Challenges in predicting positron annihilation lifetimes in lead halide perovskites: correlation functionals and polymorphism

This study demonstrates that the choice of electron-positron correlation functional, particularly the use of the non-local weighted density approximation (WDA), is critical for accurately predicting positron annihilation lifetimes in lead halide perovskites, revealing that previous discrepancies in theoretical predictions and experimental interpretations of cation vacancies stem from the sensitivity of these materials to the specific approximation used.

The Gist

Imagine you are trying to find a specific, tiny hole in a very complex, bouncy castle made of lead, iodine, and organic molecules. This "bouncy castle" is a halide perovskite, a super-hot material used to make solar cells and LEDs.

To find the holes (which scientists call vacancies), researchers use a special detective tool called Positron Annihilation Spectroscopy.

Here is the simple story of what this paper is about, using some everyday analogies.

1. The Detective and the Bouncy Castle

Think of a positron as a tiny, invisible detective sent into the material.

• The Job: The detective's only goal is to find empty spaces (vacancies) where atoms are missing.
• The Clue: When the detective finds a hole, they get stuck there for a moment before disappearing (annihilating). The longer they stay stuck, the bigger the hole is.
• The Goal: Scientists measure exactly how long the detective stays (the "lifetime") to figure out what kind of hole they found.

2. The Problem: The "Translator" is Broken

The problem isn't the detective; it's the translator the scientists use to interpret the detective's report.

In the world of quantum physics, to calculate how long a positron stays in a hole, you need a mathematical formula called an Electron-Positron Correlation Functional (EPCF). Think of this formula as a translator that converts the raw physics into a number (the lifetime).

• The Issue: The authors of this paper found that in these specific perovskite materials, the translator is very unreliable.
• The Analogy: Imagine you ask three different translators to tell you how long a guest stayed at a party.
  • Translator A says: "They stayed for 300 seconds."
  • Translator B says: "They stayed for 400 seconds."
  • Translator C says: "They stayed for 600 seconds!"
  • The Reality: The guest actually stayed for 450 seconds.
  • The Result: Because the translators disagree so wildly, the scientists can't be sure which "hole" the detective actually found. Is it a small scratch? A big crater? Or a giant cavern?

3. The "Big Empty Room" vs. The "Small Closet"

The paper focuses on two types of holes in the perovskite:

1. Lead Vacancies: Missing lead atoms. These are like small closets.
2. Methylammonium Vacancies: Missing organic molecules. These are like huge, empty ballrooms.

The researchers discovered that the "translator" (the math formula) works okay for the small closets, but it completely falls apart for the huge ballrooms.

• The Metaphor: It's like trying to measure a swimming pool with a ruler meant for a coffee cup. The math gets confused because the "density" of the material changes so drastically in these big empty spaces.
• The Finding: When they used a more advanced, "non-local" translator (called WDA), the results for the big ballrooms changed drastically compared to the older, simpler translators. In fact, the advanced translator suggested that the positron behaves very differently in these big holes than everyone previously thought.

4. The "Shape-Shifting" Material

There is a second twist. These perovskite materials aren't rigid like a brick; they are more like soft clay or jelly.

• Even though they look like a perfect cube from a distance, up close, the atoms are wiggling and shifting around. This is called polymorphism.
• The Analogy: Imagine a room where the furniture is constantly rearranging itself. Sometimes the "hole" is in the corner; sometimes it's in the middle.
• The researchers built a giant computer model to simulate this "wiggling" material. They found that even though the material is messy, the size of the holes doesn't change that much. However, the math formula used to measure them still gives wildly different answers depending on which version you pick.

5. The Big Conclusion

The paper is essentially a warning label for scientists in this field.

• The Old Way: Scientists were comparing their experimental data (what they measured in the lab) with theoretical data (what the computer said) to identify defects.
• The New Reality: The authors say, "Stop! You can't trust the computer yet." Because the "translator" (the math formula) changes the answer by hundreds of picoseconds (trillionths of a second), you can't be sure if the experiment is measuring a lead vacancy, an organic vacancy, or just the normal material.

In short:
The paper says, "We have a great tool (positrons) to find holes in solar cell materials, but our math is currently too shaky to tell us exactly what kind of hole we found. We need to fix our math formulas before we can trust the results."

They are calling for a new, better "translator" that can handle the weird, bouncy, and empty nature of these materials so we can finally build better, more efficient solar cells.

Technical Summary

1. Problem Statement

Positron annihilation lifetime spectroscopy (PALS) is a primary experimental tool for characterizing vacancy defects in semiconductors. However, applying this technique to hybrid and inorganic lead halide perovskites (e.g., MAPbI$_3$, CsPbI$_3$) faces significant challenges:

• Discrepancy in Theory vs. Experiment: Previous theoretical predictions of positron lifetimes in these materials often disagree with each other and fail to consistently match experimental data.
• Ambiguity in Defect Identification: Experimental lifetimes (often $<500$ ps) are difficult to assign to specific defects (e.g., A-site vs. Pb-site vacancies) or the pristine lattice because the theoretical range of predicted lifetimes is too wide.
• Theoretical Limitations: Standard approximations for the electron-positron correlation functional (EPCF) used in Density Functional Theory (DFT) calculations may be insufficient for materials with large voids (like methylammonium vacancies) and strong electronic density variations.
• Polymorphism: The high-temperature cubic phase of perovskites exhibits "polymorphism" (local disorder), which has not been thoroughly investigated regarding its impact on defect formation energies and positron lifetimes.

2. Methodology

The authors employed a two-component Density Functional Theory (2CDFT) approach to simulate positron lifetimes.

