Abstract
Photovoltaic devices based on metal halide perovskites are rapidly improving in efficiency. Once the Shockley–Queisser limit is reached, chargecarrier extraction will be limited only by radiative bimolecular recombination of electrons with holes. Yet, this fundamental process, and its link with material stoichiometry, is still poorly understood. Here we show that bimolecular chargecarrier recombination in methylammonium lead triiodide perovskite can be fully explained as the inverse process of absorption. By correctly accounting for contributions to the absorption from excitons and electronhole continuum states, we are able to utilise the van Roosbroeck–Shockley relation to determine bimolecular recombination rate constants from absorption spectra. We show that the sharpening of photon, electron and hole distribution functions significantly enhances bimolecular charge recombination as the temperature is lowered, mirroring trends in transient spectroscopy. Our findings provide vital understanding of bandtoband recombination processes in this hybrid perovskite, which comprise direct, fully radiative transitions between thermalized electrons and holes.
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Introduction
The past 5 years have seen intense research into the nature of hybrid lead halide perovskites used as active layers in photovoltaic devices^{1,2,3}. Improvements in fabrication processes have led to rapid increases in lighttoelectrical power conversion efficiencies beyond 22%, placing these semiconductors at the centre of potentially cheap and sustainable energy production^{4,5}. Such rapid advances have brought into question the fundamental physical mechanisms that set the ultimate limits to essential properties such as chargecarrier mobilities^{6}, exciton binding energies^{7} and intrinsic (bimolecular) recombination between electrons and holes.
At the heart of a solar cell is the absorbing layer, which is the only layer by design in which incident light should excite charge carriers. Following absorption of photons, excited electrons and holes are only of use if they can be extracted at electrodes to contribute to a photocurrent. Such extraction is impeded by competing chargecarrier recombination processes, which can occur through a variety of mechanisms^{8}. The fundamental physical processes governing the dynamics of charge carriers in the absorbing layer hence play a dominant role in the conversion process of photons into storable energy.
Previous transient spectroscopic experiments have revealed the value of rate constants associated with different chargerecombination processes in hybrid perovskites, based on the measurements of transient absorption^{9,10,11,12}, photoluminescence^{13,14,15} and terahertz photoconductivity^{16,17} dynamics. The significance of each process varies with the density of charge carriers in the material and thus the intensity of incident light. Under lowlevel illumination, monomolecular (Shockley–Read–Hall^{18}) processes mediated by trap sites such as elemental vacancies, substitutions or interstitials^{19} will dominate chargecarrier recombination. However, with increasing chargecarrier density, bimolecular (radiative) recombination between unbound electrons and holes will begin to take over, while at even higher densities, manybody Auger recombination will contribute^{2}.
Any analysis of chargecarrier recombination mechanisms therefore has to be placed in the context of solar irradiation levels (AM1.5) at which Auger recombination was found insignificant^{17}. Instead, charge recombination under AM1.5 results from a tradeoff between traprelated monomolecular and intrinsic bimolecular electronhole recombination^{2}, with the latter dominating when monomolecular recombination lifetimes exceed 10 μs. Reductions in trap concentration through rapidly improving material processing protocols mean that this limit is already being approached^{20,21}, opening the prospect of singlejunction solar cells operating near the Shockley–Queisser limit^{22}.
Such developments have brought into sharp focus the fundamental nature of bimolecular recombination between electrons and holes in the continuum states of the conduction and valence bands, which will operate in defectfree metal halide perovskites under solar illumination. Despite the importance of this process, relatively little is known about its mechanism to date. While studies have shown bimolecular recombination in hybrid perovskites to be highly nonLangevin^{16,23,24,25,26}, there is currently no consensus on suitable alternative models. Some studies have suggested that direct bandtoband transitions govern bimolecular recombination, including fully theoretical work, which however currently cannot account for the complexities of Coulomb correlations^{27}, and experimental work examining the switchover point of luminescence intensity between the trapmediated recombination regime at low chargecarrier densities, and the bimolecular regime at higher densities^{28}. However, other studies have hypothesised a range of more exotic influences, including: Rashba spinorbit coupling^{29}, a mixture of indirect and direct band gap states^{30}, and relativistic effects^{31}. Identification of the correct mechanism can be further complicated by the potential presence of electronic transitions between multiple and nonparabolic bands and an energy dependence of the associated matrix elements, which have not been evaluated with certainty to date.
