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Understanding the Implication of Carrier Diffusion Length in Photovoltaic Cells
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be obtained from cells made using these crystals (up to 3 mm thick), and to get such values of IQE, at least one of the charges has to move from one side of the crystal to the other with relatively small losses. According to the definition of charge diffusion length given in eq 2, this rationale is correct, and indeed at least one charge has a diffusion length longer than the crystal thicknesses used. (In ref 2, it was shown that both electrons and holes possess these long diffusion lengths.) However, to put these results in perspective, it is instructive to look back at some comparably long diffusion lengths in other semiconductors that are not necessarily very high quality in terms of high diffusion coefficients/mobilities and long lifetimes. In order to further elaborate this point, we consider two specific examples. (1) Dye-Sensitized Solar Cell (DSC). The DSC most commonly contains a layer of sintered TiO2 nanoparticles (usually some tens of nm in size) about 10 μm thick (a few μm for a solid state cell) that acts to accept photogenerated electrons from the dye molecules and transport them to the (conducting glass) contact. Values of IQE close to 100% are commonly obtained from these cells. Therefore, the diffusion length of electrons in this TiO2 layer is >10 μm. However, in common with other transition-metal oxides, TiO2 is normally considered to be a low-mobility semiconductor. The electron mobility (and lifetime) in TiO2 vary enormously depending on (a) whether it is rutile or anatase and (b) the morphology of the film (single crystal at one extreme and the nanoporous films typically used in dye cells at the other). For porous anatase, Könenkamp has derived values of electron mobility and lifetime3 from which values of Ld of ∼10 nm can be calculated using eqs 1 and 2. It should be noted that the commonly used (in DSCs) P25 TiO2 has an average particle diameter of 20−25 nm, which is considerably larger than the ∼6 nm used in ref 3, and therefore from that aspect, a larger Ld is expected (mainly due to the larger number of grain boundaries/unit length of sample). On the other hand, P25 is a mixture of ∼25% rutile and 75% anatase (1:3 ratio). Both electron mobilities and lifetimes in rutile are much smaller (by at least an order of magnitude for each parameter as measured in single crystals) than those in anatase.4,5 This will therefore compensate to a greater or lesser extent the mismatch in particle size between the 6 nm in ref 3 and the 20−25 nm in P25. It is difficult to estimate this compensation as it is unlikely to be simply a 1:3 ratio. The conduction band of anatase, which is more negative than the rutile phase (by ∼0.2 eV), causes the electrons to preferentially accumulate in the rutile phase. However, this will also depend on the degree of electron trapping in the two phases. (2) Single-Crystal CdS(e) Photoelectrochemical Cells. An even better example than the DSC, and one that takes us further back in time, is the use of single-crystal Cd chalcogenide and
he purpose of this Viewpoint is to dispel a commonly held misconception when comparing diffusion lengths and discuss how variation in the measuring techniques can bring about differences in the measured values. The diffusion length, Ld, of electrons or holes in a semiconductor is defined by the average distance the relevant charge moves in the semiconductor. It is influenced by the average distance the relevant charge moves in the semiconductor (for example, in photovoltaic cells, which is the topic of interest in this Viewpoint) and recombination/extraction from the semiconductor. Diffusion implies movement of charge carriers directed by a concentration gradient. If the movement is due to an electric field, then the term “drift” is used. The diffusion coefficient (D) and the equivalent term in the presence of a field, mobility (μ), are related to each other by the Einstein relation
D=μ
kT q
(1)
and Ld = (Dτ )1/2
(2)
where τ is the charge lifetime from generation until recombination or extraction. If the cell is not operating at open circuit, that is, charge is extracted, then the lifetime will clearly be less because of removal of the extracted charge. However, this is no longer an intrinsic property of the absorbing semiconductor itself but depends also on the interfaces present between the semiconductor and charge extraction phases. The charge lifetime refers to the minority charge carrier lifetime for semiconductors that are clearly either n-type or ptype; differentiation into majority and minority carrier lifetimes is not obvious for an intrinsic semiconductor such as the intrinsic semiconductor in a p-i-n cell. We will return to this important point later. Very often, the two terms (D and μ) are considered simply as the “diffusion coefficient”. It is not always known if the charges in question move in a field or not or to what extent both diffusion and drift are involved. For this reason, it is better to use the term diffusion/drift, unless it is known that only one or the other mechanism is operative. In this Viewpoint, we will be discussing mostly charge movement in a field-free semiconductor and therefore will mostly be using the term “diffusion”. As an example, we discuss here two recent back-to-back papers that focus on long diffusion lengths in methylammonium lead halide single crystals.1,2 Charge carrier diffusion lengths in single crystals of CH3NH3PbI3 of ∼51 and 1752 μm in full sun and >3 mm in weak light2 have been reported in these papers. In ref 1, CH3NH3PbBr3 crystals were also studied with diffusion lengths of ∼10 μm reported. In particular, the rationale for the claim of >3 mm initially appeared to be based on the fact that high internal quantum efficiencies (IQEs) can © 2015 American Chemical Society
Published: October 15, 2015 4090
DOI: 10.1021/acs.jpclett.5b02052 J. Phys. Chem. Lett. 2015, 6, 4090−4092
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The Journal of Physical Chemistry Letters
interface causes holes to move to the interface while electrons move into the semiconductor interior where they diffuse for most of the distance until they reach the ohmic contact, which is the electron sink that constitutes the driving force (diffusion gradient) for the electrons to move directionally over this fieldfree region. Once the electrons have moved roughly a micrometer from the barrier interface (which they do mainly by drift), the rest of their lifetime is spent in a region that contains essentially no holes. This is the important issue in such cells: most of the electron journey to the ohmic contact is spent in a region where there is no counter charge for the electron to recombine with. This scenario, of course, depends on the ability of the holes to be extracted efficiently at the barrier interface. The efficiency of this extraction will depend on the field at that interface, the mobility of the holes (the higher the mobility, the faster they reach the interface), the rate of hole transfer, and the energetics of that interface. The initial charge extraction is rather different in the cases of the DSC (it is assumed to be only diffusive) and in the PEC and perovskite cells (where it is normally mainly drift). However, regardless of the mechanism of this initial extraction, in a macroscopic (i.e., thickness much greater than the light absorption depth) sample, most of the carrier movement occurs by diffusion of one of the carriers from the side where the light is absorbed to the other side, without encountering any appreciable concentration of the counter charge. In a nonideal system, electrons can be trapped on their way to the ohmic contact, although even this need not limit the current extraction, for example, if the traps are eventually filled, although a high electron trap density could lead to a barrier to electron flow. The above two examples actually describe the diffusion length of majority carriers, electrons in both cases because both the TiO2 and CdS(e) are n-type semiconductors. There is a difference, however, between these two examples and the perovskites (or specifically the Pb-based perovskites; the Snbased ones are generally fairly highly doped) in that the Pb perovskites as normally made tend to be ambipolar and rather intrinsic in the absence of a charge-defining interface. In accordance with this property, low hole (slightly p-type) concentrations on the order of 1010 cm−3 have been reported.1,2 Hence, we infer that there is no real meaning to the terms “majority” and “minority” carriers in the pristine perovskite system, although reference to these terms may be valid in selected regions of the perovskite, depending on the interfaces present. We note that in a thin-film p-i-n cell where the thickness of the “i” layer is comparable to the light absorption depth, all (or almost all) of the charges move in a field, and the correct term for the distance that the charges move is the drift length. It is interesting to note that long estimated diffusion lengths in semiconductors with poor or mediocre “conventional” diffusion lengths never created any concern or excitement in the past. It was recognized that in such cases, the diffusion length could be large, but this was not equated with the more commonly understood meaning of diffusion length in a material where both charge carriers were present. However, it seems that this distinction needs to be made clear, particularly for the halide perovskites, which already have long “conventional” diffusion lengths. Diffusion/drift lengths are best measured by measuring the mobility (or diffusion coefficient) and charge lifetime through eq 2 above. Such measurements were made in refs 1 and 2, and
some other semiconductor photoelectrodes in photoelectrochemical cells (PECs).6−8 In these examples, single crystals of a range of semiconductor crystals, for example, CdSe, CdSe0.65Te0.35, or GaAs, typically ∼1 mm thick, were contacted with an ohmic contact on one side (usually Ga−In) and an electrolyte, typically polysulfide, polyselenide, or polyiodide, to give a liquid junction Schottky contact on the other. Again, high values of IQE were obtained. Because the cells were illuminated through the electrolyte side and the semiconductors were ntype, this means that the electron diffusion length was greater than the (∼1 mm) thickness of the crystals. In ref 6, a polycrystalline pressed pellet of CdSe, 1.5 mm thick, was also compared with a CdSe single crystal and gave 70% of the current obtained, showing that these apparent very large diffusion lengths were not limited in principle to single crystals. What these two examples (and the two single crystal perovskite papers) have in common is that these long diffusion lengths are for charges (electrons in most cases, holes in one case in ref 2) that, for most of their lifetime, essentially do not encounter any counter charge with which they could recombine. For the DSC case, electrons are selectively and rapidly injected into the TiO2, and because there are essentially no holes in the TiO2, the electrons will have a long lifetime in that phase (assuming they are not lost by other pathways such as from the TiO2 to the electrolyte or solid hole conductor). As long as the “ohmic” contact (which need not necessarily be a perfect ohmic contact) acts as a good sink for electrons, then the diffusion gradient set up mainly at the semiconductor− ohmic contact interface will drive the electrons in the TiO2 to the ohmic contact. For the other examples (e.g., CdS(e)/polysulfide or the perovskite single crystals), the mechanism is different, but the logic is the same. Scheme 1 shows such a generic single-crystal Scheme 1. Macroscopic Single Crystal of a Semiconductor with a Transparent, Barrier-Forming (Schottky) Contact at One Side (Where Light Is Incident) and an Ohmic Contact at the Othera
a
The space charge layer formed in the semiconductor is very narrow (typically hundreds of nm and roughly comparable with the light absorption depth) compared to the macroscopic crystal thickness (typically in the mm range) and is exaggerated in this scheme for clarity.
cell comprised of a light-absorbing, macroscopic single crystal sandwiched between an “ohmic” contact on one side and a barrier-forming contact (either liquid junction or solid) on the other. Light is incident on the barrier contact side, and the barrier contact is at least partially transparent to the incident light. Most of the light is absorbed in a fraction of a micrometer of the barrier contact/semiconductor interface (true for most direct band gap semiconductors), and most of the semiconductor is essentially field-free. However, the field at this 4091
DOI: 10.1021/acs.jpclett.5b02052 J. Phys. Chem. Lett. 2015, 6, 4090−4092
The Journal of Physical Chemistry Letters
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ACKNOWLEDGMENTS We acknowledge the support of the US−Israel Binational Science Foundation. We thank Juan Bisquert and David Cahen for helpful comments on the manuscript.
we compare them here for the methylammonium lead iodide perovskite. Space-charge-limited currents, measured by dark I− V plots, were used to extract the mobility (and trap density) in both references. Mobilities of 2.5 cm2 V−1 s−1 were reported for both electrons and holes in ref 1, while ref 2 found 160 and 25 cm2 V−1 s−1 for holes and electrons, respectively (all values compared here are approximately averaged from the range of values given). It may be noted that while the basic measurement method was the same for both references, the contacts to the crystals were not identical, and this difference may affect the extracted values of mobility. The charge carrier lifetime measurements in ref 1 were carried out by time-resolved photoluminescence, TRPL (1 μs bulk lifetime), and in ref 2 by transient photovoltage and impedance spectroscopy, IS (90 μs (1 sun), 200 μs (0.1 sun). Here, the method used is particularly important. While TRPL gives the lifetime in a region where both charge carriers are present (although it can be modified by the presence of charge extraction layers), photovoltage decay and IS measurements will probably be affected by the presence of the contacts and are likely to be more characteristic of a semiconductor containing a single charge for most of the measured lifetime than TRPL. Lifetime measurements in thick samples may also be subject to photon recycling, where radiative recombination results in a photon that is reabsorbed, thereby giving an anomalously high apparent lifetime.9 Combining the mobilities and lifetimes, these numbers lead to diffusion lengths of ∼5 μm in ref 1 and 175 μm (full sun) and 3 mm (weak light) in ref 2. The 3 mm value under weak light conditions appears to be extrapolated with the assumption of uniform excitation of the crystal. Such a situation would be valid for high-energy photons (X-ray and γ-ray) that are assumed to be homogeneously generated throughout the crystal thickness. While such an assumption may be true for high-energy radiation, it is certainly not applicable for normal visible light with about 4 orders of magnitude difference between the optical absorption depth and the crystal thickness. On the basis of the difference in measured lifetimes under 1 and 0.1 sun illumination given above, the diffusion length under the weaker light should be ∼250 μm (still a large number but more than an order of magnitude less than the 3 mm). Whether the much higher numbers in ref 2 are due, wholly or in part, to better-quality crystals or to differences in the measurements discussed above can be validated by measuring crystals of both groups by identical methods. We recognize the important contributions of these two and other groups in estimating the diffusion length of perovskite crystals. While the attempt to correlate longer diffusion length to crystal quality is reasonable, one needs to consider the differences in the measurements as well as assumptions made in the analysis when comparing the results.
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REFERENCES
(1) Shi, D.; Adinolfi, V.; Comin, R.; Yuan, M.; Alarousu, E.; Buin, A.; Chen, Y.; Hoogland, S.; Rothenberger, A.; Katsiev, K.; et al. Low TrapState Density and Long Carrier Diffusion in Organolead Trihalide Perovskite Single Crystals. Science 2015, 347, 519−522. (2) Dong, Q.; Fang, Y.; Shao, Y.; Mulligan, P.; Qiu, J.; Cao, L.; Huang, J. Electron-Hole Diffusion Lengths > 175 mm in Solutiongrown CH3NH3PbI3 Single Crystals. Science 2015, 347, 967. (3) Könenkamp, R. Carrier Transport in Nanoporous TiO2 Films. Phys. Rev. B: Condens. Matter Mater. Phys. 2000, 61, 11057−11064. (4) Forro, L.; Chauvet, O.; Emin, D.; Zuppiroli, L.; Berger, H.; Lévy, F. High Mobility n-Type Charge Carriers in Large Single Crystals of Anatase (TiO2). J. Appl. Phys. 1994, 75, 633−635. (5) Xu, M.; Gao, Y.; Moreno, E. M.; Kunst, M.; Muhler, M.; Wang, Y.; Idriss, H.; Wöll, C. Photocatalytic Activity of Bulk TiO2 Anatase and Rutile Single Crystals Using Infrared Absorption Spectroscopy. Phys. Rev. Lett. 2011, 106, 138302/1−138302/4. (6) Miller, B.; Heller, A.; Robbins, M.; Menezes, S.; Chang; Thomson, J., Jr. Solar Conversion Efficiency of Pressure Sintered Cadmium Selenide Liquid Junction Cells. J. Electrochem. Soc. 1977, 124, 1019−1021. (7) Parkinson, B. A.; Heller, A.; Miller, B. Effect of Cations on the Performance of the Photoanode in the n-GaAs/K2Se-K2Se2-KOH/C Semiconductor Liquid Junction Solar Cell. J. Electrochem. Soc. 1979, 126, 954−960. (8) Licht, S.; Tenne, R.; Dagan, G.; Hodes, G.; Manassen, J.; Cahen, D.; Triboulet, R.; Rioux, J.; Levy-Clement, C. High Efficiency nCd(Se,Te)/S2‑ Photoelectrochemical Cell Resulting from Solution Chemistry Control. Appl. Phys. Lett. 1985, 46, 608−610. (9) Ahrenkiel, R. K. Measurement Of Minority-Carrier Lifetime By Time-Resolved Photoluminescence. Solid-State Electron. 1992, 35, 239−250.
Gary Hodes* Weizmann Institute of Science, Rehovot 76100, Israel
Prashant V. Kamat
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University of Notre Dame, Notre Dame, Indiana 46556, United States
AUTHOR INFORMATION
Notes
Views expressed in this Viewpoint are those of the authors and not necessarily the views of the ACS. The authors declare no competing financial interest. 4092
DOI: 10.1021/acs.jpclett.5b02052 J. Phys. Chem. Lett. 2015, 6, 4090−4092
