Influence of intrinsic defects on the structure and dynamics of the mixed Pb–Sn perovskite: first-principles DFT and NAMD simulations

Abstract

The mixed tin (Sn) and lead (Pb) perovskite compositions have shown great potential in perovskite photovoltaic devices due to the significantly enhanced material stability and prolonged carrier lifetime, compared to the pure Sn halide perovskites. In spite of the increasing interest, the behaviors of photo-generated charges and of the intrinsic point defects, such as the metal cation vacancies (V_Sn and V_Pb) and the interstitial halogen (i_I), have not been well understood in this class of materials. We report first-principles density functional theory (DFT) calculations combined with ab initio non-adiabatic molecular dynamics (NAMD) simulations on the static and dynamic structures of MA₂SnPbI₆ with and without these intrinsic defects. We discuss the nature of the defect states and unveil the influence of the intrinsic point defects on the structure, optoelectronic properties, and charge carrier dynamics of MA₂SnPbI₆. The i_I defect significantly shortens the carrier lifetime by creating mid-gap states that provide new recombination pathways. In comparison, the vacancy defects have much weaker influence on the carrier lifetime. Both V_Sn and V_Pb produce the defect states just below the valence band maxima (VBMs), and do not alter the band gap. They affect the carrier lifetime through changing the energy dispersions of VBMs and the conduction band minima (CBMs). We suggest that excess cations should be used in the synthesis of perovskites to avoid the appearance of interstitial halogen defects.

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1 Introduction

Perovskite solar cells (PSCs) have been intensively studied in the past decade because of their high power-conversion-efficiency (PCE), low cost and easy synthesis.^1–7 The first PSC was designed by Miyasaka et al. with a PCE of only 3.8%.³ Subject to the rapid advances over the past ten years, the certified PCE increased quickly to 29.5%.⁸ These developments have demonstrated the great potential of PSCs as the next-generation low-cost solar technology. Unfortunately, the most efficient PSCs all contain Pb. The Pb-induced pollution and poisoning are serious environmental concerns, which have hampered the commercialization of the technologies based on these materials. Developing alternative strategies to minimize the impact of toxic Pb on the environment under the premise of maintaining the photoelectric activity of perovskite materials has become an important topic in perovskite photovoltaics.^2,7,9–11

To reduce or eliminate the content of Pb, scientists have suggested to develop PSCs containing less Pb and Pb-free PSCs.^12–16 One idea is to develop Pb-free PSCs by replacing Pb with other elements. The earliest suggestion was to replace Pb with Sn and Ge, group 14 elements with comparable ionic radii.^17,18 Later, transition metals such as Cu, Mn, Fe, Co, and Ni were identified as promising alternatives to Pb, owing to their rich chemistries and multiple oxidation states. However, these Pb-free compounds suffer from poor efficiency and stability.

Another idea is to replace two divalent Pb(II) ions with monovalent and trivalent ions. Experimentalists have successfully synthesised double perovskites by using heterovalent substitution with monovalent Cu, Ag, Au or even alkali-metals, and trivalent In, Sb, and Bi cations.^19–32 However, these kinds of Pb-free perovskite materials possess serious flaws in optoelectronic properties, which make them less likely candidates for PSCs. For example, the most easily synthesized Ag–Bi and Ag–Sb double perovskites^19,20,23,33 have an indirect band gap, which will lead to a large amount of solar energy dissipated into heat after the carrier excitation.³⁴

The PCE of Ag–Bi double perovskite-based PSCs is much lower than that of Pb-based, and even Sn-based PSCs. It is about 1.25% in its maximum.³³ In addition, because AgI is highly sensitive to light, this kind of material cannot be filled with iodide cations, resulting in a wide band gap. The bulk phase of the In(I)–Bi(III) based perovskite has been predicted theoretically.³⁵ It has a direct band gap and a carrier effective mass similar to that of the traditional Pb-bearing materials. However, the stability of monovalent In⁺ is lower than that of Ag⁺ and Tl⁺, and it has been reported experimentally that Cs₂In(I)In(III)X₆ perovskites are stable only at high temperature, while low temperature can cause a rotation between In(I) octahedra and In(III) octahedra, which leads to further degradation.³⁶

