# Lattice-distortion couplings in antiferroelectric perovskite $\rm AgNbO_{3}$
and comparison with $\rm PbZrO_{3}$

Huazhang Zhang<EMAIL_ADDRESS>Theoretical Materials Physics, Q-MAT,
Université de Liège, B-4000 Sart-Tilman, Belgium  Department of Physics,
School of Science, Wuhan University of Technology, Wuhan 430070, People’s
Republic of China  Konstantin Shapovalov  Theoretical Materials Physics,
Q-MAT, Université de Liège, B-4000 Sart-Tilman, Belgium  Safari Amisi
Theoretical Materials Physics, Q-MAT, Université de Liège, B-4000 Sart-Tilman,
Belgium  Laboratoire de Physique des Solides et des Interfaces, Institut
Supérieur Pédagogique de Bukavu, Democratic Republic of the Congo  Philippe
Ghosez<EMAIL_ADDRESS>Theoretical Materials Physics, Q-MAT,
Université de Liège, B-4000 Sart-Tilman, Belgium

(August 28, 2024)

###### Abstract

Lead-free antiferroelectric perovskite $\rm AgNbO_{3}$ is nowadays attracting
extensive research interests due to its promising applications in energy
storage. Although great progress has been made in optimizing the material
performance, fundamental questions remain regarding the mechanism stabilizing
the antiferroelectric $Pbcm$ phase. Here, combining structural symmetry
analysis and first-principles calculations, we identified crucial anharmonic
couplings of oxygen octahedra rotations and cation antipolar motions which
contribute significantly to lowering the energy of the $Pbcm$ phase. The
stabilization of this phase shows close similarities with the stabilization of
the $Pbam$ phase in $\rm PbZrO_{3}$ except that in $\rm AgNbO_{3}$ the
octahedra rotations are the primary distortions while the antipolar cation
motions appear to be secondary. The appearance and significant amplitude of
the latter are explained from the combination of hybrid-improper and triggered
mechanisms.

Keywords

Antiferroelectricity; silver niobate; lead zirconate; lattice-distortion
coupling

##  I Introduction

Antiferroelectric (AFE) materials exhibit characteristic double hysteresis
loops, making them attractive for a wide range of applications [1, 2, 3, 4, 5,
6]. Among the limited number of perovskite antiferroelectric oxides, $\rm
AgNbO_{3}$ (ANO) has attracted a lot of research interest because it is lead-
free and can exhibit large polarization under electric field (52 $\rm\mu
C/cm^{2}$ at 220 $\rm kV/cm$ [7]). Recently, extensive studies have focused on
the energy storage properties of ANO-based materials and remarkably high
energy storage density has been achieved [8, 9, 10, 11, 12, 13]. The excellent
energy storage performance of ANO is closely related to its AFE property,
which involves a reversible phase transition between nonpolar AFE state and a
highly polarized ferroelectric (FE) state upon application and removal of an
electric field. Understanding the mechanism behind the AFE properties of ANO
and identifying the key interactions stabilizing the AFE state are fundamental
and meaningful topics, which can provide valuable guidance for the further
optimization of its properties.

The exact structure and the AFE nature of ANO are currently subjects of
ongoing discussion. In fact, controversies exist regarding its complex series
of phase transitions [14, 15, 16, 17, 18, 19]. According to the X-ray and
neutron diffraction reported by Sciau et al. [17], the phase sequence of ANO
from high-temperature to low-temperature is: cubic $C$ phase ($Pm\bar{3}m$,
above $852$ K), tetragonal $T$ phase ($P4/mbm$, $660\sim 852$ K), orthorhombic
$O$ phase ($Cmcm$, $626\sim 660$ K), and orthorhombic $M$ phase ($Pbcm$, below
$626$ K). Although dielectric measurements suggested that there may be several
different $M$ phases [14, 15, 13], Sciau’s results showed that the average
structures of all these possible low-temperature $M$ phases share the same
$Pbcm$ space group [17]. The $Pbcm$ structure is centrosymmetric, in line with
its AFE property. However, it was also noted that ANO exhibits a weak
remanence on the polarization hysteresis loops [20, 13]. This phenomenon may
be caused by various reasons, such as cation disordering, polar clusters, or
defects [18, 21, 13]. A previous structural study also suggested that a polar
$Pmc2_{1}$ structure might be the low-temperature phase of ANO [22], but the
subsequent density functional theory (DFT) calculations have shown that this
$Pmc2_{1}$ structure is not stable [23]. Instead, the $Pbcm$ phase remains the
lowest energy phase in literature and is free of any dynamical instability
[23].

The emergence of a $Pbcm$ phase is rather uncommon in perovskite oxides [24,
25]. The most common structure for perovskites with small tolerance factors
($t<1$) is $Pnma$ [26], a phase which is favored by a trilinear coupling
between cation antipolar motions and in-phase and antiphase oxygen octahedra
rotations [27]. Then, among antiferroelectrics, the phase that attracted the
most attention is $Pbam$, as seen in, e.g., the prototypical antiferroelectric
$\rm PbZrO_{3}$ (PZO), where it is stabilized by a trilinear coupling between
cation displacements, oxygen octahedra rotations and their modulations [28,
29, 30, 31]. In ANO, however, the stabilization mechanism of the $Pbcm$
antiferroelectric state has not been fully clarified.

In this study, we investigate the stabilization mechanism of the $Pbcm$
antiferroelectric state in ANO through structural symmetry analysis and first-
principles calculations. We identify the cooperative trilinear and biquadratic
couplings between cation antipolar motions and oxygen octahedra rotations that
help stabilizing the $Pbcm$ phase. We also conduct a comparison between ANO
and the protyptical antiferroelectric PZO. Our findings suggest a combination
of hybrid-improper and triggered mechanisms for the appearance of the cation
antipolar motions in ANO.

##  II Calculation details

First-principles DFT calculations were performed using the Abinit package [32,
33, 34, 35] with the plane-wave-pseudopotential approach. We employed the
generalized gradient approximation (GGA) with the revised Perdew-Burke-Enzerh
parameterization for solids (PBEsol) [36] and optimized norm-conserving
pseudopotentials from the PseudoDojo server [37, 38]. For comparison, we also
checked some results using the local density approximation (LDA) functional
[39]. The energy cutoff for the plane-wave expansion was 50 $\rm Ha$ for ANO
and 60 $\rm Ha$ for PZO. The $k$-point grids were equivalent to or denser than
$6\times 6\times 6$ grid for the five-atom cubic perovskite. The electronic
self-consistent cycles were considered to be converged until the potential
residual is smaller than $10^{-18}$ $\rm Ha$. Structural relaxations were
performed based on the Broyden-Fletcher-Goldfarb-Shanno minimization
algorithm, using the convergence criteria of $10^{-6}$ $\rm Ha/Bohr$ for the
forces and $10^{-8}$ $\rm Ha/Bohr^{3}$ for the stresses. The phonon
dispersions were calculated according to density functional perturbation
theory as implemented in Abinit and analyzed with the Anaddb program [40]. The
symmetry-adapted lattice-distortion modes were analyzed using ISODISTORT [41].
The energy expansions were build with the help of INVARIANTS [42].

##  III Mode couplings in ANO

The structural parameters of ANO in its $Pbcm$ phase are reported in Table 1,
where we compare the results of our GGA-PBEsol and LDA calculations with the
experimental data from low-temperature measurements in literature [17]. The
GGA-PBEsol calculated lattice parameters show very good agreement with the
experimental values, with discrepancies less than 0.5%, while the LDA tends to
underestimate the lattice constants. It is therefore expected that the GGA-
PBEsol functional is more accurate in capturing the structural characteristics
of ANO.

