Ferroelectrics Characterization and Modeling Part 11 potx

The result obtained here has been applied to the structural phase transition of the ferroelectric crystal BaTiO3.. Self-consistent anharmonic theory and its application to the isotope e

where x n is the coordinate of the nth atom and <x n> is the averaged equilibrium position of

the n-site Ti atom along the x –axis, as shown in Fig.1,Δ r is the distance between <x n> and

the minimum position, D is potential depth, 2Dα2 is the classical spring constant in the

harmonic approximation, and CF is the coefficient of the long-range order interaction

Replacing the interatomic distance a n′ with the atomic position x n is expected to result in a

good approximation of the nearest interaction in the neighbourhood Then, Eqs.(17) and (29)

are rewritten as follows:

k

2C 6

,

S

f T

x

V c

f

n n S

( )

( )

x

V c

x

V c

f

n n S

n n S

2 2

The thermal average of V n is calculated as

B

k

exp C

2

2 2 3

S

g y T

Self-Consistent Anharmonic Theory and Its Application to BaTiO 3 Crystal 341 Here

S S B

y e

y y y

y e y g

, y y y

y e

y y y

y e y f

, c

c

, c

c

n n

n

n n

b b

b

b b S

n S

S S

n S

S S

n S S

n S

+

−+

≡

+

−+

+

−+

α

α α

γ ζ

1 2 2 2

2 1 2 2 2

1 2 2 2

1 2 2 2 2 4 4

2 2

44

11

The potential parameters D, α, Δr , and CF listed in Table1 were determined with reference

to the results of the first-principles calculations within the density functionl theory

Δ 0.02833 0.02833

(in Eq 42)

Table 1 (Y Aikawa et al., 2009, Ferroelectrics 378)

Ultrasoft pseudopotentials (D Vanderbilt, 1990)were used to reduce the size of the wave basis Exchange-correlation energy was treated with a generalized gradient approximation (GGA-PBE96) Y Iwazaki evaluated the total energy differences for a

plane-number of different positions of Ti atoms positions along the x-axis (Fig.6) with all other

atoms fixed at the original equilibrium positions, which are denoted by open circles in Fig.7

The solid lines in these figures indicate the theoretical values obtained using Eq.(44) with the fitting parameters listed in Table1

Ti

O Ba

Fig 6 Perovskite crystal structure of BaTiO3

Fig 7 Atomic potential of Ti in ferroelectric phase of BaTiO3 denoted by open circles were

obtained by first principles calculations, the solid line indicate theoretical values given by

eq.(44) (Y Aikawa et al., 2009, Ferroelectrics 378)

4.2 Ferroelectricity of barium titanate

When the softening occurs close to the Curie point, the solution λS increases rapidly This

increase implies that the second-order variational parameter B S tends to zero, the square of

the angular frequency 2

T

λ ∝

Self-Consistent Anharmonic Theory and Its Application to BaTiO 3 Crystal 343

The instability of the ferroelectrics in terms of the oscillator model can be explained as

follows:as the temperature approaches the Curie temperature Tc, ΩS2 changes to zero from

a positive value according to displacive ferroelectrics (B>0); ΩS2 changes to zero from a

negative value according to the order-disorder model (B<0) The former is termed the

propagation soft mode, and the latter, the non-propagation soft mode

The relation between the dielectric constant and the frequency of an optical mode as

expressed by Lyddane, Sachs and Teller (R.H.Lyddane et al.,1941) is

where Ωt denotes the frequency of transverse optic modes From eqs (50) and (51), the

relation between ε and λS is given by:

0

,

S

C T

where C is a constant The temperature dependence of λS is calculated by Eq.(49) Fig.8

shows the dielectric constant along the c axis measured as a function of temperature for a

single crystal (W J Merz, 1953) The solid line in Fig.8 is fitted according to the theoretical

calculation performed using Eq.(52) and the potential parameters listed in Table1

Fig 8 Temperature dependence of the dielectric constant of single crystal of BaTiO3 along

the c axis The solid line is calculated by Eq (52), and the open circles are experimental

values (Y Aikawa et al., 2009, Ferroelectrics 378)

