Beneficial effects of tensile strain on charge carrier lifetime in metal halide perovskites containing halogen vacancies†

Zhiguo Wang ^a, Pingzhi Zhang ^a, Wei Wei *^b and Wei Li *^a
aSchool of Chemistry and Materials Science, Hunan Agricultural University, Changsha, P. R. China. E-mail: weili@hunau.edu.cn
^bInstitute of Theoretical Chemistry, College of Chemistry, Jilin University, 2 Liutiao Road, Changchun 130023, P. R. China. E-mail: weiweiww@jlu.edu.cn

First published on 19th September 2023

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Introduction

In recent years, hybrid organic–inorganic perovskites have attracted intense attention for energy conversion applications due to their low-cost and high efficiency.^1–4 The power conversion efficiency of perovskite solar cells (PSCs) has rapidly evolved from an initial 3.5% to 25.2%⁴ over the past decade. The prominent performance of perovskite materials originates from their superior optoelectronic properties such as a tunable bandgap,⁵ high optical absorption coefficient,⁶ small exciton binding energies,⁷ large carrier diffusion lengths,⁸ and low trap density.⁹ Despite achieving remarkable progress, the record efficiency of PSCs is still below the Shockley–Queisser theoretical efficiency limit.¹⁰ Further improvement in the performance of PSCs requires a fundamental understanding of photophysical properties and efficiency limiting factors. This could lead to design strategies for more efficient devices. Nonradiative recombination dissipates carriers’ energy as heat, constituting the dominant pathway for charge and energy loss in solar cells. Reducing the nonradiative recombination pathway is key to achieving high-performing PSCs.

Intrinsic point defects are abundant in solution-processed perovskite materials. Typically, defects introduce sub-gap states capable of trapping free charge carriers, creating nonradiative recombination centers, and facilitating electron–hole (e–h) recombination, as illustrated in Fig. 1. There are numerous studies devoted to the investigation of structural deficiencies in PSCs.^11–19 Recently, the unique features of perovskite materials have been explained using defect tolerance (DT) based on the theoretical observation that deep level defects have a high formation energy and shallow level defects are easy to form.²⁰ Shallow defects, which are considered electronically benign, have energy levels lying close to band edges and do not form nonradiative recombination centers. However, it has been argued that hybrid perovskites do suffer from trap-assisted e–h recombination,²¹ as supported by the observation of monomolecular decay components in photoluminescence spectroscopy,²² defying the previous assumption that they are DT. Shallow level defects in hybrid perovskites are not always benign,²³ especially given the strong dependence of defect levels on thermal flucutuations.²⁴

Halide-related defects in hybrid perovskites are very common. Migration of halide ions requires a small energy barrier and usually results in under-coordinated Pb ions.²⁵ As a result, halogen vacancies are one of the major defects in perovskite films. Halogen vacancies are believed to be the origin of the photocurrent hysteresis phenomenon in PSCs.²⁶ First-principles calculations demonstrated that defects in MAPbI₃ with low formation energies create only shallow levels.²⁷ Theoretical calculations have shown that I_V is a shallow defect with a trapping level near the conduction band minimum (CBM).^27–29 Our previous work pointed out that the oxidization state of I_V has a strong influence on the nonradiative recombination rate.³⁰ Neutral I_V defects can accelerate the nonradiative charge recombination to some extent. Passivation of the halogen vacancies holds significant potential for improving the efficiency and stability of PSCs.³¹ Considerable progress has been made on this topic. A variety of passivation strategies, including compositional and additive engineering,³² chemical surface treatment,³³ exposure to environmental factors such as oxygen,³⁴ humidity,^35,36 and UV light,^37,38 dimensional tuning,^39,40 have been developed for defect management in PSCs. Along with these strategies, strain engineering is a feasible way to tune the optoelectronic properties of hybrid perovskites, owing to their dynamical flexibility and lattice softness. This initiates a number of experimental and theoretical works on the roles of strain in PSCs.^41–49 Recent experiments have identified the presence of significant tensile strain in hybrid perovskites, which can be achieved by adopting strategies such as mixing large A-site cations⁵⁰ and light illumination.⁵¹ Although the influence of tensile strain on the optoelectronic performance of defect-free hybrid perovskites was investigated,⁵² the precise relationship between tensile strain and defect passivation in PSCs has not been fully understood from an atomistic level of theory.

