Origin of the Temperature-Induced Gap Bowing of Formamidinium-Methylammonium Lead Iodide Perovskites: Role of Cationic Rattlers

By combining temperature and pressure-dependent photoluminescence measurements on FA-MA lead iodide perovskite single crystals, this study identifies that the pronounced temperature-induced gap bowing in low-temperature phases is primarily driven by an anomalous electron-phonon coupling mechanism involving mixed vibrational modes of inorganic cage tilting and formamidinium "rattler" librations.

The Gist

The Big Picture: Why Does the Color Change?

Imagine you have a special kind of solar cell material (a perovskite) that acts like a gatekeeper for light. It has a specific "energy gate" (called a band gap) that determines what color of light it can absorb or emit.

Usually, when you heat up a material, this gate gets slightly bigger or smaller in a predictable, straight-line way. But the researchers in this paper discovered something weird happening with a specific mix of these materials (a combination of Formamidinium and Methylammonium lead iodide).

When they cooled these materials down, the "gate" didn't just move in a straight line; it curved dramatically, like a rollercoaster dip. This is called "gap bowing." Until now, scientists didn't know why this curve happened. This paper solves the mystery.

The Two Suspects: The Expanding Room and the Bouncing Ball

To understand the gate's movement, the scientists realized there are two main forces at play, like two people pushing a heavy door:

1. Thermal Expansion (The Room Getting Bigger): When things get hot, they expand. When they get cold, they shrink. In a crystal, the atoms are like a room. When the room shrinks (gets cold), the "door" (the energy gap) changes size just because the walls are moving closer together.
2. Electron-Phonon Interaction (The Bouncing Ball): This is a fancy way of saying that atoms in the crystal are constantly vibrating (like a ball bouncing on a trampoline). These vibrations bump into the electrons, changing the energy of the "door."

The scientists used a clever trick: they squeezed the crystals with high pressure (like a hydraulic press) while cooling them down. This allowed them to separate the two forces. They found that while the "room shrinking" (thermal expansion) played a part, it wasn't the main culprit. The real troublemaker was the bouncing ball (electron-phonon interaction).

The Villain: The "Rattler"

Here is the most interesting part. In some crystals, there are heavy atoms that just sit in the middle of a cage of other atoms. In this specific material, the Formamidinium (FA) cation acts like a rattler.

• The Analogy: Imagine a large, empty box (the inorganic cage) with a small, loose marble (the FA cation) inside it.
• The Normal Behavior: Usually, the marble just vibrates gently.
• The "Rattler" Behavior: In this specific material, at low temperatures, the marble starts to rattle violently against the walls of the box. It's not just vibrating; it's colliding with the walls in a very specific, synchronized way.

The paper claims that this "rattling" creates a strange, negative force that pulls the energy gate down, causing that dramatic curve (bowing) in the graph. It's as if the marble is hitting the walls so hard that it actually pushes the door open wider than expected, but in the opposite direction of normal physics.

The Stage: The "Stripe" Dance

Why does this rattling only happen in certain mixes of the material? The paper suggests it's about the dance floor layout.

• The Setup: The material has a phase (a specific arrangement of atoms) that looks like a mosaic or a striped pattern. Imagine a floor made of tiles where every other stripe is rotated 90 degrees.
• The Trigger: In these "stripe domains," the crystal structure is a bit messy and disordered. This specific "striped" arrangement forces the FA cations to rattle in sync with the walls of their cages.
• The Result: This synchronized rattling is what triggers the weird "negative" force that bends the energy gap. If the material is pure (all one type of atom) or has a very different mix, the stripes don't form, the rattling doesn't happen, and the curve disappears.

The Conclusion

The scientists successfully proved that the strange curve in the energy gap of these mixed crystals is caused by FA cations acting like rattles inside their cages. This happens specifically when the crystal forms a striped, mosaic-like structure at low temperatures.

They didn't just guess this; they measured how the material reacted to pressure and temperature, calculated the forces, and matched the "rattling" frequency to the specific vibrations of the Formamidinium atoms.

In short: The material's energy gap bends strangely because, at low temperatures, the internal atoms start "rattling" against their cages in a specific striped pattern, creating a unique force that changes how the material handles light.

