# Size dependent solid-solid crystallization of halide perovskites

Paramvir Ahlawat<EMAIL_ADDRESS>Yusuf Hamied Department of
Chemistry, University of Cambridge

(March 25, 2024)

###### Abstract

The efficiency and stability of halide perovskite-based solar cells and light-
emitting diodes directly depend on the intricate dynamics of solid-solid
crystallization[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23]. In this study, we employ a multi-scale approach using
random phase approximation, density functional theory, machine learning
potentials, reduced charge force fields, and both enhanced sampling biased and
brute-force unbiased molecular dynamics simulations to understand the solid-
solid phase transitions in cesium lead iodide perovskite. Our simulations
uncover that the direct phase transition from the non-perovskite to the
perovskite involves the formation of stacked-faulted and low-dimensional
intermediate structures. Through extensive large-scale all-atom simulations
encompassing up to 650,000 atoms, we observe that solid-solid crystallization
may require the formation of a sufficiently large critical nucleus to grow
into a faceted perovskite crystal. Based on simulations, we determine that
utilizing (100)-faceted seeded crystallization could offer a promising path
for manufacturing high-performance and stable perovskite solar cells.

Halide perovskites, crystallization, solid-solid nucleation, MD simulations,
enhanced sampling,

††preprint: APS/123-QED

##  I Introduction

Halide perovskites[24, 25, 26, 27, 28] are photo-active materials with tunable
band gap ($1.1-2.3$ eV) to manufacture record efficiency solar cells
(PSCs)[29, 30, 11, 31, 32, 33, 12]. Despite their superior optoelectronic
properties for making low-priced efficient photovoltaics and light emitting
diodes, their wide-spread industrialization is still suffering because of the
poor reproducibility and stability using current manufacturing processes.

Figure 1:  Experimentally known polymoprhs of CsPbI3: Figure shows the
different phases of CsPbI3 with Pb-I structures as octahedra with red color
iodides on corners and Cs atoms as light-green spheres.

Where one of the key problem is the lack of suitable and reproducible
synthesis recipes to produce defect free highly stable thin-films of halide
perovskites. Therefore, it is of key importance to comprehend the details of
their formation process which can assist to design better experimental
methodologies. In particular, wide range of experiments have established that
the solid-solid phase transitions dominate the crystallization process and
control the micro-structure evolution of perovskite thin-films. In this work,
we perform molecular dynamics simulations and try to understand the details of
crystallization of the widely studied and thermally stable cesium lead iodide
(CsPbI3).

CsPbI3 has four experimentally known polymorphs[34, 35, 36, 37, 1, 38], often
recognized as $\delta$, $\gamma$, $\beta$ and $\alpha$-phases visualized in
Figure 1. Among them, $\delta$-phase is the most stable form, and a yellow-
colored photo-inactive material consists of edge-sharing lead-iodide octahedra
surrounded by cesium cations. On the other hand, $\gamma$, $\beta$ and
$\alpha$-phases are black coloured thermally stable photo-active perovskites
made of corner-sharing lead-iodide octahedra with different tilted angles,
however meta-stable compared to its $\delta$-form. The perovskite phases have
suitable band-gap of $\sim 1.7$ to $\sim 1.9eV$ for making tandem solar cells
with the current leading silicon technology (efficiency $\sim$ 26%) and
theoretically can reach efficiencies beyond 30%[29]. However, to construct a
competitive low priced and stable $>30\%$ efficient and stable perovskite-
silicon or all perovskites tandem solar cells [39, 40, 41, 42, 43, 32, 44, 45,
31, 46] involving CsPbI3: one of the main quests is to synthesize/crystallize
defects-free corner-sharing phases of CsPbI3 and eliminate/prevent the
formation of edge-sharing $\delta$-phase.

Figure 2:  Solid-solid phase transition by enhanced sampling: Figure shows the
simulated transformation of edge-sharing $\delta$-phase to corner-sharing
$\alpha$-phase of CsPbI3. Pb-I structures are represented as yellow octahedra
with orange color iodides on coners, and Cs atoms are depicted with blue
spheres.

Absolutely gigantic and strenuous experimental efforts[14] have been employed
to reach the present day synthesis recipes. A wide range of processes and
uncountable additives have been tried to improve the crystallization process.

Now, it is common to utilize additives like dimethylammonium[47, 48, 49, 6],
phenylethylammonium[50], bromide[51, 52, 53, 54] and chloride[55, 11] to make
efficient and optimistically stable black-phase of cesium lead iodide.
Nevertheless, existing manufacturing processes continue to struggle with
issues of reproducibility and stability, prompting ongoing efforts to enhance
them through extensive experiments aimed at achieving progressively higher
efficiencies and industrial scale stable PSCs. Therefore, it is of key
importance to understand and optimize the synthesis routes to ensure the
stability and reliability of perovskite-based devices in practical
applications. A comprehensive atomic-level picture of the nucleation and
growth processes can help understand the current experimental recipes’
limitations and, in turn, assist to design improved, reproducible and even
newer methodologies for making PSCs and LEDs.

Towards this goal, we perform molecular dynamics (MD) simulations of the
crystallization of cesium lead iodide. A simulation study of the full-scale
experiment is challenging and herculean task. Here, we focus on understanding
the solid-solid phase transitions in perovskites which are experimentally
known to controls the synthesis process of perovskite thin-films. We explore
two key aspects that have not been extensively investigated in the context of
phase transitions in halide perovskites.

