on boundary problem study of the thin acoustoelectric oscillator

J. SLECHTA: Study of the Thin Acoustoelectric Oscillator 591

phys. stat. sol. (a) 16, 591 (1973)

Subject classification: 7 and 14.3.4

School of Mathematics a d Physics, University of East Anglia, Norwich

On Boundary Problem Study of the Thin Acoustoelectric Oscillator

BY J. SLECHTA

Linearised boundary value problem for the thin amustoelectric oscillator is presented. The frequency dependence of the total electrical current is derived. It is shown that the spectrum of the self-oscillations does not depend directly on specific numerical values of the boundary conditions but on their more general features. A set of effective boundary conditions is found which gives a spectrum in which the first and second harmonics are missing.

1. Introduction

Much experimental work has recently been devoted to the study of the properties of the thin acoustoelelectric oscillator [l to 31. It is found experimentally that for the case of a thin A-E oscillator with low conductivity the first and second harmonics of the basic mechanical frequencies are usually absent. How- ever, above a certain threshold voltage it starts to oscillate in the 3rd harmonic. As the voltage is increased the 6-, 9-, 12th. . . harmonics progressively appear. For a given voltage above the threshold the amplitudes of oscillations are found to saturate and the resulting steady state amplitudes decrease with increasing harmonic number [l, 31.

Most of the existing theories of the active A-E oscillator have been concerned with the thick oscillator behaviour. A simple linear theory was developed by Maines and Paige [4], a non-linear theory was developed by Gurevich and Laikht- man [5] .

I n the author's recent paper [6] a non-linear theory was developed, which can be applied to the thin A-E oscillator (with small 8 = 1, see [S]). It was shown that in the steady state of the thin oscillator the 3rd harmonic is dominant and the 6-, 9-, 12-th . . . harmonics appear through nonlinear harmonic generation. The absence of the first and second harmonics, however, appears to have nothing to do with the non-linearities. The strong 1-dependence of the behaviour of the A-E oscillator suggests that the boundaries play an important role. I n the present paper we shall discuss the linearised set of basic equations (see Section 2) from that point of view.

592 J. SLECHTA

To determine uniquely solutions of the linearised aet of equations we need six independent boundary conditions. The set of equations is essentially of macros- copic character. A difficulty arises, in that the equations include the space charge density as a variable ; the boundary conditions for this variable are not defined in standard electromagnetic theory. The boundary conditions should strictly involve a detailed consideration of the contacts a t the oscillator boundaries. This would make the problem even more complicated, and in any case the question of contact is a delicate one in its own right.

I n this paper we adopt a different approach. We shall consider the basic set of the linearised equations as a differential operator and study its spectral properties under different effective boundary conditions. The boundary conditions for the space charge we shall choose heuristically and we shall study the sensitivity of the spectrum to variations in them. By that means we may obtain useful infor- mation about the effects which a more realistic set of boundary conditions would have on the self-oscillation spectrum of the A-E oscillator. In particular we shall demonstrate that the self-oscillation spectrum does not depend on the numerical values of the boundary conditions, but on their more general properties. It has been possible to obtain a set of effective boundary conditions which cause the first and second harmonics to be missing in the self-oscillation spectrum.

The self-oscillations of the A-E oscillator are not observed directly. They are studied by analysing the frequency spectrum of the total electrical current flowing in the oscillator system. First of all we find the complete solution of the basic equations with given boundary conditions and derive a formula for the total current. We shall then use this formula to discuss the observed frequency spectrum.

2. Basic Equations and Their Solution

We consider a piezoelectric crystal with two flat parallel surfaces. Let thick- ness of the crystal in the direction perpendicular to these surfaces be 1. We shall assume the crystal to have been cut in such a way that the acoustoelectric effect in this direction can be considered to be linear.

We shall assume that the crystal is placed in a simplified electrical circuit with a constant source of electric field, say a battery of voltage V,, and an effective impedance 2.

