Modeling MDMO-PPV/PCBM bulk heterojunction solar cells

Now that the bimolecular recombination mechanism has been established, what can be said about the carrier densities and field distributions in a device? This subsection deals with the modeling of the well known MDMO-PPV/PCBM system, which will be inves-tigated in detail.

Figure 2.9 shows the current-voltage characteristics of a 120-nm-thick MDMO-PPV/PCBM (1:4 by weight) BHJ solar cell. In this graph the effective photocurrent den-sity J_ph, obtained by subtracting the dark current from the current under illumination, is

∗This value of P is taken at the maximum power point (Va= 0.685), as discussed on page 38.

10

Figure 2.8: The experimental FF (symbols) corresponding to different annealing temperatures as a function of the ratio of hole to electron mobility. The solid line denotes simulations using Eq. (2.24), while the dashed line corresponds to simulations using Eq. (2.23).

plotted as a function of effective applied voltage V0−^Va, where V0is the compensation voltage defined by J_ph(Va =V0) =0.^[1]In this way, V0−^Vareflects the internal electric field in the device. It should be noted that Va =0.884 V is very close to the open-circuit voltage (0.848 V).

For low effective voltages V0−^Va the photocurrent increases linearly with effective voltage, and subsequently tends to saturate. Mihailetchi et al.^[1]demonstrated that this low voltage part can be described with an analytical model developed by Sokel and Hughes^[36]for zero recombination (see subsection 4.1.1), as indicated by the dashed line in Fig. 2.9. The linear behavior at low effective voltage is the result of a direct compe-tition between diffusion and drift currents. In the model by Sokel and Hughes, all free charge carriers are extracted at higher effective voltage and the photocurrent saturates to qGL, where G = PGe−h is the generation rate of free charge carriers. The fact that the experimental photocurrent does not saturate, but gradually increases for large ef-fective voltages has been attributed to the field dependence of the generation rate G.

The two parameters governing the field- and temperature-dependent generation rate, the electron-hole pair distance a and the decay rate k_f, can be determined by equating the high field photocurrents to qGL. The value of a determines the field at which the dissociation efficiency saturates, and hence a can be determined independently of k_f. It is evident from Fig. 2.9 that the calculated photocurrent fits the experimental data over the entire voltage range. For comparison, in Fig. 2.10 the experimental and calculated JLare also shown in a conventional linear plot focusing on the fourth quadrant. The ex-cellent agreement between experimental and calculated data now enables one to further analyze the device behavior under different bias conditions in more detail.

0.01 0.1 1 10 1

10

Jph

[A/m

2 ]

V 0

-V a

[V]

Figure 2.9: Photocurrent density J_ph as a function of effective applied voltage (V0−^Va). The symbols represent experimental data of MDMO-PPV/PCBM devices at room temperature. The solid line denotes the simulation, while the dashed line represents the result of Sokel and Hughes.

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -40

-20 0 20 40

JL

[A/m

2]

V a

[V]

Figure 2.10: The current density under illumination of an MDMO-PPV/PCBM device (symbols) and the numerical result (line).

Table 2.2:Overview of the parameters used in the fit to the data shown in Figs. 2.9 and 2.10.

Parameter Symbol Numerical value

effective band gap E_gap^eff 1.29 eV

electron mobility µn 2.0×10⁻⁷m²/Vs

hole mobility µp 2.0×10⁻⁸m²/Vs

eff. density of states Ncv 2.5×^1025m−3 generation rate of bound pairs G_e−h 2.7×^{1027m−3s−1} dielectric constant hεi 3.0×10⁻¹¹F/m

e/h pair distance a 1.3 nm

decay rate k_f 2.5×^105s−1

Figure 2.11: The device at short-circuit, (a) shows the potential (solid line) and the electron and hole current densities (dashed and dotted lines, respectively), (b) shows the electron and hole densities (dashed and solid lines, respectively) and the net generation rate (dotted line).

The device at short-circuit

The calculated potential, current densities, carrier densities, and net generation rate at short-circuit are depicted in Fig. 2.11. The potential under illumination is virtually equal to the potential in dark (not shown). The number of photogenerated charges is not suf-ficient to change the potential significantly. Apart from band bending near the contacts, a consequence of the large amount of charge, the field is constant throughout the device and, concomitantly, so is the dissociation rate of bound electron-hole pairs. As a result, the current densities exhibit a linear dependence on the position in the device. The only exception to this is found near the contacts where both electron and hole densities are high, especially near the top contact (x=0), and recombination becomes important.