At first, we study how the corner-sharing photo-active perovskite black phase
originates from the edge-sharing yellow phase. Secondly, we aim to unravel the
large scale crystallization mechanism of solid-solid transitions in
perovskites. To address these fundamental questions, we employ a unique
combination of simulation methods, including random phase approximation (RPA),
density functional theory (DFT), machine learning potentials (MLPs), umbrella
sampling, and large-scale seeded simulations utilizing reduced charge inter-
atomic potentials (RCPs). Through biased simulations using RCPs and MLPs for
the direct phase transition from the non-perovskite to perovskite phase, we
identify activated pathways characterized by the formation of intermediary
stack-faulted structures and edge-dislocations. In our exploration of larger
size simulations, we uncover that the solid-solid transitions could adheres to
nucleation process. This implies that the formation of a critical nucleus may
become essential for facilitating the growth into stable and faceted crystals.
We sheds light on the intricate details of the nucleation from one phase to
another, providing valuable insights into the underlying fundamental
mechanisms.

##  II Direct non-perovskite ($\delta$) phase to perovskite phase transition

In conventional experimental procedures, the thermodynamically stable
$\delta$-phase initially crystallizes and subsequently undergoes
transformation into the black-phase through annealing at elevated
temperatures. However, direct conversion processes have historically yielded
less stable and less efficient perovskite solar cells. Therefore, first we
start with the MD simulations of the direct conversion of edge-sharing
$\delta$-phase of CsPbI3 to the corner-sharing photo-active phase. Previously
Grünwald, Limmer and co-workers[56, 57] have conducted notable co-existing
simulations, revealing a liquid-like interface mediating the phase transition
between edge-corner-sharing polymorphs of CsPbI3. Another significant
contribution by Pramchu et al.[58] employed variable-cell nudged-elastic band
calculations to investigate reverse phase transitions from the corner-sharing
to the edge-sharing phase of CsPbI3, primarily aiming to gain insights into
the degradation process of smaller systems using DFT calculations.

However, to understand the crystallization of perovskite black phases, it is
necessary to reveal formation pathway of meta-stable perovskite phases from a
direct heating the $\delta$-phase as per experimental procedure. Here, we
perform MD simulations of the direct phase transition, initiated solely from
the $\delta$-CsPbI3. First-order phase transitions between crystalline phases
are often associated with substantial energy barriers[59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 63, 74, 75], potentially occurring on time
scales ranging from seconds to minutes[76, 77, 78, 79, 80]. Simulating these
experimental crystallization processes within feasible times and resources
poses computational challenges.

To address this, we employ biased enhanced sampling methods[81, 61, 82, 83,
84, 85, 86], specifically umbrella sampling[81, 87, 62] and OPES[84, 88].
These methods explore the free energy surface by applying a bias potential to
the potential energy surface. The bias potential is crafted as a function of
collective coordinates, commonly referred to as collective variables (CVs).
The choice of suitable CVs is crucial[89, 64, 90, 91, 92] for addressing
complex geometries in multi-species systems, such as the process under
investigation. In this study, we introduce a locally ordered CV[93, 94] based
on the local crystallographic information of perovskite
($s^{\alpha}(r_{i}^{Pb})$)[95, 96, 10, 97] for species ${\alpha}$ = (Cs, Pb,
and I), centered around positions $r_{i}^{Pb}$ of Pb atoms.

$s^{\alpha}(r_{i}^{Pb})=\frac{1-\left(\frac{s(q)_{i}^{\alpha}}{s(q)^{\alpha}_{cut}}\right)^{-c}}{1-\left(\frac{s(q)_{i}^{\alpha}}{s(q)^{\alpha}_{cut}}\right)^{-d}}$
(1)

Where $s(q)_{i}^{\alpha}$ is defined as:

$s(q)_{i}^{\alpha}=1+\frac{1}{N}\sum_{j=i}^{N_{\alpha}}\frac{sin(qr_{ij})}{qr_{ij}}$
(2)

Where $N_{\alpha}$ are the number of species within a distance cutoff
$r^{\alpha}_{cut}$. Taking inspiration from Ornstein–Zernike[98], Kirkwood[99]
equations and others[100, 101]: the final local symmetric order centering Pb
positions $r_{i}^{Pb}$ is calculated by multiplying these individual local
contributions[94] from different species:

$\xi_{i}=\prod_{\alpha=Cs,Pb,I}s^{\alpha}(r_{i}^{Pb})$ (3)

The final expression for the CV becomes the sum for all Pb species:

$S_{p}=\sum_{i=1}^{N_{Pb}}\xi_{i}$ (4)

Using $S_{p}$, we conduct biased simulations employing a reduced charge
potential of CsPbI3[57]. We successfully observe the crystallization of
perovskite structure, presented in Figure 2 and supplementary movie SM1.
Additional simulation details can be found in the Methods section. Our
observations reveal that the solid-solid phase transition is a complex, multi-
step process, generating various stack-faulted structures, anti-phase domains,
and $<110>$-edge dislocations[102, 103] during the transition from full edge-
sharing to corner-sharing structures of CsPbI3.

Figure 3:  Relative Energies of different polymorphs using RPA+HF and a range
of DFT functionals

To further scrutinize these observations, we employ an alternative CV based on
permutation invariant vector (PIV) distances[104, 105]. Initially, PIV
distances are calculated for the starting configuration concerning the
reference structures of the edge-sharing ($\delta$) and corner-sharing
($\alpha$) phases of CsPbI3. Subsequently, we utilize these indices to define
the functional form of the CV as follows:

$S_{PIV}=f(r|r_{\alpha})/(f(r|r_{\alpha})+f(r|r_{\delta}))$ (5)

Where $f(r|r_{\alpha})$ is the PIV distance from the corner-sharing phase,
$f(r|r_{\delta})$ is the PIV distance from edge-sharing phase, $r$ are the
coordinates of the system, $r_{\alpha}$ are the coordinates of the reference
$\alpha$-phase phase and $r_{\delta}$ are the coordinates of the reference
$\delta$-phase. Using this functional form in Equation 5, we perform biased
simulations, and again successfully observe a similar solid-solid phase
transition pathway in CsPbI3.

