The influence of hydrostatic pressure on DADP-type antiferroelectrics

We study the effects of hydrostatic pressures on deuterated ND 4 D 2 PO 4 type antiferroelectric crystals. Within the four-particle cluster approximation the free energy of the crystals is calculated, and an equation for the transition temperature and explicit expressions for static dielectric permittivities of the crystals are derived. Model pressure dependences of the effective dipole moments of unit cells are proposed. Some peculiarities of the dipole moment formation in these hydrogen bonded crystals are elucidated. In particular, we show the importance of mutual polarization of the dipole moments. The theory parameters providing a satisfactory description of the experimental data for pressure dependences of transition temperature and dielectric characteristics of the considered crystals are found.


Introduction
Ammonium dihydrogen phosphate NH 4 H 2 PO 4 (ADP) and ammonium dihydrogen arsenate NH 4 H 2 AsO 4 (ADA) are those members of the KH 2 PO 4 family of hydrogen bonded crystals which undergo antiferroelectric phase transitions, and it is still not clear what is the origin of this phase transition: Is it triggered by the ordering of protons on hydrogen bonds [1][2][3][4][5], or are the whole H 2 PO 4 groups ordering units [6,7]?
High pressure studies are a tool which can shed some light on this problem, in particular, give some information about the role of hydrogen bonds and especially of their geometric parameters in the phase transition and a polarization mechanism in crystals.
Experimentally, the influence of external pressure on these crystals is studied much less than on ferroelectric representatives of the family. However, it is known that transition temperature T N in NH 4 H 2 PO 4 , ND 4 D 2 PO 4 , NH 4 H 2 AsO 4 and ND 4 D 2 AsO 4 decreases with hydrostatic pressure linearly at low pressures [8,9] (see table 1). The transition temperatures in arsenates are higher than in phosphates, but the coefficients (∂T N /∂p) 0 in ND 4 D 2 PO 4 (DADP) and ND 4 D 2 AsO 4 (DADS) are nearly equal. At higher pressures in NH 4 H 2 PO 4 the dependence of T N (p) becomes nonlinear, and at p = 33 kbar the transition temperature in this crystal turns to zero (the ordered phase disappears) [10].
Under the uniaxial pressure p = −σ 3 the transition temperature in undeuterated NH 4 H 2 PO 4 decreases, too, and this decrease is faster than under hydrostatic pressure [11]. Table 1. Transition temperature at ambient pressure and the initial slope (∂T N /∂p) 0 of some antiferroelectrics. The data are taken from [9]. In antiferroelectrics of the NH 4 H 2 PO 4 type, a pronounced first order phase transition takes place, accompanied by a significant temperature hysteresis and jumps of both the transverse and longitudinal static dielectric permittivities. The transverse permittivity in NH 4 H 2 AsO 4 and ND 4 D 2 AsO 4 is much higher than in NH 4 H 2 PO 4 and ND 4 D 2 PO 4 , respectively.
Transverse permittivity of NH 4 H 2 PO 4 , NH 4 H 2 AsO 4 and ND 4 D 2 AsO 4 decreases with pressure in the paraelectric phase and slightly increases in the antiferroelectric phase [9,10]. It should be noted that in arsenates this decrease in ε 1 at T > T N is fairly fast (in the deuterated form faster than in the undeuterated one).
The influence of pressure on the crystal structure of the KH 2 PO 4 family of antiferroelectrics has almost not been studied. However, it is known that the transition temperature in NH 4 H 2 PO 4 and ND 4 D 2 PO 4 , as well as in KH 2 PO 4 and KD 2 PO 4 , is a linear function of the H-site distance δ -a separation between two possible positions of a hydrogen on the bond O-H. . .O -and turns to zero in NH 4 H 2 PO 4 at δ c = 0.2Å [12][13][14]. Furthermore, at equal δ with an accuracy of 0.01Å the transition temperatures in In the paraelectric phase, an NH 4 H 2 PO 4 crystal is isomorphic to KH 2 PO 4 [15][16][17]. This fact, along with the similarity of temperature and pressure dependences of several physical characteristics of the two crystals, implies the necessity to look for a unified description of high pressure effects in them.
