When defects become ‘dynamic’: halide perovskites: a new window on materials?

Abstract

Although Pb Halide perovskites (HaPs) can be prepared as organic electronic materials, they resemble top-quality inorganic semiconductors, especially with respect to their low defect densities, as derived from optical and electronic transport studies. Among causes for such low defect densities were ‘defect-tolerance’ (proposed) and ‘self-healing’ (experimentally identified). We show that HaPs are likely an example of a class of materials that cannot support static bulk defect densities significantly above thermodynamically-dictated densities. The reasons are (a) the free energy to form HaPs (from binary halides) is less than the formation energies of (static) defects in them and (b) the small kinetic stabilization of such defects. We summarize the evidence for such a situation and conclude that higher defect densities in polycrystalline films likely result from the (expected) smaller defect formation energy at surfaces and grain boundaries than in the bulk. This situation directly limits the options for doping such materials, and leads to the counter-intuitive conclusion that a low free energy of formation (from the binaries) can lead to self-healing and, consequently, to low densities of static defects, to be distinguished from dynamic ones. The latter can be benign in terms of (opto)electronic performance, because of their relatively short lifetimes. We propose that the conditions that we formulated can serve as search criteria for other low defect density materials, which can be of interest and beneficial, also for applications beyond optoelectronics.

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Background

For (opto)electronic functions of semiconductors, structural defects (in the bulk and/or at their surfaces/interfaces) are often of decisive importance (see the ESI,† i). Free electronic charge carriers with energies at/near conduction or valence band extrema are prone to interact with structural defects. Defects can trap (localize) charge carriers and/or facilitate their mutual annihilation (i.e., electron–hole recombination). Defects, charged or neutral, may act (alongside lattice vibrations) as scattering centers for carriers during their transport and thus, reduce carrier mobility; all these processes are usually detrimental for (opto)electronics.^1,2

We associate defects in what we will call ‘classical’ semiconductors (like Si, GaAs, with tetrahedral coordination), with missing or extra atoms, where the extra ones can also be extrinsic, i.e., different from the atoms making up the semiconductor. All such defects often have electronic states inside the bandgap, E_G. For example, B or P in Si form p- or n-Si, respectively, via formation of states that have energies close to the valence band maximum, VBM, or conduction band minimum, CBM of Si, respectively, the so-called shallow defects. For a material such as CuInSe₂, intrinsic defects, primarily In on a Cu site, In_Cu, and Cu vacancies, V_Cu, determine its doping. The electronic states associated with these dopants affect the electronic carrier density, as usually observed in a (logarithmic) increase of conductivity with carrier concentration over 5–7 orders of magnitude.³

In general, one will try to minimize defects, which in extremum, becomes a very demanding task. Importantly, we can prepare materials that are spatially heterogeneous in terms of doping, with a p–n junction as the best-known example. Spatial heterogeneity is possible only because the relevant dopants (i.e., defects) do not move on practical time-scales over significant distances, around the operating temperatures of the doped material: the defects are ‘kinetically stabilized’. Such defects cannot move to annihilate each other, or be annihilated by reaching the surface, as the ‘activation energy’, E_a for their migration is too high compared to k_BT, with k_B Boltzmann's constant and T the operating temperature, in K. Increasing the temperature can decrease the defect density in a process called annealing, which means here that defects can be eliminated, allowing the system to get closer to its thermodynamic equilibrium state. That state will always contain some defects because of the entropy associated with their presence (cf.Fig. 1(i)). Beyond the thermodynamically-dictated minimal defect density (which depends on the enthalpy of formation of the defect and on temperature), additional defect formation implies that the enthalpy of defect formation is smaller than that related to any form of decomposition of the material.

