The pseudo-spin lattice coupled mode model by adding third- and  fourth-order phonon anharmonic interactions and extra spin-lattice  interaction term has been considered for CsH2PO4 (abbreviated CDP) and deuterated CsH2PO4 (abbreviated DCDP) crystals. Expressions for shift and width of  response function, vibrational normal mode frequency, dielectric  constant and loss tangent have been evaluated. Double time temperature  dependent Green’s function method has been used for derivation. Fitting  the values of model parameters in expressions, the temperature  dependence of soft mode frequency, dielectric constant and loss tangent  have been calculated. Theoretical results are in agreement with  experimental results reported by Blinc et al16

Mamgain, Anubhuti 

Upadhyay, Trilok  Chandra

English

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Indian Journal of Pure & Applied Physics Vol. 54, December 2016, pp. 771-776       Temperature dependence of ferroelectric mode frequency, dielectric constant and loss tangent in CsH2PO4 and deuterated CsH2PO4 crystals  Anubhuti Mamgain* & Trilok Chandra Upadhyay Physics Department, H N B Garhwal University, Srinagar (Garhwal) 246 174, India Received 8 February 2016; revised 22 August 2016; accepted 18 September 2016 The pseudo-spin lattice coupled mode model by adding third- and fourth-order phonon anharmonic interactions and extra spin-lattice interaction term has been considered for CsH2PO4 (abbreviated CDP) and deuterated CsH2PO4 (abbreviated DCDP) crystals. Expressions for shift and width of response function, vibrational normal mode frequency, dielectric constant and loss tangent have been evaluated. Double time temperature dependent Green’s function method has been used for derivation. Fitting the values of model parameters in expressions, the temperature dependence of soft mode frequency, dielectric constant and loss tangent have been calculated. Theoretical results are in agreement with experimental results reported by Blinc et al16. Keywords: Ferroelectrics, Green function, Dielectric constant, Loss tangent 1 Introduction There has been considerable interest in the investigation of ferroelectric materials. Ferroelectric materials show peculiar behaviour. They are used in memory devices- as transducers, gas sensing materials, electro-optic materials and pyroelectric detectors in laser devices etc. Potassium dihydrogen phosphate, barium titanate, lithium niobate are examples  of ferroelectrics. Cesium dihydrogen phosphate (CsH2PO4) crystal is a KDP-type ferroelectric. It undergoes phase transition at 154 K. The lattice parameters of CsH2PO4 are a=7.918Å, b=6.387 Å, c=4.885 Å and β=107.60. It is an order-disorder type ferroelectric crystal. In this crystal the O-H...O bond forms a double well potential. In the paraelectric phase each proton or hydrogen ion can occupy either of two equilibrium positions along the O-H...O bond, but it is ordered in one of them in the ferroelectric phase. Proton motion between two potential minima along a given O-H...O bond is taken to be via tunnelling. This crystal belongs to space group P21/m in monoclinic system. The transition temperature becomes nearly double when H is replaced by deuterium. This is called isotope effect. It shows that protons play important role in transition mechanism in CsH2PO4 crystal. On deuteration (CsD2PO4) transition temperature shifts to 267 K showing large isotope effect. Experimental studies on CDP and DCDP crystals have been made by many workers. In all phosphate group ferroelectric crystals the transition temperature have two fold increase when they are deuterated (in KDP it becomes 213 K from 123 K,in RbH2PO4 it becomes 218 K from 147 K while in case of RbH2AsO4 the Tc of its deuterated form shifts  178 K from 110 K etc.). Blinc1 first suggested that isotope effect is due to tunnelling of hydrogen ions in double well potential in asymmetric O-H...O bonds  in these crystals below transition temperature.  