Temperature Dependence of Bandgap in Lead‐Halide Perovskites with Corner‐Sharing Octahedra

Abstract

Understanding the molecular origin of the atypical temperature dependence of bandgap (i.e., bandgap renormalization) in lead‐halide perovskites with corner‐sharing octahedra is of fundamental interest and a prerequisite for their applications in fabricating high‐performance optoelectronic devices. The bandgap renormalization is attributed to the lattice thermal expansion and electron–phonon interactions. However, it remains controversial whether the thermal expansion has a negligible effect on the band‐edge structure of corner‐sharing [PbX₆]⁴⁻ (X = halide anion) octahedra in perovskites, relative to the electron–phonon interactions. In this review, this issue is clarified by focusing upon the most recent theoretical advances in the field of investigating the bandgap renormalization in perovskites. The timely research progress on the modulation of bandgap renormalization through the structural engineering in perovskites is also outlined. Finally, a vision about the directions for further in‐depth research in this field is provided, with the intent of promoting a more profound understanding and controlling of bandgap renormalization in perovskites in the future.

Introduction

Lead-halide perovskites with corner-sharing octahedra, featuring outstanding optoelectronic properties, have shown great potential in several optoelectronic applications, such as light-emitting diodes, phosphors, lasers, and photodetectors.^[] For an optoelectronic device, the bandgap and its temperature dependence, namely, bandgap renormalization, are fundamental attributes of semiconductor materials. Irrespective of perovskite composition, a ubiquitous peculiarity of perovskites is that they exhibit an atypical temperature dependence of the direct bandgap: the bandgap decreases in energy with decreasing temperature, opposite to that exhibited by most of covalent bonded semiconductors, for which the bandgap increases with temperature decrease.^[]

The bandgap renormalization phenomena in perovskites can be explained in terms of their electronic structures. The conduction band minimum (CBM) of perovskites lies in the hybridized antibonding orbitals of the Pb 6p orbitals and the outer p orbitals of halide (5p for I, 4p for Br, and 3p for Cl), which tend to be p-like because of the high-density states from the Pb contribution. On the other hand, the valence band maximum (VBM) lies in the hybridized antibonding states of the Pb 6s orbitals and the same halide p-orbitals as that for the CBM.^[] Due to this special electronic structure, the origin of temperature dependence of bandgap in perovskites can be ascribed to the thermal expansion and electron–phonon interactions in the corner-sharing [PbX₆]⁴⁻ (X = halide anion) octahedra. It is worth noting that, similar to the lead-halide perovskites, PbS or PbSe semiconductor materials also exhibit a peculiar redshift of bandgap with decreasing temperature. Zunger et al. showed that this anomaly in PbS resulted from the occurrence of the filled Pb s-band below the top of the valence band, setting up coupling and level repulsion at the L point in the Brillouin zone.^[] From this aspect, the opposite temperature dependence of bandgap in perovskites as compared to that of most covalently bonded semiconductors such as CdSe can be rationalized from the antibonding properties of the 6s² lone pair of Pb coupled to halide p orbital, as an essential factor determining the electron–phonon interactions in the corner-sharing [PbX₆]⁴⁻ framework.

The profound understanding of the bandgap renormalization in perovskites is highly necessary to further strengthen their potential for diverse applications.^[] By far, there are a significant number of review articles covering the synthesis, optoelectronic properties, and related applications of perovskites. However, a comprehensive review with a specific focus on the molecular origin of the atypical temperature dependence in perovskites is still lacking. Rather than being exhaustive, this review aims to summarize the recent advances in explaining the unusual bandgap renormalization behaviors in perovskites. We start by the discussion of how to extract the bandgap energy value from the measured optical absorbance and photoluminescence (PL) spectroscopy. We then highlight the dominant contribution of electron–phonon interactions to the bandgap renormalization, and discuss the prevailing theoretical models concerning the electron–phonon interactions. We also survey the recent researches reporting the tunable bandgap renormalization behavior obtained through the structural engineering in perovskites. Finally, we envision the further research directions in the topic of bandgap renormalization in perovskites.

Extraction of Bandgap Energy Value

The temperature dependence of bandgap is typically estimated from the measurements of optical absorbance and PL spectroscopy of lead-halide perovskites.^[] For single/polycrystalline perovskites, it is convenient to directly extract the bandgap energy value from a Tauc plot of absorption spectrum (Figure ).^[] Alternatively, for the direct bandgap perovskites, the PL spectrum, arising from the radiative recombination of band-edge exciton, can be fitted using a Gaussian (Figure ), Lorentzian, or Gaussian–Lorentzian cross-product function to determine the PL peak energy position.^[] Although one cannot acquire the absolute value of bandgap from the line-shape fit of PL spectrum, the shift of the PL peak energy with temperature is, to a large extent, equivalent to the shift of bandgap.^[]