In this work, we show unambiguously that bimolecular electronhole recombination in lead iodide perovskites can be understood in terms of the inverse process to absorption, analogous to the case of GaAs^{32,33,34} and silicon^{35,36}. With the help of insights gained from ab initio calculations regarding the validity and applicatibility of Elliott’s theory to CH_{3}NH_{3}PbI_{3}, we are able to correctly unravel the contributions to absorption from bound electronhole pairs (excitons) from those of unbound electrons and holes in the continuum states of the semiconductor. This approach allows us to demonstrate a clear correlation between the temperaturedependent changes in absorption spectra of charge carriers and the rate of bimolecular chargecarrier recombination.
Results
Absorption modelling
To probe whether bimolecular recombination between unbound electrons and holes is the inverse of absorption, we need to separate accurately the contributions to the absorption from continuum states and from Coulombically bound electronhole pairs (excitonic states). Such separation is usually achieved through Elliott’s theory^{37}, however, previous attempts for metal halide perovskites have found significant deviations from experimental data for this theory at energies sufficiently above the band gap^{12,38,39}, which calls into question the rationale for using this model. Below, we perform an analysis of absorption spectra for the most frequently investigated metal halide perovskite, CH_{3}NH_{3}PbI_{3}, in combination with highaccuracy GW ab initio calculations to unravel the causes of such effects.
To determine the temperaturedependent absorption spectra of CH_{3}NH_{3}PbI_{3} (Fig. 1a), reflectiontransmission measurements were performed on thin films by use of a Fourier transform infrared (FTIR) spectrometer. Thin films were fabricated through dualsource vapour deposition (Methods), which yields smooth, uniform surfaces^{40}, as evidenced from the observation of Fabry–Perot oscillations (Supplementary Fig. 1b) and sharp absorption onsets devoid of scattering effects that would otherwise complicate the modelling. Figure 1b shows the absorption spectra of CH_{3}NH_{3}PbI_{3} at 4 K along with a fit based on Elliott’s theory^{37} within the first 100 meV above the band edge (full details are included in Supplementary Note 5). Within this energy range, this theory is found to reproduce accurately the spectral shape of the absorption, however, at higher energies significant deviations occur.
Elliott’s theory accounts for the enhancement of light absorption through Coulombic attraction between electrons and holes in the conduction and valence bands of a semiconductor^{37}. As Fig. 1b illustrates, such interactions significantly increase the absorption coefficient near the band gap energy E_{G} beyond that expected for a free electronhole picture. Here, Coulombic attraction leads to additional absorption into bound excitonic states (α_{X}(E), blue dashed line) that manifest themselves as a series of hydrogenic peaks at energies below E_{G}. Importantly, even above E_{G}, Coulomb interactions substantially enhance the absorption coefficient for electronhole continuum states (α_{C}(E), red dashed line), by a factor ξ(E) beyond that expected for free electrons and holes (α_{Free}(E), dotted line). ξ(E) is the Coulombenhancement factor derived from the values of the band gap and binding energy (see Eq. (11) in the Supplementary Information) and is therefore not a free (independent) parameter of the model. As a result, the continuum absorption spectrum differs from the classical squareroot dependence expected for uncorrelated charges in a direct semiconductor (α_{Free}), resulting instead in a steplike onset at the band gap (α_{C})^{37,39}.
The correct disentanglement of these excitonic and continuum states is key to unravelling the mechanisms of bimolecular recombination, which arises only from interactions between unbound electrons and holes. As discussed in more detail below, we also need to account for the possibility that the Coulomb interactions between charge carriers are screened, which may occur at sufficiently high chargecarrier densities (above the Mott transition^{41}). In addition, our model accounts for spectral broadening caused by electron–phonon interactions and energetic disorder^{42} through convolution with a suitable function (Supplementary Note 5). We note that photon absorption also depends on the electric dipole matrix element, which is assumed to be independent of photon energy in Elliott’s theory^{37} and hence in our modelling of the experimental data as shown in Fig. 1.
