Kundu, K K

Munshi, T K

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lndiaHJ. Phys. 71A (3), 377-385 (1997)IJP A— an international journalDisturbances in piezo-quartz transducers subjected to a flow of current in the semiconducting boundary layerT K MunshiDepartment of Physics, Kharagpur College,P.O. Kharagpur, Dist. Mi dn a pore, West Bengal-721 305K K KunduDepartment of Physics, Gty College,Calc itta-700 009andR K MahalanabisDepartment of Mathematics, jadavpur University,CalcUta-700 032Abstract : The mechanical responses in piezoelectric plate transducers (quartz) with rigid backing when the transducer is subjected to a flow of current in the semiconducting boundary layer have been investigated in the present study. The semiconducting layer is thinner than the quartz p la t by at least one order of magnitude. It is found that the order of disturbance in case of a thin layer of intrinsic semiconductor InSb (at the temperature T' = 77 K) covering the piezo-quartz is very high compared to n-type As-doped germanium of an extrinsic semiconductor at T = 300 K.Keywords : Piezoelectricity, Electron stream, Mechanical disturbance.PACS Nos. : 77.65 .-j, 77.65. Dq, 73.60. F1. I n t r o d u c t i o nT he s t u d i e s  i n  t h e  d i s t u r b a n c e s  o f  a p i e z o e l e c t r i c  m a t e r i a l  f r o m  t h e  s t a n d  p o i n t  of  m e c h a n i c s  of71A(3M9© 1997 IACS378 T K Munshi, K K Kundu and R K  Mahalanabiscontinuous media have been initiated by Cl-41. The effect of an electron stream or of a conducting layer on the electromechanical disturbances have engaged a number of researchers C5-7] by a number of decades The present study seeks to investigate the response in piezoelectric transducers (quartz) subjected to a flow of current in the semiconducting boundary layer. These types of problems are very much used in different bra’ ches of Acoustic Engineering particularly in piezoelectric amplifiers C8,91 and in different biomedical applications CIO],2. The Problem and the Fundamental Equations Let us consider a piezoelectric transducer, piezo—quartz (X—cut) acted on by an electron \stream flowing outside it. In order to avoid the difficulty of getting the very small distance between the electron stream and the piezo-quartz, we cover the surface of the latter with a semi-conducting layer in which motion of eldctric charges is 'produced by applying a voltage. The layer will play the role of an electron (or hole) guide. The choice of the semiconductor depends on the required density of carriers. We have considered the case of germanium (low concentration of carriers) of the n-type extrinsic semiconductor with arsenic at room temperature T' = 300K and a thin layer of intrinsic semiconductor InSb (high mobility of electrons) at the temperature of liquid nitrogen T' = 77K covering the piezo—quartz. The semi-conducting layer is thinner than the quartz plate by at least one orderDisturbances in piezo-quartz transducers etc 379of magnitude. The transducer is riqidly backed at both ends C7.,81. The influence of internal and external damping is disregarded [11,123. Theinfluence will be greater than in the quartz itself as a result of interaction between the semi-conducting layer and the piezo-quartz. We have taken X-axis is along the normal to the surface of the transducer in the upward direction, Y-axis is along the length and Z-axis alonq with these two axes form a rectangular system .We now, proceed to derive the differential e q u a t io n  to calculate the mechanical response with the fundamental equations.The equations representing plane compressional wave propagation in the x-direction in a piezoelectric material C2,53 are given byT = c - hD (1)6xD = Etf + he (2)6xwhere, c - elastic constant; If - the displacement in the x direction;T - normal stress in the thickness direction; E - electric field strength; D — electric