Metal-halide perovskites are a structurally, chemically, and electronically diverse class of semiconductors with applications ranging from photovoltaics to radiation detectors and sensors. Understanding neutral electron–hole excitations (excitons) is key for predicting and improving the efficiency of energy-conversion processes in these materials. First-principles calculations have played an important role in this context, allowing for a detailed insight into the formation of excitons in many different types of perovskites. Such calculations have demonstrated that excitons in some perovskites significantly deviate from canonical models due to the chemical and structural heterogeneity of these materials. In this Perspective, I provide an overview of calculations of excitons in metal-halide perovskites using Green’s function-based many-body perturbation theory in the GW + Bethe–Salpeter equation approach, the prevalent method for calculating excitons in extended solids. This approach readily considers anisotropic electronic structures and dielectric screening present in many perovskites and important effects, such as spin–orbit coupling. I will show that despite this progress, the complex and diverse electronic structure of these materials and its intricate coupling to pronounced and anharmonic structural dynamics pose challenges that are currently not fully addressed within the GW + Bethe–Salpeter equation approach. I hope that this Perspective serves as an inspiration for further exploring the rich landscape of excitons in metal-halide perovskites and other complex semiconductors and for method development addressing unresolved challenges in the field.

Excitons are charge-neutral quasiparticles (QPs) that are formed in semiconductors and insulators upon absorption of photons. Their formation, diffusion, lifetime, and recombination are key for understanding the fundamental processes underlying light-harvesting and energy-conversion in materials.1–3 The ability to control these properties is a central prerequisite for designing materials for tailored applications, including photovoltaics,4–6 sensing,7 lighting,8–10 and photocatalysis.11,12 Metal-halide perovskites are a class of materials with great chemical and structural diversity that have been explored in all of these application areas. Most notably, they have been employed as absorber materials in solar cells, leading to power conversion efficiencies exceeding 25%.13 The chemical formula of metal-halide perovskites is ABX3, where A is a monovalent (molecular) cation, e.g., Cs+, CH3NH3+ (MA = methylammonium), and (CH(NH2))2+ (FA = formamidinium), B is a divalent metal, e.g., Pb+, Sn+, and X is a halide, e.g., I, Br, and Cl. The rich diversity of this material class is based on the ability to tune their photophysics through chemical substitution at the A, B, and X sites.

The exciton binding energy, i.e., the binding energy of photoexcited electron–hole pairs, has been studied intensively in metal-halide perovskites, such as MAPbI3, since it is key for understanding energy- and charge-transport properties and the design of appropriate device architectures. Furthermore, this knowledge is crucial for the interpretation of spectroscopic information. In typical inorganic semiconductors, excitons are well-represented by the Wannier–Mott model,14 which describes a hydrogenic series of bound electron–hole states with binding energy EB = −μ/m0ɛ2· 1/n2 Ry, where μ is the reduced effective mass of the electron–hole pair, m0 is the free electron mass, ɛ is the dielectric constant, and n is the principal quantum number.

Experimental measurements of the exciton binding energy of semiconductors typically rely on the Wannier–Mott model. The exciton binding energy of several MA and FA lead-halide perovskites, for example, was determined to be on the order of 14–25 meV at room temperature and ∼16 meV at low temperatures, from the difference between the 1s and 2s excitonic states, determined by magnetoabsorption experiments.15,16 Additionally, the exciton binding energy has been determined from optical absorption measurements by using Elliott’s model, which allows for a separation of contributions to absorption from bound excitons and band-to-band transitions.17 Using this method for the low-temperature phase of MAPbI3, Davies et al. found an exciton binding energy of 16 meV, in very good agreement with earlier magnetoabsorption experiments.18 Similar to the Wannier–Mott model, the Elliott model assumes electronic transitions between parabolic and isotropic valence and conduction bands, and contributions of bound excitons to the absorption spectrum are described as a series of hydrogenic energy levels.

Accurate first-principles calculations based on density functional theory (DFT) and Green’s function-based many-body perturbation theory in the GW and Bethe–Salpeter Equation (BSE) approximation show that for chemically and structurally heterogeneous metal-halide perovskites, such as double and layered perovskites, the canonical assumptions of the Wannier–Mott model break down.19–21 The main reasons for this breakdown are the anisotropy and non-parabolicity of the valence band maximum and conduction band minimum and a highly non-uniform dielectric screening. Furthermore, even for those metal-halide perovskites for which the Wannier–Mott model holds, electron–phonon and exciton–phonon coupling effects significantly contribute to electronic-structure properties and exciton binding energies.22–24 However, the pronounced temperature-induced structural dynamics of many metal-halide perovskites sharply distinguish them from conventional semiconductors and suggest that anharmonic effects need to be taken into account in first-principles calculations of electronic and excitonic properties of these materials.

This Perspective aims to demonstrate that first-principles electronic structure calculations are indispensable for understanding excited-state properties of metal-halide perovskites. I will focus on the GW and BSE approaches, which allow for systematically studying the effects that the tremendous structural and chemical diversity have on the electronic and excited-state structure of metal-halide perovskites, and highlight their successes and current limitations to point out areas where further method development is needed. However, this Perspective does not claim to be a complete review of either these methods nor their application to metal-halide perovskites. Instead, I will highlight pertinent results with a focus on free (as opposed to self-trapped) excitons and the goal to guide other researchers in the field in interpreting experimental findings, understand the applications and limitations of textbook models for excitons, and inspire further exploration of metal-halide perovskites as a material platform for studying exciton physics and the development of new methods for addressing unresolved challenges in the field.

This Perspective is structured as follows: In Sec. II, I will provide a brief overview of the GW and BSE approaches in the forms in which they have primarily been used to calculate exciton photophysics in metal-halide perovskites. Section III focuses on calculations of bandgaps (Sec. III A) and excitons in a wide variety of metal-halide perovskites, covering ABX3 (Sec. III B), double (Sec. III C), and layered (Sec. III D) perovskites and an outline of recent developments for taking into account the coupling of excited electron–hole pairs to lattice vibrations. Finally, in Sec. IV, I will share my perspective on developments in the field and challenges that lie ahead for addressing the fate of excitons after their initial formation.

