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Lead halide perovskites: Crystal-liquid duality, phonon glass electron crystals, and large polaron formation

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Science Advances  13 Oct 2017:
Vol. 3, no. 10, e1701469
DOI: 10.1126/sciadv.1701469

Abstract

Lead halide perovskites have been demonstrated as high performance materials in solar cells and light-emitting devices. These materials are characterized by coherent band transport expected from crystalline semiconductors, but dielectric responses and phonon dynamics typical of liquids. This “crystal-liquid” duality implies that lead halide perovskites belong to phonon glass electron crystals, a class of materials believed to make the most efficient thermoelectrics. We show that the crystal-liquid duality and the resulting dielectric response are responsible for large polaron formation and screening of charge carriers, leading to defect tolerance, moderate charge carrier mobility, and radiative recombination properties. Large polaron formation, along with the phonon glass character, may also explain the marked reduction in hot carrier cooling rates in these materials.

INTRODUCTION: THE CRYSTAL-LIQUID DUALITY

Lead halide perovskites have emerged as superstars among materials for photovoltaics and light emission (16). A large number of studies have explored the mechanistic origins of their electronic properties, such as long carrier lifetimes, long carrier diffusion lengths, and exceptional defect tolerance (711). The unique properties of both hybrid organic-inorganic lead halide perovskites (HOIPs) and their all-inorganic counterparts result from the framework-like structure consisting of a sublattice of corner-sharing PbX63− (X = I, Br, or Cl) octahedral with overall stoichiometry PbX3 and a sublattice of A+ cations in the cuboctahedral voids, as shown schematically in Fig. 1 (12). Here, A+ = Cs+, Rb+, CH3NH3+ (MA), or NH2(CH)NH2+ (FA). The framework structure leads to intrinsic softness and dynamic disorder. The former is shown by their Young’s moduli being approximately 10× lower than those of Si or GaAs (13, 14). The latter is reflected by broadening and anharmonicity in far-infrared (IR) and low-frequency Raman spectra, disorder in nuclear magnetic resonance (NMR) spectra and x-ray or neutron scattering, and liquid-like responses in dielectric function or femto- to picosecond responses (1526).

Fig. 1 Structure of CH3NH3PbX3 perovskite.

It consists of a framework of corner-sharing lead (gray) halide (purple) octahedrals and the CH3NH3+ cation (black, blue, and white) in the nanocage.

For each lead halide perovskite, there are three main structural phases. In the low-temperature orthorhombic phase (T < 144.5 K and 370 K for MAPbBr3 and CsPbBr3, respectively), the PbBr63− octahedron undergoes Jahn-Teller distortion and the cation motion is restricted (24, 27, 28). For MAPbBr3 (CsPbBr3), the perovskite structure transitions to the tetragonal phase at T = 149.5 to 237 K (370 to 420 K) and the cubic phase at T > 237 K (420 K) (24, 27, 28). Similar phase transitions are present in I- and Cl-based perovskites. These transitions are accompanied by increasing dynamic disorder, particularly the orientational motions of organic cations in HOIPs (26).

The coexistence of a crystalline-like response in coherent band transport of charge carriers and a liquid-like response in phonon activities has prompted Zhu et al. (25) to call HOIPs “crystalline liquids” for the cubic and tetragonal phases. We can view the sublattice of organic cations as a confined liquid, with each polar molecular cation confined in a nanoscopic pore; thus, an HOIP in the high-temperature cubic phase is similar to a plastic crystal (29), which refers to a crystalline lattice with long-range translational order but local rotational disorder (30). More precisely, the liquid-like behavior may generally represent the highly disordered and anharmonic motions of the strongly coupled vibrational modes of both PbX3 and A+ sublattices (31). Here, we discuss three important consequences of the crystal-liquid duality: (i) The contrasting phonon and electron dynamics mean that lead halide perovskites belong to the family of solids called phonon glass electron crystals (PGECs) (32); (ii) the band-like charge transport, coupled with the liquid-like polarizability, means that a charge carrier can effectively form a large polaron (33), whose screened Coulomb potential reduces its scattering with charged defects and with other charge carriers; and (iii) large polaron formation, coupled with the phonon glass characteristics, may explain the much reduced cooling rates of hot carriers in HOIPs (25, 34).

PHONON GLASS ELECTRON CRYSTAL

PGEC refers to a class of solids where charge transport is ballistic and phonon transport is diffusive. This ensures high electric conductivity but low thermal conductivity, the most desirable combination for efficient thermoelectrics. Since the initial proposal by Slack in 1995 (35), the PGEC concept has guided much research in thermoelectrics (32, 3639). A PGEC typically has large electronic bandwidth for efficient charge transport, whereas the phonon bands are disrupted by efficient scattering with local vibrations. At the limit of phonon disorder when the mean free path for phonon scattering is less than half of the phonon wavelength, we can essentially view nuclear motions in the solid as a collection of local vibrations (32, 39), as is likely the case in HOIPs (31, 40, 41). Thermal transport occurs diffusively from the coupling of these local vibrations to their neighbors via damped oscillations.

