Limits to Electrical Mobility in Lead-Halide Perovskite Semiconductors

Semiconducting polycrystalline thin films are cheap to produce and can be deposited on flexible substrates, yet high-performance electronic devices usually utilize single-crystal semiconductors, owing to their superior charge-carrier mobilities and longer diffusion lengths. Here we show that the electrical performance of polycrystalline films of metal-halide perovskites (MHPs) approaches that of single crystals at room temperature. Combining temperature-dependent terahertz conductivity measurements and ab initio calculations we uncover a complete picture of the origins of charge-carrier scattering in single crystals and polycrystalline films of CH3NH3PbI3. We show that Fröhlich scattering of charge carriers with multiple phonon modes is the dominant mechanism limiting mobility, with grain-boundary scattering further reducing mobility in polycrystalline films. We reconcile the large discrepancy in charge-carrier diffusion lengths between single crystals and films by considering photon reabsorption. Thus, polycrystalline films of MHPs offer great promise for devices beyond solar cells, including light-emitting diodes and modulators.


Terahertz spectroscopy setup
shows the experimental setup for the optical pump terahertz probe spectroscopy (OPTPS) in both transmission and reflection modes. An amplified laser (with central wavelength of 800 nm, 5 kHz repetition rate, 35 fs pulse width and 4 W average power) was split into three arms: gate beam, THz beam and pump beam. The THz pulse was generated by a spintronic emitter due to inverse spin Hall effect. S1 The pump beam was up-converted to 400 nm by a β-barium-borate (BBO) crystal and a short-pass filter was used to filter out the residual 800 nm light. Two optical choppers were used to reduce the repetition rates of the THz and pump beam to 2.5 kHz and 1.25 kHz, respectively. The resultant THz signal was detected by electro-optic sampling using a set of a (110) ZnTe crystal, a quarter-wave plate, a Wollaston prism and a pair of balanced photodiodes. A silicon beamsplitter was used to direct the reflected THz signal from the sample to another detection arm. The photoluminescence (PL) spectrum of the sample was induced by the 400 nm pump beam and collected by a fiber-coupled CCD spectrometer (Horiba iHR320). The sample was mounted on a cold-finger cryostat which was cooled down by liquid helium and able to vary the sample temperature from 5 K up to 370 K.  Figure S1: Schematic of the experimental setup for OPTPS. An amplified laser pulse is split into three beam paths: gate beam, pump beam and THz beam. The THz signal is generated by a spintronic emitter and detected by a ZnTe crystal via electro-optic sampling. The PL spectrum is collected by a fiber-coupled CCD spectrometer.
S-4 2 Derivation of charge-carrier electrical mobility