Table 1: Lattice parameters (Å) and atomic positions (reduced coordinates) of ANO in the orthorhombic $Pbcm$ phase. The $x$, $y$, $z$ coordinates are in the direction of orthorhombic axes $a$, $b$, $c$, respectively. | GGA-PBEsol | LDA | Exp. (1.5 K) [17]
---|---|---|---
Lattice | $a$ | $b$ | $c$ | $a$ | $b$ | $c$ | $a$ | $b$ | $c$
parameter (Å) | 5.5215 | 5.6019 | 15.506 | 5.4424 | 5.5362 | 15.390 | 5.5436 | 5.6071 | 15.565
Error (%) | -0.40 | -0.09 | -0.38 | -1.83 | -1.26 | -1.12 |  |  |
| $x$ | $y$ | $z$ | $x$ | $y$ | $z$ | $x$ | $y$ | $z$
Ag1 (4c) | 0.7423 | 0.25 | 0 | 0.7439 | 0.25 | 0 | 0.7449 | 0.25 | 0
Ag2 (4d) | 0.7369 | 0.2210 | 0.25 | 0.7378 | 0.2226 | 0.25 | 0.7420 | 0.2110 | 0.25
Nb (8e) | 0.2436 | 0.2210 | 0.1250 | 0.2445 | 0.2270 | 0.1250 | 0.2436 | 0.2212 | 0.1249
O1 (8e) | -0.0438 | 0.0445 | 0.1082 | -0.0473 | 0.0479 | 0.1067 | -0.0348 | 0.0371 | 0.1098
O2 (8e) | 0.5445 | 0.4620 | 0.1417 | 0.5462 | 0.4603 | 0.1436 | 0.5360 | 0.4689 | 0.1396
O3 (4c) | 0.3167 | 0.25 | 0 | 0.3234 | 0.25 | 0 | 0.3109 | 0.25 | 0
O4 (4d) | 0.1872 | 0.2752 | 0.25 | 0.1794 | 0.2787 | 0.25 | 0.1944 | 0.2678 | 0.25

As shown in Fig. 1(a1), the unit cell of the $Pbcm$ phase can be viewed as a
$\sqrt{2}\times\sqrt{2}\times 4$ multiple of the five-atom perovskite
elemental unit cell, with the translational vectors $a$, $b$, $c$ directed
along the pseudocubic $[110]$, $[1\bar{1}0$], $[001]$ directions. By symmetry
analysis, the atomic distortion of the $Pbcm$ phase with respect to the cubic
reference structure can be decomposed into symmetry-adapted lattice-distortion
modes of this reference [Fig. 1(b), Table 2]. The dominating modes are: (1)
$R_{5}^{-}$ mode, the rotations of oxygen octahedra around the $b$-axis that
correspond to $a^{-}a^{-}c^{0}$ rotation pattern in Glazer’s notation [43]
[Fig. 1(a2)]; (2) $T_{2}$ mode, a complex octahedra rotation pattern
alternating between in-phase and antiphase rotations around the $c$-axis [Fig.
1(a3)] that we label hereafter as the $a^{0}a^{0}c^{+/-}$ rotation pattern;
(3) $\Delta_{5}$ mode, primarily the antipolar motions of the cations Ag and
Nb in the direction of $b$-axis [Fig. 1(a4)]; (4) $M_{5}^{-}$ mode, the
antipolar motions of Ag and Nb in the direction of $a$-axis [Fig. 1(a5)]. The
above four modes account for 48.88%, 41.13%, 9.02% and 0.86% of the total
lattice distortion, respectively, while the remaining modes account for the
remaining 0.11%. The fact that most of the distortions are associated to the
oxygen octahedra rotations is consistent with the small tolerance factor of
ANO ($t=0.965$) since octahedra rotations are typical in perovskites with
tolerance factor smaller than 1 [26].

Figure 1: Structure and symmetry-adapted lattice-distortion modes (respect to the cubic reference structure) of ANO in the $Pbcm$ phase. (a) Schematics of the $Pbcm$ crystal structure and the dominant modes of the cubic reference structure giving rise to the $Pbcm$ phase. (b) Mode amplitudes in the $Pbcm$ structure, respect to the cubic reference, as optimized with the GGA-PBEsol functional. Table 2: Symmetry-adapted lattice-distortion mode decomposition of the $Pbcm$ phase of ANO. The structure is obtained by relaxing fully the atomic positions and lattice parameters using GGA-PBEsol functional. The mode amplitudes are reported in the so-called “parent-cell-normalized” values of ISODISTORT. | $R_{5}^{-}$ | $T_{2}$ | $\Delta_{5}$ | $M_{5}^{-}$ | $R_{4}^{-}$ | $T_{4}$ | $X_{1}^{+}$
---|---|---|---|---|---|---|---
Amplitude (Å) | $0.5202$ | $0.4772$ | $0.2235$ | $0.0691$ | $0.0177$ | $0.0164$ | $0.0005$
Percentage (%) | $48.88$ | $41.13$ | $9.02$ | $0.86$ | $0.06$ | $0.05$ | $<0.01$
Figure 2: (a) Structural and energetic relationships of various metastable
phases. (b) Phonon dispersion curves of ANO in the cubic parent phase. (c)
Comparisons of the energies of the metastable phases (respect to the cubic
phase taken as reference) with either fixed lattice parameters of the cubic
parent phase or fully relaxed lattice parameters. The notations for
orthorhombic space groups $Pbnm$ and $Ibmm$ are chosen to keep consistency
with the crystallographic coordinate system of this work, but it should be
noted that these space groups are commonly referred to as $Pnma$ and $Imma$ in
the literature, respectively, as they are equivalent.

We can gain more insight into the stabilization mechanism of the $Pbcm$ phase
by investigating the structural and energetic relationships among different
phases of ANO, paying specific attention to the phases with different oxygen
octahedra rotations. In each calculation, we introduce octahedra rotations to
the prototypical cubic $Pm\bar{3}m$ phase and relax the atomic distortions and
lattice parameters while preserving the symmetry of the distortions in order
to obtain the corresponding metastable phase. Specifically, we consider three
types of octahedra rotation patterns around the $c$-axis, namely
$a^{0}a^{0}c^{+}$, $a^{0}a^{0}c^{+/-}$, and $a^{0}a^{0}c^{-}$, which are
corresponding to the $M$ $(1/2,1/2,0)$, $T$ $(1/2,1/2,1/4)$, and $R$
$(1/2,1/2,1/2)$ points in reciprocal space, respectively, as well as the
octahedra rotations around $b$-axis corresponding to $a^{-}a^{-}c^{0}$
rotation pattern in Glazer’s notation.

Figure 2(a) schematically illustrates the structural and energetic
relationships among various phases. When only one rotation around the $c$-axis
is present, the three phases $P4/mbm$ ($a^{0}a^{0}c^{+}$), $I4/mcm$
($a^{0}a^{0}c^{+/-}$) and $I4/mcm$ ($a^{0}a^{0}c^{-}$) are very close in
energy [Fig. 2(a)], which is consistent with the almost non-dispersive nature
of the lowest $M-T-R$ branch of the phonon dispersion curve calculated in the
cubic phase [Fig. 2(b)]. This implies that the rotations of oxygen octahedra,
while naturally strongly correlated within the planes perpendicular to the
rotation axis $c$, are weakly correlated between adjacent planes.
Consequently, the rotations between adjacent planes being either in-phase or
antiphase do not result in significant energy differences.