First-principles calculations of excitons in halide perovskites have, with a few notable exceptions,25 primarily been carried out using Green’s function-based many-body perturbation theory within the GW and BSE approaches. A typical workflow for the calculation of excitons, schematically shown in Fig. 1, starts with a DFT calculation using the Kohn–Sham (KS) or generalized Kohn–Sham (gKS) approach,26 yielding approximate electron addition and removal energies, which are then perturbatively corrected using the GW approach. The GW method relies on approximating the electronic self-energy Σ as a product of the one-particle Green’s function G and the screened Coulomb interaction W, requiring the calculation of the inverse of the frequency-dependent dielectric matrix ɛ(q, ω). The GW electron addition and removal energies and dielectric matrix are then used within the BSE approach to calculate electron–hole interactions, yielding neutral excitation energies and excitonic wavefunctions. The solutions of the BSE can then be used to construct the complex dielectric function and calculate properties that may be readily compared with experimental results, such as absorption spectra and exciton binding energies. Spin–orbit coupling (SOC) can be accounted for self-consistently by using two-component spinors in the GW and BSE approaches.27 Note that I will adopt a notation neglecting spin in Secs. II A and II B, in which I will provide a brief overview of the GW and BSE methods. More in-depth discussions and review articles will be referenced where appropriate.

FIG. 1.

Workflow of a typical GW + BSE calculation as described in the text. Figure adapted from Ref. 28.

FIG. 1.

Workflow of a typical GW + BSE calculation as described in the text. Figure adapted from Ref. 28.

Close modal

The GW approach allows for the calculation of electron addition and removal energies, i.e., charged excitations.29–31 It has been used to calculate the bandstructures and bandgaps of a wide range of functional systems, encompassing molecules, nanostructured materials, and extended solids, often in very good agreement with results from (angle-resolved) photoemission spectroscopy.31–48 

The central concept underlying the success of the GW approach is that the removal and addition of an electron from/to a system of N interacting electrons can in many materials be described by a non-interacting quasiparticle (QP).29 These QPs are electrons or holes whose interaction with other electrons is screened by a cloud of oppositely charged particles upon addition or removal of an electron, leading to characteristic particle-like peaks in the spectral function with a broadening proportional to the inverse of the lifetime of the QP.49 In the following, I will adopt a notation assuming periodic boundary conditions and plane wave basis sets, where k and q denote reciprocal space vectors and G are reciprocal lattice vectors. Other types of implementations of the GW + BSE approaches exist, and an overview of available codes can be found in Ref. 49. The QP energies, EnkQP, and wavefunctions, ϕnkQP, can, in principle, be calculated by solving
22m2+vext+vHϕnkQP(r)+d3rΣ(r,r,EnkQP)ϕnkQP(r)=EnkQPϕnkQP(r).
(1)
Here, vext is the potential arising from the interaction of the electrons with the nuclei, vH is the electrostatic Hartree potential, and Σ is the non-local, energy-dependent self-energy operator that describes the dynamical screening of electrons and holes by the polarizable cloud arising from the Coulomb interactions of the N ± 1 particle system. Σ is related to the one-particle Green’s function G through a set of coupled integro-differential equations known as Hedin’s equations.30 In the GW approach, the three-body part of Hedin’s equations—known as the vertex function—is neglected and Σ is calculated as the frequency-space convolution of G and the screened Coulomb interaction W.
In many practical applications of the GW approach, G and W are constructed non-self-consistently from KS or gKS eigenvalues, EnkgKS, and eigenfunctions, ϕnkgKS, obtained by solving the KS or gKS equations of DFT,26,
22m2+vext+vHϕnkgKS(r)+d3rvxc(r,r)ϕnkgKS(r)=EnkgKSϕnkgKS(r),
(2)
self-consistently. The screened Coulomb interaction W is calculated from WGG(q,ω)=εG,G1(q,ω)v(q+G), where v is the Coulomb interaction and εG,G1 is the inverse of the dielectric matrix in the random phase approximation (RPA). The inverse of the RPA dielectric matrix, ɛG,G(q, ω) = δG,Gv(q + G)χGG(q, ω), is constructed from the RPA polarizability using the expression by Alder and Wiser,50,51
χG,G(q,ω)=nnkMnn(k,q,G)Mnn*(k,q,G)121Enk+qgKSEnkgKSωiη+1Enk+qgKSEnkgKS+ω+iη,
(3)
where the sums run over k-points and occupied and empty bands, respectively; η is a broadening parameter; and Mnn(k,q,G)=ϕnk+qgKS|ei(q+G)r|ϕnkgKS are plane wave matrix elements. The frequency-dependence of the RPA polarizability can be evaluated explicitly, approximated using plasmon-pole models, or neglected altogether.33,52,53
With this, QP energies in the GW approximation can be calculated in a perturbative fashion by assuming that ϕnkQPϕnkgKS and solving
EnkGW=EnkgKS+ϕnkgKS|Σ(EnkGW)vxc|ϕnkgKS.
(4)
Note that if vxc is a (semi)local exchange–correlation potential, Eq. (2) takes the form of the regular KS equation.

This approach is dubbed the G0W0 method. It has been used with much success for a wide range of materials, but results are known to exhibit a pronounced dependence on the exchange–correlation functional used in the initial DFT calculation.41,42,54–58 This starting point dependence has been explored for halide perovskites by several groups and will be addressed in more detail in Sec. III A. In addressing the starting point dependence, different levels of self-consistency in the evaluation of the self-energy Σ can be used. The reader is referred to Ref. 49 for a detailed discussion of (partially) self-consistent GW.

