# Missed ferroelectricity in methylammonium lead iodide

Wen-Yi Tong  Theoretical Materials Physics, Q-MAT, CESAM, Université de Liège,
B-4000 Liège, Belgium  Jin-Zhu Zhao<EMAIL_ADDRESS>Guangdong Provincial
Key Laboratory of Quantum Engineering and Quantum Materials,Guangdong-Hong
Kong Joint Laboratory of Quantum Matter, School of Physics and
Telecommunication Engineering,South China Normal University, Guangzhou 510006,
P. R. China  Center for Computational Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, P. R. China  Philippe
Ghosez<EMAIL_ADDRESS>Theoretical Materials Physics, Q-MAT, CESAM,
Université de Liège, B-4000 Liège, Belgium

###### Abstract

Methylammonium lead iodide, as related organometal halide perovskites, emerged
recently as a particularly attractive material for photovoltaic applications.
The origin of its appealing properties is sometimes assigned to its potential
ferroelectric character, which remains however a topic of intense debate.
Here, we rationalize from first-principles calculations how the spatial
arrangement of methylammonium polar molecules is progressively constrained by
the subtle interplay between their tendency to bond with the inorganic
framework and the appearance of iodine octahedra rotations inherent to the
perovskite structure. The disordered tetragonal phase observed at room
temperature is paraelectric. We show that it should a priori become
ferroelectric but that iodine octahedra rotations drive the system toward an
antipolar orthorhombic ground state, making it a missed ferroelectric.

Organometal halide perovskites (OMHPs) of chemical formula ABX3 – with A an
organic molecule, B a metal atom and X a halogen atom – possess the same
aristotype cubic structure as conventional inorganic perovskites but
incorporate at the A-site an organic molecule, possibly polar and which can
potentially rotate. Due to their appropriate bandgap [1, 2], high absorption
coefficients [3] and exceptionally long carrier lifetimes and diffusion
lengths [4, 5, 6], OMHPs, and in particular lead-based compound like
methylammonium lead iodide (CH3NH3PbI3, or MAPbI3), produced recently a
revolution in the field of photovoltaics [7, 8, 9]. Despite the tremendous
progress regarding device efficiency, the physical origin of the extraordinary
performance of MAPbI3 solar cells is still poorly understood. Ferroelectric
(FE) domains, acting as internal p-n junctions that efficiently separate and
transport the photoexcited electron-hole pairs [10, 11, 12], are believed to
be an important factor explaining their efficiency.

Definitive consensus and explanation regarding the FE character of MAPbI3 and
related perovskites are however still missing [13]. On the one hand,
experimental fingerprints of ferroelectricity were reported in single
crystals, relying on quasielastic neutron scattering [14] or dielectric
measurements [15, 16]. Some FE hysteresis loops were also measured at low
temperature (200K) [17, 18]. Nevertheless, no clean spontaneous polarization
measurement has been reported so far for MAPbI3 at room temperature, due to
the relatively high electrical conductance and low stability under high
applied voltage bias [19]. FE/ferroelastic domains were observed in the
tetragonal phase of MAPbI3 polycrystalline films [20, 21, 22] as well. On the
other hand, some independent reports [23, 24, 25, 26] questioned or even
excluded the possibility of a FE character which remains therefore under hot
debate.

Most popular ferroelectrics, like BaTiO3 or PbTiO3, belong to the family of
inorganic perovskites, suggesting that ferroelectricity might be inherent to
the perovskite structure. However, it is worth noticing that most ABO3
perovskites with a small Goldschmidt tolerance factor $t$ [27] are not FE due
to a typical competition between polar lattice distortions and
antiferrodistortive (AFD) rotation motions of the BO6 octahedra [28, 29]. The
latter motions usually dominate for $t<1$ in ABO3 compounds and suppress the
FE instability, giving rise to a common non-polar $Pnma$ ground state that
arises from the joint condensation of in-phase and out-of-phase rotation
motions, according to an $a^{-}b^{+}a^{-}$ rotation pattern in Glazer’s
notations [30].

With its small tolerance factor ($t=0.912$ [31]), MAPbI3 seems perfectly in
line with this generic behavior. Stable at high-temperature in a cubic
$Pm\bar{3}m$ phase, it exhibits on cooling two consecutive phase transitions
arising from the condensation of AFD distortions: a first transition at 327 K
to a tetragonal phase a priori of $I4/mcm$ symmetry showing an AFD distortion
pattern $a^{0}a^{0}c^{-}$ and a second transition at 162K to an orthorhombic
ground state of $Pnma$ symmetry with the typical distortion pattern
$a^{-}b^{+}a^{-}$. As such, MAPbI3 seems to behave like a conventional non-FE
perovskite. However, while there is a broad consensus on the non-polar $Pnma$
character of the ground state [25, 32, 33], the exact assignment of the
tetragonal phase to either the non-polar $I4/mcm$ or instead a polar $I4cm$
space group with potential FE character is still unclear.

Figure 1: Preferred orientations of MA+ molecules under different phases.
Herein, balls represent the projection of the C-N bond axis from C to N atom.
In the cubic phase, 8 blue balls and 24 light-blue balls correspond to
$<$111$>$\- and $<$100$>$-oriented cations, respectively. For the tetragonal
phase, preferred orientations of the organic ions reduce to four (green
balls), lying in the $<$101$>$ directions. When it goes to the orthorhombic
phase, the orientation of MA+ molecules is fixed. Tetragonal phases in the
same energy region within various polarization states, as well as the
antiferroelectric orthorhombic phase are displayed in the bottom part. Their
lattice constants are given in the basis of corresponding superlattices. Note
that different tetragonal phases are entitled according to their net
polarization directions. Molecules of all of them lie in $<$101$>$.

