What are defects?

Defects are imperfections in the order of the lattice structure of a solid. Together with the bandgap and its electronic structure, the defects in a semiconducting material control its optoelectronic properties. Defects can promote charge transport by generating extra free charge carriers (doping defects) or can impede it by trapping (which can affect the optical absorption and emission from the material) or scattering the carriers. Such imperfections can be intrinsic, such as missing, extra, or misplaced atoms, or extrinsic (i.e., different from the constituents of the pure material).^{Reference Pankove12,Reference Queisser and Haller13} Getting control over defects is a central condition for putting a semiconducting material to use for an intended optoelectronic function, and achieving such control presents a challenge for every new material.

What, then, is unusual about HaPs in terms of defects?

In terms of the need for controlling defects to achieve or improve given functions, HaPs are no exception. Thus, understanding creation/annihilation, the density, dynamics, electrical, and optical activity of defects are all critical to understanding how HaPs behave in devices. For example, their doping efficiency will translate into resulting junctions being p–n (n–p) or p–i–n (n–i–p) type, crucial information for interpreting something as basic as current–voltage characteristics.

Although HaPs can be prepared from solution, under normal pressure, in a low (= near-room) temperature process, the steep absorption onsets and small Urbach energies^{Reference Wolf, Holovsky, Moon, Löper, Niesen, Ledinsky, Haug, Yum and Ballif14} (vide infra), long carrier lifetimes and diffusion lengths,^{Reference Stranks, Eperon, Grancini, Menelaou, Alcocer, Leijtens, Herz, Petrozza and Snaith15} and high degree of crystallinity (sharp XRD peaks)^{Reference Yang, Zhou, Zeng, Jiang, Padture and Zhu16,Reference Zhu, Fu, Meng, Wu, Gong, Ding, Gustafsson, Trinh, Jin and Zhu17} all point to low densities of structural, electrically, or optically active defects, in contrast to what one would expect from the conditions of preparation. The same low densities can be reached for other semiconductors, but only with sophisticated technologies, much time, and effort. Results from calculations, made within the static defect picture^{Reference Motti, Meggiolaro, Martani, Sorrentino, Barker, Angelis and Petrozza10,Reference Shi, Yin, Hong, Zhu and Yan18} (see next section), yield low-formation energies for most intrinsic point defects in HaPs, with shallow charge/discharge energy levels, which leads to interpretations that these defects interfere little with carrier-transport and device operations. The same approach shows that other point defects (antisites and some interstitials) can form deep defects and those have been argued to be primarily responsible for charge recombination. Given the high-formation energies of the latter group of defects (antisites and some interstitials), their density should be low, which would make their effect small to minimal.

Defect densities in HaP single crystals and polycrystalline films

Measuring defect densities is nontrivial. The studies that used techniques accepted as state of the art for “classical” semiconductors^{Reference Schroder19} yield ultralow values of 10⁹–10¹¹ cm^–3 for near-room-temperature-grown single crystals. While it is possible to achieve such densities for Si or GaAs, this is only for float-zone-grown crystals or molecular beam epitaxy (MBE)-grown films, respectively

For CsPbBr₃, single crystals can be grown from the melt at ~600°C or from solution at ~60°C. The defect densities are 4–5 orders of magnitude higher for the former than for the latter.^{Reference Saidiminov, Haque, Almutlaq, Sarmah, Miao, Begum, Dursun, Cho, Murali, Mohammed, Wu and Bakr20,Reference He, Matei, Jung, McCall, Chen, Stoumpos, Liu, Peters, Chung, Wessels, Wasielewski, Dravid, Burger and Kanatzidis21} Figure 1 shows a plot of defect density versus defect formation energy, calculated for Schottky defects, with ~1.65 eV defect formation energy (the value found from Br isotope tracer studies in PbBr₂)^{Reference Williams and Barr22} at 300 K (RT) and 600 K (as a representative value for the melt-grown sample, slowly cooled down over hours). The plot shows the orders of magnitude higher densities at 600 K, explaining the previously noted experimental results; we note that this HaP shows stronger defect stabilization and slower self-repair than the corresponding HaPs with organic cation, instead of Cs.^{Reference Ceratti, Rakita, Cremonesi, Tenne, Kalchenko, Elbaum, Oron, Potenza, Hodes and Cahen23}

