Abstract
The optoelectronic properties of metalhalide perovskites (MHPs) are affected by lattice fluctuations. Using ultrafast pumpprobe spectroscopy, we demonstrate that in stateoftheart mixedcation MHPs ultrafast photoinduced bandgap narrowing occurs with a linear to superlinear dependence on the excited carrier density ranging from 10^{17} cm^{−3} to above 10^{18} cm^{−3}. Timedomain terahertz spectroscopy reveals carrier localization increases with carrier density. Both observations, the anomalous dependence of the bandgap narrowing and the increased carrier localization can be rationalized by photoinduced lattice fluctuations. The magnitude of the photoinduced lattice fluctuations depends on the intrinsic instability of the MHP lattice. Our findings provide insight into ultrafast processes in MHPs following photoexcitation and thus help to develop a concise picture of the ultrafast photophysics of this important class of emerging semiconductors.
Introduction
Understanding the dynamics of photogenerated carriers in semiconductors has been essential to unleash their maximum performance in optoelectronic applications such as lightemitting diodes and solar cells. In emergent metalhalide perovskite (MHP) semiconductors, Wanniertype excitons are the initial photogenerated species, since the small effective electron mass (~0.1–0.15 m_{e})^{1,2} and high optical permittivity (ε_{opt} ~ 4–6.5)^{3} yield a Bohr radius larger than the perovskite’s lattice constant. Based on the Wannierexciton approximation, the exciton binding energy (R_{b}) can be estimated to ~54 meV, assuming ε_{opt} = 5 and using the hydrogen model^{2}. However, experimentally significantly lower values have been determined, for instance R_{b} ~ 5–15 meV was found for the classic CH_{3}NH_{3}PbI_{3} (MA) perovskite, based on analysis of the steadystate absorption spectra in the framework of Elliot theory^{4}, and R_{b} ~ 10–12 meV was determined independently from the dielectric response from magnetooptical measurements on MA^{1}, yielding a frequency of 2.44–2.93 THz, which is in the range of optical phonon energies (~0.5–7 THz)^{5}. Thus, the corresponding excitonic dielectric response (ε_{X} ~ 11 ± 4) is in the region of optical phonon dielectric responses^{6}, indicating that the lattice screening is dynamic in nature due to the “soft” crystal lattice of MHPs^{7}.
Longitudinal optical (LO) phonons dominate the exciton screening, since the measured R_{b} can be explained when considering strong Fröhlich electronphonon coupling, resulting from the polar ionic crystal structure of MHPs, which screens efficiently the electronhole mutual attraction potential^{8,9}. Organic cations contribute to the exciton screening via vibrational modes (~20 meV)^{10}. Although cation polarization operates at much lower frequencies (~100 GHz)^{11}, they can cause dynamic screening by coupling with the inorganic LOphonons (~3–15.8 meV)^{12}. Due to efficient thermallyassisted dissociation, excitons are barely observed in MHPs at room T (thermal energy ~25 meV). In fact, ultrashort pumpprobe measurements have revealed that the timescale of exciton dephasing is only ~20 fs^{13}. Consequently, free carriers are the primary photogenerated species, as also indicated by the quadratic carrier density dependence of spontaneous photoluminescence intensity^{14}.
However, the transport of photogenerated carriers is limited by localization effects. First, static disorder resulting from thinfilm imperfections reduce carrier mobility by scattering^{15}. Second, strong Fröhlich coupling drives lattice distortions that tend to slow carriers^{16}. The Fröhlich model not only reproduces qualitatively the changes of the carrier mobility depending on different inorganic lattices, but also it yields values only slightly larger than the experimental values^{17}, suggesting that Fröhlich coupling is the main mechanism limiting carrier mobility in MHPs. However, the Fröhlich model predicts a polaron mobility dependence on temperature of ~T^{−0.46} (for a range from 200 to 300 K), independent of multiphonon coupling^{18}. This result is inconsistent with the ~T^{−3/2} dependence observed experimentally^{19}, which subsequently has been ascribed to large atomic displacements (lattice fluctuations)^{11,20}. The thermallyactivated lattice fluctuations^{21} retard polaron transport by introducing nonlinearFröhlich coupling^{22}, while the random potential fields caused by lattice fluctuations favor quantum Anderson localization^{20}. Both are potential mechanisms that can be responsible for the T^{−3/2}dependence.
The energetic barrier for reorientation and rotation of the organiccation dipole moments is as small as ~20 meV^{23}. Consequently, organic cations can reorient and rotate around their latticesites at room T (resulting in dynamic disorder), which prolongs carrier lifetimes^{24,25} by localizing electrons and holes in different regions^{26}, implying a tradeoff exists between carrier lifetime and carrier mobility^{27}. The carrier localization induced by dynamic disorder depends also on lattice vibrations. Theoretical studies using density functional theory^{28,29} and spectroscopic studies^{30,31} have respectively demonstrated that lattice distortions compete with dynamic disorder, leading to the formation of ferroelectric polarons^{32}.
In this study, we performed transient absorption (TA) pumpprobe measurements on thin films of the stateoftheart triplecation lead mixed halide perovskite (Cs_{0.05}FA_{0.8}MA_{0.15})Pb(I_{0.85}Br_{0.15})_{3} or CsFAMA. This perovskite composition has consistently delivered a power conversion efficiency in excess of 20% in conjunction with excellent photostability in perovskite solar cells^{33,34}. Carrier densitydependent TA spectra were measured at room T across carrier densities from N ~ 1.5 × 10^{17} cm^{−3} to N ~ 3.5 × 10^{18} cm^{−3}. We developed a model to analyze the highenergy part of the TA spectra, which revealed weak photoinduced exciton screening and a linear to superlinear dependence of photoinduced bandgap renormalization (BGR) on N. This anomalous BGR cannot be explained by carriercarrier interactions or thermal effects, but can be rationalized by photoinduced lattice fluctuations, which was subsequently supported by timedomain terahertz (tdTHz) spectroscopy measurements.
