Ultrafast acoustic phonon scattering in CH₃NH₃PbI₃ revealed by femtosecond four-wave mixing

The hybrid organic-inorganic perovskite (HOIP) semiconductors have emerged in recent years as low-cost, solution-processable alternatives to traditional inorganic semiconductors, offering good optical and transport properties for applications such as solar cells,¹ optical sources,^2–4 photodetectors,⁵ and field-effect transistors (FET).^6,7 An understanding of the scattering processes that govern the charge carrier dynamics in HOIP semiconductors is essential to optimize the performance of such devices. For instance, electron-phonon scattering determines the rate of relaxation of hot carriers. The time scale of this process must be well-characterized to evaluate the potential for improvement of solar cell efficiencies through hot carrier extraction. Scattering with phonons and defects determines the mobility of charge carriers, dictating the efficiency of current extraction in a variety of optoelectronic devices. Insight into carrier scattering processes has been gained in recent years using time-resolved microwave conductivity,^8,9 THz spectroscopy,^10–12 photocurrent,¹³ photoluminescence,^14,15 and transient absorption.^16–18 Such techniques only probe the influence of these scattering processes on the average properties of the carrier distribution (e.g., mobility and carrier temperature). Four-wave mixing (FWM) techniques provide a direct time-domain probe of carrier scattering.^19–24 Since the fastest scattering process leads to dephasing of electron-hole pairs (with dephasing time T₂), FWM enables a determination of the nature and strength of the dominant interactions affecting charge carriers (e.g., carrier-carrier, carrier-phonon, and carrier-defect coupling) in quantitative terms through the measured time scale for coherence decay.

Previous low-temperature four-wave mixing measurements on the prototypical HOIP (CH₃NH₃PbI₃) revealed weak carrier-carrier scattering in the low carrier density regime (≤10¹⁶ cm³),²² in stark contrast to inorganic III-V and group IV semiconductors in which carrier-carrier scattering is much stronger than all other carrier interactions.²⁰ This surprising result was attributed to disorder tied to the frozen fluctuations of the methylammonium cations,²² a static version of the dynamic disorder responsible for large polaron formation and associated screening of the Coulomb interaction at higher temperatures.²⁵ Here, we report temperature-dependent spectrally resolved four-wave mixing (FWM) studies with the aim to determine the dominant residual scattering processes in the absence of carrier-carrier scattering. Our experiments indicate that scattering with phonons dominates for temperatures above 30 K. The rate of phonon scattering we observe for low-energy band carriers is surprisingly high (T₂ = 125 fs at a temperature of 100 K). Simulations of the carrier kinetics incorporating scattering with polar optical and acoustic phonons provide quantitative agreement with our experimental results. These calculations indicate that the rapid acoustic phonon scattering process we observe is due in part to the strong spin-orbit coupling in this material, which enhances the density of states in the vicinity of the band edge. For temperatures below 30 K, the measured scattering rate was attributed to ionized impurities. An estimate of the density of ionized impurities of 1.7 × 10¹⁷ cm⁻³ was extracted from the dependence of the carrier scattering rate on electron kinetic energy. Our findings provide insight into electron-phonon interactions and the impact of strong spin-orbit coupling on the fundamental carrier scattering processes in the hybrid perovskite family of materials.