The search for PGECs over the past 20 years has focused on semiconductors consisting of crystalline frameworks that determine their electronic band structures, but with pores enclosing weakly bound atoms or ions. A representative PGEC is the so-called type I clathrate compound, whose crystal structure consists of networked cages of tetrakaidecahedrons and dodecahedrons (Fig. 2A), each housing a group 1 or 2 guest atom “rattling” in the cage, as shown in Fig. 2 for Ba8Ga16Ge30 (32). The motion of a rattling ion with respect to the cage with opposite charge can be represented by a varying dipole moment (Fig. 2B). The rattling in the nanocages introduces efficient scattering for the phonons, particularly acoustic phonons that are mainly responsible for thermal transport.

Fig. 2 Structure of a PGEC, Ba8Ga16Ge30.

(A) Two tetrakaidecahedron cages along the z direction consisting of anions and guest Ba2+ cation rattling against the anionic cage. (B) The cage (outer circles) and the symmetry-broken off-center guest atom compose an effective electric dipole moment (thick arrows), which is the vector sum of each dipole (thin arrows). Reprinted figure with permission from Takabatake et al. (32). Copyright 2014 by the American Physical Society.

It is clear from the above definition that a lead halide perovskite crystal belongs to the PGEC material family. Here, the PbX3 sublattice forms the crystalline framework, whereas the A+ cations “rattle” and/or “rotate” in the cuboctahedral cages (Fig. 1). The “phonon glass” character has come from various spectroscopic studies. Early NMR measurement on CH3NH3PbX3 crystals revealed rapid rotational motion of CH3NH3+ cations within the perovskite framework (15). The liquid-like behavior in the tetragonal and cubic phases is particularly apparent from the temperature-dependent dielectric function, which can be well described by the Kirkwood-Fröhlich equation originally developed for polar liquids (16). More recently, direct probe of phonon activities by neutron scattering (17, 19, 20), x-ray scattering (18, 21, 26), far-IR (22), terahertz (THz) (23), Raman (17, 23, 24), two-dimensional IR (42), and time-resolved optical Kerr effect (TR-OKE) (25, 43) spectroscopies all revealed dynamic disorder, large anharmonicity, and overdamped vibrational motions. This dynamic behavior has been confirmed in molecular dynamics simulations (31, 4446). Here, we show two examples in time and frequency domains, respectively.

As the first example, Fig. 3 compares TR-OKE responses from three single-crystal lead halide perovskites—MAPbBr3, FAPbBr3, and their all-inorganic counterpart, CsPbBr3—at room temperature. Note that at 295 K, both MAPbBr3 and FAPbBr3 are in the most disordered cubic phase, whereas CsPbBr3 is in the most crystalline orthorhombic phase. In a TR-OKE experiment (47, 48), shown schematically in the inset of Fig. 3A, a polarized laser pulse (pump) induces anisotropy in the refractive index and, thus, transient birefringence in the sample. After a controlled time delay, a probe laser pulse measures the decay in transient birefringence based on polarization rotation. This technique has been used in the past to probe liquids, particularly diffusive reorientation dynamics on picosecond time scales (47, 48). In a crystalline solid, the TR-OKE transient is typically dominated by the ultrafast and nearly instantaneous electronic polarization, plus weaker coherent oscillations attributed to impulsively stimulated Raman excitation of phonons (49). The TR-OKE responses in Fig. 3 are more liquid-like than crystalline solid–like. Whereas the transient from CsPbBr3 is dominated by the electronic response (laser cross-correlation) with a tail on the subpicosecond time scale, the transients from MAPbBr3 and FAPbBr3 feature broad and picosecond responses typical of diffusive motions in liquids. Note that, in the orthorhombic phase of CsPbBr3, the crystalline behavior is revealed by the presence of coherent oscillation in the TR-OKE signal, which is particularly enhanced when the excitation photon energy moves closer to the bandgap (43).

Fig. 3 TR-OKE probes liquid-like motions in lead halide perovskites.

TR-OKE transients from single-crystal (A) CsPbBr3, (B) MAPbBr3, and (C) FAPbBr3. The pump-probe cross-correlation is depicted as gray [full width at half-maximum (FWHM), 70 fs]. The inset in (A) illustrates the experimental setup. From Zhu et al. (25). Reprinted with permission from AAAS.

