Physics Research International

Volume 2016, Article ID 9475740, 10 pages

http://dx.doi.org/10.1155/2016/9475740

## Study of Dielectric Properties and Ultrasonic Attenuation in KDP-Type Ferroelectrics

USIC, HNB Garhwal University, Srinagar, Garhwal, Uttrakhand 24 6174, India

Received 12 October 2015; Accepted 19 January 2016

Academic Editor: Sergey B. Mirov

Copyright © 2016 Vijay Singh Bist and Narayan Singh Panwar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The soft mode dynamical model has been used to study dielectric properties and ultrasonic attenuation in KDP-type ferroelectric crystals. The model Hamiltonian proposed by Blinc and Zeks has been modified by considering lattice anharmonicity up to fourth-order. The correlations appearing in the dynamical equation have been evaluated using double-time thermal retarded Green’s functions method and Dyson’s equation. Without any decoupling, the higher order correlations, appearing in the dynamical equation, have been evaluated using the renormalized Hamiltonian. The expressions for collective frequencies, width, dielectric constant, ultrasonic attenuation, and tangent loss have been calculated. The dielectric properties and ultrasonic attenuation strongly depend on the relaxational mode behavior of stochastic motion of H_{2}PO_{4} group in KDP-type ferroelectrics. By fitting model values of physical quantities, the temperature dependence of and for different value of four-body coupling coefficient and dielectric constant and loss tangent has been calculated. The calculated and observed results have been found in good agreement.

#### 1. Introduction

The oscillations of atoms in solids are responsible for different characteristics, such as specific heat, optical, dielectric, and electrical properties. The anharmonicity in solids is responsible for existence of thermal expansion, temperature variation of elastic constants, lattice thermal conductivity, deviation of specific heat from Dulong-Petit law at high temperature, existence of ferroelectricity in certain materials, and so forth. Many attempts have been made theoretically and experimentally to find the explanation of these phenomena in terms of anharmonicity. Extensive reviews [1, 2] are available discussing the contribution of anharmonicity to various properties of crystals.

In order-disorder type ferroelectrics, as Potassium Dihydrogen Phosphate (KH_{2}PO_{4}), the transition is associated with the tunneling of proton through a barrier between two positions of minimum potential energy in double well potential in the hydrogen bond at the transition temperature [3]. Busch [4] was the first to show that KDP (KH_{2}PO_{4}) exhibits a phase transition at low temperature. KDP is prototype of a family of crystals with bridging hydrogen bonds and its physical properties have been extensively studied [3, 5–10].

Kaminow and Damen [11] first observed the soft mode associated with the ferroelectric phase transition of the KDP-type crystal at 122.3 K by measuring the low frequency Raman scattering in configuration. Since then soft mode which is connected to the susceptibility along the crystalline -axis through the Lyddane-Sachs-Teller relation [12] has been extensively studied by Scott [13] and interpreted using the pseudospin model [14–16] and its modifications in [3, 17]. In these theories, such a particular mode of proton motions along hydrogen bonds in plane is coupled to other ion modes, bearing an electrical dipole moment along -axis is considered to play an essential role for the ferroelectric transition, and therefore little attention has been paid to the modes other than () soft mode. In KDP crystal, however, there are four tunneling protons in a primitive unit cell of the paraelectric (hereafter referred to as PE) phase and consequently four normal modes are belonging to (), a doubly degenerate , and modes [18]. () mode, which is both infrared and Raman active, provides valuable information, because this mode and () mode reflect the nature of collective proton motion which triggers the phase transition, and, moreover, in contrast to (), it is directly related to the polarization along the crystalline -axis.

At first, Pak [19] employed Green’s function methods in the order-disorder type ferroelectrics and, however, did not consider the anharmonic interactions. The phonon anharmonic interactions have been found very important in explaining dielectric, thermal, and scattering properties of solids by many authors [6–9, 20, 21] in the past. Pak’s theory was further developed by Ramakrishnan and Tanaka [22], who calculated the excitation spectrum of the system but did not consider the anharmonic interactions. Their attempt, however, established the superiority of Green’s function method over the other methods. Ganguli et al. [23] modified Ramakrishnan and Tanaka theory by considering anharmonic interaction. Their treatment explains many features of order-disorder ferroelectrics. However, due to insufficient treatment of anharmonic interactions, they could not explain quantitatively good results and could not describe some very interesting properties, like dielectric properties, ultrasonic attenuation, relaxation rate, and so forth.

In the present study, the four-particle cluster model Hamiltonian with the phonon anharmonicity up to fourth-order has been taken to theoretical study of dielectric properties and ultrasonic attenuation in KDP-type crystals, using double time Green’s functions method and Dyson’s equation. Proton Green’s function and phonon Green’s function have been solved for the collective system. Expressions for collective mode frequency shifts, widths, transition (Curie) temperature, and the expectation value of the proton collective mode components at site (,) have been derived and discussed in KDP-type crystals. By fitting model values of physical quantities, the temperature dependence of and for different value of four-body coupling, dielectric constant, and loss tangent has been calculated. The calculated and observed results have been found in good agreement.

