Longitudinal relaxation of ND 4 D 2 PO 4 type antiferroelectrics . Piezoelectric resonance and sound attenuation

Within the framework of the modified proton model with taking into account the interaction with the shear strain ε6, a dynamic dielectric response of ND4D2PO4 type antiferroelectrics is considered. Dynamics of the piezoelectric strain is taken into account. Experimentally observed phenomena of crystal clamping by high frequency electric field, piezoelectric resonance and microwave dispersion are described. Ultrasound velocity and attenuation are calculated. Character of behaviour of attenuation in the paraelectric phase and the existence of a cut-off frequency in the frequency dependence of attenuation are predicted. At the proper choice of the parameters, a good quantitative description of experimental data for longitudinal static dielectric, piezoelectric and elastic characteristics and sound velocity for ND4D2PO4 and NH4H2PO4 is obtained in the paraelectric phase.


Introduction
Ferroelectric compounds of the MD 2 XO 4 (M=K, D 4 ; x=P, As) type crystallize in the 4•m class of the tetragonal syngony (space group I 42d with non-centrosymmetric point group D 2d ) in the paraelectric phase and possess piezoelectric properties.When appropriate electric fields and shear stresses are applied, one can explore the role of piezoelectric coupling in the phase transition and in the formation of physical characteristics of the crystals.Theoretical investigations of the role of piezoelectricity in the KH 2 PO 4 type ferroelectricity were initiated in [1], where the Slater theory [2] was modified by taking into account the splitting of the lowest ferroelectric energy level of the proton subsystem due to the strain ε 6 .
Important results for strained ferroelectric compounds of the KH 2 PO 4 type were obtained in [3][4][5][6][7][8][9][10][11].In [3,4] the proton ordering model was modified by taking into account the ε 6 contributions to the proton subsystem energy linear in strain.The obtained Hamiltonian contains a deformational molecular field and takes splitting of lateral proton configurations into account.Later [5][6][7] all possible splittings of proton configuration energies by the strain ε 6 were taken into account.In [5] a phase transition in the strained K(H 0,12 D 0,88 ) 2 PO 4 crystal was explored; its thermodynamic, longitudinal dielectric, piezoelectric, and elastic characteristics were calculated; the effect of the stress σ 6 on the calculated quantities was studied.Similar calculations of thermodynamic, longitudinal and transverse dielectric, piezoelectric, and elastic characteristics of KH 2 PO 4 type ferroelectrics were performed in [6][7][8] with tunneling taken into account.A good description of experimental data for the KH 2 PO 4 ferroelectrics and NH 4 H 2 PO 4 antiferroelectrics in the paraelectric phase was obtained.In [9][10][11], the effect of longitudinal electric field on the physical characteristics of K(H 0,12 D 0,88 ) 2 PO 4 and KH 2 PO 4 was studied; a satisfactory quantitative agreement with the available experimental data was obtained.
We should also mention the paper [12], where the mechanism of spontaneous strain ε 6 formation in the KH 2 PO 4 type ferroelectrics and the role of proton interactions with acoustic lattice vibrations in this process were explored.
In [5][6][7][8][9][10][11], the dynamic properties of KH 2 PO 4 type ferroelectrics were not studied with taking into account the piezoelectric coupling.Such a problem, however, is very important.Due to the effect of tunneling suppression in KH 2 PO 4 family crystals found in [13][14][15], and due to the principal difficulties arising at calculations of dynamic characteristics in the presence of tunneling, this problem should be approached by neglecting tunneling.In [16][17][18][19], within the framework of the modified proton ordering models, the thermal, longitudinal and transverse dielectric, piezoelectric, and elastic characteristics of the KH 2 PO 4 family ferroelectrics were calculated.The relaxational phenomena in these crystals were explored; sound velocity and attenuation were obtained.It was shown that for a proper choice of the theory parameters, the experimental data for longitudinal dynamic characteristics of these crystals should be taken into account.
Description of dynamic dielectric characteristics of the ND 4 D 2 PO 4 type antiferroelectrics [20][21][22] was restricted to the static limit and high-frequency relaxation.The attempts to explore the piezoelectric resonance within a model that does not take into account the piezoelectric coupling are pointless.The traditional proton ordering model for the ND 4 D 2 PO 4 type antiferroelectrics does not allow one to describe the difference of the behaivor of free and clamped crystals in the static limit or the effect of crystal clamping produced by high-frequency field.It seems natural to calculate the dynamic characteristics of the ND 4 D 2 PO 4 type antiferroelectrics using the proton ordering model proposed in [5,6,18] in a wide frequency range from 10 3 kHz up to 10 12 Hz, including the piezoelectric resonance region as well.
In the present paper, following the approach developed in [23,24], within the framework of the modified proton ordering model with taking into account the coupling with shear strain ε 6 , we calculate the longitudinal dynamic dielectric, piezoelectric, and elastic characteristics of the ND 4 D 2 PO 4 type antiferroelectrics and explore their temperature and frequency dependences.The effect of crystal clamping produced by a high-frequency longitudinal electric field is studied.Sound velocity and attenuation in these crystals are also calculated.

