modeling the polarization in ferroelectric materials: a novel analytical approach
TRANSCRIPT
Solid-State Electronics 47 (2003) 1479–1486
www.elsevier.com/locate/sse
Modeling the polarization in ferroelectric materials:a novel analytical approach
Vincent Meyer a,*, Jean-Michel Sallese a, Pierre Fazan a, Delphine Bard b,Franc�ois Pecheux c
a LEG, Swiss Federal Institute of Technology, Lausanne, Switzerlandb LEMA, Swiss Federal Institute of Technology, Lausanne, Switzerland
c ERM-PhASe, CNRS, Illkirch, France
Received 16 September 2002; accepted 12 March 2003
Abstract
We propose for the first time a fully analytical formulation of the polarization in ferroelectric materials that takes
into account history effects. Our approach is based on the Preisach theory of the hysteresis loops. A symmetric ex-
ponential decay distribution of the dipoles thresholds has been introduced. Consequently, an exact analytical and
continuous solution of the polarization could be derived, taking the electric field history into consideration. Experi-
mental data on PZT capacitors show relevant agreement with the model for both saturated and minor loops. This
confirms the validity of this approach, which consequently represents a very interesting candidate for efficient ferro-
electric compact models to be used either in memory or in analog design applications.
� 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Hysteresis; Ferroelectric; Compact model; Preisach; Analog; Memory
1. Introduction
Recently, there has been a growing interest in using
ferroelectric materials for both memory [1,2] and analog
applications [3,4] by taking advantage of the hysteretic
behavior of the polarization versus electric field char-
acteristic. Many attempts have been made to model the
behavior of ferroelectric materials while taking the his-
tory effect into account [5,6]. In particular, efforts have
been made to reproduce the saturated hysteresis loops
[5]. However, such models fail to correctly predict non-
saturated minor loops, which are essential for analog
applications. Yet, building analog circuit architectures
based on the ferroelectric memory effect, either in ca-
pacitor (FeCaps) [1] or in field effect transistors (Fe-
FETs) [2–4] leads to unexplored design opportunities.
* Corresponding author. Fax: +41-21-693-36-40.
E-mail address: [email protected] (V. Meyer).
0038-1101/03/$ - see front matter � 2003 Elsevier Science Ltd. All ri
doi:10.1016/S0038-1101(03)00104-7
As a result, a real need arises, namely to propose a
simple and efficient model of the hysteresis behavior in
ferrolectric materials allowing the correct description of
both saturated and non-saturated loops.
The hysteresis theory developed by Preisach [8] in
1935 defines the hysteretic behavior through a double
integral of a distribution function over all the possible
switching thresholds of an elementary hysteresis opera-
tor. This formalism is still in use since it provides an
accurate description of both saturated and internal
minor loops.
In this work, we derive a new compact-model for the
description of the P–E hysteresis behavior, based on the
Preisach representation. We propose two simple empir-
ical expressions of the distribution function. Both pro-
vide analytical solutions to the Preisach integral and give
relevant results. This makes this model a serious candi-
date for an efficient compact-model to be implemented
in an electrical simulator for ferroelectric materials
based devices.
ghts reserved.
1480 V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486
2. Hysteresis loop modeling
2.1. Elementary hysteresis dipole
Hysteretic transducers have time-dependent input
and history-dependent output [9]. In the case of a fer-
roelectric material, the simplest hysteretic system is a
single dipole. This system has two states, ‘‘+1’’ and
‘‘)1’’, corresponding respectively to the opposite polar-
izations, in the context of ferroelectrics, namely ‘‘up’’
and ‘‘down’’. The two switching thresholds are called afor the down to up and b for the up to down transitions,
with a > b. A schematic representation of the loop be-
havior of the elementary hysteresis dipole operator ccab is
given in Fig. 1, where the arrows indicate the way the
loop is swept. In our case, the input variable is the
electric field E and the output variable Pdip ¼ ccab � E is
the dipole polarization. However, experimental data
reveal that a ferroelectric material cannot be modeled by
this basic single dipole since the polarization is a smooth
function of the electric field, meaning that the material
switches progressively from one saturation polarization
to the opposite.