Terahertz spectroscopy is a powerful technique to study photoconductivity^{19,35,36,37,38,39}. Fluencedependent THz photoconductivity spectra obtained from the tdTHz spectra show that carrier localization increases with N, possibly due to photoinduced lattice fluctuations. We found that the THz photoconductivity spectra present a broadband Lorentzlike resonance, which is governed by sizeconstrained localization possibly involving Anderson backscattering. The extracted dc carrier mobility (ω/2π = 0) decays exponentially with N, consistent with a carrier mobility limited by Anderson localization and a Mott mobility edge, indicating photoinduced enhancement of lattice fluctuations. This result is in agreement with the anomalous BGR observed in our TA experiments. The Anderson localization mainly affects the dc carrier mobility (ω/2π = 0) at high N, while its contribution to the carrier mobility in the THz range (ω/2π ~ 0.1–3 THz) at low N is less pronounced, suggesting that the T^{−3/2}dependence of carrier mobility obtained from lowfluence THz experiments is mainly due to nonlinearFröhlich coupling. The magnitude of photoinduced lattice fluctuations depends on the intrinsic instability of the MHP lattice. Our findings are related to lightinduced fluctuations also observed earlier by other techniques^{25,40}.
Results
Photoinduced transient absorption
CsFAMA films (thickness d ~ 300 nm) were prepared by the solutionprocessing method on spectroscopicgrade quartz substrates (see methods). Figure 1a shows the steadystate groundstate absorption spectrum, analyzed in the framework of the Elliott model, indicating the absorption (coefficient) consists of a linear combination of excitonic absorption (α_{X}) and absorption from continuum states (α_{C}) as also reported earlier^{4,41}. According to the Elliott theory, α_{C} is enhanced by a factor ξ due to Coulomb attraction between electrons and holes according to^{4}:
where R_{b0} is the exciton binding energy, and x = E − E_{g}, where E_{g} is the band gap. ξ approaches unity when R_{b0} → 0 (L ‘Hospital’s rule), which implies the Coulomb enhancement of α_{C} does not exist any longer once excitons (electronhole pairs) are entirely screened. Using ξ, the Elliott formula can be expressed also by^{4}:
where A is a fitting parameter related to the transfer matrix elements, δ is the Dirac delta function, \(\sqrt{x}\) is the normalized density of states in the conduction band assuming a parabolic shape (valid below ~1.82 eV in our sample, see Fig. 1a). The first term accounts for α_{X}, while the second term accounts for ξα_{C}. To describe the roomtemperature absorption spectra, a hyperbolicsecant broadening function accounting for thermal and inhomogeneous broadening was convoluted with Eq. 2 (see SI), yielding R_{b0} ~ 11 meV and E_{g} ~ 1.677 eV (Fig. 1a). The decomposed α_{X} and ξα_{C} are plotted in Fig. 1a alongside α_{C}. Clearly, the highenergy part of the absorption spectrum is very sensitive to ξ, i.e., the exciton screening.
Next, we focus on the highenergy part of the TA spectra to explore the photoinduced BGR and underlying manybody effects (Fig. 1b). When analyzing the photoinduced absorption at the highenergy part (PIAH) of the spectra, we can safely neglect α_{X} (Fig. 1a) and photoinduced broadening (see SI). Therefore, the photoinduced absorption of PIAH may be described by^{41}:
where ΔR_{b} and ΔE_{bgr} are the screened exciton binding energy and the photoinduced BGR due to manybody effects, respectively. Here, a negative ΔE_{bgr} represents bandgap narrowing. \({f}_{e}=1/[1 + {e}^{({E}{E}_{F})/{k}_{B}{T}_{e}}]\) is the FermiDirac distribution function accounting for the occupation probability of electrons in the conduction band, where E_{F} is the quasiFermi level, k_{B} is the Boltzmann constant, and T_{e} is the absolute electron temperature. Since the effective mass of holes is similar to that of electrons^{42}, we assume that the occupation probability of holes in the valence band is symmetric to that of electrons in the conduction band. For PIAH recorded 5 ps after photoexcitation, (1 − f_{e}) is close to 1 due to hotcarrier cooling^{43}. In this case, Eq. (3) can be further simplified (see SI):
where ξ_{0} represents ξ(R_{b0}, x) and \({\xi }_{0}{e}^{2\pi \sqrt{{R}_{b0}/x}}\le 1\).
The photoexcitation was varied across the linear response regime to avoid saturation effects typically occurring at high excitation density (Supplementary Fig. 3). A blue shift of the photobleach peak induced by band filling is observed (inset in Fig. 1b), implying that the photoinduced bandgap narrowing should be smaller than the BursteinMoss shift ΔE_{BM}, which can be estimated from the broadening of the TA photobleach peak (~38 meV for N = 3.5 × 10^{18} cm^{−3})^{44}. Hence, an upper limit of ΔR_{b}/R_{b0} ~ 10% can be determined from Eq. (4) using ΔE_{bgr} = −38 meV, indicating that the photoinduced exciton screening is weak. The same consideration was suggested in earlier works when fitting the entire TA spectra^{45,46}. In the framework of the hydrogen model, the excitonic dielectric response ε_{X} at the ground state can be evaluated by: R_{b0} = 13.6 m_{r}/(m_{e}ε_{X}^{2}) eV, where 13.6 is the Rydberg constant and m_{r} ~ 0.12m_{e}^{1,2} is the reduced electron mass, yielding ε_{X} ~ 12.18. According to the Mott transition criterion of a_{B}/λ_{D} = 1.19^{47}, where a_{B} = 0.0592m_{e}ε_{X}/m_{r} is the exciton’s Bohr radius and λ_{D} = [ε_{X}E_{T}/(8πΝq^{2})]^{1/2} is the Debye screening length, the estimated Mott density in our sample is N_{M} ~ 2 × 10^{16} cm^{−3} at room T, implying complete exciton screening in the entire excitation regime. The quadratic carrier density dependence of the spontaneous photoluminescence intensity supports the conclusion of complete exciton screening^{14}. This indicates that photogenerated excitons in MHPs are dynamic in nature, consequently, they can be monitored by absorptionbased measurements (responses to ε_{X}), but not by photoluminescence measurements (responses to ε_{s}). We note that the weak exciton screening observed here could indicate the presence of Mahantype excitons^{48}. Mahantype excitons are known to survive at carrier densities much larger than the nominal N_{M}^{49}. However, Mahantype excitons have only been reported in CH_{3}NH_{3}PbBr_{3} crystals until now, indicated by the observation of strong excitonic peaks even at very high N (~10^{19} cm^{−3})^{48}.
