Fast Diffusion of Native Defects and Impurities in Perovskite Solar Cell

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Fast diffusion of native defects and impurities in perovskite solar cell material CH3NH3PbI3 Dongwen Yang, Wenmei Ming, Hongliang Shi, Lijun Zhang, and Mao-Hua Du Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.6b01348 • Publication Date (Web): 01 Jun 2016 Downloaded from http://pubs.acs.org on June 7, 2016

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Chemistry of Materials

Fast diffusion of native defects and impurities in perovskite solar cell material CH3NH3PbI3

Dongwen Yang1, Wenmei Ming2, Hongliang Shi2,3, Lijun Zhang1, and Mao-Hua Du2 1

College of Materials Science and Engineering and Key Laboratory of Automobile Materials of MOE, Jilin University, Changchun 130012, China

2

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

3

Key Laboratory of Micro-Nano Measurement, Manipulation, and Physics (Ministry of

Education), School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China

AUTHOR INFORMATION Corresponding Author Mao-Hua Du ([email protected]) Lijun Zhang ([email protected])

1 ACS Paragon Plus Environment


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ABSTRACT CH3NH3PbI3-based solar cells have shown remarkable progress in recent years but have also suffered from structural, electrical, and chemical instabilities related to the soft lattices and the chemistry of these halides. One of the instabilities is ion migration, which may cause current-voltage hysteresis in CH3NH3PbI3 solar cells. Significant ion diffusion and ionic conductivity in CH3NH3PbI3 have been reported; their nature, however, remain controversial. In the literature, the use of different experimental techniques leads to the observation of different diffusing ions (either iodine or CH3NH3 ion); the calculated diffusion barriers for native defects scatter in a wide range; the calculated defect formation energies also differ qualitatively. These controversies hinder the understanding and the control of the ion migration in CH3NH3PbI3. In this paper, we show density functional theory calculations of both the diffusion barriers and the − , and Ii− ) and the Au impurity in formation energies for native defects ( VI+ , MA i+ , VMA

CH3NH3PbI3. VI+ is found to be the dominant diffusing defect due to its low formation energy and the low diffusion barrier. Ii− and MA i+ also have low diffusion barriers but their formation energies are relatively high. The hopping rate of VI+ is further calculated taking into account the contribution of the vibrational entropy, confirming VI+ as a fast diffuser. We discuss approaches for managing defect population and migration and suggest that chemically modifying surfaces, interfaces, and grain boundaries may be effective in controlling the population of the iodine vacancy and the device polarization. We further show that the formation energy and the diffusion barrier of Au interstitial in CH3NH3PbI3 are both low. It is thus possible that Au can diffuse into CH3NH3PbI3 under bias in devices (e.g., solar cell, photodetector) with Au/CH3NH3PbI3 interfaces and modify the electronic properties of CH3NH3PbI3.


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I.

Introduction

Solar cells based on hybrid organic-inorganic halide perovskites, in particular, methylammonium (MA) lead iodide (CH3NH3PbI3) and related mixed halides (e.g., CH3NH3PbI3-xClx), have recently undergone rapid development.1 The power conversion efficiency (PCE) of the halide perovskite solar cells has exceeded 20%.2 MAPbI3 is fundamentally different from conventional inorganic photovoltaic (PV) materials in many aspects. MAPbI3 has a high density of defects3 (which is a characteristic of the soft-lattice halide); nevertheless exhibits efficient carrier transport4-7. This can be traced back to the special chemistry of Pb2+ and more generally the chemistry of the ns2 ion (which has outer electron configuration of ns2).8 The presence of the ns2 cations in a halide may lead to more delocalized valence and conduction band states by enhancing cation-anion hybridization.8, 9 Significant ion migration is another interesting property of MAPbI3,10-18 which leads to new phenomena (such as hysteresis in current-voltage curves,18-21 giant dielectric constant,21, 22 switchable photovoltaic effect,10 photon-induced phase separation,23 etc.) that are not typically encountered in conventional inorganic PV materials. The ion diffusion under the built-in electric field in the solar cell leads to charge accumulation near the interfaces that block the ion diffusion. The resulting screening should reduce the built-in electric field, thereby impairing the solar cell performance.21 The ion migration in MAPbI3 can also be traced back to the chemistry of the ns2 ion as schematically shown in Figure 1. Enhanced Born effective charges (relative to nominal ionic charges), strong lattice polarization, and large static dielectric constant have been found in many halides containing the ns2 ion, including MAPbI3.8, 24-27 The response of the MA+ molecular dipole to the electrical field further increases the static dielectric constant of MAPbI3 (~60-70)1,



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28-30

. The effect of a large static dielectric constant is two-fold, i.e., on one hand, it provides

strong screening to charged defects and impurities, thereby suppressing carrier scattering and trapping; on the other hand, it reduces the formation energies of charged defects and impurities because the electrostatic potential applied on the crystal lattice by these charged species is strongly screened. A low defect or impurity formation energy implies that the energy cost for structural distortion is small. Consequently, the defect and impurity diffusion barriers may also be small.