5 Isotope effect

There have been some reports of the isotope effects on displacive-type phase transition, as

determined experimentally (T Hidaka & K Oka, 1987) In classical approximation (A D B

Woods et al., 1960; W Cochran, 1960), T C is expected to shift to a higher temperature in

heavy-isotope-rich materials and vice versa However, the experimental results are

completely opposite to the expected results It has been long considered that the origin of

these phenomena in BaTiO3 may be related to the quantum mechanical electron-phonon

interaction (T Hidaka, 1978, 1979)

However, it seems to be problematic to introduce the quantum mechanical electron-phonon

interaction to interpret the ferroelectric phase transition in BaTiO3, because the phase

transition is a phenomenon in the high-temperature region in which there is scarcely any

quantum effect In order to discuss such a phenomenon in the high-temperature region, K

Fujii et al have proposed a self-consistent anharmonic model that is applied to the phase

transition (K Fujii et al., 2001), and the authorhas extended it to derive the ferroelectric

properties of BaTiO3 (Y Aikawa et al., 2009) In this section the isotope effect of T C is

explained through this theory, and the theoretical result is compared with experimental

data

5.1 Theory

Postulating that atomic potential is independent of atomic mass, eq (33) is rewritten as

, V

V T

2

2 2

2 B

6δk

a

where ζ is the mass-dependent part in T C as

( ) ( )

2 2

4

1

nn S nn nn S

c c

in Fig.2 The force constants shown in Fig.2 are derived from the second-order derivative of

interatomic potential with respect to interatomic distance

It is, however, difficult to estimate the force constants because estimate various interactions

between atoms exist The author did attempt to estimate them so as not to contradict the

results of neutron diffraction experiments; as (α/γ, β/γ, η/γ) = (0.1, 0.09, 0.81) as

derived in 3.2

5.2 Numerical calculation and comparison with experiments

It was also shown that the soft mode is the Slater mode, which is the lowest frequency optic

mode at k = 0 under this condition Using this force constants, the ratio of T C (yBa xTi O3 that

is replaced with isotope elements) to T C (natural 137.33Ba 47.88Ti 16O3) is obtained by

calculating eq.(54) using x = 46-50, y =134-138 as parameters The results are shown in

Fig 9

Self-Consistent Anharmonic Theory and Its Application to BaTiO 3 Crystal 345

0.97 0.98 0.99 1.00 1.01

Fig 9 x-y phase diagrams of the ratio of Tc (yBa xTi O3) to Tc (137.33Ba 47.88Ti 16O3)

(Y.Aikawa et al., 2010 Jpn J Appl Phys 49 09ME11)

In Fig.10, the solid curve shows the theoretical values of the transition temperature for the isotope effects of Ti calculated using eq (54), and the experimental values are represented

by open circles It appears that the theoretical values in the solid curved line are roughly in agreement with the experimental values represented by the open circles as shown in the figure

Fig 10 Comparison between the theoretical and experimental values in terms of

x-dependence of the ratio of Tc (137.33 Ba xTi 16O3) to Tc (137.33Ba 47.88Ti 16O3) (Y.Aikawa et al.,

2010 Jpn J Appl Phys 49 09ME11)

In the case of harmonic approximation, as the heavy Ti isotope is introduced, the Curie temperature rises, and vice versa for the light Ti isotope (T Hidaka & K Oka, 1987), because only the coefficient ( )n 2

that T C is expect to shift to the lower temperature in the heavier Ti isotope

The instability temperature or the transition temperature for the trial potential represented

by an anharmonic oscillator has been derived from the variational method at finite temperature where the normal coordinates were introduced in this work to reflect the crystal symmetry in the softening phenomenon The result obtained here has been applied

to the isotope effect of the ferroelectric crystal BaTiO3 The transition temperature T C given

by eq (53) has been applied after substituting the actual values obtained for the force constants into ζ given by eq.(54) As a result, the author has been able to probe that the

transition temperature T C of barium titanate consisting of heavy-isotope Ti is lower than that

of barium titanate consisting of light-isotope Ti

6 Conclusion

The instability temperature or the transition temperature for the trial potential represented

by an anharmonic oscillator has been derived from the variational method at finite temperature where the normal coordinates were introduced in this work to reflect the crystal symmetry in the softening phenomenon