Figure 1. Calculated defect density as a function of formation energy of defects at T = 300 K and 600 K, assuming (static) Schottky defects. The pale purple band is an aid to the eye for the densities of these defects, with a formation energy ~1.65 eV, the experimental value determined from Br isotope tracer studies in PbBr₂.^{Reference Williams and Barr22} The densities are higher at higher temperatures, which fits the higher densities found for high-temperature (~600 K) melt-grown than for solution-grown (near RT) CsPbBr₃ crystals. Courtesy of Y. Rakita, Weizmann Institute of Science.

For thin polycrystalline films of most HaPs, representative defect densities are between 10¹³ cm^–3 for evaporated and 10¹⁵–10¹⁶ cm^–3 for solution-grown films; also these values are low considering the preparation methods.^{Reference Brenner, Egger, Kronik, Hodes and Cahen1,Reference Babu, Giribabu and Singh2}

Self-repair of defects

Before we explain how such low-defect densities can be possible, let us first look at how these can be maintained. This question is connected with the vexing one: Are the materials intrinsically stable or metastable, and if they are metastable, over what time scale?

Defects can occur in the bulk and at the surfaces (and interfaces) of a semiconductor. The latter complicate probing the bulk, which most computations deal with, because nearly all methods to check for defects include, often in a disproportionate way, the surface of the sample, which, from the point of view of an ordered lattice, is a defect. Localized defective points/regions are more likely to form at surfaces than in the bulk where the formation energies are often larger. As every material has surfaces, and contacts require surfaces to form interfaces, the need to passivate surfaces has accompanied the development of every semiconductor, before its possible use in a device. Here, we will not discuss HaP surface defect passivation as it is an often-reviewed topic.^{Reference Kim, Baillie and Huang24–Reference Wang, Bai, Tress, Hagfeldt and Gao27} However, comparing bulk and surface defect creation and their possible repair can help understand defect formation in HaPs.

Ceratti et al. used two-photon confocal microscopy and probed the stability in the interior (bulk) of APbBr₃ (A = CH₃NH₃⁺, [i.e., methylammonium (MA)], HC(NH₂)₂⁺ [i.e., formamidinium (FA), or Cs⁺]) single crystals, by creating localized damage with strong excitation (up to tens of times solar intensity equivalent), and observing resulting changes in photoluminescence intensity.^{Reference Ceratti, Rakita, Cremonesi, Tenne, Kalchenko, Elbaum, Oron, Potenza, Hodes and Cahen23} They found that over seconds to hours, depending on the A cation, luminescence intensity was restored, indicating self-repair. While self-repair was also reported earlier for CuInSe₂, the parent compound of copper indium gallium selenide (CIGS), Cu(In,Ga)Se₂ and used in commercial solar cells, it is much more efficient in the HaPs.^{Reference Rakita, Lubomirsky and Cahen28} In the analogous one-photon, instead of two-photon experiment, the (near) surface is probed and kinetic effects dominate, because at the surface materials can be lost or irreversibly changed (e.g., oxide formation, due to reaction with air^{Reference Ceratti, Rakita, Tenne, Goldia, Cremonesi, Kalchenko, Elbaum, Oron, Potenza, Hodes and Cahen29}).

We stress that self-repair of defects is an actual process, different from an outcome, such as defect tolerance.^{Reference Berry, Buonassisi, Egger, Hodes, Kronik, Loo, Lubomirsky, Marder, Mastai, Miller, Mitzi, Paz, Rappe, Riess, Rybtchinski, Stafsudd, Stevanovic, Toney, Zitoun, Kahn, Ginley and Cahen30} The latter was proposed to be the result of the HaP property in which the valence band maximum (and to some extent, the conduction band minimum [as explained elsewhere])^{Reference Zakutayev, Caskey, Fioretti, Ginley, Vidal, Stevanovic, Tea and Lany31,Reference Brandt, Stevanović, Ginley and Buonassisi32} has an antibonding character. Another important difference is that in defect-tolerant materials, defects are present (they exist in the lattice), but they do not affect the material's (opto)electronic properties. The presence of significant defect densities in the bulk of high-quality crystals, even if their effects are tolerated by the material, is something that awaits experimental proof.