Figure 1. A flow chart that illustrates the effects of the ns2 ion in MAPbI3 on the performance of the MAPbI3 solar cell. The electrical experiments and density functional theory (DFT) calculations on TlBr, a halide with an ns2 ion (Tl+), strongly support the concept illustrated in Figure 1. DFT calculations showed strong lattice polarization, low formation energies of vacancies, and low vacancy diffusion barriers in TlBr,31, 32 in excellent agreement with experiments33, 34. These 4 ACS Paragon Plus Environment

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characteristics have also been found in MAPbI3 1, 28,5, 3, 35-37 In particular, several experiments showed that ions can diffuse at room temperature,10, 11, 15-17 which leads to the current-voltage hysteresis. However, the nature of the ion migration in MAPbI3 remains controversial. Ref. 11 shows that the ionic conductivity in MAPbI3 is due to the diffusion of iodine ions and no diffusion of the MA ions are observed; Ref. 15 shows the observation of MA diffusion but no evidence of iodine diffusion; Ref. 16 suggests the diffusion of both MA and iodine ions. Regardless of which ionic species is responsible for the ionic conductivity, the experimentally measured activation energies for ionic conductivity are within a relatively small range, i.e., 0.36 eV in Ref. 15, 0.4 eV in Ref. 38, and 0.43 eV in Ref. 11. The ionic conductivity due to the diffusion of a particular type of charged defect is proportional to the product of the defect concentration and the defect mobility. The activation energy ( ∆Ea ) is equal to the sum of the defect formation energy ( ∆H ) and the defect diffusion barrier ( ∆Eb ) because

σ ion

[ D ] exp ( −∆Eb / kT )

exp  − ( ∆H + ∆Eb ) kT  = exp ( −∆Ea / kT ) ,39

(1)

where [ D ] is the defect concentration and is given by

[ D ] = N0 exp ( −∆H / kT )

(2)

under thermal equilibrium, k is the Boltzmann constant, and T is the temperature. Eq. 1 holds if the defect is sufficiently mobile such that the thermal equilibrium of the defects is maintained. The defect diffusion barrier should be lower than the measured activation energy of ionic conductivity (~0.4 eV). By measuring the temperature- and time-dependent photocurrent following forward and reverse biasing of the TiO2/MAPbI3 solar cell in the dark, Eames et al. obtained the rate at which the cell relaxes to equilibrium and estimated the activation energy for the relaxation of the device,



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which is 0.60-0.68 eV.12 This value is higher than the activation energy of conductivity measured directly on a MAPbI3 sample. 11, 15, 38 It is possible that the measurement on the TiO2/MAPbI3 solar cell is complicated by other rate-limiting kinetic processes and/or polarization at the interfaces in the solar cell. DFT calculations have been performed by several groups on the diffusion barriers of defects in MAPbI3.12-14 These DFT calculations all showed that, among the three types of vacancies, the diffusion barrier of the iodine vacancy ( VI+ ) is the lowest, followed by those of − ) and Pb vacancy ( VPb2 − ). However, the quantitative results on diffusion MA vacancy ( VMA

barriers scatter over a wide range. For example, the VI+ diffusion barrier was reported to be 0.58 eV in Ref. 12, 0.08 eV in Ref. 14, 0.33 eV-0.45 eV in Ref. 13, and 0.28 eV-0.45 eV in Ref. 18. Such discrepancy is significant and unusual. The diffusion barrier of 0.58 eV for VI+ calculated in Ref. 12 appears too high (higher than the measured activation energy of ionic conductivity). On the other hand, Azpiroz et al. (Ref. 14) showed a much smaller VI+ diffusion barrier of 0.08 eV, which leads to their conclusion that VI+ diffusion is too fast to account for the slow electric − response of MAPbI3 and VMA and VPb2 − may be responsible for the observed hysteresis in MAPbI3. − The calculated diffusion barriers of VMA , VPb2 − , and Ii− (iodine interstitial) have also shown

significant discrepancies.12-14 The diffusion barrier of the MA interstitial ( MA i+ ) has not been reported. Note that the diffusion barrier calculation in Ref. 12 is based on cubic α-MAPbI3 (hightemperature phase), while those in Refs. 13, 14, and 18 are based on tetragonal β-MAPbI3 (roomtemperature phase). Also, Ref. 13 used van der Waals functionals40 while Refs. 12, 14, and 18 did not.


