Introduction: What I Talk About When I Talk About
First Principles Calculations of Halide Perovskites
Metal-halide perovskites are complicated substances that have a lot of different molecular and electrical features. Quantum physics controls these qualities, which means that to fully understand them, you need to solve the Schrödinger equation for electrons and atoms. But it’s hard to find and store these kinds of answers because the interactive many-particle problem gets exponentially more complicated. This chapter gives a quick overview of density functional theory (DFT) and Green’s function-based many body perturbation theory. These are two theories that are commonly used in physics, chemistry, and material science to solve Schrödinger’s equation for N-particle systems.
Conventional halide perovskites are crystals that have metal-centered corner-sharing octahedra and the formula ABX3 or A2BB′X6. They can make hundreds of different molecules by changing the chemicals at the A, B, B′, and X sites. Adding big A site cations creates more structural, chemical, and qualitative diversity, giving rise to a range of structural shapes with broken octahedral connections and lower electronic dimensionality. Perovskites that are fully inorganic and those that are organic and inorganic both have complex structural phase diagrams that change with temperature and pressure.
Many perovskites have soft lattices, molecular cations can have complex rotating dynamics, and halogen anions can easily move through the crystal lattice. Their structure dynamics are controlled by anharmonic effects. It is important to take relativistic effects into account when figuring out the electronic structure of Pb-based and other halide perovskites because they have heavy elements that are greatly affected by spin-orbit interactions.
A lot of halide perovskites have pretty small band gaps that are way too small for usual DFT models to capture. To accurately guess the structure and electronic features of halide perovskites, you need to use advanced first-principles methods and create big models of the structures. It is very hard to meet these requirements using computers, even on very big, fast supercomputers. Most computer studies of halide perovskites have to find a balance between how accurate they are and how much they cost to run.
Structural Properties
This part talks about DFT, a first-principles method that is often used to figure out the structure features of halide perovskites. It talks about the numerical and physical models that are used in real-world DFT calculations, the main parts of zero-temperature (static) DFT calculations, and the limits of these setups. This part then talks about finite-temperature (dynamical) structural features, like phase changes and the dynamics of ion movement, in part 8.2.4.1.
A Short Introduction to Density Functional Theory
In 1964, Hohenberg and Kohn proved that the ground-state density n(r) holds all the information needed to figure out the ground-state features of a system of N electrons interacting with the potential of M nuclei. This was the first step towards developing DFT. The electronic Hamiltonian for this system is Ĥ = T̂ + V̂ ext + Ŵ. Here, T̂ is the kinetic energy operator, V̂ ext is the interaction between electrons and the nuclei’s external potential, and Ŵ is the interaction between electrons.Go ahead.
There are two parts to the Hohenberg–Kohn theorem. The first part says that for any given electron–electron interaction Ŵ, there is a one-to-one relationship between the external potential vext, the ground-state wavefunction |Ψ0▩, and the ground-state density n(r).This means that |Ψ0▩ is a unique function of the density, and that all ground-state observables and ground-state energies are also density functions.Hey.
By minimising the total energy functional, one can use the variational principle to find the exact ground-state density and energy that relate to vext. The equation for E[n] is E[n] + ∫ d3 rvext(r)n(r). F[n] = ▨Ψ[n]|T̂ + Ŵ |Ψ[n]▩ is a universal function that doesn’t depend on vext(r).Hey.
The many-electron problem is, however, mostly hard to solve because it’s hard to figure out the universal functional F[n]. In 1965, Kohn and Sham came up with the best way to figure out F[n]. They reformulated the energy functional of a system of N interacting electrons as E[n] = Ts[n] + EH[n] + Eext[n] + Exc[n]. Here, Ts is the kinetic energy of a system of N electrons that don’t interact with each other, EH is the classical electrostatic Hartree energy, and Exc is the exchange-correlation (xc) functional.
DFT Calculations in Practice
Approximations
The Kohn-Sham equations are used to figure out the molecular features of perovskites. This can be done by using both numbers and actual close-ups. The choice of pseudopotential, reduction of the basis set, and finite sampling of the first Brillouin zone are some of the numerical assumptions used to apply Kohn-Sham DFT in electronic structure coding. Approximations that are based on physical facts, like the choice of Exc, are also used.
