EQEm is still very low. This indicates that no significant breakthrough can be achieved in improving the EQEm of bilayer HSC devices by relying solely on modifications to the D/A interface. For example, phenyl-C61-butyric acid (PCBA) is known to be a strong electron acceptor with a high electron affinity, yet a net electron charge transfer from ZnO to PCBA is expected to occur [45]. The EQEm in P3HT/ZnO bilayer HSCs modified by a SAM of PCBA can reach 8e9% [45], which represents a big improvement over the EQEm of unmodified devices (i.e., 3%). Nevertheless, a higher EQEm in bilayer HSCs has not been reported when relying solely on enhancing the dissociation of excitons through interfacial modification, not including the enhanced photon absorption caused by the modification material itself. This result agrees with our theoretical prediction in Fig. 2. 3.2. Photon absorption The illumination intensity I exponentially increases in the polymer region according to the equation I ¼ I0exp(ax) [46], where I0 is the incident intensity and x is the illumination depth. Based on this, the light attenuation I/I0 was calculated at different illumination depths x in the polymer region and with different values of a using MATLAB software (Version 7.0). The dependence of I/I0 on x and a is depicted in Fig. 3a. Typically, the max a of the conjugated polymer is 105 cm1 [2,44,47], and the illumination depth (exciton generation range) of about 400 nm is much larger than the exciton diffusion length of 5e20 nm) [1,2,18]. Consequently, by enhancing the value of a in the polymer, the illumination depth can be shortened (Fig. 3a). As the value of a reaches 107 cm1, the effective region for photon absorption falls to 5 nm (Fig. 3a), which means that all of the photon-generated excitons are located within the diffusion length of the D/A interface. This means that the EQEm should increase with a, which agrees with the dependencies of EQEm on a with different values of kdis presented in Fig. 3b. These values were calculated by MATLAB based on Eq. (12) using values determined from previously reported experimental results [5,20,24,26,30,36e37,39e41]: i.e., d ¼ 80 nm, l ¼ 200 nm, De ¼ 2.5  105 cm2/s, Dz ¼ 3  103 cm2/s, tz ¼ 300  1012 s, te ¼ 1 ms and kext ¼ ∞. We see from this that even if a reaches 107 cm1, the excitons generated by l illumination (for which a(l) ¼ maximum a) remain near the D/A interface due to the greatly shortened illumination depth. However, if kdis is very low (i.e., 102 cm/s) then the excitons cannot dissociate quickly, which still results in a low EQEm (Fig. 3b). As expected, the EQEm of a device is enhanced when the high concentration of excitons near the D/A interface undergoes rapid dissociation as a result of an increase in kdis. The theoretical EQEm can exceed 80% when kdis

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F. Wu et al. / Organic Electronics 47 (2017) 108e116

Fig. 3. (a) Schematic illustration of the light attenuation in the polymer region with different a in polymer. I0 is the incident intensity, and I is the intensity at the illumination depth of x nm. (b) Dependence of the calculated EQEm response on a and kdis, other parameters for calculation are the same as those in Fig. 2.

approaches 105 cm/s when a ¼ 105 cm1, and can approach 100% when kdis approaches 105 cm/s when a ¼ 107108 cm1. Focus should therefore be given to the synthesis of high-absorption conjugated polymers that are compatible with nanocrystals and suitable for use in HSCs.

3.3. Exciton diffusion The generation of photocurrent following the formation of excitons in the polymer due to photo-excitation requires the effective diffusion of excitons toward the D/A interface, which is where exciton dissociation occurs [1,2,18,19,17,28]. Normally, only those excitons that are generated far from the D/A interface, but within the range of Lz, can contribute to the photocurrent generation [2]. The Lz of a conjugated polymer therefore plays an important role in determining the photovoltaic performance. In recent years, the power conversion efficiency of a polymer/fullerene derivative solar cells has reached levels of 8e10% [42,43], but further increases have been limited. This is due in part to the very short Lz (in the range of 5e20 nm) of organic semiconductors, which is typically much smaller than the optical absorption length (as discussed in Section 3.2). More efficient exciton diffusion is therefore desirable, as this would improve device performance and allow the use of less complex device architectures. The diffusion coefficient Dz and lifetime tz of the excitons are two parameters that are normally important in describing the diffusion properties of excitons in a polymer [2,48]. Specifically, the exciton diffusion length Lz is dependent on Dz and tz according to the following equation: L2z ¼ Dztz [2,48]. Most studies into the Lz in organic semiconductors, however, have focused on the measurement methods used rather than finding a correlation between the photovoltaic properties and the Lz values [49,50]. Furthermore, few studies have investigated the effects of Dz and tz on the EQEm performance, respectively. Here, we aim to deepen our understanding of the effects of Dz and tz on EQEm in bilayer HSCs using our analytical model.

