Structural and Chemical Features Giving Rise to Defect Tolerance of

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Structural and Chemical Features Giving Rise to Defect Tolerance of Binary Semiconductors Rachel C. Kurchin, Prashun Gorai, Tonio Buonassisi, and Vladan Stevanovi# Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.8b01505 • Publication Date (Web): 26 Jul 2018 Downloaded from http://pubs.acs.org on July 26, 2018

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Chemistry of Materials

Structural and Chemical Features Giving Rise to Defect Tolerance of Binary Semiconductors Rachel C. Kurchin,†,k Prashun Gorai,‡,¶,k Tonio Buonassisi,§ and Vladan Stevanović∗,‡,¶ †Dept. of Materials Sci. & Eng., Massachusetts Institute of Tech., Cambridge, MA, USA ‡Dept. of Metallurgical and Materials Eng., Colorado School of Mines, Golden, CO, USA ¶National Renewable Energy Laboratory, Golden, CO, USA §Dept. of Mechanical Eng., Massachusetts Institute of Tech., Cambridge, MA, USA kThese authors contributed equally to this work. E-mail: [email protected] Abstract Defect tolerance, or the resilience of electronic transport properties of a crystalline material to the presence of defects, has emerged as a critical factor in the success of hybrid lead halide perovskites as photovoltaic absorbers. A key aspect of defect tolerance is the shallow character of dominant intrinsic defects. However, while qualitative heuristics to identify other defect-tolerant materials have been proposed, in particular the presence of a partially oxidized ns2 cation such as Pb, no compelling comprehensive understanding of how these shallow defects arise has yet emerged. Using modern defect theory and defect calculations, we conduct a detailed investigation of the mechanisms and identify specific features related to the chemical composition and crystal structure that give rise to defect tolerance. We find that an ns2 cation is necessary but not sufficient to guarantee shallow cation vacancies in an s-p system, and that a

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compound’s crystal structure can ensure shallow anion vacancies in a variety of ways. Specifically, the crystal site symmetry can enforce weak interactions between the orbitals that form the defect states, thus ensuring that those defect states are shallow. We substantiate our findings by computing defect formation energies in several known as well as hypothetical materials, and conclude by discussing prospects for identifying semiconductors that satisfy these criteria.

Introduction In recent years, conventional wisdom about materials selection and synthesis for photovoltaics (PV) has been turned on its head by the emergence and rapid rise in efficiency of halide perovskites, typified by methylammonium lead iodide (MAPbI3 , or MAPI). 1–12 In contrast to all previous materials to surpass 20% power conversion efficiency such as Si, GaAs and CdTe, these materials are fabricated via solution processing rather than vacuumbased vapor deposition techniques. This makes them extremely attractive as candidates for inexpensive industrial scale-up, and should such an endeavor succeed, it could have a revolutionary impact on the PV industry. However, concerns remain surrounding lead toxicity as well as the long-term stability of these compounds, motivating a search for “perovskiteinspired” materials that might share some of the extraordinary electronic properties of the perovskites without suffering these drawbacks. 13–18 A key factor enabling the extraordinary performance of MAPI and related compounds is their very low non-radiative recombination rates, which are a consequence of the fact that the dominant (low-energy) intrinsic defects are all shallow in character. 19–22 More precisely, the defect states of all low-energy defects either appear as resonances inside the bands of the host material or are sufficiently close to its band edges. This helps to make them “defect-tolerant” – that is, their performance is resilient to the presence of these defects because the defect states do not contribute significantly to non-radiative recombination rates. However, while a qualitative explanation of this phenomenon and the descriptors to screen materials for defect 2


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A(s) C(p) band gap

A(p) C(s) Vc states Figure 1: Schematic electronic structure that gives rise to defect tolerance. According to the prior heuristic understanding of this phenomenon, a predominantly antibonding and bonding character of the valence and conduction band minima, respectively gives rise to shallow anion and cation vacancies (anion dangling bonds at the cation vacancy site are marked by X). tolerance have been proposed, a more complete physical and quantitative understanding of the properties underlying this behavior has remained elusive. In prior work, 23 the notion was advanced that defect tolerance is enabled by the presence of partially oxidized cations and the resulting antibonding character of the states at the top of the valence band. This antibonding interaction drives the valence band maximum (VBM) to energies higher than those of the interacting atomic orbitals as shown in Figure 1. The rationale for this can be understood if one considers a vacancy as the archetypal defect; vacancies also tend to be among the lowest-energy native defects in many compounds, particularly in binary systems. 24–27 The dangling bonds formed by (for example) a cation vacancy should have energies comparable to those of the participating anion atomic orbitals. Thus, it is reasonable to infer that the anion dangling bonds, which are resonant inside the valence 3



