Introduction

Organic-inorganic lead trihalide perovskite solar cells (PSCs) have been in the limelight of photovoltaic research1,2,3, as exemplified by the outstanding certified power conversion efficiency (PCE) that exceeds 25% to date4. Even in the polycrystalline form, the PSC thin films exhibit many remarkable photo-physical properties, such as high absorption coefficient5, long carrier lifetimes6, and low impurity scattering rate7,8. For photovoltaic applications, a particularly attractive feature of hybrid perovskites is that both electrons and holes are active in the photoconduction process6,7,8,9,10. From the theoretical point of view, the two types of carriers are expected to exhibit comparable effective mass, intrinsic mobility, recombination lifetime, and diffusion length11,12,13. In real materials, however, the balance between electrons and holes is usually broken by thin-film deposition conditions, defect structures, ionic disorders, and other sample-dependent parameters14,15,16,17, which may affect the photo-carrier extraction in functional devices. A thorough understanding of the dynamics of individual charge carriers is thus imperative for continuous improvements of PSC performance towards commercial applications.

The spatiotemporal dynamics of electrons and holes in optoelectronic materials are widely studied by optical measurements such as time-resolved photoluminescence (TRPL) and transient absorption spectroscopy (TAS)8,9,10,18,19. The diffusion length can then be deduced by fitting the results to a diffusion model8,9,10. It should be noted that TRPL and TAS probe optical excited states and are often dominated by transitions with large oscillator strength. In contrast, the transport of photoexcited carriers is electrical and quasi-static in nature. In order to directly evaluate photoconduction, it is common to measure the photocurrent across electrical contacts, such as scanning photocurrent microscopy (SPCM)16,20,21. The spatial resolution of SPCM is diffraction-limited and the temporal response is dominated by the carrier transit time and extrinsic metal-semiconductor Schottky effect. In recent years, noncontact methods such as time-resolved microwave conductivity (TRMC)6,7,16,22,23,24,25 and time-resolved THz spectroscopy (TRTS)17,26 are developed to probe the photo-carrier dynamics. These far-field techniques, however, do not offer spatially resolved information such as diffusion patterns.

In this article, we directly probe free-carrier diffusion dynamics in Cs-doped formamidinium (FA) lead trihalide (FACsPbI3) thin films by laser-illuminated microwave impedance microscopy (iMIM), a unique optical-pump-electrical-probe technique with 20-nm spatial resolution and 10-ns temporal resolution for the electrical detection27. The steady-state iMIM experiment addresses the most important photo-physical process in solar cells, i.e., the transport of photo-generated mobile carriers under the continuous illumination of ~1 Sun. The time-resolved iMIM (tr-iMIM) experiment detects the free-carrier lifetime that is also highly relevant for photoconduction. By depositing a hole or electron transport layer (HTL/ETL) underneath FACsPbI3, we show that the two decay constants in tr-iMIM measurements are associated with the lifetimes of electrons and holes. The spatiotemporal imaging allows us to determine the diffusion coefficients, steady-state carrier density, and mobility of individual carriers. Interestingly, while the lifetime and mobility of electrons and holes differ substantially, their products and thus the diffusion lengths are comparable to each other, which is likely due to the unique defect structures and charge trapping events in PSC thin films. Our results highlight the origin of nearly balanced diffusion dynamics of electrons and holes in hybrid trihalide perovskites, which is highly desirable for photovoltaic applications.

Results

The PSC thin film in this study, hereafter labeled as Sample A, is 5% Cs-doped FA lead triiodide deposited on cover glasses (see Methods). Compared with methylammonium (MA) based perovskites, FAPbI3 exhibits superior stability at elevated temperatures and an ideal band gap for sunlight absorption28. The Cs-doping further stabilizes the room-temperature photo-active α-phase by reducing the Goldschmidt tolerance factor29,30,31,32. Perovskite films were deposited using the typical anti-solvent-assisted spin-coating procedure. The samples were then capped by spin-coating 15 mg ml−1 PMMA (Mw ~ 120,000) film in chlorobenzene solution. For iMIM measurements, we chose a film thickness of d = 250 nm that is greater than the absorption length, such that light is fully absorbed, but much less than the carrier diffusion length, such that the photoconductivity is uniformly distributed in the vertical direction within the relevant time scale in our experiment. External quantum efficiency (EQE) spectra were also measured (Supplementary Fig. 1), showing good photoresponse across the solar spectrum. PSC devices made from the same material but with thicker film (550 nm) demonstrated a PCE above 20% under the standard air mass (AM) 1.5 illumination (Supplementary Fig. 1).