Mobility measured in transmission
When measuring the electrical mobility in the transmission mode of OPTPS, the ratio between the change of THz signal and transmitted THz signal, ∆T /T , is defined as where the subscripts "on" and "off" mean photoexcitation and non-photoexcitation, respectively. The photo-induced field change, ∆E(z), satisfies the wave equation given below, where k 0 = ω/c is the wave vector of the THz field in vacuum and k = nk 0 is the wave vector inside the sample, where n is the refractive index of the sample in THz frequency range. In the case of MAPbI 3 , n = √ ε stat = √ 30 which is an average of previously reported values measured in the THz-GHz frequency range. S2-S4 Z 0 = ε −1 0 c −1 is the vacuum wave impedance and ∆σ(z) is the photoconductivity of the sample which varies with the depth z following the Beer-Lambert law, where ∆σ 0 = ∆σ(0) is the photoconductivity at the front surface of the sample and α is the absorption coefficient of the sample at the photoexcitation wave length 400 nm, which can be measured by a Fourier-transform infrared (FTIR) spectrometer. Conventionally, the photoconductivity can be expressed in terms of ∆T /T as S5 S-5 whereñ s is the complex refractive index of the substrate, which in our experiment is the z-cut quartz disc. L = 1/α is the thickness of the photoexcited layer of MAPbI 3 thin film. Equation (S4) can be used for converting the experimentally measured −∆T /T to photoconductivity ∆σ as shown in Figure 2a in the main text. Combining Equations (S2) and (S3), the general solution can be expressed in terms of the electrical mobility µ as S6 where n 1 and n 2 are refractive indices of the medium surrounding the sample, which in the case of the MAPbI 3 thin film are air and quartz, respectively. r 1 = (n − n 1 )/(n + n 1 ) and r 2 = (n − n 2 )/(n + n 2 ) are the reflection coefficients at the front and back surfaces of the thin film respectively, and a = [1 − r 1 r 2 exp(2ikL)] −1 describes the multiple internal reflections due to Fabry-Perot effect. The effective photon fluence Φ eff is defined as where N is the number of charge carriers induced by the pump beam, with E being the energy of pump pulse at wavelength λ = 400 nm. R pump and T pump are the reflectivity and transmissivity of the sample at 400 nm respectively, which can be measured by FTIR spectroscopy. ϕ ≤ 1 is the ratio of free charge-carriers created per photon absorbed, which in the case of MAPbI 3 is assumed to be unity.
Distinct from the pump photon fluence (which is a mere measurement of the pump beam), the effective photon fluence, Φ eff , while taking into account the conversion ratio between absorbed photons and induced charge carriers, is also related to the photoconductive part S-6 of the sample, which is determined by the overlapping area between the pump beam and the THz beam. In other words, Φ eff is essentially the photo-induced charge-carrier density weighted by the THz electric field strength at each pump beam spatial position. Therefore, to calculate this effective "absorption intensity", we need to use the following weighting function, wheren pump is the two-dimensional (2D) areal charge-carrier density induced by the pump beam, which is proportional to its photon density by a factor of ϕ and g THz is the normalised THz areal spread function. We assume thatn pump and g THz follow 2D Gaussian distribution, where σ pump and σ THz are the beam waists of the pump and THz pulses respectively. r is the radial distance from the center of the Gaussian distribution. It can be shown that 2π 0 ∞ 0n pump rdrdθ = N and 2π 0 ∞ 0 g THz rdrdθ = 1. We representn pump and g THz as blue and red curves respectively in Figure S2, where their beam waists, σ i , are related to their full-width-at-half-maximum (FWHM) by, Combining Equations (S8)-(S10), we obtain the expression for the effective area, According to the thin-film approximation (TFA), in which the thickness of the photoexcited layer of the sample, L = 1/α, is much smaller than the smallest wavelength in the  Figure S2: Gaussian profiles of the pump-induced charge-carrier density and normalised THz intensity distribution. The yellow area represents the effective charge-carrier density within the overlapping area between the pump and THz beams. The inset is a 2D illustration of the overlapping between the pump and THz beams.
THz pulse, S7 we assume the unexcited part of the MAPbI 3 thin film to be the same as the substrate. Therefore, assuming n 0 ≈ n 2 and r 2 ≈ 0, Equation (S5) can be simplified as where ∆S = ∆σ(z)dz is the sheet conductivity that can be expressed in terms of the measured changed of THz signal ∆T /T , which leads to our final expression for the effective mobility in transmission, The photoconductivity term ∆T pp /T pp is measured at the instant of photoexcitation, t = 0, which is at the peak of the photoconductivity decay curve shown in Figure 2 in the S-8 main text. The change in charge-carrier density due to carrier recombination, diffusion and photon reabsorption, which happens at later times, is therefore not relevant in the calculation of charge-carrier mobility. It is also worth noting that the mobility expression given by Equation (S15) is slightly different from the conventional one which uses the value of ∆T /T at the peak of THz pulse (i.e. ∆T /T peak ) to calculate the mobility value. S8-S10 Here, we took the ratio of the peak-to-peak values between ∆T and T instead, which is denoted as ∆T pp /T pp . This is because as shown in Figure S3a, the peaks of T and ∆T do not overlap exactly with each other, which means if we took ∆T /T at the peak of THz pulse, the mobility would be underestimated. Figure S3b shows the mobility values calculated from ∆T pp /T pp and ∆T /T peak to demonstrate that by using ∆T /T peak , and thus not taking into account the shift between the ∆T and T peaks, the mobility is reduced by 25-40%. Compared with the previous mobility measurements taken by Milot et al. using ∆T /T peak (blue dots), S10 our mobility values calculated in the same way (green circles) are lower, which is consistent with the fact that our polycrystalline film has a smaller grain size (30-580 nm) than the other sample (∼ 1 µm).