In this work, we investigated nonradiative e–h recombination in MAPbI₃ perovskites containing iodine vacancies under tensile strain using time-dependent density functional theory (TD-DFT) and nonadiabatic molecular dynamics (NA-MD). We found that the I_V creates a trap state near the CBM, which is attributed to the hybridization of dangling states of under-coordinated Pb ions across the vacancy site. The applied tensile strain increases the Pb–Pb distance and correspondingly weakens the overlap of Pb-p orbitals, making the trap state shallower. We also showed that tensile strain decreases the fluctuation of the inorganic sublattice due to the suppressed transverse displacement of iodine perpendicular to the Pb–I–Pb orientation, resulting in a reduced nonadiabatic coupling (NAC) magnitude. Additionally, tensile strain causes a downshift in both valence band maximum (VBM) and CBM levels, arising from the decreased antibonding hybridization of Pb-s and I-p orbitals due to the elongated Pb–I bonds. The VBM undergoes a larger downshift than the CBM since the latter shows less anti-bonding character, leading to a larger VBM–CBM bandgap. The abovementioned aspects suppress the nonradiative e–h recombination through and bypassing the trap state, prolonging the charge carrier lifetime by more than 10 times in strained systems relative to unstrained systems. Our results demonstrate how tensile strain can be used to passivate iodine vacancies and extend the charge carrier lifetime in MAPbI₃ perovskites, paving the way to further enhance the performance of PSCs and other related devices.

Simulation methods

The unstrained I_V model was built by removing an axial iodine atom from the MAPbI₃ supercell structure, which is based on a 2 × 2 × 2 expansion of the cubic unit cell. Tensile strain was introduced along the a/b/c axis with magnitudes of 2.5% and 5%, respectively. DFT calculations were carried out using the Vienna Ab initio Simulation Package (VASP).^53,54 Geometry optimization, electronic structure, and adiabatic MD were investigated. The Perdew–Burke–Ernzerhof (PBE) functional of the generalized gradient approximation (GGA) was used to describe the electron exchange–correlation energy.⁵⁵ An energy cutoff of 400 eV and a 3 × 3 × 3 Γ-centered K-point mesh were adopted. The structural optimization stops when the forces on the atoms are less than 2 × 10⁻² eV Å⁻¹. The energy convergence criterion was set to 10⁻⁶ eV. Based on the optimized ground state structure, the systems were heated to room temperature (RT) using velocity rescaling. A 10 ps production MD run was then followed in the microcanonical ensemble using a 1 fs time step. The Hefei-NAMD code⁵⁶ was used to calculate the NAC along the MD trajectories. Specifically, 2000 initial configurations for NA-MD simulations were randomly sampled from 10 ps trajectories. NA-MD simulations were performed using the decoherence-induced surface hopping (DISH) algorithm,⁵⁷ as implemented in the PYXAID code.^58,59

The low-cost bare DFT functionals, such as PBE, usually underestimate the band gap values of most semiconductors due to their self-interaction error.⁶⁰ Hybrid DFT functionals, such as HSE06,⁶¹ can partially correct this issue and have been used to obtain relatively accurate electronic structures for semiconductors. It should be noted that spin–orbit coupling (SOC) is significant in perovskite materials due to the presence of heavy elements such as Pb and I. Generally, inclusion of SOC decreases the band gap by about 1 eV.⁶² In order to achieve the correct bandgap, including SOC, the use of hybrid DFT functionals or GW theory is required. Unfortunately, both hybrid functionals and the GW theory are computationally expensive, especially for the NA-MD simulation that requires thousands of electronic structure calculations. Fortunately, PBE is able to provide a band gap in agreement with experiment due to error cancellation. Therefore, all calculations are performed with the PBE functional. We also performed HSE06 hybrid functional (with 25% Hartree–Fock exchange) plus SOC calculations for the 0 K structure to test the validity of the PBE functional for describing defect levels of iodine vacancies. The HSE06 + SOC level of theory has been widely used for the accurate determination of defect levels in hybrid perovskites.^63–65

Results and discussion

The optimized zero-Kelvin (0 K) structures of MAPbI₃ perovskites containing an iodine vacancy defect under no strain (I_V), 2.5% tensile strain (I_V@2.5%), and 5% tensile strain (I_V@5%) are shown in Fig. 1. The vacancy site is denoted by a red dashed circle. Formation of I_V leads to the presence of under-coordinated Pb ions, potentially creating sub-gap states. Previous works demonstrated that two possible structures exist for the halogen vacancy defect, namely, the non-dimer and Pb-dimer configurations.^30,66,67 The Pb-dimer structure, accompanied by the formation of Pb–Pb covalent bond, can be stabilized by a negatively charge. In this work, we only focus on the neutral I_V defect since the Pb-dimer configuration should be insensitive to tensile strain. The Pb–Pb distance across the vacancy site for the unstrained system was calculated to be 6.2 Å. Such a large Pb–Pb distance corresponds to the non-dimer configuration. Nevertheless, the non-dimer structure is still able to moderately hybridize the dangling states of undercoordinated Pb ions, as we will demonstrate below. Upon applying tensile strain, the Pb–Pb distances are increased to 6.6 and 6.9 Å for 2.5% and 5% strained systems, respectively. A longer Pb–Pb distance is pivotal in resulting in insufficient overlap of Pb-p states across the vacancy site, which favors the formation of a shallower defect level.