The Bethe–Salpeter Equation (BSE) is a framework for the two-particle Green’s function that allows for the calculation of polarizabilities including electron–hole interactions.59 In a product basis of occupied and unoccupied QP states, the BSE takes the same block-diagonal form as Casida’s equations of linear-response time-dependent DFT,60,
RCC*R*AsBs=ΩsAsBs.
(5)
In inorganic semiconductors and insulators, the off-diagonal elements of this matrix eigenvalue equation (corresponding to de-excitations) are often neglected, leading to the BSE in the Tamm–Dancoff approximation,61,62
EckGWEvkGWAvcks+vckvck|K|vck=ΩsAvcks.
(6)
The indices v and c are referring to valence and conduction band QP states, but are commonly taken as (generalized) Kohn–Sham orbitals based on the same assumption underlying Eq. (4). In the following, the superscripts gKS are dropped for clarity. Ωs are the neutral excitation energies; Avck are exciton wavefunction coefficients represented in the basis of valence and conduction band states, allowing for an expression of the exciton wavefunction in real-space,
Ψ(re,rh)=vckAvcksϕck(re)ϕvk*(rh).
(7)
Electron–hole interactions are described by the kernel K, which is frequency-dependent in general, but commonly approximated as static and composed of a screened direct interaction Kd and a bare exchange interaction Kx, which are defined (in real-space) as
vck|Kd|vck=d3rd3rϕc*(r)ϕc(r)W(r,r)ϕv*(r)ϕv(r)
(8)
and
vck|Kx|vck=d3rd3rϕc*(r)ϕv(r)v(r,r)ϕv*(r)ϕc(r).
(9)
Solution of Eq. (6) allows for the calculation of the imaginary part of the macroscopic transverse dielectric function,
ε2(ω)=16πe2ω2s|e0|v|s|2δ(ωΩs),
(10)
where |0⟩ and |s⟩ correspond to the ground and excited state s, respectively; e is the light polarization vector; and v is the velocity operator, allowing for a direct comparison with experimental optical absorption spectra. The effect of electron–hole interactions on the optical spectra can be directly evaluated by comparing Eq. (10), with the corresponding expression in the independent-particle picture,
ε2(ω)=16πe2ω2vck|evk|v|ck|2δ(ωEckGW+EvkGW).
(11)
The exciton binding energy is then defined as EB=EgapGWΩ1, where Ω1 is the energy of the first excited state. Furthermore, Eq. (7) can be used to evaluate the spatial distribution of the excitonic wavefunction.

In this Perspective, I will focus on the metal-halide perovskite families depicted in Fig. 2: In Sec. III B, I will discuss excitonic properties of the single perovskites with chemical formula ABX3 [Fig. 2(a)], which encompass quintessential perovskites such as MAPbI3 and have widely tunable bandgaps through chemical substitution at the A, B and X sites. Figure 2(b) depicts the crystal structure of the so-called double halide perovskites (or elpasolites),63 A2BB′X6, with alternating B and B′ sites, which can feature metals with oxidation states between +1 and +4.64–67 Note that the B′ site can also be a vacancy in the vacancy-ordered perovskites.68 Thousands of stable halide double perovskites have been predicted,69,70 and their fundamental properties and potential for applications are widely studied71–73 due to their tremendous physico-chemical versatility. I will discuss the impact of the chemical heterogeneity introduced by the alternating B and B′ sites in Sec. III C. In Sec. III D, I will focus on excitons in layered quasi-two-dimensional (quasi-2D) perovskites shown in Fig. 2(c). In these materials, the reduction of the dimensionality of the inorganic sublattice by incorporation of large organic molecules, such as butylammonium (BA) or phenethylammonium (PEA), affords a wide range of functional bulk materials that exhibit features of quantum and dielectric confinement.74–77 

FIG. 2.

Schematic of halide perovskite structures referred to in this Perspective. (a) Single halide perovskites ABX3. (b) Double halide perovskites A2BB′X6. (c) Quasi-2D halide perovskites with the Dion–Jacobson structure and formula A′BX4 (left) or Ruddlesden–Popper structure and formula A′2BX4. Both types of quasi-2D perovskites can also feature alternating B and B′ metal sites in their perovskite layers.

FIG. 2.

Schematic of halide perovskite structures referred to in this Perspective. (a) Single halide perovskites ABX3. (b) Double halide perovskites A2BB′X6. (c) Quasi-2D halide perovskites with the Dion–Jacobson structure and formula A′BX4 (left) or Ruddlesden–Popper structure and formula A′2BX4. Both types of quasi-2D perovskites can also feature alternating B and B′ metal sites in their perovskite layers.

Close modal

QP band energies enter the calculation of excitons as energy differences directly in Eq. (6) and indirectly in the calculation of the screened electron–hole interaction kernel [Eq. (9)], which depends on the screened Coulomb interaction, which, in turn, relies on the RPA polarizability in Eq. (3). An underestimation of the bandgap will, therefore, lead to not only a redshift of the entire absorption spectrum but also a material-dependent underestimation of the exciton binding energy due to an overestimation of the dielectric screening.

The GW approach is generally seen as the gold-standard technique for bandgap predictions of a wide range of semiconductors and insulators.31 Halide perovskites have been somewhat notorious, albeit not singular, in challenging this assertion with reported GW bandgaps often underestimating experimental results by several 100 meV depending on the material. Moreover, both DFT and GW bandgaps reported for one and the same material sometimes differ by similar amounts. This is illustrated in Fig. 3, which shows the spread of bandgaps reported in the literature for the low-temperature orthorhombic (<160 K), the tetragonal (<315 K), and the high-temperature cubic phases of MAPbI3. The temperature-dependence of the optoelectronic properties of MAPbI3 has been studied experimentally by several groups, for example, by using photoluminescence and transient absorption spectroscopy by Yamada et al.,78 by measuring temperature-dependent absorbance and transmittance by Foley et al.79 and Milot et al.,80 and by combining optical absorption measurements with Elliott theory by Singh et al.81 and Davies et al.18 While these measurements result in slightly different bandgap predictions due to differences in measurement methods and sample-growth effects, temperature-dependent trends are consistent and the spread in the bandgap data is comparably small.