In the bulk of the device, the hole density is roughly one order of magnitude higher than the electron density. This is a result of the difference in mobility between electrons and holes. Since the holes are slower, they pile up in the device. If the difference in mobility becomes large enough and the hole mobility is indeed quite low, then the hole

Table 2.3: An overview of voltage, current density, average dissociation probability, and relative number of free carriers lost due to recombination at short-circuit, maximum power, and open-circuit conditions.

Va JL hPi rec. loss [V] [A/m²] [%] [%]

short-circuit 0 28.3 57 2

maximum power 0.685 19.6 46 14

open-circuit 0.845 0 42 91

density will become so large that the potential under illumination differs from the po-tential in dark, due to space charge (see subsection 4.1.2). However, with a mobility difference of only a factor of ten, the overall carrier densities are rather low as compared to other devices such as LEDs (at operating voltages) or FETs. This is because the field in the device is quite large at short-circuit and carriers are readily extracted. For this reason space charge effects only play a minor role, leading to a nearly constant field in the device.

The average dissociation rateh^Piof bound electron-hole pairs is equal to 57%. This implies that a significant improvement to the device performance could be made by fa-cilitating this dissociation. The number of charge carriers lost due to bimolecular recom-bination can be computed in the following way. The average net generation rateh^Ui^can be obtained by integrating Eq. (2.2a), which yields

h^Ui = ¹_L Z _L

0 U(x)dx= ^Jn(0) −^Jn(L)

qL . (2.32)

The total number of generated charges is calculated from the average dissociation rate and hence the recombination rate is known. It turns out that at short-circuit conditions only 2% of the free charge carriers are lost due to bimolecular recombination and sub-sequent decay. The low loss of charge carriers (with the mobilities of this system) is a consequence of the high field strength, which ensures good charge extraction and low carrier densities at short-circuit. Since the carrier densities are low, bimolecular recom-bination is weak and hence the recomrecom-bination lifetime of the charge carriers is relatively long. On the other hand, due to the high field strength, the time needed for the charge carriers to exit the device is quite small, and therefore only few charge carriers are lost.

An overview of dissociation rate and loss of carriers at various applied voltages is given in Table 2.3.

The device at maximum power output

The maximum power output occurs at Va= 0.685 V for the simulated device. At this bias, the field in the device is smaller than at short-circuit, resulting in a smaller dissociation

0 20 40 60 80 100 120

Figure 2.12: The device at open-circuit, (a) shows the potential (solid line) and the electron and hole current densities (dashed and dotted lines, respectively), (b) shows the electron and hole densities (dashed and solid lines, respectively) and the net generation rate (dotted line). Note that the scales of part (a) are changed as compared to Fig. 2.11(a).

efficiency being 46%, see Table 2.3. Another consequence of the reduction of the field strength is the increase of lost free carriers; as much as 14% of all free carriers recombine and subsequently decay. This increase of losses is for the largest part a consequence of a less favorable extraction of charges. First, the time it takes for a charge carrier to exit the device is longer as compared to short-circuit, due to a lower electric field strength in the device. This leads to larger charge carrier densities, thereby increasing the probability of bimolecular recombination of charge carriers. Furthermore, the dissociation probability of a bound electron-hole pair slightly decreases, leading to larger loss of carriers once bimolecular recombination has taken place.

The device at open-circuit

At open-circuit conditions, it is evident that the field in the device is smaller than at short-circuit, see Fig. 2.12. Since there is no net current, there exists a balance between drift and diffusion of charge carriers. Therefore, the field in the device cannot be zero and, consequently, Vocis smaller than E_gap^eff .^[37]

The current densities are almost zero throughout the device. The reason for the cur-rent densities not being zero everywhere lies in the field dependence of the generation rate. In case of a constant generation rate the current densities would be zero every-where. From Fig. 2.12 it can be seen that the densities are almost symmetrical. In fact, if the generation rate would be constant, then from Eq. (2.3),

1

Integration yields

Using Eqs. (2.5a) and (2.6b), we have

n(L−^x) =p(x), (2.36)

showing that the carrier densities are indeed symmetrical.

Another striking feature of Fig. 2.12 is the fact that the carrier densities are much higher at open-circuit than at short-circuit. This is because of the much lower field which makes extraction of charge carriers more difficult. This also results in a much smaller net generation rate, in fact, 91% of all free charge carriers are lost due to recombination. ^∗ Only in the immediate vicinity of the contacts there is a significant net generation of charge carriers. This is not due to an increase in generation due to a higher electric field, but due to the fact that at the contact there is an enormous difference in charge carriers densities and the free carriers have no counter part to recombine with.