Origin of Pronounced Nonlinear Band Gap Behavior in Lead-Tin

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Origin of Pronounced Nonlinear Band Gap Behavior in Lead-Tin Hybrid Perovskite Alloys Anuj Goyal, Scott McKechnie, Dimitar Pashov, William Tumas, Mark van Schilfgaarde, and Vladan Stevanovic Chem. Mater., Just Accepted Manuscript • Publication Date (Web): 05 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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Origin of Pronounced Nonlinear Band Gap Behavior in Lead-Tin Hybrid Perovskite Alloys Anuj Goyal,†,‡ Scott McKechnie,¶ Dimitar Pashov,¶ William Tumas,‡ Mark van Schilfgaarde,∗,¶ and Vladan Stevanović∗,†,‡ †Colorado School of Mines, Golden, CO 80401, USA ‡National Renewable Energy Laboratory, Golden CO 80401, USA ¶King's College London, London, WC2R 2LS, UK E-mail: [email protected]; [email protected]

Abstract Mixed lead-tin hybrid perovskite alloy CH3 NH3 (Pb1−x Snx )I3 attracted significant attention lately because of the reduction of its band gap below both end compounds, which makes it a promising bottom cell material in all-perovskite tandem solar cells. The effect is a consequence of a strongly nonlinear dependence of the alloy band gap on chemical composition. Here, we use electronic structure calculations at different levels of theory (DFT, hybrid DFT and QSGW, with and without spin-orbit interactions) to investigate the presently elusive origin of this effect. Contrary to current conflicting studies, our results show that neither spin-orbit interactions, nor the composition induced changes of the crystal structure and ordering of atoms contribute to the nonlinearity of the band gap. We find that the strong nonlinearity is primarily a consequence of chemical effects, i.e., the mismatch in energy between s and p atomic orbitals of Pb and Sn, which form the band edges of the alloy. These results unravel the nature of the band gap bowing in Sn/Pb hybrid perovskite alloys, and offer a relatively simple way to estimate evolution of the band gap in other hybrid perovskite alloys.

Introduction

FA-83%

I-50%

Hybrid halide perovskites emerged as potentially revolutionary photovoltaic (PV) materials due to a combination of high power-conversion efficiencies and relatively low processing cost. 1–3 Initially, the bulk of research was focused on single-junction devices, while more recently tandem configurations are being explored. 4–6 However, achieving high efficiency tandem solar cells requires top cell materials with band gaps in the range of 1.7 – 1.9 eV and bottom cell gaps between 0.9 – 1.2 eV. 4–7 Hence, the realization of all-perovskite tandems necessitates having materials with band gaps below the range offered by the standard stoichiometric perovskite compounds (1.3 – 2 eV). Recent synthesis of a methylammonium lead/tin halide perovskite alloy MA(Pb1−x Snx )I3 with a band gap below 1.2 eV, revealed that perovskite alloys are potentially suitable for bottom cell applications. 8,10–14 Interestingly, the alloy band gap is lower than the gap of both end compounds (1.6 and 1.3 eV for MAPbI3 and MASnI3 , respectively), which is a consequence of the strong nonlinear dependence on chemical composition. Furthermore, figure 1 shows the lowest band gap hybrid perovskite alloys achieved so far by Pb-Sn alloying. 5,6,8,9 Herein we focus on the MA(Pb1−x Snx )I3 alloy system and investigate the presently elusive mechanism be-

(FA,Cs)Pb(I,Br)3 FA-83%

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Band Gap (eV)

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Chemistry of Materials

FA-80%

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(FA,MA)PbI3 FA-70%

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(FA,MA)(Pb,Sn)I3 FA-60%

Pb-40%

(FA,Cs)(Pb,Sn)I3

MA(Pb,Sn)I3 Pb-50%

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Year Figure 1: Small and large band gap perovksite alloys (along with respective site compositions) utilized in recent years in some of the high efficiency all perovksite tandem solar cells (Refs. 5,6,8,9).

hind the pronounced nonlinear band gap dependence on chemical composition that causes the reduction below both the end compounds. We start from the standard description of the evolution of electronic properties with

ACS Paragon Plus Environment 1

Chemistry of Materials (a) CB antibond

antibond

antibond Sn (p)

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antibond

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Pb (p)

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[110]

MAPbI3

MA(Pb0.75Sn0.25 )I3

MA(Pb0.5Sn0.5 )I3

MA(Pb0.25Sn0.75 )I3

MASnI3

Figure 2: (a) Schematic summarizing the origin of the band gap bowing in MA(Pb1−x Snx )I3 . Shaded regions represent the valence and conduction bands with thick lines showing the molecular orbital picture of the formation of electronic bands in the alloy. (b) Atomic structures used to model MA(Pb1−x Snx )I3 alloys. Selected compositions are shown along different 23 crystallographic directions. Pb, Sn and I are represented by grey, red and purple spheres, respectively.

alloy composition x: Eg (x) = (1−x)Eg (x = 0)+xEg (x = 1)−bx(1−x), (1) with the nonlinearity parameterized in the usual manner by bowing parameter b. As shown experimentally, 11 for 0.50≤x≤0.75 the bowing term in Eq. (1) approaches 250 meV causing the reduction of the alloy band gap to ∼ 1.15 eV. Previous studies offer conflicting explanations for this effect. Im et al. attribute the overall band gap reduction to the combination of spin-orbit and steric effects, which are due to the compositioninduced structural changes. 13 Snaith and co-workers rule out spin-orbit coupling, based on their calculations on FA(Pb1−x Snx )I3 alloys, and propose the short-range ordering of Pb and Sn atoms as the most likely cause. 5 To resolve this apparent controversy, we also employ electronic structure calculations to study how the band gap evolves with chemical composition in the MA(Pb1−x Snx )I3 alloy. We contrast and compare de-

