Because it is good at balancing accuracy and processing cost, density functional theory (DFT) is an important tool in materials science. It has been used for more than what it was made for, like figuring out the enthalpies of chemical reactions, getting forces from the Hellmann-Feynman theorem, and doing models of molecular dynamics from scratch. DFT can also be used to get phonon spectra, electron-phonon couplings, and dielectric constants. This gives atomic-level details that help us understand what we see and guess what materials will be like in the future.
A lot of research has been done on metal-halide perovskites using this method to help us understand different parts of their physics and chemistry. This chapter is mostly about what we can learn about methylammonium lead iodide (MAPbI3) by using DFT methods on it. MAPbI3 is the most well-known and widely studied metal halide perovskite. It is divided into several sections, such as learning about chemical bonds in MAPbI3, lattice dynamics, spin-orbit coupling (SOC), charge carrier generation, transport, and recombination, and the effects of intrinsic point defects on charge carrier lifetimes and device performance.
This chapter gives an overview of recent work that has been done to study halide perovskite solar cell materials using DFT calculations. It stresses how important it is to understand the role of organic MA molecules in improving the usual picture of ionic and covalent bonding in inorganic semiconductors.
Structure and bonding
The general chemical formula for stoichiometric perovskite materials is ABX3. They have structures that can be thought of as a grid of octahedra that share corners and have X anions. The “aristotype” or “ideal” perovskite structure is cube-shaped, but there are also lower symmetry structures, or “hettotypes,” that are similar. These can be described mainly in terms of unit cells that contain multiple octahedra that are tilted with respect to each other, either at the top or the bottom. MAPbI3 has three different hettotypes: an orthorhombic structure at low temperatures, a tetragonal structure at middle temperatures (like room temperature), and a cubic structure at high temperatures.
Ionic materials can be thought of as hard, charged spheres that are held together by electrostatic forces. This model should not work as well for mixed halide perovskites like MAPbI3, though.
Partially covalent electronic bonds can be seen in the PbI3 sublattice. This is shown by partial charge analysis, which gives values about half of the formal values for Pb and I. The covalency is also clear from the valence band, which is made up of antibonding states that come from the Pb-6s and I-5p orbitals. The MA and PbI3 units have charges that are very close to integers. This means that the bond between these parts is mostly ionic, though the MA molecule’s aspherical shape makes it hard to give it an ionic radius.
In the orthorhombic phase, the four MA molecules in the unit cell have their C-N axes lined up roughly along the [101] and ½101 lines. They are arranged head-to-tail in an antiferroelectric way. It has been calculated, though, that there are a lot of local minima within a few tens of meV. There is a difference between orthorhombic and cubic structures with the same MA ordering. The orthorhombic phase has stronger hydrogen bonds because its octahedra are tilted.
It’s not clear how the MA molecules are arranged in the higher temperature tetragonal or cubic phases. Researchers have tried to figure out the potential energy surface connected to MA orientations in these phases and the interactions that control its shape. Some people have different ideas about how many local minima there are and what part hydrogen bonds and VdW interactions play.
In the pseudocubic phase, it has been found that the molecule is orientated along the h100i, h110i, and h111i lines, which match to local minima. It was also noticed that moving the MA molecule by 0.3 to 0.6 ̺ (defined by the molecule’s geometric centre of mass being at the ideal A site) helped keep the structure stable by making the hydrogen bonds between the N end of the molecule and the I ions stronger.
The main goal of the work is to use first principles equations to describe perovskite solar cell materials. The pseudocubic phase is thought to be a ferroelectric shape, and supercells are used to study how MA molecules interact with each other, which causes their angles to become (statically) disordered. A 2 3 2 3 2 supercell has the least amount of energy when all of its MA molecules are lined up face-to-face. Due to weaker limits on the structure, the energy gained by distorting the artificial lattice is a lot bigger than what was predicted from calculations using a single unit cell.
