Piezoelectric resonance in Rochelle salt: The contribution of diagonal strains

Physica B 407 (2012) 4550–4556

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Piezoelectric resonance in Rochelle salt: The contribution of diagonal strains A.P. Moina n Institute for Condensed Matter Physics, 1 Svientsitskii Street, 79011 Lviv, Ukraine

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 June 2012 Received in revised form 9 July 2012 Accepted 15 August 2012 Available online 23 August 2012

Within the framework of two-sublattice Mitsui model with taking into account the shear strain e4 and the diagonal strains e2 and e3 , a dynamic dielectric response of Rochelle salt X-cuts is considered. Experimentally observed phenomena of crystal clamping by high frequency electric field, piezoelectric resonance, and microwave dispersion are described. Analytical expressions for the resonant frequencies of these cuts, associated with the shear vibration mode of e4 and with the extensional in-plane modes of e2 , e3 , are derived. It is shown that the lowest resonant frequency is always associated with the e4 shear mode. & 2012 Elsevier B.V. All rights reserved.

Keywords: Vibrations Rochelle salt Permittivity Resonance Mitsui model

1. Introduction Crystals of Rochelle salt are a curious system. In contrast to most of the known ferroelectrics, in Rochelle salt the ferroelectric phase exists only in a temperature interval between two second order phase transitions at 255 and 297 K. Spontaneous polarization P1 is accompanied by shear strain e4 . The ferroelectric phase is monoclinic (P21 11); both paraelectric phases are orthorhombic (P21 21 21 ); all phases are non-centrosymmetric and piezoelectric. Dynamic dielectric response of Rochelle salt exhibits several dispersions. Those are: related to domain walls motion [1–3] (below 1 kHz), piezoelectric resonance [3–5] (between 10 kHz and 10 MHz), microwave relaxation [6], and the submillimeter (100–700 GHz) resonances [7]. While the very high frequency dispersions are relatively well studied, the presence of the piezoelectric resonance dispersion in Rochelle salt is acknowledged at best. Quantitative data are available mostly for the temperature dependence of the first resonant frequency [4,5,8], whereas very little information was obtained [4] about details of the temperature or frequency behavior of the permittivity. Behavior of Rochelle salt is usually described within a twosublattice Ising model with an asymmetric double-well potential (Mitsui model [9–11]). The pseudospin dynamics is considered within the Bloch equations or Glauber approach methods. Rochelle salt is a perfect example of a system, whose dynamic dielectric response cannot be correctly described without taking into account the deformational effects. Their influence is revealed,

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in particular, in the phenomena of piezoelectric resonance and crystal clamping by a high-frequency measuring field, none of which can be obtained within theories based on underformable versions of the Mitsui model [10,12]. Such theories yield a diverging relaxation time at the Curie points and, as a result, incorrect temperature behavior of the microwave permittivity near the phase transitions. In Ref. [13] the dynamic dielectric permittivity of Rochelle salt square X-cuts has been calculated, using the model with piezoelectric coupling [14] with the shear strain e4 . The standard methods of description of the lattice strain dynamics [8] based on Newtonian equations of motion have been combined with the Glauber approach to pseudospin dynamics. Evolution of the dielectric permittivity from the static free crystal value via the piezoelectric resonances to the clamped crystal value and to the microwave relaxation has been described. Recently [15], it has been pointed out that boundary conditions in Ref. [13] were not set properly, which resulted in the underestimated values of the resonant frequencies; a correct equation for the resonant frequencies of X-cuts related to the shear e4 mode has been obtained in Ref. [15]. In the paraelectric phases in Rochelle salt the longitudinal field E1 excites only the shear mode e4 , as d14 is there the only non-zero piezoelectric coefficient associated with E1. However, in the ferroelectric phase it can also excite the extensional modes associated with the diagonal strains e1 , e2 , and e3 via the non-zero coefficients d11, d12, d13. The contributions of the extensional modes to the dielectric permittivity of Rochelle salt at frequencies far from the piezoelectric resonance region are not expected to be crucial, due to smallness of d1i (i ¼ 1,2,3) in comparison with d14. The presence of these modes, however, changes the permittivity in the resonance region, at least giving rise to additional resonance peaks, and this should be explored.