Next, we examine the electronic structures of MAPbI₃ perovskites containing I_V under tensile strain. The projected density of states (PDOS) and band structures of the three systems, obtained from the PBE and HSE + SOC level of theory, are shown in Fig. S1 (ESI†). The charge density distribution of the band edge states for the 0 K structures are shown in Fig. 2. It can be seen that the PBE functional yields the similar features to HSE + SOC in the electronic structure. Generally, band edge states consist primarily of Pb and I atomic orbitals. In particular, the VBM is attributed to the antibonding hybridization of Pb-s and I-p orbitals,⁶⁸ whereas the CBM is primarily associated with the nonbonding Pb-p orbitals with a minor contribution from antibonding hybridization of Pb-s/I-p. Organic cations have no contribution to the band edge states. However, they play important roles in stabilizing the Pb–I octahedral structure and neutralizing the system.³⁰ Both the VBM and CBM are delocalized over the entire structure, Fig. 2. The VBM–CBM overlap is small since they are separated in space, which ensures a small wavefunction overlap and a low charge recombination rate. For the PBE level, the calculated VBM–CBM gaps are about 1.81, 1.90, and 1.91 eV for I_V, I_V@2.5%, and I_V@5% systems, respectively. HSE + SOC leads to minor changes in the fundamental gaps, ∼0.1 eV, compared to the PBE level. For the PBE and HSE + SOC levels, both VBM and CBM energies undergo a downshift upon increasing tensile strain, with VBM showing the larger variation, Table S1 (ESI†). This is because the VBM has a stronger anti-bonding hybridization of Pb-s/I-p than the CBM, and hence, the former is more sensitive to the tensile strain-induced structural changes.

The unstrained I_V defect produces a trap state, which is close to the CBM owing to the moderate hybridization of Pb-p dangling states across the vacancy site, forming a π-bond. A shorter Pb–Pb distance favors the formation of a deeper trapping state due to the stronger hybridization of Pb-p states. Reducing the hybridization of Pb-p dangling states can mitigate the detrimental trap states. In order to understand how tensile strain regulates the position of the defect level, we present the band structure from the highly symmetric point X to Γ using PBE and HSE + SOC in Fig. 3. The black, purple, and green lines represent I_V, I_V@2.5%, and I_V@5% systems, respectively. Dashed lines denote the position of the trap state. Generally, tensile strain shifts the defect level upward and makes it shallower. Charge density plots suggest that the trap state becomes more delocalized with increasing tensile strain. This is because the application of tensile strain increases the Pb–Pb distance, which correspondingly decreases the overlap of Pb-p states across the vacancy site. The reduced hybridization of dangling states of undercoordinated Pb ions weakens the pp bonding state, and further restricts the formation of Pb–Pb “wrong” bonds. It should be noted that both PBE and HSE + SOC levels provide a similar trend in the change of the defect level upon applying tensile strain. However, the position of the defect level relative to the band edge states differs between PBE and HSE + SOC in all considered systems, as supported by the data presented in Table S1 (ESI†). We thus applied the “scissor operator” method to PBE energies in order to match HSE + SOC results during NA-MD simulations. Such a procedure has been widely demonstrated in previous theoretical works.^59,69 Passivation of the trap state under tensile strain is expected to influence the nonradiative recombination dynamics, as we will discuss below.