Carrier scattering processes were studied in a thin film of CH₃NH₃PbI₃ in the low-temperature orthorhombic phase using FWM in the two pulse self-diffraction geometry.²⁰ The thin films were prepared using a sequential deposition procedure described previously.^21,22,27 The film thickness was 265 nm. The substrate used (c-cut sapphire) was chosen to enable accurate temperature control in optical experiments in the transmission geometry using a sample-in-vacuum helium flow-through cryostat. The FWM technique probes the decay of quantum coherence excited on the optical transitions in the sample using a short, coherent laser pulse. The decay of the macroscopic polarization density on the interband transitions occurs (with a relaxation time T₂) as a consequence of carrier scattering processes that interrupt the phase of the oscillating electric dipoles generated by individual electron-hole pairs. Since the scattering of either carrier type (electron or hole) leads to loss of coherence, this technique provides a sensitive probe of the fastest scattering process involving charge carriers. In the FWM technique, two laser pulses $E1→(t)$ and $E2→(t−τ)$ propagating with wave vectors $k1→$ and $k2→$ are used to excite the sample. The decay of the so-called self-diffraction signal along $2k2→−k1→$ vs interpulse delay τ indicates the time for loss of coherence on the optical transitions in the sample. A prism compressor was used for dispersion compensation, resulting in a pulse duration at the sample of 55 fs. The interpulse delay was varied at 12 Hz using a light-weight retroreflector mounted on a speaker. The four-wave mixing signal along $2k2→−k1→$ was spectrally resolved by passing through a 0.25 m monochromator and detected using a photomultiplier tube. The carrier density determined by the sum of the powers in the two excitation beams, reflection losses, and the measured spot size was 2 × 10¹⁶ cm⁻³.

The results of FWM experiments on a thin film of CH₃NH₃PbI₃ are shown for two different temperatures in Fig. 1(a). The FWM signal was fit to a photon echo response²⁸ convoluted with the laser pulse duration (55 fs) to obtain the interband dephasing time (T₂) at each temperature. The corresponding fits are indicated by the red curves. The spectrally resolved FWM signal is shown for a range of temperatures in Fig. 1(b). The weak oscillations in the results for 30 K are tied to excitonic quantum beats, as discussed previously.²¹ These quantum beats disappear, and the FWM signal decays more rapidly as the temperature increases, reflecting faster carrier scattering at higher temperatures. The FWM signal decay time was below the resolution of our experiments for temperatures above 110 K. The dephasing rate (⁠$1T2$⁠) extracted from fits to the FWM signal is shown in Fig. 2(a) as a function of temperature and detection photon energy. $1T2$ increases with increasing carrier energy relative to the band gap, depends weakly on temperature below 30 K, and increases rapidly for higher temperatures.

The relatively constant dephasing rate below 30 K in Fig. 2(a) is attributed to scattering with defects and impurities. Taking ionized impurity scattering as the dominant process, we may apply the analysis of Conwell and Weisskopf²⁹ modified to describe interband dephasing rather than momentum relaxation. (The latter contains a directional factor 1 − cos θ with θ being the scattering angle, i.e., a forward scattering does not impact the momentum relaxation time but does reduce T₂.) The resulting dephasing rate may be expressed as $τI−1=π4NI1/3v$⁠, where N_I is the density of ionized impurities, $v=2Ekm$ is the carrier velocity, and E_k the kinetic energy. The rate of ionized impurity scattering is independent on temperature and increases with E_k. Using the measured dependence of the dephasing rate at 10 K on the carrier excess energy, we estimate the density of ionized impurities to be 1.7 × 10¹⁷ cm⁻³.

If we subtract the 10 K dephasing rate from the measured results at each value of E_k, the residual dephasing rate [Fig. 2(b)] reflects temperature-dependent scattering processes, which could in the general case involve carrier-carrier scattering and electron-phonon scattering. The rate of carrier-carrier scattering was previously shown to be negligible in CH₃NH₃PbI₃ for carrier densities up to at least 5 × 10¹⁶ cm⁻³,²² well above the 2 × 10¹⁶ cm⁻³ excitation density used in these experiments. This result was surprising because it contrasts with the situation in typical III-V inorganic semiconductors such as GaAs and InP, in which strong carrier-carrier scattering occurs down to carrier densities on the order of 10¹³ cm⁻³.³⁰ This finding was attributed to carrier localization within the disorder potential induced by the frozen fluctuations of the methylammonium cations. This intrinsic disorder potential, which is static in the low-temperature phase but leads to dynamic disorder and large polaron formation at higher temperatures, is tied to the hybrid nature of the organic-inorganic perovskites and has no analog in inorganic semiconductors. With negligible carrier-carrier scattering, the strong increase in the dephasing rate with temperature above 30 K is attributed to carrier scattering with phonons. The overall rate of phonon scattering is faster than that expected for a simple parabolic band, given known parameters for the phonon coupling strength and the value of the optical phonon energies.^31,32 We show below that both the ultrafast scattering rate and the constant rate vs carrier excess energy are well described by acoustic phonon scattering taking into account the impact of the Rashba effect on the band edge density of states.