A key distinction between inorganic ABO3 perovskites and MAPbI3 is the
presence of polar molecules at the A site, which introduces the possibility of
an order-disorder FE transition like in PVDF [34]. As anticipated in previous
studies [35], not only the displacive mechanism of inorganic perovskites
(polar soft mode) but also the order-disorder arrangement of the polar MA+
molecule can contribute and collaborate to make MAPbI3 FE. Several studies
have highlighted the complexity of MA+ arrangement [36, 37, 14, 38, 39, 40,
41, 42, 43, 44] and its important role on controlling the electronic/optical
properties of MAPbI3 [45, 46, 37, 11, 47, 48, 49, 50]. Moreover, MA+ dipole
direction is believed to closely correlate with AFD rotations and tilts of
PbI6-octahedra through hydrogen bonding [51, 32, 52, 33, 39]. Exploring how
the complicated interplay between MA+ molecules and the PbI6-octahedra network
might impact FE properties is thus essential. However, no global picture
describing the evolution of molecule orientations, polar distortion and AFD
motions in all three phases has been reported yet.

Here, using first-principles density functional theory (DFT) calculations, we
systematically investigate the FE properties in MAPbI3 going step by step from
the cubic to the tetragonal and then orthorhombic phases. We rationalize the
impact of AFD distortions on the preferential orientations of the MA+
molecules. From this, we explain the non-FE character of the orthorhombic
phase and to which extent the tetragonal phase can be considered as FE. Our
work intents to reconcile previous observations, providing a simple unified
and comprehensive picture regarding the influence of AFD motions on the
potential FE character of MAPbI3.

## RESULTS

Cubic Phase. MAPbI3 is a typical ABX3 perovskite but that includes at the A
site an ionized molecule MA+ which, according to the difference of
electronegativity of C and N atoms, is moreover polar (with its dipole moment
pointing from C to N). This polar molecule breaks by itself the symmetry of
the aristotype cubic perovskite structure. At high temperature, the MA
molecule can however rotate relatively freely and MAPbI3 is reported in a
cubic $Pm\bar{3}m$ symmetry [53, 25]. This is nevertheless an average symmetry
resulting from the random orientation of the MA molecules. The disordered and
dynamical character of the $Pm\bar{3}m$ phase makes it more hardly accessible
from static DFT calculations with periodic boundary conditions than other
static cases.

In order to investigate the energy landscape within which the MA molecule is
rotating and its preferential orientations, we consider first a $1\times
1\times 1$ cubic cell with Pb and I atoms fixed at high-symmetry positions,
and relaxed the molecule position for various orientations. The results are
illustrated in Fig. 2(a), summarizing the evolution of the energy for the C-N
axis of the molecule rotating from the $<$100$>$ to the $<$110$>$ and then the
$<$111$>$ directions of the cubic unit cell. Consistently with previous
reports [50, 52], we observe two close local energy minima $E_{<100>}$ and
$E_{<111>}$ associated with preferential orientations of the MA+ molecule
along the $<$100$>$ and $<$111$>$ directions, respectively. The optimized
lattice parameters are 6.317 Å (6.305 Å) for the $<$111$>$ ($<$100$>$) state,
in good agreement with the experimental value of 6.31 Å [54]. Similarly to
Yin’s calculations [55], $E_{<111>}$ is $\sim$ 6 meV per formula unit (f.u.)
lower than $E_{<100>}$. However, the relative stability of both states can be
reversed when properly relaxing the octahedra cage ($E_{<111>}-E_{<100>}=12$
meV/f.u.), in line with what was observed in some alternative studies [56, 57,
58]. In practice, these states can be considered as nearly energetically
equivalent.

Figure 2: MA+ molecule’s orientation in cubic phase. (a) Evolution of energy
when the axis of MA+ molecules is oriented along different principal
directions ($1\times 1\times 1$ cubic cell). The lowest-energy configuration
with $<$111$>$-oriented molecules is taken as energy reference. (b) Sketch of
the two cubic states of lowest energies. Molecules in the left and right
configurations are pointing toward $[111]$ and $[100]$ directions
respectively. The three bonds formed between nitrogen-end-type hydrogen atoms
and iodine atoms are displayed.

The atomic configurations associated with the $E_{<111>}$ and $E_{<100>}$
minima are represented in Fig. 2(b). Both the off-centering of the molecule
and its slight angular deviation from the ideal $[100]$ direction in the
$<$100$>$ state (see right panel of Fig. 2(b) and up panels of Fig. 3(b))
highlight the formation of hydrogen bonds between the molecule and the
inorganic backbone [32, 33, 39], i.e. bonds between the three nitrogen-end-
type hydrogen atoms (HN) and iodine atoms, as well as the trend to make these
three HN-I bonds equivalent. We notice that properly reproducing this in the
calculation is not necessarily straightforward and requires some special care
during the atomic relaxation (i.e. properly rotating the molecule about the
C-N axis). As summarized in Fig. 1, the molecule will therefore preferably
align along one the of eight $<$111$>$ directions or along one of the six
$<$100$>$ directions with, in that case, four possible distinct
configurations, associated with small canting driven by HN atoms bonding with
different I.