scriptions of the alloy electronic structure obtained at different levels of theory including density functional theory (DFT), hybrid DFT and the most advanced and accurate out of the three, the many-body quasiparticle self-consistent GW approach (QSGW ) 15 . All calculations are performed both with and without spin-orbit coupling (SOC). We also assume a random distribution of Pb and Sn atoms in the alloy and use the special quasi-random structure (SQS) 16 to model the atomic configuration. As expected, different levels of theory produce different absolute band gaps, but somewhat surprisingly, all levels of theory agree in the amount of bowing (the value of b), which is also in good agreement with experimental results. Contrary to previous work 13 , we find that the spin-orbit interactions only affect the linear term and that the nonlinearity (magnitude of b), does not depend appreciably on the spin-orbit coupling. Moreover, the good agreement between our calculations and experimental results, indicates that structural changes


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do not contribute appreciably to the bowing, i.e. the composition induced deformations of the pseudo-cubic crystal structure of the alloy and/or ordering of Pb and Sn over the B-sites. We show that the large value of b is primarily a result of the energy mismatch between Pb and Sn atomic orbitals, which form the valence and conduction bands. As schematically depicted in Fig. 2(a), the band edges in pure MASnI3 are less strongly bound than those of pure MAPbI3 and this leads to differences in the absolute band edge positions (band offsets) as shown. This is mainly because Sn-s and Sn-p atomic orbitals are less strongly bound than the corresponding Pb states (see Table 1), which is true at all levels of theory. Hence, the valence band maximum in the alloy is derived from interactions between Sn-s and I-p orbitals; while the conduction band minimum is derived from the Pb-p and I-p orbitals. As a result, the band gap becomes lower than that of either end compound. In addition to this dominant chemical effect, there is a small contribution from lattice strain: as the chemical composition of the alloy is varied, there is a change in the lattice constant, along with local relaxations, that modify the band edge shifts. Lowering of the alloy band gap below both end compounds is an effect that has been observed, discussed and debated previously for classic semiconductor alloys, such as Ga(As,N) 17 or size-mismatched II-VI alloys. 18 However, the Pb-Sn hybrid perovskite alloy is rather distinct and unique in that the origins of the band gap lowering are almost exclusively chemical in nature (differences in energy of atomic orbitals between Sn and Pb) as opposed to a more common combination of chemical and elastic effects (lattice mismatch, local relaxations, etc.). Lastly, our results reveal a broader relationship between the band gap bowing and the band offsets, and offer a relatively simple way of estimating the band gap evolution with chemical composition in hybrid perovskite alloys based on the trends in the energies of atomic orbitals forming the band edges of pure end compounds.

Results In Figure 3(a) we show the dependence of the DFTGGA, hybrid DFT (HSE exchange-correlation functional), and QSGW calculated band gaps on alloy composition both with and without spin-orbit coupling, with all calculations assuming the pseudo-cubic crystal structure and random distribution of Pb and Sn atoms. The band structure of the alloy at different compositions is shown in the Fig. S1 of the supplementary information (SI). The bowing parameter b is extracted from a least squares fit of equation (1) using band gap values from electronic structure calculations; the fits are shown as continuous lines in Fig. 3(a). For different levels of theory b ranges from b=0.86 in GGA to b=1.33 in HSE and the most accurate QSGW finds b to be 1.08.

The inclusion of spin-orbit coupling leads to a large reduction in the absolute band gaps, where, as expected, the Pb-rich (Sn-poor) side is more strongly affected due to the larger atomic number of Pb compared with Sn. However, the inclusion of the spin-orbit interaction, for a given level of theory, has little effect on the bowing as evidenced by the small change in b. At first glance, the inclusion of spin-orbit coupling in Fig. 3(a) appears to account for the lowering of the alloy band gap values below the values of the two end compounds. However, this turns out to be an artifact owing to errors in the absolute band gap values of the two end compounds. Some evidence for this is provided by the bowing parameter values in Fig. 3(a), which show a weak dependence on spin-orbit coupling. To demonstrate this further, we analyze the effects of SOC and separate the bowing from the linear composition average by combining in Eq. (1) the nonlinear term from first-principles calculations with the linear term evaluated using experimental band gaps of pure MAPbI3 and MASnI3 . 11,13,14 In this way, we eliminate the errors in the theory in Eq. (1). The combined Eg (x) are plotted in Fig. 3(b) and show fair agreement for the HSE, good agreement for the GGA, and agreement within the resolution of the measurement for QSGW. The agreement includes both the amount of bowing as well as the concentration at which the minimal band gap occurs. The sources of error in QSGW band gaps are discussed later, but overall, all levels of theory exhibit relatively good agreement with experimental data indicating that SOC has only a weak effect on the band gap bowing. These results thus rule out spin-orbit coupling as the primary cause for the band gap reduction in MA(Pb1−x Snx )I3 alloys below the end compounds, as previously suggested. 13 On the other hand, spin-orbit coupling does contribute to the linear term in Eq. (1). This is evident from Fig. 3(a) and is mainly a consequence of the renormalization of the band gaps of pure end compounds. Because the SO matrix element (which scales roughly as Z 2 ) is stronger for Pb (Z = 82) than Sn (Z = 52), the SOC will affect MAPbI3 more than MASnI3 . As a consequence, the SOC affects the concentration xmin where the band gap reaches the minimum value, but not the bowing that determines the reduction of the band-gap below the linear composition average. The magnitude of the band gap renormalization in the end compounds owing to relativistic effects can be roughly measured by considering the shifts in the atomic p energies, p − p1/2 . In Table 1, the reduction for Pb is shown to be approximately three times that found in Sn. It is important to note that, among other effects, this simple measure does not account for electrostatic shifts due to charge exchange between ions. Nevertheless, it can be used as a simple guide to understand the band gap shifts due to relativistic effects. To accurately predict xmin , both the linear and quadratic contributions to Eq. (1) must be well described. For this purpose, comparing different lev-