As proposed by Jiang [6], a good description of the
real behavior of the hysteresis systems can be obtained
by considering a discrete set of elementary dipoles. He
assumed that the system was composed of 260 dipoles
which switching thresholds distributions ai and bi
(16 i6 260) can be measured experimentally. Then,
knowing these latter distributions and the electric field
history, it is possible to determine the amount of up and
down dipoles and the resulting polarization.
Beyond the advantage of simplicity offered by this
approach, there are still three major drawbacks related
Fig. 1. Representation of the elementary hysteresis operator.
to its discrete nature. First, the accuracy of the model is
directly related to the choice of the dipoles number.
Whereas the choice of a low number of elementary di-
poles is unacceptable in terms of accuracy, a high
number requires an excessive computing time. Secondly,
and even more detrimental for analog applications, the
discrete nature of the model will generate discontinuities
in the output variables. Finally, the dipole threshold
distributions ai and bi have to be determined by specific
complex and time consuming measurements which
number grows with the square of the number of dipoles.
2.2. Continuous dipole threshold distribution
The aim of this work is to propose the replacement of
the discrete set of dipoles by a continuous distribution of
the dipole thresholds. This has been done in keeping
with the approach established by Preisach [8]. Some
further development has also been found in the work
done by Mayergoyz [9], who explained the full signifi-
cance of the Preisach work. According to this theory, the
contribution of the ferroelectric hysteresis to the polar-
ization is directly related to the lða; bÞ distribution
function and to the state of the dipoles in the (a; b) half-plane delimited by the inequality a > b. This can be
written in term of a double integral:
Pferro ¼Z Z
a>blða; bÞccab � Edadb ð1Þ
where lða; bÞ represents the density of dipoles which
threshold values are ða; bÞ, and ccab represents the ele-
mentary dipole hysteresis operator. Applied to the
electric field, the latter gives the description of the his-
tory of the whole system. It represents a two-dimension
map of the dipoles state (up or down) depending on the
input variable history. For simplicity, we will omit to
discuss the linear contribution to the polarization
(Plin ¼ e � E), which is still taken into account in the
model.
Given integral (1), two points need to be investigated.
The first concerns the definition of the dipoles states
two-dimension map. The second is the determination of
the lða; bÞ distribution, which defines the shape of the
hysteresis loop.
2.3. Hysteresis operator evaluation
The successive changes in the electric field define a
frontier dividing the ða; bÞ triangular half-plane in two
contiguous domains [10]. Increasing the electric field Ewill cause the bottom half of the triangle, defined by
a < E, to be turned to the up state. Conversely, de-
creasing the electric field E will cause the right part of the
triangle, corresponding to b > E, to be turned to the
down state. The successive changes in E define the seg-
Fig. 2. Example of a frontier between the up and down zones in
the (a; b) plane. The slicing of the up zone in trapezes is also
illustrated for a better comprehension of the double integral
boundaries in the calculations found in Appendix A.
V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486 1481
ments composing the indented frontier as it is illustrated
in Fig. 2. This allows us to determine the value of ccab � Eover the (a; b) half-plane.