The previous results are obtained using a $1\times 1\times 1$ periodic cell so
that all periodically-repeated molecules are artificially aligned. In such a
case, the structure is therefore naturally polar and shows very similar
spontaneous polarizations of 9.79 and 9.91 $\mu$C/cm2 in the $<$111$>$ and
$<$100$>$ states respectively. This spontaneous polarization can be seen as
the sum of two contributions: the first one, labelled “order-disorder”
($P_{\mathrm{o-d}}$), arises from the alignment of the intrinsic dipole moment
of the MA+ cations, while the second one, labelled “displacive”
($P_{\mathrm{dis}}$), comes from the relative polar motion of cations and
anions respect to centrosymmetric positions. $P_{\mathrm{o-d}}$ is estimated
to 5.66 (5.37) $\mu$C/cm2 for the molecule along $<$111$>$ ($<$100$>$)
direction, a value comparable to that of 5.56 $\mu$C/cm2 reported by Stroppa
et al.[35] and roughly 50% of the total polarization ($P_{\mathrm{tot}}$).
This highlights that both order-disorder (i.e. alignment of MA molecule) and
displacive (i.e. polar soft mode) mechanisms equally contribute to produce a
polarization when enforcing a polar phase in cubic MAPbI3 but this does not
guarantee that such a state will naturally emerge.

Considering larger supercells and orienting differently the molecules from
site to site either along the $<$111$>$ or $<$100$>$ directions, we can
generate various non-polar configurations that appear at energies similar to
the polar one, as long as neighboring molecules don’t try to bond with the
same iodine atoms. This means that, except through their interactions with I
atoms, the molecules behave rather independently and there is no clear
tendency for them to align from site to site. Looking more globally at the
energy landscape in Fig. 2(a), it appears relatively flat, in agreement with
previous calculations [56]. The energy barriers between distinct minima are
small ensuring that the molecule can freely rotate at sufficiently high
temperature [59]. All this supports the view of a disordered cubic phase in
which molecules rotate rather independently and without strong tendency to
align and form a polar state.

Tetragonal Phase. Consistently with its small tolerance factor, MAPbI3
undergoes on cooling slightly above room temperature ($T=327$ K) a first
structural phase transition arising from the condensation of AFD rotations of
PbI6-octahedra according to the $a^{0}a^{0}c^{-}$ pattern. The emergence of
these rotations lowers the symmetry of the system from $Pm\bar{3}m$ to
$I4/mcm$. This distorts the originally cubic environment at the A site and is
therefore expected to affect the behavior of the MA+ molecules [52, 60]. In
inorganic perovskites, octahedra rotations typically reduce the natural
tendency to polar distortion and emergence of displacive ferroelectricity. It
is however questionable if it could here force the polar molecules to align
and favor instead order-disorder ferroelectricity.

Figure 3: The role of PbI6-octahedra rotations on reorientation of MA+
molecules from cubic to tetragonal phase. For the cases with (a) $[111]$\- and
(b) $[100]$-oriented molecule, stretched (red) and compressed (blue) HN-I
bonds due to $a^{0}a^{0}c^{-}$ PbI6-octahedra rotations are circled. The
octahedron rotation factor $p=-1$ here.

As highlighted in the previous Section, the preferential orientations of the
MA+ molecules are mainly driven by the formation of three equivalent hydrogen
bonds. In Fig. 3, we illustrate how the bonds are affected by the octahedra
rotations. When the molecule is aligned along the $[111]$ direction, as
sketched in Fig. 3(a), anticlockwise AFD motions typically induce a
compression of one HN-I bond and the stretching of another one, forcing the
molecule to rotate from the $[111]$ to the $[101]$ direction. This
reorientation is generic to any $<$111$>$ directions although the specific
final orientation of the molecule [$a^{\prime},b^{\prime},c^{\prime}$] depends
on both its initial direction [$a,b,c$] ($a,b,c=\pm 1$) and the clockwise or
anticlockwise character $p$ of the octahedron rotation ($p=\pm 1$) according
to the following relationship:

$\begin{pmatrix}a^{\prime}\\\\[5.16663pt] b^{\prime}\\\\[5.16663pt]
c^{\prime}\end{pmatrix}=\begin{pmatrix}\frac{1-cp}{2}&0&0\\\\[5.16663pt]
0&\frac{1+cp}{2}&0\\\\[5.16663pt]
0&0&1\end{pmatrix}\begin{pmatrix}a\\\\[5.16663pt] b\\\\[5.16663pt]
c\end{pmatrix}$ (1)

When the molecule is aligned instead in the $<$100$>$ directions, it prefers
to stay perpendicular to the axis of PbI6-rotations. Then, as shown in Fig.
3(b), appearance of PbI6-rotations stretch one HN-I bond forcing the MA+
molecule to rotate from the $[100]$ direction to the $[101]$ direction again.
This time, the final direction of the molecule can be obtained as
$[\pm\frac{k-1}{2},\pm\frac{k+1}{2},kp]$ and depends of its original direction
$[\pm\frac{k-1}{2},\pm\frac{k+1}{2},0]$ ($k=\pm 1$) and the character of the
octahedron rotation $(p=\pm 1$).

This justifies that, related to hydrogen bonding effects, the
$a^{0}a^{0}c^{-}$ PbI6-rotations in the tetragonal phase produce the
reorientation of MA+ cations along $<$101$>$ directions, in agreement with
experimental X-ray diffraction measurements on single crystals [26]. However,
the previous discussion points out that all $<$101$>$ orientations are not
equivalent. As illustrated in Fig. 1, a distinct set of four orientations will
be favored according to the distortion produced by the local clockwise ($p=1$;
dark green) or anticlockwise ($p=-1$; light green) character of the
PbI6-rotations. This means that the appearance of the $a^{0}a^{0}c^{-}$ AFD
rotation pattern, which alternates clockwise and anticlockwise rotations from
site to site along the three directions, will prevent the development of a
polar state with all molecules aligned along the same direction and will favor
instead configurations in which molecules at neighboring sites are either
perpendicularly oriented or anti-aligned (populating alternatively light and
dark green directions in Fig. 1).