Fig. 1 (i) General scheme for thermodynamically-imposed point defect density (n) of a system at a finite temperature, if |ΔG_r| ≫ ΔH^defect_f. As defect density increases ‘enthalpy’ (ΔH^defect_f) is invested for each defect, but ‘entropic energy’ (TΔS^defect_conf.) is gained, so that the overall ‘free energy’ of the system (ΔG = ΔG_r + ΔG^defect = ΔH − TΔS) has a minimum at a finite defect density (n_eq). (ii) Estimates of the equilibrium defect densities, n_eq, at T = 300 K for common intrinsic point defects: Schottky-pair (a pair of oppositely charged ionic vacancies) or Frenkel defect (displacement of atom into interstitial site, creating a vacancy), where (see the ESI,† x). The pre-exponential factor contains the lattice parameter (a) and the number of lattice/interstitial sites that can become a defect (N),³⁴ and was chosen for cubic MAPbX₃ (with a ∼ 0.63 nm and N ∼ 6). Similar results are obtained for other HaPs. The pale-blue and gray bars are guides to the eye for the defect formation energy (ΔH^defect_f ∼ 160 kJ mol⁻¹), using the value for V_Br in PbBr₂,³¹ and for MAPbX₃ dissociation (ΔG^eqn (1)_r ∼ 10–20 kJ mol⁻¹).²⁵

Here we show how the very low bulk defect densities of halide perovskites, can be readily understood within the context of energies of activation and formation of (de)composition and possible defects, which, thus, suggests a new path to defect management in materials.

Halide perovskites (HaPs)

HaPs are materials with ABX₃ stoichiometry (A, B mono-, di-valent cations, resp.; X halide anion) that, with B = Pb can perform close to, or like those made with known high-quality semiconductors. This is so, even though they can be prepared by low temperature synthesis, a route that normally implies poor semiconductor quality.

Energies, densities and cross-sections for scattering of electrically-active defects can be deduced from experiments such as Space Charge Limited Current, SCLC (on a device structure), Thermally Stimulated Current, TSC, Deep Level Transient Spectroscopy, DLTS, and others.^4–9 The use of such methods on HaPs, especially MAPbBr₃ and MAPbI₃ (where MA = CH₃NH₃⁺) yields remarkably low densities of ∼10¹⁰ cm⁻³ for (low temperature, solution-grown) single crystals and at most ∼10¹⁶ cm⁻³ and usually less (down to 10¹³ cm⁻³ for vacuum evaporated MAPbI₃) for polycrystalline thin films.^10,11 The difference between these two densities can be attributed to surface and/or grain-boundary defects, as for the mm- or larger-sized single crystals, the densities, deduced from the measurements, should be due mainly to bulk defects. Similar low densities of defects (∼10¹⁰ cm⁻³) in other semiconductors are possible (e.g., ultra-pure Si, or epitaxially-grown GaAs),⁹ but require significant efforts/advanced apparatus, involving increased temperatures; HaPs seem to be much more forgiving than classical semiconductors to specific growth or deposition paths.

Low cross-sections of interaction between electronic charge carriers and defects is one way to explain the low densities, deduced for HaPs. Very shallow in-gap defect states (within a range of a few k_BT of the VBM or CBM) or states with levels in the bands, the so-called resonances, in which carriers are delocalized within the crystal, should not interact with charge carriers, observable in a measurement that probes trap density; they will be ‘invisible’ in experiments used to determine defect densities (and energies).^12,13 In HaPs and Pb-chalcogenides (PbX, X = S, Se or Te), the valence band may have ‘anti-bonding’, rather than the usual ‘bonding’ character,^14,15 which can, theoretically, lead to such very shallow or resonant defect states.

Apart from that this idea that awaits experimental proof, attributing shallow defects in HaPs to an ‘anti-bonding’ VBM is limited to intrinsic defects and should not apply to extrinsic ones. Another issue with this concept is the low doping efficiency in HaPs, in sharp contrast with most ‘classical’ semiconductors. Even if shallow defects would not interact strongly with free charges, they should dope the material. However, extrinsic doping, with e.g., Bi³⁺ for Pb²⁺ in MAPbBr₃ single crystals, should, at 10¹⁹ cm⁻³ doping, result in much more than the observed ∼2 orders of magnitude increase in conductivity.^16,17 Similar examples of limited changes in HaP conductivity were also reported for exposure to I₂ or O₂.^18,19 Making the reasonable, but yet to be proven assumption that some Bi³⁺ replaces Pb^2+ 17 and considering the measured trap density of an intrinsic MAPbBr₃ crystal (∼10¹⁰ cm⁻³), such doping is very inefficient (see the ESI,† ii). Similar low doping efficiencies are seen in chalcopyrites,²⁰ amorphous Si or, earlier doping efforts in organic semiconductors.^21,22

What may explain low trap densities and inefficient ‘doping’ is thermodynamic and kinetic instability of defects with respect to the free energy and activation energy for material decomposition.