An increasing number of studies have shown that it is difficult to eliminate Pb from high-performance PSCs. The more stable and better performing photoelectric materials are those still containing Pb, but in reduced amounts.^18,37–46 In such compounds, Pb is partially replaced by other elements, including Sn, Ge, Sr, or other alkaline-earth-metals. At present, the practical scheme is to adopt a mixed Sn–Pb composition, and to decrease the Pb content as much as possible, in order to reduce the environmental toxicity of Pb while maintaining the working stability of the device. For instance, Ramirez et al. adopted a 1 : 1.5 stoichiometric Sn : Pb ratio and reported a relatively high PCE of 10.5%.⁴⁷ Kapil et al. produced a spike structure of the Pb–Sn mixed perovskite and reached a 17.6% PCE.⁴⁸ Lanzetta et al. explained the surface-oxidation mechanism of Pb–Sn based perovskites.⁴⁹ Long et al.⁵⁰ discovered that the presence of Sn enhances the carrier lifetime of the traditional Pb-bearing perovskites.

When Sn(II) is exposed to air, it oxidizes spontaneously to form Sn(IV). Such a process leads to formation of various intrinsic defects,^42,51,52 which have a severe impact on the optoelectronic properties of the material.^53–63 The most common intrinsic defects generated in Pb–Sn perovskites include the Pb and Sn vacancies (V_Pb and V_Sn), and the interstitial iodide (i_I), which can be formed by the following chemical processes.

MASn(II)I₃ + MAPbI₃ − 2e⁻ − Pb²⁺ = MASn(IV)I₃²⁺ + MA[V_Pb]I₃²⁻;

MASn(II)I₃ + MASn(II)I₃ − 2e⁻ − Sn²⁺ = MASn(IV)I₃²⁺ + MA[V_Sn]I₃²⁻;

MASn(II)I₃ + I⁻ − 2e⁻ = MASn(IV)I₃²⁺ + i_I⁻.

In this work, we study the structure, and the static and dynamic properties of the mixed Pb–Sn perovskite, and elucidate how the structure, and the optoelectronic and photoelectric properties are influenced by the intrinsic defects that are formed by the oxidation of Sn(II). We perform both first-principles density functional theory (DFT) calculations and non-adiabatic molecular dynamics (NAMD) simulations to obtain the fundamental insights, guiding the construction of high performance PSCs. We generate a microscopic understanding of the characteristics of the materials with and without the intrinsic defects, and study the nature of the defect states, which usually play a decisive role in determining carrier recombination and diffusion lengths in solar cells, and explain why the photophysical properties of mixed Sn–Pb perovskites are enhanced considerably compared to those of Sn-based crystals.

2 Computational details

The first-principles DFT simulations were performed with the Vienna Ab Initio Simulation Package (VASP)^64–66 to study the geometric and electronic structures of the pristine and defective MA₂SnPbI₆ perovskite series. Unless otherwise specified, the generalized gradient approximation exchange-correlation functional of Perdew–Burke–Ernzerhof (PBE)⁶⁶ was adopted in the calculations. Since PBE tends to underestimate bandgaps, we employed the HSE06 functional^67,68 for benchmark calculations. HSE06 includes a fraction of screened Hartree–Fock exchange, α = 0.25, improving the discontinuity in the Kohn–Sham potential derivative for integer numbers of electrons.

The spin–orbit coupling (SOC) effect is taken into account in the electronic structure calculations.

The electronic constituents are 4d 5s 5p for Sn, 5d 6s 6p for Pb, 5p 6s for I, 2s 2p for C and N, and 1s for H. For all the bulk-like systems, we adopted the Γ-point only for the HSE06 + SOC calculations, 4 × 4 × 4 Γ-centered k-point grid generated by the Monkhorst–Pack scheme for detailed properties obtained with PBE + SOC, and 12 × 12 × 12 linear scheme for the band-structure calculation with PBE + SOC. The projector augmented wave pseudopotentials with the cutoff energy of 500 eV were employed.

Considering the interaction between the hydrogen atoms and high-electronegativity groups, the PBE with the DFT-D3 dispersion correction of Grimme with zero-damping^69–71 was applied to optimize the geometric structures. During the optimization of the geometries, all structures were allowed to relax to ensure that each atom was in mechanical equilibrium without any residual force larger than 10⁻⁴ eV Å⁻¹.

Once the geometry optimization was completed, the repeated velocity rescaling was performed to equilibrate the structures at 300 K for 5 ps. Then, 8 ps adiabatic molecular dynamics trajectories were obtained within the microcanonical ensemble with the molecular dynamics time step set to 1 fs. To perform the NAMD simulations, we computed the non-adiabatic couplings d_ij using the time-domain numerical method⁷²via the overlap between orbitals i and j at sequential time steps .