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Occasionally, especially in the case of very small or thin nanostructured perovskites, the estimation of bandgap information through the line-shape fit of PL spectrum using the Gaussian or Lorentzian function becomes ambiguous because the PL spectrum presents an asymmetric line-shape with a long low-energy PL tail. The tail is rationally ascribed to the radiative recombination of trap states (Figure ) and the inhomogeneity in size distribution of the perovskites nanomaterials.^[] Under such circumstances, for the purpose of more accurate estimation of the bandgap shift with temperature, researchers adopted the strategy of fitting the absorption coefficient near the band-edge using the Elliot model to extract the bandgap energy value.^[] For instance, the absorption coefficient near the band-edge of 2-monolayer-thick (2-ML-thick) CsPbBr₃ nanoplatelets (NPLs) was modeled by Chen et al. based on Elliot's theory of Wannier excitons to identify the dominant 1s excitonic peak and the lowest band of the continuum transition, respectively (Figures ). As such, they were able to extract the continuum absorption onset energy (i.e., the bandgap energy), and also the position of 1s excitonic transition E_1s, at different temperature points.^[]

Microscopic Origin of Bandgap Renormalization

Over the last few years, the empirical Varshni model (E_g(T) = E₀−αT²/(β + T), where E₀ is the value of bandgap at 0 K, α is a constant, and β is related to the Debye temperature) and the semiempirical Bose–Einstein oscillator model (vide infra) have been widely used to describe the observed bandgap shift with temperature in lead-halide perovskites.^[] However, the empirical or semiempirical models are unable to fully disclose the physical origin of the bandgap renormalization in perovskites. Under the premise that no phase transition occurs in the temperature range studied, the derivative of bandgap E_g over temperature contains two terms: one accounts for thermal expansion (TE) effects and the other corresponds to the renormalization directly caused by electron–phonon (EP) interactions, as explicitly expressed in Equation ().^[]1$\frac{\mathrm{d}{E}_{\mathrm{g}}}{\mathrm{d}T}={\left(\frac{\partial E_{\mathrm{g}}}{\partial T}\right)}_{\text{TE}}+{\left(\frac{\partial E_{\mathrm{g}}}{\partial T}\right)}_{\text{EP}}$An expansion of the unit cell volume increases both the ionization potential and electron affinity, thus shifting the energies of VBM and CBM relative to the vacuum level.^[] The thermal expansion contribution can be estimated by the product of the volumetric expansion coefficient α_ν, the bulk modulus B₀, and the pressure coefficient of bandgap dE_g/dP^[]2${\left(\frac{\partial E_{\mathrm{g}}}{\partial T}\right)}_{\text{TE}}=\frac{\mathrm{d}{E}_{\mathrm{g}}}{\mathrm{d}V}\cdot \frac{\mathrm{d}V}{\mathrm{d}T}=-\frac{\mathrm{d}V}{\mathrm{d}T}\cdot \left|\frac{\mathrm{d}P}{\mathrm{d}V}\right|\cdot \frac{\mathrm{d}{E}_{\mathrm{g}}}{\mathrm{d}P}=-{\alpha }_{\mathrm{v}}\cdot B_0\cdot \frac{\mathrm{d}{E}_{\mathrm{g}}}{\mathrm{d}P}$To suppress the electron–phonon interactions, it is necessary to experimentally determine the pressure coefficient of bandgap dE_g/dP by freezing the lattice at T = 0 K. Although the lattice can never be completely frozen even at 0 K because of the zero-point vibration energy,^[] this approximate calculation method provides an experimental approach to investigate solely the bandgap change with increasing or decreasing lattice constant. Recently, Mannino et al. used literature values of dE_g/dP to estimate the variation of bandgap due to the thermal expansion for CsPbBr₃ single crystals. Their calculation result showed that ${(\frac{\partial E_{\mathrm{g}}}{\partial T})}_{\text{TE}}$ was at the order of 2.21 × 10⁻⁴eV K⁻¹.^[] Similarly, Francisco-Lópe et al. used the value of dE_g/dP measured at T = 300 K to estimate the contribution of thermal expansion to the bandgap shift in MAPbI₃ (MA = CH₃NH₃) single crystals, and claimed that thermal expansion should be treated on equal footing with the electron–phonon interactions for the correct interpretation of temperature effects on their electronic structure. Nevertheless, the value of pressure coefficient of bandgap dE_g/dP used in their calculation was not experimentally measured at T = 0 K. Accordingly, the thermal expansion contribution may be overestimated because the measured value of dE_g/dP inevitably incorporates the contribution of electron–photon interactions due to the dependence of the harmonic frequencies of lattice on structural changes in the high-pressure experiment.