A correct determination of the exciton binding energy E_{X} is a prerequisite for the accurate extraction of the continuum states. We determine the value of E_{X} to be 20 ± 2 meV for the lowtemperature (0–160 K) orthorhombic phase from fits based on Elliott’s theory, which is in good agreement with the lowtemperature value of 16 ± 2 meV determined from magnetoabsorption studies^{7}. Figure 1c shows that the binding energy declines only slightly with temperature within each structural phase, apart from a rapid drop by ~5 meV that occurs at the phase transition into the tetragonal structure. These changes can be understood in terms of temperaturedependent trends in the dielectric function at low frequencies; in particular, at the orthorhombictetragonal phase transition a stepchange in permittivity^{43} and hence exciton binding energy occurs. The trends with temperature of the binding energy are similar to those reported in previous studies^{44,45,46} and the value of E_{X} ~8 meV at room temperature is commensurate with efficient exciton dissociation, as suggested in other studies^{7,16,44,45,47,48}.
Careful modelling of the bandedge absorption also allows us to extract values for the band gap energy that can be compared with those determined from ab initio calculations discussed below. Figure 1d illustrates that the extracted band gap energy E_{G} generally increases with temperature, in contrast with trends for typical inorganic semiconductors such as GaAs^{49}. This phenomenon has previously been attributed to a reverse ordering of the bandedge states^{1,50,51,52}. At the transition between the orthorhombic and tetragonal phases (160 K), E_{G} displays a noticeable discontinuity of 100 meV as previously observed^{17} and deriving from changes in octahedral tilts^{53}.
Theoretical calculations
As mentioned above, the use of Elliott’s theory to model the absorption edge of CH_{3}NH_{3}PbI_{3} has proven to be controversial, given that several studies (including the present) found that satisfactory agreement could only be obtained for energies near the band edge^{12,38,39}. It has been postulated that nonparabolicity of the conduction and valence bands could be to blame^{28,44,54}, however, this would have to be severe in order to account for the observed discrepancies. Here we resolve this controversy by showing that the divergence mostly arises from a breakdown of several assumptions that Elliott’s theory is based on, which are that (i) only one valence and one conduction band are involved in the transitions; (ii) the bands are spherical (i.e. parabolic and isotropic); and (iii) the electric dipole transition matrix element is independent of photon energy. We examine the validity of these assumptions in the context of CH_{3}NH_{3}PbI_{3} by conducting highly accurate firstprinciples calculations that can take into account potential nonparabolicity and higherlying electronic transitions.
Figure 2a shows the electronic band structure of CH_{3}NH_{3}PbI_{3} calculated from first principles using the GW approximation. Details for all calculations can be found in Supplementary Note 4 and in previous articles^{55,56}. We calculate a quasiparticle band gap (V1 ↔ C1) of 1.57 eV, which is in good agreement with the band gap energy we determine at 4 K (1.66 eV, see Fig. 1d). The small difference between the calculated and measured band gaps derives from the absence of electron–phonon interactions in our calculations. Interestingly, our calculations also reveal the presence of another electronic transition (V2 ↔ C1) with an onset energy of 1.95 eV, which is relatively close to the band edge and therefore has the potential to influence the shape of the absorption onset.
From the calculated GW electronic band structure, we are also able to extract the quasiparticle electron and hole effective masses, as shown in Table 1. Effective masses are calculated as the inverse of the second derivatives of the energy with respect to the wave vector at the Γpoint, with derivatives evaluated numerically from finite differences. The reduced effective mass of 0.11m_{e} is in excellent agreement with experimental values^{7}. The accuracy of these calculations in terms of reproducing closely the experimental values of both band gap energy and effective masses highlights the power of the GW method, in comparison with traditional densityfunctional theory (DFT) that tends to underestimate effective masses and band gap energies (Supplementary Note 4).