In deuterated crystals since mass of dipole increases and so tunnelling diminishes. Due to this transition temperature largely increases. Theoretical studies on CDP and DCDP crystals were initiated by Ganguli  et al.2 who have considered pseudo spin model. They have used Green’s function method and obtained ferroelectric mode frequency, dielectric constant  and Tc etc. for these crystals and for similar crystal KDP. Extensive experimental studies on CDP and  its deuterated form (DCDP) have been carried out  by many experimentalists. Zhang et al.3 have established relationship between dielectric response and potential structure in CDP and DCDP crystals. Brilingas et al.4 have measured dielectric constant of these crystals at different frequencies and discussed relaxational dynamics of these crystals. Schuele and Schmidt5 have made NMR study on CDP and DCDP crystals. Raman spectra study on these crystals has been made by Wada et al.6, X-ray structure analysis study has been made by Hagiwara and Nakamura7. —————— *Corresponding author (E-mail: anubhuti.mamgain01@gmail.com) INDIAN J PURE & APPL PHYS, VOL 54, DECEMBER 2016   772 Nakamota et al.8 have done dielectric measurements on CDP crystal at high pressures. Hosseini et al.9 have made study of effect of surfactants in synthesis of these crystals as proton conductive membrane. Ahn  et al.10 have grown CDP and DCDP crystals and made characterization study on these crystals. Schur et al.11 have made lattice dynamics simulation study on these crystals. Iwata et al.12 have made neutron diffraction study on these crystals. Luspin et al.13 have made elastic properties study on CDP and DCDP crystals. Boysen et al.14 have done study of high- temperature behaviour of CDP and DCDP crystals under both ambient and high pressure conditions and found the result that CDP undergoes a polymorphic phase transition at 228±2 °C under atmospheric pressures. Kim et al.15 worked on characterization of the dynamics in the protonic conductor CsH2PO4 by 17O solid-state NMR spectroscopy and first principles calculations: correlating phosphate and protonic motion. Earlier researchers2 have not considered third-order phonon anharmonic interaction term and extra spin-lattice term. They have decoupled the correlations at an early stage. So that they could not produce better and convincing results. In this paper we have considered third-and fourth-order phonon anharmonic interactions terms as well as pseudospin ( xiS ) and phonon ( kA ) interaction term. In the present study we shall fit model values of physical quantities in the expressions obtained for CDP and DCDP crystals. Temperature dependences of soft mode frequency, dielectric constant and loss tangent will be calculated for CDP and DCDP crystals. Theoretical variation of normal mode frequency has been compared with variation obtained by correlating with experimental dielectric measurements on CDP and DCDP crystal by Blinc et al.16 Theoretical variations of dielectric constant and loss tangent have been compared with experimental data of Blinc et al.16 for CDP and DCDP crystals.  