The most energy-efficient way for MA molecules to be arranged in the tetragonal phase is so that their projection onto the ab plane is along the h110i direction. The tetragonal phase is less symmetrical than the cubic phase. This means that there are two different chemical settings for the h001i orientation, which leads to two different hydrogen bonding modes (α and ²).
Several studies show that adding a dispersion term to a GGA functional makes the orthorhombic phase of MAPbI3’s lattice parameters better. This is also true for the cubic and tetragonal phases. On the other hand, a good GGA function (SCAN) in the orthorhombic phase can give a similar correct structure picture.
The energies of snapshots from MD calculations were compared to those of high-level RPA calculations on the same structures. This showed that VdW functionals do not improve the cubic phase in a consistent way. It doesn’t change the vibrating range much when VdW functionals are used, and they give the wrong picture of how the MA moves. So, the question of what the meaning of VdW interactions in MAPbI3 is still open.
Phonons, anharmonicity and MA dynamics
In different works, the harmonic phonon spectra of all three stages of perovskite solar cell materials have been worked out. The orthorhombic phase has frequencies that go up to 3000 cm21, and the modes between 300 and 1500 cm21 are caused by the movements inside MA cations. The low-frequency area has normal modes with frequencies below 60 cm21 that are mostly made up of PbI3 contributions. The modes between 140 and 180 cm21 are made up of MA cation movements and waves of the PbI3 network.
The tetragonal MAPbI3 spectrum at room temperature has also been worked out in the range of 50 to 450 cm21. It was possible to find three sharp peaks at 62, 94, and 119 cm21 and one broad feature around 250 cm21. These were linked to the Pb-I-Pb bonds bending, the Pb-I bonds stretching, and the MA cations moving around. For the cubic phase, the orders are the same.
It has also been used to look into how vibrational entropy affects phase stability through phonon modelling. It was discovered that the cubic and tetragonal phases had more low frequency intermolecular modes than the usual oxide perovskite, SrTiO3. The harmonic approximation and vibrational entropy have been used to show that MAPbI3 is intrinsically thermodynamically metastable in the tetragonal phase but only slightly stable in the orthorhombic phase.
A lot of research has been done on anharmonic effects, which are different from the rotations of MA molecules and the movements of the structure. As an example, soft (imaginary frequency) modes are seen at the R and M border points of the Brillouin zone. These modes match to the group tilted modes of the PbI3 octahedra.
In perovskite solar cell materials, phonon-phonon interactions are stronger than in solid semiconductors like GaAs and CdTe. This means that phonon lifetimes are shorter and heat movement is slowed down. For molecular dynamics (MD) simulations to fully explain anharmonic effects, they are needed. An even number of supercell expansions is needed to correctly explain octahedral tilted modes and MA relationships.
The way MA molecules spin shows that the methyl group can spin around the axis of the C-N bond, but the amine group stays still because it hydrogen bonds to the metal cages. In stages with higher temperatures, boundaries for the rotation of the MA as a whole of only a few tenths of an eV have been found. This shows that the molecular orientations of the MA are both steady and dynamically disordered. This is supported by MD models, which show that MA molecules can easily spin in the cubic phase and are a little longer in the tetragonal phase.
MD simulations have also been used to look at the dynamical relationships between MA molecules. This was partly inspired by the idea that the large dipole moment of the MA molecule might lead to the formation of ferroelectric domains. It’s easier for the MA molecules to move around in the tetragonal phase because they don’t have to deal with as many problems when they contact with the artificial frames. As the temperature goes up, the associations get weaker because the MA molecules aren’t as affected by their interactions with the artificial frames. The conclusion is that the PbI3 framework, not the electrostatic dipole-dipole interactions, control the interactions between MA molecules. This is in line with what was seen earlier and what was found in the static analysis in Ref. [37].