Hamiltonian of proton ordering model
We shall consider a system of deuterons moving on the O-D. . .O bonds in deuterated ND 4 D 2 PO 4 type crystals.The primitive cell of the Bravais lattice of these crystals consists of two neighboring tetrahedra PO 4 along with four hydrogen bonds attached to one of them (the "A" type tetrahedron).The hydrogen bonds attached to the other tetrahedron ("B" type) belong to the four structural elements surrounding it.Spontaneous polarization in these crystals is zero due to antipolar ordering of dipole moments of hydrogen bonds.External fields applied along a, b, and c axes induce non-zero net polarization.
The model Hamiltonian, with taking into account the short-range and long-range interactions, in the presence of mechanical stress σ 6 = σ xy and external electric field E 3 directed along the crystallographic axis c, consists of the "seed" and pseudospin parts.The "seed" energy of a primitive cell corresponds to the lattice of heavy ions and is explicitly independent of the configurations of hydrogen bonds.The pseudospin part of the Hamiltonian includes long-range ( Ĥlong ) and shortrange ( Ĥshort ) deuteron interactions as well as the effective interactions of deuterons with the electric field E 3 .Hence, where N is the number of primitive cells; σ qf is the operator of the z-component of a pseudospin describing the state of a deuteron in the q-th cell on the f-th bond.Eigenvalues of the operator σ qf = ±1 correspond to the two possible equilibrium positions of the deuteron on the bond.Symmetry of the effective dipole moments of the primitive cells along the c-axis per one hydrogen bond is as follows: The "seed" energy U seed is expressed in terms of the electric field E 3 and strain ε 6 .It consists of the elastic, piezoelectric, and dielectric parts where v = v kB , v is the primitive cell volume; k B is the Boltzmann constant; c E0 66 , e 0 36 , χ ε0 33 are the "seed" elastic constant, coefficient of piezoelectric stress, and dielectric susceptibility, respectively.The "seed" quantities determine the temperature behavior of the corresponding characteristics at temperatures far from the transition point T N .
The Hamiltonian Ĥlong includes the long-range interactions between deuterons and an indirect lattice-mediated deuteron interactions taken into account within the mean field approximation, as well as the linear in the strain ε 6 molecular field [3,4], induced by piezoelectric coupling Here and we took into account the fact that the single-particle deuteron distribution functions can be presented as a sum of a modulated part and uniform terms induced by the longitudinal electric field = ∓η (1) e ik z aq + η (1)z , σ q 2 4 = ±η (1) e ik z aq + η (1)z .
Having calculated the eigenvalues of the single-particle and four-particle Hamiltonians, we present the thermodynamic potential per unit cell in the form [18]: (2.9) Here and further we note ε = ε kB , w = w kB .From the conditions of thermodynamic equilibrium we obtain (in the limit w 1 → ∞) an equation for the strain ε 6 and polarization P 3 : Here we use the notations

