Text-chap2

_______________________________________________2 MATHEMATICAL TREATMENT OF FERROELECTRICS

Physicists usually treat a natural phenomenon using a simple mathematical form: one is a linear approximation and another is a non-linear expansion theory. Hooke's law, the stress - strain relation and Ohm's law, the voltage - current relation are two of the most famous linear laws in physics. These linear relations are extended into matrix or tensor relations in linear algebra. On the other hand, the Maclaurin or Taylor series are popularly used to calculate slightly perturbed physical quantities around an equilibrium state including non-linear effects. In this chapter, we will consider the tensor representation of physical properties (linear relation) and phenomenology of ferroelectricity (non-linear relation).

2.1 TENSOR REPRESENTATION OF PHYSICAL PROPERTIES (1)

Tensor Representation1)

Let us first consider the tensor for electric conductivity. The conductivity is defined so as to correlate an applied electric field E and the induced current density J as follows: J=σE.

(2.1)

Since both the electric field and the current density are first rank tensor (that is, vector) quantities, the conductivity should have a second rank tensor representation (that is, with two suffixes); this is described as J1 J2 J3

or

=

s 11 s 21 s 31

Ji = Σ σij Ej

s 12 s 22 s 32

s 13 s 23 s 33

E1 E2 E3

(2.2)

(2.3)

j

A third-rank tensor is exemplified by piezoelectric coefficients, providing a relation between the applied field E and the induced strain x: 23

24

Chapter 2 x = d E.

(2.4)

Since the E and x are first-rank and second-rank tensors, respectively, the d should have a third-rank tensor form represented as xjk = Σ d ijk Ei

(2.5)

i

The d tensor is composed of three layers of the symmetrical matrices.

1st layer (i = 1)

2nd layer (i = 2)

3rd layer (i = 3)

d111

d112

d113

d121

d122

d123

d131

d132

d133

d211

d212

d213

d221

d222

d223

d231

d232

d233

d311

d312

d313

d321

d322

d323

d331

d332

d333

(2.6)

Generally speaking, if two physical properties are represented using tensors of prank and q-rank, the quantity which combines the two properties in a linear relation is also represented by a tensor of (p + q)-rank. (2)

Crystal Symmetry and Tensor Form

A physical property measured along two different directions must be equal if these two directions are crystallographically equivalent. This consideration sometimes reduces the number of independent tensor components representing the above property. Let us again take electric conductivity as an example of a second-rank tensor. If the current density J in an (x,y,z) coordinate system is described in an (x',y',z') system as J', J and J' are related using a unitary matrix# as follows: J' 1 a11 a12 a13 J1

J' 2 J' 3

=

a21 a31

a22 a32

a23 a33

The electric field is transformed in the same way:

J2 J3

(2.7)

Mathematical Treatment of Ferroelectrics

E' 1 E' 2 E' 3

=

25

a11

a12

a13

E1

a21 a31

a22 a32

a23 a33

E2 E3

(2.8)

or E'i = Σ aij Ej

(2.9)

Then, we can calculate the corresponding σ' tensor defined by

J’ 1 J’ 2

E’ 1 =

σ'

J’ 3

E’ 2

(2.10)

E’ 3

σ' 11

σ' 12

σ' 13

σ' 21

σ' 22

σ' 23

σ' 31

σ' 32

σ' 33

σ11

σ12

σ21 σ31

.

,

a11

a12

a13

a21

a22

a23

a31

a32

a33

σ13

a11

a21

a31

σ22

σ23

a12

a22

a32

σ32

σ33

a13

a23

a33

=

# A unitary matrix without an imaginary part has the following relation: a11

a12

a21 a31

a22 a32

a 31 -1 a 11 a23 = a 12 a33 a 13

a21

a31

a22 a23

a32 a 33

For centro-symmetry, the transformation matrix is written as -1 0 0

0 -1 0

0 0 -1

and for rotation about a principal axis, cos q -sin q 0

sin q cos q 0

0 0 1

(2.11)

26

Chapter 2

or σ' ij = Σ aika jl σkl

(2.12)

k,l

When the crystal has a 2-fold axis along the z-axis, the electric conductivity should have the same tensor form in terms of the transformation:

-1 0 0

0 -1 0

0 0 1

.

=

From the condition σ' 11 σ' 21

σ' 12 σ' 22

σ' 13 σ' 23

σ' 31

σ' 32

σ' 33

σ11

σ12

σ21 σ31

σ22 σ32

σ11 σ21 σ31

σ12 σ22 σ32

σ13 σ23 σ33

.

=

-1 0

0 -1

0 0

0

0

1

σ13

-1

0

0

σ23 σ33

0 0

-1 0

0 1

(2.13)

the following equivalencies can be derived: σ31 = σ13 = σ32 = σ23 = 0 σ11 , σ22 , σ33 ? 0 σ12 = σ21

(2.14)

It is very important to note that most physical constants have a symmetric tensor form. [The proof involves thermodynamical considerations that are beyond the scope of this textbook. Refer to Ref. 1).] For a third -rank tensor such as the piezoelectric tensor, the transformation due to a change in coordinate system is represented by d'ijk = Σ aila jmakn d lmn

(2.15)

Mathematical Treatment of Ferroelectrics

27

When the crystal has a 4-fold axis along z-axis, for example, the transformation matrix is given by

0 -1 0

1 0 0

0 0 1

Considering the tensor symmetry with m and n such that d123 = d132 and d213 = d 231 (each matrix of the ith layer of the d tensor is symmetrical), we can obtain: d 111 = d 222 = d 112 = d 121 = d 211 = d 221 = d 212 = d 122 = d 331 = d 313 = d 133 = d 332 = d 323 = d 233 = d 312 = d 321 = 0 d 333 ? 0 d 311 = d 322 d 113 = d 131 = d 223 = d 232 d 123 = d 132 = -d 213 = -d 231

(2.16)

Then we get the d tensor as follows:

1st layer 0

0

d131

0 d131

0 d123

d123 0

0 0

-d123 d131

-d123

d131

0

d311

0

0

0

d311

0

0

0

d333

2nd layer 0 0

(2.17)

3rd layer

(3)

Reduction of the Tensor (Matrix Notation)

A general third-rank tensor has 33 = 27 independent components. Since dijk is symmetrical in j and k some of the coefficients can be eliminated, leaving 18 independent d ijk coefficients; this facilitates the use of matrix notation.