The Rashba effect results from the breaking of inversion symmetry in conjunction with spin-orbit coupling,^33,34 which is large in the hybrid perovskites owing to the incorporation of heavy elements (Pb, I, and Br). While inversion asymmetry is expected in 2D perovskites due to distortions of the lead-iodide octahedra induced by the competing effects of the small and long organic cations,^43,44,58 3D perovskites such as CH₃NH₃PbI₃ are believed to possess inversion symmetry in the bulk of the crystal. Nevertheless, a strong surface Rashba effect is expected in the 3D case.^51,55–57 This surface Rashba effect is caused by surface reconstruction that leads to a redistribution of the conduction and valence states, resulting in a strong surface dipole⁵⁶ with a penetration depth comparable to typical thin film thicknesses (200–300 nm).⁵⁵ Recent observations of strong Rashba coupling in single crystal CH₃NH₃Br₃ using the surface-sensitive angle-resolved photoemission spectroscopy technique⁴⁰ and in thin films and nanostructures of CH₃NH₃PbI₃^41,42,45,46 are likely dominated by surface Rashba effects. Such a surface Rashba effect would enable the development of a wide variety of spintronic devices such as spin FETs, all-optical spin switches, and spin lasers that would exploit thin film geometries rather than large single crystals.^35,37–39

The Rashba effect lifts the degeneracy of the spin states and provides a means to manipulate carrier spin without the need for external magnetic fields.^35–39 The energy-momentum dispersion in the presence of Rashba coupling is

where m_e is the effective mass, $k⊥=kx2+ky2$⁠, k₀ = mγ_c/ℏ², γ_c is the Rashba coupling strength, and $Ec0−ℏ2k022me≡0$⁠. The corresponding eigenstates are $ψ±(k)=12(1,±ieiϕ)T$ with the spin orientation being a function of the momentum direction. A schematic representation of the spin-split states in the conduction band is shown in the inset to Fig. 3(a). The Rashba effect shifts the band edges away from the high symmetry point in the corresponding inversion symmetric structure, leading to an indirect band gap separated from the direct gap by a relatively small energy (∼50 meV).^42,47 This feature has been invoked to explain the simultaneous presence of a large absorption coefficient and long carrier recombination time in CH₃NH₃PbI₃ thin films leading to high solar cell efficiencies.^31,48–53 Much less is known about the effect of Rashba coupling on the scattering and relaxation processes in HOIP semiconductors,^31,32,54 despite the crucial role played by these scattering processes in the operation of optoelectronic devices.

In order to explain the measured temperature dependence of T₂, the dephasing rate tied to scattering of electrons with acoustic and polar optical phonons was calculated following a model developed previously³² taking into account the Rashba splitting. The electron-phonon coupling can be expressed as

where $cks†$ creates an electron with momentum k and spin s (=↑, ↓), $bq†$ creates a phonon with momentum q, and the two terms describe phonon absorption and emission processes. Since H_ep does not change electron spin, the matrix elements between electron eigenstates, M_±±(k, k′) ≡ ⟨ψ_±(k′)|H_ep|ψ_±(k)⟩, in the presence of the Rashba effect are

The coupling between electrons and acoustic phonons is

where E₁ is the deformation potential coupling for the conduction band, ρ is the material density, ω_q = $vs$q is the acoustic phonon dispersion with $vs$ being the velocity of sound in the material, and $cL=ρvs2$ is the longitudinal elastic constant. In HOIP semiconductors, the polar coupling for electrons can be expressed as

where α_ej is the dimensionless polar coupling strength for the jth optical phonon mode.³² The scattering rate can be calculated from the Fermi’s golden rule. In our calculations, we restrict ourselves to the scattering rate of electrons. As we show below, the Rashba effect strongly enhances the rate of acoustic phonon scattering. Since the magnitude of the Rashba effect in the valence band is expected to be weaker than in the conduction band due to the dominant contribution of the I 5p orbital to the valence states,⁵⁴ electron scattering is assumed to dominate the interband dephasing process. The magnitude of the Rashba coupling strength γ_c was taken as an adjustable parameter to obtain the best fit to the experimental results.