3.3.1. Effects of exciton diffusion coefficient Dz Exciton diffusion is one of the most important photophysical properties of solar cells. Fig. 4a depicts the dependence of EQEm on Dz (¼3  1043  101 cm2/s) and kdis (¼102105 cm/s), which was calculated by MATLAB software (Version 7.0) using Eq. (12) and the following experimentally determined parameters [5,20,24,26,30,36,37,39e41]: d ¼ 80 nm, l ¼ 200 nm, De ¼ 2.5  105 cm2/s, a ¼ 1  105 cm1, tz ¼ 300  1012 s, te ¼ 1 ms, and kext ¼ ∞. It is clear from these results that an increase in Dz facilitates faster exciton diffusion to the D/A interface; however, only the excitons dissociated at the D/A interface can effectively contribute to the photocurrent [2,33]. With a small kdis (i.e., <104 cm/s), the increase in Dz results in an initial increase and then a decrease in EQEm, leading to a peak EQEm value at a certain Dz as shown in Fig. 4bed (i.e., regions IIII in Fig. 4a). This peak EQEm value also increases with kdis from 102 to 104 cm/s and the range over which the EQEm increases in the DzEQE profile gradually becomes elongated. When the kdis is sufficiently large (~105 cm/s), the EQEm increases with Dz to a saturation value of 35% (Fig. 4e); this low EQEm value being attributed to a low tz of 300  1012 s, as will be discussed in Section 3.3.2. The presence of a peak EQEm value at a certain Dz in the case of low kdis conditions can be easily understood in terms of the accumulation of excitons at the D/A interface [51] (see Fig. 4f). Prior to this peak, the increase in EQE results from more excitons being diffused to the D/A interface; however, when the Dz becomes too high at a low kdis (e.g., Fig. 4bed), this inevitably leads to an accumulation of excitons at the D/A interface. This impedes the subsequent diffusion and dissociation of excitons toward the D/A interface, thereby reducing the photocurrent. It is also evident in Fig. 4bed that the range over which the EQEm decreases in the DzEQE profile is reduced by an increase in kdis, which means that a decrease in exciton accumulation can be achieved through an improvement in exciton dissociation (Fig. 4f). When the kdis reaches values as high as 105 cm/s, rapid exciton dissociation overcomes the detrimental effects of exciton accumulation, resulting in the saturated EQEm. It can therefore be concluded from Fig. 4 that the photon-to-current conversion is critically related to the value of Dz [18], and only in devices with a highly effective D/A interface for exciton dissociation (i.e., a kdis value approaching 105 cm/s) can an increase in Dz effectively contribute to improving the device's EQEm [33]. Otherwise, if kdis is much lower than 105 cm/s, an increase in Dz will in fact reduce device performance. 3.3.2. Effect of exciton lifetime tz The calculated dependence of EQE on tz (¼1010106 s) with different kdis (¼102105 cm/s) values, as shown in Fig. 5, was calculated by MATLAB software (Version 7.0) using Eq. (12) and the following previously reported experimental values [5,20,24,26,30,34e38]: d ¼ 80 nm, l ¼ 200 nm, a ¼ 1  105 cm1, De ¼ 2.5  105 cm2/s, Dz ¼ 3  103 cm2/s, te ¼ 1 ms and kext ¼ ∞. Although both Dz and tz contribute to Lz via the equation L2z ¼ Dztz [2,48], the profiles of EQE with respect to tz and Dz show asymmetrical characteristics. There is no response in the tzEQEm profile shown in Fig. 5, which adopts a pattern of increase followed by decrease, whereas the calculated results show that a larger kdis promotes a higher EQEm for a given tz. Moreover, a larger tz always yields a larger EQEm, and this property is independent of kdis. The asymmetrical dependencies of the calculated EQEm responses with respect to De and te can also be seen in Fig. S2 of the supplementary data. The different dependencies of EQEm on tz and Dz can also be explained by Fig. 4f, which shows that changes in tz and Dz have different effects on the exciton transport properties of the polymer. In the case of a large tz and small Dz, the excitons can slowly diffuse