In our recent papers [18][19][20] we developed an approach which allows us to describe consistently the influence of hydrostatic and uniaxial p = −σ 3 pressures (pressures which do not lower the crystals symmetry) on deuterated ferroelectric crystals of the KH 2 PO 4 family. This approach is based on the model of strained KH 2 PO 4 type crystals proposed earlier by Stasyuk et al. [23,24]. In [21,22] we also studied the influence of external pressures on the ND 4 D 2 PO 4 type antiferroelectrics.
In the present paper we slightly change the model (namely, in the way we model the pressure dependence of the Slater energies) and propose a new simple way to describe the polarization processes in antiferroelectric crystals. These changes allow us to describe the experiment for the pressure dependences of dielectric characteristics of a ND 4 D 2 AsO 4 crystal 1 [9], reduce the number of free parameters and to reproduce within the proton ordering model the universality of the transition temperature vs H-site distance dependence in several crystals of this family [12][13][14].

The model of strained DADP type antiferroelectrics
We consider a system of deuterons moving on hydrogen O-D. . . O bonds in the ND 4 D 2 PO 4 type crystals to which an external hydrostatic pressure and electric field E i (i = 1, 2, 3) along one of the crystallographic axis a, b, c are applied.
The calculations are performed within the four-particle cluster approximation which allows one to take into account the strong short-range correlations between deuterons adequately. The four-particle cluster Hamiltonian of such a system, with the effective long-range interactions (dipole-dipole as well as indirect deuterondeuteron interactions via lattice vibrations) taken into account in the mean field approximation, reads [21,22]: Two eigenvalues of Ising spin σ qf = ±1 are assigned to two equilibrium positions of a deuteron on the f -th bond in the q-th cell. The parameters of short-range configurational interactions V = (ε−w 1 )/2, U = (ε + w 1 )/2, Φ = 2ε − 8w + 2w 1 are the functions of the Slater energies ε = ε s − ε a , w = ε 1 − ε a , w 1 = ε 0 − ε a , where ε a , ε s , ε 1 , and ε 0 are the energies of the lateral, up-down, single-ionized and double-ionized deuteron configurations, respectively. Let us mention that the energy of lateral configurations in these antiferroelectric crystals is the lowest.
The fields z i qf include the effective cluster fields ∆ i qf created by the sites neighbouring to the qf -th site, external electric field E i , and the long-range deuterondeuteron interactions taken into account in the mean field approximation The influence of external pressure on the considered system is modelled on the basis of the ideas of [23,24] and [25] in the following way.
It is known that the Slater energies ε, w, w 1 and the components of the longrange interaction matrix J f f ′ (qq ′ ) are proportional to the square of the separation between two possible positions of a deuteron on a bond -D-site distance δ. Assuming the linear δ = δ 0 + δ 1 p.
dependence of the D-site distance δ of ND 4 D 2 PO 4 and ND 4 D 2 AsO 4 crystals on hydrostatic pressure (according to [26,27] the variation of δ with pressure in KH 2 PO 4 and KD 2 PO 4 is linear, indeed, only that δ 0 and δ 1 are temperature dependent), we expand ε, w, w 1 , and J f f ′ (qq ′ ) in powers of pressure up to the linear terms. However, there can be other mechanisms of the pressure influence on the energy parameters of the model. Those mechanisms we take into account by expanding ε, w, w 1 , and J f f ′ (qq ′ ) in powers of components of lattice strain tensor up to the linear terms. We obtain (in our earlier works we did not take into account the influence of the pressure changes in the D-site distance on the Slater energies), and Here S = S 11 + 2S 12 + 2S 13 + S 22 + 2S 23 + S 33 ; S ij are the elastic compliances of the crystals. To avoid an explicit dependence of the Hamiltonian on pressure p, we express the latter in terms of the resulting strain j ε j . It is understood that above the transition point δ k1 = δ k2 , and ψ 1 f f ′ (qq ′ ) = ψ 2 f f ′ (qq ′ ). In the above expansions we take into account only the diagonal components of strain tensor ε 1 , ε 2 , ε 3 and neglect piezoelectric strains ε 4 , ε 5 , ε 6 which must arise if an electric field is applied (piezomodules d 14 , d 25 , d 36 are different from zero).