Mobility measured in reflection
In the case of an OPTPS reflection measurement, the ∆T /T in Equation (S1) is replaced by the reflected THz signal ∆R/R, The solution to Equation (S2) is then modified as S11 where the blue and red curves represent T and ∆T , respectively. ∆T has been scaled up by a factor of 10 to show the shift between T and ∆T peaks more clearly. (b) Comparison of mobility values calculated using ∆T pp /T pp and ∆T /T peak , which are represented by the red and green circles, respectively. The blue dots represent the previously reported mobility values of another MAPbI 3 thin film sample whose grain size is larger. S10 where t 1 = 2n 1 /(n+n 1 ) is the transmission coefficient at the front sample surface. Assuming the incident field E inc is unity, the reflected field E r is given by Under the condition of TFA, a simplified expression for ∆R/R is obtained, S7 where n 1 = 1 for the single crystal and a = 1 since there is no internal reflection as observed in the thin film sample. Correspondingly, the photoconductivity of the MAPbI 3 single crystal is given by S-10 where n = √ 30 is the refractive index and α = 3×10 5 cm −1 is the absorption coefficient of the MAPbI 3 single crystal. S2,S12 Using Equation (S20), the experimentally measured −∆R/R shown in Figure 2b can be converted to photoconductivity ∆σ.
According to Equation (S13), the mobility in reflection is expressed as This equation also applies to the MAPbI 3 single crystal, since its photoexcited layer is still thinner than the THz wavelength required to satisfy the TFA condition. As illustrated in Figure S4b, since we put a sapphire substrate in front of the single crystal to improve the thermal contact, there will be some reflection loss of the pump beam at the air-sapphire and sapphire-crystal interfaces. As a result, the actual intensity of the pump beam at the single crystal surface, I pump , is less than the initial intensity I pump , which reduces the number of charge carriers given by Equation (S7), such that where f < 1 is the power transmission coefficient which can be calculated according to Fresnel equation: in the configuration shown in Figure S4b, f ≈ 0.68. Finally, the effective mobility measured in reflection is expressed as below, where n 1 = n air and f = 1 for the MAPbI 3 thin film on quartz ( Figure S4a) and n 1 = n sapphire for a single crystal with sapphire at the front.
Previous discussion has shown that the shift between T and ∆T peaks requires using ∆T pp /T pp instead of ∆T /T peak to calculate the mobility. When it comes to calculating the mobility from the reflection measurements, the same principle applies, since both the  Figure S4: (a) THz and pump pulses are incident normally on the MAPbI 3 thin film which is deposited on a quartz substrate. (b) THz and pump pulses are incident normally on the sapphire substrate first which is inserted in front of the MAPbI 3 single crystal. The orange area represents the photoexcited layer within the MAPbI 3 thin film or single crystal, where the photoconductivity ∆σ follows the Beer-Lambert law. E inc , E t and E r represent the incident, transmitted and reflected THz electric field respectively.
MAPbI 3 thin film and single crystal exhibit a shift between their R and ∆R peaks as illustrated in Figure S5. While the shift for the thin film is small enough that the mobility calculated using the old method might not deviate so much from the exact value, the much larger shift for the single crystal makes it critical to use ∆R pp /R pp to extract the mobility value. On top of that, by measuring the traces of the THz pulse and change of THz in the time domain, the noise that comes from the fluctuation of the THz peak is reduced, which gives a better signal-to-noise ratio (SNR). Figure S6a shows the mobility values of the MAPbI 3 thin film measured in reflection using ∆R pp /R pp and ∆R/R peak , which again shows that our new way of extracting mobility overcomes the issue with the shift between R and ∆R. The less scattered data points of ∆R pp /R pp (red triangles) also indicate a better SNR. More importantly, in the case of the MAPbI 3 single crystal, the mobility measured in reflection using ∆R pp /R pp is significantly improved, as shown in Figure S6b. Moreover, it is observed that ∆R also shifts with temperature, which makes ∆R/R peak even less reliable and the temperature-dependence trend represented by the green triangles in Figure S6b less conclusive as well. Figure S7 shows the mobility of the MAPbI 3 thin film measured in transmission and S-12 Amplitude (arb.u.) Figure S5: ( reflection, which are calculated by Equations (S15) and (S23), respectively. The consistency between the transmission and reflection measurements verifies the validity of the mobility measurement of the MAPbI 3 single crystal, which was done in reflection mode only.
It should be noted that the calculation of mobility using Equations (S15) and (S23) needs S-13 to be done within the low photoexcitation regime where −∆T /T (or −∆R/R) changes linearly with fluence. As illustrated in Figure S8, when the fluence is below 40 µJcm −2 , the peak of the transient absorption spectrum, −∆T /T , increases linearly with fluence and this is the range from which we extract the mobility as reported in the main text. Above 40 µJcm −2 , the photoexcitation peak starts to saturate and the mobility value extracted in this range will be an underestimate of the true mobility value.