FIG. 3.

Comparison of bandgaps of orthorhombic (green), tetragonal (light blue), and cubic (red) MAPbI3 reported in the literature, calculated with DFT22,58,82–86 and the GW approach22,58,82–84,86 and as determined experimentally.18,78–81 Black symbols correspond to the average and standard deviation of the data. Experimental values are corresponding to measurements at 4 K and just above the phase transitions at 160 and 315 K. DFT bandgaps all correspond to calculations with (semi)local functionals and include SOC self-consistently. GW bandgaps all correspond to G0W0 calculations using (semi)local starting points and also include SOC. For the cubic phase, I only report values obtained using the primitive single formula unit cell.

FIG. 3.

Comparison of bandgaps of orthorhombic (green), tetragonal (light blue), and cubic (red) MAPbI3 reported in the literature, calculated with DFT22,58,82–86 and the GW approach22,58,82–84,86 and as determined experimentally.18,78–81 Black symbols correspond to the average and standard deviation of the data. Experimental values are corresponding to measurements at 4 K and just above the phase transitions at 160 and 315 K. DFT bandgaps all correspond to calculations with (semi)local functionals and include SOC self-consistently. GW bandgaps all correspond to G0W0 calculations using (semi)local starting points and also include SOC. For the cubic phase, I only report values obtained using the primitive single formula unit cell.

Close modal

In contrast, calculated bandgaps, regardless of the level of theory, show a significantly larger spread. This problem has been studied in detail by several groups and can be traced back largely to three main sources: 1. the choice of structural models used for calculations; 2. technical differences between how different codes implement the GW approach, including differences in pseudopotentials; and 3. the starting-point dependence of the GW approach.

Choice of structural model and inclusion of electron–phonon coupling. The choice of structural models can have a significant impact on the calculated bandgaps of halide perovskites. This becomes apparent from the spread of DFT bandgaps shown in Fig. 3 since technical implementation and pseudopotentials play a less pronounced role at the DFT level and all bandgaps reported in Fig. 3 were calculated using qualitatively similar exchange–correlation functionals (PBE and LDA) and include SOC self-consistently. The main difference in these bandgaps is, therefore, due to differences in the structural models used in these calculations. For the tetragonal phase of MAPbI3, this was shown early by Quarti et al. who reported DFT bandgaps of tetragonal structural models of MAPbI3 with bandgaps differing by up to 0.5 eV depending on the chosen orientation of the MA molecules in the tetragonal unit cell.85 Similar effects were also shown in Ref. 58 comparing the bandgaps of a perfectly cubic structure of MAPbI3 with that of a molecular-dynamics-based structure featuring significant distortions.22 The spread in the DFT bandgaps of the orthorhombic phase is smallest, in line with the frozen orientations of the MA molecules in the low-temperature phase. However, significant discrepancies in bandgaps calculated at equilibrium and experimental lattice parameters were reported even for all-inorganic halide perovskites.87 

An interesting technical consequence of the choice of a structural model that is particularly pronounced for halide perovskites with molecular A sites, such as MA, is that for certain molecular orientations, the strong coupling between the molecules and the inorganic metal-halide framework can induce spurious polar distortions, such as an off-centering of the metal ion. In such non-centrosymmetric structures, strong SOC leads to a splitting of Kramer’s degenerate valence- and conduction-band edges in reciprocal space, an effect known as Rashba–Dresselhaus splitting.88–91 While external electric fields could be applied to align the MA molecules such as to allow for a significant macroscopic polarization,92 no conclusive experimental evidence supports a significant Rashba–Dresselhaus effect in bulk MAPbI3 or similar systems and the effect is likely small at elevated temperatures because of the on-average centrosymmetric structure of MAPbI3.18,93

Bandgap renormalization due to electron–phonon coupling effects has been studied for a range of metal-halide perovskites, often with a focus of reconciling the discrepancy between the experimentally observed minor temperature-dependent bandgap variations with the large bandgap differences between low- and high-temperature phases predicted by first-principles calculations.94 In particular for studies of the cubic phase of many halide perovskites, it is important to highlight the reference structure with respect to which the effect of electron–phonon coupling is calculated. Molecular-dynamics simulations of many halide perovskites at elevated temperatures exhibit strong local deviations from cubic symmetry over time. The average bandgap of such locally distorted structures is up to 0.7 eV larger than the bandgap of a static, perfectly cubic structural model due to the presence of octahedral tilts and rotations in the molecular dynamics snapshots.95 Similarly, the inclusion of polymorphism, i.e., static structural disorder, leads to a significant blueshift of the bandgap with respect to the monomorphic, i.e., perfectly cubic, reference phase.96,97

Calculations of electron–phonon coupling, including quantum zero-point corrections and temperature dependence of bandgaps and other optoelectronic properties, have primarily relied on perturbative approaches in which electron–phonon matrix elements are calculated from DFT98 or, more recently, the GW approach99 or supercell approaches, such as the special displacement method by Zacharias et al.100–102 in which atomic displacements capture dynamic disorder. For the cubic phase of a wide range of single perovskites, the special displacement method in combination with a dielectric-constant-dependent range-separated hybrid functional103 was used to calculate bandgap renormalization due to electron–phonon interactions and found to be below 0.1 eV for Sn- and Pb-based perovskites at 300 K.104 However, these calculations were not including anharmonic contributions to electron–phonon interactions, which are known to be substantial in many halide perovskites.105–107 More recent work combining self-consistent phonon theory with the special displacement method might offer a computationally efficient way to consider anharmonic electron–phonon coupling effects in bandgap calculations of halide perovskites.108 

Technical differences between GW implementations. Technical implementations of the GW approach differ primarily in three aspects: the type of the basis set and whether/which pseudopotentials are used to represent the core electrons, how the frequency-dependence of the screened Coulomb interaction W is evaluated, and differences in how the divergence of the Coulomb potential v(q + G) = 4πe2/|q + G| is evaluated when q → 0 as G = 0. GW bandgaps from codes that eliminate the latter two differences are in excellent agreement with each other.109 However, in practice, it may be difficult to assess the exact technical details underlying reported QP bandgaps since many codes feature more than one scheme for dealing with these numerical problems. It should also be noted that GW calculations rely on several interdependent convergence parameters, which have to be evaluated simultaneously, placing a significant computational burden on these types of calculations. These problems are not exclusive to the halide perovskites, and differences between reported GW bandgaps of some notoriously hard-to-converge systems have only been reconciled in recent years.110 Assuming the same level of convergence and comparable technical implementation, GW bandgaps calculated with different codes usually do not differ by more than 0.1 eV.

