GW Band Structures and Carrier Effective Masses of CH3NH3PbI3

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GW Band Structures and Carrier Effective Masses of CH3NH3PbI3 and Hypothetical Perovskites of the Type APbI: A = NH, PH, AsH and SbH 3

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Marina R Filip, Carla Verdi, and Feliciano Giustino J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b07891 • Publication Date (Web): 19 Oct 2015 Downloaded from http://pubs.acs.org on October 20, 2015

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The Journal of Physical Chemistry

GW Band Structures and Carrier Effective Masses of CH3NH3PbI3 and Hypothetical Perovskites of the Type APbI3: A = NH4, PH4, AsH4 and SbH4 Marina R. Filip, Carla Verdi, and Feliciano Giustino∗ Department of Materials, University of Oxford, Parks Road OX1 3PH, Oxford, UK Phone: (+44) 1865 612790 E-mail: [email protected]

∗

To whom correspondence should be addressed

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Abstract Solar cells based on organic-inorganic lead-halide perovskites are currently one of the fastest improving photovoltaic technologies. Understanding the fundamental electronic and optical properties of CH3 NH3 PbI3 and related metal-halide perovskites represents a key step in the future development of perovskite optoelectronic devices. Here we study the quasiparticle band structures, band gaps and effective masses of CH3 NH3 PbI3 and the hypothetical perovskites NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 within the GW method, using Wannier interpolation. We find that the quasiparticle band gaps of the hypothetical perovskites decrease as the size of the cation increases, obtaining values of 1.9 eV (NH4 PbI3 ), 1.8 eV (PH4 PbI3 ), 1.6 eV (AsH4 PbI3 ) and 1.4 eV (SbH4 PbI3 ). The same trend is followed also by the electron and hole effective masses of these compounds, all of which have values below 0.3 electron masses. By estimating the ideal short circuit current, the open circuit voltage, and the theoretical limit for the power conversion efficiency of a solar cell based on these compounds, we find that PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 could improve the performance of solar cells based on CH3 NH3 PbI3 .

Introduction Metal-halide perovskites have been attracting the attention of the research community during the last three years primarily in the context of emerging photovoltaic technologies. 1–5 Solar cells based on metal-halide perovskites show impressive power conversion efficiencies, with the current certified record exceeding 20%. 6 In addition to applications in photovoltaic devices, metal-halide perovskites are now actively investigated for integration in light emitting devices, 7–10 lasers, 9,11 field effect transistors 12,13 and potential applications in spintronics. 14 In photovoltaic devices, methylammonium lead iodide (CH3 NH3 PbI3 ) and mixed halides have been used as light sensitizers as well as hole 15 and electron conductors 16 in a variety of architectures, 2,17 ranging from the dye-sensitized solar cell configuration (DSC 18 ) 19 to

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meso-superstructured 16 and planar heterojuction devices. 20–22 The fast development of photovoltaic devices based on metal halide perovskites has been largely due to the improvement of the crystallinity 23 and morphology control 24 of the perovskite light absorber, and to the exploration of mixed halides, 16,25 mixed metal 26,27 and mixed cation perovskites, 28,29 in an effort to improve light absorption and charge carrier transport. For example, it has been shown recently that the replacement of the CH3 NH3 cation by formamidinium [CH(NH2 )2 ] reduces the optical band gap of the lead-iodide perovskite from 1.5-1.6 eV to 1.48 eV, 29 enabling more efficient light absorption. High efficiencies have recently been achieved by using mixed CH3 NH3 /CH(NH2 )2 lead-iodide-bromide perovskite; this solved stability problems associated with CH(NH2 )2 PbI3 . 30 This development route motivates the search for novel perovskite compounds, specifically designed for better light absorption in the visible range. 31 In a previous study 31 we have shown that the idea of replacing CH3 NH3 by other cations is attractive, since it enables finetuning of the band gap, without interfering with the chemistry of the inorganic network. At the same time we demonstrated how in silico design could assist the experimental discovery of novel perovskite light sensitizers. In order to establish reliable computational methods for the design of novel perovskite light absorbers, it is important to first understand the fundamental physical properties of CH3 NH3 PbI3 , and how to correctly calculate these properties from first principles. Experimental studies have revealed remarkable electronic and optical properties of CH3 NH3 PbI3 , such as a direct band gap of 1.5-1.7 eV, 32–34 exciton binding energy below 50 meV, 32,35–37 high carrier diffusion lenghts of the order of a µm, as well as long carrier lifetimes and low carrier effective masses. 35,38–41 The electronic band structure, as well as the band gap and character of the valence and conduction band states of CH3 NH3 PbI3 have already been elucidated from first principles. 42–48 However, a complete theoretical understanding of the optical and transport properties of CH3 NH3 PbI3 is still to be achieved. CH3 NH3 PbI3 is a perovskite with the inorganic lead-iodide components forming a net-