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Ionic conductivity depends not only on the defect diffusion barrier but also on the defect formation energy as shown in Eq. 1.39 Thus, which defects are abundant in MAPbI3 is also an important question. Furthermore, if a particular ion can diffuse, is it due to the vacancy or the interstitial? These questions can in principles be answered by calculating defect formation energies in bulk MAPbI3. However, the calculated defect formation energies in MAPbI3 differ qualitatively. Between the donor defects VI+ and MA i+ , Yin et al. showed that MA i+ has lower formation energy than VI+ 35 whereas Buin et al. showed the opposite41. Between the acceptor − − 35 defects VMA and Ii− , Yin et al. showed that Ii− has lower formation energy than VMA whereas

Buin et al. showed that the formation energies of the two defects are close to each other and their relative stability depends on the chemical potentials.41 The discrepancies in calculated defect formation energies and diffusion barriers seriously hinder the understanding and the control of the ion migration in MAPbI3. In this paper, we show the DFT42 calculations on defect formation and diffusion in bulk β-MAPbI3 (the room-temperature phase with a tetragonal structure). Defects on surfaces and at interfaces are not included in our calculations. We performed calculations on four low-energy − vacancy and interstitial defects, i.e., VI+ , VMA , Ii− , MA i+ . The diffusion barriers of these defects

are calculated. The vibrational frequencies and vectors at the transition state are checked to confirm the saddle point in the defect diffusion path. The calculated diffusion barriers show that the iodine ions are faster diffusers than the MA ions. Further analyses of defect formation energies shows that VI+ is the dominant diffusing defect in MAPbI3. While Ii− and MA i+ also have low diffusion barriers, their formation energies are relatively high. The hopping rate of VI+ is calculated by considering the contribution of the vibrational entropy within the harmonic



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approximation. For defects with low diffusion barriers, the generation of the defects is controlled primarily by thermodynamics, not by kinetics. The mobile defects such as VI+ in MAPbI3 can equilibrate with their defect reservoirs and maintain their thermal-equilibrium concentrations even at room temperature. We suggest that chemically modifying surfaces, interfaces, and grain boundaries, which are the reservoirs of vacancies, may be effective in controlling the population of the iodine vacancy and the device polarization. If the formation energies and diffusion barriers for native defects are generally low in a material, it is likely that they are also low for impurities. Previous DFT calculations showed that the proton migration in MAPbI3 has a small barrier of 0.29 eV.43 Studies on the diffusion of other impurities have not been found in the literature. In this paper, we study the formation and the diffusion of the interstitial Au (Aui) in MAPbI3. Au is often used as the back contact for the MAPbI3 solar cells. A hole transport material (HTM) between MAPbI3 and the Au electrode is typically used for hole collection and electron blocking. However, due to the cost and the stability issues of the HTM, the hole-conductor-free MAPbI3 solar cell, in which the Au electrode is directly deposited on the MAPbI3 layer, has also been extensively investigated.44-49 Furthermore, photodetectors based on the Au/MAPbI3/Au configuration have shown promise.50 Although the Au electrode does not react with MAPbI3, air-stable Au iodides do exist.51, 52

It is possible that Au atoms can diffuse into MAPbI3 and modify the electronic structure of the

MAPbI3/Au interface. An energy-dispersive X-ray spectroscopy (EDS) study observed a small amount of Au in MAPbI3 sandwiched between two Au electrodes.16 Our calculations indeed show that the Aui has low formation energy and a low diffusion barrier (0.3 eV). It is likely that many metal impurities are mobile in MAPbI3 at room temperature.


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II.

