1 Introduction

The crystallography of cesium lead halide perovskites dates back to the 1950s, when CsPbX3 (X = Cl, Br, I) crystals were determined to adopt the perovskites structure, and transitions to the ideal cubic phase at high temperature were reported [1]. In the 1970s, Hirotsu et al. found that the low-temperature phase transitions are associated with the condensation of zone boundary phonon modes related to the rotation of Pb\(\text {X}_6\) octahedra [2, 3]. Large thermal displacements are a feature of the cesium lead halide structures [4]. However, the atoms do not simply vibrate around their ideal crystallographic positions. Mizusaki et al. measured high ionic conductivity in CsPbX3 samples. They proposed the migration of X ion vacancies with an energy barrier of 0.25−0.39 eV [5]. The energy barrier is relatively small compared to the activation energy of vacancy migrations for other ionic compounds, e.g. 0.93 eV for oxygen migration is SrTi\(\text {O}_3\) [6]. Later studies confirmed these materials to be predominately halide ion conductors [7,8,9,10,11].

More recently, significant attention has been given to the study of organic-inorganic halide perovskites because of their high power conversion efficiencies in solar cells [12]. Ion migration is also an intrinsic property of these perovskites. There is growing experimental evidence that ion diffusion has critical impact on the long-term operation stability of metal halide materials and devices [13,14,15]. Several studies indicate the low formation energy of native defects in CsPbX3 due to the their weak chemical bond strength [16,17,18,19]. This corresponds to a high equilibrium defect density (\(10^{16}-10^{18} \mathrm {cm}^{-3}\)) [20,21,22] that supports ionic-electronic conduction [10, 22,23,24]. Haruyama et al. calculated iodine vacancy-assisted diffusion barriers of 0.32−0.45 eV [25]. Using both experimental data and theoretical calculations, Eames et al. proposed vacancy-assisted iodide transport with an energy barrier of 0.58−0.68 eV [23]. Yang et al. calculated an energy barrier for the vacancy-assisted diffusion of iodine ions of 0.26 eV [26]. Despite the range of values, which are sensitive to both the model setup and choice of Hamiltonian (e.g. exchange-correlation functional), there is qualitative agreement that the halide ions move through halide vacancy defect sites. However, since the diffusion profiles depend on direction and phase (perovskite tilt system), it is not easy to compare different studies directly.

Interestingly, the ion transport behavior also changes under device conditions. Illumination provokes different diffusion kinetics compared to dark conditions [27, 28]. Kim et al. reported evidence that illumination with energies above the bandgap increases the ion conductivity by orders of magnitude [29]. First-principles calculations agree that a large number of point defects favour ionized charge states (e.g. V\(_{\mathrm {I}}^+\)). [22] Light irradiation may promote the formation of metastable charge states due to the capture of photogenerated electrons and holes [30,31,32, or even the formation of metastable defect pairs [33].

There are additional contributions at play under operating conditions. The chemical bond between the lead and halogen (X = I, Br, Cl) largely determines the bandgap in these systems. The bandgap can be engineered to achieve high efficiency and stability by forming mixed-anion materials. Nevertheless, halide phase segregation happens under illumination for wider bandgap materials formed by mixing iodine and bromine on the X site. Halide ions redistribute to form a more stable phase to minimize the Gibbs free energy. A number of models have been reported to explain the thermodynamics and kinetics at this process [34, 35]. Changes in defect behaviour due to internal strains and external pressure have been reported [36, 37]. By applying external pressure to structure, the mobility of halide ions is altered, as tracked through the kinetics of Br/I phase separation. Internal strains, which is naturally produced as the crystal grows, has been reported to accelerate non-radiative losses [38, 39].

In this study, we focus on ion transport via charged halide vacancy defects in inorganic perovskites. A systemic investigation is performed to understand the factors impacting ion migration behaviour including mechanical, chemical, and electrical stimulii. We chose the cubic phase of lead halide perovskites as a model system to isolate and quantify competing effects and trends. We carried out first-principles density functional theory (DFT) calculations of the activation energies \(E_{a}\) of X ion migrations. We cover changes in the ion diffusion behavior as a function of: (a) charge state; (b) halide species; (c) hydrostatic strain. Finally, a set of guidelines are proposed for tuning ion transport in these materials.

2 Results and discussion

For each of the three CsPbX3 halides, we calculated the 2D potential energy surface for V\(_\mathrm{X}^{0}\) migration on a real-space grid. This approach provides access to both the energetic cost of ion migration and the corresponding pathway through the crystal. The diffusion pathway of an iodide vacancy, between equivalent halide sites, is shown in Fig. 1. The energy profiles are created by calculating the total energy of the supercell containing migrating ions along the diffusion path and identifying the appropriate saddle point. Vacancy migration reveals a curved path with the saddle point away from the neighboring B site. The curved path has been found from oxide-ion conduction in inorganic perovskites [40] and is also feature in halide perovskites crystal [23, 25].