In order to further assess the validity of this, we consider
$\sqrt{2}\times\sqrt{2}\times 2$ supercells with various possible orientations
of the four independent molecules. All identified low-energy structures
correspond to configurations with molecules at neighboring sites either
perpendicular or anti-aligned as illustrated in Fig. 1. According to the
relative orientations of the molecules, some of these structures are polar and
can develop a polarization along distinct directions. Due to partial
compensation of dipole moments from cell to cell, the polarization is reduced
compared to the case with all molecules aligned but it includes again almost
equal $P_{\mathrm{dis}}$ and $P_{\mathrm{o-d}}$ contributions and the latter
can be estimated from the orientation of a rigid dipole moment associated with
each molecule (see Supplementary Material Table. SI for the polarization
details of tetragonal phases).

The lowest energy configuration that we found corresponds to a polar state
T-[111] with all molecules sharing the component along the same direction of
$c$-axis and giving rise to a net polarization of 7.74 $\mu$C/cm2 oriented
nearly along the [111] direction. But various other polar states – some of
them previously discussed in the literature – with their net polarization
along either the [101] direction [11, 61] (+3.51 meV/f.u., T-[101]) or the
[001] direction [37, 35] (+6.72 meV/f.u., T-[001) or the [100] direction
(+6.94 meV/f.u., T-[100]) and even an antipolar state (+5.97 meV/f.u.,
T-[000]) appear very close in energy. So, although the lowest energy phase is
polar, this highlights a energy landscape with many local minima almost at the
same energy, compatible with the view of a partly dynamical tertagonal phase
at room temperature [62, 14], with molecules hopping from one direction to
another and no very strong tendency to ferroelectricity again.

Figure 4: Sketch of six structural degrees of freedom considered in our energy
analysis.

To better clarify the interplay between distinct structural distortions, we
investigate how the energy is evolving from an artificial “cubic” phase (C)
taken as reference to the lowest-energy polar tetragonal T-[111] phase. For
the reference phase, the unit cell is fixed cubic, the Pb and I atoms as well
as the center of mass of the MA+ molecules are located at high-symmetry
positions and the MA+ molecules oriented each along the [110]-type direction
that is the closest to the corresponding direction in the T-[111] phase.
Strictly speaking, symmetry of this structure is not cubic. It even shows (see
the center of Fig. 4) a net polarization along the [010] direction but
constitutes a suitable reference for our purpose. The total distortion from
this reference to the T-[111] phase is then spread into six independent
contributions illustrated in Fig. 4: the tetragonal macroscopic strain
relaxation of the unit cell ($\varepsilon$), the a0a0c- rotation of the PbI6
octahedra ($R$), rotations of MA+ molecules about either in
(M${}^{\mathrm{i}}_{\mathrm{rot}}$) or out-of
(M${}^{\mathrm{o}}_{\mathrm{rot}}$) $a$-$b$ plane and displacive polar motions
in (M${}^{\mathrm{i}}_{\mathrm{dis}}$) or out-of
(M${}^{\mathrm{o}}_{\mathrm{dis}}$) $a$-$b$ plane. Then a simple model energy
is written as:

$E(\mathrm{T})=E(\mathrm{C})+\sum_{i=1}^{6}F_{i}+\sum_{i\neq
j}F_{ij}+\sum_{i\neq j\neq k}F_{ijk}$ (2)

with $i,j,k$ running over the six distortions and $F$ energy contributions
fitted separately on first-principles calculations including related
individual and combined distortions. The deviation of energy between first
principles and the calculated $E$(T-[111]) from Eq. (2) is less than 0.03%,
proving its reliability.

$F_{i}$ and $F_{ij}$
---
j i | $\varepsilon$ | $R$ | M${}^{\mathrm{o}}_{\mathrm{dis}}$ | M${}^{\mathrm{o}}_{\mathrm{rot}}$ | M${}^{\mathrm{i}}_{\mathrm{dis}}$ | M${}^{\mathrm{i}}_{\mathrm{rot}}$
$\varepsilon$ | 46.12 | -79.17 | 2.88 | 1.69 | 27.13 | -3.44
$R$ |  | -14.79 | 53.42 | 35.95 | 63.40 | -92.20
M${}^{\mathrm{o}}_{\mathrm{dis}}$ |  |  |  -8.49 | -39.43 | 13.75 | 11.21
M${}^{\mathrm{o}}_{\mathrm{rot}}$ |  |  |  | -5.23 | 18.19 | -2.72
M${}^{\mathrm{i}}_{\mathrm{dis}}$ |  |  |  |  | -79.23 | -46.59
M${}^{\mathrm{i}}_{\mathrm{rot}}$ |  |  |  |  |  | -5.11
Table 1: Values (meV/f.u.) of the fitted $F_{i}$ and the $F_{ij}$ terms. Most
significant negative terms are highlighted in red. Light-gray-highlighted data
are in the cubic regime without rotations.

The first- and second-order $F$ values are reported in TABLE 1. Starting from
the C phase and looking at $F_{i}$ contributions, the dominant term to
decrease the energy is $F$[M${}^{\mathrm{i}}_{\mathrm{dis}}$] (-79.23
meV/f.u.) that corresponds to the off-centering of the MA+ molecules driven by
the formation of HN-I bonds as previously discussed and really existing in the
$<$110$>$ case in Fig. 2(a). It appears here as a polar motion because the
molecule arrangement in the C phase is already polar but it would take the
form of an antipolar distortion in a C phase with antipolar alignment of the
molecules.

