Unique ion diffusion properties in lead-free halide double perovskites: A first-principles study

Journal of Power Sources 412 (2019) 689–694

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Unique ion diffusion properties in lead-free halide double perovskites: A first-principles study

T

Jian Xua, Jian-Bo Liua,∗, Bai-Xin Liua, Bing Huangb,∗∗ a b

Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing, 100084, China Beijing Computational Science Research Center, Beijing, 100193, China

H I GH L IG H T S

G R A P H I C A L A B S T R A C T

fast ion diffusion contributes to • The the performance instability of HDPs. Ag and halogen ion diffusion are • Both fast-diffusing species in Cs AgInX 2

• •

6

HDPs. Ag ion diffusion through the self-interstitial mechanism has a much lower barrier. Applying a suitable external strain may effectively reduce the aging of HDPs.

A R T I C LE I N FO

A B S T R A C T

Keywords: ab initio calculations Lead-free halide double perovskites Ion diffusion Performance stability Solar cells

Pb-free halide double perovskites (HDPs) are proposed as potential candidates for various optoelectronic applications to replace the mainstream hybrid organic-inorganic halide perovskites, e.g., CH3NH3PbI3. While it is known that ion diffusion is a critical problem to affect the structural and electronic stability of CH3NH3PbI3, the mechanism of ion diffusions in HDPs is still unclear and highly desired to be revealed. In this study, taking Cs2AgInX6 (X = Cl, Br) HDPs as prototypes, for the first time we suggest that the fast ion diffusion of the dominant defects may play an important role in the performance stability of HDPs. Importantly, we find that the Agi+ diffusion in a multi-ion concerted fashion has a much faster diffusion rate, compared to the VAg− and VX+ diffusion in a single-ion fashion. It is revealed that HDPs exhibit quite different diffusion properties from CH3NH3PbI3. Furthermore, we demonstrate that the diffusion rate of Agi+ in HDPs can be effectively suppressed by applying an epitaxial strain, which opens a promising way to enhance the performance stability of perovskite materials for various device applications.

1. Introduction Hybrid organic-inorganic halide perovskite CH3NH3PbI3 (MAPbI3) is currently the standout in the photovoltaic (PV) applications, and its highest power conversion efficiency now exceeds 22%. However, two major concerns exist in MAPbI3: the intrinsic instability against heat,

∗

light and moisture [1,2]; and the toxic Pb element. Recently, Pb-free halide double perovskites (HDPs) [3–7] with a formula of A2BIB′IIIX6 have emerged as potential candidates for PV absorbers to overcome the toxicity and instability issues inherent within MAPbI3. Besides of the conventional PV applications, HDPs have also been found to be promising functional materials in many other fields such as X-ray detectors,

Corresponding author. Corresponding author. E-mail addresses: [email protected] (J.-B. Liu), [email protected] (B. Huang).

∗∗

https://doi.org/10.1016/j.jpowsour.2018.12.015 Received 24 January 2018; Received in revised form 18 August 2018; Accepted 6 December 2018 0378-7753/ © 2018 Elsevier B.V. All rights reserved.

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Gaussian smearing width was set to 0.05eV. To calculate the diffusion behaviors and defect properties of HDPs, a 160-atom supercell with a 1 × 2 × 2 Γ-centered k-mesh were employed during the PBE + SOC geometry optimizations. We use a Γ-only k-mesh in the HSE + SOC calculations for the defective supercells.

