Reliable frequency-temperature characteristics of the new quartz-crystal resonators vibrating in thickness-shear modes

97

Sensors and Actuators A, 40 (1994) 97-101

Reliable frequency-temperature characteristics of the new quartzcrystal resonators vibrating in thickness-shear modes Makoto Takeuchi, Mltsuo Nakazawa and Hiroshl Matsuzaka Shmhu

Unwergv, 500 Wakasato, Nagano-shr 380 (Japan)

Toyosaka Moruzurru and &o-o Tokyo Instuute of Technology, 2-12-l bokayama,

Nnyama Megwo-ku,

Tokyo IS2 (Japan)

(Recewed October 6, 1992, m rewed form September 10, 1993)

Abstract Thrs paper describes the turning pomts of frequency-temperature charactenstxx for rotated Y-cut quartz-crystal resonators wbratmg in b modes in the temperature range O-300 ‘C In the htgh-temperature region, the new experimental results are compared wth the theoretical values calculated by use of the ldeahzed thickness-wbration theory

1. Introduction

Smgle-crystalline alpha-quartz IS one of the most famous matenals used for resonators [l, 21, because it has excellent properties m the thermal, elastic, optical, plezoelectric and chermcal fields Therefore, it IS an mcreastngly noticeable trend that quartz-crystal resonators have recently been investigated for use as chemical sensors [3] This paper describes the frequency-temperature charactenstlcs for rotated Y-cut quartz b-mode resonators, which can be useful to decide the cut angles for resonators reliable enough to be used as chermcal sensors m the high-temperature region For the temperature range of interest, Kahan has already presented calculated values of the turnover temperatures for doubly rotated quartz resonators [4] An important expemnent has recently been carned out by the authors [5] In this paper, our calculated values are compared with the experunental results m the high-temperature region, thus leading us to propose new relations between the cut angles and the turning points of the frequency-temperature charactenstlcs for rotated Y-cut resonators vlbratmg in b modes Up to now httle information has been brought forward about such expernnental results as ours

as possible Usually, a crystallographic axis IS an axis of symmetry, and hereon, the axes of the rectangular Cartesian coordinate system for the rotated Y-cut plates are comcldent Hnth the crystallographic axes Hence, we define a thm quartz-crystal plate whose electrrcal, mechanical and opt& axes are m the dlrectlons of the x,, x, and x, axes, respectively, and further we denote the direction normal to the mam plane by polar coordinates (s, 0, 4) as shown m Fig 1, dlustratmg a thm quartz-crystal plate cut m any dlrectlon Now, the quartz-crystal resonator cut m anv. duectlon vibrates plezoelectncally and elastically through the electrodes of the mam planes The balance equations Optical x3 A

2. Theoretical analysis of frequency-temperature characteristics

for the crystal resonator

The crystallographic axial dlrectlons are chosen so as to make the speclficatlon of crystal faces as simple

0924-4247/94/$0700 0 1994 Elsevler Sequoia All rights reserved SSDI 0924-4247(93)00752-P

Electrxal Fig 1 A thm quartz-crystal plate

Mechanical

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for the quartz-crystal

$2 -

resonator

are as follows

_!2t$

(1)

I

for I = 1, 2, 3, where I = tnne, x, = components of rectangular coordmates, p = density, U, = components of displacement vector U, and TJ,=components of stress tensor T Accordmg to the thickness-mbratlon theory [l, 71 we assume the components of the displacement vector as the followmg U,=U, exp[l(lr*r-wt)]

(2)

for I= 1, 2, 3, where U,=amphtude for rth component of the displacement vector, J = ( - l)l”, w = angular frequency ( = 21$), k = wave vector and r = posltlon vector of marked particle m the plate The resonance frequency equations for thlcknessvibration modes m a thm ideal crystal plate are represented by eqns (l), (2) and Hooke’s law as

(“)

f= 5

l/z

P

0

(3)

for m=a, b, c, and n= 1, 3, where f = frequency, y0 = thickness of crystal plate and cc”‘)= elgenvalues for each thickness-vlbratlon mode m The ccrn) are solutions of Iqlwm,m, - cCm)$l= 0