Model

Thermodynamics of formation vs. decomposition

The stability of a system is defined by its free energy with respect to any dissociation reaction (here denoted by subscript ‘r’), e.g., ΔG_r = ΔH_r − TΔS_r, where ΔH_r and ΔS_r are the enthalpy and entropy of the dissociation reaction, respectively. A compound will be called ‘stable’ if ΔG_r is negative; spontaneous dissociation will occur eventually if ΔG_r is positive. Thus, ΔG_r is the free energy, resulting from the total enthalpy, ΔH_r and entropic energy, TΔS_r for the formation of the material from its constituents (see the ESI,† iii). ΔS_r mostly results from vibrational, rather than configurational, entropy²³ and for compounds with low ΔG_r, adding mixing entropy may be important.

In “classical” semiconductors ΔG_r is dominated by the enthalpy (ΔH_r); the entropic energy (TΔS_r) plays a minor role in stabilizing the material, and can often be destabilizing. In long-chained molecules, especially proteins, entropy plays a major role in stabilizing and defining their final conformation.²⁴ For semiconductors, though, it is very unusual to have entropic stabilization.

A recent review²⁵ about the thermochemistry and calorimetry of HaPs clearly shows the difference between HaPs and “classical” semiconductors in terms of ΔG_r, ΔH_r and TΔS_r. Considering the dissociation of MAPbX₃ (MA = CH₃NH₃), eqn (1) describes the energetically most favorable path for dissociation into constituents. Below we write the chemical equation for the reverse, formation, reaction, to be consistent with common practice, where a negative sign of formation energy, −|ΔG_r|, refers to a spontaneous reaction (see the ESI,† iv):

PbX_2(s) + MAX_(s) → MAPbX_3(s) (1)

The dissociation reaction is the opposite, i.e., reading eqn (1) from Right to Left (←), instead of from Left to Right (→). Overall, ΔG_r (at 300 K) for MAPbI₃ formation from its binaries is around ∼−10 kJ mol⁻¹ (∼4 k_BT at 300 K), slightly higher for MAPbBr₃ and MAPbCl₃ (∼−6.5 k_BT and ∼−5.5 k_BT, respectively)²⁵ (see the ESI,† v). These results are also consistent with an earlier study of ours,²⁶ showing that the total (free + activation) energy, needed to form MAPbI₃ from its binaries (in isopropanol solution) is <20 kJ mol⁻¹. This implies that any energy input >∼20 kJ mol⁻¹ will locally decompose MAPbI₃ into MAI and PbI₂ (following eqn (1)).

With regard to the contributions of ΔH_rvs. TΔS_r to ΔG_r, for MAPbI₃ and MAPbBr₃ (and less so for MAPbCl₃) ΔH_r of eqn (1) is positive or barely negative. To have negative ΔG_r (and a stable compound), the entropic part must overcome the positive enthalpy (see the ESI,† vi). It is still possible, however, that the system will be kinetically stable, even with negative |ΔG_r|, because thermodynamics tells us only what is possible and a reaction can be so slow that on a given time scale no reaction can be observed. For HaPs, nevertheless, the ease by which they can form tribochemically from their binaries (both organic²⁷ and inorganic²⁸), is, though, incompatible with kinetic stabilization. Adding to that result their ready formation by co-evaporation as well as from solution, strongly indicates that HaPs are stable thermodynamically, rather than kinetically.

Defect thermodynamics

Next we consider defect formation and define ΔG^defect as the free energy, including the enthalpy of defect formation, ΔH^defect_f, and TΔS^defect_conf., the configurational (also known as ‘mixing’) entropic energy that favors an increase in the number of defects. As noted, ΔG_r values of HaPs are clearly different from those of ‘classical’ semiconductors, for which mostly |ΔG_r| ≫ 10·k_BT (see the ESI,† vii) and inequality

|ΔG_r| ≫ ΔH^defect_f (2)

holds. If, in addition, the activation energies for decomposition and for defect formation and annihilation are large, compared to k_BT, then we can use the models of defect chemistry and physics that serve us so well to describe and predict properties of semiconductors (and other materials). It is important to keep in mind that introducing point defects in a crystalline system increases its configurational degrees of freedom, increasing the system's entropy and, thus, its overall stability. Fig. 1(i) illustrates that this logic implies that defects are thermodynamically unavoidable, meaning that for a given temperature there is a finite defect density, n_eq, for which the free energy of defect formation is at a minimum. n_eq, depends exponentially on ΔH^defect_f as . Any defect density >n_eq must be kinetically stabilized.