thermal neutral scintillators for nuclear monitoring applications, white emitting phosphors, and ultraviolet detectors [8–12]. Despite significant research efforts, HDPs are found to exhibit structural stability issues and thus only very few HDPs, e.g., Cs2AgBiX6 (X = Cl, Br), Cs2AgInCl6, Cs2AgSbCl6, Cs2NaBiCl6 and Cs2KBiCl6, had been synthesized until now [13–20]. Cs2AgInX6 compounds were predicted to have direct bandgaps that are suitable for optoelectronic applications [13,19]. The absorption coefficient of Cs2AgInBr6 is estimated to be even higher than that of silicon [13]. However, up to date, all the attempts to synthesize high-quality Cs2AgInBr6 thin-films have failed [7,13]. It is therefore highly desirable to understand the origins of the stability issues in those HDP thin-films. A major challenge for the large-scale applications of perovskite solar cells is their performance stability against various conditions, such as heat, electric stress, light, moisture, oxygen or adjacent layers. Among all these possible situations, ion diffusion has been inferred to play important roles and has recently attracted considerable attention. It is known that fast ion migration may cause the current-voltage hysteresis and impair the long-term stability of the system, thus leading to the aging of perovskite solar cells, as found in MAPbI3 [21–24]. Recent experimental results have revealed that reducing the activation energy for ion migration can accelerate degradation of perovskite films (i.e., decrease the device stability) [25]. Despite the importance of ion diffusion issues, it is still unclear about the ion diffusion behaviors in HDPs and the relations to their performance stability. Thus, it is of great interest to explore the main diffusing ions and their diffusion mechanism in HDPs, in order to improve their performance stability. In this study, taking Cs2AgInX6 (X = Cl, Br) HDPs as prototypes, we have systematically investigated the ion diffusion properties in HDPs by using ab initio calculations. Interestingly, in Cs2AgInX6 HDPs, the most dominant defects (VAg−, Agi+ and VX+) with low formation energies are found to have low diffusion energy barriers, making them fast diffusers in HDPs materials. More importantly, compared to the VAg− and VX+ diffusion through the vacancy-mediated mechanism, Agi+ diffusion through the self-interstitial mechanism has a much lower barrier. Therefore, the fast intrinsic diffusion properties of Ag and halogen ions in HDPs may lead to the performance instability of HDPs. On the other hand, for the intrinsic defects with higher formation energies, e.g., VCs− and VIn0 in Cs2AgInX6, their diffusion coefficients are as low as 10−20 cm2s−1, indicating that they are not responsible for the instability and aging of HDP materials. Our study provides a comprehensive understanding of the defect properties and ion diffusion mechanisms in leadfree HDPs. Furthermore, we propose that an external compressive biaxial strain could be applied to suppress the ion diffusion in HDPs and further enhance the performance stability of HDPs.

2.2. Defect thermodynamics The defect formation energy can be evaluated as [34,35]

Ef

[X q ] = Etot [X q ] − Etot [P ] +

∑ ni μi + q (εVBM + EF) i

(1)

[X q ]

where Etot is the total energy of a supercell containing the defect X in charge state q, and Etot [P ] is the total energy of the corresponding pristine bulk lattice. ni is the number of atoms removed from (ni > 0) or added to (ni < 0) the supercell, and μi denotes the corresponding chemical potential. Δμi is referenced to the corresponding elemental phase. εVBM and EF are the energy levels corresponding to the valence band maximum (VBM) and the Fermi level measured from the VBM, respectively. The potential-alignment term (i.e. qΔV) is applied to correct the finite-size effects in the calculations of charged defects. This term is determined by aligning the core levels of atoms far from the defect center in defective cells to that of the pristine bulk [36,37]. The charge-state transition levels (CTLs) are defined as [38–40]

ε(q1/ q2) =

Etot [X q1] − Etot [X q2 ] − εVBM q2 − q1

(2)

