Field-dependent sound attenuation in barium titanate, strontium. _ titanat~ and potassium tantalat~ type perovskit~s

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Indian Journal of Pure & Applied Physics VoL 40, October 02, pp. 701-708 4- Field-dependent sound attenuation in barium titanate, strontium. _ titanat and potassium tantalat type perovsits ----.

1 Indian Journal of Pure & Applied Physics VoL 40, October 02, pp Field-dependent sound attenuation in barium titanate, strontium. _ titanat and potassium tantalat type perovsits I, ' - D S,Lingwal; 'U C Naithani,& B SjSemwai "I,", :='..' ',-, ", Department 0(' Phy 'ic HNB Garhwal Uni. rsi ty/'campus, Pauri (Garhwal ), Uttaranchal I --./ Received 14 January 02: revised I April 02; accepted 3 April 02 ;' A theoretical expression for attenuati on constant in para-electri c phase is described. Taing electric field as parameter. variation of attenuation constant -with frequency and temperature is also discussed by considering third and fourth order anharmonic interactions. The attenuation constant increases wi th increasing temperature and frequency in presence of an electricfield. In the vi cinity or the Curie temperature, the attenuation constant increases anomalously. The results obtained are co mpared with the results of oth;;-;i..j 1 Introduction The study of the propagation behaviour of ultrasonic waves near a phase transition, gives va luable information about both the static and dynamic aspects of phase transition. Ultrasonic measurements provide a sensitive tool for the study of the phase transition in solids l, These studies have played an increasingly important role in characterizing the behaviour of a system near the cooperative phase transitions (cooperating-ordering of the dipoles which gives ri se to the spontaneous polarization, is destroyed by thermal agitation above the Curie temperature) and critical points. One advantage of the ultrasonic measurements is that, the static and dynamic properties of the system can be simultaneously measured. Low frequency acoustic velocities provide precise information about the equilibrium adiabatic properties of the system and the effects of the te mperature, pressure and external fields can be readil y studied. Ultrasonic attenuation data provide information about the dynamic behaviour and from the frequency and temperature dependences, much can be learned about the mechanisms involved. Theoretically, new ways of describing the critical phenomena in terms of fluctuation correlations have been of great importance. :rhe large variations in the velocity and the strength of the attenuation near a transition are usually referred to as 'anomalous'. Such special variations are now descri bed as 'critical '. As revealed experimentally, as well as theoretically, the soft or the ferroelectric mode plays an essential role 111 di splacive ferroelectrics. As the temperature approaches the Curie te mperature Tn the soft mode frequency Q becomes vanishingly sma ll (Q-T- TJ, resulting in an increase in its amplitude, which should influence the acoustic mode via the phonon-phonon interactions and is expected to give an anomalous 2 behaviour of sound near T e,. The phenomenological theory of sound attenuation near the transition point has been developed by Landu & Kalatniov'. The degeneracy of the soft mode with the longitudinal acoustic mode for large wave numbers gives the characteristic behaviour of the soft mode in di splacive ferroelectrics 4 lie SrTi0 3. Tani & Naoyui 5 have also extended the phenomenological theory of sound near the transition point to displaci ve ferroelectrics and they have shown the anomalously increasing behaviour of the attenuation constant in the vicinity of the Curie temperature, The idea of coupling of elastic strains and the soft phonon modes was suggested by many worers where the coupling is described in terms of the electro-strictive coupling, which is quadratic in polarization and linear in strain. The velocity of the transverse wave does not show any appreciable critical behaviour. In general, it is agreed that, the main contribution to low frequency attenuati on is from an interacti on of soft phonons with each other and with acousti c phonons. In the vicinity of the Curie temperature, whe n soft mode frequency tends

