**2. Effects of optical interference on** *JSC*

*JSC* is directly related to the absorption ability of organic materials. It is believed that increasing the light harvesting ability of the active layer is an effective method to increase *JSC*. In order to increase *JSC*, some optical models (Pettersson, 1999; Peumans et al., 2003) have been built to optimize the active layer thickness. However, only optimizing the thickness for better light absorption is difficult to improve *JSC*. This is because that *PCE* depends not only on the light absorption, but also on exciton dissociation and charge collection. In polymer solar cells, a blend layer consisting of conjugated polymer as the electron donor and fullerene as the electron acceptor is always used as the active layer. For a well blended layer, the length scale of *D* and *A* phases is smaller than the exciton diffusion length (typically less than 10 nm), so that most of the generated excitons can diffuse to the *D/A* interface before they decay. Even if all the excitons can reach the *D/A* interface, not all of them can be dissociated into free carriers. The exciton-to-free-carrier dissociation probability is not 1 and depends on some factors such as electric field and temperature. When the active layer thickness is increased to optimize the light absorption, the electric field in the blend layer decreases, which lowers down the exciton-to-freecarrier probability and makes charge collection less effective simultaneously. As a result, *JSC* may become low, although the thickness has been optimized for better light absorption. Thus to obtain a higher *JSC*, both the optical and the electric properties should be considered at the same time.

Some previous works (Lacic et al., 2005; Monestier et al., 2007) studied the characteristic of *JSC*. However, they neglected the influence of exciton-to-free-carrier probability, which is important for polymer solar cells. Another study (Koster et al., 2005) considered this factor, but they neglected the optical interference effect, which is a basic property for the very thin organic film. All the above studies are based on the numerical method, and it is not easy to solve the equations and understand the direct influence of various parameters on *JSC*. In this part, a model predicting *JSC* is presented by using very simple analytical equations. Based on this model, the effects of optical interference on *JSC* is investigated. Besides, the carrier lifetime is also found to be an important factor. By considering the optical interference effect and the the carrier lifetime, it is found that when the lifetimes of both electrons and holes are long enough, the exciton-to-free-carrier dissociation probability plays a very important role for a thick active layer and *JSC* behaves wavelike with the variation of the active layer thickness; when the lifetime of one type of carrier is too short, the accumulation of charges appears near the electrode and *JSC* increases at the initial stage and then decreases rapidly with the increase of the active layer thickness.

### **2.1 Theory**

2 Solar Cells – New Aspects and Solutions

Besides the serious optical interference effect, *JSC* also suffers from the non-ideal free carrier generation, low mobility and short carrier lifetime. In order to reduce the exciton loss and guarantee the efficient carrier transport, the optimal interpenetrating network, or to say, the optimal morphology is desired in the bulk HJ structure. In order to achieve an optimal morphology, a thermal treatment is usually utilized in the device fabrication, especially for the widely used P3HT:PCBM solar devices. It is found that the sequence of the thermal treatment is critical for the device performance (Zhang et al., 2011). The polymer solar cells with the cathode confinement in the thermal treatment (post-annealed) show better performance than the solar cells without the cathode confinement in the thermal treatment (pre-annealed). The functions of the cathode confinement are investigated in this chapter by using X-ray photoelectron spectroscopy (XPS), atomic force microscopy (AFM), optical absorption analysis, and X-ray diffraction (XRD) analysis. It is found that the cathode confinement in the thermal treatment strengthens the contact between the active layer and the cathode by forming Al–O–C bonds and P3HT-Al complexes. The improved contact effectively improves the device charge collection ability. More importantly, it is found that the cathode confinement in the thermal treatment greatly improves the active layer morphology. The capped cathode effectively prevents the overgrowth of the PCBM molecules and, at the same time, increases the crystallization of P3HT during the thermal treatment. Thus, a better bicontinuous interpenetrating network is formed, which greatly reduces the exciton loss and improves the charge transport capability. Meanwhile, the enhanced crystallites of P3HT improve the absorption property of the active layer. All these aforementioned effects together can lead to the great performance improvement of polymer solar cells. Besides the thermal treatment sequence, temperature is another very important parameter in the annealing process. Various annealing temperatures have also been tested

The contents of this chapter are arranged as the following: Section 2 introduces the effects of the optical interference on *JSC* in polymer solar cells by considering the non-ideal free carrier generation, low mobility and short carrier lifetime at the same time; Section 3 investigates the influence of the sequence of the thermal treatment on the device performance with emphasis on the cathode confinement in the thermal treatment; based on the optical interference study and the proper thermal treatment sequence, the overall device optimization is presented in Section 4. At last, a short conclusion is given in Section 5.

*JSC* is directly related to the absorption ability of organic materials. It is believed that increasing the light harvesting ability of the active layer is an effective method to increase *JSC*. In order to increase *JSC*, some optical models (Pettersson, 1999; Peumans et al., 2003) have been built to optimize the active layer thickness. However, only optimizing the thickness for better light absorption is difficult to improve *JSC*. This is because that *PCE* depends not only on the light absorption, but also on exciton dissociation and charge collection. In polymer solar cells, a blend layer consisting of conjugated polymer as the electron donor and fullerene as the electron acceptor is always used as the active layer. For a well blended layer, the length scale of *D* and *A* phases is smaller than the exciton diffusion length (typically less than 10 nm), so that most of the generated excitons can diffuse to the *D/A* interface before they decay. Even if all the excitons can reach the *D/A* interface, not all of them can be dissociated into free carriers. The exciton-to-free-carrier

to find the optimized annealing condition in this chapter.