Pseudopotentials are used for simplifying the numerically difficult problem of accurately representing the nodal structure of wavefunctions close to the atomic nuclei and dealing with relativistic effects in heavy elements. Their choice can already strongly affect DFT bandgaps. However, GW bandgaps are known to be particularly sensitive to the choice of pseudopotential, requiring the inclusion of complete shells of semicore electrons in the valence electron space for accurate results.35,37,111 For halide perovskites, the pseudopotential was shown to account for differences of up to 0.7 eV in the GW bandgap, primarily due to differences in the valence configurations of different pseudopotentials.58,84,86

Starting-point dependence of G0W0. In most applications of the GW approach, the one-shot G0W0 method detailed in Sec. II is used to calculate QP energies as perturbative corrections to KS or gKS eigenvalues. It is well understood that this method carries a dependence on the (g)KS eigensystem used for constructing the zeroth-order Green’s function G0 and screened Coulomb interaction W0. In halide perovskites, this starting point dependence can be severe with standard (semi)local approximations to DFT, leading to G0W0 bandgaps underestimated by several 100 meV.58,84,112 The GW starting point dependence can be addressed by including (partial) self-consistency (and vertex corrections)83,95 and by using material-dependent DFT starting points, such as optimally tuned113,114 or dielectric-dependent hybrid functionals,103 or meta-GGAs, such as the TASK functional, that are constructed to yield improved bandgaps but come at a smaller computational cost than hybrid functionals.115,116

At low temperatures, the optical absorption spectra of MAPbI3 and other ABX3 perovskites exhibit a characteristic peak at the absorption onset, which has been attributed to a bound exciton. For MAPbI3, the binding energy of this exciton has been determined to be ∼16 meV for the low-temperature orthorhombic phase, through absorption measurements at high magnetic fields.15,16 Similar results were obtained by using Elliott’s model.17,18,81 First-principles calculations of exciton binding energies of this material and other ABX3 perovskites, however, predict results by up to a factor of three higher than experiments.22,24 Despite this consistent overestimation of the exciton binding energies, the line shape of optical absorption spectra of CsPbX3 (X = Cl, Br, I) calculated within the GW + BSE approach was shown to be in very good agreement with experimental spectra.24 Furthermore, the authors of Refs. 22 and 24 showed that exciton binding energies extracted from GW + BSE calculations and those estimated using the Wannier–Mott model, by calculating EB = −μ/m0ɛ2 Ry, where the reduced effective mass μ and the macroscopically averaged electronic contribution to the dielectric constant ɛ were calculated using the GW approach, are in excellent agreement for a range of ABX3 perovskites. In other words, excitons in these ABX3 perovskites were shown to be well-described by the Wannier–Mott model owing to the highly isotropic and parabolic valence and conduction band edges of these materials.

The discrepancy between first-principles and experimental exciton binding energies has been related to neglecting electron–phonon and exciton–phonon coupling effects. The calculation of exciton–phonon coupling effects from first-principles is computationally challenging.117 Phenomenologically, the importance of exciton–phonon coupling in halide perovskites has been justified by determining an effective dielectric constant ε=2EB/μ from the measured exciton binding energy EB and charge-carrier effective masses (which are in very good agreement with those extracted from first-principles calculations).15 For MAPbI3, this results in a value intermediate between the low (ɛ0 ∼25.7) and the high (ɛ = 5.6) frequency dielectric constant for the orthorhombic phase. How coupling to phonons contributes to this effective dielectric screening has not been conclusively determined using first-principles calculations. However, several papers have examined possible mechanisms: the authors of Ref. 22 attributed the overestimation of the measured exciton binding energies to polaronic effects, i.e., the coupling of free electrons and holes to phonons. On the other hands, calculations based on model dielectric functions, including the effect of infrared active phonons, concluded that exciton screening is the dominant effect responsible for the reported overestimation.23 By taking dynamical phonon screening into account directly in the BSE electron–hole interaction kernel, the authors of Ref. 24 showed that, while screening from phonons significantly reduces calculated exciton binding energies, it does not completely reconcile calculated and measured values, indicating that polaronic and other effects likely play an important role as well. Note that calculated values for the static dielectric constant of orthorhombic MAPbI3 are significantly smaller than experimental values of ɛ0 ∼60 at room temperature,118 while molecular-dynamics simulations predict ε0=4050.119 