Motivating from these exciting observations, we further investigate direct
solid-solid phase transitions with the machine learning potentials (MLPs). In
this work, Nequip MLPs[106] are built from dataset of forces, energies and
stresses acquired by performing DFT driven MD simulations of CsPbI3 in the
isothermal–isobaric ensemble.

However, the accuracy of DFT calculations may depend on the exact exchange-
correlation functional (XC) adopted in the procedure. Therefore, we first
conduct an elementary comparison of widely employed XC functionals against
reference first-principles many-body electronic structure method of random
phase approximations (RPA+HF)[107, 108, 109, 110, 111] which can account for
dynamic electronic screening and long-range dispersion interactions. Figures
3A-C show the relative ground state energies of different perovskite
polymorphs against the thermodynamically stable $\delta$-phase of CsPbI3. The
best results are obtained for the PBE+D3(BJ)[112, 113, 114, 115],
r2SCAN+rVV10[116], optB86b[117] functionals that include van der Waals
interactions. We note that although acceptable relative energies are obtained
with LDA[118, 119], however this might be due to error cancellations where
exchange energy is generally underestimated, and the binding energy is
overestimated. To choose the best, we perform an additional calculation of
ground state relative energies between $\gamma$-phase and manually constructed
polymorph of CsPbI3 based on the framework of Pb-I face-sharing octahedra[23].
This structure is prepared by replacing formamidnium cations with cesium into
hexagonal phase of formamidnium lead iodide (FAPbI3), illustrated in Figure
3D. Numerous experimental investigations[120, 121, 122] have consistently
demonstrated that the introduction of cesium into hexagonal-FAPbI3 rapidly
form the black perovskite phase, and prevent the formation of its hexagonal
face-sharing counterpart.

Figure 4:  Classifications of trajectories with SOAP and ACE

Consequently, one anticipates that a face-sharing structure doped with cesium
would exhibit instability and higher energy levels compared to the
$\gamma$-perovskite phase, a phenomenon corroborated by the ground state
energies presented in Figure 3D. Taking into account the comprehensive set of
results presented in Figure 3, we opt for the utilization of the r2SCAN+rVV10
functional, considering its suitable depiction of relative energies across
various structural configurations. It is worth noting that Shuxia Tao and co-
workers[123, 124] have also previously utilized SCAN+rVV10[125] to investigate
point defects in various perovskites. Subsequently, we perform DFT based ab-
initio MD (AIMD) simulations[126, 127] using the selected functional. Nequip
MLPs are generated based on the dataset acquired from the AIMD simulations.
The Methods section provide details of these simulations. Using MLPs, we again
perform a series of biased simulations to further scrutinize the phase
transformation dynamics. Remarkably, the observed pathway is similar to the
transformation sequence previously portrayed in Figure 1. This noteworthy
consistency between the outcomes obtained through distinct simulation
methodologies, RCPs and MLPs underscores the robustness and reliability of our
findings.

To further understand the structural transformations, next we perform a
detailed analysis of simulated trajectories using smooth overlapping of atomic
positions (SOAP)[128, 129, 130] and atomic cluster expansion (ACE) [131, 132,
133] descriptors.

Figure 5: Qualitative free energy profile for direct solid-solid phase
transition.

SOAP and ACE, known for capturing detailed atomic variations, are complemented
by principal component analysis (PCA) and uniform manifold approximation and
projection (UMAP)[134] for dimensionality reduction. This combined approach
enhances our ability to comprehend intricate molecular relationships and
dynamics. The resulting Figure 4 represents our simulated trajectories,
highlighting clusters of similar structural conformations and uncovering
underlying patterns in the simulated trajectories. Beyond the discerned
mixtures of corner-sharing and edge-sharing structures highlighted in Figure
4, our findings suggest the potential existence of distinct tilted perovskite
configurations (see clusters that are not circled and highlighted in Figure 4)
on the free energy surface, see Figure 5. It is noteworthy to acknowledge the
prior simulations conducted by Wiktor, Erhart and co-workers, have employed
Glazer notations to rationalize the presence of diverse locally tilted
octahedra during tilt-based phased transformations in cesium lead
bromide[135]. Whereas, we show that the wide range of tilted-perovskite phases
could exist on the free energy surface during solid-solid crystallization and
can be quantified using machine learning techniques. Our observations broadens
the understanding of the diverse structural landscapes in halide perovskites
and can be easily extended to wide range of perovskite materials, in
particular in oxide perovskites where tilt-based phase transformation play a
crucial role in determining their electronic properties for designing
electronic devices.

Nonetheless, to achieve a realistic material simulation of phase transition
dynamics, it is important to extend beyond the collective atomic motions of
approximately 1000s of atoms. It is crucial for capturing the detailed
macroscopic mechanisms and dynamics of solid-solid crystallization on a larger
scale ensuring a direct atomic visualization of experimental processes, in
contrast to small-scale simulations. This transition sets the stage for the
subsequent section, wherein we try to delve into the investigation of the
macroscopic aspects of solid-solid phase transitions.

To comprehend the macroscopic dynamics of solid-solid phase transitions, we
initiated biased simulations on relatively large-scale structures comprising
up to 35,000 atoms. As illustrated in Figure 2, our observations reveal that
the phase transition progresses through the sliding of octahedra. To emulate
this on large scales, we adopt an initial configuration of the $\delta$-phase,
characterized by the relatively larger dimensions along the X and Z
directions, however relatively smaller dimensions along the Y-direction,
approximately measuring 250 × 24 × 250 Å, illustrated in Figure 6A. Once
again, using enhanced sampling simulations with a multi-species structure
factor as CV, we excitingly witness the emergence of corner-sharing perovskite
structures, illustrated sequentially in Figures 6A-B-C-D. Upon analyzing the
trajectory data and Figure 6B, we again observe that the morphological
evolution occurs through the mixed formation of stack-faulted or edge-
dislocated configurations. This phenomenon is directly observable in Figures
6B-D, where significant amount of layers featuring mixed edge-sharing and
corner-sharing arrangements are crystallized during the phase transition.
Supplementary movie SM2 provides a detailed atomic-scale depiction of this
phase transformation.