2 Model Hamiltonian For ferroelectric CDP and DCDP crystals, the pseudo-spin lattice coupled mode model (earlier used by Ganguli et al.2) by adding third-and fourth order phonon anharmonic interactions and other terms. The pseudo-spin lattice coupled mode model is expressed as: ( )∑ ∑ ∑ ∑−++−Ω−= ++i ij ik ikkziikkkkkkzjziijxi ASVBBAASSJSH ω412121  ... (1) we shall add:  ( ) ( ) ( ) ( )∑ ∑ ∑−+=ijk kkkk ikkxiikkkkkkkk ASVAAAAkkkkVAAAkkkVH43214321321 4321432132 ,,  ... (2) where Ω is proton tunnelling frequency between double well potential O-H...O bonds, xiS  is x-component of the spin variables, ijJ  is exchange interaction constant, ziS  is z-component of pseudospin variable S, ikV  is spin- lattice interaction constant kA  and kB  are position and momenta co-ordinates. ( )3213 ,, kkkV and ( )43214 ,,, kkkkV  are the third and fourth order phonon anharmonic interactions term. The third term in Eq. (2) describes an indirect coupling between the tunnelling motion of one proton and other proton. This term describes the modulation of distance between the two equilibrium sites in the O-H...O bonds. This modulates Ω by non polar optic phonons. We shall consider the total Hamiltonian for our study of CDP and DCDP crystals:  21 HHH +=  ... (3)  3 Green’s Function, Shift and Width  The method of statistical double time temperature dependent Green’s function was proposed by Zubarev19. Accordingly, for any pair of operators,  this is:  ( ) >>=<<− zjziij SSttG ;, ( ) [ ] ><−−= zjzi SStti ;,θ   ... (4)  where θ is unit step function, θ =1 for t>t′ or zero otherwise. Differentiating Green’s function Eq. (4) with respect to times t and t′, respectively, twice with the help of model Hamiltonian Eq. (3), Fourier transforming and putting in the Dyson,s Equation form, one obtain:  ( ) ( ) ( ) ( ) ( )ωωωωω 000 ~ ijijijij GPGGG +=  ... (5)  where, ( ) ( )220 4Ω−><Ω=ωpiωxiSG  ... (6) and,   ( ) [ ] 22222,,2,~><Ω>><<+><Ω><=xijixiiSFFSBFPpipiω  ... (7) MAMGAIN & UPADHYAY: DIELECTRIC CONSTANT OF CsH2PO4 AND DEUTERATED CsH2PO4 CRYSTALS   773 where,  ∑∑ ∑∑ ∑∑ ∑∑Ω+Ω−Ω−Ω−=i ij i ij i ik i ikzikikxikikxiziijzjxiiji SAVSAVSSJSSJF 42 ∑∑ ∑∑ ∑ ∑+−−−ik ij ik ij ik ikzikikxikikxiziijkikzjxiijkik SAVSAVSSJAVSSJAV22222121  ... (8)  The Green function (GF) can be now be written as:  ( ) [ ])(~ 22 ωωpiω PSGxi−Ω−><Ω=   ... (9)  where,   >′<><+Ω=Ω )](),([4~ 22 tStFSi yiixi   ... (10) and,   ( ) ( )>><<><Ω=,,);( tFtFSP jixipiω   ... (11)  Solving Eq. (10) we obtain:  bcba −+=Ω 222~    ... (12)  Where  ><=Ω=><=xiziSJcbSJa2.  The second term of Eq. (7) is solved by decoupling of higher order Green′s functions >>′<< ji FF , . The decoupling is done as follow:  >>><<<+>>><<<+>>><<<+>>><<>>=<<<CABDDABCCBADDBACCDAB;;;;; We obtain value of ( )ωP of Eq. (11). We resolve ( )ωP  into real and imaginary parts using formula:  )(11lim xiximxompiδ±=+→  The real part is called shift ( )ω∆  and imaginary part is called width ( )ωΓ . These are obtained, respectively, as:  ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )22422422242222222222222222222222224~6~2~~~8~42~~~,,,,kkkkzikikkkkkxikikkikkikkkkkziikkikkkkkxiikkikbaSNVSNVbaNVNVcSaVNVbSbVaNVcbaωωδωωωδωωωωωδωωωωδωωωωω−Ω><+−Ω><+Ω−+Ω−+−><+Ω−+−><+Ω−+Ω−+Ω−=∆   ...