Electronic band structure and charge carrier dynamics
Local and semi-local DFT estimates often don’t take into account the band gaps of semiconducting and insulating materials. But the first studies on MAPbI3 that used LDA or GGA functionals found band gaps that are very close to the values that were measured in experiments. The valence band maximum comes from I-5p and Pb-6s orbitals, while the conduction bands are mostly based on Pb-6p orbitals. The estimated band gap is greatly changed by spin-orbit coupling (SOC), which lowers it by about 1 eV and fixes the problem caused by (semi-)local functionals underestimating the band gap.
Most of the band gap decrease is due to the strong breaking of the conduction bands drawn from lead. To get a band gap that is closer to what was found in the lab, you need to use hybrid functionals or GW corrections along with SOC. Standard DFT, which doesn’t take SOC into account, says that MAPbI3 has a straight band gap at the R point in the Brillouin zone in the cubic phase and at the Γ point in the tetragonal and orthorhombic phases. However, SOC along with MA molecules breaking the inversion symmetry splits the conduction and valence bands valley into two, two-fold, degenerate valleys. This creates gaps that are slightly spaced apart in k space. It is thought that this so-called Rashba-Dresselhaus breaking lowers the rate of radiation recycling in the medium.
SOC also changes the spread around the band ends, which changes how charges move. SOC lowers the effective masses, and the band gaps are not at all shaped like a normal curve. But when the sun shines, the carrier densities are too low for thermalised carriers to explore the nonparabolic part of the bands.
The band structure tells us a lot about how electricity affects an object, but it does so by assuming that electrons are separate from matter. The Bethe-Salpeter equation has been used to figure out what part excitonic effects play in the production of charge carriers in light. The ways that MA molecules are arranged have a big effect on their electronic structure and qualities that are connected to it, like the dielectric function and absorption coefficients.
Molecular dynamics simulations on the cubic phase show that the valence band maximum (VBM) moves around by about 0.34 eV and up to 0.16 eV in the tetragonal phase. Changes in the positions of the conduction band minimum (CBM) and VBM are about equally responsible for these changes. The effect on the exciton binding energy, on the other hand, is not nearly as strong.
Because of how its electrons and phonons are connected, MAPbI3 is thought to be very soft. It has been shown that orthorhombic MAPbI3 has strong electron-phonon coupling with three longitudinal optical (LO) phonon modes, which are thought to cause very fast relaxation of the electrons within the band. New advances in first principles studies have been used to figure out how MHPs scatter light and how far they can move. The main type of scattering is phonon scattering from polar optical modes, which involves stretching the Pb-I bond in the orthorhombic phase.
Making big polarons is another thing that could happen when charge carriers combine with polar optical phonons. The most well-known result is that a charge carrier’s effective mass goes up because of the drag it experiences when it interacts with the lattice. Some people have said, though, that the polaron’s bigger effective mass might make spreading less likely. It has also been suggested that polaron formation could explain why MAPbI3 has low carrier cooling when exposed to high fluence irradiation. This is because the finite sized polaronic states might overlap when there are more charge carriers.
Researchers have used molecular dynamics to find that big polaron creation also stops radiation recombination because the electron and hole wave functions seem to be spread out in space. Some writers think that charge carriers self-trap to make small polarons. However, this would mean that the charge carriers’ movement would increase with temperature because they move through a process called “hopping,” which has not been seen in real life.
Intrinsic point defects
Most of the time, low temperature solution-based methods are used to make perovskite-based materials for solar cells. This is likely to cause a lot of point flaws. These flaws create locally centred defect states inside the band gap. These states trap charges and act as nonradiative recombination centres, which shortens carrier lives and lowers the efficiency of the device. Even so, halide perovskites have very long carrier lives and large mean free paths, which is even more impressive when you consider how slowly the carriers move.
Native point flaws in MAPbI3 can be gaps, antisites, or interstitials. DFT-based models have been used to study all of these in great detail. Many studies have used computers to look into how these kinds of flaws form, how stable they are, and what their electronic qualities are. But because different levels of math are used, the flaw traits are very different from one report to the next.