With a conclusive computational framework in place, we probe the possibility of nonparabolicity contributing to deviations from Elliott’s theory. We compute the joint density of states using the quasiparticle eigenvalues and contrast this with the trend expected for a simple parabolic onset based on the calculated reduced effective mass. As Fig. 2b shows, the density of states associated with the V1 ↔ C1 transition (onset 1.57 eV) is remarkably parabolic for photon energies of up to ~250 meV above the gap. However, the second transition (1.95 eV; V2 ↔ C1) makes a significant contribution to the joint density of states and will give rise to additional absorption, if it has significant oscillator strength. Hence, we conclude that the parabolic approximation assumed in Elliott’s model will hold for photon energies within ~250 meV of the band edge, but additional electronic transitions will lead to deviations already within the first ~200 meV.
In addition, we examine the validity of the assumption that the electric dipole transition matrix element is constant with photon energy in CH_{3}NH_{3}PbI_{3}. It has long been known for inorganic semiconductors such as silicon or germanium that significant deviations may occur from this simplifying approximation^{57}. We therefore calculate the matrix element at a number of different kpoints in the Brillouin zone relating to different energy transitions (Fig. 1c). Grey circles are the calculated values at different kpoints with the average value shown as a blue line. Our results clearly demonstrate that the matrix element decreases significantly with increasing energy for both the V1 ↔ C1 and V2 ↔ C1 transitions; therefore Elliott’s approximation of constant matrix element will only hold close to the band edge. In addition, we find that the magnitude of the V2 ↔ C1 transition is approximately 1 order of magnitude lower than that of the V1 ↔ C1 transition, which means that its contribution to absorption above 1.95 eV is not as significant as in the unweighted joint density of states.
Our rigorous evaluation of all factors contributing to the absorption allows us to derive the absorption coefficient entirely from ab initio calculations (Supplementary Note 4), as shown in Fig. 2d for both DFT and GW approximations. An accurate calculation of the absorption coefficient within the GW approximation requires the inclusion of the offdiagonal terms of the GW selfenergy, which is computationally prohibitive for the case of CH_{3}NH_{3}PbI_{3}. Instead, in Fig. 2d we show two limits of the GW absorption spectra obtained using the dipole matrix elements calculated from DFT, both before and after scaling with respect to the quasiparticle transition energies (Supplementary Information). Figure 2d shows that the absorption coefficient is sensitive to scaling, and can vary by up to a factor of 4, while the shape of the spectrum does not vary significantly. Moreover, it can be shown that the fsum rule in these two cases is either overestimated, if the matrix elements are scaled, or underestimated in the unscaled case, consistent with the respective magnitudes of the absorption coefficient shown in Fig. 2d. For this reason, we propose that the true absorption coefficient is bracketed by the scaled and unscaled GW spectra, which give good agreement with the experimental data (α_{Free}, Fig. 1b). This result indicates that the inclusion of band nonparabolicity, higherlying transition and the functional dependence of the matrix element are fundamental for the accurate evaluation of the absorption coefficient in CH_{3}NH_{3}PbI_{3}. Hence, these ab initio calculations have enabled us to individually explore each assumption that Elliott’s theory is built on, allowing us to probe its validity for hybrid perovskite semiconductors. Overall, our analysis conclusively reveals that deviations from Elliott’s theory for CH_{3}NH_{3}PbI_{3} are a reflection of a breakdown in the assumptions the model is based on, for energies somewhat higher than the band edge. Our evaluation of these factors however suggest that within ~100 meV, the simplified assumptions of Elliott’s theory holds, allowing us to perform an analysis of bandedge chargecarrier recombination based on the continuum states extracted from this model.
Bimolecular recombination
With a comprehensive picture of the contributions to the absorption onset in place, we proceed to link bimolecular recombination of electrons and holes to the reverse process of light absorption by continuum states. Figure 3a shows a schematic diagram linking the Fermi–Dirac distributions of electrons and holes with absorption and emission processes as mediated by the Bose–Einstein functions of the photon radiation field. To validate such links, we first evaluate the temperaturedependent bimolecular recombination rate constants that would be expected according to the van Roosbroeck–Shockley^{58} relation for absorption by electrons and holes in the presence and absence of Coulomb correlations. We then compare the calculated rate constants with those obtained from transient spectroscopy, showing excellent agreement for both absolute values and temperature trends.