A.P. Moina / Physica B 407 (2012) 4550–4556

components of the stress tensor. Relevant to our case are the displacements Z2 and Z3 , giving the strains

2. The model Our calculations are based on the modification of the Mitsui model of Ref. [16] that takes into account both the shear strain e4 and the diagonal strains. It considers motion of pseudospins, corresponding to certain ordering units in a real crystal, in two interpenetrating sublattices with asymmetric double well potentials. The system behavior is described in terms of the following linear combinations of the mean values of the pseudospins belonging to different sublattices: /s1 S þ /s2 S sinh g ¼ , 2 cosh g þ cosh d /s1 S/s2 S sinh d ¼ s , 2 cosh g þcosh d

x

ð1Þ

the equations for which were obtained within the mean field approximation [14,16]. Here     JþK JK g¼b x2c4 e4 þ m1 E1 , d ¼ b sþD ; 2 2

b ¼ 1=kB T, kB is the Boltzmann constant; m1 is the effective dipole moment. c4 e4 is the internal field created by the piezoelectric coupling with the shear strain e4 . A longitudinal electric field E1 is applied. The parameters J, K are the Fourier-transforms (at k ¼ 0) of the constants of interaction between pseudospins belonging to the same and to different sublattices, respectively. They, along with the asymmetry parameter D, are assumed [16] to be linear functions of the diagonal strains J 7 K ¼ J0 7 K 0 þ 2

3 X

ci7 ei , D ¼ D0 þ

i¼1

3 X

c3i ei :

ð2Þ

i¼1

The stress–strain relations and polarization have been obtained in the following form [16]:

si ¼

3 X

0 0 cE0 ij ½ej aj ðTT j Þ

j¼1

0 s4 ¼ cE0 44 e4 e14 E1 þ 2

e0 E þ P 1 ¼ e014 e4 þ w11 1

c4 v

m1 v

1 þ 2 1  2 1 c x  ci s  c3i s 2v i 2v v

x,

x:

ði ¼ 13Þ,

ð3Þ ð4Þ

Here si are the stress tensor components in Voigt notations; cE0 ij and a0i are the ‘‘seed’’ elastic constants and thermal expansion coefficients, representing the contributions from the host lattice, which forms the asymmetric potentials for the pseudospins [16]; v is the unit cell volume of the model.

e2 ¼

We consider vibrations of a thin rectangular Ly  Lz plate of a Rochelle salt crystal cut in the (100) plane (X-cut) induced by time-dependent electric field E1 ¼ E1t expðiotÞ. This field gives rise to the shear strain e4 at all temperatures, as well as to the diagonal strains e1 , e2 , e3 in the ferroelectric phase. We take into account the in-plane vibrational modes, allowed by the system symmetry, and neglect the out-of-plane mode associated with e1 . Dynamics of the strains will be described, using classical (Newtonian) equations of motion [17] of an elementary volume @2 Zi X @sik ¼ , @xk @t 2 k

ð5Þ

where r ¼ 1:767 g=cm3 is the crystal density; Zi are displacements of an elementary volume along the axis xi; sik are

@Z2 , @y

e3 ¼

@Z3 , @z

e4 ¼

@Z2 @Z3 þ : @z @y

ð6Þ

Dynamics of the pseudospin subsystem will be described within the Glauber approach [18], where the kinetic equations for the time-dependents x and s read [14]   d 1 1 1 a x ¼ x tanh ðg þ dÞ þ tanh ðgdÞ , dt 2 2 2   d 1 1 1 a s ¼ s tanh ðg þ dÞtanh ðgdÞ : ð7Þ dt 2 2 2 Here a is the parameter setting the scale of the dynamic processes in the pseudospin subsystem; its value is usually found by fitting the theoretical curves of the microwave permittivity to experiment [20]. The form of Eq. (7) is not affected by inclusion of the diagonal strains into consideration. At small departure from equilibrium, the dynamic variables x, s, and ej (or Zi ) can be presented as sums of the equilibrium values and of the fluctuational deviations, while the deviations are taken to be in the form of harmonic waves

x ¼ x~ þ xt ðy,zÞexpðiotÞ, s ¼ s~ þ st ðy,zÞexpðiotÞ,

Zi ¼ Z~ i þ Zit ðy,zÞexpðiotÞ: Eqs. (3)–(7) are expanded in terms of these deviations up to the linear terms. For the equilibrium quantities we obtain Eqs. (1), (3), and (4). For the fluctuational parts, from Eqs. (3)–(4) we get the constitutive equations

si ðy,zÞ ¼

3 X

1 þ~ 1  1 ~ cE0 ij ejt ðy,zÞ ci xxt ðy,zÞ ci sst ðy,zÞ c3i st ðy,zÞ, v v v j¼1

0 s4 ðy,zÞ ¼ cE0 44 e4t ðy,zÞe14 E1t þ2

e0 E þ P1t ðy,zÞ ¼ e014 e4t ðy,zÞ þ w11 1t

c4 v

m1 v

xt ðy,zÞ,

ð8Þ

xt ðy,zÞ

ð9Þ

(for i¼ 1–3). Taking these into account, from Eq. (7), and Eq. (5) we obtain the following system: a