The non-adiabatic coupling is proportional to the overlap of the wavefunctions between the initial (i) and final (j) states, −iℏ〈ϕ_i|∇_R|ϕ_j〉, and the nuclear velocity, dR/dt. The final 6000 configurations of the microcanonical molecular dynamics trajectories were selected as initial geometries for the following NAMD simulations of the nonradiative charge relaxation and electron–hole recombination with the Python extension for Ab Initio Dynamics code.^73,74 Since the structures had direct bandgaps located at the Γ-point, the NA couplings were computed for the Γ-point only.

3 Results and discussion

3.1 Geometric structures

At first, we optimize all the possible structures of pristine MA₂SnPbI₆ to determine which kind of topological arrangement of the metal cations gives the highest stability. For this reason, we build a few cubic perovskite-phase MA₂SnPbI₆ structures with different arrangements of Pb and Sn atoms, as shown in Fig. 1. Then, we optimize them and calculate their free energies. The results in Table 1 prove that the structure in Fig. 1(c) with the face-to-face arrangement of Pb and Sn atoms possesses the best thermal stability. This minimum unit cell is able to generate all internal point defects. Thus, this configuration is adopted in the later study at an affordable computational cost.

Fig. 1 Optimized structure of mixed Pb–Sn perovskite MA₂SnPbI₆.

Table 1 Geometric parameters and free energies for the MA₂SnPbI₆ structures without defects

Structures	(a)	(b)	(c)
a/Å	12.25	11.74	11.16
bÅ	12.23	12.25	12.20
c/Å	12.28	12.28	12.21
Free energy/eV	−417.302	−417.303	−417.360

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The influence of V_Pb, V_Sn, and i_I on the structure of MA₂SnPbI₆ is then explored. All the defective structures are relaxed with the lattice parameters fixed at the defect-free values to simulate point defects in a bulk crystal. To describe the influence of the defects on the system's geometry, we show the topological structure parameters of the defect-adjacent atoms in the flawed structures in Fig. 2 and Table 2, which demonstrate that when V_Pb or V_Sn appears, the average Sn–I bond lengths are longer than those without any defect. Thus, the atoms around the vacancies tend to move backwards the defect sites, so that the [SnI₆]⁴⁻ octahedra are strongly distorted to accommodate these inert defects. However, it has weaker effect towards the structure of [PbI₆]⁴⁻ octahedra. In addition, the vacancy defects flatten the metal-halide framework, as manifested in the increase of the average ∠M–X–M angles. For the i_I structure, the atoms around i_I are pulled away from the point defect by the adjacent framework, as reflected by the shorter bond lengths, smaller bond angles and distortion of M–X octahedral coordination.

Fig. 2 Optimized defect-bearing structures of the MA₂SnPbI₆ perovskites: (a) V_Pb, (b) V_Sn and (c) i_I.

Table 2 The main geometric parameters and defect formation energy (DFE) of MA₂SnPbI₆ with and without defects. d̄i denotes the average distance for the vacancy-sharing atoms in the flawed crystal

Structures	Pristine	VSn	VPb	iI
d Sn–I/Å	3.134	3.175	3.099	3.110
d Pb–I/Å	3.185	3.074	3.127	3.145
∠Sn–I–Sn/°	154.2	161.1	156.4	147.2
∠Pb–I–Pb/°	153.5	165.6	167.8	153.2
min(∠M–X–M)°	148.6	146.6	148.2	108.2
d̄ i /Å	—	3.092	3.095	3.157
DFE/(eV per cell)	—	+0.294	+0.128	+0.045

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The formalism of DFE allows calculation of defect and impurity equilibrium structures and concentrations, and the relative stability of the different charge states of a given defect in the crystals.⁷⁵ We also calculated the relative DFEs of those point defects (see Table 2). The DFEs of charged defects are affected by the Fermi energy and the valence band level. The DFEs are expressed as⁷⁶ DFE = E_defected − (E_perfect + ∑ε_i,dopants) + q(E_f + VBM + ΔV_corr) + V_corr, in which, E_f represents the Fermi energy, and ΔV_corr means the VBM potential correction with the bulk flawless phase reference. q denotes the charge state of a defect: for a neutral defect, q = 0, and if the defect structure has a positive charge, q = 1. V_corr is set for the electrostatic interaction correction between charged point defects in the periodically repeated supercells. In this work, the Makov–Payne approach⁷⁷ with the monopole approximation, , is adopted to calculate this term, in which the constant α represents the Madelung factor,^78–80 and ε is the dielectric constant obtained from the density functional perturbation theory.^81,82 The calculated DFEs demonstrate that i_I is generated most easily, followed by V_Pb and V_Sn. When a Sn(II) cation gets oxidized to Sn(IV), vacancy formation in the adjacent Pb(II) atoms is more likely than that in the Sn(II) atoms.

