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# Parallel Iterated Extended and Sigma-Point Kalman Smoothers

###### Abstract

The problem of Bayesian filtering and smoothing in nonlinear models with
additive noise is an active area of research. Classical Taylor series as well
as more recent sigma-point based methods are two well-known strategies to deal
with these problems. However, these methods are inherently sequential and do
not in their standard formulation allow for parallelization in the time
domain. In this paper, we present a set of parallel formulas that replace the
existing sequential ones in order to achieve lower time (span) complexity. Our
experimental results done with a graphics processing unit (GPU) illustrate the
efficiency of the proposed methods over their sequential counterparts.

Index Terms—  parallel computing, nonlinear estimation, iterated extended
Kalman smoother, sigma-point smoother

##  1 Introduction

In recent years, the rapid advancements in hardware technologies such as
graphics processing units (GPUs) and tensor processing units (TPUs) allow
compute-intensive workloads to be offloaded from the central processing units
(CPUs) by introducing parallelism [1, 2, 3]. There is a wide variety of areas
that can benefit from parallelization [4], one of which is state estimation.

State estimation is a common task that arises in various areas of science and
engineering [5, 6, 7]. It aims at combining the noisy measurements and the
model to estimate the hard-to-measure states. A frequent and classical method
for solving this problem is based on Bayesian filtering and smoothing [5]
which inherently provides a sequential solution with linear complexity in the
number of time steps.

In order to tackle the computational burden of Kalman type of filters and
smoothers, [8, 9] provide sub-linear computational methods by taking advantage
of the sparse structures of the matrices appearing in the batch forms of the
problems. In other works, using an ensemble formulation of Kalman filter has
been used to speed up the matrix computations through parallelization [10,
11]. The primary focus of these works was the efficient computation of the
covariance matrices either by introducing sparse or sample covariance matrices
rather than considering the temporal state-space structure per se. While in
the aforementioned works, parallelization of the sub-problems in the area of
Bayesian filtering and smoothing were considered, [12] presented a general
parallelizable formulations specifically designed for parallelizing state-
estimation problems in the temporal direction. Moreover, for the special case
of linear Gaussian model, parallel equations for computing Kalman filter and
Raugh–Tung–Striebel smoother solutions were derived.

Overcoming the computational burden in the case of nonlinear dynamical systems
with additive Gaussian noise is also of paramount importance. In these types
of models, various linearization approaches can be used. Taylor series
expansion based iterated extended Kalman smoother (IEKS) methods [13, 14, 15]
and sigma-point based methods [5] are well-established techniques in
literature. Iterated sigma-point methods have been proposed, for example, in
[16, 17]. Despite the capabilities of the aforementioned methods in state
estimation in nonlinear Gaussian models, they lack a framework which enables
the computations in a more efficient way when using parallelization.

The contribution of this paper is to present a set of parallelizable formulas
for filtering and smoothing in nonlinear Gaussian systems, in particular, IEKS
and sigma-point based methods using a scan algorithm [18, 12]. The proposed
methods reduce the linear span complexity of the state estimation methods to
logarithmic with respect to the number of measurements.

This paper is organized as follows: Section 2 briefly reviews the generic
parallel framework for Bayesian filters and smoothers. Sections 3 and 4 are
concerned with presenting the formulation of the problem and proposing our
method. Section 5 analyzes the efficiency and the computational complexity of
the proposed method through one numerical example, and Section 6 concludes the
paper.

##  2 General Parallel Framework for Bayesian Filters and Smoothers

It is shown in [12] that the computation of sequential Bayesian filtering and
smoothing can be converted to general parallel formulas in terms of
associative operations. This allows for the use of the parallel scan method
[18] which is a common algorithm used to speed-up sequential computations, for
example, on GPU-based computing systems. In the rest of this section, we
review the general parallel algorithms provided in [12] which we then extend
to nonlinear Gaussian models.

Given a state space model of the following form:

$x_{k}\sim p(x_{k}\mid x_{k-1}),\quad y_{k}\sim p(y_{k}\mid x_{k}),$ (1)

the goal of the filtering problem is to find the posterior distributions
$p(x_{k}\mid y_{1:k})$ for $k=1,\ldots,n$. This distribution is a
probabilistic representation of the available statistical information on the
state $x_{k}\in\mathbb{R}^{n_{x}}$ given the measurements
$y_{1:k}=\\{y_{1},\ldots,y_{k}\\}$ with $y_{k}\in\mathbb{R}^{n_{y}}$. Having
acquired the filtering results for $k=1,\ldots,n$, and using all the $n$
measurements, the Bayesian smoother can be used to compute the posterior
distributions $p(x_{k}\mid y_{1:n})$. The following strategies are used in
[12] so as to particularize $a_{k}$ and the binary associative operator
$\otimes$ which provide a parallel framework for solving the aforementioned
sequential filtering and smoothing problem.

### Filtering.

Given two positive functions $g_{i}^{\prime}(y),g_{j}^{\prime}(y)$ and two
conditional densities $f_{i}^{\prime}(x\mid y),f_{j}^{\prime}(x\mid y)$, the
authors of [12] proved that the binary operation
$(f_{i}^{\prime},g_{i}^{\prime})\otimes(f_{j}^{\prime},g_{j}^{\prime})=(f_{ij}^{\prime},g_{ij}^{\prime})$
defined by

$\begin{split}f_{ij}^{\prime}(x|z)=\frac{\int{g_{i}^{\prime}(y)f_{j}^{\prime}(x\mid
y)}f_{i}^{\prime}(y\mid z)dy}{\int{g_{j}^{\prime}(y)}f_{i}^{\prime}(y\mid
z)dy},\\\
g_{ij}^{\prime}(z)=g_{i}^{\prime}(z)\int{g_{j}^{\prime}(y)f_{i}^{\prime}(y\mid
z)dy},\end{split}$ (2)

is associative and by selecting $a_{k}=(f_{k}^{\prime},g_{k}^{\prime})$ as
follows:

$\begin{split}f_{k}^{\prime}(x_{k}\mid x_{k-1})&=p(x_{k}\mid
y_{k},x_{k-1}),\\\ g_{k}^{\prime}(x_{k-1})&=p(y_{k}\mid x_{k-1}),\end{split}$
(3)

where $p(x_{1}\mid y_{1},x_{0})=p(x_{1}\mid y_{1})$ and $p(y_{1}\mid
x_{0})=p(y_{1})$, the Bayesian map $\binom{p(x_{k}|y_{1:k})}{p(y_{1:k})}$ can
be rewritten as the $k$-th prefix sum, $a_{1}\otimes\dots\otimes a_{k}$.

