The basic fitting ofΔn andΔk is simple. We first model the reflectance and trans-mittance of the ground state using the transfer matrix and the knownnandkof the material. We then model the excited stateRand T by introducing a small change in thenandk. By using standard error minimization algorithms, we can match the modelledΔRandΔT with the measuredΔRandΔT, which gives us theΔnand Δk. However, rather than using these slow error minimization algorithms, we devel-oped a faster method for fitting theΔnandΔk. Because the photoinduced changes are typically small in comparison to the ground state, we can use linear approxima-tion to directly calculate theΔn andΔk from theΔRandΔT. Computationally this can be orders of magnitude faster than the brute-force algorithms, which is very important when other factors such as the sample thickness also need to be adjusted for accurate results.
When the photoinduced changes inRandT are small, we can assume that they linearly depend on theΔnandΔk:
R(t)R0+RnΔn(t) +RkΔk(t) (4.5)
T(t)T0+TnΔn(t) +TkΔk(t) (4.6) whereRn,Rk,TnandTkare derivatives ofRandT overnandk, respectively, and R0andT0are transmittance and reflectance in the ground state (steady-state). From these equations,ΔnandΔkcan be solved as
Δn= TkΔR−RkΔT
RnTk−TnRk (4.7)
Δk= RnΔT −TnΔR
RnTk−TnRk . (4.8)
where ΔR = R(t)−R0 and ΔT = T(t)−T0. The 4 derivatives required by the above equations are acquired by modelling the transmittance and reflectance with the transfer matrix or an equivalent method. By applying a smallΔnto the model,
we acquireRnandTn as
Rn= R(n+Δn,k)−R(n,k) Δn
Tn= T(n+Δn,k)−T(n,k) Δn
(4.9)
and similarly with a smallΔk
Rk= R(n,k+Δk)−R(n,k) Δk
Tk=T(n,k+Δk)−T(n,k)
Δk .
(4.10)
TheΔnandΔk are typically within the range of 0.0001 to 0.01 depending on the pump power and other factors, so we can use for exampleΔk=0.0001 in the above equation.
The main concern in using the TFI model for fitting theΔnandΔk – whether one uses the linear approximation or a traditional fitting algorithm – is getting the sample thickness correct. Figure 4.2 shows how even a minor difference of few nanometers can significantly throw off theΔn fit. Therefore, it is recommended to use the computationally fast linear approximation to check a range of thicknesses until an optimal fit is found: one that is the least discontinuous and makes the most sense from the perspective of the KK relations. In the case of Figure 4.2, the bandgap bleaching of perovskite in the visible range causes theΔnto be negative in NIR (sim-ilarly to Figure 3.5), while theΔkchange in NIR is too small to make a significant difference from the KK perspective. However, theΔk fit is much less affected by these problems than the Δn fit. It should also be noted that the sharp spikes in theΔnfit are caused by the determinantRnTk−TnRk approaching zero, and these spikes also appear in the standard fitting algorithms. These artifacts should be re-moved from the finalΔnspectra.
Similar issues with the fitting may be encountered if the steady-state refractive index is off or the carrier distribution is not uniform. A non-uniform carrier distri-bution can also be accounted for by splitting the film into multiple layers, as will be explained later in the diffusion modelling Section 4.4. However, relying on a
WAVELENGTHNM
NANDK
NNM KNM NNM KNM
Figure 4.2 ΔnandΔk fit done with two different film thicknesses, 520 and 525 nm, using linear ap-proximation. TheΔkhas been multiplied by 5 for clarity.
uniform carrier distribution will increase the risk of error. Roughness, for instance, can cause problems because it has a greater impact on the TR signal of inhomoge-neous carrier distributions than that of a uniform distribution, as will be shown later in Section 4.4.5.
One should also first model the steady-state reflectance of the sample before at-tempting modelling the transient signals. This not only provides a starting point for fine-tuning the sample thickness in the model but also reveals if the sample is coherent enough to be modelled this way at all. The sample may for instance have too large variations in thickness to have a clear interference pattern, or reproduc-ing the interference pattern would require an unreasonable refractive index or film thickness.
Overall, the measurements and data analysis can be carried out in the following steps:
1. Measure the steady-state absorption and reflectance spectra and do an initial estimation of the sample thickness with the transfer matrix model.
2. Carry out pump-probe measurements in transmittance (ΔT) and reflectance (ΔR) modes in otherwise identical conditions.
3. Do necessary group velocity compensation to align the delay times of the mea-sured spectra.
4. Model the R and T with transfer matrix and acquire the 4 derivatives using Equations (4.9) and (4.10).
5. Calculate theΔnandΔk at any delay time and wavelength of interest using Equations (4.7) and (4.8).
4.2.1 Multi-layered samples and films with low refractive index
Samples with multiple thin layers can be modelled with the same transfer matrix method. In these scenarios, the excitation wavelength is usually selected to only ex-cite one of the layers to avoid overlapping transient signals from different materials.
Therefore only the active layer will experience a photoinducedΔñ=Δn+iΔk, and the other layers have no change in their complex refractive indexes unless there is a charge transfer between the different layers, which requires case-by-case modelling.
However, adding supplementary layers to a system will alter the TFI, and there-fore the transient signals, even if the active layer remains the same in thickness and other properties. In fact, films with low refractive index (<2), which generate only minor TFI by themselves because their refractive index is too close to the substrate (glass or plastic), may produce an interference pattern if coupled with another mate-rial with high refractive index. Figure 4.3 presents an example where a hypothetical material with a refractive index of 1.7 is deposited on another layer withn=2.75.
The transient signals are altered because the reflection from the active layer – bot-tom layer interface becomes stronger as the difference in refractive indexes increases.
Such cases could be, for instance, a polymer film deposited on a metal oxide such as TiO2or Al2O3.