Charge Carriers in Hybrid Organic–Inorganic Lead Halide Perovskites

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Charge Carriers in Hybrid Organic−Inorganic Lead Halide Perovskites Might Be Protected as Large Polarons

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To understand how charge screening and localization can result in a quasi-particle “protected” from scattering and trapping, we briefly review polaron physics.11 For an excess charge in a deformable solid, electron−phonon coupling may be sufficiently strong to result in a self-trapped polaron, that is, the electron (hole) dressed by the nuclear polarization. For strong electron−phonon coupling typical of ionic and highly polar crystalline solids, there are two driving forces for polaron formation, the long-range Coulomb potential (VLR) between the excess electron (hole) and the ionic lattice and the shortrange deformation potential (VSR) due to the change in local bonding by the excess charge. The former is given by

he perovskite fever has been fueled not only by the exceptional performance of hybrid organic−inorganic lead halide perovskites (hereafter referred to as hybrid perovskites) in solar cells but also by the ease with which the material can be made. The latter has essentially removed any barrier for an unprecedented number of research laboratories to enter the race. A large number of mechanistic studies in the past 3 years have revealed a few apparent puzzles in hybrid perovskites: • Polycrystalline thin films show remarkably long carrier diffusion length (LD > 1 μm) and long lifetime (τ ≥ 1 μs) and relatively modest mobilities (μ ≈ 1−100 cm2 V−1 s−1),1−4 despite the unavoidable presence of defects/ traps. • Electron−hole recombination rates are as low as those of the best-known single-crystalline inorganic semiconductors and are five orders of magnitude lower than that predicted by the Langevin model,2 as verified by our Hall mobility measurement (unpublished reference). • Temperature dependence of carrier mobility (μ ∝ T −3/2 ) from microwave conductivity,4 THz spectrosco5,6 py, and our Hall mobility measurement (unpublished reference) suggests that charge transport is dominated by acoustic phonon scattering, rather than impurity scattering, even in disordered films and crystals. • The exceptionally low carrier scattering rate is also observed for hot carriers whose lifetimes are order(s) of magnitude longer than those in conventional inorganic semiconductors, as revealed by transient absorption7−10 and by our time-resolved two-photon photoemission measurements (unpublished reference). These puzzles seem to suggest that charge carriers in hybrid perovskites are somehow protected from scattering with defects, longitudinal optical (LO) phonons, and other carriers and are not susceptible to trapping. Before we propose the nature of this protection, let us first review the unique structure of these materials. The building blocks of a hybrid organic−inorganic lead halide perovskite structure are the Pb halide octahedra, PbX64− (X = I, Br, or Cl), that are corner-shared to form a 3D crystalline network with the stoichiometry of PbX3−. The voids in the inorganic crystalline network are filled (and charge-balanced) with organic cations (A+), most commonly CH3NH3+. There are two important consequences for such a hybrid crystalline structure: (1) the crystalline structure can be viewed as two interpenetrating sublattices, the inorganic PbX3− sublattice and the CH3NH3+ sublattice; and (2) both the valence and conduction bands relevant to charge transport are formed by the inorganic PbX3− sublattice. Thus, charge transport in hybrid perovskites occurs primarily within the PbX3− sublattice, with the organic CH3NH3+ sublattice serving as a medium that modulates the electrostatic landscape experienced by the charge carriers, leading to charge screening and localization. © XXXX American Chemical Society

⎡ 1 1 ⎤ e2 − V LR (r) = −⎢ ⎥ εr(0) ⎦ |r|ε0 ⎣ εr (∞)

