Visualizing Buried Local Carrier Diffusion in Halide Perovskite Crystals via Two-Photon Microscopy

Halide perovskites have shown great potential for light emission and photovoltaic applications due to their remarkable electronic properties. Although the device performances are promising, they are still limited by microscale heterogeneities in their photophysical properties. Here, we study the impact of these heterogeneities on the diffusion of charge carriers, which are processes crucial for efficient collection of charges in light-harvesting devices. A photoluminescence tomography technique is developed in a confocal microscope using one- and two-photon excitation to distinguish between local surface and bulk diffusion of charge carriers in methylammonium lead bromide single crystals. We observe a large dispersion of local diffusion coefficients with values between 0.3 and 2 cm2·s–1 depending on the trap density and the morphological environment—a distribution that would be missed from analogous macroscopic or surface measurements. This work reveals a new framework to understand diffusion pathways, which are extremely sensitive to local properties and buried defects.

This clear solution was filtered through 0.22-μm-pore-size PTFE filter and then divided into 5 vials. The vials were placed on a hot plate at room temperature and then slowly heated up to about 60°C at which the growth of MAPbBr3 single crystals was achieved.

Details of the 1P optical setup:
Confocal time-resolved one-photon photoluminescence images and diffusion were measured using a confocal microscope setup (PicoQuant,MicroTime 200.) The excitation laser, a 405-nm pulsed diode (PDL 828-S"SEPIA II", PicoQuant, pulse width of around 100 ps), was directly focused onto the perovskite surface with an air objective (100x, 0.9 NA). The emission signal was separated from the excitation light (405 nm) using a dichroic mirror (Z405RDC, Chroma). The photoluminescence was then focused onto a SPAD detector for single-photon counting (time resolution of 100 ps) through a pinhole (50 μm), with an additional 410-nm longpass filter.
Repetition rates of 10 MHz were used for the maps and the diffusion profiles. The lateral spatial resolution is ~550 nm. We note that the use of local excitation and detection will lead to carrier lifetimes that are apparently shorter than bulk measurements due to diffusion away from the local region within the collection time.
For the 1P low fluence measurements, a power of 30 nW was used which corresponds to a fluence of 1.3 μJ.cm -2 . We can estimate the carrier concentration to be around 10 17 cm -3 (see discussion below) if we take into account the absorption depth of 100 nm in the present configuration 3,4 (405 nm excitation wavelength). For the high fluence measurements, a power of 300 nW was used which corresponds to a fluence of 13 μJ.cm -2 .

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The raster scanning was performed using a galvo mirror system while both the objective and the sample remain at a fixed position. In the case of regular local TRPL measurements with this setup, both the excitation and the emission are scanned through the mirror system. On the other hand, only the emission path was scanned to create the diffusion profiles from the main text, while the excitation was decoupled and fixed at the center of the sample (x=0).

Impact of the optical resolution to the width of the 1P TRPL beam:
To evaluate the contribution of the optical resolution to the width of the PL Gaussian at t=0, we use the Abbe diffraction formula. In such a model, the distance between the maximum of the intensity and the first minimum is expressed as L=0.61*λ/NA, where NA=0.9 is the numerical aperture of the used objective lens. For the diffusion measurement, we evaluate the standard deviation σ of such intensity profile, under a Gaussian approximation of the profile. Therefore, the value of this variance can be estimated as σreso ≃ 0.65 * L. This yields the values of σreso≃180 nm under excitation at 405 nm and ≃240 nm for emission at 540 nm.

1P estimation of the TRPL decay:
Figure S1: 1/e Method to obtain the lifetime of the TRPL curve. The TRPL decay was estimated by measuring the delay at which the PL intensity has decreased a factor e from the initial intensity. After a prolonged illumination of the sample, or when a higher fluence of 13 μJ.cm 2 is used, we observe a broadening of the initial PL width σx(0) with respect to the lower fluence measurements.
Additionally, σx(t) stays almost constant on a timescale of around 1.5 ns. After this delay, it starts increasing with time indicating that the regular diffusive behaviour is progressively recovered.
Interestingly, the value of the obtained diffusion coefficient in these conditions is 0.33 cm 2 .s -1 , very similar to the one obtained without these intense excitation conditions. Therefore, the regular diffusion is still happening but another phenomenon induces a local change in the photophysical