2 7ff2 lndlan J PURE & APPL PHYS, VOL 40, OCTOBER 02 to zero, onl y interacti o ns with lo ng itudinal sound re ma llls. Barium titanate. strontium titanate and potassium tantalatc have attracted uni versal attention of researchers and have found wide practi cal applicati ons. Several worers 6. l o have derived expression for atte nuation constant in di spl ac ive fe rroelectrics. DeOl'ani el al. 9 have also di scussed the attenuati on constant, quantitati vely, in these perovsites for zero fie ld case. Naithani et al. 11 have di scussed attenuati on co n s tlilt with e lectric fie ld, qualitati vely, T he aim of the present wor is to study the attenuation constant in BaT io, SrTi0 3 and KTa0 3 above the phase transiti on temperature with the vari ati on of frequency and te mperature taing e lectric fi e ld as a parameter and to corre late the results with the resul ts of other worers. For quantitati ve purposes, the authors consider that the Curie temperature changes with the e lectric fie ld as 12 15: 6.T = 1.9 x x E... (1 ) where E is measured in V/cm. 2 Hamiltonian and Green's Function The authors consider the G reen's functi on for the acousti c phonon as: G, (I-I') = «a '\ (I) ; {lot ' (I')»... (2a) T he modi fied S il verman Hamiltonian used in the present study is exactl y similar to Eq. (7) (for zero defect case) of the previous studi. A si mil ar type of Hamiltoni an have also been used in the past by Gairola & Semwal 17. T hi s transformed Hamiltoni an is expressed as: It {/ H r = "" L- (A "+ A" + B"+ Ba ) 4 + I h()aoo Ao\ Ao + I ft B"()A OIO AO\ Ao + Ilt B (,l.:)aoloa "\A\ + I lt -4>(l,2,3)A ",,A"JA" 2 A\3 l.2.u + I ft \jf ( l, 2,})A"" A " l A " 2 A"u + hv A ".j" l.2.u + he{ I A() A\ A"\ + I B\ )A)'\ A\ A"".ic + Ie ( l. 2,,)A"u A\l A\3 l ID (l,2, 3)A"uA"2A",, } I It W""g A''.,-2gE L It F() A"\ A\ -4gE L ItB"() AO\ A A o l gE I h\jf () A"\ A\ AO - 2 g E I h <I> ( l, 2, 3) AO l A\lA"U l ge L I \jf ( l,2, 3 ) AO l A"lA"U l. 2,3-8g E ItVA"3" + hagf-- 2gE 1 I h-b)' () A)'\A\ + 4gF- L h B" ()A"\A".;, + 2g2E2 L hb" ()A"\A\+ 24 glf-h VA"l"... (2b) Tn brief, F(K) and B)' (K) represents the secondorder anharmo ni c terms, while V, \jf and (D represent the third order anharmoni c terms, whil e A(K), B i. (K); C( l h.3) and D( l, 2, 3 ) represents the secondorder and third-order e lectric moment terms, respecti vely. T he notati ons used are the same and in the same sense, as used in Ref. 8. Writing the equati on of motion for the G reen's f un cti on [Eg. (2a)] with the help of the modified Hamiltoni an [Eq. (2b) 1 Fouri er transforming and writing it in the Dyson's equati on form, o ne obtain s 8 : :; (/... (3) :; (/ Where r(w) is the damping constant and uh is the f ie ld-dependent stabilized acou tical frequency and IS g Iven as: == n 2 uh = W "2 +2 " (W)...(4) where