Ignoring at first octahedra rotations ($R$), we notice the strong cooperative
coupling of M${}^{\mathrm{i}}_{\mathrm{dis}}$ with
M${}^{\mathrm{i}}_{\mathrm{rot}}$ that will produce a reorientation of the
molecules along the [100]-type directions ($<$100$>$-C phase in Fig. 5) in
agreement with the related minimum in Fig. 2(a). We can obtain another low-
energy configuration for the molecules along the [111]-type directions
($<$111$>$-C phase in Fig. 5), but our model with few rigid degrees of freedom
connected to the ground state is not appropriate for full relaxations and this
requires considering only the part of M${}^{\mathrm{i}}_{\mathrm{dis}}$
related to the translation along the molecule. All these can be interpreted as
molecular motions driven by H-bonding formation.

Then, $F[R]$ is negative in Table 1 (-14.79 meV/f.u.) highlighting that AFD
rotations of PbI6 octahedra are spontaneously unstable, in line with the small
tolerance factor of MAPbI3. We first observe a strong coupling of these
rotations with the macroscopic tetragonal strain that significantly helps
stabilizing an AFD phase ($F[R,\varepsilon]$= -79.17 meV/f.u.). Further the
significant coupling of $R$ and M${}^{\mathrm{i}}_{\mathrm{rot}}$ strongly
favors orientation of the molecules along the [100]-type rather than the
[111]-type directions. Interestingly, while
$F$[M${}^{\mathrm{i}}_{\mathrm{dis}}$, M${}^{\mathrm{i}}_{\mathrm{rot}}$] and
$F$[M${}^{\mathrm{o}}_{\mathrm{dis}}$, M${}^{\mathrm{o}}_{\mathrm{rot}}$]
appear globally competitive in absence of octahedra rotations (favoring
molecule orientations along either [100]- or [111]-type directions), thanks to
different third-order terms listed in Supplementary Material Table SII(a),
they become cooperative in presence of $R$ and $\varepsilon$ and molecules
thus favor [101]-type orientations ($<$101$>$-T phase in Fig. 5). To explore
the driving force for the cooperation, we fit some related fourth-order terms
that are not taken into account in Eq. (2). From Supplementary Material Table
SII(b), the terms within $\varepsilon$ trend to increase the total energy,
which nicely confirms that the reorientation of the molecules along the
$<$101$>$ directions in the tetragonal phase results from the AFD motions.
Note that, this appears mainly driven by local steric and geometric effects
compatible with a partial disorder from site to site. The combination of
distortions gives rise here to the T-[111] lowest energy phase but a very
similar decomposition can be obtained for the other T phases in Fig. 1
highlighting no strong tendency to favor a particular polar alignment.

Figure 5: Connection between the reference C phase and the T-[111] phase. When
including successively the distinct degrees of freedom listed in Fig. 4, the
reference C phase with $<$110$>$-oriented molecules gradually transforms to
the T-[111] lowest-energy phase with $<$101$>$-oriented molecules. Light-gray
and light-orange backgrounds refer to phases without and with $R$ distortion.
Dominant negative and positive energy contributions are highlighted
respectively in red and blue.

Orthorhombic Phase. Below 162 K, MAPbI3 adopts an orthorhombic $Pnma$
structure with a PbI6 rotation pattern $a^{-}b^{+}a^{-}$. Comparing to the
tetragonal phase including a single out-of-phase rotation about $c$-axis, this
arises from the appearance of two more AFD distortions: out-of-phase rotations
about $a$-axis and in-phase rotations about $b$-axis. Our calculations confirm
that this phase is significantly lower in energy than the tetragonal phases
and the proper ground state of MAPbI3. Similar to what happens in the
tetragonal phase, the two new AFD distortions will further constraint the
direction of the MA+ molecules.

Figure 6: Atomic configuration in the $a$-$c$ plane for two consecutive layers
along $b$-axis in the $a^{-}b^{+}a^{-}$ orthorhombic phase. MA+ molecules are
displayed according to their expected original positions (along
$[10\overline{1}]$ or $[\overline{1}01]$ directions) in the phase with
$a^{-}b^{0}a^{-}$ rotation pattern. Presence of the additional
$a^{0}b^{+}a^{0}$ distortion then produces the compression (blue circles) and
stretching (red circles) of some HN-I bonds and forces molecules‘
reorientation according to the yellow arrows. The final structure is
necessarily antipolar.

In the tetragonal phase, we have seen that the out-of phase rotations along
$c$-axis impose the MA+ molecule axis to be aligned along generic directions
defined as $[\pm\frac{k-1}{2},\pm\frac{k+1}{2},kp]$. Similarly, the appearance
of a new out-of-phase rotation along $a$-axis will force alignment along
$[k^{\prime}p,\pm\frac{k^{\prime}-1}{2},\pm\frac{k^{\prime}+1}{2}]$ directions
(taking, without loss of generality, the convention that $p$ is the same for
both rotations in a given cell). Combining these two constraints, the only
possibility is $k=-1$ and $k^{\prime}=1$ so that the molecules can only be
aligned along generic directions $[p,0,-p]$, with $p$ switching from $+1$ to
$-1$ when moving from one unit cell to the next one along $a$, $b$ or $c$
direction. This gives rise to an antipolar configuration in which molecules at
neighboring sites are oppositely aligned (see Fig. 6).