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work of corner sharing octahedra, and the CH3 NH3 cations located at the center of the inorganic cuboctahedral cavity. 49 The structure of CH3 NH3 PbI3 exhibits two phase transitions, at 160 K (from an orthorhombic to a tetragonal structure) and 330 K (from a tetragonal to a cubic structure). 33,49,50 The CH3 NH3 cations have a fixed orientation at low temperature, when the crystal lattice is orthorhombic, belonging to the P nma 62 symmetry group. 33 Above the first transition temperature the orientation of the CH3 NH3 cations becomes structurally disordered 51,52 and the Pb and I exhibit broad ellipsoids around the equilibrium atomic positions, as reveald by X-ray and neutron diffraction experiments. 53 The disordered character of the CH3 NH3 cations and the Pb-I perovskite cavity at high temperature is confirmed by molecular dynamics simulations, 54 indicating band gap variations due to structural disorder over a range approximately 0.3 eV wide, 55 as well as the possible formation of ferroelectric domains due to dipole twinning effects. 56,57 Moreover, structural optimizations of CH3 NH3 PbI3 reveal that the geometry of the crystal lattice is strongly dependent on the orientation of the CH3 NH3 cation, 58 and the starting point of the lattice relaxation. 43,58,59 The complex nature of the crystal structure of CH3 NH3 PbI3 , has led to some discrepancies in the description of the electronic and optical structure properties from first principles. As recently shown in Ref., 58 the scalar relativistic and fully relativistic band gap calculated within density functional theory (DFT) 60 varies significantly and may even become indirect as a consequence of the orientation of the CH3 NH3 cation and its interaction with the inorganic network. 58 Furthermore, it is now well known that the spin-orbit coupling (SOC) is essential in the correct description of the electronic band structure of CH3 NH3 PbI3 . 44,45,61 In addition, depending on the symmetry properties of the structure, the conduction and valence band edges may exhibit a Rashba-Dresselhaus spin-orbit splitting, 62–64 complicating the topology of the band edges. 47,65,66 The quasiparticle band gap of CH3 NH3 PbI3 (1.6-1.7 eV) has been calculated within the GW method 67 including SOC in several studies, 46,47,68 obtaining good agreement with

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optical band gaps measured at low temperature (1.65 eV 32 ), as well as band gaps measured from photoemission and inverse photoemission spectroscopy (1.7 eV 34 ). Recently, Ref. 47 pointed out that the quasiparticle band gap calculated within G0 W0 +SOC is systematically underestimated with respect to experiment due to the small DFT/LDA band gap, and showed that this underestimation can be corrected by considering self-consistency within quasiparticle self-consistent GW (QSGW ). 69 In our recent study, 48 we presented a simpler approach to the self-consistency requirement by which we calculated a quasiparticle band gap of 1.7 eV for CH3 NH3 PbI3 at the same computational cost as a “one-shot” G0 W0 calculation. In this study, we build on the results obtained in Ref., 48 and extend that study by obtaining the GW band structure of CH3 NH3 PbI3 using maximally localized Wannier functions, 70,71 in a similar approach as the one used in Ref. 66 for CH(NH2 )2 SnI3 and CH(NH2 )2 SnBr3 . Using this method we are able to calculate accurate electron and hole effective masses within the GW approximation. We obtain 0.23 and 0.22 electron masses for the hole and electron effective masses, respectively. Our electron and hole effective masses obtained from GW are in agreement with previous calculations and experimental data in Refs. 35,46,47,72–74 To the best of our knowledge, the detailed calculation of effective masses from the Wannier interpolation of the quasiparticle eigenvalues has not been reported so far for CH3 NH3 PbI3 . However, we note that three other papers report carrier effective masses obtained from the quasiparticle eigenvalues of CH3 NH3 PbI3 , 46,47,72 using different approximations for the interpolation of the quasiparticle band structure. In the second part of this study we use the self-consistent scheme described in Ref. 48 to predict the band gaps of four hypothetical perovskite structures identified in Ref. 31 as potential candidates for perovskite light sensitizers: NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 . Furthermore, we obtain the Wannier interpolated GW band structures and effective masses for all four hypothetical perovskites. The manuscript is organized as follows. In the “Methodology” Section we give a brief description of the computational methodology employed in this study, including the standard

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GW method, the “self-consistent scissor” GW (SS-GW ) scheme proposed in Ref., 48 as well as the Wannier interpolation method. In the “Computational setup” Section we describe the computational details at each step of the study, including the crystal structure optimizations, the ground state DFT calculations used as a starting point for the GW calculations, the setup used for the calculation of the quasiparticle band gaps within G0 W0 and SS-GW , as well as the interpolation of the quasiparticle band structures and calculation of the effective masses. In the “Results” Section we discuss the main results obtained in this study, while in the “Discussions” Section we comment on these results in the broader context of photovoltaic applications.