(1)

where r is the vector between an electron and an ionic site, e is the electron charge, ε0 is the vacuum permittivity, and εr(0) and εr(∞) are the static and high-frequency dielectric constants, respectively. The difference between the two terms in the square brackets ensures that the fast electronic contribution to the polarizbility is eliminated from this expression, leaving behind only the nuclear contribution. This long-range Coulomb potential is insignificant when the two dielectric constants are similar, εr(0) ≈ εr(∞), as is the case for crystalline Si or GaAs, and becomes important for ionic solids where εr(0) can be more than twice the value of εr(∞). Depending on whether VSR or VLR dominates, the result is either a small or large polaron, respectively. In the former, the size of the polarization cloud, called the coherence length (Lcoh), is smaller than the unit cell dimension (a), Lcoh < a; in the latter, Lcoh > a. The difference between a small and a large polaron is not just in their size but more importantly in their transport. The motion of small polarons occurs incoherently, with a low mobility (μ ≪ 1 cm2 V−1 s−1) that increases with temperature (∂μ/∂T > 0) due to thermally activated hopping from one localized site to another. In contrast, the transport of a large polaron is coherent, and the mobility decreases with temperature (∂μ/∂T < 0); in this regard, large polaron transport resembles the coherent transport of a free electron (hole) in the conduction band. Compared to a free band electron, a fascinating consequence of the heavier mass of a large polaron is the much-reduced scattering with phonons or defects. The reduced scattering rate partially compensates for the mass of the large polaron to yield a sizable mobility (μ > 1 cm2 V−1 s−1).11 Keeping the basics of polaron physics in mind, we now attempt to rationalize the experimental observations outlined at the beginning. The methylammonium lead halide perovskite, MAPbX3, is characterized by a large difference between the static and the high-frequency dielectric constants, εr(0) ≫ εr(∞).12−14 Thus, the long-range Coulomb potential in eq 1

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DOI: 10.1021/acs.jpclett.5b02462 J. Phys. Chem. Lett. 2015, 6, 4758−4761

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The Journal of Physical Chemistry Letters should play an important role in MAPbX3, more so than that in other ionic solids,11 leading to large polaron formation. While phonon modes in both the PbX3− sublattice and the CH3NH3+ sublattice can localize a charge carrier, Raman spectroscopy and theoretical analysis clearly show that motion associated with CH3NH3+ is much faster.15 Indeed, many authors have suggested that motion of the organic cations, particularly the reorientation of CH3NH3+ and the associated dipole (see the green arrow in Figure 1),16 can stabilize and localize charge

recombination of two oppositely charged large polarons requires the removal of their “protective” shield, and this process slows down charge recombination. The removal of the polarization shields may be thermally activated. Interestingly, Savenije et al. reported an activation energy of 75 meV for radiative recombination in the CH3NH3PbI3 perovskite,6 but Herz and coworkers reported no thermal activation.5 • The coherent (band) transport is signified by the inverse temperature dependence of mobility, as observed in GHz and THz spectroscopies4−6 and in our Hall effect measurement (unpublished reference). Remarkably, the ∼T−3/2 temperature dependence of GHz−THz mobility4,6 or Hall effect mobility (unpublished reference) is expected only in defect-free semiconductors, where scattering by acoustic phonons dominates.26 Thus, the coherent transport of large polarons in hybrid perovskites is slowed down by acoustic phonon scattering, rather than impurity scattering. • The enhanced effective mass of the large polaron provides a “protective shield” against scattering with phonons (particularly high-energy LO phonons) not only for band edge carriers but also for hot carriers. In the latter case, reduced phonon scattering leads to slow hot carrier cooling. Measurements based on transient absorption spectroscopy have suggested the slow cooling of the initial hot carrier distribution (from the excess photon energy) in CH3NH3PbI3 perovskite with hot carrier lifetimes on the order of 102 fs.7−9,27 This time scale is consistent with the formation time of the large polaron, for example, due to the reorientation/wobbling of the CH3NH3+ cations.18,20 Beyond this initial and slowed cooling process, we found that the loss of remaining excess energy in the hot carriers after the first few hundred femtoseconds slowed by another 3 orders of magnitude (unpublished reference). In related work, Yang et al. reported slow cooling of hot carriers at high excitation densities (≥5 × 1017 cm−3).10 What is the ef fective mass of the large polarons in hybrid perovskites? Magneto-absorption measurement by Nicholas and co-workers28 revealed an effective excitonic reduced mass of ∼0.1me, where me is the bare electron mass, in CH3NH3PbI3 perovskite, as expected from the Mott−Wannier exciton of the weakly interacting electron and hole in the conduction and the valence band, respectively. However, given the extremely short lifetime of the exciton inferred in this work, such an absorption measurement provides information only on the nascent and bare electron−hole pair, before they are dressed by nuclear polarization. In carrier transport, we are interested in dressed electrons/holes after they form large polarons, that is, on the time scale longer than ∼102 fs. To our knowledge, there have been no reliable measurements of effective carrier mass from transport measurements. Here, we provide an estimate of the effective carrier mass (m*) in hybrid perovskites. Charge carrier mobility can be expressed either via a momentum relaxation time, τ, also called a scattering time (not to be confused with the carrier lifetime), or via a mean-free path, λ, as follows, μ = eτ /m* = eλ / 2kT ·m* , where kT is a thermal energy. Thus, we can express the effective mass through the mobility, mean-free path and kT as26