The appearance of the additional in-phase rotations along $b$-axis will
further distort the structure, contracting the H${}^{1}_{\rm N}$-I bond (blue-
circled in Fig. 6) and expanding the H${}^{2}_{\rm N}$-I and H${}^{3}_{\rm
N}$-I bonds (red-circled in Fig. 6). This will force the molecules to rotate
nearly 45∘ either clockwise or anti-clockwise and align their axis closely
along $<$100$>$ directions. Due to their reversed rotations, molecules at
neighboring sites in the same $a$-$c$ plane are perpendicularly oriented.
Taking into account the in-phase character of the rotations along $b$-axis,
the molecules will adopt in consecutive $a$-$c$ planes generic directions
corresponding respectively to $[\frac{p-1}{2},0,-\frac{p+1}{2}]$ and
$[\frac{p+1}{2},0,-\frac{p-1}{2}]$. This is associated with a pattern in which
first-neighbor molecules along $a$ and $c$ are perpendicularly aligned (second
neighbors being aligned) while first-neighbor molecules along $b$-axis are
anti-aligned.

In the $Pbnm$ phase, due to the combination of the AFD motions, the MA+
molecules are therefore forced to take an antipolar arrangement in which
planes with opposite net polarizations in $[101]$ and
$[\overline{1}0\overline{1}]$ directions alternating along $b$-axis. This
demonstrates unambiguously that, in line with the literature [25, 32, 33],
this phase is antipolar and cannot be FE.

## DISCUSSION

In contrast to inorganic ABO3 perovskites, MAPbI3 possesses MA+ polar ions at
the A site. As such, its FE character, rather than displacive, is closely tied
to the arrangement of these permanent dipoles, which is itself determined by
the interplay of two main phenomena. On the one hand, the local positions and
orientations of MA+ dipoles are guided by the natural tendency of nitrogen-
end-type hydrogen atoms to form three equivalent HN-I bonds with the
surrounding inorganic lattice. On the other hand, like traditional inorganic
ABO3 perovskites with small tolerance factors, MaPbI3 shows AFD octahedra-
rotation instabilities that give rise to consecutive phase transitions from
cubic to $a^{0}a^{0}c^{-}$ tetragonal and then $a^{-}b^{+}a^{-}$ orthorhombic
distorted phases. These cooperative AFD distortions modulate the shape of the
A-site cavities forcing then some specific orientations of the molecules at
each atomic site.

At high temperature, in absence of AFD distortions the A-site cavities keep a
cubic symmetry. Due to their tendency to H-bonding, the molecules show
preferential orientations along either $<$100$>$ or $<$111$>$ directions but
the energy landscape is sufficiently flat for the molecules to rotate
relatively freely and independently as typically observed experimentally [59]
and also confirmed from molecular dynamics simulations [36, 57]. This gives
rise to a dynamically-disordered non-polar average structure which can be seen
as a disordered paraelectric state.

At $T=327$ K, MaPbI3 then evolves to a tetragonal $I4/mcm$ phase arising from
the condensation of a $a^{0}a^{0}c^{-}$ pattern of octahedra rotations. These
AFD motions distort the A-site cavities forcing a partial reorientation of the
MA+ molecules along the $<$101$>$ directions, although leaving them some
degrees of freedom. At a given site, four preferential orientations
perpendicular to each others are allowed while at neighboring sites they are
either perpendicular or anti-aligned. In this tetragonal phase, the T-[111]
structure owing the lowest energy is polar, with a spontaneous polarization
arising evenly from the partial alignment of the MA+ dipoles and from the
further polarizability of the lattice. This latter effect can however be seen
as a pseudo-proper effect rather than a driving force for the appearance of
the polarization since it would translate into an antipolar contribution for
antipolar molecular arrangements. Consequently to the presence of PbI6
rotations, this polar state does not correspond to a naïve configuration of
perfectly aligned dipoles but is one in which MA+ molecules at neighboring
sites are perpendicularly oriented giving rise to what we could call a weakly
polar configuration. According to this, the polarization reversal is possible
so that the T-[111] structure is not only polar but even FE. But, polarization
switching involves individual molecular rotation motions of 90∘ rather than a
complete 180∘ switching of their orientation. Beyond this T-[111]
configuration, alternating the possible $<$101$>$ orientations, we identified
also various other polar and antipolar configurations at very close energies,
highlighting the absence of strong correlation in the molecule orientations
from one site to another. At room-temperature, the tetragonal phase is still
observed to be disordered [62, 14] and random orientation of the MA+ molecules
along equivalent $<$101$>$ directions preserves, on average, a non-polar
character so that the system can still be seen as being in the paraelectric
disordered regime.

From this, MAPbI3 should a priori exhibit on cooling an order-disorder
transition toward the FE T-[111] phase at a given Curie temperature
$T_{\text{FE}}$. However, at $T$= 162K, appearance of additional PbI6
octahedra rotations bring the system into a $Pnma$ orthorhombic phase. This
phase is significantly lower in energy than the T-[111] phase and the
combination of in-phase and out-of-phase AFD motions further distorts the
A-site cavities in such a way that the MA+ molecules are now forced to align
along $<$100$>$ directions according to a well-defined antipolar pattern. The
ground state of MAPbI3 is therefore clearly not FE, although the rearrangement
of MA+ dipoles at ferroelastic domain walls might possibly contribute to
provide some polar character.