(4)

where ($kl=

where cu, /3, and y are the first-, second-, and thirdorder temperature coefficients of frequency, respectwely, and AT= T- To

c,,M E

+

(m,P,m,)-‘(e,m,)(mkek,)

3. Results and discussion Now let us consider that a plane wave IS propagated with the wave vector k in a thm quartz-clystal plate, as shown in Fig 1 If we standardize the elastic stiffness constants, plezoelectnc strain constants, dielectric constants, and density of the plate to room temperature To=20 “C, substitute these values into eqns (3), (4), (5), and change the values for the angles 8 into 6 = 30” m eqn (6), we finally obtam the a; p and y for the quartz-crystal resonators which will show reliable frequency-temperature behavlour in the high-temperature region Figure 2 shows the first-order frequency-temperature coefficients of a, b and c modes m the rotated Y-cut plates (4=30”) It 1s seen that there IS a singular point m the vlcmlty of 6= 67” and 4=30”, and the replacements of the frequency-temperature coefficients for b and c modes occur around the same angle In Fig 2 we find the AT-cut near 8=125 25” and 4=30” On the other hand, we see the BT-cut m the same Figure around the angles 0= 41” and 4=30” It 1s possible for this theory to decide precisely the angles for the AT- and BT-cuts Slmdar calculations can be presented on the /3-e and ~0 characteristics for a, b and c modes m the rotated

(5)

and c$, = elastic stflness constants at constant electric field for I,], k, 1= 1, 2, 3, m,, m4 =jth or qth component of unit normal vector perpendicular to the plate for 1, 4 = 1, 2, 3, ehl,=plezoelectnc stress constants for h, 1,) = 1, 2, 3, I$, = dielectric constants at constant strain for q, J = 1, 2, 3, and S,,= Kronecker’s delta for I, I= 1, 2, 3 As shown m Fig 1, we obtam the followmg relationships m the polar coordinate system

50’

m,=en ec0s4 m,=sin Osin 9 m3=cos 8 The vlbratlonal frequencies for the quartz-crystal resonators are generally functions of temperature T, depending on the azimuth 4 and colatltude 0 m Fig. 1 Hence a Taylor expansion for f around the reference temperature To 1s approximately expressed as follows f(T) =f(T,)[l + aAT+ 1/2/3AT*+ 1/6yAT3]

(7)

Fig 2 The first-order frequency-temperature coefficients of the tinckness wbrataons m thm quartz-clystal plates, as a funchon of the polar angle 0

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Y-cut plates (I$= 30”), as shown in Figs 3 and 4, respectively We shall also use +,, and 0, to denote the angles of rotations, where the angles 4 and 0, through which the rotations take place are defined as positwe when seen counterclockwlse by the observed lookmg back toward the orlgm from the posltlve ends of the axes of rotation There are the followmg relations between 4//e and &,/t$ ++ &=30” and B- 0,= 90” For rotated Y-cut plates, &=O” Figure 5 shows the plate onentatlons wth respect to the crystallographic quartz axes Figure 6 shows the f&-T (turnover temperature) relationships for rotated Y-cut resonators, the crosses

300 7.80 260 240 7.20 7.00 180 160

0

e

140

Fag 6 Calculated and observed temperatures of zero temperature coefficient of frequency vs the onentatlon angles for rotated Ycuts wbratmg m b modes The symbol + shows an expenmental value obtamed by Nakazawa ef al [S] and the crosses show the expenmental values appearmg m refs 6 and 7 The two solid hnes show the probable repons mcludmg the results calculated by use of the ordmary constant values of a quartz-crystal resonator [l] The broken hne presents the experlmental results Fig 3 The second-order frequency-temperature coefficients of the thtckness vlbratlons m thm quartz-crystal plates, as a fun&on of the polar angle 0

Fig 4 The third-order frequency-temperature cocffiaents of the thickness nbratlons m thm quartz-crystal plates, as a funchon of the polar angle 6