1 Though the expression obtained here has the same form as the Landau expansion, the transition temperature and the expansion coefficients can be represented by the characteristic constants of the potentials between atoms From the fact that the coefficient of the second order term in the trial potential is expressed by the form such

as B R( )(k TC−T), the author has proposed the equations to determine the soft mode by

imposing the condition that its k -dependent part takes the minimum value The result

obtained here has been applied to the structural phase transition of the ferroelectric crystal BaTiO3 The dispersion relations derived from the dynamical matrix has been compared with that from the neutron diffraction experiment The force constants between atoms have been fitted so as to reproduce the experimental results for the dispersion relations The determination equations given by eq.(40) has been applied after substituted the actual values obtained for the force constants into γR(k) given by eq.(38) As a result, the author

has been able to probe that the lowest frequency mode at Γ point corresponded to the S2mode causing the structural phase transition in the BaTiO3 crystal

2 The author has shown that the ferroelectric properties of BaTiO3 result from the equilibrium condition of free energy by using the anharmonic oscillation model and the elemental parameters derived using first-principles calculations

Self-Consistent Anharmonic Theory and Its Application to BaTiO 3 Crystal 347

3 The result obtained here has been applied to the isotope effect of the ferroelectric crystal BaTiO3 The transition temperature T C given by eq (53) has been applied after substituting the actual values obtained for the force constants into ζ given by eq (54)

As a result, the author has been able to probe that the transition temperature T C of barium titanate consisting of heavy-isotope Ti is lower than that of barium titanate consisting of light-isotope Ti

7 References

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pp 6301-6312

18

Switching Properties of Finite-Sized Ferroelectrics

L.-H Ong1 and K.-H Chew2

1School of Physics, Universiti Sains Malaysia, Minden, Penang,

2Department of Physics, University of Malaya, Kuala Lumpur,

Malaysia

1 Introduction

The characteristic property of ferroelectric materials, which is the reversal of polarization by

an external electric field, is of technological importance in device applications, particularly

in nonvolatile ferroelectric random access memories (NV-FeRAMs) These binary coded NV-FeRAMs can be fabricated by using ferroelectric materials in which the polarization direction can be switched between two stable states when a minimum electric field is applied To fabricate good quality NV-FeRAMs to meet the demands of the current market, the ability to achieve low coercive field E c(the minimum external field required to reverse the direction of remnant polarization), short switching time t s and high packing density

(Scott, 2000; Dawber et al., 2005) in the memory chips are great challenges These

challenging factors are closely knitted with the underlying physics on the switching properties of ferroelectric materials Though the subject of interest has been elucidated both theoretically and experimentally over the past sixty years and the achievements are enormous, but the challenging factors mentioned are still current Auciello, Scott and Ramesh (1998) have explicitly outlined four main problems in NV-FeRAMs fabrication which are related to basic physics Firstly, what is the ultimate polarization switching speed? Secondly, what is the thinnest ferroelectric layer which sustains stable polarization? Thirdly, how do switching parameters, such as coercive field, depend on frequency? Lastly, how small can a ferroelectric capacitor be and still maintain in ferroelectric phase? These are fundamental problems which should be tackled through continuous experimental and theoretical efforts

From the theoretical perspective on this area of studies, a few models (Duiker and Beale,

1990; Orihara et al., 1994; Hashimoto et al., 1994; Shur et al., 1998) were proposed to study the

switching properties of ferroelectric thin film based on the Kolmogorov-Avrami theory of crystallization kinetics (Avrami, 1939, 1940 and 1941) In these models, the authors focused

on statistics of domain coalescence Tagantsev et al (2002) proposed another model based on the experimental work of a few groups (Lohse et al 2001; Colla et al., 1998a; Ganpule et al.,

2000) Their model also focuses on statistics of domains nucleation Another approach, which is different from the classical nucleation reversal mechanism, is based on the Landau-type-free energy for inhomogeneous ferroelectric system as discrete lattices of electric

dipoles (Ishibashi, 1990) However, all these models neglect the surface effect, which is shown to have influence on phase transitions of ferroelectric films, and as the films get thinner the surface effect becomes more significant