Concerning the thermal atomic fluctuations, Pb and I atoms show decreased displacement velocity upon applying tensile strain, as presented in Fig. 4. This is somewhat surprising and can be understood from the following reasons. It is known that hybrid perovskites feature strong vibrational anharmonicity owing to the transverse motion of halides, which manifests as displacements perpendicular to the Pb–X–Pb orientation.^70,71 Applying tensile strain decreases the rocking motions of iodine in the transverse orientation, leading to a smaller dynamical disorder in the inorganic lattice. This is further supported by the decrease in the transverse atomic displacement of iodine under tensile strain, Fig. 4. In contrast, the longitudinal displacement, which shows smaller amplitude than the transverse displacement, increases with the application of tensile strain. This is not surprising because tensile strain increases the lattice volume, allowing for more space for Pb–I bond stretching. Our previous work has also shown that the motion of the inorganic lattice in MAPbI₃ perovskites increases under compressive strain.⁷² This is qualitatively similar to the mechanism of negative thermal expansion (NTE),⁷³ as observed for hybrid perovskites, in which higher temperature enhances the transverse motion of halide atoms, leading to lattice contraction. We emphasize that the transverse halide motion is more important in determining the optoelectronic properties of hybrid perovskites, as demonstrated by Egger and co-workers.⁷¹ NA coupling depends on wavefunction overlap and nuclear velocity according to its definition.⁷⁴ One may expect a decreased NA coupling in strained systems because of smaller nuclear velocity arising from the weaker fluctuation of the inorganic sublattice, given the similar VBM–CBM wavefunction overlap under different strains.

Distributions of Pb–Pb distance (d_Pb–Pb) across the vacancy site and CBM/trap energy gap (E_CBM-trap) extracted from MD trajectory are shown in Fig. 5. The standard deviation (σ) and averaged value (μ) for both quantities are listed. Generally, the E_CBM-trap is correlated with d_Pb–Pb. For d_Pb–Pb, the unstrained I_V system features an ensemble averaged of 5.833 Å and a standard deviation of 0.381 Å. Tensile strain elongates the averaged d_Pb–Pb value, in agreement with the trend reported for the static structure. Interestingly, standard deviation decreases with increasing tensile strain, further evidencing the suppressed fluctuation of neighboring Pb atoms near the vacancy site. For the E_CBM-trap distribution, the unstrained I_V system shows a standard deviation of 0.094 eV and an average value of 0.193 eV. A smaller E_CBM-trap indicates that the defect level is closer to the CBM, and hence, the defect level is shallower and less detrimental. The largest E_CBM-trap for the unstrained I_V system is up to ∼0.33 eV, suggesting that neutral iodine vacancies can create a deep defect level transiently. The short d_Pb–Pb, arising from the large amplitude thermal fluctuations of the inorganic lattice, contributes to the appearance of transient deep defect levels. Deep level defects can form nonradiative recombination centers, which are harmful for solar cell performance. Applying tensile strain decreases the averaged value and fluctuation of E_CBM-trap. This is due to the decreased thermal fluctuation of neighboring Pb atoms. Overall, the depth of the defect level is inversely related to change of Pb–Pb distance. These results demonstrate the beneficial roles of tensile strain in passivating the defect states in hybrid perovskites.

To characterize the phonon modes involved in nonradiative e–h recombination processes, we calculated the spectral densities obtained by performing Fourier transforms (FT) of the autocorrelation functions of the time-dependent Kohn–Sham levels, as shown in Fig. 6. NA coupling is directly related to the second derivative of the energy along the nuclear trajectory, and can thus be characterized by the intensity of FT modes. It can be seen that the spectra for VBM, CBM, and trap energy fluctuation exhibit exclusively low-frequency signals below 400 cm⁻¹. In particular, the low-frequency modes below 100 cm⁻¹ can be purely ascribed to translations/vibrations of the [PbI₆]⁴⁻ octahedron, according to Raman spectroscopy characterization.^75,76 Librations of MA cations dominate the modes in the range of 150–250 cm⁻¹. These modes contribute to electronic transition by coupling to the fluctuation of the inorganic sublattice. The slight difference in the position of contributing modes for VBM and CBM fluctuations can be explained by the similarities in the distribution of the two states. The defect level fluctuation couples to higher-frequency modes, covering a broader energy range in the strained systems compared to the unstrained I_V system. This is because trap state charges are localized on neighboring Pb in the unstrained system, and are more delocalized in strained systems. Therefore, the trap level is more likely to be affected by the motion of atoms away from neighboring Pb atoms in the strained systems. Interestingly, an unusual decrease in low-frequency peak intensities is observed upon applying tensile strain, which is consistent with the reduced dynamical disorder of the inorganic lattice mentioned above.