However, a crucial insight emerges from our large-scale simulations. As
demonstrated in Figure 6D, the process gives rise to several defects,
particularly observed in the neighbouring regions where mixed-corner-edge-
sharing layers have crystallized. Despite the subsequent conversion of these
mixed configurations to full corner-sharing perovskite structures during the
biased simulations, the formation of defects is evident, and defects prevailed
in the final configuration. These are easily recognizable by incomplete or
broken octahedra (and empty spaces) in Figure 6D. In opposite to that, where
parts of the $\delta$-phase that undergo direct conversion to the perovskites
instead of mixed corner-edge-sharing structures, exhibit fewer defects, and
can be readily visualized in Figures 6A, B and Supplementary movie SM2. From
this comprehensive analysis, we propose that one of the mechanism of
underlying defect formation in halide perovskites may strongly involve the
formation of long-range intermediate structures during the crystallization
process.

Simulating structural transformations arising from biased simulations
involving more than 35,000 atoms surpasses the capabilities of current code
implementations and available computational resources. As a result, in the
following section, we undertake brute force MD simulations to investigate
various aspects of solid-solid crystallization in greater depth.

Figure 6:  Solid-solid phase transition by enhanced sampling: Figure shows the
simulated transformation of edge-sharing $\delta$-phase (A) to corner-sharing
perovskite-phase of CsPbI3. Selectively Pb-I structures are represented as
blue octahedra to guide the eye.

##  III Large scale seeded crystallization

Figure 7:  Seeded simulations: (A) Snapshot depicting the starting
configuration with a seed size of 3.7 nm, and (B) illustrates its time evolved
configuration. (C) Presents a snapshot of the initial structure with a seed
size of 7.0 nm, while (D) showcases the evolved structure. To enhance clarity,
middle slices from the 3D box’s center are shown here to guide the eye. The
representation is focused on Pb-I octahedra for clear distinction between the
two phases.

Earlier experiments[136, 137, 79, 76, 138] and simulations[139, 140, 72]
investigating solid-solid phase transitions in materials other than
perovskites, have noticed that following the formation of initial smaller
nuclei in a foreign crystalline solid, the subsequent solid-solid
crystallization process may obey classical nucleation pathways[141, 142, 62,
143, 144] on larger length scales. Seeking a fundamental macroscopic level
understanding in perovskite, we perform unbiased brute force seeded
simulations[145, 146, 147, 148, 149, 150, 151], a sophisticated approach that
could allows us to explore and probe underlying mechanisms governing
macroscopic solid-solid phase transitions in halide perovskites.

Figure 8:  Seeded simulations: Figure shows the time evolution of amount of
perovskite phase calculated using lq4.

To start with, we prepare different structural configurations by incorporating
spherical structures (seeds) of perovskite phase into extensive yellow-phase
structures, details are provided in the Methods section. We select perovskite
seeds with varying radii ranging from 3.0 to 7.0 nm which could offers a
comprehensive exploration of nucleation and growth processes at different
initial conditions. Starting from equilibrated configurations with different
initial seeds we carry out brute force MD simulations using RCPs and monitor
the time evolution of fraction of perovskite in these systems. To assess the
quantity of perovskite within our simulations, two distinct metrics are
employed. Firstly, the relative intensity of the corresponding (100) X-ray
peak of perovskite-phase is utilized, allowing for tracking and comparison
with experimental data. Additionally, a local-bond order parameter lq4[152,
153, 93], is employed to precisely quantify the number of formula units
corresponding to different phases within the simulation box. To compute the
X-ray peaks for these extensive structures, we employ a simplified Debye form
of structure factor, more simulation details are provided in the Methods
section.

Figure 9:  Seeded simulations: Figure shows the time evolution of amount of
perovskite phase calculated using structure factor.

From simulations, we observe that as the fraction of perovskite decreases, the
inserted seed becomes subcritical and completely transform into
$\delta$-phase. Conversely, as the fraction increases, the cluster transitions
to a post-critical state. This approach allows us to ascertain whether the
inserted cluster is critical, similar to the methodologies employed in earlier
seeded simulations involving ice[144] and carbon-graphite-diamond
transitions[72]. Figures 7(A) and 7(C) depict the initial configurations for
seed sizes of 3.7 nm and 7.0 nm, respectively, serving as the starting points
for our simulations. Subsequently, Figures 7(B) and 7(D) show the concluding
configurations for the respective seeded configurations. First, the visual
analysis of these figures directly reveals that the smaller perovskite seed
experiences significant contraction, as illustrated in Figure 7(B), ultimately
transforming into the $\delta$-phase characterized by an increase in edge-
sharing structure.

Figure 10:  Seeded simulations with 7 nm seed: Figure illustrates the time
evolution of the normalized (100)-peak in the structure factor at different
temperatures for a seed radius of 7.0 nm.

Conversely, the larger seed (7.0 nm) exhibits substantial growth, resulting in
the formation of a very larger perovskite crystal structure. Figures 8 and 9
illustrate the perovskite fraction over time for various simulated perovskite
seeds.

Interestingly, a noteworthy observation arises at the boundary between the two
phases, where the atoms exhibit liquid-like behavior. This underscores a
dynamic moving interface between the coexisting phases, suggesting that solid-
solid phase transition is facilitated by the propagation of a liquid-like
front[154, 76, 144, 57]. The interplay between liquid-like behavior and
crystallization at the phase boundary adds a layer of complexity to the
understanding of solid-solid crystallization in perovskites.