In a doped semiconductor where carriercarrier interactions are predominant, ΔE_{bgr} should follow a power law dependence on N according to: ΔE_{bgr} = −E_{ex} − E_{c}∝N^{k}, where −E_{ex} and −E_{c} are the bandgap narrowing due to exchange correlation and electronimpurity interactions, respectively^{50}, and k ~ 1/3 if electronimpurity interaction is insignificant^{51}. However, Fig. 1c shows k = 1 for Δα_{1.82eV}, which appears to be universal in MHPs (Supplementary Fig. 4). This behavior is independent of photon energy, since all TA spectra can be rescaled to similar line shapes (inset in Fig. 1c). Because exciton screening depends either sublinearly (Wanniertype exciton^{48}) or linearly (Mahantype exciton^{49}) on N, Eq. (4) suggests k ≥ 1 for ΔE_{bgr}. Such anomalous BGR cannot be explained by carriercarrier interactions. Electronphonon coupling and thermallyinduced lattice expansion can be responsible for the observed k ≥ 1 dependence as they both contribute to the BGR by:^{52} ΔE_{bgr} = −E^{’}_{ex} − E^{’}_{c} + E_{ep}, where E^{’}_{ex} and E^{’}_{c} account for the exchange correlation and electronimpurity interactions, respectively, if electronphonon coupling (E_{ep}) is included. We believe that photoinduced thermal effects can be neglected since, first, heat accumulation should be negligible, because of the low repetition rate of the pump laser system (3 kHz) used here and thus long offtimes between excitation pulses (~333 μs), much longer than the reported heat transport time from the MHP film to the substrate (<9 µs)^{53}. Second, the estimated temperature increase of the lattice caused by a single excitation pulse is only about ~0.32 K for N = 3.5 × 10^{18} cm^{−3} and E_{ph} = 2.25 eV (see SI), calculated by using the heat capacity of the MHP lattice (~170–190 J/K^{−1} mol^{−1} at room T^{54}). Third and importantly, E_{g} in MHPs increases with the lattice temperature by ~0.3 meV/K^{4,55}, which reduces the k value rather than increasing it.
Electronphonon coupling is important in determining the bandgap in MHPs. As mentioned above, unlike classical semiconductors (Si, GaAs, etc.), whose bandgap reduces with temperature, electronphonon coupling widens the bandgap of MHPs according to^{56}:
where F(ω) is a spectral function related to the phonon spectra and n_{B} is the Bose–Einstein distribution of optical phonon modes. Despite E_{ep}(T) being positive, lattice fluctuations can effectively reduce F(ω,T) and thus introduce a negative component^{56}, yielding a temperaturedependent E_{g} with reduced slope^{55}. Thus, the k ≥ 1 BGR found here is most likely due to an increase of the contribution from photoinduced lattice fluctuations, supported by lightinduced fluctuations as revealed by various other techniques^{25,40}. The BGR can include a polaronic contribution, since optical spectroscopy measures the bandgap of polarons, so the BGR is possibly linked to polaron formation. However, we have no clear evidence of polaron formation, only indirect evidence is found when comparing the hotcarrier cooling dynamics with the rise of the timeresolved terahertz signal. Figure 2a reveals that the terahertz photoconductivity reaches only half of its maximum, when hotcarrier cooling concludes, which has been ascribed to reduction of the carrier scattering rate due to polaron formation^{57}.
This scenario can help to clarify why the lowenergy photoinduced absorption (PIAL, Fig. 2b) at t ~ 0.3 ps was ascribed to BGR^{46} and polaron formation^{27}, respectively. Figure 2b shows that the amplitude of PIAL scales with N, once again indicating weak photoinduced exciton screening, since a strong exciton screening would significantly reduce the PIAL amplitude at high N. This dependence coincides with the linear dependence of PIAH on N, suggesting that PIAL is determined by BGR and polaron formation. However, the amplitude of PIAL increases with the carrier’s excess energy (Fig. 2b), which is inconsistent with the dependence of PIAH on photon energy (inset in Fig. 1c). This discrepancy is likely caused by band filling. Since the PIAL decay mimics hotcarrier cooling^{46}, the band filling is predominant in determining PIAL’s amplitude during the hotcarrier cooling process. Therefore, if carriers at the same N (similar ΔE_{bgr}) have different excess energy (significant difference in T_{e}), higher excess energy results in less band filling and thus a larger amplitude of PIAL.
Photoinduced THz absorption
Photoinduced lattice fluctuations affect the carrier photoconductivity in MHPs^{12,22}. Measurements of the tdTHz spectrum without prior photoexcitation (E_{THz}) were performed followed by measurements of the photoinduced tdTHz spectra (ΔE_{THz}) recorded at 5 ps after photoexcitation (Supplementary Fig. 7). The differential tdTHz spectra in the frequency domain (ΔE_{THz}/E_{THz}) were obtained by Fast Fourier Transformation (Supplementary Fig. 7), and subsequently the photoinduced change of the photoconductivity (Δσ) was calculated by^{58}:
where ε_{0} is the vacuum dielectric constant, c the speed of light, n_{sub} ~ 2.13 the refractive index of the quartz substrate in the terahertz region, d is the sample thickness, and ω is the angular frequency.