Computational Methods

Calculations on defect formation and diffusion shown in this paper are based on DFT as implemented in VASP code. 53 The room-temperature phase of MAPbI3, which has a tetragonal structure and experimental lattice constants of a = 8.849 Å and c = 12.642 Å54, was used in the calculations. The diffusion barriers were calculated using nudged elastic band (NEB) method in conjunction with the climbing image method.55-57 The hopping rate can be calculated using

f = f 0 exp ( −∆Eb / kT ) ,

(3)

The prefactor f0 is given by N

f 0 = ∏ν j j =1

N −1

∏ν

j'

,

(4)

j '=1

which takes into account the change of the vibrational entropy during hopping within the harmonic approximation.58, 59 Here, ν j and ν j ' are the vibrational frequencies of jth and j’th ion at the initial and the transition states, respectively. The calculations of the diffusion barriers and the hopping rates of defects were performed using Perdew-Burke-Ernzerhof functionals without the spin-orbit coupling (PBE-non-SOC).60 The defect formation energy is given by:

∆H = (E D − E host ) − ∑ ni (µi + µibulk ) + q(εVBM + ε f ) ,61

(5)

i

where E D and Ehost are the total energies of the defect-containing and the host (i.e. defect-free) supercells. Formation of a defect involves exchange of atoms with their respective chemical reservoirs. The second term in Eq. (5) represents the change in energy due to such exchange of atoms, where ni is the difference in the number of atoms for the i’th atomic species between the



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defect-containing and defect-free supercells. µi is the relative chemical potential for the i’th atomic species, referenced to its bulk µiref .61 The third term in Eq. (5) represents the change in energy due to exchange of electrons with their reservoir. ε VBM is the energy of the valence band maximum (VBM) and ε f is the Fermi energy relative to the VBM. The PBE-non-SOC calculations were used to obtain the formation energy differences between donor defects and between acceptor defects. The formation energy of a Au i+ is calculated using bulk ∆H ( Au i+ ) = (E Au − E host ) − ( µ Au + µ Au ) + (εVBM + ε f )

(6)

where E Au is the total energy of the supercell for Au i+ . µ Au is the relative chemical potential for bulk 61 . ∆H ( Au i+ ) was calculated using the Au, referenced to the bulk Au chemical potential µ Au

both the PBE-non-SOC method and Heyd-Scuseria-Ernzerhof (HSE) hybrid functionals62, 63 including the spin-orbit coupling (HSE-SOC). The purpose of using the HSE-SOC method for the Au impurity is to correct the band gap and to check whether the Au impurity introduces any deep levels in the band gap. The HSE-SOC calculation with 43% Fock exchange produces a band gap of 1.50 eV for MAPbI3, in good agreement with the experimental value of 1.51-1.52 eV.54, 64 Note that the HSE energy was calculated based on the PBE structure8, 65 since the structural relaxation using HSE functionals is extremely slow. Computational details, such as the cut-off energy, the effect of the van der Waals functionals, the supercell sizes, etc., can be found in the Supporting Information.


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III.

Results and Discussion A. Diffusion barriers of native defects

− Table I shows the calculated diffusion barriers for VI+ , VMA , Ii− , and MA i+ , which are

low-energy defects,35, 41 in MAPbI3. Vacancy diffusion involves an ion moving from one lattice site to its adjacent vacant lattice site [for example, see Figures 2(a)-(c) for the hopping of VI+ ]. However, interstitials in MAPbI3 do not diffuse by hopping along different interstitial sites because the perovskite structure is a close-packed structure, lacking sufficient space for an ion to move between two interstitial sites. Instead, the diffusion of an interstitial uses the ion on the regular lattice site as a bridge. Ii− shares a lattice site with another iodine ion (forming a splitinterstitial) in its ground state;8, 65 it simply rotates around the Pb ion to move to the adjacent lattice site forming another split-interstitial [see Figures 2(d)-(f)]. The diffusion of MA i+ involves a kick-out mechanism, i.e., the MA i+ moves from an interstitial site into an adjacent lattice site by kicking out the MA ion that was on the lattice site, creating a new MA i+ on the adjacent interstitial site, as shown in Figures 2(g)-(i) .

− Table I. The diffusion barriers (in eV) for VI+ , VMA , Ii− and MA i+ on the ab plane and along

the c axis in β-MAPbI3. VI+

− VMA

Ii−

MA i+

ab plane

0.26

0.62

0.19

0.38

c axis

0.34

0.89

0.33

0.48



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Figure 2. Diffusion paths of VI+ , Ii− , and MA i+ on the ab plane: the initial (a), the transition (b), and the final (c) states of VI+ ; the initial (d), the transition (e), and the final (f) states of Ii− ; the initial (g), the transition (h), and the final (i) states of MA i+ . In (g)-(i), the two MA

ions involved in the kick-out process of the MA i+ diffusion are labeled (1) and (2).


