CEJP 1 (2003) 91{99

In° uence of the Sample Orientation in Sn2P2S6 Crystals on the Hydrostatic Piezoelectric Coe± cients Stanislav Pano·s¤, Dagmar Pano·sov¶a Department of Physics, Technical University of Liberec, H¶alkova 6, 461 17 Liberec 1, Czech Republic Received 29 July 2002; revised 16 December 2002 Abstract: The in®uence of the sample orientation on the e¬ective value of the hydrostatic (i) piezoelectric coe¯ cients dh of Sn2 P2 S6 crystals has been studied. The hydrostatic (1) (3) (1) piezoelectric coe¯ cients dh and d’h , were measured, dh = (244 § 3) pC/N and (3) (3) d’h = (92 § 1) pC/N. The hydrostatic piezoelectric coe¯ cient dh for orthogonal axis (3) system was calculated to be dh = (87 § 2) pC/N. The optimal orientation of the sample has been found as (XYl)-20¸-cut. Maximal value of the e¬ective hydrostatic piezoelectric (1) coe¯ cient dh equals 260 pC/N. Double rotated samples were also studied. The orientation of the samples insensitive to the pressure has been found. The theoretical mean value of hydrostatic piezoelectric coe¯ cient (dh )mean corresponding to randomly oriented Sn2 P2 S6 grains in a poled composite has been calculated to be (dh )mean = 136 pC/N. c Central European Science Journals. All rights reserved. ® Keywords: Sn2 P2 S6 , Hydrostatic piezoelectric coe± cient. PACS (2000): 77.65.Bn, 77.84.-s

Introduction

¤

Sn2 P2 S6 is uniaxial semiconductive ferroelectric material. The second-order ferroelectric phase transition (2/m to m) is observed near the temperature TC = 337 K [1 - 2]. Due to relatively low transition temperature, the temperature stability of the properties in this crystal is rather poor which restricts potential applications of this material. The higher transition temperature can be achieved by Ge doping (TC = 361 K) [3]. Sn2 P2 S6 crystals exhibit high values of pyroelectric ¯gure of merit comparable or superior to those of the most widely used pyroelectric materials [4]. High values of the hydrostatic E-mail: [email protected]

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S. Panoµ s, D. Panoµ sov´a / Central European Journal of Physics 1 (2003) 91{99

piezoelectric coe±cients are also interesting for many applications. These crystals are promising materials for designing e®ective piezo- and pyrotransducers. It is possible to de¯ne three components of the hydrostatic piezoelectric coe±cient (i) dh (i = 1, 2, 3) [5]: (1)

dh = d11 + d 12 + d 13

(1)

(2) dh (3) dh

= d21 + d 22 + d 23

(2)

= d31 + d 32 + d 33

(3) (i)

We have measured the hydrostatic piezoelectric coe±cients dh for several samples with di®erent orientation. The question was which orientation gives the maximal value of dh .

In° uence of the sample orientation on dh (i)

The e®ective values of the measured hydrostatic piezoelectric coe±cients dh depend on the orientation of the sample. The following analysis has been done for crystal symmetry m (Sn2 P2 S6 in ferroelectric phase). All calculations assume an orthogonal system of axes. Directions [100] and [010] of the monoclinic system coincide axis x and y of the orthogonal system. Direction [001] is not perpendicular to the x-axis, it includes angle 91.2º (see Fig. 1) [6]. For rotation around axis x (rotation angle ¬ ) the e®ective hydrostatic piezoelectric coe±cients are:

0(1)

dh

0(2) dh 0(3) dh

(1)

= d11 + d 12 + d 13 = d h ; = (d31 + d 32 + d 33 ) ¢ sin ¬ = (d31 + d 32 + d 33 ) ¢ cos ¬

(4) (3) = dh ¢ sin ¬ (3) = d h ¢ cos ¬

;

(5)

:

(6)

For rotation around axis y (rotation angle ­ ) the e®ective hydrostatic piezoelectric coe±cients can be expressed in the following way:

0(1)

dh

= (d11 + d 12 + d 13 ) ¢ cos ­ ¡ =

0(2) dh 0(3) dh

=

(1) dh (2) dh

¢ cos ­ ¡

(3) dh

(d31 + d 32 + d 33 ) ¢ sin ­

¢ sin ­

=0

(7) (8)

= (d31 + d 32 + d 33 ) ¢ cos ­ + (d11 + d12 + d 13 ) ¢ sin ­ (3) (1) = dh ¢ cos ­ + d h ¢ sin ­

(9)

For rotation around axis z (rotation angle ® ) the e®ective hydrostatic piezoelectric coe±cients are:

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93

Fig. 1 Orientation of the orthogonal system of the axis x, y, z and monoclinic axis [100], [010] and [001]. Angle ­ equals 91.2¸.

0(1)

dh

0(2) dh 0(3) dh

(1) = (d 11 + d 12 + d13 ) ¢ cos ® = dh ¢ cos ® ;

= ¡ (d 11 + d 12 + d13 ) ¢ sin ® = ¡ = (d 31 + d 32 + d33 ) =

(1) dh

¢ sin ® ;

(3) dh :

(10) (11) (12)

The value of e®ective hydrostatic piezoelectric coe±cient can be higher than the high(i) est value of hydrostatic piezoelectric coe±cients dh only in cases described by equations (7) and (9).

Measurement of the hydrostatic piezoelectric coe± cients (1)

(3)

The hydrostatic coe±cients dh and dh were measured using static d.c. technique inside the pressure chamber. The hydraulic press could produce a static pressure up to 60 MPa. A disadvantage of this method is change in temperature during increasing (or decreasing) pressure. It is not possible to wait for stabilising the temperature because of the conductivity of the material. The charge produced at the sample’s electrodes has been measured with a KEITHLEY 6517 Electrometer.

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S. Panoµ s, D. Panoµ sov´a / Central European Journal of Physics 1 (2003) 91{99

The in°uence of pressure p and temperature £ on the charge formed on the piezoelectric sample could be expressed by the formula [7]: (i)

dQ (i) = A ¢ d h (p; £) ¢ dp + A ¢ º

(i)

(p; £) ¢ d£;

(13)

(i)

i i where i is 1, 2 or 3, d h = @P is hydrostatic piezoelectric coe±cient, º (i) = @P is @p @£ pyroelectric coe±cient and A is area of the electrodes. A component of the polarisation Pi is equal to Qi /A, where Qi is the charge measured at the electrodes. We suppose that the value of the electrode area A is independent of pressure and temperature. The value of hydrostatic pressure equals to diagonal components of the stress tensor (p = ¡ T 11 = ¡ T22 = ¡ T33 ). (i) Hydrostatic piezoelectric coe±cient dh and pyroelectric coe±cient º (i) are pressure and temperature dependent. If the pressure p and the temperature £ are less than critical ones (i.e., far from the phase transition), we can express these dependencies by linear equations [7]. In our case, for Sn2 P2 S6 , we can reduce the formulas and write the ¯tting function of measured data as:

¯ Qi Q0 (i) ¯ ¢ (p ¡ = + dh ¯ p0;£0 A A where Q0 is an initial charge.

p0 ) + º

¯ ¯

(i) ¯

p0;£0

¢ (£ ¡

£0 ) ;

(14)

Experiment The samples were rectangular plates with major faces of (100) orientation (4.5 x 4.5 x 0.71 mm3 ) or (001) orientation (4.5 x 1.5 x 0.25 mm3 ). The plates have been cut from crystal prepared by vapor transport technique using SnI 2 as a transport agent. After polishing circular gold electrodes were thermally evaporated. All samples have been heated above the Curie temperature and annealed (at the temperature of 393 K for 30 minutes). Electric ¯eld (Ed:c .= 1300 V/cm) was applied on each sample during slow cooling from the temperature 393 K to the room temperature. The samples were illuminated by the white light after pooling to stabilise intrinsic electric ¯eld of the samples. (1) (3) It is necessary to know values of dh and dh to determine orientation of the sample with the maximal e®ective value of dh . The values of hydrostatic piezoelectric coe±cient (1) (3) dh and dh can be directly measured or calculated from the piezoelectric coe±cients di¸ (see eq. (1)-(3)). We have measured the charge produced by increasing and decreasing hydrostatic pressure (see Fig. 2). The measurement is not an isothermal process. The temperature changes more during decreasing pressure, so we used to apply only the increasing pressure part of data. We ¯tted measured data as a function of pressure and temperature using equation (14). (1) (3) Calculation of the values of hydrostatic piezoelectric coe±cients dh and d’h have been