The calculated Δσ shows a weak groundstate bleach of two transverseoptical phonon modes (Fig. 3a, b, see also Supplementary Fig. 10), indicating the contributions from phononic responses are secondary, consistent with the observations reported in earlier THz works^{19,36}. At low N, Re(Δσ) is virtually frequency independent (Fig. 3a) and Im(Δσ) is close to zero (Fig. 3b). These spectral signatures indicate that the photoconductivity is mediated by free charges that undergo highrate Drude scattering^{38}. At high N, Re(Δσ) turns downward at the low frequency side and Im(Δσ) shows a zerocrossing in the high frequency side. This nonDrude behavior can be assigned to a broadband Lorentzlike resonance centered out of our THz window (Fig. 3c), which is a direct response to the photoinduced lattice fluctuations, or a characteristic of the carrier localization^{59}. In fact, the former has been observed in organic crystals^{60}, supported by the THz lattice fluctuations in MHPs revealed by Raman spectroscopy^{61}, however, centered at the lowenergy side. The latter is common in sizeconstraint or disordered semiconductors^{59}.
If the broadband Lorentzlike resonance is a direct response to the photoinduced lattice fluctuations, then the photoconductivity response can be described by the Drude–Debye model (DD model, see SI):
where σ_{0} is the peak photoconductivity provided by the lattice fluctuations. For experimental THz photoconductivity, m^{*} = m_{r} should be constraint in the fit, because tdTHz probes both electrons and holes. Clearly, our Δσ can be well reproduced by the DD model (Fig. 3a, b). The fits yield τ_{s} ~ 1.47–1.04 fs (Ndependent, see Supplementary Table 1 for details), σ_{0} ~ 0.68–12.29 S/cm (Ndependent), τ_{r} ~ 261 fs (global). The resonance peak can be estimated to ~16–39 meV by ω^{2}_{0} = 1/(τ_{r}τ_{0}) using τ_{s} < τ_{0} < 6.37 fs (see SI). This energy range equals ~3.9–9.5 THz and ~130–390 cm^{−1} and is related to the isolated cation modes^{62}. However, these fluctuations are infraredinactive and thus they cannot be resonant with the THz radiation directly^{62}.
The photoconductivity response of a localized system is usually described by the Drude–Smith model (DS model) which considers a series of backscattering events^{36}:
where q is the elementary charge and τ_{s} is the carrier’s momentum scattering time. c_{j} is a parameter (−1 ≤ c_{j} ≤ 0) that is related to the backscattering probability of the jth scattering event. Here, the scattering time for each scattering event is considered constant. c_{j} = −1 implies full backscattering (i.e., no conduction), while c_{j} = 0 implies no backscattering (Drude conduction). A negative c_{j} decreases the dc photoconductivity (ω = 0) by shifting the Drude peak to higher frequency. Generally, j is considered only once, since the transport direction of carriers cannot be retained after collision. This approach works well for the case of c_{1} ~ −1, e.g., isolated nanoparticles with zero dc photoconductivity^{63}. In the case of MHPs, their high dc conductivity indicates moderate localization (Fig. 3a), and thus we assume the jth scattering event to dominate backscattering. The DS model fits the Δσ except for Im(Δσ) at high N (dashdotted lines in Fig. 3a, b), which could be caused by the simple assumption that carriers don’t change transport direction after any nonbackscattering events.
The DS fits yield τ_{s} = 1.76–1.59 fs (Ndependent), j = 122 (global), and c_{j} = −0.11–0.18 (Ndependent), indicating that backscattering occurs on a timescale of τ_{b} ~ 122τ_{s}, yielding a localization length of L ~ 3.72 nm calculated by \({L}^{2}\cong {\tau }_{s}{\tau }_{b}\sqrt{2{E}_{T}/{m}^{* }}\)^{64} using τ_{s} = 1.76 fs, E_{T} = 25 meV, m^{*} = 2m_{r} ~ 0.24 m_{e}. Comparison of the film thickness (d ~ 300 nm) and grain size (>300 nm) with L yields d/L ~ 80. The results of c_{j} = −0.11–0.18 and d/L ~ 80 are consistent with the previous assumption of moderate sizeconstrained localization. The likely explanation for j = 122 is the quantum Anderson localization^{20}. Due to the random electrostatic potential induced by lattice fluctuations, Anderson backscattering can occur after many scattering events with a backscattering time τ_{b} ≫ τ_{s}^{20}. Since Anderson backscattering results in substantial reduction of the dc photoconductivity (Fig. 3c), the dc photoconductivity of localized states is governed by carrier hopping (Fig. 3c), which breaks the Anderson localization in the dephasing time τ_{d}^{20}. The Drude–Anderson model (DA model) is given by^{64}:
where 0 ≤ ϕ ≤ 1 is the fraction of localized carriers^{20} and 0 < ϕ_{b} ≤ 1 is a parameter accounting for the hopping dephasing. Here, ϕϕ_{b} is the effective fraction of localized carriers and ϕ_{b}τ_{b} is the effective backscattering time. The DA fits the experimental Δσ similarly well as the DD model, since they are mathematically equivalent (see SI). The fits yield τ_{s} ~ 1.9–1.76 fs (Ndependent), ϕϕ_{b} ~ 0.23–0.42 (Ndependent), and ϕ_{b}τ_{b} ~ 245.6 fs (global).
Both the DS model and DA model are backscattering models that cannot describe the case of localization due to an energy barrier characterized by a barrier height (E_{h}) and hopping time (τ_{h}) (scheme in Fig. 3). The thermallyassisted hopping time can be expressed as τ_{h} = τ_{s}exp(E_{h}/k_{B}T). When E_{h} → 0, τ_{h} reduces to the Drude scattering time τ_{s}. In this case, the photoconductivity response can be expressed by the Drude–Lorentz model (DL model):
where 0 ≤ ϕ ≤ 1 is the fraction of localized carriers. Since ωτ_{s} ≪ 1 is satisfied in the THz window, the DL model becomes mathematically equivalent to the DA model, when ϕ = ϕϕ_{b}, ω^{2}_{h} = 1/(τ_{s}ϕ_{b}τ_{b}), and τ_{h} = τ_{s} (see SI) however, the two models have different physical implications. The DL fits are similar to the DD fits in Fig. 3a, b except, they yield τ_{s} ~ 1.68–1.31 fs, ϕ ~ 0.1–0.22, E_{h} ~ 22.6 meV. The localized pairs with energy E_{p} + E_{h} are usually the surface/interface plasmons caused by carrier accumulation at grain boundaries and the film surface^{65}. However, plasmon resonance can be ruled out, because its peak position is proportional to \(\sqrt{N}\), yet no obvious peak shift can be observed here when N increases (Fig. 3a). Therefore, we hypothesize that the broadband Lorentzlike resonance is caused by sizeconstrained localization due to Anderson backscattering.