Plotting n_eq (considering Schottky-pair or Frenkel type defects, common for ionic compounds), on a log scale vs. ΔH^defect_f for cubic MAPbX₃ (Fig. 1(ii)), shows that the experimentally-derived n_eq ∼ 10⁹–10¹¹ cm⁻³ for single crystalline HaPs^9,10,29 corresponds to: ΔH^defect_f ∼ 150 kJ mol⁻¹ (1.60 eV) (see the ESI,† viii). Because for HaPs |ΔG_r| ∼ 10–20 kJ mol⁻¹ the inequality in eqn (2) does not hold.

Experimental defect data relevant to HaPs

While there are quite some computational theory studies on point defects in HaPs,³⁰ the limited comparison with actual experimental data and the continuing improvement in methodologies (as well as the implicit use of a static defect model), lead us to focus on the (very few) experimental data on defects, relevant for HaPs, to test the suggested non-‘classical’ nature of defects in these materials.

A combined impedance and isotope tracer study on PbBr₂, the only Pb-halide for which this combination of experimental data exists till now, showed that Br⁻ dominates ion conduction (Fig. 2(ii)).³¹ The activation energy, E_a, for Br⁻ migration, derived from experiments that measure only ionic conductivity, is ∼30 kJ mol⁻¹, while ΔH^defect_f, deduced from radioactive ⁸²Br isotope tracing, is ∼160 kJ mol⁻¹ (see the ESI,† ix). If we use this value for V_Br in MAPbBr₃ (for lack of other options), then obviously, eqn (2) is not satisfied, which would imply that the ‘classical’ view of static point defects does not hold for this and similar HaPs. In the following we consider this possibility further:

Fig. 2 (i) Temperature dependence of Pb and Br diffusion coefficients in PbBr₂, derived from isotope tracer and impedance experiments (reproduced with permission from ref. 31). The energies for activation (E_a = 0.31 eV = 30 kJ mol⁻¹) and formation, (ΔH^defect_f = 1.65 eV = 159 kJ mol⁻¹) were deduced from the ionic and isotopic diffusion coefficients, D_ionic(T), (from impedance measurements) and D_iso.(T), respectively (see the ESI,† xii). (ii) Schematic representation of defects formed in (top) a ‘rigid’ lattice (e.g., ‘classical’ semiconductor), where |ΔG_r| ≫ ΔH^defect_f (kinetically stabilized material – blue dots) or (bottom) a ‘soft’ lattice with low energy for dissociation of the material, where |ΔG_r| ≪ ΔH^defect_f, e.g., in HaPs viaeqn (1) (thermodynamically-stabilized – yellow dots).

The probability of finding defects in a lattice, P, can be represented by the ratio between defect density, n_(defects), with the potential atomic sites that can become defects, N_(sites). Thermodynamically, the origin of the defects does not matter, as formation of defects will always follow the most probable path, i.e., that of least resistance (with minimal energy requirement). Assuming a low activation energy for dissociation or a low barrier for diffusion (at most a few times k_BT), formation of a defect in a periodic structure can lead to (i) a point defect that rapidly diffuse in space or (ii) dissociation/re-formation of the material:

As illustrated in Fig. 2(ii), the probability of finding defects in ‘classical’ semiconductors will follow eqn 3(i) only if eqn (2) is satisfied. Then formation of ‘classical’ defects will not generate dissociation of the material, and with sufficient kinetic stabilization (usually the case in ‘classical’ semiconductors), the lifetime of such defects can be eons.^32,33 Therefore, point defects in ‘classical’ semiconductors are usually present at densities well above their thermodynamic lower limit (n_eq in Fig. 1).

