Effect of hydrostatic pressure on dynamic dielectric characteristics of CsH$_2$PO$_4$ ferroelectric

Based on the pseudospin model of the deformed CsH$_2$PO$_4$ crystal within the Glauber method, the equation for the time-dependent mean value of the pseudospin is obtained, which is solved in the case of small deviations from the equilibrium state. Using the solution of the equation, we find expressions for the longitudinal dynamic dielectric constant and relaxation time. Based on the proposed parameters of the theory, the temperature and frequency dependences of the dynamic dielectric constant and the temperature dependence of the relaxation time are calculated and investigated. A detailed numerical analysis of the obtained results was performed. The influence of hydrostatic pressure on the dynamic characteristics of CsH$_2$PO$_4$ is studied.


Introduction
A ferroelectric with hydrogen bonds CsH 2 PO 4 (CDP) is an example of a crystal where the effect of pressure is significant. In this crystal there are two structurally equivalent types of hydrogen bonds of different lengths (figure 1b). Longer bonds have one equilibrium position for protons, and shorter bonds have two equilibrium positions. They connect the groups PO 4 in chains along the -axis (figure 1a); therefore, the crystal is quasi-one-dimensional.
At room temperature in the absence of pressure, the crystal is in the paraelectric phase and has monoclinic symmetry (space group P2 1 /m) [1,2]. In this case, the protons on the short bonds are in two equilibrium positions with equal probability. Below = 153 K, the crystal passes into the ferroelectric phase (space group P2 1 ) [3,4] with spontaneous polarization along the crystallographic axis, and protons with a higher probability occupy the upper position (figure 1a). Based on dielectric studies [5,6] it was established that at pressures = = 0.33 GPa and cr = 124.6 K, double hysteresis loops appear, i.e., there is a transition to the antiferroelectric phase. Using the neutron diffraction study [7], it was found that in the antiferroelectric phase, the unit cell of the CDP crystal doubles along the a-axis, since there are two sublattices in the form of planes bc, which are polarized antiparallel along -axis and alternate along a-axis. The symmetry remains monoclinic (space group P2 1 ). Protons on hydrogen bonds are arranged in adjacent sublattices in an antiparallel manner.
The effect of hydrostatic pressure on the phase transition temperature and dielectric properties of ferroelectrics Cs(H 1− D ) 2 PO 4 was studied in [5,6,[8][9][10][11][12]. The molar heat capacity of CDP was measured in [13], and was also calculated on the basis of lattice dynamics modelling in [14,15]. The important role of proton tunneling on bonds was shown based on the first-principle calculations [16] and using theoretical calculations on the basis of quasi-one-dimensional pseudospin model [17].
Using ab initio calculations, piezoelectric coefficients, elastic constants, and molar heat capacity of CDP [18,19] were obtained. The theoretical description of the dielectric properties of CDP at different values of hydrostatic pressure was carried out in [20,21] on the basis of the pseudospin model. However, in these works, the interaction parameters do not depend on the deformations of the lattice. As a result, it is impossible to obtain piezoelectric and elastic characteristics of the crystal, and the critical pressure does not depend on temperature.
In [22], the temperature dependences of lattice strains 1 , 2 , 3 , 5 were measured. It also proposes a quasi-one-dimensional Ising model for the CDP crystal, in which the interaction parameters are linear functions of these strains. Based on this model, the temperature behavior of ( ) was explained. However, this model does not consider the crystal as two sublattices and does not make it possible to describe the ferroelectric-antiferroelectric transition at high pressures.
In [23][24][25][26], there is proposed a two-sublattice pseudospin model of the deformed crystal CDP, in which the interactions between the nearest pseudospins in the chain are considered in the mean field approximation. In this case, the interaction parameters are linear functions of the strains . As a result, the temperature dependences of spontaneous polarization, dielectric constant, piezoelectric coefficients and elastic constants are calculated, and the influence of hydrostatic and uniaxial pressures and longitudinal electric field on these characteristics is studied.
In this paper, based on the model [25], the dynamic dielectric characteristics of CDP in the presence of hydrostatic pressure are calculated.

