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**Authors :**Mayank Joshi, Trilok Chandra Upadhyay -
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**Volume & Issue :**MAM – 2013 (Volume 1 – Issue 05) -
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#### Electric field on Dielectric & acoustic Properties of Ferroelectric crystal PbHPO_{4}

Mayank Joshi & Trilok Chandra Upadhyay Physics Department,

H.N.B. Garhwal University Srinagar (Garhwal) Uttarakhand-246174 Email-Mayankphysics@Gmail.Com

Abstract

The third and fourth order phonon anharmonic interactions and external electric field terms are added in the two-sublattice pseudospin model for PbHPO4 crystal. By using double time thermal Green's function method modified model, theoretical expressions for soft mode frequency, dielectric constant, shift, width and tangent loss are evaluated for PbHPO4 crystal. Temperature and field variations of soft mode frequency, dielectric constant and loss tangent are calculated numerically. Present theoretical results agree with experimental results of Smutney and Fousek for dielectric constant of PbHPO4

Introduction

Due to their promising application in the field of electronics and technology ferroelectric crystals are continuously being attracted to both physicists and material scientists. Memory devices, infrared and pyroelectric detectors, transducers, display devices piezoelectric devices are some common uses of these materials [1].

Lead hydrogen phosphate (PbHPO4) crystal and its isomorphs (PbAsPO4, CaHPO4, BaHPO4, CaHPO4 etc.) from an interesting group of quasi-one dimensional hydrogen bonded ferroelectric crystals. In PbHPO4 the direction of spontaneous polarization is almost parallel to the direction of the H-bonded O- H…O projecting on the (010) plane unlike in KH2PO4. The PO4 groups are bound to one another by the O-H…O bonds in the from of a one dimensional chain along c-axis[2] Raman spectroscopic studies showed the value of tunneling integral for PbHPO4 crystal very small although very large changes of curie temperature and Curie-Weiss constant occur on deuteration [3]. According to Cochran[3] the frequency of some of the normal mode of vibration of crystal called soft mode becomes zero at the transition temperature. It is this soft

mode that largely determines the dielectric and scattering properties in ferroelectric crystals.

The acoustic experiments of Litov and Garland [4] clearly showed that external electric field has pronounced effect on temperature dependence of ultrasonic attenuation and elastic measurements on ferroelectric crystals. Baumgartner[5]. Choi and Lockwood[6] and Silverman[7] have carried out experiments on effects of electric field on dielectric measurements in KDP and BaTiO3.

Ohno and Lockwood[8] have measured Raman Spectra in PbHPO4 crystal in the temperature range 100K to 350K. Tezuka et al[9] have made experimental studies of soft mode spontaneous polarization in PbHPO4 crystal. Ratajezak et al [10] have studied structural and second harmonic generation studies in PbHPO4 crystal. Mahadevan et al[11] have grown PbHPO4 crystal and measured its dielectric constant and loss.

Wesselinowa[12] has considered pseudospin-lattice coupled mode modal with third and forth order phonon interaction terms to study dynamical structure factor of central peak in PbHPO4 , crystal using Tsernikov's Green's function. This model is over simplified for the case of PbHPO4 crystal since it does not explain salient features of these systems such as low value of tunneling integral etc. Their work is quite different to our approach. Chaudhuri et al [13] have used a two sub lattice pseudo spin-lattice coupled mode model with a fourth order phonon anharmonic term. But they failed to consider third order phonon anharmonic term which is very important. Moreover they decoupled the correlation at an early stage so that some important cross terms disappeared from their calculated results.

In the present study a two sub lattice pseudospin lattice coupled mode (PLCM) Model along with third and fourth order phonon anharmonic interactions is considered for PbHPO4 crystal using Green's function method[15] the field dependent shift,

width, soft mode frequency dielectric constant and loss tangent have been calculated for,

values for PbHPO4 crystal, temperature

H anh

V (3) k

1

k k k

, k2

, k3

A A A

k k k

1 2 3

dependence of the above quantities have been calculated in the presence of electric field and

1 2 3

2

V (4) k , k , k , k A A A A ,

compared with experimental results of Smutny and Fousek[16].