Chemistry of Materials (b) Comparison with experiment

Band gap (eV)

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7 Figure 3: (a) Computed band gaps for different MA(Pb1−x Snx )I3 alloy compositions using QSGW, HSE, and DFT (GGA), with and without spin-orbit coupling (SOC). Continuous lines represent least-squares fits for b in eq. (1) to the calculated data. (b) Comparison with the experimental data from Refs. 11,13,14 is made by adding computed bowing to the measured band gaps of the pure end compounds.

els of theory is useful. As is well known, DFT does not reliably predict semiconductor band gaps. As others have found 13,19 , our GGA calculations overestimate band gaps in both pure compounds and the alloy, while adding SOC leads to the expected underestimation. The HSE hybrid functional, empirically constructed to yield good semiconductor band gaps, performs better than GGA, but still underestimates band gaps by ∼0.30.35 eV after SOC is added. It is well known that QSGW, a true ab initio theory, slightly overestimates semiconductor band gaps, and the error is thought to be connected mainly to missing ladder diagrams in the RPA approximation to W . It has also been established empirically 20,21 that a “hybrid” of QSGW and the LDA (80% QSGW + 20% LDA) largely accounts for this error in many compounds. With the hybrid scaling, the discrepancy with measured band gaps becomes rather small, see Fig. 3(a), albeit with a larger discrepancy on the Pb-rich side. The

residual discrepancy with experiment may be connected in part with our choice of structure to model the true disordered compound (the potential energy surfaces are complicated and small structural changes can shift the band gap values). In addition, there is a phonon contribution which can be significant on the scale of the residual discrepancy, and is discussed briefly in the methodology section. Concerning the influence of the crystal structure on the alloy band gap, pure MASnI3 crystallizes in the pseudo-cubic perovskite structure at room temperature, whereas pure MAPbI3 exists in the tetragonal form, which transforms to the pseudo-cubic at temperature around 330 K. 11 In all our calculations we choose pseudo-cubic as the crystal structure for both the pure and alloy compositions, which would, under the assumption that alloying reduces phase transition temperature, correspond to a situation with T . 330 K. As shown in Fig. 2, the pure MASnI3 , MAPbI3 , as well as the al-


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Table 1: Relativistic atomic energy levels. The s and p refer to the standard scalar-relativistic energy levels (without S.O.), while s1/2 , p1/2 and p3/2 are the fully-relativistic Dirac energy levels. The difference in energy p − p1/2 shown in the last row is a measure of the shift in the band gap due to relativistic corrections (beyond scalar-relativistic).

s s1/2 p p1/2 p3/2 p − p1/2

Pb -12.4 -12.3 -3.8 -4.9 -3.4 1.1

Atomic energy levels (eV) Sn I N C -10.9 -17.7 -18.5 -13.7 -10.8 -17.7 -18.5 -13.7 -4.0 -7.3 -7.3 -5.5 -4.2 -8.0 -7.3 -5.5 -3.8 -7.0 -7.3 -5.5 0.3 0.7 0.0 0.0

Discussion We start by comparing atom and orbital projected densities of states (DOS) for different alloy compositions computed using QSGW, see Fig. 4. In both pure comx = 1.0

H -6.4 -6.4

1.47 Sn (p) Sn (s) Sn (p)

x = 0.75

1.41 Sn (p) Pb (p)

Sn (s)

loy structures at various compositions retain the overall pseudo-cubic geometry after the DFT relaxations. As expected the supercell volume increases with increasing Pb content from pure MASnI3 to MAPbI3 , however, in our calculations we do not observe any spontaneous pseudo-cubic to tetragonal structural transformation even for composition x < 0.5 in MA(Pb1−x Snx )I3 , as reported from experiments. 11 At temperatures below the ambient conditions MASnI3 also distorts to tetragonal symmetry 22 . Our calculations show that alloying in the tetragonal structure is energetically more favorable by ∼10-50 meV per formula unit relative to the alloys in the pseudo-cubic structure for the same compositions. We observe smaller band gap reduction for alloys in the tetragonal structure, with DFT-GGA computed band gap bowing of b = 0.67 (see SI, Fig. S2). However, if the two sets of results are combined, those pertinent to cubic structure for Sn-rich compositions and those corresponding to the tetragonal structure for Pb-rich side, the band gap evolution would closely resemble results shown in Fig. 3. Finally, the good overall correspondence between calculated and measured bowing of the alloy band gaps shown in Fig. 3(b), suggests that the compositioninduced phase transition observed in experiments, and associated steric effects have little influence on the band gap evolution. Further, we performed additional alloy calculations in which Sn atoms are placed adjacent to each other (i.e., clustered together), and find that the enthalpy of these clustered structures is higher by ∼15 - 30 meV/f.u. (see SI, Fig. S3) than that of the random SQS structures. The increased stability of a random distribution indicates that the SQS structures are a good representation and provides further evidence that the band gap bowing is not a result of the ordering of Pb and Sn atoms into clusters, as previously proposed 5 . This is in line with the experimental results of Leijtens et al. 23 , where the Pb-Sn alloy is found to be more stable than the pure Sn compound and is attributed to a reduced probability of finding multiple adjacent Sn atoms (that are prone to oxidation) in Pb-Sn alloys.