In defect-repairing materials, defects disappear or at least their density decreases after defects are formed. For self-repair to be effective, a low baseline defect density is required. How such low defect density is possible in HaPs is explained in detail in Reference 28. Briefly, the small negative free energy of formation of the HaPs for which this was determined, especially MaPbX₃ (X = Br or I), leads to a situation where the energy required to form a defect is higher than that to decompose the material (in this case, into its binaries MA halide (MAX) and lead halide (PbX₂)).^{Reference Rakita, Lubomirsky and Cahen28} While such energy balance is a necessary condition to obtain the low defect density (and self-repair), it is not a sufficient one. This is illustrated by the fact that the same condition holds true for the case of CuInSe₂. The other condition that needs to be met is that the activation energy for the formation and decomposition process must be smaller than the energy required to form a defect. The latter condition holds for MA- and FA-HaPs, but not for CuInSe₂.

What could be defects in HaPs?

Shortly after the discovery of the remarkable photovoltaic behavior of Pb halide perovskites, results of calculations that consider point defects that we discussed briefly above, started to appear. The basis is the model developed for materials such as oxides, which considers the point defects that exist or are formed as static in the lattice. This model is a useful and powerful way to describe and understand defect chemistry and physics, with significant predictive power in classical semiconductors, and it was applied successfully also to lead dihalides.^{Reference Schoonman33} The condition for validity is that the defects live long enough so that they can be considered static for measurements carried out on the materials and for the functions that the materials are meant to fulfill in devices. The results from the calculations were then interpreted to rationalize the low electronic carrier densities (associated with low defect densities and low optically active defect densities; see the section on the Urbach tail).

The relevance, or at least the completeness, of this view was called into doubt by two recent studies. Cohen et al. found,^{Reference Cohen, Egger, Rappe and Kronik34} by combining molecular dynamics and density functional theory (DFT), that the concept of a defined energy of the defect state in HaPs is problematic (Figure 2), because of the strong lattice vibrations (which also lead to anharmonicity). Rakita et al. argued that there are two types of defects in HaPs: the “classical” static ones, for which the previously discussed calculations were made, and “dynamic” ones, with ~1 ps lifetimes.^{Reference Rakita, Lubomirsky and Cahen28} The density of the former is close to the thermodynamically dictated one, explaining the low values, deduced from experimental data, while the densities of the latter, calculated from hopping probabilities, approach ~10²¹ cm^–3.

Figure 2. Density functional theory-based molecular dynamics calculations, taking into account the structural dynamics in Cs-lead bromide perovskite and GaAs, showing the point defects V_Br (Br vacancies in CsPbBr₃) and As_Ga (As on a Ga site in GaAs) properties. Theconduction band minimum (CBM), valence band maximum (VBM), and the specific defect level are shown as eigenvalues (as a function of time along the molecular dynamics trajectory) for (a) CsPbBr₃ and (b) GaAs. Reprinted with permission from Reference 34. © 2019 American Chemical Society.

In addition, there are defects at the surfaces, interfaces, and grain boundaries (GBs), which were argued to be the ones that dominate in thin polycrystalline films. Simple calculations show how this can be—if we assume the defect fraction per unit surface/interface area to be 0.1–1 at.%, then for a polycrystalline HaP film with average grain size of ~300 nm, the surface defect density varies from 3 × 10¹⁶–3 × 10¹⁷/cm³ (i.e., especially for medium-low bulk defect densities, surface/interface defects can easily dominate polycrystalline HaP films). As noted earlier, surface defect activity can be diminished by surface passivation.