band, would lead to the shallow defect behavior. Similarly, anion interstitial defects would introduce states that are, to first order, close in energy to the anion atomic orbitals and would also imply similar behavior. Analogous arguments can be made for the conduction band in relation to the cation vacancies and/or interstitial defects. For defect tolerance, one would then look for the predominantly bonding character of the conduction band minimum (CBM), which would be a consequence of the strong interaction between the cation(p) and empty (next-shell) anion(s) atomic orbitals, as illustrated in Figure 1. However, finding a material with (i) partially oxidized cations, (ii) sufficiently strong interactions between the cation(s) and anion(p) to ensure the pronounced antibonding character of the VBM, and (iii) sufficiently strong interactions between the cation(p) and the next-shell anion(s) to ensure the pronounced bonding character of the CBM, might be challenging. Also, in addition to being fairly qualitative, all these arguments pertain to the chemistry of materials and largely ignore the influence of the crystal structure, known to be of importance in specific compounds such as TlBr. 28 However, using these relatively simple criteria, which mainly center around the partially oxidized cations (such as In+ , Tl+ , Sn2+ , Pb2+ , Sb3+ , or Bi3+ ), has led to surprisingly successful identification of candidate materials. 14 A subset of such compounds, including BiI3 , SbSI, and MA3 Bi2 I9 , were then synthesized and characterized, both experimentally and computationally. 29–32 While many of these compounds show preliminary promise for PV applications, with the lifetimes of photo-generated carriers in the nanosecond range (previously identified as a “rule-of-thumb” threshold for a compound meriting further research work 33 ), none achieved lifetimes near the order of magnitude that has been observed in MAPI and other hybrid perovskites (hundreds of nanoseconds). 34,35 It is important to note that our focus here is on defect tolerance, which pertains to non-radiative recombination rates of the photo-generated charge carriers. In the context of the radiative recombination rates, which are much less detrimental for PV performance, the spin-orbit Rashba-Dresselhaus splitting 36,37 that is known to be relevant in MAPI is certainly

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a key factor. In this work, we endeavor to extend our understanding of the chemical and structural origins of defect tolerance, drawing on and systematically testing prior ideas from the literature as well as introducing novel ones. We use modern defect theory to compute formation energies of native point defects and the associated defect energy levels in a variety of compounds and find that both the chemistry and crystal structure influence the shallow or deep nature of defects and impact the defect tolerance of materials. Our analysis, focused mainly on binary s-p systems and vacancies as dominant defects, shows that the presence of a partially oxidized cation is a necessary but not sufficient condition to ensure the shallow nature of cation vacancies. Good energy alignment of the cation(s) and anion(p) atomic orbitals is also required to maximize the strength of interactions, which can also be modulated by the crystal structure. Concerning the anion vacancies, finding binary chemistries with good alignment of empty anion(s) and cation(p) has proven difficult because the empty (next-shell) anion(s) states are typically much higher in energy. However, we show here that the crystal structure can remedy an unfavorable orbital energy alignment and play a decisive role in determining the shallow vs. deep nature of anion vacancies. Similar arguments can in principle be made for antisites and interstitials. (While interstitials can have more complicated behavior that may lead to deep defects, they are rarely the dominant (i.e. lowest-energy) defects in binary systems, which are the focus of this study.) Finally, we formulate and validate a set of design criteria for defect tolerance and discuss their extension to ternary chemistries and application in searching for novel defect-tolerant materials.

Point defects and their energy levels According to Shockley-Read-Hall statistics, 38,39 shallower (i.e. closer to the band edges) defects contribute exponentially less to trap-assisted recombination rates. Defect-tolerant materials are, therefore, characterized by shallow native defects. To ascertain the shallow

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vs. deep nature of defects, one must determine the defect energy levels. In modern theory of defects in semiconductors, these defect energy levels are the thermodynamic charge transition levels rather than one-electron energies, typically calculated in density functional theory (DFT). 40 Defect charge transition levels can be determined by calculating defect formation energies as functions of the Fermi energy. Equation 1 describes how to compute the formation energy of a point defect, (e.g. a vacancy VA ) in charge state q:

∆HVA ,q = (EVA ,q − Ehost ) + qEF + µA + Ecorr

(1)