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95

Fig. 2 Dependence of the charge produced on (100) oriented sample on pressure. (1)

done for several cycles. The hydrostatic piezoelectric coe±cient dh equals (244§3) pC/N (3) and d’h equals (92§1) pC/N. Contribution of temperature changes to the measured charge is 1-2 orders lower than contribution of pressure changes, so values of pyroelectric coe±cient º (i) have been calculated with high deviation and are not presented in this paper.

Discussion (1)

(3)

The determined values of hydrostatic piezoelectric coe±cients dh and d’h are relevant to the monoclinic system of the axis. The values most often cited in the literature are relevant to the orthogonal system of the axis. The hydrostatic piezoelectric coe±cient (3) (1) (3) dh (relevant to the orthogonal system) can be calculated from dh and d’h using the (3) equation (9). The calculated value is dh = (87§2) pC/N. For samples with di®erent orientation, the values of e®ective hydrostatic piezoelectric coe±cients depend on the orientation. Higher values can be reached only for samples rotated around axis 2. The dependence of the e®ective hydrostatic piezoelectric coe±cients on the angle ­ is shown in Fig. 3. Simple analysis equations (7) and (9) gives the angle ­ 1 = (-20§1)º for maximal values of e®ective hydrostatic piezoelectric coe±cient (1) (3) d’h and the angle ­ 2 = (70§1)º for maximal values of e®ective d’h . We extended our calculations to double rotated samples. We assume a ¯rst rotation (3) around z-axis and second rotation around y-axis. Value of d’h is invariant to rotation (1) around z-axis. The hydrostatic piezoelectric coe±cient d"h for double rotated sample we can describe using equations (7), (10) and (12) by:

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Fig. 3 In®uence of the sample orientation on the value of e¬ective hydrostatic piezoelectric (1) (3) coe¯ cients d’h and d’h .

00(1)

dh

(1) = dh ¢ cos ® ¢ cos ­ ¡

(3) d h ¢ sin ­

(15) (1)

Value of the e®ective hydrostatic piezoelectric coe±cient d"h for new sample orien(1) tation was calculated. The dependence of d"h on the angles ® and ­ is shown in Fig. 4. The function (15) is a periodical function. The values of angles ­ and ® from the interval (0; 2º ) describe all sample orientations in the space. The analysis of the function (1) (15) gives maximal value of e®ective hydrostatic piezoelectric coe±cient d"h for these sample orientations: [­ = 340º, ® = 0º] or [­ = 200º, ® = 180º]. Function (15) (1) gives the minimal value of e®ective hydrostatic piezoelectric coe±cient d"h for these orientations: [­ = 160º, ® = 0º] or [­ = 20º, ® = 180º]. (1) The values of e®ective hydrostatic piezoelectric coe±cient d"h are in maxima positive (1) and in minima negative. The absolute magnitudes of d"h equal in maxima and minima. In these orientations is x0 -axis parallel (positive value) or antiparallel (negative value) to spontaneous polarisation direction and electrodes are on surfaces orthogonal to x0 -axis. The sample orientations at ­ = 90º and ­ = 270º are invariant to rotation around z-axis. (1) The orientations with zero value of e®ective hydrostatic piezoelectric coe±cient d"h represent physical minimum of equation (15). These orientations occur for angle ® and angle ­ suitable to the equation (16): µ

(1).