Fig. 1
figure 1

Schematic of the cubic perovskite crystal structure (2 unit cells) is shown in the left panel. The vacancy-mediated diffusion pathway is indicated by the dashed arrow. Figure for Pb and I are colored black and purple. The right panel shows the calculated potential energy surface for iodine vacancy migration V\(_\mathrm{I}^{+}\) in CsPbI3 on a 41 \(\times\) 41 grid in a (100) plane using DFT (PBEsol) 

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2.1 Charge state

Vacancy mediated defect diffusion will be influenced by the charge state of the species, which depends on their formation energy. Under equilibrium conditions where the Fermi level is close to the centre of the band gap, the +1 charge state is favoured for halide vacancies in CsPbX3 [16,17,18,19]. When formed in charge-neutral combinations through Schottky or Frenkel ionic disorder, these charged defects have low energy to form, thus enabling high concentrations [22, 41].

Here we considered other possible charge states of V\(_\mathrm{X}\) which could be formed by illumination (carrier capture) or device conditions (Fermi level changes). To study ion migration behavior under light illumination, the calculations are repeated for each defect in different charge states. We note that while the negative and positive charges have an even number of electrons (closed shell), the neutral charge state has an odd number that requires spin-polarised calculations.

Table 1 Energy barrier (eV) associated with vacancy migration in cubic CsPbX3 for three defect charge states. The lowest barrier is found for the most stable (+) charge state
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The activation energies for the transport of three X vacancies in different charge states (−, 0, +) are listed in Table 1. The corresponding diffusion path considered is illustrated in Fig. 1. The barrier height of the V\(_\mathrm{X}^{+}\) migration corresponds to the lowest saddle point between the minimum energy configurations. The activation energy increased as the charge state of the defect was reduced, and the same trend was observed across all halide species. e.g. from 0.34 eV (V\(_\mathrm{I}^{+}\)) to 0.50 eV (V\(_\mathrm{I}^{-}\)). This trend, under electron rich conditions, is related to the change in Coulomb repulsion due to the partial occupancy of the vacancy site by one (charge state 0) or two (charge state −) electrons. These findings can be linked to a reduction in mobility when defects trap charge, which can facilitate electron-hole recombination [42]. However, we note that other factors that could effect ion conductivity under illumination, such as the change in defect concentrations through photolysis [43].

2.2 Halide species

We next consider the relative stability and diffusion behavior of different halide species. The activation energy for iodide, bromide and chloride migration is 0.36 eV, 0.35 eV and 0.35 eV, respectively (Fig. 2). These values are very similar and must contain a cancellation of energy terms. While iodide has the largest ionic radius, it is also the most polarizable species. The longest interatomic separation for the lead iodide bond (\(\text {r}_{\text {Pb}-\text {I}}\) = 3.14 Å), compared to bromide (2.94 Å) and chloride (2.81Å), is associated with the weakest bond strength [44]. There is a further difference in the dielectric screening: the high-frequency dielectric constant increases as a function of the halide mass due to the decrease in bandgap [45, 46]. This effect alone should reduce the activation energy for the motion of charged defects down the series.

Fig. 2
figure 2

Calculated potential energy surface for vacancy-assisted V\(_\mathrm{X}^{+}\) halide migration in CsPbX3. This is the lowest energy path extracted from the 2D potential energy surface. The corresponding barriers are given in Table 1

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High device performance with thermal and compositional stability has been observed for certain mixed halide perovskites [47, 48]. Here we consider the dilute incorporation of Br in CsPbI3 (i.e. a Cs8Pb8I22Br1 supercell) to compare iodide and bromide migration in comparison to CsPbI3 (Fig. 3). The energy barrier for Br to overcome in the vicinity of the vacancy is reduced by 0.2 eV compared to the pure iodide structure. Bromide has a smaller size, which provides more space for migration in an iodide structure, consistent with other reports [49]. The barrier undergoes a mild increase by 0.03 eV for I migration close to the Br substitution. The local strain caused by the shorter Pb−Br−Pb bonds is expected to be the dominant factor here.

Fig. 3
figure 3

Diffusion models considered in mixed halide systems. Pathway for iodine vacancy migration (purple circle) with: a a bromine ion in Cs8Pb8I22Br1 (reduced by 0.20 eV); b iodine ion migration in Cs8Pb8I22Br1 (increased by 0.03 eV); c reference iodine ion migration in pure CsPbI3

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2.3 Hydrostatic strain

Strain can be caused in halide perovskite films by substrate effects, compositional inhomogeneity, or from thermal processing [50]. For instance, it can arise from the mismatched thermal expansion of the film and substrates. Large thermal expansion coefficients have been reported for these soft crystals [51]. Moreover, rather than being fixed, strain has been found to evolve under visible light illumination [52, 53]. While the understanding of strain distributions for crystals and films continues to evolve in the community, we note that the effect of pure hydrostatic pressure has been experimentally probed for a range of metal halide perovskites [37, 54].

Here, we simulated the role of mechanical perturbations using a model of CsPbX3 that is subjected to uniform expansion and compression. Calculations were performed at effective hydrostatic pressures ranging from − 2.0 (dilation) to 2.0 GPa (compression). All types of charge states and halide species were considered. The mapping between cell volume (V) and pressure (P) was performed using calculated bulk moduli of 17.34 GPa (CsPbI3), 21.70 GPa (CsPbBr3) and 24.76 GPa (CsPbCl3) using the standard relationship \(B_{0}=-V_{0} \frac{d P}{d V}\). The bulk moduli were obtained from fitting the energy-volume data to the third-order Birch-Murnaghan equation-of-state [55].