The spatiotemporal iMIM setup with a focused laser beam illuminating from below the sample stage is illustrated in Fig. 1a. In the tip-scan mode, the laser is focused by one set of piezo-stage and the second piezo-scanner carries the tip to scan over the sample27. In the sample-scan mode, the first set of piezo-stage aligns the center of the laser spot to the tip, whereas the sample is set in motion by the piezo-scanner33,34. In both configurations, one can fix the relative position between tip and sample and perform time-resolved (tr-iMIM) measurements27. Here the laser output is modulated by an electro-optic modulator (EOM), which is driven by a 7-kHz square wave from a function generator such that steady-state photoconductivity is reached in the laser-ON state and zero photoconductivity in the laser-OFF state. The same waveform also triggers a high-speed oscilloscope for iMIM measurement. The temporal resolution of our setup is ~10 ns (see Methods). The microwave electronics detect the tip-sample impedance, from which the local conductivity can be deduced35. The optical excitation in our setup is diffraction-limited, whereas the electrical imaging has a spatial resolution of 20–50 nm compared to the tip diameter. Quantification of the iMIM signals by finite-element analysis (FEA)36 is detailed in Supplementary Fig. 2.

Fig. 1: Photoconductivity mapping on FACsPbI3 and diffusion analysis.
figure 1

a Schematic of the iMIM setup with either the tip-scan or sample-scan mode. The tr-iMIM configuration is shown inside the dash-dotted box. The FACsPbI3 thin film deposited on a glass substrate and encapsulated by a PMMA layer (Sample A) is also illustrated. b Tip-scan iMIM images when the sample is illuminated by a 446-nm diode laser at PC = 100 mW/cm2. c Photoconductivity map based on the iMIM data and FEA simulation. The dashed lines are various linecuts for the calculation of average signals. d Image of the laser spot taken by a CCD camera. e Line profiles of averaged photoconductivity and laser intensity, from which the diffusion length can be extracted. The solid black and dashed gray lines represent the best curve fitting and upper/lower bounds, respectively. All scale bars are 10 μm.

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Figure 1b shows the iMIM images when Sample A was illuminated by a 446-nm (hν = 2.78 eV) diode laser with the intensity at the center of the laser spot PC = 100 mW/cm2, i.e., on the order of 1 Sun. The granular features are due to topographic crosstalk with the polycrystalline sample surface33. It is nevertheless evident that the photoresponse is continuous across many grain boundaries (GBs). Based on the iMIM response (Supplementary Fig. 2), we can replot the data to a conductivity map (Fig. 1c) with high conversion fidelity. For comparison, the optical image of the laser spot acquired from a charge-coupled device (CCD) camera shows a much smaller spatial spread in Fig. 1d. To improve the signal-to-noise ratio and minimize the topographic artifact, we averaged eight line profiles shown in Fig. 1c. The resultant curve, plotted in Fig. 1e, is clearly broader than the Gaussian beam profile (\({e}^{-{r}^{2}/{w}^{2}}\), w ~ 2 µm). Assuming that the carrier mobility μ is independent of charge density n within the range of our experiment, the measured photoconductivity profile σ(r) is proportional to the steady-state density profile n(r) as

$$\sigma \left(r\right)=n\left(r\right)q\mu$$
(1)

where q is the elemental charge. Here n(r) can be described by the diffusion equation27,37,38

$$n\left(r\right)-{L}^{2}{\nabla }^{2}n\left(r\right)=\frac{\eta }{d}\frac{{P}_{c}\tau }{h\nu }{e}^{-{r}^{2}/{w}^{2}}$$
(2)

where \(L=\sqrt{{{{{{\rm{D}}}}}}\uptau}\) is the diffusion length, D the diffusion coefficient, τ the lifetime, and η ~ 1 the incident photon-to-current conversion efficiency (IPCE). The analytical solution to Eq. (2) is

$$n\left(r\right)\propto \int_{-{{\infty }}}^{{{\infty }}}{K}_{0}\left({r}^{{\prime} }/L\right){e}^{-{\left(r-{r}^{{\prime} }\right)}^{2}/{w}^{2}}d{r}^{{\prime} }$$
(3)

where K0 is the modified Bessel function of the second kind. By fitting the iMIM data to Eq. (3), we obtain a diffusion length L = 5.1 ± 0.6 µm, consistent with values reported in the literature for thin-film PSCs9,10,16,46. As the laser power increases, L decreases to ~4 µm at PC ~ 103 mW/cm2 and saturates at 3.5 µm for PC ~ 104 mW/cm2 (Supplementary Fig. 3).