The model of CDP crystal
To calculate the dynamic characteristics of CDP, we use the model [25], which considers a system of protons on O-H ... O bonds with two-minimum potential as a system of pseudospins. The primitive cell has one chain, marked in figure 1 as "A". To describe the transition to antiferroelectric phase at high pressures, paper [25] considers an extended primitive cell formed by two chains ("A" and "B"). All chains "A" form a sublattice "A", and all chains "B" form a sublattice "B". Each chain in the primitive 23701-2 cell contains two adjacent tetrahedra PO 4 (type "I" and "II") together with two short hydrogen bonds (respectively, "1" and "2" , and in the presence of electric field 2 = Hamiltonian of the model of CDP is given by [25]:ˆ= seed +ˆs hort +ˆl ong +ˆ+ˆ , where is the total number of restricted primitive cells. The first term in (2.1) is "seed" energy, which relates to the heavy ion sublattice and does not explicitly depend on the configuration of the proton subsystem. It includes elastic, piezolectric and dielectric parts, expressed in terms of the electric field 2 and strains maintaining the symmetry of crystal 1 = , 2 = , 3 = , 5 = 2 : Parameters 0 , 0 2 , 0 22 are the so-called "seed" elastic constants, "seed" coefficients of piezoelectric stresses and "seed" dielectric susceptibility, respectively; is the volume of a restricted primitive cell. In the paraelectric phase, all coefficients 0 ≡ 0.
The other terms in (2.1) describe the pseudospin part of Hamiltonian. In particular, the second term in (2.1) is Hamiltonian of short-range interactions: In (2.3), A,B 1,2 are -components of pseudospin operator that describe the state of the bond "1" or "2" of the chain "A" or "B", in the -th cell, ì is the lattice vector along -axis. The first Kronecker delta corresponds to the interaction between protons in the chains near the tetrahedra PO 4 of type "I", where the second Kroneker delta is near the tetrahedra PO 4 of type "II". Contributions into the energy of interactions between pseudospins near tetrahedra of different type are identical. Parameter , which describes the short-range interactions within the chains, is expanded linearly into a series with respect to strains : The termˆl ong in (2.1) describes long-range dipole-dipole interactions and indirect (through the lattice vibrations) interactions between protons which are taken into account in the mean field approximation: where such notations are used: The parameter 1 describes the effective long-range interaction of the pseudospin that with the pseudospins inside the sublattice, and parameter 2 -with the pseudospins of the other sublattice.
The fourth term in (2.1) describes the interactions of pseudospins with the external electric field: where is y-component of effective dipole moments per one pseudospin. The termˆ in Hamiltonian (2.1) takes into account the dependence of effective dipole moments on the mean value of pseudospin : where (f =1, 2, 3, 4) are a brief notation of pseudospins A 1 , A 2 , B 1 , B 2 , respectively. Here, we use corrections to dipole moments 2 instead of because of the symmetry considerations, and the energy should not change when the field and all pseudospins change their signs.
The termˆ , as well as long-range interactions, are taken into account in the mean field approximation:ˆ Our theory does not explicitly take into account the tunneling effects because the tunneling of protons on the bonds and strains of the lattice being taken into account simultaneously greatly complicates the calculations. In our model, the tunneling parameter is partially taken into account in the form of the other renormalized parameters, such as the interaction parameter 0 and the effective dipole moment . Explicit consideration of tunneling could somewhat improve the agreement of the calculated characteristics with the experimental data.
In the two-particle cluster approximation for short-range interactions, the thermodynamic potential is given by: where = 1 B , B is Boltzmann constant,ˆ( 2) ,ˆ( 1)A ,ˆ( 1)B are two-particle and one-particle Hamiltonians:ˆ( where such notations are used: Symbols Δ are the effective fields created by the neighboring bonds from outside the cluster. From the condition of the minimum of the thermodynamic potential / Δ = 0, the system of equations for these 23701-4 Effect of hydrostatic pressure fields, as well as for the order parameters are obtained: Here, the following notations are used: Minimizing the thermodynamic potential with respect to the strains, an additional system of equations for the strains was obtained in [25]: In the case of ferroelectric ordering 1 = 2 = , = 2 = . Then, (2.14), (2.15) are given by: In the presence of hydrostatic pressure 1 = 2 = 3 = − , 4 = 5 = 6 = 0. In [25], the expression for the longitudinal component of polarization 2 is also obtained, which in the case of ferroelectric ordering is given by:

Dynamic dielectric properties of mechanically clamped CDP crystal. Analytical results
Dynamic dielectric properties are studied in the absence of the electric field and shear stresses. To study the dynamic properties of the CDP crystal, an approach is used which is based on the ideas of Glauber's stochastic model. Based on this approach, we obtain a system of equations for time-dependent unary correlation functions of pseudospins: where the parameter sets the time scale of the dynamic processes in the sytem; ( ) is the local field acting on the -th pseudospin in the -th cell; this is the factor at /2 in the initial Hamiltonian. In order to obtain a closed system of equations for unary correlation functions, we use the two-particle cluster approximation. In this approximation, the local fields ( ) are the coefficients at /2 in two-particle and one-particle Hamiltonians (2.13), (2.12). In the two-particle approximation, they are given by:

23701-5
and in the two-particle approximation they are given by: Here, the effective fields ,¯are given by expressions (2.14). As a result of (3.1), we obtain a system of equations for the mean values of pseudospins = in the two-particle approximation and in the one-particle approximation where the following notations are used: We limit ourselves to solving the equations (3.4) and (3.5) to the case of small deviations from the equilibrium state. To do this, we present and the effective fields ,¯as the sum of two termsequilibrium values and their deviations from the equilibrium state (mechanically clamped crystal): Here, Δ is the effective cluster field, and Δ is the deviation from its equilibrium value, is the parameters of long-range interactions.
We expand the coefficients and in series over /2 limited to linear terms: where the following notations are used: Substituting (3.6), (3.7) in (3.4), (3.5), we obtain a system of differential equations for unary distribution functions: Since the equilibrium state holds (2.18), it is easy to see that (1 − (0) )˜= (0) . Then, the system of equations (3.8) is given by Let us write it more in detail, taking into account (2.17): Excluding the parameter Δ , we obtain the equation for the time-dependent order parameter: Solving this equation, we obtain the time-dependent mean values of pseudospins, and thus we find longitudinal dynamic dielectric susceptibility of mechanically clamped CDP crystal: where is the static susceptibility of the pseudo-spin subsystem: is relaxation time: Dynamic dielectric constant CDP is given by:

Comparison of theoretical results with experimental data. Discussion of the obtained results
The theory parameters are determined in [25] from the condition of agreement of calculated characteristics with experimental data for temperature dependences of spontaneous polarization 2 ( ) and dielectric permittivity 22 ( ) at different values of hydrostatic pressure [6], spontaneous strains [22], molar heat capacity [13] and elastic constants [27]; as well as the agreement with ab initio calculations of the lattice contributions to molar heat capacity [18] and dielectric permittivity at zero temperature [19].
It should be noted that the temperature dependences of the dielectric constant 22 at different values of hydrostatic pressure were also measured in [12]. However, they do not agree with experimental data [6]. It is quite possible that another crystal sample was used there, which was grown under different conditions. In addition, in [12] there are no data for the temperature dependences of spontaneous polarization at different pressures, as well as there are no data for dielectric characteristics at zero pressure. Therefore, we used experimental data [6] to determine the model parameters. To describe the dynamic dielectric permittivity, we used the data [28] because they are measured at frequencies corresponding to the range of dispersion. In the case of [12], the frequencies were much lower (up to 1MHz). At these frequencies, the dielectric constant in the paraelectric phase behaves as static, and in the ferroelectric phase there is a large contribution to the permittivity associated with the reorientation of the domain walls. However, our theory does not take into account the reorientation of the domain walls and cannot describe the dielectric permittivity in the ferroelectric phase at low frequencies.
Parameters of short-range interactions 0 and long-range interactions 0 1 ("intra-sublattice"), 0 2 ("inter-sublattice") mainly fix the phase transition temperature from paraelectric to ferroelectric phase at , it is necessary to use experimental data for the shift of the phase transition temperature under hydrostatic and uniaxial pressures as well as the data for temperature dependences of spontaneous strains , piezoelectric coefficients and elastic constants. Unfortunately, only the data for the spontaneous strains and hydrostatic pressure effect on the dielectric characteristics are available. As a result, the experimental data for strains and dielectric characteristics can be described using a great number of combinations of parameters 1 , 2 . Therefore, for the sake of simplicity, we chose 2 to be proportional to 1  1 . The effective dipole moment in the paraelectric phase is found from the condition of agreement of the calculated curve 22 ( ) with experimental data. We consider it to be dependent on the value of hydrostatic pressure p, that is = 0 (1 − ), where 0 = 2.63 · 10 −18 esu·cm, = 0.4 · 10 −10 cm 2 /dyn. The correction to the effective dipole moment = −0.43 · 10 −18 esu·cm is found from the condition of the agreement of calculated saturation polarization with experimental data. In [28], the experimental data for the dielectric permittivity 22 are almost twice as large as in [6]. This may be due to different growing conditions of the samples and different features of measurement the dielectric permittivity. Therefore, to describe the experimental data [28] for 22 , we assume that these samples are characterized by a more effective dipole moment per pseudospin. Namely, 0 = 5.2 · 10 −18 esu·cm, = −1.8 · 10 −18 esu·cm.
The "seed" dielectric susceptibility 0 22 , coefficients of piezoelectric stress 0 2 and elastic constants 0 are found from the condition of the agreement of theory with experimental data in the temperature regions far from the phase transition temperature . Their values are obtained as follows:  [7]. As can be seen from (3.13), the dynamic dielectric susceptibility is determined by the behavior of the static dielectric susceptibility of the pseudospin subsystem and the relaxation time in the system. Their temperature dependences at different values of pressure are shown in figure 2.
The calculated relaxation time tends to infinity at = . It is associated with the relaxation  frequency characteristic of this crystal (soft relaxation mode) = (2π ) −1 , which conditionally separates the region of low-frequency and high-frequency dispersion. The inverse time −1 , as well as the relaxation frequency decrease while approaching the phase transition temperature and tending to zero at a temperature = .
At frequencies , the real part of the dynamic dielectric constant 22 behaves as static, and the imaginary part 22 is close to zero at all temperatures, except for a narrow area near . This can be seen in the frequency dependences 22 ( ) for different Δ = − in the frequency range < 10 7 Hz ( figure 3, 4), as well as on the temperature dependences 22 ( ) at low frequencies and temperatures far from (figure 5).
At frequencies ≈ , there is a relaxation dispersion, which is manifested in a rapid decrease of the real part of the dielectric constant 22 with increasing frequency, as well as in large values of the The increase in the relaxation time and the decrease in the relaxation frequency as we approach the temperature = , is manifested in the shift of the region of dispersion to lower frequencies on the frequency dependence 22 ( ) ( figure 3,4), as well as in the expansion of the "depression" of 22 and of the peak on 22 when approaching the temperature = ( figure 5). Since → 0 at = , this "depression" [as well as the peak on 22 ( )] exists at all frequencies; at low frequencies, it is very narrow. The value of permittivity 22 at the minimum point (at = ) is equal to the lattice contribution 0 22 . Hydrostatic pressure ℎ lowers the phase transition temperature and reduces the value of −1 at a fixed Δ = − , (figure 2, curves h 1 , h 2 , h 3 ) compared to −1 without pressure (curve h 0 ). As a result, the dispersion region shifts to lower frequencies (figure 3,4, curves h 1 , h 2 ), compared to the case of no pressure (curves h 0 ). While on the temperature dependence of the permittivity, the "depression" on 22 and the peak on 22 (figure 6, curves h 1 , h 2 ) broaden under pressure, compared with the case of no pressure (curves h 0 ).
In addition, at low frequencies in the paraelectric phase (figure 3) and at temperatures far from the permittivity 22 in figure 6 in the presence of pressure is greater than without pressure, because it is close to static in this area. The static permittivity at Δ = const increases with pressure (figure 2b). In the ferroelectric phase at low temperatures, the pressure reduces the dielectric constant.

Conclusions
It is established that the dynamic dielectric constant of CDP at low frequencies behaves as static; at frequencies commensurate with the inverse relaxation time, there is a relaxation dispersion; and at high frequencies, only the lattice contribution is manifested in permittivity. The region of the longitudinal dispersion in CDP shifts to low frequencies when the temperature approaches the phase transition point, which is associated with a significant increase in relaxation time when approaching the temperature . A satisfactory agreement of theoretical results with experimental data is obtained.
The effect of hydrostatic pressure on the dielectric properties is manifested in a decrease of the phase transition temperature, in an increase of the static dielectric constant in the paraelectric phase and in an increase of the relaxation time. This leads to an increase of the dynamic permittivity at pre-relaxation frequencies and to the shift of the dispersion region to lower frequencies. In the ferroelectric phase at low temperatures, the pressure reduces the dielectric constant. The dynamic permittivity is monodisperse.
As you can see, the dielectric constant in the dispersion region is quite sensitive to the pressure. Thus, the CDP crystal can be used as a high-frequency filter in which the absorption and transmission spectra can be adjusted by pressure.