Soft lattice and entropic stabilization

Until recently, all high-quality optoelectronic materials were rigid ones with high elastic constants (bulk modulus), quite different from HaPs, which have a softer lattice structure. The(bulk) Young's moduli for HaPs, though higher than for organic semiconductors, are close to an order of magnitude lower than for classical semiconductors.^35–37 The soft nature of the HaP structure originates from weaker electrostatic interactions within the Pb-halide sublattice. These weak interactions lead to low-activation energies for atom displacements, making the bonds highly polarizable, which increases configurational entropy-driven stabilization.^{Reference Rakita, Lubomirsky and Cahen28} Such stabilization is important in HaPs, with enthalpies of formation from their binaries close to zero or barely positive.^{Reference Ciccioli and Latini38}

Another corollary of the soft lattice is that the deformation potential, dE_G/dP, where E_G is the bandgap and P the applied pressure, is low (and positive), for HaPs. This small positive deformation potential, which results from the anti-bonding nature of the valence band, allows the HaPs to tolerate defect-induced strain fields, much better than classical semiconductors because the soft lattice of the HaPs can relax, while it will break for classical semiconductors. One result is that defect formation in HaPs from internal strains will be smaller than usual.^{Reference Rakita, Kirchartz, Hodes and Cahen39}

Defects: Shallow, deep, both, or none?

Results of electrical measurements have been used to arrive at energies and densities of defect levels in the gap of HaPs, using known models for defect levels in semiconductors. While in several cases, deep energy levels have been deduced in this manner,^{Reference Levine, Vera, Kulbak, Ceratti, Rehermann, Márquez, Levcenko, Unold, Hodes, Balberg, Cahen and Dittrich40,Reference Azulay, Levine, Gupta, Kulbak, Bera, San, Simha, Cahen, Millo, Hodes and Balberg41} we currently lack clear direct evidence for such. As it is likely that surface/interface defects can create such levels, it is possible that their densities or optical cross sections are low.

Things are different for shallow defects, where optical, photoemission, and scanning surface spectroscopic data show evidence for such defects, more so in the bromides than the iodides. Still, a powerful method to look for tail states (a ubiquitous type of shallow defect levels, namely measuring the extension of optical absorption below the bandgap, or the tail) generally shows low levels. Tail states are characterized by the Urbach parameter, E_U, composed of two components—a static component, which exists even at 0 K, and reflects static, often structural disorder, and a temperature-dependent component, associated with the lattice dynamics. E_U depends on the material and sample preparation. Top-quality GaAs samples show the lowest E_U, ~7 meV; for good photovoltaic (PV)-quality semiconductors, 10–20 meV is typical, while higher values reaching >100 meV are also possible, for instance for statically disordered amorphous Si.

Most iodide and iodide-rich HaPs have E_U (@RT) ~ 15 meV. The temperature-dependent component is the cause for most of this energy, consistent with the soft HaP lattice (i.e., strong electron–phonon coupling). Most remarkably, the ~3.8 meV static component of E_U for polycrystalline MAPI films is the lowest known value for any semiconductor.^{Reference Ledinsky, Schönfeldová, Holovský, Aydin, Hájková, Landová, Neyková, Fejfar and Wolf42} For HaP single crystals, this component may be even lower. This result is consistent with Rakita et al.'s conclusions previously noted (i.e., dynamics heavily dominate HaP defect chemistry and physics).

For MAPbBr₃, E_U is close to 30 meV—maybe a clue to the observed higher voltage loss (deviation of highest V_oc from theoretically best possible value) for Br-HaP-based PV cells than for their iodide counterparts. For mixed HaPs (double/triple A cations and mixed halides I, Br), some compositional disorder as well as extra structural disorder is expected to increase E_U (ignoring possible mixed-phase formation). Still, such mixed systems with low Br content yield MAPI-like values,^{Reference Hoke, Slotcavage, Dohner, Bowring, Karunadasa and McGehee43–Reference McMeekin, Sadoughi, Rehman, Eperon, Saliba, Hörantner, Haghighirad, Sakai, Korte, Rech, Johnston, Herz and Snaith45} while high Br content renders MAPbBr₃-like E_U values.^{Reference Sadhanala, Deschler, Thomas, Dutton, Goedel, Hanusch, Lai, Steiner, Bein, Docampo, Cahen and Friend46}