**2. Effects of optical interference on** *JSC*

### **2.1.1 Exciton generation**

The active layer in polymer solar cells absorbs the light energy when it is propagating through this layer. How much energy can be absorbed depends on the complex index of refraction *nni* of the materials. At the position z in the organic film (Fig. 1 (a)), the time average of the energy dissipated per second for a given wavelength of incident light can be calculated by

$$\mathcal{Q}(z,\mathcal{A}) = \frac{1}{2} c\varepsilon\_0 a\_j n \left| \overline{\mathbf{E}}(z) \right|^2 \tag{1}$$

where *c* is the vacuum speed of light, <sup>0</sup> the permittivity of vacuum, *n* the real index of refraction, the absorption coefficient, 4 / , and *E(z)* the electrical optical field at point *z*. *Q z*(, ) have the unit of <sup>3</sup> *W m*/ . Assuming that every photon generates one exciton, the exciton generation rate at position *z* in the material is given by

$$\mathbf{G}(z,\lambda) = \frac{\mathbf{Q}(z,\lambda)}{h\chi} = \frac{\lambda}{hc}\mathbf{Q}(z,\lambda) \tag{2}$$

where *h* is Planck constant, and is the frequency of incident light. The total excitons generated by the material at position *z* in solar spectrum are calculated by

$$\mathbf{G}(z) = \int\_{300}^{800} \mathbf{G}(z, \lambda) d\lambda \tag{3}$$

Effects of Optical Interference and Annealing on the

and (b) treating the multilayer as a virtual layer.

transmission coefficients for the whole multilayer are:

Because in the final layer, *Em* <sup>1</sup>

Where

 

where

the relationship as

 *j jj* 2 / 

Performance of Polymer/Fullerene Bulk Heterojunction Solar Cells 5

optical electric fields in the substrate (subscript *0*) and the final layer (subscript *m+1*) have

*S I LI*

 

Fig. 1. Multilayer structure in a polymer solar cell. (a) the optical electric field in each layer

0 21 <sup>11</sup> <sup>0</sup>

*E S*

*S E*

1 <sup>11</sup> <sup>0</sup>

( 1) ( 1)

*Em* <sup>1</sup> *<sup>t</sup> S E*

 

*r*

In order to get the optical electric field *Ej* (z) in layer *j*, *S* is divided into two parts,

1

*j*

*v*

1

*j v v v jj*

*S I LI* 

 

'

0 1 11 12 1

*E E SS E*

*S S EE E*

12 22 <sup>1</sup> 0 1 <sup>1</sup>

*<sup>m</sup> m m*

*<sup>v</sup> m m*

*n d* is phase change the wave experiences as it traverses in layer *j*. The

( 1) ( 1)

(8)

*v v v mm*

is 0, it can be derived that the complex reflection and

(9a)

(9b)

' " *S SLS jjj* (10)

(11a)

Here the integration is performed from 300 nm to 800 nm, which is because that beyond this range, only very weak light can be absorbed by P3HT: PCBM active layer. In inorganic solar cells, *Q z*(, ) is usually modeled by

$$Q(z, \mathcal{A}) = \alpha I\_0 e^{-\alpha z} \tag{4}$$

<sup>0</sup>*I* is the incident light intensity. Here, the optical interference effect of the materials is neglected. But in polymer solar cells, the active layers are so thin compared to the wavelength that the optical interference effect cannot be neglected.

### **2.1.2 Optical model**

In order to obtain the distribution of electromagnetic field in a multilayer structure, the optical transfer-matrix theory (TMF) is one of the most elegant methods. In this method, the light is treated as a propagating plane wave, which is transmitted and reflected on the interface. As shown in Fig. 1 (a), a polymer solar cell usually consists of a stack of several layers. Each layer can be treated to be smooth, homogenous and described by the same complex index of refraction *nni* . The optical electric field at any position in the stack is decomposed into two parts: an upstream component *E* and a downstream component *E* , as shown in Fig. 1 (a). According to Fresnel theory, the complex reflection and transmission coefficients for a propagating plane wave along the surface normal between two adjacent layers *j* and *k* are

$$r\_{jk} = \frac{\overline{n\_j} - \overline{n\_k}}{\overline{n\_j} + \overline{n\_k}} \tag{5a}$$

$$t\_{jk} = \frac{2\overline{n\_j}}{\overline{n\_j} + \overline{n\_k}}\tag{5b}$$

where *jk r* and *jk t* are the reflection coefficient and the transmission coefficient, *nj* and *nk* the complex index of refraction for layer *j* and layer *k*. So the interface matrix between the two adjacent layers is simply described as

$$I\_{jk} = \frac{1}{t\_{jk}} \begin{bmatrix} 1 & r\_{jk} \\ r\_{jk} & 1 \end{bmatrix} = \begin{bmatrix} \frac{\overline{n\_j} + \overline{n\_k}}{2\overline{n\_j}} & \frac{\overline{n\_j} - \overline{n\_k}}{2\overline{n\_j}} \\ \frac{\overline{n\_j} - \overline{n\_k}}{2\overline{n\_j}} & \frac{\overline{n\_j} + \overline{n\_k}}{2\overline{n\_j}} \end{bmatrix} \tag{6}$$

When light travels in layer *j* with the thickness *d*, the phase change can be described by the layer matrix (phase matrix)

$$L\_j = \begin{bmatrix} e^{-i\beta\_j} & \mathbf{0} \\ \mathbf{0} & e^{i\beta\_j} \end{bmatrix} \tag{7}$$

where *j jj* 2 / *n d* is phase change the wave experiences as it traverses in layer *j*. The optical electric fields in the substrate (subscript *0*) and the final layer (subscript *m+1*) have the relationship as

$$
\overline{\left[\frac{\overline{E\_0^+}}{E\_0^-}\right]} = \overline{S\left[\frac{\overline{E\_{m+1}^+}}{E\_{m+1}^-}\right]} = \overline{\left[\begin{matrix} S\_{11} & S\_{12} \\ S\_{12} & S\_{22} \end{matrix}\right]} \overline{\left[\frac{\overline{E\_{m+1}^+}}{E\_{m+1}^-}\right]} = \left(\prod\_{v=1}^m I\_{(v-1)v} L\_v\right) \bullet I\_{m(m+1)}\tag{8}
$$

Fig. 1. Multilayer structure in a polymer solar cell. (a) the optical electric field in each layer and (b) treating the multilayer as a virtual layer.