28

Chapter 2

So far all the equations have been developed in full tensor notation. But when calculating actual properties, it is advantageous to reduce the number of suffixes as much as possible. This is done by defining new symbols, for instance, d21 = d211 and d14 = 2d 123 : The second and third suffixes in the full tensor notation are replaced by a single suffix 1 to 6 in matrix notation, as follows:

Tensor notation 11 22 33 23,32 31,13 12,21 ___________________________________________________________ Matrix notation

1

2

3

4

5

6

In terms of these new symbols the array (2.6) is rewritten as: d11 (1/2)d 16 (1/2)d 15 (1/2)d 16 d 12 (1/2)d 14 (1/2)d 15 (1/2)d 14 d 13 d21 (1/2)d 26 (1/2)d 25 (1/2)d 26 d 22 (1/2)d 24 (1/2)d 25 (1/2)d 24 d 23

(2.18)

d31 (1/2)d 36 (1/2)d 35 (1/2)d 36 d 32 (1/2)d 34 (1/2)d 35 (1/2)d 34 d 33

The last two suffixes in the tensor notation correspond to those of the strain components; therefore, for consistency, we make the following change in the notation for the strain components.

x11

x12

x31

x12 x31

x22 x23

x23 x33

(1/2)x 6

(1/2)x 5

---> (1/2)x 6 x2 (1/2)x 5 (1/2)x 4

x 1

(1/2)x 4 x3

(2.19)

The reason for the (1/2)s in the substitution (2.19) is due to the cancellation with (1/2)s in (2.18). Then, we have xj = Σ d ij Ei (i = 1, 2, 3; j = 1, 2, ..., 6) or

(2.20)

Mathematical Treatment of Ferroelectrics

x1

29

d11

d21

d31

E1

d12

d22

d32

E2

d13

d23

d33

E3

x4

d14

d24

d34

x5 x6

d15 d16

d25 d26

d35 d36

x2 x3

=

(2.21)

Concerning the stress components, the (1/2)s are unnecessary.

X 11

X 12

X 31

X 12 X 31

X 22 X 23

X 23 X 33

--->

X1

X6

X5

X6 X5

X2 X4

X4 X3

(2.22)

The marix notation has the advantage of compactness over the tensor notation, and it makes it easy to display the coefficients on a plane diagram. However, it must be remembered that in spite of their form, the d ij's do not transform like the components of a second-rank tensor. An example of a piezoelectric matrix for the point group 4 is written as

0

0

0

d14

d15

0

0 d31

0 d31

0 d33

d15 0

- d 14 0

0 0

(2.23)

In the solid state theoretical treatment of the phenomenon of piezoelectricity or electrostriction, the strain xkl is expressed in terms of the electric field Ei or electric polarization P i as follows: xkl = Σ d ikl Ei + Σ M ijkl Ei Ej = Σ g iklP i + Σ QijklP iPj

(2.24)

Here, dikl and g ikl are called the piezoelectric coefficients, and Mijkl and Qijkl the electrostrictive coefficients. Since the E and x are first-rank and second-rank tensors, respectively, d and M should be third-rank and fourth-rank tensors, respectively. Using a similar reduction of the notation for the electrostrictive coefficients Mijkl, we get the following equation corresponding to Eq. (2.24):

30

Chapter 2 x1 x2 x3

=

x4 x5 x6

+

d11 d12 d13

d21 d22 d23

d31 d32 d33

d14 d15 d16

d24 d25 d26

d34 d35 d36

E1 E2 E3

M 11

M 21

M 31

M 41

M 51

M 61

E 12

M 12

M 22

M 32

M 42

M 52

M 62

E 22

M 13

M 23

M 33

M 43

M 53

M 63

E 32

M 14 M 15 M 16

M 24 M 25 M 26

M 34 M 35 M 36

M 44 M 45 M 46

M 54 M 55 M 56

M 64 M 65 M 66

E 2 E3 E 3 E1 E 1 E2

(2.25)

Tables 2.1 and 2.2 summarize the matrices d and M for all crystallographic point groups.1)

Example Problem 2.1_________________________________________________ Suppose that a shear stress is applied to a square crystal and the crystal is deformed as illustrated in Fig. 2.1. Calculate the induced strain x5 ( = 2x31). F 1 deg 3

F 1

Fig. 2.1 Shear stress and strain configuration.