In conclusion, MAPbI3 can be seen as a missed FE in which the expected T-[111]
FE tetragonal ground state is finally avoided by the appearance of AFD motions
enforcing instead an antipolar orthorhombic ground state. The latter shows
frozen orientations of dipoles, as confirmed experimentally [25]. Depending on
how large is really $T_{\text{FE}}$, a FE T-[111] phase could eventually be
stabilized in a narrow range of temperature above 162K. As previously
mentioned, the molecules still rotate at room temperature [62, 14], locating
$T_{\text{FE}}$ definitively below it. It has been reported that molecules’
rotation significantly slows down at 190K [63] but no specific fingerprint of
a FE phase transition has been observed at that temperature and molecules do
not appear totally frozen yet. It could possibly happen that when the system
finally reaches a temperature at which a more static polar configuration could
emerge, this helps initiating the final transition to the orthorhombic ground
state that suppresses the FE transition. Although it does not seem to ever
stabilize a FE state, we like to emphasize that, even in the paraelectric
regime, MAPbI3 can likely show fluctuating polar nanoregions – which can
become even more present below 190K – and that poling it in a field might
eventually bring it into a metastable FE state. This view can explain the
controversial experimental observations on its FE character and contribute to
clarify its puzzling properties. We mention finally that these conclusions
reveal specific to MAPbI3 and should not necessarily be seen as a generic
property of similar hybrid perovskites. Nevertheless, the method and strategy
we used here are versatile and might reveal appropriate to address the
potential FE nature of related hybrid perovskites with other A-site molecules.

## Methods

Ab initio calculations. The calculations are performed within DFT using the
full-potential projector augmented wave (PAW) method [64], as implemented in
the Vienna ab initio Simulation Package (VASP) [65]. The exchange-correlation
energy is estimated within the PBEsol [66] generalized gradient approximation.
For the tetragonal phase, we do further checks using the non-empirical
strongly constrained and appropriately normed (SCAN) [67] meta-generalized
gradient approximation (meta-GGA), which was shown to touch the accuracy of
hybrid functional for diversely bonded molecules [68]. The relaxed states with
different net polarizations are robust and indeed energetically comparable to
PBEsol. We use a kinetic-energy cutoff of 500 eV. The k-point mesh is adapted
to the supercell under investigation. For the reference $1\times 1\times 1$
cubic structure, a $8\times 8\times 8$ $\Gamma$-centered Monkhorst-Pack grid
is adopted for the geometry optimization and self-consistent calculations. For
the tetragonal and orthormbic phases, $\sqrt{2}\times\sqrt{2}\times 2$ and
$\sqrt{2}\times 2\times\sqrt{2}$ supercells are used with $6\times 6\times 4$
and $6\times 4\times 6$ grids, respectively. The convergence criterion for the
electronic energy is $10^{-6}$ eV and the structures are relaxed until the
Hellmann–Feynman forces on each atom are less than 1 meV/Å. The Berry-phase
method [69] is employed for the evaluation of the macroscopic polarization,
properly eliminating the quantum indeterminacy [70].

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### Acknowledgments

W.-Y.T. acknowledges the support from F.R.S.-FNRS Belgium. J.-Z.Z acknowledges
the support from the Startup Funding for Outstanding Young Scientist of South
China Normal University and the financial support of China Scholarship Council
(Grant No. 202006755025). The authors acknowledge access to the CECI
supercomputer facilities funded by the F.R.S-FNRS (Grant No. 2.5020.1) and to
the Tier-1 supercomputer of the Federation Wallonie-Bruxelles funded by the
Walloon Region (Grant No. 1117545).

### Author contributions

P.G. and J.-Z.Z. conceived the idea and supervised the work. W.-Y.T carried
out first-principles calculations and did the data analysis. W.-Y.T. and
J.-Z.Z. contributed to the interpretation of the results. W.Y.T and P.G. co-
wrote the paper. All the authors reviewed and modified the manuscript.


11institutetext: Brandeis University, Waltham MA 02453, USA
11email: {zhengzhou<EMAIL_ADDRESS>22institutetext: Swarthmore
College, Swarthmore PA 19081, USA
22email<EMAIL_ADDRESS>
# BrainActivity1: A Framework of EEG Data Collection and Machine Learning
Analysis for College Students

Zheng Zhou 11 0000-0001-9313-2106 Guangyao Dou 11 0000-0001-8011-9658 Xiaodong
Qu 1122 0000-0001-7610-6475

###### Abstract

Using Machine Learning and Deep Learning to predict cognitive tasks from
electroencephalography (EEG) signals has been a fast-developing area in Brain-
Computer Interfaces (BCI). However, during the COVID-19 pandemic, data
collection and analysis could be more challenging than before. This paper
explored machine learning algorithms that can run efficiently on personal
computers for BCI classification tasks. Also, we investigated a way to conduct
such BCI experiments remotely via Zoom. The results showed that Random Forest
and RBF SVM performed well for EEG classification tasks. The remote experiment
during the pandemic yielded several challenges, and we discussed the possible
solutions; nevertheless, we developed a protocol that grants non-experts who
are interested a guideline for such data collection.

###### Keywords:

Brain-Machine Interface Machine Learning Ensemble Methods Remote BCI
Interpretable AI

##  1 Introduction

Previous research in Computer Science, Neuroscience, and Medical fields has
implemented EEG-based Brain-Computer Interfaces (BCI) in several ways, [23,
13, 34, 1, 25, 21, 33, 35, 19, 14], such as diagnosis of Alzheimer’s, emotion
recognition, mental workload, motor imagery tasks [15, 22, 26, 3, 5, 4].
Machine learning, deep learning, and transfer learning algorithms have
demonstrated the great potential in such biomarker data analysis [24, 39, 38,
17, 37, 28, 20, 40, 27, 7, 41, 31, 6, 2, 16].

Figure 1: College Students using EEG Devices

However, EEG datasets’ size is still relatively small compared with peers in
Computer Vision and Natural Language Processing [23, 36, 13].