These CTLs correspond to the Fermi-level positions at which X changes its charge state. 3. Results and discussion The Cs2AgInX6 (X = Cl, Br) compounds possess the standard HDP structures in space group of Fm3¯m, where the [AgX6] and [InX6] octahedral alternates along the three crystallographic axes and form the rock-salt type ordering [4,5]. Our calculated cubic lattice parameters and the bandgaps of Cs2AgInX6 are in good agreement with previous calculations or the experimental values (see Table S1 in the Supplementary Material). The schematic diagram of Cs2AgInX6 supercell used for the migration calculations are shown in Fig. 1. To simulate the single interstitial or vacancy defects, one Cs, Ag, In, or X ion is either added to or removed from this supercell. Generally speaking, two major factors need to be determined for the understanding of the ion diffusion issues in a material: (1) the defect densities that are related to the defect formation energies and (2) the diffusion barriers of the dominant diffusing ions. Recently, the intrinsic defect properties of Cs2AgInX6 (X = Cl, Br) have been examined by advanced first-principles calculations [41]. The stoichiometric Cs2AgInCl6 and Cs2AgInBr6 can only grow in very narrow areas of chemical potentials to avoid the formation of any possible competitive secondary phase (see Fig. S1 in the Supplementary Material). Importantly, it is found that the Ag-rich and Br-poor chemical conditions are the ideal chemical potential region to grow n-type Cs2AgInBr6 without deep in-gap levels and the unwanted secondary phases, while in Cs2AgInCl6, its carrier type strongly depends on the pinned Fermi level inside the bandgap and their conductivity are predicted to be rather poor. Here, for the sake of simplicity, our study focus on the Agrich conditions in Cs2AgInX6. The calculated formation energies of intrinsic defects as a function of the Fermi level in Cs2AgInX6 are depicted in Fig. 2, in which one representative Ag-rich chemical potential point is adopted in both materials. It is shown that the most dominant acceptors are VAg− while the most dominant donors are Agi+ and VX+ in both Cs2AgInBr6 and Cs2AgInCl6 under Ag-rich conditions. The concentrations of these dominant defects under equilibrium growth condition, obtained by the self-consistent calculations [42], are estimated to be above 1018 cm−3 in Cs2AgInBr6 and 1016 cm−3 in Cs2AgInCl6 at

2. Computational methods 2.1. Computational details Our first-principle calculations were carried out using density functional theory (DFT) with a plane-wave basis set and projector augmented wave (PAW) method [26,27] as implemented in the Vienna Ab initio Simulation Package (VASP) [28]. The exchange-correlation energy was treated both with the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [29] and with the screened Heyd−Scuseria−Ernzerh (HSE06) hybrid density functional [30,31]. The spin-orbit coupling (SOC) effect is included in all calculations. In our calculations, the PBE + SOC method is used for geometry optimizations, and then the HSE + SOC method is used for selfconsistent static calculations, which was proved to be a reliable method to balance the computational cost and the calculation accuracy [4,24,32,33]. A kinetic-energy cutoff of 400 eV was tested to be sufficient for plane-waves expansion to achieve good convergence. The conjugate gradient algorithm was used until the total energy was converged to 10−5 eV and a force criteria of 0.02 eV Å−1 was allowed. The 690

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Fig. 1. The top and side views of A2BIB′IIIX6 (A = Cs, B]Ag, B′ = In, X = Cl, Br) supercell structures used for the migration calculations.

D = fa2ve

−

ΔEa kB T

(3)