flux density in the x—direction; e — dielectric constant and h — the piezoelectric constant of the material.The equation of motion isp <5*Jf/<5t2 = 6T/6n (3)where p is the density of the material.The current in the semi-conducting layer is expressed asJ = qncfiEc + qDn 6n^/6x (4 )380 T K Munshi. K K Kundu and R K Mahalanabiswhere, J = electron current density, n. totalnumber of electrons in the conducting band, E£ « total electric field in the semiconductor,q « electron charge (absolute value), p -  mobility ofelectrons and Dn = Diffusion constant « kT'p/q (5) where T' is temperature in absolute value.The interaction constant between the disturbed electric field E in the semiconductor and that E in the® >v>transducer is defined as E = a (6)Then the total field in the semiconductor isexpressed by the equation E E + AE (7)— owhere Eq = constant outer field and AE - perturbed field, which is composed of the perturbed fi^ld in semiconductor and in the transducer. So that \AE =  E -  otE ( 8 )Denoting now by qn -  p the electric density inthe semiconductor, the velocity of the electronstream can be related to the electric field by theequation -p v = up E • (9)ec c ec cBearing in mind that, p = p +fp , v = v +v, E’ Kec oc ^ef c o ’ c= E +AE and, -v = pE we get o  O o  „-v =pAE = p(Es~aE) (10)where, f - a coefficient less than 1. The fraction f accounts for a division of space charge between the conduction band and bound states in the energy qap. The continuity condition of the charges in semiconductor is div j - 6p /<5t = 0 (IDO Cwhich after linearization assumes ‘the form <5p/<51 + f vQ6pe/<5 x +PQe^v/6 x - f D ^S^p^/6 (12)The divergence condition in the transducer isdivD * «6E/«5x (13)where E is the sourcelss field of the action on the transducer of the electron stream in the seeiconductor.Now from Eqs.(2) and (13) it follows that<5E/<5x * «5E/<5x + h <52{/«5x2Therefore • E ■ E + h 6f/6x (14)Next from the continuity condition of tangential field E in the transducer and the semiconductor, we find thatEs-o»E = E - h 6Z/6xor, E * [Eg+h 6?/6x]/l+a (15)Again divergence condition in the semiconductor isc <5E /<5x = -p (16)s s' ^eTherefore 6E/6x = ~[ pei)/«s+ha 6*l[/6x2] /1+a (17)where r) is the ratio of the cross sectional area of the semiconducting layer to that of the transducer. Now from Eqs.(2), (3) and (17) we get<52{/<5t2-a262{ /6xz-t\cps n / p l l + a > c s = 0 (18)z zwhere, a = Cc—h c/1 +al/pFrom Eqs.(lO), (12) and (17) we getPe + fv o6 p e /6 x  + P o e f|r)Pe /tf  s  ( 1+a)- fD <5*p /<5x2 + p _uah62?/<5x2/(l+a) = 0  (19)n e oeNow. eliminating pg from Eqs.(IB) and (19), we get[ <5/<5t + fv S / S x + p (1+a) - fD <5*/<5x2]o s  nx [6*?/<5t* - a*6*?/6x2]+ p^h/l+oJ^oiTje i*( /6x2/tgp = 0Disturbances in pievr-quartz transducers etc 3gj( 20 )382 TKMimshi, K K Kundu amt R K htahaUmabisZm Solution of tho ProblemTaking Laplace transform of Eq. (20) and can*id»rinq f - exp(«K) be a trial solution end for seall value, of t it takes the forefDne*-fvoe-<p + - 0 (2I)where p is the Laplace transforms paraeeter.Then the general solution is{  = A exphs^x) * B cxpts^x) (2 2 )where m and «2 are the roots of the Eq. (21) and A, B are constants to be determined froe boundary conditions.For a rigidly backed transducer we assume the stresses and displacements at the ends 1 are continuous. To obtain the solution of the problems we attach to the extremities x = 0 and x * X twomechanical systems and denote the corresponding quantities by the symbols 1 and 2. Accordingly, the boundary conditions become(F ) = (F) , t o  o " V x * I F , x - ‘V x ^ ’ xwhere F and F are the values of F(stress)t  Z andand are the values of £ (displacement) at the two boundaries, i.e., at x=C and x=X respectively.From (1) we get T + hD = c6?/6x. Applying Gauss Law at the surface of the transducer we get D=Q/yz and F=Tyz. From above two, we get F'+hS - yzc<5f/6xy and z are the dimension of the piezoquartz bar.Voltage across the transducer is given by/* Edx = - ( ( ^ - ( V ) ^  * V(say)=-h((?