Methodology The GW approximation We calculate the quasiparticle band gap within GW many body perturbation theory as: 75–79

Enk = ǫnk + Z(ǫnk )hnk|Σ(ǫnk ) − Vxc |nki,

(1)

where ǫnk are the Kohn-Sham eigenvalues of the states with band index n and crystal momentum k, Enk are the corresponding quasiparticle eigenvalues, Σ(ω) is the frequency −1 dependent self energy, Z(ω) = 1 − ∂ReΣ/∂ω is the quasiparticle renormalization and Vxc is the exchange correlation potential calculated from DFT. In the G0 W0 approximation the self energy is approximated by Σ = iG0 W0 , where G0 is the single particle Green’s function calculated from the Kohn-Sham eigenvalues and wave functions, and W0 is the screened Coulomb potential. 75 The frequency dependent dielectric matrix is calculated within the random-phase approximation (RPA) 80,81 using the Godby-Needs plasmon-pole model. 75,82 In the case of CH3 NH3 PbI3 we start from the eigenvalues and eigenfunctions obtained from DFT including spin-orbit coupling. As a consequence of the small Kohn-Sham band

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gap, the dielectric screening is overestimated, leading to an underestimation of the quasiparticle band gap obtained from G0 W0 . 47,48 In order to mitigate this effect, we employ a simple self-consistent scissor scheme (SS-GW ) proposed in Ref. 48 In this approach, we first calculate the G0 W0 quasiparticle correction to the band gap, apply it to the original KohnSham hamiltonian as a scissor correction, and repeat the G0 W0 calculation. We iterate this process until the quasiparticle correction vanishes. Formally we can write this approach as follows:

Enk = ǫnk + Z(ǫnk )hnk|Σ(ǫnk ) − (Vxc + VSC )|nki,

(2)

where VSC = ∆Pc is the non-local scissor operator which acts only on the conduction states, via the projection operator Pc , and ∆ is the magnitude of the scissor shift.

Wannier interpolation In order to interpolate the eigenvalues calculated from DFT and G0 W0 on an arbitrary k-point mesh we construct the basis set of maximally localized Wannier functions: 70,71,83 V |wnR i = (2π)3

Z

N X

(k) Umn |ψmk ie−ik·R dk,

(3)

BZ m=1

where U (k) is a unitary matrix, and ψn (k) are the Kohn-Sham Bloch eigenstates. 83 Using the unitary matrix U (k) we rewrite the Hamiltonian in the basis of the maximally localized Wannier functions, and express it in real space as: 83,84

H (W) (R) =

1 X −ik·R †(k) e U H(k)U (k) , N0 k

(4)

where Hnm (k) = ǫnk δnm (for DFT) or Hnm (k) = Enk δnm (for GW ), and N0 is the number of unit cells corresponding to the k-mesh. Due to the localization of the Wannier functions, (W)

Hnm (R) decays rapidly with R. This property enables the calculation of interpolated eigen-

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(W)

values at any point k′ by performing the Fourier transform of Hnm (R) and diagonalizing the resulting matrix: 83,84

ǫk′ = diag

X

′

(W) eik R Hnm (R),

(5)

R

Effective masses We calculate the effective mass tensor by approximating the second derivatives of the valence and conduction band edges with respect to the wave vector k at the Γ point: 85

m∗ij = ~2 (∂ 2 ǫ/∂ki ∂kj )−1 ,

(6)

where the second-order partial derivatives are calculated with respect to the wave vector k along the three crystallographic directions indexed by i and j. We calculate these derivatives numerically using the finite difference method in the first order, and obtain the electron and hole effective masses by diagonalizing the effective mass tensor m∗ij for the conduction and valence band, respectively. Finally, we calculate the isotropic hole and electron effective masses by averaging over the eigenvalues of the valence and conduction band effective masses.

Short-circuit current, open-circuit voltage and maximum theoretical efficiency For a light absorber with an optical band gap Eg , the maximum short circuit current density, JSC , can be estimated by assuming that all photons with energy larger than Eg are absorbed and always produce an electron-hole pair upon absorption: 86,87

JSC = e

Z

∞ Eg

S(E) dE, E

(7)

where e is the electronic charge, S(E) is the power radiated by the sun on Earth per unit area and per unit of photon energy, and E is the energy of an incident photon. 8


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The open circuit voltage can be expressed in terms of the band gap of the light absorber and a variable parameter, called the loss-in-pontential Eloss as: 87

eVOC = Eg − Eloss .