Interestingly, when the octahedra rotations around the $c$-axis coexist with
the $a^{-}a^{-}c^{0}$ rotations, we find that the energies of the relaxed
$Pbnm$ ($a^{-}a^{-}c^{+}$, equivalent to $Pnma$ but hereafter will be referred
to it as $Pbnm$ to keep consistency with the crystallographic coordinate
system), $Pbcm$ ($a^{-}a^{-}c^{+/-}$) and $R\bar{3}c$ ($a^{-}a^{-}a^{-}$)
phases are no longer so close to each other. In particular, the energy
lowering from $I4/mcm$ ($a^{0}a^{0}c^{+/-}$) to $Pbcm$ ($a^{-}a^{-}c^{+/-}$)
is notably larger than in the other two cases [i.e. from $P4/mbm$
($a^{0}a^{0}c^{+}$) to $Pbnm$ ($a^{-}a^{-}c^{+}$), and from $I4/mcm$
($a^{0}a^{0}c^{-}$) to $R\bar{3}c$ ($a^{-}a^{-}a^{-}$)]. As we will show
below, such comparatively larger energy lowering is caused by condensation of
additional non-polar modes trilinearly coupled to the octahedra rotations. In
addition to this, $R\bar{3}c$ phase allows a further lowering of its symmetry
towards $R3c$ by introducing a polar mode $\Gamma_{4}^{-}$ along the total
rotation axis in the $a^{-}a^{-}a^{-}$ pattern (pseudocubic $[111]$
direction), resulting in a large energy lowering. Nevertheless, the energy of
the $R3c$ phase is still a bit higher than that of the $Pbcm$ phase.

The impact of strain relaxation is examined in Fig. 2(c), where, in addition
to the above results, we report the energies of the phases obtained at fixed
lattice parameters of the cubic parent phase. It can be seen that fixing the
lattice parameters does not qualitatively change the energetics of the phases:
the energy reduction from $I4/mcm$ ($a^{0}a^{0}c^{+/-}$) to $Pbcm$
($a^{-}a^{-}c^{+/-}$) is as pronounced as with strain relaxation, with the
$Pbcm$ phase still slightly below the $R3c$ phase in energy. This rules out
the strain-phonon coupling to be a dominant factor for stabilizing the $Pbcm$
phase.

We then turn to analyze in more details how the lattice-distortion modes are
coupled in ANO. We will mainly focus on the $Pbcm$ phase and take the $Pbnm$
and $R3c$ phases for comparisons. We will adopt Landau-type energy expansion
in terms of symmetry-adapted lattice-distortion modes for the understanding of
the mode couplings, and neglect strain relaxation which does not play a
dominant role.

For the $Pbcm$ phase, we consider the aforementioned four dominating modes,
i.e. $R_{5}^{-}$, $T_{2}$, $\Delta_{5}$ and $M_{5}^{-}$. Taking these modes as
order parameters and restricting to the lowest necessary fourth order, the
Landau energy expansion for the $Pbcm$ phase is written as

$\begin{split}E_{Pbcm}=&E_{\rm
cubic}+A_{R}Q_{R}^{2}+B_{R}Q_{R}^{4}+A_{T}Q_{T}^{2}+B_{T}Q_{T}^{4}+A_{\Delta}Q_{\Delta}^{2}+B_{\Delta}Q_{\Delta}^{4}\\\
&+A_{M^{\prime}}Q_{M^{\prime}}^{2}+C_{RT\Delta}Q_{R}Q_{T}Q_{\Delta}+C_{T\Delta
M^{\prime}}Q_{T}Q_{\Delta}Q_{M^{\prime}}\\\
&+D_{RT}Q_{R}^{2}Q_{T}^{2}+D_{R\Delta}Q_{R}^{2}Q_{\Delta}^{2}+D_{T\Delta}Q_{T}^{2}Q_{\Delta}^{2},\end{split}$
(1)

where $E_{\rm cubic}$ is the energy of the cubic reference structure, $Q_{R}$,
$Q_{T}$, $Q_{\Delta}$ and $Q_{M^{\prime}}$ are the amplitudes of the
$R_{5}^{-}$, $T_{2}$, $\Delta_{5}$ and $M_{5}^{-}$ modes, respectively, and
the $A$, $B$, $C$, $D$ parameters are the coefficients of the energy
expression, with subscript indicating the associated modes. For comparisons,
we also write the energy expansions for the $Pbnm$ phase

$\displaystyle E_{Pbnm}=$ $\displaystyle E_{\rm
cubic}+A_{R}Q_{R}^{2}+B_{R}Q_{R}^{4}+A_{M}Q_{M}^{2}+B_{M}Q_{M}^{4}+A_{X}Q_{X}^{2}$
(2)
$\displaystyle+C_{RMX}Q_{R}Q_{M}Q_{X}+D_{RM}Q_{R}^{2}Q_{M}^{2}+D_{RX}Q_{R}^{2}Q_{X}^{2}+D_{MX}Q_{M}^{2}Q_{X}^{2},$

where $Q_{R}$, $Q_{M}$, and $Q_{X}$ are the amplitudes of the $R_{5}^{-}$,
$M_{2}^{+}$, and $X_{5}^{-}$ dominant modes, respectively, and for the $R3c$
phase

$\displaystyle E_{R3c}=$ $\displaystyle E_{\rm
cubic}+A_{R^{\prime}}Q_{R^{\prime}}^{2}+B_{R^{\prime}}Q_{R^{\prime}}^{4}+A_{\Gamma}Q_{\Gamma}^{2}+B_{\Gamma}Q_{\Gamma}^{4}+D_{R^{\prime}\Gamma}Q_{R^{\prime}}^{2}Q_{\Gamma}^{2},$
(3)

where $Q_{R^{\prime}}$ and $Q_{\Gamma}$ are the amplitudes of the $R_{5}^{-}$
and $\Gamma_{4}^{-}$ modes, respectively.

Table 3: Fitted coefficients for the Landau energy expansions of the $Pbcm$ [Eq. (1)], $Pbnm$ [Eq. (2)] and $R3c$ [Eq. (3)] phases of ANO at fixed cubic lattice parameters. The coefficients are fitted using the units of energy in meV/f.u. and mode amplitudes in Å. The values in brackets are instead the coefficients fitted using relative mode amplitudes (i.e. with mode amplitudes normalized to 1 in the relaxed structure of each phase), which also correspond to the energy contributions (meV/f.u) of the related terms to the energy lowering of each phase. The mode amplitudes in each phase can be easily derived from the ratio between the coefficients and the corresponding energy contribution in brackets.  Phase | Fitted coefficients (Energy contributions)
---|---
$Pbcm$ |  $A_{R}=-670.5$ ($-113.9$) |  $B_{R}=+976.3$ ($+28.2$) |  $C_{RT\Delta}=-691.8$ ($-34.3$) |  $D_{RT}=+1721.4$ ($+85.3$)
|  $A_{T}=-682.6$ ($-199.2$) |  $B_{T}=+1099.1$ ($+93.6$) |  $C_{T\Delta M^{\prime}}=-4692.4$ ($-55.5$) |  $D_{R\Delta}=-1257.0$ ($-10.6$)
|  $A_{\Delta}=-443.3$ ($-22.0$) |  $B_{\Delta}=+21447.7$ ($+53.0$) |  |  $D_{T\Delta}=-1959.4$ ($-28.4$)
|  $A_{M^{\prime}}=+2878.0$ ($+27.7$) |  |  |
$Pbnm$ |  $A_{R}=-670.5$ ($-137.4$) |  $B_{R}=+976.3$ ($+41.0$) |  $C_{RMX}=-1443.2$ ($-17.3$) |  $D_{RM}=+1859.9$ ($+63.1$)
|  $A_{M}=-717.2$ ($-118.8$) |  $B_{M}=+1151.4$ ($+31.6$) |  |  $D_{RX}=+1079.9$ ($+0.9$)
|  $A_{X}=+1547.3$ ($+6.6$) |  |  |  $D_{MX}=+1654.3$ ($+1.2$)
$R3c$ |  $A_{R^{\prime}}=-670.5$ ($-299.4$) |  $B_{R^{\prime}}=+901.3$ ($+179.7$) |  |  $D_{R^{\prime}\Gamma}=-2194.2$ ($-60.1$)
|  $A_{\Gamma}=-768.8$ ($-47.1$) |  $B_{\Gamma}=+14264.8$ ($+53.6$) |  |
Figure 3: Evolution of the energy of ANO when condensing different symmetry
adapted mode in the cubic phase (taken as energy reference) in the absence or
eventually in the presence of other modes as indicated by the legends. Mode
amplitudes of 100% correspond to the amplitude in the relaxed (a, b, d, e)
$Pbcm$, (c) $Pnma$ and (f) $R3c$ phases with cubic lattice parameters.