In [23,24] Stasyuk et al. showed that if one takes into account the lattice strains and neglects the changes with pressure in the dipole moment of hydrogen bond µ = eδ, then application of external pressure to ferroelectric crystals of the DKDP type leads to renormalization of the created by long range interactions mean fields by terms proportional to the mean values of quasispins η qf = σ qf and strains ε i . The same follows from (4) in the mean field approximation if δ 1 /δ 0 = 0.
Taking into account the symmetry of the mean values of quasispins σ qf = and effective dipole moments presenting η i qf as a sum of two terms: a modulated spontaneous part and a uniaxial one induced by the external electric field are the basic vectors of the reciprocal lattice, e ik Z Rq = ±1), we exclude the cluster fields ∆ i qf from the expressions for z i qf , the symmetry of which is the same as of the corresponding η i qf . For this purpose, we use the condition that the mean values σ qf , calculated within the four-particle and single-particle cluster approximation, that is, with Hamiltonian (1) and witĥ must coincide. We obtain where the averages η i qf are The quantities ν a (k) where k = k Z , 0) and ν c (0) are the eigenvalues of the long-range interaction matrix Fourier-transforms The strains ε i and the order parameter η = lim E 1 →0 η i are to be determined from the thermodynamic equilibrium conditions where f is the free energy of a crystal when no electric field is applied ij are the so-called "seed" elastic constants of a crystal, describing the elasticity of a "host" lattice -a fictitious lattice without deuterons; The system of equations for η and ε i reads: where is a contribution of the deuteron subsystem to crystal elasticity. Since determination of this contribution lies beyond the scope of the present paper, we may safely consider instead of (9) the system where c ij are elastic constants of the whole crystal, being determined from an experiment.
The temperature of the first order phase transition is found from the following condition for the values of thermodynamic potential g(η, the order parameter and lattice strains obeying the system (10).

Dielectric properties of strained DADP type antiferroelectrics
It is assumed that the polarization of a crystal is triggered by the deuteron ordering, therefore, Different from the zero macroscopic polarization of antiferroelectric crystals is determined by those parts of the mean values of quasispins which are induced by the external electric field Differentiating (12) with respect to E i one can easily obtain the expressions for the corresponding static dielectric susceptibilites of a clamped (ε i = const) crystal of the ND 4 D 2 PO 4 type where we use the notations and, thereupon, the static dielectric permittivities Let us note that in the paraelectric phase at ambient pressure and at w, w 1 → ∞, coincides with the corresponding expressions of [2,3] and [28,29].
