# Influence of external field direction on polarization rotation in
antiferroelectric squaric acid H2C4O4

A. P. Moina  $|$

(Received August 10, 2021, in final form October 2, 2021)

###### Abstract

Using the previously developed model we explore the processes of polarization
rotation in antiferroelectric crystals of squaric acid by the electric fields
directed arbitrarily within the $ac$ plane. Except for some particular
directions of the field, the two-step polarization reorientation at low
temperatures is predicted: first, to the noncollinear phase with perpendicular
sublattice polarizations and then to the collinear ferroelectric phase.
However, when the field is directed along the axis of spontaneous sublattice
polarizations, the intermediate noncollinear phase is absent; when the field
is at 45∘ to this axis, the field for transition to the ferroelectric phase
tends to infinity. The ground state proton configurations and the directions
of the sublattice polarization vectors are determined for all field
orientations. The $T$–$E$ phase diagrams are constructed for the fields
directed along the diagonals of the $ac$ plane and for the above discussed
particular directions of the field.

Key words: antiferroelectricity, electric field, phase transition, phase
diagram, squaric acid

###### Abstract

Ç âèêîðèñòàííÿì çàïðîïîíîâàíî¿ ðàíøå ìîäåë äîñëäæóþòüñÿ ïðîöåñè îáåðòàííÿ
ïîëÿðèçàö¿ çîâíøíìè åëåêòðèíèìè ïîëÿìè, îðíòîâàíèìè äîâëüíèì èíîì â ïëîùèí
$ac$, â àíòèñåãíåòîåëåêòðèíèõ êðèñòàëàõ êâàäðàòíî¿ êèñëîòè. Ïåðåäáààòüñÿ, ùî
çà âèíÿòêîì äåêëüêîõ îñîáëèâèõ íàïðÿìêâ ïîëÿ, ðåîðíòàöÿ ïîëÿðèçàö¿ ïðè íèçüêèõ
òåìïåðàòóðàõ âäáóâàòüñÿ â äâà åòàïè: ñïåðøó äî íåêîëíåàðíî¿ ôàçè ç
ïåðïåíäèêóëÿðíèìè ïîëÿðèçàöÿìè ïäãðàòîê, à ïîòì äî êîëíåàðíî¿
ñåãíåòîåëåêòðèíî¿ ôàçè. Îäíàê, êîëè ïîëå îðíòîâàíå óçäîâæ îñ ñïîíòàííèõ
ïîëÿðèçàöé ïäãðàòîê, ïðîìæíà íåêîëíåàðíà ôàçà âäñóòíÿ. Êîëè æ ïîëå ñïðÿìîâàíå
ïä êóòîì 45∘ äî ö¿ îñ, òî âåëèèíà ïîëÿ, ïðè ÿêîìó âäáóâàòüñÿ ïåðåõä äî
ñåãíåòîåëåêòðèíî¿ ôàçè, ïðÿìó äî íåñêíåííîñò. Äëÿ óñõ íàïðÿìêâ çîâíøíüîãî ïîëÿ
âèçíàåí ïðîòîíí êîíôãóðàö¿ îñíîâíîãî ñòàíó òà íàïðÿìêè âåêòîðâ ïîëÿðèçàö¿
ïäãðàòîê. Ïîáóäîâàíî $T$–$E$ ôàçîâ äàãðàìè äëÿ ïîëâ, ñïðÿìîâàíèõ óçäîâæ
äàãîíàëåé ïëîùèíè $ac$, à òàêîæ äëÿ ïîëâ çãàäàíèõ îñîáëèâèõ íàïðÿìêâ.

Ключов слова: àíòèñåãíåòîåëåêòðèê, åëåêòðèíå ïîëå, ôàçîâèé ïåðåõä, ôàçîâà
äàãðàìà, êâàäðàòíà êèñëîòà

##  1 Introduction

The squaric acid H2C4O4 is a classical two-dimensional antiferroelectric. The
crystal is tetragonal, $I4/m$, in the paraelectric phase and monoclinic,
$P2_{1}/m$, in the antiferroelectric one. The hydrogen bonded C4O4 groups form
sheets, parallel to the $ac$ plane and stacked along the $b$-axis. Below the
transition at 373 K, a spontaneous polarization arises in these sheets, with
the neighboring sheets polarized in the opposite directions [1, 2, 3]. Protons
on the hydrogen bonds in squaric acid move in double-well potentials, so each
of the protons can occupy one of the two sites on the bond: closer to the
given C4O4 group or to the neighboring one. A double bond is formed between
those two adjacent carbon atoms, closer to which two protons on the hydrogen
bonds sit. The sublattice polarization is mostly electronic, while the direct
contribution of ions and displaced protons is significantly smaller [4].

External electric field is known to switch a negative sublattice polarization
in antiferroelectrics and induce thereby the transition from the
antiferroelectric (AFE) to a ferroelectric (FE) phase, visualized in an
experiment as the classical double $P-E$ hysteresis loops. In contrast to the
uniaxial antiferroelectrics, where the sublattice polarizations are collinear
to one predetermined axis, the pseudo-tetragonal symmetry of the squaric acid
crystal lattice and of its hydrogen bond networks allows the sublattice
polarizations in the fully ordered system to be directed along two
perpendicular axes. As a result, in the AFE phase one of the sublattice
polarizations can be rotated by 90∘ by the external field to go along the
other allowed axis, at what a noncollinear ferrielectric phase with
perpendicular sublattice polarizations (NC90 [5]) is induced. Hysteresis loop
measurements and Berry phase calculations [4] have given evidence for such a
rotation. Further calculations [6] have indicated that the 90∘ rotation is
possible at different orientations of the field within the $ac$ plane. It is
also predicted [4, 6] that application of higher fields along the diagonals of
the $ac$ plane can lead to the second rotation of the negative sublattice
polarization by 90∘ and to the induction of the collinear ferroelectric phase.
Experimentally, the transition to the FE phase in squaric acid has not been
detected so far, because of the dielectric breakdown of crystal samples at the
fields yet below 200 kV/cm [4, 7]. The general scheme of the sublattice
polarization rotation and the sequence of the observed/predicted phases is
given in figure 1.

Recently, we developed the deformable two-sublattice proton-ordering model for
squaric acid [8] that described the effects associated with the diagonal
lattice strains. Subsequently [5], this model was modified for description of
the squaric acid behavior in external electric fields applied within the plane
of the hydrogen bonds. The dipole moments associated directly with protons and
with the $\pi$-bonds that can be switched by proton rearrangement were
included into the model. The field-induced polarization rotation in this
picture occurs via flipping of one of the two protons in each molecule to the
other site along the same hydrogen bond and a simultaneous switching of the
$\pi$-bond to those carbon atoms, close to which the protons would sit after
the flipping.

The temperature–electric field $T$–$E_{1}$ phase diagram of squaric acid has
been constructed [5] for the field $E_{1}$, directed along the $a$ axis. The
model calculations confirm the two-step process of polarization reorientation
[4, 6] at low temperatures, with the negative sublattice polarization being
switched twice, by 90∘ at each transition. The system behavior is best
characterized by the introduced noncollinearity angle $\theta$, which is the
angle between the sublattice polarizations. The collinear ferroelectric (FE)
phase and two noncollinear phases with almost antiparallel (AFE) and
perpendicular (NC90) polarizations of the sublattices are identified. The
diagram also contains the crossover region, where the noncollinearity angle
varies continuously between, nominally, 180∘ and 90∘. The supercritical line,
formed by the loci of the maxima of the dielectric permittivity isotherms, is
used to mark the crossover between these two phases [5].