Solution Since x5 = 2x31 = tan θ = θ and 1° = π /180 rad., x5 = 0.017. ___________________________________________________________________

Mathematical Treatment of Ferroelectrics Table 2.1 Piezoelectric coefficient matrix.*

31

32 Table 2.1 (Continued) Piezoelectric coefficient matrix.*

Chapter 2

Mathematical Treatment of Ferroelectrics Table 2.2 Electrostrictive coefficient matrix.*

33

34 Table 2.2 (Continued) Electrostrictive coefficient matrix.*

Chapter 2

Mathematical Treatment of Ferroelectrics

35

Example Problem 2.2_________________________________________________ For a cube-shaped specimen, tensile stress X and compressive stress -X were applied simultaneously along the (1 0 1) and (1 0 1) axes, respectively (Fig. 2.2). When we take the prime -coordinates as illustrated in Fig. 2.2, the stress tensor is represented as X 0 0 0 0 0 0 0 -X . Using the transformation matrix A

cosθ 0 -sinθ

0 1 0

sinθ 0 cosθ ,

calculate A .X .A -1 , and verify that the above-stress is equivalent to a pure shear stress in the original (non-prime) coordinates.

X 3 3’ 1’

-X 2, 2’ 1

Fig. 2.2 Application of a pair of stresses X and - X to a cube of material Solution Using θ = - 45o , we can obtain the transformed stress representation:

A . X. A-1 =

0 0 X

0 0 0

X 0 0

.

(P2.2.1)

The off-diagonal components X13 and X31 have the same magnitude X, and represent a pure shear stress. Note that a shear stress is equivalent to a combination

36

Chapter 2

of extensional and contractional stresses. Only an extensional stress applied along a diagonal direction 1' may exhibit an apparently similar diagonal distortion of the crystal. However, precisely speaking, without the contraction along the 3' direction, this is not exactly equivalent to the pure shear deformation. ___________________________________________________________________

Example Problem 2.3_________________________________________________ Barium titanate shows a tetragonal crystal symmetry (point group 4mm) at room temperature. Therefore, its piezoelectric constant matrix is : 0

0

0

0

d15

0

0 d31

0 d31

0 d33

d15 0

0 0

0 0

(a) Calculate the induced strain under an electric field applied along the crystal c axis. (b) Calculate the induced strain under an electric field applied along the crystal a axis. Solution x1 x2

0 0

0 0

d31 d31

E1 E2

0

0

d33

E3

x4 x5

0 d15

d15 0

0 0

x6

0

0

0

x3

=

(P2.3.1)

is transformed into x1 = x2 = d 31 E3 x3 = d 33 E3 x4 = d 15 E2 x5 = d 15 E1 , E6 = 0

(P2.3.2)

(a) When E3 is applied, elongation in the c direction (x3 = d33 E3, d33 > 0) and contaction in the a and b directions (x1 = x2 = d 31E3 , d 31 < 0) are induced. (b) When E1 is applied, shear strain x5 (= 2x31) = d15 E1 is induced. Figure 2.3(a) illustrates a case of d 15 > 0 and x5 >0. ___________________________________________________________________

Mathematical Treatment of Ferroelectrics

37

E

E

Polarization

3 2

(a)

(b)

1

Fig. 2.3 (a) Orientation of the shear strain in the point group 4mm, and (b) [111] axis strain in the point group m3m.

Example Problem 2.4_________________________________________________ Lead magnesium niobate exhibits a cubic crystal symmetry (point group m3m) at room temperature and does not show piezoelectricity. However, large electrostriction is induced under an applied electric field. The relation between the strain and the electric field is given by

x1

M 11

M12

M12

0

0

0

M 12

M11

M12

0

0

0

M 12

M12

M11

0

0

0

x4

0

0

0

M44

x5 x6

0 0

0 0

0 0

0 0

x2 x3

=

0 M 44 0 M

E 12 E 22 E 32 E 2 E3

0 0 44

(P2.4.1)

E 3 E1 E 1 E2

in a matrix representation. Calculate the induced strain under an electric field applied along the [111] direction.

Solution The electric field along the [111] direction , E[111] , is represented as (E[111]/√3, E[111]/√3, E[111]/√3). Substituting E1 = E2 = E3 = E[111]/√3 into Eq. (P2.2.1), we obtain

38

Chapter 2 x1 = x2 = x3 = (M 11 + 2M 12 )E[111]2/3 (= x11 = x22 = x33), x4 = x5 = x6 = M 44 E[111]2/3 (= 2x23 = 2x31 = 2x12). (P2.4.2)

The distortion is illustrated in Fig. 2.3(b). The strain x induced along an arbitrary direction is given by x = Σ xij l i lj

(P2.4.3)

where li is a direction cosine with respect to the i axis. Therefore, the strain induced along the [111] direction, x[111]// , is given by x[111]// = Σ xij (1/√3)(1/√3) = [x1 + x2 + x3 + 2(x4/2 + x5/2 + x6/2)]/3 = (M 11 + 2M 12 + M 44) E[111]2/3. (P2.4.4) On the other hand, the strain induced perpendicular to the [111] direction, x[111] , is calculated in a similar fashion as x[111] = (M 11 + 2M 12 - M 44/2)E[111]2/3. (P2.4.5) Figure 2.11(b) shows the distortion schematically. It is important to note that the volumetric strain (∆V/V) given by x[111]// + 2 x[111] = (M 11 + 2 M 12 ) E[111]2

(P2.4.6)

is independent of the applied field direction. ___________________________________________________________________

2.2 PHENOMENOLOGY OF FERROELECTRICITY (1)

Landau Theory of the Phase Transition

A thermodynamical theory explaining the behavior of a ferroelectric crystal can be obtained by considering the form of the expansion of the free energy as a function of the polarization P. We assume that the Landau free energy F in one dimension is represented formally as: F(P,T) = (1/2)α P2 + (1/4)β P4 + (1/6)γ P6 + ...