### Smoothing.

Similarly [12], for any conditional densities $f_{i}^{\prime}(x\mid y)$ and
$f_{j}^{\prime}(x\mid y)$ the binary operation $f_{i}^{\prime}\otimes
f_{j}^{\prime}\coloneqq\int{f_{i}^{\prime}(x|y)f_{j}^{\prime}(y|z)}dy$ is
associative and by selecting $a_{k}=p(x_{k}\mid y_{1:k},x_{k+1})$ with
$a_{n}=p(x_{n}\mid y_{1:n})$, the Bayesian smoothing solution can then be
calculated as $p(x_{k}\mid y_{1:n})=a_{k}\otimes a_{k+1}\otimes\dots\otimes
a_{n}$.

Having considered the aforementioned general formulations, in this paper, we
aim to extend the element $a_{k}$ and the binary associative operator
$\otimes$ to linear approximations of non-linear Gaussian systems,
specifically, to the extended Kalman filter and smoother, and sigma-points
methods.

##  3 Problem formulation

We consider the following model:

$\begin{split}x_{k}&=f_{k-1}(x_{k-1})+q_{k-1},\\\
y_{k}&=h_{k}(x_{k})+r_{k},\end{split}$ (4)

where $f_{k-1}(.)$ and $h_{k}(.)$ are nonlinear functions. The $q_{k}$ and
$r_{k}$ are the process and measurement noises, which are assumed to be zero-
mean, independent Gaussian noises with known covariance matrices, $Q_{k}$ and
$R_{k}$, respectively. Furthermore, the initial state is Gaussian $x_{0}\sim
N(m_{0},P_{0})$ with known mean $m_{0}$ and covariance $P_{0}$. This paper is
concerned with the computing approximate posterior distributions of the states
$x_{0:n}=\\{x_{0},x_{1},\ldots,x_{n}\\}$ given all the measurements
$y_{1:n}=\\{y_{1},x_{1},\ldots,y_{n}\\}$ in parallel form, or more precisely,
the corresponding filtering and smoothing distributions.

Since the filtering and smoothing problems are not solvable in closed-form in
the general non-linear case, one needs to resort to approximations. Here we
follow the Gaussian filtering and smoothing frameworks [5] and form linear
approximations of the system (4) in the following form:

$\begin{split}f_{k-1}(x_{k-1})&\approx F_{k-1}x_{k-1}+c_{k-1}+e_{k-1},\\\
h_{k}(x_{k})&\approx H_{k}x_{k}+d_{k}+v_{k},\end{split}$ (5)

where $F_{k}\in\mathbb{R}^{n_{x}\times n_{x}}$, $c_{k}\in\mathbb{R}^{n_{x}}$,
$H_{k}\in\mathbb{R}^{n_{y}\times n_{x}}$, $d_{k}\in\mathbb{R}^{n_{y}}$,
$e_{k}\in\mathbb{R}^{n_{x}}$ and $v_{k}\in\mathbb{R}^{n_{y}}$ are zero mean
Gaussian noises with covariance matrices $\Lambda_{k}$ and $\Omega_{k}$,
respectively.

There are different strategies to effectively select the parameters of (5). In
this paper, we will consider two such strategies widely-used in the Gaussian
filtering literature, namely iterated sigma-point and extended Kalman
smoothers [14, 15, 16]. In these approaches, the linearized-filter-smoother
method is repeated $M$ times, with the linearization parameters leveraging the
results of the previous smoothing pass instead of the previous step. We can
therefore see our successive linear approximations as being parametrized by
the following vectors and matrices:

$F_{0:n-1}^{(i)},c_{0:n-1}^{(i)},\Lambda_{0:n-1}^{(i)},H_{0:n-1}^{(i)},d_{1:n}^{(i)},\Omega_{1:n}^{(i)}.$
(6)

In the rest of this section, we will discuss how to acquire the linearized
parameters of (6) using these methods. Also, for the sake of notational
simplicity, we drop the index $i$ from these parameters.

### Iterated sigma-point method.

In this approach, we select the parameters $(F_{k-1},c_{k-1},\Lambda_{k-1})$
and $(H_{k},d_{k},\Omega_{k})$ using sigma-point-based statistical linear
regression (SLR) method [16] as follows. First, we select $m$ sigma points
$\mathcal{X}_{1,k}^{(i)},\ldots,\mathcal{X}_{m,k}^{(i)}$ and their associated
weights $w_{1,k}^{(i)},\ldots,w_{m,k}^{(i)}$ according to the posterior
moments $\bar{x}_{k}^{(i-1)}$ and $\bar{P}_{k}^{(i-1)}$ of the previous
iteration, which are the best available estimates for the means and
covariances of the smoothing distribution. Then, in order to find the
parameters $(F_{k-1},c_{k-1},\Lambda_{k-1})$, transformed sigma-points are
obtained as $\mathcal{Z}_{j}=f_{k-1}(\mathcal{X}_{j,k-1}^{(i)})$ for
$j=1,\ldots,m$, and the linearization parameters are then given by:

$\begin{split}F_{k-1}&=\Psi^{\top}{\bar{P}_{k-1}}^{-1},\\\
c_{k-1}&=\bar{z}-F_{k-1}\bar{x}_{k-1},\\\ \Lambda_{k-1}&=\Phi-
F_{k-1}\bar{P}_{k-1}F_{k-1}^{\top}.\end{split}$ (7)

If we now write $\bar{x}=\bar{x}_{k-1}$ and $w_{j}=w_{j,k-1}^{(i)}$, the
required moment approximations for Equation (7) are [19]:

$\begin{split}\bar{z}&\approx\sum_{j=1}^{m}w_{j}\mathcal{Z}_{j},\\\
\Psi&\approx\sum_{j=1}^{m}w_{j}(\mathcal{X}_{j}-\bar{x})(\mathcal{Z}_{j}-\bar{z})^{\top},\\\
\Phi&\approx\sum_{j=1}^{m}w_{j}(\mathcal{Z}_{j}-\bar{z})(\mathcal{Z}_{j}-\bar{z})^{\top}.\end{split}$
(8)

Similarly, reusing Equations (8) with $\bar{x}=\bar{x}_{k}$,
$w_{j}=w_{j,k}^{(i)}$, and $\mathcal{Z}_{j}=h_{k}(\mathcal{X}_{j,k}^{(i)})$
the parameters $(H_{k},d_{k},\Omega_{k})$ can be calculated as follows:

$\begin{split}H_{k}&=\Psi^{\top}{\bar{P}_{k}}^{-1},\\\
d_{k}&=\bar{z}-H_{k}\bar{x}_{k},\\\ \Omega_{k}&=\Phi-
H_{k}\bar{P}_{k}H_{k}^{\top}.\end{split}$ (9)

The iterated posterior linearization smoother (IPLS) [16] now consists in
iterating Equations (7) and (9) with updated approximate means and covariances
of the posterior distribution at each iteration.

### Iterated extended Kalman smoother.

In this case, $\Omega$ and $\Lambda$ are selected as zeros, and
$(F_{k-1},c_{k-1})$ and $(H_{k},d_{k})$ are obtained by analytical
linearization at the previous posterior (smoother) mean estimate of $x_{0:N}$.
This approach is recognized as Gauss–Newton method when computing the MAP
estimates [14] and it can also be extended to correspond to
Levenberg–Marquardt method [15]. Here, we aim to obtain the linearized
parameters according to this method which will be used in the next section to
get parallel formulas.

By expanding $f_{k-1}(x_{k-1})$ and $h_{k}(x_{k})$ in the first-order Taylor
series utilizing the previous posterior means $\bar{x}_{k}$, the parameters of
(6) are:

$\begin{split}F_{k-1}&=\nabla f(\bar{x}_{k-1}),\\\
c_{k}&=f(\bar{x}_{k-1})-F_{k-1}\bar{x}_{k-1},\\\ H_{k}&=\nabla
h({\bar{x}_{k}}),\\\ d_{k}&=h({\bar{x}_{k}})-H_{k}\bar{x}_{k},\end{split}$
(10)

where $\nabla f$ and $\nabla h$ are the Jacobians of $f$ and $h$,
respectively. Please note that in this paper computation of parameters in (7)
and (9), and (10) is performed offline, which means that we have all
measurements as well as the results of previous trajectory, that is,
$\bar{x}_{1:n}$ and $\bar{P}_{1:n}$ for all $n$ data points.

Having obtained the linearized parameters, the remaining task is to find the
parallel formulas which will be discussed in the next section.

##  4 The proposed method

Probability densities for the model of form (4) with linearization parameters
of form (6) can be formulated as follows:

$\begin{split}p(x_{k}\mid x_{k-1})&\approx
N(x_{k};F_{k-1}x_{k-1}+c_{k-1},Q^{\prime}_{k-1}),\\\ p(y_{k}\mid
x_{k})&\approx N(y_{k};H_{k}x_{k}+d_{k},R^{\prime}_{k}),\end{split}$ (11)

where $Q_{k-1}^{\prime}=Q_{k-1}+\Lambda_{k-1}$ and
$R_{k}^{\prime}=R_{k}+\Omega_{k}$. The goal here is to obtain the parallel
nonlinear Gaussian filter and smoother for the model (11). To meet this goal,
similar to the method used in [12], we define $a_{k}$ and binary operator
$\otimes$ for our new linearized model.

### Nonlinear Gaussian filtering.

Aiming to specify the element $a_{k}$ for obtaining parallel filtering
equations according to (3), we apply Kalman filter update step to the density
$p(x_{k}\mid x_{k-1})$ with measurement $y_{k}$. The results of the matching
terms are as follows:

$\begin{split}f_{k}^{\prime}(x_{k}\mid x_{k-1})&=p(x_{k}\mid y_{k},x_{k-1})\\\
&=N(x_{k};A_{k}x_{k-1}+b_{k},C_{k}),\end{split}$ (12)

where:

$\begin{split}A_{k}&=(I_{n_{x}}-K_{k}H_{k})F_{k-1},\\\
b_{k}&=c_{k-1}+K_{k}(y_{k}-H_{k}c_{k-1}-d_{k-1}),\\\
C_{k}&=(I_{n_{x}}-K_{k}H_{k})Q_{k-1}^{\prime},\\\
K_{k}&=Q_{k-1}^{\prime}{H_{k}}^{\top}S_{k}^{-1},\\\
S_{k}&=H_{k}Q_{k-1}^{\prime}H_{k}^{\top}+R_{k}^{\prime}.\end{split}$ (13)

It is worth noticing that in order to find parameters of (13) at $k=1$ and
given $m_{0}$ and $P_{0}$, conventional formulations of the Kalman filter
method with the linearized parameters are applied directly for prediction and
update steps.

Also, using the information form of Kalman filter [20], the distribution
$g_{k}^{\prime}(x_{k-1})=p(y_{k}\mid x_{k-1})\propto
N_{I}(x_{k-1};\eta_{k},J_{k})$ can be obtained as follows:

$\begin{split}J_{k}&=(H_{k}F_{k-1})^{\top}S_{k}^{-1}H_{k}F_{k-1}\\\
\eta_{k}&=(H_{k}F_{k-1})^{\top}S_{k}^{-1}H_{k}(y_{k}-H_{k}c_{k-1}-d_{k}).\end{split}$
(14)

Equations (13) and (14) provide the parameters of element
$a_{k}=(A_{k},b_{k},C_{k},\eta_{k},J_{k})$ in the filtering step, and they can
be computed in parallel. Also, given $a_{i}$ and $a_{j}$ with the mentioned
parameters, the binary associative operator $a_{i}\otimes a_{j}=a_{ij}$ can
then be calculated with the following parameterization [12, lemma 8]:

$\begin{split}A_{ij}&=A_{j}(I_{n_{x}}+C_{i}J_{j})^{-1}A_{i},\\\
b_{ij}&=A_{j}(I_{n_{x}}+C_{i}J_{j})^{-1}(b_{i}+C_{i}\eta_{j})+b_{j},\\\
C_{ij}&=A_{j}(I_{n_{x}}+C_{i}J_{j})^{-1}C_{i}A_{j}^{\top}+C_{j},\\\
\eta_{ij}&=A_{i}^{\top}(I_{n_{x}}+J_{j}C_{i})^{-1}(\eta_{j}-J_{j}b_{i})+\eta_{i},\\\
J_{ij}&=A_{i}^{\top}(I_{n_{x}}+J_{j}C_{i})^{-1}J_{j}A_{i}+J_{i}.\end{split}$
(15)

The proof for Equations (15) can be found in [12].