The term (EVA ,q − Ehost ) represents the difference in the total energy between the defectfree host crystal with no net charge (Ehost ) and the host crystal with an A atom removed and the charge q exchanged with the reservoir of charges described by the Fermi energy EF . The next two terms on the right hand side are the thermodynamic terms to account for the energy associated with the exchange of charge and elemental species. The former is described by the qEF term, while the latter by the chemical potential µA of the species A. These quantities are typically calculated from first principles using periodically repeated supercells, an approach which suffers from the artifacts arising from finite-size effects. Thus, we need to include energy corrections to account for this and better approximate the dilute limit of defect concentration we seek to represent. Various corrections are grouped into the Ecorr term, and are briefly discussed in the next section. A more detailed description can be found elsewhere. 41,42 For a given material, one calculates the defect formation energies (∆H) for all defects of interest in all plausible charge states. The results are typically shown in a plot like the ones in Figure 2 as well as later in this work. The x-axis of the plot is the Fermi energy, which spans from the VBM (conventionally set to 0) to the conduction band minimum (CBM). Since the only EF -dependence in Equation 1 is linear, one can read the charge state off the plot according to the slope of each line. Typically, only the lowest-energy state for each EF 6


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(c) SnI2

(a) InI

Sn(2)

3

∆HD,q (eV)

I(3)

4

2

1

IIn

+2

VIn

+1

-3 -1

ISn(1)

2

ISn(2)

1.0

2.0

I(1)

I(2)

-2

SnI(3)

+3 +1

VSn(2) -2 VSn(1)

-1

0

Sn(1)

3

0

InI

(e) SbI3

0

VI(3)

VI +1

1

VTl

0

-1

+1

ITl

1.0

EF (eV)

2

ISb

VI

+1

0 0

2.0

VSb

0.5

-1

4

+3

3

VPb

2

1.0

2.0

+1 -1

-2

0 0

1.0

EF (eV)

1.5

2.0

4

BiI

-1

3

VI

IPb

-1

-3

(f) BiI3

PbI

1

-2 -1

0

-1

1

-3

5

2

-1

+1

3

(d) PbI2

4

TlI

SbI

4

-3

VI(1,2)

1.0

(b) TlI

3

5

SnI(1,2)

1

-2

VI

0

∆HD,q (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3 2.0

+3

2 1

+1

0

+1 0

VI IBi -1 -3 1.0

EF (eV)

VBi 2.0

Figure 2: Calculated formation energies (∆HD,q ) as functions of Fermi energy (EF ) of native defects (vacancies and antisites) in the ground-state structures of 6 binary iodides that contain partially oxidized ns2 cations (In+ , Tl+ , Sn2+ , Pb2+ , Sb3+ , or Bi3+ ). The corresponding crystal structures are shown above the defect diagrams. SnI2 has multiple Wyckoff positions for each elemental species; multiple defect lines of the same defect type represent different Wyckoff positions. value is shown on the plot, which means that a change in slope of the line for a given defect represents the value of EF where the most energetically favorable charge state changes, or more familiarly, the charge transition level of that defect. These transition levels typically 7



occur nearer the VBM for acceptor defects and nearer the CBM for donor defects. Generally, a defect is considered shallow if its charge transition levels occur within a few kB T of the relevant band edges, where kB is the Boltzmann constant. A defect is also shallow if it has no charge transition levels inside the gap, that is, its transition levels are resonances inside the corresponding bands.

Computational Methods To calculate the formation energy of the native defects, we perform first-principles defect calculations with density functional theory (DFT). The generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) 43 is utilized in the projector augmented wave (PAW) formalism as implemented in VASP. 44 The formation energies are calculated using the supercell approach. 41 The total energies of the defect supercells are calculated with a plane-wave energy cutoff of 340 eV and a Γ-centered Monkhorst-Pack k-point grid to sample the Brillouin zone. The defect supercells are relaxed and the formation energy calculated as following the methodology in Ref. 41. Defect formation energies of all vacancies and antisites in charge states q = -3, -2, -1, 0, 1, 2, and 3 are calculated. Vacancies and antisites derived from all unique Wyckoff positions in the crystal structure are considered. The elemental chemical potential µi is expressed relative to the reference elemental phase such that µi = µ0i + ∆µi , where µ0i is the reference chemical potential under standard conditions and ∆µi is the deviation from the reference. ∆µi = 0 eV corresponds to i-rich conditions. The reference chemical potentials are obtained by fitting to a set of measured formation enthalpies, using a procedure similar to Ref. 45. The region of phase stability of a given compound sets the bounds on the values of ∆µi . The underestimation of band gap in DFT is remedied by applying individual valence and conduction band edge shifts as determined from GW quasi-particle energy calculations. 41 GW calculations are performed with VASP using the standard set of PAW pseudopotentials supplied with the VASP distribution. Following the methodology in Ref. 46, we use DFT 8