One indication that things may be different in HaPs comes from our experimental evidence for ‘self-healing’ in Br-based HaPs,³⁵viz. after inflicting damage, the system can return to its original state (see the ESI,† xi). We connected this healing to the materials’ dynamic disorder, also termed ‘local polar fluctuations’.³⁶ Therefore, in materials such as HaPs, where ΔH^defect_f ≫ |ΔG_r|, we suggest to view defects differently in terms of position, lifetime, and, likely, also in energy.³⁷ As illustrated in Fig. 1(ii), the probability of finding a dynamic defect in HaPs, if the condition for eqn 3(ii) (ΔH^defect_f ≫ |ΔG_r|) holds, with ΔG_r ∼ 10–20 kJ mol⁻¹ is ∼ few %, where N_atoms ∼ 2 × 10²² cm⁻³ (see the ESI,† x), defects become ‘dynamic’, meaning, the lifetimes of the resulting defects must be much shorter than those of static ones (see below). If conditions for eqn 3(i) (ΔH^defect_f ≪ |ΔG_r|) would apply, then , but such defects should be viewed as static.

Interaction between defects and free charge carriers

We now consider the question of whether the defect's lifetime is long enough for free charges to interact with them. The point is that there is some characteristic lifetime for a defect, sufficiently long for a free charge to sense the defect's presence as different from the rest of the periodic bulk.

When discussing lifetimes, we distinguish between two types of defects: (1) those that, to be eliminated, require mass diffusion over distances, sufficient to allow them to be expelled to the surface/interface; (2) defects that can form and recombine spontaneously within the bulk (without, or with minimal mass diffusion within nearest-neighbor distances), such as those that form as a result of decomposition (e.g., viaeqn (1)).

Referring to the latter, we consider ΔG^eqn (1)_r of HaPs (the defects illustrated in yellow in Fig. 2(ii)). These defects are the products of decomposition (viaeqn (1)), and will be referred to as ‘dynamic’ defects. The minimal interaction time for free charge carriers with defects should be many natural vibrations of the material. In HaPs, with a lowest (optical) phonon lifetime of ∼ps,³⁸ this requires defects that exist for times ≫ps. In the next section we consider the lifetime of defects.

Dynamic and static defects

Considering the entropic stabilization of the material discussed earlier, lattice vibrations are usually viewed as the main contributors to the entropy of solids.²³ Using the Debye frequency, ω_D, which relates to the ‘attempt frequency’ for a reaction to occur, we can estimate an ‘effective’ defect lifetime:with N̄ being an integer number of vibrations until a hoping event occurs, τ_D the Debye lifetime (= 1/ω_D) and, for the current discussion, E_a = E^eqn (1)_a is the activation energy for eqn (1) to occur. From previous work, we estimate E^eqn (1)_a for MAPbI₃ as <10 kJ mol⁻¹,²⁶ leading to the lifetime of a ‘dynamic defect’ of just several (<50) cycles of vibrations (which is sub-ps, taking³⁸ω_D ∼160 cm⁻¹).

Following Almond and West,³⁹ we find that this estimate of a defect's lifetime, or (1/τ_(defect)), is equivalent to the hopping rate, ω_p, that determines ion diffusion:

where . In eqn (5)E_a = E^{(self diffusion)}_a refers to the activation energy for (ion) self-diffusion, meaning the rate at which defects diffuse and, eventually, can be eliminated at surfaces, or as clusters. Considering diffusion activation energies for HaPs as low as ∼30 kJ mol⁻¹, the typical hopping lifetime is of the order of μs – already long enough for a free charge carrier to interact with a defect, which, in this respect, can be considered ‘static’. Still, in HaPs, similar to other solid electrolytes with E_a for self-diffusion <∼50 kJ mol⁻¹, defects may diffuse within seconds–hours, leaving behind a ‘defect-free’ bulk material, consistent with the ‘self’ healing’, observed in the Br-HaPs.³⁵ In ‘classical’ semiconductors, where E_a is of the order of several eV (in Si: E_a(S⁰_i – self diffusion) ∼440 kJ mol⁻¹; E_a(O⁰_i) ∼240 kJ mol⁻¹),⁴⁰ at room temperature a single hopping event will, effectively, never occur without external stimulation. Thus, such defects are ‘kinetically-stabilized’.

