Piezoelectric Resonance in KH2PO4 Type Crystals Revisited

Within the framework of proton ordering model with taking into account the piezoelectric interaction with the shear strain $\varepsilon_6$, a dynamic dielectric response of KD2PO4 type ferroelectrics is considered. An expression for the piezoelectric resonance frequencies of the rectangular thin plate of the crystal cut in the (001) plane is obtained.


Introduction
In our previous papers [1,2] we explored the dynamic dielectric response of square thin plates cut from the KH 2 PO 4 family crystals in the planes (001), perpendicular to the axis of spontaneous polarization. Using the modification of the proton ordering model [3] that includes the piezoelectric coupling with the shear strain ε 6 , within the framework of the Glauber approach [4] and the four-particle cluster approximation, we obtained expressions for the dynamic dielectric permittivity of the crystals, which took into account the dynamics of the shear strain ε 6 . In the low-frequency limit these expressions coincided with the static permittivities of mechanically free crystals, whereas in the microwave region they coincided with the dynamic permittivities of clamped crystals, exhibiting a relaxational dispersion.
In the intermediate region, the obtained permittivities had numerous peaks associated with the piezoelectric resonances. However, while solving the partial differential equations for the strain in [1,2], the boundary conditions were not set correctly. Instead of demanding that the entire edges of the plate were mechanically free, we considered the plate free only at its vertices. It resulted in the underestimated values of the resonant frequencies. In the present paper we shall correct these errors.
We shall not repeat here the details of the previous calculations, which were correct. The system Hamiltonians, most of the used notations, as well as derivation of the dynamic dielectric permittivities of clamped crystal (the pseudospin subsystem dynamics), can be found in [1,2].
where ε z q f ′ (t ) is the local field acting on the f ′ th deuteron in the qth cell, which can be found from the system Hamiltonian (see [1]); α is the parameter setting the time scale of the dynamic processes in the pseudospin subsystem.
Taking into account the symmetry of the distribution functions from (1) we obtain for them a closed system of equations The used here notations can be found in [1]. Dynamics of the deformational processes is described using classical Newtonian equations of motion of an elementary volume, which for the relevant to our system displacements u 1 and u 2 ( Here ρ is the crystal density, σ 6 is the mechanical shear stress, which, being the function of η (1) , E 3 , and ε 6 , is found from the constitutive equations derived in [1].
At small deviations from the equilibrium we can separate in the systems (3) and (4) The fluctuational part of (3) is then reduced to the system of linear first-order differential equations with constant coefficients, solving which we get the notations introduced here can be found in [1]. Substituting (5) into Eqs. (4), we obtain where k 6 is the wavevector ?????-2 Differentiating the first and second equations of (6) with respect to y and x, correspondingly, remembering that we neglect the diagonal strains ε 1 = ∂u 1 /∂x and ε 2 = ∂u 1 /∂y, and adding the two obtained equations, we arrive at the single equation for the strain ε 6 Boundary conditions for ε 6 (x, y) follow from the assumption that the crystal is simply supported, that is, it is traction free at its edges (at x = 0, x = L x , y = 0, y = L y , to be denoted as Σ) In our previous consideration [1] this condition was fulfilled at the corners of the crystal plate only, but not along all its edges. Substituting (10) into the constitutive relations, we obtain the explicit boundary conditions for the strains in the following form where e 36 (αω) = e 0 36 + Solution of (7) with the boundary conditions (11) is with ω 0 kl given by Using the expression, relating polarization P 3 to the order parameter η (1) and strain ε 6 (see [1]), we find that where is the dynamic dielectric susceptibility of a clamped crystal.
Now we can calculate the dynamic dielectric susceptibility of a free crystal χ σ 33 (αω) as ?????-3 obtaining χ σ 33 (ω) = χ ε 33 (αω) + R 6 (ω) where In the static and the high frequency limits from (18) we obtain the static susceptibility of a free crystal [3] and the dynamic susceptibility of a mechanically clamped crystal, exhibiting relaxational dispersion in the microwave region. Thus, eq. (18) explicitly describes the effect of crystal clamping by highfrequency electric field.
In Comparing (14) to the expression obtained previously [1] for a square L ×L plate cut in the (001) plane  we can see that the incorrectly set boundary conditions [1] led to the first resonance frequency, being 2 times smaller than the one given by (14).
However, the low and high frequency limits of the susceptibility calculated in [1] (the static value and the clamped values with the relaxational dispersion in the microwave region) were correct.
The used values of the model parameters can be found in [1]. As one can see, in the paraelectric phase the first resonance frequency of a rectangular 0 • Z-cut of a KH 2 PO 4 crystal, calculated from (14), accords well with experimental data. The discrepancy between the theory and experiment in the ferroelectric phase is obviously caused by the contributions of the domain effects into the elastic constant of the crystal, which are not considered in the present model. Note that the first resonant frequency has a sharp minimum at the transition point, owing to the similar behavior of the elastic constant c E 66 .

Resonant frequencies of NH 4 H 2 PO 4 type crystals
We consider vibrations of a 0 • Z-cut of an antiferroelectric NH 4 H 2 PO 4 type crystal, produced by an external time-dependent electric field E 3t = E 3 e iωt . Taking into account the system Hamiltonian, the symmetry of the proton distribution functions for the case of antiferroelectric ordering [2,5], and following the procedure, described in the previous section, we obtain an expression for the dynamic dielectric permittivity of a free crystal, which is formally the same as for the case of ferroelectric ordering (18). However, the elastic constant is different  (14), respectively. However, the functions F (1) (ω) and other auxiliary quantities used in these formulae as well as in (20) differ from those from the previous section and can be found in [2,5].
In fig. 2 we compare the calculated frequency constants (the resonant frequencies multiplied by the sample edge length L x ω 0 kl ; the size-independent quantity) of a square 0 • Z-cut of a NH 4 H 2 PO 4 crystal to the available experimental data. As one can see, a very good agreement is obtained. The fitting procedure and values of the model parameters were given in [2,5].

Conclusions
Within the proton ordering model with taking into account the shear strain ε 6 we explored a dynamic response of ferroelectric and antiferroelectric crystals of the KH 2 PO 4 family to an external harmonic electric field E 3 . Dynamics of the pseudospin subsystem is described within the stochastic Glauber approach. Dynamics of the strain ε 6 is obtained from the Newtonian equations of motion of an elementary volume, with taking into account the relations between the order parameter of the pseudospin subsystem and the strain. Corrected expressions for the piezoelectric resonance frequencies of simply supported rectangular 0 • Z-cuts of these crystals are obtained. They are shown to yield a good quantitative agreement with experimental data for KH 2 PO 4 and NH 4 H 2 PO 4 crystals.
The ultimate goal of the present studies will be to generalize the obtained expression for the dynamic permittivity to the case of the Rb 1−x (NH 4 ) x PO 4 type proton glasses, in order to explore their dynamic dielectric response. It is known [9,10] that, just like their pure constituents, these mixed systems are piezoelectric, and their dynamic dielectric permittivity has a piezoelectric resonance dispersion. As our preliminary calculations show, the experimentally obtained resonant frequencies of such mixed crystals [11] are well described by the obtained here expression for the resonant frequencies, provided that the corresponding elastic constant c E 66 of such a system is known.