>x-(f) ^ +QX/«yzDisturbances in piezthquartz transducers etc 3B3orwhere CQ ■ *yx/X is called the static capacitance of the transducer.Taking into consideration iepulsive voltage function V * VQ<5(t> idiere 6(t> is the Dirac delta function, we get 7 » VQ/p and by using conditions of continuity of forces and displaceeents as stated above, we getTo get its inverse transfora, the mechanical disturbance in the transducer, we proceed [12,131f = V VfD n . erf (Va"!"€)/2hVa'N o o n #+ QV?!T {h/yze - l/hCQl exp (-a t) t ~ ±''Z/2. Vn (23)4. DiscussionHere we have considered two examples of seeiconductor covering the piezo-quartz. Let us consider i) a thin layer of the intrinsic seei-conductor InSb (at the teeperature of liquid nitrogen, 77K) covering the piezo-quartz and ii) the case of n-type As-doped germanium covering the piezo—quartz at 300K. The standard values of the material constants are taken from [2,3,81, The variation of the mechanical disturbance produced in the piezo—quartz transducer subjected to a flow of current in the semiconducting boundary layer withwhere,+ Q VfDn {h/yze - l/hC^/2 Vp+aa' = fV*/4D + p ur)/e (1+a) o n a e ' sCerf - error function!384 T K Miutshi, K K Kundu and R K Mahalanabistime is found to be parabolic in nature (Fig.l) and_Qis the order of 10 m in case of (i) and in thecase (ii), the variation of the disturbance is found-12to be of the order of 10 ■.The disturbances consist of two parts - an exponential function and an error function of time. Here Me have considered two examples of semiconductor covering the piezo-quartz. It is interesting to note that when a piezo-quartz is covered by a thin layer of intrinsic semiconductorFigure 1. T h .  v a r i a t i o n  o f  d i s t u r b a n c e  when c o v i  r e d  b y  a  t h i n  l a y e r  o f  i n t r i n i i c  e e m i  c o n d u c t o r  I n S b  a t  7 7 K .Figure 2. T h e  v a r i a t i o n  o f  d i s t u r b a n c e  w h e n  c o y «- r e d  by a thin layer of g e r m a n i u m  w i t h  a r s e n i c  a t  B O O K .InSb at the temperature of liquid nitrogen the values ofthe mechanical disturbance increases almost 10 times than when a thin layer of- germanium of the n-type extrinsic with asrsenic at room temperature covering the piezo-quartz.Disturbances in piezo-quartz transducers etc 385References[13 W P Mason P i e z o e l e c t r i c  C r y s t a l s  and t h e i rA p p l i c a t i o n s to U l t r a s o n i c s (New York; D. VanNostrand) p84 (1950)[23 W G Cady P i e z o e l e c t r i c i t y  (New YorksMcGraw-Hi11) (1959).C33 M Redwood J .  A c o u s t .  S o c .  Am. 33 527 (1961).[43 R D Hindi in on the E q ua t i o n  o f  Motion o fP i e z o e l e c t r i c  C r y s t a l s ;  Problems on Continuum Hechanicsg Muskhe1i s h v i 1i A n i v .  Volume (S.I.A.M) Philadelphia, Pensylvania) p.282 (1961).[53 S Kaliski P r o c .  V i b r .  P r o b l . 6 323 (1965).[63 D L White J. A p p l .  P h y s . 33 2547 (1962).[73 A R Hutson and D L White J .  A p p l .  P h y s . 33 40(1962).[83 T.Musha J. P h y s .  S o c .  Japan 18 1326 (1963).[93 L D Ivanov S o v .  P h y s .  -  C r y s t a l l o g r . (USA)36 466 (1991).[103 S N Kurilkina, N P Kazakov and K G Yurchenko T e c h .  P h y s . (USA) 38 50 (1993).[113 D F Gibbs F e r r o e l e c t r i c s  (UK) 82 133 (1988).[123 T K Munshi, K K Kundu and R K Mahalalanis J .A c o u s t .  S o c .  Am. 96 2836 (1994).[133 V I Alshits et al. Have Motion (Netherlands) 19 113 (1994).7IA(3>20

Disturbances in Piezo-Quartz Transducers Subjected to a Flow of Current in the Semiconducting Boundary Layer

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Disturbances in Piezo-Quartz Transducers Subjected to a Flow of Current in the Semiconducting Boundary Layer

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