The liquid-like behavior is also evident in the frequency domain, as shown by low-frequency Raman spectra of Yaffe et al. (24) for CH3NH3PbBr3 and CsPbBr3 crystals across the two phase transitions with increasing temperature (Fig. 4). Although the low-temperature orthorhombic phases (blue) of both crystals feature well-resolved phonon modes (≤100 cm−1), mainly of the lead bromide sublattice, the spectra become very diffuse and are dominated by a central peak in the tetragonal (green) and cubic (red) phases, indicating anharmonic and local polar fluctuations over broad time scales (24). Note that the phase transition temperatures of CsPbBr3 are much higher than those of CH3NH3PbBr3; at room temperature, the former is in the low-temperature orthorhombic phase, whereas the latter is in the high-temperature cubic phase.

Fig. 4 Low-frequency Raman spectra of hybrid and inorganic lead halide perovskite crystals.

MAPbBr3 (A) and CsPbBr3 (B) in the orthorhombic phase (blue), tetragonal phase (green), and cubic phase (red). Note that the phase transition temperatures for CsPbBr3 are much higher than those for MAPbBr3. The central peak characteristic of dynamic disordered, highly anharmonic, and dissipative vibrations grows with temperature in both materials. a.u., arbitrary units. Reprinted figure with permission from Yaffe et al. (24). Copyright 2017 by the American Physical Society.

The phonon glass characteristics, along with the respectable charge carrier mobilities reported to date, suggest that lead halide perovskites are promising candidates for thermoelectric materials (31, 40, 41, 5052). Experimental measurements (5356) and dynamics simulations (31, 41) revealed that thermal conductivities in HOIPs and their all-inorganic counterparts range from 10−1 to 100 W m−1 K−1, which are one to two orders of magnitude lower than those in conventional inorganic semiconductors but are rather similar to those of glasses or liquids (57). Kovalsky et al. (54) measured the thermal conductivity of MAPbI3 that is a factor of 1.5 lower than that of CsPbI3. The Debye model (that is, phonon gas model) effectively describes the thermal conductivity of both perovskites, consistent with the phonon glass characters, including the greater phonon disorder in MAPbI3 than that in CsPbI3. This phonon glass character and slow acoustic phonon transport may also contribute to the slow cooling of hot carriers, as discussed later.

DYNAMIC SCREENING AND LARGE POLARON FORMATION

The PGEC “crystal-liquid” duality shown in lead halide perovskites is likely critical to their success as optoelectronic materials through dynamic screening and large polaron formation. Because lead halide perovskite single crystals and thin films are typically grown from solutions at room temperature, a high density of defects is inevitable. However, these materials show exceptional defect tolerance, which leads to low charge carrier trapping and low nonradiative recombination rates (58) that are responsible for long carrier lifetimes and diffusion lengths. Zhu and Podzorov (33) posited that the key to understanding this lies in the realization that charge carriers in lead halide perovskites are not free carriers but are large polarons.

A polaron is an excess electron or hole dressed by polarization of nuclear coordinates in a crystalline lattice and is ubiquitous to polar and polarizable solids (5963). For lead halide perovskites, we must consider charge formation and transport within the soft and ionic lattice with crystal-liquid duality. Given the exceptionally high polarizability of a soft lattice, a polaron must form for an excess charge. Two fundamental interaction potentials are responsible for polaron formation (63). The first is the long-range Coulomb potential (VLR) between an excess charge and lattice ions, and the second is the short-range deformation potential (VSR) that relates electronic energy to strain. If VSR dominates, the result is a small polaron, with coherence length (Lcoh) smaller than the unit cell dimension (a). When VLR becomes more significant, Lcoh is larger than a and the result is a large polaron. A large polaron is delocalized over multiple unit cells and its transport is coherent and band-like, with carrier mobility (μ) decreasing with increasing temperature (T), that is, dμ/dT < 0. In contrast, a small polaron is localized to a unit cell and its transport occurs via thermally activated hopping, that is, dμ/dT > 0.

The polarons in lead halide perovskites must be large polarons, because both transport and spectroscopic measurements showed dμ/dT < 0 in broad temperature windows. Figure 5 shows the temperature-dependent charge carrier mobility obtained from Hall effect measurement on single-crystal CH3NH3PbBr3 (64). In both tetragonal and cubic phases, the observed temperature dependences (dμ/dT < 0) establish coherent transport, but the different scaling laws, μ ∝ T−1.5 and μ ∝ T−0.5, may suggest the dominance of different scattering mechanisms in the cubic and tetragonal phases, respectively (64). Similar scaling laws have been found from spectroscopic measurements on CH3NH3PbI3 thin films (6568).

Fig. 5 Temperature dependence of Hall mobility in single-crystal MAPbBr3.