To quantify the coefficients in the energy expansions, we constructed first-
principles data sets for each of the $Pbcm$, $Pbnm$ and $R3c$ phases by
introducing relevant lattice-distortion modes with different mode amplitudes
to the prototypical $Pm\bar{3}m$ cubic phase at fixed cubic lattice
parameters, and performed the global least-squares fittings of all the energy
data points. For all the three phases, the fitting shows good quality
($R^{2}=0.9975$), indicating that these energy expressions describe well the
associated potential energy surfaces. The obtained coefficients are summarized
in Table 3.

It is worth noticing that the energy expansion of the $Pbcm$ phase [Eq. (1)]
includes two trilinear coupling terms, i.e. $C_{RT\Delta}Q_{R}Q_{T}Q_{\Delta}$
and $C_{T\Delta M^{\prime}}Q_{T}Q_{\Delta}Q_{M^{\prime}}$. We plotted in Figs.
3(a, b) the associated energy curves. In Fig. 3(a), it is seen that the
antipolar $\Delta_{5}$ mode, when condensed alone in the cubic phase, can only
give rise to a shallow and symmetric double well. However, combined with the
$R_{5}^{-}$ and $T_{2}$ modes, the energy curve becomes highly asymmetric,
highlighting the effect of the trilinear coupling term
$C_{RT\Delta}Q_{R}Q_{T}Q_{\Delta}$. In Fig. 3(b), the potential energy curve
associated with the $M_{5}^{-}$ mode is initially symmetric and single-well
shaped when $T_{2}$ and $\Delta_{5}$ modes are zero. However, when the
amplitudes of $T_{2}$ and $\Delta_{5}$ modes are non-zero, the potential well
becomes asymmetric with the bottom of the well shifting to a non-zero
amplitude of $M_{5}^{-}$ mode. This is a consequence of the trilinear coupling
term $C_{T\Delta M^{\prime}}Q_{T}Q_{\Delta}Q_{M^{\prime}}$.

In the $Pbnm$ phase, there exists a similar trilinear term
$C_{RMX}Q_{R}Q_{M}Q_{X}$ (Eq. [2)], which represents the coupling of octahedra
rotations $R_{5}^{-}$, $M_{2}^{+}$, and cation antipolar motion $X_{5}^{-}$.
This trilinear coupling has been well-known in literature, as the $Pbnm$
symmetry is the most commonly observed in perovskites [44]. Fig. 3(c) shows
the associated energy curves in ANO. By itself, the $X_{5}^{-}$ mode is
characterized by a single energy well. The introduction of $R_{5}^{-}$ and
$M_{2}^{+}$ modes leads to the asymmetric shift of the single well due to the
trilinear coupling.

Having a closer look at the $Pbcm$ and $Pbnm$ phases of ANO, one can see
similarities between the modes involved in the respective trilinear couplings
of $R_{5}^{-}$, $T_{2}$ and $\Delta_{5}$, and of $R_{5}^{-}$, $M_{2}^{+}$ and
$X_{5}^{-}$. The first mode, $R_{5}^{-}$, comprises the same $a^{-}a^{-}c^{0}$
rotations present in both phases; because of this we fixed the coefficients
$A_{R}$ and $B_{R}$ to have the same value in the energy expansions for the
two phases [Eqs. (1, 2)]. The second mode, $T_{2}$ in $Pbcm$ and $M_{2}^{+}$
in $Pbnm$, essentially comprises oxygen octahedra rotations around the
$c$-axis, the main difference being the relative signs of the rotations. In
the $M_{2}^{+}$ mode, all oxygen octahedra in a $c$-directed column have the
same rotation sign, while in $T_{2}$ the octahedra in a $c$-directed column
form a ${+}{+}{-}{-}$ rotation pattern. The $M_{2}^{+}$ and $T_{2}$ modes
having the same mode amplitude show the same absolute rotation angle and have
very similar energetics: indeed, as one can see in Tab. 3, $A_{T}$ and
$A_{M}$, as well as $B_{T}$ and $B_{M}$ have almost identical values, lying
within 5% from each other.

The main difference in the energetics comes from the third mode, which is
$\Delta_{5}$ in $Pbcm$ and $X_{5}^{-}$ in $Pbnm$. At a first glance, from the
point of view of crystalline structure, these modes are very similar: they
both consist of $b$-directed antipolar cation motions manifesting in the same
kind of displacement of the Ag cations lying within the same plane normal to
the $c$-axis. In the $X_{5}^{-}$ mode, the Ag planes form a ${+}{-}$
displacement pattern, while in the $\Delta_{5}$ mode they form a
${+}{0}{-}{0}$ pattern. Interestingly, in the $Pbcm$ phase, the non-zero Ag
displacement planes of the $\Delta_{5}$ mode are centered between the _in-
phase_ layers of the oxygen octahedral ${+}{+}{-}{-}$ rotation pattern of the
$T_{2}$ mode [Fig. 1(a)], locally reminiscing the distortions observed in the
$Pbnm$ phase. This points towards similar origins for the trilinear coupling
of $R_{5}^{-}$, $T_{2}$ and $\Delta_{5}$ that we report here and the well-
known trilinear coupling of $R_{5}^{-}$, $M_{2}^{+}$ and $X_{5}^{-}$. Despite
this similarity, the energy lowering from $I4/mcm$ to $Pbcm$ phase is
significantly larger than from $P4/mbm$ to $Pbnm$ [Fig. 2(a)]. One reason for
this is that inducing the $X_{5}^{-}$ mode by itself costs energy,
consistently with the positive sign of its quadratic coefficient in the energy
expansion ($A_{X}>0$) and by the single-well nature of the related energy in
Fig. 3(c). On the contrary, the $\Delta_{5}$ mode is, by itself, unstable in
the cubic phase ($A_{\Delta}<0$), as apparent from the double-well energy
profile in Fig. 3(a), allowing for larger trilinear-coupling-mediated energy
gains.