Because in the final layer, *Em* <sup>1</sup> is 0, it can be derived that the complex reflection and transmission coefficients for the whole multilayer are:

$$r = \frac{\overline{E\_0}}{\overline{E\_0}^+} = \frac{S\_{21}}{S\_{11}}\tag{9a}$$

$$t = \frac{\overline{E\_{m+1}^{+}}}{\overline{E\_{0}^{+}}} = \frac{1}{S\_{11}}\tag{9b}$$

In order to get the optical electric field *Ej* (z) in layer *j*, *S* is divided into two parts,

$$S = \stackrel{\circ}{S\_j} \stackrel{\circ}{L\_j} \stackrel{\circ}{S\_j} \tag{10}$$

Where

4 Solar Cells – New Aspects and Solutions

Here the integration is performed from 300 nm to 800 nm, which is because that beyond this range, only very weak light can be absorbed by P3HT: PCBM active layer. In inorganic solar

<sup>0</sup> (, ) *<sup>z</sup> Qz I e*

<sup>0</sup>*I* is the incident light intensity. Here, the optical interference effect of the materials is neglected. But in polymer solar cells, the active layers are so thin compared to the

In order to obtain the distribution of electromagnetic field in a multilayer structure, the optical transfer-matrix theory (TMF) is one of the most elegant methods. In this method, the light is treated as a propagating plane wave, which is transmitted and reflected on the interface. As shown in Fig. 1 (a), a polymer solar cell usually consists of a stack of several layers. Each layer can be treated to be smooth, homogenous and described by the same

is decomposed into two parts: an upstream component *E* and a downstream component *E* , as shown in Fig. 1 (a). According to Fresnel theory, the complex reflection and transmission coefficients for a propagating plane wave along the surface normal

*j k*

*n n*

*n n*

2 *<sup>j</sup>*

*n n*

where *jk r* and *jk t* are the reflection coefficient and the transmission coefficient, *nj* and *nk* the complex index of refraction for layer *j* and layer *k*. So the interface matrix between the

1 2 2 1

When light travels in layer *j* with the thickness *d*, the phase change can be described by the

*j*

*i j i <sup>e</sup> <sup>L</sup>*

0

*jk jk j k j k*

*t r nnnn*

2 2

*j j*

*n n*

*jk j j*

*r n n*

0

*e*

*j*

*j k j k*

*nnnn*

*n*

*j k*

*j k*

*jk*

*jk*

*t*

1

*jk*

*I*

*r*

 

wavelength that the optical interference effect cannot be neglected.

(4)

. The optical electric field at any position in the stack

(5a)

(5b)

(6)

(7)

cells, *Q z*(, )

**2.1.2 Optical model** 

complex index of refraction *nni*

between two adjacent layers *j* and *k* are

two adjacent layers is simply described as

layer matrix (phase matrix)

is usually modeled by

$$\boldsymbol{S}\_{\boldsymbol{j}} = \left( \prod\_{v=1}^{j-1} I\_{\{v-1\}v} \boldsymbol{L}\_v \right) \bullet \boldsymbol{I}\_{\boldsymbol{j}(j-1)} \tag{11a}$$

Effects of Optical Interference and Annealing on the

the probability of electron-hole pair dissociation,

Braun gives the simplified form for dissociation rate

pair binding energy described as <sup>2</sup> *Uq a B r* /(4 )

temperature, *F* the electric field and *<sup>r</sup>*

temperature keeps constant.

and the carrier lifetime

lifetime

**2.1.5** *JSC* **expression equations** 

Performance of Polymer/Fullerene Bulk Heterojunction Solar Cells 7

dissociation probability has been taken into consideration in polymer solar cells [13, 16]. The geminate recombination theory, first introduced by Onsager and refined by Brau later, gives

> ( ) (, ) ( ) *D D X*

where *Xk* is the decay rate to the ground state and *kD* the dissociation rate of a bound pair.

*kF k*

2

3

and 3 2 /(8 ) <sup>0</sup> *r B b qF kT*

the dielectric constant of the material. In equation

(, ) *SC J qP F T GL* (20)

 

(18)

(19)

. *T* is the

, the electric field *F*

*k F PFT*

/ ( ) 1

electron and hole drift lengths (which is the product of carrier mobility

where *G* is the average exciton generation rate in the active layer.

(a) (b) (c)

*D R*

*U kT B B*

*<sup>b</sup> k F ke b*

where *a* is the initial separation distance of a given electron-hole pair, *UB* is electron-hole

(19), *Rk* is a function of the carrier recombination. For simplification, we treat *Rk* as a constant. Thus, the dissociation probability *P* only depends on the electric field *F* when the

*JSC* is determined by the number of carriers collected by the electrodes in the period of their

their transit time (case I as in Fig 2 (a)), all generated free carriers can be collected by the

Fig. 2. Energy band diagrams under short circuit condition. (a) Case I: thickness is shorter than both drift lengths, (b) Case II: thickness is longer than hole drift length but shorter than electron drift length, c) Case III: thickness is longer than both hole and electron drift lengths.

electrodes. Considering the exciton-to-free-carrier dissociation probability *P*, *JSC* is

under short circuit condition. If the active layer thickness *L* is shorter than the

) or in other word, the lifetimes of both types of carriers exceed

 <sup>0</sup> 

$$\mathbf{S}\_{j}^{\prime} = \left( \prod\_{v=j+1}^{m} I\_{\{v-1\}v} L\_{v} \right) \bullet I\_{m\{m+1\}} \tag{11b}$$