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The diffusion of VI+ and Ii− both involves a simple rotation of a Pb-I bond (see Fig. 2). Thus, their diffusion barriers are both low (0.19 eV – 0.34 eV) as shown in Table I, enabling iodine ion diffusion at room temperature. The calculated diffusion barriers for MA i+ are higher than those of VI+ and Ii− but are still low enough (0.38 eV – 0.48 eV) for room-temperature − diffusion. The diffusion barriers for VMA are significantly higher (0.62 eV – 0.89 eV) especially − is the least mobile defect among the four defects studied here. along the c axis (0.89 eV). VMA

Brivio et al. calculated phonon density of states for the tetragonal MAPbI3, which show no imaginary modes.66 For each diffusion barrier calculation, we calculated the vibrational frequencies at the transition state based on the dynamical matrix approach. Ions within a sphere of radius R from the diffusing ion were included in the dynamical matrix calculation. We tested

R = 3 Å, 4 Å, 5 Å, and 6 Å for VI+ diffusion on the ab plane. For each radius, no imaginary frequencies were found for the initial and the final states and only one imaginary frequency was found at the transition state. The value of the imaginary frequency for VI+ roughly converges at R = 5 Å. We further calculated vibrational frequencies for other defects using R = 5 Å and found only one imaginary frequency at the transition state for each case. The corresponding vibrational vectors of the diffusing ions are approximately along the diffusion path. These calculations show that the saddle points along the diffusion paths of the defects have indeed been found. A detailed comparison between the calculated defect diffusion barriers in this work and those in the literature can be found in the Supporting Information. − The results in Table I suggest that VI+ , VMA , Ii− and MA i+ can all diffusion at room

temperature. Which defect is the primary contributor to the ionic conductivity in MAPbI3 also depends on the defect formation energy (as shown in Eq.1), which is discussed below. 13 ACS Paragon Plus Environment


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B. Which defects are main diffusers Here, we address two important questions regarding ion diffusion in MAPbI3, i.e., (1) which ion is the main diffusing ion; and (2) for the main diffusing ion, is it the vacancy or the interstitial that diffuses. To answer these questions, one needs to know both the diffusion barriers and the formation energies of the defects. The first-principles calculations of the formation energies in MAPbI3 are controversial as discussed in Section I. The problem is further complicated by the dependence of the defect formation energy on chemical potentials. To clarify the controversies, we calculated the formation energy differences between the two donor defects − ( VI+ and MA i+ ) and between the two acceptor defects ( VMA , and Ii− ) instead of calculating defect

formation energies at different chemical potentials. These two formation energy differences are found to be only weakly dependent on the chemical potentials and are estimated (at the PBEnon-SOC level) to be

and

∆H ( MA i+ ) − ∆H (VI + ) ≈ 0.46 eV ,

(7)

− ∆H ( Ii− ) − ∆H (VMA ) ≈ 0.38 eV .

(8)

The detailed derivation of Eqs. (7)-(8) can be found in Section S3 in the Supporting Information. − Clearly, VI+ and VMA are much more stable and more abundant than MA i+ and Ii− ,

respectively. At the typical growth temperature of 100 °C, these energy differences can give rise to very large differences (several orders of magnitudes) in defect concentrations, assuming that the thermal equilibrium is reached and the defect density can be calculated using Eq. 2. As shown in Eq. 1, the activation energy ( ∆Ea ) of the ionic conductivity is equal to the sum of the formation energy ( ∆H ) and the diffusion barrier ( ∆Eb ) of the diffusing defect. ∆H + ∆Eb for − VI+ , MA i+ , VMA , and Ii− are estimated by using the calculated defect diffusion barriers in Table I


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and by using Eqs. (7)-(8). The results are shown in Table II. The Fermi level of MAPbI3 is largely determined by the compensation between the lowest-energy donor and acceptor defects, i.e., VI+ and VPb2 − , respectively; in other words, pinned by ∆H (VI+ ) and ∆H (VPb2− ) .41 Thus, at the − Fermi level, ∆H (VI+ ) < ∆H (VMA ) . The results in Table II clearly show that VI+ is the dominant

diffusing defect due to its low formation energy and low diffusion barriers. Ii− and MA i+ should also be mobile at room temperature due to their low diffusion barriers but they should have much lower concentrations than that of VI+ due to their higher formation energies.

Table II. The sum of the formation energy and the diffusion barrier (on the ab plane and − − , and Ii− . Note that ∆H (VI+ ) < ∆H (VMA along the c axis) ( ∆H + ∆Eb ) for VI+ , MA i+ , VMA ) (see

text).

