Self-regulation mechanism for charged point defects in hybrid halide perovskites
Aron Walsh, David O. Scanlon, Shiyou Chen, Xingao Gong, Su-Huai Wei
SSelf-regulation mechanism for charged point defects in hybridhalide perovskites
Aron Walsh* ∗ Centre for Sustainable Chemical Technologies and Department of Chemistry,University of Bath, Claverton Down, Bath BA2 7AY, UK
David O. Scanlon* † University College London, Kathleen Lonsdale Materials Chemistry,Department of Chemistry, 20 Gordon Street, London WC1H 0AJ, UK andDiamond Light Source Ltd., Diamond House,Harwell Science and Innovation Campus,Didcot, Oxfordshire OX11 0DE, UK
Shiyou Chen
Key Laboratory of Polar Materials and Devices (MOE),East China Normal University, Shanghai 200241, China
X. G. Gong
Key Laboratory for Computational PhysicalSciences (MOE) and Surface Physics Laboratory,Fudan University, Shanghai 200433, China
Su-Huai Wei
National Renewable Energy Laboratory, Golden, CO 80401, USA (Dated: December 1, 2014) a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov bstract Hybrid halide perovskites such as methylammonium lead iodide (CH NH PbI ) exhibit unusu-ally low free carrier concentrations despite being processed at low-temperatures from solution. Wedemonstrate, through quantum mechanical calculations, that the origin of this phenomenon is aprevalence of ionic over electronic disorder in stoichiometric materials. Schottky defect formationprovides a mechanism to self-regulate the concentration of charge carriers through ionic compensa-tion of charged point defects. The equilibrium charged vacancy concentration is predicted to exceed0.4% at room temperature. This behaviour, which goes against established defect conventions forinorganic semiconductors, has implications for photovoltaic performance. NH PbI (denoted here as MAPI), which consists of a singly-charged closed-shell methy-lammonium cation (CH NH +3 or MA) at the centre of a PbI –3 cage formed of corner sharingoctahedra. The same structure is adopted by the chloride and bromide perovskites, withsolid-solutions on the anion sub-lattice also reported.[7, 8]The defect chemistry and physics of inorganic perovskites have been well studied foralmost a century, but they remain a complex case in solid-state science, with contribu-tions from electronic disorder (delocalised and localised charges) and ionic point defects,as well as extended dislocations and grain boundaries.[9, 10] In contrast, little is knownabout the hybrid perovskites. Preliminary reports have demonstrated the shallow natureof common point defects, which can contribute to effective electron and hole generation orrecombination.[11–13] An anomaly is that despite exceptionally low defect formation ener-gies, the measured carrier concentrations of thin-films are also remarkably low, in the regionof 10 –10 cm − ,[14, 15] and bulk electron-hole recombination is highly suppressed. In com-parison, for pristine semiconductors (e.g Si and Ge) values of 10 –10 cm − are common dueto a combination of high-purity materials and large point defect formation energies, while insolution processed multi-component materials (e.g. Cu ZnSnS ) values of 10 –10 cm − arefrequently observed due to lower purity samples and smaller defect formation energies.[16]For inorganic perovskites, Schottky disorder is a dominant type of defect, which is associ-ated with the formation of stoichiometric amount of anion and cation vacancies, which canbe distributed randomly in a crystal.[17] It is found, for example, in SrTiO and BaTiO .Following the notation of Kr¨oger and Vink, for methylammonium lead iodide, we can con-sider both ‘full’ Schottky disorder nil → V / MA + V // Pb + 3 V I • + MAPbI (1)and ‘partial’ disorder with respect to the methylammonium iodide nil → V / MA + V I • + MAI (2)or lead iodide nil → V // Pb + 2 V I • + PbI (3)3 ABLE I. Calculated reaction energies (∆ E = (cid:80) products E − (cid:80) reactants E ), independent equilibriumconstants (300 K) and concentrations for Schottky disorder in CH NH PbI . For partial disorderthe chemical potentials are taken to be pinned to the formation of PbI and CH NH I, respectively.The values of K C (%) are normalised to the site fraction of vacancies, and n refers to the associatedvacancy defect concentration.Reaction ∆ H S (eV per defect) K C n (cm − ) nil → V / MA + V // Pb + 3 V I • + MAPbI × nil → V / MA + V I • + MAI 0.08 3.82 2 × nil → V // Pb + 2 V I • + PbI × sub-lattices. Reactions 1, 2 and 3 are charge-neutral (self-compensated), e.g. [V / MA ]+[V // Pb ] ≡ I • ] for 1. While the individual isolated point defects have a net charge, the sum ofthese charges is zero for a macroscopic sample, and does not involve the generation ofelectron or hole carriers. Reaction 1 preserves the overall stoichiometry of the material,but Reactions 2 and 3 result in the loss of MAI ( µ CH NH + µ I = ∆ H f (CH NH I) ) and PbI ( µ Pb + 2 µ I = ∆ H f (PbI ) ), respectively, and can be associated with non-stoichiometry.The equilibrium concentration of lattice vacancies arising from Schottky disorder can becalculated by applying the law of mass action to Reaction 1:[V / MA ][V // Pb ][V I • ] = K C = K ◦ C exp (cid:18) − ∆ H S k b T (cid:19) , (4)where K C represents the fraction of the lattice sites ( K ◦ C ) that are vacant due to the reactionenthalpy (∆ H S ), and contributions from the changes in vibrational entropy are neglected.The computed reaction energies – a combination of the formation energy of the each ofthe individual charged point defects – are summarised in Table 1. The local structures aredrawn in Figure 1. The Schottky formation energy of 0.14 eV per defect is remarkably low,and corresponds to an equilibrium vacancy concentration of 0.4 % at room temperature. Incomparison, the reported Schottky formation energy for BaTiO is 2.29 eV per defect,[10]which results in ppm equilibrium vacancy concentrations. Interestingly, the reactions forpartial Schottky formation are most favourable with respect to the loss of MAI, which at0.08 eV per defect suggests that up to 4 % of the CH NH and I sublattice will be vacant(in an open system). Such non-stoichiometric behaviour goes beyond the non-interacting4 a) (b) (c) V MA-1 V Pb-2 V I+1
FIG. 1. Calculated local structure around the charged CH NH +3 , Pb and I – point defect vacanciesin CH NH PbI that contribute to Schottky ionic disorder. The nominal vacancy site (missingchemical species) in shown in black for each case. The dipole response to defect formation isdriven by a complex combination of molecular reorientation and octahedral distortions (see Ref.[18] for dynamic structural analysis). Note that under conditions of charge and mass conservation[V / MA ] + [V // Pb ] −− I • ]. point defect limit; hence, inter-defect correlations will be important to consider in futurequantitative models.The defect chemistry of this hybrid halide perovskite is unusual. It is common for wideband gap materials to favour ionic disorder (self-compensated arrangements of charged pointdefects) and low band gap materials to favour electronic disorder (a distribution of carriersin the valence and conduction bands). Here, ionic disorder is favoured despite the fact thatelectrons and holes are facile to form, with all of the defects investigated here being shallowdonors (V I • ) or acceptors (V Pb // and V MA / ).[11] The self-regulation of equilibrium electronand hole concentrations will be provided by the formation of charge-compensating latticevacancies.One factor behind this behaviour is the lattice energy: in comparison to metal oxideperovskites, for halides the electrostatic potential of all lattice sites is reduced due to thelower formal oxidation states.[19] The electrostatic contribution to the vacancy formationenergy varies with the square of the charge, e.g. q = 4 for oxides (O − ); 1 for halides (I − ).Hence for the metal halides both anion and cation vacancies can form with high probabil-ity due to a decrease in the chemical bond strength. While point defects, formed due to5onfigurational entropy, facilitate significant equilibrium hole densities of up to 10 cm − inthin-film photovoltaic absorbers such as Cu ZnSnS ,[20] Schottky disorder in CH NH PbI limits their formation as it provides a route to minimise the free energy of the crystal without generating charge carriers.It is now well established that carrier diffusion lengths in hybrid perovskite thin filmsare long ( > µm ).