The up and down domains correspond respectively to
ccab � E ¼ þ1 and ccab � E ¼ �1. Thus, we can rewrite the
polarization as
Pferro ¼ �Z Z
up
lða; bÞdadb �Z Z
down
lða; bÞdadb
ð2Þ
Referring to Psat as the total polarization, correspondingto all the dipoles in the up state, we obtain:
Pferro ¼ 2 �Z Z
up
lða; bÞdadb � Psat ð3Þ
2.4. Determination of the distribution function
At this point, we introduce some fundamental as-
sumptions regarding the distribution function lða; bÞ.First, we assume that the distributions along the a and baxis are independent, allowing the density function l(a; b) to be written as a product of two functions, each
one depending only on a single variable:
lða; bÞ ¼ gðaÞ � hðbÞ ð4Þ
Secondly, we assume that both gðaÞ and hðbÞ distribu-tions can be defined by a common symmetric function f ,i.e.:
gðaÞ ¼ f ða � Eþc Þ ð5Þ
hðbÞ ¼ f ðb � E�Þ ð6Þ
cwhere Eþc and E�
c represent two parameters correspond-
ing respectively to the positive and negative coercive
fields in experimental data. Under these assumptions,
integral (3) becomes:
Pferro ¼ 2 �Z Z
up
f ða � Eþc Þf ðb � E�
c Þdadb � Psat ð7Þ
In this section, we propose a simple method to extract
the distribution function f from a saturated hysteresis
profile. Assuming that the ferroelectric is saturated in
what we call the down state and increasing the electric
field E (see Fig. 2a), we can deduce the boundaries of the
up zone:
PferroðEÞ
¼ 2 �Z E
�1
Z a
�1f ðb
�� E�
c Þdb�� f ða � Eþ
c Þda � Psat
ð8Þ
Derivation with respect to E gives:
dPferrodE
¼ 2 � f ðE � Eþc Þ �
Z E
�1f ðb � E�
c Þdb ð9Þ
However, this formulation is not straightforward for the
extraction of the function f . Noting that the integral
over the whole domain can be split in the form of a
product:Z þ1
�1
Z þ1
�1f ða � Eþ
c Þ � f ðb � E�c Þdadb
¼Z þ1
�1f ða
�� Eþ
c Þda�2
¼ Psat ð10Þ
relation (9) can be rewritten as
f ðE � Eþc Þ ¼
1
2� dPferro
dE�
ffiffiffiffiffiffiffiPsat
p��
Z þ1
af ðb � E�
c Þdb��1
ð11Þ
Then, knowing the derivate of the total polarization
with respect to the applied electric field for the increasing
branch of a saturated loop, it is possible to reconstruct
the function f ðxÞ assuming f ð�1Þ ¼ 0. Note that the
termRþ1
a f ðb � E�c Þdb can indeed be calculated from
previous evaluation of the function f ðxÞ that is knownfor x > a � Eþ
c through the self consistent calculation of
relation (11).
3. Measurements and simulations
The measurements have been carried out using the
Sawyer–Tower configuration on patterned Pt/PZT/Pt
circular capacitors of 1 mm diameter. The 240 nm thick
PZT is Sol–Gel deposited and its composition is 45% of
Zr and 55% of Ti. First, single saturated loops have been
1482 V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486
measured at low frequency (we suppose quasi static
measurements) to obtain a set of data for the parameters
extraction procedure. Then, additional hysteresis loops
have been measured on the same samples, including
unsaturated loops. Concerning simulations, the linear
part of the polarization (Plin ¼ e � E) has been added to
the ferroelectric contribution (Ptot ¼ Pferro þ Plin).
3.1. Distribution function extraction
The distribution lða; bÞ was extracted according the
method discussed in Section 2 by considering the in-
creasing branch of a saturated hysteresis loop, with the
ferroelectric polarization increasing from �Psat to þPsat.Note that the extraction methodology is specific to this
model and to the assumptions we made. As a conse-
quence, the extracted dipole threshold density might
have no meaning if the basis hypotheses are changed.
Fig. 3 shows a hysteresis curve and its corresponding
extracted distribution as a function of the electric field,
considering different fitting function candidates.
Fig. 3. Extraction of the distribution function and illustration of the
simulation: line) using four candidates for the empirical distribution fu
functions: line), with (a) Gaussian, (b) Lorentzian, (c) simple and (d)
According to the shape of the density function, we
first tried to fit the extracted function f ðxÞ by using
Gaussian and Lorentzian functions. This choice seems
adequate since we are concerned with statistics where
these functions are mostly encountered. The general
form of these functions is given in Table 1 and the
number of fitting parameters contained in each distri-
bution function candidate is also reported.