(3)

k k k k

1 2 3 4

1 2 3 4

k k k k

1 3 4

#### Model Hamiltonian

Mitsui[17] and Blinc and Zeks [18] proposed a two-sublattice pseudospin model, which was applied to the case of PbHPO4 and isomorphous crystals . For PbHPO4 type crystals we have, extended their two-sublattice pseudospin-lattice coupled mode model4 by adding third and fourth order phonon anharmonic interaction terms [14] as well as external electric field term which is expressed as

1i

2i

ij

1i

2i

2i

2i

H 2S x S x J S z S z S z S z

V(3) (k1,k2,k3) and V(4) ( k1,k2,k3,k4) are third and fourth-order atomic forces constants given by Born and Huang[20].

#### Greens functions, Width and Shift Equations

For the evaluation of expressions, soft mode frequency, dielectric susceptibility, dielectric constant and loss tangent we consider the evaluation of Green's function[15]

We consider the Green's function

i ij

G (t t) Sz (t);Sz (t)

K S z S z 2E(S z S z )

ij 1i 1j

ij 1i 2i

ij

1i 2i

i

i ( t t ) S z ( t );S z ( t ) ,

k

1 A A B

4 k k k k

B k

(1)

1i 1 j

(4)

in which (t- t') is unity for t<t' and zero

In Eq.(1) above is proton tunneling frequency between O-HO double well potential,

Jij is exchange interaction between neighboring lattice dipoles, and kij that in same lattice is dipole moment of O- HO bond, E is external electric field, is

phonon frequency, Ak and Bk are position

otherwise. The angular bracket is ensemble average.

The Green's function (GF) is differentiated twice first with respect to time t and then with respect to t' , Fourier transforming the Green's function and putting in the or of Dyson's equation,

G ij G 0 G 0 P G 0

and momentum operators and Sx and Sz are components of spin variable.

(5)

ij ij ij

1 i

Chaudhari et al[13] have modified above model by following Kobayashi [19] by adding pseudospin-lattice interaction terms

where G 0 ()is unperturbed Green's function

ij

S

given as

H V

S z A

V

S z A

0

x ij

(2)

s p

ik 1i k

ik

ik 2i k

ik

G ij

P(

(6)

2

4 2

In Eqs(2) above Vik is spin lattice interaction constant.

We add the third and the fourth-order phonon anharmonic interaction terms[14] as

and ~ )is polarization operator given by

~

P() f

(7)

2

F1i (t); Fj1 (t')

where

4

a

where f

i F, S1 j

y

S x1i

~

and (8)

1 () 2 2 ~ 2

(16)

i1

1i

1 j

1 j

ij

V 2 N a 2

F (t) 2 S x S z S z S x/p>

ik k

2 2 2 2

2K S X 1J S Z 1J 2V S x A

2V S x A k

(17)

1 j

(9)

The Green's function as

ik 1i k

G()

ik 1i

is then obtained

4 2 E 2 a2

3 2 2 ~ 2

(18)

)P(

G() G0 ()1 G0 (

~ )1

2V 2 S x 2 ~ 2

ik

1i

k

kk

k

i

(10)

4 2 ~ 2 2 4 2 2

This gives the values of Greens function (4)

k

(19)

and width

k k

S x

1 2 3 4 ,

ij

2 2 2i

G0

1i ij

~

, (11)

(20)

1

where ~ 2 42

(12)

S x f

1

(21)

a 4

4~

~ ~

~ F (t); F (t) ,

V 2 N a2

P x i j

ik k~ ~ ~ ,

S1

(13)

2

(22)

4

where <<Fi(t);Fj(t')>> are higher order Green's

4V 2 S x

2 ~ 2

functions. They are evaluated by decoupling

ik 1i

k k k ' k

3 2 ~ 2 2 4 2 2

them using decoupling scheme

<abcd>=<ab><cd>+<ac><bd>+<ad><bc>

~

k

(23)

k k

P()after evaluation is resolved into its real

2 2 E 2 a 2 ~ ~

and imaginary parts using formula

4 4~

1 1

(24)

lim

i x ~

m0 x im

x

In Eqs.(19) and (23)

