Temperature-Dependent Intensity Modulated Two-Photon Excited Fluorescence Microscopy for High Resolution Mapping of Charge Carrier Dynamics

We present a temperature-dependent intensity modulated two-photon excited fluorescence microscopy technique that enables high-resolution quantitative mapping of charge carrier dynamics in perovskite microcrystal film. By disentangling the emission into harmonics of the excitation modulation frequency, we analyze the first and second order charge carrier recombination processes, including potential accumulation effects. Our approach allows for a quantitative comparison of different emission channels at a micrometer resolution. To demonstrate the effectiveness of the method, we applied it to a methylammonium lead bromide perovskite microcrystal film. We investigated the temperature-dependent modulated imaging, encompassing the exciton dissociation-association and charge carrier trapping-detrapping equilibrium. Additionally, we explored the potential freezing out of traps and the phase transition occurring at low temperatures.


S1. Optical setup
Figure S1 illustrates the optical setup used in this work.A mode locked oscillator (OS) of Ti-Sa laser is used as the light source.Broad band laser pulses were with the spectrum width from 700 nm to 900 nm and the pulse duration of about 10 fs.Due to the group velocity dispersion induced by the optical elements in the whole setup, a pair of chirp mirror pairs (CMP) is used to compensate the group velocity.A time dependent phase change was added in each arm of a Mach-Zehnder interferometer by two acousto-optic modulators (AOM) to modulate the average intensity of laser beam.The driven frequencies of radio wave in the AOM are 55 MHz and 54.95 MHz respectively with the difference of 50 KHz.After the third beam splitter, one part is used as the reference which can reflect the fluctuations in the laser intensity.Another part is sent to an inverted microscope (M).A dichroic mirror is equipped in the microscope to reflects light longer than 650 nm and transmits the shorter wavelengths.A reflective objective (RO) with a numerical aperture of 0.65 is used to tightly focus the laser beam onto the sample.The intensity of two-photon induced photoluminescence (PL) is detected by an avalanche photodiode (APD).The bandwidth of APD is about 5 MHz.A temperature-controlled stage (Linkam Scientific Instruments, LTS420E-P) was used to vary the temperatures of sample at 298, 273, 253, 233,  213, 193, 173, 153,

Figure S2
The SEM image of the MAPbBr 3 perovskite crystals.

S2. Excitation fluence calculation
The focus diameter in the two-photon microscopy with the relective objective NA=0.5 is: The excitation fluence is: The transition of photons per per pulse is: The excitation density calculated here indicated the predominant first-order 6.655 * 10 13 pℎo  3 *  recombination process in MAPbBr 3 film compared with the reported results 3,4 .The averaged distance between two photons is about 114.5 nm, and this is close to the reported diffusion lengths 100 50 ± nm of solution processed MAPbBr 3 film 5 .
S3 Accumulation effect in the first order recombination process 6 The effect of the accumulation effect can be described as following: describes the population of systems in excited state.And the time evolution of the excited state P(σ) population is given by the kinetic equation which consists of the relaxation and excitation transition with light intensity.
where counts the exciting laser pulses and is the interval between the consecutive pulses.The  ≡   0  0 first term at the right-hand side of (S1) describes the population decay bringing the system back to the ground state.
is the amount of decay during the pulse interval and is the population depletion  =  0   time which is equal to the population depletion time.The second term describes the probability of twophoton excitation by the intensity modulated pulses with , refers to a pulse in   = (1 + cos (Ω)) 2  the laser pulse train, and the delta function warrants that .R is the amount of excitation per  =  *  0 pulse ( .And accounts for the depletion of the ground state due to the 6.655 * 10 13 pℎo accumulation.
The Fourier transform of the solution of kinetic equation can be performed as: ⑼ In the steady state approximation, the Taylor expansion is performed.
where and .
The modulus of the Fourier transformed complex numbers show modulation amplitudes at first, second, third and fourth harmonic frequencies generated by the first-order recombination with charge accumulation effect are denoted , , , and , respectively (Equation S5-S8).1 2 3 4 The phase of the Fourier transformed complex numbers show phase at first, second, third and fourth harmonic frequencies generated by the first-order recombination with charge accumulation effect are denoted , , and , respectively (Equation S9-S12).
And the amplitude ratios as: ) The   4 AE3H and AE4H appear with the π phase shift (opposite sign) relative to AE1H and AE2H when the ground state recovery time is from 14.25 ns to 80 ns.Consequently, they will partially offset the F3H&S3H and F4H&S4H.While only AE3H appears with the π phase shift (opposite sign) relative to AE1H, AE2H and AE4H when the ground state recovery time is larger than 80 ns, meaning that AE3H will partially offset the F3H&S3H.MSa/s sampling rate) from first-order recombination process (e), second order recombination process (f), first order recombination process with charge accumulation effect (g), and the sum of the three processes (h).The , and are set to 1, the ratio is set as , is  1  2  1 1:2 1:

S4. Intensity modulation spectroscopy model simulation
1 4 1:2:3:4 set as , is set as .And the amplitudes are added 1 to 1: is set to 0.01 and the population depletion time is set  0 as 200 ns.A small noise was added before operating FFT to avoid the interference.

S5. The relative errors of A1H, A2H, A3H and A4H for four-character regions performance at different initial given time scales and temperatures.
Figure S4 shows the relationships between the initial ground state recovery time (the starting value for the iterative Trust-Region fitting procedure) and the results (Fitted time scale (ns)) at 300 K for the four characteristic regions.It turns out that in all four regions the convergence of τ is very robust and independent on the initial guess.Therefore 14.25 ns was used as the starting value of τ fitting in all pixels.A thorough analyses of the relative errors of the fitting parameters in the four regions is provided in the supplementary material.The relative errors for A1H, A2H and A3H are small, while the larger relative error of A4H at low temperature probably originates from the fluctuation of the cooling panel in the temperature control system.Since A4H is weak compared to other harmonics, the error for this amplitude does not significantly influence the following analyses.