3.2. Saturated hysteresis loops
We propose to extract the set of fitting parameters
from saturated loops, since the physical parameters de-
scribing the ferroelectric materials are connected with
these latter. The relative error is defined as the normal-
ized surface between measured and simulated curves.
This constitutes our fit criterion. The distribution func-
tion f is extracted from measurements through relation
(11) and compared with each fitting function using the
best parameters obtained from the hysteresis curves. As
we can see on Fig. 3a, the Gaussian distribution gives a
fitting of the saturated hysteresis loop (experimental data: dots;
nctions (distribution extracted from experiment: dots; candidate
double exponential absolute decay functions.
Table 1
Fitting function candidates
Function name Equation Parameters
Gaussian f ðxÞ ¼ A � e�x2=2r2 A;rLorentzian f ðxÞ ¼ ð2A=pÞ � ðr=4x2 þ rÞ A;rSimple exponential absolute decay f ðxÞ ¼ A � e�jxj=r A;rDouble exponential absolute decay f ðxÞ ¼ A1 � e�jxj=r1 þ A2 � e� xj j=r2 A1, A2, r1, r2
V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486 1483
correct description of the hysteresis loop, with a relative
error of 18.4%. However, the corresponding distribution
function does not match the points extracted from ex-
periment, suggesting that this function is not adapted to
our model and our assumptions. A better concordance is
obtained by using the Lorentzian function (Fig. 3b) for
both the hysteresis loop (relative error is 10.4%) and the
extracted distribution function, even if we can still ob-
serve a mismatch at the top of this latter. However, there
is still a major drawback in using either of these func-
tions concerning the evaluation of the integral (7), which
has to be done numerically since no analytical expres-
sion could be obtained. This last point is a serious
handicap for future use in compact simulation tools,
where compute-time and accuracy are critical.
Then, we must seek for a function that both accu-
rately fits the experimental data and leads to an ana-
lytical solution to relation (7). According to the shape of
the extracted density function, a possible candidate is
given by the function of the general form:
f ðxÞ ¼ A � e�jxj=r ð12Þ
This is shown in Fig. 3c where the simple exponential
absolute decay function gives a very nice fitting of the
saturated hysteresis loops (relative error is 13.8%). In
addition, the threshold distribution function is also
better fitted as compared to the Gaussian function, and
nearly as good as the Lorentzian functions. At the same
time, and that is perhaps the most important, the ex-
ponential distribution function has an exact analytical
solution to the integral (7). This result is presented in
Appendix A. Note that the evaluation of the integral in
the different domains of the (a; b) plane (see AppendixA) only requires elementary functions. To our knowl-
edge, no similar analytical result of ferroelectric hyster-
esis calculation based on relation (1) has been proposed
yet.
These results can be further improved by using a
linear combination of exponential absolute decays for
the function f :
f ðxÞ ¼ A1 � e�jxj=r1 þ A2 � e�jxj=r2 ð13Þ
Not only the integration of the double exponential
absolute decay function (Fig. 3d) better describes the
hysteresis loop, with a relative error down to 8.7%, but a
very good concordance is now obtained with the ex-
tracted distribution as well, suggesting that this function
gives a coherent description of the whole system, in
agreement with the assumptions we made. As for the
function (12), exact analytical solutions to relation (7)
can also be obtained (see the Appendix A).
As listed in Table 1, the simple exponential absolute
decay function only requires two fitting parameters,
namely A and r. These are connected in a unique man-
ner, with the two phenomenological parameters char-
acterizing the hysteresis curve, namely the saturation
polarization (Psat) and the remnant polarization (Pr). Anexact relation can be found between the model param-
eters and the physical quantities by resolving a simple
two-unknown system. Positive and negative coercive
fields are taken into account in the definition of the re-
lation (7). Concerning the double exponential decay,
two more fitting parameters are needed (see Table 1).