(2.26)

The coefficients α, β, γ depend, in general, on the temperature. Note that the series does not contain terms in odd powers of P because the free energy of the crystal will not change with polarization reversal (P --> -P). The phenomenological formulation should be applied for the whole temperature range over which the material is in the paraelectric and ferroelectric states.

Mathematical Treatment of Ferroelectrics

39

The equilibrium polarization in an electric field E satisfies the condition: (∂F/ ∂P) = E = α P + β P3 + γ P 5

(2.27)

To obtain the ferroelectric state, the coefficient of the P2 term must be negative for the polarized state to be stable, while in the paraelectric state it must be positive passing through zero at some temperature T0 (Curie-Weiss temperature): α = (T - T0 )/ε0 C

(2.28)

where C is taken as a positive constant called the Curie-Weiss constant and T0 is equal to or lower than the actual transition temperature TC (Curie temperature). The temperature dependence of α is related on a microscopic level to the temperature dependence of the ionic polarizability coupled with thermal expansion and other effects of anharmonic lattice interactions. Refer to the discussion in Section 1.2. Second-order transition When β is positive, γ is often neglected because nothing special is added by this term. The polarization for zero applied field is obtained from Eq. (2.27) as [(T - T0 )/ε0 C] PS + β PS3 = 0

(2.29)

so that either PS = 0 or P S2 = (T0 - T)/β ε0 C. For T > T0, the unique solution PS = 0 is obtained. For T < T0 the minimum of the Landau free energy is obtained at: ______________ PS = √(T 0 - T)/(β ε0 C). (2.30) The phase transition occurs at TC = T0 and the polarization goes continuously to zero at this temperature; this is called a second-order transition. The relative permittivity ε is calculated as:

Then,

1/ε = ε0/(∂P/∂E) = ε0 (α + 3β P2)

(2.31)

ε = C/(T - T0) C/[2(T - T0 )]

(2.32)

(T > T0) (T < T0)

Figure 2.4(a) shows the variations of PS and ε with temperature. It is notable that the permittivity becomes infinite at the transition temperature. Triglycine sulphate is an example of a ferroelectric exhibiting the second-order transition.

40

Chapter 2 Permittivity ε

Permittivity ε Spontaneous Ps

Inverse permittivity 1/ ε

Spontaneous Ps

Tc Temperature (Curie Temp.)

Inverse permittivity 1/ ε

Tc Temperature (Curie Temp.)

(a)

(b)

Fig. 2.4 Phase transitions in a ferroelectric: (a) second-order and (b) first-order. First-order transition When β is negative in Eq. (2.26) and γ is taken positive, the transition becomes first order. The equilibrium condition for E = 0 in Eq. (2.33) leads to either PS = 0 or Eq. (2.34). [(Τ − Τ0 )/ε0 C] PS + β PS3 + γ PS5 = 0 _________________ PS2 = [- β + √ β2 - 4γ(T - T0 )/ε0 C] /2γ

(2.33) (2.34)

The transition temperature TC is obtained from the condition that the free energies of the paraelectric and ferroelectric phases are equal: i.e., F = 0, or: [(Τ − Τ0 )/ε0 C] + (1/2) β P S2 + (1/3) γ PS4 = 0

(2.35)

Therefore: TC = T 0 + (3/16)(β2 ε0 C/ γ)

(2.36)

Note that the Curie temperature TC is slightly higher than the Curie-Weiss temperature T0, and that a discrete jump of PS appears at TC. Also, the permittivity exhibits a finite maximum at TC for a first-order transition [Fig. 2.4(b)]. Barium titanate is an example of a ferroelectric that undergoes a first-order phase transition. The free energy curves are plotted for the second- and first-order phase transitions at various temperatures in Fig. 2.5. In the case of β > 0, the phase transition is not associated with a latent heat, but with a jump of the specific heat; thus, this is called a second-order transition. On the other hand, in the case of β < 0, the transition exhibits a latent heat, and is called a first-order transition, where the permittivity shows a maximum and a discontinuity of the spontaneous polarization appears at TC.

Mathematical Treatment of Ferroelectrics

Free Energy T>TC

41

T>T1

Free Energy

TC
T=TC

T=TC

T
T0
T
P (b)

Fig. 2.5 Free energy curves plotted for the second- (a) and first-order (b) phase transitions at various temperatures.

Example Problem 2.5_________________________________________________ Verify the difference between the Curie and Curie-Weiss temperatures as expressed by: TC = T 0 + (3/16)(β2 ε0 C/ γ) for a first-order phase transition, where the Landau free energy is expanded as F(P,T) = (1/2)α P 2 + (1/4)β P4 + (1/6)γ P6, α = (T - T0 )/ε0 C. Hint The potential minima are obtained from (∂F/ ∂P) = E = α P + β P3 + γ P5 = 0 .

(P2.5.1)

There are generally three minima including P = 0 (F = 0). At the Curie temperature, the free energy at the non-zero polarization must be equal to zero (F = 0). Thus we obtain another condition: F = (1/2)α P2 + (1/4)β P4 + (1/6)γ P6 = 0 .

(P2.5.2)

42

Chapter 2

Solution Equations (P2.5.1) and (P2.5.2) are reduced for non-zero polarizations to α + β P2 + γ P4 = 0 , α + (1/2) β P2 + (1/3) γ P4 = 0 .

(P2.5.3) (P2.5.4)

Note that Eq. (P2.5.3) is valid for all temperatures below TC, but Eq. (P2.5.4) is only valid at T = TC. Eliminating the P terms from these two equations, we obtain α + β (- 4 α/ β) + γ (- 4 α/ β)2 = 0 .

