Dynamical model for phase coexistence in proton glass V. Hugo Schmidt Citation: AIP Conference Proceedings 436, 192 (1998); doi: 10.1063/1.56272 View online: https://doi.org/10.1063/1.56272 View Table of Contents: http://aip.scitation.org/toc/apc/436/1 Published by the American Institute of Physics Dynamical Model for Phase Coexistence in Proton Glass V. Hugo Schmidt Physics Department, Montana State University, Bozeman, MT 59717 Abstract. We describe a model for static and dynamic behavior of KH2PO4 (KDP) type crystals in the ferroelectric phase, and for proton glass crystals of this type in the temperature and composition range in which the ferroelectric and paraelectric phases coexist. Model predictions are compared with experimental results for ferroelectric RbH2AsO4 (RDA) and for the proton glass Rbl_• (RADA). The model is based on the statistical mechanics and dynamics of a fundamental excitation in the ferroelectric phase, namely chains consisting of nonpolar H~.AsO4 groups with an H3AsO4 group at one end and an HAsO4 group at the other end. INTRODUCTION Proton glasses offer a rich variety of phenomena, and are of particular interest because their dynamical processes can be understood in terms of simple excitations. We reported coexistence of ferroelectric and paraelectric phases in Rbl.x(NH4)xHxAsO4 (RADA) based on dielectric permittivity (1-3) and spontaneous polarization (3) measurements. We have employed other methods and studied other crystals as well, and have discussed the nature of such coexistence (4,5). This paper develops a model which explains ome of the phenomena observed in the coexistence r gion of the phase diagram. DESCRIPTION OF MODEL This model for static and dynamic properties of coexistence is based on the statistics of the most prevalent excitations in ferroelectric domains in KDP-type crystals. We choose Rbl.x(Ni-Ia)xH2AsO4 (RADA) as the crystal with which we compare model and experimental results in this work. In such crystals, these excitations are chains consisting of an H3AsO4 group and an HAsO4 group connected by nonpolar H2AsO4 groups. A nonpolar H2AsO4 group has energy eo, whereas a polar H2AsO4 group has zero energy, according to the Slater model (6). Each AsO4 arsenate ion is connected by four O-H..-O "acid" hydrogen bonds to neighboring arsenate ions. Most arsenate CP436, First-Principles Calculations for Ferroelectrics edited by Ronald E. Cohen 9 1998 The American Institute of Physics 1-56396-730-8/98/$15.00 192 ions have two protons close (hence the H2AsO4 designation for such groups), in these four bonds in which protons occupy off-center positions. If both bottom (relative to the c axis) protons are close, the As 5+ ion is pushed upward and the group has a positive electric dipole moment ~t along c. If both top protons are close, the moment is negative. The remaining four H2AsO4 proton arrangements produce dipole moments in the ab plane and contribute to dielectric permittivity in that plane. They are called nonpolar because they have no moment along c, the axis for ferroelectric polarization. A chain of such nonpolar groups must terminate with a "Takagi" group of energy ~z, typically about 5~0 in KDP-type crystals. Takagi (7) incorporated them into Slater's model and obtained better agreement with experiment. The reason for existence of such chains can be understood by following the dynamics of creation and growth of such a chain. Creation occurs within, say, a positive ferroelectric domain by means of one intrabond proton transfer, t t 2AsO 4 + H2AsO 4--YH sAsO 4+ HAs0 4. (1) The proton is transferred to a site close to one of the two top oxygens of an H2AsO4 group, making it an H3AsO4 group and changing one of the two groups just above it to an HAsO4 group. This Takagi group pair has an apriori statistical weight of 2, relative to 1 for the original polar ion at the bottom of this short chain. The chain can grow upward if a proton moves close to the top of the HAsO4 group, thereby converting it to a nonpolar H2AsO4 group because it has one close proton each at the top and bottom. The HAsO4 group has in effect moved to one of two possible positions higher along the c axis, thereby again doubling the a ptqori statistical weight of the chain, to 4. The chain energy has now increased from 2a~ to 2~+e0. We ignore the possibility that the last proton moves away from the HAsO4 group, because that would entail a large energy increase, of order el. In general, if no ammonium ions are encountered, the Boltzmann factor W,s in the Slater limit for a chain ofn links, involving n proton transfers, is W,,s= Yexp{-[2c~ + (n-1) c@k l' }. (2) For each step that the chain lengthens, the Boltzmann factor increases by the factor 2exp(-g0/kT). We pointed out long ago (8) that the Slater model ferroelectric transition temperature Tc(Slater) = cr (3) is exactly the temperature atwhich growth of the chain by one step does not change the Boltzmann factor, thereby allowing such chains to grow without limit and destroy the ferroelectric order. This perfect agreement of the chain model with the Slater model hinges on being at the Slater limit, for which ~-+oo. For finite Takagi energy e~, a large number of chains 193 of finite length will destroy the ferroelectric order at a temperature below the Slater transition temperature. APPLICATION OF MODEL TO RDA As a first step in developing the chain model, without the complication of randomly placed ammonium ions, we apply it to pure RbHzAsO4 (RDA) and compare results with experiment and with Takagi model predictions. If Wn is the probability that a chain of n links has its bottom end at a given arsenate anion site, and one notes that each link destroys the dipole moment of one arsenate anion (more exactly, of one fornmla unit), then the ferroelectric order parameter p is given by p I-Z.W,, ]-Z(nW,~s)'XW~s., (4) These sums, and all sums over n in this work, run from zero to infinity. The Wns for n=0 is 1, and the others are provided by Eq. (2). Upon evaluating these sums, we find p(c'hailV- (a-he) (a. bc).,/(a-bc9 (a+ bc) + b:/2], (s) whereas the Takagi result is p(Takagi) --[(a-b) (a + b) f @a. (6) Here. a=l-2exp(-e0/kT), b=2exp(-2l/kT), and c=exp(-80/2kT). The transition temperature Tc is the lowest temperature at which p becomes zero. The transition temperatures for these models are given by 7c(chain)--d,a-bc- l-2exp(-eo/k Tc)-2exp{-(el+ O.58o)/k Tc} O, (7) ]'c(Takagi) --m-b l-2exp(-go, k 2~.)