Atom Projected Density of States

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x = 0.5

1.6 Sn (s)

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Pb (p)

1.78 Sn (p) Pb (s)

Sn (s)

x=0

Pb (p)

2.19 Pb (p) Pb (s) Pb (p)

17 Figure 4: Atom projected densities of states from QSGW +SOC calculations, showing projections onto Pb-6s and 6p orbitals (grey), and Sn-5s and 5p orbitals (red) in MA(Pb1−x Snx )I3 alloy for different compositions x. The position of the VBM is marked by a dotted line and that of CBM by a solid line. The zero of energy is placed at the VBM for each composition.

pounds and in the alloy, the valence band is composed of admixtures of the B-site cation (Pb and/or Sn) s atomic orbitals and Iodine-5p atomic orbitals, while the conduction band results from the p-p interactions between the same atoms. Because of the energy difference between Pb-s and Sn-s as well as the Pb-p and Sn-p orbitals shown in Table 1, both the valence band maximum (VBM) and the conduction band minimum (CBM) of MASnI3 will be less bound than the VBM and CBM of MAPbI3 . Evidence is provided by the alloy



and is qualitatively consistent with the QSGW results. The alignment of band edges between different calculations is done as described in the computational details section. DOS

CBM

Sn (5p)

x=1

x = 0.75

Pb (6p)

x = 0.5

DOS

x = 0.25

DOS at small Sn concentrations (see x=0.25 in Fig. 4), which shows Sn-derived states at the VBM while the CBM remains mainly of the Pb-p character. A similar, but somewhat weaker, effect on the behavior of the CBM can be observed at the other end of the composition range. While the VBM remains Sn-s/I-p character for both pure MASnI3 and low-Pb alloy, the CBM changes its character from Sn-p/I-p to Pb-p/I-p. Spinorbit coupling enhances the differences in the energy of p-orbitals between Sn and Pb (see Table 1), which then lowers the band gap (more in Pb-compound) and leads to better description of the conduction band offsets between the end compounds. This discussion about respective atomic orbital contribution to VBM and CBM is also consistent with the partial densities of states from GGA+SOC calculations calculated on larger supercells, that are provided in the SI, Fig. S4. Hence, for all alloy compositions the band gap forms between the Sn-s/I-p derived valence band maximum and the Pb-p/I-p derived conduction band minimum, which ultimately leads to the band gap bowing and its reduction below each end compound. We conclude that the main reason for the observed reduction of the band gap is the band offset between the end compounds. Analysis of the charge densities that correspond to the VBM and CBM wavefunctions further supports this finding, by clearly showing the localization of the VBM around Sn atoms and the CBM around Pb atoms, consistent with the DOS picture (see SI, Fig. S5). In addition, our analysis of the formation of point defect states at either end, i.e., substitution of Pb by Sn and vice versa, clearly shows formation of deep defects in both cases, filled defect states above the VBM in the case of Sn substitution for Pb in MAPbI3 and empty defect states below the CBM when replacing Sn by Pb in MASnI3 . The origins of the band gap bowing in semiconductor alloys are typically classified in three categories 17,18,24 : (a) volume deformation potential effects, determined by the changes in the electronic structure of pure compounds due to the changes in the volume, (b) chemical effects, due to intermixing of different types of atoms bringing atomic orbitals at different energies, and (c) effects due to local relaxations and/or lattice distortions that appear as a consequence of the broken symmetry. MA(Pb,Sn)I3 alloys are unique in that the dominant contribution to the band gap bowing is chemical in nature and originates from the differences in the energies of Pb and Sn s and p orbitals. Both the volume deformation potential and local relaxation effects appear to be of second order. To further support this result we analyze in Fig. 5 the volume contribution to the band gap bowing relative to the chemical contribution by overlaying the GGA+SOC calculated evolution of the VBM and CBM of pure Pb and Sn compounds calculated at different lattice constants, which correspond to alloy compositions with x = 0, 0.25, 0.5, 0.75, and 1. We use the GGA+SOC for this purpose because of computational convenience as the GGA+SOC requires less resources

x=0

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Eg Eg

VBM

Sn (5s)

Pb (6s)

MAPbI3

MASnI3

Figure 5: Band edge energies (VBM as circles and CBM as squares) of pure MAPbI3 (grey) and MASnI3 (red) calculated at different lattice constants, which corresponds to marked alloy compositions. At compositions x=0 and 1, atom projected density of states (DOS) show the dominating orbital (s or p) contributions to the band edges. Reduction in the band gap of alloy compositions is captured by the effect of strain on the band edge energies of pure compositions. Data for the analysis is taken from GGA+SOC calculations.

It is clear that changes in the lattice constant do affect positions of the individual band edges of pure compounds and that their volumetric band gap deformation potentials both imply reduction of the band gap with decreasing the lattice constant. This observation is consistent with the positive band gap deformation potentials in hybrid perovskites 28 and supports recent experimental 29 and theoretical 30,31 studies proposing decrease in lattice constant of hybrid perovskites in the cubic phase as a possible strategy to achieve lower band gaps. This follows from the fact that by decreasing the lattice constant the overlap between B-site and X-site atomic orbitals increases, which increases the energy of both VBM and CBM (composed of antibonding states) but at a faster rate for VBM such that it ultimately results in the decrease of the band gap. However, in the (Pb,Sn) alloy the lattice constant is in between those of the pure compounds, which would lead to opposite trends of the band gap upon forming the alloy (band gap decrease for MAPbI3 and increase for MASnI3 ). Therefore, based on the deformation potentials alone these effects would not lead to the observed bowing. From Fig. 5 it is apparent that the band offsets dominate the lowering of the band gap. If the minimal gap of the alloy is estimated just from band offsets shown at the opposite ends of Fig. 5, the resulting gap would be very similar to to the minimal gap calculated from GGA+SOC, see Fig. 3(a) (bottom panel). The deformation potential and local relaxation effects also contribute to the changes in the band gap with composition and help es-


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Chemistry of Materials A-site: (MA,FA)PbI3 MAPbI3