[3] The effective carrier mobility has been estimated to be ca. 20 cm V − s − .[21] The contribution of Schottky disorder to electron transport (carrier life-time), must therefore not be detrimental. This can be understood by the low charge andhigh dielectric constant of these materials, which limits the cross-section associated withionised impurity scattering. Clustering of the vacancies into charge neutral combinationwill further suppress this process. In addition, due to the absence of mid-gap defect states,Shockley-Read-Hall electron-hole recombination is not expected for this type of disorder.The behaviour reported here has important implications for the application of hybridperovskites in photovoltaics: (i) The stoichiometric hybrid perovskites can simultaneouslybe highly defective and electronically benign, with the low carrier concentrations ensuringeffective Fermi-level splitting for operation of a p − i − n photovoltaic device.[22] The combi-nation of high carrier mobility and built-in electric fields can efficiently drift photo-generatedelectrons and holes towards the p and n contacts; if the background carrier concentrationwas too high the i region would not be fully depleted. (ii) Empirically a synthesis routerich in MAI precursors has been adopted,[23] which increases the chemical potentials of MAand I; hence, suppressing the partial Schottky disorder proposed in Reaction 2. The ther-modynamic balance for disproportionation into the binary iodides is delicate, but should bepreventable with appropriate encapsulation, which avoids loss of the more volatile MAI com-ponent. (iii) The high concentration of vacancies on all sites will facilitate mass transport,supporting the ionic conductivity evidenced in impedance spectroscopy,[15] and supportingit as one of the possible causes of hysteresis[23] in the current-voltage behaviour and thegiant dielectric constant at low frequencies.[24] The perovskite crystal structure can supportvacancy-mediated diffusion on each of the lattice sites.[25]In summary, the unusual defect chemistry of CH NH PbI identified here is key to itssuccess as an intrinsic photovoltaic material. The first report by Weber on this mate-rial in 1978,[7] concluded ‘the compounds show intense colour, but there is no significantconductivity’, which our model can now explain. If the self-regulation predicted for the sto-6chiometric material could be overcome, either through extrinsic doping or kinetic controlof non-stoichiometry, the extension of hybrid perovskites to a wider range of photovoltaicarchitectures would be possible. Theoretical Methods
The total energy of bulk and defective CH NH PbI was calcu-lated in a 96 atom pseudo-cubic perovskite (2 × ×
2) supercell with a Γ-centred 3 × × k -point grid (400 eV plane wave cut-off), using a set-up previouslyreported.[19, 26] Lattice-dynamics calculations were performed to ensure the absence ofimaginary zone-centre phonon frequencies in the structures and supercells considered. Themain approximation is the supercell size, in particular relating to the long-range order ofthe methylammonium ions.The formation energy of the individual charged defects is defined as E defectivesupercell − E bulksupercell .The total energy of the charged defective systems were corrected[27] to account for: align-ment of the electrostatic potential between the bulk and the defective supercells; finite sizeeffects in the calculation of charged impurities; band filling by defect levels resonant in theband edges. The static dielectric constant of 24.1 was employed in the calculations,[28] whichinclude the electronic and vibrational response of the system, but excludes the rotationalresponse of the dipolar molecules, which can occur at lower frequencies.All structures and energies were calculated using Kohn-Sham density functional the-ory (in the code VASP ). Interactions between the core and valence electrons is describedwithin the PAW method[29] including scalar-relativistic corrections. Spin-orbit couplingwas not included, which is not expected to affect the defect structures – the unoccupied Pb6p conduction bands are most perturbed – but it will be essential for quantitative defectspectroscopy.The PBEsol exchange-correlation functional was employed.