To gain further insights into electron–vibrational interactions, we computed the pure-dephasing function using the second-order cumulant approximation^77,78 based on optical response theory.⁷⁹ The simulated pure-dephasing function is shown in Fig. 7. The dephasing function characterizes the loss of quantum coherence in electronic subsystems due to coupling to phonons. Loss of coherence slows down quantum transition dynamics, as demonstrated by the quantum Zeno effect.⁸⁰ The decoherence time is equivalent to pure-dephasing time. In this work, we determined the pure-dephasing times (τ) by fitting the decay curve using a Gaussian function, exp[−0.5(t/τ)²]. The pure-dephasing times of the CBM to VBM transition for all investigated systems are quite similar, i.e., on a 12 fs time scale. Quantum coherence is longest, 43.1 fs, for the CBM to trap transition in the I_V@5% system, Table 1; this is because these states have the smallest energy gap and have a similar chemical environment, i.e., both are composed of Pb-p orbitals and delocalized over the entire structure, leading to correlated fluctuations of energy levels. The VBM trap transition generally has shorter quantum coherence because the two states are localized on different atoms.

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Finally, we carried out NA-MD simulations of charge trapping and recombination dynamics in all investigated systems. Generally, the nonradiative carrier lifetime can be up to nanoseconds or microseconds. It is impractical to model charge trapping and recombination processes over such a long timescale at the ab initio level. To extend the timescale of NA-MD simulation, we adopted a procedure developed previously.³⁰ Namely, we first computed the rates for CBM to VBM, CBM to trap, and VBM to trap transitions using the DISH algorithm, Table 1; then we constructed coupled differential equations based on the transition rates for each pairwise states, and finally integrated these equations to obtain the time-dependent populations for each state. The essential details are described in ref. 30. Time-evolution of ground state population is shown in Fig. 8. The characteristic timescales are obtained through exponential fitting. The abovementioned results have demonstrated that iodine vacancies introduce charge traps near the CBM, which is capable of trapping holes. Once holes are trapped, they will recombine with CB electrons (trap-assisted e–h recombination). The CB electrons can also recombine with VB holes directly, bypassing the trap state (direct e–h recombination). The data suggest that the excited state population in the unstrained system survives for about 350 ns. The nanosecond carrier time can be explained by the relatively small NAC and rapid quantum decoherence, Table 1. Tensile strain has a significant impact on charge recombination dynamics. Applying 2.5% tensile strain extends the carrier lifetime by an order of magnitude. Further applying 5% tensile strain slows down the charge recombination by a factor of 2. The elongation of charge carrier lifetime in MAPbI₃ containing iodine vacancies under tensile strain is due to several reasons. First, the trap-assisted e–h recombination is suppressed under tensile strain because the defect level is shallower and nonradiative recombination centers are partly eliminated. Second, direct e–h recombination is slowed down because NA coupling for the CBM to VBM transition is reduced owing to the weaker fluctuation of Pb and I atoms that support charge carriers. Interestingly, Ghosh et al. reported an extended excited state carrier lifetime with increasing tensile strain in defect-free hybrid perovskites.⁵² This suggests that the influence of tensile strain on charge recombination is independent of halogen vacancies. A longer carrier lifetime is an indication of smaller nonradiative charge and energy loss, which is beneficial for the performance of PSCs.

Conclusions

To recapitulate, we have investigated the nonradiative charge recombination dynamics in MAPbI₃ perovskites with iodine vacancies under different tensile strains. Our calculations suggest that tensile strain elongates the Pb–Pb distance around the vacancies, which weakens the Pb–Pb interactions. The trap state charge density is delocalized in strained systems, making the defect state shallower. The shallower trapping state is beneficial to the suppression of trap-assisted e–h recombination. In addition, applying tensile strain increases the band gap owing to the larger downshift of the VBM compared to the CBM since the latter shows less anti-bonding character, leading to insensitivity of the CBM relative to the structural change. We also found that the thermal fluctuation of the inorganic lattice is weakened on applying tensile strain, resulting in a decreased electron–vibrational interaction and NA coupling. This can be explained by the suppressed rocking motions of the iodine atoms in the direction perpendicular to the Pb–I–Pb orientation. The larger band gap and weaker NA coupling reduce the rate of direct e–h recombination. The suppression of both direct and trap-assisted recombination extends the lifetimes of charge carriers by about an order of magnitude under 2.5% tensile strain. These findings suggest that tensile strain can passivate halogen vacancies and extend the carrier lifetime in metal halide perovskites. This work demonstrated the beneficial effects of tensile strain in hybrid perovskites, paving the way for the enhancement of material performance and optimization of optoelectronic devices.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

W. L. acknowledges financial support from the National Natural Science Foundation of China (No. 22373033) and the Science and Technology Innovation Program of Hunan Province (No. 2021RC3089).

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Footnote

† Electronic supplementary information (ESI) available: Comparison of PDOS obtained using PBE and HSE + SOC. Calculated energy levels of the VBM, trap, and CBM. See DOI: https://doi.org/10.1039/d3tc02828c

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