d x þa1 xt þ a2 st þa32 e2t þ a33 e3t þ a34 e4t ¼ a02 E1t , dt t

a

d st þ b1 xt þb2 st þb32 e2t þ b33 e3t þb34 e4t ¼ b02 E1t , dt

r

3. Vibrations of X-cuts of Rochelle salt

r

4551

r

 2  @2 Z2t @2 Z2t @2 Z2t @ Z2t @2 Z3t þcE0 ¼ cE0 þ cE0 þ 22 23 44 2 2 2 @y@z @y@z @y @z @t   2c4 @xt 1 @st þ ~ @xt   c2 x þ ðc2 s~ þ c32 Þ , þ v @z v @y @y  2  2 2 @2 Z3t @2 Z2t E0 @ Z2t E0 @ Z3t E0 @ Z3t þ c ¼ c þc þ 23 33 44 @y@z @y@z @z2 @y2 @t 2   2c4 @xt 1 @st þ ~ @xt   c3 x þ ðc3 s~ þ c33 Þ : þ v @y v @y @z

ð10Þ

The expressions for its coefficients, being the functions of the equilibrium x, s, and ei , are given in the Appendix. Hereafter it is implied that the deviations are functions of y and z. Solving the two first equations of Eq. (10) with respect to xt at Z2t ¼ Z3t ¼ 0, substituting the result into Eq. (9), and differentiating that with respect to the field, we find the dynamic dielectric permittivity of a clamped crystal, exhibiting relaxational dispersion in the microwave region e0 þ 4p ee11 ðoÞ ¼ e11

bm21 2v

F 1 ðaoÞ,

e0 ¼ 1 þ4pwe0 : e11 11

ð11Þ

4552

A.P. Moina / Physica B 407 (2012) 4550–4556

This expression is same as for the model [13] without the diagonal strains. See the Appendix for the introduced notations. In the regime of a mechanically free crystal, the two first equations of Eq. (10) give

bm1

xt ðy,zÞ ¼

F 1 ðaoÞE1t bc4 F 1 ðaoÞe4t ðy,zÞ

2

b

b

 F x42 ðaoÞe2t ðy,zÞ F x43 ðaoÞe3t ðy,zÞ, 2 2

bm1

st ðy,zÞ ¼

eit 9S  ei0 ¼ d1i ðoÞE1t ,

F s1 ðaoÞE1t bc4 F s1 ðaoÞe4t ðy,zÞ

2

b

b

s

with d1i ðoÞ ¼

s

 F 42 ðaoÞe2t ðy,zÞ F 43 ðaoÞe3t ðy,zÞ 2 2

ð12Þ

(notations are in the Appendix). Hence, the field-dependent part of polarization is   2 X e0 þ bm1 F ðaoÞ E þ P 1t ðy,zÞ ¼ w11 e1i ðoÞeit ðy,zÞ, ð13Þ 1 1t 2v i ¼ 2,3,4 with the dynamic piezoelectric coefficients given by e12,3 ðoÞ ¼

bm1 2v

F x42,3 ðaoÞ,

e14 ðoÞ ¼ e014 

bm1 c4 v

F 1 ðaoÞ:

ð14Þ

The observable dynamic dielectric permittivity at constant stress es11 ðoÞ is expressed via the derivative from the polarization averaged over the sample volume Z Ly Z Lz 4p @ es11 ðoÞ ¼ 1 þ dy dzP 1t ðy,zÞ Ly Lz @E1t 0 0 X @ ¼ ee11 ðoÞ þ 4p e1i ðoÞ eit , ð15Þ @E 1t i ¼ 2,3,4 where 1 eit ¼ Ly Lz

Z

Ly

dy

0

microwave dispersion of the dielectric susceptibility, but not in the piezoelectric resonance region, where Re c~ Eij ðoÞ and Re e1j ðoÞ (14) are very close to the corresponding static quantities (see Refs. [16,20]). Boundary conditions for eit ðy,zÞ follow from the assumption that the crystal is traction free at its edges (at y¼0, y ¼ Ly , z ¼0, z ¼ Lz , to be denoted as S): s1 9S ¼ s2 9S ¼ s3 9S ¼ s4 9S ¼ 0: Using Eqs. (8) and (12), we obtain the boundary conditions for the strains

ro2 e2t ¼ c~ E22 ðoÞ

dzeit ðy,zÞ:

eit ¼ ei0 þE1t

@2 e2t @2 e2t @2 e4t þ ½c~ E23 ðoÞ þ c~ E44 ðoÞ þ c~ E34 ðoÞ 2 @y@z @z @z2 2 2 2 @ e3t @ e3t @ e3t þ c~ E33 ðoÞ þ c~ E44 ðoÞ þ c~ E34 ðoÞ , @y@z @y2 @z2

ro2 e3t ¼ c~ E24 ðoÞ

Dkl 2 ¼

o

b 2v

ð19Þ

c~ E44 ðoÞ ¼ cE0 44 

16 "