The static dielectric permittivities of the mechanically free crystal (p =const) ε F i (0, T, p) are related to (14) by [30] ε F where d ij are the piezomodules, and c E are the elastic constants of a short-circuited crystal [31]. Our main task is to determine the pressure dependences of the effective dipole moments µ i . In the case of ferroelectrics KD 2 PO 4 and RbD 2 PO 4 , a satisfactory quantitative description of temperature curves of the spontaneous polarization and a longitudinal static dielectric permittivity at different values of hydrostatic pressure have been obtained by means of the following simple expressions [32]: found on the basis of the model assumptions about the mechanism of polarization in hydrogen-bonded crystals of this type: The effective dipole moment µ i , created by displacements of heavy ions and, perhaps, by redistribution of electronic density which are, in turn, triggered by the deuteron ordering, is proportional to δa i , where a i is the corresponding lattice constant. Using (16), for the pressure dependences of coefficients f i = µ 2 i /v we obtain In our earlier work [21,22] we used (17) and (18) to predict the possible pressure dependences of dielectric permittivities of ND 4 D 2 PO 4 , for which no experimental results were available; at that moment we were not aware of the data for ND 4 D 2 AsO 4 [9]. However, our later numerical calculations showed that for ND 4 D 2 AsO 4 (to which large dipole moments and rapid changes in dielectric permittivity with external pressure are particular [9]), expressions (17) and (18) do not yield a good description of experimental data, though qualitatively the behaviour of dielectric permittivity with pressure is reproduced. We believe that in this case in order to improve an agreement with the experimental data, as an additional mechanism of dipole moment formation, one should take into account mutual polarization of the dipoles created by the deuteron ordering.
We estimate the contribution of this mechanism to the pressure dependence of the dipole moment from the following speculations. Consider a simple model of a polarized lattice, when each unit cell possesses a dipole moment, whose size is much less than the distances between the dipoles (the distances between the nearest dipoles are the lattice constants a and c). Up to the terms linear in the external field, the transverse polarization of an antiferroelectric crystal (naturally, induced by this external field) reads: where (see (13) d 1 is the transverse dipole moment of a unit cell, v is the unit cell volume, and µ 1 is the effective dipole moment.
To find µ 1 we take into account the fact that the field acting on a dipole is different from the external one E 01 , but also includes a field created by other dipoles of a crystal. Assuming that the dipole moment of the i-th unit cell d i is proportional to the entire field (α is the polarizability tensor, n ij is the unit vector directed from the dipole d j to d i ; R ij is the distance between these dipoles). We neglect a finiteness of the crystal volume, and assume that all the dipoles, as well as the field created by them, are oriented along the external field E 01 . Then the internal field is β is the angle between vectors n ij and d j , determined by (x, y, z) are the Cartesian coordinates of the dipole d j if d i is placed in the origin. Substituting the latter expression into (21), we obtain a, b, c are lattice constants; n 1 , n 2 , n 3 are numerals, and n 2 1 +n 2 2 +n 2 3 = 0. Comparing (23) and (19) and differentiating the obtained expression with respect to pressure, we find that Here we neglect the variation ofχ 1 with pressure, which is very weak, indeed. It turns out that to find K 1 and ∂K 1 /∂p, we may consider only the values of n i = −2, −1 . . . 2 (we take into account only two surroundings of a site).
The value of 1 α 1 ∂α 1 ∂p is chosen so that it corresponds to the rate of the pressure changes in the square of effective dipole moment µ 2 1 in the case when the mutual polarization of dipoles is not taken into account, namely: Therefore, with a sufficient accuracy we can think that coefficient k f 1 of the considered antiferroelectrics reads: where µ 0 1 is the effective dipole moment of a unit cell at ambient pressure. The dielectric permittivity should be determined as a derivative of polarization with respect to the complete field E 1 = E 01 +d 1 K 1 = E 01 /(1−α 1 K 1 ). Then, instead of (13), we obtain Let us mention that the difference between (13) and (25) is larger, the stronger the crystal lattice differs from the cubic one; at a = c, K 1 ≡ 0. Similarly, for a longitudinal susceptibility we obtain and the expressions for K 3 and ∂K 3 /∂p read: The differences between k f 1 and k f 3 , calculated from (17) and (24), and from (18) and (27), respectively, are more significant, the larger effective dipole moments µ i are. Since the longitudinal moments of ferroelectrics KD 2 PO 4 and RbD 2 PO 4 are comparatively small, their pressure dependences calculated from (18) and (27) practically coincide. The main contribution to these dependences results from the term 1 α 3 ∂α 3 ∂p , that is, they are determined by the pressure changes in the D-site distances, whereas the mutual polarization of dipoles is unimportant here.