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wave functions as input to the GW calculations. The GW eigen-energies are iterated to self-consistency to remove the dependence of the G0 W0 result on the single-particle energies of the initial DFT calculation. The input DFT wave functions are kept constant during the GW calculations, which allows the interpretation of the GW quasi-particle energies in terms of energy shifts relative to the DFT Kohn-Sham energies. The individual band edge shifts for all the materials considered in this work are tabulated in the supplementary information (Table S1). The efficacy of this approach in describing the defect chemistry of semiconductors has been well documented in several previously-published works, 41,46 including our own on KGaSb4 , 47 Mg3 Sb2 , 48 ZnSiP2 , 49 and PbTe. 50 Beyond successful demonstration of this approach, further confidence in the calculated GW shifts stem from the significantly improved prediction of the band gap. See Table S2 for comparisons between GW and experimental gaps. As outlined in Ref. 41, the following corrections are included in Ecorr : (1) image charge correction for charged defects, (2) potential alignment correction for charged defects, (3) band filling correction for shallow defects, and (4) band gap correction for shallow defects. The calculation setup and analyses are performed using a software package for automation of defect calculations. 51 The calculated dielectric constants, including electronic and ionic contributions, that are used in determining the image charge correction are tabulated in the supplementary information (Table S3). The approach used in this work to determine the defect energetics (GGA-PBE for defects with individual band edge shifts from GW calculations) could artificially delocalize the otherwise localized, deep states, in particular those related to the formation of small polarons. 52 However, this is not of serious concern as polaron formation in materials with partially oxidized cations is not likely due to their significant covalent character. The absolute band edge positions for a given structure is computed for the non-polar surface with lowest surface energy. The standard approach for referencing the bulk electronic eigen-energies to the vacuum level, as described in Refs. 53 and 54, is adopted. This approach combines accurate GW calculations for the bulk electronic

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structure and the potential step ∆V (between the bulk average electrostatic potential and the vacuum) for a given surface computed with DFT.

Results Breakdown of the Heuristic Description The criteria presented previously 14 suggest that chemistry – namely, the presence of a partially oxidized ns2 cation – is sufficient to ensure defect tolerance, at least with regard to cation vacancies. This notion can inadvertently be reinforced by molecular orbital diagrams such as those shown in Figure 1, which at first glance seems to suggest that if MAPI exhibits shallow cation vacancies, so too should PbI2 , since it contains the same ions in the same oxidation states. The presence of the methylammonium (CH3 NH3 , or MA) can largely be ignored in this discussion as it does not contribute to the states near the band edges. However, one should not neglect the influence of MA on the crystal structure, which, as discussed later, plays a decisive role in defect tolerance. Figure 2 shows defect plots for six binary iodides (InI, TlI, SnI2 , PbI2 , SbI3 and BiI3 ) with partially oxidized ns2 cations. Defect plots in Figure 2 are ordered according to the groups in the periodic table these cations belong to. Results shown are generally consistent with previous work on InI 26 and BiI3 , 55 although the latter did not employ a band gap correction. It is evident from Figure 2 that all the compounds contain deep defects, i.e., those with charge transition levels deep inside the band gap, away from the band edges by more than a few kB T at room temperature. This is the case for most of the anion and cation vacancies in Figure 2. Out of the six compounds, only InI and TlI exhibit shallow cation vacancies. It is clear from these defect diagrams that the presence of ns2 cation is not a sufficient condition for ensuring shallow defects. Interestingly, we observe a trend as we move from InI→SnI2 →SbI3 and TlI→PbI2 →BiI3 : cation vacancies generally become deeper, suggesting that there are additional chemical effects, beyond ns2 cations, which are at play. 10


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a. deep cation vacancy A(s)

b. shallow cation vacancy

VC states

VC states A(s) C(p)

C(p) band gap

deep A(p)

A(p)

A(p)-nn

resonant

A(p)-nn C(s)

C(s)

Figure 3: Mechanism of cation vacancy formation, showing interaction of neighboring anion orbitals. (a) Sub-optimal orbital energy alignment leads to a deep cation vacancy state, while (b) optimal orbital energy alignment pushes the valence band maximum higher, relative to the A(p) state and makes the cation vacancy state resonant within the valence band. We discuss these chemical effects next.