FAPbI3

B-site: MA(Pb,Sn)I3 MAPbI3

MASnI3

X-site: MAPb(I,Br)3 MAPbI3

MAPbBr3 -3.69

-3.7 -4.1

-4.27

-4.1

-4.1

smallest Eg alloy -5.0 -5.7

-5.74

(a)

-5.7

-5.7

(b)

-6.0

(c)

Figure 6: Shows the band edge positions (obtained from Refs. 12,25–27) relative to the vacuum level in pure perovskites constituting the A-, B- and X-site alloys with MAPbI3 as the reference system. If the band gap in the alloy is derived by the highest valence band edge and the lowest conduction band edge of the constituent pure perovskites, then B-site emerges as the most promising site to achieve smallest band gap via alloying.

tablish a continuous change in the region between the dilute compositions near x=0 and x=1. In this paper we have focused on a particular perovskite alloy system. However, there are numerous other A-site and X-site alloys that have been studied. For example: (FA,Cs)SnI3 29 , MASn(I,Br)3 32 , and MASn(I,Cl)3 32 , which are shown to also display large and positive band gap bowing; or (FA,Cs)PbI3 29 , (Cs,Rb)SnI3 33 , MAPb(I,Br)3 34 , and CsSn(I,Br)3 35 whose band gaps to a good approximation follow the Vegard’s law. The physical mechanisms responsible for lowering the band gap in MA(Pb,Sn)I3 , discussed here, could also provide insights into the general trends of the band gap evolution with chemical composition in hybrid perovskite alloys. As shown in Fig. 6, if we consider MAPbI3 as a baseline system and analyze effects of alloying over different sublattices, the following conclusions can be drawn. As the A-site (Fig. 6(a)) do not contribute electronic states close to the band edges, substituting MA for formamidinium (FA) do not change the position of band edges significantly and hence do not results in notable lowering of the band gap. Alloying on the anion X-sublattice (Fig. 6(c)) on the other hand, lowers the energy of the VBM and increases the energy of the CBM, leading to an overall increase in the band gap upon substituting I to Br. Finally, alloying with Sn on the B-sublattice (Fig. 6(b)), aligns the band edges in such a way that they have the strongest influence on lowering the band gap, and the minimum band gap in the alloy below both end compounds is achieved.

making this and similar materials attractive for application in all-perovskite tandem solar cells. We use electronic structure calculations at different levels of theory both with and without spin-orbit interactions, which affords an accurate decoupling of the linear and non-linear contributions to the band gap as a function of composition. Contrary to current and conflicting explanations, we find that neither the spin-orbit interactions nor the composition induced changes in the crystal structure or ordering of atoms have significant contributions to the nonlinearity. We find that the pronounced band gap bowing (nonlinearity) is mainly due to chemical effects, that is, the energy mismatch between Pb and Sn atomic orbitals which form the band edges of the alloy. Both valence and conduction band edges are less bound in the pure Sn- relative to the pure Pb-perovskite which is a consequence of the less bound atomic orbitals of Sn that form the bands. This implies that the band gap of the alloy forms between the Sn-derived VBM and Pb-derived CBM, which consequently leads to the decrease of the alloy band gap below both end compounds. These results resolve the apparent controversy about the origins of the band gap reduction in Pb-Sn hybrid perovskite alloys and offer a more broadly applicable way of estimating the evolution of the band gap in this, and other perovskite alloy systems, purely based on the relative energies of the frontier atomic orbitals.

Methods Modeling alloy crystal structures

Conclusions In conclusion, we studied the origins of the strongly nonlinear dependence of the band gap with chemical composition in MA(Pb,Sn)I3 hybrid perovskite alloy. An effect that lowers the band gap below both end compounds,

Alloy structures are created using pseudo-cubic as the starting structure (obtained from Ref. 22), with Pb sites randomly substituted with Sn atoms. Structures for various alloy compositions are generated using the special quasi-random structure (SQS) method 16 , as implemented in the ATAT package 36,37 . To achieve pair cor-


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relation of less than 10% at Pb-sites in SQS structures, 96 and 144-atom supercells are used.

tour de force calculation) we adopt this simpler hybrid approach.

DFT and hybrid calculations

Phonon contributions

DFT calculations are performed within the projector augmented wave (PAW) method 38 as implemented in the VASP code 39 . The Perdew Burke Ernzerhof (PBE) exchange correlation functional 40 is used within GGA. For hybrid calculations, the functional proposed by Heyd et al., 41,42 (HSE06) is used with the exchange mixing of α = 0.25. Spin-orbit coupling (SOC) is included in the total energy calculations with both functionals by performing static self-consistent SOC calculations on relaxed GGA and HSE structures. Same supercell sizes are used in both GGA and HSE calculations, and all degrees of freedom (cell shape, volume and ionic positions) are relaxed. For QSGW calculations, DFT relaxed 48-atom supercell are employed. The plane wave energy cutoff of 340 eV, and a Monkhorst-Pack k-point sampling 43 is used.

The Fröhlich part of the electron-phonon contribution to the self-energy can be on the order of a few tenths of an eV when there is a large difference in the static and high-frequency dielectric constants, 0 and ∞ respectively. 50 Fröhlich showed, 51 that for ionic compounds there is a correction to a band edge that scales in proportion to (1/∞ − 1/0 ). The ratio 0 /∞ is known to be large in MAPbI3 (of order 10), and it is likely to be larger in MAPbI3 than MASnI3 . This is because Pb is less strongly bound than Sn and thus more polarizable, making 0 larger. In addition, the band gap of MAPbI3 is larger than that of MASnI3 , making ∞ smaller, with the net effect that 0 /∞ , and thus the Fröhlich correction is likely significantly larger in MAPbI3 . This contribution will be studied in detail elsewhere, but these indications already provide a simple explanation why QSGW describes the gaps in MAPbI3 a bit less well than in MASnI3 . In any case, corrections are small and have only a minor effect on the results.