Fig. 3 plots the result of eqn (4) and (5) as a function of the different activation energies (and temperatures), and vividly illustrates the differences in kinetic stabilization of defects in different semiconductor families. Comparing ‘dynamic’ and ‘static’ defects, there is a significant difference between the number of vibrations between a single hoping occurs, namely: N̄_(dynamic) ≪ N̄_(static), N̄_(dynamic) may decrease to only a few vibrational cycles, while N̄_(static) may be so large that on human timescales it is infinite, which we then refer to as a ‘kinetically-stabilized’ defect. The latter are the very basis for doping of semiconductors, optical defects in lasing materials and other cases where defects are crucial for the desired function of a material.

Fig. 3 (i) Number of vibrations before hopping (left axis) and time between hopping events (right axis) as a function of the activation energy for ion diffusion, following eqn (4) and (5) (dashed red line @ 300 K and shaded red dotted line @ 1200 K).³⁹ Defects in Si, Cu(In,Ga)Se₂ and HaPs are denoted by light blue bars; (ii) zoomed-in view of lower left corner of (i). The activation energies for HaPs in (ii) are divided into those for self- (ion-)diffusion, E^{(self-diffusion)}_a, and for (re)combination viaeqn (1), E^eqn (1)_a. The right axis, which takes ω_D in the THz frequency range,³⁸ is a multiplication of the left axis by 10⁻¹²; thus, it represents the time between two dissociation or diffusion events in ∼s. The pale-blue and gray bars are guides to the eye for the different activation energies (cf. text).

Effects of dynamic disorder & defects

Evidence that HaPs behave as dynamically structurally disordered materials was already derived from experimental and theoretical work.^{36,37,41–44} In such materials the average atomic position is ordered, so the FWHM of the (X-ray and neutron) diffraction peaks will be narrow and the optical absorption edge sharp,^38,45 as long as the static disorder of the material is small (both are measurements taken over time scales ≫ω_D⁻¹, ∼THz). Therefore, if: (i) a defect is formed, but is not fixed in time and space; (ii) the measurement is slower than the time-dependent delocalization; (iii) the lattice fluctuations can “absorb” the strain that the delocalized defect will induce, i.e., low activation energy for delocalization of the defect, any interaction of free carriers with traps will become comparable to the transit time range of the lattice vibration, which will prevent defect energies from being experienced as different from those of the lattice (see the ESI,† xiii). Recent work by Ledinsky et al. experimentally separates the Urbach energy in parts due to ‘static’ and ‘dynamic’ defects.⁴¹ Comparing with ‘classic’ high-quality semiconductors, MAPbI₃ shows the lowest ‘static’, but the highest ‘dynamic’ defect contribution to the Urbach energy.

The practical implication of ‘dynamic’, but benign, defects is that they limit both ‘good’ (doping) and ‘bad’ (e.g., trapping) defect levels. Since doping requires ‘static’ defects, having ‘dynamic’ defects will not contribute to carrier density in the system. This can explain the preponderance of p-i-n structures in HaP-based devices;^46–48 as the “i”-part can be very efficient for (especially) photovoltaics, this allows for good solar cells. Thus, likely, effective doping of such materials is possible primarily via their internal and external surfaces (see the ESI,† xiv). The fundamental regenerative capability of the bulk is another important reason why HaPs (see the ESI,† xv) are of great interest as radiation detectors,^49–51 which usually require bulk single crystals with low carrier and (static) defect density.

Annealing, from static to dynamic

In cases where kinetic stabilization dominates, thermal annealing occurs when the activation barrier for recombination or diffusion of defects becomes comparable to the thermal energy of the system (∼few k_BT), as also shown in the calculated plot in Fig. 3(i) (dashed line for 1200 K): upon increasing the temperature, τ_defect, which scales exponentially with , becomes comparable to the lattice vibration time, ∼τ_D. Here E_a can be referred to as both E^eqn (1)_a and E^{(self diffusion)}_a that are presented above.

