In this chapter, a model for the open-circuit voltage was introduced. By studying the
*dependency of V*oc *on incident light intensity, it has been demonstrated that the V*ocof
*BHJ solar cells is inconsistent with p-n junction models: Whereas the experimental data*

*showed that the slope S of V*ocas a function of ln(*I*)*is equal to V**t**, the p-n junction model*
*predicts a slope of nV**t**, where n ranges from 1.35 to 1.98. This phenomenon was observed*
for two different PPV derivatives as donor material. The main cause of this discrepancy
lies in the fact that the strong voltage dependence of the photogenerated current is not
taken into account.

An alternative model for the open-circuit voltage has been presented, based on the
notion that the quasi-Fermi potentials are constant throughout the device. Subsequently,
*a formula for V*ocwas derived, consistently explaining the light intensity dependence of
the open-circuit voltage of polymer/fullerene bulk heterojunction devices with Ohmic
contacts. Next, the predictions of the MIM model and its relation to other types of solar
cells have been discussed.

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**FOUR**

## Short-circuit current of bulk heterojunction solar cells

**Summary**

A typical feature of polymer/fullerene based solar cells is that the short-circuit
current density does not scale exactly linearly with light intensity. Instead, a power
*law relationship is found given by J*sc ∝ *I*^{α}*, where α ranges from 0.85 to 1. In a*
number of reports this deviation from unity is speculated to arise from the occurrence
of bimolecular recombination. In this chapter, simple analytical models are discussed,
showing that the short-circuit current should show a power law behavior on intensity,
*J*sc ∝ *I** ^{α}*, with 0.75 <

*< 1, depending on the mobility of both carriers. By applying the numerical model as outlined in chapter 2, it is demonstrated that the experimentally observed intensity dependence is indeed caused by space charge effects and does not originate from bimolecular recombination losses. This explanation is verified for an experimental model system with a mobility difference that can be tuned from one to three orders of magnitude by changing the post-production annealing treatment.*

_{α}**4.1** **Introduction**

*One of the key parameters of any solar cell is the short-circuit current density J*sc, and its
optimization is of great importance for the further improvement of organic photovoltaic
devices. What determines the short-circuit current? An important issue, in this respect,
*is the dependence of J*sc *on incident light intensity I. Several authors have reported a*
*power law dependence of J*sc*, i.e. J*sc ∝ *I*^{α}*, where α ranges typically from 0.85 to 1 for*
polymer/fullerene based solar cells.^{[1–7]} *Thus far, the deviation from α* = 1 has been
conjectured to arise from a small loss of carriers via bimolecular recombination.^{[2,4,7]}As
a first step, this section deals with three simple analytical expressions for the
photocur-rent generated by a BHJ solar cell. The concept of space-charge-limited photocurphotocur-rent,
introduced in this section, will prove to be of use in explaining the intensity dependence
*of J*sc.

**4.1.1 Uniform field approximation**

Consider the simple case of a photoconductor with non-injecting contacts and a uniform
electric field distribution. Goodman and Rose derived that, under the assumption of
negligible recombination of charge carriers,^{[8]}

*J*_{ph} =*qGL,* (4.1)

meaning that in this case all photogenerated charge carriers are simply extracted and
*the current density depends only on the generation rate G. In their derivation, Goodman*
and Rose took only drift of charge carriers into account and neglected the contribution of
diffusion. Sokel and Hughes carried this analysis one step further by including diffusion
of carriers, finding^{[9]}

*J*_{ph} =*qGL*hexp(*V/V**t*) +1

exp(*V/V**t*) −^{1} −^{2}^{V}_{V}^{t}^{i}^{,} ^{(4.2)}
*where V is the voltage drop across the active layer. For a BHJ device this voltage drop*
*is given by V*0−^{V}*a*. The result by Sokel and Hughes shows two regimes: A linear
*dependence of J*_{ph}on voltage for small biases, while reducing to Eq. (4.1) at moderately
high bias (including short-circuit conditions), see Fig. 4.1.