(P2.5.5)

Then, taking account of α = (T - T0 )/ε0 C, the Curie temperature is calculated as TC = T 0 + (3/16)(β2 ε0 C/ γ) . (P2.5.6) ___________________________________________________________________ (2)

Phenomenology of Electrostriction

In a ferroelectric whose prototype phase (high temperature paraelectric phase) is centrosymmetric and non-piezoelectric, the piezoelectric coupling term PX is omitted and only the electrostrictive coupling term P2X is introduced. The theories for electrostriction in ferroelectrics were formulated in the 1950s by Devonshire 2) and Kay.3) Let us assume that the elastic Gibbs energy should be expanded in a one-dimensional form: G1 (P,X,T) = (1/2)α P2 + (1/4)β P4 + (1/6)γ P6 - (1/2)s X2 - Q P2 X, (α = (T - T0)/ ε0 C)

(2.37)

where P, X, T are the polarization, stress and temperature, respectively, and s and Q are called the elastic compliance and the electrostrictive coefficient, respectively. This leads to Eqs. (2.38) and (2.39). E = (∂G1/ ∂P) = α P + β P3 + γ P5 - 2Q P X x = - (∂G1/∂X) = sX + QP2

(2.38) (2.39)

Case I: X = 0 When the external stress is zero, the following equations are derived: E = α P + β P3 + γ P5 x = QP2 1/ε0 ε = α + 3 β P2 + 5 γ P4

(2.40) (2.41) (2.42)

Mathematical Treatment of Ferroelectrics

43

If the external electric field is equal to zero (E = 0), two different states are derived; _______ 2 P = 0 and P = (√ β2- 4αγ - β)/2γ. (i) Paraelectric phase: P S = 0 or P = ε0 ε E (under small E) Permittivity:

ε = C/(T - T0)

(Curie -Weiss law)

Electrostriction: x = Q ε02 ε2 E2

(2.43) (2.44)

The previously mentioned electrostrictive coefficient M in Eq. (2.24) is related to the electrostrictive Q coefficient through M = Q ε02 ε2

(2.45)

_______ (ii) Ferroelectric phase: P S2= (√ β2- 4αγ - β)/2γ or P = PS + ε0 ε E (under small E) x = Q(P S + ε0 ε E)2 = QP S2 + 2 ε0 ε QP SE + Q ε02 ε2 E2

(2.46)

where we define the spontaneous strain xS and the piezoelectric constant d as: Spontaneous strain: Piezoelectric constant:

xS = QPS2 d = 2 ε0 ε QP S

(2.47) (2.48)

We see by Eq. (2.48) that piezoelectricity is equivalent to the electrostrictive phenomenon biased by the spontaneous polarization. The temperature dependences of the spontaneous strain and the piezoelectric constant are plotted in Fig. 2.6. Case II: X ? 0 When a hydrostatic pressure p (X = - p) is applied, the inverse permittivity is changed in proportion to p: 1/ε0 ε = α + 3 β P2 + 5 γ P4 + 2Qp α + 2Qp = (T - T0 + 2Qε0 Cp)/(ε0 C)

(Ferroelectric state) (Paraelectric state) (2.49)

Therefore, the pressure dependence of the Curie -Weiss temperature T0 or the transition temperature TC is derived as follows: (∂T0/∂p) = (∂TC/∂p) = - 2Qε0C

(2.50)

In general, the ferroelectric Curie temperature is decreased with increasing hydrostatic pressure (i.e. Qh > 0).

44

Chapter 2 Piezoelectric Spontaneous constant d strain x s

Tc (Curie Temp.) Temperature

Fig. 2.6 Temperature dependence of the spontaneous strain and the piezoelectric constant.

Example Problem 2.6_________________________________________________ Barium titanate has d33 = 320 x 10-12 C/N, εc (= ε3 ) = 800 and Q33 = 0.11 m4C-2 at room temperature. Estimate the spontaneous polarization PS. Solution Let us use the relation: d 33 = 2 ε0 ε3 Q33P S .

(P2.6.1)

PS = d 33/2 ε0 ε3 Q33 = 320x10-12[C/N]/{2 x 8.854x10-12 [F/m] x 800 x 0.11 [m4 C-2]} = 0.21 [C/m2] (P2.6.2) ___________________________________________________________________

Example Problem 2.7_________________________________________________ In the case of a second-order phase transition, the elastic Gibbs energy is expanded in a one-dimensional form as follows: G1 (P,X,T) = (1/2)α P2 + (1/4)β P4 - (1/2)s X2 - Q P2 X,

(P2.7.1)

where only the coefficient α is dependent on temperature, α = (T - T0)/ ε0 C. Obtain the dielectric constant, spontaneous polarization, spontaneous strain and piezoelectric constant as a function of temperature.

Mathematical Treatment of Ferroelectrics

45

Solution E = (∂G1/ ∂P) = α P + β P3 - 2Q P X

(P2.7.2)

x = - (∂G1/∂X) = sX + QP2

(P2.7.3)

When an external stress is zero, we can deduce the three characteristic equations: E = α P + β P3 x = QP2 1/ε0 ε = (∂E/∂P) = α + 3 β P2

(P2.7.4) (P2.7.5) (P2.7.6)

By setting E = 0 initially, we obtain the following two stable states: PS2 = 0 or -α/β. (i) Paraelectric phase -- T > T0 -- PS = 0 1/ε0 ε = α, then ε = C/(T - T 0)

(P2.7.7)

___________ (ii) Ferroelectric phase -- T < T0 -- P S = √(T0 - T)/ε0 Cβ

(P2.7.8)

1/ε0 ε = α + 3 β P2 = - 2α, then ε = C/2(T0 - T)

(P2.7.9)

xS = QPS2 = Q (T 0 - T)/ ε0 Cβ

(P2.7.10)

From Eqs. (P2.7.8) and (P2.7.9), the piezoelectric constant is obtained as d = 2 ε0 ε Q P S _____ = Q√ε0 C/β (T 0 - T)-1/2 (P2.7.11) ___________________________________________________________________ (3)

Converse Effects of Electrostriction

So far we have discussed the electric field induced strains, i.e. piezoelectric strain (converse piezoelectric effect, x = d E) and electrostriction (electrostrictive effect, x = M E2 ). Let us consider here the converse effect, that is, the material's response to an external stress, which is applicable to sensors. The direct piezoelectric effect is the increase of the spontaneous polarization by an external stress, and expressed as ∆P = d X.