Numerical analysis
With the help of the presented theory we are going to describe the effects of hydrostatic pressure in highly deuterated antiferroelectric crystals ND 4 D 2 PO 4 and ND 4 D 2 AsO 4 .
For a numerical estimate of the temperature and pressure dependences of the calculated above physical characteristics of the crystals, we need to set the values of the following parameters: • Slater energies ε 0 , w 0 , w 0 1 , parameters of the long-range interactions ν a (k Z ), ν a (0), ν c (0), constants f 1 ) 2 /v 0 which occur in the expressions for longitudinal and transverse permittivities, respectively. All these parameters correspond to unstrained crystals.
• elastic constants c ij , As parameters describing an unstrained state of ND 4 D 2 PO 4 , we choose those found in [21,28,29] and which provide a satisfactory agreement with the experimental data for the transition temperature, specific heat, static and dynamic dielectric permittivities of the crystal at the atmospheric pressure. The corresponding parameters for ND 4 D 2 AsO 4 were found in [32]. Everywhere, w 1 → ∞. The calculations were performed for N(H 1−x D x ) 4 (H 1−x D x ) 2 PO 4 with T N0 = 235K, for which the dependence of the transition temperature T N on hydrostatic pressure is known [8] (hereafter, it is denoted simply as ND 4 D 2 PO 4 .) The deformation potentials δ ij , ψ aj (k Z ), ψ aj (0) and ψ cj (0), as well as parameter δ 1 /δ 0 are determined by fitting the theoretical results for the pressure dependence of the transition temperature of the crystals to the corresponding experimental data 2 .
As our calculations show, the deformation potentials δ ij can be safely put equal to zero. This means that the contribution of the lattice strains to the pressure dependence of the Slater energies is negligibly small, and the latter results mainly from the pressure changes in the D-site distance δ.
However, that is not the case for the parameters of long-range interactions. Here we have to set the values of the deformation potentials ψ aj (k Z ), ψ aj (0), and ψ cj (0). The ψ aj (k Z ) enter the equation for order parameter (9), whereas the ψ aj (0), and ψ cj (0) occur only in expressions for dielectric permittivities.
We tried to set the values of ψ aj (k Z ) in a consistent way, estimating them from the relation Since the values of dielectric permittivities are not very sensitive to small changes in parameters ν a (0) = ν 0 for the sake of simplicity we assume that ψ ai (0) = ψ ci (0) = 0.
As elastic constants of ND 4 D 2 PO 4 we take the elements of a matrix inverse to the elastic compliances matrix s E ij , measured at ambient pressure [31]. As far as ND 4 D 2 AsO 4 is concerned, the corresponding experimental data are available only for an undeuterated form [33]. Taking into account the fact that elastic constants of deuterated crystals are, as a rule, higher that those of their undeuterated forms, we fix the values of deformation potentials ψ ai (k Z ) calculated with (28), and obtain an agreement with the experimental dependence T N (p) by changing c ij from c + ij of the undeuterated crystal up to ∼ 1.5c + ij . Since no experimental data for the elastic constants of the crystals in the ordered phase are available, we calculated the transition temperature of these crystals at different pressures and different but close to c + ij trial values of c − ij , choosing those which provide the best fit to the experimental data.
The values of the theory parameters which yield the best agreement of the theoretical results with the available experimental data for the temperature and pressure dependences of thermodynamic and dielectric characteristics of the considered crystals are given in tables 2 and 3. Further experimental pressure studies of these crystals will help us to find more precise values of the theory parameters.
In numerical calculations we minimize the thermodynamic potential with respect to the order parameter η and find the strains ε i from the three latter equa- Table 2. The theory parameters for two crystals, ψ ai = ψ ai (k Z ).
crystal crystal   (10); the transition temperature is determined from condition (11). Results of the numerical calculations are given in the figures below.