~and

are

k k

(14)

The real part is called shift

()

and the

phonon frequency and phonon half width which are obtained by solving phonon Green's Function(in a similar way)

imaginary part is called half width ()

G (t t') A (t), A

(t')

kk '

k k1

We therefore obtain shift and width as

() 1 2 3 4

(15)

i (t t') Ak (t), Ak1 (t)

(25)

X 1 n

48

k

V (3) (k , k , k ,k ) 2

2 ~ 2

~

which gives G

ij'

kk '

k

2ik

1 2 3 4

k1

n

k 2

n

k 2

n

k 3

n

k 3

n

k 4

(26)

X ~

~ ~

k 2 k 3

~

~ ~

k 2 k 3

k1

k1

where

(29)

k

k

~ 2 ~ 2

2 k k

In Eq(28) and (29)

~

ki and P

~ 2

k

k

b)

Ak

(27a,

nki Coth k

stand for principal part.

BT

Phonon shift

k

Re P0 (k,)

2

In Eq.(8)is solved by using means field approximation for co-relation i.e. second term

in Eq.(12) is evaluated using mean field approximation, i.e. correlations are finite, i.e.

1

2

18Pk1k2

V (3) (k , k

,k)

S

z

k1

k 2

k1

k 2

1i

x

S

1i

1 ~

tanh , (30)

k1

k 2

n

n

~ ~

~ ~

a b 2~ 2

~ ~

k1 k2

2

2

k2 k1 2

2

n

n

k1 k 2

k1 k 2

k1 k 2

which gives

~ 2 a 2 b2 bc(first frequency)

V (4) (k , k

, k ,k 2 k1k 2k 3

(31)

1 2 3

~ ~ ~

S

S

1

,

2

1 n n

n n

n n

k1 k 2 k 3

~ ~ ~

k1

k1 k 2 k 3

where a 2 J 0

(32)

z K 0 z

k1 k 2

k 2 k 3

k 3 k1

2 ~

~ ~ 2

k 2 k 3

b 2; (33)

S

S

1

2

31 nk 2

nk1

nk 2

nk 3

nk 3

nk1

and

c 2J 0

x K x

~ ~ ~

(34)

k1 k 2 k 3

k1

2 ~

~ ~ 2

k 2 k 3

Therefore, the Green's function finally takes the from

+ highter terms }

(28)

S x

Gij 2

1i ij

2

Phonon width

(35)

2i

k Im P(k,)

2

~ 2 2

(36)

9 V (3) (k , k ,k ) 2 k1k 2

In Eq.(35) and (36) and are liven

1 2 ~ ~

by Eq.(20) and (15) respectively.

n

~

k1

n

k1 k 2

k1

k1

k1

~ ~ ~

Solving Eq.(36) one gets

Â½

k

k

k 2 k1

n

n ~

~

~

~ 2

1 ~ 2

~ 2

1 ~ 2

~ 2 2

2 x

k 2 k1

k1 k1

k1 k1

(37)

2

2

8 Vik

S1i

The Curie temperature is given by

Tc

3

The dissipation of power in dielectric material is called tangent loss which expressed as

2

2k B tanh 1

4 J'

(38a)

tan

(43)

Where 2 2J K 2 2 42

(38b)

Where

''

and '

are imaginary and real

2V 2~ 2

parts of dielectric constant

k

k

J * 2J K ~

ik k

2

(38c)

~ 4 4

2

tan

2 2

k

(44)

#### Acoustic Attenuation

Where

and are half width and soft

Acoustic attenuation is a measure of the energy loss of sound propagation in media.The

mode frequency given by Esq. (20) and (37) respectively. By using model values of various

acoustic attenuation provides an important

quantities

in expression for

~

clue about phase transition. At Tc is abruptly increase showing anomalous behavior. According To Tani & Tsuda[20] attenuation

()

,,, ,,and tan for PbHPO4 crystal from literature their electric field and temperature dependences ,and tan

is given by (39)

near transition temperature are calculated, which are shown in figs.1, 2, 3and 4.

#### Figures For crystal

Where ()is acoustic width and is sound

velocity.