These can be adjusted to suit the ferroelectric material of
the device under test, still allowing an efficient extraction
of the parameters. Note that although there is a direct
relation between the physical and the simulation pa-
rameters, all those latter have been adjusted for slightly
better results.
Finally, it appears that all four functions give a rel-
atively good description of the saturated loop. However,
taking into account the relative error rating as indicated
on the figures, we conclude that the double exponential
absolute decay provides the best fitting results (8.7%),
followed by the Lorentzian (10.4%) and the simple ex-
ponential absolute decay (13.8%), while the Gaussian is
the worst (18.4%). These results are encouraging, since
the Lorentzian, and the Gaussian both require the nu-
merical evaluation of (7), whereas the exponential ab-
solute decay functions provide an analytical solution,
thus being very interesting candidates for use in electri-
cal simulators.
This point constitutes a major improvement in
ferroelectric compact modelling since it eliminates
the counting of individual dipoles (like in [6]) and the
numerical integration of functions (like in [7]), both
methods leading to equivalent calculations requirements
and lacking accuracy. For comparison, with a given set
of parameters, the measured time needed to simulate a
complete saturated hysteresis curve (400 experimental
points) on a 450 MHz clocked computer is around 120 s
for the Gaussian function, 40 s for the Lorentzian, both
requiring numerical integration of Eq. (7), and less than
-200 -100 0 100 200
-40
-20
0
20
40Simple ExponentialRel.Err.= 7.4 %
Ec+ = 49.0
Ec- = -49.3
= 32.00
Pol
ariz
atio
n(
C/c
m2)
Electric field (kV/cm)
σ
µ
Fig. 4. Illustration of the matching between simple exponential
decay fitting function and experimental data for saturated and
minor hysteresis loops. (experimental data: dots; simulation
with simple exponential absolute decay function: line). Note
that the fitting was carried out on the saturated loop.
µ
σ
c+
c-
20
10
0
-10
-20
-100 0 100
Fig. 5. Illustration of overlapped saturated and minor hyster-
esis loops (experimental data: dots; simulation with simple ex-
ponential absolute decay function: line). Note that the fitting
was carried out on the saturated (largest) loop. The minor in-
ternal loops are still correctly depicted by the model.
1484 V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486
10 ms for both exponential absolute decay functions,
since both use analytical results of (7).
More recently, Chen et al. [7] also proposed a com-
pact model for the hysteresis loop that was implemented
in SPICE. However, their approach still requires the
numerical resolution of a differential equation that rep-
resents a serious drawback in terms of computation time.
3.3. Minor hysteresis loops
Based on the same parameters extraction method, we
also used the single and double exponential functions to
predict complex loops. This includes non-saturated minor
loops, corresponding to the partial switching of the ferro-
electric material. First, a saturated loop followed im-
-200 -150 -100 -50 0 50 100 150 200
-20
-10
0
10
20 Simple ExponentialRel.Err.= 6.2 %
Ec+ = 50.1
Ec-
= -42.0= 24.8
Pol
ariz
atio
n(
C/c
m2 )
Electric field (kV/cm)(a) (
σ
µ
Fig. 6. Illustration of a complex hysteresis path (experimental data: d
absolute decay functions. Simple and double exponential functions le
cedure was carried out on saturated loop (outer part).
mediately by a minor loop has been measured. Fig. 4
shows the experimental curve and the fit obtained with the
simple exponential absolute decay function. The agree-
ment is good (7.4% relative error). This is also the case on
the minor loop even though the function parameters have
been extracted from the saturated hysteresis loop. The
double exponential function gives slightly better results
(7.2% relative error), which are not shown here. Another
case of interest concerns minor closed loops.