[30] PBEsol is a revision ofthe PBE functional, specifically tailored for solids, and has been shown to yield structuraldata in agreement with experiment.[26] This functional reproduces the structure of common‘London dispersion’ correction functionals without the addition of an empirical potential.Due to the ionic nature of the hybrid perovskite system, secondary polarisation is a minoreffect.We acknowledge membership of the UK’s HPC Materials Chemistry Consortium, which isfunded by EPSRC (EP/L000202/1) and the Materials Design Network. A.W. acknowledgessupport from the ERC (Grant 277757) and EPSRC (EP/K016288/1 and EP/M009580/1).7.O.S. acknowledges support for the NSF of China (11450110056). S.C. and X.G. are sup-ported by Special Funds for Major State Basic Research, and NSFC (61106087, 91233121).The work at NREL is funded by the U.S. Department of Energy under Contract No. DE-AC36-08GO28308. ∗ Electronic mail:[email protected] † Electronic mail:[email protected][1] M. D. McGehee, Nature , 323 (2013).[2] M. M. Lee, J. Teuscher, T. Miyasaka, T. N. Murakami, and H. J. Snaith, Science , 643(2012).[3] S. D. Stranks, G. E. Eperon, G. Grancini, C. Menelaou, M. J. Alcocer, T. Leijtens, L. M.Herz, A. Petrozza, and H. J. Snaith, Science , 341 (2013).[4] G. Xing, N. Mathews, S. Sun, S. S. Lim, Y. M. Lam, M. Gr¨atzel, S. Mhaisalkar, and T. C.Sum, Science , 344 (2013).[5] D. Bhachu, D. Scanlon, E. Saban, H. Bronstein, I. Parkin, C. Carmalt, and R. Palgrave, J.Mat. Chem. A
In Press (2015), 10.1039/C4TA05522E.[6] K. K. Bass, R. E. McAnally, S. Zhou, P. I. Djurovich, M. E. Thompson, and B. C. Melot,Chem. Commun. , 15819 (2014).[7] D. Weber, Z. Fur Naturf. , 1443 (1978).[8] F. Hao, C. C. Stoumpos, D. H. Cao, R. P. Chang, and M. G. Kanatzidis, Nature Photon. ,489 (2014).[9] J. Maier, Physical chemistry of ionic materials: ions and electrons in solids (John Wiley &Sons, 2004).[10] D. M. Smyth,
The defect chemistry of metal oxides (Oxford University Press, Oxford, 2000).[11] W.-J. Yin, T. Shi, and Y. Yan, Appl. Phys. Lett. , 063903 (2014).[12] M. L. Agiorgousis, Y.-Y. Sun, H. Zeng, and S. Zhang, J. Am. Chem. Soc. , 14570 (2014).[13] A. Buin, P. Pietsch, O. Voznyy, R. Comin, A. H. Ip, E. H. Sargent, and B. Xu, Nano Lett. , 6281 (2014).[14] C. C. Stoumpos, C. D. Malliakas, and M. G. Kanatzidis, Inorg. Chem. , 9019 (2013).[15] L. M. Peter (Private Communication).
16] A. Luque and S. Hegedus,
Handbook of photovoltaic science and engineering (John Wiley &Sons, 2011).[17] F. A. Kr¨oger,
The Chemistry of Imperfect Crystals: Volume 2 , 2nd ed. (North-Holland, Am-sterdam, 1974).[18] J. M. Frost, K. T. Butler, and A. Walsh, APL Mater. , 081506 (2014).[19] J. M. Frost, K. T. Butler, F. Brivio, C. H. Hendon, M. van Schilfgaarde, and A. Walsh, NanoLett. , 2584 (2014).[20] S. Chen, A. Walsh, X.-G. Gong, and S.-H. Wei, Advan. Mater. , 1522 (2013).[21] T. Leijtens, S. D. Stranks, G. E. Eperon, R. Lindblad, E. M. Johansson, I. J. McPherson,H. Rensmo, J. M. Ball, M. M. Lee, and H. J. Snaith, ACS Nano , 7147 (2014).[22] E. Edri, S. Kirmayer, S. Mukhopadhyay, K. Gartsman, G. Hodes, and D. Cahen, NatureCommun. , 3461 (2014).[23] H. J. Snaith, A. Abate, J. M. Ball, G. E. Eperon, T. Leijtens, N. K. Noel, S. D. Stranks,J. T.-W. Wang, K. Wojciechowski, and W. Zhang, J. Phys. Chem. Lett. , 1511 (2014).[24] E. J. Juarez-Perez, R. S. Sanchez, L. Badia, G. Garcia-Belmonte, Y. S. Kang, I. Mora-Sero,and J. Bisquert, J. Phys. Chem. Lett. , 2390 (2014).[25] M. S. Islam, J. Mater. Chem. , 1027 (2000).[26] F. Brivio, A. B. Walker, and A. Walsh, APL Mater. , 042111 (2013).[27] C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C. G.Van de Walle, Rev. Mod. Phys. , 253 (2014).[28] F. Brivio, K. T. Butler, A. Walsh, and M. van Schilfgaarde, Phys. Rev. B , 155204 (2014).[29] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169 (1996).[30] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin,X. Zhou, and K. Burke, Phys. Rev. Lett. , 136406 (2008)., 136406 (2008).