The first computer studies of flaws in MAPbI3 mostly used GGA exchange correlation functionals, which incorrectly matched the band gap seen in experiments. Charge carriers get stuck in the cubic phase of MAPbI3 when it has defects with low formation energies, like interstitial MA and I (MAi, Ii), lead and iodide vacancies (VPb, VI), and others. It takes a lot of energy to make deep trap flaws, mostly antisites (iodine-on-MA (IMA) and iodine-on-lead (IPb)) and Pb interstitials (Pbi).
The authors said that the so-called “defect tolerance” of mixed perovskites is a big reason why they have a long carrier diffusion length. A later study came to the same conclusions, but it pointed out that defect shapes can change a lot based on the charge state, which changes the type of defect. More research looked into how defect formation energies are affected by chemical potentials. They found that the iodide void (VI) is also a stable defect in a lot of different growth conditions, along with MAi and VPb. Ii and Pbi act as deep traps.
More recently, it was shown that models with GGA functionals can only give valid structures and energies inside halide perovskites.
The chemical surroundings has a big effect on the flaw chemistry of perovskite solar cell materials. Spin-orbit coupling and many-body interactions of electrons are needed to get around these problems. It has been found that high-level computer methods are the most accurate way to describe the electronic features of MAPbI3. Most experiments use the vapour pressure of I2 as the controlling factor, and computer studies have found the flaw formation energy in various settings, including those with high, medium, and low I.
It looks like I with different charge states are the main cause of deep transition levels in all growing situations. When I is present in large amounts, the positively charged Ii ðI1i Ú is stable, and its change to a neutral charge state happens deep inside the band gap. There are other growing conditions that make Ii with a negative charge (I2i) steady. At these conditions, the (0/2) transition level is close to the VBM and can trap holes.
Lead gaps are the other stable flaws that happen during all growth processes. It is deep in the band gap, but the (0/2 2) transition state doesn’t catch many electrons because the cross section for this process isn’t very high. Other stable flaws with low formation energy, like MA interstitials, have their transition level inside the bands, so they can’t catch charges.
In mixed perovskites, iodine interstitials are the best places for charge to get trapped through recombination. Point flaws can also move around while the device is working, which is another way they can affect it. A lot of research has also been done on how they affect the stability and power conversion efficiency of halide perovskites. The pushed elastic band method was used to figure out the energetic limit for flaws in MAPbI3.
Most of the research agrees that the lowest barriers to movement are for iodide, then MA, and finally Pb. However, the activation energies used in each paper are different.
Conclusion
In this chapter, we take a look at what we can learn about the classic hybrid halide perovskite, MAPbI3, from first principles formulas. Different kinds of bonding can change the structure and phase stability of a material. It also talks about lattice dynamics, anharmonic effects, MA rotations, spin-orbit coupling, scattering processes that affect charge carrier transport, and the electronic and transport features of defects.
Most of the research on structure and bonds has been done on how the MA cation and the artificial framework interact with each other. The bending of the octahedra has a lot to do with where and how the MA molecule is positioned within the empty spaces between them. This is mostly because the NH3 group and iodide ions form hydrogen bonds. Some people say that Van der Waals contacts are very important, but this is still up for debate.
Phonon spectra show that the organic and inorganic parts are connected at low frequencies, which is important for understanding how the grid moves. The cubic structure has imaginary modes at the zone borders that show it is at an energy saddle point. The structure that was observed in the experiment is most likely a dynamical average of lower symmetry structures. It has been shown that the relationships between MA orientations are not caused by direct dipole-dipole interactions, but by interactions between molecules and the artificial sub-lattice.
It has been shown that spin-orbit coupling has a big impact on the structure of the bands, changing things like the band gap size, the spread of particles around the band edges, and Rashba-Dresselhaus splits. However, what the formation of polarons means is still unknown.
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