Instead of the high-pressure experiment, thermal expansion effect can be accounted for by calculating the band structure of the system as a function of the equilibrium lattice at different temperatures.^[] On the basis of the density functional theory calculations using the temperature- dependent lattice parameters of the tetragonal phase of MAPbI₃ single crystals, Saidi et al. found that the contribution from lattice expansion to the bandgap shift was an order of magnitude smaller than that from electron–phonon interactions.^[] Thereafter, motivated by Saidi's study, researchers preferred to ascribe the bandgap shift of lead-halide perovskites solely to the electron–phonon interactions of corner-sharing [PbX₆]⁴⁻ octahedra, neglecting the contribution from thermal expansion.

Differing from the case of thermal expansion, the estimation of contribution from electron–phonon interactions is extremely complicated. It needs a detailed analysis of the electron–phonon interactions. In 1951, Fan presented a theory of the temperature dependence of electronic band structures based on the self-energy of an electron worked out to second order in the electron–phonon interactions, which only evaluated the intraband self-energy terms, assuming the interband terms as unimportant.^[] On the other hand, Antoncik et al.^[] and later authors (e.g., Keffer)^[] pointed out that there should be a Debye–Waller correction to the lattice potential. Antoncik's paper expressed a feeling that his theory (“Debye–Waller”) and Fan's (“self-energy”) were in some sense equivalent.^[] Afterward, it had been increasingly recognized by the semiconductor research community that Debye–Waller and self-energy corrections were in fact both needed, and the two effects were intimately related (Baumann^[] and Schluter^[]).

As a consequence, in 1976, Allen et al. developed a general second-order adiabatic theory which rigorously gave both the self-energy term and the Debye–Waller correction.^[] For notational convenience, the authors assumed a primitive lattice with identical atoms of mass M having small thermal displacements ${\mathit{u}}_{\mathit{l}}$ about their average position l, and assumed that the total lattice potential could be written as a sum of potentials from single atoms. The electron–atom interaction $\mathit{V}(\mathit{r}-\mathit{l}-{\mathit{u}}_{\mathit{l}})$ is Taylor-expanded about the positions ${\mathit{u}}_{\mathit{l}}=0$. The neglect of displacement ${\mathit{u}}_{\mathit{l}}$ gives a Hamiltonian H₀ which is solved for the one-electron states |nk> and band energies ε_nk, where n and k are band index and wave vector, respectively (note that the spin index is suppressed). Moreover, as a reasonable approximation, they only considered the effects of the two leading correction terms in powers of ${\mathit{u}}_{\mathit{l}}$ as follows^[]3$H_1=\sum _{\mathit{l}}{\mathit{u}}_{\mathit{l}}\cdot {\nabla }_{\mathit{l}}\mathit{V}(\mathit{r}-\mathit{l})$4$H_2=\frac{1}{2}\sum _{\mathit{l}}{\mathit{u}}_{\mathit{l}}{\mathit{u}}_{\mathit{l}}:{\nabla }_{\mathit{l}}{\nabla }_{\mathit{l}}\mathit{V}(\mathit{r}-\mathit{l})$In principle, the factors ${\mathit{u}}_{\mathit{l}}$ are operators whose time dependence can be Fourier analyzed in terms of exp(iω_qt), where ω_q is phonon frequency. However, to order (m/M)^1/2 (electron to atom mass ratio) the adiabatic approximation, they rationally neglected the time dependence of ${\mathit{u}}_{\mathit{l}}$ and treated ${\mathit{u}}_{\mathit{l}}$ as a classical parameter. This treatment renders the calculation of the perturbed electron energy E_nk({${\mathit{u}}_{\mathit{l}}$}) of the electron (nk) in the presence of a configuration {${\mathit{u}}_{\mathit{l}}$} of static lattice displacements. To second order in ${\mathit{u}}_{\mathit{l}}$, the result reads5$E_{n\mathit{k}}(\{{\mathit{u}}_{\mathit{l}}\})={\epsilon }_{n\mathit{k}}+⟨n\mathit{k}|(H_1+H_2)|n\mathit{k}⟩+\sum _{n^{\prime }{\mathit{k}}^{\prime }}\frac{{|⟨n^{\prime }{\mathit{k}}^{\prime }|H_1|n\mathit{k}⟩|}^2}{{\epsilon }_{n\mathit{k}}-{\epsilon }_{n^{\prime }{\mathit{k}}^{\prime }}}$Then, if defining the temperature-dependent band energy E_nk(T) as the energy E_nk({${\mathit{u}}_{\mathit{l}}$}) thermally averaged over the ensemble of displacement ${\mathit{u}}_{\mathit{l}}$, it is written as^[]6$E_{n\mathit{k}}(T)={\epsilon }_{n\mathit{k}}+\frac{1}{2}\sum _{\mathit{l}}⟨n\mathit{k}|{\nabla }_{\mathit{l}}{\nabla }_{\mathit{l}}V|n\mathit{k}⟩:\overline{{\mathit{u}}_{\mathit{l}}{\mathit{u}}_{\mathit{l}}}+\sum _{{\mathit{}}^{{\mathit{ll}}^{\prime }}}\sum _{n^{\prime }{\mathit{k}}^{\prime }}\frac{⟨n\mathit{k}|{\nabla }_{\mathit{l}}\mathit{V}|n^{\prime }{\mathit{k}}^{\prime }⟩⟨n^{\prime }{\mathit{k}}^{\prime }|{\nabla }_{{\mathit{l}}^{\prime }}V|n\mathit{k}⟩}{{\epsilon }_{n\mathit{k}}-{\epsilon }_{n^{\prime }{\mathit{k}}^{\prime }}}:\overline{{\mathit{u}}_{\mathit{l}}{\mathit{u}}_{{\mathrm{l}}^{\prime }}}$where the bar above denotes thermal averaging and the first-order correction from H₁ is dropped because ${\mathit{u}}_{\mathit{l}}$ vanishes in the harmonic approximation.