Boundary Problem Study of the Thin Acoustoelectric Oscillator 593

and the definitions and material relations

1 D = E E ' + e S , 5 = q p n ' ,

T = c X - e E ' , o,=qpn,,

I O ' I

n = n ' - n

form the basic equations of the problem and have been used as the starting point in all works concerned with the acoustoelectric phenomenon (see, for example [S to 81). The notation is identical with the one used in [S] equation (1) and (2).

To define the system uniquely we need to know six boundary conditions for the whole of the time after switching on the circuit. We shall use the following boundary conditions :

T(0, t ) = T,(t) = c S(0: t ) - e E'(0, t ) , T(Z, t ) = Ta(t) = c # ( I , t ) - e E'(Z, t ) , v'(0, t ) = 0 9

1

0 vo = 2 J ( t ) - J E(x, I ) dx ,

I D(0, t ) = D(1, t ) , (3)

an an - (0, t ) = - - ( I , t ) . i3X ax

T,(t) and Ta(t) are known functions of time. The physical meaning of the first four conditions is obvious. The first two

mean that we control the stress on both ends of the crystal (usually, TI = T, e 0) the third one means that we put the zero potential level on one of the ends of the crystal and the fourth one is a formulation of the Kirchhoff's laws for the circuit. The Z is an effective impedance of the external circuit. We shall discuss the physical meaning of the latter two boundary conditions later. It is important in our calculation to remember that the boundary conditions, both electrical and mechanical, consist of two parts. The first is the simple condition imposed and controlled by the experimenter, e.g. zero stress a t the surface; the second repre- sents the effects of random disturbances, including thermal noise, however small they are. For each boundary condition both parts are to be added together. At this point it may seem that the complication caused by including the noise term is unnecessary. But we shall see later that in the case of a self-oscillating system its presence may give rise to several interesting features.

In this paper we shall concentrate on the development of the oscillations im- mediately after switching on, while the non-linear term still plays a small part. After linearisation, Laplace transformation and introducing a new variable E ( z , t ) = El($, t ) - E, where E,, is the steady part of the electrical field intensity

594

from (1) and ( 2 ) we get

J. SLECHTA

as E C ~ E

ax e ax s2 u - v:- + 2- - = 0 ,

(4)

where

In the derivation of the system (4) we replaced E by e El a new auxiliary function h = anlax.

by e n and introduced

When we substitute i o instead of s we get linearised part of (3) in [6]. The boundary conditions now have the form

The impedance of the external circuit is considered to be of the form Z(s) = = R,ff -+ s L,ff f C,,ls where R,ff, Leff and Ceff are effective resistance, in- ductance and capacitance of the external circuit.

The solution of (4) with the boundary conditions using the standard theory of a system of simultaneous ordinary differential equations of the first order (e.g.


191) can be shown to be of the form

where

and

596

and

J. SLECHTA

After substitution i w instead of s into (6) to (8) and the limit 4 +. 0 we get (7) to (9) in [6].

The coefficients vi(s) can be determined from the boundary conditions. We assume the initial conditions are zero.

From the first two boundary conditions (5) we obtain 1 4

- [ V o - o o Z E , ] + t:(l - ekil)qi e s i = l

1 --(% + s )

(9) & Z 4 n

From the other boundary conditions (6) and (14) we get the system of four algebraic equations

P6 =

4

i = l 2 k: ( 1 - A t ) (1 + ekii")yi = 0 ,

4 ki ( A t - 1) (1 - ekiz) qi = 0 .

i = l

When we introduce the matrix

( A , + 4 1- 4 X , ) k, (ekiL (A + A) + A X , ) ki ki ( 1 - Ai) (1 + ekil) ki ( A t - 1 ) (1 - earl)

(Aji)

and (1 - ekil) xi =

then

where B = det ( A i j ) and Bij = subdet ( A i j ) ,

where subdet ( A i i ) means the subdeterminant of the matrix (A i i ) joined to the term Aij .


At this point we would like to discuss some interesting features which are re- vealed when we include in our calculation, the random contribution of the boundary conditions. Let us assume that we had accepted the usual procedure and did not include noise. Then some typical conditions imposed would be TI = T, = = 0. So all the right sides of (10) would have been zero. Then (10) would degenerate into the usual boundary conditions for the string equation with no coupling to the space charge waves. On the other side when we include the noise parts of the external influence, the system of equations (10) becomes non-trivial even if the directly controlled parts of the boundary conditions are zero, and provide us with a new condition for the self-oscillations which we shall discuss later on. I n the present study we shall be interested only in the random contri- butions to the boundary conditions.