At the down interface in layer *j*, the upstream optical electric field is denoted as

$$
\overline{E\_j^+} = t\_j^+ \bullet \overline{E\_0^+} = \frac{S\_{j11}^+}{S\_{j11}^+ S\_{j11}^+ + S\_{j12}^+ S\_{j21}^+ e^{i2\beta\_j}} \overline{E\_0^+} \tag{12}
$$

Similarly, at the up interface in layer *j*, the downstream optical electric field is

$$
\overline{E\_{j}^{-}} = t\_{j}^{-} \bullet \overline{E\_{0}^{+}} = \frac{S\_{j21}^{\cdot}}{S\_{j11}^{\cdot}} e^{i2\beta\_{j}} \overline{E\_{j}^{+}} \tag{13}
$$

The optical electric field ( ) *E z <sup>j</sup>* at any position *z* in layer *j* is the sum of upstream part ( ) *E z <sup>j</sup>* and downstream part ( ) *E z <sup>j</sup>* 

$$\overline{E\_j}(\mathbf{z}) = \overline{E\_j^+}(\mathbf{z}) + \overline{E\_j^-}(\mathbf{z}) = (t\_j^+ e^{i\beta\_j} + t\_j^- e^{-i\beta\_j}) \overline{E\_0^+} \tag{14}$$

### **2.1.3 Light loss due to the substrate**

Because the glass substrate is very thick compared to wavelength (usually 1mm>> wavelength), the optical interference effect in the substrate can be neglected. Here only the correction of the light intensity at the air/substrate and substrate/multilayer interfaces is made. As shown in Fig. 1 (b), the multilayer can be treated as a virtual layer whose complex reflection and transmission coefficients can be calculated using above equations. Then the irradiance to the multilayer is

$$I\_{\mathcal{S}} = T \left( \sum\_{i=0}^{\infty} \{ \boldsymbol{R}^\* \boldsymbol{R} \}^i \right) I\_0 = \frac{1 - \boldsymbol{R}^\*}{1 - \boldsymbol{R} \boldsymbol{R}^\*} I\_0 \tag{15}$$

*gI* is described as

$$I\_g = \frac{1}{2} c \varepsilon\_0 n\_g \left| E\_0^+ \right|^2 \tag{16}$$

It can be derived that

$$\left| E\_0^+ \right| = \sqrt{\frac{2(1 - \text{R}^\*) \,^\*I\_0}{\varepsilon\_0 c n\_g (1 - \text{R} \,^\*I)}}\tag{17}$$

#### **2.1.4 Free carrier generation**

When the excitons are generated, not all of them can be dissociated into free carriers. The dissociation probability depends on the electric field and temperature. Recently, the

*j vv v m m*

*S I LI*

 

*j j i*

*EtE E*

 

1

*m*

*v j*

At the down interface in layer *j*, the upstream optical electric field is denoted as

Similarly, at the up interface in layer *j*, the downstream optical electric field is

and downstream part ( ) *E z <sup>j</sup>*

irradiance to the multilayer is

*gI* is described as

It can be derived that

**2.1.4 Free carrier generation** 

*g*

*i*

**2.1.3 Light loss due to the substrate** 

( 1) ( 1)

" 11 0 0 '" '" <sup>2</sup> 11 11 12 21

*S*

*j*

*jj jj*

" 21 2

*S EtE eE S*

0 " 11 *<sup>j</sup> j i*

The optical electric field ( ) *E z <sup>j</sup>* at any position *z* in layer *j* is the sum of upstream part ( ) *E z <sup>j</sup>*

<sup>0</sup> () () () ( ) *j j i i E z E z E z te te E jj j j j* 

Because the glass substrate is very thick compared to wavelength (usually 1mm>> wavelength), the optical interference effect in the substrate can be neglected. Here only the correction of the light intensity at the air/substrate and substrate/multilayer interfaces is made. As shown in Fig. 1 (b), the multilayer can be treated as a virtual layer whose complex reflection and transmission coefficients can be calculated using above equations. Then the

*<sup>R</sup> I T RR I I*

1 <sup>2</sup> *g g I cnE* 

\* \* 0 0 \* <sup>0</sup> <sup>1</sup> ( ) <sup>1</sup> *i*

0 0

\*

(1 ) *<sup>g</sup>*

2(1 ) \*

*cn RR*

0 \* 0

When the excitons are generated, not all of them can be dissociated into free carriers. The dissociation probability depends on the electric field and temperature. Recently, the

*R I <sup>E</sup>* 

*RR*

2

0

(15)

(16)

(17)

*j j j j*

*SS SSe*

(11b)

(12)

*j*

(13)

 (14)

''

dissociation probability has been taken into consideration in polymer solar cells [13, 16]. The geminate recombination theory, first introduced by Onsager and refined by Brau later, gives the probability of electron-hole pair dissociation,

$$P(F,T) = \frac{k\_D(F)}{k\_D(F) + k\_X} \tag{18}$$

where *Xk* is the decay rate to the ground state and *kD* the dissociation rate of a bound pair. Braun gives the simplified form for dissociation rate

$$k\_D(F) = k\_R e^{-lI\_\beta / k\_B T} \left[ 1 + b + \frac{b^2}{3} + \dotsb \right] \tag{19}$$

where *a* is the initial separation distance of a given electron-hole pair, *UB* is electron-hole pair binding energy described as <sup>2</sup> *Uq a B r* /(4 ) <sup>0</sup> and 3 2 /(8 ) <sup>0</sup> *r B b qF kT* . *T* is the temperature, *F* the electric field and *<sup>r</sup>* the dielectric constant of the material. In equation (19), *Rk* is a function of the carrier recombination. For simplification, we treat *Rk* as a constant. Thus, the dissociation probability *P* only depends on the electric field *F* when the temperature keeps constant.