Longitudinal dynamic permittivity of ND 4 D 2 PO 4 type crystals
The dynamic characteristics of the ND 4 D 2 PO 4 type crystals will be explored within the framework of the dynamic model of these crystals based on the stochastic Glauber approach [26], where the time dependence of the deuteron distribution functions is described by the following equation where α is the time constant that effectively determines the time scale of the dynamic processes in the system; ε z qf is the local field acting on the f -th bond in the q-th cell in the presence of the field E 3 .The fields can be determined from the Hamiltonian (2.8) where The right hand sides in (3.2) can be written as where When an electric field E 3 along the c-axis is applied, the deuteron distribution functions possess the following symmetry z Substituting (3.3) into the system (3.1) and taking into account the symmetry of the distribution functions (3.6), we obtain the following system of equations for the time-dependent deuteron distribution functions in the presence of the field E 3 : Expressions for the coefficients cq11 , . . ., cq88 are given in [18].In the one-particle approximation, we obtain the following system of equations We shall consider the vibrations of a thin square plate with sides l of a ND 4 D 2 PO 4 type crystal cut in the [001] plane, produced by an external time-dependent electric field E 3t = E 3 e iωt .For the sake of simplicity we shall neglect the diagonal strains ε i (i = 1, 2, 3), which, in fact, are also created in the crystal.
The shear strain ε 6 is determined by the displacements u x = u 1 and u y = u 2 , namely The classical equations of motion of an elementary volume, describing the dynamics of deformational processes in ND 4 D 2 PO 4 type crystals, read where ρ is the crystal density.
Let us expand the coefficients (3.4) in series over the time-dependent terms.Taking into account (3.11) and eliminating ∆ ct from the system (3.7)-(3.8),we obtain a system of equations for the time-dependent distribution functions for a mechanically free crystal The expressions for coefficients of this system are given in [18].
We look for the solutions of the systems (3.12) and (3.13) in the form of harmonic waves (2) Solving the system (3.12) with taking into account (3.15), we find that a6 (ω) + βδ 16 F (1) where s + (iω)r a + (iω)r (1) and the expressions for r 2 , . . ., r 1 are presented in [18].Taking into account (3.13) and (3.16), we obtain the following wave equations for u 1E and u 2E : where the wavenumber is We look for the solutions of (3.18) in the form As a result, We set the boundary conditions in the following form Using expressions (2.11) and (3.17), we find that where e 36 (ω) = e 0 36 + Using the relation between polarization P 3 and the order parameter η (1) and strain ε 6 (2.11), as well as (3.17), we find where The longitudinal dielectric dynamic permittivity of a ND 4 D 2 PO 4 type crystal can be calculated using the relation where then from (3.27) we find that Thereafter, longitudinal dynamic dielectric permittivity of the ND 4 D 2 PO 4 type crystals is It should be noted that at ω → ∞ R 6 (ω) → ∞ and χ σ 33 (ω) → χ ε 33 (ω).

Sound attenuation and velocity in ND 4 D 2 PO 4 type crystals
We consider propagation through the ND 4 D 2 PO 4 type crystals of a sound wave, whose length is much smaller than sample dimensions.Then, all the dynamic variables, namely, the order parameter and elementary displacements depend only on the spatial coordinate which is the direction of sound propagation.For the thin bars cut along [001] we should consider a transverse ultrasound wave polarized along [010].Among the derivatives ∂ui ∂xj only ∂u2 ∂x is different from zero; therefore, instead of (3.12) and (3.13) we can write Solving the system (4.1),we obtain the wavenumber that coincides with the one found above Using (4.2), we can calculate the ultrasound velocity and attenuation where α 06 is the constant frequency and temperature independent term, describing contributions of other mechanisms to the observed attenuation.

Longitudinal static dielectric, piezoelectric, and elastic characteristics of ND 4 D 2 PO 4 type crystals
In the static limit ω → 0 in (3.26), (3.23), and (3.19), we obtain the isothermal static dielectric susceptibility of a mechanically clamped crystal, coefficient of piezoelectric stress, and elastic constant and constant field in the antiferroelectric phase in the following form e 36 = e 0 36 + 2 ) Here we use the notation In the paraelectric phase, from (5.1)-( 5.3) one easily obtains (5.4) (5.5) (5.11)