Literature values of electrical mobility in MAPbI 3
As mentioned in the main text, a wide range of electrical mobilities has previously been reported for MAPbI 3 single crystals and thin films using different experimental methods, which is summarised in Table S1. Depending on which experimental method is used, the measured mobility can refer to the mobility of a single carrier species (i.e. either electron mobility µ e or hole mobility µ h ), or the sum of both µ e and µ h .
S-14 S-15 Table S1: Comparison of electrical mobilities of MAPbI 3 single crystals and polycrystalline thin films measured using different experimental methods. µ = µ e + µ h denotes the total charge-carrier mobility, where µ e represents the electron mobility and µ h represents the hole mobility.

S-16
3 Effect of grain-boundary scattering on electrical mobility

Theoretical modeling
Since one crucial difference between the MAPbI 3 polycrystalline thin film and the single crystal is the existence of grain boundaries, it is important to take into account their effect on the charge-carrier mobility as a function of temperature. This was done through the ab initio transport calculation of carrier mean free path λ transport in Equation (3) of the main text. However, it is also interesting to look at the mean free path calculated at the most representative energy, which we determined in an earlier work S25 to be (3/2)k B T away from the band edge. This is the energy at which the carrier contributes most to the transport properties. The carrier mean free path thus becomes where v nk is the carrier velocity and τ nk is the electron-phonon lifetime for a state of band n and momentum k. The mean free path average for electrons and holes is performed for states nk such that their energy is within ∆ = 1.4 meV of the dominant energy (3/2)k B T from the band edge. The comparison between the transport mean free path and the mean free path computed at the most representative energy is shown in Figure S9 with solid and dashed lines, respectively. The most representative mean free path closely follows the transport one but slightly underestimates it, leading to a predicted grain size of around 60 nm. If one uses the correct transport mean free path, the predicted grain size is instead around 100 nm as can be seen in Figure S10 where various other grain sizes are also shown for comparison. We highlight that the grain size g is the only fitting parameter since the probability of reflection is fixed to R = 0.5 and can be used for other perovskite materials. S26,S27 However, for S-17 completeness, we show in Figure S11 the change of room-temperature electrical mobility as a function of grain size g and reflection probability R. It is noted that the mobility depends moderately on those two parameters. Specifically, the mobility changes from 10 cm 2 V −1 s −1 with g = 25 nm and R = 0.75, to 70 cm 2 V −1 s −1 with g = 200 nm and R = 0.25.   Figure S11: Room-temperature mobility of MAPbI 3 as a function of grain size g and reflection probability R.

Grain-size measurement of polycrystalline MAPbI 3 thin film
In order to measure the grain size in the MAPbI 3 thin film, a top-down scanning electron microscope (SEM) image was taken over an area of 9 µm×8 µm as shown in Figure S12a.
An imaging processing program (ImageJ S28 ) was used to analyze the lateral size of each grain within this area, and a histogram showing the distribution of grain sizes is shown in Figure S12b, from which the average grain size is found to be (580 ± 212) nm.
Since the SEM only gives the upper limit of the grain size and some sub-grain features might be hidden, S29 we performed an X-ray diffraction (XRD) measurement on the thin film to obtain a better estimate of the grain size. Figure S12c shows the Gaussian-fit to the XRD spectrum given in Figure 1f in the main text, from which the width of each diffraction peak is extracted and used for calculating the grain size g according to Scherrer equation, S30 S-20 where K = 0.93 is the shape constant, λ source = 0.154 nm is the wavelength of the X-ray source, β is the FWHM of each diffraction peak and θ is the Bragg angle. The resultant grain size averaged over three diffraction peaks, (110), (220) and (330), is (31 ± 10) nm.
As mentioned in the main text, the XRD peaks might have been broadened further due to strain effects, which places a lower bound on the grain size. Therefore, based on the SEM and XRD measurements, we estimate the grain size of the MAPbI 3 thin film to be between 30-580 nm.