Figure 2a shows a typical tr-iMIM decay curve (averaged over 242,880 cycles) on Sample A illuminated by the 446-nm laser at PC = 100 mW/cm2 before t = 0. As the iMIM-Im signal scales with photoconductivity in our measurement range (Supplementary Fig. 4), we will just present the raw data in the following analysis. From Eq. (1), the decay of tr-iMIM-Im signal provides a direct measure of the lifetime of mobile carriers in the conduction or valence band. The relaxation fits nicely to a biexponential function \(y={A}_{1}{e}^{-t/{\tau }_{1}}+{A}_{2}{e}^{-t/{\tau }_{2}}\), with τ1 ~ 0.7 µs and τ2 ~ 10 µs. As shown in the inset of Fig. 2a, we observed the same τ1 and τ2 when using 517-nm and 638-nm lasers (Supplementary Fig. 5), suggesting that the time constants are intrinsic to the sample and independent on the laser wavelength. In contrast, the TRPL data on Sample A (Fig. 2b) exhibit two different times τf ~ 150 ns and τs ~ 0.7 µs, which will be discussed in the next section. By parking the tip at various locations of the film and measuring the decay curves, we also show that the tr-iMIM response is spatially uniform on the sample surface within statistical errors (Supplementary Figs. 6 and 7).

Fig. 2: Time-resolved iMIM and carrier lifetime.
figure 2

a Typical tr-iMIM relaxation curve of Sample A. The sample was illuminated by the 446-nm laser at PC = 100 mW/cm2 before t = 0 μs. The inset shows the two lifetimes under excitation lasers with wavelengths of 446, 517, and 638 nm. b TRPL data of Sample A. The blue curve is a biexponential fit to the TRPL data. The inset is a close-up view, showing the two decay time constants from the fitting.

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In order to shed some light on the tr-iMIM data, we studied the carrier dynamics in perovskite thin films with electron or hole transport layers6,7,9,10,22,23. For the HTL sample, hereafter referred to as Sample B, a 20-nm PTAA (poly-triaryl amine) was spin-coated on the substrate before the same 250-nm FACsPbI3/30-nm PMMA film was deposited. The PTAA layer rapidly extracts photo-generated holes from FACsPbI3 within a sub-10-ns time scale22,23,39. Similarly, a 40-nm ETL TiO2 layer for the extraction of electrons was coated on the substrate before the FACsPbI3/PMMA deposition for Sample C. Control experiments have been conducted to ensure that the PL is quenched in both Samples B and C due to the extraction of holes and electrons, respectively (Supplementary Fig. 8). Note that the charge dynamics in the transport layers (~300 nm below the surface) would not affect the iMIM results due to the shallow probing depth of 50–100 nm. The tr-iMIM data in Samples B and C under the 446-nm laser illumination with PC = 100 mW/cm2 are shown in Fig. 3a. It is evident that only the fast process with τ1 ~ 0.7 μs survives in Sample B and the slow process with τ2 ~10 μs in Sample C. The observation strongly suggests that the two time constants in Sample A are associated with the lifetime of electrons and holes in FACsPbI3.

Fig. 3: Results and analysis on HTL/ETL samples.
figure 3

a tr-iMIM signals on the HTL Sample B (upper panel) and ETL Sample C (lower panel). The layer structures of each sample are illustrated in the insets. b Diffusion maps of Sample B and c Sample C under the illumination of 446-nm laser at PC = 100 mW/cm2. Scale bars are 10 μm. d Tabulated parameters for the calculation of electron/hole mobility values by two methods, i.e., μdiff from the Einstein relation and μpc from photoconductivity analysis.