### Nonlinear Gaussian smoothing.

Assume that the filtering means $x^{*}_{k}$ and covariance matrices
$P^{*}_{k}$ for the model (11) have been acquired as described above. We now
get the following parameters for the smoothing step:

$\begin{split}p(x_{k}\mid
y_{1:k},x_{k+1})&=N(x_{k};E_{k}x_{k+1}+g_{k},L_{k})\\\ \end{split}$ (16)

for $k<n$:

$\begin{split}E_{k}&=P_{k}F_{k}^{\top}(F_{k}P^{*}_{k}F_{k}^{\top}+Q_{k-1}^{\prime})^{-1},\\\
g_{k}&=x^{*}_{k}-E_{k}(F_{k}x^{*}_{k}+c_{k}),\\\
L_{k}&=P^{*}_{k}-E_{k}F_{k}P^{*}_{k},\end{split}$ (17)

and for $k=n$:

$\begin{split}E_{n}&=0,\\\ g_{n}&=x^{*}_{n},\\\ L_{n}&=P^{*}_{n}.\end{split}$
(18)

In the smoothing step, the parameters $a_{k}=(E_{k},g_{k},L_{k})$ can be
calculated in parallel. Now, given two elements $a_{i}$ and $a_{j}$, the
binary associative operator defined by $a_{i}\otimes a_{j}=a_{ij}$ can be
parametrized as follows [12, lemma 10]:

$\begin{split}E_{ij}&=E_{i}E_{j},\\\ g_{ij}&=E_{i}g_{j}+g_{i},\\\
L_{ij}&=E_{i}L_{j}E_{i}^{\top}+L_{i}.\end{split}$ (19)

##  5 Experimental results

In this section, we evaluate the performance of the proposed methods on a
simulated coordinated turn model with a bearings only measurement model [21]
which was also used in [15]. To this end, we compare the effective average run
time of the parallel versions of the extended (IEKS) and cubature integration
[5] based sigma-point iterated smoothers (IPLS) with $M=10$ iterations, as
described in Section 4, with their sequential counterparts both on a CPU
(Intel® Xeon® running at 2.30GHz) and on a GPU (Nvidia® Tesla® P100 PCIe 16 GB
with 3584 cores). For our experiments we leverage the JAX framework [22] which
implements the Blelloch parallel-scan algorithm [18] natively111The code to
reproduce the experiments can be found at the following address..

In Figures 1(a) and 1(b) we observe that while the total computational cost of
the parallel implementation of the iterated smoothers is higher than that of
their sequential counterparts (Figure 1(a)), the parallelization properties of
our proposed algorithms prove beneficial on a distributed environment such as
a GPU (Figure 1(b)). Moreover, as outlined by the medallion in Figure 1(b),
our experiments indeed exhibit the theoretical logarithmic span complexity -
derived in [12] for a linear Gaussian state space model - up to the
parallelization capabilities of our GPU (3584 cores).

$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$n$
number of time steps run time (in seconds) parallel IEKSsequential
IEKSparallel IPLSsequential IPLS

(a) CPU run time

$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$n
cores$n$ number of time steps run time (in seconds)

(b) GPU run time

Fig. 1: Run time comparison of the parallel and sequential versions of the
IEKS and IPLS on CPU (a) and GPU (b)

##  6 conclusion

In this paper, parallel formulations for two kinds of nonlinear smoothers,
namely, iterated sigma-point-based smoothers and iterated extended Kalman
smoothers, have been presented. The proposed algorithms have the capability of
diminishing the span-complexity from linear to logarithmic. Furthermore, the
experimental results, which were conducted on a GPU, showed the benefits of
the proposed methods over classical sequential methods.

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# Anomalous structural behavior and antiferroelectricity in BiGdO3: Detailed
temperature and high pressure study

Rajesh Jana<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur
741246, Nadia, West Bengal, India.  Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Apurba Dutta  CMP Division, Saha Institute of Nuclear Physics,
HBNI, 1/AF, Bidhannagar, Kolkata-700064, India  Pinku Saha  Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur Campus, Mohanpur 741246, Nadia, West Bengal, India.  Kapil Mandal
Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur Campus, Mohanpur 741246, Nadia, West Bengal, India.
Bishnupada Ghosh  Department of Physical Sciences, Indian Institute of Science
Education and Research Kolkata, Mohanpur Campus, Mohanpur 741246, Nadia, West
Bengal, India.  Amreesh Chandra  Department of Physics, Indian Institute of
Technology Kharagpur, Kharagpur-721302, India  I. Das  CMP Division, Saha
Institute of Nuclear Physics, HBNI, 1/AF, Bidhannagar, Kolkata-700064, India
Goutam Dev Mukherjee<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur
Campus, Mohanpur 741246, Nadia, West Bengal, India.

###### Abstract

A detailed temperature and pressure investigation on BiGdO3 is carried out by
means of dielectric constant, piezoelectric current, polarization-electric
field loop, Raman scattering and x-ray diffraction measurements. Temperature
dependent dielectric constant and dielectric loss show two anomalies at about
290 K (Tr) and 720 K (TC). The later anomaly is most likely due to
antiferroelectric to paraelectric transition as hinted by piezoelectric
current and polarization-electric field loop measurements at room temperature,
while the former anomaly suggests reorientation of polarization. Cubic to
orthorhombic structural transition is observed at about 10 GPa in high
pressure x-ray diffraction studies accompanied by anisotropic lattice
parameter changes. An expansion about 30 % along $a$-axis and 15 % contraction
along $b$-axis during the structural transition result in 9.5 % expansion in
unit cell volume. This structural transition is corroborated by anomalous
softening and large increase in full width half maximum (FWHM) of 640 cm-1
Raman mode above 10 GPa. Enhancement of large structural distortion and
significant volume expansion during the structural transition indicate towards
an antiferroelectric to ferroelectric transition in the system.

