Structural and electronic properties of hybrid perovskites for high-efficiency thin-film photovoltaics from first-principles

The performance of perovskite solar cells recently exceeded 15% solar-to-electricity conversion efficiency for small-area devices. The fundamental properties of the active absorber layers, hybrid organic-inorganic perovskites formed from mixing metal and organic halides [e.g., (NH₄)PbI₃ and (CH₃NH₃)PbI₃], are largely unknown. The materials are semiconductors with direct band gaps at the boundary of the first Brillouin zone. The calculated dielectric constants and band gaps show an orientation dependence, with a low barrier for rotation of the organic cations. Due to the electric dipole of the methylammonium cation, a photoferroic effect may be accessible, which could enhance carrier collection.

Progress in the performance of hybrid pervoskite solar cells has rapidly advanced over the last five years.^1–6 They represent the convergence of inorganic thin-film and dye-sensitised solar cells.⁷ The conversion efficiencies have quickly surpassed both those of conventional dye-cells, as well as next-generation thin-film absorbers such as Cu₂ZnSnS₄.^8,9 Despite their high-performance for small area (∼0.2 cm²) cells, the underlying material properties are largely unknown, which could help with producing more robust large-area devices.

Perovskite refers to the crystal structure of the mineral CaTiO₃, which is adopted by a large family of ABX₃ materials, with two notable examples being SrTiO₃ and BaTiO₃ (Figure 1). They are well known for their phase complexity, with accessible cubic, tetragonal, orthorhombic, trigonal, and monoclinic polymorphs depending on the tilting and rotation of the BX₆ polyhedra in the lattice.¹⁰ Phase transitions are frequently observed under the influence of temperature, pressure, and/or applied electric field.

While oxide perovskites (ABO₃) are formed from divalent A^II (1b site – cuboctahedral) and tetravalent B^IV (1a site – octahedral) metals, halide perovskites (e.g., ABI₃) can be formed from monovalent A^I and divalent B^II metals. For example, the application of CsSnI₃ in solar cells has recently been demonstrated.¹¹ Hybrid organic-inorganic perovskites are produced by replacing one of the inorganic cations by an isovalent molecule, e.g., CsSnI₃ → (NH₄)SnI₃, where

$NH 4+$ (A) is the ammonium cation. Recently, the methylammonium (MA) cation (i.e., CH₃NH₃⁺) has been widely applied, resulting in the highest-performance perovskite-structured photovoltaic absorbers.^2,3 More generally, a large series of hybrid perovskites have been reported, which vary in their dimensionality and the orientation of the underlying perovskite lattice.^12–15 

In this letter, we assess the properties of the two archetypal hybrid perovskites (NH₄)PbI₃ and (CH₃NH₃)PbI₃ using density functional theory (DFT) for the ground-state properties and density functional perturbation theory (DFPT) for the dielectric and optical response functions. We provide insights into the key properties required for device models and screening procedures and that underpin their utility in photovoltaic cells.

Characterisation of the crystal structures of inorganic perovskites is difficult, and hybrid perovskites are even more challenging. Recent analysis of high-quality (MA)PbI₃ crystals identified cubic (⁠

$Pm3¯m$⁠), tetragonal (I4/mcm), and orthorhombic (Pnam) phases from X-ray diffraction, while transmission electron microscopy suggested a pseudo-cubic behaviour that is consistent with variability in the octahedral titling and/or rotations.¹⁶ For our investigation, we take a cubic basis, starting from the

$Pm\bar{3}m$

$Pm3¯m$ lattice, and investigate the potential energy landscape associated with the molecular orientation.

The crystal structures (internal forces and external pressure) were optimised at the level of DFT, with the exchange-correlation functional of Perdew, Burke, and Ernzerhof revised for solids (PBEsol).¹⁷ The Pb 5d orbitals were treated as valence and scalar-relativistic effects are included. Further calculations were made using a non-local hybrid exchange-correlation functional (HSE06).¹⁸ Calculations were performed using the VASP code,¹⁹ a 500 eV plane-wave cut-off, and reciprocal space sampling of 6 × 6 × 6 k-point density. Internal structural parameters were converged to within 5 meV/Å, and the phonon frequencies were checked at the zone-centre to ensure that no imaginary modes were present. The high-frequency (ε_∞) and static (ε₀) dielectric constants were computed using DFPT²⁰ based upon a tightly converged electronic wavefunction (within 10⁻⁹ eV) and a denser grid of 10 × 10 × 10 k-points.

For (A)PbI₃ there is no strong orientation dependence of the ammonium ion owing to its approximately spherical topology. A local minimum structure is found where the four hydrogens are directed towards interstitial positions. For (MA)PbI₃ the behaviour is more subtle owing to a molecular dipole (of strength 2D) associated with the methylammonium ion (i.e., [CH₃]^{δ +} − [NH₃]^{δ −}). We identified three local minima, where the dipole is aligned along of the ⟨100⟩, ⟨110⟩, and ⟨111⟩ directions, relative to the origin of the cubic lattice (Figure 1). The total energy difference is within 15 meV per atom, with the ⟨100⟩ orientation being most stable. The absence of a significant barrier to rotation (<40 meV) is consistent with labile movement under standard conditions. Indeed, ²H and ¹⁴N spectra have confirmed that MA cation rotation is a rapid process at room temperature.²¹ 