(2.51)

On the contrary, since an electrostrictive material does not have a spontaneous polarization, it does not generate any charge under stress, but does exhibit a change in permittivity [see Eq. (2.49)]:

46

Chapter 2

∆(1/ ε0 ε) = 2QX

(2.52)

This is the converse electrostrictive effect. The converse electrostrictive effect, the stress dependence of the permittivity, is used in stress sensors.4) A bimorph structure which subtracts the static capacitances of two dielectric ceramic plates can provide superior stress sensitivity and temperature stability. The capacitance changes of the top and bottom plates have opposite signs for uniaxial stress and same sign for temperature change. The response speed is limited by the capacitance measuring frequency to about 1 kHz. Unlike piezoelectric sensors, electrostrictive sensors are effective in the low frequency range, especially DC. The sensor applications of piezoelectrics will be discussed in Section 7.2 of Chapter 7.

(4)

Temperature Dependence of Electrostriction

Several expressions for the electrostrictive coefficient Q have been given so far. From the data obtained by independent experimental methods such as 1) electric field-induced strain in the paraelectric phase, 2) spontaneous polarization and spontaneous strain (x-ray diffraction) in the ferroelectric phase, 3) d constants from the field-induced strain in the ferroelectric phase or from piezoelectric resonance, 4) pressure dependence of permittivity in the paraelectric phase,

Fig. 2.7 Temperature dependence of the electrostrictive constants Q33 and Q13 in Pb(Mg1/3Nb 2/3)O3.

Mathematical Treatment of Ferroelectrics

47

nearly equal values of Q were obtained. Figure 2.7 shows the temperature dependence of the electrostrictive coefficients Q33 and Q31 for the complex perovskite Pb(Mg1/3Nb 2/3)O3 , whose Curie temperature is near 0°C.5) It is seen that there is no significant anomaly in the electrostrictive coefficient Q through the temperature range in which the paraelectric to ferroelectric phase transition occurs and piezoelectricity appears. Q is almost temperature independent.

2.3 PHENOMENOLOGY OF ANTIFERROELECTRICITY (1)

Antiferroelectrics

The previous sections dealt with the case in which the directions of the spontaneous dipoles are parallel to each other in a crystal (polar crystal). There are cases in which antiparallel orientation lowers the dipole-dipole interaction energy. Such crystals are called anti-polar crystals. Figure 2.8 shows the orientation of the spontaneous electric dipoles in an anti-polar state in comparison with a non-polar and a polar state. In an anti-polar crystal, where the free energy of an antipolar state does not differ appreciatively from that of a polar state, the application of an external electric field or mechanical stress may cause a transition of the dipole orientation to a parallel state. Such crystals are called antiferroelectrics. Figure 2.9 shows the relationship between E (applied electric field) and P (induced polarization) in paraelectric, ferroelectric and antiferroelectric phases. In a paraelectric phase the P-E relation is linear; in a ferroelectric phase there appears a hysteresis caused by the transition of the spontaneous polarization between the positive and negative directions; in an antiferroelectric phase, at low electric field, the induced polarization is proportional to E, and when E exceeds a certain value Ecrit , the crystal becomes ferroelectric (electric field induced phase transition), and the polarization shows hysteresis with respect to E. After removal of the electric field, the crystal returns to its anti-polar state, and hence, no spontaneous polarization can be observed as a whole. This is called a double hysteresis curve.

stripe type (a) Nonpolar

(b) Polar

checker board type

(c) Anti-polar

Fig. 2.8 Schematic arrangement of the spontaneous dipoles in non-polar, polar and antipolar materials.

48

Chapter 2

Polarization

Polarization

Electric field Ec

Electric field

(a) Paraelectric

Polarization

(b) Ferroelectric

Electric field Et

(c) Antiferroelectric

Fig. 2.9 Polarization vs. electric field hysteresis curves in paraelectric, ferroelectric and antiferroelectric materials. (2)

Phenomenology of Antiferroelectrics

We will discuss here the introduction of electrostrictive coupling in Kittel's free energy expression for antiferroelectrics.6,7) The simplest model for antiferroelectrics is the "one-dimensional two-sublattice model." It treats the coordinates as one-dimensional, and a superlattice (twice the unit lattice) is formed from two neighboring sublattices each having a sublattice polarization Pa and Pb . The state Pa = Pb represents the ferroelectric phase, while Pa = - Pb , the antiferroelectric phase. For the electrostrictive effect, ignoring the coupling between the two sublattices, the strains from the two sublattices are QPa 2 and QPb 2 , respectively (assuming equal electrostrictive constants Q for both sublattices). The total strain of the crystal becomes x = Q(Pa 2 + Pb 2)/2.