3 LINGWAL et al.:sound ATTENUATION 703 ffi,"= 0\"2 +8 (J) a 8 E 2 (2g " () -B"(» and il«(j) = t., «(J) +AE2... (4.1)... (4.2) t., «(J) and f«(j) &re identified as the expressions for the shift and width of acoustical phonons, in the absence of the field, in the lowest approximation. t.,«(j) depends upon third and fourth order anharmonic coefficients in presence of electric moment terms and the constant A depends upon third order anharmonic terms only. For displacive ferroelectrics, damping constant f,«(j) is expressed as:... (5) where f A«(J) and f E«(J) are the damping constants which are dependent upon anharmonic contributions and electric field terms, respectively, and their values are obtained as: I A (w) = -7r 1 F (K) 12 ( w / ffi )'2:, (N ± No ) ± [8 ( w - Q =+= w ) - 8 ( w - Q ± w ) 1 + 4n I W()1 2 (co\ 1 ffi \ ) < (1 + NLo +2No N \ ) [0(co-2Q- ffi \ )-0(co+2Q+ ffi \ )] +(l+n 02-2No N \ )[0(co-2Q + ffi \ ) -0(co+2Q- ffi \ )]> +n 8 3 ( " /( - - " ) ( 1 N N-,) co,i CO J<2 W, I W J<2 < + 0 l L!l>(KI,K 2,-K:)<I> (KI,K2K) K lx2 + N ' 1 N \ 2 + No N \2) [o(co-q- ffi l - ffi \2) -o(co + Q+ ffi o, l+ ffi \ 2)] and f E (co) = 2n e[ B"( ) -4g " (KW (co'v ffi \) I( N \ ± No) [o( ffi -Q+co\ )- o( ffi +Q+co\ )] +n 2 [A()-2gF()P(coV ffi,o) [o(co- ffi,o) - o(co+ ffi,o)] +2n 2 L [C( l.c ) l, 2... (6) -28 <I> ( l, 2, - W(coO' 1 co\21 ffi '\l ffi\2) XL (N "'2 ± N 'I)[(co-ffio'l+ ffi \2) ± - o(co+ ffi o l ± ffi \2 )] Here 0 3 = = 0,1"'1(0,2"'2 OU-"} + O2-,'} OU" '2) N\ = <AA\A\ > = coth (1 I 2 It ul") and N o = < A Oo A Oo = coth (1/2 It Q) with = (K B ]) -I. It = hl2n... (7)... (8)... (9)... (to)... (11) Q is effective soft mode frequency and temperature dependence of Q is given as l8 Q2 - (T - TJ. In the expression (6) and (7) F(), " (); <1>( l> 2, 3) and C( l, 2, 3) are third-order, fourth-order and electric moment coefficients, respectively. The notations used here are the same and in the same sense as in Ref Theoretical Considerations 3.1 Temperature and field-dependence of attenuation constant The expression for the attenuation constant is given as l l : a = f, I c... (12) where c is the sound velocity and f is the damping constant: f is also defined as the width of frequency response of acoustic phonon mode. The value of f«(j) is given by Eq. (5). Here, the attenuation constant can be expressed as:... (13) In the absence of an external electric field (zero field case), a = aa «(J), which have already been discussed, qualitatively and quantitatively, in the authors previous papers There, the temperature dependence of a A «(J) is expressed as: a A «(J) = [A I «(J) +{A 2 «(J)+A3 «(J)/ (T-T,. ) 112

4 704 INDIAN J PURE & APPL PHYS, VOL 40, OCTOBER 02 +A 4 (w)/(t- T.)'I2 }T+/As(w)+A 6 (w)/(t- TJ312 +A 7 (w)/ (T- T J2 }T2]... (14) where A I is the temperature-independent term and A i's (i = 2-4) and A/ (j = 5-7) are the coefficient of T and r- respectively Ai' s and A/ are dependent upon higher order anharmonic terms. It is clear from above relation that, as T 7:., attenuation constant increases anomalously in agreement with the results of Tani 2.4.' 9 and Pytte. In the low temperature range for the reduced temperature, Eq. (14) will become as:... (15) which is same as obtained by Tani '9. Thus, in the low temperature range, the law a A(w) oc T/(T-TJ312 can be valid to a good approximation to study these properties quantitatively. Consider the effect of electric field on the attenuation constant. The electric field-dependent contribution to a(w) is given by ue(w) (= IE(w)/ C). At any temperature well above To> the temperature dependence of ue(w) is given by:... (16) where C' is the coefficient of temperature and electric field in Eq. (7) and it has different values for BT, ST and KT, respectively. In the present study, only the net critical relaxation attenuation (ae), associated with the critical fluctuations above phase transition temperature To in presence of field is concerned. By plotting u.: versus TI(T-TJ' 12 in presence of different field strengths, a straight line was obtained, in all cases (BT, ST and KT) showing that, present quantitative results are in good agreement with previous theoretical results 9. ". In a previous paper of the authors ", on equating only fourth-order anharmonic interactions and comparing (a = Uo+ aj to attenuation relation, they obtained, a c ::::a2w2. Now, comparing this attenuation (u.:) with the attenuation constant due to fourthorder anharmonic coefficients only (i n Eq. 14 here), one gets: Q2 w 2 = A 3 (w)t/(t - Tj /2 or or A3 (w) = a2wo 2 x 10-3 (T - TJ' 12 / T where a2 is a field-independent constant and has been calculated by best fit of data in Ref.22 as a2=4.1xlo-'4 S2m l (at 398 K and 10 MHz). On comparing field-independent coefficient of T/(T TJ'12 of both Eqs (14) and (16), one can get: C I = A3(W) The calculated values of C I at zero biasing and at its phase transition temperature are: C' = 0.21 X 10 4 S2 m 1 HZ2 8 1/12 (for BT) C I =2. x 10 4 S2 m IHz 2 8 1/2 (for ST) C' =2.67 x 10 4 S2 m IHz (for KT) Tn Tables 1-3, Eq. (1) has been used to find the change in Curie temperature at different field strengths. In the absence of applied electric field, the values of the Curie temperature for BT, ST and KT are 395 K, 37 K and 13 K, respectively. 3.2 Frequency variation of field-dependent attenuation constant The frequency dependence of attenuation constant is given by: a A (w) = a l + a 2 w 2 + a 3(O... ( 17) where w 2 dependence arises due to fourth-order anharmonic interactions, the w dependence arises due to third-order anharmonic interaction terms. If one considers third order anharmonic interactions then Eq. (17) can be rewritten as:... (18) Garland el al. 21 has an expression for attenuation constant as: On comparing Eqs (18) and (19), we get:... ( 19)... ()... (2 1) Eq. (21) shows that, the critical attenuation varies linearly with w. Considering only fourth-order anharmonic interactions:... (22)