###### pacs:

77.80.-e, 62.50.-p, 64.70.Nd

##  I Introduction

Ferroelectricity is the property of a certain class of materials which exhibit
reversible spontaneous polarization in the presence of an electric field.
Since its discovery ferroelectric (FE) materials have been extensively studied
for their novel properties viz. spontaneous polarization, piezoelectricity,
pyroelectricity, coupling with magnetic order (multiferroic), etc. On the
other hand, antiferroelectric materials are less studied compared to FE
materials. Using a phenomenological approach, C. Kittel proposed
antiferroelectric (AFE) behaviour in 1951 Kittel . An AFE state can be defined
as the lines of ions in which two neighbouring lines are oppositely polarized.
Therefore, there will be no net polarization even with the application of the
electric field and they will not be piezoelectric in contrast to the
ferroelectric materials. However, a certain high electric field (EF) will
force the antiparallel dipoles to reorient in parallel fashion and an electric
field induced AFE to FE transition can be achieved since both the FE and AFE
phases have similar free energy. Shirane et al. for the first time reported
the AFE behavior in PbZrO3 below 230 K Shirane . Thereafter,
antiferroelectricity was observed in PbHfO3 and in other materials - NaNbO3 ,
AgNbO3, Pb(Mg0.5W0.5)O3, Pb(Yb0.5Nb0.5)O3 Shirane2 ; Shirane3 ; Francobe ;
Smolenskii ; Yamamoto . In recent times antiferroelectric materials have
generated considerable renewed research interest owing to their promising
ultrahigh energy density, large strain, giant electrocaloric effect Zhang ;
Chauhan ; Bao ; Zhang2 ; Zhao ; Tian ; Liu ; Pirc  etc. AFE materials also
possess some indispensable properties for industrial applications such as -
low dielectric loss, low coercive field, low remanant polarization, and fast
discharge rates.

Bi-based perovskite oxides BiMO3 where M is a transition metal ion have drawn
special research interest in designing new multiferroics. In these materials
ferroelectricity arises due to 6s lone pair electrons of Bi3+ ion which shift
towards oxygen octahedra to form local dipole structure. On the other hand,
presence of transition metal ion with partially filled d shell at the B site
makes long range magnetic ordering. Multiferroic properties have been observed
several FE materials such as - BiFeO3, BiCrO3, BiNiO3, BiCoO3 and BiMnO3
Kiselev ; Niitaka ; Cai ; Belik ; Kimura  etc. Besides ferroelectricity, AFE
behavior has also been evidenced in some Bi based oxides. For example, Kim et
al.Kim  and Kan et al.Kan  reported AFE signature in BiCrO3 thin films and
rare‐earth doped BiFeO3 respectively. Baettig et al. theoretically predicted
antiferroelectric arrangement in centrosymmetric BiMnO3 originated from
stereo-chemical activity of Bi3+ 6s2 lone pair electrons Baettig . In BiFeO3
and BiMnO3 two outer 6s electrons of Bi ion, instead of making chemical bond,
shift away from Bi ion towards oxygen octahedra and produce local dipoles
which may align in ferroelectric or antiferroelectric arrangement Khomskii ;
Seshadri ; Ravindran . In a recent study, Saha et al. suggested dielectric
relaxation and antiferromagnetic interaction in BiGdO3 (BG) Saha . However, as
per authors knowledge no detailed study on structural and electronic
properties of this material is available in literature. In this context we
have synthesized and carried out a detailed investigation on BiGdO3 by means
of dielectric constant, piezoelectric current, polarization-electric field
loop, Raman scattering and x-ray diffraction measurements at ambient as well
as high pressures. Two anomalies have been observed in the temperature
dependent dielectric constant profile of BG at about 290 K and 720 K.
Piezoelectric current and polarization-electric field loop measurements at
room temperature indicate antiferroelectric behaviour in BG below 720 K. High
pressure x-ray diffraction (XRD) and Raman studies show cubic to orthorhombic
structural transition at 10 GPa accompanied by a large volume expansion. We
speculate that pressure induced large increase in structural distortion and
significant volume expansion may lead to AFE to FE transition.

##  II experiment

Polycrystalline BG sample is synthesized using conventional solid-state
reaction method. Stoichiometric amount of 99.99 % pure (Sigma Aldrich) Bi2O3
and Gd2O3 are thoroughly mixed using agate mortar and pestle for several hours
in 2-propanol medium. Pellets of well mixed powder are then calcined at 9300C
for about 9h. This calcination process is repeated to improve crystalline
quality of the sample. Formation of pure single phase of the sample is
confirmed by the collecting x-ray diffraction pattern using Rigaku SmartLab
x-ray diffractometer with 9 KW rotating anode and Cu Kα x-ray source.
Temperature dependent dielectric constant measurements are carried out on
sintered BG pellets of 8 mm dia and 2 mm thickness after polishing both the
faces of the pellets with 0.25 $\mu$m silver paste. It is first washed with
acetone to clean its surfaces and then dried at 1200 for 20 min to remove
moisture. A N4L PSM 1735 impedance analyzer is used to explore temperature
dependence dielectric constant in the frequency range 100 Hz to 1 MHz.
Temperature is calibrated using a K type thermocouple and a digital voltmeter.
Piezoelectric current measurements are carried out using toroid anvil cell and
300 ton hydraulic press. Detailed experimental procedure for this measurement
can be found elsewhere Jana ; Jana2 . Room temperature polarization-electric
field hysteresis loop measurements are carried out using Radiant Technology’s
Precision 10 KV HVI-SC loop measurement system. High pressure x-ray
diffraction and Raman spectroscopic measurements are carried out using a
piston type diamond anvil cell (easyLab Co., UK) with a culet size of 300
$\mu$m . The sample along with pressure marker are loaded in the 100 $\mu$m
central hole of a pre-indented steel gasket. A mixture of methanol and ethanol
is also added in the 4:1 ratio in order to maintain hydrostatic condition
inside the sample chamber. High pressure Raman spectra are taken using Horiba
Jobin-Yvon Lab RAM HR-800 spectrometer with 1800 g/mm grating. The samples are
excited with 488 nm line of Ar-ion laser and pressure inside the sample
chamber are determined using Ruby fluorescence technique Mao . XRD
measurements at high pressures are carried out in EXPRESS beamline in the
Elettra synchrotron light source, Italy, using a monochromatic wavelength 0.5
Å. In XRD studies, pressure is determined from isothermal P-V equation of sate
of silver Dewaele , small quantities of which was mixed with sample before
loading in diamond anvil cell. X-ray beam is collimated to about 30 $\mu$m and
diffracted x-rays are collected using a MAR 3450 image plate detector aligned
normal to the incident beam. Collected two dimensional patterns are converted
to intensity versus 2$\theta$ profile using FIT2D software Hammersley . Full
profile fitting of the XRD patterns are carried out using GSAS Toby  after
indexing using the software CRYSFIRE Shirley .

