Monitoring Charge Carrier Diffusion across a Perovskite Film with Transient Absorption Spectroscopy

We have developed a new noninvasive optical method for monitoring charge carrier diffusion and mobility in semiconductor thin films in the direction perpendicular to the surface which is most relevant for devices. The method is based on standard transient absorption measurements carried out in reflectance and transmittance modes at wavelengths below the band gap where the transient response is mainly determined by the change in refractive index, which in turn depends on the distribution of photogenerated carriers across the film. This distribution is initially inhomogeneous because of absorption at the excitation wavelength and becomes uniform over time via diffusion. By modeling these phenomena we can determine the diffusion constant and respective mobility. Applying the method to a 500 nm thick triple cation FAMACs perovskite film revealed that homogeneous carrier distribution is established in few hundred picoseconds, which is consistent with mobility of 66 cm2 (V s)−1.

: Initial distribution of charge carrier density (a) without roughness and (b) with roughness. The distribution is stretched in the roughness layer because of lower relative volume of perovskite at the surface. The charge carrier density at the surface is 2 · 10 18 cm −3 .
The roughness was modelled as a nanocone which was also split into multiple layers for the transfer matrix. The refractive index n in each layer was calculated as S2 where V f rac is the relative volume of perovskite and n f ilm and n air refractive indexes of perovskite and air.
Diffusion and charge carrier density distribution was simulated using Matlab pdepefunction 1 in one dimension. The simulation assumed a flat film, so the charge carrier distribution in the roughness layer was estimated by calculating the total volume of perovskite as function of depth and stretching the distribution accordingly ( Figure S1). The total volume with and without roughness layer was kept approximately the same: Thickness of bulk perovskite was 527 nm without roughness and 496.5 nm with roughness, and height of nanocone (roughness) was 90 nm. Cone with this height has the same volume as 30 nm flat cylinder, 30 nm + 496.5 nm ≈ 527 nm, so the total volume is roughly the same with and without roughness. 90 nm roughness is reasonable in these kind of perovskite films ( Figure S2). Figure S2: An example of a typical FAMACs perovskite thin film cross-section.

Modelling steps
We have provided Matlab codes for each modelling step as part of supporting info. The modelling steps are: 1) First we roughly approximated the thickness of the film from steady state transmittance and reflectance. At this point we assumed flat film without surface roughness. Then we fine-tuned the thickness for transient simulations by simulating the transient reflectance and transmittance and comparing them to measured transient data around 1 ns delay time.
We used the spectral change in refractive index acquired in our previous paper 2 as the initial estimation for photoinduced change in refractive index. Thickness mainly determines the position of the signal peaks and zero-spots in transient reflectance signal, so thickness can be determined by aligning those.
2) We estimated the change in complex refractive index by using a transfer matrix based fitting program. We again used measured data at 1 ns delay time for the fitting. imental data at long delay times but not at short delay times (depending on wavelength, Figure S4), so we added a roughness layer to the model and increased its thickness until the simulated and measured signals matched at short delay times as described in the main article. We kept the average thickness the same while adding more roughness, and the charge carrier density distribution in the roughness layer is shown in Figure S1.  absorbance Figure S7: Sample absorbance. The small peak around 950 nm is caused by reflectance (thin film interference). Measurement reference was air.

Time-resolved terahertz spectroscopy
Time-Resolved Terahertz Spectroscopy is a pump-probe technique which is described in detail in the reviews 4,5 and our particular setup in article by Hempel et al. 6 A pump pulse (with wavelength of 402 nm and photon flux of 4 · 10 12 photons/cm 2 per pulse) photo-excites a charge carrier concentration ∆n in semiconductor. The electric field E(f ) of the following terahertz pulse oscillates at terahertz frequencies f and includes a drift of the photo-excited charge carriers. The corresponding current I(f ) = eµ(f )∆nE(f ) can be described by a frequency-dependent mobility µ (f ), which is the sum of the electron mobility and of the hole mobility. It is a complex quantity as its phase describes the phase S9 delay between driving terahertz field and oscillating current.
During the interaction, energy is transferred from the terahertz pulse to the charge carriers, which can be understood as free carrier absorption of the terahertz radiation. The change in the transmission ∆T /T of terahertz probe pulse is measured in the TRTS setup and can be analyzed for the mobility µ (f ) of the photo-excited charge carriers by the so-called "thin film approximation" in Equation (2).
The photo-excited sheet carrier concentration ∆n S of 4 · 10 12 electrons/cm 2 per pulse is calculated for the photon flux of the pump pulse and its reduction by the reflection at the sample, which is shown in Figure S8. The reactive index n of the glass substrate at frequencies of 0.5 THz is 2.55. Figure S8: Reflection of the sample measured by a PerkinElmer UV-vis spectrometer.
The sum mobility of photo-excited electrons and holes is shown in Figure S9. The real part has only a slight frequency-dependence. Therefore, it can be extrapolated to DCvalue (f = 0) of 65 ± 10cm 2 (Vs) −1 . Such a DC-value is relevant quantity for application S10 in solar cell or LEDs, which are operated at DC-conditions. This value is in line with our previous publication on MAPbI3 7 (40 cm 2 (Vs) −1 ) and the results of other TRTS groups 8 (ca. 20 − 80 cm 2 (Vs) −1 ). Figure S9: Sum mobility of photo-excited electrons and holes measured by time-resolved terahertz spectroscopy. The real part is extrapolated to the DC-value, which is the relevant quantity for application in solar cells or LDEs.

Transient absorption spectroscopy equipment
The samples were excited by laser pulses generated by Libra F, Coherent Inc., coupled with