The continuum Landau free energy for a ferroelectric film, extended by Tilley and Zeks (Tilley and Zeks, 1984), incorporates the surface parameter δ (extrapolation length) Positive

δ models a decrease in local polarization at surface, and negative δ an increase, with a smaller absolute value of δ giving a stronger surface effect This model has been used to elucidate phase transition and dielectric properties of FE thin films with great success

(Wang C L et al., 1993; Zhong et al., 1994; Wang Y G et al., 1994; Wang C L and Smith, 1995; Ishibashi et al., 1998; Tan et al., 2000; Ong et al., 2001; Ishibashi et al., 2007) and the

results of phase transition and dielectric properties of ferroelectric thin films obtained are well accepted In this chapter, we outline the progress of theoretical and experimental work

on switching phenomena of ferroelectric thin films, and the main focus is on the results of switching properties of ferroelectric thin film obtained from the Tilley – Zeks continuum

Landau free energy and Landau-Khalatnikov (LK) dynamic equation (Ahmad et al., 2009;

Ong and Ahmad, 2009; Ahmad and Ong, 2009; 2011a; 2011b) The surface effects, represented by ±δ, on properties of polarization reversal, namely, coercive field E c and switching time t will be discussed (Ahmad et al., 2009) For positive s δ, E and c t decrease S

with decreasing δ while for negative δ, E c andt s increase with decreasing δ Strong surface effects represented by smaller δ are more profound in thin ferroelectric films As

the film size increases, the delay in switching at the centre relative to switching near the edges is more remarkable for systems of zero or small polarization at the edges (δ≅ ) It is 0found that the dipole moments at the centre and near the edges switch almost together in

small-sized systems of any magnitudes of δ (Ong et al., 2008a; 2008b) We also elucidated

the phenomena of polarization reversal of second-order ferroelectric films, particularly the characteristics of hysteresis loops by an applied sinusoidal field It is shown that at a constant temperature, the size of hysteresis loops increases with increasing film thickness

for δ > 0 and the reverse is true for δ < 0 For a film of fixed thickness, the size of hysteresis loop decreases with increasing temperature for cases of δ > 0 and δ < 0 We have

demonstrated that the effect of magnitude of the applied field on the hysteresis loops is similar to the experimental results (Ong and Ahmad, 2009) Our numerical data also show that switching time t S is an exponential function of the applied field and the function implies that there is a definite coercive field in switching for various thicknesses of FE films (Ahmad and Ong, 2011b) Lastly, since in reality, ferroelectric thin films are fabricated on conductive materials (such as SrRuO3) as electrodes, hence, we shall include the effects of misfit strain on switching phenomenon of epitaxial film of BaTiO (Ahmad and Ong, 32010a) and conclude with some remarks

2 Ferroelectric thin film and Tilley-Zeks model

The behaviour of ferroelectric thin films is significantly different from that of the bulk The arrangement of atoms or molecules at the surface is different from that of the bulk material Due to the process of surface assuming a different structure than that of the bulk, which is known as surface reconstruction, polarization at the surface is not the same as that in the bulk; and it affects the properties of the material This so called surface effect may have little influence on the properties of material if the material is thick enough However, when the

Switching Properties of Finite-Sized Ferroelectrics 351

material becomes thinner, the surface effect becomes significant The demand by current

technological applicants on material thickness of ferroelectric thin film is now in the range of

nano-scale Hence, surface effect in ferroelectric thin film is a significant phenomenon and

how it can affect switching must be understood

The Landau free energy for a ferroelectric film, extended by Tilley and Zeks (Tilley and

Zeks, 1984), incorporates the surface parameter δ (also named the extrapolation length) and

for convenient, we named it TZ model Positive δ models a decrease in local polarization at

the surface, and negative δ an increase; with a smaller absolute value of δ giving a stronger

surface effect This is important since both forms of behavior have been observed in

experiments on different materials With this surface parameter δ, the inherent material

properties at the surfaces of a ferroelectric film which can be either of the two cases are

explained This Landau free energy is given by

where S is the area of the film with plane surfaces at z= ± /2L and P±=P L(± / 2) α is

temperature dependent, taken in the form α α= 0(T T− C0) with T C0 the critical temperature of

the bulk material and the constantsα0, β and κ are positive ε0 is the dielectric permittivity of

the material and the κ term inside the integral in Eq (1), represents the additional free energy

due to spatial variation of P Whereas the κ term outside the integral in Eq (1), represents the

free energy due to the surface ordering

3 Phase transition in ferroelectric thin film

We (Ong et al., 2000; 2001) reinvestigated the TZ model and obtained much simpler

expressions, compared with previous work (Tilley and Zeks, 1984), for the polarization

profiles of ferroelectric thin films in Jacobian elliptic functions for both positive and negative