Charge-carrier dynamics analysis
As described in the main text, we incorporate the effect of photon reabsorption and carrier diffusion into the charge-carrier dynamics of MAPbI 3 to extract the intrinsic recombination rates according to Equation (1), where µ is the electrical mobility, k B is the Boltzmann constant and T is the sample temperature. The monomolecular recombination rate k 1 was obtained from a time-resolved photoluminescence (TRPL) measurement which is shown in Figure S13. According to the stretched exponential fit to the transient measured at the lowest photoexcitation fluence, the single crystal shows a longer average charge-carrier lifetime (54 ns) than the thin film (12 ns).
The Auger recombination rate k 3 was neglected (i.e. k 3 = 0) in the low-fluence regime.
In order to solve Equation (1) carrier recombination process. The update of n(t, z) was repeated for a total time period of 1600 ps, covering the entire time window shown in Figure 2. The photoconductivity was then determined by ∆σ(t) = n(t)eµ where n(t) is the sum of charge-carrier densities across all of the slices, which was fitted globally to the experimentally measured photoconductivity decay dynamics shown in Figure 2. As a result, the bimolecular recombination rate k 2 was extracted as 2.6 × 10 −10 cm 3 s −1 for the thin film and 8.7 × 10 −10 cm 3 s −1 for the single crystal.
To demonstrate that our photon-reabsorption model describes the charge-carrier dynamics in MAPbI 3 more accurately and is able to extract the intrinsic bimolecular recombination rate, we compare its result with that obtained from a simple model which neglects the effect of photon reabsorption and is described by the following rate equation, where G refers to the charge-carrier generation rate which arises only from the initial photoexcitation. Since the effect of photon reabsorption is neglected in this simple model, the diffusion term given in Equation (1) in the main text is omitted. Figure S14 shows the theoretical fit to the photoconductivity decay dynamics using this simple model. While the fit to the MAPbI 3 thin film data looks reasonably good, the fit to the MAPbI 3 single crystal data is much worse than that obtained from the photon-reabsorption model shown in Figure 2. This is because the single crystal is much thicker than the thin film, which makes the effect of photon reabsorption much more prominent and the neglect of photon reabsorption can lead to inaccurate analysis of the charge-carrier dynamics. In particular, as pointed out by Crothers et al., S31 photon reabsorption can mask the intrinsic charge-carrier recombination rate significantly, which makes the simple model governed by Equation (S27) invalid. single crystal, due to the significant effect of photon absorption, the charge-carrier decay dynamics ( Figure S14b) appears to decay slower than that of the thin film ( Figure S14a), which results in a smaller bimolecular recombination rate if using the simple model. However, the recombination rate as an intrinsic property of the material should be independent of the sample thickness, which means the value of k 2 extracted from the simple model has been masked by the effect of photon reabsorption. Therefore, it is crucial to take into account the effect of photon reabsorption when analysing the charge-carrier dynamics, especially for thick samples such as a single crystal. Table S2: Comparison of bimolecular recombination rates, k 2 , extracted from the photonreabsorption model and the simple model (no photon absorption). The monomolecular recombination rates, k 1 , were obtained from the TRPL measurements.  Figure S14: Photoconductivity decay dynamics of (a) MAPbI 3 thin film measured in transmission and (b) MAPbI 3 single crystal measured in reflection. The circles represent the experimental data identical to what is shown in Figure 2 in the main text. The curves represent the theoretical fits obtained from the simple model governed by Equation (S27).