(8)

Finally, using the maximum short circuit current estimated in Eq.(7) and the open circuit voltage in Eq. (8) we can estimate a maximum theoretical limit for the efficiency as a function of the band gap of the light absorber: 86–88

η(Eg ) =

F F × JSC VOC . Psun

(9)

where F F is the fill factor, defined as the ratio between the area under the J(V ) curve and the product JSC VOC , and Psun is the total incident power, calculated as Z ∞ S(E)dE, 86–88 using solar spectrum data reported by the National Renewable Psun = 0

Energy Laboratory (NREL). 89

Computational setup DFT and GW calculations Crystal structure optimization. Unless otherwise specified, in the case CH3 NH3 PbI3 we use the experimental crystal structure measured for the low temperature orthorhombic phase and reported in Ref., 33 thus avoiding complications arising from the disordered orientation of the CH3 NH3 cation. All other structural optimizations are performed within the local density approximation (LDA) to density functional theory (DFT) as implemented in Quantum ESPRESSO. 90 For Pb, I, C, N, P, As and H we use ultrasoft pseudopotentials 91,92 with nonlinear core-correction 93 for Pb and I, while for Sb we use a norm conserving pseudopotential. 94 The charge density and electron wave functions are expanded in plane wave basis sets with cutoffs of 200 Ry and 40 Ry, respectively, and the Brillouin zone is sampled 9



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using an unshifted 4 × 4 × 4 k-point mesh. Ground state calculations. We obtain the ground state DFT/LDA band gaps including spin-orbit coupling for CH3 NH3 PbI3 , NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 , and use them as starting point for our quasiparticle calculations. For these calculations we use normconserving Troullier-Martins 94 pseudopotentials. For Pb and I we generate fully-relativistic norm conserving pseudopotentials including semicore d states, as indicated in Ref. 48 For C, H, N, P, As, and Sb we use non-relativistic or scalar-relativistic pseudopotentials as available in the Quantum ESPRESSO library. For the ground state calculations we use a kinetic cutoff of 150 Ry and sample the Brillouin zone using an unshifted 6×6×6 k-point grid. GW calculations. We calculate the quasiparticle band gaps of the hypothetical leadiodide perovskites within the G0 W0 approximation including spin-orbit coupling 95–97 as implemented in the Yambo code. 98 For all G0 W0 calculations of the quasiparticle band gap we use the parameters reported in Ref. 48 for the case of CH3 NH3 PbI3 , which we summarize here for clarity: 150 Ry plane wave cutoff for the ground state calculation, 30 Ry plane wave cutoff for the exchange self energy, 6 Ry plane wave cutoff for the polarizability, 1000 bands, 1 Ry plasmon energy, and 2 × 2 × 2 Γ-centered k-point mesh. As it was pointed out for the case of CH3 NH3 PbI3 and CsPbI3 in Ref., 48 G0 W0 band gaps are understimated with respect to experimental values, due severe band gap underestimation in DFT+SOC calculations. It is therefore reasonable to assume that the same problem would occur for the G0 W0 band gaps of the hypothetical perovskites. For this reason we employ the SS-GW method described in Ref. 48 As discussed in Ref., 48 a significant difference between the SS-GW band gap and the G0 W0 band gap is an indication that self-consistent GW would be required for the calculation of the quasiparticle band gaps of these materials. We employ the SS-GW scheme in order to calculate the quasiparticle band gaps of the hypothetical perovskites NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 , SbH4 PbI3 . In Fig. 1(a-d) we show the convergence of the SS-GW band gap for each of the four hypothetical perovskites. In all four cases the SS-GW band gap converges within 5 or 6 iterations, after which the

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Figure 1: Convergence of the quasiparticle band gaps calculated within the SS-GW scheme for SbH4 PbI3 (a), AsH4 PbI3 (b), PH4 PbI3 (c) and NH4 PbI3 (d). The large empty circles represent the quasiparticle band gaps corresponding to the G0 W0 calculation performed at each iteration. The vertical dashed blue lines represent quasiparticle calculations, while the horizontal dashed blue lines are the scissor corrections performed at each step of the SSGW scheme. The inset in each panel shows the quasiparticle corrections obtained at each iteration step.