Figure 1. Structure of hybrid organic−inorganic lead halide perovskite. (Gray) Pb; (purple) halide; (green) CH3NH3+ molecular cation that carries a dipole moment (schematically shown by the arrow).

carriers in the hybrid perovskite lattice, with the polarization cloud spanning many unit cells.14,17,18 Molecular dynamics (MD) simulations have shown time scales as short as ∼0.1 ps for charge localization, with a stabilizing energy on the order of ∼0.1 eV.18,19 Two-dimensional infrared spectroscopy identifies two prominent CH3NH3+ motions, a fast (∼0.3 ps) wobbling motion and a slower (3 ps) rotational motion.20 The former is likely responsible for the ∼0.1 ps charge carrier localization seen in ab initio MD simulations.18 The collective motion of the CH3NH3+ cations has been proposed to screen the electron−hole Coulomb potential and favor the dissociation of excitons into free carriers,21 to localize the electron and the hole in spatially distinct locations to inhibit recombination17,18 and, similarly, to form ferroelectric-like domains.14 We hypothesize that the large polaron provides the necessary protection and can explain the four puzzles in hybrid perovskites outlined at the beginning of this Viewpoint. • The coherent transport, combined with a large effective mass that drastically reduces scattering, explains the long carrier diffusion length (>1 μm) and lifetime (≥1 μs), as well as the modest intrinsic charge carrier mobilities (μ < 100 cm2 V−1 s−1) inferred by many groups1−3,22,23 and quantitatively established in our direct Hall effect measurements to be 1−60 cm2 V−1 s−1 for polycrystalline and single-crystal samples (unpublished reference). • The second-order e−h recombination rate constant, γ, has been found to be in the range of 10−9−10−11 cm3 s−1 from a variety of measurements on hybrid perovskites.2,4,6,8,24,25 These values are approximately 5 orders of magnitude lower than that predicted by the Langevin model based on a simple Coulomb capture radius: γL = eμ/ε, where ε = εr· ε0 is the dielectric permittivity.2,4 The 4759

DOI: 10.1021/acs.jpclett.5b02462 J. Phys. Chem. Lett. 2015, 6, 4758−4761

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The Journal of Physical Chemistry Letters m* =

(eλ)2 2kTμ2

carrier properties attractive for solar cells or other optoelectronics. While our large polaron hypothesis may unlock the key puzzles that have emerged from an exceptionally large number of physical measurements over the past 3 years, it also raises many challenging questions. For instance, we do not know the effective mass or the coherence length of the large polaron. Quantitative determination of these microscopic properties is a challenging task. Unlike the free electron Drude model,26 where absorption scales inversely with the square of the light frequency, the large polaron features a low-frequency cutoff (three times the polaron binding energy) in the absorption spectrum.11,33 While time-resolved studies in the THz34 and IR35 spectral regions have shown absorption onsets, it is not clear if these are signatures of large polarons. In particular, our transient absorption measurement (unpublished reference) revealed an absorption onset at ∼0.23 eV, which coincides with ∼3× the activation energy (75 meV) in radiative recombination reported by Savenije et al.6 In addition to these experimental/theoretical challenges in the physics of hybrid perovskites, whether the concept of a large “bulk” Fröhlich polaron can be developed into a general design principle for hybrid semiconductors remains an open question.