To use numbers to figure out solid-state materials, the first thing that needs to be done is to use the crystal structure’s translational symmetry and solve the Kohn-Sham equations with periodic boundary conditions. You can make the Kohn-Sham orbitals bigger by using a base of plane waves. This turns them into a matrix eigenvalue problem that can be solved with standard diagonalisation schemes for matrices. An important way to get close to the numbers is to use a finite plane-wave basis set, a finite k-point mesh, and pseudopotentials.
The selection of k-vectors in the first Brillouin zone is another number choice. There are different ways to sample the irreducible wedge, but in real-world math, the density of the k-mesh also needs to be carefully converged. The third decision is how to deal with vext, which is the electron-nuclei potential in the Kohn-Sham equations. This can be changed to a “pseudopotential” or “effective core potential.” The complicated nodal structure of the orbitals close to the core makes the computer solution of the Kohn-Sham equations easier. This is made possible by pseudopotentials.
Pseudopotentials are used because it has been observed that only the valence electrons are important for many structure and optical features. They are also used as a practical way to account for relativistic effects. When mixing different approximations, you need to be careful because each pseudopotential is made with a different xc approximation of DFT.
With these numbers, the Kohn-Sham equations can be solved to any level of accuracy. If you know the ground-state number n(r), you can figure out the total energies and forces acting on nuclei. This lets you do things like shape optimisation or molecular dynamics models.
The xc potential, vxc[n] = 𝟠Exc[n]∖𝜠n(r), is very important in many DFT uses. Based on the idea that the xc energy per electron, exc, is about the same as ehom xc in a system where the electron density changes slowly, the local density approximation (LDA) is used. They know what the exchange payment is and give it to you.
Generalised gradient estimates are better than the LDA at calculating energies and forces because they include information about how fast the electron density changes in space in the form EGGA. You can make the function f(n(r), ∇n(r)) by meeting exact limits or fitting parameters to big sets of thermochemical data.
People often use semi-local functionals to figure out ground-state features and for large-scale tasks like training machine-learning models and collecting a lot of data at once. However, they have well-known flaws, such as not being able to predict solids’ basic band gaps, giving a wrong picture of where charges are located and how they are spread across subsystems, and severely misrepresenting polarizabilities, hyperpolarizabilities, and charge transfer.
Adding accurate (Fock-like) exchange (EXX) is one way to get better results than semi-local functions. It is possible to make global mixed functions by combining a set fraction 𝛀 of EXX with semilocal exchange and correlation. It is possible for EXX and semilocal exchange to control the total xc energy at different electron-electron distances r, which expands the idea of global hybrid functionals.
Meta-generalized gradient approximations (GGAs) are another well-known way to get around the problems with semi-local functionals. When it comes to kinetic energy density and sometimes terms ∇2n(r), meta-GGAs rely on them. In recent years, meta-GGAs have been created with the goal of meeting many of the exact requirements of the exact xc functional. This lets us predict total energies more accurately and improve band gaps while keeping the processing cost about the same as semilocal xc approximations.
It’s not as easy to figure out the xc potential for xc functionals that depend on the orbitals directly as it is for the simple functional derivative of Exc with respect to the electron density. Many real-world examples of hybrid and other orbital-dependent functionals are based on the so-called “generalised Kohn-Sham framework.” This framework lets the potential in single-particle equations be nonlocal.
Calculations of Structural Properties
The Kohn-Sham equations are used to figure out overall energies and the shapes of electrons, but there needs to be a way to figure out the forces acting on atomic nuclei as well. The Born-Oppenheimer potential energy surface, also known as the clamped-ion energy, is equal to the Kohn-Sham ground-state energy plus the energy of the electric interaction between the nuclei. Each centre must no longer be affected by any forces in order for the system to be in balance.