where f is the diffusion correlation factor, a is the hop distance, v is the attempt frequency, kB is the Boltzmann constant, and T is the temperature. A typical v of 1012 Hz for ionic species at room temperature is used in this work. Our calculated ΔEa and D of VAg−, Agi+, VX+, VCs− and VIn0 migrations in Cs2AgInX6 (X = Br, Cl) are summarized in Table 1. Our calculations confirm that the SOC effects and the use of HSE have negligible effect on the predicted ΔEa (see Fig. S4 in the Supplementary Material). From Table 1, we find that in the diffusion of ions through the vacancy-mediated mechanism, VAg− and VX+ have much higher D above the order of 10−12 cm2s−1 in Cs2AgInX6 (X = Br, Cl), as compared to VCs− and VIn0. Owing to the fact that the VBM of Cs2AgInX6 is formed by the strong antibonding between cationic Ag(d) and anionic X(p) orbitals, Ag-X bonds are weak enough to be broken during the Ag and X ions diffusion. Here, we have investigated the ion diffusion in HDPs through not only the vacancy-mediated mechanism, presumably the most common mechanism in most materials with perovskite structure [21], but also the previously neglected self-interstitial mechanism. In Cs2AgInBr6 (Cs2AgInCl6), the diffusion coefficient of Agi+ migration is estimated to be 10−5 cm2s−1 (10−6 cm2s−1), which is three (six) orders of magnitude higher than the value of 10−8 cm2s−1 (10−12 cm2s−1) for VAg−. Therefore, we conclude that compared to the diffusion through the vacancy-mediated mechanism, Ag ion diffusion through the selfinterstitial mechanism has a much faster diffusion rate. It is noted that the diffusion coefficient above the order of 10−12 cm2s−1 is relatively high, comparable to that of lithium superionic conductor materials, such as Li10GeP2S12 [48,49], and Li6.4La3Zr1.4Ta0.6O12 [50]. The next to clarify is the origin of the different diffusion coefficients of Ag ion through those two diffusion mechanisms, i.e., the diffusion of Agi+ and VAg−. For the diffusion of Agi+, two Ag ions participating in the diffusion process hop into their nearest sites simultaneously, i.e., in a concerted migration fashion. As shown in Fig. 3b, Ag 2 ion on the site II hops into the nearest unoccupied site III, and meanwhile Ag 1 ion on the site I jumps into the site II. During the multi-ion concerted migration, the ions located at the high-energy sites migrate downhill, cancelling out a part of the energy barrier felt by other uphill-climbing ions. Moreover, the Coulomb repulsion between Ag 1 and Ag 2 ions accelerate the bond breaking of Agi+ with its neighboring X ions. For the diffusion of VAg−, the migration distance of each Ag ion in VAg− migration almost doubles that in Agi+ migration, therefore the strain energy caused by the local structural distortion is expected to be higher in VAg− migration. As a result, Agi+ migration in a multi-ion concerted fashion has a significantly lower energy barrier than that of VAg− migration in a single-ion fashion. Since the Shannon ionic radii R of Br− (1.96 Å) is larger than that of − Cl (1.81 Å), the calculated cubic equilibrium lattice parameters of

Fig. 2. Calculated formation energies of all the intrinsic defects as a function of the Fermi level in (a) Cs2AgInBr6 under ΔμAg = 0 eV, ΔμIn = −1.5 eV, ΔμCs = −3.0 eV, ΔμBr = −0.88 eV, and in (b) Cs2AgInCl6 under ΔμAg = 0 eV, ΔμIn = −2.0 eV, ΔμCs = −3.13 eV, ΔμCl = −1.06 eV. The Fermi level is referenced to the host VBM. The slope of the line segments indicates the defect charge states, and the kinks (solid dots) denote the transition energy levels. The intrinsic defects expect Cs, Ag, In, Cl/Br vacancies and interstitials are plotted as grey dashed lines. The vertical dashed lines indicate the pinned Fermi level positions.

room temperature, respectively. Moreover, these three dominant defects are all shallow with the (0/1-) or (1+/0) CTL resonant in the valence band or the conduction band. Our calculations show that the predicted CTLs are not susceptible to the SOC effects and the use of HSE instead of PBE (see Fig. S2 in the Supplementary Material). Next we investigate the dominant diffusing ions and their diffusion mechanisms in Cs2AgInX6 (X = Cl, Br) HDPs. The climbing image nudged elastic band (CI-NEB) method [43] has been employed to determine the diffusion barriers (ΔEa) and the migration paths for the most dominant defects, i.e., VAg−, Agi+ and VX+ in HDPs. For comparison purposes, we have considered another two diffusers VCs− and VIn0 in Cs2AgInX6. The energy profiles along the migration paths of VAg−, Agi+, and VX+ in Cs2AgInX6 are depicted in Fig. 3. Besides, the atomic configurations at the initial, transition, and final states during the diffusion processes are also shown in the right panels of Fig. 3. It is worth noting that the chosen diffusion paths are the most energetically feasible one among all the possible paths for every defect location. The schematic diagram of the diffusion paths of VCs− and VIn0 are shown in Fig. S3 in the Supplementary Material. The diffusion coefficient (D) of defects can be estimated from the calculated ΔEa by following the Arrhenius law [44–47]:

691

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Fig. 3. Energy profiles along the migration paths for (a) Agi+, (c) VAg− and (e) VX+ in Cs2AgInX6 (X = Cl, Br). The atomic configurations at the initial, transition, and final states during the diffusion processes of Agi+, VAg−, and VX+ are shown in (b), (d) and (f), respectively. The initial and final configurations are equivalent and well converged.

resulting in a lower ΔEa. For the intrinsic defects with higher formation energies, e.g., VCs− and VIn0 in Cs2AgInX6, they have diffusion coefficients as low as 10−20 cm2s−1. Especially, VIn0 migration has the lowest diffusion coefficients below the order of 10−30 cm2s−1. It suggests that In sub-lattices are immobile; therefore unlikely to cause the hysteresis in the current-voltage curve and the aging of double perovskite solar cells. The generality of our findings is further substantiated by the investigation of other HDPs, such as Cs2AgBiBr6 (see Fig. S5 in the Supplementary Material). It is interesting to compare the ion diffusion mechanism in HDPs and in the organic-inorganic halide perovskite such as MAPbI3. In MAPbI3, it is found that MA and iodine vacancies and interstitials can migrate in the perovskite sensitizers, while Pb ion with a very high ΔEa cannot diffuse [22,23]. The fast ion migration in MAPbI3 thin-films can explain the accelerated degradation of MAPbI3 into PbI2, because MA+ and I− ions can migrate more easily from the MAPbI3 thin-films, producing PbI2 [25]. In Cs2AgInX6 (X = Cl, Br) HDPs, B-site In3+ with high-valence has very high migration barrier, while another B-site Ag+ with low-valence has comparably low diffusion barrier. Most importantly, interstitial defects in Cs2AgInX6 can diffuse directly by hopping along different ideal interstitial sites in a multi-ion concerted migration fashion, which is not the case in MAPbI3 [23,24]. Large structural distortions exist in the lowest energy structures for MA and iodine interstitials in MAPbI3. They deviate from the ideal interstitial sites and form a MA-MA pair (MA interstitial) or double bridges between Pb ions (iodine interstitial) at the minimum energy positions. The comparison of ΔEa in Cs2AgInCl6 and Cs2AgInBr6 (Fig. 3)

Table 1 HSE + SOC calculated diffusion barriers (ΔEa) and the diffusion coefficients (D) of defect migrations in Cs2AgInX6 (X = Br, Cl). Defects

X = Br −

VAg Agi+ VX+ VCs− VIn0

D (cm2s−1)

ΔEa (eV)

0.31 0.15 0.24 0.99 1.68

X = Cl 0.55 0.17 0.41 1.12 1.88

X = Br

X = Cl −8

7.11 × 10 1.70 × 10−5 2.88 × 10−7 9.43 × 10−20 1.15 × 10−30

4.71 × 10−12 7.87 × 10−6 2.30 × 10−10 6.38 × 10−22 4.30 × 10−34

Cs2AgInBr6 (11.20 Å) is larger than that of Cs2AgInCl6 (10.65 Å). Accordingly, there is more freedom for structural relaxation in Cs2AgInBr6 to release strain energies during diffusion, in turn giving rise to a higher diffusion coefficients of VAg−, Agi+, VX+, VCs− and VIn0 in Cs2AgInBr6 than that in Cs2AgInCl6. The fast migration of these three dominant defects (VAg−, Agi+, and VX+) is detrimental for the HDP PV performances, as it may destroy the stability of [AgX6] and [InX6] octahedrons and lead to irreversible distortion of the perovskite structures. Generally, we find that in Cs2AgInX6 (X = Cl, Br) HDPs, the most dominant defects (VAg−, Agi+ and VX+) with low formation energies all have low ΔEa, making them fast diffusers and thus leading to the performance instability of HDPs. Qualitatively, lower defect formation energy implies less energy cost for the structural distortion, thus 692