Considering that the intrinsic defects in actual system are often scattered in the lattice, we build a structure expressed as MASn_0.03125Pb_0.96875I₃, which possesses a large unit cell constructed by setting the minimum unit cell of MA₂SnPbI₆ shown in Fig. 1(c) in the center and surrounding it by the high-stability MAPbI₃ (see Fig. S1†). Then we investigated the influence of three kinds of point defects on the topological structure and calculate DFEs. The data shown in Table 2 and S1† tell that the influence of internal defects on MA₂SnPbI₆ and MASn_0.03125Pb_0.96875I₃ is consistent. However, the defect-induced deformation on the MASn_0.03125Pb_0.96875I₃ lattice is slightly weaker than that on the MA₂SnPbI₆ lattice because of the tolerance of a large amount of stable MAPbI₃ in the former. The tolerance effect of the surrounding protective layer on the bond length of the defect-sharing atoms is less than 3%, indicating that the deformation on the defect-sharing structure is not affected by the lattice size. Therefore, to achieve a better balance between the computational cost and accuracy, we adopt the MA₂SnPbI₆ structure for later study.

3.2 Electronic and optoelectronic properties

Before NAMD calculations and carrier recombination analysis, we first make a static energy band analysis of the electronic structure. The band structure and projected density-of-states (pDOSs) of MA₂SnPbI₆ with and without the defects are plotted in Fig. 4 and S3–S5.†

The calculated band gap of MA₂SnPbI₆ is between those of MAPbI₃ (1.57 eV) and MASnI₃ (1.28 eV) by HSE06 + SOC, attributed to the hybridization of inert Pb-6s orbitals (with a minor 6p contribution) and active Sn-5s orbitals (with a minor 5p contribution) in the VBM. The CBM of MA₂SnPbI₆ is also composed of Pb-6p, Sn-5p and I-5p. i_I creates a characteristic deep defect state (DDS) lying in the band gap near the Fermi level, which is 0.43 eV above the VBM, and 0.26 eV below the CBM. This DDS is composed of the I-5p orbital of the interstitial I⁻, and its energy dispersion E(k) is nearly flat.

The characteristic defect states created by V_Pb and V_Sn are located below the VBMs, and they thus do not change the width of the band gap. However, the vacancies create several impurity states under the VBM and change the energy dispersions, indicating that they will play a role in the carrier migration and recombination. The pDOS in Fig. S3† shows that V_Sn weakens the contribution of Sn-5s to the VBM and CB edge states, while V_Pb had no such severe influence. The contribution of Pb-6s orbitals to the pDOS starts at lower energies compared to Sn-5s, and therefore, the influence of V_Pb on the VBM near the Fermi level is weaker than that of V_Sn. The charge distributions of the pristine and flawed MA₂SnPbI₆ are shown in Fig. S4.†

The relative mobility of the charge carriers can be characterized by the carrier effective masses.^83,84 The effective masses are calculated by fitting the energy dispersions of the VBM and the CBM to quadratic functions as , where m_xx, m_yy, and m_zz correspond to the components in the x, y, and z directions, respectively. The results calculated for MA₂SnPbI₆ with and without defects are shown in Fig. 3(b).

Fig. 3 (a) Band gaps calculated by PBE, PBE + SOC and HSE06 + SOC. (b) Effective masses and (c) PCEs calculated by PBE + SOC for the perfect and defective MA₂SnPbI₆. Comparison is made with the pure Pb-based and Sn-based perovskites.

Fig. 4 Band structure of MA₂SnPbI₆ (a) without and (b–d) with intrinsic defects calculated by PBE + SOC. The red lines highlight the bands created by the intrinsic defects.