##  III results and discussions

Temperature dependence of dielectric constant and dielectric loss of BG in the
range 77 K to 755 K at a few selected frequencies are shown in Fig. 1.
Dielectric constant and dielectric loss show two anomalies in their
temperature profiles. On increasing temperature from 77 K, dielectric constant
slowly increases and exhibit a broad peak around 290 K (Tr) for all
frequencies in the range 10 KHz to 1 MHz. At this temperature a peak in the
dielectric loss has also been observed at all frequencies except for 1 MHZ
frequency. Upon further increasing temperature, dielectric constant start to
increase rapidly above 550 K with a clear peak at about 720 K (TC). A strong
frequency dispersion in the dielectric constant is observed. The absence of
peak position dependence on frequency indicates non-relaxor type behaviour
which is in contrast to the previous literature report describing BG as a
relaxor type ferroelectric Saha . Temperature dependent dielectric loss also
increases sharply around TC. Above anomalous behavior in dielectric constant
and dielectric loss around TC are the signatures of ferro/antiferroelectric to
paraelectric phase transition while the anomaly around Tr could be due to
reorientation of polarization. However, x-ray diffraction analysis establish
that BG crystallizes in Fm-3m centrosymmetric cubic structure. The structural
symmetry does not allow it to be a conventional ferroelectric material. In
order to check further about the presence of ferroelectric order we have
carried out piezoelectric current measurements at room temperature (298 K)
using TA apparatus Jana ; Jana2 . The obtained data are plotted in Fig. 2(a),
which shows a small fluctuating current even with reversing the direction of
electric field. A ferroelectric material should show piezoelectric effect and
the polarity of piezoelectrically induced voltage must be reversed with
reversal of applied electric field. In this case we do not observe any
polarity change with change in applied poling field direction indicating the
absence of ferroelectric order in the sample. Similar sudden increase in
dielectric constant near the transition temperature and non-relaxor type of
behavior have been observed in several antiferroelectric materials like –
PbZrO3, NaNbO3, PbHfO3, Cs2Nb4O11 Shirane ; Shirane3 ; Samara ; Smith ; Liu2 .
Therefore , one can speculate that BG may be antiferroelectric below 720 K.
Electric field (E) dependent polarization (P) measurement is the best
experimental way to confirm ferro/antiferroelectric behaviour in a material.
We have carried out room temperature polarization-electric field loop
measurements and the corresponding data are shown in Fig. 2(b). A
ferroelectric material should exhibit saturation in polarization at high
electric field. However, in our study we have observed unsaturated elliptical
loop with very small area up to the highest applied field of 50 KV/cm which
exhibits lossy capacitor behaviour Evans . We could not apply higher field
above 50 KV/cm due to dielectric breakdown. The lossy dielectric response can
be explained by considering the phase difference between polarization and
applied electric field. For an ideal linear capacitor, polarization and
electric field are in phase and one would get a linear behaviour in
polarization-electric field loop. On the other hand an ideal resistor produces
circular polarization-electric field loop as current and voltage are in phase.
Therefore, when linear capacitor and ideal resistor are connected in parallel
combination, it gives lossy dielectric behaviour with elliptical loop. In
several materials antiferroelectric behaviour is confirmed by the field
induced double hysteresis loop under high electric field. However, in our
study, we do not observe this type of behaviour. Probably the critical value
for electric field (EC) to induce AFE to FE transition is greater than the
breakdown fields at room temperature. Therefore, our experimental
observations, such as, peak in the temperature dependent dielectric constant,
non-appearance of piezoelectricity and absence of hysteresis nature in
polarization-electric field loop suggest that the sample is most likely
antiferroelectric below 720 K.

In order to investigate structural behavior of this sample at high pressures,
we have conducted anle dispersive x-ray diffraction measurements under
pressure. Ambient and a few selected XRD patterns with increasing pressure up
to 22.5 GPa are shown in Fig. 3. Ambient XRD analysis of BiGdO3 shows a cubic
Fm-3m phase having lattice parameters of 5.47551(4) Å. We have not observed
any remarkable changes in the diffraction patterns except peak broadening up
to 6.4 GPa. As the pressure increases Bragg peaks shift to higher 2$\theta$
values due to lattice contraction. At about 7.2 GPa onset of a new Bragg peak
around 2$\theta$ = 10º in between first two reflection lines is observed and
it becomes prominent at about 10 GPa as shown by the vertical arrows in Fig.
3. On further increase of pressure the intensity of the new peak gradually
increases while the intensity of its two neighbouring lines decreases rapidly
and these three peaks finally merge at about 18 GPa. We do not observe any
further change other than broadening of all the peaks up to 22.5 GPa. The XRD
pattern above 9 GPa could not be fitted with ambient Fm-3m cubic structure.
Therefore, the appearance of the new reflection line at 10 GPa definitely
indicate towards a structural transition in the sample. On indexing, the high
pressure phase is found to be orthorhombic with Pbca symmetry having lattice
parameters a = 6.87935(2) Å, b = 5.30753(4) Åand c= 4.48219(3) Å. We have
carried out full structural refinement using Rietveld refinement process
taking starting atomic coordinates from similar structure material Redhammer .
Rietveld refined XRD pattern of BG with Pbca structure at 12.1 GPa and 16.5
GPa are shown in Fig. 4 (b) and (c). The obtained structural parameters for
12.1 GPa XRD data are presented in Table I. Fig. 5 shows the pressure
evolution of unit cell lattice parameters for both cubic and orthorhombic
structures. In the Fig. 5(a) one can see that structural transition is
accompanied by large expansion (about 30 %) along $a$-axis, 15 % contraction
along $c$-axis, while no significant change is observed along $b$-axis. As a
result of this, unit cell volume is expanded by about 9.5 % (Fig. 5(b)). In
the inset of Fig. 5(b) a gradual decrease in the c/a ratio in orthorhombic
phase is exhibited. P-V data in both the phases could be fitted with 3rd order
Birch-Murnaghan equation of state Birch ; Murnaghan