Discussion
Based on the finding of Anderson localization, Fig. 4a shows the Drude mobility (µ_{s}) calculated from the momentum scattering time (µ_{s} = eτ_{s}/m^{*}), together with the carrier mobility calculated from the frequencyaveraged photoconductivity (µ_{avg} = σ_{avg}/eN), and the dc mobility calculated by µ_{dc} = (1 − ϕϕ_{b})µ_{s}. The decrease of µ_{s} with N may be related to the photoenhanced nonlinearFröhlich coupling introduced by lattice fluctuations^{12,22}. µ_{dc} is smaller than µ_{s} due to the Anderson localization. Interestingly, µ_{dc} decreases faster than µ_{s}, since 1 − ϕϕ_{b} decreases with N (Fig. 4b), indicating photoenhanced Anderson localization and thus lattice fluctuations, consistent with the conclusion drawn from µ_{s}. We note that µ_{avg} is higher than µ_{dc}, because it partially accounts for the nondc mobility contributed by localized carriers. However, µ_{avg} has a similar Ndependence as µ_{s} and unlike µ_{dc}, indicating that µ_{avg} is dominated by µ_{s}. While Anderson localization results in a carrier mobility obeying the experimentally observed T^{−3/2}dependence^{20}, this finding suggests that the T^{−3/2}dependence of the carrier mobility determined from σ_{avg} is primarily caused by nonlinearFröhlich coupling, especially for lowfluence measurements^{19,38}. It is noteworthy that the carrier mobility obtained from microwave photoconductivity may be mainly determined by Anderson localization, because the microwave is closer to ω = 0^{66}.
The DA model also fits the data for MAPbI_{3} (MA) and FA_{0.83}MA_{0.17}PbI_{2.49}Br_{0.51} (FAMA) (Supplementary Fig. 11). The ratio of µ_{dc}/µ_{s} = 1 − ϕϕ_{b} decreases exponentially with N (Fig. 4b). According to the Mott mobility edge theory^{67}, µ_{dc}/µ_{s} decreases exponentially according to:
where E_{M} is the mobility edge determined by lattice fluctuations^{67}. Since after hotcarrier cooling (i.e., t = 5 ps), E_{F} and T depend only weakly on N, the exponential decay of 1 − ϕϕ_{b} results in an increase of E_{M} and thus an increase of lattice fluctuations, consistent with the conclusion drawn from the k ≥ 1 dependence of the BGR observed in the TA spectra. Consequently, the product ϕϕ_{b} is a measure of lattice fluctuations. We note that ϕϕ_{b} at N = 0 is nonzero due to the presence of thermallyactivated lattice fluctuations. The large value of ϕϕ_{b} (~0.46) in FAMA indicates that FAMA is intrinsically instable, possibly due to the larger cation size of FA^{+}, and/or the dynamic disorder of FA^{+} reorientation and rotation. However, incorporating only 5% Cs^{+} into the FAMA structure greatly improves the lattice stability, as evidenced by the small background of ϕϕ_{b} (~0.21) in CsFAMA. This improvement has previously been ascribed to the smaller size of Cs^{+} which results in a lattice structure closer to the Goldschmidt tolerance range^{68}, indicating that the contribution of dynamic disorder from FA^{+} to the lattice fluctuations is secondary.
By comparing the exciton binding energy R_{b0} with the Drude scattering time τ_{s} (Fig. 4c), we find that τ_{s} has a positive correlation with R_{b0}, which appears reasonable since both τ_{s} and R_{b0} are affected by electronphonon coupling. However, this is no longer valid if the unscreened exciton binding energy is very different^{9}. τ_{s} of all samples decreases with the population of excited states, as discussed above, which can be a result of the photoenhanced nonlinearFröhlich coupling introduced by lattice fluctuations^{12,22}. This is supported by the larger ϕϕ_{b} for excited states and the anticorrelation between τ_{s} and ϕϕ_{b} (Fig. 4c). In fact, the same behavior was proposed theoretically^{69} and experimentally observed in PL spectra^{70}, but it has been ascribed to the polaron destabilization related to the wave function overlap between neighboring polarons^{70}. The photoinduced lattice fluctuations reported here could be related to lightinduced fluctuations observed by other techniques^{25,40}, with magnitudes depending on the intrinsic instability of the MHP lattice (Fig. 4c). The Anderson localization scenario suggests coexistence of bandlike and hopping transport which challenges the standard bandlike carrier transport model for perovskites^{24}. However, the precise nature of carrier transport in MHPs remains an open question. Here, we emphasize that the observation of photoinduced lattice fluctuations, which is the main finding of our work, is independent of the Anderson localization. In fact, bandlike transport models (Supplementary Tables 1–3) indicate a photoinduced increase of the dielectric response (DD model) or carrier localization (DS model and DL model). To some extent, both can be related to photoinduced lattice fluctuations.
In conclusion, triplecation MHP films (CsFAMA) were investigated by TA and tdTHz spectroscopy. We developed a model to analyze the highenergy part of the TA spectra, which takes into account photoinduced exciton screening and bandgap narrowing. The bandgap narrowing was found to increase linearly or superlinearly with carrier density, which cannot be explained by carriercarrier interactions, but can be rationalized by photoinduced lattice fluctuations, as supported by the increased carrier localization in the terahertz photoconductivity spectra. Anderson backscattering is the most likely mechanism responsible for the increased carrier localization. The photoinduced lattice fluctuations aggravate the randomness of the quantum electrostatic potential and thus increase carrier localization. Finally, the comparison of Anderson localization between different MHP systems revealed that the magnitude of photoinduced lattice fluctuations correlates with intrinsic instability of the MHP lattice. Our findings provide specific insight into the excitedstate photophysics of stateoftheart MHPs, and they aid the development of a concise picture of the ultrafast physics of this important class of semiconductors.