When Sn atoms are present in the structure, the effective electron mass of the CBM (m̄c) is very close to or even slightly lower than that of the pure Pb-based perovskite (approx. 5%⁸⁵), indicating higher electron carrier mobility. Introduction of V_Pb and i_I can inhibit the electron mobility to a certain extent, and V_Sn has no obvious effect on it. In addition, these intrinsic defects have a significant impact on the hole mass in the VBM (m̄v), especially i_I.

V_Sn and V_Pb insert defect states below the VBM, and increase the energy dispersion of the VBM without changing the band gap magnitude. Since the VBMs arise mainly from the Sn-5s orbitals, V_Sn reduces the VBM dispersion more significantly. In other words, vacancy defects, such as V_Sn and V_Pb, make the VBM more localized.

The band gap is a key property of a light-to-energy conservation device, putting limits on its efficiency. Photons with a lower energy than the band gap cannot be absorbed efficiently, while electron–hole pairs generated above the band gap usually relax to the band edges, so that only part of the absorbed energy can be turned into work. In this sense, the absorption spectrum directly determines the PCE value. Hereafter, we adopt the spectral-limited maximum efficiency method as the PCE descriptor, going beyond the band gap, as suggested by Yu and Zunger^86,87 to calculate the number of photons absorbed by each structure. The efficiency, η, is obtained as . Here, n_tot is the total number of photons emitted by the sun at AM1.5,⁸⁸ ph_abs(E) is the photon absorptivity characteristic of the material, and n_ph(E) represents the number of sun photons at the energy, E, in the unit of eV. In this model, no absorption occurs below the direct bandgap, E^dir_g, since absorption through an indirect transition requires a change in the electron momentum assisted by phonons.

The calculated PCEs of the materials under investigation corresponding to the same optoelectronic device thickness (shown in Fig. S6†) are shown in Fig. 3(c) and S5 and S6.† The calculated results show that the band gap of pristine MA₂SnPbI₆ is closer to the Shockley–Queisser limit value of 1.37 eV (ref. 89) than the MAPbI₃ and MASnI₃ band gaps. Therefore, it exhibits higher PCE. All three point defects decrease the PCE. V_Pb has the weakest impact on the electronic structure and PCE. V_Sn decreases the PCE more than V_Pb. The DDS induced by i_I greatly changes the band structure and has the most serious impact on the PCE.

3.3 Electron–hole recombination and hot electron relaxation

In order to quantitatively describe the influence of the intrinsic defects on the carrier lifetimes, we have performed a series of NAMD simulations on the photexcited carrier relaxation and electron–hole recombination processes for the perfect and defective structures. Considering the computational effort, the NAMD simulations are based on the PBE functional without SOC since the band gaps calculated by PBE without SOC are much closer to those obtained by HSE06 + SOC than the band gaps obtained by PBE + SOC, and NAMD calculations based on HSE06 + SOC are too computationally demanding. The calculated non-adiabatic coupling (NAC) matrices are shown in Fig. S7.†

Fig. 5 shows the pure-dephasing functions calculated using the second order cumulant expansion as . This optical response function characterizes the vibrationally induced pure-dephasing process for a pair of states entangled in a coherent superposition. It can be obtained directly or via the second-order cumulant expansion,⁹⁰ which allows notably better convergence. Pure-dephasing is associated with fluctuations of electronic energy levels due to coupling to phonons. The fluctuations are characterized by the energy gap autocorrelation function (ACF). The un-normalized ACF is defined as C(t) = 〈ΔE(t)ΔE(0)〉. It represents an ensemble average of the correlation of the energy gap fluctuation. A rapid decay of the ACF implies fast pure-dephasing. The ACFs characterize memory of the energy gap fluctuation.

Fig. 5 Calculated pure-dephasing function of MA₂SnPbI₆ with and without defects. The result comparison is made with MAPbI₃.

Generally, shorter pure-dephasing time leads to slower dynamics, as exemplified by the quantum Zeno effect.⁹¹ The pure-dephasing times depend most strongly on the magnitude of the phonon-induced fluctuation of the corresponding electronic energy gap.⁹² Fig. S8(a)† shows the un-normalized ACF of the energy gap fluctuation, and the ACF initial values characterize the fluctuation magnitude. The fluctuation is the largest for the pristine system, resulting in the fastest decay of the pure-dephasing function. The i_I defect exhibits slower pure-dephasing, because the corresponding energy gap and its fluctuation are the smallest.