Refined Description - Role of Orbital Alignment The existing explanation of defect tolerance, based only on the presence of partially oxidized cations, does not account for the interactions of the dangling bonds and associated structural relaxation upon forming a defect. Figure 3a shows a schematic of how cation vacancy states form as a result of the interactions between the anion dangling bonds. Depending on the strength of these interactions, determined by the crystal structure as well as the strength of the chemical bonds of the anions with the rest of the crystal, the resulting antibonding state could rise above the VBM to form a deep defect level. Therefore, to decrease the likelihood of this happening, the mere presence of an ns2 cation is not sufficient. The chemistry that would facilitate defect tolerance (with regard to cation vacancies) also needs to ensure that there is good energy alignment (i.e. a small difference in energy) between the valence cation(s) and the anion(p) orbitals, as shown in Figure 3b. The alignment would help increase the anion(p)-cation(s) repulsion, which would push the VBM to higher energies relative to the cation(s) level, and further above the dangling anion(p) orbitals that interact to form the cation vacancy state. Their antibonding interaction is then not strong enough to

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shallow anion vacancy VA states

a. CsCl structure

Tl

C(p)-nn

Br

resonant

C(p)

resonant

band gap

A(p)

b. low anion coordination

C(p)

band gap

A(p)-nn

Pb

A(p)

I C(s) triply-degenerate Pb(p)

Figure 4: Two structural features that give rise to shallow anion vacancies. (a) Symmetry of the CsCl structure leads to independent cation and anion sublattices such that no dangling cation bonds interact at the anion vacancy state. (b) Low coordination of the anions promote large separation between the neighboring cations such that the anion vacancies are shallow. push that state above the VBM. In the case of the six binary iodides studied here, the energy difference between the cation(s) and anion(p) atomic orbitals increases as one moves from the +1-charged cations in TlI and InI to the +3 cations in BiI3 and SbI3 . This increasing energy misalignment explains the effect observed in Figure 2, wherein cation vacancies get shallower from right to left. Thus, to increase the chances for realizing shallow cation vacancies, both criteria (partially oxidized cation and good energy alignment of relevant atomic orbitals) need to be satisfied. An analogous conclusion can be reached for anion vacancies, where the energy alignment between the next-shell, unoccupied anion(s) and cation(p) is of interest. Good energy alignment and strong interactions between unoccupied anion(s) and occupied cation(p) would push the CBM to lower energies. The cation(p) dangling bonds left upon forming the anion vacancy may form shallow defect levels or resonant states inside the conduction band. However, good energy alignment of the unoccupied anion(s) and cation(p) orbitals is unlikely to be realized, because the energy differences between the anion(p) and the nextshell, unoccupied anion(s) are usually much larger than the differences between cation(s) and cation(p). Consequently, good alignment of orbitals forming the VBM usually means misalignment of those forming the CBM in a binary compound. As discussed next, these

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a. TlBr (CsCl structure): I-rich

4

Tl

3

HD,q (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

+1

VTl

0

1.0

EF (eV)

+2

VTl

VBr

-1 2.0

hypo PbI2

InI

3.0 0

1.0 EF (eV)

IPb

+1

IIn

VI

-3

VI

-2

+1

ITl

-1 0

d. PbI2 (Cu2O structure): I-rich

+2 TlI

+2

-2

c. InI (CsCl structure): I-rich

TlBr

Br

BrTl

1

b. TlI (CsCl structure): I-rich

+1

-2

-1

-2

VI

VPb

VIn

2.0

0

0.5

1.0 EF (eV)

1.5

0

0.5

1.0 EF (eV)

1.5

Figure 5: Defect energetics of cation and anion vacancies and antisites in the CsCl structure of (a) TlBr, (b) TlI, and (c) hypothetical InI, and the Cu2 O structure of (d) hypothetical PbI2 . Representative CsCl and Cu2 O structures are shown as insets in (a) and (d). The defect formation energy (∆HD,q ) is calculated at the I-rich growth conditions in all cases. With the exception of ITl in TlI, all the vacancies and antisites in these structures are shallow defects. In contrast, the anion vacancy (VI ) in the native structures of TlBr (rocksalt), TlI (Figure 2b), InI (Figure 2a), and PbI2 (Figure 2d) are deep defects. limitations can be effectively overcome by a favorable crystal structure, which influences the strengths of these orbital interactions.