þ d12 ðoÞ c~ E44 ðoÞ 16

þ fcj x~ F x2i ð

 Þ þ ðcj

ao

s

s~ þ c3j ÞF 2i ðaoÞg:

p2 ð2kþ 1Þ2 L2y

Their frequency variation is perceptible only in the region of the

L2y þ c~ E33 ðoÞ

p2 ð2l þ 1Þ2 L2z

!# ro2

,

1

þ d13 ðoÞ c~ E22 ðoÞ

p2 ð2kþ 1Þ2 L2y

p2 ð2l þ 1Þ2 L2z þ c~ E44 ðoÞ

p2 ð2l þ 1Þ2 L2z

!# ro2

,

ð21Þ " Dkl 4 ¼

#

X 8 16d14 ðoÞ o2 þ Q p p , p2 ð2kþ 1Þð2l þ 1Þ mn p2 mn km ln ðoð4Þ Þ2 o2 kl

ð22Þ

and mn ~E ~E Q mn ¼ ½c~ E22 ðoÞ þ c~ E23 ðoÞDmn 2 þ ½c 33 ðoÞ þ c 23 ðoÞD3 ,

Dð23Þ ðoÞ ¼ c~ E22 ðoÞ kl ð17Þ

p2 ð2k þ1Þ2

p2 ð2k þ 1Þð2l þ1Þ Dð23Þ ðoÞ kl

"

2

ð20Þ

1

"

2bc4 F 1 ðaoÞ, v

i ¼ 2,3,4,

p2 ð2k þ 1Þð2l þ1Þ Dð23Þ ðoÞ kl

 d12 ðoÞðc~ E23 ðoÞ þ c~ E44 ðoÞÞ

The frequency-dependent elastic constants c~ Eij ðoÞ in Eq. (16) are given by the model expressions F x4i ðaoÞ,

X kl pð2kþ 1Þy pð2l þ 1Þz Di sin sin , Ly Lz kl

 d13 ðoÞðc~ E23 ðoÞ þ c~ E44 ðoÞÞ

Dkl 3 ¼

@2 e2t @2 e2t þ ½c~ E22 ðoÞ þ c~ E23 ðoÞ @y@z @y2 @2 e2t @2 e3t E E E þ ½c~ 24 ðoÞ þ2c~ E34 ðoÞ þ ½2c~ 24 ðoÞ þ c~ 34 ðoÞ 2 @z @y2 2 2 @ e3t @ e3t þ c~ E34 ðoÞ þ ½c~ E23 ðoÞ þ c~ E33 ðoÞ @y@z @z2  2  2 @ e @ e 4t 4t þ c~ E44 ðoÞ þ : ð16Þ @y2 @z2

v

@2 e2t @2 e3t @2 e3t þ c~ E44 ðoÞ þ c~ E33 ðoÞ , 2 2 @z @y @z2

obviously satisfying the boundary conditions (18), with

ro e4t ¼ c~ E24 ðoÞ

Þ ¼ cE0 ij 

@2 e2t @2 e2t @2 e3t þ c~ E44 ðoÞ þ ½c~ E44 ðoÞ þ c~ E23 ðoÞ , 2 2 @y @z @y2

@2 e2t @2 e3t þ ½c~ E23 ðoÞ þ c~ E33 ðoÞ ro2 e4t ¼ ½c~ E22 ðoÞ þ c~ E23 ðoÞ @y@z @y@z  2  2 @ e @ e 4t 4t þ : þ c~ E44 ðoÞ @y2 @z2

@2 e2t @2 e2t @2 e2t þ c~ E44 ðoÞ þ c~ E24 ðoÞ @y@z @y2 @z2 2 2 @ e @ e @2 e3t 4t 3t , þ c~ E24 ðoÞ þ ½c~ E44 ðoÞ þ c~ E23 ðoÞ þ c~ E34 ðoÞ @y@z @y2 @y2

c~ Eij ð

oÞe1j ðoÞ. Here sEij ðoÞ are the elements of a

ro2 e3t ¼ ½c~ E23 ðoÞ þ c~ E44 ðoÞ

Lz 0

bc4

E j ¼ 1 sij ð E to c~ ij ð Þ

matrix inverse o of Eq. (17). The system of second-order partial differential equations (16) will be solved numerically using the finite element method. However, we can obtain some analytical results, if we take into account the fact that in Rochelle salt c22, c33, cE44 , c23 b9cE24 9, 9cE34 9 [21]. Neglecting in Eq. (16) the terms proportional to cE24 , cE34 (in the paraelectric phases cE24 ¼ cE34 ¼ 0 exactly) we get a partially split system, with the two first equations depending on e2t and e3t only

ro2 e2t ¼ c~ E22 ðoÞ

c~ Ei4 ðoÞ ¼

ð18Þ

Its solutions are the series (see Ref. [19] for details)

