Spatial and temporal imaging of long-range charge transport in perovskite thin films by ultrafast microscopy

Charge carrier diffusion coefficient and length are important physical parameters for semiconducting materials. Long-range carrier diffusion in perovskite thin films has led to remarkable solar cell efficiencies; however, spatial and temporal mechanisms of charge transport remain unclear. Here we present a direct measurement of carrier transport in space and in time by mapping carrier density with simultaneous ultrafast time resolution and ∼50-nm spatial precision in perovskite thin films using transient absorption microscopy. These results directly visualize long-range carrier transport of ∼220 nm in 2 ns for solution-processed polycrystalline CH3NH3PbI3 thin films. Variations of the carrier diffusion coefficient at the μm length scale have been observed with values ranging between 0.05 and 0.08 cm2 s−1. The spatially and temporally resolved measurements reported here underscore the importance of the local morphology and establish an important first step towards discerning the underlying transport properties of perovskite materials.

. Spatial resolution of the imaging system. In situ characterization of the imaging system's spatial resolution when two beams are in focus and give the signal maximum. The pump wavelength is at 400 nm and the probe wavelength is at 580 nm. The beam sizes are measured by scanning the pump/probe beams over a CdTe quantum wire that has a small cross-section (< 20 nm) using either a piezo electric stage (step size ~50 nm) or a galvo scanning mirror (step size ~64 nm).

Supplementary Figure 5. Time resolved PL decay of CH 3 NH 3 PbI 3 perovskite film.
Sample were excited using 470 nm pulse laser with a repetition rate of 4 MHz. The excitation fluence was set at 50 nJ/cm 2 .

Factors that limits the resolution of TAM measurements
For the diffusion coefficient measurements, the critical resolution limit of TAM is limited by signal to noise rather than diffraction limit. From the equation used to obtain the diffusion coefficient (eq. 4 in the main text), One may perform a sensitivity analysis through differentiating the above expression and immediately find that This shows that the error comes from the uncertainty of the Gaussian profiles measured at different time ( t   and 0   ), which is determined by the signal-to-noise of the detection system. A similar discussion on the resolution of such imaging approach can also be found in a prior work by Akselrod et al. 2 , where a photoluminescence microscopy imaging technique was employed.
There are two mains sources of noise contributing to transient absorption experiments: laser fluctuation noise and electronic noise from the detection system (detector and lock-in amplifier, for instance). Noise due to laser intensity fluctuation can be effectively eliminated by using heterodyne lock-in detection with MHz modulation 3 where the intensity of the excitation beam (or additional local oscillator) is modulated by an acoustic-optical modulator. Subsequently, a lock-in amplifier referenced to this modulation frequency can sensitively extract the induced signal. The fluctuation of laser intensity (1/f noise) usually occurs at low frequency of dc to 10 kHz. When f is in the MHz range, the laser intensity noise becomes near the quantum shot noise limit, which is always present because of the Poisson distribution of the photon counts at the detector. The pixel dwell time should be significantly longer than the modulation period to allow for reliable demodulation for each pixel. Such a modulation scheme has been successfully applied to transient absorption microscopy 3 .

Estimation of diffusion length
In the case where diffusion coefficient D is a constant, the diffusion equation is written as: ) , , N is the dimension being studied. N=1,2,3 corresponds to 1D, 2D and 3D diffusion respectively. The above solution has spherical symmetry and satisfies the normalization condition It can also be shown that the first moment and the second moment of the diffusing species' population density distribution has the following relationship in the spatial coordinates In the TAM imaging case, the normalized diffusion pattern reflects a free, in-plane 2D diffusion process because the signal is integrated over the out-of-plane direction.
Therefore, Dt dr r r t r n r d r t r n t r For two time delays, t 1 and t 2 , the variance of the ensemble distribution is given by