In the present paper, the model [5] is used without any further modifications
to explore the polarization rotation in squaric acid by the electric fields of
different directions within the $ac$ plane. In particular, it is interesting
to check whether the field-induced phase transition to the collinear FE phase
can be lowered down to more experimentally realizable values of fields than it
was predicted for the field $E_{1}$. Herein below, we give a brief outline of
the model; all the details can be found in [5]. In section 3 the low-
temperature field direction–field magnitude phase diagram is constructed; the
$T$–$E$ diagrams are presented for some specific field directions.

Figure 1: (Colour online) The scheme of the two-step polarization rotation in
squaric acid crystals at low temperature. Solid arrows show the actual
polarization vectors $\mathbf{P}_{\pm}$ of the two sublattices in each phase:
antiparallel in AFE, perpendicular in NC90, parallel in FE phase. Dotted
arrows show the two other possible directions of these vectors, allowed by the
crystal structure in the fully ordered system. The electric field $\mathbf{E}$
is noncollinear to the axes of sublattice polarizations.

##  2 The model

There are two formula units in the low-temperature monoclinic unit cell of
squaric acid. In our model, the unit cell consists of two C4O4 groups and four
hydrogen atoms ($f=1,2,3,4$, see figure 2) attached to one of them (the A type
group). All hydrogens around the B type groups are considered to belong to the
A type groups, with which the B groups are hydrogen bonded. Unless it is
specified otherwise, the tetragonal axes $a$ and $c$ are used to identify the
directions and angles within the plane of hydrogen bonds.

Figure 2: Crystal structure of squaric acid as viewed along the $b$ axis.
Figure is taken from [9, 8, 5]. Two adjacent layers are shown, with black and
open circles each. The A and B type C4O4 groups are indicated (see text for
explanation), and the hydrogen bonds are numbered, $f=1,2,3,4$.

The motion of protons in the double-well potentials is described by Ising
pseudospins, whose two eigenvalues $\sigma_{yqf}=\pm 1$ are assigned to two
equilibrium positions of the $f$-th proton. Here, $y$ stands for the layer
index, and $q$ is the index of the A type C4O4 group. Two interpenetrating
sublattices are formed by these layers with alternating polarization, to be
denoted hereafter as the plus and minus sublattices. It is assumed that the
crystal is placed into electric fields $E_{1}$ and $E_{3}$ directed along the
$a$ and $c$ axes.

These fields break the equivalence of the hydrogen bonds linking the C4O4
groups along the $a$ and $c$ axes (with $f=1,3$ and $f=2,4$). Moreover, in
presence of the fields the pseudospin mean values of the two sublattices
differ not only by their signs, as in an antiferroelectric without the
external field, but also by their absolute values. It means that we have four
independent order parameters [5]

$\langle\sigma_{yqf}\rangle=\eta_{f\pm}\,,$

two for each sublattice

$\eta_{1\pm}=-\eta_{3\pm};\quad\eta_{2\pm}=-\eta_{4\pm}\,,$ (2.1)

the latter condition being imposed by the system translational symmetry. It
should be mentioned that there is an experimental evidence for a slight non-
equivalence of the hydrogen bonds, linking the C4O4 groups along the $a$ and
$c$ axes, even in the absence of the field [1]. This non-equivalence, however,
is not taken into account by the present model, which is pseudotetragonal in
the paraelectric phase.

The total system Hamiltonian [8, 5] includes ferroelectric intralayer long-
range interactions, ensuring ferroelectric ordering within each separate
layer, antiferroelectric interlayer interactions responsible for alternation
of polarizations in the stacked layers, the short-range configurational
interactions between protons, which also include the interactions with
external electric fields.

The short-range Hamiltonian describes the four-particle configurational
correlations between protons sitting around each C4O4 group. The usual scheme
[10, 11, 8, 5] of degenerate levels of four lateral (the energy ${\cal
E}_{a}$), two diagonal (${\cal E}_{s}$), eight single-ionized with three or
only one close protons (${\cal E}_{1}$), and two double-ionized with four or
zero protons (${\cal E}_{0}$) configurations is assumed, with ${\cal
E}_{a}<{\cal E}_{s}\ll{\cal E}_{1}\ll{\cal E}_{0}$. Only the groups with
protons in the lateral and single-ionized configurations have dipole moments
in the $ac$ plane; the degeneracy of their energy levels is removed by the
electric fields $E_{1}$ and $E_{3}$ (see table 1).

Our modelling of dipole moments of the H2C4O4 groups is based on the results
of the Berry phase calculations [4], which have shown that the sublattice
polarization in this crystal is formed by the electronic contributions of
switchable $\pi$-bond dipoles and directly by displacements of protons along
the hydrogen bonds. Positions of the $\pi$-bonds are determined by the proton
configurations around the given C4O4 group: in the ground state lateral
configurations, the bond is formed between the two neighboring carbons, near
which protons sit on the hydrogen bonds (see figure 3a), and also between the
carbons and the adjacent oxygens, next to which there is no proton (meaning
that the protons on these bonds sit in the minima close to the neighboring
C4O4 groups). For the depicted lateral proton configuration, the vector of the
proton contribution to the dipole moment goes along the $a$ axis, while the
total dipole moment is at the angle $\piup/2-\varphi_{0}$ to this axis. The
three other lateral configurations and their dipole moments can be obtained
from the scheme of figure 3a by rotation by a multiple of 90∘.

Figure 3: (Colour online) Dipole moments of the lateral (a) and single-ionized
(b) proton configurations (configurations 1 and 7 of table 1, respectively),
as deduced from the results of [4]. ${\bm{\mu}}^{H}_{1}=(2\mu^{H},0,0)$,
$\bm{\mu}^{\pi}_{1}=(2\mu^{\pi}_{\parallel},0,-2\mu^{\pi}_{\perp})$;
${\bm{\mu}}^{H}_{7}=(\mu^{H},0,-\mu^{H})$,
$\bm{\mu}^{\pi}_{7}=(\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp},0,-\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp})$.
Directions of the dipole moments associated with protons and with electrons
are shown by blue and red arrows, respectively; the green arrows are the total
dipole moment of the configuration; the vector lengths are nominal.
$\varphi_{0}$ is the angle between the total dipole moment of configuration 1
and the $-c$ axis.

No Berry phase calculations for the single-ionized configurations have been
performed yet. We assume [5] that the angle between the protonic and
electronic polarization vectors and the ratio of their absolute values are the
same as in the case of the lateral configurations. For the lateral
configuration 1 we have, according to figure 3a,
${\bm{\mu}}^{H}_{1}=(2\mu^{H},0,0)$,
$\bm{\mu}^{\pi}_{1}=(2\mu^{\pi}_{\parallel},0,-2\mu^{\pi}_{\perp})$. For
configuration 7, the proton dipole moment is directed at 45∘ to the
crystallographic axes, ${\bm{\mu}}^{H}_{7}=(\mu^{H},0,-\mu^{H})$, as it is
easily determined from figure 3b, and from the above assumptions it follows
that
$\bm{\mu}^{\pi}_{7}=(\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp},0,-\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp})$.
The dipole moments of the other single-ionized configurations can be obtained
by rotation of the dipole moments of configuration 7. The dipole moment
vectors and energies $\widetilde{{\cal E}_{i}}$ of all configurations in
presence of the electric fields $E_{1}$ and $E_{3}$ are summarized in table 1.