To find the visual frequencies, you need to find the eigenvalues of the Hessian of the total energy, where MI is the mass of nucleus I. The Hellmann-Feynman theorem can be used to find the first and second derivatives of the total energy with respect to nuclear coordinates. This theorem connects the derivative of the energy with respect to a parameter 𝼆 to the expectation value of the derivative of the Hamiltonian with respect to 𝼆.
The forces FI that act on the nuclei can be written as FI = − 𝜦E(R) 𝼦RI = −∫ d3 r n(r) 𝜦vext(r) 𝜦RI − 𝜦EN (R) 𝼦RI (8.17). It’s easy to figure out the Hessian: 𝜦2E(R) RIRJ = ∫ n(r) 𝜦RJ 𝜦vext(r) 𝜦RI − 𝜦EN (R) 𝜦RI (8.18). You can use density functional perturbation theory (DFPT) to figure out the first term of Eq. (8.18), which shows the ground-state density’s linear reaction to a change in the nuclear shape. This creates a set of self-consistent equations for the changed system, which can be used to figure out phonon frequencies, phonon dispersion relations, and other features that are related.
Zero-Temperature Calculations for Halide Perovskites
It is very important to use the right structural models when comparing estimated structural features to actual results. This is especially true for halide perovskites, where the rotational dynamics of the molecular cation and the twists and tilts of the metal-halide sublattice interact in complex ways. Because zero-temperature DFT models don’t take temperature into account, they can make it hard to figure out how to understand temperature-dependent structure and optical features. Large anharmonic displacements can also have a big effect on some features of halide perovskites, but these are not taken into account in DFPT or frozen phonon models. So, the low-temperature phase should be used to check the structure features that were calculated at zero temperature against the results of the experiment.
Take the low-temperature structure of CH3NH3PbI3 (MAPbI3) with Pnma space group symmetry as an example. We know that methylammonium (MA) molecules have a preferred orientation at low temperatures because we have done X-ray diffraction and neutron scattering studies on them. Another way to find out which xc function best describes the structure features of this model is to use a DFT-based shape optimisation.
It turns out that the LDA tends to underestimate the lattice parameters by about 2%, while the PBE tends to overestimate them by about the same amount. The comparison also shows that the results depend on small details of how the computations were set up, like the limits on the symmetry and convergence factors. It has been said that dispersion interactions are very important for accurately describing the structural features of halide perovskites. The dispersion-corrected functional optB86b-vdW results are very close to what the experiment showed for orthorhombic MAPbI3.
It is possible to improve the shape at zero temperature for the high-temperature phases of halide perovskites as well, like the tetragonal I4/mcm phase of MAPbI3 and the cubic Pm3m phase of the same material. But for these high-temperature phases, picking the right structural model is very important and not easy because halide perovskites have very disharmonic structural dynamics.
The research looks into the atomic theory of halide perovskites, in particular MAPbI3, which has been looked at for its structure. It is hard to see in X-ray diffraction data that the cubic phase of perovskites has non-thermal static distortions that are local. It might be better to use polymorphic networks instead of a single cubic unit cell to describe the electronic structure and excited states of these on-average cubic phases. These networks have a spread of low-symmetry octahedral tilts, rotations, and ionic displacements.
Perovskites’ cubic phase can show static distortions that aren’t caused by heat. These distortions are local, so they’re hard to see with X-ray diffraction. It might be better to use polymorphic networks instead of a single cubic unit cell to describe the electronic structure and excited states of these on-average cubic phases. These networks have a spread of low-symmetry octahedral tilts, rotations, and ionic displacements.
This study shows how hard it can be to pick a good structure model for MAPbI3 in its cubic form. On average, this phase has Pm3m symmetry, which is a type of space group that can be shown by a single formula unit cell. If periodic border conditions are put on such a unit cell, however, it doesn’t matter which way the organic cation is orientated; an intentionally ordered structure is always made. Optimising the structure’s shape causes the Pb ion to move off-center and the P4mm to be symmetrical overall, which is not what the experiment showed.
Using a bigger supercell and spreading out the MA molecular orientations randomly can help you make a more accurate structural model. There are also octahedral tilts that happen when this structure is relaxed, but the normal structure is more like cubic symmetry than a single unit cell.