F. Wu et al. / Organic Electronics 47 (2017) 108e116

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Fig. 4. (a) Dependence of the calculated EQEm response on Dz (¼3  1043  101 cm2/s) and kdis (¼102105 cm/s), other parameters for calculation are the same as those in Fig. 2. (bed) are the corresponding region of IIV in (a), which are divided by different kdis value. (f) Illustration of the exciton diffusion and dissociation with slow Dz and fast Dz in polymer.

Fig. 5. Dependence of the calculated EQEm response on tz and kdis, other parameters for calculation are the same as those in Fig. 2.

to the D/A interface for gradual dissociation. This avoids the detrimental effects of exciton accumulation at the D/A interface by providing a fast Dz (as discussed in Section 3.4.1), with the larger tz meaning that excitons have a longer time to decay to their ground states before reaching the D/A interface [52]. The possibility of excitons generated far from the D/A interface diffusing to the interface for dissociation is therefore increased, resulting in a larger EQEm. This point can be further demonstrated by the calculated results in Fig. 5, which depict the case for a larger kdis of 105 cm/s and a larger tz of 106 s, which facilitates the diffusion of excitons generated far from the D/A interface to the interface [18]. Their subsequent rapid dissociation results in a high EQEm that can exceed 50%, which is clearly much higher than the saturation EQE m value of 35% achieved in Section 3.3.1 by increasing Dz. Thus, increasing tz can potentially lead to a high EQEm in HSC devices. From the above discussion, it is apparent that extending Lz is indeed an important consideration in optimizing the design of HSCs and polymer solar cells, because this can be achieved by increasing Dz and tz [18]. For example, the use of bilayer and long

conjugated chromophores is beneficial in phenyl-cored thiophene dendrimer synthesis, where long-range energy transfer is required for better exciton diffusion [18]. The non-conjugated or conjugated parts, which do not participate in energy transfer, should be minimized during the synthesis of such materials. However, it should be noted that an increase in Dz is harmful to device performance when the kdis is lower than 104 cm/s, whereas increasing tz is always beneficial regardless of how large kdis becomes. In addition, high EQEm values can be produced by a combination of a long Lz (i.e., the capacity of excitons to diffuse to the D/A interface) and a large kdis based on the calculations of Figs. 4 and 5. This means that as long as just one of these factors is poor, a desirable EQEm cannot be achieved. This line of thought can explain the phenomenon whereby only a slight increase in EQEm is observed in nanorod-array bulk HSCs when compared to an equivalent bilayer HSC. That is, as the interfacial properties of polymer/nanoarrays are not normally as ideal as expected (e.g., low kdis of 102 cm/s) [9e17,26], then even if the capacity for exciton diffusion to the D/A interface is enhanced in bulk HSCs they still cannot provide an effective number of electrons to notably enhance the EQEm. 3.4. Electron diffusion Once excitons have dissociated into free electrons and holes, the generation of photocurrent requires the effective diffusion of electrons toward the conducting glass electrode [1,2]. Normally, only those electrons with an electron diffusion length, Le, comparable to the thickness of the nanocrystal can possibly be collected, and so the efficiency of electron collection from a nanocrystalline layer will depend on the degree of competition between recombination and the transport of electrons toward the conducting glass electrode [53]. This competition can be conveniently described by the diffusion length, Le [50e53], which is the average distance an