The predicted lattice constants for (MA)PbI₃ vary from 6.26 to 6.29 Å (Table I) depending on the molecular orientation. These compare well to the value of 6.26 Å obtained from powder diffraction measurements¹⁶ and 6.33 Å from computations.²² Distortions of the PbI₆ octahedra are observed for both materials, even within the cubic basis. The presence of the molecule lowers the internal lattice symmetry of the parent perovskite structure and allows octahedral tilting to take place. The inherent structural “softness” of Pb(II) is well documented, arising from the stereochemical activity of the 6s² lone pair electrons.²³ 

The electronic band gap is determined by the states at the valence band maximum (VBM) and conduction band minimum (CBM). For both materials, the VBM is formed of an anti-bonding Pb s/I p combination, while the CBM is formed of empty Pb p orbitals, consistent with the formal electronic configuration of 5d¹⁰6s²6p⁰. From the topology of the band-edge wave functions (Figure 2) isotropic hole and electron band conduction is expected. The strong dispersion of the bands in reciprocal space, confirmed from E(k) plots with

$mn*$ and

$m^*_p$

$mp*$ < 0.3 m_e, is consistent with the reports of effective bipolar electrical conductivity.²⁴ A dedicated study of the defect physics is required to fully assess the origin of intrinsic conductivity, but partial occupancy of the organic cation sites is likely, which would correspond to p-type doping.

The magnitude of the band gap determines the onset of optical absorption and is closely related to the maximum voltage achievable in a photovoltaic device. The gap increases from 1.20 eV for (A)PbI₃ to 1.38 eV for (MA)PbI₃ (Table I). In all cases, the band gap is strongly direct and determined at the boundary of the first Brillouin zone (⁠

$R:2πa[12,12,12]$⁠). The increase is related to the significant expansion of the cell due to the larger cation: there is no evidence of M or MA contributing electronically to the frontier orbitals. In agreement with our results, a previous investigation of Sn-based hybrids showed that while the organic cations maintain overall charge neutrality, implying electronic mixing with the framework, they do not contribute the upper valence or lower conduction band states responsible for conductivity.¹⁵ Indeed, the σ/σ* bonds of A and MA are found at least 5 eV below the highest occupied state and 2.5 eV above the lowest empty state, respectively.

Although the application of hybrid perovskites in solar cells has predominately focused on (MA)PbI₃, our results suggest that (A)PbI₃ would also be effective for sensitisation towards longer wavelengths, and an (A)_x(MA)_1−xPbI₃ alloy could be used to tune the absorption onset. It should be noted that the same trends calculated with PBEsol are observed using the hybrid HSE06 functional; however, the predicted band gaps are larger. Optical absorption measurements place the band gap of (MA)PbI₃ at 1.5 eV,¹⁶ while PBEsol and HSE06 predict values close to 1.4 eV and 2.0 eV, respectively. The calculated band gaps for materials formed by ns² cations such as Pb(II) are reasonably described without non-local electron exchange.^25,26 This originates here from a cancellation of errors with the neglect of spin-orbit splitting in the I 5p valence and Pb 6p conduction bands.²⁷ 

Due to the pseudo-cubic lattice symmetry, the dielectric response of the hybrid materials is anisotropic (Table I). The high-frequency optical constants (5.6–6.5) are close to those of other absorber materials, e.g.,

$ε∞CdTe=7.1$⁠;²⁸ however, the static dielectric response is much larger. The large dielectric constants of pervoskite materials are associated with the structural flexibility that can support ferro-, anti-, and para-electric ordering of dipoles associated with the organic and/or inorganic sublattices. (A)PbI₃ is predicted to have low-frequency constants from 18.1 to 18.6, while (MA)PbI₃ exhibits a stronger screening of 18.0–37.3, which can be understood through the additional response from the molecular dipole in (MA)PbI₃. An isotropic average of the tensor across each orientation results in an effective dielectric constant of 25.7 for (MA)PbI₃, which is significantly larger than most tetrahedral semiconductors (e.g.,

$\epsilon _{0}^{CdTe} = 10.4$

$ε0CdTe=10.4$⁠)²⁸ and will affect electron transport in a photovoltaic device, with stronger screening of any macroscopic electric fields across the absorber layer.

An order-disorder transition associated with the molecular dipole of MA, in a perovskite analogue, has recently been reported to give rise to reversible switching in the dielectric response.²⁹ While no evidence has been reported of this effect in hybrid halide perovskites, it is expected that they will also display some degree of amphidynamic behaviour arising from the collective motion of the constituent polar molecules. Tuning of the molecular dipoles represents a potential route for further improving the performance of this class of materials.

In summary, we have reported key properties of the hybrid perovskite layers used in thin-film solar cells. These materials combine low-energy direct band gaps with static dielectric constants larger than 18. Due to the low barrier for rotation of the organic cations, no long-range ordering of the molecular dipoles associated with CH₃NH₃ is expected; however, this could change in the presence of an external electric field. A transition from para- to ferroelectric order would enhance electron-hole separation. If the electric field was light induced, e.g., in a photovoltaic device, this could give rise to a novel photoferroic effect.

We thank L. M. Peter, K. T. Butler, and P. J. Cameron for useful discussions, and acknowledge membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC Grant No. EP/F067496. F.B. is funded through the EU DESTINY Network (Grant No. 316494). A.B.W. has received funding from EPSRC Grant No. EP/J017361/1, and A.W. acknowledges support from the Royal Society and the ERC (Grant No. 277757).