-2exp(-gl.,k Tc) =0. (8) The chain model result for p differs from the Takagi result in that p(chain) has finite slope at To(chain), while the Takagi model predicts infinite slope at Tc(Takagi) as expected for a second-order t ansition. This defect of the chain model occurs because intersection of chains is neglected, and this neglect becomes erious near the transition. Neither model predicts the measured weakly first-order nature of the transition. Senko (9) added a long-range interaction to the Takagi model, which was shown by Silsbee, Uehling, and Schmidt (10) to change the transition to first order if this long-range interaction is large enough. To make numerical and graphical comparison of these model predictions with experiment, we use the experimental value Tc=109.75 K given by Fairall and Reese (11) for RDA, and the value el/kTc=4.20 provided by them, which corresponds to 194 eJk=460.95 K. Because they included both the above-mentioned long-range interaction, and the tunneling effect introduced by Blinc and Svetina (12), their value for e0/kTr is much too low to provide the measured Tc if only it and their El/kT~ are used in the Takagi model. Accordingly, we instead use for eo/k the value 79.41 K which when used in the Takagi model together with eJk=460.95 K gives the measured T~. With these values for e0 and el, which force the Takagi result for Tc to coincide with the experimental value, we obtain for Tr given in Eqs. (3), (7), and (8) for the various models the values T~(experimental)=T~(Takagi) = 109.75 K, T~(chain) =111.00 K, To(Slater) =114.56 K. APPLICATION OF MODEL TO RADA The reasonable agreement of the chain model result with the Takagi result encourages us to apply an extension of this chain model to proton glass crystals in the range of x for which coexistence of ferroelectric and paraelectric/proton glass phases occurs. This extension provides for a reduction in the chain energy wherever the chain passes by an ammonium ion. Although each cation site has probability x of containing an ammonium instead of a rubidium ion, this is not a mean field model because we specifically take into account the preference for chains to choose paths with neighboring ammonium sites. Taking this preference into account yields the non-mean- field result that even at zero temperature, the crystal contains both ferroelectric and proton glass phase regions for all x between 0 and xc, the concentration at which ferroelectricity disappears at zero temperature. To take the effect of ammonium ions into account, we use an interaction we introduced previously (13) in an extension of the Slater model. This is the "cross- cation interaction" which lowers the energy by ~, whenever the two "acid" hydrogens opposite each other across an ammonium ion occupy sites consistent with an antiferroelectric ather than a ferroelectric configuration. There are two such hydrogen pairs for each ammonium ion, so the potential for the ammonium ion to destroy ferroelectric order is not exhausted until two acid hydrogens adjacent to the ammonium ion have moved away from their ordered-phase positions. Each acid hydrogen is bonded to two oxygens, and each of these oxygens can form an N-H--.O bond to a different ammonium ion if such an ion is in that cation site. If the chain has n displaced hydrogens and runs past m ammonium ions, its unnormalized Boltzmann factor Wnu, the mixed-crystal nalog of Wns in Eq. (2), is Wnmu : [2n(] -y) 2n-mym('Jn) .[/(2n-lJvl) .Ira .]_]f(Untr~. (9) 195 Here y is the effective x, taking into account he above-mentioned saturation of the tendency of ammonium ions to destroy ferroelectfic order, and given by y=x-O. 5 Z(m W~m~) / VW .... (lo) Here and in the following equations, Z signifies a sum over n from zero to infinity, and a sum over m from zero to 2n. After solving Eqs. (9) and (1 0) self-consistently for y, the ferroelectric order parameter is found from the analog of Eq (4), namely p- l -Z (nW.~) = J -Z(nW., .~)/z-~V ..... (11) The factor f(U~) in Eq. (9) depends on the chain energy U~, given by (12) One might think that f(U~m) should have the Boltzmann form seen in Eq. (2), but use of that form in this calculational method would give completely erroneous results for mixed crystals. The reason for such erroneous results is that sufficiently large m can give negative U~m. If we would construct a finite-size crystal with a given ammonium ion distribution, the ground state corresponding to zero temperature would contain a number of such chains with negative nergies of various magnitudes. Instead, we are looking at all the possible chains that can begin at one given anion site, in a crystal of infinite extent. If we would employ Boltzmann statistics, then at zero temperature only the lowest energy state (of energy negative infinity!) would be occupied. A better choice for f(U~) would be the Fermi-Dirac distribution function, even though we are dealing with distinguishable particles so that there is no physical necessity for using Fermi-Dirac statistics. A simpler choice which also avoids the above low-temperature catastrophe would be to choose f = l./br Unm_O. (13) The f=l choice is consistent with the unnormalized probability W00u=l which must be used in the above sums, and simply means that no given configuration of a chain beginning at a single site can have more than single occupancy. The possibility of negative U~ in Eq. (1 2) justifies the above statement that a mixed crystal cannot be in a completely ordered ferroelectric state. If in Eq. (1 2) we would replace m by its mean-field (m-f) value 2nx, then at zero temperature the crystal would have complete ferroelectric order for x