Fig 5 Plate onentatlons axes

wltb respect to crystallographs

quartz

represent the experimental values [6, 71 In ref 3 the solutions were expanded to the hatched area and the boundary hnes around the area were decided from the equations contaming some uncertamty elements, such as the size and shape effects, the amblgmtles of the manufacture for cut angles, non-linear effects for temperature, etc Recently an expernnent was conducted by Nakazawa et al [5] The result IS represented m Fig 6 by the symbol + A thm circular rotated Y-cut resonator was set at f?,= -40 3” and the turnmg point of the frequency-temperature characterlstlc is 232 “C, as seen m Fig 6 The frequency constant IS 2 4920 MHz mm for the resonator vlbratmg us b mode, as seen m Fig 7 Figure 7 shows the relations between the cut angles and the frequency constants of the rotated Y-cut resonators vlbratmg m b mode Figure 8 shows an experimental result of the frequency-change ratlo to temperature charactenstlcs of a rotated Y-cut (t&= -40 3”) sample vlbratmg m b mode These results may lead to the conclusion that the calculated value 1s ddferent from the expenmental one m the higher-temperature regon It IS noticed in Fig 6 that our calculation IS vahd m the temperature range &120 “C So when we tly to predict the tummg pomt of the frequency-

We have found that our theoretical analysis 1s valid in this temperature range As seen m Frg 6, there are new and important relatlonshlps found between the temperatures and the cut angles m the high-temperature region These results can be applied to chemical sensors and resonators, etc [8]

Acknowledgements 2441,

/

,

-52

,

-48

,

,

-44

1

,

-40

-36

(degrees)

e.

Ftg 7 Calculated and observed frequency constants vs the onentatlon angles for rotated Y-cut resonators vlbratmg m b modes The symbol + shows an expertmentalvalue for 0, = - 40 3”

600

,

,

,

,

,

,

1

/

,

,

,

/

‘Ihs work was supported by San-e] Denshl Kogyo Co Ltd and Mlyota Co Ltd , Japan The authors wish to thank Dr Arthur Ballato of the USA, Dr fiyoshl Takahashl of Tokyo Institute of Technology, Professor Hiroshl Takahashl and Mr Atsushi Arakl of Shmshu University, and Dr Hltohlro Fukuyo, Japan, for their help and suggestions

400 100t

z x

B :

-loo-400 -

H

9

References

O-

Y

-600 -800 -

”

6 E

-lO”O-

it-

Izoo-

I: -1400-1600-

Tenlperature

IOC)

Fig 8 Frequency change ratio vs temperature characterlstln of a rotated Y-cut (&,= -40 3”) resonator vlbratmg m b mode (experimental values)

temperature charactenstlcs for the high-temperature repon (e g ,200 “C), we have to take mto conslderatlon the new relationships between the temperatures and the cut angles rather than the calculated ones

I Koga, M Aruga and Y Yoshmaka, Theory of plane waves m a plezoelectnc clystalhne medmm and the determmatron of the elastic and plezoelectrlc constants of quartz, Phys Rev, 109 (1958) 1467-1473 A Ballato, Doubly rotated tluckness mode plate vibrations, m W P Mason and R N Tburston (eds ), Pl9szcal Aco~ncs, Vol 13, Acadernlc Press, New York, 1977, Ch 5, pp 115-181 M Nakazawa, M Takeuchl, X Guanghua, T Mowmm and H Nnyama, A theoretIca study of the quartz crystal resonators vlbratmgmb-mode andhavmgrehablefrequency-temperature cbaractenstlcs m the repon of h@er temperature than 200 “C, Chzna?rapan Sc~nhfic Symp on Ekcbrmu Sensog Techno&y, Harbtn, Churn, 1992, pp 38-47 A Kahan, Turnover temperatures for doubly rotated quartz, Pruc 36th Frequency Contml Symp, PhnWelphua, PA, USA, 1985, pp 170-180 M Nakazawa, M Takeuclu, I-i. Yamaguchr and A Ballato, Tbm rotated Y-cut quartz resonators viiratmg m b-mode over a wide temperature range, I993 IEEE Frequency Con& Symp, Salt Lake Crty UT, USA, 1993, pp 541-547 T Shmada, llreory and Prachce for Quan~ Cytal Oscdlator, Ohm Co., Tokyo, 1955. pp 202-203 (m Japanese) I Koga, plezorlecmary and H@ Fquency, Ohm Co, Tokyo, 1937, pp 30-66 (m Japanese) M Nakazawa, M Takeucln, H Yamaguchi, A Ballato and T Lukaszcck, A theoretical study of quartz crystal resonators for chemical sensors, wbratmg m c mode, &oc 1992 IEEE Frequency Control Symp , Hershey, PA, USA, 199.2, pp 610613