Table 1: Proton configurations and their energies $\widetilde{{\cal E}_{i}}$ in presence of the electric fields $E_{1}$ and $E_{3}$; $W_{3}=\mu^{H}E_{1}+\mu^{\pi}_{\parallel}E_{1}-\mu_{\perp}^{\pi}E_{3}$; $W_{1}=-\mu^{H}E_{3}-\mu^{\pi}_{\parallel}E_{3}-\mu_{\perp}^{\pi}E_{1}$. Directions of the dipole moments associated with protons and with electrons are shown by blue and red arrows, respectively.  $i$ |  | $\widetilde{{\cal E}_{i}}$ | $i$ | $\widetilde{{\cal E}_{i}}$
---|---|---|---|---
1 |  | ${\cal E}_{a}-2W_{3}$ | 7 | ${\cal E}_{1}-W_{1}-W_{3}$
2 |  | ${\cal E}_{a}+2W_{1}$ | 8 | ${\cal E}_{1}+W_{1}-W_{3}$
3 |  | ${\cal E}_{a}+2W_{3}$ | 9 | ${\cal E}_{1}+W_{1}+W_{3}$
4 |  | ${\cal E}_{a}-2W_{1}$ | 10 | ${\cal E}_{1}-W_{1}+W_{3}$
5 |  | ${\cal E}_{s}$ | 11 | ${\cal E}_{1}+W_{1}+W_{3}$
6 |  | ${\cal E}_{s}$ | 12 | ${\cal E}_{1}-W_{1}+W_{3}$
15 |  | ${\cal E}_{0}$ | 13 | ${\cal E}_{1}-W_{1}-W_{3}$
16 |  | ${\cal E}_{0}$ | 14 | ${\cal E}_{1}+W_{1}-W_{3}$

To go from the representation of proton configuration energies to the
pseudospin representation we use the standard procedure [12, 13, 8, 5], and
the four-particle cluster approximation is employed for the obtained
Hamiltonian. The mean field approximation (MFA) is used for the long-range
interlayer and intralayer interactions [8, 5].

Eventually, the thermodynamic potential per one unit cell is obtained in the
following form [5]

$\displaystyle g=U_{\textrm{seed}}-\frac{1}{\beta}[\ln D_{+}+\ln
D_{-}]-\frac{1}{2\beta}\left[\ln(1-\eta^{2}_{1+})+\ln(1-\eta^{2}_{2+})+\ln(1-\eta^{2}_{1-})+\ln(1-\eta^{2}_{2-})\right]$
$\displaystyle{}+\nu\frac{(\eta_{1+}-\eta_{1-})^{2}}{4}+\nu\frac{(\eta_{2+}-\eta_{2-})^{2}}{4}+\nu^{\prime}\frac{(\eta_{1+}+\eta_{1-})^{2}}{4}+\nu^{\prime}\frac{(\eta_{2+}+\eta_{2-})^{2}}{4},$
(2.2)

where

$\displaystyle D_{\pm}$ $\displaystyle=a+\cosh(z_{1\pm}+z_{2\pm})+2b(\cosh
z_{1\pm}+\cosh z_{2\pm})+\cosh(z_{1\pm}-z_{2\pm}),$ $\displaystyle z_{f\pm}$
$\displaystyle=\frac{1}{2}\ln\frac{1+\eta_{f\pm}}{1-\eta_{f\pm}}\pm\beta\nu\frac{\eta_{f+}-\eta_{f-}}{2}+\beta\nu^{\prime}\frac{\eta_{f+}+\eta_{f-}}{2}+\beta\frac{\mu_{f1}E_{1}}{2}+\beta\frac{\mu_{f3}E_{3}}{2},$
$\displaystyle a$ $\displaystyle=\exp(-\beta\varepsilon),\quad b=\exp(-\beta
w).$ (2.3)

The “seed” energy [8]

$U_{\textrm{seed}}=\frac{v}{2}\sum_{ij=1}^{3}c_{ij}^{(0)}u_{i}u_{j}-v\sum_{ij=1}^{3}c_{ij}^{(0)}\alpha_{i}^{(0)}\left(T-T_{i}^{0}\right)u_{j}$
(2.4)

corresponds to the deformable host lattice of heavy ions that forms the
double-well potentials for the motion of protons. It contains elastic and
thermal expansion contributions associated with uniform diagonal lattice
strains $u_{1}$, $u_{2}$, and $u_{3}$; $c_{ij}^{(0)}$ and $\alpha_{i}^{(0)}$
are the “seed” elastic constants and thermal expansion coefficients;
$T_{i}^{0}$ determine the reference point of the thermal expansion of the
crystal, which can be chosen arbitrarily; $v$ is the unit cell volume at this
reference point. Inclusion of $U_{\textrm{seed}}$ into the Hamiltonian allows
us [8] to correctly describe the regular thermal expansion of squaric acid in
the paraelectric phase and the hydrostatic pressure dependence of the lattice
constants.

The dipole moments $\mu_{f1}$ and $\mu_{f3}$ are as follows

$\mu_{11}=-\mu_{31}=\mu^{H}+\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp},\quad\mu_{21}=-\mu_{41}=\mu^{H}+\mu^{\pi}_{\parallel}+\mu^{\pi}_{\perp},$
(2.5)

and

$\mu_{13}=-\mu_{33}=-\mu^{H}-\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp},\quad\mu_{23}=-\mu_{43}=\mu^{H}+\mu^{\pi}_{\parallel}-\mu^{\pi}_{\perp}.$
(2.6)

The short-range Slater-Takagi type energy parameters

$\varepsilon={\cal E}_{s}-{\cal E}_{a},\quad w={\cal E}_{1}-{\cal E}_{a},\quad
w_{1}={\cal E}_{0}-{\cal E}_{a},$

according to the model [8], are considered to be quadratic functions of the
H-site distance $\delta$, and it has been assumed in equation (2.3) that
$w_{1}\to\infty$. In its turn, the distance $\delta$ is taken to vary
according to its experimentally observed [1] above the transition linear
temperature dependence

$\delta=\delta_{0}[1+\delta_{T}(T-T_{\textrm{N0}})],$ (2.7)

where $T_{\textrm{N0}}$ is the transition temperature at zero electric field.
It yields

$\varepsilon=\varepsilon^{\prime}_{0}[1+\delta_{T}(T-T_{\textrm{N0}})]^{2},\quad
w=w_{0}[1+\delta_{T}(T-T_{\textrm{N0}})]^{2}.$ (2.8)

The parameters $\nu$ and $\nu^{\prime}$ from (2.2) are linear combinations of
the Fourier transforms, taken at the center of the Brillouin zone and at ${\bf
k}_{2}=(0,b_{2},0)$, of the eigenvalues of the matrices of the long-range
(dipole-dipole and lattice mediated) interactions. Here $b_{2}$ is the basis
vector of the reciprocal lattice [8, 5]. For them, both the dependence of the
dipole moments on $\delta$ and the variation of the interactions with the
changes in the equilibrium distances between protons (dipoles) are taken into
account [8, 5]