The results show that as unit cell sizes get bigger, the agreement with the experiment gets better, and the gap between the three lattice factors gets smaller. This is because certain MA orientations become less important as the unit cell size increases. The average structure of an infinitely large supercell would have Pm3m symmetry. However, local structural distortions and certain MA directions can cause big differences from the idealised cubic symmetry.
It is important to note that the choice of structural model has a big effect on both the finding of structural properties and the estimates of optical properties. People have come up with more realistic ways to make static models of the high-temperature structure of MAPbI3 and other perovskites.
Structural Dynamics
Molecular Dynamics: From Classical Force Fields to DFT Accuracy
Molecular dynamics (MD) simulations can be used to model how materials change over time. These simulations treat atoms as simple things moving according to Newton’s equations. The Born-Oppenheimer model is used, which says that because nuclei have a much bigger mass than electrons, their motions can be separated. The MD models begin with a set of starting places and speeds for each atom. These can be picked so that the mean kinetic energy matches a certain temperature T. Next, the forces FI on each atom are found. These forces are then used to combine Newton’s equation of motion.
The Hellmann–Feynman formula can be used with DFT to figure out the energies and forces between atoms. However, the modelling can only be as accurate as the Born–Oppenheimer approximation and the computational and physical approximations talked about in Section 8.2.2.1. In real life, DFT doesn’t scale well with the number of atoms because it takes too long to simulate systems with more than a couple dozen atoms for many qualities that are important.
On the other end of the accuracy scale are models of molecular dynamics that use classical force fields. For force fields to work, a semi-empirical parametrisation of a functional expression for the potential energy surface is needed. This expression usually has bond lengths, bond angles, and other structure factors that are fit to experimental or computational data sets. They let you do molecular dynamics simulations that get faster as the number of atoms goes up. You can use them on systems with millions of atoms, and the simulations can last a long time.
There is a way out of this catch: machine-learning methods that use DFT (or other first principles) data to create classical force fields. Recent years have seen a huge increase in the use of machine learning in material science, physics, quantum chemistry, and other related fields. It is now being used for everything from finding new materials and predicting crystal structures to creating new DFT models and completely replacing the first principles method.
Perovskites and the Breakdown of the Harmonic Approximation
The harmonic approximation says that the crystal lattice’s potential energy is equal to ΔRI(t), where ΔRI(t) are the nuclei’s changes in position from their initial equilibrium position R0 over time. These changes are quantised because the crystal lattice is symmetrical in terms of translation. The quantised movements of the lattice are what we call phonons. In the harmonic approximation, phonons are vibrational modes that can move on their own. They are also called normal modes of the lattice. That being said, this estimate doesn’t work for all bromide perovskites.
In both organic-inorganic and fully inorganic perovskites, big anharmonic effects have been seen in experiments and computer models. You can find out about anharmonic effects in first principles simulations by taking the vibrational density of states from molecular dynamics calculations. You can find the total vibrating density of states by adding up the contributions of each atom to the Fourier transform of fI(t), where m is the mass of atom I.
It is possible to get a model in reciprocal space by Fouriertransforming the mass-weighted velocity field into it. The k-resolved vibrational density of states for a perfectly harmonic solid should be the same as a phonon band structure. This means that the dynamics of the lattice can be broken down into a group of separate harmonic oscillators. But it’s still hard to find a full first-principles explanation of these effects in the anharmonic limit.
A Primer on Ion Migration
Halide perovskites are mixed electronic and ionic conductors that are very important in the breaking down, phase separation, and growth of materials. It is necessary for material breakdown, phase separation, and growing processes for ions to move, especially chlorine ions. Molecular dynamics simulations can directly model how ions move through the crystal structure, but they need to be run for a long time on big computer cells.
The activation energy of important ion movement processes can also be calculated as a way to get atomistic information about how ions move. The activation energy Ea is made up of the flaw formation energy Ef and the energy barrier forion migration. The energy barrier forion migration is the energy difference between the starting and final states that have the largest energy shift. There are different ways to figure out the structure and energy of transition states, which are shaped like saddle points in the potential energy environment.