δ Variation of Eq (1) about the equilibrium form P(z) shows that this satisfies the

Euler-Lagrange equation (Ong et al., 2001)

2 3

It follows from Eq (3) that if the extrapolation length δ is positive, P(z) decreases near the

surface, and if it is negative, P(z) increases In consequence, the critical temperature T C of

the film is reduced below T C0 for positive δ and increased for negative δ The first

integration of Eq (2) leads to

where G is the constant of integration The extremum of P(z) is at z = 0 so that the central

value P(0) is a solution of the quadratic equation corresponding to dP/dz = 0:

P is always positive for both signs of δ, in the case of δ> , there is a temperature 0

interval in which G and therefore 2

1

P are negative

The P integral resulting from Eq (4) can be expressed by inverse elliptic functions so that

ultimately P(z) is expressed in terms of an elliptic function The detailed forms depend on

positive Since positive δ leads to a decrease of P(z) at the surface of the film we have the

inequalities 0<P z( )<P1<P2 The central value P(0) is the maximum value of P(z) and is in

fact equal to P1 The expression for P(z) is

1

in standard notation for elliptic functions The modulus λ is given by λ=P1/P2; K K= ( )λ

is the complete elliptical modulus and the scale length ξ is

2 1/2 2

In the case of δ< , the analytical work is complicated because the expression for 0

polarization profile depends on temperature range In the temperature interval T0≤ ≤T T C

in which G < 0 and P12≤ ≤0 P22≤P z2( ) In the interval where G is negative, P(z) takes the

form,

2 1

( )( / , )

ε κζ

Similarly, the modulus λ is given by λ=P1/P2; K K= ( )λ is the complete elliptical

modulus These simpler analytical expressions, describing the polarization profile of a

Switching Properties of Finite-Sized Ferroelectrics 353

ferroelectric film with either positive or negative δ, are important in helping the study of

switching properties of ferroelectric film as well as size and surface effects on properties of

phase transition In the case of negative δ, there is no size induced phase transition; but for

positive δ, we have found an expression for the minimum thickness L C of ferroelectric film

to maintain ferroelectric properties This minimum thickness is a function of δ and

temperature T as shown below:

ξ is the correlation length at critical temperature given by ξC=ξ0/ T C0−T C and ξ0 is the

zero temperature of correlation length T C and T C0 are the critical temperatures of the film

and bulk, respectively Eq (11) provides hints to the experimentalists that minimum

thickness of ferroelectric film is dependent on the critical temperature, as well as δ; hence

these values are material dependent

We have also presented new thermodynamic functions, the entropy and specific heat

capacity, for ferroelectric films with both cases of ± These thermodynamic functions δ

provide useful information that the phase transitions in both cases of ± are stable (Ong et δ

al., 2001) The reports in the literature on the claims of possible surface state in the case of

negative δ (Tilley and Zeks, 1984) and film transition can be first order even if its bulk is

second order (Qu et al., 1997) had caught our attentions and after careful investigation, we

found that there is no surface state in the negative δ case and the film transition is always

second order as in the bulk transition

4 Formalism for switching in ferroelectric thin films

Theoretical and experimental work on switching phenomena of bulk ferroelectric began

about half a century ago The interest in this research area has further been extended to

ferroelectric thin films; and the interest has not waned even up to these days because of the

advancement in thin film fabrication technology, where higher quality and more reliable

ferroelectric thin films can be fabricated; thus making the applications of ferroelectric thin

films in microelectronic devices and memories (Uchida et al., 1977; Ganpule et al., 2000) more

reliable Current theoretical and experimental researches in polarization reversal in

ferroelectric thin films are focused on phenomena related to effects of size and surface in

thin films on switching time and coercive field (Ishibashi and Orihara, 1992; Wang and