#### 2. The Model Hamiltonian and Green’s Function

For KDP crystal, the four-particle cluster model Hamiltonian [24] along with third- and fourth-order phonon anharmonic interaction terms is expressed aswhere the first two terms constitute the original pseudospin model Hamiltonian and the third is the quadrupole contribution (the four-body interaction). is the tunneling operator which measures the tunneling power of the proton between the hydrogen double well, the tunneling frequency, and the half of the difference of the occupation probabilities for the proton in the two equilibrium positions of hydrogen bond. is the two-body coupling coefficient and is the same for every pair of protons in KDP and the four-body coupling coefficient, and refers to the four hydrogen bonds in the PO_{4} group in KDP. In the last fourth terms is bare phonon frequency, and are displacement and momentum operators, is proton-lattice interaction term, and and are the third- and fourth-order anharmonic coefficients.

##### 2.1. Collective Proton Wave Width and Shift

The correlations appearing in the proton response function can be evaluated using double time thermal retarded Green’s function [25] using the symmetrical decoupling scheme, after applying Dyson’s treatment:where the angular brackets denote the average over the large canonical ensemble and is the Heaviside step function having propertiesDifferentiating (2) twice with respect to time “,” using Hamiltonian (1) taking Fourier transformation, one obtains where is higher order Green’s functions:with is calculated by differentiating (5) twice with respect to “” using Hamiltonian (1) and then taking Fourier transformation; one obtainsand higher order Green’s functionswithSubstituting the value of from (7) into (4) and using Dyson’s equation, one obtainswhere the renormalized frequency is withHigher order Green’s functions are calculated using symmetrical decoupling scheme, the cross combinations are not considered because they do not contribute significantly, and (10) can be written aswhere is the proton renormalized frequency of the coupled system, which on solving self-consistently takes the formThe real and imaginary parts of (12) are obtained by using the formula and represent collective proton mode frequency shift and width given aswith

##### 2.2. Collective Phonon Half Width and Mode Frequency Shift

The acoustic phonon frequency width and shift are obtained analogously from acoustic phonon Green’s function: Differentiating (18) twice with respect to time “,” using Hamiltonian (1) taking Fourier transformation, one has whereHigher order Green’s functions are evaluated without any decoupling and using renormalized Hamiltonian. Putting the evaluated value of higher order Green’s function in (19), one getsThe real part of is obtained by using (14) and the collective mode frequency shift is obtained aswithwhere is occupation number and , being Boltzmann’s constant and being the absolute temperature. Calculating (20) self-consistently and approximating, the collective mode frequency is given byEquation is obtained using model Hamiltonian (1), and and are obtained without decoupling and using the renormalized Hamiltonian The imaginary part of is obtained by using (19) and collective phonon half width is obtained as

##### 2.3. Order Parameter Values of , , and

The expectation values of the proton collective mode component at site “” have been obtained by Blinc and Zeks [26] as In PE phase (), (26) represent a system of equations for the average value of the collective mode components. The solution of this system will, however, be stable only if they minimize the free energy, that is, if , and soin the ferroelectric phase () [27]. Consider

##### 2.4. Dielectric Constant and Tangent Loss

Following Kubo [28] and Zubarev [25] the real part of dielectric constant is given by where is the effective dipole moment per unit cell and is the number of unit cells in the sample.

The dielectric loss is defined as the ratio of imaginary and real parts of dielectric constant and can be written asThus retarded phonon Green’s function is enough to determine the dielectric constant and loss tangent. Using (29) and (22), the real part of dielectric constant can be written asFor the experimental range of frequencies (as well as for KDP crystals), (31) can be reduced towhere is given by (20) and is given by .

The tangent loss is given bywhere is given by (25). For , (33) becomeswhere is harmonic and defect contribution, and are due to three and four phonon anharmonic interaction terms of the lattice.

##### 2.5. Ultrasonic Attenuation

The expression for ultrasonic attenuation constant is given bywhere damping constant is given by (17) and is the ultrasonic velocity. For small limit, we obtain the collective proton mode frequency width, for , from (17) expressed aswhere is polarization relaxation time and is given by Litov and Garland [29] as Similar expressions for frequency shift and ultrasonic attenuation have been derived by Zurek [30], using the Landau-Khalatnikov theory. If the collective mode damping is neglected, a discontinuity in sound velocity is obtained at , since is proportional to () for the effect of factor is to smooth out the discontinuity.

The ultrasonic attenuation peaks for and (i.e., ) goes to zero as For temperature such as , the attenuation will have the formwhereThe dependence of ultrasonic attenuation on applied frequency and the collective mode frequency () may be expected to apply also to the displacive phase transition [31], for which the coupling is linear to the strain and bilinear to the soft optical normal mode coordinates, independent of whether the phonon instability occurs at , , or at a general point in the Brillouin zone. Though the linear dependence of () on has been assumed for , this dependence presumably breaks down sufficiently close to .

##### 2.6. Transition Temperature

In the PE phase , and the stability limit of PE phase is determine by the temperature where approaches zero as Considerusing (13), where with effective exchange coupling constantas well as transition temperature

#### 3. Comparison with Experiments and Discussion

##### 3.1. Numerical Calculations

The parameters in our calculation are listed in Table 1. The calculated values of and for KDP-type crystals for different values of four-body coupling coefficient , collective phonon mode frequency , transverse dielectric constant , observed dielectric constant , and tangent loss along -axis and -axis for KDP-type crystals are listed in Tables 2 and 3. Their variations with temperature are shown in Figures 1–4.