Phenomenological explanation of spontaneous polarisation and onset ferroelectric Phase transition in RbH2AsO4 (RDA) crystal

By applying two-time thermal dependent Zuberav’s statistical, retarded Green function approach and modified earlier simple PLCM model Hamiltonian by adding some extra terms into it, like third-order and fourth-order, phonon anharmonic interactions, direct spin-spin terms, extra spin-lattice terms, and four body interaction terms, for theoretical investigation of thermal dependent spontaneous polarization and ferroelectric phase transition in the first-order phase, of RbH2AsO4crystal. It undergoes a ferroelectric phase transition at 109.9K. With the help of Dyson’s equation in the Mean-Field Approximation (MFA), theoretical formulae are obtained for electrical permittivity, tangent delta, Cochran’s mode frequency, spontaneous polarization, and response function. Model values are fitted for the above physical parameters to obtain variations with temperature. A comparison of theoretical finding has been made with the experimental finding reported by Blinc et al [12], and Zolototrubov et al[8].


Introduction
Rubidium dihydrogen arsenate crystal is an inorganic and prototype of an order-disorder class of ferroelectric crystals. Another type is displacive one like BaTiO3, SrTiO3, LiTiO3, etc. RbH2AsO4 crystal is an H-bonded isomorphic crystal of dihydrogen, phosphates, and arsenates of potassium, rubidium, and cesium (KH2PO4, RbH2AsO4, RbH2PO4, CsH2AsO4, and CsH2PO4), etc. and these type of crystals are formed a very interesting and famous group called KH2PO4 family by Schmidt [1]. The investigation of RbH2AsO4 crystal is of obvious interest aroused because of their peculiar properties, which are widely used in computers and electronics as Laser technologies, memory devices, optical devices, pyroelectric detectors, gas sensing material, detectors, and modulators. So, their investigation is useful both from theoretical as well as technological applications. RbH2AsO4 crystal is abbreviated as RDA. Rubidium dihydrogen arsenate crystal is characterized by covalently bonded AsO4 units by bridging H-bonds and ionic bonding between rubidium cations and AsO4 anions. It consists of two interpenetrating bodycentred lattices of AsO4 tetrahedral and two Rb lattices, in which the rubidium and arsenate are separated by the c-axis. It undergoes a phase transition at Tc = 109.9K. On deuteration, Tc shifts to 163K due to the exchange of proton with hydrogen shows a large isotopic effect. It shows orthorhombic structure when (T<Tc) and H22 type tetragonal structure above Tc.
In the theoretical approach presented here, we have considered in this work the direct spin-spin interaction term, extra spin-phonon interaction term, (third-order and fourth-order lattice anharmonic interactions terms), and four body interaction term in the (PLCM) model for explaining the physical properties of RbH2AsO4 crystal. The phonon anharmonic interaction terms and four body interaction term in the PLCM model are the important terms for explaining the physical quantities like permittivity, tangent delta, Cochran's mode frequency, and polarization for KH2PO4 type crystals. Determination of expressions by modified model, using thermal statistical two times Green function technique. The above quantities have been calculated by using Zuberav's statistical approach and the Green function technique for different temperatures in RbH2AsO4 crystal. We have compared our theoretical calculated findings with the experimental finding of Blinc et al. [12] and Zolototrubov et al [8].

Modified Hamiltonian
For explaining the ferroelectric phase transition, and spontaneous polarisation in RDA crystal, [19] model Hamiltonian is modified by adding extra interaction terms into it as and  , , k k k k V represents third-order and fourth-order lattice's phonon anharmonic interaction terms, ' ' ij J is the four body interaction constant and this interaction plays an important role in the Oxygen Hydrogen-Oxygen bonds. The x S A k i and 2 x S A i k terms represent an indirect coupling between proton-proton tunnelling motion, and also represent the variation of space between two sites at an equilibrium distance of Oxygen Hydrogen Oxygen bonds in H bonded RbH2AsO4 crystal and Bij term represent the coupling constant Therefore resultant H for RDA crystal as;

Response Function (Shift and Width), and Dyson's equation
The two-time temperature-dependent statistical method of retarded Green's function technique by Zubrerav [38] is expressed as; , : where angular bracket in equation (3) <<…. >> represents the Green function notation, <….> represents the average grand canonical ensemble, and -iθ(t-t ' ) represent unit step function, and its value is 1 for t>tʹ or zero otherwise. Now we write the equation of the Green function given by Equation (3) by using H in Eq. (2); double time differentiation in terms of t ʹ and t, and writing in the following form.
We, obtain and where, Fi (t) and Fi (t ' ) are given as; Where G 0 (ω) in Eq. (5) is the zeroth-order Green's function, and are obtained as In Eq. (7) P (ω) is the response function and Ω is the normalized pseudo-spin frequency in the lowest order approximation. Different product of operator's likes The parameter  was arbitrarily chosen as +1 or -1. If both x S and z S are (Bose) operators,  remained as +1, and if they are (Fermi) operators,  is to be -1. Equation (7)  . Therefore we resolve   P  into its real Δ(ω) and imaginary Γ(ω) parts called to shift and width. We, therefore, obtain the quantities of Δ (ω) and Γ (ω) is expressed as : : : : ....
: : : : : : : : : The above obtained shift    16 and 17]. This is because we have considered direct spin-spin interaction terms, extra spin-lattice terms, four body interaction terms, and four spin coupling terms in model Hamiltonian. We aim to take into account all , Now, from equations (14) and (15) Here : h , and P represents the principal part of the above equation.

Permittivity (ε) and Tangent delta (δ)
Following by Zuberav's [20], the susceptibility (  ) is expressed as The electrical permittivity    associated with the susceptibility of permittivity (  ) as     In equation (15) we have obtained as