Charge-carrier diffusion length calculation
Having obtained the charge-carrier mobility and recombination rates, we can calculate the carrier diffusion length according to Equation (2), where D is the diffusion constant given by Equation (S26) and R represents the total recombination rate of k 1 , k 2 and k 3 , which is expressed as where n is the charge-carrier density, which depends on the photoexcitation intensity and has a significant effect on carrier diffusion length. Figure S15 shows the diffusion lengths as a function of carrier density, which were calculated at different values of k 1 , with the bimolecular recombination rates, k 2,film = 2.6 × 10 −10 cm 3 s −1 and k 2,crystal = 8.7 × 10 −10 cm 3 s −1 , extracted from the photon-reabsorption model. The electrical mobilities used for thin film and single crystal are 33 and 59 cm 2 V −1 s −1 respectively, which were obtained from roomtemperature photoconductivity measurement ( Figure 2 in the main text). The arrows indicate the diffusion lengths obtained at certain carrier densities, n AM1.5 , corresponding to 1 sun illumination (1 kWm −2 AM1.5-filtered), which is the usual operating condition for perovskite solar cells. According to Ref. S32, under 1 sun illumination with the assumption of no charge extraction, the carrier density is estimated by setting ∂n/∂t = 0 in Equation (1) given in the main text. The Auger recombination k 3 and the diffusion term D(∂ 2 n/∂z 2 ) are neglected for a simplistic approximation. The resultant quadratic equation can be solved with known charge-carrier generation rate G, which is determined from the following equation,

S-27
where h is the Planck constant, c is the speed of light in vacuum, z is the film depth and α(λ) is the absorption coefficient of MAPbI 3 at wavelength λ. f solar is the ASTM G173-03 Global Tilt reference spectrum for the solar spectral irradiance distribution, which satisfies f solar (λ)dλ = 1 kWm −2 . An average value G = 4 × 10 21 cm −3 s −1 is extracted for a typical MAPbI 3 thin film, S32 from which the carrier density n AM1.5 and diffusion length can be determined.
As shown in Table S3, the diffusion lengths of the MAPbI 3 single crystal extracted from the simple model is significantly longer than that of the thin film, whereas the difference in diffusion lengths obtained from the photon-reabsorption model is much smaller. In particular, when k 1 is less than 10 7 s −1 , such difference is more prominent, since the difference in n AM1.5 between the photon-reabsorption model and the simple model is larger as shown in Figure S16.  S-30

Photoconductivity decay dynamics of thin film measured in reflection
We also measured the photoconductivity decay dynamics of MAPbI 3 thin film at room temperature in reflection ( Figure S17) to show its consistency with the transmission measurement described in the main text (Figure 2a). According to Equation (S23), we extracted the electrical mobility of thin film in reflection as (29 ± 3) cm 2 V −1 s −1 , which agrees with the value measured in transmission, i.e. (33 ± 2) cm 2 V −1 s −1 . The bimolecular recombination rate of thin film measured in reflection is 2.9 × 10 −10 cm 3 s −1 , which is also consistent with the transmission measurement.  Figure S17: Photoconductivity decay dynamics of MAPbI 3 thin film measured in reflection at room temperature. The photoexcitation fluence ranges from 4.57 to 71.0 µJcm −2 . S-31