$P={3/2}B_{0}[(V_{0}/V)^{7/3}-(V_{0}/V)^{5/3}]\times[1-3/4(4-B^{\prime})\\{(V_{0}/V)^{2/3}-1\\}]$
(1)

Where, B0 and V0 are the bulk modulus and volume at ambient pressure
respectively. B′ is the first order pressure derivative of bulk modulus. For
the first region up-to 9 GPa our best fit gives V0 = 164.29(2) Å3, B0 =
67.42(3) GPa, and B′ = 4.99(6). In high pressure region these values become V0
= 184.61(7) Å3, B0 = 66.14(5) GPa, and B′ = 4.69(6). From the calculated bulk
modulus it is established that despite large volume expansion the
compressibility of the sample remains almost same after cubic to orthorhombic
transition.

As a complementary study, we have carried out high pressure Raman studies on
BG to probe structural behavior more deeply. Factor group analysis for BiGdO3
with Fm-3m structure predicts five (A1g \+ Eg \+ 3T2g) Raman active modes
associated with oxygen vibration Rousseau ; Saha  . High pressure Raman
spectra at selective pressure values are shown in Fig. 6. At ambient pressure
we have observed six Raman modes at 104, 140, 241, 368, 586 and 640 cm-1. The
observed Raman modes are found to be at higher values than the previously
reported values Saha . One reason could be that high quality sample is used in
the present study, whereas a trace of secondary phase was present in the
earlier study Saha . On increasing pressure two prominent Raman modes 104 cm-1
(M1) and 640 cm-1 (M2) harden up to 10 GPa. Then the high frequency M2 mode
starts softening above 10 GPa where the systems transform to orthorhombic
structure as observed in high pressure x-ray diffraction studies. Upon further
increase in pressure the intensity of M1 mode rapidly decreases and broadening
in the M2 mode significantly increases. In Fig. 7(a), we have shown high
pressure behavior of M2 Raman mode. Upon increasing pressure it linearly
increases with a slope of 3.58(0.06) cm-1/GPa up to 10 GPa followed by a
drastic fall which is corroborated by the sudden volume expansion while system
transform to orthorhombic phase around this pressure. Above 15 GPa this mode
frequency again increases linearly up to 18.3 GPa with a slope 2.94(0.18)
cm-1/GPa. At higher pressures error in mode fitting parameters increases
drastically due to peak broadening. Emergence of large octahedral distortion
is also reflected with rapid increase in the full width at half-maximum (FWHM)
of the M2 mode above 10 GPa (Fig. 7(b)). At about 25 GPa M1 mode is found to
have disappeared completely.

Shirane et al. and Swaguchi et al. did not observe any electric field induced
AFE to FE transition in the first reported antiferroelectric PbZrO3 in its
polycrystalline form at room temperature Shirane ; Sawaguchi . The double
hysteresis behavior was only seen near the Curie temperature (533 K) and using
a linear extrapolation of threshold electric field (EC), it was calculated
that 360 KV/cm field is needed to trigger the force field transition at 298 K
in PbZrO3 polycrystal Shirane ; Sawaguchi ; Tan . Since the value of EC is
larger than the dielectric breakdown strength, field induced AFE to FE
transition can not be observed at room temperature in ceramic PbZrO3 sample.
Later, Fesenko et al. was able to detect the AFE to FE transition in the broad
range of temperature (103 K to 433 K) in PbZrO3 single crystal Fesenko . In
NaNBO3, double hysteresis feature was also observed in only single crystal
sample and EC is found to increase with decreasing temperature Tan ; Cross .
Thus, polycrystalline antiferroelectric materials may or may not exhibit field
induced transition at room temperature. Most probably this is the reason
behind non observation of double hysteresis loop in our polycrystalline
sample. With appropriate doping or making a thin film may reduce the EC Cross2
. To explain the presence of antiferroelectricity and transition to
paraelectric state in cubic BiGdO3, we need to discuss about the lone pair
mechanism. Analyzing crystal chemistry, Volkova et al. showed that presence of
lone pair in Bi3+ compound only confirm the local distortion in cation
polyhedra and emergence ferroelectricity requires high degree of stereo-
chemical activity of the one pair which increases with decreasing temperatures
Volkova . They have also suggested that external perturbation of temperature
and pressure will strongly affect the lone pair activity by changing its shape
and position. Hence, one can anticipate that an
antiferroelectric/ferroelectric to paraelectric transition in Bi3+ compound at
high temperatures. This same mechanism is applicable for reorientation of
polarization at Tr. Ram Seshadri and Nicola A. Hill reported that
ferroelectricity and highly distorted monoclinic structure in BiMnO3 arises
due to lone pair activity Seshadri . Lone pair driven
ferroelectricity/antiferroelectricity were also observed in other materials -
AgSbSe2, BiAlO3, Pb2MgWO6 Aggarwal ; Kaczkowski ; Seshadri2 . In a recent
study, pair distribution function (PDF) analysis revealed that stereo-chemical
activity of the 6s2 lone pair produces signiﬁcant lattice anharmonicity and
local distortion in AgBiS2 while on average the system showed cubic Fm-3m
structure Rathore . This local distortion can be detected by local probe like
Raman spectroscopy. Appearance of significant asymmetric Raman modes suggest
that the local distortions are indeed present in cubic BiGdO3. Bao et. al.
have shown that La doping in Pb(Yb0.5Nb0.5)0.92Ti0.8O3 compound increased c/a
ratio and reduced [BO6] octahedral distortion which in turn stabilize the AFE
phase Bao . Conversely it might happen that decrease in $c/a$ ratio will
stabilize FE phase increasing [BO6] octahedral distortion. In our HP XRD study
c/a ratio has been observed much less than 1 in the orthorhombic phase, which
further decreases with pressure. Raman measurements also revealed large
enhancement of structural distortion as evidenced from large increase in FWHM
of M2 Raman mode above 10 GPa. Field induced AFE-FE transition generally
accompanied by large increase in polarization and unit cell volume Chauhan ;
Liu ; Sawaguchi ; Pan . In our study 9.5 % volume expansion might be related
to antiferroelectric to ferroelectric transition due to pressure induced large
structural distortion.