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quasiparticle correction is below 10 meV. As shown in Fig. 1(a-d), the quasiparticle band gap increases by up to 0.5 eV upon employing the SS-GW scheme, confirming the requirement for a self-consistent approach in the calculation of the quasiparticle band gap. Dielectric constant calculation. We obtain the high-frequency dielectric constant for all five compounds considered in this study by performing finite-electric field calculations within the Berry-phase technique. 99,100 We use a Γ-centered 4×4×4 Brilouin-zone mesh with a finite electric field of 0.001 a.u. directed along the b axis (the axis with the largest lattice constant). In all five cases we calculate the dielectric constant within scalar-relativistic DFT/LDA. In order to contain the computational effort we use the same ultrasoft pseudopotentials and cutoff parameters as for the crystal structure optimization.

Wannier interpolation The Wannier interpolation of all the DFT and GW band structures is performed using the Wannier90 code. 83 We construct our set of maximally localized Wannier functions for the interpolation of the DFT band structure of CH3 NH3 PbI3 in the vicinity of the band gap using the Pb-p and I-p states as the initial guess, leading to 72 and 24 maximally localized Wannier functions for the valence and conduction bands respectively. This choice is consistent with the dominant I-p character of the valence, and the mixed Pb-p and I-p character of the conduction band of CH3 NH3 PbI3 in the vicinity of the gap. 31 To obtain accurate effective masses it is important that valence and conduction states are wannierized simultaneously. We use a 4 × 4 × 4 Γ centered k-point mesh for the Brillouin zone integration in Eq. (4) in order to interpolate the Kohn-Sham eigenvalues. We check that the effective masses are accurate up to 2% with respect to direct DFT calculations. Therefore, we build our set of maximally localized Wannier functions on a 4 × 4 × 4 k-point grid, and use them to interpolate the DFT, G0 W0 and SS-GW eigenvalues onto arbitrary k-point grids. In order to interpolate the G0 W0 eigenvalues using Wannier functions we recalculate the quasiparticle eigenvalues for CH3 NH3 PbI3 , NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 12


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using a 4 × 4 × 4 k-point grid centered at Γ. To contain the computational effort we reduce the plane wave cutoff of the ground state calculations to 100 Ry, the number of bands to 400, the polarizability cutoff to 4 Ry, and the exchange cutoff to 20 Ry. The G0 W0 and SS-GW band gaps obtained with this computational setup are within 0.13 eV from the fully converged values reported in Table 1. We further test the convergence of the electron and hole effective masses obtained from the Wannier interpolation of the G0 W0 eigenvalues for CH3 NH3 PbI3 by increasing the number of bands to 480 and the polarizability cutoff to 6 Ry. The hole effective mass changes by 2% while the electron effective mass changes by 1%, therefore confirming that 400 bands and a polarizability cutoff of 4 Ry are sufficient for the convergence of the effective masses. For the calculation of the second derivatives we calculate the eigenvalues at a distance of 1% of 2π/a from the Γ point in reciprocal space (where a is the smallest of the three lattice parameters in each case). This choice is sufficient to calculate the effective masses with a precision of 0.5%. A plot of the dependence of the electron and hole effective masses of CH3 NH3 PbI3 on the distance from Γ point in reciprocal space is shown in Fig. S1 of the Supporting Information. Furthermore, for the case of CH3 NH3 PbI3 we analyze the parabolicity of the bands in the direction of the lattice vectors. We obtain that the electronic bands calculated within SS-GW are parabolic for up to one third of the high-symmetry directions Γ-X and Y-Γ-Z, as shown in Fig. S2 of the Supporting Information.

Results Crystal structures The crystal structures of the four hypothetical perovskites are optimized using as a starting point the crystal structure of CH3 NH3 PbI3 reported by Ref., 33 and by replacing the CH3 NH3 with our proposed cations. The details of the structural optimization calculations are reported elsewhere. 31 13



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In Table S1 of the Supporting Information we indicate the lattice parameters and unit cell volumes obtained for each of the four hypothetical perovskites, as reported in Ref. 31 The volume of the unit cell increases in order to accommodate cations of increasing size. NH4 PbI3 departs from this trend slightly, in that its unit cell volume is marginally higher than that of PH4 PbI3 . The optimized unit cells belong to the P nma space group, with the Wyckoff positions of the asymmetric unit cell listed in Table S2 of the Supporting Information.