To complement with visual observations, simulation movies SM3 and SM4 are
provided in the SI for the dynamical insights into the temporal evolution of
these systems. Importantly, it is crucial to note that these simulations are
conducted at significantly lower temperatures of 600, 650 and 700K than the
calculated melting temperature 800K of CsPbI3, see the Methods section for
details. This deliberate choice ensures the liquid-like front does not results
from the melting of the crystal.

In the following sections, we delve into the mechanistic details of the solid-
solid nucleation mechanism. Figure 10 offers an insightful perspective into
the growth profile of a larger seed across different temperatures. The impact
of temperature on growth rate is evident, with a pronounced acceleration at
higher temperatures beyond 650K, similar to the earlier reported
experiments[155]. However, this heightened growth rate is simultaneously
accompanied by the emergence of defects within the crystal structure.
Subsequently, we scrutinize these configurations and find the formation of
different type of point defects, as portrayed in Figure 11. For clarity, we
selectively depict connected Pb atoms, highlighting displaced or defected Pb
atoms with cross marks on the evolving perovskite lattice.

Furthermore, we conduct room-temperature relaxation of these final structures
at first-principle level of calculations. Performing direct ab-initio
simulations for systems of this magnitude is utterly unfeasible. We achieve
this by employing MLPs through the neuroevolution potential (NEP) at
r2SCAN+rVV10 level and using GPUs accelerate MD (GPUMD) code for molecular
dynamics simulations of these large scale configurations. Detailed procedures
are explained in the Methods section. These structurally relaxed
configurations can serve as valuable input for the comparisons with
transmission electron microscopy (TEM) experiments and spectroscopic
experiments, thereby offering direct visualizations for tracking the phase
transition, formation of defects and help experiments to obtain atomic-level
details of the heterogeneities during crystallization process.

Moreover, a significant finding in our investigation is related to the dynamic
evolution of the perovskite phase. We discern a distinct discrepancy in the
evolution rates of different facets of the perovskite phase. Specifically, the
(100)-facet displays accelerated growth compared to its counterparts, as
evident in Figure 7 and Figure 11. These figures illustrates the rapid
expansion of the (100)-facet boundary, originating from an initial spherical
seed. However, when considering the boundary of the (110)-facet and the
$\delta$-phase, interestingly, we observe the formation of a coherent phase
between them. Figure 11 provides direct evidence of this phenomenon, where the
green color interface highlights the (110)-facet of the grown crystal, while
the yellow color interface represents the (100)-facets.

Figure 11:  Seeded Simulations. Figure displays a configuration of the
ultimately crystallized structure. For clarity, only a slice from the center
of the seed is presented, featuring connected Pb atoms exclusively to guide
the eye. Figure 12:  Solid-solid co-existing simulations of
(100)${}_{\delta}\|(100)_{\alpha}$: Figures A to E illustrate the temporal
progression of structural changes during co-existing simulations. For clarity,
only magenta-colored Pb-I octahedra are displayed alongside yellow cesium
atoms. These visualizations were generated using VMD software[156].

To further investigate this growth process of respective (100)-perovskite-
facet, we perform solid-solid co-existing simulations of utilizing NEP MLPs
and GPUMD code. Detailed methodologies and simulation set-ups are provided in
the Methods section. Brute force MD simulations are conducted at temperatures
of 600K, 650K, and 700K, revealing a layer-by-layer transformation[23] of the
$\delta$-phase into the perovskite phase at different temperatures. This whole
transformation is illustrated in Figure 12 and Supplementary Movie SM5.
Further characterization of the layer-by-layer transformation is evaluated by
calculating the change in the number of Pb species belonging to the perovskite
phase using the previously defined lq4. Figure 13 shows the time evolution of
the perovskite phase across eight repeated MD runs started with the variable
momenta. The step-increase in the number of perovskite Pb atoms indicates that
upon the onset of perovskite crystallization at the co-existing interface, a
complete layer of the $\delta$-phase transitions into the perovskite phase.
Additionally, we computed the free energy profile of this process using the
equation F = kBT log(p($lq4$)). The free energy profile of the co-existing
simulations at T=700K is depicted in Figure 14 and eventually indicate a
layer-by-layer nature of the transformation, as evidenced by the individual
free energy minima separated by barriers on the order of kBT.

Figure 13:  Time evolution of local bond order parameter (lq4$>$0.5) for
solid-solid co-existing simulations of (100)${}_{\delta}\|(100)_{\alpha}$
Figure 14:  Free energy profile of the solid-solid co-existing simulations of
(100)${}_{\delta}\|(100)_{\alpha}$

Now, we investigate scenarios involving sub-critical nuclei. In such
instances, we observe a distinctive behavior in which the (110)-facet of the
perovskite seed undergoes conversion to the [010]-facet of the $\delta$-phase
at an accelerated rate compared to other facets, as highlighted in Figure 7 B.
Furthermore, we also perform co-existing simulations of the (110)-perovskite
facet with the $\delta$-phase at 700 K are aimed to understand the interface
dynamics. Interestingly, seven time repeated simulations lasting up to 2
microseconds did not show any significant increase in the perovskite
structure. This stability contrasts with the behavior observed for the (100)
perovskite facet, where co-existence with the $\delta$-phase led to a complete
conversion to perovskite structure by formation of a dynamical interface with
a liquid-like character as shown in Figure 12C and D. In the case of the (110)
facet, however, a distinct interfacial behavior emerged. Here, the (110) facet
formed a well-defined and stable boundary with the $\delta$-phase, lacking the
mobile characteristics of the boudary atoms observed with the (100) facet.
This difference in interfacial dynamics between the two facets suggests a
potential influence of crystallographic orientation on the stability and
transformation behavior of the perovskite phase.

