Diffusion constant fitting with the thin-film interference model

Now that we have both the TFI model and the diffusion model set up, we can de-termine the diffusion constant from the transient signals. We can simply model the carrier distributions at each delay time after introducing the starting distribution based on the sample absorption at the excitation wavelength, and then use the TFI model to produce the correspondingΔR(λ,t)andΔT(λ,t).

The full workflow chart of the TFI model for diffusion fitting is presented in Figure 4.6, except for the determination of roughness effects, which is covered in the next section (4.4.5). There is a fairly large number of different variables that need to be resolved before performing the diffusion fit, but most of these were cov-ered previously. As explained in Section 3, we first need to model the steady-state spectra in order to confirm that thenandk are correct (that the sample is uniform and suitable for reflectance modelling), and to acquire a starting estimation for the

Figure 4.6 The TFI based diffusion fitting requires three preliminary datasets: 1) the complex refractive index and the thickness of the film (including the roughness if necessary), 2) the photoinduced Δn^andΔkspectra, and 3) the recombination rates and surface recombination velocity.

film thickness. The pump-probing should then be started by picking an excitation wavelength that produces as uniform initial carrier distribution as possible, in other words, a wavelength where the sample has low absorption but can still be excited.

This way the Δñ is easiest to fit (Section 4.2) by minimizing the TFI from non-uniform carrier distribution, and as a reminder, the transmittance and reflectance mode measurements need to be performed under identical conditions (pump power, measurement spot, etc) to minimize errors. During theΔñfit, the sample thickness may also need to be optimized. Then the excitation wavelength is changed to where the sample absorption is higher in order to increase the transient response to the carrier diffusion. Additionally, the diffusion simulation requires knowledge of the

recombination rates and surface recombination velocity. If no literature values are available, pump-probing the sample with different intensities can reveal these recom-bination rates. In an optimal case, the carrier lifetime is longer than the time it takes to reach a uniform carrier distribution, but if not, one should focus the analysis on wavelengths where the diffusion effects are at their strongest and the recombination is least relevant. Some examples of this are given in Section 4.4.6. Nonetheless, once all the recombination parameters, sample thickness, andΔñare figured out, the dif-fusion constant is the only remaining factor for fitting the modelled ΔR(λ,t) and ΔT(λ,t)to the measured transient spectra. The relationship between the diffusion constant and the carrier mobilityμis

μ= q D

k_BT_a (4.21)

wherek_B is the Boltzmann’s constant,T_ais the temperature, andq is the electrical charge of the particle, which equals 1 for electrons and holes.

Any changes in theΔk/Δnratio of the charge carriers during the measurement may overlap with the changes in TR caused by diffusion, which greatly compli-cates the fitting process. These effects depend entirely on the sample type, including whether or not they render the diffusion constant fitting impossible.

4.4.5 Effect of surface roughness on the transient reflectance signal

Roughness means that the film has variations in the thickness, and it can be included in the reflectance modelling in two different ways: 1) by averaging the reflectance signals of the different thicknesses, or 2) by introducing a gradient in the refractive index between the air and the film. The former is for large-range variations and the latter is for microscopic variations, where the film surface features are too small for the probe wavelength to differentiate them. In this section I describe how to model these microscopic variations and how they may affect the transient signals.

Since the microscopic roughness is just a gradual change in refractive index be-tween two mediums, any such appropriate gradient can be used. We decided to use nanopyramids with an approximation of the effective refractive indexn_{e f f}:

n_{e f f} =

V_{f rac}·n₂²+ (1−V_{f rac})n₁² (4.22)

(a) (b)

Figure 4.7 The distribution of charge carriers immediately after excitation (a) without roughness and (b) with roughness. The latter distribution is stretched at the perovskite surface to have the same total number of charge carriers in both cases.

whereV_{f rac} is the relative volume of the nanopyramid (or its slice) and n₁and n₂ are the refractive indexes of the two media (air and perovskite). To increase the accu-racy of the model, the nanopyramids can be split into multiple slices for the transfer matrix. This nanopyramid approach with then_{e f f} allows for easy adjustment of the roughness layer thickness (nanopyramid height), because it maintains approxi-mately the same interference pattern when the total perovskite volume is kept the same: for instance, a 430 nm perovskite layer could be split into a 400 nm perovskite layer and a 90 nm roughness layer (volume of a pyramid is one third of the volume of a cube), without significantly shifting the steady-state TFI. Considering that the film thickness always needs to be optimized for modelling the transient signals, this method greatly simplifies the introduction of roughness. However, if the excited car-rier distribution is not uniform, the carcar-rier distribution in the roughness layer needs to be stretched to maintain the same total number of charge carriers, as presented in Figure 4.7 (a-b). Note that this "stretch" only affects then₂of Equation (4.22), not n₁.

Figure 4.8 shows an example of how adding roughness to the TFI model affects the TR signal: In this case the roughness weakens the signal response to inhomoge-neous carrier distribution and diffusion[41]. While this can greatly complicate the diffusion analysis, roughness does not always significantly change the TR response, as it depends greatly on a multitude of different factors: essentially any factor that affects the TFI can also influence the effect from roughness. When modelling the

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MEAS 2 ROUGHSIM 2 FLATSIM 2 MEAS 4 ROUGHSIM 4 FLATSIM 4

Figure 4.8 Effect of surface roughness on transient reflectance and transmittance simulations at 880 nm [41]. The roughness thickness was 90 nm while the rest of the perovskite film on a glass substrate was 496.5 nm thick. The transmittance signal was multiplied by 4 for clarity.

diffusion of charge carriers, for instance, one should first optimize all the other pa-rameters involved, and then tune the roughness to see if it makes a difference while keeping the total volume of the film the same.

4.4.6 Sensitivity of the transient reflectance and thin-film interference features to the carrier distribution

The TR signal is naturally sensitive to the carrier concentration at the sample sur-face at wavelengths where the sample has a high absorption coefficient. However, due to the high sensitivity to surface effects, the TR signal at these wavelengths may be greatly influenced by surface traps and other effects that can mask the diffusion response. The interference pattern at wavelengths where the sample is transparent, on the other hand, is slightly less sensitive to the exact surface conditions and may therefore provide a better opportunity for diffusion studies. Of course, using a vast measurement range and combining the analysis of these different features can reveal additional information. However, the TFI features are not all equally sensitive to

(a)

Figure 4.9 Some parts of the TFI pattern are more sensitive to the differences in the carrier distribution than the others. (a) The carrier distributions as a function of time and (b) the TR spectra in NIR at 0 and 1000 ps delay times. (c) The TR response to the diffusion is sensitive up to 1 ns at 875 nm wavelength whereas (d) at 1050 nm wavelength the TR loses responsiveness to the carrier distribution after 200 ps. A 540 nm perovskite film (ñ≈ 2.4+i0.013^and Δ^ñ=−0.001+i0.0001) on glass was used in this simulation.

diffusion: although the exact TFI pattern depends entirely on the sample structure, some of the features highlighted in Figure 4.9 (b) are more optimal for diffusion fitting than others. Especially in the case where the excited carriers have a short life-time, TFI features where the TR signal crosses 0 (Figure 4.9 (c)) have their diffusion dependence maximized while the relative impact of recombination is minimized.

Meanwhile in Figure 4.9 (d) the diffusion would overlap with the recombination, if there were any, and the signal in Figure 4.9 (c) is also better at distinguishing the minor changes in distribution after 300 ps as presented by Figure 4.9 (a).

4.4.7 Validity of the surface carrier model in comparison to the thin-film