Smith, 1996; Chew et al., 2003)

From the literature, several theoretical models based on a Landau-typed phase transition

have given good explanations on switching behaviours of mesoscopic ferroelectric

structures (Ishibashi and Orihara, 1992; Wang and Smith, 1996); and some of the predictions

concerning size on switching behaviours by Landau-typed models agree well with

experimental observations However, the detailed understanding of surface effect on

ferroelectric films under the applied electric field is still inadequate, but understanding of

surface effect is important for the overall understanding on the switching behaviours of

ferroelectric films Thus, we extended the TZ model for ferroelectric thin films given in Eq

(1) by adding in the energy expression a term due to electric field (–EP)

and minimization of Eq (12) by variational method shows that the polarization satisfies the

Euler Lagrange (EL) equation

2 3

The Landau-Khalatnikov dynamic equation is used to study switching behaviours in

ferroelectric thin films (Ahmad et al., 2009; Ong et al., 2008a; 2008b; 2009; Ahmad and Ong,

2011b), and it is simplified to the form as follow:

2 3

where γ is the coefficient of viscosity which causes a delay in domain motion and τ is the

time In this equation, the kinetic energy term m P∂2 /∂ is ignored, since it only τ2

contributes to phenomenon in the higher frequency range The applied electric field E can

be a static step field or a dynamic field We obtained the equilibrium polarization profile

( )

P z from the elliptic function derived Eq (1), and this profile is symmetric about the film

centre z = 0.0 The initial polarization profile of the ferroelectric film at e = 0.0 is obtained

from solving Eq (4) for the elliptic functions derived by Ong et al (2001) In all our

simulations, the initial polarization in the film is switched from its negative value By

solving Eq (15) using the Runge-Kutta integration by finite difference technique, we

obtained the reversal of polarization The reversal of polarization is studied by applying a

stepped electric field and the hysteresis loops are obtained by sinusoidal field

respectively, as these fields are usually used in experiments The applied stepped field is

1 for 0( )

τ is the time taken when the field is switched off and E0 is the maximum applied electric

field The sinusoidal field in the reduced form is

Switching Properties of Finite-Sized Ferroelectrics 355

0sin( r)

where e0 is the amplitude and ω is the angular frequency The dimensionless formulations

used in the calculations are obtained by scaling Eqs (12) to (16) according to the way discussed

in our articles All parameters listed in the equations above are scaled to dimensionless

quantities We let ζ=z/ξ0 with 2

0 0/ T C

ξ =κ α and ξ0 corresponds to the characteristic length of the material Normally, ξ0is comparable to the thickness of a domain wall l is the

dimensionless form of thickness L scaled to ξ0 We have temperature T scaled as t T T= / C0,

5 Polarization evolution in ferroelectric films

Surface condition due to δ and size of ferroelectric films affect the switching profiles of

ferroelectric films A ferroelectric film with zero δ means the surface polarization is zero

at both surfaces of the film When a positive electric field E is applied on ferroelectric

films with zero and non-zero δ, various stages of switching profiles are shown in Fig 1

and Fig 2, respectively, for temperature T=0.6T C0 The starting equilibrium polarization

profile is set at negative at time t = 0; and the profile is switched over to positive state by

the applied electric field E until it is completely saturated In a thin ferroelectric film

(Fig 1), switching at the surface and at the centre is almost the same irrespective of either

zero or non-zero δ

Fig 1 Polarization profiles during switching, at various time in term of fraction of the total

time τS"to reversed the profile, at temperature T=0.6T C0, applied field E=0.83E C, for

thickness ζ = 3.3: (a) δ = 0; (b) δ = 2.0 The number at each curve represents time taken to

reach the stage in term of fraction of "

S

τ (Ong et al., 2008b)

For thick films, surface switching takes place relatively faster than the interior of the films

(Fig 2); the reversal of polarization begins near the surfaces first, and then goes on to the

0.0 0.1 0.3 0.6 0.7 0.75 0.8 0.85 0.9 1.0

(b)(a)