Fig. 4
figure 4

Change in activation energy for halide diffusion in CsPbX3 as a function of defect charge state and hydrostatic pressure. Negative pressure corresponding to a unit cell expansion. Points for Cl, Br and I are colored blue, green, coral

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The energy barrier of halide ion migration increases linearly across the pressure range considered (Fig. 4). When the interatomic separations are decreased, there is less available space for the diffusing species, and the activation energy is increased. Similar slopes are found in each case, but some non-linearities appear owing to the compounding of electrostatic and steric factors. We note that similar barrier heights have been reported by by Smolders et al. [54]. for CsPbBr3, but a distinct change in anistropic diffusion behaviour was predicted above 1 GPa due to static octahedral tilting patterns that emerge.

3 Conclusions

Ion migration is a fundamental process that contributes to the unusual behaviour of metal halide perovskite materials and devices. Solid-state ion transport is facilitated by charged defects and thus intimately connected with charge-carrier recombination, transport rates, electrical instability, and thereby both device efficiency and lifetime [56, 57]. In general, ion migration should be suppressed in high-performance photovoltaic devices.

To understand the role of various stimuli, we separately simulated the effect of electrical, chemical, and mechanical changes on vacancy-mediated halide diffusion in CsPbX3. We found that the activation energies for vacancy-mediated diffusion in the stable charge states (V\(_\mathrm {I}^+\), V\(_{\mathrm {Br}}^+\), V\(_{\mathrm {Cl}}^+\)) are similar in the three halides. Our analysis showed that excess electrons increase the barrier for ion diffusion in each case. The same change is observed for materials placed under hydrostatic pressure (reduction in the unit cell volume).

A degree of freedom beyond the choice of halide, is the A-site cation. By changing the ratio of caesium, methylammonium, and formamidinium, the unit cell volume can be altered changing the diffusion kinetics [36]. The introduction of larger species such as dimethylammonium can further perturb the order of the crystal [58]. The leading effect of altering the A-site composition will be the unit cell volume, while the change in local structure and strains will also be important, especially for multi-cation mixtures. Indeed, the A-site cation can influence the kinetics and thermodynamics [59]. Recent research has shown that cation mixing can control the defect density to achieve stable and efficient photovoltaic devices [60].

While here we considered diffusion in the model cubic crystals, in reality, other effects can contribute to the physical response of these materials. The perovskite polymorphs accessible at room temperature vary between halides, resulting in distinct octahedral tilting patterns that can exhibit dynamic spatial fluctuations that will alter the diffusion behaviour. Beyond bulk factors, surfaces and extended defects such as grain boundaries will also influence the behaviour in polycrystalline films and devices, which should be considered in future studies. A deeper understanding of defect transport, trapping and accumulation is critical to improving the resistance of halide perovskites to environmental conditions.

4 Computational methods

4.1 Structure model

The cubic crystal structure of CsPbX3 (X = I, Br, Cl) is used, which corresponds to space group Pm3m. We considered ion transport in a 2 \(\times\)2 \(\times\)2 supercell which contains 40 atoms. The optimized unit cell lattice constants are 6.25 Å, 5.87 Å, and 5.62 Å for CsPbI3, CsPbBr3, and CsPbCl3, respectively. Based on the supercell expansion, one X ion is removed to obtain the vacancy defects, which corresponds to a defect fraction of 1/24. No internal degrees of freedom are considered. The individual jump distance of the halide ion vacancy is a/\({\sqrt{2}}\) and it has eight equivalent neighbors in three dimensions.

4.2 Density functional theory

The underlying total energies were calculated from DFT using VASP [61, 62], with the projector augmented-wave (PAW) [63] method. The Perdew-Burke-Ernzerhof exchange-correlation functional revised for solids (PBEsol) [62] was used to optimize structure. For all PBEsol calculations, the plane-wave kinetic energy cutoff was set to 700 eV, while convergence criteria of \(10^{-5}\) eV and \(10^{-2}\) eVÅ\(^{-1}\) for total energy and forces on each atoms, respectively, were employed.

The halide vacancies were modelled within periodic boundary conditions using the supercell approach [64]. For the positively and negatively charged defects, a homogeneous background charge was included to ensure charge neutrality and convergent electrostatic summations. The neutral halide vacancies result in an odd-electron count. Spin-polarised calculations were therefore performed, and this was found to be essential to describe the ground-state behaviour of V\(_\mathrm{Br}\) and V\(_\mathrm{Cl}\). These display localised F-centre like states, while the wavefunction associated with V\(_\mathrm{I}\) is more delocalised with no spin polarisation. As this study concerns relative energies for different defect configurations, no charge state correction schemes were applied. The low defect charge states considered (+/0/-) and supercell dimensions of \(> 10\) Å  in each direction ensure that finite-size corrections are small in magnitude [65].