#### Dielectric constant and Loss

The response of a ferroelectric crystal to the external electric field is expressed dielectric susceptibility which is related to

Green's function as

Calculated electric field and temperature dependences for PbHPO4 crystal

lim

0

2 N 2 G ij i

(40)

The dielectric constant is related to electrical susceptibility as

Fig 1. Soft mode frequency in PbHPO4 crystal (Present

1 4

(41)

calculation, Experimental results)

By putting value of Green's function from Eq.(39) and (40) we obtin

8N2 S x 2 2 2 2 2 4221

1

(42)

Fig.2. Dielectric constant in PbHPO4 crystal (Present calculation, Experimental results)

)

Fig.4 Acoustic Attenuation in PbHPO4 crystal((Present calculation, Experimental results

#### Conclusion

In the present work the effect of electric field on the dielectric properties of PbHPO4 crystal has been studied .The two sub-lattice pseudo spin-lattice coupled mode model is extended by adding third and forth order phonon anharmonic interaction terms and external electric field term. With the help of double-time Green's function method, theoretically the field dependent expression for shift, width, soft mode frequency dielectric constant and loss tangent have been obtained. By fitting model values of physical quantities appearing in the expressions derived, field and temperature dependences of soft mode frequency, dielectric constant width shift and loss tangent have been calculated. Theoretical results have been compared with experimental result of Smutney & Fousek. [16] Previous workers have not considered the third order interaction an electric field terms in their calculation. Chaudhari et al [13] have not considered third-order-phonon anharmonic interaction terms in their model. This term is essential to explain linear temperature dependence (A) of soft mode frequency square

(i.e.

AT BT )

since third order

Fig 3. Tangent loss in PbHPO4 crystal (Present calculation, Experimental results)

anharmonic term gives linear temperature dependence. Therefore, our calculation provides much better results to fit the experimental data. Secondly Chaudhuri et al

[13] has decoupled the cor-relations in the early stage while we have decoupled them at a proper stage. As a result some important interactions disappeared from their calculations. If third order-phonon anharmonic terms are neglected from our expressions, these at once reduce to the expressions of previous workers. Present study shows that the electric field has pronounced effect on ferroelectric and dielectric properties of PbHPO4-type crystal. The soft mode frequency increase while dielectric constant and losstangent decrease with increase in electric field strength. It can be seen from our expressions

that our frequency ~is same with the initial

frequency of Chaudhuri et al [See Eqs(31)]. However, our soft mode frequency ~contains extra terms [given in Eqs (15)]. Our soft

mode frequency contains extra terms

R.K.Osterheld, R.Sussott.,

Ferroelectrics 6, (1974) 179.

[3] W.Cochran, Adv. Phys.18, 157 (1969).E.Litov and C.W. Garland,

Ferroelectrics 21, 12 (1987).

H.Baumgartner, Heli. Physica Acta 24,

in ~ and applying in

650 (1950).

k k k

(Eq.27a)].These extra terms are

B.K.Choi and D.J. Lockwood,

1 2 3

given V (3) (k , k ,k ) 2

in k

[Eqs.Ferroelectrics 72,303 (1977).

(28)]and

V (3) (k , k

,k3

2

)term given

B.D.Silverman, Phys.Rev.B.125 1961 (1962).

1

2

in k (Eq.29-)]. . These terms differentiate

our expressions with the expressions given in the work of Chaudhuri et al[13]. Attenuation increase with electric field strength both below and above Tc in PbHPO4 crystal. This is in agreement with experimental results of Litov & Garleand21. So far we could not find experimentally data for electric field dependence for PbHPO4 crystal. This Is quite similar to other ferroelectric crystal like KH2PO4 Experimentally.

#### Acknowledgements

Authors are grateful to Prof. B.S. Semwal (HNBGU,) Prof. R. P. Tandon(Delhi Univ.), Prof. Shyam Kumar (Kurusharta University Prof K. K. Verma (RML Avadh Univ. University, Faizabad), Prof. N .S Negi (H P University) , Prof S K Singh (Panjab University) and Prof R. P. Gairola (HNBG University) for their kind suggestions and encouragements.

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