(2)

Note that the mean-free path is related to the momentum relaxation time via thermal velocity, ν, as λ = τν ≈ τ 2kT /m* . The inverse temperature dependence of carrier mobility (∂μ/∂T < 0) observed in hybrid perovskites at around room temperature from spectroscopy4−6 and from Hall effect measurement (unpublished reference) is consistent with the coherent band-like transport of large polarons. For such a coherent transport to take place, the mean-free path of a charge carrier must be much longer than the lattice constant. We can assume that λ is at least ∼10 lattice constants, or ∼10 nm, as suggested by simulations.17,18 The experimental steadystate charge carrier mobility can be taken from our Hall measurements in single crystals (unpublished reference), μ ∼ 10−60 cm2 V−1 s−1. With these numbers, eq 2 leads to the effective mass at room temperature in the range of m* ≈ 10me−300me. This simple calculation shows that in hybrid perovskites, the charge carriers are indeed rather heavy, hundreds or thousands times heavier than the calculated single-particle band masses (∼0.1me).29−31 This estimate supports the notion of a heavy (large) polaron in these materials. The heavy large polarons may also explain the modest mobility in hybrid perovskites. As pointed out by Brenner et al. elsewhere in this issue,3 the charge carrier mobilities in hybrid organic−inorganic lead halide perovskites are surprisingly low, given the very small single-particle band mass (∼0.1me) and absence of significant scattering with defects and LO phonons. In a prototypical inorganic semiconductor with similar band mass, such as Si,26 the charge carrier mobilities are 1−2 orders of magnitude higher than those in hybrid perovskites. This difference can only be explained by the much heavier large polaron in hybrid perovskites. Large polarons are common to a broad range of polar solids, such as inorganic perovskites, ferroelectric materials, metal oxides, and ionic crystals,11 but they may possess unique properties in the hybrid organic−inorganic lead halide perovskites. As discussed earlier, the crystal structure of the hybrid perovskite consists of two interpenetrating sublattices, the inorganic PbX3− sublattice for carrier transport and the MA+ molecular sublattice more susceptible to geometric deformation, including the motion of the permanent dipole of MA+ in each inorganic cage. Thus, besides the conventional displacement polarization, hybrid perovskites may exhibit an additional polarization mechanism that locally screens the charge carrier via formation of a cloud of dipoles that align themselves with the Coulomb’s field. Such a spatially distinct polaron, with an electron/hole in the PbX3− sublattice and the polarization cloud in the MA+ sublattice, can be viewed as a bulk analogue of an interfacial Fröhlich polaron, which is formed at the interface between, for instance, a semiconductor and a polarizable dielectric material, a situation that is common for charge carriers in the accumulation channel of organic field effect transistors.32 A charge carrier moving in the semiconductor along the interface with a gate dielectric polarizes this dielectric, leading to a quasi-particle with an increased effective mass and a reduced mobility that is roughly inversely proportional to the dielectric permittivity of the gate insulator. Similarly, polarons can form at the internal “interfaces” of the two sublattices in hybrid perovskites, giving rise to the charge

X.-Y. Zhu*,† V. Podzorov*,‡ †

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Department of Chemistry, Columbia University, New York, New York 10027, United States ‡ Department of Physics and the Institute for Advanced Materials and Devices for Nanotechnology (IAMDN), Rutgers University, Piscataway, New Jersey 08854, United States

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (X.-Y.Z.). *E-mail: [email protected] (V.P.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS X.-Y.Z. acknowledges support by the U.S. Department of Energy, Office of Science - Basic Energy Sciences, Grant ER46980 for experimental research on perovskites. Partial support to X.-Y.Z. by the National Science Foundation, Grant DMR 1420634 (Materials Research Science and Engineering Center) during the writing of this Viewpoint is also acknowledged. V.P. acknowledges support by the Department of Physics and the Institute for Advanced Materials and Devices for Nanotechnology (IAMDN) at Rutgers University, as well as the National Science Foundation, Grant DMR 1506609 (Condensed Matter Physics). X.-Y.Z. and V.P. thank their group members at Columbia and Rutgers Universities for their excellent work that precipitated the ideas presented here and Profs. Louis Brus, David Cahen, Leeor Kronik, Anvar Zakhidov, and Yuri Gartstein, as well as Dr. Yaroslav Rodionov for fruitful discussions.

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