Note that the abovementioned perturbation calculation (i.e., the Allen–Heine–Cardona [AHC] theory) gives rise to both the Debye–Waller (second term on the right side of Equation ()) and self-energy (third term) corrections, whose Feynman diagrams are shown in Figure .^[] The Debye–Waller term represents the simultaneous interactions of an electron with two phonons in the branch j having the opposite wave vectors q and –q. This two-phonon interaction process is calculated using the first-order perturbation theory. The self-energy term describes the virtual phonon emission and absorption processes as if electron–phonon interaction is taken twice in second-order perturbation theory.^[]

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For the calculation of Debye–Waller and self-energy terms of electron–phonon interactions, it is strictly necessary to count with a satisfactory description of the electronic states as well as the phonon spectrum of crystal lattice. In the former case, the empirical pseudopotential method has demonstrated to be a powerful tool for the calculation of the temperature dependence of direct bandgaps in covalent semiconductors based on the AHC theory.^[] Specifically, within the pseudopotential approximation, the electronic states corresponding to the eigenvectors Ψ_nk with band index n and wave vector k can be obtained from the following secular equations:^[]7a$\sum _{{\mathit{G}}^{\prime }}\left[\right[\frac{{\hslash }^2}{2m}{(\mathit{k}+\mathit{G})}^2-{\epsilon }_{n\mathit{k}}]{\delta }_{\mathbf{G}{\mathbf{G}}^{\prime }}+\sum _{\kappa }{\mathit{V}}_{\kappa }(\mathit{G}-{\mathit{G}}^{\prime })S_{\kappa }(\mathit{G}-{\mathit{G}}^{\prime })]C_{n\mathit{k}}({\mathit{G}}^{\prime })=0$7b${\Psi }_{n\mathit{k}}=\frac{1}{{\Omega }_{{}_c}^{{}^{1/2}}}\sum _{\mathit{G}}{C}_{n\mathit{k}}(\mathit{G})e^{i(\mathit{k}+\mathit{G})\cdot \mathit{r}}$7c$\sum _{\mathit{G}}{|C_{n\mathit{k}}(\mathit{G})|}^2=1$where G, G′ are reciprocal-lattice vectors, C_kn(G) are the properly normalized Fourier coefficients of Ψ_nk, Ω_c is the volume of the unit cell, and V_κ(G) and S_κ(G) are the local pseudopotential form factor and the structure factor of the atom κ in the unit cell, respectively.

In addition to the electronic states calculation based on empirical pseudopotential method, Cardona et al. employed the rigid-ion model to calculate the phonon spectrum of semiconductors to obtain an expression for the electron–phonon renormalization of the electronic band energies as a function of temperature. All phonon modes of the branch j with wave vector q and frequency ω_jq contribute to the renormalization of electronic band energies, which then reads as8$\Delta E_{n\mathit{k}}(T)=\sum _{j\mathit{q}}\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}\left(n_{j\mathit{q}}(T)+\frac{1}{2}\right)=\sum _{j\mathit{q}\kappa }\frac{A(n,\mathit{k},j,\mathit{q},\kappa )}{{\omega }_{j\mathit{q}}\cdot M_{\kappa }}\left(n_{j\mathit{q}}(T)+\frac{1}{2}\right)$where $n_{j\mathit{q}}={(e^{\beta \hslash {\omega }_{j\mathit{q}}}-1)}^{-1}$ is the Bose–Einstein phonon occupation factor with β = 1/K_bT. The coefficients $\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}$ in front of the bracket depict the dependence of electron–phonon matrix element on the frequency ω_jq of phonon mode involved and the mass M_κ of vibrating atom (an average mass for modes with more than one atom vibrating). In essence, these coefficients are the sum of two contributions: the Debye–Waller correction and the self-energy term. Alternatively, in order to obtain numerical estimates of the temperature dependence of the electronic states, Equation () can be rewritten by transforming the summation into an integral over the phonon frequencies^[]9$\Delta E_{n\mathit{k}}(T)=\underset{0}{\overset{\infty }{\int }}\mathrm{d}\omega \cdot g^{2}F(n,\mathit{k},\omega )\cdot \left(n_{j\mathit{q}}(T)+\frac{1}{2}\right)$where10$g^{2}F(n,\mathit{k},\omega )=\sum _{j\mathit{q}}\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}\delta (\omega -{\omega }_{j\mathit{q}})$