It has been used a lot to figure out the activation energies of ion movement in halide perovskites using the nudged elastic band (NEB) method. In the NEB method, pictures with N-1 dimensions and atomic coordinates are made between the starting and ending states. The spring constant k tells us how strong the forces between these images are. The forces are given by Fsi = k(|Ri+1 − Ri| − |Ri − Ri−1|)𝟏̂i. It is the sum of the spring force along the local tangent and the true force that is not parallel to the local tangent that acts on each picture. Minimising Fi should lead the rubber band to the way with the least amount of energy.
The energy needed to form a flaw X in a charge state q is given by Ef(Xq) = E(Xq) – E(pristine) – ∑ i ni 𝼇i + qEF + Ecorr. The total energy of the system with defect is E(Xq), and the total energy of the system without defect is E(pristine). The number of atoms of type i that have been added to or taken out of the system to create the defect is ni. The chemical potentials of these species are 𝼇i, and the Fermi energy is EF. There are also correction terms called Ecorr that take into account the limited size of the supercell and the limited number of k-point samples.
Choice of xc functional
It is very important to choose the right functional for figuring out the energy barriers for flaw formation in halide perovskites, since the total energy changes between similar supercells make them less responsive. Reliable total energies for xc functionals are enough and don’t vary much from estimates that are harder to compute. But how the flaw transition values are calculated depends on how good the xc estimate is that is picked.
It’s very important to pick the right structural model for structures that work at high temperatures, like halide perovskites with molecular A-site cations. The complex relationship between the organic and inorganic sublattices and the A-site cation’s rotational and vibrational degrees of freedom can cause distortions that aren’t real, which can cause mistakes in the calculations of ion movement and flaw formation energies.
It is thought that anharmonic effects will lower the energy hurdles for ions to move, but no one has looked into how big these effects are in different halide perovskites. Large-scale molecular dynamics studies have shown that the amounts of Br vacancy defects can change by up to 1 eV at room temperature. However, it is still not known how much these changing defect properties affect the movement of carriers and the optoelectronic properties of halide perovskite.
Optoelectronic Properties
The text talks about the optoelectronic properties of halide perovskites and focusses on two types of electronic excitations: charged excitations and neutral excitations. Charged excitations change a N electron system into a N + 1 or N – 1 system because photons excite them, while neutral excitations keep the number of electrons the same and are measured by optical absorption. Inverse photoemission studies can be used to measure the band structure and band gap of a solid, which are features that have to do with charged excitations. The book also talks about the Bethe-Salpeter Equation (BSE) method in Green’s function-based many-body perturbation theory and the most important qualitative and numeric features of halide perovskite band structures. It is also talked about the results for halide perovskites, with a focus on the bound excitons.
Electronic Band Structures
What Can DFT Tell Us About Band Gaps of Solids?
The space between the bands A solid’s egap is an excited state feature that has to do with taking or adding an electron from a neutral (N-electron) solid. It is the difference between the ionisation energy I(N) and the electron affinity A(N). The ground-state energy differences can be used to figure it out. These three ground-state DFT calculations of the total energies of the N − 1, N, and N + 1 electron systems can be used to figure out Egap.
When used in real life, Egap stands for the energy difference between the most occupied (𝼀VBM) and least occupied (𝼀CBM) electrical states. This equality is not true when 𝼀CBM and 𝼀VBM come from a Kohn–Sham DFT calculation, though. It has been found that the Kohn–Sham band gap difference (EKS gap = 𝼀CBM − 𝼀VBM) is the constant that causes the Kohn–Sham potential to jump when an electron is added to the N-electron system. This is called the derivative discontinuity Δxc.
The fact that Kohn–Sham DFT can’t predict basic band gaps isn’t because of bad xc approximations; it’s because the Kohn–Sham potential has a derivative break. In a generalised Kohn–Sham scheme, hybrid functionals and meta-GGAs can lead to band gaps that are more in line with experiments. The xc potential in generalised Kohn–Sham theory is not a multiplication potential; instead, it is an integral operator. This means that all electrons see the same mean-field potential. So, the derivative discontinuity is included in the generalised Kohn–Sham band gap in a limited way.