S-9
The relative error between the experimental amplitudes A1H, A2H, A3H and A4H and the corresponding model amplitudes can be expressed as, (S16-S19) These relative errors for four regions at different given initial time scale are presented in Figure S5.And these relative errors for four regions at different temperatures are presented in Figure S6.), and (d) the averaged ground state recovery time τ of accumulation effect   in the localized spot region.The shaded regions represent the 0.5*SD.The SD are calculated based on the 9, 16, 22, and 6 pixels in localized spot regions (LOR).
For the emission behaviors in localized spot region, charge carriers recombine more complicate.It is observed the first-order and second-order PL emission ( ) (Figure 5 a-b) for the localized spot  1 &  2 region increase from 300 K to 190 K (stage 1), then decrease from 190 K to 150 K (stage 2), and last increase from 150 K to 110 K (stage 3), which are closely related to the morphology shapes in three stages (Figure S8).The increase of in stage 1&3 could cause by the similar reasons that PL  1 &  2 increase in these two relatively stable morphology geometries.While the decrease of in stage 2  1 &  2 indicate the morphology change hinder the first-order and second-order PL emission ( ) a lot.
The decreased ground state recovery time and the increased amplitude (Figure 5 c-d) in the  1 tetragonal phase with decreased temperature may be due to the increased small spot effect.During the phase transition from tetragonal to orthorhombic at around 150 K [7][8][9][10][11] , the accumulation effect PL in the localized spot regions significantly decreases, indicating that the small spot effect may disappear and alternatively additional defects may be induced due to the possible morphology change.The populations of the free carriers and the excitons in a semiconductor at any given temperature is usually approximated by the Saha-Langmuir equation (1) [12][13][14][15] : where x is the ratio between the free charge carriers over the total excitation density, n is the excitation density ( ), h is the Planck constant, E b is the exciton binding energy 7 * 10 13  -3 (25, 55, and 85 meV), T is the temperature, k B is the Boltzmann constant and µ is the reduced mass of the exciton.
133 and 113 K to explore the phase transition of sample.The temperature values are rounded in this work.A scanner (MAD CITY LABS INC, MCLS02957) is used to scan the sample film in 2 µm scale.Since the two-photon absorption depends on the square of the intensity, the excitation is smaller than the focal spot of the laser beam.The two-photon excited photocurrent map with a µm spatially resolution is established based on the setup in our previous work 1 .

FigureFrom
Figure S3 (a) Simulated results for the four harmonic amplitudes ( , , , and ) 1 2 3 4 from the first-order recombination with charge accumulation effect with different population depletion time (10 ns to 200 ns).(b) Simulated results for the four harmonic phases ( , , and )  1  2  3  4 from the first-order recombination with charge accumulation effect with different population depletion time (10 ns to 200 ns).(c) Simulated results for the three amplitude ratios ( , and ) between  21  31  41 with different population depletion time (10 ns to 200 ns). is set as 0.01. 0 From the figure, the four amplitudes , , , and are increasing while the four 1 2 3 4 phases , , and are decreasing with increased population depletion time.Besides,  1  2  3 4 AE3H and AE4H appear with the π phase shift (opposite sign) relative to AE1H and AE2H when the ground state recovery time is from 14.25 ns to 80 ns.Consequently, they will partially offset the F3H&S3H and F4H&S4H.While only AE3H appears with the π phase shift (opposite sign) relative to AE1H, AE2H and AE4H when the ground state recovery time is larger than 80 ns, meaning that AE3H will partially offset the F3H&S3H.

Figure S4
Figure S4 Simulated time-domain signals of corresponding harmonic signals and total sum signals from the first-order recombination process (a), second order recombination process (b), first order recombination process with charge accumulation effect (c), and the sum of the three processes (d).Simulated frequency-domain signals after the FFT of corresponding time-domain signals (0.1 s for 2MSa/s sampling rate) from first-order recombination process (e), second order recombination process (f), first order recombination process with charge accumulation effect (g), and the sum of the three

Figure S5 ( a )
Figure S5 (a) The relationships between the Startpoint of ground state recovery time used in the fitting and the resulting time at 300 K taken separately for four characteristic regions.The shaded regions represent the 0.5*SD (standard deviation).(b) Time-domain signals of experimental data (position [2 µm, 24 µm] at 300 K) with red line vs the fitted results with blue dotted line.(c) Frequency-domain signals after the FFT of corresponding time-domain signals (0.1 s for 2 MSa/s sampling rate) of experimental data with red line vs the fitted results with blue dotted line.A small noise was added before operating FFT to avoid the interference effect.

S7.
Figure S9 Temperature dependence of the averaged ground state recovery time τ of accumulation effect in the four intensity regions.The shaded area represents half of the standard deviation (SD).The SDs are calculated based on the 9, 16, and 22 pixels in the HR, MR, and LR, respectively.

300 K 270 K 250 K 230 K 210 K 190 K 170 K 150 K 130 K 110 K
Table S1 the averaged ground state recovery time , , , and τ of the four characteristic regions at  1  2  1 different temperatures are shown here.

Table S2 the ratios between the total excitations
1  1  2 ( + + ) in medium PL region, low PL region,

S9. Modelling of the fraction of free charges over the total excited state population. Figure S10 simulation
of the ratio of the free charge over the total excited state population (