The function g²F(n, k, ω) is a temperature-independent electron–phonon spectral function. It corresponds to the density of phonon states weighted by appropriate electron–phonon matrix elements.

Theoretically, the coefficients $\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}$ can be calculated, provided a good pseudopotential description of the electron bands and an available lattice-dynamical model. However, in practice, it remains computationally not trivial to calculate the coefficients $\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}$ as this requires summation over all phonon modes with full pseudopotential description of electronic bands of a semiconductor.^[] The complexity in calculation imposes onerous restrictions on the precise prediction or analysis of the temperature-dependent bandgap variation due to electron–phonon interactions. For example, it was evidenced that the 14-parameter shell model^[] described the phonon dispersion and eigenvectors of CuCl quite well, while a satisfactory calculation of the CuCl band structure within a local-empirical pseudopotential framework was not available.^[] As an alternative, Cardona and co-workers constructed a simplified model, namely, the Bose–Einstein oscillator model, to approximate the $\frac{\partial E_{n\mathit{k}}}{\partial n_{j\mathit{q}}}$ coefficients by effective electron–phonon interaction parameters A_i for phonons with average frequency ω_i. The electron–phonon correction to the bandgap then reads11${[\Delta E_g(T)]}_{EP}=\sum _i\frac{A_i}{M_i\cdot \hslash {\omega }_i}\cdot \left(\frac{1}{e^{\hslash {\omega }_i/K_{b}T}-1}+\frac{1}{2}\right)$where A_i represents the relative weight of oscillator; M_i denotes the oscillator atomic mass for the effective phonon energy ћω_i. Equation () can be further approximated to model the electron–phonon interaction behavior for N = 2 phonon modes (acoustic and optical) as12$E_{\mathrm{g}}(T)=E_0+\frac{A_{\text{ac}}}{M_{\text{ac}}{E}_{\text{ac}}}\left(\frac{1}{e^{\hslash {\omega }_{\text{ac}}/K_{\mathrm{b}}T}-1}+\frac{1}{2}\right)+\frac{A_{\text{opt}}}{M_{\text{opt}}{E}_{\text{opt}}}\left(\frac{1}{e^{\hslash {\omega }_{\text{opt}}/K_{\mathrm{b}}T}-1}+\frac{1}{2}\right)$where E₀ is unrenormalized bandgap energy, A_ac and A_opt represent the relative weight of oscillators, ћω_ac and ћω_opt are the acoustic and optical phonon energies, and M_ac and M_opt are the atomic masses of the oscillator.

The Bose–Einstein two-oscillator model is usually used to analyze the experimental data concerning the temperature dependence of bandgap in lead-halide perovskites. For instance, Saran et al. lately demonstrated that the Bose–Einstein two-oscillator model provided an excellent fit to the temperature-dependent PL emission peak positions obtained for CsPbBr₃ and CsPbI₃ nanocrystals (NCs) across the temperature range studied (Figure ).^[] The parameters including the relative contributions of electron coupling with acoustic and optical phonon branches, and the unrenormalized bandgap energy could be derived through fitting the experimental data to the two-oscillator model. It was found that both the acoustic and optical phonons were heavily involved in the NC's electron–phonon interactions and opposed each other, with the former resulting in a redshift while the latter causing the blueshift of the bandgap energy with decreasing temperature (Figure ).^[]

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For the lead-halide perovskites, it is allowed to use a two-oscillator model to roughly account for the effect of electron–phonon interactions to the temperature dependence of bandgap.^[] This is justified by the following characteristics of perovskites: 1) the acoustic mode vibrations are dominated by the displacement of heavy Pb atoms with phonon energy of ω_ac (purely Pb-like phonons), while optical mode vibrations are dominated by displacement of light halogen atoms with phonon energy of ω_opt (purely X-like phonons); 2) the CBM and VBM of CsPbX₃ are formed by states whose characters are dominated by the p-band of Pb, the s-band of Pb, and p-band of halide ion in the [PbX₆]⁴⁻ octahedra, respectively; 3) the phonon modes of corner-sharing [PbX₆]⁴⁻ octahedra dominate the electron–phonon interactions. Nevertheless, multivalue solutions may sometimes appear in the fitting of the temperature-dependent bandgap by the Bose–Einstein two-oscillator model, which will produce unphysical values of the mean phonon frequency and/or electron–phonon coupling constant.^[]