Z

Substituting Eq. (12) into the two last equations of Eq. (10) (we get Christoffel equations by this step) and differentiating them with respect to y and z, we obtain the three following equations for the strains e2t , e3t , e4t (details of calculations can be found in Ref. [19])

2

P4

p2 ð2k þ 1Þ2

"  c~ E44 ðoÞ

L2y

þ c~ E44 ðoÞ

p2 ð2kþ 1Þ2 L2y

p2 ð2l þ1Þ2

þ c~ E33 ðoÞ

L2z

# ro2

p2 ð2l þ 1Þ2 L2z

# ro2

A.P. Moina / Physica B 407 (2012) 4550–4556

½c~ E23 ðoÞ þ c~ E44 ðoÞ2

p2 ð2kþ 1Þ2 p2 ð2l þ 1Þ2 L2y

L2z

and it is associated with the extensional modes of e2 and e3 . Solutions of Eq. (25), to be denoted as oð23Þ , exist at all temperakl tures, but the resonances are present in the ferroelectric phase only, as the corresponding modes are not excited in the paraelectric phases. On the other hand, R4 ðoÞ-1 both at o ¼ oð23Þ and kl at o ¼ oð4Þ given by Eq. (23). These resonances originate from the kl shear vibrational mode associated with e4 and persist in the paraelectric phases. Such a division is, however, artificial, as the modes are coupled. All three strains calculated numerically from the complete system (16) have resonances at the same frequencies.

,

and oð4Þ given by equation kl vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u E uc~ 44 ðoð4Þ Þp2 ð2kþ 1Þ2 ð2l þ 1Þ2 ð4Þ kl t : okl ¼ þ r L2y L2z

ð23Þ

Also pkm ¼

1 m þk þ 1

or

pkm ¼

1 , mk

if k and m are of the same or of different parities, respectively. kl ~ Note that e20 ¼ e30 ¼ 0 and Dkl 2 ¼ D3 ¼ 0 at x ¼ 0. It means that in the paraelectric phases the extensional modes associated with the strains e2 and e3 are not excited by the field E1, which is in accordance with the symmetry considerations for the orthorhombic point group P21 21 21 . Finally, the permittivity is X s e11 ðoÞ ¼ ee11 ðoÞ þ 4p e1i ðoÞd1i ðoÞRi ðoÞ, ð24Þ

4. Numerical analysis The dynamic permittivity and resonant frequencies of Rochelle salt X-cuts to be evaluated here are expressed via the equilibrium values of the dynamic variables: the order parameters x~ and s~ and the strains e~ i (i¼1–4). Those quantities are calculated within the model [16], described in Section 2. They are found from Eqs. (1) and 7 (3). The values of the model parameters c3i , c4 , a, and others were chosen in Refs. [16,20] by fitting the theoretical dependences of the transition temperatures in Rochelle salt on hydrostatic and various uniaxial pressures, as well as the theoretical temperature dependences of several dielectric, piezoelectric, and elastic characteristics of the crystal to experimental data. The fitting procedure, adopted values of the model parameters, as well as the obtained description of the experiment (including the relevant to the present paper elastic constants) are discussed in Refs. [16,20] in great detail. To calculate the dynamic permittivity no additional theory parameter needs to be determined apart from those of Refs. [16,20], but the sample dimensions should be specified. In the present paper we shall use Ly ¼1.60 cm, Lz ¼2.45 cm Rochelle salt X-cut, for which experimental data for the dielectric permittivity in the resonance region are available [4]. Eqs. (16) for the fieldinduced strains eit ðy,zÞ with the boundary conditions (18) are

i ¼ 2,3,4

with Ri ðoÞ ¼ 1 þ

X

4Dkl i 2 ð2k þ 1Þð2l þ1Þd

p

k,l

oÞ

1i ð

:

In the zero (o-0, Ri ðoÞ-1) and high frequency (o-1, Ri ðoÞ-0) limits from Eq. (24) we obtain the static free permittivity (see Ref. [20]) and the clamped dynamic permittivity (11). Thus, Eq. (24) explicitly describes the effect of crystal clamping by high-frequency electric field. In the intermediate frequency region, the permittivity has a resonance dispersion with numerous peaks at frequencies where Re½Ri ðoÞ-1. The resonances are of two types. The first type of resonances is given by R2,3 ðoÞ-1, that is by equation

Dð23Þ ðoÞ ¼ 0, kl

ð25Þ ε2t(y,z)

ε3t(y,z)

20

20

15

15

15

10

10

10

5

5

5

0

0

z(mm)

20

0

5 10 y(mm)

15

0

5

10

15

0

20

20

20

15

15

15

10

10

5

5

0

0

0

5

10

15

0

5

10

15

5 10 y(mm)