4. Conclusions

We have performed both a theoretical analysis and expernnents with respect to the zero temperature coefficients of frequencies versus the onentation angles for rotated Y-cut crystals nbratmg m b modes From these results we have obtamed new relatlonshlps between the cut angles and the turnmg points of the frequency-temperature charactenstlcs m the temperature range O-300 “C

Biographies IUUWONakazawa was born m Nagano Prefecture, Japan, in 1938 He recerved the B S degree m electrical engmeermg from Shmshu Umverslty, Japan, m 1961, and the M S and Ph D degrees 111electrical engmeermg from Tokyo Institute of Technology, Tokyo, Japan, m 1963 and 1966, respectively

101

He was appomted lecturer m 1966 and associate professor m 1967 m the Faculty of Engmeermg, Shmshu University From September 1980 to July 1981, he was working at Prmceton Umverslty and the Electromcs Research and Development Command, USA, as a visiting fellow At present his pnnclpal concerns are with the theoretical analysis and design of high-stablhty quartzcrystal resonators, superconducting matenals and senSot-S

Dr Nakazawa 1sa member of the Institute of Electrical Engineers of Japan, The Institute of Electromcs, Informatlon and Commumcatlon Engineers of Japan, the Japan Society of Applied Physics and Physlcal Society of Japan, and he IS a Senior Member of IEEE Makoto Takeuchl was born m Nagano prefecture, Japan, m 1953 He received the B S degree from Yokohama National Unlveruty, Japan, m 1975 and received the M S degree from Tokyo Institute of Technology, Tokyo, Japan, m 1978 He worked at Shin-Etu Chemical Co Ltd from 1983 to 1985 and from 1985 up to now ISwith San-El Denshl Co, Ltd Now he is m the doctorate course at Shmshu Umverslty HIS prmclpal Interests are the electric wire and high-T, superconductor Hucwhl Matsuzaka was born m Nagano Prefecture, Japan, m 1946 He received the BE and M E degrees from Shmshu Umverslty, Japan, m 1969 and 1971, respectively

He 1s now working at Yamada Selsakusho Co, Ltd , Japan, and 1s a vice director of the Engmeermg Department HIS pnnclpal Interest LSIC packagmg technology Toyosaka Monmr was born m Gumma Prefecture, Japan, m 1942 He received the B E , M E and Ph D degrees, all m electronics engineering, from Tokyo Institute of Technology, Tokyo, Japan, m 1964, 1966 and 1969, respectively From 1973 to 1985, he was an associate professor of the Department of Electrical and Electronics Engmeenng, a professor at the International Cooperation Center for Saence and Technology from 1986 to 1988, and he IS now a professor at the Department of Electrical and Electromc Engmeermg, Tokyo Institute of Technology From 1978 to 1979, he was at Carnegie-Mellon Umverslty, Pittsburgh, PA, USA as a vlsltmg associate professor He has been engaged m research m the fields of semiconductor thm films, heteroJunctlons and surface acoustic wave devices I& present Interest IS bloelectromc devices Hwo-o Nuyama was horn m Japan m 1941 He recewed the Ph D degree from Tokyo Institute of Technology HIS present status IS a professor of the International Cooperation Center for Science & Technology, Tokyo Institute of Technology He IS a member of the Chemical Engmeenng Society of Japan, Catalysis Society of Japan, Chermcal Society of Japan, and the Petroleum Institute HIS research fields are catalysis and reaction engmeermg