At the end, we try to analyze and compute the associated free energy barrier
with the seeded crystallization process. We achieve this by gathering multiple
histogram data derived from X-ray peak intensity across multiple seeded
simulated runs, and combining them into a single run.

Utilizing the combined data, we calculate the free energy using the equation F
= kBT log(p($q_{100}$)), where p($q_{100}$) represents the probability
distribution of the (100)-perovskite peak, as explained in prior discussions
and detailed in the Methods section. Figure 15 reveals a qualitative free
energy barrier, highlighting the necessity to overcome an energy barrier on
the order of 10 kBT for solid-solid nucleation at 700K. Excitingly, our
collective observations pave the way for quantifying the rates of solid-solid
nucleation process at different conditions utilizing a modified classical
nucleation theory framework[157, 79]:

Figure 15:  Seeded simulations: Figure presents the qualitative free energy of
crystallization across various seeded systems.

$\Delta G=-V\rho\Delta\mu+S\gamma+E_{strain}+E_{defect}$ (6)

Here, S represents the surface area, and V denotes the volume of the
perovskite phase. Additionally, $\rho$ signifies the number density of the
product nuclei, $\Delta\mu$ denotes the chemical potential difference between
the bulk phases, $\gamma$ represents the surface tension, and $E_{strain}$
signifies the positive misfit strain free energy per unit volume, which
depends on the elastic modulus of the solids. Finally, $E_{defect}$ refers to
the energy of pre-existing defects per unit volume V.

##  IV  Discussion and Conclusion

In conclusion, our study offers new insights into the atomistic mechanism for
the solid-solid phase transitions in CsPbI3 perovskites. Numerous X-ray
crystallography experiments investigating the synthesis process have already
established the active and crucial involvement of edge-sharing crystalline
intermediate structures in the overall crystallization of perovskite thin-
films. From simulations of direct phase transition from edge-sharing non-
perovskite to perovskite structure, we find the formation of intermediate
mixed edge-corner sharing structures that commonly identified as stacking
faults or dislocation in wide range of perovskites. This knowledge is
complemented by earlier findings from transmission electron microscopy (TEM)
experiments[158], which have identified analogous intermediate edge-sharing
stack-faulted domains in halide perovskites. Most importantly, photoemission
electron microscopy based experimental investigations[159] have further
elucidated the presence of edge-dislocations and stack-faulted structures is
harmful to the stability of halide perovskite under operational conditions.
Recent experiments also revealed their possible photoplastic[160, 161, 162]
nature of such structures, and signifying their capability to undergo movement
under the influence of light during solar cell operations, therefore
highlighting them as possible notorious degradation centers for PSCs.

From our results and analysis, one can conclude that the synthesis
methodologies that involve the direct heating of the $\delta$-phase to
perovskite can give rise to the formation of these defected intermediate
structure and potentially stable on free energy surface. Hence, a careful
attention is required in the development of large-scale industrial processes
for producing PSCs those involving the direct annealing of the $\delta$-phase
to perovskite. Secondly, our extensive investigation through large-scale
seeded simulations, have unveiled crucial finding of the preferred occurrence
of nucleation on a macroscopic level, characterized by a continuous transition
marked by the formation of well-defined faceted perovskite crystals.
Significantly, our simulations reveal the accelerated growth of the
(100)-facet, which efficiently cut through the $\delta$-phase.

We observed that (100)-oriented seeds promoting the formation of the (100)
facet could strategically optimize the perovskite crystallization process.
Earlier experiments[3, 7, 8] on FAPbI3 demonstrated that the controlled solid-
solid phase transitions have produced highly efficient and stable PSCs.
Building on our current study and observations of layer-by-layer
transformation via a highly mobile interface, we propose two industrial scale
synthesis methodologies to crystallize defect-free, and highly crystalline
thin-films for PSCs and perovskite LEDs. First, strategies of surface vapor
treatment-based can be devised to produce highly crystalline thin-film PSCs by
a top to bottom layer-by-layer transformation as demonstrated by earlier
experiments on FAPI3. Alternatively, molecular additives and chlorine can be
employed, for instance, to initially create (100)-oriented domains of quasi-2D
perovskite or Cs2PbI2Cl2 serving as seeds for the uniform conversion of the
$\delta$-phase to the complete perovskite structure. Therefore, simulations
can ultimately contribute to the production of stable perovskite electronics.

Moreover, we provide valuable insights into how a potentially stable boundary
may form between the (110)-facet of the perovskite crystal and its
corresponding $\delta$-phase. Specifically, it may occur when the perovskite
crystal reaches a critical size. Drawing inspiration from previous work by
Haibo Zeng and co-workers[155], who engineered co-existing
$\alpha$/$\delta$-CsPbI3 thin-films to obtain all-inorganic white LEDs[163],
our study provides a platform to understand the impact of different process
conditions for example effect of temperatures on free energy barriers for
stabilizing the $\alpha$/$\delta$-interface.

Therefore, we open avenues for enhancing the efficiency and stability of PSCs
and perovskite LEDs building from all-atom MD simulations.

Our study has been limited. Specifically, we have not fully explored the
formation and significance of more complex structures, such as Cs4PbI6[14],
which could play a crucial role in the synthesis of perovskites. Therefore,
looking ahead, our research road-map involves exploring the influence of non-
stoichiometric complex structures, seed shapes and commonly used additives on
phase transitions through precise finite temperature molecular dynamics
studies of nucleation, utilizing large scale machine learning potentials
derived from accurate quantum chemical calculations[164, 165, 166, 167, 168,
169, 170]. Accurate calculations of free energy barriers for the multi-step
crystallization process will serve as a crucial parameter for calculating the
rates of phase transitions to determine the actual kinetic stability of these
meta-stable perovskites: one of the key unsolved problem for halide
perovskites.