Refined Description – Role of Crystal Structure Iodine (anion) vacancies in all six binary iodides shown in Figure 2 introduce deep defect levels, suggesting that the existing criteria (partially oxidized cation) do not suffice at all to enable shallow anion vacancies. Indeed, there are many other examples of compounds with deep anion vacancy states, which often are detrimental to their electronic performance, such as in CdTe, 56 ZnO, 57 SnS, 27 and others. 32,49 Materials known to have shallow anion vacancies, particularly halides, tend to exhibit one of two key structural features, which are discussed in this section and shown in Figure 4. Crystal Site Symmetry. Shi and Du have previously noted that Br vacancy in TlBr in the CsCl and rocksalt structures are shallow and deep defects, respectively. 28 The shallow anion vacancy of TlBr in the CsCl structure is a consequence of the cubic coordination of atoms and the symmetry properties of the p atomic orbitals that form the conduction

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band edge. Namely, in the 8-fold cubic coordination, the p-orbitals on the central atom point to the face centers and not to the vertices of the coordination cube. This implies stronger orbital overlap between the second neighbors, which then leads to almost exclusive cation nature of the states near the conduction band edge, which is evident from the partial charge density isosurface shown in Figure 4a. Thus, when an anion vacancy is created, no cation dangling bonds are exposed and therefore, the anion vacancy introduces only resonant states inside the conduction band (Figure 4a). We confirm the shallow nature of the Br vacancy by calculating the formation energies of the native defects (vacancies, antisites) in TlBr in the CsCl structure (Figure 5a). Note that this structure also exhibits a shallow Tl (cation) vacancy, consistent with the condition of partially oxidized cation presence (Tl) and good energy alignment between the orbitals forming the valence band edge. We have also confirmed that interstitials are not the dominant defects in the CsCl structure of TlBr (Figure S1). On the other hand, in the 6-fold octahedral coordination present in the rocksalt structure, the p-orbitals of the central atom point directly to the vertices of the coordination octahedron; a significant contribution to the states close to the conduction band edge arises from this interaction (sigma bonding). If the central atom is removed, the dangling bonds of the nearest neighbors point to the center of the octahedron and can interact relatively strongly among themselves to form deep defect states. The strength of this interaction will entirely depend on the size of the anion (central atom) relative to the spatial extent of the dangling bonds. In case of large anions and relatively small cations, the anion vacancies can still be shallow despite the octahedral coordination of anions, as in the case of Cu3 N. 23 These arguments can also be extended to tetrahedral coordination of atoms. The scenario is qualitatively similar to the octahedral coordination when it comes to orbital interactions. However, an important difference is that in the tetrahedral coordination, the coordinating atoms are, in general, closer to each other and the size of the anion has less influence on the overlap of the dangling bonds left upon forming the vacancy. Hence, in Cu2 O, for example,

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0

E - Evacuum (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−2 TlBr (225) TlI (63) −4

InI (63)

TlBr (221)

InI (221)

TlI (221) intolerant tolerant −6

−8

Figure 6: Calculated absolute band edge positions w.r.t. vacuum of TlBr (rocksalt, CsCl), TlI (orthorhombic, CsCl), and InI (orthorhombic, CsCl). Band edge positions include band edge shifts calculated with GW method (see Computational Methodology). the tetrahedral coordination of oxygen (smaller ion than nitrogen) leads to O vacancies that are so deep that their charge transition levels occur inside the valence band. 58,59 This is likely the main reason for the ubiquitously deep nature of anion vacancies in almost all tetrahedrally coordinated semiconductors. To test the robustness of this crystal site symmetry effect, we consider both TlI and InI in the CsCl structure. While TlI in the CsCl structure is a known polymorph, 60 InI in the CsCl structure is simply a hypothetical material that serves as a test of our hypothesis. The formation energies of native defects in TlI and InI in the CsCl structure are presented in Figs. 5b and 5c. As expected, iodine (anion) vacancies in both of these materials become shallow. (In their native structures, iodine vacancies in TlI and InI (Figure 2) exhibit negative-U behavior. In the CsCl structures (Figure 4), the iodine vacancies still exhibit negative-U behavior; however, the +1/-1 charge transition occurs inside the conduction band.) The corresponding cation vacancies are shallow as well, consistent with the previous discussion. In contrast, the iodine vacancies in the native structures of both TlI and InI (Figs. 2a and b) are deep defects. This confirms the hypothesis that a suitable crystal structure can lead to shallow vacancies. In the case of using chemistry to achieve shallow cation vacancies, we argued that the valence band maximum moves up relative to the cation(s) state. In this case, the conduction 15