Details of the 2P optical setup:
For the 2P diffusion profiles, an optical fibre (25 μm core) was used for the raster scanning of the detection only. The depth excitation is adusted by a motor that controls the position of the objective lens with micrometre steps. The collected photons were then sent onto a single-photon avalanche photodiode (SPAD), and their arrival times were recorded with a time-resolution of 100 ps. The 2P pulsed excitation is achieved using an Optical Parametric Oscillator (OPO, 150 fs pulses), set at a wavelength of 1200-nm (below the MAPbBr3 bandgap) and a 100x air objective lens (NA = 0.95). The lateral spatial resolution of this optical microscope is estimated to be around 1.2μm in FWHM (and 0.5 μm in variance), and the vertical spatial resolution is around 1.5 μm (see below).
The derivation for the 2P depth resolution can be estimated by taking into account the convolution between the PL spatial beam profile and the product of the point spread function (PSF) of the lens operating at the two wavelengths (excitation and emission) 6,7 . In that configuration, the PL intensity depth profile depends on the excitation wavelengths, λem and λex respectively, 6 and on the ratio β = λem/λex. If the 2P excitation wavelength and photoluminescence wavelengths are the same (β = 1), the diffraction limited depth profile of a two-photon (2P) confocal microscope is given by 6 : where n is the index of refraction of the imaged medium, λex is the excitation wavelength in vacuum and NA=nsin(θ) is the numerical aperture of the objective lens.
In the typical two-photon configuration (where the emission wavelength is shorter than the excitation wavelength), this formula overestimates the resolution (FWHM). Indeed, the shortest resolution is obtained when the excitation wavelength is slightly longer than the PL emission wavelength (since the depth response increases with λex).
In practice, the depth-of-focus increases when the focal volume is focused deeper into the sample due to aberrations caused by refraction at the surface of the sample 8 . A correction has to be applied to the position of the focal point Δ set experimentally to obtain the actual depth-of-focus (DOF) 8 : where n is the refractive index of the medium and NA is the numerical aperture of the objective lens. In our case, this formula provides an estimation of the depth resolution to be around 1.5 μm.
Lateral steps of 500 nm have been used in the sample. By changing the depth-of-focus, we create this profile for different depths (z), with a step of 1 μm. On each location, the integration time for the total TRPL measurement is set to 20 seconds. For time-resolved collection, we spectrally-filtered the PL using a linear variable longpass filter to only get the red part of the spectrum and therefore minimize the influence of reabsorption and reemission on the measured diffusion. At each (x,z) coordinate, a 2P TRPL measurement was performed with a pulse repetition rate of ∼8 MHz. We focus our study on a total time window of 7 ns that allows observing most of the intensity decay as well as the influence of the trapping dynamics on such decay. We used here two different fluences: a lower one of 584 μJ.cm -2 and a larger one of 1300 μJ.cm -2 .
We note that in both the 1P and the 2P measurements, both the sample and the objectives remain immobile. For the diffusion measurements, the excitation location is also fixed at a particular position. The only part that is modified during the raster scanning is the emission optical 12 path. In the case of the 1P setup, the emission optical beam is raster scanned using a rotating galvo mirrors system. This system tilts the optical beam allowing us to change the area of the sample that is probed. In the 2P Setup, the raster scanning fibre (25 μm core) is implemented in the collection arm of the microscope to monitor the PL emission as a function of distance (x) from the 2P-excitation spot (x=0) and coupled to a single-photon avalanche photodiode (SPAD). 13
In the 2P configuration, the used fluences are 580 μJ.cm -2 and 1300 μJ.cm -2 , with a pulse duration of 150 fs and an estimated beam vertical width of 1.5 μm. For these two fluences, we can estimate the pulse peak energy density to be respectively of 4 and 9 GW.cm -2 Assuming a β coefficient of μJ.cm -2 (higher fluence, HF, orange) highlighting the transition between a pure monomolecular to more slightly more bimolecular regimes.

Optical resolution of the 2P setup:
In order to characterise the detection resolution of our 2P setup, we carry out a 2D (x,y) map of the laser reflection at the surface of a MAPbBr3 crystal. This map is shown as inset in 15 Figure SI 1. We perform a Gaussian fit on the laser profile ( Figure S7 and measure a σlaser∼ 0.5μm. This corresponds to a FWHM ∼ 1.2μm, which is consistent with a diffraction-limited resolution.

TRPL decays and Accuracy of the Gaussian diffusion fittings.
We fit each PL profile over a time window of 7 ns, highlighted in red in Figure S5. Figure   S8 a-d shows a selection of these PL profiles and their respective fit as a function of depth and fluence at t=0 ( Figure S8 a and b ) and t=7 ns ( Figure S8 c-d). We note a good agreement of the data and the fit across the analysis window (0 ns to 7 ns). From the fit, we extract the timedependent outward diffusion σx(t). Figure S8

Figure S10: Comparison of 2P PL profiles measurements between regions in MAPbBr3 and
MAPbI3 crystals. To reject the possibility that the asymmetry in the PL beam (cf. Figure 4

Derivation of the diffusion equation:
Let us first look at the differential diffusion equation in 1D: Where c is the physical quantity that diffuses as a function of time and D is the corresponding diffusion coefficient. In this article, c will be the carrier density.
In this equation, we have neglected the losses of carriers that mostly originates from the first order Shockley Read Hall 10 non-radiative recombination. As stated elsewhere 11 , these losses will not affect the shape of the Gaussian beam can, therefore, be neglected as long as we consider the normalised distribution of carriers at every instant t. A particular solution above of this equation is of the form: This particular solution corresponds to an infinitely thin distribution of charge carriers for t=0 and x=0. At longer times, it corresponds to a Gaussian beam: In the case of another initial distribution c0(x), the general solution can be written as the convolution product of the particular solution and the initial distribution of charge carriers:

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In the case of a monochromatic (Laser) plane wave focused into a microscope objective, the initial distribution of carriers should be an Airy diffraction function. In the following, we will approximate this distribution with a Gaussian function of standard deviation σ0: The full solution of the diffusion equation can be therefore expressed as: which is the product of convolution of two Gaussian functions. Given the mathematical properties of the convolution product of two Gaussian functions, we know that distribution of charge carriers at each instant t will still be a Gaussian with a standard deviation σ1 according to the next equation: Due to the rotational symmetry of the studied system, this expression can be generalised to the 2D and 3D case by just considering each dimension as independent. As seen in Figure S12, a simple iterative simulation for an arbitrary Gaussian case in 1D, 2D and 3D confirms the relationship between the standard deviation and D, σ 2 (t) = σ 2 0 +ADt with A = 2. A simulation with a Gaussian 23 squared, more relevant to the 2P excitation configuration, gives A ∼ 2.05, very close to the Gaussian case.  Additionally, we have also included in our simulation a phenomenological first-order term in the diffusion equation to account for the loss of charge carriers due to the presence of traps: