Excitonic Effects in Methylammonium Lead Halide Perovskites

Perspective Cite This: J. Phys. Chem. Lett. 2018, 9, 2595−2603

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Excitonic Effects in Methylammonium Lead Halide Perovskites Xihan Chen, Haipeng Lu, Ye Yang,* and Matthew C. Beard*

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Chemistry and Nanoscience Center, National Renewable Energy Laboratory, Golden, Colorado 80401, United States ABSTRACT: The exciton binding energy in methylammonium lead iodide (MAPbI3) is about 10 meV, around 1/3 of the available thermal energy (kBT ∼ 26 meV) at room temperature. Thus, exciton populations are not stable at room temperature at moderate photoexcited carrier densities. However, excitonic resonances dominate the absorption onset. Furthermore, these resonances determine the transient absorbance and transient reflectance spectra. The exciton binding energy is a reflection of the Coulomb interaction energy between photoexcited electrons and holes. As such, it serves as a marker for the strength of electron/hole interactions and impacts a variety of phenomena, such as, absorption, radiative recombination, and Auger recombination. In this Perspective, we discuss the role of excitons and excitonic resonances in the optical properties of lead-halide perovskite semiconductors. Finally, we discuss how the strong light−matter interactions induce an optical stark effect splitting the doubly spin degenerate ground exciton states and are easily observed at room temperature. reflecting an exciton binding energy that is only ∼10 meV, much smaller than the available thermal energy. However, excitonic effects play an important role in defining the optical properties, which will be discussed in the following order. (1) Exciton resonances dominate the bandedge absorption onset; (2) the transient absorption (TA) and transient reflection (TR) spectra are dominated by bleaching of the excitonic resonance; (3) the bi- and trimolecular carrier recombination exhibit dependence on the exciton binding energy because they are modulated by the same Coulomb interaction; (4) the bandedge exciton states can be entangled with photons via the optical stark effect; (5) exciton bleach dynamics under circularlypolarized excitation can follow the total angular momentum depolarization dynamics of carriers.

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ead-halide perovskites represent an intriguing class of solution-deposited semiconductors with great potential for optoelectronic applications. Recent interest in these systems stems from their performance in solar cells, with the highest certified power conversion efficiency of a polycrystalline leadhalide perovskite thin-film solar cell approaching 23%, exceeding that of solar cells fabricated from multicrystalline silicon and CdTe.1 Studies have attributed this success in solar cells to their photophysical properties such as strong light absorption, long charge-carrier diffusion lengths2−5 and low intrinsic surface recombination.6−12 Lead-halide perovskite systems have also shown promise in other optoelectronic applications such as lasers,13−16 light-emitting diodes,17−20 photodetectors21−24 and optospintronics.25−28 In contrast to the rapid progress in solar cells, the aforementioned optoelectronics are more or less still lingering at the stage of “proof-of-concept” and require deeper understanding of their optoelectronic properties. In this Perspective, we describe how excitonic effects modulate the opto-electronic behavior in the prototypical methylammonium lead iodide (MAPbI3) perovskite semiconductor. Absorption of photons in semiconductors lifts electrons out of the valence band into the conduction band, generating freecharge carriers, i.e., negatively charged electrons in the conduction band and positively charged holes in the valence band. Coulombic interactions between the photoexcited electrons and holes give rise to “excitonic effects” that modulate the optoelectronic behavior of semiconductors. It is important to note that the linear absorption strength of any particular optical resonance does not indicate the presence of excited states but rather the strength of the light-matter coupling at that resonance frequency.29 Excitons are fundamental quasi-particles that form bound electron−hole pairs due to the Coulomb interaction. At room temperature and under 1-sun illumination intensity, very few excitons exist in thin films of MAPbI3 © 2018 American Chemical Society

Coulombic interactions between the photoexcited electrons and holes give rise to “excitonic effects” that modulate the optoelectronic behavior of semiconductors. Excitons are generally ascribed to two categories, Frankel and Wannier. The former are tightly bound electron−hole pairs with small Bohr radii and usually exist in wide-gap semiconductors and organic materials with small dielectric constants, while the latter are weakly bound and mostly found in semiconductors with large dielectric constants. In the case of the lead-halide perovskite semiconductors, we only Received: February 16, 2018 Accepted: May 1, 2018 Published: May 1, 2018 2595

DOI: 10.1021/acs.jpclett.8b00526 J. Phys. Chem. Lett. 2018, 9, 2595−2603

The Journal of Physical Chemistry Letters

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consider Wannier excitons. Similar to any two-particle problem, the exciton Schrodinger equation can be solved by separating the motion of the electron−hole pair into center-of-mass motion and electron−hole relative motion.30,31 Considering the center-of-mass motion, the exciton is treated as a neutral particle freely moving inside the semiconductor, and the kinetic energy from this center-of-mass motion, ER, is given by

Eex = Eg + E R + Er ≈ Eg +

⎛1⎞ (ℏKc)2 − R ex ⎜ 2 ⎟ ⎝n ⎠ 2M

(3)

where Eg is the band gap energy. As n increases, the exciton energy approaches the continuum (Figure 1A, shaded area). The energy difference between the lowest energy exciton state and lowest conduction band continuum state is equal to Rex, also defined as the exciton binding energy. Note that these equations define the available exciton states and their energies but not the optical transitions. Optical transitions from the ground state to exciton states significantly modify the absorption spectrum, particularly below and near the bandedge. To conserve both energy and momentum, optical transitions to exciton states are only allowed when the photon and exciton dispersion curves (green arrow and blue parabolas, Figure 1A, respectively) intersect. It should be noted that the green arrow in Figure 1A exaggerates the photon momentum, and the real intersection must be much closer to the vertical axis, where Kc ≈ 0, and thus according to eq 3, the exciton resonances are only observed for photon energies below the bandgap energy (Figure 2A). Note that both exciton and free-carrier states (shaded region in Figure 1A), exist for Kc ≠ 0 but those states cannot be populated by light

(ℏKc)2 (1) 2M where M is the sum of electron and hole effective masses, and Kc is the sum of electron and hole momentum, resulting in a parabolic energy dispersion curve (blue curves, Figure 1a) in a ER =

Figure 1. Exciton absorption in direct band gap semiconductors. (A) Schematic diagram of bound (excitons) and ionized (free-carriers) electron−hole pair energy levels. The vertical and horizontal axes correspond to the energy (E) and the electron−hole center-of-mass momentum (Kc). The origin point (0,0) indicates the ground state with zero energy and zero momentum. The blue parabolic curves and the shaded region represent the kinetic energy bands of the exciton states and continuum free-carrier states, respectively. The green arrow indicates the optical transition for a photon with energy ℏω and momentum q. Since the momentum, q, of photon is very small only those states along the Kc = 0 axis are optically allowed producing the discrete exciton lines in the absorption spectrum. The gap between the lowest exciton state and bottom of continuum is defined as exciton binding energy, Rex. (B) Hypothetical bandedge absorption spectrum according to eq 4. The lowest three exciton states are resolved (with arbitrary broadening). The exciton states merge continuously into the continuum at the bandedge. The energy difference between the first exciton peak and free carrier absorption edge corresponds to the exciton binding energy. The red curve represents the free carrier absorption spectrum in the absence of excitonic effect. Lower plot is an expansion of the upper plot.

two-particle diagram. On the other hand, the relative electron− hole motion is determined by the Coulomb potential, which resembles the relative electron−proton motion in a hydrogen atom. Thus, the relative motion part (Wannier equation) is mathematically equivalent to the Schrodinger equation for the hydrogen atom, with quantized energy eigenvalues: ⎛1⎞ Er = −R ex ⎜ 2 ⎟ ⎝n ⎠

Figure 2. Steady-state and transient spectra of lead-halide perovskites. (A) Absorption coefficient as a function of energy for MAPbBr3 (bluetraces) and MAPbI3 films (red-traces). The band edge absorption for both samples are modeled by the Elliott equation (gray-traces). The cyan and green-traces correspond to the components arising from exciton and free-carrier resonances, respectively. (B) Typical transient absorption (TA) spectra of lead-halide perovskite thin films. The transient spectrum of MAPbI3 is truncated at 2.3 eV here. (C) Typical transient reflection (TR) spectra of the MAPbBr3 and MAPbI3 perovskite single crystals. The transient spectra are all recorded at 10 ps delay with low excitation intensities. This figure is adopted from ref 33 with perimission from American Chemical Society (Copyright 2015) and refs 6 and 8 with permission from Nature Publishing Group (Copyright 2017 and 2015).

(2)

where Rex is exciton Rydberg energy; n is the principle quantum number. The negative sign in eq 2 indicates that formation of an exciton bound state releases energy. The total energy of excitons is expressed as 2596



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Figure 3. (A) Comparison of bandedge absorption for GaAs (ref 39), MAPbI3 (Red), MAPbBr3 (Blue). (B) Binding energy of range of semiconductors versus the bandgap (data from ref 31) of the semiconductor. Blue squares are for MAPbI3 and MAPbBr3.

absorption. Photon energies that are greater than the bandgap only produce occupation of free-carrier states. Furthermore, the optical resonances require both angular momentum and parity to be conserved. To satisfy these requirements, only the transitions to exciton states with l = 0 (s-like states) are allowed for semiconductors with p-like conduction band and s-like valence band, such as II−VI and III−V semiconductors, as well as, the opposite (p-like conduction band and s-like valence band), as the case for lead-halide perovskite semiconductors.31 Applying these selection rules and taking into account absorption into transitions to free-carrier states, the bandedge absorption spectrum can be described by the Elliott equation:31,32

α(ℏω = Eg ) = αf ·

+ R ex ∑ n=1

⎤ 4π 2 ⎥ · δ ℏ ω − + ( E R / n ) g ex ⎥⎦ n3

ℏω − Eg

(5)

where αf is the absorption coefficient in the absence of Coulombic effects. Since αf varies as ℏω − Eg , the absorption coefficient mathematically approaches a constant value at the bandedge. Elliott’s formula (eq 4) can successfully deconvolve the bandedge absorption into exciton and continuum contributions (Figure 2A). Such an analysis has been done for methylammonium lead-bromide (MAPbBr3) and MAPbI3, including for single crystals and polycrystalline films. The exciton binding energies are determined to be ∼10 meV and ∼40 meV for MAPbI3 and MAPbBr3, respectively, with values that slightly vary from sample-to-sample.6,8,26,33−37 Compared with the serial exciton peaks expected in the ideal case (Figure 1B), only one exciton peak is resolved because of the relatively small exciton binding energy and large spectral broadening. The spectral-broadening for these samples is due in large part to phonon scattering, which is confirmed by temperature dependent absorption spectra. As the temperature is lowered, the 1S-exciton resonance and free-carrier absorption become spectrally isolated and the exciton resonance sharpens.34,38 To further illustrate the impact of the exitonic effect on the absorption spectrum, we compare MAPbI3 and MAPbBr3 to that of GaAs39 by dividing the absorption coefficient, α, by the respective exciton binding energy and then plotting versus the photon energy divided by the respective bandgap energy (Figure 3). Represented in this fashion, the intensity is proportional to the intraband matrix element, and we find that the lead-halide perovskite semiconductors exhibit similar behavior to that found in GaAs. The bandedge states for both GaAs and the lead halide perovskites arise from p and s atomic orbitals (although inverted from one another), therefore it is not surprising the intraband matrix elements are similar. As the photon energy increases, Elliott’s model deviates from the experimental data due to nonparabolicity of the bands, as well as, contributions from higher energy bands. The binding energies for the lead halide peroskites are similar to those found in III−V semiconductors (Figure 3B compares the binding enegy vs the bandgap for a variety of semiconductors).31 Again, we find that the dependence of the binding energy indicates similarity between the Pb-halide system and III−V semiconductors.

⎡ ⎛ π e πx ⎞ α(ℏω) = Pcv ⎢θ(ℏω − Eg ) ·⎜ ⎟ ⎢⎣ ⎝ sinh(πx) ⎠ ∞

1/2 2πR ex

(4)

where the frequency dependence of Pcv is approximated as a constant and related to the interband transition matrix element, ℏω is the photon energy, θ(ℏω − Eg) is the Heaviside step function, x is defined as R ex /(ℏω − Eg ) , and δ denotes a delta function. The first term describes the absorption of continuum states, and the second term is for the series of optically allowed exciton resonances. A prototypical bandedge absorption spectrum is sketched in Figure 1B by substituting arbitrary parameters into eq 4 and introducing arbitrary line broadening. In this hypothetical case, the first three exciton peaks (denoted as 1S, 2S, and 3S) are resolved below the continuum edge. However, all excitonic transitions contribute near the bandedge and even though their amplitudes decrease as n increases, the absorption over a small energy range reaches a constant and merges into the continuum at the bandedge. In the absence of excitonic effects, the free-carrier absorption spectrum would be proportional to the square root of energy (red curve, Figure 1B) and right at the bandgap energy the density of states goes to zero. However, this is not the case in the presence of Coulomb interactions and leads to a step-like band edge absorption. We can simplify the first term in eq 4 for photon energies at or near the bandedge to 2597



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decomposition and determine the exciton binding energy more accurately.33 Except under extremely high excitation intensity,37,46 the TA-bleach and TR antisymmetric peaks are linearly proportional to the photoinduced carrier density8,33 as phasespace filling exhibits a linear dependence on the carrier density.42,43,47−49 Therefore, the carrier dynamics in the bulk (TA) and near the surface (TR) can be monitored by measuring the kinetics of TA-bleach and TR antisymmetric peaks. Under low excitation intensity, monomolecular recombination dominates the carrier lifetime, and usually it occurs with assistance of defects which are generally assigned either to the bulk (point defects and grain boundaries), or exposed surfaces. The monomolecular recombination is most important when considering photovoltaic applications because PV devices operate under low excitation intensities such that the steady state carrier densities are in the range of 1015 to 1016 cm−3. The defect density is usually controlled by extrinsic factors, and in this sense the monomolecular rate constant (A-coefficient) should vary from film-to-film depending on how the samples are prepared. When considering the carrier lifetime in these low carrier concentration regimes, carrier diffusion must also be considered. In the case of thin films, diffusion of carriers to the surfaces followed by surface recombination can dominate the carrier lifetimes, and thus, surface recombination must be taken into consideration. Furthermore, when the excitation energy is varied, carriers can be generated close to or further from the surface and thereby modulate the measured carrier lifetimes. By considering surface and bulk recombination as well as diffusion of carriers to the surfaces, we find a general rule of thumb: if the measured carrier lifetime is shorter than ∼0.5 μs, then that likely indicates the carrier lifetimes are limited by surface recombination, while if the measured carrier lifetime is greater ∼0.5 μs, then that indicates carrier lifetimes are limited by bulk recombination.6

In phase-space, exciton states are built from a linear combination of conduction and valence band free-carrier states. An experimental complication with using eq 4 to extract the exciton binding energy occurs when the binding energy is small in comparison to the line width broadening (as in the case here for MAPbI3).35 In this case, the exciton line width merges with the continuum and the spectral decomposition presented above can be problematic. In this case, it is useful to use modulation spectroscopies that are able to further deconvolve the exciton and free-carrier contributions.40 Here we show that both transient absorption (TA) and transient reflectance (TR) can enable extraction of the binding energy because they modulate the excitonic line width and not the continuum. As detailed in our previous paper, the TA-bleach is dominated by reduction of the exciton resonance due to phase-space filling, rather than by bleaching of the continuum states (state-filling), under low and moderate excitation intensity.37 State-filling of exciton transitions bleaches the exciton resonance intensity due to the presence of excitons (as in the case of quantum dots41). However, as discussed above, excitons are not photoexcited when the photon energy exceeds the bandgap, and excitons in MAPbI3, even when they are created initially, should be thermally ionized into free carriers at room temperature.38 Nevertheless, the exciton resonance is clearly bleached even in the absence of actual excitons. This paradox can be explained by the phase-space filling phenomena.37 Briefly, in phase-space, exciton states are built from a linear combination of conduction and valence band free-carrier states. The exciton resonance strength includes the interband transition matrix element (Pcv, in eq 4). Thus, the presence of free-carriers can reduce the strength of interband transitions (state-filling) as well as the exciton resonance (phase-space filling) because of the exclusion principle.42−45 We find that the spectral width and peak-position of the TAbleach for both samples (Figure 2B) coincide with those of the exciton resonance, and not the continuum band, in the corresponding steady-state absorption spectra. Near the bandedge, bleaching of contiunnum states is offset by a photoinduced absorption that occurs due to bandgap renormalization. The presence of free-carriers can also reduce the exciton binding energy due to increased dielectric screening, and, thus, the exciton resonance can also be reduced by the presence of free-carriers due to the reduction in binding energy (see eq 4). However, this should be negligible for low excitation intensity.42,43 High excitation intensities can also introduce additional spectral broadening, which is not important for low excitation intensity.37 In contrast to the single peak in the TA-spectra, a pair of negative and positive features near the bandedge is observed in the TR-spectra. The features are antisymmetric about the exciton resonance energy (Figure 2C). The TR-spectra are related to the TA-spectra through a Kramers−Kronig transformation, and we find that the TR antisymmetric peaks also originate from bleaching of exciton resonance.6,8 Given that there is spectral overlap in the linear absorption spectrum between the exciton resonance and free-carrier absorption for MAPbI3, modeling the linear and transient-spectra simultaneously can greatly reduce the uncertainty of the spectral

If the measured carrier lifetime is shorter than ∼0.5 μs, then that likely indicates the carrier lifetimes are limited by surface recombination, while if the measured carrier lifetime is greater than ∼0.5 μs, then that indicates carrier lifetimes are limited by bulk recombination. As the excitation intensity increases, bi- and trimolecular recombination events begin to dominate the carrier lifetime. Since these are governed by the same Coulomb interaction, the trends in bi- and trimolecular should also reflect the trends in exciton binding energy. The bimolecular recombination rate constants (B coefficient) for both MAPbBr3 and MAPbI3 films are in line with values of the B coefficient for other direct band gap semiconductors (∼10−9−10−10 cm3s−1). Therefore, we can attribute the bimolecular recombination to radiative recombination. Note that the value of the B coefficient for MAPbI3 implies that the radiative lifetime would be ∼6 μs at 1 Sun intensity (∼1015 cm−3). For semiconductors, the B coefficient is proportional to the product of the optical transition energy (approximated as Eg) and the absorption coefficient at the 2598



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optical transition energy, α(ℏω = Eg).50−52 As shown in Figure 2A, the ratio of α(ℏω = Eg) for MAPbBr3 to that for MAPbI3 is approximately 3.1, which is primarily determined by the ratio of exciton binding energies. Taking into account the ratio of Eg between two samples, the ratio of B-coefficient for MAPbBr3 to that for MAPbI3 is around 4.5, qualitatively consistent with experimental data (Table 1), which indicates that this ratio is

the product of two correlation factors, geh·gx, where geh is the electron−hole correlation factor, and gx describes the correlation between an incoming electron/hole and an already correlated electron−hole pair.55 Unfortunately, it is difficult to quantify geh and gx. However, geh is also the radiative recombination enhancement factor,55 and therefore is proportional to the exciton binding energy. Given the neutral entity of the correlated electron−hole pair, gx is expected to be smaller than g eh because of screening. Thus, g eh (MAPbBr 3 )/ geh(MAPbI3) can be estimated as 3.1 according to the ratio of exciton binding energies between these two materials, while the ratio of geh·gx is likely to fall in the range from 3.1 to 9.5 (3.1 × 3.1). The measured ratio of C coefficients between these two lead halide perovskites is determined to be 4, closer to the lower boundary of the ratio of Coulomb enhancement factor. The different energy band gaps can be another factor that affects the C coefficients. Because momentum conservation during the Auger process can be satisfied more easily in a narrower band gap semiconductor, the C coefficient has been found to increase with decreasing Eg.56 Therefore, in contrast to the Coulomb enhancement, the momentum conservation requirement decreases the ratio of the C coefficient for MAPbBr3 compared to that for MAPbI3, which may explain the overestimation of the C coefficient ratio based solely on Coulomb enhancement. Apparently, more rigorous analysis is required to quantitatively rationalize the difference in Ccoefficients between the different lead-halide perovskites. Another interesting observation that is worth consideration, is that the C-coefficient in lead halide perovskites is at least 2 orders-of-magnitude larger than that found in GaAs (C ∼ 10−30 cm6 s−1)56 and Si (C ∼ 10−31cm6 s−1),54,57 two important semiconductors for solar cells. The Eg of MAPbI3 is similar to that of both GaAs and Si, and in particular, MAPbI3 and GaAs possess many similar photophysical properties. A fundamental understanding of thr difference in C coefficient between these materials could be of importance to applications that require higher carrier density such as LEDs and lasers. In this section, we will focus on another type of excitonphoton interaction; the optical Stark effect (OSE). The consequence of the OSE is the entanglement of the exciton

Table 1. List of Carrier Recombination Rate Constants for MAPbBr3 and MAPbI3 Filmsa B (cm3 s−1) MAPbBr3 MAPbI3

C (cm6 s−1) −10

4.9 ± 0.2 × 10 1.5 ± 0.1 × 10−10

13.5 ± 0.3 × 10−28 3.4 ± 0.1 × 10−28

a

The rate constants B and C represent the bi- and tri-molecular recombination rate constants, respectively.33.

3.3. Since the absorption is enhanced by the Coulomb attraction due to the excitonic effect, the radiative rate is also enhanced accordingly by the same affect. This estimation further confirms that, in lead-halide perovskites, the radiative recombination is responsible for the bimolecular recombination. The trimolecular recombination is intuitively attributed to three-particle Auger recombination, and its rate constant (referred to as the C coefficient) is also listed in Table 1. We find that the C coefficient of MAPbBr3 is larger than that for MAPbI3 by a factor of ∼4; the larger exciton binding energy can also partially account for the enhancement of its C coefficient. Due to the Coulomb interaction, a carrier attracts the opposite charge to its vicinity and repels those with identical charges, so the otherwise homogeneous carrier distribution in space actually displays microscopic inhomogeneity. As a result, the possibility of a carrier finding two other carriers with opposite charges at the same location is increased, and thus Auger recombination will be enhanced. This Coulomb-enhanced Auger recombination has been theoretically investigated and experimentally confirmed in silicon, and the enhancement factor is a three-particle (eeh or ehh) correlation factor, gt.53,54 To simplify the correlation function, gt can be approximated as

Figure 4. Optical Stark effect (OSE) in MAPbI3 films. (A) Exciton state energy shift due to OSE. The diagram shows energy levels of ground state (| 0⟩) and the 2-fold degenerate exciton states (|±1⟩). These quantum states are labeled by the azimuthal quantum number of the total angular momentum. The interaction of the exciton state and a photon shifts the exciton state toward higher energy and is governed by optical selection rules. The doubling degenerate exciton state can be lifted by the OSE. (B) TA spectra recorded during OSE. The pump and probe pulses are circularly polarized, and the polarization notation is described in the main text. The spectra are recorded at zero pump−probe delay (Δt = 0 ps). The pump photon energy is 1.55 eV with a detuning Δ of 81 meV. The probe is a white-light continuum with photon energy from 2.76 to 1.55 eV. The black curves are the fitting functions described by eq 6 in main text. This figure is adapted from ref 26 with permission from Nature Publishing Group (Copyright 2016). 2599



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TA spectroscopy can monitor the energy shift via an induced spectral change ΔA(ℏω). As δE ≪ E0, ΔA(ℏω) can be approximately described by the following equation:26

state with a nonresonant pump photon, and this light−matter interaction only persists as long as the photon field is present.58−60 Unlike absorption, OSE is not associated with carrier or exciton generation. As shown in Figure 4A, when the pump-photon energy (ℏω) is lower than the exciton transition energy (E0), a mixed exciton-photon state results with energy that is shifted toward higher energies. Applying a perturbation to the exciton−photon interaction in a two-level system, the energy shift (δE) can be expressed as61

δE =

ΔA(ℏω) = −

1 ∂A(ℏω) ·δE 2 ∂ℏω

(7)

where A(ℏω) is the steady state absorption spectrum (Figure 2), and a positive (negative) δE indicates blue (red) shift of the spectrum. Equation 7 suggests that the profile of ΔA(ℏω) resembles the first derivative of the absorption spectra, while the amplitude of the ΔA(ℏω) linearly depends on the pump intensity via eq 6. ΔA(ℏω) spectra recorded at zero pump− probe delay are shown in Figure 4B for two pump−probe polarization configurations. For a σ+ pump, a σ+ probe detects a spectral change while a σ− probe shows no spectral change. The TA-spectra are then modeled by eq 7, and the best fits find δE for the exciton state |+1⟩ as 1.1 meV. To simulate ΔA(ℏω), the first derivative of the exciton absorption spectrum (deconvolved from the linear absorption using eq 4), rather than use of the total absorption spectrum is needed. Thus, the OSE operates on exciton states and not on the continuum. A strong OSE has also been observed at room temperature in two-dimensional lead-halide perovskite layers.62 A circularly polarized pump (σ+ or σ−) with photon energy resonant with or above the exciton resonance can generate carriers with polarized total angular momentum that selectively bleach the exciton |+1⟩ or |−1⟩ resonance due to phase-space filling. Thus, the total angular momentum depolarization (referred to here as spin-depolarization) of carriers can be measured by probing the kinetics of the corresponding exciton bleach. Because phonon-mediated carrier-cooling can efficiently depolarize the carrier’s angular momentum, we have excluded this effect by tuning our pump pulse to be resonant with the lower energy side of the exciton resonance. The exciton resonance bleach kinetics for MAPbBr3 and MAPbI3 samples (Figure 5) are measured by TA spectroscopy with circularly polarized pump and probe pulses. Pump σ+ exclusively bleaches the absorption of the exciton |+1⟩ state, and the probe σ+ and σ− selectively detects the bleach of exciton |+1⟩ and |−1⟩ state, respectively. The spin depolarization will lead to the decay of |+1⟩ bleach and the formation of |−1⟩ bleach simultaneously, which is justified by the similar decay and

(μ0X )2 ⟨F ⟩2

(6) Δ where μ0X is the transition dipole moment, ⟨F⟩ is the timeaveraged electric field of the pump light, and Δ is the detuning of the photon frequency, ℏω, from E0. Since ⟨F⟩2 is proportional to pump light intensity, δE should be linear with increasing pump intensity. OSE is implicitly governed by optical selection rules through μ0X, and thus a large exciton absorption, implying large μ0X, is a prerequisite of a strong OSE. For MAPbI3, the bright exciton state has 2-fold of degeneracy, denoted by the azimuthal quantum number of the total angular momentum |±1⟩ (Figure 4A). According to the optical selection rules, using a circularly polarized pump-light can selectively interact with one of the degenerate exciton states. For example, a nonresonant circularly polarized pump with an angular momentum quantum number of +1, referred to as σ+, can only interact with an exciton state with angular momentum |+1⟩ while leaving the other |−1⟩ exction state unperturbed.

In contrast to the Coulomb enhancement, the momentum conservation requirement decreases the ratio of the C coefficient for MAPbBr3 compared to that for MAPbI3, which may explain the overestimation of the C coefficient ratio based solely on Coulomb enhancement.

Figure 5. Total angular momentum depolarization dynamics probed by the TA exciton state bleach kinetics. The polarized exciton state bleach kinetics for (A) MAPbI3 film, (B) MAPbBr3 film, and (C) MAPbBr3 single crystal. The kinetic traces are labeled by the pump-probe polarization configuration. The black curves are the single-exponential fittings. In panel A, the fast component of the blue curve due to the OSE is not shown. 2600



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formation time constants extracted from single exponential decay (black curves, Figure 5). The decay and formation kinetics merge at the halfway point between their respective initial amplitudes, indicating that the spin of carriers is completely depolarized at this stage. Our results indicate that the spin depolarization in MAPbI3 is twice as fast as that in MAPbBr3. The scattering with defects or grain boundaries through the Elliot−Yafet (EY) mechanism has been inferred as the primary cause of the depolarization.63 However, we compare the spin depolarization dynamics between single crystals of MAPbBr3 and a polycrystalline film (Figure 5B and 5C). The single crystal sample should have fewer grain boundaries and fewer bulk defects than the polycrystalline films. Our results suggest no significant difference in spin depolarization time between these two samples, implying that scattering with defects or grain boundaries may not be the major depolarization channel. Recently, the lifetime of exciton spin polarization has been determined to be longer than 1 ns in lead-halide perovskite films at the temperature of 4 K, under which condition most phonons are frozen out, yet the phonon scattering via EY mechanism still could not satisfactorily describe the experimental data.27 Therefore, further investigations are needed to explore the cause of the fast spin depolarization. In this Letter, we have discussed the excitonic effect in methylammonium lead-halide perovskite semiconductors. The Coulomb interaction plays an important role in defining their photophysical properties and can influence optoelectronic applications. We have shown that the lead-halide perovskite system behaves as a direct bandgap semiconductor with particular similarities to III−V semiconductors. As pointed out in this Letter, more theoretical or experimental investigations should be conducted to quantitatively examine the excitonic effect on radiative and Auger recombination, on photon-matter entanglement and on spin-depolarization. From the material aspect, learning how to control excitonic effects by adjusting the composition, structure, morphology, and tuning of quantum confinement should be fruitful avenues for exploration.

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AUTHOR INFORMATION

ORCID

Xihan Chen: 0000-0001-7907-2549 Haipeng Lu: 0000-0003-0252-3086 Matthew C. Beard: 0000-0002-2711-1355 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

We gratefully acknowledge support through the Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy Sciences, Office of Science within the U.S. Department of Energy through Contract Number DE-AC3608G028308. Spin-depolarization measurements were performed with funding through an LDRD project. We acknowledge helpful discussions with Jao van de Lagemaat, Joseph Berry, Joseph Luther, and Kai Zhu.

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Excitonic Effects in Methylammonium Lead Halide Perovskites

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