It may seem that the band tails, as long as they remain shallow (i.e., relatively low E_U values), should not affect recombination greatly, since recombination is associated with deeper levels (which is why, from theoretical studies of static defect levels for HaPs, electronically inactive defects are predicted for most native defects). However, shallow defects can affect cell performance via trapping or, for even shallower levels, such as those from tail states, by pinning quasi-Fermi levels, preventing maximum V_oc being reached. Furthermore, theV_ocloss also increases if E_U starts to exceed the thermal energy, kT. This is because then the reverse saturation current (J ₀) of the cell will be dominated by the semiconductor's thermal emission, due to the band tailing, rather than by what is expected for a black body at the operational temperature, thus leading to a reduction in V_oc.^{Reference Jean, Mahony, Bozyigit, Sponseller, Holovský, Bawendi and Bulović47} Nayak et al. showed that if E_U in, for example, MAPI were moderately large, say 40 meV, this would result in a V_oc loss of nearly 400 mV. Such loss would be on top of the loss, referenced to the bandgap energy, dictated by the Shockley–Queisser (SQ) model. We note that some of today's HaP cells show total V_oc losses that are <100 mV (in addition to the SQ loss), comparable to those of today's commercial cells.^{Reference Nayak, Mahesh, Snaith and Cahen48–Reference Liu, Krückemeier, Krogmeier, Klingebiel, Márquez, Levcenko, Öz, Mathur, Rau, Unold and Kirchartz50}

A clear and direct correlation between E_U and V_oc loss in HaP (and other) cells from the maximum one, expected from the SQ model, has been well documented.^{Reference Ledinsky, Schönfeldová, Holovský, Aydin, Hájková, Landová, Neyková, Fejfar and Wolf42,Reference Jean, Mahony, Bozyigit, Sponseller, Holovský, Bawendi and Bulović47,Reference Nayak, Mahesh, Snaith and Cahen48,Reference Wolf, Holovsky, Moon, Löper, Niesen, Ledinsky, Haug, Yum and Ballif51} While Ledinsky et al. deduced from this observation that the V_oc loss is directly dictated by the E_U value of the absorber layer, Miller et al. reported that, although their cells had MAPbI₃ films with uniformly low E_U (15–18 meV), PV performance varied greatly (V_oc less so, but still from 800 to 970 mV), with a highest cell efficiency of 8.4% (i.e., the cells were rather mediocre). The authors deduced the presence of two defect levels, at ~1.35 eV and at mid-gap, 0.75 eV, which may explain their low V_oc values.^{Reference Miller, Eperon, Roe, Warren, Snaith and Lonergan52} We note this to stress that correlating E_U and voltage loss will work only for semiconductors with low/shallow recombination levels.

Eliminating defects in HaPs

Control over type and density of defects in a semiconductor is of paramount importance for its use in optoelectronics. As noted already, static bulk defect densities in HaPs are typically low, close to their thermodynamically dictated minimum value.^{Reference Rakita, Lubomirsky and Cahen28} The net carrier densities in HaP single crystals are governed by these static defect densities, while the dynamic defects with ~ps lifetime are virtually transparent for charge-carrier transport. It is reasonable to assume that in the rapidly crystallized polycrystalline HaP films, it will be defects on external surfaces and internal ones (GBs) that control net carrier densities. Such behavior is not unique, and it has been shown that surface adsorption and desorption of O₂ can control carrier densities in polycrystalline films of chalcogenide semiconductors.^{Reference Cahen and Noufi53} Furthermore, doping with Bi³⁺ to control bulk carrier densities has shown this process to be inefficient.^{Reference Nayak, Sendner, Wenger, Wang, Sharma, Ramadan, Lovrinčić, Pucci, Madhu and Snaith4}

Much of the nonempirical HaP research after 2016 focused on managing the defects, both surface and bulk, to harness the full potential of these materials in devices. Several articles discuss strategies for, and results on, surface/interface passivation.^{Reference Qiu, He, Ono and Qi7–Reference Jin, Debrove, Keshavarz, Scheblykin, Roeffaers, Hofkens and Steele11,Reference Kim, Baillie and Huang24–Reference Wang, Bai, Tress, Hagfeldt and Gao27} Such approaches work quite well for reducing defects (mostly extrinsic ones) at GBs, and at the contacts, the HaP/ETL and HaP/HTL interfaces. Incorporating additives (both organic and inorganic) in the precursors for film growth can reduce defect formation and/or passivate defects on polycrystalline HaP films. The extent to which such passivation is effective depends on the nature of the additives, manner of preparation, and choice of precursors. Successful experiments on devices made with polycrystalline thin films improved overall performance (i.e., efficiency and stability) of the dark (diode), and photovoltaic parameters of devices. Interfacial engineering also can enhance electronic coupling between the HaP and its contacts, by suppressing putative deep, interface defect states.^{Reference Qiu, He, Ono and Qi7}