band minimum moves down to help make the anion vacancies shallow. Because the chemistry is the same between polymorphs, this means a change can be observed in the absolute band positions – computed band positions of TlBr, TlI, and InI in both defect-intolerant and defect-tolerant structures are shown in Figure 6, confirming that the conduction band minimum is energetically lower in defect-tolerant structures. Low Anion Coordination. Low (2-fold) anion coordination 28 can also promote large spatial separation between the neighboring cations and can facilitate formation of only shallow anion vacancies. This is especially true if the anion is large; in addition to the size, the particularly strong sigma bonding between cations and anions will prevent large relaxations of the neighboring cations toward the anion vacancy and the orbital overlap between the two cations will tend to remain relatively weak. To test whether this feature, present in MAPI and other hybrid (and non-hybrid) perovskites, does in fact contribute to its defect tolerance, we devise another hypothetical material and compute its defect formation energies. We consider PbI2 , which exhibits deep defect states in its native structure (Figure 2d), in an inverted Cu2 O structure, i.e., with iodine occupying two-fold coordinated Cu sites and Pb occupying tetrahedrally coordinated oxygen sites. In contrast to ground-state PbI2 , this hypothetical structure exhibits vacancies with no transition levels inside the band gap (Figure 5d). This serves as a further example that the 2-fold anion coordination benefits shallow anion vacancies, and that this effect is not unique to the perovskite structure.

Discussion Thus far, we have shown that the prior heuristics for defect tolerance are incomplete. The presence of a partially oxidized cation is important in ensuring shallow cation vacancies in s-p systems but the interactions between the corresponding orbitals needs to be relatively strong, which can be achieved by a combination of good energy alignment and the appropriate

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crystal structure (local coordination). Furthermore, symmetry and coordination effects in the atomic structure have a strong impact on the behavior of anion vacancies. In this section, we will further explore the relationships between these criteria and discuss their potential extensions.

Synergies and Trade-offs Between Structural and Chemical Features The structural and chemical effects discussed in preceding sections appear to operate largely independently of each other – for example, InI and TlI have favorable orbital alignment but non-ideal structure, and as expected, exhibit shallow cation vacancies and deep anion vacancies (Figure 2). For a complementary example, consider WO3 , which has a perovskitelike ReO3 structure. According to the previous heuristics, WO3 has an almost ideal structure for shallow anion vacancies. However, WO3 does not posses a partially oxidized cation (W6+ is fully oxidized), which we have established to be a critical factor for shallow cation vacancies. Consistent with our hypothesis, we find that the anion (O) vacancy is shallow while the cation (W) vacancy is deep, as shown in Figure 7. These results are also consistent with recently published defect calculations of WO3 . 61 However, the orbital energy alignment and structural effects may not be entirely independent. For example, ground-state PbI2 has deep cation vacancies, while hypothetical cubic PbI2 as well as MAPI exhibit shallow VPb , suggesting that structure may also influence the nature of the cation vacancies. In the case of cubic PbI2 , we see that a non-ideal chemistry (suboptimal Pb-I orbital energy alignment) can exhibit defect tolerance in a more favorable crystal structure. Namely, the tetrahedral coordination of Pb and shorter Pb-I distances in the hypothetical structure strengthens the orbital interactions that form the valence band edge. Consequently, the antibonding valence band edge is pushed to higher energies relative to the octahedrally-coordinated Pb in the native structure, which then gives rise to the shallow nature of the cation vacancies. 17



It is important to also mention that in compounds with partially oxidized cations, stereochemically active lone pairs may oppose formation of certain potentially desirable structures such as those with symmetric, linear anion coordination. In particular, it has been shown that compounds with better energy alignment between cation(s) and anion(p) are more prone to these distortions. 62 This, in addition to the difficulty of achieving energy alignment in both the conduction and valence bands in binary compounds, suggests that the phase space of ternary and multinary compounds is likely a more fruitful area in which to search for new defect-tolerant materials. In materials with multiple cation or anion species, the larger combinatorial space may afford a possibility to achieve good energy alignment and good anion coordination environment despite the structural distortions induced by the lone pairs. The example of MAPI suggests a particular extension of this work to ternary systems, specifically those that exhibit “pseudobinary” behavior – namely, where one of the elements does not contribute to states near the valence or conduction band edge. In MAPI, the MA cation serves as a “filler” to stabilize the pervoskite crystal structure; this concept, in principle, could be extended to other systems. In ternary (or multinary) compounds where multiple cation (or anion) species contribute to the band-edge states, pairwise energy alignments and relative spatial orientation of orbitals would need to be considered in order to evaluate these criteria.