Double perovskites are a family of materials with tremendous chemical diversity, leading to a wide range of properties. These quaternary materials have a chemical formula of A2BB′X6, and many of them adopt a cubic structure with Fm3̄m symmetry, as shown in Fig. 2(b). The electronic structure of these materials is strongly dependent on the metal and halide sites and can thus be tuned significantly by chemical substitution at these sites, encompassing indirect and direct bandgap semiconductors with a wide range of bandgaps.67,120,121 Figure 4(a) shows the bandgaps of a representative selection of perovskites with X = Cl: one single perovskite CsPbCl3 and seven double perovskites, namely, Cs2InBiCl6, Cs2AgInCl6, Cs2AgSbCl6, Cs2NaBiCl6, Cs2KBiCl6, Cs2NaBiCl6, and Cs2NaInCl6. Note that of these seven double perovskites, only one has not been synthesized (Cs2InBiCl6). However, the isoelectronic MA2TlBiBr6 has been made.122 Several important lessons can be drawn from Fig. 4(a): First, the bandgaps (calculated with the G0W0@PBE + SOC approach) span a wide range, between less than 1 eV and more than 5 eV. Consequently, exciton binding energies as calculated with the GW + BSE method also span two orders of magnitude, ranging from a couple of meV for Cs2InBiCl6 to ∼2 eV for the wide-gap semiconductors Cs2NaBiCl6 and Cs2KBiCl6. However, the exciton binding energy does not scale with the size of the bandgap. For example, the largest bandgap material in this selection Cs2NaInCl6 does not have the highest exciton binding energy. Second, a comparison of the first-principles results with those obtained using the Wannier–Mott model shows that the latter only predicts the exciton binding energies of a subset of halide perovskites well: CsPbCl3, Cs2InBiCl6, Cs2AgInCl6, and Cs2AgNaCl6 exhibit a hydrogenic exciton fine structure, whereas Cs2AgBiCl6, Cs2AgSbCl6, Cs2KBiCl6, and Cs2NaBiCl6 do not. This is also reflected in the excitonic wavefunctions, shown for the 1s excitonic state of CsPbCl3 and Cs2AgBiCl6 in Fig. 4(b).21 

FIG. 4.

(a) G0W0@PBE bandgaps (black dots), G0W0@PBE + BSE exciton binding energies (blue bars), and exciton binding energies calculated on the basis of the Wannier–Mott model (orange bars) of eight perovskites distinguished by their B/B′ site. (b) Radial probability density of the 1s exciton (top) and head of the RPA dielectric matrix (bottom) for CsPbCl3 and Cs2AgBiCl6; the reciprocal space extent of the exciton is marked by a vertical line. (c) Absorption spectra calculated within the independent particle approximation (w/o e–h) and by solving the BSE (w/e–h) for Cs2InBiCl6, Cs2AgBiCl6, Cs2AgInCl6, and Cs2NaBiCl6. (a)–(c) Reproduced from Ref. 21 (d) DFT-PBE band structure (top) and transition matrix elements (bottom) of Cs2AgInCl6. Orange, blue, green, and cyan denote the Cl 3p, Ag 4d, In 5s, and Ag 5s orbital character, respectively. Reprinted with permission from the work of Meng et al. J. Phys. Chem. Lett. 8, 2999–3007 (2017). Copyright 2017, ACS. (e) G0W0@PBE band structure (top) and G0W0@PBE + BSE absorption spectrum of the vacancy-ordered perovskite Cs2SnBr6. Reprinted with permission from the work of Cucco et al. ACS Mater. Lett. 5, 52–59 (2023). Copyright 2023, ACS.

FIG. 4.

(a) G0W0@PBE bandgaps (black dots), G0W0@PBE + BSE exciton binding energies (blue bars), and exciton binding energies calculated on the basis of the Wannier–Mott model (orange bars) of eight perovskites distinguished by their B/B′ site. (b) Radial probability density of the 1s exciton (top) and head of the RPA dielectric matrix (bottom) for CsPbCl3 and Cs2AgBiCl6; the reciprocal space extent of the exciton is marked by a vertical line. (c) Absorption spectra calculated within the independent particle approximation (w/o e–h) and by solving the BSE (w/e–h) for Cs2InBiCl6, Cs2AgBiCl6, Cs2AgInCl6, and Cs2NaBiCl6. (a)–(c) Reproduced from Ref. 21 (d) DFT-PBE band structure (top) and transition matrix elements (bottom) of Cs2AgInCl6. Orange, blue, green, and cyan denote the Cl 3p, Ag 4d, In 5s, and Ag 5s orbital character, respectively. Reprinted with permission from the work of Meng et al. J. Phys. Chem. Lett. 8, 2999–3007 (2017). Copyright 2017, ACS. (e) G0W0@PBE band structure (top) and G0W0@PBE + BSE absorption spectrum of the vacancy-ordered perovskite Cs2SnBr6. Reprinted with permission from the work of Cucco et al. ACS Mater. Lett. 5, 52–59 (2023). Copyright 2023, ACS.

Close modal

These findings can be rationalized by examining the assumptions behind the Wannier–Mott model, namely, a direct bandgap and isotropic, parabolic band edges, reflected in an isotropic reduced effective mass μ, and an isotropic dielectric constant, here evaluated using the electronic contribution to dielectric screening. Note that the relevant quantity here is the localization of the excitonic wavefunction on the length scale of the dielectric screening, shown in Fig. 4(b) for CsPbCl3 and Cs2AgBiCl6. Perovskites such as CsPbCl3 that have delocalized excitons with a small reciprocal space extent feature a hydrogenic exciton series, even though dielectric screening is strongly q-dependent in this material. In contrast, perovskites with localized excitons in real space, which see the long-range behavior of the dielectric screening, agree less well with the Wannier–Mott model.124 

The above framework is useful for predicting the character of excitons in halide double perovskites and demonstrates that care should be taken when extracting exciton binding energies from experimental data. This is also evident from the diversity of absorption spectra of this class of materials, shown for four double perovskites in Fig. 4(c). For example, Cs2AgInCl6 has a substantial exciton binding energy, calculated to be between 18021 and 250 meV,125 and has a magnitude similar to CsPbCl3.24 However, contrary to CsPbCl3, it does not feature a pronounced excitonic peak at the onset of absorption [Fig. 4(c)] because the bandstructure of the direct gap material Cs2AgInCl6 is characterized by a parity-forbidden transition at the Γ point [Fig. 4(d)] and further symmetry-disallowed transitions at higher energies,121 which lead to a series of dark excitons and a slow onset of absorption. On the other hand, the double perovskites Cs2InBiCl6, Cs2AgBiCl6, and Cs2NaBiCl6 each have (at least one) distinct excitonic peaks that can be assigned to an excitonic state, albeit with widely different character.124 