The fits to μHall (cm2 V−1 s−1) ∝ T (K)−γ give γ = 0.5 in the tetragonal phase and γ = 1.5 in the cubic phase. From Yi et al. (64). Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

A direct consequence of large polaron formation in lead halide perovskites is the efficient screening of the Coulomb potential. To understand the charge carrier properties, we consider the dielectric functions. Figure 6 shows the calculated dielectric functions for both MAPbBr3 and CsPbBr3 in the terahertz range (43). As the frequency decreases from >5 to ~1 THz, we see increases in the real part (ε1) of the dielectric function by approximately one order of magnitude due to the activation of PbBr3 longitudinal optical (LO) phonons. The positions of LO phonons are shown by the peaks in Im[−1/ε]. The values of ε1 are further increased in HOIPs by another factor of 2 as the frequency decreases to ~0.1 THz, due to the rotational motion of the dipolar methylammonium cations (69, 70). The over one order of magnitude increase of ε1 in lead halide perovskites is in stark contrast to what is known in conventional semiconductors, such as GaAs, where the corresponding increase in ε1 is only ~20% (57). The large increase in ε1 over the terahertz frequency range in HOIPs results in efficient screening of a nascent charge carrier on the subpicosecond time scale. Hence, for the resulting large polaron, all events requiring the Coulomb potential, such as scattering of a charge carrier with charged defects or other charge carriers, are inhibited. Electron-phonon coupling is also reflected in transport as localized shallow state (7174), in spectroscopy by exciplex emission and polaron–to–free carrier transition in IR absorption (75), and macroscopically by photostriction (76). The coupling of methylammonium cation rotors to charge carriers is suggested by Gong et al. (77), who observed decreasing carrier lifetime with increasing moment of inertia when the methylammonium cation is deuterated.

Fig. 6 Calculated dielectric functions of MAPbBr3 (blue) and CsPbBr3 (red).

The solid and dashed curves show the real, Re[ε] (left), and imaginary, Im[−1/ε] (right), parts, respectively. From/adapted from Miyata et al. (43). Copyright The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC) http://creativecommons.org/licenses/by-nc/4.0/.

Miyata et al. (43) provided a direct time-domain view of large polaron formation in single-crystal MAPbBr3 and CsPbBr3 using TR-OKE, with excitation photon energy above bandgap for charge injection (Fig. 7). On the basis of the initial time constants for polaron formation, τpol = 0.3 ps and 0.7 ps in MAPbBr3 and CsPbBr3, respectively, along with first-principles calculations, these authors concluded that large polarons form predominantly from the deformation of the PbBr3 frameworks, irrespective of the cation type. Although there is more liquid-like disorder in the cubic MAPbBr3 than in the orthorhombic CsPbBr3, at room temperature, the intrinsic softness and polarizability of the PbX3 framework structure enables easy formation of large polarons upon charge injection in both HOIPs and their all-inorganic counterparts. HOIPs and their all-inorganic counterparts show similarly exceptional charge carrier properties for band edge carriers and solar cell performances (7883).

Fig. 7 Phonon dynamics triggered by photo-carrier injection as a function of pump-probe delay observed by TR-OKE with across-gap excitation.

The TR-OKE responses for MAPbBr3 (red circles) and CsPbBr3 (blue circles) were obtained with excitation photon energies of 2.30 and 2.38 eV, respectively. The lines are double-exponential fits convoluted with a Gaussian function, which describes cross-correlation between the pump and the probe pulse (gray; FWHM, 70 fs). From/adapted from Miyata et al. (43) Copyright The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC) http://creativecommons.org/licenses/by-nc/4.0/.

LONG-LIVED HOT CARRIERS

One of the most striking observations in HOIPs is long-lived energetic carriers at low excitation densities (25, 34). In a conventional polar semiconductor, excess electronic energy above the LO phonon energy is lost on ultrafast time scales of 100s fs due to the ubiquitous Coulomb potential, which governs the scattering of electrons with polar LO phonons (84, 85). In GaAs, the electronic cooling rate 〈dE/dt〉 is of the order of 1 eV/ps for hot electrons with mean energy 〈dE*〉 > ELO, where ELO (=36 meV) is the LO phonon energy; this cooling process is slowed down by three to four orders of magnitude when 〈dE*〉 falls below ELO and scattering with acoustic phonons due to the deformation potential becomes dominant (84, 85).