Band gaps In Table 1 we show the band gaps obtained from DFT+SOC, G0 W0 , and SS-GW for all four hypothetical perovskites and for the experimental and optimized structures of CH3 NH3 PbI3 (calculations for CH3 NH3 PbI3 were reported in Ref. 48 ). The band gaps calculated within DFT+SOC, G0 W0 and SS-GW exhibit the same trend, as shown in the bar plot of Fig. 2 (a). Moreover, from Fig. 2 (a) we can see that by replacing of the CH3 NH3 cation by cations of the AH4 type, with A descending down the pnictogen column of the periodic table, we can tune the band gap by up to 0.5 eV. In Fig. 2 (b) we show that the band gap correlates well with the steric size of the cation (as it is defined in Ref 31 ), and decreases as the size Table 1: Calculated band gaps of NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 , SbH4 PbI3 , and CH3 NH3 PbI3 , and experimental optical gaps. The calculations for CH3 NH3 PbI3 are performed for the optimized and experimental unit cell. All values are in eV. NH4 PbI3

PH4 PbI3

AsH4 PbI3

SbH4 PbI3

CH3 NH3 PbI3 optimized experimental

DFT+SOC

0.66

0.64

0.45

0.41

0.58

0.48

G0 W0

1.45

1.38

1.12

1.03

1.32

1.20

SS-GW

1.93

1.82

1.56

1.43

1.79

1.71 1.62-1.64 32

experiment

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Figure 2: (a) Bar chart of the calculated band gaps of NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 within DFT+SOC (light blue), G0 W0 + SOC (black) and SS-GW (dark blue). (b) Band gap as a function of the steric size of the cation in the case of DFT+SOC (light blue), G0 W0 +SOC (black) and SS-GW (dark blue) calculations. The straight lines are linear fits through the data points.

of the cation increases. The largest deviation from the linear fit of the band gaps with respect to the steric sizes is of 0.15 eV, which is close to the band gap difference calculated for the experimental and optimized crystal structures of CH3 NH3 PbI3 (0.08 eV). Therefore, these small deviations could be attributed to structural distortions occuring during lattice relaxation. From Fig. 2(a) and (b) we can clearly see that the trend identified in Ref. 31 for the scalar relativistic DFT band gaps is robust. This result confirms that the band gap variation with respect to the size of the central cation in lead-iodide perovskites is purely due to the structural changes of the perovskite, and it is caused by the steric interaction between the central cation and the Pb-I network. Furthermore, in Table S3 of the Supporting Information we report the high frequency dielectric constants obtained from finite electric field calculations within scalar-relativistic DFT. For CH3 NH3 PbI3 we obtain a dielectric constant of 5.8, in good agreement with previous calculations. 46,47 For the hypothetical perovskites we obtain

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dielectric constants between 5.7 and 7, increasing as the size of the cation increases.

Quasiparticle band structures In Fig. 3 we show a comparison between the band structure of CH3 NH3 PbI3 obtained from DFT+SOC and that obtained from the Wannier interpolated SS-GW eigenvalues. The conduction bands calculated from GW and DFT have very similar dispersions. This suggests that the quasiparticle correction for the conduction band should be captured for the entire Brillouin zone by a scissor operator, with good accuracy. Indeed, this is also the premise of the SS-GW scheme, in which only the conduction states are shifted to higher energies by the scissor operator. In the case of the valence bands, in Fig. 3 we can clearly see that the GW valence band has a more pronounced dispersion than the DFT valence band. We can attempt to quantify this change in dispersion by calculating the difference in the energies of the top of the valence band states at Γ and T. We obtain 0.7 eV and 1.2 eV from DFT

Figure 3: Band structure of CH3 NH3 PbI3 obtained from DFT+SOC (gray) and SS-GW (blue) on the following path (in crystal coordinates): Γ (0, 0, 0) - X (0.5, 0, 0) - S (0.5, 0.5, 0) - Y (0, 0.5, 0) - Γ - Z (0, 0, 0.5) - U (0.5, 0, 0.5) - R (0.5, 0.5, 0.5) - T (0, 0.5, 0.5) - Z. The SS-GW band structure is obtained from the Wannier interpolation of the quasiparticle eigenvalues.

16


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Figure 4: Band structures of the hypothetical perovskites NH4 PbI3 (a), PH4 PbI3 (b), AsH4 PbI3 (c) and SbH4 PbI3 (d) calculated from DFT+SOC (gray) and SS-GW (blue). The SS-GW band structure is obtained from the Wannier interpolation of the quasiparticle eigenvalues.

and G0 W0 respectively. This observation is in line with the QSGW calculations reported in Ref. 47 The same observations can be made from the band structures of NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 calculated from DFT/LDA and SS-GW , shown in Fig. 4 (a-d).