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injected electron must travel through the nanocrystalline layer (d) before recombination; i.e., the distance electrons diffuse toward the collection electrode. This parameter is normally evaluated from the relation: Le ¼ (Dete)1/2 [53e56]. Fig. 6a and b shows the dependence of the calculated EQEm response on Le (i.e., De and te) with kdis ¼ 102 cm/s (Fig. 6a) and kdis ¼ 105 (Fig. 6b), as calculated by MATLAB software (Version 7.0) using Eq. (12) and previously reported experimental values [5,20,24,26,30,36,37,39e42] of: d ¼ 80 nm, l ¼ 200 nm, a ¼ 1  105 cm1, Dz ¼ 3  103 cm2/s, tz ¼ 300  1012 s, kdis ¼ 105 cm/s and kext ¼ ∞. The dependence of EQEm on tz and Dz is clearly symmetrical, with the calculated EQEm response to L2e exhibiting four distinct regions: (I) essentially no EQEm, (II) an exponential increase in EQEm, (III) a quadratic increase in EQEm, and (IV) a saturated EQEm. The calculated EQEm in Region I is very small (~0) when Le < 10 nm, and there is no increase in the EQEm response. This is easily explained by the fact that when Le ≪ d (d ¼ 80 nm in our calculation), most of the injected charge will be lost before it is collected, resulting in a very low EQEm of <0.05% in Region I. In contrast, when Le is comparable to d, EQEm becomes sensitive to Le because the reduced number of lost electrons increases the rate of electron collection. This, in turn, results in an exponential increase in EQEm with increasing Le in Region II. When Le > d, as in Region III, only a small fraction of the injected charge will be lost before it is collected, producing a quadratic increase in EQEm. In Region IV, where Le » d, the recombination and transport of electrons is balanced and the EQEm response becomes saturated. Note that the saturated value in this case is low due to the very low values of Dz (¼3  103 cm2/s) and tz (¼300  1012 s) that were used. A comparison of Fig. 6a and b reveals that a larger Le (i.e., larger De or te) with a higher kdis generates a higher EQEm because of the greater number of electrons being dissociated from the D/A interface, which are then able to diffuse toward the electrode. However, even with a large De, of 101 cm2/s, te of 101 s and high kdis of 105 cm/s (as in Fig. 6), the theoretical saturated EQEm is still lower than 10%. This is far below the values returned by calculations that

optimize the Dz and tz parameters (see Figs. 4 and 5), which indicates that the EQEm of a bilayer HSC cannot be significantly improved by relying solely on optimizing De and te. The dependences of EQEm on the th and Dh have shown in the similar characteristics to De and te (Fig. S3 in Supplementary Material).

Fig. 6. Dependence of the calculated EQE responses on De and te with kdis ¼ 102 cm/s (a) and 105 cm/s (b), other parameters for calculation are the same as those in Fig. 2. The black dot lines denote the value of L2e with different De and te.

Fig. 7. The EQE(l) of P3HT/ZnO and P3HT/C16SH/ZnO solar cells presented in literature [56] were used to fit our model. The inset is absorption coefficient curve of P3HT which was extracted from Ref. [10].