The results of theoretical simulations of the interband dephasing rate are shown in Fig. 3(c), together with the experimental data. For these simulations, only one optical phonon mode is included tied to the Pb-I stretching mode, which was shown previously to have the largest polar optical coupling (α_e = 1.1) with phonon energy ℏω₀ = 16.7 meV,³² and the value of the Rashba coupling strength is γ_c = 2 eV Å. The theoretical simulations provide good quantitative agreement with the experimental results, including both the temperature dependence and the lack of dependence of the phonon scattering rate on carrier energy. The contribution of each phonon scattering process to the calculated dephasing rate at a temperature of 50 K is shown as a function of electron energy in Fig. 3(b). The solid (dashed) curves indicate the scattering rates with (without) Rashba coupling. Acoustic phonon scattering dominates over polar optical phonon scattering for the full range of temperatures considered here (≤110 K).

The rate of acoustic phonon scattering is strongly enhanced for electrons’ energies near the band edge with Rashba coupling included. This strong enhancement is the origin of the ultrafast interband dephasing observed in our experiments. Our calculations indicate an enhancement of the low temperature acoustic phonon scattering rate of 6× relative to the case of a simple parabolic band without Rashba coupling. The Rashba effect also modifies the energy-dependence of the scattering rate tied to acoustic phonons, which changes from an approximate square root dependence without Rashba to constant with the Rashba effect included. This feature is tied to the constant density of states in the lower-energy spin band [Fig. 3(a)], and explains the lack of an observed dependence of the electron-phonon scattering contribution to the dephasing rate on carrier energy near the band gap. The $q$ dependence of the acoustic phonon coupling strength also contributes to the enhancement of the scattering rate as the Rashba effect leads to an increase in the average |q| for phonon scattering near the band minimum [Fig. 3(a), inset]. The value of γ_c = 2 eV Å we extract from our simulations is in good agreement with a recent calculation of the surface Rashba effect using density functional theory⁵¹ as well as recent observations in CH₃NH₃PbI₃ films in the orthorhombic phase using the circular photogalvanic effect.⁴⁶ 

In conclusion, we have applied four-wave mixing to measure electron scattering times in thin films of CH₃NH₃PbI₃. Our experiments indicate that ionized impurities provide the dominant scattering process at low temperatures, while above 30 K strong temperature-dependent acoustic phonon scattering dominates the carrier kinetics. Our experiments reveal that the rate of scattering with phonons is quite high, even for electrons very close to the band edge, and that the electron-phonon scattering rate is relatively insensitive to the carrier energy. Both of these findings were accounted for in numerical simulations taking into account the influence of the Rashba effect on the density of states near the band edge. These simulations yield a Rashba coupling strength of γ_c = 2 eV Å, believed to be dominated by inversion symmetry breaking at surfaces and interfaces in the polycrystalline thin film. Measurements on thin films with a range of substrates, sample thicknesses, and/or surface treatments would provide a useful extension to the present work by probing the degree to which the surface Rashba effect may be engineered in the 3D perovskites. The insight into the strength of electron-phonon coupling provided by FWM experiments in this work complements recent broadband transient absorption experiments revealing phonon-induced modulation of the carrier transition energies.²⁶ Our findings provide insight into carrier scattering processes in HOIP semiconductors with implications for a wide range of device applications.

This research was supported by the Natural Sciences and Engineering Research Council of Canada. Work at Washington State University was supported by Grant No. W911NF-17-1-0511 from the U.S. Army Research Office.