∆H + ∆Eb (ab plane)

∆H + ∆Eb (c axis)

VI+

∆H (VI+ ) + 0.26 eV

∆H (VI+ ) + 0.34 eV

MA i+

∆H (VI+ ) + 0.84 eV

∆H (VI+ ) + 0.94 eV

− VMA

− ∆H (VMA ) + 0.62 eV

− ∆H (VMA ) + 0.89 eV

Ii−

− ∆H (VMA ) + 0.57 eV

− ∆H (VMA ) + 0.71 eV

The experimentally measured activation energy ( ∆Ea ) for ionic conductivity ranges from 0.36 eV to 0.43 eV.15, 38, 11 It is clear from Table II that only VI+ has the activation energy that is possibly in agreement with the experimental value. The activation energies for other defects are



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too high. The formation energy of VI+ depends on the chemical potential of iodine and the Fermi level (which are unknown) and is thus difficult to determine. However, the calculated formation − - VPb2 − Schottky defect is only 0.14 eV,3 which is independent of the Fermi energy of 3 VI+ - VMA

level and the chemical potentials of the constituent elements in MAPbI3. Thus, the theoretical formation energy for VI+ should be just a few tens of meV; and the calculated activation energy for VI+ should be roughly 0.3-0.4 eV based on Table II, in good agreement with the experimentally measured activation energy for ionic conductivity in MAPbI3 (0.36 eV – 0.45 eV). Our results are consistent with the work by Yang et al.,11 which shows that the ionic conductivity in MAPbI3 is due to the migration of iodine ions. In the work by Yuan et al.,15 the photothermal induced resonant microscopy (PTIR) (sensitive to MA molecules) found significant MA ion migration while the EDS (which is used for mapping heavy ions) did not observe iodine ion migration. The PTIR is a highly sensitive probe to MA molecules and should be able to detect MA molecules in low concentrations. The EDS is often used to image heavy ion distribution. However, since the mass distribution has to vary significantly for the EDS to see the spatial variation of the ion density, it is is possible that the iodine ion redistribution under the electric field was missed by the EDS. The measured activation barrier for ionic conductivity may not correspond to the MA diffusion observed by the PTIR but to the iodine diffusion missed by the EDS. It should also be pointed out that, if MAPbI3 is undergoing decomposition during the electrical measurement due to the elevated temperature (which is the case in Ref. 16), the nonequilibrium condition may lead to the formation of more interstitials ( Ii− and MA i+ ) and their diffusion.


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C. Hopping rates of the iodine vacancy Since VI+ is the main diffusing defect as shown in Sec. III-B, we calculated the hopping rates for VI+ using Eqs. (3)-(4).58, 59 The prefactors (Eq. 4) of the hopping rates are calculated to be 0.39 and 0.47 THz for hopping on the ab plane and along the c axis, respectively. These values are about one order of magnitude lower than the thermal phonon frequency (kT/h) at room temperature, which is 6.25 THz. The ions within a sphere of radius R = 5 Å are included in the calculation of vibrational frequencies and the prefactors. Varying the radius R from 3Å to 6 Å does not change f0 by more than 0.1 THz. The calculated hopping rates for VI+ at 300 K are 1.6 × 107 s-1 on the ab plane and 8.9 × 105 s-1 along the c axis. Such high hopping rates enable the diffusion of VI+ at room temperature. VI+ may hop rapidly within a MAPbI3 grain but may require much longer time to cross grain boundaries (GBs). The diffusion of the charged defects under the built-in electric field of the solar cell causes the polarization of the device, which may hinder the charge separation and collection and reduce the open circuit voltage. The polarization phenomenon due to the ion migration can be catastrophic in electronic devices. For example, TlBr is an excellent radiation detection material but the TlBr detector breaks down quickly at room temperature due to the ion migration and the resulting space charge near electrodes that screens out the external electric field.67 The ion migration is likely the main cause of the photocurrent hysteresis in MAPbI3,18-21 but does not seem to cause device breakdown. This may be due to the fast carrier diffusion and the thin MAPbI3 layer, which enable efficient extraction of both electrons and holes by selective contacts even if the built-in electric field is reduced by the polarization.