5 LINGWAL et al.:sound iitfenuation 705 Table I-Calculated attenuation constant with respect to temperature at different field strengths tor BaTi T, K E=Y/cm C,X T/(T - TJI f E= 15 Y/cm C,X T/ (T- Tc)l f E= IOY/cm C,X T/ (T- Tc)1f Table 2-Calculated attenuation constant with respect to temperature at different field strengths for KTa E=OY/cm T,K C,x I T/(T - TJIf E= 15Y/cm T,K C, x T/(T -TJlfl E= 10Ylcm T,K C,x T/(T - TJlfl Table 3 -Calculated attenuation constant with respect to temperature at different field strengths for SrTi0 3 E=Y/cm T,K C,x T/(T-Tc)lfl E= 15 Y/cm T,K C,x T/(T _ Tc)lfl E= 10Y/cm T, K C,x T/(T-TJ lfl Comparing Eqs (19) and (22): (J..; = d mt, where 00 is in MHz. To show the field-dependence, (J..; can be rewritten as:... (23) In the above relation, az is calculated by best-fit of data (Ref. 22) and has value a 2 = 4.1xlO- 14 SZm 1 at temperature 398 K and frequency 10 MHz and gives the value of critical attenuation as: (J..; = 1.62 X 10-3 Q 00 d ml, because, soft mode frequency Q is very large as compared to the microwave frequency 00 (m/q "" 10-3 ). Here, Q is the stabilized soft mode frequency and it depends upon third and fourth-order anharmonic terms in presence of electric moment terms. In a pure