The theory of photonradiative recombination of electrons and holes was first developed by van Roosbroeck and Shockley^{58} and applies the principle of detailed balance to the thermal equilibrium rate of photon absorption, i.e., it assumes the rate of radiative recombination at thermal equilibrium at a certain frequency to be equal to the rate of generation of electronhole pairs by thermal radiation at the same frequency (Fig. 3a)^{59}. The van Roosbroeck–Shockley relation hence allows deduction of the radiative recombination rate and bimolecular recombination rate constant from steadystate absorption spectra. The theory is based on the assumption of unit quantum efficiency and is applicable to a material in which there is no kselection rule, or to a nondegenerate material in which there is a selection rule^{32,60}, and has extensively been applied to group IV, VI, and III–V direct and indirect semiconductors in the past^{60}.
The van Roosbroeck–Shockley relation describes the temperaturedependent radiative recombination rate R_{rad}(T) by the following function^{58}:
where ρ(E, T)dE is the density of photons in the material in the interval dE, and P(E, T) is the probability per unit time that a photon of energy E is absorbed and hence is proportional to the absorption coefficient. The principle of detailed balance demands that transitions between any two states occurs with equal frequency and in either direction at equilibrium and also prevents the maintenance of equilibrium by means of cyclic processes^{61}. Hence, the absorption coefficient that enters Eq. (1) is that of the continuum states and the choice of whether to include Coulombic enhancement is discussed below. Dividing the radiative recombination rate by the square of the intrinsic carrier concentration, n_{i}, we are able to obtain the bimolecular recombination rate constant, \(k_2(T) = R_{{\mathrm{rad}}}(T){\mathrm{/}}n_{\mathrm{i}}^2(T)\), which is also often referred to as the ‘Bcoefficient’ in literature on inorganic semiconductors. All associated equations are detailed in full in Supplementary Note 6. The intrinsic carrier concentration and the photon distribution are dependent on the Fermi–Dirac and Bose–Einstein distributions respectively, which vary strongly with temperature.
The correct evaluation of k_{2} through the van Roosbroeck–Shockley relation requires a choice to be made about which absorption coefficient spectrum enters, Eq. (1), as discussed in detail in Supplementary Note 6. Past studies for the case of silicon^{35,36} discussed the role of Coulomb interactions for the radiative recombination between electrons and holes in the conduction and valence band. In the presence of Coulomb correlations, absorption is strengthened by the energydependent factor ξ and the Coulombenhanced spectrum α_{C}(E) may be used, while in the absence of such interactions, α_{Free}(E) may be used in Eq. (1). In general, Coulomb correlations increase the probability of an electron and a hole to exist in the same space, and therefore such interactions should a priori be included in a calculation of k_{2}. However, at chargecarrier densities n, exceeding the Mott density n_{M}, Coulomb interactions will be effectively screened by the presence of the high background chargecarrier density, therefore the free electronhole absorption α_{Free} is the relevant quantity. The Mott density n_{M} places a quantitative value on the transition from an insulating gas of excitons at lower densities to a metallike state of an electronhole plasma at high chargecarrier densities^{41}. The electrostatic force between charged particles is a continuous function of their relative distances to each other, therefore it is intrinsically challenging to define a distinct density at which a Mott transition occurs. Hence, a wide range of expressions for the Mott density have been proposed, which depend on different model assumptions and complexities, and whose validity is also temperaturedependent. A detailed discussion and literature summary for a variety of models can be found in the Supplementary Note 7. To represent the variety of such models, we evaluate the Mott density as a function of temperature^{41}, based on different expressions and using values for the exciton binding energy shown in Fig. 1c and the reduced effective mass obtained from GW calculations (Table 1), as described in Supplementary Note 7. We find values of n_{M} to be distributed about 10^{17} cm^{−3} for 50 < T < 295 K.