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D. How to suppress ion migration Although the ion migration does not cause device breakdown in MAPbI3 cells, it does appear to be the main cause of the hysteresis in the current-voltage curve. Suppressing the ion migration is not a trivial task. Lowering the temperature can freeze the ions but it is not an ideal solution to practical applications. Introducing dopants that bind with vacancies through donoracceptor binding has been proposed for TlBr with the hope that the vacancies would form dopant-vacancy complexes and become immobile.68, 69 Unfortunately, such approach is ineffective34 because (1) the static dielectric constant in compounds with ns2 ions (such as TlBr and MAPbI3) is typically very high (see Figure 1) such that the Coulomb binding is very weak; (2) even if the dopant-vacancy complex is formed, new isolated vacancies would form automatically at room temperature through thermal equilibration with the vacancy reservoir in the system, which is surfaces, interfaces, and GBs.70 The low kinetic barriers for defect migration can enable thermal equilibration at the room temperature. Consequently, although the charged defects are swept away by the electric field, the thermal-equilibrium defect concentration can be maintained if the defect can always equilibrate with its reservoir. Since VI+ is the main diffuser in MAPbI3, one may introduce electron donors into MAPbI3 to increase the Fermi level. A higher Fermi level leads to higher formation energy for all donor defects including VI+ , thereby reducing the density of VI+ .70 But if the surfaces and GBs can supply vacancies, it would just be a matter of time for enough VI+ to be generated, which migrate responding to the electric field and eventually reach a static state in which the external electric field is screened out. One may also change the defect diffusion barriers by changing the lattice constant of MAPbI3, which may be achieved through applying strain or alloying on the MA and I sites. These may be interesting for future investigation. 18 ACS Paragon Plus Environment

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If the vacancy population in bulk MAPbI3 is difficult to manage, approaches may be taken to control the reservoir of the vacancies. At surfaces, interfaces, and GBs of MAPbI3, each Pb dangling bond corresponds to an iodine vacancy. These iodine vacancies may supply vacancies to bulk MAPbI3 or diffuse along the GBs. Therefore, passivation of the Pb dangling bonds should affect the availability of VI+ and the device polarization. Indeed, there are many reports on the elimination of hysteresis by the GB passivation (by fullerenes,71, 72 chlorine,73 or Spiro-OMeTAD74) and by reducing the area of the GBs by increasing the grain size75. The GB passivation by large molecules may also create large barriers at GBs that block the ion migration. For TlBr detectors (based on single crystals), incorporating F or Cl onto the surface by HF or HCl etching has been shown to significantly reduce the polarization phenomenon.76 This may also be related to the creation of the anion-rich surface that reduces the availability of the Br vacancy that is the fast diffuser in TlBr.31, 33, 34 These evidences suggest that chemical modification of surfaces, interfaces, and GBs in MAPbI3 may be an effective tool for managing the device polarization. The GB and surface treatments described above do not prevent ion migration; their main role is to reduce the number of available diffusing defects in the system and consequently reduce the device polarization and hysteresis. The current-voltage hysteresis observed in MAPbI3 depends not only on how fast the defect diffuses but also on the defect concentration and the scan rate. The absence of hysteresis in the device does not mean that there is no ion migration in the material.



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E. Formation and diffusion of Au-related defects We performed DFT calculations to study Au related defects following the procedures taken in Refs.8 and 65. The defect structures were optimized at the PBE level and the total energy was obtained by both the PBE and the HSE calculations based on the PBE structures. Our results show that Aui is a shallow donor and is stable at the +1 charge state. Figure 3 shows the structure of Au i+ . Au i+ that binds with two equatorial I- ions (as shown in Figure 3) is more stable than that binds with two apical I- ions by 0.09 eV at the PBE level. The structure of Au i+ features a linear I-Au-I structure with Au-I distance of 2.58 Å and the I-Au-I angle nearly 180°. This structure is very similar to the linear I-Au-I units found in Au iodides, in which the Au-I distances are 2.5560 Å in (C12H14N2)Au2I451 and 2.565 Å in AuI[S(CH2)3CH2]52.

Figure 3. Structure of Au i+ on the ab plane of MAPbI3.

The formation energy of a Au i+ , ∆H ( Au i+ ) , calculated using Eq. 6 is shown in Figure 4. The HSE-SOC calculations yield ∆H ( Au i+ ) = −0.76 eV + ε f . µ Au is taken as zero (the Au-rich limit). This is a suitable choice for MAPbI3 near the Au electrode. ∆H ( Au i+ ) calculated using 20 ACS Paragon Plus Environment