$\nu=\nu_{0}[1+\delta_{T}(T-T_{\textrm{N0}})]^{2}+\sum_{i=1}^{3}\psi_{i}u_{i},\quad\nu^{\prime}=\nu^{\prime}_{0}[1+\delta_{T}(T-T_{\textrm{N0}})]^{2}+\sum_{i=1}^{3}\psi_{i}^{\prime}u_{i}.$
(2.9)

The plus and minus sublattice polarization vectors are

$\mathbf{P}_{\pm}=(P_{1\pm},0,{P_{3\pm}}),$ (2.10)

where [5]

$\displaystyle
P_{1\pm}=\frac{\mu^{H}+\mu^{\pi}_{\parallel}}{2v}(\eta_{1\pm}+\eta_{2\pm})+\frac{\mu_{\perp}^{\pi}}{2v}(\eta_{2\pm}-\eta_{1\pm}),$
$\displaystyle
P_{3\pm}=\frac{\mu^{H}+\mu^{\pi}_{\parallel}}{2v}(\eta_{2\pm}-\eta_{1\pm})-\frac{\mu_{\perp}^{\pi}}{2v}(\eta_{2\pm}+\eta_{1\pm}).$
(2.11)

The order parameters $\eta_{f\pm}$ are found by numerical minimization of the
thermodynamic potential (2.2). At the same time, the strains $u_{i}$ are
determined from the conditions that $\partial g/\partial u_{i}=0$ (see [8,
5]).

The noncollinearity angle $\theta$ is defined as the difference of the angles
between the current plus and minus sublattice polarization vectors (2.10) at
the given temperature and field and the polarization vector
$\mathbf{\bar{P}}_{+}$ of the fully ordered plus sublattice at low fields [5].
The last vector is determined from equations (2.10), (2.11) with the values of
$\eta_{f+}=\bar{\eta}_{f+}$, corresponding to the ground state proton
configuration of the plus sublattice. As the plus sublattice we take the one,
the polarization in which is not switched by high positive fields. Since
$\bar{\eta}_{f+}$ could only be either 1 or $-1$, we obtain

$\theta=\left|\arccos\frac{\bar{\eta}_{1+}\eta_{1+}+\bar{\eta}_{2+}\eta_{2+}}{\sqrt{2(\eta_{1+}^{2}+\eta_{2+}^{2})}}-\arccos\frac{\bar{\eta}_{1+}\eta_{1-}+\bar{\eta}_{2+}\eta_{2-}}{\sqrt{2(\eta_{1-}^{2}+\eta_{2-}^{2})}}\right|.$
(2.12)

##  3 Calculations

###  3.1 Fitting procedure

The developed model allows us to explore the influence of the electric field
of any arbitrary orientation within the $ac$ plane. The field $E$ in question
with the components $(E_{1},E_{3})$ is directed at the angle $\varphi_{E}$ to
the axis $a$; $\varphi_{E}$ is assumed to increase at a counterclockwise
rotation. At $E_{1}>0$, $\varphi_{E}=\arctan E_{3}/E_{1}$.

The values of all model parameters were determined earlier [8, 5]. In
particular, they were required to provide the best fit to the experimental
temperature curves of the order parameter at ambient pressure, the temperature
and hydrostatic pressure dependences of the lattice strains $u_{i}$, and the
pressure dependence of the transition temperature $T_{\textrm{N}}$. The
adopted values of the model parameters are given in table 2. A technical
error, unfortunately, occurred in [5]; as a result, the incorrect values of
$\mu^{H}+\mu_{\parallel}^{\pi}$ and $\mu_{\perp}^{\pi}$ were given therein.
Here, the correct numbers are presented.

From figure 3 and the values of $\mu^{H}+\mu_{\parallel}^{\pi}$ and
$\mu_{\perp}^{\pi}$ from table 2 it follows that the angle $\varphi_{0}$
between the vector of the total dipole moment of configuration 1 and the $-c$
axis (or of configuration 2 and the $a$ axis) is about $56^{\circ}$ for this
set of the model parameters. Note that from the Berry phase calculations [4]
it follows that the respective angle is about $50^{\circ}$.

The parameter $\nu^{\prime}_{0}$ and the dipole moments
$\mu^{H}+\mu_{\parallel}^{\pi}$ and $\mu_{\perp}^{\pi}$ were determined by
fitting the calculated curve of the dielectric permittivity $\varepsilon_{11}$
at zero external bias field to the experimental points of [4], while
simultaneously trying to get as good as possible agreement with the experiment
for the values of the switching fields, corresponding to the first 90∘
rotation of the sublattice polarization [5]. Unfortunately, the switching
fields calculated with these parameters are still considerably higher than the
measured ones. The theoretical values of total polarization in the fully
ordered NC90 and FE phases, induced by the field $E_{1}$, are also much higher
[5] than predicted by the Berry phase calculations or measured experimentally
(in the NC90 phase) [4], and so is the value of the saturated spontaneous
sublattice polarization in the AFE phase.

Alternatively, $\mu^{H}+\mu_{\parallel}^{\pi}$ and $\mu_{\perp}^{\pi}$ can be
chosen by fitting the theory to the values of polarization, found from the
Berry phase calculations [4]. However, it will essentially spoil the agreement
with the experiment for the permittivity and even further increase the
difference between the theory and experiment for the switching fields. It
appears that all these three quantities (permittivity, polarization, switching
fields) cannot be well described by the present model with the same set of the
fitting parameters. This controversy can possibly be addressed by taking into
account the non-linear terms in the Hamiltonian, e.g., of the type
$\mu^{\prime}(\sum\sigma)^{2}\sigma E$, which is equivalent to the terms like
$EP^{3}$ in the phenomenological approach, similarly to how it was proposed
for a description of electric field effects in the KH2PO4 family crystals [14,
15].

Table 2: The adopted values of the model parameters, taken from [8, 5].

$\varepsilon^{\prime}_{0}/k_{\textbf{B}}$ | $w_{0}/k_{\textbf{B}}$ | $\nu_{0}/k_{\textbf{B}}$ | $\nu^{\prime}_{0}/k_{\textbf{B}}$ | $\psi_{1}/k_{\textbf{B}}$ | $\psi_{2}/k_{\textbf{B}}$ | $\psi_{3}/k_{\textbf{B}}$ | $c_{11}^{0}$ | $c_{12}^{0}$ | $c_{13}^{0}$ | $c_{22}^{0}$
---|---|---|---|---|---|---|---|---|---|---
(K) | ($10^{10}$ N/m2)
395 | 1100 | 79.9 | $-50$ | $-518$ | 445 | 1096 | 6.5 | 2.3 | $-3.1$ | $2.38-0.02T$

$\alpha_{1}^{0}$ | $\alpha_{2}^{0}$ | $\delta_{T}$ | $v$ | $\mu^{H}+\mu^{\pi}_{\parallel}$ | $\mu^{\pi}_{\perp}$
---|---|---|---|---|---
($10^{-5}$ K-1) | ($10^{-28}$m3) | ($10^{-29}$ Cm)
1.2 | 13.0 | 20 | 2.0 | 3.16 | 2.12

Moreover, as extensively discussed in [5], the adopted set of
$\nu^{\prime}_{0}$ and $\mu$ is not unique. The values of these parameters can
be varied within certain limits, and other acceptable sets of the values can
be found, which provide a similar agreement with the available experimental
data for the permittivity and switching fields. The calculated position of the
upper field-induced transition line between the noncollinear ferrielectric and
collinear ferroelectric phase is most sensitive to the values of these
parameters. In view of the discussed uncertainty, this position should be
treated with caution and as a qualitative theoretical prediction, rather than
a quantitative estimate.