We should emphasize that E^eqn (1)_a, which is central to our model, differs from E^{(self diffusion)}_a. For the fate of extrinsic or intrinsic (any non-stoichiometry) defects, it is E^{(self diffusion)}_a that counts, as it defines the ability of a system to remove defects to a surface/interface. This process can be illustrated by the transformation of a kinetically-stabilized system, such as an amorphous material, to a crystalline one, where E^{(self diffusion)}_a is such that atom movement may become noticeable only after centuries or more, unless we heat. In HaPs, E^{(self diffusion)}_a is more like that for low temperature solid ion conductors (≤∼30 kJ mol⁻¹), which allows defects to migrate at RT to a surface/interface within seconds.

In amorphous materials, like glassy (organic) polymers or inorganics, with large E^{(self diffusion)}_a, re-formation is usually more favorable than formation of point defects (see the ESI† xvi), leading to inefficient doping of amorphous systems (cf. a-Si).²¹ When an energy equivalent of a few times k_BT approaches E^{(self diffusion)}_a (or E_a for material decomposition, in eqn (1)), an amorphous structure becomes entropically less favorable than higher symmetry crystalline forms and the transformation is a form of thermal annealing (cf.Fig. 4 – path (3) → (4)). In Pb HaPs |ΔG_r| < E^{(self diffusion)}_a, so that ion displacement should lead to dissociation into PbX₂ and AX (following eqn (1)), followed by a fast (entropy-driven) reconstruction to crystalline HaP.

Fig. 4 Schematic representation of (1) ‘static’ (‘classical’) semiconductors, SCs, (2) ‘dynamic’ (HaP-like), (3) ‘amorphous’ (glassy) and (4) annealed systems as a function of the ln of the probability, P, of finding a defect, ln(P) (eqn (3)), of the lifetime of a defect, ln(τ_(defect) (eqn (4)), and the temperature of the system, T. The blue area represents the thermodynamic limit, above which systems are kinetically stabilized. The relative energies of ΔG_r and E_a with respect to k_BT will define whether defects in a system are ‘static’ (low density, long lifetime) or ‘dynamic’ (high density, short lifetime). The transition between a frozen (or an ‘amorphous’ (3)) state to a dynamically changing (or ‘annealing’ (4)) state, is determined by the temperature, where T_Critical is a temperature of melting or decomposition.

These four extreme cases of: (1) ‘static’ (‘classical’), (2) ‘dynamic’ (HaP-like), (3) ‘amorphous’ (glassy) and (4) annealed systems are summarized in a 3D plot (see the ESI,† xvii) of defect lifetime (eqn (4)) – probability (eqn (3)) – temperature in Fig. 4.

How different are HaPs from CuInSe₂ (& CIGS)?

Similar to HaPs, the free energy of a reaction to compose CuInSe₂ from its binaries: Cu₂Se_(s) and In₂Se_3(s) is very small, |ΔG_r| ∼ 4 kJ mol⁻¹ (or ∼2k_BT), which, at increased temperatures, leads to its phase segregation into these binaries and a limited range of existence.^52,53 Although doping chalcopyrites is more challenging than doping GaAs or Si, maximum carrier and defect densities of 10¹⁹ cm⁻³ can be reached;^20,54,55 even in PV-quality films the carrier densities, 10¹⁷ cm⁻³, are still significantly higher than those of HaPs (10¹³–10¹⁶ cm⁻³). Interestingly, similar to ‘self-healing’ of HaPs that occurs at room temperature,³⁵ defect ‘annealing’ is observed in Cu(In,Ga)Se₂ thin films, but at ∼160 °C.⁵⁶ The most important difference with HaPs is the activation energy for ion diffusion, which, at ∼1.1 eV⁵⁷ is significantly higher than for HaPs (∼0.3 eV).^31,58

These data indicate that chalcopyrites are similar to HaPs thermodynamically, but not kinetically. They also drive home the point that while a small free energy of (de)composition is a necessary condition for a compound to be at its thermodynamic limit in terms of ‘static’ defect density, it is not a sufficient one: a low activation energy for (de)composition is also required.

Compounds for which both conditions are met will be prone to react with the ambient and to external stimuli such as irradiation. However, due to their dynamic nature, they can recover from inflicted damage under mild conditions, e.g., at low temperatures.