In addition to the trilinear couplings, the energy expansion of the $Pbcm$
phase [Eq. (1)] also contains three biquadratic coupling terms:
$D_{RT}Q_{R}^{2}Q_{T}^{2}$, $D_{R\Delta}Q_{R}^{2}Q_{\Delta}^{2}$ and
$D_{T\Delta}Q_{T}^{2}Q_{\Delta}^{2}$. Among them, the latter two have negative
coefficients, i.e. $D_{R\Delta}<0$ and $D_{T\Delta}<0$ (Table 3), revealing
the cooperative nature of the biquadratic interactions between $R_{5}^{-}$ and
$\Delta_{5}$ [Fig. 3(d)] and between $T_{2}$ and $\Delta_{5}$ [Fig. 3(e)]. In
contrast, in $Pbnm$ phase all biquadratic couplings are competitive, with
$D_{RM}$, $D_{RX}$, $D_{MX}>0$, which makes another important contribution to
the different energy lowering and is ultimately favoring $Pbcm$ over $Pbnm$ in
ANO. Interestingly, we also observe a similar cooperative biquadratic coupling
$D_{R^{\prime}\Gamma}Q_{R^{\prime}}^{2}Q_{\Gamma}^{2}$ in the $R3c$ phase [Eq.
(3), Fig. 3(f)]. This cooperative biquadratic coupling in $R3c$ leads to an
even more significant energy lowering than the cooperative biquadratic
coupling in the $Pbcm$ phase (Table 3).

Figure 4: (a) Energy lowerings of $Pbcm$, $Pbnm$ and $R\bar{3}c/R3c$ phases
with respect to the cubic parent phase when different distortion modes are
progressively introduced and relaxed. (b) Effects of “switching off” some
specific couplings on the potential energy surfaces of the $Pbcm$ phase in
terms of the amplitudes of $\Delta_{5}$ and $M_{5}^{-}$ modes. At each point
on the plots in (b), the energy value were calculated by relaxing $R_{5}^{-}$
and $T_{2}$ modes at the fixed $\Delta_{5}$ and $M_{5}^{-}$ amplitudes.

To gain deeper insights on the effects of distinct couplings on the
stabilization of the $Pbcm$ phase and to make the above discussion of the
couplings more quantitative, we performed Landau-based calculations where
different distortion modes are progressively introduced and relaxed. As shown
in Fig. 4(a), when only the octahedra rotation modes are allowed to relax
while the polar/antipolar modes are fixed to zero, the $Pbnm$, $Pbcm$, and
$R\bar{3}c$ structures exhibit very similar energies, namely $\rm-120.9\
meV/f.u.$, $\rm-121.3\ meV/f.u.$, and $\rm-124.7\ meV/f.u.$, respectively. In
this case, the $Pbcm$ phase does not show any significant energy advantage
over the $Pbnm$ and $R\bar{3}c$ phases.

When we further allow the polar/antipolar modes to relax, we find that the
$Pbcm$ phase acquires a significant energy lowering. As shown in Fig. 4(a), by
introducing antipolar $\Delta_{5}$ and $M_{5}^{-}$ modes, the energy of $Pbcm$
phase is lowered by 35.59 $\rm meV/f.u.$ and 19.29 $\rm meV/f.u.$,
respectively. In comparison, the introduction of antipolar $X_{5}^{-}$ mode in
the $Pbnm$ phase only lowers the energy by 8.21 $\rm meV/f.u.$. In the
$R\bar{3}c$ phase, the condensation of the polar $\Gamma_{4}^{-}$ mode also
leads to a significant energy reduction of 48.57 $\rm meV/f.u.$, but the
resulting $R3c$ phase has an energy still slightly higher than that of the
$Pbcm$ phase. These results demonstrate the crucial role of the antipolar
$\Delta_{5}$ and $M_{5}^{-}$ modes and of their related couplings in
stabilizing the $Pbcm$ phase.

Table 4: Energy and mode amplitudes of the relaxed $Pbcm$ phase when artificially “switching off” some specific coupling terms. The mode amplitudes are normalized to 1 in the real $Pbcm$ phase (“unmodified” case). |  $E_{\rm min}$ (meV/f.u.) | $R_{5}^{-}$ | $T_{2}$ | $\Delta_{5}$ | $M_{5}^{-}$
---|---|---|---|---|---
unmodified | $-176.14$ | $1.000$ | $1.000$ | $1.000$ | $1.000$
$C_{RT\Delta}=0$ | $-146.43$ | $0.791$ | $1.011$ | $0.901$ | $0.911$
$C_{T\Delta M^{\prime}}=0$ | $-156.85$ | $1.143$ | $0.826$ | $0.839$ | $0.000$
$D_{R\Delta}=0$ | $-167.29$ | $0.862$ | $1.038$ | $0.964$ | $1.001$
$D_{T\Delta}=0$ | $-156.53$ | $1.145$ | $0.823$ | $0.836$ | $0.688$
$C_{RT\Delta}=0$, $C_{T\Delta M^{\prime}}=0$ | $-133.19$ | $1.122$ | $0.751$ | $0.697$ | $0.000$
$D_{R\Delta}=0$, $D_{T\Delta}=0$ | $-148.35$ | $1.047$ | $0.859$ | $0.769$ | $0.660$
$C_{RT\Delta}=0$, $D_{R\Delta}=0$, $D_{T\Delta}=0$ | $-127.80$ | $1.025$ | $0.800$ | $0.613$ | $0.490$
$C_{RT\Delta}=0$, $C_{T\Delta M^{\prime}}=0$, $D_{R\Delta}=0$, $D_{T\Delta}=0$ | $-123.56$ | $1.151$ | $0.679$ | $0.456$ | $0.000$

Having the Landau energy expansions, it is also interesting to investigate the
effects of “switching off” some specific couplings by setting their
coefficients zero. Fig. 4(b) presents the energy surfaces with respect to the
amplitude of $\Delta_{5}$ and $M_{5}^{-}$ modes from the unmodified Landau
energy expansion as well as from some examples of modified versions in which
some couplings are artificially “switched off”. At each point on the energy
surface plots, the $R_{5}^{-}$ and $T_{2}$ modes have been relaxed at the
given amplitudes of $\Delta_{5}$ and $M_{5}^{-}$ modes. We also list in Table
4 the energy values and the relative mode amplitudes corresponding to the
minimum points of the energy surfaces.

Several interesting features can be seen from Table 4 and Fig. 4(b). First,
“switching off” any of the trilinear couplings, i.e. $C_{RT\Delta}$ and
$C_{T\Delta M^{\prime}}$, or the cooperative biquadratic couplings, i.e.
$D_{R\Delta}$ and $D_{T\Delta}$, reduces the amplitude of the antipolar
$\Delta_{5}$ mode as compared to the unmodified case. This is because all
these couplings are associated with the antipolar $\Delta_{5}$ mode and they
are cooperative in nature. Second, when the trilinear coupling term
$C_{T\Delta M^{\prime}}$ is set to zero, the amplitude of the antipolar
$M_{5}^{-}$ mode will be zero. This is because the $M_{5}^{-}$ mode, by
itself, is stable in the cubic phase ($A_{M^{\prime}}>0$), and its
condensation in the $Pbcm$ phase is due entirely to the trilinear coupling
$C_{T\Delta M^{\prime}}$. In addition, from Table 4, we can see that the
$R_{5}^{-}$ and $T_{2}$ modes always behave in opposite manners, i.e. if one
of these two is suppressed, the other one is always enhanced. This is due to
the fact that the coupling between the $R_{5}^{-}$ and $T_{2}$ modes is
competitive ($D_{RT}>0$). Finally, if any of the cooperative coupling terms is
“switched off”, the energy of the $Pbcm$ phase (the minimum of the energy
surface) will be increased. This means that the $Pbcm$ phase would no more be
so low in energy and win over in the competition with other polymorphs such as
the $R3c$. All these results demonstrate the crucial role of these cooperative
couplings in stabilizing the $Pbcm$ antiferroelectric state.