### **2.1.5** *JSC* **expression equations**

*JSC* is determined by the number of carriers collected by the electrodes in the period of their lifetime under short circuit condition. If the active layer thickness *L* is shorter than the electron and hole drift lengths (which is the product of carrier mobility , the electric field *F* and the carrier lifetime ) or in other word, the lifetimes of both types of carriers exceed their transit time (case I as in Fig 2 (a)), all generated free carriers can be collected by the electrodes. Considering the exciton-to-free-carrier dissociation probability *P*, *JSC* is

$$J\_{\rm SC} = qP(F, T)GL \tag{20}$$

where *G* is the average exciton generation rate in the active layer.

Fig. 2. Energy band diagrams under short circuit condition. (a) Case I: thickness is shorter than both drift lengths, (b) Case II: thickness is longer than hole drift length but shorter than electron drift length, c) Case III: thickness is longer than both hole and electron drift lengths.

Effects of Optical Interference and Annealing on the

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>0</sup>

PEDOT: PSS (nm)

Depth in the multilayer (nm)

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>0</sup>

Active layer: 50nm

PEDOT: PSS (nm)

Active Layer (nm)

ITO(nm) ITO(nm)

Active Layer (nm)

effect is considered.

0.5 1 1.5 2

2.5x 1025

Exciton Generation (m-3s-1)

1

2

ITO(nm)

3

4x 1025

Performance of Polymer/Fullerene Bulk Heterojunction Solar Cells 9

such that the average generation rate becomes larger. With the further increase of the active layer, the average generation rate decreases although other light peaks enter the active layer. This is because for a thicker film, the thickness of the active layer dominates the generation rate. This evolution of exciton generation is plotted in Fig. 4 for the 500 nm wavelength.

Fig. 3. The calculated exciton generation rate in the active layer when the optical interference

0.5 1 1.5 2

Fig. 4. Evolution of exciton generation in the active layer. The light wavelength is 500 nm. It can be seen that with the increase of the active layer thickness, the first peak enters the active layer, which makes the average exciton generation rate become large. For very thick film, although other peaks can enter the active layer, the absolute values for the peaks become small, which leads to the corresponding decrease of average exciton generation rate.

2.5x 1025

0.5 1 1.5 2 2.5 3x 1025

Depth in the multilayer (nm) Depth in the multilayer (nm)

Active layer: 150nm Active layer: 200nm

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>0</sup>

PEDOT: PSS (nm)

Active Layer (nm)

Depth in the multilayer (nm)

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>

PEDOT: PSS (nm)

Active layer: 100nm

Active Layer (nm)

ITO(nm)

If *L* is longer than drift lengths of electrons and holes, that is to say that the lifetimes of both types of carriers are smaller than their transit time, the carriers are accumulated in the active layer. At steady state, *JSC* follows Ohm's law. Considering the exciton-to-free-carrier dissociation probability *P*, *JSC* is

$$\mathbf{J}\_{SC} = qP(\mathbf{F}, T)\mathbf{G}(\mu\_e \tau\_e + \mu\_h \tau\_h)\mathbf{F} = qP(\mathbf{F}, T)\mathbf{G}(\mu\_e \tau\_e + \mu\_h \tau\_h)V\_{bi} \text{ / L} \tag{21}$$

where *Vbi* is the built-in potential which is usually determined by the difference between cathode and anode work functions. This is case III as described in Fig. 2 (c).

Between case I and case III, it is case II as described in Fig. 2 (b). In this case, *L* is only longer than the drift length of one type of carrier. For P3HT:PCBM based polymer solar cells, the mobilities of holes and electrons in P3HT:PCBM (1:1 by weight) layer are 82 11 2 10 *mV s* and 7 2 11 3 10 *mV s* , respectively [Mihailetchi et al., 2006]. Because the hole mobility is one order lower than the electron mobility, holes are easy to accumulate in the active layer, which makes the electric field non-uniform. In order to enhance the extraction of holes, the electric field increases near the anode. On the other hand, in order to diminish the extraction of electrons, the electric field decreases near the cathode. The electric field is modified until the extraction of holes equal to the extraction of electrons. Goodman and Rose studied this case and gave an equation for the photocurrent [Goodman & Rose, 1971]. Considering the exciton-to-free-carrier dissociation probability *P*, *JSC* is

$$J\_{sc} = qP(F,T)GL(1+c)\frac{-c + \left(c^2 + 4(1-c)V\mu\_h\tau\_h / L^2\right)^2}{2(1-c)}\tag{22}$$

where /( ) *hh ee c* is the drift length ratio of holes and electrons. When *c<<1*, the equation is simplified to

$$J\_{sc} = qP(F,T)G(\mu\_h \tau\_h)^{1/2} V^{1/2} \tag{23}$$

### **2.2 Results and discussion**

### **2.2.1 Exciton generation profile in the active layer**

For the studied bulk HJ cell, the *D* and *A* materials are well blended and form the active layer. Because the *D* and *A* domains are very small, we can neglect the complex reflection and transmission at *D/A* interfaces, and treat the whole active layer as one homogenous material. All the optical constants (*n, k*) of the indium tin oxide (ITO), poly(3, 4 ethylenedioxythiophene):poly (styrene sulfonate) (PEDOT:PSS), P3HT:PCBM and the Al electrode are input into our program, and the exciton generation rate in polymer solar cells is calculated. If the interference effect is neglected, the exciton generation rate decreases with the increasing thickness of the active layer as described in equation (4) which makes the corresponding average exciton generation rate (total exciton generation rate divided by the thickness) become smaller. However, when the optical interference effect is considered, the modulation effect of average exciton generation rate with the thickness variation is very clear as shown in Fig. 3. At the initial stage, the average exciton generation rate increases with the increasing thickness of the active layer. This is because the first light peak does not appear in the active layer when the active layer is thin due to the interference effect. With the increase of the active layer, the first light peak approaches and enters the active layer