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through the self-interstitial mechanism (i.e., in a multi-ion concerted migration fashion) has a much lower barrier. Our work thus supports that both Ag and halogen ion diffusion are fast-diffusing species in HDPs and thus contribute to their performance instability; (iii) introducing a suitable external compressive strain is an efficient strategy to suppress the ion transport and reduce the aging of perovskite materials. Acknowledgments This work at Tsinghua University is support from the National Key Research and Development Program of China (2017YFB0702201), the National Natural Science Foundation of China (51571129, 51631005), and the Science Challenge Project (No. TZ2016004). B. Huang acknowledges the support from NSFC (Grant No. 11574024), National Key Research and Development Program of China Grant No. 2016YFB0700700, NSAF U1530401 and the Science Challenge Project (No. TZ2016003). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jpowsour.2018.12.015. References [1] M. Salado, L. Contreras-Bernal, L. Caliò, A. Todinova, C. López-Santos, S. Ahmad, A. Borras, J. Idígoras, J.A. Anta, J. Mater. Chem. A 5 (2017) 10917–10927. [2] S. Yang, W. Fu, Z. Zhang, H. Chen, C.-Z. Li, J. Mater. Chem. A 5 (2017) 11462–11482. [3] F. Giustino, H.J. Snaith, ACS. Energy. Lett. 1 (2016) 1233–1240. [4] X.G. Zhao, J.H. Yang, Y. Fu, D. Yang, Q. Xu, L. Yu, S.H. Wei, L. Zhang, J. Am. Chem. Soc. 139 (2017) 2630–2638. [5] X.G. Zhao, D. Yang, Y. Sun, T. Li, L. Zhang, L. Yu, A. Zunger, J. Am. Chem. Soc. 139 (2017) 6718–6725. [6] Z. Shi, J. Guo, Y. Chen, Q. Li, Y. Pan, H. Zhang, Y. Xia, W. Huang, Adv. Mater. 29 (2017) 1605005. [7] K.-Z. Du, W. Meng, X. Wang, Y. Yan, D.B. Mitzi, Angew. Chem. Int. Ed. 56 (2017) 8158–8162. [8] A. Bessiere, P. Dorenbos, C. W. E. v. Eijk, K.W. Kramer, H.U. Gudel, IEEE Trans. Nucl. Sci. 51 (2004) 2970–2972. [9] B.D. Milbrath, A.J. Peurrung, M. Bliss, W.J. Weber, J. Mater. Res. 23 (2011) 2561–2581. [10] J. Glodo, W.M. Higgins, E. V. D. v. Loef, K.S. Shah, IEEE Trans. Nucl. Sci. 56 (2009) 1257–1261. [11] M.-H. Du, K. Biswas, J. Lumin. 143 (2013) 710–714. [12] W. Pan, H. Wu, J. Luo, Z. Deng, C. Ge, C. Chen, X. Jiang, W.-J. Yin, G. Niu, L. Zhu, L. Yin, Y. Zhou, Q. Xie, X. Ke, M. Sui, J. Tang, Nat. Photon. 11 (2017) 726–732. [13] G. Volonakis, A.A. Haghighirad, R.L. Milot, W.H. Sio, M.R. Filip, B. Wenger, M.B. Johnston, L.M. Herz, H.J. Snaith, F. Giustino, J. Phys. Chem. Lett. 8 (2017) 772–778. [14] M.R. Filip, S. Hillman, A.A. Haghighirad, H.J. Snaith, F. Giustino, J. Phys. Chem. Lett. 7 (2016) 2579–2585. [15] E.T. McClure, M.R. Ball, W. Windl, P.M. Woodward, Chem. Mater. 28 (2016) 1348–1354. [16] A.H. Slavney, T. Hu, A.M. Lindenberg, H.I. Karunadasa, J. Am. Chem. Soc. 138 (2016) 2138–2141. [17] P. Barbier, M. Drache, G. Mairesse, J. Ravez, J. Solid State Chem. 42 (1982) 130–135. [18] W. Urland, Chem. Phys. Lett. 83 (1981) 116–119. [19] T.T. Tran, J.R. Panella, J.R. Chamorro, J.R. Morey, T.M. McQueen, Mater. Horiz. 4 (2017) 688–693. [20] J. Zhou, Z. Xia, M.S. Molokeev, X. Zhang, D. Peng, Q. Liu, J. Mater. Chem. A 5 (2017) 15031–15037. [21] C. Eames, J.M. Frost, P.R. Barnes, B.C. O'Regan, A. Walsh, M.S. Islam, Nat. Commun. 6 (2015) 7497. [22] J. Haruyama, K. Sodeyama, L. Han, Y. Tateyama, J. Am. Chem. Soc. 137 (2015) 10048–10051. [23] J.-H. Yang, W.-J. Yin, J.-S. Park, S.-H. Wei, J. Mater. Chem. A 4 (2016) 13105–13112. [24] D. Yang, W. Ming, H. Shi, L. Zhang, M.-H. Du, Chem. Mater. 28 (2016) 4349–4357. [25] J. Zhao, Y. Deng, H. Wei, X. Zheng, Z. Yu, Y. Shao, J.E. Shield, J. Huang, Sci. Adv. 3 (2017) 1–8. [26] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. [27] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. [28] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [29] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [30] J. Heyd, G.E. Scuseria, M. Ernzerhof, J. Chem. Phys. 118 (2003) 8207–8215.

