Simulating this Effect

When a photoactive material is illuminated, excitons are generated and, based on the internal field between the acceptor and donor, they split into charge carriers. In a full working device, charges would be separated by the respective electrodes providing an internal field. In this case only one electrode is present and earthed, hence there is no net current flow. Since there is no other internal field other than the LUMO/LUMO (and HOMO/HOMO) offset and any created by the ITO, the build up of charge would act to cancel these fields. The process of this can be modelled as two separate exponentials one describing the charging of the sample and the other the discharging [192, 191]. The charging of the sample can be modelled as

Vsp = 1−(1−Vc)e−t/τc (4.12)

likewise the discharge is defined as

Vsp=Vde−t/τd (4.13)

where theVsp is the normalised surface photovoltage measured, defined in the sim- ulation to be between 0 and 1, 1 being at full charge and 0 being dark conditions; τc and τd are the characteristic lifetimes of charging and discharging respectively; and Vc and Vd are the starting values of the normalised SPV of the charging and discharging segments respectively.

Figure 4.28: Simulated surface potential for a 10 Hz modulation

(blue) and a 200 Hz modulation (red) with a slow charging rate

and a fast discharge rate. Their averages over a second are shown by solid horizontal lines.

time of 0.5 ×10−3 s over a low frequency oscillation, and a second with the same values ofτcandτdbut at 50 times the frequency. Horizontal lines correspond to the average value across a full 1 second time window. For low frequencies the response approaches that of a square wave between 0 and the maximum generated voltage of the system, shown in blue. The average of this wave, assuming the maximum voltage produced by the system is equal toVmax and no back ground light, is equal to 0.5 Vmax. In the higher frequency case the average value is increased. This is because the limited time to discharge, means the discharge is not completed whilst the charging is, hence the average increases.

Figure 4.28 shows the resulting potentials if the charging and discharging rates are swapped. Notice that when the modulation frequency increases the average value of the surface potential decreases. In this case, the sample does not fully charge as the frequency of modulation rises, resulting in the decrease of the average.

Figure 4.29 shows the behaviour of the time averaged surface potential across a range of frequencies. Following from the work of Shao et al. [192], it is possible to fit this with the equation

Avsp(f) = 1 2 + τ fΓ 1 β, 1 2τ f β β (4.14)

where f is the modulating frequency,β is a stretching exponent lying between 0 and 1,τ is the characteristic lifetime and Γ is the incomplete gamma function, defined

Figure 4.29: a) Shows the average surface potential as a function of frequency for a discharge 10 times slower than the charge, with the natural fitting. b) Shows the same data but allowing the fitting to have an extra scaling factor.

as

Γ(s, x) =

Z ∞

x

ts−1e−tdt. (4.15) Note that a stretched exponential is used here for fitting as the processes involved are more likely to have multiple charge/discharge rates rather than the single value used in the simulation. Figure 4.29 a) shows the fit of this equation to the simulated time averaged surface potential. Clearly there is a large discrepancy here.

The fitting assumes that τc is infinitely small, i.e. the sample charges in- stantly, the simulation however has a finiteτc thus resulting in the average surface potential not being equal to 1 at very high frequencies, and hence the equation not fitting. However it is possible to modify the equation to get a better fit by adding a scaling factor to the equation so that

Vsp,avg(f) = 1 2 +Sc τ fΓ 1 β, 1 2τ f β β (4.16)

frequency. The results of this fitting are shown in Figure 4.29 b). The simulation conditions haveτcand τdas 0.5 x10−4 s and 0.5 x10−3 s respectively. The fitting of Equation 4.14 returns aβ of 0.5 and aτ of 0.43±0.08 x10−3 _{s. Whereas Equation}

4.16 produces a β of 1.25, with a τ of 0.561 ± 0.008 x10−3 s. The τ returned by the fitting does not match eitherτcorτd, however it is a close approximation of τd, the slower response here. τ does not equalτd because the assumption in the fitting is thatτc is infinitely small, when in reality it is small but still significant. This is shown by the scaling factor, Sc which in this case Sc = 0.82. The scaling factor is related to the relationship betweenτc andτd.

If the modulation frequency tends to infinity,VcandVdare nearly equal, and equally t tends to 0. For this reason differentiating Equation 4.12 gives

Vsp =1−(1−Vc)e− t τc d dtVsp = 1 τc (1−Vc)e −t τc _(4.17) Sincetτc =1 τc (1−Vc) (4.18)

And since, on charging for a time period t

Vd= t τc

(1−Vc) +A (4.19)

Likewise, the same approach can be taken for the discharging curve (Equation 4.13)

Vsp =Vde−t/τd d dtVsp =− Vd τd (4.20)

So after a time t the sample has discharged back to Vc, hence

Vd− Vd τd

t=Vc (4.21)

substituting forVc in 4.19 yields

t τc − Vdt τc +Vdt 2 τcτd −Vdt τd = 0 Sincet→0 τd τd+τc =Vd (4.22)

Since the starting value of the fitting is 0.5 the scale parameter Sc is related to Vd by Vd= Sc 2 + 1 2 (4.23)

this means that the relationship between the two time constants can be found, providing the scaling factor is found. So that

τd τd+τc = Sc 2 + 1 2 (4.24)

The τ returned by the fitting should most closely approximate the larger of the two time constants, though with a finite τc it will not be an exact match. By simulating the results across a wide range of τc and τd it might be possible to determine the approximate relationship betweenτc,τdandτ, however this will only be useful ifSc can be determined from the experimental data.