Methods
Materials: PbI_{2} and PbBr_{2} were purchased from TCI. FAI and MABr were purchased from Dyesol. CsI, RbI and all anhydrous solvents (DMF, DMSO, chlorobenzene) were purchased from SigmaAldrich. SnO_{2} colloid precursor was obtained from Alfa Aesar, the particles were diluted by H_{2}O and isopropanol to 2.67%. All chemicals were used without further purification.
Perovskite film fabrication: MAPbI_{3} perovskite precursor solution in a mixed solvent (DMF/DMSO = 9:1) was used. In total, 70 μl of perovskite solution was spun onto the substrates at 4000 rpm for 30 s. In total, 150 μl of chlorobenzene was dropped in the center of the substrates 22 s before the end of the spincoating process. After the rotation ceased, the substrates were immediately transferred onto a hotplate of 100 °C and annealed for 10 min. PbI_{2} (508 mg, 1.1 mmol), PbBr_{2} (80.7 mg, 0.22 mmol), FAI (171.97 mg, 1 mmol) and MABr (22.4 mg, 0.2 mmol) in 1 ml of a 4: 1 (v/v) mixture of anhydrous DMF and DMSO. This resulting precursor solution for (FA_{0.83}MA_{0.17})Pb(I_{0.83}Br_{0.17})_{3} contains a 10 mol% excess of PbI_{2} and PbBr_{2}, respectively, which was introduced to enhance device performance. The FAMA solution was filtrated through a 0.45 μm syringe filter before use. CsI was dissolved in 1 ml DMSO and 42 μl of the ~1.5 M CsI stock solution was added to 1 ml FAMA solution to get (Cs_{0.05}FA_{0.8}MA_{0.15})Pb(I_{0.85}Br_{0.15})_{3} solution. In total, 70 μl of perovskite solution was spun onto the substrates at 2000 rpm for 10 s and 4000 rpm for 30 s. In total, 150 μl of chlorobenzene was dropped in the center of the substrates 10 s before the end of the spincoating process. After the rotation ceased, the substrates were immediately transferred onto a hotplate of 100 °C and annealed for 10 min.
TA setup: our TA setup uses a commercial Ti:sapphire amplifier operating at 800 nm with a repetition rate of 3 kHz as laser source. Its pulse width (FWHM) is compressed to ~125 fs. Two optical parametric amplifiers are used to tune the laser wavelength. The whitelight probe is generated by 1300 nm laser (from TOPAS1) with a CaF2 crystal that mounted on a continuously moving stage, which enables us to generate a supercontinuum pulses with a spectral range from 350 to 1100 nm. The pump laser (from TOPAS2) is chopped to 1.5 KHz and delayed by an automated mechanical delay stage (Newport linear stage IMS600CCHA) from −400 ps to 8 ns. Pump and probe beams were overlapped on the front surface of the sample, and their spot sizes (D86 ~ 3 mm) were measured by a beam viewer (Coherent, LaserCamHR II) to make sure the pump beam was about three times larger than the probe beam (D86 ~ 1 mm). The perovskite samples are stored in a nitrogenfilled chamber to protect from degradation, and photoexcited by 475, 550, and 675 nm in this work. The probe beam was guided to a custommade prism spectrograph (Entwicklungsbüro Stresing) where it was dispersed by a prism onto a 512 pixel complementary metaloxide semiconductor linear image sensor (Hamamatsu G11608512DA). In order to account for the reflection, we first measured the transient reflection (ΔR/R), then measured the TA (ΔT/T).
tdTHz setup: our tdTHz setup uses the same Ti:sapphire amplifier as the TA setup. The THz emitter and detector are two 1 mm thick <110> oriented zinc telluride (ZnTe) crystals. All the THz related optics were placed in a closed chamber, which was continuously purged with pure nitrogen gas. Perovskite samples were excited by 550 nm laser pulses obtained from the same TOPAS2 as used in the TA experiment. A motor equipped with a circular ND filter was used to change the pump fluence in the fluencedependent experiments.
Data availability
The main data supporting the findings of this study are available within the Article and its Supplementary Information. Extra data are available from the corresponding author (M.W. or F.L.) upon request.
References
Miyata, A. et al. Direct measurement of the exciton binding energy and effective masses for charge carriers in organic–inorganic trihalide perovskites. Nat. Phys. 11, 582–587 (2015).
Tanaka, K. et al. Comparative study on the excitons in leadhalidebased perovskitetype crystals CH3NH3PbBr3 CH3NH3PbI3. Solid State Commun. 127, 619–623 (2003).
Wilson, J. N., Frost, J. M., Wallace, S. K. & Walsh, A. Dielectric and ferroic properties of metal halide perovskites. APL Mater. 7, 10901 (2019).
Davies, C. L. et al. Bimolecular recombination in methylammonium lead triiodide perovskite is an inverse absorption process. Nat. Commun. 9, 293 (2018).
Sendner, M. et al. Optical phonons in methylammonium lead halide perovskites and implications for charge transport. Mater. Horiz. 3, 613–620 (2016).
Herz, L. M. How lattice dynamics moderate the electronic properties of metalhalide perovskites. J. Phys. Chem. Lett. 9, 6853–6863 (2018).
Miyata, K., Atallah, T. L. & Zhu, X.Y. Lead halide perovskites: crystalliquid duality, phonon glass electron crystals, and large polaron formation. Sci. Adv. 3, e1701469 (2017).
MenéndezProupin, E., Beltrán Ríos, C. L. & Wahnón, P. Nonhydrogenic exciton spectrum in perovskite CH 3 NH 3 PbI 3. Phys. Status Solidi RRL 9, 559–563 (2015).
Soufiani, A. M. et al. Polaronic exciton binding energy in iodide and bromide organicinorganic lead halide perovskites. Appl. Phys. Lett. 107, 231902 (2015).