3.5. Application to experimental data In order to demonstrate the model can be applied to experimental results, we extracted the experimental absorption coefficient curve a(l) (400e700 nm) of P3HT in Ref. [10], and EQE(l) data of bilayer solar cells that was constructed by P3HT with ZnO, and hexadecanethiol (C16SH) modified ZnO in Ref. [57] using software of GetData Graph Digitizer. As shown in Fig. 7, the calculation of EQE(l) curve based on eq. (16) and experimental a(l) can fit the reported experimental data in both P3HT/ZnO and P3HT/C16SH/ ZnO devices. In the calculation and fitting, only modulation of the kdis can fit the two device EQE(l), other parameters were kept the same and the values were taken from experimental results in references. The fitting parameters are L ¼ 6 nm [57], d ¼ 30 nm [57], De ¼ 2.5  105 cm/s [24,26], te ¼ 1  103 s [24,26], Dp ¼ 1.4  102 cm/s [58], tp ¼ 1  106 s [59], Dz ¼ 3  104 cm/s [30e36], tz ¼ 300  1012 s [30e36], kext ¼ ∞ [26], kp ¼ ∞ [26], And then we obtained the value of kdis ¼ 811 cm/s in P3HT/ZnO device, while a higher value of kdis ¼ 1050 cm/s was obtained in P3HT/ C16SH/ZnO device, indicating that the extion dissociation rate in C16SH modified P3HT/ZnO interface is better than the unmodified interface. This fitting result further confirms their experimental conclusion in Ref. [54] that the charge separation in P3HT/ZnO interface was enhanced after introducing the interfacial modifier of C16SH, which was caused by fewer defect sites in C16SH modified ZnO and a stronger inter-chain interaction between ZnO and P3HT. Therefore, our fitting results demonstrate that the increased kdis should be responsible for the increased EQE in P3HT/C16SH/ZnO solar cells. Our model provides a general physical framework for interpreting of EQE and may provide a theoretical basis to use EQE spectra to gain insight into the dynamics of charge generation, charge transport, and device performance involved in the HSC devices.

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4. Conclusion The kinetic parameters that correlate to device EQEm were systematically investigated using a quantum efficiency model. This revealed a close correlation between EQEm and a number of optoelectronic processes, which can be expressed in terms of the parameters a, kdis, Dz, tz, De, and te. A higher kdis generates a higher EQEm, which reaches saturation when kdis is higher than 105 cm/s. The EQEm also increases with a, and can theoretically exceed 80% when kdis approaches 105 cm/s when a > 105 cm1. Although both Dz and tz contribute to the exciton diffusion length Lz via the equation L2z ¼ Dztz, the profiles of EQEm with respect to tz and Dz show asymmetrical characteristics. With a small dissociation rate (kdis z 102104 cm/s), an increase in Dz results in an initial increase in EQEm followed by a decrease, which is caused by the accumulation of excitons at the D/A interface. When the kdis is sufficiently high (i.e., kdis z 105 cm/s), there is no decrease in the EQEm value with an increase in Dz. In contrast, a larger tz always yields a larger EQEm, and this property is independent of kdis except at higher values of kdis, which promote a higher EQEm for a given tz. A larger Le (i.e., larger De, or te) with a higher kdis can produce a higher EQEm due to the greater number of electrons being generated from the D/ A interface, which are then able to diffuse toward the electrode. When Le » d, the recombination and transport of electrons is balanced, resulting in a saturated EQEm responses. The calculated results presented here indicate that relying solely on the modification of kdis, De, or te cannot achieve a high EQEm in bilayer HSCs, but an increase in a, Dz or tz represents a viable strategy for achieving large EQEm values, especially at high interfacial kdis values. Also, we demonstrated that our model may provide a theoretical basis to use EQE(l) to gain insight into the dynamics of charge generation, charge transport, and device performance involved in the HSC devices. These results therefore provide information crucial to the optimization of device EQEm, and which may provide greater insight into the design and optimization of materials for HSCs. Acknowledgements This work was supported by the National Natural Science Foundation of China (21607041; 11547312; 11647306), Zhejiang Provincial Natural Science Foundation of China (LQ14F040003), Science and Technology Planning Project of Zhejiang Province, China (2017C33240) and the Seed Fund of Young Scientific Research Talents of Huzhou University (RK21056). The authors also acknowledge the support provided under the “1112 Talents Project” of Huzhou City. Notation HSCs PHJ BHJ EQEm

a

Dz

tz

kdis De Dp

te tp d l Lz

hybrid solar cells planar heterojunction bulk heterojunction maximum external quantum efficiency the photon absorption coefficient the exciton diffusion coefficient the exciton lifetime the exciton dissociation rate the electron diffusion coefficient the hole diffusion coefficient the electron lifetime the hole lifetime the thickness of the nanocrystal layer the thickness of the polymer layer exciton diffusion length

Le n(x) Dn Z(x) kext kp

115

electron diffusion length the steady-state electron density the diffusion coefficient of electrons in conduction band the steady-state exciton density extraction of electrons rate extraction of holes rate

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