###  3.2 Ground state phase diagram

As it has been already mentioned, our previous calculations [5] have revealed
that the external field $E_{1}$ at low temperatures induces two phase
transitions in the squaric acid crystals, each associated with a sublattice
polarization rotation by 90∘. The predicted sequence of the phases with
increasing field is as follows: nearly antiferroelectric with antiparallel
sublattice polarizations (AFE*) — noncollinear ferrielectric with
perpendicular sublattice polarizations (NC90) — collinear ferroelectric (FE).

In order to verify whether this behavior persists for other directions of the
electric field confined to the $ac$ plane, we construct the low-temperature
$\varphi_{E}$–$E$ phase diagram of the squaric acid. The model symmetry in the
absence of the field is pseudotetragonal, and the electric field applied
within the $ac$ plane breaks this symmetry. The thermodynamic potential (2.2)
is invariant with respect to the 90∘ rotation around the $b$ axis, at which
$E_{1}\to E_{3}$, $E_{3}\to-E_{1}$, $\varphi_{E}\to\varphi_{E}+\piup/2$,
$\eta_{1\pm}\to\eta_{2\pm}$, $\eta_{2\pm}\to-\eta_{1\pm}$. It is thus expected
that the $\varphi_{E}$–$E$ diagram should also be symmetric (in terms of the
phases and transitions between them) with respect to the 90∘ rotation of the
field $\varphi_{E}\to\varphi_{E}+\piup/2$. Figure 4, where the obtained
$\varphi_{E}$–$E$ diagram is presented, proves that this is, indeed, the case.

As one can see, the two-step polarization rotation and the AFE*–NC90–FE phase
sequence are observed at all field orientations, except for eight special
directions with $\varphi_{E}=\varphi_{n}$, where
$\varphi_{n}=\varphi_{0}+\piup n/4$, $n=0,...7$. The lines of the first order
transitions between the AFE* and NC90 phases and between the NC90 and FE
phases are denoted as II and III, following the notations adopted in [5].

Figure 4: (Colour online) The $\varphi_{E}$–$E$ phase diagram of squaric acid
at 250 K. The solid lines II and III are the lines of the first order phase
transitions between the AFE* (the red region, $\theta\approx 180^{\circ}$) and
NC90 (the green regions, $\theta\approx 90^{\circ}$) phases and between the
NC90 and FE (the purple regions, $\theta=0$) phases, respectively. Vertical
dashed lines are a guide to the eye. The numbers in parentheses indicate the
ground state proton configurations in each phase in the plus and minus
sublattices, respectively (table 1).

There are, obviously, two types of these special directions. The first type is
at an even $n$, i.e., at $\varphi_{E}=\varphi_{2k}$, $k=0,1,2,3$. In this
case, the electric field is collinear to the antiparallel sublattice
polarizations of the AFE* phase (this collinearity is discussed in more detail
later in this subsection), so the squaric acid becomes effectively a uniaxial
antiferroelectric. The intermediate non-collinear phase NC90 disappears, and
polarization switching is a traditional one-step process from AFE* to FE
phase. The second type is at odd $n$, at $\varphi_{E}=\varphi_{2k+1}$, when
the external electric field is applied at 45∘ to the sublattice polarizations
of the AFE* phase. In this case, the non-collinear NC90 phase is the widest,
with the upper transition field to the FE phase at lines III tending to
infinity.

Experimentally, the low-temperature transition to the ferroelectric phase has
not been detected yet because of the dielectric breakdown of the samples [4,
7]. As it follows from the constructed $\varphi_{E}$–$E$ diagram, the field of
this transition is the lowest when its direction is close to the axis of the
sublattice polarization, $\varphi_{E}=\varphi_{2k}$. Measurements of the
polarization switching for the fields oriented in the sector between
$50^{\circ}$ and $60^{\circ}$ to the $a$ axis seem most promising in this
respect.

The $\varphi_{E}$–$E$ phase diagram, in terms of the phases, phase
transitions, and relative orientation of the sublattice polarization vectors
described by the noncollinearity angle $\theta$, is symmetric with respect to
the field rotation by $90^{\circ}$. However, the absolute orientations of the
sublattice polarization vectors for each $\varphi_{E}$ and for the
corresponding $\varphi_{E}\pm\piup/2$ at the same field magnitudes are
different. This is caused by the fact that different lateral proton
configurations become the ground state ones in the same phase.

At low temperatures, all protons within a sublattice are in the same lateral
configuration, one of the four possible (see table 1). The ground state
sublattice polarization vector, therefore, can be only oriented along one of
the four directions, depending on which lateral configuration is realized in
this sublattice. This configuration can be determined from the signs of the
order parameters $\eta_{1\pm}$ and $\eta_{2\pm}$ at low temperatures, found by
minimization of the thermodynamic potential (2.2). Positive $\eta_{f\pm}=1$ or
negative $\eta_{f\pm}=-1$ mean that the proton on the $f$th bond in the
plus/minus sublattice is localized at the site close to the considered C4O4
group or to the neighboring C4O4 group, respectively. A change of the sign of
the order parameter, therefore, means flipping of the proton at the respective
bond. The obtained ground state configurations of the plus and minus
sublattices in each region of the $\varphi_{E}$–$E$ phase diagram are
indicated by the pairs of numbers in figures 4 and 5; the respective proton
arrangements are visualized in figure 10 in the appendix.

Figure 5: (Colour online) The diagram of the ground state proton
configurations vs angle $\varphi_{E}$. The pairs of numbers in each segment or
sector indicate the proton configurations of the plus and minus sublattices,
respectively, in the AFE* (the inner circle, shades of red), NC90 (the
intermediate segments, shades of green), FE (the outer segments, shades of
purple) phases. Different shades correspond to different configurations. Bold
black solid lines: crystallographic axes $a$ and $c$. Black dashed lines: axes
of (pseudo)spontaneous polarization $\mathbf{P}_{(1)}$–$\mathbf{P}_{(3)}$ and
$\mathbf{P}_{(2)}$–$\mathbf{P}_{(4)}$.

The four possible directions of the low-temperature sublattice polarization
vectors are shown in figure 5 by the dotted lines as the axes
$\mathbf{P}_{(1)}$–$\mathbf{P}_{(3)}$ and
$\mathbf{P}_{(2)}$–$\mathbf{P}_{(4)}$. They coincide with the directions of
the total dipole moments of the lateral configurations 1–4 and oriented at the
angles $\varphi_{2k}$ to the axis $a$. Each change of the configuration number
$1\to 2\to 3\to 4\to 1$ corresponds to a counterclockwise rotation of the
polarization vector by $90^{\circ}$. Figure 10 illustrates the polarization
vectors for the two sublattices at full saturation in different regions of the
$\varphi_{E}$–$E$ phase diagram.