Schlipf, M., Poncé, S. & Giustino, F. Carrier lifetimes and polaronic mass enhancement in the hybrid halide perovskite CH_{3}NH_{3}PbI_{3} from multiphonon Fröhlich coupling. Phys. Rev. Lett. 121, 86402 (2018).
Anusca, I. et al. Dielectric response: answer to many questions in the methylammonium lead halide solar cell absorbers. Adv. Energy Mater. 7, 1700600 (2017).
Poncé, S., Schlipf, M. & Giustino, F. Origin of low carrier mobilities in halide perovskites. ACS Energy Lett. 4, 456–463 (2019).
Ghosh, T., Aharon, S., Etgar, L. & Ruhman, S. Free carrier emergence and onset of electronphonon coupling in methylammonium lead halide perovskite films. J. Am. Chem. Soc. 139, 18262–18270 (2017).
Saba, M. et al. Correlated electronhole plasma in organometal perovskites. Nat. Commun. 5, 5049 (2014).
Chen, B., Rudd, P. N., Yang, S., Yuan, Y. & Huang, J. Imperfections and their passivation in halide perovskite solar cells. Chem. Soc. Rev. 48, 3842–3867 (2019).
Meggiolaro, D., Ambrosio, F., Mosconi, E., Mahata, A. & Angelis, Fde Polarons in metal halide perovskites. Adv. Energy Mater. 10, 1902748 (2020).
Herz, L. M. Chargecarrier mobilities in metal halide perovskites: fundamental mechanisms and limits. ACS Energy Lett. 2, 1539–1548 (2017).
Frost, J. M. Calculating polaron mobility in halide perovskites. Phys. Rev. B 96, 195202 (2017).
Milot, R. L., Eperon, G. E., Snaith, H. J., Johnston, M. B. & Herz, L. M. Temperaturedependent chargecarrier dynamics in CH 3 NH 3 PbI 3 perovskite thin films. Adv. Funct. Mater. 25, 6218–6227 (2015).
Lacroix, A., de Laissardière, G. T, Quémerais, P., Julien, J.P. & Mayou, D. Modeling of electronic mobilities in halide perovskites: adiabatic quantum localization scenario. Phys. Rev. Lett. 124, 196601 (2020).
Ledinsky, M. et al. Temperature dependence of the urbach energy in lead iodide perovskites. J. Phys. Chem. Lett. 10, 1368–1373 (2019).
Mayers, M. Z., Tan, L. Z., Egger, D. A., Rappe, A. M. & Reichman, D. R. How lattice and charge fluctuations control carrier dynamics in halide perovskites. Nano Lett. 18, 8041–8046 (2018).
Motta, C. et al. Revealing the role of organic cations in hybrid halide perovskite CH3NH3PbI3. Nat. Commun. 6, 7026 (2015).
Zhu, H. et al. Screening in crystalline liquids protects energetic carriers in hybrid perovskites. Science 353, 1409–1413 (2016).
Munson, K. T., Kennehan, E. R., Doucette, G. S. & Asbury, J. B. Dynamic disorder dominates delocalization, transport, and recombination in halide perovskites. Chem 4, 2826–2843 (2018).
Ma, J. & Wang, L.W. Nanoscale charge localization induced by random orientations of organic molecules in hybrid perovskite CH_{3}NH_{3}PbI_{3}. Nano Lett. 15, 248–253 (2015).
Miyata, K. et al. Large polarons in lead halide perovskites. Sci. Adv. 3, e1701217 (2017).
Ambrosio, F., Meggiolaro, D., Mosconi, E. & de Angelis, F. Charge localization, stabilization, and hopping in lead halide perovskites: competition between polaron stabilization and cation disorder. ACS Energy Lett. 4, 2013–2020 (2019).
Gehrmann, C. & Egger, D. A. Dynamic shortening of disorder potentials in anharmonic halide perovskites. Nat. Commun. 10, 3141 (2019).
Nagai, M. et al. Longitudinal optical phonons modified by organic molecular cation motions in organicinorganic hybrid perovskites. Phys. Rev. Lett. 121, 145506 (2018).
Duan, H.G. et al. Photoinduced vibrations drive ultrafast structural distortion in lead halide perovskite. J. Am. Chem. Soc. 142, 16569–16578 (2020).
Miyata, K. & Zhu, X.Y. Ferroelectric large polarons. Nat. Mater. 17, 379–381 (2018).
Wang, K. et al. Kinetic stabilization of the solgel state in perovskites enables facile processing of highefficiency solar cells. Adv. Mater. 31, 1808357 (2019).
Dang, H. et al. Multication synergy suppresses phase segregation in mixedhalide perovskites. Joule 3, 1746 (2019).
Kumar, A. et al. Ultrafast THz photophysics of solvent engineered triplecation halide perovskites. J. Appl. Phys. 124, 215106 (2018).
LaOVorakiat, C. et al. Elucidating the role of disorder and freecarrier recombination kinetics in CH_{3}NH_{3}PbI_{3} perovskite films. Nat. Commun. 6, 7903 (2015).
Wang, M. et al. Impact of photoluminescence reabsorption in metal‐halide perovskite solar cells. Sol. RRL 351, 2100029 (2021).
Karakus, M. et al. Phononelectron scattering limits free charge mobility in methylammonium lead iodide perovskites. J. Phys. Chem. Lett. 6, 4991–4996 (2015).
Wehrenfennig, C., Eperon, G. E., Johnston, M. B., Snaith, H. J. & Herz, L. M. High charge carrier mobilities and lifetimes in organolead trihalide perovskites. Adv. Mater. 26, 1584–1589 (2014).
Wu, X. et al. Lightinduced picosecond rotational disordering of the inorganic sublattice in hybrid perovskites. Sci. Adv. 3, e1602388 (2017).
Yang, Y. et al. Observation of a hotphonon bottleneck in leadiodide perovskites. Nat. Photon 10, 53–59 (2016).
Amat, A. et al. Cationinduced bandgap tuning in organohalide perovskites: interplay of spinorbit coupling and octahedra tilting. Nano Lett. 14, 3608–3616 (2014).