(13)  and,  ( ) ( ) ( ){ } ( ) ( ){ } ( ) ( ){ }( ) ( ){ } ( ) ( ){ }( ) ( ){ } ( ) ( ){ } ( ) ( ){ }( ) ( ){ } ( ) ( ){ }kkkkkkzikikkkkkkkxikikkikkikkkkkziikkikkkkkxiikkikbaSNVSNVbaNVNVcSaVNVbSbVaNVcbaωωδωωδωδωωωδωωδωδωωδωδωδωδωδωδωδωωδωδωδωδωδωωδωδωδωδωδωδω~~~26~~~22~~~2~~~2~~~28~~~24~~~22~~~2~~~2~~2,,,,442422222222224+−−Ω><++−−Ω><+Ω+−Ω−Ω+Ω+−Ω−Ω+Ω+−Ω−><+Ω+−Ω−Ω+Ω+−Ω−><+Ω+−Ω−Ω+Ω+−Ω−Ω+Ω+−Ω−Ω=Γ  ... (14) where ( )ωkΓ  is the phonon width due to third  and fourth-order phonon anharmonicities. kω~ and kΓ  are phonon-frequency and phonon width respectively. ( )ωk∆ is phonon shift, these are given as:  ( ) ( ) ( )( )( ) ( )( ) ( )( )( )( ) ( )( ) ( ) −−−−−−+−+++−+++++×−+−−−−+−++×−==∆∑∑22222321422222213321321133221321321133221321321321212112212121212121~~~*13~~~1~~~|,,,|48~~~~~~~~~~|,,|18RekkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkknnnnnnnnnnnnkkkkVPnnnnkkkVPPωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωω  + higher terms .  ... (15)   ( ) ( ) ( )( )( ) ( )( )( ) ( )( )∑+−−−+−+−−+++×−==Γ212121122121212121~~~~~~~~~~|,,9Im 2213kkkkkkkkkkkkkkkkkkkknnnnkkkVPωωωδωωωδωωωδωωωδωωωωpiωω ( )( ) ( )( )( )∑+++×−−−−+++×+321321321133221321321~~~1~~~~~~|,,,|48 243214kkkkkkkkkkkkkkkkkkkkknnnnnnkkkkVωωωωωωωωωωδωωωωδpi  +higher terms   ...(16) INDIAN J PURE & APPL PHYS, VOL 54, DECEMBER 2016   774 4 Normal Mode Frequency By putting value of ( )ωP in Eq.(9), the value of Green′s function (GF) is finally obtained as:  ( ) ( ) ( )[ ]( ).~ˆwhere,,2ˆ221221ωωωδpiεω∆+Ω=ΩΩΓ+Ω−><=+−− iSiG ijxi  ... (17)  Solving Eq.(17) self consistently, we obtain renormalized frequency as:  21442222222,,,,62824~~~21~~~21ˆΩ><+Ω><+><+><+ Ω−± Ω+=ΩbaSNVSNVSaVSbVkkkzikikkkkxikikkkkziikkkkxiikkk δωδωδωδωωω   ... (18) The frequency Ωˆ  is the normal mode frequency which decreases with temperature from below Tc, and so is responsible for ferroelectric phase transition.  5 Dielectric Constant and Tangent Loss The response of crystal to electric field is expressed by electrical susceptibility which using Zubarev′s17 formalism is expressed as:  −=χ )(2lim 2 εωµpi iGNo ij+∈→ . .. (19)  where, N is no of dipoles having dipole moment µ in the sample using relation piχε 41 += . The expression for dielectric constant with the help of Eq. (17) and Eq. (19) can be expressed as:  ( ) ( ) ( )( )[ ] 122222222 4ˆˆ241 −ΓΩ+Ω−Ω−><−=− ωωµpipiωε xSN    ... (20)  The dissipation of power can conveniently be expressed in dielectrics as tangent loss (tanδ) as:  εεδ′′′=tan   ... (21)  Using Eq. (21) one obtains:  ( )( )22 ˆ2tan Ω−ΩΓ−= ω ωδ   ... (22)  At microwave frequencies Ω<< ˆω , above Eq. (22) reduces to: ( )2ˆ2tanΩΩΓ=ωδ   ... (23) 6 Results and Discussion By using model values2 of various physical quantities in expressions from literature, temperature dependence of normal mode frequency, width, shift and dielectric constant for CDP and DCDP crystals have been calculated and have been shown in Figs 1-6 and corresponding values are listed in Table 1.   Fig 1 – Temperature dependence of soft mode frequency Ωˆ (cm-1) in CDP crystal (— present calculation, ♦ correlated values with experimental values of Blinc et al.16)    Fig 2 – Temperature dependence of dielectric constant ( ε ) in CDP crystal (—present calculation, ♦ correlated with experimental values of Blinc et al.16)    Fig 3 – Temperature dependence of loss tangent (tanδ ) in CDP crystal (—present calculation, ♦ correlated with experimental values of Blinc et al.16) MAMGAIN & UPADHYAY: DIELECTRIC CONSTANT OF CsH2PO4 AND DEUTERATED CsH2PO4 CRYSTALS   775 The theoretical variations of normal mode frequency with temperature are compared with values obtained by correlating the experimental results of Blinc et al16. The calculated dielectric constant versus temperature and loss tangent versus temperature curves agree with experimental data of Blinc et al16. Earlier workers2 have decoupled the correlations functions at an  early stage. Therefore some important interactions disappeared from their results. Equations (15), (17) and (18) show that if