If *L* is longer than drift lengths of electrons and holes, that is to say that the lifetimes of both types of carriers are smaller than their transit time, the carriers are accumulated in the active layer. At steady state, *JSC* follows Ohm's law. Considering the exciton-to-free-carrier

*SC* (, )( ) (, )( ) / *ee hh e e h h bi J qP F T G*

where *Vbi* is the built-in potential which is usually determined by the difference between

Between case I and case III, it is case II as described in Fig. 2 (b). In this case, *L* is only longer than the drift length of one type of carrier. For P3HT:PCBM based polymer solar cells, the mobilities of holes and electrons in P3HT:PCBM (1:1 by weight) layer are 82 11 2 10 *mV s* and 7 2 11 3 10 *mV s* , respectively [Mihailetchi et al., 2006]. Because the hole mobility is one order lower than the electron mobility, holes are easy to accumulate in the active layer, which makes the electric field non-uniform. In order to enhance the extraction of holes, the electric field increases near the anode. On the other hand, in order to diminish the extraction of electrons, the electric field decreases near the cathode. The electric field is modified until the extraction of holes equal to the extraction of electrons. Goodman and Rose studied this case and gave an equation for the photocurrent [Goodman & Rose,

2 2 <sup>2</sup> ( 4(1 ) / ) ( , ) (1 ) 2(1 )

*c c cV L J qP F T GL c*

1/2 1/2 (, )( ) *sc h h J qP F T G V* 

For the studied bulk HJ cell, the *D* and *A* materials are well blended and form the active layer. Because the *D* and *A* domains are very small, we can neglect the complex reflection and transmission at *D/A* interfaces, and treat the whole active layer as one homogenous material. All the optical constants (*n, k*) of the indium tin oxide (ITO), poly(3, 4 ethylenedioxythiophene):poly (styrene sulfonate) (PEDOT:PSS), P3HT:PCBM and the Al electrode are input into our program, and the exciton generation rate in polymer solar cells is calculated. If the interference effect is neglected, the exciton generation rate decreases with the increasing thickness of the active layer as described in equation (4) which makes the corresponding average exciton generation rate (total exciton generation rate divided by the thickness) become smaller. However, when the optical interference effect is considered, the modulation effect of average exciton generation rate with the thickness variation is very clear as shown in Fig. 3. At the initial stage, the average exciton generation rate increases with the increasing thickness of the active layer. This is because the first light peak does not appear in the active layer when the active layer is thin due to the interference effect. With the increase of the active layer, the first light peak approaches and enters the active layer

*F qP F T G*

 

1

(23)

*h h*

 

*c*

(22)

is the drift length ratio of holes and electrons. When *c<<1*, the

*V L* (21)

 

cathode and anode work functions. This is case III as described in Fig. 2 (c).

1971]. Considering the exciton-to-free-carrier dissociation probability *P*, *JSC* is

dissociation probability *P*, *JSC* is

*sc*

**2.2.1 Exciton generation profile in the active layer** 

 

where /( ) *hh ee c* 

equation is simplified to

**2.2 Results and discussion** 

such that the average generation rate becomes larger. With the further increase of the active layer, the average generation rate decreases although other light peaks enter the active layer. This is because for a thicker film, the thickness of the active layer dominates the generation rate. This evolution of exciton generation is plotted in Fig. 4 for the 500 nm wavelength.

Fig. 3. The calculated exciton generation rate in the active layer when the optical interference effect is considered.

Fig. 4. Evolution of exciton generation in the active layer. The light wavelength is 500 nm. It can be seen that with the increase of the active layer thickness, the first peak enters the active layer, which makes the average exciton generation rate become large. For very thick film, although other peaks can enter the active layer, the absolute values for the peaks become small, which leads to the corresponding decrease of average exciton generation rate.

Effects of Optical Interference and Annealing on the

obtained (Fig. 7) when the average hole lifetime

[Li et al., 2005].

Performance of Polymer/Fullerene Bulk Heterojunction Solar Cells 11

Fig. 6. Relations of electric field and exciton-to-free-carrier probability with layer thickness. We have predicted *JSC* precisely for the long enough carrier lifetime case. However, for polymer solar cells, the performance is sensitive to the process and experimental conditions. This may make the carrier lifetime relatively short. For P3HT:PCBM system, because the hole mobility is one order of magnitude lower than the electron mobility, holes are easy to accumulate in the active layer and limit the photocurrent. This is the case II as described in section 2.1.5. By tuning the parameters to fit the experimental data, the best fitting curve is

dissociation probability is unity. A short lifetime *τ* may imply that there are many defects.

Fig. 7. Short hole carrier lifetime condition. Left arrow: hole lifetime is longer than its transient time; right arrow: hole lifetime is shorter than its transient time, and hole lifetime is <sup>7</sup> 6.2 10 *s* , and electron lifetime is <sup>6</sup> 1 10 *s* . Experimental data are from

is <sup>7</sup> 6.2 10 *s* and exciton-to-free-carrier

### **2.2.2** *JSC* **and the active layer thickness**

Based on the calculated exciton generation rate, it is easy to predict *JSC* when the drift lengths of both carriers are larger than the blend layer thickness. If all the generated excitons can be dissociated into free carriers, and then collected by the electrodes, *JSC* should be proportional to the total exciton generation rate and behave wavelike as shown in Fig. 5 (solid line). Monestier [Monestier et al., 2007] have found this trend based on P3HT:PCBM systems. In their experiments, the active layer thickness is varied from a few tens nanometer to 215 nm. When the thickness is 70 nm, *JSC* reaches the maximum value, and followed by a little decrease until 140 nm. When the thickness increases further, *JSC* increases again. Unfortunately, there is obvious deviation between the prediction and the experiment results, especially in the thick film as shown in Fig. 5 (solid line). Obviously, the assumption that the exciton-to-free-carrier dissociation probability is unity is not correct. The influence of dissociation probability on *JSC* must be considered.