Fig. 4. Energy profiles along the migration paths for Agi+ in compressive biaxial strain (ε) applied (a) Cs2AgInCl6 and (b) Cs2AgInBr6. The calculated ΔEa and D vs ε for Agi+ migration in Cs2AgInX6 (X = Cl, Br) are shown in the insets.

indicates that strain energy could play an important role in modulating the ion diffusion properties in HDPs. Consequently, it is expected that an external compressive biaxial strain (ε) can be applied to enhance ΔEa of dominant defects in HDPs, since compressive ε can increase the strain energy during ion diffusion. As shown in Fig. 4, for the Agi+ diffusion in Cs2AgInCl6 (Cs2AgInBr6), our calculations show that a compressive ε of 3% (2%) gives rise to a four (five) orders of magnitude reduction in the diffusion coefficient at 300 K. Especially in Cs2AgInBr6, only a 0.5% compressive ε can increase the ΔEa from 0.15 eV to 0.24 eV (see calculation details for strained systems in the Supplementary Material). As compressive ε in HDPs increases, the bond length of Ag-X (X = Br, Cl) would become shorter, indicating a stronger bonding strength between Ag and X atoms and hence Agi+ diffusion would require more energy cost, i.e., ΔEa would be higher. In practice, an external strain can be applied during the growth of HDP thin-films on a lattice-mismatched substrate. Importantly, the experiments reported by zhao et al. [25] revealed that the lattice strain was an important intrinsic source of the instability in MAPbI3 thin-films reducing the activation energy for ion migration, which then accelerated the degradation of perovskite films. 4. Conclusions In summary, using ab initio calculations, we have developed an indepth understanding of the ion diffusion issues in two HDPs, i.e., Cs2AgInX6 (X = Cl and Br). Our results show: (i) the most dominant defects (VAg−, Agi+ and VX+) with low formation energies have low diffusion energy barriers, making them fast diffusers in Cs2AgInX6 HDPs; (ii) compared to the diffusion through the vacancy-mediated mechanism (i.e., in a single-ion migration fashion), Ag ion diffusion 693

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