third-order phonon anharmonic interactions terms and extra spin-lattice interactions terms are neglected, our expressions reduce to the expressions of earlier authors2. The width and shift are contributions of present work. Equation (18) shows that soft mode frequency Ωˆ  decreases from below Tc becoming very small at transition temperature, then increases. Thus confirming Cochran′s suggested behaviour. The dielectric constant first increases from below Tc becoming large near Tc then decreases. Similarly loss tangent shows its behaviour.   7 Conclusions Present study shows that the modified pseudospin-lattice coupled mode model along  with third- and fourth-order phonon anharmonic interactions and extra spin-lattice interactions  terms explains well the temperature dependence of normal mode frequency, dielectric constant and loss tangent in CDP and DCDP crystals. Earlier authors1 did not consider terms which we have considered. Hence our results are much better and convincing, which agree with experimental results of Blinc  et al16. Present results may also be useful to other similar crystals RbH2PO4, KH2AsO4, RbH2AsO4 and CsH2AsO4.  Acknowledgements One of the author AM is thankful to UGC,  New Delhi for a Research Fellowship to H N B Grahwal University, Srinagar (Garhwal).  References 1 Blinc R, J Phys Chem Solids, 13 (1960) 204. 2 Ganguli S, Nath D & Chaudhuri B K, Phys Rev B, 21  (1980) 2937. 3 Zhang L, Kleemann W & Zhong W L, J Phys C, 14 (2002) 5187. 4 Brilingas A, Grigas J, Mizaras R & Sobiestianskas, Ferroelectrics, 11 (1995) 170. 5 Schuele P J & Schmidt V H, Phys Rev B, 39 (1989) 2549.  6 Wada M, Sawada A & Ishibashi Y, J Phys Soc Jpn, 47 (1974) 1571.  7 Nakamura E, Itch K & Hagiwara T, J Phys Soc Jpn, 52 (1983) 2626.  8 Nakamota Y, Tashima K, Endo S, Ming L C, Azumi S & Kakegawa T, J Korean Phys Soc, 46 (2005) 2. 9 Hosseini S, Daud W R & Badiei M, Bull Mater Sci, 34 (2011) 759.  10 Ahn, Mangani I R, Park C W & Kim J, J Power Sources, 163 (2006) 107.   Fig 4 – Temperature dependence of soft mode frequency Ωˆ (cm-1) in DCDP crystal (—present calculation, ♦ correlated values with experimental values of Blinc et al.16 for dielectric constant)    Fig 5 – Temperature dependence of dielectric constant (ε ) in DCDP crystal (—present calculation, ♦ correlated with experimental values of Blinc et al.16 )    Fig 6 – Temperature dependence of loss tangent (tanδ ) in DCDP crystal (—present calculation, ♦ correlated with experimental values of Blinc et al.16)  Table 1 – Model values of physical quantities for CDP and DCDP* crystals Tc (K) Ω (cm-1) J (cm-1) J* (cm-1) Vik (cm-1) ω k (cm-1) µ ×10-18 (esu) 154 80 350 450 1.72 130 2.3 267* 1.710* 399.15* 395.96* 14.16* 130* 2.3* INDIAN J PURE & APPL PHYS, VOL 54, DECEMBER 2016   776 11 Shchur Ya, Bryk T, Klevets I & Kityk A V, Comput Mater Sci, 111 (2016) 301.  12 Iwata K & Deguchi T, J Phys Soc Jpn, 63 (1994) 4044. 13 Luspin Y, Vaills Y & Hauret G, J Phys I France, 7 (1997) 785. 14 Boysen D A & Haile S M, Chem Mater, 15 (2003) 736.  15 Kim G, Griffin J M, Blanc F, Haile S M & Grey C P, JACS, 137 (2015) 3867. 16 Blinc R & Zeks B, Phys Rev Lett, 43 (1979) 231. 17 Zubarev D N, Sov Phys Usp, 3 (1960) 320.   

Temperature dependence of ferroelectric mode frequency, dielectric constant and loss tangent in CsH2PO4 and deuterated CsH2PO4 crystals

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Temperature dependence of ferroelectric mode frequency, dielectric constant and loss tangent in CsH2PO4 and deuterated CsH2PO4 crystals

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