In figure 1 we plot the dependence of the phase transition temperature of ND 4 D 2 PO 4 and ND 4 D 2 AsO 4 on hydrostatic pressure along with experimental points of [8,9]. Naturally, a perfect fit of theoretical results to the experimen-  [34]. It means that the universality of the transition temperature vs H-site distance dependence observed experimentally in [12][13][14] is reproduced in this theory, and even expanded to other crystals of this family and to uniaxial pressure p = −σ 3 . In figure 2 we present temperature curves of the transverse static dielectric permittivity ε 1 (0, T, p) of ND 4 D 2 AsO 4 at different values of hydrostatic pressure along with the experimental points of [9]. The experimental data correspond to a clamped crystal [9], therefore, the theoretical values of ε 1 (0, T, p) were calculated with (14). For ND 4 D 2 AsO 4 , as well as for other arsenates (KH 2 AsO 4 [35] and NH 4 H 2 AsO 4 [9]), the high values of the transverse dielectric permittivity in the paraelectric phase, as compared with those in the corresponding phosphates, are peculiar. The transverse permittivity in ND 4 D 2 AsO 4 decreases with pressure fairly fast, whereas a decrease of the transition temperature is comparatively slow. The coefficient k f 1 was calculated from (24), which provides a satisfactory agreement with the experimental data at temperatures far from critical. In the antiferroelectric phase ε 1 (0, T, p) slightly increases with pressure.
In figure 3 we give the temperature curves of the transverse ε 1 (0, T, p) and longitudinal ε 3 (0, T, p) static dielectric permittivities of ND 4 D 2 PO 4 at different values of hydrostatic pressure along with the experimental points [31] for ambient pressure. Unfortunately, we are not aware of any experimental work, where the pressure influence of dielecrtic characteristics of deuterated DADP was studied; therefore, the presented graphs only illustrate the plausible pressure dependences of ε 1 (0, T, p) and ε 3 (0, T, p). Theoretical values of the permittivities of a free crystal were calculated from (15), using the experimental data for elastic compliances c E 44 and c E 66 and for piezomodules d 14 and d 36 of [31]. Since in ND 4 D 2 PO 4 d 14 ≪ d 36 , the difference between the values of longitudinal permittivity of free and clamped crystals is several times larger than that between the values of the transverse one.
The quantities k f 1 and k f 3 were calculated from (24) and (27). For ε 1 (0, T, p) a fair description of experimental data is obtained, whereas for ε 3 (0, T, p) an agreement with the experiment is somewhat worse. Perhaps, this agreement can be improved if one explicitly takes into account the influence of piezoelectric strain ε 6 . As one can see, the dependences of transverse static dielectric permittivities of ND 4 D 2 PO 4 and ND 4 D 2 AsO 4 on hydrostatic pressure are similar -a strong decrease in ε 1 (0, T, p) in the paraelectric phase and a weak increase in the antiferroelectric phase. The pressure changes in the longitudinal permittivity are mainly the shifts to lower temperatures due to the lowering of the transition temperature. Besides, ε 3 (0, T, p) slightly decreases with pressure in the both phases.

Concluding remarks
In this paper we propose a theory of strained by hydrostatic pressure antiferroelectric deuterated crystals of the ND 4 D 2 PO 4 type. It should be noted that the theory can be easily generalized to the case of uniaxial pressure p = −σ 3 and other pressures which do not lower the system symmetry. In fact, for this purpose one should only change the system of equations for the lattice strains (10). Within the four-particle cluster approximation in the short-range interactions and the mean field approximation in the long-range interactions we derived an equation for the transition temperature and explicit expressions for dielectric characteristics of the crystals. A set of free parameters was found, providing a good agreement between the presented theory and the available experimental data. An importance of the D-site distance in the phase transition and a dielectric response of the crystals is shown. We have also shown that as an essential mechanism of the dipole moment formation in these crystals, one should take into account the processes of mutual polarization of dipole moments of unit cells.