We have taken elementary steps in understanding the complex crystallization of
multi-species materials and towards the design of atom by atom industrial
process design, however importantly, the first-principles-inspired all-atom
computational framework presented in this study offers promising opportunities
for investigating complex structural transformations across a diverse range of
multi-species materials. Examples include ferroelectric[171, 172, 173, 174,
175, 176, 177, 178], multiferroics[179], and superconducting[180, 181, 182,
183, 184, 185] materials with their potential to reduce global electricity
consumption[186, 187, 188, 189].

##  V Methods

###  V.1 MD simulations with RCPs

We utilized a reduced point charge[190, 191] inter-atomic potential for CsPbI3
following the descriptions provided in the references[57]. The production runs
are carried out in the isothermal-isobaric ensembles, using the Large-scale
Atomic/Molecular Massively Parallel Simulator (LAMMPS) code (version 31 Mar
2017)[192]. Non-bonded interactions were restricted by a 1.0 nm cutoff, and
electrostatic interactions were managed using the particle-particle-particle-
mesh Ewald method. Temperature control was achieved using velocity re-scaling
thermostat[193] with a relaxation time of 0.1 ps, while pressure control
utilized a Parrinello-Rahman barostat[194, 195, 196] with a relaxation time of
10 ps. OPES and umbrella sampling biased simulations were conducted using
PLUMED 2.8[197, 198], interfaced with LAMMPS. To scrutinize finite-size
effects concerning yellow to black phase transitions, we employed supercells
of varying sizes, ranging from 32, 128, 512 to 1024 formula units (f.u.) of
$\delta$-CsPbI3. In the context of seeded simulations, we started by
generating three distinctive configurations featuring spherical perovskite
seed sizes with radii of 3.7 nm, 5.0 nm, and 7.0 nm. These seeds were embedded
within $\delta$-phase structures, characterized by approximate dimensions of
16x16x16 nm (88540 atoms), 23x23x23 nm (272310 atoms), and 33x31x33 nm (675180
atoms), respectively. It is crucial to emphasize that the selection of the
simulation box size, representing $\delta$-phases, was purposefully set to be
more than at least four times the radius of the perovskite seeds. This
consideration ensures that the perovskite seeds do not interact with each
other within the confines of periodic boundary conditions in MD simulations.
Additionally, meticulous attention was given to maintaining suitable distances
between pairs of atoms, aligning with the Pb–Pb, Pb-Cs, and Pb-I distances
between the two phases. Subsequent to the observation of perovskite seed
growth and dissolution in these three initial systems, we expanded our
configurations. Additional simulations were conducted, introducing perovskite
seeds with radii of 4.0 nm, 4.5 nm, 4.8 nm, 5.0 nm, 5.3 nm, 5.5 nm, and 6.0 nm
into a simulation box with dimensions approximately 33x31x33 nm, encompassing
around total 678770 atoms. These subsequent simulations were intentionally
designed first to repeat earlier simulations, and to offer qualitative
insights into the associated free energy barrier for the process. The
structure factors were computed employing a methodology similar to previous
studies[10], and are commonly depicted, as shown in Figure 16, for different
polymorphs. To facilitate analysis of the crystallization process, we have
normalized the intensities relative to the initial configurational
intensities, thus initiating values from 1.0 in Figures 9 and 10.

Figure 16:  Debye structure factors for perovskite($\alpha$) and non-
perovskite($\delta$) phases of CsPbI3.

####  V.1.1 Melting point

To determine the melting point of CsPbI3 using the reduced charge inter-atomic
potential, we carried out co-existing simulations[87, 199] involving
crystalline seeds composed of 6400 atoms in contact with a melt containing
equal number of atoms. Subsequently, isothermal-isobaric simulations were
executed over a temperature range spanning from 720 to 850 K. The progression
of potential energy at these temperatures is illustrated in Figure 17,
indicating that perovskite melts above 800 K and grows below 800 K. It is
noteworthy that the experimental melting point of CsPbI3 is around 750 K[200].

Figure 17:  The figure shows the time evolution of potential energy in co-
existing simulations at different temperatures.

####  V.1.2 local q4 analysis

To quantify the number of formula units within the perovskite phase, whether
in direct coexistence simulations or clusters in seeding simulations, we
employ the rotationally invariant local-bond order parameter, denoted as
lq4[152, 153, 93, 197, 198]. This parameter is determined by the relative
positions of a tagged particle and its neighbors within a defined distance. In
our investigation, we restrict the calculation of lq4 to the Pb species within
the systems, considering a cutoff distance of 8 Å. Figure 18 illustrates the
histogram of lq4 values of a finite-temperature simulated trajectories for
perovskite and non-perovskite phases. Notably, a threshold can be established
at lq4 $>$ 0.5, enabling the calculation of the number of perovskite formula
units in a given system. To further validate this approach, we conduct a
simple test on a trajectory created by combining an equal number of frames
from simulated perovskite and non-perovskite phases. As depicted in Figure 19,
lq4 effectively distinguishes and counts only the number of perovskite units.
Similarly, the quantity of $\delta$-phase can be assessed by considering lq4
$<$ 0.33.

Figure 18:  Histogram illustrating the distribution of $q_{4}$ for a combined
simulated trajectory of $\alpha$ and $\delta$ phases. Figure 19: Time
evolution of local $q_{4}$ exceeding a threshold of 0.5 in a combined
simulated trajectory featuring both $\alpha$ and $\delta$ phases.