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The HTL/ETL samples also allow us to separately address the diffusion dynamics of electrons and holes. Figure 3b, c show the tip-scan photoconductivity maps of Samples B and C under PC = 100 mW/cm2, from which Le ~ 5.2 μm and Lh ~ 2.7 μm can be extracted, respectively. As tabulated in Fig. 3d, we can derive the diffusion coefficient from the diffusion equation \(L=\sqrt{{{{{{\rm{D}}}}}}\uptau}\) and carrier mobility (μe,diff = 15 cm2/V\(\cdot\)s and μh,diff = 0.3 cm2/V\(\cdot\)s) from the Einstein relation μ = (q/kBT)*D. A different method to analyze the transport properties is through the photoconductivity (Eq. 1) and density profile (Eq. 2). The calculated mobility values are μe,pc = 24 cm2/V⋅s and μh,pc = 0.3 cm2/V⋅s. The small difference between the two methods is within the error bars of the measurements. We note that in MAPbI3 and FAPbI3 thin films, mobility values measured by different techniques vary in a considerable range from 0.2 to 30 cm2/V⋅s6,7,8,9,10,14,17,23,24,25,28,40,41. As tabulated in Supplementary Fig. 9, either μe > μh or μe < μh has been reported in the literature. In our experiment, mobility values are directly calculated from the measured L and τ under an illumination intensity ~1 Sun, with no other assumptions or modeling involved. The pronounced difference between μe and μh is thus genuine. Figure 3d also indicates that the equilibrium carrier density in our experiment is on the order of 1015–1016 cm−3. Within this range, the electron/hole mobility is largely independent of the carrier concentration42. It is thus well justified to approximate the density profile by the measured photoconductivity profile in our diffusion analysis (Fig. 1e).

Finally, we briefly discuss the iMIM results at higher illumination intensities. As shown in Fig. 4a, b, the temporal evolution of HTL/ETL samples again displays the biexponential decay when PC increases beyond 100 mW/cm2 (complete data in Supplementary Fig. 10), with one of the processes substantially suppressed. For instance, while A1/A2 ~ 2 is expected in plain FACsPbI3, the electron dynamics clearly dominate in Sample B such that A1/A2 > 2 in Fig. 4c. Conversely, with electrons efficiently removed by the ETL, the hole dynamics prevail and A1/A2 < 2 in Sample C. We have also performed photoconductivity mapping on Samples B and C under various PC (Supplementary Figs. 11 and 12) and the results are plotted in Fig. 4d. As PC increases towards 103–104 mW/cm2, the contribution from the other type of carriers is no longer negligible. Consequently, in addition to the general trend of decreasing diffusion length at increasing excitation, L in Sample B decreases further at high PC, whereas L in Sample C increases slightly at high PC. Similarly, while only one type of carriers is responsible for the low-power photoconductivity, σC does not scale with PC in either sample towards 104 mW/cm2 (Fig. 4e).

Fig. 4: Power-dependent iMIM results.
figure 4

a tr-iMIM signals on Sample B and b Sample C under various laser powers. Signals below the noise level are truncated for clarity. c Power-dependent A1 and A2 in Sample B (upper panel) and Sample C (lower panel). The dash-dotted lines are A1/2 (Sample B) and 2A2 (Sample C) for comparison with the plain perovskite Sample A, which shows A1/A2 ~ 2. d Power-dependent diffusion lengths and e photoconductivity at the center of the illumination spot in both samples.

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Discussion

The spatiotemporally resolved iMIM experiments reveal rich information on organometal trihalide perovskite thin films. To begin with, we take a close look at the impact of GBs on charge transport in PSC materials, which has been under intense debate19,43,44,45,46. As summarized in a recent review47, while GBs strongly affect the current–voltage hysteresis and long-term stability of PSCs, their effect on carrier recombination and thus the open-circuit voltage is rather mild under the illumination of ~1 Sun. In a previous report33, we showed that the photoconductivity is spatially homogeneous over grains and GBs, consistent with conductive AFM and SPCM studies20,46. In this work, we further demonstrate that the carrier diffusion is not impeded by the presence of numerous GBs in all three samples. It is possible that the GBs in the current study are not strong nonradiative recombination (i.e., highly defective) centers, and there is no significant band bending at the GBs to block electron/hole movement across multiple grains48. As a result, under the normal solar-cell operation, GBs in our samples do not lead to appreciable spatial variation of transport properties such as the density and mobility of photoexcited carriers, consistent with the early theoretical prediction49. We caution that sample-to-sample variation is widely observed in the PSC research. It is still possible that GBs in other hybrid perovskite thin films exhibit strong impacts on the carrier lifetime and transport properties.