QSGW calculations Relativistic all-electron QSGW calculations 44 are performed within the full-potential linearized muffin-tin orbital (FP-LMTO) method 45,46 using the Questaal suite of programs 47 . The LDA is used as the starting point but by construction the QSGW approach is, for the most part, independent of the reference hamiltonian. Two basis functions are used per l channel and local orbitals are included for Pb (5 d) and I (high-lying s and p). Spin-orbit coupling is treated perturbatively for the alloy calculations but the free atom calculations (LDA) are carried out using both scalar Dirac and fully relativistic Dirac formulations 48 . Hybrid scaling of QSGW calculations It is known that QSGW, slightly overestimates semiconductor band gaps, and and the error is thought to be connected mainly to missing ladder diagrams in the RPA approximation to W . It has also been established empirically 20,21 that a “hybrid” of QSGW and the LDA (80% QSGW + 20% LDA) largely accounts for this error in many compounds. There is some physical justification for this empirical fact: the dielectric constant ∞ is systematically underestimated by a nearly universal factor of 0.8 in a wide range of semiconductors. This is closely connected to the fact that plasmon peaks are universally blue shifted relative to experiment when the starting hamiltonian produces a good band gap, and it has been well established that the addition of ladder diagrams largely corrects for this shift 49 . As is evident from the Kramers-Kronig relations, the blue shift in the plasmon peak causes a corresponding reduction in ∞ , which then increases W = −1 v, and the splitting between occupied and unoccupied states. Rather than include ladder diagrams in the calculation of W (q, ω) (a

Alignment of the band edges Here we describe the alignment of band edges computed at different lattice constant in Fig. 5. It is not meaningful to align bands from separate bulk calculations in a rigorous way, because the G=0 contribution to the electrostatic potential, that introduces a constant shift of the reference energy, is arbitrary. However, a physically reasonable reference can be constructed from the following thought experiment. Imagine a lattice of spheres that fill space. If each sphere is neutral, a sensible definition of the electrostatic potential is to choose 0 at the sphere boundary. As a next step, transfer charge between spheres. The electrostatic potential (the Madelung potential) is now well defined, since the reference was defined. Aligning bands in this way for two independent bulk calculations relies on an assumption, namely that if an interface were formed between two materials, the interfacial dipole with respect to this reference would be small. In practice this assumption seems to be well borne out for a range of semiconductors. At the GGA level we accomplish this by aligning the calculated electrostatic potentials averaged around the atomic cores. We observe an almost uniform jump in the average electrostatic potentials around the cores of the atoms of the same kind including H, C, N, and I, and not Pb and Sn because of the changing ratio between the Pb and Sn (see SI, Fig. S6). In a manner similar to the band alignment procedure applied to standard defect calculations 52 , we treat Pb and Sn as defects and align electronic energies of different calculations by treating this uniform jump as the offset between the absolute reference energies. This way of aligning facilitates the


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analysis and is fully consistent with the picture that can be derived from the DOS alone (see SI, Fig. S7). The zero of energy in Fig. S6 is placed at the valence band maximum of the pure MAPbI3 . In this construction, the lower edge of the p-derived DOS is around −4 eV (which consists of a mixture of I p, and (Pb,Sn) p) in QSGW calculations and around −3.5 eV in GGA calculations, and is essentially independent of composition. Supporting Information Available: 1. SI file (PDF): Band structure plots, Band effective masses, Enthalpy of mixing, Density of states plots, Partial charge density, Change in average electrostatic potential with composition. 2. DFT alloy structure files in VASP format (ZIP). This material is available free of charge via the Internet at http://pubs.acs.org/. Acknowledgement This work was supported as part of the Center for the Next Generation of Materials by Design, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. 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 National Renewable Energy Laboratory. The KCL staff was supported by the EPSRC (Grant No EP/M009602/1), and thank UK Materials and Molecular Modeling Hub for computational resources (EP/P020194/1)).

References

Sutton, R. J.; Ma, W.; Moghadam, F.; Conings, B.; Babayigit, A.; Boyen, H.-G.; Bent, S.; Giustino, F.; Herz, L. M.; Johnston, M. B.; McGehee, M. D.; Snaith, H. J. Perovskite-perovskite tandem photovoltaics with optimized band gaps. Science 2016, 354, 861–865. (6) Zhao, D.; Yu, Y.; Wang, C.; Liao, W.; Shrestha, N.; Grice, C. R.; Cimaroli, A. J.; Guan, L.; Ellingson, R. y. J.; Zhu, K.; Zhao, X.; Xiong, R.-G.; Yan, Y. Low-bandgap mixed tinâĂŞlead iodide perovskite absorbers with long carrier lifetimes for all-perovskite tandem solar cells. Nat. Energy 2017, 2, 17018. (7) Meillaud, F.; Shah, A.; Droz, C.; VallatSauvain, E.; Miazza, C. Efficiency limits for singlejunction and tandem solar cells. Sol. Energy Mater. Sol. Cells 2006, 90, 2952–2959. (8) Li, Y.; Sun, W.; Yan, W.; Ye, S.; Rao, H.; Peng, H.; Zhao, Z.; Bian, Z.; Liu, Z.; Zhou, H.; Huang, C. 50% Sn-Based Planar Perovskite Solar Cell with Power Conversion Efficiency up to 13.6%. Adv. Energy Mater. 2016, 6, 1601353. (9) Zhao, D.; Wang, C.; Song, Z.; Yu, Y.; Chen, C.; Zhao, . X.; Zhu, K.; Yan, Y. Four-Terminal All-Perovskite Tandem Solar Cells Achieving Power Conversion Efficiencies Exceeding 23%. ACS Energy Letters 2018, 3, 305–306.