##  IV Comparisons between ANO and PZO

We have analyzed the mode couplings in ANO, highlighting the crucial role of
trilinear and cooperative biquadratic couplings of octahedra rotations and
cation antipolar motions in stabilizing the $Pbcm$ antiferroelectric state. In
the following, we would like to discuss the similarities and differences
between ANO and the prototypical antiferroelectric material PZO in the
stabilization mechanisms of their respective $Pbcm$ and $Pbam$
antiferroelectric states. Previously, J. Íñiguez, et al. have provided a
detailed elucidation of the mode couplings and the stabilization mechanism for
the $Pbam$ antiferroelectric state of PZO [28]. Here, to keep consistency and
achieve quantitative comparisons, we briefly reanalyze PZO using the same
methodology that we employed for ANO. It should be noted that our results are
consistent with the previous literature [28], despite minor differences in
methodology.

Table 5: Symmetry-adapted lattice-distortion mode decomposition of the $Pbam$ phase of PZO. The structure is obtained by relaxing fully the atomic positions and lattice parameters using GGA-PBEsol functional. The mode amplitudes are reported in the so-called “parent-cell-normalized” values of ISODISTORT. | $R_{5}^{-}$ | $\Sigma_{2}$ | $S_{2}$ | $R_{4}^{-}$ | $M_{5}^{-}$ | $X_{1}^{+}$
---|---|---|---|---|---|---
Amplitude (Å) | $0.5323$ | $0.4316$ | $0.1366$ | $0.0307$ | $0.0180$ | $0.0179$
Percentage (%) | $57.91$ | $37.97$ | $3.81$ | $0.19$ | $0.07$ | $0.07$

The room-temperature antiferroelectric state of PZO is the $Pbam$ phase. The
dominant symmetry-adapted lattice-distortion modes in the $Pbam$ phase of PZO
with respect to its cubic parent phase are: (1) $R_{5}^{-}$ mode, the
$a^{-}a^{-}c^{0}$ octahedra rotations, (2) $\Sigma_{2}$ mode, primarily the
antipolar Pb motions $\uparrow\uparrow\downarrow\downarrow$ and (3) a $S_{2}$
mode. These three modes contribute 57.91%, 37.97%, and 3.81% to the total
distortion, respectively (Table 5). The $S_{2}$ mode has a complex distortion
pattern, whose physical interpretation as a modulated oxygen octahedra
rotations was recently given by Shapovalov and Stengel [31]. The Landau energy
expansion of the $Pbam$ phase of PZO can be written as

$\begin{split}E_{Pbam}=&E_{\rm
cubic}+A_{R}Q_{R}^{2}+B_{R}Q_{R}^{4}+A_{S}Q_{S}^{2}+B_{S}Q_{S}^{4}+A_{\Sigma}Q_{\Sigma}^{2}+B_{\Sigma}Q_{\Sigma}^{4}\\\
&+C_{RS\Sigma}Q_{R}Q_{S}Q_{\Sigma}+D_{RS}Q_{R}^{2}Q_{S}^{2}+D_{R\Sigma}Q_{R}^{2}Q_{\Sigma}^{2}+D_{S\Sigma}Q_{S}^{2}Q_{\Sigma}^{2},\end{split}$
(4)

where $Q_{R}$, $Q_{S}$, and $Q_{\Sigma}$ are the amplitudes of the
$R_{5}^{-}$, $S_{2}$, and $\Sigma_{2}$ modes, respectively. Notably, this
energy expansion contains a trilinear coupling among the $R_{5}^{-}$, $S_{2}$
and $\Sigma_{2}$ modes, and biquadratic couplings between any two of the three
modes. It should be kept in mind that the energy expression has been
restricted to the three most dominant modes; there are several other modes in
the $Pbam$ phase (Table 5), but these modes have been omitted due to their
minor amplitudes and negligible energy contributions. We qualified the
coefficient values of Eq. (4) by performing a least-squares fitting on a
first-principles data set ($R^{2}=0.9952$). The fitted coefficients are
summarized in Table 6. We also show in Fig. 5 the energy curves associated
with the trilinear and biquadratic couplings of the $Pbam$ phase of PZO.

Table 6: Fitted coefficients of the Landau energy expansion of the $Pbam$ phase of PZO [Eq. (4)] at fixed cubic lattice parameters. The coefficients are fitted using the units of energy in meV/f.u. and mode amplitudes in Å. The values in brackets are instead the coefficients fitted using relative mode amplitudes (i.e. with mode amplitudes normalized to 1 in the relaxed structure of the $Pbam$ phase), which also correspond to the energy contributions (meV/f.u) of the related terms to the energy lowering of this phase. The mode amplitudes in the phase $Pbam$ phase can be easily derived from the ratio between the coefficients and the corresponding energy contribution in brackets. Phase | Fitted coefficients (Energy contributions)
---|---
$Pbam$ |  $A_{R}=-922.6$ ($-277.1$) |  $B_{R}=+1073.4$ ($+96.9$) |  $C_{RS\Sigma}=-1075.5$ ($-38.7$) |  $D_{R\Sigma}=+1607.3$ ($+84.5$)
|  $A_{S}=-574.2$ ($-14.1$) |  $B_{S}=+2205.8$ ($+1.3$) |  |  $D_{S\Sigma}=+2892.7$ ($+12.4$)
|  $A_{\Sigma}=-1399.4$ ($-244.9$) |  $B_{\Sigma}=+2730.6$ ($+83.6$) |  |  $D_{RS}=+2480.1$ ($+18.3$)
Figure 5:  Evolution of the energy of PZO when condensing different symmetry
adapted mode in the cubic phase (taken as energy reference) in the absence or
eventually in the presence of other modes as indicated by the legends. Mode
amplitudes of 100% correspond to the relaxed amplitude in the relaxed $Pbam$
phase with cubic lattice parameters.

There is a certain similarity between the $Pbcm$ phase of ANO and the $Pbam$
phase of PZO in terms of the mode couplings. First, we note that the trilinear
coupling of $R_{5}^{-}$, $T_{2}$ and $\Delta_{5}$ in ANO appears to be similar
to the trilinear coupling of $R_{5}^{-}$, $S_{2}$ and $\Sigma_{2}$ in PZO:
they involve the same $a^{-}a^{-}c^{0}$ octahedra rotations around $b$-axis in
the $R_{5}^{-}$ mode, both the $T_{2}$ and $S_{2}$ can be interpreted as
octahedra rotations around $c$\- and $a$-axis, respectively, modulated along
the respective rotation axis direction [31], and both the $\Delta_{5}$ and
$\Sigma_{2}$ consist of $b$-directed displacements of cations. Second, there
is an additional trilinear coupling of $T_{2}$, $\Delta_{5}$ and $M_{5}^{-}$
modes identified in ANO, whose counterpart in the $Pbam$ phase of PZO, despite
being overlooked by the energy expansion Eq. (4), is also allowed by symmetry.
From symmetry analysis, we identified a trilinear coupling in the $Pbam$ phase
of PZO, which involves $S_{2}$, $\Sigma_{2}$ and $X_{1}^{+}$ modes, that is
similar to the trilinear coupling of $T_{2}$, $\Delta_{5}$ and $M_{5}^{-}$ in
the $Pbcm$ phase of ANO. The $X_{1}^{+}$ mode is also mainly the antipolar
motion of cations, just like the $M_{5}^{-}$ mode in the $Pbcm$ phase of ANO.
However, due to its negligible amplitude and energy contribution, the
$X_{1}^{+}$ mode is usually not considered when constructing the energy
expansion. Finally, all the biquadratic coupling terms between any two of the
modes are allowed by the symmetry for both $Pbcm$ and $Pbam$. Therefore, from
the qualitative perspective, the antiferroelectric $Pbcm$ phase in ANO and
$Pbam$ phase in PZO are similar in terms of mode couplings.