*On the basis of Eqs. (4.1) and (4.2) one expects that J*sc = *qGL and, hence, that J*sc

is proportional to the incident light intensity. From experiments it is evident that this
*is not always true; for some systems J**sc* is clearly sublinear in light intensity. The fact
that Eqs. (4.1) and (4.2) do not depend on charge carrier mobility is a consequence of
the assumption of no recombination, thereby ensuring that all carriers exit the device
and that the electric field is uniform. In this chapter, we shall see that it is exactly this
*assumption of a uniform electric field that fails when J*scshows sublinear behavior.

0.01 0.1 1

**Figure 4.1:** The photocurrent data (symbols) of an MDMO-PPV/PCBM device, as discussed in
subsection 2.4.2, together with the Sokel and Hughes result Eq. (4.2) (solid line) and the prediction
*by Goodman and Rose [dashed line, Eq. (4.1)]. At low effective applied voltages J*_{ph} ∝ *V*_{0}−^{V}*a*

*(regime I), while at V*_{0}−^{V}*a*'*0.3 V, the photocurrent saturates to qGL (regime II).*

**4.1.2 Space-charge-limited photocurrents**

*Mihailetchi et al. have demonstrated that a large difference in electron- and hole mobility,*
accompanied by a low mobility of the slowest carrier, may lead to space-charge-limited
photocurrents.^{[10]} The extraction of photogenerated carriers is governed by the mean
*carrier drift length w, which is the mean distance a carrier travels before recombination*
*occurs. When both the electron (w**n**) and hole (w**p*) drift lengths are larger than the active
layer thickness, then the charges will readily flow out without distorting the field in
*the device, see Fig. 4.2(a). However, in case w**n* ≫ ^{w}^{p}^{and w}* ^{p}* <

*L there will be a*net positive space charge near the anode, see Fig. 4.2(b). There exist three regimes in the device: Near the cathode the electron density is much larger than the hole density;

this is a small region (I). Next to this region, there exists a balance between electron and hole density, yielding a neutral region (II). Near the anode, the holes dominate the device (III), resulting in a large net space charge and concomitant large voltage drop, as indicated in Fig. 4.2(b). The large field strength in region III facilitates the extraction of holes, ensuring that the extraction current of holes and electrons is equal.

When the photocurrent is space-charge-limited, the following relation holds:^{[8,10]}

*J*_{ph}∝*G*^{0.75}*pV*0−^{V}*a*. (4.3)

Thus, fully space-charge-limited photocurrents are characterized by a square-root
*de-pendence on voltage and are proportional to I*^{0.75}, irrespective of the amount of
bimolec-ular recombination. On the other hand, non space-charge-limited devices have a linear
*dependence of J*_{ph}*on I. Therefore, it is expected that the three-quarter power intensity*
dependence gradually increases to a linear dependence if the difference between the

**mo-Figure 4.2:** (a) Band diagram of a BHJ solar cell with balanced electron and hole mobilities; both
types of charge carriers can readily flow out of the device and the field in the device is uniform.

(b) Band diagram in the case of hole accumulation and concomitant space-charge-limited behavior.

Near to the cathode, a small region dominated by electrons (I) exists next to a large neutral region (II), where electron and hole densities are comparable. Most of the potential drops across the hole accumulation layer (III) in order to facilitate the extraction of holes.

bility of electrons and holes is reduced. Since the occurrence of space charge is sensitive
to the mobility difference, it is easy to understand that various material combinations
*will give different exponents α describing the intensity dependence of J*sc.

The experimental observation of space-charge-limited photocurrents in BHJ solar
cells makes a very strong point in favor of the MIM model and cannot be explained
*by a p-n junction model. The fact that Eq. (4.3) describes the experimental data so well,*
even though Goodman and Rose originally derived it for one material instead of a blend,
shows that the assumption of an effective medium holds.