The non-hydrogenic nature of excitons in some perovskites was also observed for the structurally and chemically closely related family of vacancy-ordered perovskites, which crystallize in the same space group as the double perovskites, but instead of alternating B and B′ metal sites, feature a vacancy at one of the metal sites. Despite this structure that leads to isolated quasi-0D metal-halide octahedra, the electronic structure of vacancy-ordered perovskites, such as Cs2SnBr6, shown in Fig. 4(e) results in a hydrogenic exciton series,123 whereas the vacancy-ordered perovskites Cs2TiX6 (X = Br, Cl) and Cs2TeBr6 were shown to feature non-hydrogenic Frenkel-like excitons.20,123

The degree to which exciton binding energies of halide double perovskites calculated with the GW + BSE approach agree with experimental results is harder to evaluate than for the case of the single perovskites because the span of reported experimental values is considerably larger. For example, for Cs2AgBiBr6, one of the best studied double perovskites, values ranging from 70 to 268 meV, have been extracted from experiments.126–129 Calculated values range from 170 to 340 meV and mainly depend on the starting point and level of self-consistency used in the GW calculations.130,131 Next to these technical considerations, it is clear that electron–phonon and exciton–phonon coupling effects play an important role in double perovskites as well.126,132 While significant differences between the lattice dynamics of single perovskites, such as CsPbBr3, and double perovskites, such as Cs2AgBiBr6, have been reported based on temperature-dependent Raman spectroscopy107 and first-principles calculations,133 anharmonic lattice vibrations play an important role in this family of materials too. How lattice dynamics affect the diverse set of excitons found in halide double perovskites is at present an open question.

Layered perovskites are quasi-two-dimensional crystals in which n layers of metal-halide octahedra are spatially separated by organic molecular layers. Typical structural motifs are Ruddlesden–Popper and Dion–Jacobson perovskites, shown schematically in Fig. 2(c), with chemical formulas A2′An−1BnX3n+1 and A′An−1BnX3n+1, respectively. The two structural motifs differ in how adjacent layers are stacked on top of one another: Ruddlesden–Popper structures feature an in-plane shift of adjacent layers along half of the in-plane diagonal.134 This family of materials allows for broad compositional tunability and can be synthesized in the bulk form,135–138 exfoliated to a monolayer,139 and assembled in interfaces with other two-dimensional or layered materials.140 A wide variety of organic molecules can be used in the fabrication of these materials, allowing for variation of the interlayer distance and orientation,141 as well as functionalization, for example, by introduction of chiral142–145 or electroactive146 organic molecules. Furthermore, and similar to their 3D congeners, chemical substitution at the B sites can be used for tailoring optoelectronic properties147 and both single and double quasi-2D perovskites can be synthesized.148,149

In the monolayer limit, quantum confinement effects significantly increase the QP bandgap and exciton binding energies of layered perovskites as compared to 3D perovskites.151 However, the organic sublayer significantly contributes to the dielectric screening of excited electron–hole pairs, counteracting quantum confinement effects on the exciton binding energies to some extent. Accurate measurements of the exciton fine-structure of several Pb- and Sn-based quasi-2D perovskites were reported by Dyksik et al. using linear optical absorption measurements under high magnetic fields.152 The lowest-energy exciton in quasi-2D perovskites is dark and can, therefore, provide a non-radiative channel for carrier recombination. The excitonic fine structure of Pb-based quasi-2D perovskites was systematically studied using first-principles GW + BSE calculations by Filip et al., reproduced in Fig. 5(a)150 and in qualitative agreement with experimental results.152 These calculations also demonstrated the distinctly different spatial localization of the low-energy excitonic states, which are localized in one perovskite layer and contribute to the main excitonic peak in the absorption spectrum, and the higher-energy excitonic states, which are dark and feature photoexcited electrons and holes localized in adjacent layers [Fig. 5(b)]. By replacing several molecular A sites of varying length with Cs+ in model structures not featuring any structural distortions in the Pb-halide layer, the authors of Ref. 150 also showed the pronounced effect of the dielectric properties of the organic sublayer on exciton binding energies in these materials. Additionally, different organic A sites introduce distinct structural distortions and dynamic disorder at elevated temperatures,153,154 thus affecting optoelectronic properties further.155 

FIG. 5.

(a) Exciton fine structure of (PMA)2PbI4 and Cs2PbI4. Bright excitonic states are represented in dark red for A = PMA and dark blue for A = Cs; dark states are represented in gray. The energy zero corresponds to the QP bandgap. Labels “L” and “IL” correspond to layer and interlayer states, respectively. (b) Exciton wavefunction probability corresponding to the L and IL excited states of PMA2PbI4. (a) and (b) Reproduced from the work of Filip et al., Nano Lett. 22, 4870 (2022) with the permission of the authors. (c) Measured 1s exciton binding energy as a function of the number of sublayers n in (BA)2MAn−1PbnI3n+1. The red line corresponds to a scaling law derived using a classical model for low-dimensional systems (red dots). The parameter γ quantifies the contribution of the organic sublayer to dielectric screening of electron–hole pairs. Reproduced from Ref. 77. (d) Bandgap (black) and 1s optical gap (red) of model structures of (BA)2MAn−1PbnI3n+1 calculated using a tight-binding BSE approach. The orange and yellow shaded areas indicate the contribution of quantum and dielectric confinement effects to the bandgap. The blue pattern corresponds to the reduction of the bandgap due to the exciton binding energy. The bulk bandgap is shown in green. Reprinted with permission from the work of Y. Cho and T. C. Berkelbach, J. Phys. Chem. Lett. 10, 6189–6196 (2019). Copyright 2019, ACS.

FIG. 5.