Li, M., Fu, J., Xu, Q. & Sum, T. C. Slow hotcarrier cooling in halide perovskites: prospects for hotcarrier solar cells. Adv. Mater. 31, e1802486 (2019).
Gao, Y. et al. Impact of cesium/rubidium incorporation on the photophysics of multiple‐cation lead halide perovskites. Sol. RRL 4, 2000072 (2020).
Lim, J. W. M. et al. Hot carriers in halide perovskites: how hot truly? J. Phys. Chem. Lett. 11, 2743–2750 (2020).
Price, M. B. et al. Hotcarrier cooling and photoinduced refractive index changes in organicinorganic lead halide perovskites. Nat. Commun. 6, 8420 (2015).
Asano, K. & Yoshioka, T. Exciton–Mott physics in twodimensional electron–hole systems: phase diagram and singleparticle spectra. J. Phys. Soc. Jpn. 83, 84702 (2014).
Palmieri, T. et al. Mahan excitons in roomtemperature methylammonium lead bromide perovskites. Nat. Commun. 11, 850 (2020).
Schleife, A., Rödl, C., Fuchs, F., Hannewald, K. & Bechstedt, F. Optical absorption in degenerately doped semiconductors: Mott transition or Mahan excitons? Phys. Rev. Lett. 107, 236405 (2011).
Berggren, K.F. & Sernelius, B. E. Bandgap narrowing in heavily doped manyvalley semiconductors. Phys. Rev. B 24, 1971–1986 (1981).
Jain, S. C. & Roulston, D. J. A simple expression for band gap narrowing (BGN) in heavily doped Si, Ge, GaAs and GexSi1−x strained layers. SolidState Electron. 34, 453–465 (1991).
Allen, P. B. & Cardona, M. Theory of the temperature dependence of the direct gap of germanium. Phys. Rev. B 23, 1495–1505 (1981).
Ščajev, P. et al. Anisotropy of thermal diffusivity in lead halide perovskite layers revealed by thermal grating technique. J. Phys. Chem. C. 123, 14914–14920 (2019).
OnodaYamamuro, N., Matsuo, T. & Suga, H. Calorimetric and IR spectroscopic studies of phase transitions in methylammonium trihalogenoplumbates (II)†. J. Phys. Chem. Solids 51, 1383–1395 (1990).
Saidi, W. A., Poncé, S. & Monserrat, B. Temperature dependence of the energy levels of methylammonium lead iodide perovskite from firstprinciples. J. Phys. Chem. Lett. 7, 5247–5252 (2016).
Patrick, C. E., Jacobsen, K. W. & Thygesen, K. S. Anharmonic stabilization and band gap renormalization in the perovskite CsSnI_{3}. Phys. Rev. B 92,201205 (2015).
Bretschneider, S. A. et al. Quantifying polaron formation and charge carrier cooling in leadiodide perovskites. Adv. Mater. 30, 1707312 (2018).
Nienhuys, H.K. & Sundström, V. Intrinsic complications in the analysis of opticalpump, terahertz probe experiments. Phys. Rev. B 71, 1759 (2005).
LloydHughes, J. & Jeon, T.I. A review of the terahertz conductivity of bulk and nanomaterials. J. Infrared Milli Terahz Waves 33, 871–925 (2012).
Laarhoven, H. A. et al. On the mechanism of charge transport in pentacene. J. Chem. Phys. 129, 44704 (2008).
Yaffe, O. et al. Local polar fluctuations in lead halide perovskite crystals. Phys. Rev. Lett. 118, 136001 (2017).
PérezOsorio, M. A. et al. Vibrational properties of the organic–inorganic halide perovskite CH_{3}NH_{3}PbI_{3} from theory and experiment: factor group analysis, firstprinciples calculations, and lowtemperature infrared spectra. J. Phys. Chem. C. 119, 25703–25718 (2015).
Kužel, P. & Němec, H. Terahertz conductivity in nanoscaled systems: effective medium theory aspects. J. Phys. D: Appl. Phys. 47, 374005 (2014).
Fratini, S., Ciuchi, S. & Mayou, D. Phenomenological model for charge dynamics and optical response of disordered systems: application to organic semiconductors. Phys. Rev. B 89, 278 (2014).
Nienhuys, H.K. & Sundström, V. Influence of plasmons on terahertz conductivity measurements. Appl. Phys. Lett. 87, 12101 (2005).
Savenije, T. J. et al. Thermally activated exciton dissociation and recombination control the carrier dynamics in organometal halide perovskite. J. Phys. Chem. Lett. 5, 2189–2194 (2014).
Mott, N. The mobility edge since 1967. Adv. Phys. 20, 3075–3102 (1987).
Rehman, W. et al. Photovoltaic mixedcation lead mixedhalide perovskites: links between crystallinity, photostability and electronic properties. Energy Environ. Sci. 10, 361–369 (2017).
Sentef, M. A. Lightenhanced electronphonon coupling from nonlinear electronphonon coupling. Phys. Rev. B 95, 205111 (2017).
Niesner, D. et al. Persistent energetic electrons in methylammonium lead iodide perovskite thin films. J. Am. Chem. Soc. 138, 15717–15726 (2016).
Acknowledgements
This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No: OSR2018CARF/CCF3079, OSR2018CRG73737, OSR2019CRG84093, and OSR2020CRG94350.
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M.W. performed the timedomain terahertz measurements and data analysis. He conceived and developed the theoretical model and wrote the initial draft of the paper. M.W. and Y.G. performed the groundstate absorption and transient absorption measurements. K.W. and J.L. prepared the perovskite thinfilm samples. S.D.W. and F.L. supervised the work and revised the manuscript. All authors contributed to refining the final version of the manuscript.
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Wang, M., Gao, Y., Wang, K. et al. Photoinduced enhancement of lattice fluctuations in metalhalide perovskites. Nat Commun 13, 1019 (2022). https://doi.org/10.1038/s41467022285320
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DOI: https://doi.org/10.1038/s41467022285320
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