­ = arctg d h d (3) ¢ cos ® h

¶

+k¢º ;

(16)

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97

Fig. 4 In®uence of the sample rotations around two axis on the value of e¬ective hydrostatic (1) piezoelectric coe¯ cients d"h .

where the parameter k is an integer number. A sample in this orientation is irresponsive to the hydrostatic pressure and these orientations are unusable for construction pressure sensors. An unpoled composite (type 0-3) with random oriented Sn2 P2 S6 grains glued together by nonpiezoelectric glue (e.g. epoxy) can also be described by the expression (15). Calculation the mean value of hydrostatic piezoelectric coe±cient (dh )mean gives zero, because contributions of all grains scratch. It is in agreement with theoretical postulates. After poling, all grains have the same sign value of hydrostatic piezoelectric coe±cient. We used the following expression (17) to calculate the theoretical maximum mean value of hydrostatic piezoelectric coe±cient (dh )mean (we neglected the in°uence of the glue). ³

´

d 2h mean

=

2¼ R 2¼ R ³ 0 0

´ 00(1) 2

dh

2¼ R 2¼ R

d­ d®

Z2¼Z2¼ ³

1 = 2 ¢ (2 ¢ º ) 0

d­ d®

0

´ 00(1) 2

dh

d­ d® :

(17)

0 0

Substitution the equation (15) into the equation (17) and calculation these integrals give for the maximum mean value of hydrostatic piezoelectric coe±cient (dh )mean : ³

d2h

´

mean

=

³

´ (1) 2

dh

³

´ (3) 2

+ 2 ¢ dh 4

:

(18)

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S. Panoµ s, D. Panoµ sov´a / Central European Journal of Physics 1 (2003) 91{99

Using the equation (18) and measured values of hydrostatic piezoelectric coe±cients (3) and d’h we determined the maximal mean value of the hydrostatic piezoelectric coe±cient (dh )mean = 136 pC/N. (1) dh

Conclusion (1)

(3)

We have measured hydrostatic piezoelectric coe±cients dh and dh in Sn2 P2 S6 material. (1) (3) Measured value of hydrostatic piezoelectric coe±cient dh equals (244§3) pC/N and d’h (3) equals (92§1) pC/N. Our measured value of d’h is in good agreement with the previously (3) published value 90 pC/N [3]. Calculated value of hydrostatic piezoelectric coe±cient dh for orthogonal axis system equals (87§2) pC/N. Expressions (7) and (9) make it possible to determine orientation of the sample with the maximum value of the hydrostatic piezoelectric coe±cient. Determined direction corresponds to the spontaneous polarisation direction. The maximum value of e®ective (1) hydrostatic piezoelectric coe±cients d’h is observed for sample oriented as (XYl)­ 1 -cut (plate rotated around axis y) with the angle ­ 1 = (-20§1)º. The maximum value of (3) e®ective hydrostatic piezoelectric coe±cients d’h is observed for sample oriented as (ZYl)­ 2 -cut (plate rotated around axis y) with the angle ­ 2 = (70§1)º. Both situations represent the same state. Determined angles are in good agreement with the literature [8]. The maximum value of e®ective hydrostatic piezoelectric coe±cient is (1) (3) d’h (max) = d’h (max) = 260 pC/N. The di®erence between the maximal value of the e®ective hydrostatic piezoelectric coe±cient dh and the value of hydrostatic piezoelectric (1) coe±cient dh is approximately 7 %. We have discussed also the double rotated samples. Samples oriented as ­ -cuts (angle ­ ful¯lling expression (16)) are irresponsive to the hydrostatic pressure. We present orientations of the sample with the maximum value of the absolute magnitude of hydrostatic 00(1) piezoelectric coe±cient dh . We have calculated the theoretical mean value of hydrostatic piezoelectric coe±cient (dh )mean corresponding to random oriented Sn2 P2 S6 grains in a poled composite, (dh )mean = 136 pC/N.

Acknowledgements This work was supported by the Grant of Ministry of Education of the Czech Republic MSM 242200002.

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