• Software: Electronic densities were calculated using Quantum Espresso, and positron lifetimes were computed using ATSUP.
• Materials Studied:
  • Hybrid: Methylammonium lead iodide (MAPbI$_3$) in orthorhombic, tetragonal, and cubic phases.
  • Inorganic: CsPbI$_3$ and CsPbBr$_3$.
• Defects Modeled: Cation vacancies (Methylammonium $V_{MA}^{-1}$, Lead $V_{Pb}^{-2}$) and Iodine/Bromine vacancies (noted as positively charged and unlikely to trap positrons).
• Polymorphism:
  • For MAPbI$_3$, a large 32-formula-unit (fu) supercell was generated to model the disordered cubic phase.
  • For inorganic perovskites, the Special Displacement Method (SDM) was used to generate dynamically stable polymorphous structures.
• Electron-Positron Correlation Functionals (EPCF): The study systematically compared several approximations:
  • LDA: Boronski-Nieminen (BN-LDA), Drummond et al. (D-LDA).
  • GGA (Semi-local): B95-GGA, K14-GGA, B15-GGA (parameter-free).
  • Non-local: Weighted Density Approximation (WDA) by Callewaert et al., which accounts for strong density variations. The WDA screening charge parameter ($Q$) was varied from 0.8 to 2.0.
• Technical Details: Calculations used optB86b+vdW exchange-correlation functionals, ultrasoft pseudopotentials, and included corrections for charged defects (monopole corrections) and potential shifts ($\Delta V$) in supercells.

3. Key Contributions

• Systematic EPCF Benchmarking: This is the first comprehensive comparison of multiple EPCF approximations specifically for lead halide perovskites, highlighting that the choice of functional is far more critical here than in conventional semiconductors or metals.
• Polymorphism Analysis: The authors quantified the distribution of defect formation energies and positron lifetimes in polymorphous supercells, demonstrating that local disorder leads to narrow distributions of these properties.
• WDA Application: The study introduces the non-local WDA to perovskites, revealing that semi-local functionals fail to capture specific physical behaviors in large voids (like $V_{MA}$).
• Voronoi Volume Correlation: A robust correlation was established between the largest Voronoi volume (void size) and the calculated positron lifetime, providing a geometric predictor for lifetimes that is less dependent on the specific electronic density approximation.

4. Key Results

A. Sensitivity to Correlation Functionals

• Huge Variations: For the methylammonium vacancy ($V_{MA}$), the calculated lifetime varies by >200 ps depending on the functional used (e.g., ~360 ps with BN-LDA vs. ~570 ps with B95-GGA).
• Lead Vacancy ($V_{Pb}$): Variations are smaller but still significant (~30–70 ps).
• WDA Behavior: The WDA approximation shows a unique dependence on the screening charge $Q$.
  • Crucially, WDA predicts that the positron is more strongly bound to the Lead vacancy ($V_{Pb}$) than to the Methylammonium vacancy ($V_{MA}$), which is the reverse of the trend predicted by all semi-local functionals (GGA/LDA).
  • The WDA captures a local maximum in the positron potential at the $V_{MA}$ site, whereas semi-local functionals show a minimum.

B. Polymorphism and Defect Distributions

• In the polymorphous cubic phase of MAPbI$_3$, the standard deviation of defect formation energies and positron lifetimes is small (<2.6% for lifetimes).
• There is a linear relationship between positron binding energy and lifetime in the polymorphous phase, suggesting that local structural distortions do not drastically alter the trapping characteristics compared to the ordered phase.

C. Comparison with Experiment

• Experimental Range: Experimental lifetimes for bulk MAPbI$_3$ range from ~309 ps to ~375 ps.
• Theoretical Mismatch: No single EPCF consistently matches all experimental data points.
  • If the experimental signal is attributed to the pristine lattice, the B95-GGA or WDA ($Q \approx 1$) results align best.
  • If the signal is attributed to vacancies, the B15-GGA or WDA with higher $Q$ values align better.
• Positronium (Ps) Formation: The authors rule out the formation of ortho-positronium (Ps) or Ps$^-$ ions as the dominant mechanism for the observed lifetimes, as the calculated lifetimes for Ps would be significantly longer (>500 ps) than experimental observations.

D. Void Size Correlation

• A linear correlation exists between the largest Voronoi volume and the positron lifetime. This relationship holds for both the lattice and vacancy defects, offering a simpler geometric method to estimate lifetimes without relying solely on complex electronic structure calculations.

5. Significance and Conclusion

• Re-evaluation of Experiments: The study concludes that the wide spread in theoretical predictions prevents a definitive assignment of experimental PALS signals to specific defects in lead halide perovskites. Researchers must reconsider interpretations of experimental data, as the "best" functional is not yet identified.
• Need for Non-local Functionals: The failure of semi-local functionals to describe the $V_{MA}$ vacancy correctly suggests that non-local correlations (like WDA) are essential for materials with large, low-density voids.
• Future Directions: The authors recommend further combined theoretical and experimental studies, potentially revisiting simpler ionic materials to calibrate the WDA screening charge ($Q$) and developing functionals that account for the covalent character of organic cations.
• Polymorphism: The work establishes that while polymorphism introduces local disorder, it results in a relatively narrow distribution of defect properties, validating the use of supercell averaging for predicting bulk defect behavior.

In summary, this paper highlights that predicting positron lifetimes in perovskites is highly sensitive to the choice of electron-positron correlation functional, particularly for cation vacancies, and that current theoretical tools are not yet sufficient to unambiguously interpret experimental PALS data without further refinement of the correlation models.