Effective masses In Table 2 we report the isotropic effective masses calculated for NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 , SbH4 PbI3 , and CH3 NH3 PbI3 from the Wannier interpolation of the DFT, G0 W0 and SS-GW eigenvalues. Indeed, in all cases we note that the effective masses calculated 17



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Page 18 of 35

from DFT are lower than the ones calculated from G0 W0 and SS-GW , in agreement with the fact that the band gaps are underestimated. The effective masses calculated for the hypothetical perovskites appear to follow a similar trend to the band gaps, exhibiting smaller effective masses as the size of the cation increases. In the case of PH4 PbI3 , the electron effective mass marginally departs from this trend, having higher electron effective mass than NH4 PbI3 by 0.009 electron masses. In Table S4 of the Supporting Information we report the three diagonal elements of the electron and hole effective mass tensors as calculated from SS-GW for all five compounds. Notably, the electron effective masses are consistently more anisotropic than the hole effective masses. The reduced effective mass calculated for the orthorhombic structure of CH3 NH3 PbI3 is in good agreement with the experimental reduced effective mass, as obtained from magnetic measurements of the exciton binding energy at 2 K. 35 It is possible that the small difference Table 2: Isotropic hole (mh ), electron (me ) and reduced [µ = me mh /(me + mh )] effective masses of NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 , SbH4 PbI3 , and CH3 NH3 PbI3 calculated from the Wannier interpolation of the DFT, G0 W0 and SS-GW band structures. The experimental reduced effective mass reported in Ref. 35 is included for reference. All values are reported in units of electron mass.

mh

me

µ

calculation

NH4 PbI3

PH4 PbI3

AsH4 PbI3

SbH4 PbI3

CH3 NH3 PbI3

DFT+SOC

0.179

0.180

0.122

0.106

0.130

G0 W0

0.224

0.224

0.161

0.145

0.180

SS-GW

0.261

0.261

0.193

0.174

0.230

DFT+SOC

0.177

0.173

0.116

0.098

0.112

G0 W0

0.223

0.234

0.174

0.145

0.163

SS-GW

0.265

0.274

0.216

0.178

0.217

DFT+SOC

0.089

0.088

0.060

0.051

0.060

G0 W0

0.112

0.114

0.084

0.072

0.086

SS-GW

0.131

0.134

0.102

0.088

0.112 0.104 ± 0.003 35

experiment

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between our calculated value and the experimental reduced effective mass is due to the effect zero point renormalization on the valence and conduction band edges which is not included in our calculations. 101,102 Moreover, a direct comparison between the excitonic reduced effective mass obtained in Ref. 35 and the quasiparticle reduced effective mass shown in Table 2 is possible only within the Wannier exciton model. 103 This simplified model approximates the electron-hole pair as a hydrogenic atom, with the electron-hole interaction screened by a static dielectric constant. 103 An accurate assessment of the exciton binding energy and the reduced excitonic effective mass of CH3 NH3 PbI3 requires a thorough analysis of the electron-hole interaction within the Bethe-Salpeter equation, 76 and calls for a separate detailed analysis. Direct

comparison

with

previous

calculations

of

the

effective

masses

of

CH3 NH3 PbI3 46,61,72 is not straightforward given that the crystal structures considered in each case were obtained from structural optimizations with different starting points and different exchange correlation functionals. However, we note that our average isotropic electron effective masses are consistently smaller than the hole effective masses for DFT, G0 W0 and SS-GW, in agreement with similar trends reported in Refs. 46,61,72

Discussions In Ref. 48 we observed that the GW calculations performed for the optimized crystal structure of CH3 NH3 PbI3 overestimate the experimental optical band gaps by up to 0.15 eV, 32 and the band gap measured from photoemission spectroscopy by up to 80 meV. 34 However, in the case of the experimental crystal structures the calculated band gaps are only 80 meV and 10 meV larger than the experimental optical and photoemission band gaps respectively. This comparison indicates that there is a systematic error introduced in the calculation of the quasiparticle band gap due to structural optimization and the absence of electron-hole interactions, which we can tentatively quantify to be around 0.15 eV. It is therefore expected

19



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Page 20 of 35

that the quasiparticle band gaps calculated within SS-GW for the hypothetical perovskites are overestimated by similar amounts. In order to take this effect into account, in Table 3 we estimate the “optical gap” for each structure by substracting 0.15 eV from the SS-GW gap calculated for the relaxed structures. Using these estimated “optical” gaps we can obtain the open circuit voltage, short circuit current and a theoretical limit for the power conversion efficiency, using Eqs. (7), (8) and (9). In Table 3 we report these estimates for the hypothetical perovskites as well as for CH3 NH3 PbI3 . For the calculation of the open circuit voltage we use two values for the loss-inpotential, 0.7 eV and 0.5 eV respectively, and for the calculation of the short-circuit current we use the reference direct normal spectral irradiance reported on the NREL website. 89 The calculation of the power conversion efficiency takes into account a fill factor parameter of 80%. We have chosen this parameter in order to match the power conversion efficiency reported in Ref. 23 for CH3 NH3 PbI3 single crystals, with a loss-in-potential of 0.7 eV. Using the same setup we calculate the short circuit current, open circuit voltage, and theoretical power conversion efficiency for the hypothetical perovskites. From Table 3 we can see for a loss-in-potential of 0.7 eV the estimated power conversion efficiencies for PH4 PbI3 and AsH4 PbI3 are either similar or marginally higher than Table 3: Calculated open circuit voltages, short circuit currents, and power conversion efficiencies for the hypothetical perovskites and CH3 NH3 PbI3 . NH4 PbI3