Nevertheless, some important differences can be noticed if we inspect more
closely the relative mode contributions in the two materials. Among the three
most dominant lattice-distortion modes of ANO, the antipolar $\Delta_{5}$ mode
is remarkably less important than the other two modes, the rotational
$R_{5}^{-}$ and $T_{2}$, as its amplitude and energy contribution are notably
smaller. In this sense, the two rotational modes can be considered as the
“primary” modes defining the symmetry of the $Pbcm$ phase of ANO, and the
antipolar $\Delta_{5}$ mode as the “secondary” mode whose amplitude and energy
contribution are largely enhanced by the coexistence with the primary modes.
In PZO, the roles of rotational and antipolar modes are switched: there, the
“primary” modes are rotational $R_{5}^{-}$ and antipolar $\Sigma_{2}$, which
have larger amplitudes and give significantly larger contributions to the
energetics of the $Pbam$ phase, while the remaining rotational $S_{2}$ mode is
“secondary”. Thus, the antipolar cation motions in the antiferroelectric state
of PZO appear to be more dominant than in ANO.

The difference between ANO and PZO lies not only in the relative importance of
the antipolar modes with respect to the rotational modes, but also in how
these modes interact with each other. The nature of a specific coupling term
in the cubic phase, whether it is cooperative or competitive, can be judged by
the sign of the respective coefficient in the energy expansion. However, it is
not always straightforward to know the overall coupling effect between any two
modes in the strongly distorted low symmetry phase, since multiple coupling
terms may be involved. Here, we propose an approach that more thoroughly
evaluates the combined impact of all relevant coupling terms, thereby
ascertaining the overall coupling effect between any two modes. The detailed
derivations are presented in the Appendix. Adopting this approach, we found
that in ANO the overall coupling effect of the octahedra rotational modes,
i.e. $R_{5}^{-}$ and $T_{2}$, on the antipolar $\Delta_{5}$ mode remain
essentially cooperative in the $Pbcm$ phase (calculated based on the energy
coefficients listed in Table 3 and the mode amplitudes $Q_{R}=0.4122$ Å,
$Q_{T}=0.5402$ Å, $Q_{\Delta}=0.2230$ Å, $Q_{M^{\prime}}=0.0982$ Å):

$\begin{split}\frac{\partial Q_{\Delta}}{\partial
Q_{R}}\bigg{|}_{Pbcm}&=-\frac{C_{RT\Delta}Q_{T}+4D_{R\Delta}Q_{R}Q_{\Delta}}{2A_{\Delta}+12B_{\Delta}Q_{\Delta}^{2}+2D_{R\Delta}Q_{R}^{2}+2D_{T\Delta}Q_{T}^{2}}=+0.081>0,\\\
\frac{\partial Q_{\Delta}}{\partial
Q_{T}}\bigg{|}_{Pbcm}&=-\frac{C_{RT\Delta}Q_{R}+C_{T\Delta
M^{\prime}}Q_{M^{\prime}}+4D_{T\Delta}Q_{T}Q_{\Delta}}{2A_{\Delta}+12B_{\Delta}Q_{\Delta}^{2}+2D_{R\Delta}Q_{R}^{2}+2D_{T\Delta}Q_{T}^{2}}=+0.163>0.\end{split}$
(5)

This can also be evidenced from Figs. 3(a, d, e), where both the first-
principles calculation and the energy expansion-based simulation show that the
amplitude of the $\Delta_{5}$ mode is largely enhanced when coexisting with
$R_{5}^{-}$ or/and $T_{2}$ modes. Instead for PZO, by the same analysis, we
find that the overall coupling effect of $R_{5}^{-}$ and $S_{2}$ modes on the
antipolar $\Sigma_{2}$ mode remain essentially competitive in the $Pbam$ phase
(calculated based on the energy coefficients listed in Table 6 and the mode
amplitudes $Q_{R}=0.5481$ Å, $Q_{S}=0.1568$ Å, $Q_{\Sigma}=0.4184$ Å):

$\begin{split}\frac{\partial Q_{\Sigma}}{\partial
Q_{R}}\bigg{|}_{Pbam}&=-\frac{C_{RS\Sigma}Q_{S}+4D_{R\Sigma}Q_{R}Q_{\Sigma}}{2A_{\Sigma}+12B_{\Sigma}Q_{\Sigma}^{2}+2D_{R\Sigma}Q_{R}^{2}+2D_{S\Sigma}Q_{S}^{2}}=-0.323<0,\\\
\frac{\partial Q_{\Sigma}}{\partial
Q_{S}}\bigg{|}_{Pbam}&=-\frac{C_{RS\Sigma}Q_{R}+4D_{S\Sigma}Q_{S}Q_{\Sigma}}{2A_{\Sigma}+12B_{\Sigma}Q_{\Sigma}^{2}+2D_{R\Sigma}Q_{R}^{2}+2D_{S\Sigma}Q_{S}^{2}}=-0.042<0.\end{split}$
(6)

As can be seen from Fig. 5, coexistence with the $R_{5}^{-}$ or/and $S_{2}$
modes suppresses the amplitude of the $\Sigma_{2}$ mode.

To summarize, although very similar antipolar and octahedra rotation modes –
trilinearly coupled by symmetry – are involved in both cases, we find a
slightly different story in ANO and PZO. In ANO, the antipolar $\Delta_{5}$
mode by itself only shows a weak instability, in contrast with the strong
instability of the antipolar $\Sigma_{2}$ mode in PZO. The emergence of the
$\Delta_{5}$ mode in the antiferroelectric $Pbcm$ phase of ANO is at first
favored by the trilinear coupling with the rotational $R_{5}^{-}$ and $T_{2}$
modes. This behavior reminds us of the concept of hybrid-improper
ferroelectricity [45, 46, 47], and suggests a more hybrid-improper-like nature
of the antipolar cation motions in ANO [48]. Then, this $\Delta_{5}$ mode is
also significantly enhanced by cooperative biquadratic couplings with
$R_{5}^{-}$ and $T_{2}$ modes, which corresponds to a so-called triggered
mechanism [49]. In comparison, the $\Sigma_{2}$ antipolar cation motions in
PZO appear more proper and compete with $R_{5}^{-}$ and $S_{2}$ modes. This
observation highlights the importance of octahedra rotations in inducing the
cation antipolar motions in ANO. Usually, the octahedra rotations can be tuned
by various factors such as strain, pressure, doping, defects, etc [50, 51, 52,
53]. In this regard, investigating the control of octahedra rotations so as to
modulate the AFE property in ANO will be an interesting perspective for a
future work.