If you use the right xc model and the right material, generalised Kohn–Sham band gaps that are found by comparing eigenvalue energies can, in theory, match exact fundamental band gaps.
A Short Introduction to the GW Approach
The most up-to-date way to figure out the band structures and band gaps of objects is to use Green’s wave equation (GW). It is found by expanding the Green’s function for a single particle with perturbations. This function gives us the chance amplitude that a particle made at point r, t will be destroyed at point r′, t′. In math, the creation and destruction of particles are shown by a sum of time-ordered field operators 𝟓̂ (r, t) acting on the N-particle ground state wavefunction ΨN0. The time-ordering operator is denoted by T̂.
The poles of this Green’s function show the energies needed to add and remove electrons from a bound starting state j. The Lehmann representation of G shows this property: G(r, r′, 𝜏) = ∑ j gj(r)g∗j (r′) 𝜏 − 𝜀j + i𝜂 ⋅ sgn(𝜀j − 𝜀F) (8.26). The single-particle Green’s function is linked to the screened Coulomb interaction W, the irreducible polarisability 𝼒, and the so-called vertex function Γ, which is a three-particle object. Hedin’s equations provide a way to find this object in real life.
The GW approximation doesn’t include the vertex in the expression for the self-energy. This creates a set of coupled integral-differential equations: G(1, 2) = G0(1, 2) + ∫ d3d4G0(1, 3)Σ(3, 4)G(4, 2) (8.27), 𝼒0 = −iG(1, 2)G(2, 1) (8.28), W(1, 2) = v(1, 2) + ∫ d3d4v(1, 3)𝼒0(3, 4)W(4, 2) (8.29), and λ(1, 2) = iG(1, 2)W(1+, 2) (8.30).
For computing-efficient approaches to fully self-consistent GW, Equation (8.33) is a good place to start. In lowest-order perturbation theory, also known as G0W0, the QP states are modelled by (generalised) Kohn-Sham eigenfunctions, which leads to a set of equations. G0 and W0(r, r′, 𝜏) = ∫ d3r′′𝜖−1(r, r′′, 𝜏)v(r′′, r′) (8.35), in the random phase approximation (RPA), are used to find the self-energy Σ.
Different kinds of (partial) self-consistency have been created so that QP excitations can be dealt with in the GW way.
The Band Structure of Halide Perovskites: A Tight-Binding Perspective
The band structure of ABX3 perovskites that are perfectly cubic (Pm3m) is defined by a straight band gap at the R point of the Brillouin zone and conduction and valence bands that are very spread out. The metal-p orbitals make up the conduction band. The metal-s and halogen-p orbitals make up the valence band.
To use these results in real life, you can get information about the orbital character of a band by putting the (generalised) Kohn–Sham eigenfunctions in the plane wave basis onto spherical harmonics in a certain radius centred around each ion in the lattice. We can use this method to figure out partial charges and predicted densities of states, but they rely on the ionic sizes that are picked and should be fully understood.
The Wannier representation is a second first-principles way to talk about the electronic structure of periodic solids in terms of localised wavefunctions. The Bloch wavefunctions are localised through a set of unitary matrix changes. In real life, this can be done with a localisation method created by Marzari and Vanderbilt that makes the Wannier functions spread out as little as possible in space and gives rise to what are called maximally localised Wannier functions (MLWF). Many people have used MLWFs to learn about chemical bonds and the electronic structure of solids. They have also been used to make model Hamiltonians for a wide range of solids phenomena, from electron dynamics to magnetic interactions.
The tight-binding method is similar, but it doesn’t directly get localised orbitals from first principles results. Instead, it builds a model Hamiltonian from atomic orbitals at different lattice sites and how they interact with each other. Boyer-Richard et al. [76] were the first to suggest a tight-binding model for cubic ABX3 perovskites. This model takes into account 16 basis functions if spin–orbit coupling (SOC) is not taken into account. When using SOC, the base size needs to be twice, and the tight-binding Hamiltonian needs to have an extra term equal to L̂ ⋅ Sˆ. L̂ is the orbital angular momentum operator, and Sˆ is the spin angular momentum operator.