The following conclusions can be drawn from the analysis of the ground state
proton configurations at different field directions:

  1. 1.

The obvious one, that rotation of the field by $90^{\circ}$,
$\varphi_{E}\to\varphi_{E}+\piup/2$, leads to a $90^{\circ}$ rotation of the
sublattice polarization vectors in the same direction for all field
magnitudes.

  2. 2.

The ground state configurations are switched in a first order transition-like
manner at $\varphi_{2k+1}$ in all phases and at $\varphi_{2k}$ in the NC90
phase only. Because of the switching at $\varphi_{E}=\varphi_{2k+1}$ and the
resulting rotation of the polarization vectors by 90∘, which occur even at
vanishingly small field magnitudes, in the AFE* phase the angle between field
and the polarization axis never exceeds 45∘ and is zero at each
$\varphi_{2k}$, where the field is collinear to the sublattice polarizations.

  3. 3.

The minus sublattice configuration for the NC90 phase changes to the
antiparallel one at each $\varphi_{2k}$. This has an obvious geometric
explanation: The sign of the tangential to the AFE* phase polarization vectors
component of the electric field, which causes the 90∘ rotation of the
polarization, is changed at $\varphi_{2k}$.

The entire scheme of figure 5 stems from the low-field switching of the proton
configurations in the AFE* phase at $\varphi_{2k+1}$. It creates the initial
AFE* configuration of the system, starting from which the subsequent high-
field rotation of the minus sublattice polarization and transitions to the
NC90 and FE phases occur. Basically, this low-field switching is the change of
the orientation of the (pseudo) spontaneous polarizations caused by the
external field. In a perfect crystal, the paraelectric (tetragonal) –
antiferroelectric (monoclinic) transition is a spontaneous symmetry breaking
process in absence of external fields, and either the configurations 1 and 3
or 2 and 4 can become the ground state one below the transition. Similarly,
either of the tetragonal axes $a_{t}$ or $c_{t}$ above the transition can
become the $a$-axis of the monoclinic setting below it. At cooling the sample
from the paraelectric phase in presence of a low external field with
$\varphi_{E}$ between $\varphi_{1}$ and $\varphi_{3}$ or between $\varphi_{5}$
and $\varphi_{7}$, which component along the
$\mathbf{P}_{(1)}$–$\mathbf{P}_{(3)}$ axis is larger than the tangential one,
the (pseudo)spontaneous sublattice polarizations in the AFE* phase will be
directed along this axis, and 1 and 3 configurations will be realized.
However, the field with $\varphi_{E}$ between $\varphi_{7}$ and $\varphi_{1}$
or $\varphi_{3}$ and $\varphi_{5}$, where the tangential component to the
$\mathbf{P}_{(1)}$–$\mathbf{P}_{(3)}$ axis is larger than the longitudinal
one, changes that preferential direction for the spontaneous
polarization/deformation of the sample, so the configurations 2 and 4 become
the ground state ones, and the monoclinic axis $a$ goes along $c_{t}$.

It should be kept in mind that the present theory does not describe the
experimental situation, when the electric field is applied to the crystal,
already spontaneously polarized (antiferroelectrically) in the zero field, and
the spontaneous proton configuration does not coincide with the one favored by
the applied field. In this situation, the system finds itself in a metastable
local minimum of the system energy, and with increasing the field, a
transition to the global minimum will occur. Instead, we always consider the
behavior of the system, which is in the global minimum of its thermodynamic
potential. Then, the (pseudo)spontaneous proton configurations and the
directions of the (pseudo)spontaneous sublattice polarizations are always
those, which are dictated by the external electric field. The experimental
procedure, which would reproduce this behavior, is to apply a small by
magnitude field of the required direction to a heated crystal in the
paraelectric phase, then cool the sample down to the required temperature, and
then increase the field up to the required magnitude.

The lines II and III in figure 4 are very well approximated by the piecewise
functions $E^{II}=E_{sw}/\cos(\Delta\varphi)$ and
$E^{III}=E_{sw}/\sin(\Delta\varphi)$, where
$\Delta\varphi=\varphi_{E}-\varphi_{2k+1}$, $E_{sw}=214$ kV/cm, and each piece
is defined between $\varphi_{2k}$ and $\varphi_{2k+2}$. At
$\Delta\varphi=\pm\piup/4$, or, equivalently $\varphi_{E}=\varphi_{2k}$, the
fields $E^{II}$ and $E^{III}$ coincide, in accordance with figure 4. A semi-
qualitative explanation of these dependences is as follows. At zero
temperature in the NC90 phase, the polarizations are oriented at
$\piup/4-\Delta\varphi$ and $\piup/4+\Delta\varphi$ to the field (see figure
5). In the FE phase, the sublattice polarizations are parallel, both at
$\piup/4-\Delta\varphi$ to the field at zero temperature. In the AFE phase,
the sublattice polarizations are antiparallel, directed at
$\piup/4-\Delta\varphi$ and at $\piup-(\piup/4-\Delta\varphi)$ to the field.
The difference between the energies of the fully ordered NC90 and FE
arrangements, where two sublattices are treated as two dipoles in an external
field, is proportional to $(E\sin\Delta\varphi)$. On the other hand, the
difference between the AFE and NC90 arrangements is proportional to
$(E\cos\Delta\varphi)$. When, roughly speaking, one of this differences
becomes larger than the difference between the energies of the AFE
correlations in the same arrangements, which is field-independent, the phase
transition occurs. From here directly follow the above given dependences of
the transition fields $E^{II}$ and $E^{III}$ on $\Delta\varphi$.

The above discussion is relevant for low temperatures and for a fully ordered
system, when all protons within each sublattice are arranged in the same
lateral configuration. As a result, the sublattice polarization directions
coincide with the directions of the total dipole moments of these
configurations, and only four directions, along the
$\mathbf{P}_{(1)}$–$\mathbf{P}_{(3)}$ and
$\mathbf{P}_{(2)}$–$\mathbf{P}_{(4)}$ axes, are allowed. Thermal fluctuations
in the presence of the electric field destroy this order, and some protons
rearrange into other configurations, including the single-ionized ones, which
dipole moments are directed at $45^{\circ}$ to these axes. In this case, the
resulting average vector of the sublattice polarization can be directed
arbitrarily, as determined by relative populations of different proton
configurations within the sublattice. Herein below we discuss the
temperature–electric field behavior of squaric acid for some particular
orientations of the field.

###  3.3 Electric fields $\pm E_{1}$ and $\pm E_{3}$

These fields correspond to $\varphi_{E}=\piup n/2$. As mentioned above, due to
the symmetry of the system with respect to $90^{\circ}$ rotations around the
$b$ axis, the $T$–$E$ phase diagrams for these fields will be identical,
though the polarization vectors in all phases will be rotated counterclockwise
by $\piup n/2$ as compared to the case of the field $E_{1}$. The $T$–$E_{1}$
diagram of squaric acid was obtained earlier [5]. This diagram, overlapped
with the color gradient plot of the noncollinearity angle $\theta$, is shown
in figure 6. We detected three lines of the first order phase transitions I,
II, and III. Lines I and II terminate at the bicritical end points BCE1 and
BCE2. Line III ends at the tricritical point TCP. The line of the second order
phase transitions IV starts at the tricritical pont TCP and terminates at the
critical end point CEP, lying at the line of the first order transitions I in
a close vicinity of the bicritical end point BCE1.