Given the extensive use of PL in studying carrier dynamics, it is instructive to compare the TRPL and tr-iMIM results in our samples. In short, TRPL measured excited states such as exciton recombination via emitted photons, whereas tr-iMIM measures the decay of steady-state conductivity following optical injection of free carriers. In TRPL experiments, the signal strength depends on the radiative recombination process that emits photons, whereas the temporal evolution measures the total lifetime of certain carriers or excitons8,9,10,18,19. For the TRPL data of Sample A in Fig. 2b, the fast (τf ~ 150 ns) and slow (τs ~ 0.7 μs) processes are associated with the surface recombination and the relaxation of the shorter-lived electrons in FACsPbI3, respectively. Note that τs matches τ1 in the tr-iMIM data. After the elapse of τs, however, no more mobile electrons are available for radiative recombination with mobile holes. As a result, TRPL cannot measure the lifetime of the longer-lived carriers16. We emphasize that the extraction of one type of carriers by HTL or ETL quenches the PL process and the TRPL decay constants in these samples no longer represent lifetimes of electrons or holes in plain PSCs50. In tr-iMIM, however, both the signal strength and temporal evolution depend on photoconductivity, which is proportional to the product of carrier density and mobility. For the three time scales above, the decay on the order of 100 ns is not seen in tr-iMIM, presumably due to the small steady-state density and low mobility of surface-bound carriers. On the other hand, because of the low efficiency of radiative recombination in PSCs16, the removal of free electrons does not lead to appreciable changes in the dynamics of free holes. Consequently, the relaxation process of electrons and holes can be treated independently, as revealed by the tr-iMIM data. It should be noted that PL microscopy has also been utilized to map out the diffusion dynamics in PSC materials51,52. For the same reasons discussed above, it is not straightforward to compare photoluminescence and photoconductivity imaging results across multiple grains, which will be subjected to future experiments.

The difference between photo-physical properties of electrons and holes, as evidenced in Fig. 3d, deserves further discussions. In hybrid perovskites, deep-level defects dominate the trapping/de-trapping process and nonradiative recombination of free carriers. In general, the deeper the trap level, the longer time it takes for carriers to be de-trapped, and consequently the longer lifetime and lower mobility. Theoretical studies53,54 suggest that cation and anion vacancies create shallow energy levels, while iodine interstitials introduce deep levels in the bandgap. Interestingly, iodine interstitials can be both positively (Ii+) and negatively (Ii−) charged, which leads to spatially separated trapped electrons and holes with very low recombination efficiency. The transition energy for Ii+ (0/+) (de-trapping for electron) is calculated to be 0.48 eV below the conduction band minimum (CBM), whereas the transition energy for Ii− (0/−) (de-trapping for hole) is 0.78 eV above the valence band maximum (VBM)54. The larger de-trapping barrier for hole results in its longer lifetime and lower mobility. When photoexcited electrons are quenched, the remaining holes will be trapped and then de-trapped via Ii−, and vice versa. The trapping/de-trapping process induces delayed recombination, as manifested in the tr-iMIM decay curves. This qualitatively explains that the holes have a long carrier lifetime but lower mobility than electrons. Further theoretical work is needed to elucidate this physical picture in a quantitative manner.

As a final remark, we emphasize that in solar cells, diffusion lengths of both electrons and holes much larger than the film thickness is desirable for the effective separation of photo-carriers. Because of the unique defect properties in hybrid perovskite thin films, as well as the competition between the recombination and trapping/de-trapping process, the imbalance in mobility (μe ≫ μh) is largely compensated by the imbalance in lifetime (τe ≪ τh). As a result, the difference between Le ~ 5 μm and Lh ~ 3 μm is insignificant in our samples, which is of fundamental importance for the superior performance of PSC devices.

In summary, we systematically study the optoelectronic properties of 5%-Cs-doped FAPbI3 thin films (PCE > 20%) by imaging the carrier diffusion in real space and detecting the photoconductivity evolution in real time. For plain perovskite films, two relaxation processes are observed on the sample. By selectively removing one type of carriers, we demonstrate that the fast and slow decay constants are associated with the lifetimes of photo-generated electrons and holes, respectively. The diffusion mapping on HTL/ETL samples allows us to extract parameters such as diffusion coefficient, equilibrium carrier density, and mobility of both carriers. The imbalance in carrier lifetime is offset by the difference in mobility such that diffusion lengths of electrons and holes are comparable to each other. We emphasize that, prior to our work, separate experiments are needed to measure relaxation time (TRPL or TRTS) and mobility (transport or SPCM on doped samples) of free carriers. To our knowledge, it is the first time that diffusion length, carrier lifetime, and charge mobility can be individually addressed for mobile electrons and holes on the same batch (as-grown, HTL-coated, and ETL-coated) of samples. The spatiotemporal microwave imaging provides the most direct measurement of photo-physical properties of organometal trihalides, which is crucial for the research and development of these fascinating materials towards commercial products.