3.4. Overlapped saturated and minor hysteresis loops
On a virgin sample, without hysteresis history, closed
loops have been measured successively with increasing
electric field range (from �10 to �400 kV/cm). The
-200 -100 0 100 200
-20
-10
0
10
20Double ExponentialRel.Err.= 5.8 %
Ec+ = 50.5
Ec- = -42.1
A2/A1 = 0.19
1 = 14.35
2 = 53.63
Pol
ariz
atio
n(
C/c
m2)
Electric field (kV/cm)b)
σσ
µ
ots; simulation: line) with (a) simple and (b) double exponential
ad to a very good agreement with experiment. The fitting pro-
V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486 1485
function parameters set has been extracted from the
largest, widely saturated, hysteresis curve. Fig. 5 shows
the corresponding simulation results for the smallest
curves with the simple exponential absolute decay
function, confirming that the adequation is very good
with experimental data, even for the smallest loops, since
the shape and the amplitudes are predicted with accu-
racy over the whole range of electric field with only a
unique set of parameters, confirming the coherence of
the model. Note that the measurement bench centers
automatically the measured curves with respect to the
polarization. Thus, the simulation results have been
centered as well for consistent comparison.
3.5. Complex hysteresis loops
Finally, a sawtooth electrical field signal has been
applied with decreasing amplitude on a third sample.
The fitting parameters have been extracted from the
outer (saturated) part of the measured curve. Simple and
double exponential absolute decays are plotted on Fig. 6
for comparison, leading to a relative error as low as
6.2% (Fig. 6a) and 5.8% (Fig. 6b) respectively.
Note that for all the curves shown here, the para-
meters have been extracted on saturated curves, and ap-
plied to the complex hysteresis measurements including
minor closed or open loops. These results can still be
improved if the extraction is performed directly on the
complex loops themselves. In the case of the Fig. 6, for
example, using the simple or double exponential decay
with the best parameters set for the complete curve, the
relative error can be further reduced down to 5.7% and
4.9% respectively.
4. Conclusion
Based on the Preisach theory, a new analytical model
for the simulation of the hysteresis loops in ferroelectric
materials has been presented. We proposed to use a
simple distribution function of elementary dipoles that
lead to a unique analytical form for the total polariza-
tion, including the history effect. This approach gives
excellent agreement with experimental data where satu-
rated and unsaturated loops could be well predicted. In
addition, this model is very efficient in terms of com-
putation time and should be a good candidate for the
compact modeling of ferroelectric devices.
Acknowledgements
This work was supported by the �Commission pour laTechnologie et l�Innovation�, project CTI 3989.2, fundedby the Swiss government. The authors would like to
thank Dr. M. Lobet for his support in the project, Dr.
I. Stolitchnov for his help in measurements and Dr.
V�eeronique Adam for improvements of the manuscript.
Appendix A
The surface corresponding to the dipoles in the up
state can be considered as the sum of N adjacent trape-
zoidal domains, allowing expressing relation (7) as the
sum of N integrals. This is illustrated in Fig. 2 where the
sum of three trapezoidal areas (a, b and c) representsthe up domain (note that the triangle labeled (c) is a par-ticular case of a trapeze). In addition, the insert of Fig. 2b
shows how three coordinates, namely Eleft, Eright and Etop,
can completely define each subdomain. Then, the con-
tribution of each trapeze to the total polarization through
the evaluation of the integral (1) can be put in the form:
I ¼Z Eright
Eleft
Z Etop
bf ða
�� Eþ
c Þda
�� f ðb � E�
c Þdb ðA:1Þ
A.1. Simple exponential absolute decay
We suppose that the distribution function is given:
f ðxÞ ¼ A � e�jxj=r
Defining new intermediate quantities as
D1 ¼ Eleft � E�c ; D2 ¼ Eleft � Eþ
c ; D3 ¼ Eright � E�c ;
D4 ¼ Eright � Eþc ; D5 ¼ Etop � E�
c ; D6 ¼ Eþc � E�
c
si ¼ sgnðDiÞ ¼ þ1 or )1 whether Di is respectively pos-
itive or negative.