In contrast to GaAs, hot carriers with excess energy above the LO phonon energy are much longer-lived in HOIPs (25, 34). Using time-resolved two-photon photoemission (TR-2PPE) spectroscopy and time-resolved photoluminescence (TR-PL), Zhu and co-workers found long-lived hot carriers in HOIPs at room temperature. The excess electronic energy of the order of 100 meV, which is much higher than ELO, cools down at rates typical of acoustic phonon scattering, as shown in Fig. 8 for polycrystalline MAPbI3 and single-crystal MAPbBr3 (25, 34). A comparison of HOIP single crystals with their all-inorganic counterpart at room temperature showed long-lived hot PL in MAPbBr3 and FAPbBr3, but not in CsPbBr3 (25). This correlates well with the more liquid-like responses in MAPbBr3 and FAPbBr3 than in CsPbBr3, as observed in TR-OKE measurements in Fig. 3. TR-2PPE measurement suggests a large polaron formation time of ~0.3 ps in MAPbI3 (34), in agreement with the TR-OKE measurement for MAPbBr3 (Fig. 7); this time scale is competitive with the initial LO phonon cooling time, leading to the partial retention of excess electronic energy. Although large polarons form in both MAPbBr3 and CsPbBr3 (43), the rate of formation is not competitive with phonon cooling in the latter. More recently, the presence of long-lived hot carriers has also been observed by Guo et al. (86) in MAPbI3 thin films using transient absorption microscopy.

Fig. 8 Slow cooling of hot carriers in HOIPs.

The data points are mean hot carrier energies from TR-2PPE of MAPbI3 thin films (red) (34) and TR-PL of MAPbBr3 single crystals (blue) (25). In the latter, a background offset due to intrinsic broadening in PL spectra has been subtracted. The excess excitation energy (above bandgap) is 1.0 eV for MAPbI3 and 0.8 eV for MAPbBr3.

Monahan et al. (87) reported the observation of long-lived “warm” carriers (excess energy ~70 meV) on the 10-ps time scale in two-dimensional electronic spectroscopy of MAPbI3. These authors showed strong coupling of charge carriers to a 0.9-THz phonon mode attributed to PbI3 lattice distortions (87), in excellent agreement with conclusions of Miyata et al. (43) on MAPbBr3 and CsPbBr3 from TR-OKE measurements. Monahan et al. (87) also observed long-lived coherence on the 10-ps time scale in the 0.9-THz phonon mode. It is possible that the large polaron formed from electron coupling to the PbI3 LO phonons further isolates the quasi-particle from the phonon glass environment, leading to long-lived coherence in the 0.9-THz mode. This interpretation finds support in a recent far-IR spectroscopy study of Ivanovska et al. (88), who observed very narrow absorption features in the terahertz region in photoexcited MAPbI3. Further support for this idea can be found in the succinct analysis of Frost et al. (89), who argued that the transfer of phonon energy from the subset of optical phonons that are in quasi-equilibrium with the hot carrier to the acoustic phonon bath is inefficient due to the low thermal conductivity, a result of the phonon glass character. This analysis also explains the order-of-magnitude higher hot polaron cooling rate in CsPbBr3 than in MAPbBr3.

The slowed cooling of energetic carriers due to large polaron formation may be mechanistically different from the conventional phonon bottleneck effect. The latter occurs at high excitation densities when efficient scattering between hot carriers and LO phonons occurs, but the hot LO phonons are not cooled down by acoustic phonons at sufficiently high rate, that is, the bottleneck effect (90, 91). As a result, the hot phonons can reheat the electrons/holes. The phonon bottleneck effect has been observed in MAPbI3 thin films at high excitation densities (≥1018/cm3) (9294). A marked difference between the conventional phonon bottleneck and the large polaron mechanism is seen in the excitation density dependences. In the conventional phonon bottleneck at ρex ≥ 1018/cm3 for MAPbI3, the excess electronic energy at a given time increases with increasing ρex (9294). In contrast, the large polaron mechanism is most important at low excitation densities (ρex ≤ 1018/cm3) and the excess electronic energy decreases with increasing ρex (Fig. 9) (34). With increasing excitation density, the mutual repulsive interaction destabilizes the large polarons (63); this is essentially a depolarization effect, leading to the reduction in the magnitude of the Coulomb screening and/or enhancement of the coupling of hot polarons to the phonon glass environment. As pointed out by Frost et al. (89), the transition density of ρex ~ 1018/cm3 is essentially the critical Mott density when the large polarons start to overlap spatially.

Fig. 9 Excitation density dependence of excess electronic energy in MAPbI3.

The data points are from TR-PL in the initial 0- to 20-ps time window (time resolution) of a vapor-deposited MAPbI3 thin film. PL at three excitation photon energies, hν = 3.06 eV (blue), 2.48 eV (green), and 2.15 eV (orange). The gray curve is an exponential fit to the data points. The y axis is excess electronic energy, referenced to the asymptotic value at long times (~0.5 ns). Reprinted (adapted) with permission from Niesner et al. (34). Copyright (2016) American Chemical Society.