Fourier transform of the ACF produces the spectral density . As shown in Fig. S8(b),† the calculated spectral densities characterize the phonon modes that couple to the electronic subsystem. Low-frequency modes play a predominant role, corresponding to twisting and stretching motion of the [PbI₆]⁴⁻ and [SnI₆]⁴⁻ octahedra, according to the previous research.^93,94

To determine the time scales τ of the non-radiative electron–hole recombination, we obtain 30 ps of NAMD data, and fit the state populations to the short-time, linear approximation of exponential decay, i.e., P(t) = exp(−t/τ) ≈ 1 − t/τ. Table 3 demonstrates that the carrier lifetime of the MA₂SnPbI₆ mixed perovskite is between those of MAPbI₃ and MASnI₃. Due to the higher VBM energy, pristine MA₂SnPbI₆ possesses a smaller band gap, and therefore, exhibits a shorter carrier lifetime than MAPbI₃. On the other hand, the incorporation of Pb into the Sn-based perovskite increases the carrier lifetime compared with that of the pure Sn-based perovskite. All the defective MA₂SnPbI₆ perovskites exhibit shorter carrier lifetimes than the pristine system. This can be traced to smaller initial values of the unnormalized ACF and longer pure-dephasing times. The carrier lifetimes of the perovskites with the cation vacancies are shorter because the VBM state is mixed with the defect states formed in the VB, Fig. 6(a). The V_Sn defect influences the electronic states near the VBM more strongly than the V_Pb defect, because it is energetically closer to the VBM, and notably hybridizes the VBM and deeper VB state, Fig. S3.† As a result, V_Sn changes the nonradiative charge recombination between the VBM and CBM more significantly than V_Pb. Changes in the spatial distributions of charges near the VBM and the CBM, Fig. S4,† are an important reason for variation of the carrier lifetime in different structures.⁹⁶

Table 3 Calculated time scales of the carrier recombination for MA₂SnPbI₆ with and without defects. Comparison is made with MAPbI₃ and MASnI₃

Structure	SnPb	VPb	VSn	iI	all-Pb	all-Sn^95
τ/ns	54.88	20.11	5.72	0.441	159.6	8.37

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Fig. 6 Electron–hole recombination process from the CBM to the VBM for MA₂SnPbI₆ with and without defects.

The i_I-bearing system exhibits much faster nonradiative carrier recombination than the other systems studied in this work, Fig. 6. The DDS plays a decisive role here. The i_I creates DDS, Fig. 4(a), splitting the band gap into two smaller gaps, 0.43 eV + 0.26 eV by PBE + SOC. Nonradiative transitions across small energy gaps are fast, resulting in quick carrier quenching. 32% of active carriers recombine within the 30 ps NAMD simulation and 14% of them are captured by the interstitial DDS, Fig. 6(b).

To unveil the effect of these point defects on the carrier relaxation process, we simulate the hot electron relaxation dynamics from the low-lying CB edge states to the CBM, as shown in Fig. 7. Considering hot charge carriers at energies away from the band edges can provide advantages in solar energy applications, such as faster charge separation and transport, generation of additional carriers, and reduced trapping,⁹⁷ here we include all the relaxation effects from CBM + 1, CBM + 2 and CBM + 3 to the CBM.

Fig. 7 Hot electron relaxation process from CBM + 1, CBM + 2 and CBM + 3 to the CBM for MA₂SnPbI₆ with and without defects.

The CB edges are mainly composed of Sn-5s 5p and Pb-6s6p orbitals (see Fig. S3†). Therefore, V_Pb and V_Sn significantly weaken the contribution of Pb and Sn to CB edges, leading to the increase of the relaxation time. The calculated carrier relaxation times τ for the V_Pb- and V_Sn-bearing structures are 2.02 and 1.29 ps, respectively. Because the i_I defect creates the mid-gap states and hardly affects the deep CB states, the carrier relaxation time of the i_I-bearing structure (τ = 0.97 ps) is nearly the same as that of the defect-free structure (τ = 1.01 ps).

4 Concluding remarks

We have performed first-principles DFT calculations combined with NAMD simulations on the static and dynamic characteristics of MA₂SnPbI₆ with and without intrinsic point defects, and compared its properties with those of its pure counterparts MASnI₃ and MAPbI₃. Meantime, in order to discuss the stability of defects in the realistic system as accurately as possible, we also calculated the geometric structures and defect formation energies of MASn_0.03125Pb_0.96875I₃ with a large supercell. This study unveils the nature of defect states and reveals how the intrinsic defects in the mixed Pb–Sn perovskite frameworks affect the electronic and geometric structures, the electron-vibrational coupling, the hot electron relaxation times, and the times of the nonradiative electron–hole recombination process at the ambient temperature. The following conclusions can be extracted from this study.

