Letter pubs.acs.org/JPCL
Decreasing Radiative Recombination Coefficients via an Indirect Band Gap in Lead Halide Perovskites Thomas Kirchartz*,†,‡ and Uwe Rau† †
IEK5-Photovoltaics, Forschungszentrum Jülich, 52425 Jülich, Germany Faculty of Engineering and CENIDE, University of Duisburg-Essen, Carl-Benz-Str. 199, 47057 Duisburg, Germany
‡
S Supporting Information *
ABSTRACT: High absorption coefficients imply also high radiative recombination coefficients due to detailed balance. It is therefore worthwhile to investigate whether a combination of two transitions with different absorption coefficients (such as a direct and an indirect band gap) could be used to reduce radiative recombination while at the same time retaining the high absorption coefficient. We show here that a longer radiative lifetime helping charge collection can indeed be achieved, while an increase in open-circuit voltage by adding an indirect band gap below the direct one is impossible. We also show that the absorption coefficient in CH3NH3PbI3 could indeed consist of a direct and an indirect contribution; however, the indirect one seems to dominate luminescence and therefore radiative recombination. Thus, the condition that the direct gap is mainly responsible for absorption and emission would not be valid for CH3NH3PbI3. Therefore, we would not expect any benefit of an indirect gap in the radiative limit. However, there may be benefits for charge collection but not open-circuit voltage if nonradiative recombination is dominant.
S
principle have two beneficial effects in photovoltaics, namely, better charge collection and a higher open-circuit voltage, both of which would eventually lead to a higher power conversion efficiency. However, such a combination of indirect and direct band gap with a distance of tens of millielectronvolts would lead to an absorption edge that is less abrupt than for instance a simple direct band gap (without an indirect one). Because the step function has previously27,28 been shown to be the mathematically ideal shape for a solar cell absorptance in the radiative, high-mobility limit, a positive effect of the indirect− direct character of the absorption edge becomes questionable at least close to the radiative limit. It is therefore of general interest to the community to better understand the implications of having a combination of direct and indirect band gaps for photovoltaic performance in general and in the specific case of MAPI and related materials. If this combination of direct and indirect band gap defined a generic way of increasing photovoltaic performance under certain circumstances, it would become a feature for which theoretical and experimental material screening efforts could be looking and optimizing their selection metrics. This is particularly interesting because one of the most widely used29 selection metrics for photovoltaic material screening uses the energetic distance between the direct gap and any lower-energy forbidden transition (like an indirect gap) as a negative factor to rate the material’s anticipated photovoltaic potential. This
ome metal halide perovskites have exceptionally good optoelectronic properties that make the materials attractive for applications in photovoltaics,1−4 light-emitting devices,5−11 photonics,12 and sensors.13 Among the most remarkable properties are the extremely long charge-carrier lifetimes14−17 that are observed in thin films of CH3NH3PbI3 (MAPI) and related compounds despite the fact that the films are typically deposited at low temperatures. In combination with suitable contact materials, these long lifetimes can lead to both high open-circuit voltages and high luminescence quantum efficiencies,18−22 which implies that the material is useful for both photovoltaics and light-emitting applications. Following initial reports on the sharp absorption edge23 of the lead-based metal halide perovskites, the material was generally assumed to have a direct band gap. This seems to be a sensible assumption given that the absorption coefficient within tens to hundreds of millielectronvolts above the band gap is high when compared to that of other direct band gap materials relevant for photovoltaics such as GaAs, InP, or Cu(In,Ga)Se2.1,23 However, recent reports24,25 cast doubt on this initial assessment. Both theoretical and experimental work shows that at least the room-temperature phase of the most widely used lead halide perovskite MAPI has an indirect band gap just below the direct band gap. This has been used as an apparent explanation for the long lifetimes and the high open-circuit voltages observed. The idea is that the direct band gap would provide the high absorption coefficient and the high photocurrent,26 while the recombination of charge carriers at the lower-lying indirect band gap would provide the long charge-carrier lifetimes. Slow recombination and long lifetimes can in © XXXX American Chemical Society
Received: January 30, 2017 Accepted: March 2, 2017
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is independent of EFn. If the energy of the indirect band is well below that of the direct one, the difference ΔEC = EdC − EiC becomes positive, and we may have many more electrons in the indirect band than in the direct one. Therefore, we may write the total electron concentration n = nd + ni as
implies that current selection metrics would actively discard materials with a combination of an indirect gap and a direct gap just above. The problem can be divided into three main questions, namely, (i) what happens in the radiative limit, (ii) what happens in the case when Shockley−Read−Hall recombination dominates, and (iii) how do these findings relate to the specific case of MAPI. In the case of the radiative limit, one might first look at the high mobility limit of the radiative efficiency limit. This is essentially the case of the Shockley−Queisser limit30 but for an arbitrary absorptance instead of a step-function like absorptance. For this situation, it has been shown previously that if the band gap of the absorber is at the peak of the efficiency versus band gap curve of the Shockley−Queisser limit, then any change away from the step-function would automatically reduce efficiency.27,28 The Shockley−Queisser limit for nonstep-function-like absorbers has been discussed in quite some detail for cases such as band gap inhomogeneities,31 organic absorber materials,32,33 charge-transfer states, or fluorescent collectors.34,35 Therefore, we will focus on the sofar much less-thoroughly explored situation of the radiative limit for finite mobilities,36 where the indirect−direct band gap combination can indeed have a positive effect. In this context of understanding the effect of band gap type and therefore absorption coefficients on photovoltaic efficiency, one advantage of the radiative limit is that radiative lifetimes or recombination coefficients are related to the absorption coefficient α.37 Thus, we can directly relate how any change in the absorption coefficient by adding an indirect band gap at a certain energy below the direct band gap would affect the recombination coefficients and rates. We first assume that the exchange between direct and indirect band is much faster than recombination. Despite the fact that slow carrier cooling has been reported for lead halide perovskites, this is still a sensible assumption. Thermalization at charge densities relevant for photovoltaic operation without concentration ( 1, which depends exponentially on the energy difference between the two gaps. However, this increase in lifetime would matter only for charge collection not for the open-circuit voltage because any increase in lifetime has to be paid for by a lower equilibrium charge carrier density, thereby always keeping the recombination rate at constant voltage the same. It was shown in ref 25 that applying pressure to a MAPI film changes the band structure from indirect (without pressure) to direct (with pressure) and that the nonradiative charge carrier lifetime does indeed become shorter when applying pressure. This would indicate that there is at least some beneficial effect of the indirect band gap on nonradiative lifetime. However, because the equilibrium concentration n0 of electrons should change with pressure as well, it is not clear whether the ratio n0/τnr is indeed getting smaller when going from a direct to an indirect band gap. Only then, there would be a positive effect on Voc. It is well-known that the difference between the opencircuit voltage Voc and the radiative open-circuit voltage Voc,rad is proportional to the logarithm of the electroluminescence quantum efficiency QEL [i.e., Voc,rad − Voc = −ln(QEL) × kT/ q].59,60 While there was no electroluminescence measured in ref 25, the authors report that the photoluminescence quantum yield is reduced in the case of the indirect band gap. This suggests that Voc suffers rather than benefits from the indirect band gap if taken relative to its radiative limit. Finally, we need to study the actual absorption coefficient of MAPI to find evidence for the strength of the direct and indirect transitions to see whether those are comparable to the
Figure 3. (a) Experimental absorption coefficient of MAPI (open circles) taken from ref 17 and an attempt to fit an absorption coefficient based on the sum (black solid line) of an absorption coefficient based on an indirect (violet, dashed line) and a direct (green, dotted line) transition according to eq 9. (b) Product of absorption coefficient and blackbody spectrum ϕ bb for the experimental absorption coefficient (open circles), as well as the indirect, direct, and total absorption coefficient from panel a. The fit shows that there is a subgap part of the absorption coefficient that is not captured by the direct−indirect combination. The rest fits quite well. However, the integral ∫ α(E)ϕbb(E)dE relevant for radiative recombination is dominated by the indirect transition (shaded in blue).
α(E)ϕbb(E). The absorption coefficient is taken from ref 17 and is based on a combination of photothermal deflection spectroscopy and photoluminescence data. In addition, Figure 3 shows fits to the experimental α based on the theoretical relations for direct αd ∝
E − Egd and indirect transitions
αi ∝ (E − Egi )2 . Here, Egd and Egi are the direct and indirect band gap, respectively. As already observed in Figure 1f of ref 25, the absorption coefficient of MAPI has a region between 1.555 and 1.655 eV that is quite well-described by the equation for an indirect band gap, while for higher energies the absorption coefficient fits better to the square-root-like behavior expected for a direct transition (see also Figure S3). At lower energies a small band tail is visible both in our data and in ref 25. If we combine the part above 1.555 eV via
αeff
⎧ ⎛ i ⎞2 ⎪ ⎜ E − Eg ⎟ α for Egi < E < Egd ⎪ 0,i⎜ kT ⎟ ⎝ ⎠ ⎪ =⎨ ⎪ ⎛ E d − E i ⎞2 ⎛ E − E d ⎞1/2 g g ⎪α ⎜ g ⎟ ⎜ ⎟ + α for E > Egd 0,d ⎜ ⎟ ⎪ 0,i⎜⎝ kT ⎟⎠ kT ⎝ ⎠ ⎩
(9)
we obtain the shaded areas as the contributions of direct and indirect band gap to the integral ∫ α(E)ϕbb(E)dE that determines Beffn0p0 (see also Figure S3). It is clear that if this 1269
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split as suggested here and similarly in ref 25 is sensible, the indirect transition provides the main contribution to Beff. Therefore, the situation in MAPI is in no way comparable to the situation described in Figures 1 and 2 that could be helpful in principle. In conclusion, we have shown that having an indirect band gap with a small absorption coefficient close to a direct band gap with high absorption coefficient could indeed increase the efficiency of a solar cell in the radiative limit in the presence of low mobilities. For higher mobilities, the efficiency in the radiative limit cannot increase by going away from a stepfunction like absorptance. In the limit of nonradiative recombination being dominant, there is at least no obvious reason why the indirect band gap would help. However, this would be a point that requires further investigation. Finally, we observe that in the special case of MAPI, the indirect band gap seems to be the transition that dominates the integral ∫ α(E)ϕbb(E)dE and therefore radiative recombination and luminescence. The fact that indirect band gap just below the direct one can be beneficial for charge collection is not a novel phenomenon but rather an implementation of a general principle that has been previously used in photovoltaics, e.g., in organic bulk heterojunction solar cells. Here, the use of two different molecules allows charge carrier recombination to be slowed at the cost of reducing the energy of the charge-separated electron−hole pair relative to the photogenerated exciton. Thus, combinations of two different transitions with largely different absorption coefficients can be a way of compensating low mobilities, but for materials that have high absorption coefficients, low defect densities, and reasonably high mobilities, it would likely be best to avoid weakly absorbing transitions below the main absorption edge.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00236. Simulations of efficiency as a function of mobility for lower built-in voltages or higher doping densities; details on the fit of the absorption coefficient using eq 9 (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Thomas Kirchartz: 0000-0002-6954-8213 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS T.K. and U.R. acknowledge support from the DFG (Grant KI1571/2-1 and RA 473/7-1). We also thank Tom Markvart (Prague), Jean-Francois Guillemoles (IRDEP, CNRS), and Urs Aeberhard (Jülich) for discussions on the topic of the Letter.
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