Fig. 5. Long carrier lifetime condition: the lifetimes of both carriers are always longer than their transient time. Experimental data are extracted from the work (Monestier et al. 2007).

In the previous work, Mihailetchi [Mihailetchi et al., 2006] exactly predicted photocurrent of P3HT:PCBM solar cells by assuming the same e-h separation distance (*a*) and decay rate ( *Xk* ). By fitting the experimental data, they obtained e-h separation distance of *a*=1.8 nm, room temperature bound pair decay rate of 4 1 2 10 *Xk s* for a 120 nm active layer, and the dissociation probability is close to 90%. We use the same data and derive the parameter 8 1 3.9662 10 *Rk S* (equation 19). The dissociation probability is calculated according to section 2.1.4. The results are shown in Fig. 6. Obviously, the exciton-to-free-carrier probability becomes lower with the increase of the active layer thickness. Using the results to correct *JSC*, another *JSC* curve is obtained and also shown in Fig. 5 (dash line). It can be seen that the predicted *JSC* is exactly in accordance with the experimental results. This confirms the validity of our model. In the previous work, Monestier [Monestier et al., 2007]] modeled *JSC* and found that the predicted *JSC* is larger than the experimental data, especially for the thickness larger than 180 nm. They attributed this to the thickness dependence of optical constants. Here, according to our model, it is found that the deviation should come from the low exciton-to-free-carrier probability for thick active layers.

Based on the calculated exciton generation rate, it is easy to predict *JSC* when the drift lengths of both carriers are larger than the blend layer thickness. If all the generated excitons can be dissociated into free carriers, and then collected by the electrodes, *JSC* should be proportional to the total exciton generation rate and behave wavelike as shown in Fig. 5 (solid line). Monestier [Monestier et al., 2007] have found this trend based on P3HT:PCBM systems. In their experiments, the active layer thickness is varied from a few tens nanometer to 215 nm. When the thickness is 70 nm, *JSC* reaches the maximum value, and followed by a little decrease until 140 nm. When the thickness increases further, *JSC* increases again. Unfortunately, there is obvious deviation between the prediction and the experiment results, especially in the thick film as shown in Fig. 5 (solid line). Obviously, the assumption that the exciton-to-free-carrier dissociation probability is unity is not correct. The influence

Fig. 5. Long carrier lifetime condition: the lifetimes of both carriers are always longer than their transient time. Experimental data are extracted from the work (Monestier et al. 2007). In the previous work, Mihailetchi [Mihailetchi et al., 2006] exactly predicted photocurrent of P3HT:PCBM solar cells by assuming the same e-h separation distance (*a*) and decay rate ( *Xk* ). By fitting the experimental data, they obtained e-h separation distance of *a*=1.8 nm, room temperature bound pair decay rate of 4 1 2 10 *Xk s* for a 120 nm active layer, and the dissociation probability is close to 90%. We use the same data and derive the parameter 8 1 3.9662 10 *Rk S* (equation 19). The dissociation probability is calculated according to section 2.1.4. The results are shown in Fig. 6. Obviously, the exciton-to-free-carrier probability becomes lower with the increase of the active layer thickness. Using the results to correct *JSC*, another *JSC* curve is obtained and also shown in Fig. 5 (dash line). It can be seen that the predicted *JSC* is exactly in accordance with the experimental results. This confirms the validity of our model. In the previous work, Monestier [Monestier et al., 2007]] modeled *JSC* and found that the predicted *JSC* is larger than the experimental data, especially for the thickness larger than 180 nm. They attributed this to the thickness dependence of optical constants. Here, according to our model, it is found that the deviation should come

from the low exciton-to-free-carrier probability for thick active layers.

**2.2.2** *JSC* **and the active layer thickness** 

of dissociation probability on *JSC* must be considered.

Fig. 6. Relations of electric field and exciton-to-free-carrier probability with layer thickness.

We have predicted *JSC* precisely for the long enough carrier lifetime case. However, for polymer solar cells, the performance is sensitive to the process and experimental conditions. This may make the carrier lifetime relatively short. For P3HT:PCBM system, because the hole mobility is one order of magnitude lower than the electron mobility, holes are easy to accumulate in the active layer and limit the photocurrent. This is the case II as described in section 2.1.5. By tuning the parameters to fit the experimental data, the best fitting curve is obtained (Fig. 7) when the average hole lifetime is <sup>7</sup> 6.2 10 *s* and exciton-to-free-carrier dissociation probability is unity. A short lifetime *τ* may imply that there are many defects.

Fig. 7. Short hole carrier lifetime condition. Left arrow: hole lifetime is longer than its transient time; right arrow: hole lifetime is shorter than its transient time, and hole lifetime is <sup>7</sup> 6.2 10 *s* , and electron lifetime is <sup>6</sup> 1 10 *s* . Experimental data are from [Li et al., 2005].

Effects of Optical Interference and Annealing on the

how they affect the device performance are still not well studied.

effects contribute to improve the device performance.