6 706 INDIAN J PURE & APPL PHYS, VOL 40, OCTOBER 02 harmonic approximation, this frequency is purely imaginary and only the presence of anharmonicity and electric moment terms stabi lizes this frequency. The variation of soft-mode frequency with temperature is approximated as 8 Q2 - K (T- 7;J Using this relation, the authors calculated the va lue of K = 4. J7xlO 'O MH/ e 1 (for BT), K = 2.94xlO " MH z2 e 1 (for ST); K = 5.08x MH z 2 e ' (for KT), where, maximum range of temperature for BT is taen as 423 K, for ST as 60 K and for KT as 40 K. Calculated values of soft-mode frequency and attenuation constant are tabulated in Table 4. x E al ;;:., rj a Table 4 - Calculated soft-mode ftequency and critical attenuation constant at different field strengths for BT. ST and KT E, Y/cm BT (Q),x HYMHz ST (Q),xHYMHz KT(Q),xlo"MH z BT(<J,;),x HY (w) \.83 ST (<J,;),x IO\ w) KT(<J,;),x IOZ(w) SERlES 1... SERIES 2-0- SERIES) SflUHS4 -+- SF.RIES 5 E., V:"cm o 100 Fig. 2 - Attenuation constant versus frequency in SrTi0 3 (Series I: E = 8 Y/cm, Series 2: E = 6 V/cm. Series 3: E = 4 Y/cm; Series 4 : E = 2 Y/cm, Series 5: E = 0 Y/cm) 60 r- 0., x C m if 60 -Q- SERlES 1... SERlES 2 -tr- SERIES 3 --()- SERIES 4 -.-SERlES 5 E,.V/cnl o O I o f(mhz) Fig. I - Attenuation constant versus frequency in BaTi0 3 (Series I: E = 0 Y/cm, Series 2: E = 6 Y/cm, Series 3: E = 4 Y/cm; Series 4 : E = 2 Y/cm, Series 5: E = 0 Y/cm)... - <:> ; -; r:j 16 12, 10 Fig. 3 - ",/ /' E. 1V:(" -0- ST'Rlr:s! II... SEIES2 & -o-seli.ij:::\ Sr:,Rlt.S 4... SERIES S 0 O--L-1±O-----4O-i f (IIoIH-l) Attenuation constant versus frequency in KTa03 (Series I: E = 8 Y/cm, Series 2: E = 6 Y/cm; Series 3: E = 4 Y/cm; Series 4: E = 2 Y/cm, Series 5: E = 0 Y/cm)

7 LINGW AL et al.:sound AlTENUATION Sli...{}-SERlIS r:rjhs 2 --&-SERlES :\ E, V:'cm 10 t 5 2() OO---L--2O--4a---L--O70 T :(l- T. )I.' Fig. 4 - Attenuation constant versus temperature in BaTi0 3 (Series I: E = 10 Y /cm, Series 2: E = 15 Y/cm, Series 3: E = Y/cm) 0 - '7 " S I -0- SrR.IES l... SERIES 2 -fr- SERll'S () F. V.. \ m If) 15 OO----2O i Fig. 5 - Attenuation constant versus temperature in SrTi0 3 (Series I: E = 10 Y /cm, Series 2: E = 15 Y /cm, Serics 3: E = Y /cm) Fig. 6 - Attenuation constant versus temperature in KTaO} (Series I: E = 10 Y/cm, Series 2: E = 15 Y/cm, Series 3: E = Y/cm) 4 Results and Discussion It is well nown that, the study of the propagation behaviour of ultrasonic waves near a phase transition gives valuable informati on about both the static and dynamic aspects of phase transition. The low frequency ultrasonic velocity yields information about the static aspects, while the absorption data give information about the dynamic aspects. There have been several experiments on the propagation of ultrasonic waves near the ferroelectric phase transition temperature in di splacive-type ferroelectrics. In recent years, much attention has been paid to the dynamical aspects of phase transitions accompanying sma ll di spl acements of atoms in perovsite crystals. In the structural phase transitions, there exist anomalously large fluctuations of the order parameter, which correspond to the excitation of certain phonon modes. These fluctuations cause anomalous absorption of the sound waves as well as shift in the ve locity due to coupli ng of elastic strains and the soft phonon modes. The acollstic modes are scattered by the anharmonicity and applied electric field and thus giving a large attenuation of sound wave and a simultaneous decrease in sound velocity.

708 INDIAN] PURE & APPL PHYS, VOL 40, OCTOBER 02 Green's function technique and Dyson's equation treatment have been used to obtain an expression for the attenuation constant of sound in BT, ST and