##  IV Conclusion

We have synthesized and studied detailed structural and electronic behaviour
of BiGdO3. Temperature dependent dielectric constant and dielectric loss show
two anomalies at about 290 K (Tr) and 720 K (Tm). The later anomaly most
likely due to antiferroelectric to paraelectric transition as hinted by
piezoelectric current and room temperature polarization-electric field loop
measurements, while the former anomaly suggest reorientation of polarization.
We attributed both anomalies to the change in degree of stereochemical
activity of the Bi3+ lone pair with increasing temperatures. Cubic to
orthorhombic structural transition has been observed at about 10 GPa in high
pressure x-ray diffraction studies accompanied by anisotropic lattice
parameters changes. An expansion about 30 % along a axis and 15 % contraction
along b axis during the structural transition result 9.5 % expansion in the
unit cell volume. This structural transition is corroborated by anomalous
softening of the high wave number Raman mode frequency. Enhancement of
structural distortion and large volume expansion indicate towards an
antiferroelectric to ferroelectric transition above 10 GPa. Further studies
are needed to understand ferroelectric behaviour if any, at high pressures.

##  V Acknowledgments

The authors gratefully acknowledge the financial support through project grant
MoES/16/25/10-RDEAS from Ministry of Earth Sciences, Government of India. The
authors also acknowledge Department of Science and Technology (DST), Indo-
Italian Program of Cooperation and Elettra Synchrotron Light Source for
financial and laboratory support for synchrotron based x–ray diffraction
measurements.

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Table 1:  Rietveld refined crystal structural parameters of BiGdO3 obtained from x-ray diffraction data at 12.1 GPa in space group Pbca (No. 61) with lattice parameters a = 6.8642(7) Å, b = 5.3113(6) Å, c = 4.4578(6) Å, and V = 162.522(34) Å3; Goodness of ﬁt: $\chi^{2}$ = 1.49, wRp = 3.01%, Rp 2.9% atom | site | x | y | z | Uiso
---|---|---|---|---|---
Bi1 | 8c | 0.56991(2) | 0.94666(2) | 0.92461(4) | 0.04696(3)
Bi2 | 8c | 0.37076(7) | 0.41722(9) | 0.38707(1) | 0.05351(4)
Gd1 | 8c | 0.31035(7) | 0.32019(8) | 0.17058(3) | 0.05075(4)
Gd2 | 8c | 0.32073(5) | 0.45330(3) | 0.76125(2) | 0.04806(1)
O1 | 8c | 0.18480(12) | 0.33960(13) | 0.03870(13) | 0.02543(2)
O2 | 8c | 0.31180(17) | 0.49640(18) | 0.05820(14) | 0.02440(4)
O3 | 8c | 0.30250(14) | 0.23630(16) | 0.81630(19) | 0.02648(6)
O4 | 8c | 0.56110(21) | 0.33650(24) | 0.78680(12) | 0.02552(6)
O5 | 8c | 0.43320(16) | 0.48060(17) | 0.69320(18) | 0.02495(4)
O6 | 8c | 0.44760(26) | 0.20280(29) | 0.58680(21) | 0.02582(2)
Figure 1: (Colour online) Temperature dependent dielectric constant and
dielectric loss at a few selected frequencies in the temperature range 77K -
450K shown in (a) and (c); range 450K - 755K in (b) and (d) Figure 2: (Colour
online) (a) Piezoelectric current measured with reducing pressure after poling
the sample under positive and negative electric field of 20KV/cm at 5.5 GPa.
(b) Polarization-electric field measurements up to 50KV/cm at room temperature
and ambient pressure. Figure 3: (Colour online) High pressure XRD spectra
(x-ray wavelength = 0.5 Å) at selected pressures along with pressure calibrate
silver peaks which are marked by Ag. Appearance of a new Bragg reflection line
is observed at 10 GPA (Shown by arrow). Figure 4: (Colour online) (a) Rietveld
refined XRD pattern of BiGdO3 at ambient pressure, 12.1 GPa and 16.5 GPa. The
red stars are the observed data points, the black line shows the fit to the
data, Bragg positions of the sample are marked with vertical ticks and the
Bragg peaks of Ag marked by the ticks at a slightly lower level, and the
difference is marked by the blue line. Figure 5: (Colour online) (a) Pressure
evolution of unit cell lattice parameters before and after transition in cubic
and orthorhombic structure respectively. (b) Pressure-volume data fitted to
two different 3rd order Birch-Murnaghan equation of state for two different
regions of pressure. Inset figure shows c/a ratio with pressure in the
orthorhombic phase. Figure 6: (Colour online) High pressure Raman spectra at
selected pressure values up to 50 GPa. Deconvolution of six Raman modes shown
for ambient pressure data Figure 7: (Colour online) Pressure evolution of 640
cm-1 (M2) mode (a) frequency; and (b) FWHM with error bar up to 18.3 GPa.
Error bar is less than data point symbol for the mode frequency. Red line in
the figure (a) is the linear fitting with slope m = 3.58(0.06) cm-1/GPa below
10 GPa and m = 2.94 (0.18) cm-1/GPa above 15 GPa