PL-facilitated temperature correction of MAPbI 3 single crystal
To measure the mobility of the single crystal as a function of temperature, we mounted the crystal on a cold-finger cryostat (Oxford Instruments, MicrostatHe) to get it down to low temperature. Unlike the thin film, it was much more challenging to cool down the single crystal to low temperature due to its poor thermal conductivity. Therefore, we inserted a sapphire disc (2 mm in thickness and 13 mm in diameter) between the crystal and the sample holder to improve the thermal contact and dissipate the heat generated by the pump beam more efficiently. As a result, we were able to cool down the crystal to a much lower temperature. However, because the temperatures between the cold-finger cryostat and the crystal can be slightly different, the true temperature of the crystal could not be determined accurately. Thankfully, with the help of a gas-exchange cryostat (Oxford Instruments, Op-tistatCF2), the temperature of the single crystal could be measured more accurately and reach the minimum of 5 K. This is because as the single crystal was surrounded by helium gas, it did not have the problem with thermal contact. Therefore, we measured the PL spectra of the single crystal in the gas-exchange cryostat at various temperatures and used this set of PL spectra as a reference to correct the crystal temperature measured using the cold-finger cryostat.
Figures S18 and S19 show the PL spectra of the single crystal measured in the gasexchange cryostat and cold-finger cryostat, respectively. Figure S20 shows the PL of the thin film measured in the cold-finger cryostat. By fitting a two-Gaussian function, the PL spectra were resolved into two peaks, P1 and P2, which eventually converge to the same wavelength at high temperature.
S-32  Figure S18: PL spectra of the single crystal measured by gas-exchange cryostat at different temperatures. The crystal was photoexcited by a pulsed laser at 398 nm with a 10 kHz repetition rate and a fluence of 0.2 µJcm −2 . The blue curves represent the PL data, whereas the red curve represents the Gaussian fit which consists of two peaks represented by the black curve and the green curve, respectively.
S-33  Figure S19: PL spectra of the single crystal measured by cold-finger cryostat at different temperatures. The crystal was photoexcited by a pulsed laser at 400 nm with a 5 kHz repetition rate and a fluence of 30.8 µJcm −2 . The blue dots represent the PL data, whereas the red curve represents the Gaussian fit which consists of two peaks represented by the black curve and the green curve, respectively.
S-34  Figure S20: PL spectra of the thin film measured by cold-finger cryostat at different temperatures. The thin film was photoexcited by a pulsed laser at 400 nm with a 5 kHz repetition rate and a fluence of 30.8 µJcm −2 . The blue dots represent the PL data, whereas the red curve represents the Gaussian fit which consists of two peaks represented by the black curve and the green curve, respectively.
S-35 Figure S21 shows the central wavelengths of P1 and P2 for the single crystal as a function of temperature, which was measured by the gas-exchange cryostat. There is an abrupt redshift of the peak wavelengths at the orthorhombic-tetragonal phase transition at 160 K, which is consistent with what has been reported previously on MAPbI 3 thin films. S10 Moreover, both P1 and P2 show similar temperature dependence throughout the entire temperature range, although the phase shift of P1 at 160 K is more significant compared to the gradual redshift of P2 at temperatures 115-165 K.  Figure S21: Central wavelength of single crystal's PL spectrum measured by the gasexchange cryostat as a function of temperature. The black and green triangles represent the central wavelengths of P1 and P2 respectively.
Since P1 and P2 converge to the same wavelength above 160 K and P1 is more prominent in the cold-finger cryostat measurement of the single crystal (see Figure S19), we decided to take P1 as the reference PL peak for correcting the single crystal's temperature. As shown in Figure S22a, before temperature correction, P1 of the single crystal measured by the cold-finger cryostat (pink triangle) deviates from the gas-exchange measurement (blue triangles), whereas the thin film measurement (black circles) shows a good agreement due to its good thermal contact with the cold-finger cryostat. In Figure S22b, the cold-finger S-36 cryostat temperature is matched with the gas-exchange measurement, from which we can determine the single crystal's real temperature accurately. One might have noticed that the relative intensity of P1 and P2 is slightly different between the gas-exchange measurement ( Figure S18) and the cold-finger measurement ( Figure S19) of the single crystal at 140 K and 150 K. This is attributed to the different excitation fluence between these two sets of measurements, which differs by two orders of magnitude. In addition, the extra sapphire substrate in the cold-finger cryostat configuration may also affect the photon reabsorption and emission in the MAPbI 3 single crystal, which results in different intensity ratios between P1 and P2. Nevertheless, the more crucial feature in the PL spectra measurement is the peak positions of P1 and P2, whilst the relative intensity is less informative. As shown in Figure Figure S22: (a) Central wavelength of P1 before temperature correction. (b) Central wavelength of P1 after temperature correction. The blue triangles represent the P1 positions of the single crystal measured by the gas-exchange cryostat, which is used as the true temperature reference. The black circles represent the P1 positions of the thin film measured by the cold-finger cryostat, which shows a good agreement with the single crystal's gas-exchange measurement. Hence, there is no need to correct the temperature of the thin film measurements. The pink and red triangles represent the cold-finger cryostat measurements of the single crystal before and after temperature correction respectively. S-38