Figure 6: (Colour online) The $T$–$E_{1}$ phase diagram of the squaric acid,
overlapping the $T$–$E_{1}$ color contour plot of the noncollinearity angle
$\theta$. Solid and dashed lines indicate the first and second order phase
transitions, respectively. The open square $\square$, star ☆, and full circles
$\bullet$ indicate the critical end point, tricritical point, and bicritical
end points, respectively. Blue full triangles $\blacktriangle$ are the
experimental points [4].

The phase denoted as AFE* (the red region) is non-collinear antiferrielectric,
but very close to the initial AFE phase. Here, $\theta\sim 180^{\circ}$, as
only a slight rotation of the minus sublattice polarization takes place in
this region at non-zero fields, discernible mostly at its upper right
boundary. The purple region on the phase diagram is the collinear field-
induced ferroelectric phase (FE) with $\theta=0$. The true paraelectric phase
with zero sublattice polarizations, just like the true AFE phase, exists only
at $E=0$. The phase between the transition lines II, III, and IV (the green
and blue regions in figure 6) is the non-collinear ferrielectric phase NC90.
In this phase, the noncollinearity angle $\theta$ is mostly close to 90∘, only
rapidly decreasing to zero at increasing temperature within a narrow (blue)
stripe along the second-order phase transition line IV, denoted as NC22*.

In the region NC135* between the BCE1 and BCE2 points (orange to yellow in
figure 6) a crossover between the AFE* and NC90 phases occurs. Here, $\theta$
gradually changes from $\sim 180^{\circ}$ to $\sim 90^{\circ}$: the minus
sublattice polarization rotates continuously with increasing field and becomes
perpendicular to the plus sublattice polarization. As discussed above, this
continuous rotation in the NC135* and NC22* regions is a statistically
averaged effect, possible only in the presence of thermal fluctuations.

As one can see in figure 6, the calculated fields of switching between the
AFE* and NC90 phases, which occurs either as a phase transition at lines II
and I or as a crossover within the region NC135* between the points BCE1 and
BCE2, are essentially higher than the ones observed experimentally [4].

###  3.4 Diagonal electric fields $E_{1}\pm E_{3}$

These are the fields directed along the diagonals of the tetragonal $ac$
plane: $\varphi_{E}=\piup/4+\piup n/2$. For the sake of convenience, we denote
them as $E_{1}\pm E_{3}$. For all these fields, the $T$–$E$ phase diagrams are
identical because of the the symmetry of the system with respect to
$90^{\circ}$ rotations around the $b$ axis.

Figure 7: (Colour online) The same as in figure 6 for the fields $E_{1}\pm
E_{3}$. The tricritical point TCP is split to the critical end point CEP2 and
bicritical end point BCE3. Graphs in the right-hand column are the closeups of
the phase diagram in the vicinities of BCE1/CEP1 and BCE3/CEP2. Blue full
triangle $\blacktriangle$ is the experimental point obtained for $E_{1}+E_{3}$
(taken from the Electronic supplementary material to [4]).

As one can see in figure 7, the $T$–$(E_{1}\pm E_{3})$ phase diagram topology
is mainly similar to that of the $T$–$E_{1}$. The fields of the low-
temperature switching to the noncollinear phase NC90 (line II) are higher than
predicted for the field $E_{1}$, but the fields of the transition to the
collinear FE phase (line III) are much lower. The increase of the switching
field $E^{II}$ for $E_{1}\pm E_{3}$ as compared to the case of $E_{1}$ is in a
qualitative agreement with the experimental observations (see [4] and the
Electronic supplementary material to it). The theoretical values of $E^{II}$
are, however, significantly higher than the measured value.

There are also some qualitative differences between the phase diagrams,
predicted for the fields $E_{1}$ and $E_{1}\pm E_{3}$. The obvious difference
is that the tricritical point TCP from the $T$–$E_{1}$ diagram is here split
into the system of the bicritical end point BCE3, terminating the line III of
the first order transitions, and the critical end point CEP2, where line IV of
the second order transitions starts (see figure 7). In the emerged wedge
between the lines III and IV, a new phase NC22* is stabilized. This is a
noncollinear ferrielectric phase with a small angle $\theta$ between the
sublattice polarizations (see figure 7). It becomes clear then that in the
$T$–$E_{1}$ diagram (figure 6), the region NC22* is a prototype of the NC22*
phase, not fully stabilized at the chosen values of the model parameters.

Such a splitting of a TCP into a system of a BCE and CEP points is a well
known phenomenon in the theory of metamagnetic transitions, occurring, for
instance, in the two-sublattice Ising model for some sets of the model
parameters, in particular, at different ratios between the competing intra-
and inter-sublattice interactions [16]. Even though we use the same set of the
model parameters for all $\varphi_{E}$, changing the field from $E_{1}$ to
$E_{1}\pm E_{3}$ is, technically, equivalent to changing the values of the
dipole moments $\mu^{H}+\mu^{\pi}_{\parallel}$ and $\mu^{\pi}_{\perp}$, as can
be seen from equations (2.3) and (2.5), (2.6). No attempt has been made to
determine the direction of the field, at which this splitting of the TCP point
appears in the phase diagram. Moreover, the splitting can appear even in the
$T$–$E_{1}$ diagram, if other values of the parameters are used. This is
another ramification of the uncertainty in the choice of the model parameters.

There are indications that for the two-sublattice Ising and similar models,
the splitting of the TCP point is an artefact of the used mean field
approximation (see e.g., [17] and references therein) and is never realized in
experiment. Nonetheless, it clarifies some peculiarities of the system
behavior in a situation, when the splitting does not occur with the present
set of the model parameters, like the discussed earlier case of the field
$E_{1}$.

###  3.5 Fields directed at $\piup/4$ to the sublattice polarization axes

Those are the fields directed at $\varphi_{E}=\varphi_{2k+1}$ to the $a$ axis
(e.g. $\varphi_{7}\approx 11^{\circ}$ for the used set of the model
parameters). Since at these $\varphi_{E}$, the switching of AFE proton
configurations extensively discussed in subsection 3.2 takes place, in order
to avoid confusion, we shall imply that the case of
$\varphi_{E}=\varphi_{2k+1}-\Delta\varphi$, $\Delta\varphi\to 0$, is
considered.

The phase diagram at low fields is very similar to the $T$–$E_{1}$ one. The
AFE, NC90, and FE phases are observed. The BCE1 and BCE2 points are closer
here than in the field $E_{1}$. No second, high-field transition from NC90 to
the NC22* or FE phase at low temperatures is detected below 109 kV/cm.

Figure 8: (Colour online) The same as in figure 6 for the fields directed at
$\varphi_{E}=\varphi_{2k+1}$ to the $a$ axis.