(a) Exciton fine structure of (PMA)2PbI4 and Cs2PbI4. Bright excitonic states are represented in dark red for A = PMA and dark blue for A = Cs; dark states are represented in gray. The energy zero corresponds to the QP bandgap. Labels “L” and “IL” correspond to layer and interlayer states, respectively. (b) Exciton wavefunction probability corresponding to the L and IL excited states of PMA2PbI4. (a) and (b) Reproduced from the work of Filip et al., Nano Lett. 22, 4870 (2022) with the permission of the authors. (c) Measured 1s exciton binding energy as a function of the number of sublayers n in (BA)2MAn−1PbnI3n+1. The red line corresponds to a scaling law derived using a classical model for low-dimensional systems (red dots). The parameter γ quantifies the contribution of the organic sublayer to dielectric screening of electron–hole pairs. Reproduced from Ref. 77. (d) Bandgap (black) and 1s optical gap (red) of model structures of (BA)2MAn−1PbnI3n+1 calculated using a tight-binding BSE approach. The orange and yellow shaded areas indicate the contribution of quantum and dielectric confinement effects to the bandgap. The blue pattern corresponds to the reduction of the bandgap due to the exciton binding energy. The bulk bandgap is shown in green. Reprinted with permission from the work of Y. Cho and T. C. Berkelbach, J. Phys. Chem. Lett. 10, 6189–6196 (2019). Copyright 2019, ACS.

Close modal

Despite continuing increases in computing power, first-principles calculations of the complex interplay between the exciton physics and static and dynamic disorder in these layered perovskites remain computationally challenging. Calculations using the BSE approach in combination with tight-binding models have led to many insights into these materials before fully first-principles GW + BSE calculations were computationally feasible.156 These approaches rely on intuitive physical models of quantum well structures75 and have successfully been applied to other quintessential 2D materials, such as monolayer transition metal dichalcogenides.157 Tight-binding BSE calculations have allowed for understanding of the evolution of exciton binding energies in (BA)2MAn−1PbnI3n+1 with increasing layer thickness n [see Fig. 5(c)]77 and the non-hydrogenic character and transition of optoelectronic properties from quasi-2D to 3D behavior [see Fig. 5(d)].19 However, these approaches require parameterization of the tight-binding model with input from first-principles calculations or experiments, limiting their predictive power. Another recently developed approach separates the GW polarizability of a composite organic–inorganic supercell into smaller building blocks158 and may allow for accurate yet computationally efficient computational modeling of larger supercells, including thermal effects.

GW + BSE calculations of the excited-state structure and properties of systems of unprecedented size have become possible in the last couple of years due to rapid and significant developments in adapting computer codes to new hardware architectures.47 Despite these continuing advances, halide perovskites remain a fascinating, yet challenging family of materials, in which strong spin–orbit coupling, low bandgaps, and large band dispersion present a significant challenge for state-of-the-art electronic structure methods. For example, there is currently no computationally feasible method available that would allow for a rapid and accurate high-throughput screening of bandgaps of halide perovskites. Advances in method development, e.g., of new DFT exchange–correlation approximations113–116,159,160 and the GW approach48,161–163 possibly in conjunction with machine-learning techniques164–168 may help overcome computational bottlenecks. However, at the time of writing and in the foreseeable future, careful validation of calculated bandgaps has to remain the norm in the field.

At the same time, a clear challenge has emerged: to accurately simulate the coupled motion of excited states and atomic nuclei. While methods for calculating electron–phonon and exciton–phonon coupling, e.g., based on density functional perturbation theory169 or many-body perturbation theory99 are available, the pronounced phonon–phonon coupling that has been observed in many halide perovskites at elevated temperatures94,106,153,170 is a challenge that has yet to be resolved within the framework of Green’s function-based many-body perturbation theory. Understanding the coupling of excited states to the structural dynamics could provide important insights into the fate of excitons after their formation, i.e., on radiative and non-radiative recombination processes and on structural reorganization due to photoexcitation, that could lead to the self-trapping of excitons, changes in the symmetry of defects and energetics of ion migration, and the pronounced Stokes shift experimentally observed for many perovskites.

Advancements in machine-learned force fields have made the numerical modeling of anharmonic lattice dynamics feasible and have led to important insights into phase transitions,171 phonon–phonon coupling,172 and other structural-dynamical properties of these materials.173–175 At the same time, methods for simulating exciton dynamics have been developed primarily for understanding exciton transport in molecular crystals.176–180 However, only a few attempts at calculating coupled lattice and excited-state dynamics have been undertaken due to the computational burden of such calculations, which, in principle, would involve the calculation of excited-state forces and nonadiabatic coupling coefficients between ground and excited states at each molecular dynamics time step. Nonadiabatic dynamics based on GW + BSE have been carried out for transition metal dichalcogenide monolayers by Jiang et al.;181 however, the calculation of excited-state forces from GW + BSE was circumvented by these authors. An analytical expression for excited-state forces from GW + BSE was suggested in 2003 by Ismail-Beigi and Louie182 and shown to lead to excited state geometries in very good agreement with quantum-chemical methods and experiment for two small molecules. The accuracy of GW + BSE excited-state forces was later confirmed by Caylak and Baumeier for a wider range of more complex molecules using excited-state gradients obtained numerically using finite differences.183 Recently, an analytical Z-matrix formulation of the problem has been used to calculate excited-state derivatives with respect to an applied electric field, which significantly reduces the computational cost as compared to previous methods and can, in principle, also be adapted for calculating gradients with respect to nuclear positions.184 At the time of writing, none of these methods has been applied to periodic solids, a challenge that will require extensive method development and benchmarking. Halide perovskites will provide an interesting set of model systems for testing new first-principles methods describing exciton dynamics due to their pronounced anharmonicity at elevated temperatures and wide range of excitonic properties.

The author acknowledges valuable discussions with M. R. Filip, J. B. Neaton, and R.-I. Biega and funding from the Dutch Research Council (NWO) through Grant Nos. OCENW.M20.337 and VI.Vidi.223.072.

The author has no conflicts to disclose.

Linn Leppert: Conceptualization (lead); Funding acquisition (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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