##  V Conclusions

In this study, we have emphasized the crucial role of trilinear and
biquadratic couplings of octahedra rotations and cation antipolar motions in
stabilizing the $Pbcm$ phase in ANO. These anharmonic couplings contribute
significantly to lower the energy, making the $Pbcm$ phase energetically more
favorable compared to other competing variants. We have also conducted a
comparative discussion between the antiferroelectric phases of ANO and PZO,
which are distinct but similarly combine antipolar and octahedra rotation
motions trilinearly coupled by symmetry. Unlike in the $Pbam$ phase of PZO,
the antipolar motions in $Pbcm$ phase of ANO appear to be secondary. Their
significant amplitude in the $Pbcm$ phase arises from the combination of
hybrid-improper (i.e. trilinear coupling with primary octahedra rotations) and
triggered (i.e. cooperative biquadratic coupling with primary octahedra
rotations) mechanisms. This contrasts with PZO in which antipolar motions are
primary and compete with octahedra rotations.

###### Acknowledgements.

This work was supported by the European Union’s Horizon 2020 research and
innovation program under grant agreement No. 964931 (TSAR) and by F.R.S.-FNRS
Belgium under PDR grant T.0107.20 (PROMOSPAN). H.Z. acknowledges the
International Postdoctoral Exchange Fellowship (PC2020060) and the Research
IPD-STEMA Program. S.A. acknowledges Erasmus+International Credit Mobility
program 2023 of Université de Liège and the PRD-CCD ARES 2019-2024 project: Le
coltan du Kivu: Capacité de traitement physico-chimique et études
d’applications (Grant No. 32210). The authors acknowledge the use of the CECI
supercomputer facilities funded by the F.R.S-FNRS (Grant No. 2.5020.1) and of
the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the
Walloon Region (Grant No. 1117545).

## Appendix: Evaluation of overall coupling effect between distortion modes

Having the energy expansion and the related coefficients, it is relatively
easy to judge from the sign of the coefficients of specific coupling terms
which modes are initially competitive or cooperative in the cubic reference.
However, it is not always so straightforward to determine the overall coupling
effect between different modes in the low-energy phases, especially when
multiple coupling terms are involved. In this Appendix, we present an approach
to comprehensively assess the whole effect of all relevant coupling terms,
thereby evaluate the overall impact of one mode on the other.

Let us consider a typical example in a relatively general sense, where three
modes $P$, $Q_{1}$ and $Q_{2}$ are mutually coupled and the energy expansion
is given by

$\begin{split}E=E_{0}+AP^{2}+BP^{4}+CQ_{1}Q_{2}P+D_{1}Q_{1}^{2}P^{2}+D_{2}Q_{2}^{2}P^{2},\end{split}$
(7)

where $A$, $B$, $C$, $D_{1}$ and $D_{2}$ are energy coefficients, and $E_{0}$
includes all the energy contributions irrelevant to $P$. This energy expansion
contains one trilinear coupling term, and two biquadratic terms. The goal is
to reveal how $P$ will be affected if small perturbation is applied on $Q_{1}$
or $Q_{2}$ near the energy minimum point.

The $P_{0}$ (the relaxed value of $P$ at given $Q_{1}$ and $Q_{2}$) minimizes
the energy and must satisfy the condition $\frac{\partial E}{\partial
P}|_{P_{0}}=0$, which yields

$\begin{split}2AP_{0}+4BP_{0}^{3}+CQ_{1}Q_{2}+2D_{1}Q_{1}^{2}P_{0}+2D_{2}Q_{2}^{2}P_{0}=0.\end{split}$
(8)

This equation gives an implicit expression of the function
$P_{0}=P_{0}(Q_{1},Q_{2})$. Taking the partial derivatives with respect to
$Q_{1}$ and $Q_{2}$, respectively, on both side of Eq. (8), we have

$\begin{split}2A\frac{\partial P_{0}}{\partial
Q_{1}}+12BP_{0}^{2}\frac{\partial P_{0}}{\partial
Q_{1}}+CQ_{2}+4D_{1}Q_{1}P_{0}+2D_{1}Q_{1}^{2}\frac{\partial P_{0}}{\partial
Q_{1}}+2D_{2}Q_{2}^{2}\frac{\partial P_{0}}{\partial Q_{1}}=0,\\\
2A\frac{\partial P_{0}}{\partial Q_{2}}+12BP_{0}^{2}\frac{\partial
P_{0}}{\partial Q_{2}}+CQ_{1}+2D_{1}Q_{1}^{2}\frac{\partial P_{0}}{\partial
Q_{2}}+4D_{2}Q_{2}P_{0}+2D_{2}Q_{2}^{2}\frac{\partial P_{0}}{\partial
Q_{2}}=0.\end{split}$ (9)

Solving $\frac{\partial P_{0}}{\partial Q_{1}}$ and $\frac{\partial
P_{0}}{\partial Q_{2}}$ from Eq. (9), and evaluating at the energy minimum
point (denoted as “min$E$”, where $P_{0}=P_{00}$, $Q_{1}=Q_{10}$,
$Q_{2}=Q_{20}$), we can get

$\begin{split}\frac{\partial P_{0}}{\partial Q_{1}}\bigg{|}_{{\rm
min}E}&=-\frac{CQ_{20}+4D_{1}Q_{10}P_{00}}{2A+12BP_{00}^{2}+2D_{1}Q_{10}^{2}+2D_{2}Q_{20}^{2}},\\\
\frac{\partial P_{0}}{\partial Q_{2}}\bigg{|}_{{\rm
min}E}&=-\frac{CQ_{10}+4D_{2}Q_{20}P_{00}}{2A+12BP_{00}^{2}+2D_{1}Q_{10}^{2}+2D_{2}Q_{20}^{2}}.\end{split}$
(10)

The partial derivatives of $P_{0}$ with respect to $Q_{1}$ and $Q_{2}$ provide
the evaluation of the overall impact of $Q_{1}$ and $Q_{2}$ on $P$ at the
energy minimum point, respectively, with a positive value meaning a
cooperative effect, and negative meaning competitive.

In the specific case of ANO, replacing $P$ by $Q_{\Delta}$, $Q_{1}$ by
$Q_{R}$, and $Q_{2}$ by $Q_{T}$ and considering an additional term $C_{T\Delta
M}Q_{T}Q_{\Delta}Q_{M}$ in the energy expansion of the $Pbcm$ phase of ANO,
Eq. (5) can be derived following the same procedure.

In the specific case of PZO, replacing $P$ by $Q_{\Sigma}$, $Q_{1}$ by
$Q_{R}$, and $Q_{2}$ by $Q_{S}$, Eq. (10) is directly turned into Eq. (6).

As a final remark, we point out that the equations can be further simplified
if the mode amplitudes are relatively evaluated (renormalized so as to make
$P_{00}^{\prime}=Q^{\prime}_{10}=Q^{\prime}_{20}=1$ at the energy minimum
point):

$\begin{split}\frac{\partial P_{0}^{\prime}}{\partial
Q^{\prime}_{1}}\bigg{|}_{{\rm
min}E}&=-\frac{C^{\prime}+4D^{\prime}_{1}}{2A^{\prime}+12B^{\prime}+2D^{\prime}_{1}+2D^{\prime}_{2}},\\\
\frac{\partial P_{0}^{\prime}}{\partial Q^{\prime}_{2}}\bigg{|}_{{\rm
min}E}&=-\frac{C^{\prime}+4D^{\prime}_{2}}{2A^{\prime}+12B^{\prime}+2D^{\prime}_{1}+2D^{\prime}_{2}},\end{split}$
(11)

where the primed coefficients have the physical meaning of their respective
energy contributions in the studied phase (corresponding to the values in
parentheses in Tabs. 3, 6). In cases where only the qualitative nature of the
coupling effect is of concern, rather than its quantitative strength, this
simplified form is easier to use. We notice that using such primed equivalent
to Eq. (5) and (6) would not change our conclusions.

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