Further Contributions to Defect Tolerance It is worth noting that shallow defect states are not the only contributing factor to defect tolerance. A large dielectric constant 14,63–65 is also important as it can reduce the capture cross-section of the charged defects and inhibit non-radiative recombination. The role of the polar organic cation in the large dielectric constant may be a reason why MAPI seems to outperform its inorganic analogues such as CsPbI3 . 66 The calculated dielectric constants for materials considered in this study are shown in Table S3. Within this dataset, the materials 18


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8 WO3: O-rich

-2 WO

6

ΔHD,q (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


O

VW

-1

4

W

-3 OW -6

2

VO

0 0

0.5

+2

1.0 EF (eV)

-3

1.5

2.0

Figure 7: Calculated formation energy of vacancy and antisite defects in WO3 , which adopts a perovskite-like ReO3 structure (inset) with fully oxidized W6+ . While the anion (O) vacancy is shallow, the cation (W) vacancy introduces deep gap states providing further evidence for the importance of partially oxidized cation in giving rise to shallow cation vacancies. with the largest dielectric constants (cubic TlBr and TlI) are indeed defect-tolerant, which is in agreement with the criteria previously proposed by us 31,67 as well as others 63 and discussed in the commentary by Walsh and Zunger. 68 However, in contrast, hypothetical InI (cubic) and PbI2 (Cu2 O structure) have small dielectric constants and yet exhibit defect-tolerant behavior. It is evident that a large dielectric constant is a contributing factor but not the sole feature giving rise to defect tolerance of semiconductors. Small effective mass, particularly for minority carriers, can enhance transport for a fixed value of carrier lifetime and has also been used previously as a criterion in materials screening efforts. 14,16,18,64,69 Prior work on screening for transparent p-type conducting oxides 70–73 has shown that strong cation(s)-anion(p) hybridization, which is facilitated by their energy alignment (one of the criteria proposed in this work), is also beneficial in creating dispersive bands and hence, low effective masses and higher charge carrier mobilities. Other possibilities, such as engineering defect complexes to neutralize deleterious effects of certain defects in isolation, have also been discussed. 68 19



In this endeavor to identify real, synthesizable materials that satisfy these criteria, stability and a viable synthetic pathway are also critical considerations. Concern has been raised in the literature particularly about compounds including the In+ species, as it is easily converted to the fully oxidized In3+ state, leading to difficulty in synthesizing the desired phase and also likely associated with high defect concentrations in these materials. 74,75 Similar effects have been reported for Sn2+ oxidizing 76 to Sn4+ , although this effect can be suppressed. 77–79 It is also important to note that the discussion in this paper applies to systems where small polaron formation is not likely. In solids with pronounced ionic character, such as ZnO and MgO, self-trapping of charge carriers (holes in particular) due to local lattice distortions may also create deep states in the band gap. These small polaron states would also impede the band transport of carriers in a semiconductor. As we mainly consider materials with partially oxidized ions, which exhibit significant covalent character, we do not expect these effects to be of importance. That said, development of quantitative understanding of chemical and structural features leading to small polaron formation is still in its nascent stages.

Conclusions Understanding the physical mechanisms underlying defect tolerance is critical to the discovery of novel optoelectronic materials amenable to production via low-cost manufacturing processes, such as solution-based methods. One of the foremost characteristics of defecttolerant materials is the presence of only shallow intrinsic defects. Previously, it was suggested that the presence of partially oxidized ns2 cations (In+ , Tl+ , Sn2+ , Pb2+ , Sb3+ , Bi3+ ) could lead to shallow defect states. However, when compounds meeting this criterion were synthesized and characterized, their performance was found to be severely lacking compared to MAPI and related compounds, leading to a re-examination of the computational screening criteria. In this study, we find that a partially oxidized ns2 cation is necessary but not

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sufficient for shallow defects in s-p semiconductors. The presence of such a cation, if its highest occupied s-level is well-aligned with the p-level of the anion, can lead to shallow cation vacancies. However, shallow anion vacancies require crystal structures that either enforce separate cation and anion sublattices through symmetry (CsCl structure) or have low anion coordination. With this improved understanding of structural and chemical features giving rise to defect tolerance, materials search and design efforts can be better targeted towards materials likely to exhibit good carrier transport even in the presence of defects. Such materials are far more probable to perform well in devices manufactured through inexpensive, scalable techniques, enabling cheaper PV technologies.

Acknowledgement P. G. and V. S. acknowledge support from NSF Suschem grant CBET-1605495. T. B. acknowledges support from NSF SusChem grant CBET-1605547. The research was performed using computational resources sponsored by the Department of Energy’s Office of Energy Efficiency and Renewable Energy and located at the NREL. R.C.K. acknowledges the financial support of a MIT Energy Initiative Total Energy Fellowship and the Blue Waters Graduate Fellowship.

Supporting Information GW band edge shifts, comparison of experimental and calculated band gaps, calculated dielectric constants, defect formation energies in cubic TlBr

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Graphical TOC Entry shallow anion vacancy CsCl structure

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Tl

C(p)-nn

resonant

C(p)

Br A(p)

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A(p)-nn

Structural and Chemical Features Giving Rise to Defect Tolerance of

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