Comparison of numerical calculations with experimental data
Let us now evaluate the found above longitudinal dielectric, piezoelectric, and elastic characteristics of the NH 4 H 2 PO 4 (ADP) and ND 4 D 2 PO 4 (DADP) crystals and compare them with the corresponding experimental data.It should be noted that the developed theory is valid, strictly speaking, only for highly deuterated ND 4 D 2 PO 4 type crystals.The experimentally established relaxational character of ε * 33 (ω, T ) dispersion [27][28][29] in these crystals, according to [13][14][15] is most likely related to suppression of tunneling by the short-range interactions.Therefore, proton tunneling for the NH 4 H 2 PO 4 type crystals will be neglected.Since the majority of experimental studies were performed for the paraelectric phase, we shall also restrict our calculations to temperatures T > T N .
good quantitative description of the experimental points is obtained.At T = T N the coefficients d 36 and e 36 are finite and decrease with temperature increasing.The coefficients d 36 and e 36 of KH 2 PO 4 at T = T c are about one order of magnitude larger than the corresponding values in the ADP crystal and decrease with temperature increasing much faster than the coefficients d 36 and e 36 of ADP [16].
In figures 4 and 5 we plot the temperature dependences of the constants of piezoelectric stress h 36 and piezoelectric strain g 36 of ADP and DADP crystals.The experimental data are well described by the proposed theory.The constants h 36 and g 36 are practically temperature independent.The temperature dependences of the h 36 and g 36 constants of KH 2 PO 4 are also weak, with their values being nearly three times smaller than the values of h 36 and g 36 of ADP.Even though the dielectric permittivities of ADP and DADP along the c-axis are relatively small, the values of the constants of piezoelectric strain and piezoelectric stress in this direction are rather significant.
The temperature dependences of the calculated isothermal elastic constants c E 66 and c P 66 of ADP (a) and DADP (b) well agree with the corresponding experimental data (see figure 6).The elastic constants c E 66 of ADP and DADP, in contrast to those of KH 2 PO 4 , are finite at T = T N and hardly depend on temperature.
The calculated frequency curves of real and imaginary parts of dielectric permittivity ε * 33 (ω, T ) and experimental points of [29] are presented in figure 7 for ADP at ∆T = 28 K and in figure 8 for DADP at ∆T = 64 K.In the frequency range of 10 6 − 10 8 Hz a resonance dispersion is observed.At ω → 0 we obtain a static dielectric permittivity of a free crystal.The dashed line corresponds to the low-frequency part of the clamped permittivity.Above the resonances, the permittivity corresponds to a clamped crystal and has a relaxational character.Theoretical results and experimental points for the temperature dependences of real and imaginary parts of complex dielectric permittivity ε * 33 (ω, T ) of ADP and DADP at frequencies where the effect of crystal clamping by a high-frequency field takes place are given in figures 9, 10, respectively.As one can see, the experimental data of [27,29] are quantitatively well described by the proposed theory.At the transition temperature the real and imaginary parts of permittivity ε * 33 (ω, T ) of ADP have finite maxima at all frequencies.With ∆T increasing the values of ε 33 (ω, T ) and ε 33 (ω, T ) slightly decrease at all frequencies.
In the temperature curves of ε 33 (ω, T ) and ε 33 (ω, T ) of DADP a maximum is observed at T = T N at frequencies below the dispersion frequency and there is a shallow minimum at higher frequencies.With ∆T increasing at dispersion frequencies the values of ε 33 (ω, T ) and ε 33 (ω, T ) increase, reaching a maximum, which shifts to higher ∆T with frequency increasing.
The calculated frequency dependences of ε * 33 (ω, T ) along with the experimental points are presented in figure 11 for ADP and in figure 12 for DADP.A good quantitative description of   crystals are shown in figure 13, 14, respectively.At T = T N the attenuation α 6 is finite and slightly decreases with temperature increasing.Below 10 8 Hz attenuation α 6 is small, whereas at further increase of frequency up to 10 11 Hz α 6 it rapidly increases and saturates.Such high values of α 6 at saturation mean that sound does not propagate in the crystal.In contrast, in the KH 2 PO 4 type crystals, the attenuation rapidly increases at temperatures close to T = T c .
In figure 15 we plot the calculated temperature dependence of the sound velocity v 6 for ADP(a) and DADP(b) crystals.The sound velocity is practically independent of temperature and frequency, except for the frequency region where the dispersion of the clamped dielectric permittivity is observed; in this region the sound velocity v 66 rapidly increases and saturates.

Concluding remarks
In this paper, using the modified proton ordering model for the KH 2 PO 4 family crystals, with taking into account the linear in the strain ε 6 contribution to the proton system energy, without tunneling, within the framework of the four-particle cluster approximation, we develop a theory of dynamic longitudinal dielectric, piezoelectric, and elastic properties of the ND 4 D 2 PO 4 type antiferroelectrics.Sound velocity and attenuation in these crystals are also calculated.Numerical analysis of the dependences of the found characteristics on the values of the theory parameters is performed.Optimum sets of the model parameters and "seed" quantities for ND 4 D 2 PO 4 and NH 4 H 2 PO 4 crystals are found.They permit a satisfactory description of the available experimental data.
The piezoelectric coupling (ψ 6 = 0) being taken into account gave rise to understandable differences between static dielectric permittivities of mechanically free ε σ 33 and clamped ε ε 33 crystals.In the ADP type crystals, the permittivity ε σ 33 is ≈ 18% larger than ε ε 33 , and this difference is practically temperature independent.The isothermal elastic constants c P 66 and c E 66 in ADP and DADP crystals are different, just like in the KH 2 PO 4 type crystals, but they have no peculiarities at T = T N .The sound attenuation coefficient α 6 in the ADP type antiferroelectrics is finite and has a weak temperature dependence, whereas in the KDP type ferroelectrics it has an anomalous behavior in the phase transition region.
The obtained results for the ADP crystals are compared with the calculations performed in [6,7].It is established that tunneling practically does not affect the static dielectric, piezoelectric, and elastic characteristics of ADP.

. 3 )
Equating the right hand sides of (3.2) and (3.3) and taking into account the fact that σ qf = ±1,

Figure 4 .
Figure 4.The temperature dependences of the constant of piezoelectric stress h36 of NH4H2PO4

Figure 5 .
Figure 5.The temperature dependences of the constant of piezoelectric strain g36 of NH4H2PO4

Figure 7 .
Figure 7. Frequency curves of real and imaginary part of dielectric permittivity of free and clamped (dashed line) NH4H2PO4 crystals at ∆T = 28 K, -[29].