(1) The oxidation of Sn(II) results in the generation of many intrinsic defects such as V_Sn, V_Pb and i_I in the MA₂SnPbI₆ material. i_I can be most easily generated, followed by V_Pb and V_Sn. When one Sn(II) atom is oxidized, its adjacent Pb(II) atoms are prone to forming a defective vacancy, prior to Sn(II) atoms, and once a defect appears nearby, the [SnI₆]⁴⁻ octahedra are always distorted more severely than [PbI₆]⁴⁻ octahedra. It has been noted that Sn–Pb mixed perovskite materials possess an important feature: when Sn(II) is oxidized to Sn(IV), an adjacent Pb-vacancy forms, which will partially tolerate the negative impact of Sn(IV) on the photoelectric properties of Sn–Pb mixed perovskite materials.

(2) V_Pb, V_Sn and i_I affect the crystal structure differently. V_Pb and V_Sn cause elongation and distortion of the MX octahedra around the defects. This is reflected in a displacement of the octagon geometric centers. The M–X bond lengths of the center ion increase to offset somewhat the distortion of the octahedral structure. i_I makes the two adjacent MX octahedra repel each other, manifested in the reduction of the M–X bond lengths. Additionally, the vacancy defects flatten the M–X framework, while i_I distorts it severely.

(3) The band gap width and the carrier lifetime of MA₂SnPbI₆ lie between those of MAPbI₃ and MASnI₃ because the orbital hybridization of Sn and Pb affects the energy level of its VBMs, which are composed of the Sn-5s5p and Pb-6s6p orbitals. MA₂SnPbI₆ exhibits higher PCE than MAPbI₃ and MASnI₃ because its band gap width is much closer to the Shockley–Queisser limit. The tolerance of Pb-vacancy defects by the mixed Pb–Sn perovskite is also reflected in the electronic structure. The CBMs of MA₂SnPbI₆ are predominantly composed of the Sn-5p orbitals so that removing a Pb atom has a nearly negligible impact on its CB-related properties. Therefore, we can conclude that the existence of Pb atoms can inhibit the degradation arising from Sn oxidation in MA₂SnPbI₆. This is of great significance for maintaining the photoelectric activity of Sn-bearing perovskites.

(4) The point defects shorten the carrier lifetime of the material in the following order, V_Pb, V_Sn and i_I, with i_I giving the fastest carrier losses. The i_I defect creates deep defect states in the middle region of the band gap, which introduce new carrier recombination pathways, leading to a significant decrease of the carrier lifetime and the PCE. 32% of the carriers are lost within 30 ps due to the nonradiative electron–hole recombination in the presence of i_I and 14% of them are captured by the deep defect states created by the i_I defect. In order to avoid formation of the i_I defects, we suggest to decrease the anion content, or increase the cation content, to prevent the generation of i_I during nucleation of the crystal. Both V_Sn and V_Pb do not create defect levels in the band gap. However, they alter the energy dispersions of both the VBM and the CBM, changing the carriers' mobility and lifetime. The V_Sn defect weakens hybridization of the Pb and Sn orbitals, such that the Sn-5s and 5p orbitals contribute to the VBM very weakly around the defect. V_Pb has less impact on the material than V_Sn, because Pb atoms contribute less to the VBM edge states.

(5) Contrary to their influences on the electron–hole recombination process, the three kinds of point defects exhibit an impact on the hot carrier relaxation process with the order of i_I, V_Sn and V_Pb. i_I hardly affects the relaxation and both V_Pb and V_Sn slow down the relaxation, attributed to the defect-induced different changes on the energy dispersions of CB edge states.

This study establishes the microscopic mechanisms underlying the simultaneous improvement of the perovskite stability and the charge carrier lifetime by the hybrid cation strategy. It reveals the nature of the defect states and the influence of the intrinsic defects on the structures and optoelectronic properties, as well as the charge carriers' mobility and lifetimes. The general and fundamental principles established by analysis of the simulation results are useful for design of advanced materials for solar energy and optoelectronic applications.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Financial support from the National Natural Science Foundation of China (Grant Nos 21833006 and 22173074) and the USA National Science Foundation (Grant No. CHE-1900510) is acknowledged.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1ta09027e

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