Xi ¼ expð�jDij=rÞThe result of (A.1) is given by the following relation:
I ¼ A2½r2½s1ð1� X1Þðs2 � s6ð1� X6ÞÞ� s3ð1� X3Þðs4 � s6ð1� X6ÞÞ þ s5ð1� X5Þðs4 � s2Þþ 1=4ðð1þ s1s2ÞðX1X2 � X5Þ � ð1þ s3s4ÞðX3X4 � X5ÞÞ�� ½D5ð1� s1s3Þ � D2ð1� s1s2Þs5 þ D4ð1� s3s4Þs5��
ðA:2Þ
A.1.1. Double exponential absolute decay
We suppose that the distribution function is given:
f ðxÞ ¼ A1 � e�jxj=r1 þ A2 � e�jxj=r2
Defining new intermediate quantities as
rþ ¼ r1 � r2
r1 þ r2
; r� ¼ r1 � r2
r2 � r1
D1 ¼ E1 � E�c ; D2 ¼ E1 � Eþ
c ; D3 ¼ E2 � E�c ;
D4 ¼ E2 � Eþc ; D5 ¼ E3 � E�
c ; D6 ¼ Eþc � E�
c
1486 V. Meyer et al. / Solid-State Electronics 47 (2003) 1479–1486
hi ¼ HðDiÞ, where H is the Heavyside step function.
Yi ¼ expð�jDij=r1Þ and Zi ¼ expð�jDij=r2Þ
The result of (A.1) is now given by the following relation
(A.3):
I ¼ ½A1r1ðY5 þ 2h5ð1� Y5ÞÞ þ A2r2ðZ5 þ 2h5ð1� Z5ÞÞ�� ½ð1� 2h1ÞðA1r1ð1� Y1Þ þ A2r2ð1� Z1ÞÞ� ð1� 2h3ÞðA1r1ð1� Y3Þ þ A2r2ð1� Z3ÞÞ�� ½A1r1ðA1r1=2ððh1 þ h2 � 1ÞðY1Y2 � Y6Þ� ðh3 þ h4 � 1ÞðY3Y4 � Y6ÞÞ þ A2r
þððh1 þ h2 � 1ÞZ1Y2� ðh3 þ h4 � 1ÞZ3Y4 þ ðh3 � h1ÞY6 þ ðh4 � h2ÞZ6ÞÞþ A2r2ðA2r2=2ððh1 þ h2 � 1ÞðZ1Z2 � Z6Þ� ðh3 þ h4 � 1ÞðZ3Z4 � Z6ÞÞ þ A1r
þððh1 þ h2 � 1ÞY1Z2� ðh3 þ h4 � 1ÞY3Z4 þ ðh3 � h1ÞZ6 þ ðh4 � h2ÞY6ÞÞ�� A1A2r
�½r2ððh1 � h2ÞY1Z2 � ðh3 � h4ÞY3Z4þ ðh3 � h1ÞZ6 � ðh4 � h2ÞY6Þ þ r1ððh1 � h2ÞZ1Y2� ðh3 � h4ÞZ3Y4 þ ðh3 � h1ÞY6 � ðh4 � h2ÞZ6Þ�þ ðA2
1r1Y6 þ A22r2Z6Þ½ðh1 � h2ÞD2 � ðh3 � h4ÞD4
� ðh3 � h1ÞD6� � 2ðA1r1 þ A2r2Þ½A1r1ðh2ðY1 � Y6Þ� h4ðY3 � Y6ÞÞ þ A2r2ðh2ðZ1 � Z6Þ � h4ðZ3 � Z6ÞÞ�þ ðA1r1Þ2½h2ðY1Y2 � Y6Þ � h4ðY3Y4 � Y6Þ�þ ðA2r2Þ2½h2ðZ1Z2 � Z6Þ � h4ðZ3Z4 � Z6Þ�� 2A1A2r
þ½h4ðr1ðY4Z3 � Z6Þ þ r2ðY3Z4 � Y6ÞÞ� h2ðr1ðY2Z1 � Z6Þ þ r2ðY1Z2 � Y6ÞÞ� ðA:3Þ
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