(1) Lee, M. M.; Teuscher, J.; Miyasaka, T.; Murakami, T. N.; Snaith, H. J. Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites. Science 2012, 338, 643–647.

(10) Zuo, F.; Williams, S. T.; Liang, P.-W.; Chueh, C.C.; Liao, C.-Y.; Jen, A. K.-Y. Binary-Metal Perovskites Toward High-Performance PlanarHeterojunction Hybrid Solar Cells. Adv. Mater. 2014, 26, 6454–6460.

(2) Snaith, H. J. Perovskites: The Emergence of a New Era for Low-Cost, High-Efficiency Solar Cells. J. Phys. Chem. Lett. 2013, 4, 3623–3630.

(11) Hao, F.; Stoumpos, C. C.; Chang, R. P. H.; Kanatzidis, M. G. Anomalous Band Gap Behavior in Mixed Sn and Pb Perovskites Enables Broadening of Absorption Spectrum in Solar Cells. J. Am. Chem. Soc. 2014, 136, 8094–8099.

(3) Yin, W.-J.; Yang, J.-H.; Kang, J.; Yan, Y.; Wei, S.-H. Halide perovskite materials for solar cells: a theoretical review. J. Mater. Chem. A 2015, 3, 8926–8942. (4) McMeekin, D. P.; Sadoughi, G.; Rehman, W.; Eperon, G. E.; Saliba, M.; Höorantner, M. T.; Haghighirad, A.; Sakai, N.; Korte, L.; Rech, B.; Johnston, M. B.; Herz, L. M.; Snaith, H. J. A mixed-cation lead mixed-halide perovskite absorber for tandem solar cells. Science 2016, 151– 155. (5) Eperon, G. E.; Leijtens, T.; Bush, K. A.; Prasanna, R.; Green, T.; Wang, J. T.-W.; McMeekin, D. P.; Volonakis, G.; Milot, R. L.; May, R.; Palmstrom, A.; Slotcavage, D. J.; Belisle, R. A.; Patel, J. B.; Parrott, E. S.;

(12) Ogomi, Y.; Morita, A.; Tsukamoto, S.; Saitho, T.; Fujikawa, N.; Shen, Q.; Toyoda, T.; Yoshino, K.; Pandey, S. S.; Ma, T.; Hayase, S. CH3 NH3 Snx Pb(1x) I3 Perovskite Solar Cells Covering up to 1060 nm. J. Phys. Chem. Lett. 2014, 5, 1004–1011. (13) Im, J.; Stoumpos, C. C.; Jin, H.; Freeman, A. J.; Kanatzidis, M. G. Antagonism between SpinâĂŞOrbit Coupling and Steric Effects Causes Anomalous Band Gap Evolution in the Perovskite Photovoltaic Materials CH3 NH3 Sn(1x) Pbx I3 . J. Phys. Chem. Lett. 2015, 6, 3503–3509.


Chemistry of Materials 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

(14) Anaya, M.; Correa-Baena, J. P.; Lozano, G.; Saliba, M.; Anguita, P.; Roose, B.; Abate, A.; Steiner, U.; Grätzel, M.; Calvo, M. E.; Hagfeldt, A.; Míguez, H. Optical analysis of CH3 NH3 Snx Pb1x I3 absorbers: a roadmap for perovskite-on-perovskite tandem solar cells. J. Mater. Chem. A 2016, 4, 11214–11221. (15) van Schilfgaarde, M.; Kotani, T.; Faleev, S. Quasiparticle Self-Consistent GW Theory. Phys. Rev. Lett. 2006, 96, 226402. (16) Zunger, A.; Wei, S.-H.; Ferreira, L. G.; Bernard, J. E. Special quasirandom structures. Phys. Rev. Lett. 1990, 65, 353–356. (17) Mattila, T.; Wei, S.-H.; Zunger, A. Localization and anticrossing of electron levels in GaAs1−x Nx alloys. Phys. Rev. B 1999, 60, R11245–R11248. (18) Moon, C.-Y.; Wei, S.-H.; Zhu, Y. Z.; Chen, G. D. Band-gap bowing coefficients in large sizemismatched II-VI alloys: first-principles calculations. Phys. Rev. B 2006, 74, 233202.

Page 10 of 12

(25) Brivio, F.; Butler, K. T.; Walsh, A.; van Schilfgaarde, M. Relativistic quasiparticle selfconsistent electronic structure of hybrid halide perovskite photovoltaic absorbers. Physical Review B 2014, 89, 155204. (26) Butler, K. T.; Frost, J. M.; Walsh, A. Band alignment of the hybrid halide perovskites CH 3 NH 3 PbCl 3 , CH 3 NH 3 PbBr 3 and CH 3 NH 3 PbI 3. Materials Horizons 2015, 2, 228–231. (27) Ma, F.; Li, J.; Li, W.; Lin, N.; Wang, L.; Qiao, J. Stable α/δ phase junction of formamidinium lead iodide perovskites for enhanced nearinfrared emission. Chemical Science 2017, 8, 800– 805. (28) Frost, J. M.; Butler, K. T.; Brivio, F.; Hendon, C. H.; van Schilfgaarde, M.; Walsh, A. Atomistic Origins of High-Performance in Hybrid Halide Perovskite Solar Cells. Nano Lett. 2014, 14, 2584– 2590.

(19) Castelli, I. E.; García-Lastra, J. M.; Thygesen, K. S.; Jacobsen, K. W. Bandgap calculations and trends of organometal halide perovskites. APL Mater. 2014, 2, 081514.