To theoretically calculate the contribution of electron–phonon interactions to the bandgap shift in semiconductors, the AHC theory, the afterward theoretical framework developed by Cardona et al. using the empirical pseudopotential method, and the recent ab initio techniques are commonly used in the community.^[] It is noteworthy that the AHC theory is based on the quadratic approximation, namely, truncating at second order in atom displacement ${\mathit{u}}_{\mathit{l}}$, as depicted in Equation (). Particularly, for systems where the low-order expansion of Equation () is insufficient to depict the temperature dependence of electronic levels, researchers can turn to evaluate the band energy E_nk(T) directly using a Monte Carlo (MC) approach as $E_{n\mathit{k}}(T)=\frac{1}{N}\sum _i^N{E}_{n\mathit{k}}({\mathit{u}}_{\mathit{l}})$, where N is the number of the sampling points, and the atomic configurations (${\mathit{u}}_{\mathit{l}}$) are randomly drawn from the vibrational density of states.^[] The MC approach has an associated statistical uncertainty that can be reduced by using a large number of sampling points. Although the MC approach allows one to include all higher-order terms neglected in the AHC theory, this is at the expense of a larger computational cost especially to control finite size effects.^[]

Recently, Saidi, Poncé, and Monserrat found that the commonly applied AHC theory significantly overestimated the bandgap changes of cubic phase of MAPbI₃ (Figure ). They also showed that the failure of the AHC theory was not due to the rigid-ion approximation, neglect of spin–orbit coupling (SOC), or due to intrinsic errors in the exchange–correlation functional, but was due mainly to the approximation to second order in atomic displacement. Furthermore, their study concluded that an excellent agreement with experiment for bandgap renormalization was only obtained when including all high-order terms (MC approach) in the electron–phonon interactions and SOC (Figure ). Consistently, using the MC scheme in conjunction with Perdew–Burke–Ernzerhof (PBE) functional and PBE + SOC by averaging over 20–60 different configurations sampled according to the vibrational density of states, Saidi and Kachmar calculated the theoretical values of bandgap shift with temperature for the orthorhombic and tetragonal phases of MAPbI₃, the results of which showed an excellent agreement with the experimental data (Figure ).^[] Moreover, their calculation results were justified by the long ab initio molecular dynamics simulations carried out for 30 ps to monitor the bandgap of MAPbI₃.

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Modulation of Bandgap Renormalization Behavior

For perovskite MAPbI₃, it has been well established that the antibonding properties of the 6s² lone pair of Pb coupled to I (5p) orbital and the Pb (6p)–I (5p) directly determine the electron–phonon interactions in the corner-sharing [PbI₆]⁴⁻ framework of MAPbI₃.^[] These are general rules for both organic–inorganic hybrid and all-inorganic APbX₃, as the bandgap in all cases is defined by antibonding Pb and halogen orbitals, whereas states that derive from the A-cations are well removed in energy from the bandgap region.^[] Besides the orbital characters of Pb and halogen, the crystal structure of lead-halide perovskites, namely, the spatial arrangements of inorganic or organic cations and octahedral units, also act as an important factor to influence the bandgap renormalization deriving from electron–phonon interactions in [PbX₆]⁴⁻. In other words, the bandgap renormalization behavior can be tuned through the structural engineering in lead-halide perovskites.

For example, Alberti et al. showed that, in APbBr₃ (A = MA⁺, FA⁺, and Cs⁺) perovskites, the bandgap renormalization behaviors were varied along with the two structural phase transitions, i.e., orthorhombic–tetragonal and tetragonal–cubic transformations. In particular, they observed a discontinuous variation of bandgap at the orthorhombic–tetragonal phase transition temperature point of CsPbBr₃, because the octahedral tilting of [PbBr₆]⁴⁻ in orthorhombic phase shifted the VBM away from the vacuum level toward more negative energies, and the CBM toward more positive energies (Figure ).^[] It was obvious that the magnitude of bandgap renormalization was altered as the A-cation changed from Cs⁺ to MA⁺ and to FA⁺ (Figure ). This is understandable because the electron–phonon interactions in [PbBr₆]⁴⁻ is correlated with the phonon–phonon interactions between the organic A-cation and the inorganic octahedra. Besides the tunability of temperature dependence of bandgap through the engineering of A-cation, the bandgap renormalization can be varied with the anion type of lead-halide octahedra. For instance, Saran's investigation on the steady-state PL emission spectra of CsPbX₃ NCs (X = Cl, Br, I) over a range of temperatures from 300 to 5 K revealed that CsPbCl₃ NCs displayed an initial blueshift followed by a redshift after undergoing a structural phase change in the crystal at ≈175–200 K, while the CsPbBr₃ and CsPbI₃ remained the redshift trend of their bandgap with decreasing temperature.^[] Similarly, Saxena et al. found that, apart from those discontinuous temperature points corresponding to the crystal phase transitions, MAPbX₃ (X = I, Br) showed a redshift of bandgap with decreasing temperature in the range of 300–175 K, while MAPbCl₃ exhibited an opposite blueshift. They quantified the contribution of electron–phonon interactions to the bandgap renormalization by using Fröhlich's theory of large polarons and showed that the effect of electron–phonon interactions on bandgap shift in MAPbCl₃ was almost double as compared to MAPbBr₃, thus explaining the opposite blueshift of bandgap for the former.^[]