PH4 PbI3

AsH4 PbI3

SbH4 PbI3

optical gap (eV)

1.78

1.67

1.42

1.28

1.64

1.63 23

VOC (V) (Vloss = 0.7 eV)

1.08

0.97

0.72

0.58

0.94

0.92 23

VOC (V) (Vloss = 0.5 eV)

1.28

1.17

0.92

0.78

1.14

JSC (mA/cm2 )

17.4

20.4

28.3

32.4

21.3

22.4 23

η(%) (Vloss = 0.7 eV)

17.0

17.8

18.4

16.9

18.1

18.0 23

η(%) (Vloss = 0.5 eV)

20.1

21.5

23.5

22.7

21.9

20


CH3 NH3 PbI3 calc. exp.

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CH3 NH3 PbI3 . The short circuit current improves by up to 52% upon replacing the CH3 NH3 cation by SbH4 , but the open circuit voltage can drop by up to 38% for a band gap as small as the one of SbH4 PbI3 , explaining the small differences in the power conversion efficiencies. For a smaller loss-in-potential of 0.5 eV, the efficiencies estimated for SbH4 PbI3 and AsH4 PbI3 increase by 4% and 7% respectively as compared to the efficiency estimated for the CH3 NH3 PbI3 in the same conditions. This observation suggests that by reducing the losses it is possible to capitalize on the improved short circuit current achieved using lower gap absorbers.

Conclusions In this paper we use the GW method and Wannier functions to calculate the quasiparticle band gaps, band structures and effective masses of CH3 NH3 PbI3 and four other hypothetical lead-iodide perovskites: NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 . We find that the quasiparticle correction for the hypothetical perovskites is increased by up to 0.5 eV when employing the self-consistent scissor correction, obtaining band gaps of 1.9, 1.8, 1.6 and 1.4 eV for NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 respectively. We show that the quasiparticle valence bands in all five cases have a more pronounced dispersion than the Kohn-Sham valence bands. In all five cases the DFT effective masses are underestimated with respect to the quasiparticle effective masses by up to 50%. The effective masses of the five lead-iodide perovskites analyzed follow the same trend as their band gaps and decrease as the size of the central cation increases. Based on the quasiparticle band gaps and effective masses calculated in this study, we find that AsH4 PbI3 and SbH4 PbI3 are promising candidates for light absorbing materials in photovoltaic applications due to their optimum band gaps for absorption in the visible (1.6 and 1.4 eV), as well as their low effective masses (0.17-0.22 electron masses). In particular, due to the optimum band gap of SbH4 PbI3 , the short circuit current of a single-crystal

21



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perovskite solar cell could be improved by up to 50%. The estimated theoretical efficiency limit for solar cells based on CH3 NH3 PbI3 could also be improved by increasing the open circuit voltage and replacing CH3 NH3 with AsH4 and SbH4 , gaining up to two points in efficiency due to stronger light absorption. We find that the short circuit current, the open circuit voltage and the maximum theoretical efficiency obtained in the case of PH4 PbI3 are very similar to CH3 NH3 PbI3 . This suggests that by replacing CH3 NH3 with PH4 , similar performance should be achieved under the same conditions. At the same time, the tetrahedral symmetry of the PH4 cation allows for the reduction of the hysteresis effects associated with the dipolar nature of CH3 NH3 . We hope that the results presented in this work will stimulate experimental synthesis and further investigation of the structural, electronic and optical properties of NH4 PbI3 , PH4 PbI3 , AsH4 PbI3 and SbH4 PbI3 .

Acknowledgement This work was supported by the European Research Council (EU FP7/ERC grant No. 239578), the UK Engineering and Physical Sciences Research Council (Grant No. EP/J009857/1), the Leverhulme Trust (Grant RL-2012- 001), and the Graphene Flagship (EU FP7 grant No. 604391). The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility 104 and the ARCHER UK National Supercomputing Service under the ‘AMSEC’ Leadership project in carrying out this work. Structural models were rendered using VESTA 105

Supporting Information The Supporting Information contains crystal structure data, additional effective mass data and the dielectric constants for the compounds studied in this work. This information is available free of charge via the Internet at http://pubs.acs.org.

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GW Band Structures and Carrier Effective Masses of CH3NH3PbI3

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