(29) Prasanna, R.; Gold-Parker, A.; Leijtens, T.; Conings, B.; Babayigit, A.; Boyen, H.-G.; Toney, M. F.; McGehee, M. D. Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics. J. Am. Chem. Soc. 2017, 139, 11117–11124.

(20) Chantis, A. N.; van Schilfgaarde, M.; Kotani, T. Ab Initio Prediction of Conduction Band Spin Splitting in Zinc Blende Semiconductors. Phys. Rev. Lett. 2006, 96, 086405.

(30) Filip, M. R.; Eperon, G. E.; Snaith, H. J.; Giustino, F. Steric engineering of metal-halide perovskites with tunable optical band gaps. Nat. Commun. 2014, 5, 5757.

(21) Deguchi, D.; Sato, K.; Kino, H.; Kotani, T. Accurate energy bands calculated by the hybrid quasiparticle self-consistent GW method implemented in the ecalj package. Jpn. J. Appl. Phys. 2016, 55, 051201.

(31) Meloni, S.; Palermo, G.; Ashari-Astani, N.; Grätzel, M.; Rothlisberger, U. Valence and conduction band tuning in halide perovskites for solar cell applications. J. Mater. Chem. A 2016, 4, 15997–16002.

(22) Stoumpos, C. C.; Malliakas, C. D.; Kanatzidis, M. G. Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and NearInfrared Photoluminescent Properties. Inorg. Chem. 2013, 52, 9019–9038.

(32) Stoumpos, C. C.; Kanatzidis, M. G. The Renaissance of Halide Perovskites and Their Evolution as Emerging Semiconductors. Acc. Chem. Res. 2015, 48, 2791–2802.

(23) Leijtens, T.; Prasanna, R.; Gold-Parker, A.; Toney, M. F.; McGehee, M. D. Mechanism of Tin Oxidation and Stabilization by Lead Substitution in Tin Halide Perovskites. ACS Energy Lett. 2017, 2, 2159–2165. (24) Wei, S.-H.; Zunger, A. Band gaps and spin-orbit splitting of ordered and disordered Alx Ga1−x As and GaAsx Sb1−x alloys. Phys. Rev. B 1989, 39, 3279–3304.

(33) Jung, Y.-K.; Lee, J.-H.; Walsh, A.; Soon, A. Influence of Rb/Cs Cation-Exchange on Inorganic Sn Halide Perovskites: From Chemical Structure to Physical Properties. Chem. Mater. 2017, 29, 3181–3188. (34) Noh, J. H.; Im, S. H.; Heo, J. H.; Mandal, T. N.; Seok, S. I. Chemical Management for Colorful, Efficient, and Stable InorganicâĂŞOrganic Hybrid Nanostructured Solar Cells. Nano Lett. 2013, 13, 1764–1769. (35) Sabba, D.; Mulmudi, H. K.; Prabhakar, R. R.; Krishnamoorthy, T.; Baikie, T.; Boix, P. P.; Mhaisalkar, S.; Mathews, N. Impact of Anionic Br


Page 11 of 12 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

Chemistry of Materials Substitution on Open Circuit Voltage in Lead Free Perovskite (CsSnI3−x Brx ) Solar Cells. J. Phys. Chem. C 2015, 119, 1763–1767.

(36) van de Walle, A.; Asta, M.; Ceder, G. The alloy theoretic automated toolkit: A user guide. Calphad 2002, 26, 539–553. (37) van de Walle, A.; Tiwary, P.; de Jong, M.; Olmsted, D.; Asta, M.; Dick, A.; Shin, D.; Wang, Y.; Chen, L.-Q.; Liu, Z.-K. Efficient stochastic generation of special quasirandom structures. Calphad 2013, 42, 13–18. (38) Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953–17979. (39) Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6, 15–50.

(49) Bruneval, F.; Vast, N.; Reining, L.; Izquierdo, M.; Sirotti, F.; Barrett, N. Exchange and Correlation Effects in Electronic Excitations of Cu2O. Phys. Rev. Lett. 2006, 97, 267601. (50) Lambrecht, W. R. L.; Bhandari, C.; van Schilfgaarde, M. Lattice polarization effects on the screened Coulomb interaction W of the GW approximation. Phys. Rev. Mater. 2017, 1, 043802. (51) Fröhlich, H. Electrons in lattice fields. Adv. Phys. 1954, 3, 325–361. (52) Lany, S.; Zunger, A. Assessment of correction methods for the band-gap problem and for finitesize effects in supercell defect calculations: Case studies for ZnO and GaAs. Phys. Rev. B 2008, 78, 235104.

(40) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (41) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 2003, 118, 8207–8215. (42) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Erratum: âĂĲHybrid functionals based on a screened Coulomb potentialâĂİ [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 2006, 124, 219906. (43) Monkhorst, H. J.; Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 1976, 13, 5188–5192. (44) Kotani, T.; van Schilfgaarde, M.; Faleev, S. V. Quasiparticle self-consistent GW method: A basis for the independent-particle approximation. Phys. Rev. B 2007, 76, 165106. (45) Dreyssé, H., Ed. Electronic Structure and Physical Properties of Solids; Springer: Berlin, 2000; p 114. (46) van Schilfgaarde, M.; Kotani, T.; Faleev, S. V. Adequacy of approximations in GW theory. Phys. Rev. B 2006, 74, 245125. (47) http://www.questaal.org. Our GW implementation was adapted from the orig inal ecalj package, now at https://github.com/tkotani/ ecalj/. (48) Ebert, H. Fully relativistic treatment of core states for spin-dependent potential. J. Phys.: Condens. Matter 1989, 1, 9111.



Graphical TOC Entry MAPbI3

Energy relative to

vacuum (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

-4

MASnI3

CBM

band gap

-5 -6

alloy

VBM


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Origin of Pronounced Nonlinear Band Gap Behavior in Lead-Tin

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