You can use the same methods to talk about the electronic structure of quaternary halide double perovskites A2BB′X6 and other more complicated halide perovskites. The tight binding models for these materials are more complex than those for single cubic perovskites. This is because the basic unit cell of the double perovskite lattice is rhombohedral, the atomic orbital basis is at least twice as big, and many of the materials that are relevant have d-orbital derived valence bands, which adds more parameters.
Slavney et al. [78] proposed a qualitative model that can predict where the valence and conduction band edges will be in reciprocal space. The model shows that the double perovskite stoichiometry can create a wide range of electronic structures by carefully combining the s and p orbitals of the B and B′ site atoms. The positions and spread of the band edges are mostly determined by the frontier orbitals of the B and B′ site ions.
Toward Predictive Band Structure Calculations for Halide Perovskites
To get a general idea of a material’s electronic structure, tight-binding models are helpful. But to accurately guess the material’s optical qualities, first-principles methods must be used. Standard DFT approximations aren’t perfect for figuring out band gaps, but it’s a good idea to start looking into the electronic structure of new materials using standard xc functionals because they are fast and usually give you data that you can use to train machine-learning models or feed into tight-binding or other semiempirical methods.
It is very important to choose the right structural models in order to predict the band gaps and band structures of halide perovskites. When B-sites are off-center and octahedral tilts and rotations happen, they strongly shift the band gaps of Pb- and Sn-based single perovskites to the blue. For instance, the band gap of MAPbI3 can be different by several hundred meV depending on the size of the supercell and the orientation of the molecular cations within the Pb-I sublattice. This can be found using the code described in Section 8.2.2.2 and the PAW formalism as implemented in the Vienna Ab-initio simulation package (VASP).
Not only does the choice of structural model change the size of the band gap, but it also changes other important aspects of the band structure. In the band structure of metal-halide perovskites, a strong Rashba/Dresselhaus splitting appears, along with strong spin–orbit interactions and a structure that is not symmetrical around the centre. When used in a generalised Kohn–Sham framework, hybrid functionals make the forecasts of standard (semi)local xc approximations a lot better. For many solids, including some all-inorganic halide perovskites, global and RSH functions can be parameterised based on first-principles considerations. They also do a good job of reproducing test band gaps.
The Green’s Equation (GW) method is thought to be the best way to predict band structure and band gap, and it can also be used as a starting point for figuring out visual features in the BSE method. It was Umari et al. who reported the first G0W0 calculations for halide perovskites in 2014. They used a method where SOC was only used to build the zeroth-order Green’s function G0 and not in the screened Coulomb interaction W0.
The G0W0 method, on the other hand, depends too much on the (generalised) Kohn–Sham eigensystem that was used to make G0 and W0. It is important to be extra careful with halide perovskites that have small band gaps, like those that have a strong SOC or a lot of band dispersion. It’s still not possible to use GW self-consistency or tuning methods for global and RSH functions because they are too hard to implement in high-throughput apps. Even though electronic structure methods like DFT are useful for designing materials and making estimates, they can’t be used without first being tested in the real world.
Optical Properties
We talk about the BSE method and how it can be used to study the excited-state structure of halide perovskites. At the time of writing, time-dependent DFT has only rarely been used to figure out the optical properties of halide perovskites. It can be used to figure out the optical properties of solids, though.
A Short Introduction to the Bethe–Salpeter Equation Approach
The First Rules of Behaviour Atomistic Theory of Halide Perovskites (BSE) is a way to figure out the two-particle correlation function, which shows how an excited electron-hole pair, or exciton, moves through a material. In the Green’s function approximation, K(3, 5; 4, 6) = v(3, 6)𝛿(3, 4)𝛿(5, 6) − W(3+, 4)𝛿(3, 6)𝛿(4, 5) is the kernel. This can be used to find the eigenvalues of a matrix, assuming that the screened Coulomb interaction doesn’t depend on frequency.