(2.53)

However, since antiferroelectricity originates from the coupling between the sublattices, it is appropriate to consider the sublattice coupling also for the

Mathematical Treatment of Ferroelectrics

49

electrostrictive effect. The coupling term for the electrostriction Ω is introduced in the following form: G1 = (1/4) α (Pa 2+Pb 2 ) + (1/8) β (Pa4 +Pb 4) + (1/12) γ (Pa6 +Pb 6) +(1/2) η PaPb - (1/2) χT p 2 +(1/2) Qh (Pa 2 +Pb 2 +2Ω Pa Pb )p (2.54) in which hydrostatic pressure p is employed, and χT is the isothermal compressibility, Qh and Ω are the electrostrictive constants. Introducing the transformations PF = (Pa + Pb )/2 and PA = (Pa - Pb )/2 leads to the following expression: G1 = (1/2) α (P F2 + PA 2 ) + (1/4) β (PF 4 + PA 4 + 6P F2 PA 2) + (1/6) γ (P F 6+PA 6+15P F4 PA 2+15P F2 PA 4)+(1/2) η (P F 2 - PA 2 ) - (1/2) χ T p 2 + Qh [PF 2 +PA 2 + Ω (PF 2 - PA 2)]p (2.55) The dielectric and elastic equations of state follow as ∂G1/∂PF = E = PF [α + η + 2 Q (1 + Ω)p + β PF 2 + 3β PA 2 + γ PF 4 + 10γ P F 2PA 2 + 5γ PA 4] (2.56) ∂G1/∂PA = 0 = PA [α − η + 2 Q (1 - Ω)p + β PA 2 + 3β PF 2 + γ PA 4 + 10γ P F 2PA 2 + 5γ P F4 ] (2.57) 2 2 ∂G1/∂p = ∆V/V = - χ T p +Qh (1+Ω)PF + Qh (1 - Ω)PA (2.58) Hence, the induced volume change in the paraelectric phase can be related to the induced ferroelectric polarization by the following formula: (∆V/V) ind = Qh (1 + Ω)P F,ind2

(2.59)

Below the phase transition temperature (this temperature for antiferroelectrics is called Neel temperature) the spontaneous volume strain and the spontaneous antiferroelectric polarization are related as (∆V/V)S = Qh (1 - Ω)PA,S 2

(2.60)

Even if the perovskite cystal shows Qh > 0, the spontaneous volume strain can be positive or negative depending on the value of Ω (Ω < 1 or Ω > 1), that is, if the inter-sublattice coupling is stronger than the intra-sublattice coupling, a volume contraction is observed at the Neel point. This is quite different from ferroelectrics, which always show a volume expansion at the Curie point. Figure 2.10 illustrates the spontaneous strains in a crystal schemetically for Ω > 0. When Pa and Pb are in

50

Chapter 2

the parallel configuration (ferroelectric phase), the Ω-term acts to increase the strain xS, when they are in the anti-parallel configuration (antiferroelctric phase), the Ωterm acts to decrease the strain. This phenomenological theory explains well the experimental results for the antiferroelectric perovskite crystal PbZrO3 and others.8) Figure 2.11 shows the strain in the antiferroelectric ceramic Pb 0.99 Nb 0.02[(Zr0.6Sn 0.4)0.94 Ti0.06 ]0.98O3 as a function of an applied electric field.9) The large change in the strain associated with the field -induced transition from the antiferroelectric to ferroelectric phase can be estimated to be (∆V/V) = Qh (1 + Ω)PF,S2 - Qh (1 - Ω)PA,S2 = 2 Qh Ω PF,S2

(2.61)

Here, we assume that the magnitudes of Pa and Pb do not change drastically through the phase transition.

(a) Ferroelectric Arrangement x = Q (1+Ω) (Pa + Pb )2/4

x = Q Pa2

x = Q P b2

(b) Antiferroelectric Arrangement x = Q (1-Ω) (Pa - Pb )2/4

Fig. 2.10 Intuitive explanation of the sublattice coupling with respect to electrostriction (for Ω > 0).

Mathematical Treatment of Ferroelectrics

51

Fig. 2.11 Field induced strain in a Pb(Zr,Sn)O3 based antiferroelectric.

CHAPTER ESSENTIALS_________________________________ 1.

Tensor representation: when two physical properties are represented using tensors of p-rank and q-rank, the quantity which combines the two properties in a linear relation is also represented by a tensor of (p + q)-rank.

2.

A physical property measured along two different directions must be equal if these two directions are crystallographically equivalent. This consideration reduces the number of the independent tensor components representing the above property. d ijk' = Σ a il a jm a kl d lmn

(a il, a jm, akl : transformation matrices)

l,m,n

d ijk' = d ijk 3.

Shear strain: x5 = 2 x31 = 2 φ, taken as positive for smaller angle.

4.

Phenomenology: (1/4) β P 4 > 0 --> second-order phase transition (1/4) β P 4 < 0 --> first-order phase transition

5.

Electrostriction equation: x = Q PS2 + 2 Q ε0 ε P S E spontaneous strain piezostriction

+ Q ε0 2 ε2 E2 electrostriction

52

Chapter 2 The electrostrictive Q constant is insensitive to temperature changes.

6.

Piezoelectric equations: x = sE X + d E P = d X + ε0 εX E

7.