###  V.2 RPA and DFT calculations

In this study, all DFT and RPA calculations were executed using the VASP
code[201, 202]. The projected Aagmented wave functions (PAW) with PBE
potentials[203] were employed for DFT calculations, and LDA potentials were
used for LDA functional. RPA+HF results were also obtained using PBE PAW
potentials, given their comparability with those derived from PAW GW
potentials[111]. We systematically compared various exchange-correlation (XC)
functionals and dispersion methods, including LDA[118, 119], PBE[204],
PBEsol[205], r2SCAN[206], PBE+D3(BJ)[114, 115, 113, 112], r2SCAN+rVV10[116],
optB86b[117] and RPA+HF[109, 107, 108, 110, 111]. Each polymorph was
represented using 20-atom supercells, and a cutoff energy of 600 eV was
established. Different k-point grids of 4x10x2, 6x6x4, 6x6x4, 4x6x4, and 6x6x4
for DFT calculations and for RPA+HF calculations k-point grids of 2x6x1,
2x3x2, 2x3x2, 2x3x2, 2x3x3 were used for $\delta$, $\gamma$, $\beta$,
$\alpha$, and manually constructed face-sharing phase, respectively. AIMD[126]
simulations using k-points were conducted to generate the training dataset,
utilizing the r2SCAN+rVV10 functional. These simulations involved 40-atom
supercells for each polymorph and were carried out within the isothermal-
isobaric (NPT) ensemble, employing a 2fs time step. Pressure and temperatures
were controlled using the Parrinello-Rahman barostat and Langevin
thermostat[207], respectively.

###  V.3  Machine Learning Potentials

####  V.3.1 Nequip MLPs

The Nequip[106] MLPs are derived from AIMD simulations data based on
r2SCAN+rVV10 DFT level of theory, as described in above sections. For the
training process, we employed three interaction layers, implementing a radial
cutoff of 8 Åand embedding interatomic distances in a basis comprising eight
Bessel functions. Feature representations were selected up to rotation orders
of l=2. The loss function incorporated forces, energies, and stresses obtained
from DFT calculations, and optimization was executed using Adam with a
learning rate of 0.003 and a batch size of 10. The training procedure was
conducted on a V100 GPUs.

####  V.3.2 NEP-GPUMD simulations

For large-scale simulations using MLPs, we utilized the neuroevolution
potentials (NEPs)[208, 209] model in GPUMD code[210, 211], as extensively
discussed recent publications[212, 213]. NEPs feature a neural network
architecture where local atomic environments are delineated by radial and
angular components. The radial component of the atomic environment descriptor
is designed through linear combinations of Chebyshev basis functions, while
the three-body angular part is similarly constructed from Legendre
polynomials. In this study, we tried two different setups, first with a radial
and angular cutoffs at 6Åand 5Å, respectively, and second with using radial
and angular cutoffs cutoffs at 8Åand 4Å, respectively, both with radial and
angular expansions at orders 12 and 8. The neural network architecture
consisted of hidden layer comprising 50 neurons, and the training process
involved 100,000 generations. The ultimate model for CsPbI3 was trained based
on forces, energies, and stresses derived from r2SCAN+rVV10 DFT calculations.
In order to perform co-existing solid-solid simulations, we prepare
configurations comprising different polymorphs with a minimal lattice mismatch
between them. Specifically, for the case of (100)${}_{\delta}\|(100)_{\alpha}$
, a supercell consisting of 2520 atoms (with dimensions of 88.046 x 37.734 x
37.734 Å) representing the $\alpha$-phase is combined with a 2560-atom
supercell (with dimensions of 83.472 x 38.320 x 35.522 Å) representing the
$\delta$-phase. The final configuration is attained by averaging the lattice
parameters, resulting in dimensions of 175.0 x 37.847 x 36.448 Å. For the case
of (100)${}_{\delta}\|(110)_{\alpha}$ , we first create the (110)-facet of the
$\alpha$-phase utilizing ase utilities. Subsequently, it is combined with a
supercell of the $\delta$-phase, ensuring minimal mismatch between the two
structures. All structures are provided in the supplementary materials.

## Acknowledgements

This research is funded by Swiss National Science Foundation through Post.Doc
mobility fellowship $P500PN\\_206693$. I am sincerely thankful to Professors
Daan Frenkel, Michael Graetzel, Nitin Padture, Ali Alavi, Volker Heine, and
Carlos Vega for helpful discussions, encouragements, and useful guidance. I
thank Professors Erin Johnson, Michael Sprik, Gabor Csyani, and Alessandro
Laio for helpful insightful discussions. I am greatly thankful to Dr. Felix
Musil for discussion and help regarding SOAP kernels. I thank Dr. Michele
Invernizzi for discussions on enhanced sampling simulations. I am greatly
thankful to Dr. Jiahuan Zhang, Dr. Zaiwei Wang, Dr. Chen Li and Dr. Pepe
Marquez for highly insightful discussions on experimental data of phase
transitions between $\delta$-$\alpha$ phase transitions. I thank Ben Shi and
Dr. Michele Simoncelli for carefully reading the manuscript and providing
helpful comments. I am indebted and greatly thankful to Yusuf Hamied
Department of Chemistry, Professor Angelos Michaelides for hosting and
providing necessary computational resources through Rogue-GPU cluster, Nest-
CPU cluster, UK Materials and Molecular Modelling Hub YOUNG CPU/GPU cluster,
which is partially funded by EPSRC (EP/T022213/1, EP/W032260/1 and
EP/P020194/1) and ARCHER2 UK National Supercomputing Service.

## Data availability

All parts of this study are easily reproducible: all kinds of input/output
files, starting configurations, notified output configurations, plumed input
files, lammps input files, RPA/DFT input files and MD trajectories are
available on open access server Zenodo: 10.5281/zenodo.10938457, Github:
https://github.com/ahlawat-paramvir/plumed2, https://github.com/ahlawat-
paramvir/CsPbI3_phase_transitions and also from author.

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