**3.1 Experimental** 

suspension (through 0.45

Performance of Polymer/Fullerene Bulk Heterojunction Solar Cells 13

tended to use the cathode confinement and carry out the thermal treatment after the cathode deposition, what are the functions of the cathode confinement in the thermal treatment and

In this part, the effects of cathode confinement on the performance of polymer solar cells are investigated. It is shown that a better device performance can be achieved by using the cathode confinement in the thermal treatment. The experimental analysis indicates that by capping the cathode before the thermal treatment, the Al-O-C bonds and P3HT-Al complexes are formed at the interface between the P3HT:PCBM active layer and the cathode, which leads to a better contact and thus improves the charge collection capability. More importantly, the cathode confinement in the thermal treatment greatly improves the active layer morphology. It is shown that the cathode confinement in the thermal treatment can effectively inhibit the overgrowth of the PCBM molecules, and at the same time increase the crystallization of P3HT. Thus, a better morphology is achieved, which effectively reduces the exciton loss and improves the charge transport capability. Meanwhile, the enhanced P3HT crystallites improve the absorption property of the active layer. All these

Fig. 8 shows the layer structure of our polymer solar cells and the chemical structures of P3HT and PCBM. All the devices were fabricated on the ITO-coated glass substrates. Briefly, after being cleaned sequentially with detergent, de-ionized water, acetone, and isopropanol in an ultrasonic bath for about 15 mins, the dried ITO glass substrates were treated with oxygen plasma for about 3 mins. Then the filtered PEDOT:PSS (Baytron P VP AI 4083)

nm layer under ambient condition, and dried at 120oC in an oven for about one hour.

P3HT:PCBM solution dissolved in 1,2-dichlorobenzene with a weight ratio of 1:0.8 was then spun coated on the PEDOT:PSS layer in the glove box to form a 100 nm blend layer. A 100 nm Al cathode was further thermally evaporated through a shadow mask giving an active device area of 20 mm2. In order to investigate the effects of the cathode confinement on the device performance in the thermal treatment, two different types of devices were investigated: the devices without the cathode confinement in the thermal treatment (anneal the devices before the cathode deposition, pre-anneal) and the devices with the cathode confinement in the thermal treatment (anneal the devices after the cathode deposition, post-

Fig. 8. Layer structure of the polymer solar cells investigated in this work.

*m* filter) was spun coated on top of the ITO surface to form a ~50

These defects increase the exciton-to-free-carrier probability. More important, the transport process becomes the dominant limiting factor for *JSC*, and the exciton-to-free-carrier process becomes relatively unimportant. Then it seems that the assumption of exciton-to-free-carrier probability as unity can satisfy the need of the prediction. In Fig. 7, we can see that there are two regions in the fitting curve. The left region is determined by equation (20). In this region, the lifetimes of both carriers are longer than their transient time. The solid line in the right region is determined by equation (22). In this region, hole lifetime is shorter than its transit time and electron lifetime is longer than its transit time.

If it is assumed that the drift length ratio of hole and electron is very small, then the equation (23) can be used to predict *JSC*. As shown in Fig. 7 (dash line), it can predict *JSC* very well, which means *c<<1*.

### **2.3 Summary**

In this part, the exciton generation rate was calculated by taking the optical interference effect into account. Based on the calculated exciton generation rate, the dependence of *JSC* on the active layer thickness was analyzed and compared with experimental data. Because of the optical interference effect, the total exciton generation rate does not monotonously increase with the increase of the active layer thickness, but behaves wavelike which induces the corresponding variation of *JSC*. The carrier lifetimes also influence *JSC* greatly. When the lifetimes of both electrons and holes are long enough, dissociation probability plays an important role for the thick active layer. *JSC* behaves wavelike with the variation of the active layer thickness. When the hole lifetime is too short (drift length is smaller than device thickness), accumulation of charges appears near the electrode and *JSC* increases at the initial stage and then decreases rapidly with the increase of the active layer thickness. The accordance between the predictions and the experimental results confirms the validity of the proposed model. These results give a guideline to optimize *JSC*.

### **3. Effects of annealing sequence on** *JSC*

The detail of the interpenetrating network, or to say, the morphology is essentially important for the performance of polymer solar cells. In order to achieve an optimal morphology, a thermal treatment is usually utilized in the device fabrication. The thermal treatment can be carried out after and before the electrode deposition. Both the methods can greatly improve the device performance. The functions of the thermal treatment have been extensively investigated, and it has been shown that the morphology will be rearranged through the nanoscale phase separation between donor and acceptor components during the thermal treatment. By carefully optimizing the thermal treatment condition, an optimal interpenetrating network can be formed, which greatly improves the charge transport property. Besides, the thermal treatment can also effectively enhance the crystallization of P3HT, which will increase the hole mobility and the optical absorption capability. Due to the importance of the thermal treatment for P3HT:PCBM solar devices, great efforts have been devoted into the study of the thermal annealing process in the past few years. How the thermal annealing ambient, thermal annealing temperature and thermal annealing time affect the device performance has been well studied. However, only very few studies paid attention to the role of cathode in the thermal treatment. As is known, the thermal treatment can be done before and after the cathode deposition and both methods can greatly improve the device performance. The unique difference between them is whether there is cathode confinement in the thermal treatment or not. Although most of the previous studies have tended to use the cathode confinement and carry out the thermal treatment after the cathode deposition, what are the functions of the cathode confinement in the thermal treatment and how they affect the device performance are still not well studied.

In this part, the effects of cathode confinement on the performance of polymer solar cells are investigated. It is shown that a better device performance can be achieved by using the cathode confinement in the thermal treatment. The experimental analysis indicates that by capping the cathode before the thermal treatment, the Al-O-C bonds and P3HT-Al complexes are formed at the interface between the P3HT:PCBM active layer and the cathode, which leads to a better contact and thus improves the charge collection capability. More importantly, the cathode confinement in the thermal treatment greatly improves the active layer morphology. It is shown that the cathode confinement in the thermal treatment can effectively inhibit the overgrowth of the PCBM molecules, and at the same time increase the crystallization of P3HT. Thus, a better morphology is achieved, which effectively reduces the exciton loss and improves the charge transport capability. Meanwhile, the enhanced P3HT crystallites improve the absorption property of the active layer. All these effects contribute to improve the device performance.
