# The electric field and defect dependence of the curie temperature in [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3] ferroelectric Perovskites.

IntroductionFerroelectric have been a subject of considerable interest because of their wide range of practical applications in various fields like ceramic industry, optical communication, memory display, holography, storage media, etc. The Curie temperature of a ferroelectric crystal has great importance as it is useful in studying various properties of the crystal. The Dielectric and Acoustical [1] properties show anomalous behaviour at a temperature near the Phase transition temperature. The Curie temperature is affected by the external electric field. So different impurity and electric field values can change the Curie temperature '[T.sub.c]' of the same material.

The Barium Calcium Titanate ([Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3]) is an important ferroelectric material. It is a solid of a family composed of Barium Titanate (BaTi[O.sub.3]) and Calcium Titanate (CaTi[O.sub.3]) with its Curie temperature varying over wide range. Barium Calcium Titanate ([Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3]) is an ferroelectric material because of the significant variation in its physical and structural properties due to Ca substitution. The Ca substitution increases the dielectric constant by a less order as compared to the variations occur due to substitution of Sr atoms replacing Ba atoms in the crytstal BaTi[O.sub.3]. A current review on [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3] is available in the current literature [2].

The defect dependence of the Curie temperature in ferroelectric crystals is discussed by some workers [3] considering the change in mass and harmonic force constants between the impurity atoms and the host lattice atoms. They have obtained a general expression and not taken a specific crystal. But our work is different from their work [3] as the crystal [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3] (BCT) is subjected to an external electric field.

In the present work we have theoretically studied the electric field and defect dependent Curie temperature in anharmonic Ba1-xCaxTi[O.sub.3(BCT).] Ferroelectric perovskites in paraelectric phase by using double time Green's function technique. Double times thermal Green's function is used to obtain thermally averaged correlation functions and hence the observable quantities with the help of the modified model Hamiltonian taking into account anharmonic effects up to fourth order with substitutional defects and electric moment terms. In this formulation the real part of the dielectric constant [[epsilon].sup./] will lead to an expression for the change in Curie temperature ([T.sub.c]) by the presence of electric field.We make use of unitary transformation discussed in previous studies [4] which render the most significant first order dipole moment to effect the Curie temperature ([T.sub.c]) with the applied electric field. The calculated results are compared with the results of other workers [3-5].

Calculations

The real part of the dielectric constant and microwave loss (tano) are related [3] to the Green's function

G([omega]+I[epsilon]) = <<[A.sub.o.sup.o](t);[A.sub.o.sup.o]([t.sup./])>>[sub.[omega]+i[epsilon]] =[G.sup./]([omega])i[G.sup.//]([omega]), (1)

as [[epsilon].sup./]([omega])-1= -8[[pi].sup.2][M.sub.[mu].sup.2](0)[G.sup./]([omega]) (2)

and tan[delta] = [G.sup.//]([omega])/[G.sup./]([omega]) (3)

Writing the equation of motion for the Green's Function (1) with the help of the modified Hamiltonian [4], Fourier transforming and writing it in the Dyson's equation form, one obtains

G([omega]+i[epsilon])= [[omega].sub.o.sup.o]/[pi][[[omega].sup.2]-[v.sup.2]([omega]) + i[GAMMA]([omega])], 5(a)

where [v.sup.2]([omega]) = - [([[omega].sub.o.sup.o]).sup.2] + 4 [[omega].sub.o.sup.o] D(0,0) + [[omega].sub.o.sup.o] [E.sup.2](96[g.sup.2]V-24g[D.sub.1.sup./]) + 4 [[omega].sub.o.sup.o]Q+[DELTA]([omega]), 5(b)

where [DELTA]([omega]) is shift of frequency response and has been earlier discussed in our previous studies [4].

In the presence of electric field, in a defect ferroelectric crystal, the real part of the Green's function 5(a), gives Cochran's temperature dependent frequency Q(T) as the self-consistent solution of the equation,

[[OMEGA].sup.2] = - [([[omega].sub.o.sup.o]).sup.2] + 4[[omega].sub.o.sup.o] D(0,0) + [[omega].sub.o.sup.o] [E.sup.2](96[g.sup.2]V-24g[D.sub.1.sup./]) + 4 [[omega].sub.o.sup.o]Q+[DELTA]([SIGMA]) (6)