The long-lived hot carriers in HOIPs may provide two potential advantages in solar cells. The first is the realization of the so-called hot carrier solar cell, with a theoretical power conversion efficiency (η) as high as 66% (9597). For comparison, a conventional solar cell harvests carriers only at the band edges and the loss of excess electronic energy (from the absorption of above-bandgap photons) is partially responsible for the Shockley-Queisser limit of η ~ 33% (98). On the basis of the partial retention of hot carrier energy, Niesner et al. (34) estimated an upper limit of η ~ 38% from MAPbI3 when hot carriers are harvested. The second advantage results from the excess kinetic energy. A portion of carriers generated from the absorption of sunlight are hot polarons. The higher kinetic energy of hot polarons may allow for easier transport across energetic barriers at domain boundaries or interfaces, leading to a higher photocurrent seen in HOIP solar cells.

DEFECT TOLERANCE

Efficient screening and large polaron formation in lead halide perovskites are key to the most significant advantage of lead halide perovskite as a semiconductor, that is, defect tolerance (99102). The defect tolerance leads to long carrier lifetimes and diffusion lengths responsible for their exceptional performance in solar cells and light-emitting devices (15, 103106). Quantitative measurements of PL intensity (79) or photoconductivity (58) as a function of excitation density revealed low first-order nonradiative recombination rates. The latter is illustrated by the scaling of photoconductivity (σ) with excitation density (G, proportional to incident photon flux, F), σ ∝ Gα or Fα, in the four-probe measurement of Chen et al. (58) in either vapor-grown polycrystalline MAPbI3 thin film or single-crystal MAPbBr3 (Fig. 10). For excitation density varying over three orders of magnitude, the data can be well described by α = 1/2 expected for radiative recombination, as opposed to α = 1 for trap-mediated nonradiative recombination. The low nonradiative recombination rate exists in both HOIPs and their all-inorganic counterparts; for band edge carriers, the different large polaron formation rates on the picosecond time scales are not expected to affect the properties of band edge carriers with nanosecond to microsecond lifetimes (79).

Fig. 10 Excitation density–dependent photoconductivity shows little charge carrier trapping.

(A) Photoconductivity measured in a vapor-grown 100-nm-thick polycrystalline CH3NH3PbI3 film as a function of absorbed photon density. (B) Photoconductivity measured as a function of incident photon flux in a macroscopic single-crystal CH3NH3PbBr3 with a (dark) Hall mobility μHall ≈ 60 cm2 V−1 s−1 (for holes). Dashed lines are the power law fits with exponent α = 1/2: σ ∝ Gα or Fα. Reprinted with permission from Chen et al. (58). Copyright (2016) Nature Commun.

We can understand the defect tolerance from two perspectives. From a carrier perspective, a free band carrier is rapidly protected on the pico- or subpicosecond time scale to form a large polaron (43), whose screened Coulomb potential drastically reduces its scattering with a charged defect or a trapped charge of opposite sign on an originally neutral defect. The latter scenario means that the well-known Shockley-Read-Hall (SRH) mechanism (107, 108) for nonradiative recombination is diminished. From a defect perspective, the high polarizability in the soft and ionic lattice, including ion mobility on long time scales, can lead to the sufficient screening and compensation of charged defects in a lead halide perovskite lattice (69, 109111). As a result, the traps are energetically shallow and electrostatically screened (100, 102), thus diminishing the role of these traps in SRH recombination.

The conventional wisdom to reduce defects in semiconductor technology is to achieve perfection, as exemplified by single-crystal silicon with remarkably low defect densities. However, this requirement cannot be met by emerging semiconductors made from molecular, hybrid, or nanocrystalline materials, where room temperature and solution synthesis/processing conditions inevitably result in a high density of defects that are detrimental to their performance. The exceptional defect tolerance in lead halide perovskites suggests that dynamic screening in intrinsically “soft” and polarizable materials provides a design principle to achieve defect tolerance, thus turning imperfection to perfection.

LARGE POLARON TRANSPORT

The “electron crystal” nature of lead halide perovskites is established by the band-like transport of charge carriers, particularly with charge mobility (μ) scaling with temperature (T) by μ ∝ T−1.5, as determined directly by Hall effect measurements for MAPbBr3 in the cubic phase (64), as well as by microwave, gigahertz, and terahertz spectroscopies on polycrystalline MAPbI3 (6568). The decreasing mobility with temperature (dμ/dT < 0) unambiguously establishes coherent transport, as expected from large polarons, in stark contrast to thermally activated transport of small polarons (63).

For lead halide perovskites, the valence and conduction bands are determined mainly by the PbX3 sublattice, with the A+ cation sublattice modulating the band structure electrostatically and through electron-phonon coupling. Electronic structure calculations predict large valence and conduction band dispersions that yield low electron/hole masses of 0.1 to 0.15 me (112114). This low band mass is confirmed by magneto optical absorption measurement, which gives the reduced e-h (exciton) mass from the Landau levels (115). Whereas the experimentally determined μ ∝ T−1.5 scaling law agrees with prediction from acoustic phonon scattering in nonpolar semiconductors (116), analysis based on acoustic phonon scattering yields charge carrier mobilities (μ) of the order of 103 cm2 V−1 s−1 (51, 117119), only slightly lower than those predicted from band masses and still more than one order of magnitude higher than experimental values.