15

0

5

10

15

0

10

5

5

0

0

0

0

122.5 kHz

0

5

10

15

0

5

10

15

0

5

10

15

0

20

20

20

15

15

15

10

10

10

10

5

5

5

5

0

0

0

20

20

20

15

15

15

10

10

5 0

15

0

10

0

10

5

0

5

5

5

5

0

10

10

0

5

5

5

15

10

10

10

10

10

15

15

5

15

10

20

20

0

20

15

15

15

15

15

20

15

20

20

10

10

20

15

15

5

5

ε3t(y,z)

20

20

0

66.2 kHz

0

ε2t(y,z)

ε4t(y,z)

z(mm)

ε4t(y,z)

4553

170.3 kHz

0

5

10

15

0

5

10

15

0

5

10

15

0

20

20

20

15

15

15

10

10

10

10

5

5

5

5

5

0

0

0

0

0

5

10

15

238.7 kHz

0

5

10

15

0

5

10

15

0

5

10

15

0

0.302 MHz

0

5

10

15

0.397 MHz

0

5

10

15

0.875 MHz

0

5

10

15

46.1 MHz

0

5

10

15

Fig. 1. Spatial distributions of the strains eit ðy,zÞ=E1t of the Rochelle salt X-cut at 293 K and different frequencies. Grayscales are different for each graph. Ly ¼1.60 cm, Lz ¼2.45 cm. The frequencies are chosen to illustrate the extrema multiplication.

4554

A.P. Moina / Physica B 407 (2012) 4550–4556

solved numerically with the finite element method package FreeFemþþ [22]. In Fig. 1 we show the spatial distribution of the fluctuational parts of the strains eit ðy,zÞ=E1t . The distributions have a single extremum at the sample center at low frequencies; then the extrema multiply. The reason for this multiplication can be seen already in the expression (20) for the strains: the major contributions to the sum in Eq. (20) are from the terms with the values of k and l, yielding the smallest DðiÞ ðoÞ in the denominators of Dkl i in Eqs. (21) and (22). kl With increasing o, these k, l increase; thus the number of extrema of the sin pð2k þ1Þy=Ly sin pð2l þ 1Þz=Lz function that would occur within y A ½0,Ly  and z A ½0,Lz  increases the well. At high frequencies, eit ðy,zÞ are zeros in most of the plate, only going to the boundary values given by Eq. (18) within very 1000

ε'11

500 0 -500 -1000

105

107

109

ν (Hz)

Fig. 2. Frequency dependence of dielectric permittivity of a Rochelle salt X-cut at 293 K. The line: a theory. Symbols: experimental points taken from Ref. [4]—’, [6]—X, [23]—J. The line and ’ are for Ly ¼1.60 cm, Lz ¼ 2.45 cm. The other symbols correspond to samples of different sizes.

ε'11

300

narrow strips near the sample edges. It illustrates the effect of crystal clamping by a high-frequency electric field. In Fig. 2 we plot the frequency dependence of dynamic permittivity of the Rochelle salt X-cut within the entire frequency range of the current model applicability. That range does not include the region of the domain-related dispersion below 1 kHz [1,3] or the submillimeter (100–700 GHz) region of resonant dispersion [7]. The experimental data shown by open symbols are for the frequencies outside the piezoelectric resonance regions of the respective samples used in the measurements, so the dimensions of those samples are irrelevant. The obtained dependence is analogous to the experimental [3] and to the previously obtained theoretical [13,15] ones: from the static permittivity of a free crystal at low frequencies, via the piezoelectric resonance region (104–107 Hz for this sample dimension), to the clamped crystal value, and, eventually, to a relaxational dispersion in the microwave region. A fairly good agreement with experiment is obtained. We shall be interested in the permittivity behavior in the piezoelectric resonance region. Fig. 3 shows that most of the resonant frequencies, calculated numerically using the complete set of Eqs. (16), are very well reproduced by the resonant frequencies (23) and (25) of the simplified system (19). Therefore we can use Eqs. (23) and (25) to analyze the resonant behavior of the Rochelle salt X-cuts. The temperature dependence of a few lowest resonance frequencies is shown in Fig. 4. At all temperatures the lowest is the resonance frequency nð4Þ ¼ oð4Þ =2p associated with the kl kl shear mode at k ¼ l ¼ 0. The lowest frequency nð23Þ ¼ oð23Þ =2p of the kl kl extensional modes (at k ¼ l ¼ 0) is higher at any temperature and at any sample dimensions and always remains finite. The shear mode frequencies nð4Þ , on the other hand, go to zero at the Curie kl temperatures at all k and l. The agreement with experiment is quite

ε'11

600

200 400

100

200

0

0

-100 -200

-200

0

100

200

ν (kHz)