Methods

Materials

All solvents were purchased from Sigma-Aldrich and used as-received without any other refinement. Formamidinium iodine (FAI) was purchased from Greatcell Solar. Lead iodide (PbI2) was from TCI Corporation. Spiro-OMeTAD was received from Merck Corporation. Cesium iodine (CsI) and PTAA were purchased from Sigma-Aldrich. Patterned fluorine-doped tin-oxide-coated (FTO) glass (<15 Ω/square) and indium tin-oxide-coated (ITO) glass were obtained from Advanced Election Technology Co., Ltd.

Sample preparation

The perovskite films in this work were deposited on top of cover glasses or ITO glass. The substrate glasses were cleaned extensively by deionized water, acetone, and isopropanol. For the HTL deposition, the PTAA (Sigma-Aldrich) was dissolved in toluene with a concentration of 5 mg mL−1 and spin-coated on the substrates at 5000 rpm for 30 s. The spun PTAA films were annealed at 100 °C for 10 min. For the ETL deposition, a compact titanium dioxide (TiO2) layer of about 40 nm was deposited by spray pyrolysis of 7 mL 2-propanol solution containing 0.6 mL titanium diisopropoxide bis(acetylacetonate) solution (75% in 2-propanol, Sigma-Aldrich) and 0.4 mL acetylacetone at 450 °C in air. The FA0.95Cs0.05PbI3 precursor solution was prepared by dissolving 0.4 M Pb2+ in dimethyl sulfoxide (DMSO) and dimethylformamide (v/v = 3/7) mixed solvent. Perovskite films were deposited using a three-step spin-coating procedure with the first step of 100 rpm for 3 s, the second step of 3500 rpm for 10 s, and the last step of 5000 rpm for 30 s. Toluene (1 mL) was applied on the spinning substrates at 20 s of the third step. After spin coating, the substrates were annealed at 170 °C for 27 min. The encapsulated perovskite films were capped with PMMA (Mw about 120,000) film by spin-coating 15 mg ml−1 PMMA in chlorobenzene solution at 4000 rpm for 35 s.

iMIM and tr-iMIM setup

The sample-scan iMIM was performed on a modified ParkAFM XE-100 platform with bottom illumination. The tip-scan iMIM was performed in a customized chamber (ST-500, Janis Research Co.) with positioners and scanners (AttoCube Systems AG). During the measurements, we kept the samples in a vacuum by pumping the chamber below 10−4 mbar. The PtIr tips were purchased from Rocky Mountain Nanotechnology LLC, model 12PtIr400A for diffusion mapping and 12PtIr400A-10 (ultra-sharp tips) for the point measurements in Fig. 2. For tr-iMIM, the diode laser was modulated by an EOM (M350-160–01 EOM, Conoptics Inc.) with a power supply of 8-ns rise/fall time. The EOM was driven by a 7-kHz square wave from a function generator (DG5071, RIGOL Technologies USA Inc.) with <4-ns rise/fall time. The tr-iMIM signals were measured by a 600-MHz oscilloscope (DS6062, RIGOL Technologies USA Inc.) with a 5-GSa/s sampling rate.

Finite-element analysis

FEA was performed using the AC/DC module of commercial software COMSOL4.4. The tip for diffusion mapping is relatively blunt due to the extensive contact-mode scans on the sample surface. We modeled it as a truncated cone with a half-angle of 15° and a diameter of 200 nm at the apex. The ultra-sharp tip was mostly used for point measurements in Fig. 2 and the apex was well preserved. It was thus modeled as a truncated cone with a half-angle of 38° and a diameter of 20 nm at the apex. In the simulation, the dielectric constants of PMMA and FACsPbI3 are 3 and 62, respectively, consistent with that reported in the literature33,55. The FEA software computes the real and imaginary parts of the tip-sample admittance (proportional to iMIM-Re/Im signals) as a function of the conductivity of the perovskite layer, using the values at σ = 0 as the reference.