Figure 3b displays the temperature dependence of the bimolecular recombination rate constant k_{2} derived from the unbound electronhole absorption coefficient spectra through the van Roosbroeck–Shockley relation, for the case of Coulombenhanced absorption (α_{C}(E), white triangles) and screened Coulomb interactions (α_{Free}(E), red triangles), i.e., above and below the Mott transition. At room temperature, Coulomb enhancement of the continuum absorption increases the bimolecular recombination constant by about a factor 4 with respect to the case of screened interactions. This discrepancy increases with decreasing temperature as Coulombenhancement effects become more prominent. For both cases, the value of k_{2} increases by roughly 1 order of magnitude between room temperature (295 K) and 50 K. Our analysis reveals that this increase is directly linked with the sharpening of the Fermi–Dirac and Bose–Einstein distribution functions with decreasing temperature, with changes in absorption (and hence refractive index) making only a very minor contribution by comparison. As the photon distribution function and the thermal spread of electrons and holes across the conduction and valence bands narrow, bimolecular recombination is substantially enhanced.
To validate our overall approach, we compare these trends with values obtained from terahertz photoconductivity transients, as reported in a previous study^{17} and corrected for selfabsorption effects^{62,63,64,65,66} to yield the intrinsic values of k_{2} (see Supplementary Note 8 for details). Figure 3b reveals that temperature trends observed in transient spectroscopy agree very well with those derived from the van Roosbroeck–Shockley relation. A comparison in terms of absolute values shows that these transient spectroscopic data are closest to the case of Coulombenhanced (unscreened) bimolecular recombination, although most points fall within the intermediate regime (shaded region). This result is not surprising, given that a clear observation of bimolecular processes in transient spectroscopy requires excitation with laser pulse fluences for which these recombination events begin to dominate over traprelated recombination that is prominent at lower chargecarrier densities. As suggested previously^{2} and described in detail in Supplementary Note 6, this case applies at chargecarrier densities above ~10^{15}–10^{17} cm^{−3} for typical CH_{3}NH_{3}PbI_{3}, which may potentially come near the Mott density (Figure S7d). However, a recent study^{67} reported the direct observation of excitonic Rydberg states following excitation with fluences 1 order of magnitude higher than the highest fluence utilised for the transients against which we reference our calculated values of k_{2} here. We therefore conclude that the most adequate scenario for comparison is with values obtained from the Coulombenhanced (unscreened) absorption coefficient, for which Fig. 3b also indicates best agreement. Hence, our analysis clearly supports the notion of an inverse absorption process leading to bimolecular recombination in CH_{3}NH_{3}PbI_{3}.
Our findings have direct impact on the use of hybrid perovskites in photovoltaic applications because they inherently connect the requirement for strong light absorption near the band edge with the presence of significant electronhole recombination. However, we note that this fundamental link in no way inhibits the efficient operation of photovoltaic devices based on CH_{3}NH_{3}PbI_{3}. While faster bimolecular chargecarrier recombination will shorten chargecarrier diffusion lengths, it also mandates higher absorption coefficients, and hence absorber layers can be made thinner to compensate. Chargecarrier diffusion lengths for thin hybrid perovskite films are already in the micron regime^{13}, which exceeds their typical thickness in solar cells of a few hundred nanometres, thus allowing excellent chargecarrier extraction to electrodes. Therefore, while the mechanism we identify here poses a fundamental hurdle to a reduction of bimolecular recombination, it does not preclude the development of highefficiency photovoltaic cells based on lead halide perovskites. In addition, our findings demonstrate that bimolecular (radiative) recombination rate constants will depend to some extent on light intensity levels. Under solar illumination^{2}, chargecarrier densities in CH_{3}NH_{3}PbI_{3} are near 10^{15}–10^{16} cm^{−3} and therefore below the Mott density, leading to Coulombenhanced electronhole recombination. However, implementation in solar concentrators could potentially lead to chargecarrier densities approaching 10^{18} cm^{−3}, for which bimolecular recombination rate constants are likely to be reduced as a result of screening.