0

100

200

ν (kHz)

Fig. 3. Frequency dependences of the dynamic dielectric permittivity of a Rochelle salt X-cut at 275 K (left) and 293 K (right). The solid line is calculated with eit found from Eq. (16). n and ’ are the resonant frequencies nð4Þ and nð23Þ of the simplified system (19). Ly ¼ 1.60 cm, Lz ¼ 2.45 cm. Dotted lines are a guide to the eye. kl kl

ν kl(23)(kHz)

ν kl(4) (kHz)

(0.1)

200

400 (0.1)

(1.0) 100

(1.0) (0.0) (0.1)

200

(0.0) (0.0)

0 240

T (K) 260

280

300

(1.0)

(0.0) 0 240

Τ (Κ) 260

280

300

Fig. 4. Temperature dependences of the lowest resonance frequencies of the Rochelle salt X-cut with Ly ¼ 1.60 cm, Lz ¼ 2.45 cm. Lines: the theory. Solid lines: nð4Þ ; dashed kl lines: nð23Þ . Symbols: experimental points of Ref. [4]. The numbers in parentheses are the (k,l) values. kl

A.P. Moina / Physica B 407 (2012) 4550–4556

ε'11

ε'11

4000

500

0

0

-4000 290

4555

-500 295

300

305 T (K)

240

260

280

300

T (K)

Fig. 5. Temperature dependences of the dielectric permittivity of the Rochelle salt X-cut at 60 kHz (left) and 800 kHz (right). Lines: the theory. : experimental points of Ref. [4]; n: the resonant frequencies of the simplified system (19) given by Eqs. (23) and (25). Ly ¼ 1.60 cm, Lz ¼ 2.45 cm.

good in the ferroelectric phase and gets worse in the upper paraelectric phase. It is, apparently, caused by a similar misfit for the elastic constant cE44 (see Eq. (23) and Ref. [20]). The temperature variation of the dynamic dielectric permittivity of the Rochelle salt X-cut near T C2 is shown in Fig. 5 (left). There are two distinct and well resolved resonance peaks below and above the Curie point. With increasing frequency, these peaks move away from the Curie point. They are associated with the e4 shear mode and given by nð4Þ 00 . The disagreement between theory and experiment for nð4Þ 00 seen in Fig. 4 resulted in a more than 2 K difference between the theoretical and experimental temperatures of the resonant peak above T C2 . It has not been observed experimentally [4] that apart from the two discussed peaks, in the close vicinity of the transition temperature the permittivity has a multitude of other peaks, which are higher order resonances of the e4 mode (nð4Þ at k þ l 4 0). kl Obviously, as seen in Fig. 4, if the measuring frequency of the permittivity is below the lowest resonant frequency of the extensional mode nð23Þ 00 ðT C Þ at the Curie points (which can be found from Eq. (25) at cE44 -0), all resonances are associated with the shear e4 mode. Above nð23Þ 00 ðT C Þ, the resonances associated with the extensional modes appear in the ferroelectric phase. It is seen in the temperature curve of the permittivity plotted in a wider range (Fig. 5, right) that the resonances form two packets around each Curie temperature, with the density of resonances increasing at approaching the Curie points. It can be shown that the packet widths increase with frequency, because of the shape of the nð4Þ ðTÞ curves (see Fig. 4) and appearance of the nð23Þ resonances. kl kl Eventually, at some sufficiently high frequency the packets overlap. One can also see that the presented curve is, actually, the curve of the clamped permittivity, decorated with numerous resonances.

analyzed; the lowest resonance is found to be associated with the shear mode at all temperatures. The resonances associated with the extensional modes appear above a certain threshold frequency and in the ferroelectric phase only, which is consistent with the symmetry considerations. In the close vicinities of the Curie temperatures, the permittivity has a multitude of overlapping peaks, which are higher order resonances of the e4 mode. Appendix The coefficients of the system (10) are the following: a1 ¼ 1 þ b a32,3 ¼

JþK l1 , 4

a2 ¼ b

KJ l2 , 4

b ~ þ  ½xl1 c2,3 l2 ðc2,3 s~ þ c32,3 Þ, 2

a34 ¼ bc4 l1 ,

bm1

a02 ¼ 

JþK b1 ¼ b l2 , 4

2

l1 ,

b2 ¼ 1 þ b

JK l2 , 4

b

~ c þ l ðc s~ þ c b32,3 ¼  ½xl 2 2,3 1 32,3 Þ, 2,3 2 b34 ¼ bc4 l2 , 2

b02 ¼

bm1 2

l2 ,

l1 ¼ 1x~ s~ , l2 ¼ 2x~ s~ : 2

The notations introduced in Eq. (11) are iaol1 þ j3 , DðoÞ

F 1 ðaoÞ ¼

j1 ¼ 2

bJ 2

j3 ¼ l1 þ

DðoÞ ¼ ðiaoÞ2 þ ðiaoÞj1 þ j2 ,

l1 , j2 ¼ 1

bðKJÞ 4

2

bJ 2

l1 b2

K 2 J 2 2 2 ðl1 l2 Þ, 16

2

ðl1 l2 Þ:

5. Concluding remarks

The notations introduced in Eq. (12) are

Vibrations of X-cuts of Rochelle salt crystals and their influence on the dynamic dielectric permittivity are analyzed using the modified Mitsui model that takes into account the shear strain e4 and the diagonal strains e1 , e2 , e3 [16]. The system dynamics is described within the frequency range, starting from 1 kHz (above the dispersion associated with the domain wall motion) via the piezoelectric resonance region and the microwave relaxational dispersion up to about 1011 Hz. The changes in the calculated spatial distributions of the strains with increasing frequency visualize the effect of crystal clamping by the high-frequency electric field. Explicit expressions for the resonant frequencies, associated with the shear mode of e4 and with the extensional in-plane modes of e2 , e3 , of such cuts are derived. They are obtained at neglecting the outof-plane mode associated with e1 , as well as the elastic constants cE24 and cE34 . The temperature behavior of the resonant frequencies is

F x4j ðaoÞ ¼

~ c þ ðc s~ þ c Þl  j4j þ ðiaoÞ½xl 1 j 3j 2 j , DðoÞ ~ c þ þðc s~ þ c Þj þ ðiaoÞ½xl ~ c þ ðc s~ þ c Þl  xl 1 j 2 j 3j 3j 1 5 j j F s4j ðaoÞ ¼  , DðoÞ F s1 ðaoÞ ¼ 

l2 ð1 þ iaoÞ

, DðoÞ   j4i ¼ ciþ x~ j3  ci s~ þ c3i l2 ,

j5 ¼ l1 

bðK þ JÞ 4

2

2

ðl1 l2 Þ:

References [1] J.F. Araujo, J. Mendes Filho, et al., Phys. Rev. B 57 (1998) 783. [2] A.V. Shyl’nikov, N.M. Galijarova, et al., Kristallografiya 31 (1986) 326.

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[3] Y.M. Poplavko, V.V. Meriakri, P. Pereverzeva, V.V. Alesheckin, V.I. Molchanov, Fiz. Tverd. Tela (Leningrad) 15 (1974) 2515. [Sov. Phys. Solid State 15 (1974) 1672]. [4] M.R. Leonovici, I. Bunget, Ferroelectrics 22 (1979) 835. [5] H. Mueller, Phys. Rev. 58 (1940) 565. [6] F. Sandy, R.V. Jones, Phys. Rev. 168 (1968) 481. [7] A.A. Volkov, G.V. Kozlov, S.P. Lebedev, JETP 52 (1980) 722. [8] W.P. Mason, Phys. Rev. 55 (1939) 775. [9] T. Mitsui, Phys. Rev. 111 (1958) 1259. [10] B. Zeks, G.C. Shukla, R. Blinc, Phys. Rev. B 3 (1971) 2306. [11] B. Zeks, G.C. Shukla, R. Blinc, J. Phys. 33 (Suppl. N4) (1972) c2-68. [12] R.R. Levitskii, I.R. Zachek, T.M. Verkholyak, A.P. Moina, Condens. Matter Phys. 6 (2003) 261. [13] A.P. Moina, R.R. Levitskii, I.R. Zachek, Phys. Rev. B 71 (2005) 134108. [14] R.R. Levitskii, I.R. Zachek, T.M. Verkholyak, A.P. Moina, Phys. Rev. B 67 (2003) 174112.

[15] A. Andrusyk, Piezoelectric effect in Rochelle salt, in: M. Lallart (Ed.), Ferroelectrics—Physical Effects, InTech, 2011. [16] A.P. Moina, R.R. Levitskii, I.R. Zachek, Condens. Matter Phys. 14 (2011) 43602. [17] W.P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, Van Nostrand, New York, 1950. [18] R.J. Glauber, J. Math. Phys. 4 (1963) 294. [19] A.P. Moina, Preprint arXiv:1201.3482 [cond-mat.mtrl-sci]. [20] A.P. Moina, Condens. Matter Phys. 15 (2012) 13601. [21] K.-H. Hellwege, A.M. Hellwege (Eds.), Numerical Data and Functional Relationships in Science and Technology, Landolt-Bornstein, New Series. Group III: Crystal and Solid State Physics, vol. 16, Pt. b, Springer-Verlag, Berlin, 1982. [22] /http://www.freefem.org/ffþþS. [23] U. Schneider, P. Lunkenheimer, J. Hemberger, A. Loidl, Ferroelectrics 242 (2000) 71.