8 708 INDIAN] PURE & APPL PHYS, VOL 40, OCTOBER 02 Green's function technique and Dyson's equation treatment have been used to obtain an expression for the attenuation constant of sound in BT, ST and KT in presence of an external electric field by using a model Hamiltonian in presence of higher order anharmonic and electric moment terms. The anharmonic coefficients and higher order electric moment terms gi ve their contributions to various scattering processes. There are two scattering mechanisms present in the crystal; one is the various three and four-phonon scattering amongst phonons due to higher order anharmonicity and the other is the electric field induced scattering due to the electric moment terms in the presence of field. These two distinct scattering processes operate simultaneously, having the relaxation times (1"\)" and (1"\)E for a particular acoustic phonon mode a and the scattering rates are additive. It is shown that, the acoustic width in the phonon frequencies are temperature dependent giving the temperature dependences of the attenuation constant. These temperature dependences are the direct consequence of anharmonicity. It is clear from Eq. (12) that, the fielddependence of attenuation constant is a clear consequence of the field-dependence of fe(w) [Eq. (7)] and hence of r(w) [Eq. (5)]. r E(w) varies directly as the square of electric field strength and thus, the attenuation constant increases with the increase of applied biasing field. Figs 1-3 show the variation of attenuation constant with frequency for different electric fields in case of BT, ST and KT. It is evident from figures that, attenuation constant increases linearly with frequency, which is in good agreement with the results obtained theoreticaily and experimentally by many worers Figs 4-6 show the variation of critical attenuation constant with temperature at different electric fields for BT, ST and KT res pectivly. According to the present results, as temperature approaches towards Curie temperature, attenuation constant increases. The experiments on the field-dependent sound attenuation in displacive ferroelectrics in the vicinity of the Curie temperature T" seems not to have been done except in BaTi03 ceramics by Heuter & Neuhaus 6, many years ago. They have measured the attenuation of 10 mega cycle sound as a function of temperature and applied electric field, and found an anomalous increase of the attenuation near T" K. Thus, the results are in good agreement with those of Heuter & Neuhaus. The expressions predict the variation of attenuation constant, of sound with the electric field, showing that, at a constant temperature, the sound attenuation constant increases in the presence of electric field from the value obtained in the absence of electric field. References Jones C K & Holm J K. Phys Lell A, 26 ( 1968) Tani K, 1 Phys Soc lpn, 29 (1970) Landau L D & Kalatniov I M, Ool Aad Nall. SSSR. 96 (1954) Tani K, 1 Phys Soc lpn. 26 (1969) Tani K & Naoyui T, 1 Phys Soc lpn, 26 (1969) Heuter N P & Neuhaus D P, 1 Acousl Soc Alii, 27 (1955) aluni G N & Naithani U C, Solid slale ionic, (North Holland), 22 (1986) Naithani U C & Semwal 8 S, Pramana, 11 ( I (78) Deorani S C, Naithani U C & Semwal 8 S Pramalla, 35 (1990) 181; Inl 1 Phy Chem Solids, 51 (1990) Naithani U C & Semwal 8 S, 1 Physiqlle. 42 ( 1981 ) C Naithani U C, Lingwal D S, Asho Kumar. Deorani S C & Semwal 8 S, Sri Lana 1 Phys, 1 (00) Merz Walter J, Phys Rev, 91 (1953) Lingwal D S, Naithani U C & Semwal 8 S, II/dian 1 Pure & Appl Phys, 39 (0 1) Lingwal D S, Naithani U C & Semwal 8 S, Indian 1 Pure & Appl Phys, 39 (0 I) Lingwal D S, Naithani U C & Se mwal 8 S'. Indian 1 Pllre & Appl Phys. 39 (0 I ) Zubarev D N, Sov Phys Usp, 3 (1960) Gairola R P & Se mwal 8 S, 1 Phys Soc lpn. 42 ( 1977) 975; 43 (1977) ahadur Rita & Sharma P K, Phys Rev B, 12 ( 1975) Tani K, Phys Lell A, 25 ( 1967) 400; 29 ( 1970) 594. Pytte E, Phys Rev B. I ( 1970) Garland C W & Pars G, Phys Rev S, 29 ( 1984) See Landolt 80m Stein, Seri es Ill, Vol. 3 (Springer Verl ag, New Yor), Deorani S C, Naithani U C & Semwal 8 S, Indian 1 Pllre & Appl Phys, 29 (1991 ) 526.

Electric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites

Indian Journal of Pure & Applied Physics Vol. 49, February 2011, pp. 132-136 Electric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites Dushyant Pradeep, U C Naithani