###  3.6 Fields collinear to the sublattice polarizations

Those are the fields directed at $\varphi_{E}=\varphi_{2k}$ (e.g.
$\varphi_{0}\approx 56^{\circ}$). Being collinear to the sublattice
polarization vectors in the AFE phase, they have no tangential component to
rotate these vectors within the $ac$ plane. The squaric acid crystal then
behaves like a uniaxial antiferroelectric, where the field-induced switching
occurs by flipping the minus sublattice polarization along the same direction.
The switching at low temperatures is thus a one-step process from the AFE* to
the FE phase. At all temperatures and field magnitudes
$\eta_{1\pm}=-\eta_{2\pm}$. In this case, the noncollinearity angle $\theta$
cannot be used to distinguish between different phases, as it always equals
either $180^{\circ}$ or 0. Instead, we use the traditional AFE ordering
parameter $q=\eta_{2+}-\eta_{2-}$. The obtained $T$–$E$ phase diagram,
overlapping the color contour plot of $q$ is shown in figure 9.

The line II and BCE2 point are obviously absent from the diagram. At higher
temperatures, the intermediate collinear ferrielectric phase FI is observed
between the AFE* and FE phases. The FI phase is, apparently, an analog of the
NC22* phase/region, observed in the phase diagrams for the fields,
noncollinear to the polarization axis. In the AFE* and FI phases
$\eta_{f+}=-\eta_{f-}$, in the FE phase $\eta_{f+}=\eta_{f-}$.

The system of the bicritical end point BCE1 and critical endpoint CEP1,
observed at the $T$–$E_{1}$ and $T$–$(E_{1}\pm E_{3})$ phase diagrams,
collapses here into a single tricritical point TCP1, at which the line I of
the first order transitions ends, and the line IV of the second order
transitions starts. Line IV continues up to the other tricritical point TCP2,
at which the line III of the first order transitions begins. Not far from the
TCP2 point there is a triple point TP, at which the AFE*, FE, and FI phases
are in equilibrium. The line V of the first order transitions, separating the
AFE* and FI phases, is very short, starting from the triple point TP and
ending at the bicritical end point BCE. A downward jump of $q$ occurs, when
temperature is increased across line V. The phase diagram, overall, is highly
reminiscent of the one obtained by the Landau expansion method for uniaxial
antiferroelectrics [18] (figure 1b therein), especially in the region of the
TP-TCP2-BCE points.

Figure 9: (Colour online) The $T$–$E$ phase diagram of the squaric acid,
overlapping the $T$–$E$ color contour plot of the AFE ordering parameter $q$
for the field collinear to the polarization axis ($\varphi_{E}=\varphi_{2k}$).
The right-hand panel: a closeup of the phase diagram in the vicinity of the
triple point TP. Solid and dashed lines indicate the first and second order
phase transitions, respectively. The stars ☆, full circles $\bullet$, and open
triangles $\triangle$ indicate the tricritical points, bicritical end point,
and the triple point, respectively.

##  4 Concluding remarks

Using the previously developed [5] deformable two-sublattice proton ordering
model, we explore the processes of polarization rotation in antiferroelectric
crystals of squaric acid under the inluence of external electric fields,
oriented arbitrarily within the plane of the hydrogen bonds. The unique
structure of the two-dimensional hydrogen bond networks in squaric acid
permits 90∘ rotation of the sublattice polarization. Application of the model
to the case of the electric field $E_{1}$ has revealed [5] that unlike in the
conventional uniaxial antiferroelectrics, the polarization rotation in squaric
acid is a two-step process, as it was predicted earlier [4]. At low
temperatures the crystal is switched from the antiferroelectric phase AFE*,
first to the noncollinear phase NC90 with perpendicular sublattice
polarizations and only then to the collinear ferroelectric phase FE.

From the low-temperature $\varphi_{E}$–$E$ phase diagram constructed in the
present paper where $\varphi_{E}$ is the angle between the field and the $a$
axis, it follows that the polarization rotation is the same two-step process
for all $\varphi_{E}$ except for a few particular directions with
$\varphi_{E}=\varphi_{n}$, where $\varphi_{n}=\varphi_{0}+\piup n/4$,
$n=0,...7$. For these special directions, the field is either collinear to the
axes of the sublattice polarization in the AFE* phase (even $n$,
$\varphi_{E}=\varphi_{2k}$) or directed at $\piup/4$ to these axes, (odd $n$,
$\varphi_{E}=\varphi_{2k+1}$). For $\varphi_{E}=\varphi_{2k}$, the crystal
behaves like a uniaxial antiferroelectric, undergoing a single-step
polarization switching to the FE phase without the intermediate noncollinear
phase. On the other hand, for $\varphi_{E}=\varphi_{2k+1}$, the transition
field from the NC90 to the FE phase tends to infinity, meaning that the
transition never occurs. The dependences of the first-order transition fields
$E^{II}$ and $E^{III}$ between the AFE* and NC90 and between the NC90 and FE
phases, respectively, on $\varphi_{E}$ are given by the piecewise functions
$E^{II}=E_{sw}/\cos(\Delta\varphi)$ and $E^{III}=E_{sw}/\sin(\Delta\varphi)$,
where $\Delta\varphi=\varphi_{E}-\varphi_{2k+1}$, and each piece is defined
between $\varphi_{2k}$ and $\varphi_{2k+2}$.

Experimentally, the low-temperature transition to the ferroelectric phase with
increasing field has not been detected yet because of the dielectric breakdown
of the samples. As follows from the constructed $\varphi_{E}$–$E$ diagram, the
field of this transition is the lowest, when its direction is close to the
axis of the sublattice polarization, $\varphi_{E}=\varphi_{2k}$, so it is most
likely to be experimentally observed at this field orientation.

The $T$–$E$ phase diagrams are constructed for the electric fields, directed
along the diagonals of the $ac$ plane, for the fields collinear to the axes of
the sublattice polarization in the AFE* phase, $\varphi_{E}=\varphi_{2k}$, as
well for the fields directed at $\piup/4$ to these axes
$\varphi_{E}=\varphi_{2k+1}$.

## Appendix

$\varphi_{5}<\varphi_{E}<\varphi_{7}$

$\varphi_{7}<\varphi_{E}<\varphi_{1}$

$\varphi_{1}<\varphi_{E}<\varphi_{3}$

$\varphi_{3}<\varphi_{E}<\varphi_{5}$

Figure 10: (Colour online) The ground state proton configurations realized in
different phases at different orientations of the electric field (see figure
5). The polarization vectors $\mathbf{P}_{+}$ and $\mathbf{P}_{-}$ of the plus
and minus sublattices are shown by the green arrows. The numbers in
parentheses are the numbers of the respective lateral proton configurations
from table 1. The hydrogen bond indices $f=1,2,3,4$ are shown.

## Note added in proof

The recently published results [19] of of the revisited hysteresis loops
measurements in the squaric acid crystals with improved dielectric strength
appear to be much better described by the present theory than the earlier
experimental data of [4]. The difference between the theoretical and
experimental values of the switching field $E^{II}_{1}$ at room temperature
reduces twice. The agreement with the experiment for the polarization values
in the NC90 phase induced by the field $E_{1}$ is significantly improved as
well. It renders a simultaneous quantitative description of experimental data
for the permittivity, polarization, and switching fields by the present model
with the same set of the fitting parameters though not attained yet, but much
more feasible than it was previously thought.

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