In antiferroelectrics, consideration of sublattice couplings is essential to understanding the stable sublattice polarization configuration and the significant jump in strain associated with an antiferroelectric - ferroelectric phase transition induced by an external electric field. ___________________________________________________________________

CHAPTER PROBLEMS 2.1

The room temperature form of quartz belongs to class 32. (1) Find that the piezoelectric matrix (d ij) is given by

d11

-d11

0

d14

0

0

0 0

0 0

0 0

0 0

-d14 0

-2d11 0

Notice that the piezoelectric tensor must be invariant for a 120° rotation around the 3-axis and for a 180°rotation around the 1-axis. The transformation matrices are

-1/2 -√3/2 0

√3/2 -1/2 0

0 0 1

1 0 0

0 -1 0

0 0 -1

and

,

respectively. (2) The measured values of the d ij for right-handed quartz are

-2.3 2.3 0 0 0 0

0 0 0

-0.67 0 0

0 0 0.67 4.6 0 0

x 10

-12 [C/N]

Mathematical Treatment of Ferroelectrics

53

(a) When a compressive stress of 1 kgf/cm2 is applied along the 1-axis of a quartz crystal, find the polarization generated. (kgf = kilogram force = 9.8 N) (b) When an electric field of 100 V/cm is applied along the 1-axis, find the strains generated. Hint Let us initially use the following equation for a 180o rotation (with respect to the a-axis): d ijk' = Σ a il a jm a kl d lmn (a il, ajm , akl : transformation matrices) l,m,n

Considering that a11 = 1, a22 = -1, a33 = -1, we obtain the following relationship: d111 '=d111 d112 '=- d112 d113 '=- d113 d121 '=- d121 d122 '=d122 d123 '=d123

d131 '=- d131

d132 '=d132

d133 '=d133

d211 '=- d211

d212 '=d212

d213 '=d213

d221 '=d221

d222 '=- d222

d223 '=- d223

d231 '=d231

d232 '=- d232

d233 '=- d233

d311 '=- d311

d312 '=d312

d313 '=d313

d321 '=d321

d322 '=- d322

d323 '=- d323

d331 '=d331

d332 '=- d332

d333 '=- d333

or

d111 0 0

0 d122 d123

0 d123 d133

0 d212 d231

d212 0 0

d231 0 0

0 d312 d331

d312 0 0

d331 0 0

54

Chapter 2 Next, a 120o rotation is considered such that a11 = -1/2, a12 = √3/2, a21 = -√3/2, a 22 = -1/2, a 33 = 1: d 111' = Σ a 1l a1m a 1n d lmn l,m,n

= Σ a11 a 1m a1n d 1mn + Σ a 12 a1m a 1n d 2mn m,n

m,n

= (-1/2) 3 d 111 + (-1/2)(v3/2)2 d 122 + 2(-1/2)(v3/2) 2 d 212 = - (1/8) d 111 - (3/8) d 122 - (3/4) d 212 = d 111 d 122' =

Σ a11 a 2m a2n d 1mn + Σ a 12 a2m a 2n d 2mn m,n

m,n

= (-1/2)(-v3/2) 2 d 111 + (-1/2) 3 d 122 + 2(v3/2)(-v3/2)(-1/2) d 212 = - (3/8) d 111 - (1/8) d 122 + (3/4) d 212 = d 122 Continuing the calculations for d123, d212, d231, d312, d331 , we can obtain all the necessary equations for deriving the final matrix form. 2.2

In the case of a first-order phase transition, the Landau free energy is expanded as in Example Problem 2.5. Calculate the inverse permittivity in the vicinity of the Curie temperature, and verify that the slope (?(1/ε)/∂Τ) just below TC is 8 times larger than the slope just above TC. Hint In a first-order phase transition, PS satisfies the following equation in the temperature range of T < TC : α + β Ps 2 + γ Ps 4 = 0 . The permittivity is given by 1/ε0 ε = α + 3 β Ps 2 + 5 γ Ps 4 . Thus, 1/ε0 ε = α + 3 β Ps 2 + 5 (- α - 3 β Ps 2) = - 4 α - 2 β Ps 2 _______ Since α = (T - T 0 )/ε0 C, P S2 = (√ β2 - 4αγ - β)/2γ and

Mathematical Treatment of Ferroelectrics

55

(T - T0)/ ε0 C = (3/16)(β 2/ γ) - (TC - T)/ ε0 C, we can obtain 1/ε0 ε = - 4 α - 2 β Ps 2 = - 4 [(3/16)(β 2/ γ) - (T C - T)/ ε0 C] ______________________________ 2 + (β / γ) - (β/γ)(√ β 2- 4γ [(3/16)(β2/ γ) - (T C - T)/ε0 C] Considering (TC - T) << 1, obtain the approximation of this equation.

REFERENCES 1) 2) 3) 4)

5) 6) 7) 8) 9)

J. F. Nye: Physical Properties of Crystals, Oxford University Press, London, p.123, p.140 (1972) A. F. Devonshire: Adv. Phys. 3, 85 (1954) H. F. Kay: Rep. Prog. Phys. 43, 230 (1955) K. Uchino, S. Nomura, L. E. Cross, S. J. Jang and R. E. Newnham: Jpn. J. Appl. Phys. 20, L367 (1981); K. Uchino, S. Nomura, L. E. Cross, S. J. Jang and R. E. Newham: Jpn. J. Appl. Phys. 20, L367 (1981); K. Uchino: Proc. Study Committee on Barium Titanate, XXXI-171-1067 (1983) J. Kuwata, K. Uchino and S. Nomura: Jpn. J. Appl. Phys. 19, 2099 (1980) C. Kittel: Phys. Rev. 82, 729 (1951) K. Uchino: Solid State Phys. 17, 371 (1982) K. Uchino, L. E. Cross, R. E. Newnham and S. Nomura: J. Appl. Phys. 52, 1455 (1981) K. Uchino: Jpn. J. Appl. Phys. 24, Suppl. 24-2, 460 (1985)

56

Chapter 2

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