Surfaces, a limit on dynamic defect effects

It is important that we consider also the law of mass action and external impurities: if a material, like an HaP decomposes into its binaries,^59,60 volatile constituents (e.g., halogens or organics for HaPs) can be released.⁶¹ These can also react with ambient impurities (e.g., O₂).⁶² Even upon material loss or gain (introduction of extrinsic impurities, such as H₂O, O₂ or dopants), extrinsic point defects are still highly improbable to exist inside the bulk of HaP-like materials. However, due to low activation and formation energies, even if thermal or external factors (e.g., supra-bandgap illumination, electric field) create defects, these should be eliminated at surfaces/grain-boundaries or segregate, so that the bulk of the system will end up in a thermodynamically stable state, i.e., with the defect state described above.³⁵ This is consistent with the narrow range of existence (with a narrow chemical potential window) of HaPs, as demonstrated both experimentally^60,63 and theoretically,^64,65 leading to phase segregation of PbX₂ or AX₂.

At the same time a ∼0.5–1% change in the ratio of precursors significantly affects device performance.⁶³ That result can be understood if, as has been argued, what dominates the performance of HaP-based devices is related to surface, grain boundary and interface effects, where formation and activation energies of defects are smaller than in the bulk⁶⁶ (see the ESI,† xviii).

Summary

We showed how in HaPs defect densities, their dynamics, and their formation and activation energies connect. We explain this remarkable behavior of HaPs, based on their low formation energy and low kinetic barrier for formation from, and decomposition into their constituents and by a relatively high defect formation energy, but low activation energy for defect self-diffusion.

Low formation energy from constituents, as well as entropic stabilization, suggest that whatever defect may exist in the structure momentarily, cannot be described within the commonly-used picture of a static defect, because its effective lifetime (and thus its interaction with free charge carriers) is so small that, effectively, it is non-existing for interaction with electronic charge carriers. We then connect these energetic considerations to measured defect densities.

For the Halide Perovskites the driving force to restore them from their binary constituents is mostly entropic. Based on the tribochemical^27,28 and self-healing^35,67 experimental results, we postulate that, at room temperature, entropy drives formation and regeneration of partially-organic and fully-inorganic HaPs. Since entropy dominates the material's stability, mixed HaPs (e.g., (Cs,MA,FA)Pb(I,Br)₃) should be further stabilized, as suggested before,⁶⁸ due to the additional component to the material's entropy.⁶⁸ The basis for this experimentally observable extra stabilization is the mixing entropy ∼ k_BT·ln(X_i) ≤ 2–4 kJ mol⁻¹ (X_i = # of configurations). Usually this is insignificant, but if the system is only just stable, additional stabilization becomes very important.

We suggest to assess the potential of a system to have (HaP-like) benign ‘dynamic’ defects as follows:

– Free energy of formation with significant entropic stabilization.

– Activation energy for self-diffusion that is ≤∼30 kJ mol⁻¹ (or ∼0.3 eV, ≤10 ∼ k_BT@RT).

– If tribochemistry (mechanical grinding) of constituents results in a material with sharp diffraction peaks, there is a good chance that the system is entropically-stabilized with low activation energy for formation, which will result in low (static) bulk defect density.

– Generally, the lower the formal valency (e.g., monovalent halide vs. divalent oxides) and effective local (atomic) charge densities (large vs. small ion radius), the weaker the inter-atomic bonds and the smaller the electrostatic (Madelung energy); both these effects lead to lower enthalpy of formation and lower activation energy for atomic displacement.

– Systems that can decompose into constituents that do not require change in phase or oxidation state are likely to have low activation and formation energies for (de)composition.

– The higher the atoms’ coordination number, e.g., corner-sharing polyhedra vs. tetrahedra, the more likely the compound is to benefit from enhanced vibrational and configurational entropy (see the ESI,† vi).

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank Davide Ceratti, Gary Hodes, Leeor Kronik, Omer Yaffe (Weizmann Inst.), Antoine Kahn, Sigurd Wagner (Princeton Univ.), David Egger (Regensburg) and Jean-Franois Guillemoles (IPVF, Paris) for fruitful discussions, Juan Bisquert (U Jaume II) for constructive criticism, and the Minerva (Munich) Centre for Self-Repairing Systems for Energy & Sustainability, at the Weizmann Institute, for partial support of this work.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9mh00606k

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