Technical Summary

Technical Summary: Origin of the Temperature-Induced Gap Bowing in Formamidinium-Methylammonium Lead Iodide Perovskites

Problem Statement
While the temperature dependence of semiconductor band gaps is generally described by two terms—thermal expansion (TE) and electron-phonon (EP) interaction—the origin of the pronounced "gap bowing" (a non-linear, parabolic decrease in band gap energy at low temperatures) observed in mixed-cation formamidinium-methylammonium lead iodide ($FA_xMA_{1-x}PbI_3$) perovskites has remained elusive. Previous studies on pure $MAPbI_3$ and $CsPbCl_3$ indicated that TE and EP terms contribute differently depending on the cation, but the specific mechanism driving the anomalous gap bowing in intermediate FA/MA compositions (specifically in the low-temperature orthorhombic and pseudo-tetragonal phases) was unknown. Understanding this behavior is critical for optimizing the performance of optoelectronic devices, as the band gap dictates solar cell efficiency and emission color.

Methodology
The authors employed a comprehensive experimental approach combining temperature- and pressure-dependent photoluminescence (PL) measurements on high-quality single crystals of $FA_xMA_{1-x}PbI_3$ across the full composition range ($x \in [0, 1]$).

• Sample Synthesis: High-quality single crystals were grown using the inverse solubility method.
• PL Measurements: Temperature-dependent PL spectra were recorded from ~365 K down to 6 K using 633 nm excitation with low power to avoid degradation.
• High-Pressure Experiments: To disentangle TE and EP contributions, the authors performed high-pressure PL measurements at cryogenic temperatures (down to 4 K) using a diamond anvil cell (DAC) with helium as the pressure-transmitting medium. This allowed for the determination of the gap pressure coefficient ($dE_g/dP$) at specific low temperatures.
• Data Analysis: The band gap temperature dependence was analyzed by calculating the first derivative of the gap energy with respect to temperature ($dE_g/dT$). The total derivative was modeled as the sum of the TE term (derived from $dE_g/dP$, volumetric thermal expansion coefficient $\alpha_V$, and bulk modulus $B_0$) and the EP term. The EP term was fitted using an Einstein-oscillator model, where contributions are represented by oscillators with specific amplitudes ($A_i$) and frequencies ($\omega_i$).

Key Contributions and Results

1. Disentanglement of TE and EP Terms: By experimentally determining the gap pressure coefficient at cryogenic temperatures, the authors successfully separated the thermal expansion contribution from the electron-phonon interaction. They found that while the TE term varies significantly with temperature and composition, it is insufficient to explain the observed gap bowing.
2. Identification of Anomalous EP Coupling: The primary contribution to the gap bowing in the low-temperature phases (Ortho and Tetra-III) for FA concentrations between 20% and 90% is an anomalous electron-phonon coupling term. This term is characterized by an additional Einstein oscillator with a negative amplitude ($A \approx -105$ meV) and a frequency of approximately 16 meV (corresponding to ~135 cm$^{-1}$).
3. Role of FA Rattler Modes: The frequency of this anomalous oscillator matches the low-frequency libration modes of the formamidinium (FA) cation. The authors identify this as a "FA-rattler" mode, analogous to the "Cs-rattler" modes previously identified in Cs-containing perovskites. This mode involves a mixed vibrational character combining inorganic-cage phonons (octahedral tilting) with low-frequency FA librations.
4. Phase Diagram and Structural Correlation: The anomalous coupling is observed specifically in the low-temperature Ortho and Tetra-III phases. The authors correlate this behavior with the formation of ferroelastic stripe domains featuring alternating octahedral tilt-axis patterns (90-degree domain walls). These domains are hypothesized to form at the morphotropic phase boundaries (MPBs) between the out-of-phase ($a^0a^0c^-$) and in-phase ($a^0a^0c^+$) tilt patterns, a region of enhanced dynamic disorder.
5. Quantitative Fitting: The gap temperature dependence for intermediate compositions is perfectly described only when the model includes both a "normal" EP term (positive amplitude, ~1.5 meV) and the "anomalous" FA-rattler term (negative amplitude). The negative amplitude of the anomalous term overcompensates for the TE and normal EP effects, resulting in the observed negative slope (bowing) of the gap derivative.

Significance
The paper claims to have elucidated the origin of the pronounced temperature-induced gap bowing in mixed-cation lead iodide perovskites, a phenomenon that had previously lacked a clear explanation. The study demonstrates that this behavior is driven by a specific, anomalous electron-phonon coupling mechanism linked to FA-rattler modes, which are activated in the presence of ferroelastic stripe domains in the low-temperature phases.

The authors emphasize that correctly accounting for this gap temperature dependence is essential for the design and performance optimization of optoelectronic devices, particularly photovoltaics and light-emitting devices, as the band gap directly determines device efficiency and emission characteristics. The work bridges the gap between structural dynamics (cation rattling and domain formation) and electronic properties (band gap renormalization) in hybrid perovskites.