The Tamm-Dancoff estimate is often used to figure out excitons in objects. To solve for each exciton state S, you need to find (𝜀QP a − 𝜀QP i)ΩS XS ia + ∑ jb ◨ia|K|jb▩ = ΩS XS ia, where ΩS is the excitation energy and XS ia is the exciton wavefunction that goes with it. In terms of the productstate basis of occupied and unoccupied orbitals, the electron-hole interaction kernel is written as ▨ia|K|jb◩ = vijab − Wiajb.
Most of the time, the eigenvalue difference in Equation (8.39), which is based on QP energies estimated using the GW method, is correct. The exciton binding energies can be found by comparing Equation (8.41), which says that 𝼖IP2 (𝼏) = 16𝼋e2 𝼏2 ∑ S |e ⋅ ▨0|v|S▩|2𝛿(𝼏 − ΩS) (8.41),. When it comes to crystals, the dielectric function can be turned off by local field effects like off-diagonal matrix elements of the dielectric matrix G,G′ ().
Neutral Excitations in Halide Perovskites
It’s hard to figure out the neutral excitations of halide perovskites using the GW +BSE method because the structures are complicated and the band gaps aren’t taken into account properly. MAPbI3 and other related perovskites have valence and conduction bands that are very spread out. This means that excitons are strongly spread out in space and calculations need to be done on very dense k-point grids.
When these studies are done on MAPbI3 and other single-halide perovskites, they show three things: First, the excitons are well described by the Wannier–Mott model that uses hydrogen. This means that the exciton binding energies are given by EX = 𝼇∗2𝼀2∞ ⋅ 1∖n2 (in atomic units), where 𝼇 is the effective mass of the electron–hole pair and is found by taking the second derivative of the GW energy of the conduction and valence band with respect to the wave vector, 𝼀∞ is the optical dielectric constant, and n is the main quantum number. This model also does a good job of describing the exciton wavefunctions XSia from Equation (8.39).
Second, the exciton binding energies are up to three times higher than what the experiment shows. This is because of polaronic and phonon screening effects. The GW+BSE method described in Section 8.3.2.1, which doesn’t take these effects into account, does, however, very well match the lineshape of the excitonic feature found in the experiment.
In experiments, exciton binding energies are often found by using the Elliott model on absorption spectra. This model assumes that optical changes are caused by a single set of (parabolic and isotropic) valence and conduction bands. In Elliott theory, the absorption coefficient is made up of X, which is the absorption from bound excitons, and C, which is the absorption from electron–hole continuum states. X takes the form of a Rydberg-like series at energies −EX below the band gap EG.
If excitons are found in halide perovskites that are more structurally and chemically different from single ABX3 perovskites, the Wannier–Mott and Elliott models need to be changed or even don’t work at all. For instance, the different types of dielectric screening in quasi-2D perovskites with Ruddlesden–Popper and Dion–Jacobson structures have a big effect on the excitons that move through them.
For 3D halide perovskites, like the halide double perovskite Cs2AgBiBr6, they may stray a lot from the Wannier–Mott model because their chemicals are not all the same. This makes their electronic structure and dielectric screening very uneven. The exciton binding energy is 35% higher than the first-principles model when Elliott’s formula for optical absorption is used to fit the first-principles absorption spectrum.
Concluding Remarks: First Person Singular
This chapter talks about how first-principles computer modelling can help us understand complicated materials like halide perovskites. DFT and Green’s function-based methods are useful and beautiful in how simple they are. They take the quantum physics “many-body problem” and turn it into a problem that can be solved and saved on any computer with the right tools. This is why these tools, especially DFT, have been so useful in many science areas. This chapter uses cases to show how these methods can help us understand the structure and optical features of halide perovskites at the atom level, figure out what the results of experiments mean, and separate out different physical effects. But the chapter also says not to use computer systems that don’t take into account the basic and useful limits of the theories and methods they use. Researchers are still needed to set up, validate, and understand smart first principles computer studies, even with the help of new methods, high-performance computing tools, and machine-learning-based approaches.
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