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It should be mentioned that previous studies in the literature revealed that the temperature dependence of bandgap in low-dimensional nanomaterials of lead-halide perovskites was closely related to the sizes of nanomaterials. To exemplify this, Chen et al. summarized the recently reported representative examples of CsPbBr₃ perovskites exhibiting distinct variation trends of spontaneous emission (or lasing, reflectance) spectra, as shown in Table .^[] As a typical example, the PL spectra of ultrasmall CsPbBr₃ QDs with an average radii of ≈1.3 nm were reported to present a monotonous blueshift trend with decreasing temperature in the range of 225–19 K, contrary to the redshift usually observed in bulk-like CsPbBr₃ NCs. Also, Ai et al. studied the PL properties of CsPbBr₃ QDs embedded in glasses, and they found that the CsPbBr₃ QDs with average radii of ≈4.8 nm showed a redshift trend of PL peak with decreasing temperature (240–40 K), while the CsPbBr₃ QDs with average radii of ≈3.3 nm exhibited an initial blueshift followed by redshift trend (Figure ).^[] Analogously, Chen et al. reported that the bandgap of 2-ML-thick CsPbBr₃ NPLs exhibited an initial blueshift and then redshift trend with temperature decrease (Figure ). Theoretical analyses based on the Bose–Einstein two-oscillator model uncovered that the more enhanced weight of contribution of electron–optical phonon interaction in CsPbBr₃ 2-ML NPLs than in bulk-like CsPbBr₃ NCs was responsible for the blueshift–redshift crossover of bandgap in the NPLs (Figure ). They explained that, owing to the breaking translational periodicity in the thickness direction of CsPbBr₃ 2-ML NPLs, the electron and phonon structures, and consequently the bandgap renormalization deriving from electron–phonon interactions were apt to change remarkably relative to the CsPbBr₃ NCs counterparts.^[] The correlation of bandgap renormalization with the nanosizes of perovskites was also reported for CsPbI₃,^[] MAPbBr₃,^[] and MAPbI₃.^[] For example, the temperature-dependent PL measurements showed the spectral blueshift of the PL peak for the small MAPbBr₃ NCs (3.1 ± 0.2 nm) with decreasing temperature from 300 to 20 K, contrasting the redshift observed for the large- (9.2 ± 0.5 nm) and middle-sized (5.1 ± 0.3 nm) NCs.^[]

Table 1 Summary of the recently reported variation trends of spontaneous emission (or lasing, reflectance) spectra in CsPbBr₃ perovskites with dimensions ranging from 3D bulk crystals, quasi-3D bulk-like NCs to 2D NPLs, 1D nanowires (NWs), and to 0D QDs. Reproduced with permission.^[] Copyright 2021, Wiley-VCH

Dimension	Size	Variation trend with decreasing temperature
bulk crystals	–	Redshift (300–10 K)
NCs	15 nm (edge length)	Redshift (300–3 K) (reflectance spectrum)
NCs	11 nm (edge length)	Roughly constant (380–220 K)-redshift (220–80 K)
QDs	1.3 nm (radius)	Blueshift (225–19 K)
QDs embedded in phosphate glasses	3.3 nm	Blueshift (240–140 K)-redshift (140–40 K)
4.2 nm	Blueshift (240–180 K)-redshift (180–40 K)
4.8 nm (radius)	Redshift (240–40 K)
NWs	12 nm × 5 μm (diameter × length)	Redshift (295–5.8 K)
NWs	372 nm × 22 μm (width × length)	Redshift (295–78 K) (spontaneous emission peak)
Blueshift (295–195 K)–redshift (195–78 K) (lasing peak)
Microplatelets	150 nm × 7–15 μm (thickness × lateral size)	Redshift (300–10 K)
NPLs	3.4 nm × 18.2 nm (thickness × lateral size)	Redshift (300–16 K)