With the efficiency of perovskite photovoltaic cells rapidly increasing, the Shockley–Queisser limit is not too far from reach. We have examined the physical processes behind bimolecular electronhole recombination in CH_{3}NH_{3}PbI_{3}, which will dominate over trapassisted monomolecular recombination once this limit is approached. Our direct comparison between bimolecular recombination rate constants derived from the van Roosbroeck–Shockley relation, with those extracted from transient spectroscopic measurements, demonstrates that these processes can be seen as the inverse to absorption. The established validity of the van Roosbroeck–Shockley relation further demonstrates that photon emission derives from fully radiative^{2,28,62} bandtoband transitions, which in this case involve the recombination of thermalized electrons and holes across a direct band gap. We demonstrate that the sharpening of photon, electron and hole distribution functions enhances the bimolecular radiative rate by roughly an order of magnitude as the temperature is lowered from room temperature to ~50 K. In addition, our highly accurate GW ab initio calculations have clearly identified the salient factors defining the shape of the absorption onset, such as nonparabolicity, higherlying transitions and the correct functional dependence of the matrix element. We further show that screening of the Coulomb interactions between electrons and holes above the Mott transition will reduce bimolecular recombination constants, meaning that such values have to be considered in the context of the chargecarrier densities present. Our findings thus provide a fundamental understanding of the electronic processes in these hybrid perovskites, which will allow for guided exploration of alternative stoichiometries with desirable photovoltaic properties.
Methods
Perovskite thinfilm fabrication
Dualsource thermal evaporation was used to grow thin films of CH_{3}NH_{3}PbI_{3} on zcut quartz substrates under high vacuum as reported previously^{68}. CH_{3}NH_{3}I and PbI_{2} were placed in separate crucibles, and the substrates were mounted on a rotating substrate holder to ensure a uniform film was deposited. The temperature of the substrates was kept at 20 °C throughout the deposition and the chamber was allowed to reach a high vacuum (10^{−6} mbar). Once the deposition rate for CH_{3}NH_{3}I and PbI_{2} had stabilised, the substrates were exposed to the vapour for 2 h at a nominal growth rate of 0.6 Å s^{−1}. The deposition rates of both CH_{3}NH_{3}I and PbI_{2} were monitored using a quartz crystal microbalance to ensure that a 1:1 molar ratio was achieved in the final composition of the film. The thickness of the thinfilm CH_{3}NH_{3}PbI_{3} was measured by crosssectional scanning electron microscope (SEM) images to be 435 ± 5 nm (Supplementary Information Fig. 2).
Reflectiontransmission measurements
A FTIR spectrometer (Bruker Vertex 80v) was used in conjunction with a gas exchange cryostat (Oxford Instruments OptistatCF2) to measure the temperaturedependent reflectance and transmittance of CH_{3}NH_{3}PbI_{3}. The system was set up with a tungsten halogen lamp as the illumination source, a CaF_{2} beamsplitter and a silicon detector. A schematic of the spectrometer, and the reflection and transmission spectra at 4 K are included in Supplementary Note 1.
Data availability
Data supporting this publication is available from the corresponding author on request. The calculations were performed using the opensource software projects Quantum ESPRESSO, Yambo and Wannier90, which can be freely downloaded from www.quantumespresso.org, www.yambocode.org and www.wannier.org, respectively.
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Acknowledgements
This work was supported by the UK Engineering and Physical Sciences Research Council UK through grant no. EP/L024667/1, EP/P006329/1, EP/P033229/1 and M0205171/1. M.R.F., C.V. and F.G. acknowledge the support from the Leverhulme Trust (grant RL2012001) and the Graphene Flagship (Horizon2020 grant no. 696656—Graphene Core 1), the use of the University of Oxford Advanced Research Computing (ARC) facility (http://dx.doi.org/10.5281/zenodo.22558) and the use of the DECI13 and DECI14 resources Abel and Cartesius based in Norway and the Netherlands, with support from the PRACE AISBL.
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C.L.D. performed the FTIR experiments, data analysis and participated in the experimental planning. M.R.F. and C.V. carried out the firstprinciples calculations. J.B.P. prepared the samples and provided support with FTIR experiments. T.W.C. carried out the numerical selfabsorption corrrections. A.D.W. conducted temperaturedependent photoluminescence measurements. R.L.M. provided support with the analysis. The project was conceived, planned and supervised by F.G., M.B.J. and L.M.H.. C.L.D. wrote the first version of the manuscript and all authors contributed to the discussion and preparation of the final version of the article.
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Davies, C.L., Filip, M.R., Patel, J.B. et al. Bimolecular recombination in methylammonium lead triiodide perovskite is an inverse absorption process. Nat Commun 9, 293 (2018). https://doi.org/10.1038/s41467017026702
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DOI: https://doi.org/10.1038/s41467017026702
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