From our previous studies[4], the value of Dielectric constant can be given as

[[epsilon].sup./]([omega]) = C/[T-[T.sub.c]+[xi][T.sup.2]+(1/[gamma]){[DELTA]([v.sub.D.sup.2]) + [[omega].sub.o.sup.0][E.sup.2](96[g.sup.2]V-24g[D.sub.1.sup./])}], (7)

where C = 8[pi][M.sub.[mu].sup.2](0) [[omega].sub.o.sup.o]/[gamma] 7(a)

[T.sub.c] = [([omega].sub.o.sup.o]).sup.2]/[gamma], 7(b)

And [xi] = [[gamma].sub.2]/[gamma]. 7(c)

In the above equations, D(0,0) is harmonic force constant change due to defect. V and [D.sub.1.sup./] are electric moment terms and [xi] is nonlinearity constant.

The T and [T.sup.2] dependence of soft mode frequency is due to Third- and Fourth-order anharmonicities respectively.

Equation (7) can be rewritten as,

[[epsilon].sup./]([omega]) = C/[T-T-[T.sub.c.sup./]+[xi][T.sup.2]], (8)

where [T.sub.c.sup./] = [T.sub.c] + [DELTA]([T.sub.c]) 8(a)

and [DELTA]([T.sub.c])= 1.9 X [10.sup.-3] E, 8(b)

E (V/cm)is the applied electric field.

The above expression shows the change in Curie temperature depends both on impurities and external field. Also it is evident that [DELTA]([T.sub.c]) is a function of mass change due to defect, anharmonic force constants and the applied electric field.

Variation of Curie temperature with Electric Field and Defect concentration (x)

Impurity dependent Curie temperature [T.sub.c] in [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3] (BCT) for different values of defect concentrations (x = 0, 0.05, 0.10 & 0.15) have been taken from our previous study [6]. We have calculated field and impurity dependent Curie temperature in [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3]. The variation of the Curie temperature with applied electric field for different values of x is shown in Fig.1.

Discussion

The dependence of the Cochran's frequency gives the dependence of the Curie temperature, in the paraelectric phase. Our results show the comparative variation of [T.sub.c] in BCT with different values of x and electric field. Figure shows the variation of the Curie temperature with applied electric field. [T.sub.c] increases with increase in electric field. Also taking any electric field as reference [T.sub.c] also increases with increase in defect concentration. The results obtained in our study are in good agreement with previous experimental [7] and theoretical studies [3,4,7].Recently we have discussed various physical properties of [Ba.sub.1-x][Ca.sub.x]Ti[O.sub.3] (BCT) ferroelectric perovskites [6,8-11]

[FIGURE 1 OMITTED]

References

[1] Ashish Kukreti, Ashok Kumar & U.C. Naithani, Indian Journal Of Pure and Applied Physics. 46, 276(2008).

[2] See e.g., J. Noh Hyun, Sung-Gap Lee, Young-Hie Lee & Sung-Pil Nam. J. Caramic Proc. Research. 9, 267(2008); F. V. Motta, A.P.A. Marques, M.T. Escote, D.M.A. Melo, A.G. Ferreira, E.Longo, E.R. Leite & J. Verela. J.Alloys and Comp, 465, 452(2008) and references therein.

[3] R. Bahadur & P.K. Sharma, Physical Review,12, 448(1975).

[4] U.C. Naithani & B.S. Semwal, Pramana, 11, 423(1978).

[5] Ashok Kumar & U.C. Naithani, Journal of Mountain Research, 3, 63(2008).

[6] Dushyant Pradeep, Ashok Kumar & U.C. Naithani, Indian Journal of Pure & Applied Physics, 48, 220(2010).

[7] W.J. Merz, Physical Review, 91, 51(1953).

[8] Dushyant Pradeep, Ashok Kumar and U C Naithani, Indian J Pure and Appl. Phys., 48, 276(2010).

[9] Dushyant Pradeep, Ashok Kumar and U C Naithani, International J Pure and Appl. Phys., vol. 6, No.1, 83(2010).

[10] Dushyant Pradeep, Ashok Kumar and U C Naithani, International J Pure and Appl. Phys., vol. 6, No.2, 165(2010).

[11] Dushyant Pradeep, Ashok Kumar and U C Naithani, International J Pure and Appl. Phys., vol. 6, No.2, 171(2010).

* Dushyant Pradeep and U.C. Naithani

Department of Physics, H.N.B. Garhwal University (a Central University), Campus Pauri (Garhwal), Uttarakhand--246 001, India.

* Corresponding Author E-mail: dushyant_pradeep@yahoo.co.in