It has been suggested that scattering with optical phonons is more important than that with acoustic phonons in limiting mobility in HOIPs (120, 121). The scattering of charge carriers with optical phonons has been traditionally treated within the polaron theory of Fröhlich (61), which was subsequently extended by Feynman (62). Here, a unit-less parameter (αe-ph) is introduced to describe the strength of electron LO phonon couplingEmbedded Image(1)where ε and ε0 are the optical and static dielectric constants, respectively; ke is the Coulomb’s law constant; Embedded Image is the Planck’s constant; ωLO is the LO phonon frequency; and m* is the band mass of electron or hole. To the first-order approximation, the renormalized carrier mass, that is, polaron mass (mp), is given byEmbedded Image(2)

On the basis of the mean LO phonon frequencies and the dielectric constants for HOIPs, several groups (22, 45, 58) estimated the electron-phonon coupling constants of αe-ph = 1.7 to 2.4, which corresponds to intermediate coupling.

From Eq. 2, we obtain mp in the range of 1.3 to 1.4 m*. The corresponding polaron radii, Rp = h½/(2cmLO)½, are in the range of 3 to 5 nm for HOIPs (22); thus, the diameter of the large polaron is ~10× the unit cell dimension. We estimate the polaron binding energy from the polaron radius to be of the order of 10 to 100 meV, depending on the dielectric constant used (see Fig. 6). The large polarons discussed here are in contrast to the tightly bound small polarons from the computational study of Neukirch et al. (122). These small polarons are likely minority trapped charges, not the charge carriers responsible for transport.

The LO phonons are highly populated in HOIPs at room temperature because of their low frequencies (22, 43). Under this condition, it is appropriate to use the general equations of Feynman-Ōsaka (62, 123) to calculate electron and hole polaron mobilities. Miyata et al. (43) obtained μe = 149.8 cm2 V−1 s−1 and μh = 79.2 cm2 V−1 s−1 in CH3NH3PbBr3; the corresponding values in CsPbBr3 are μe = 48.2 cm2 V−1 s−1 and μh = 41.3 cm2 V−1 s−1. Similarly, Sendner et al. (22) calculated average electron-hole mobilities of μ = 50, 150, and 200 ± 30 cm2 V−1 s−1 for MAPbCl3, MAPbBr3, and MAPbI3, respectively, and Frost (124) calculated electron and hole polaron mobilities in MAPbI3 of 136 and 94 cm2 V−1 s−1, respectively. The calculated mobility values for HOIPs are slightly higher than those reported from experimental studies. For single-crystal MAPbBr3 and MAPbI3 at room temperature, the reported μ values are 10 to 115 cm2 V−1 s−1 and 24 to 164 cm2 V−1 s−1, respectively (8, 9, 58, 64, 125). Whereas a large polaron is formed mainly from the deformation of the PbX3 sublattice, a moving large polaron is further slowed down from dielectric loss to the disordered vibrations at lower frequencies attributed to cation motions. In HOIPs, this additional dielectric loss channel occurs at ~0.1 THz, where the rotational motions of the dipolar MA+ cations are activated, leading to another step increase by ~2× in ε1 (70). This loss process is called “dielectric drag” for ion movement in polar liquids (126). In related work, Anusca et al. (70) suggested that the two-tiered dielectric responses led to a “hyperpolaron” as a result of the additional screening of a Fröhlich polaron by the liquid-like MA+ cation motions.

RADIATIVE RECOMBINATION

The bimolecular radiative recombination rate constants are of the order k2 = 10−10 cm3 s−1 in both HOIPs and their all-inorganic counterparts (58, 79, 127). There have been arguments that these rate constants are about five orders of magnitude lower than that predicted from the Langevin model. The Langevin model (128) is typically used to describe bimolecular recombination in low-mobility and disordered systems (129), where the recombination rate can be calculated from the Coulomb capture radius (rc), that is, the e-h distance for which the Coulomb potential is equal to the thermal energy. However, the Langevin model does not actually apply to lead halide perovskites because of the band-like transport with modest (~10 to 100 cm2 V−1 s−1), not low (<1 cm2 V−1 s−1), mobilities.