Similar to the cases of APbX₃-type 3D perovskites, the modulation of bandgap renormalization behavior was recently reported in 2D layered perovskites (such as 2D Ruddlesden–Popper perovskites), featuring metal halide slabs separated by the organic layers with a dielectric constant that is smaller than that of the inorganic layer. As a typical example, in the 2D layered (n-BA)₂(MA)_n − 1Pb_nI_3n + 1 (BA = C₄H₉NH₃, and MA = CH₃NH₃) microplates, Li et al. found that the emission peak exhibited first a blueshift and then redshift for n = 2 or 3 samples with a decrease in temperature from 290 to 77 K, and that for the n = 1 sample even presented a monotonous blueshift trend. However, a monotonic blueshift was observed for n = 4 and 5 samples, which is consistent with the emission peak shift in the case of the 3D perovskites (Figure ).^[] It should be noted that, in the 2D layered perovskites, the inorganic octahedral slabs are usually sandwiched by two insulating ligand layers to form multiple quantum wells. In other words, due to the weak electronic coupling between the octahedral slabs, the 2D layered (n-BA)₂(MA)_n − 1Pb_nI_3n + 1 perovskites with n = 2 can be viewed as the superlattice self-assembly of (MA)Pb₂I₇ perovskite NPLs with two monolayers of [PbI₆]⁴⁻ octahedra and with BA cations as the surface chelating ligands. Consequently, it is not surprising that the bandgap renormalization in the (n-BA)₂(MA)_n − 1Pb_nI_3n + 1 perovskites with n = 2 resembled that in the 2-ML-thick CsPbBr₃ NPLs, both exhibiting an initial blueshift followed by redshift trend with decreasing temperature (Figure , ). Another feature of the 2D layered perovskites is their phonon–phonon interactions between the octahedral slabs and the organic ligand layers. Thus, the bandgap renormalization in 2D layered perovskites is tunable through the engineering of organic ligand layers. As observed in the 2D layered (iso-BA)₂(MA)_n − 1Pb_nI_3n + 1 microplates, the emission peak exhibited a monotonous redshift trend for the n = 1 sample,^[] which is opposite to the monotonous blueshift trend in the (n-BA)₂(MA)_n − 1Pb_nI_3n + 1 microplates with n = 1. In a word, all the factors that influence electron–phonon interaction in [PbX₆]⁴⁻ octahedra, including the 2D arrangement of corner-sharing [PbX₆]⁴⁻, the strong quantum confinement effect, the phonon–phonon interactions between the octahedral slabs and the organic ligand layers, and the reduced dielectric screening due to the low dielectric constant of organic ligands, account for the bandgap renormalization behavior in the 2D layered perovskites.

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Conclusion and Perspectives

Lead-halide perovskites are the rising star in the field of new materials for optoelectronics. Researchers adopted lately the viewpoint of interpreting the abnormal bandgap renormalization in perovskites as due to the electron–phonon interactions in [PbX₆]⁴⁻ octahedra. Recent results from first-principle calculations also indicated that lattice thermal expansion had negligible effects on the bandgap renormalization in comparison to the electron–phonon interactions. However, there are still a lot of unanswered questions and challenges to be solved, and a fundamental understanding of the microscopic origin of the bandgap renormalization in perovskites is still evolving.

First, the temperature dependence of bandgap in nanostructured perovskites remains elusive because factors including the defects distribution, surface lattice reconstruction, surface electronic states and phonon modes, and quantum and dielectric confinement add complexity to the calculation or prediction of electron–phonon interactions. To clarify this puzzle, more sophisticated theoretical models and material characterization techniques are required in the future. Moreover, the structural engineering strategies in perovskites, including the phase transition (i.e., the tilting of [PbX₆]⁴⁻ octahedra), the reduction of materials dimensionality from quasi-3D CsPbBr₃ NCs to 2D NPLs or 0D QDs, and the tuning of n value in the 2D layered perovskites (n-BA)₂(MA)_n − 1Pb_nI_3n + 1, have been reported lately as potential methods to modulate the bandgap renormalization behavior. However, some other regulation methods such as ion doping, A-site cation substitution, ligand engineering, surface passivation, etc. have not been widely explored. Last but not the least, given that the practical optoelectronic devices may be working in a complex environment, it will be necessary to investigate the bandgap renormalization in perovskites in the presence of external fields such as electric and magnetic fields, ultrasound field.

Acknowledgements

This work was supported by the NSFC (grant nos. 22135008, U1805252, 12074380, and 12104455), the Key Research Program of the Chinese Academy of Sciences (grant no. ZDRW-CN-2021-3), the CAS/SAFEA International Partnership Program for Creative Research Teams, Youth Innovation Promotion Association of CAS (grant no. 2022306), NSF of Fujian Province (grant nos. 2021J01524 and 2022J05092), and the China Postdoctoral Science Foundation (grant no. 2021M703220).

Conflict of Interest

The authors declare no conflict of interest.

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