Reducing GB density by increasing the grain size has been used to deal with GB-related defects and several methods have been explored to achieve this, including thermal/solvent annealing, hot-casting, tuning precursor molarity, and additives. Additives in the precursors for the desired HaP have been reported to have a multifaceted impact on the perovskite devices. They not only improve grain sizes, but also passivate presumably the Pb- and I-related defects at the GBs and improve the overall stability of the devices. Lately, two-dimensional halide perovskites (2D HaPs) have been extensively used as interfacial layers between the three-dimensional HaP absorber and its contacts, as the 2D-modified ones show improved overall cell performance, which was ascribed to better stability against moisture, as well as defect passivation.^{Reference Ran, Xu, Gao, Huang and Dou8,Reference Ono, Liu and Qi9,Reference Kim, Baillie and Huang24,Reference Yang, Chen, Xu, Zhang, Liu, Liu and Yuan25}

Summary and outlook

The question that this article tries to bring to the fore is if, in terms of defects, dopants, and their densities, HaPs are business as usual, with “classical” point defects as the usual suspects, or are one example of a new type of material with a new type of behavior. Taking such an approach a step further, should we look for material systems, which, like HaPs, have low static and high dynamic defect densities, where our “classical” view of point defects may not describe (well) the nature and densities of optoelectronically relevant defects? As a corollary, we can ask if we are attempting to move too far too fast with insufficient data (as much remains unclear or unexplored) to explain/interpret/analyze experimental data with existing defect models, and as work progresses, all will fall in place, with one or another of the models? These are not trivial questions. Our present model about the nature and density of point defects has worked well for classical semiconductors, crystalline solid electrolytes, and many inorganic–optical materials. Will the lattice in HaPs (and HaP-like materials) turn out to be a boojum (one type of Lewis Carroll's imaginary snark that can cause “soft and sudden vanishing away”)?^{Reference Mermin54,Reference Carroll55} In such case the kinetics of defects and their thermodynamics, especially as far as their stability is concerned, should be considered on equal footing. This would be the case for both theoretical calculations and interpretations of experimental results, such as exploring possibilities to dope the materials. Alternatively, possibly the static point defect model will join other venerable models, such as the Debye–Hückel one model for dilute ionic solutions or the free electron model for ordered (semi)conductors, that we can adapt for use well beyond the assumptions of the original model?

To answer some of these questions, the predictive power of the results of theoretical calculations, pertaining to defects in HaPs (single defects as well as defect complexes), should be scrutinized. This process may be complicated by the fact that most of the existing experimental data on defects appear to be related to defects at GBs and surfaces/interfaces.

To bring clarity, there is a need to develop computational ability that can handle the complexities of such defect dynamics and related corrections. Additionally, we need to design experiments that can distinguish bulk from non-bulk effects as clearly as possible, which implies the need to use single crystals. Such studies will be boosted when epitaxial thin films become available. Also, we are still lacking halide isotope tracer experiments, which may not be practical for iodides, but seemingly possible for bromides.

Another approach is to try to “fool” the system (e.g., by introducing large enough point [or other] defect complexes), so that steric hindrance can stabilize them. A different challenge is to find ways to prove the existence or absence of defect tolerance, not as an outcome, but as a process (i.e., where are the defects that the system tolerates and can we find and identify them?).

By leaving the reader with these questions, we trust that we have been able to convey some of the fascination with, and excitement about this aspect of HaP research, results of which may well have implications in other areas of materials research.

*A boojum is one type of Lewis Carroll’s imaginary snark that can cause “soft and sudden vanishing away” (Figure 3).^{Reference Carroll55}

Figure 3. Back cover illustration by Henry Holiday of Lewis Carroll's The Hunting of the Snark: An Agony in Eight Fits.

N. David Mermin introduced the concept into the natural sciences (see Reference Reference Mermin54).