The perceived low values in radiative recombination rate constants that are partially responsible for long carrier lifetime and diffusion length have also led to discussions on the indirect character of bandgap in HOIPs. A particular argument attributes the indirect bandgap character to large Rashba splitting (130133). Because of strong spin-orbital coupling in the presence of heavy atoms, the conduction band and the valence band may be spin-polarized. In the presence of local electric field resulting from electrostatic ordering (110, 134) and static or dynamic symmetry breaking (130132), electrons with different spins (ms = ±1/2) experience opposite forces from electromagnetic interaction. This leads to the splitting of conduction band and (to a lesser extent) the valence band in momentum space, thus introducing indirect bandgap character. Experimentally, the indirect bandgap is believed to lie at 47 to 75 meV below the direct bandgap in MAPbI3, as obtained from temperature-dependent PL (65, 135) and pressure-dependent absorption and PL (136). Numerical simulations also addressed the disparate roles of direct and indirect bandgaps in absorption and emission (137).

Here, we present an alternative mechanism to explain the indirect bandgap character. As argued by Chen et al. (58), a model more appropriate than the Langevin model to describe radiative recombination in lead halide perovskites is the van Roosbroeck–Shockley theory (138), which obtains the radiative recombination rate from detailed balance under thermal equilibrium in the dark (138, 139). The van Roosbroeck–Shockley theory gives k2 only approximately one order of magnitude higher than experimental values; this difference can be well accounted for by polaron formation (58). The equilibrium carrier concentration is increased upon polaron formation by the stabilization of a charge carrier and the increased density of states resulting from deviation from parabolic dispersion. These effects decrease k2 from the van Roosbroeck–Shockley theory by approximately one order of magnitude, leading to good agreement with experimental values (58). For a polaron stabilization energy within 2 to 3 kT at room temperature, there is a dynamic equilibrium between large polarons and free carriers. The electron and hole large polarons are expected to be located in spatially separate regions, due to their opposite effects on the bending of Pb-X-Pb (43), as well as their opposite effects in dipolar polarization (58, 110, 134); in this sense, the indirect character of the bandgap is in real space, not in the momentum space. As a result, the radiative recombination rate from the energetically stabilized large polarons is expected to be much lower than that from free band carriers. This can give rise to thermally activated PL (65, 135). The lower transition strength for the radiative recombination from the e-h polaron states can also appear in absorption due to electron-phonon coupling in the absorption process (136, 140). Besides the large polaron–free carrier equilibrium, there is an equilibrium between carriers and excitons with binding energies of the order of kT at room temperature (5, 10, 69, 141, 142) and the carriers and excitons may be located in spatially separate regions in a heterogeneous environment of a polycrystalline thin film (143). All these effects may contribute to the apparent thermal activation in PL measurements.

SUMMARY AND OUTLOOK

In this perspective, we attempt a unified view of the optoelectronic properties of lead halide perovskites from their unique crystal-liquid duality. We show that lead halide perovskites belong to the PGEC family due to the binary structure consisting of a cage-like PbX3 sublattice and a more weakly bound A+ cation sublattice. This PGEC character gives rise to the remarkable contrast in the transport properties of carrier and phonons. In particular, the thermal transport in lead halide perovskites, particularly HOIPs, is diffusive because of the liquid-like dynamic disorder in the phonon manifold, whereas charge transport is ballistic and band-like in the PbX3 sublattice. The partial localization of a band carrier in the PbX3 phonon environment leads to the formation of a large polaron, whose screened Coulomb potential reduces its scattering with charged defects and other charge carriers. The effective Coulomb screening can be found in the dielectric function, which, in HOIPs, increases by more than an order of magnitude as frequency decreases from optical to the terahertz region. We can account for the modest mobility of the large polarons by the low frequencies of PbX3 LO phonons that are highly populated at room temperature and to additional dielectric loss to the disordered vibrations of the cation sublattice. Large polaron formation and the phonon glass character may also explain the long lifetimes of hot carriers in HOIPs. The understanding presented here suggests a new design principle for high performance and defect-tolerant semiconductors from PGEC type of structures with liquid-like dielectric functions.

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REFERENCES AND NOTES

Acknowledgments: X.-Y.Z. thanks V. Podzorov, S. Jin, and F. De Angelis for fruitful collaborations; X. Roy, M. Bonn, A. Walsh, and J. Frost for insightful discussions; and D. Niesner, X. Wu, H. Zhu, M. Tuan Trinh, J. Wang, and P. Joshi for the experimental work, which precipitated the ideas presented here. Funding: X.-Y.Z. acknowledges the NSF (grant DMR 1420634) (Materials Research Science and Engineering Center) for support during the writing of this perspective. The experimental works presented in Figs. 3 and 6 to 9 were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (grant ER46980). K.M. acknowledges the financial support of the Japan Society for the Promotion of Science. Author contributions: K.M., T.L.A., and X.-Y.Z. contributed to the reviewing of literature and writing of this perspective. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.
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