Our research questions are: 1\. Can we develop a larger dataset with data from
a larger audience? for example, college students? 2\. Can we design easy-to-
use BCI experiments to collect EEG data with consumer-grade devices for
college students? 3\. Can we develop a step-by-step guide for such a data
collection and machine learning analysis process?

##  2 Methods

As [18, 32, 30, 11] mentioned, several affordable (less than three hundred
dollars) non-invasive consumer-grade EEG headsets are commercially available.
As shown in Figure 1, we have pilot-tested several clinical and non-clinical
EEG devices with college students. The Top left cell in Figure 1 showed an
example of the wearing of a clinical device while others demonstrated the
consumer-grade devices. Muse Headset were used as an example for
demonstration, followed by more details in the 6-page full-length page.

Figure 2: Data Collection App Selection

As shown in Figure 2, first, we installed the data collection software or
application. The following method section was formulated based on the example
of using Muse Headset. We have used this device since 2016. First, the end
user’s OS version must match the version of the EEG recording application. If
the end-user had a newer Macbook or the latest version of Muse headset, they
ought to use the Mind Monitor Application to record EEG signals. Otherwise,
the end-user may still use the Muselab application for EEG recording.

Figure 3: Data Collection Flow Chart, User and Coach Figure 4: Data Analysis
Flow Chart Figure 5: Subject 1 Task 2 Prediction for all Six Sessions

Two individuals, the EEG coach, and the end-user, were usually required to
complete the data acquisition of such non-invasive EEG signals via Muse
headset and Muse Recording software, As shown in Figure 3. First, the EEG
Coach introduced the Muse Headset and Muse recording application to the end-
user. Then the EEG coach explained the details of a specific experiment. Next,
the EEG coach and the end-user started the experiment to collect the data. The
data was saved as a MUSE file. The end-user then summarized the experimental
feedback to the EEG coach. Together with the research team, the EEG coach
generated visual feedback to the End-user. Such feedback may contribute to the
development of future experiments.

If the end-user was interested, they could learn the data analysis themselves,
which took two hours on average for students who major in computer science.

Most EEG recording applications came with a toolset to convert the recording
files to TXT or CSV files. Afterward, we could pick the subset of data we
planned to use for further analysis; we recommended starting with the absolute
value of the EEG signals.

We implemented several machine learning algorithms commonly used in the field
[8, 9, 10, 12] from the scikit-learn [29]. For example, Linear Classifiers,
Nearest Neighbors, Decision Tree, and Ensemble Methods.

As shown in the Figure 4, once we had six TXT files for all six sessions of
data, we first executed a Matlab program - preprocess.m - to identify the
noises and turn all the TXT files into separate CSV files. Then, we ran the
Clean.ipynb to exclude these data in the Pandas Dataframe for further
analysis. Next, we executed the TMV.ipynb to train and cross-validate existing
machine learning algorithms such as the Random Forest, SVM, and KNN. Then we
selected the top two best-performing algorithms and used these two algorithms
to perform Time Majority Voting. In our case, these two algorithms were Random
Forest and RBF SVM. Lastly, the TMV.ipynb would generate a visualization of
task predictions for end-users. Figure 5 and a heatmap for all six sessions
(Fig 6). These two figures are examples of task one’s prediction results for
all six sessions of subject 1 in the TCR Experiment.

##  3 Results

As shown in Figure 5, sessions designed task were clearly recognized. All six
sessions of subject 1’s task 2 showed consistent patterns. Such visual
feedback was provided to the EEG coach and end-user who collected this set of
data. From the experiment notes, we ran into a signal issue behind the right
ear of the end-user throughout the session, then we recorded this case and
potential solutions to improve future experiments.

As shown in Figure 6, the X-Axis is the designed tasks, the Y-Axis is the
predicted tasks. The diagonal means the designed tasks matched the prediction.
Such visual feedback also helped the research team and the end-users better
understand what task pairs were easy to be confused with each other.

Figure 6: Subject 1’s TMV Heatmap for all Six Sessions

##  4 Discussion

This paper proposed an approach for non-export independent researchers to
collect EEG-based BCI data with affordable non-clinical devices. When
performing test trials with Muse headsets, we provided a general guideline, as
shown in figure 2 and 3, which showed promising result in many EEG data
collections performed by naive EEG end-users. This general guideline
demonstrated a decision tree for non-expert researchers to acquire data
collection hardware and software. We also presented our data collection flow,
which formed a closed loop between the researchers and the experimental
subjects. In addition, we elaborated our data-cleaning analysis procedures.

Significant progress has been made in user-training for EEG-based BCI studies,
while the framework proposed in this paper serves as a stepping stone for
further improved training programs in future research. However, some
limitations were identified along the course of our project. Our project
spanned from pre-pandemic to post-pandemic time. We found that in-person data
collection trials were significantly more efficient than trials that took
place virtually during the global pandemic. We ought to explore more
strategies and updated methods that could grant us the efficiency when data
collection has to be completed in a virtual environment.

Even though as much detail and trial and error experience we managed to
include in our guideline, there are chances that individual cases develop
distinct issues. Our future work includes research/EEG coach-based student
study/work community, in which they can learn and discuss their experience
with non-clinical devices collecting EEG-based data and possibly establish
solutions to various issues after the encounter.

##  5 Conclusion

This paper investigated the data collection for EEG-based BCI to develop
larger datasets. We explored the possibility of collecting EEG data from
college students with affordable devices. The results demonstrated that the
proposed framework could simplify the process and contribute to developing a
larger EEG dataset.

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