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the PBE-non-SOC method is also shown in Figure 4. The PBE-non-SOC calculation misplaces both the VBM and the CBM as discussed in details in Ref. 65. We corrected the band gap to 1.50 eV using the HSE functionals with 43% Fock exchange. Despite the band gap difference, the absolute formation energies calculated using PBE and HSE functionals usually do not differ significantly. This is why the deep defect levels calculated using PBE and HSE methods are usually close to each other on the absolute scale although their positions relative to the VBM are different due to the different VBM in PBE and HSE calculations.77-79 It can be seen in Figure 4 that ∆H ( Au i+ ) calculated using PBE is lower than that calculated using HSE by 0.24 eV. This is likely because the structure is not relaxed at the HSE level. With structural relaxation, the HSE calculated ∆H ( Au i+ ) is expected to decrease and move closer to the PBE result. In any case, ∆H ( Au i+ ) at the Au-rich limit is very low, even negative in p-type MAPbI3. In MAPbI3 solar

cells, MAPbI3 is typically semi-insulating with its Fermi level near midgap. The results in Figure 4 suggest that the formation of Au i+ should not be difficult near the MAPbI3/Au interface. In the ground-state structure of Au i+ (Figure 3), the Au ion binds with two equatorial Iions. For Au i+ to further diffuse on the ab plane or along the c axis, Au i+ needs to hop out of the ab plane and binds with two apical I- ions. The Au i+ hopping between the two sites requires the overcoming of a small kinetic barrier of 0.3 eV based on our NEB calculation. This barrier is close to the diffusion barrier of VI+ (Table I). Such a low diffusion barrier suggests that Au i+ can − diffuse in MAPbI3 at room temperature. While diffusing in MAPbI3, Au i+ may fill VMA and VPb2 − ,

both of which are electron acceptors, forming less mobile Au 0MA and Au −Pb , respectively. The



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− reaction energy for VMA + Au i+ → Au MA is calculated to be 0.08 eV at the PBE level, which is

very low. The formation of these Au-related defects should raise the Fermi level of MAPbI3.

Figure 4. Formation energy of Au i+ obtained by both PBE-non-SOC and HSE-SOC calculations. Note that the HSE calculations were performed based on the PBE structure without further relaxation. The VBM and the CBM in both the PBE and the HSE calculations are shown.

The work function of MAPbI3 is usually lower than that of Au. The resulting band bending at the MAPbI3/Au interface produces a barrier for Au to diffuse into MAPbI3. Thus, the Au diffusion into MAPbI3 is unlikely unless there is an external field that reduces or eliminates the barrier. This can happen when measuring the I-V curves of a MAPbI3 solar cell and in Au/MAPbI3/Au photodetectors in which a large bias is applied to collect the photo-generated free carriers. Indeed, a small amount of Au was observed by EDS in MAPbI3 under bias between two Au electrodes.16


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IV.

Conclusions

DFT calculations are performed to investigate the generation and the diffusion of native − defects (i.e., VI+ , MA i+ , VMA , and Ii− ) in CH3NH3PbI3, which have seen many controversies in

both experimental and theoretical studies. The diffusions of iodine and MA ions have both been reported based on different experimental techniques. Previous theoretical calculations showed drastically different defect diffusion barriers and formation energies. To clarify the controversies, we calculated the diffusion barriers and the formation energies for vacancies and interstitials of iodine and MA ions. We find that VI+ , MA i+ , and Ii− all have low diffusion barriers; thus, both iodine and MA ions can diffuse in CH3NH3PbI3 at room temperature. However, VI+ has significantly lower formation energy than MA i+ and Ii− and therefore is the primary contributor to the ionic conductivity in CH3NH3PbI3. The defect hopping rate for VI+ is further calculated (taking into account the contribution of the vibrational entropy), confirming VI+ as a fast diffuser. We suggest that chemically modifying the surfaces, interfaces, and grain boundaries may be effective in controlling the population of the iodine vacancy and the device polarization. We further show that the formation energy and the diffusion barrier of Au interstitial in CH3NH3PbI3 are both low, indicating that Au may diffuse into CH3NH3PbI3 under bias in devices (e.g., solar cell, photodetector) with Au/CH3NH3PbI3 interfaces.

Supporting Information 1. Computational details; 2. Comparison of calculated defect diffusion energies in this work and those in the literature; 3. Calculated formation energy differences between the two donor defects − ( VI+ and MA i+ ) and between the two acceptor defects ( VMA and Ii− ).



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ACKNOWLEDGMENT We are grateful for the helpful discussion with Bin Yang. The work at Jilin University was supported by the Recruitment Program of Global Experts (the Thousand Young Talents Plan). The work at ORNL was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Part of the calculations was performed in the high performance computing center of Jilin University.

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Fast Diffusion of Native Defects and Impurities in Perovskite Solar Cell

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