Further, we shall be looking for the spectral o content of the total current which is a directly experimentally accessible quantity. I n formulating the w analysis of the system, we express the boundary conditions, which are in general complicated stochastic functions of time, as

W

T,(t) = J T , ( o ) e-imt d o , i = 1 , 2 , (13) -W

where T,(o) are assumed to be known functions of o. W

qJ&) = J qJr(w, 8) do I -W

where from (12) and (13) we have

The formulae describe starting of oscillations from the noise.

3. Total Electrical Current

The total electrical current, a characteristic which can be directly experimentally analysed, is given in terms of our solution by the formula

where the first term is the ohmic term and the second term is a non-ohmic one due to the active oscillations, when they exist.

From (15), (9), and (12) we can easily show that the total current is given by the formula

W

J ( s ) = ~ OoE0 - [ J ( o , s) dw , ' J

- w where

598 J. SLECHTA

The w-component J(w, t ) of the total current, the physical characteristic which is directly analysed by means of an oscilloscope is given by the formula

J ( o , t ) = 2 [Res J(w, s ) ] ~ = ~ ~ esrt , sr

when s, are poles of J ( w , s) in the variable s. Here we shall discuss the a c components of the current response (19) only,

which is determined through (16) by the poles of the second part of (18). When we look a t (14) and (18) we see there are three types of poles :

a) The poles given by the zero points of the equation B(s) = 0. We shall see they form a discrete set and we shall denote them s, = st) + i sf). They give components of (19) which correspond to the current response of the self-oscillations. We shall discuss the self-oscillation spectrum later on. . b) s = i w - this pole gives the component of (19) which has exp (i (0 t ) time dependence, so it neigher grows nor dies out in time. When we put directly controlled part of T, equal zero and of TI equal to L(t) , which means the boundary I = 0 is stimulated by an external acoustic source (usually L(t) = a exp { i o t } ) then we get from (la), (9) and (8) a component with the amplitude proportional t o exp {k,(i w) I } . The exp {kl(i w ) I } is the White amplification factor [lo].

From that we see the pole s = i w corresponds to the functioning of the system as an amplifier. However, the amplitude of the current response connected with that pole gives a sharp narrow peak a t the frequencies in the vicinity of 5%). Tf sp) = 0 then the peaks degenerate into &function in w = siz), otherwise they are finite. c) Poles given by the equations I = E 214 n (cut + s) will be discussed in the next paragraph (see (21)).

4. Self-Oscillations

The complex frequencies of self-oscillations are determined by the roots of the non-linear algebraic equation

B(s ) = 0 . (20) I ts solution depends through Xi on the impedance of the external circuit. The dcpendence is extremely strong especially in the vicinity of the impedances 2, given by the equation

when s, is a solution of (20). But due to the smallness of A (usually w lop2) and large modules of kt the 2-dependent term X i in (20) becomes small compared with the other terms, especially A i and so the strong dependence of s, on 2 is only in the extremely close vicinity of 2, given in (21). 2, form only isolated poles in the complex plane of 2. We shall concentrate on the solution of (20) outside the circular vicinity of 2,. It is a complicated equation in the form of a


determinant. In the case of the A-E oscillator with large 8, exp (k3 2) x 0 and exp (k4 1 ) = 00 and the equations degenerate into the simple case discussed in [a]. However in the case of the thin oscillator (small 8) one has to take into account more terms. We found the dominant ones are those containing k3(s). The solution is then

We note that the spectrum given by (23) depends predominantly on the nature of the boundary conditions. It is determined by the form of the determinant B of the matrix (11). The spectrum changes only if that matrix changes; e.g. for the case of anlax (0, t ) = an(& t)/ax as the last condition in (3) (see later) we get the same expressions (23) and (24) however G3i = k'4, (1 - A,) (1 - exp [k, 11) and GN = %(A, - 1) (1 - exp [-k4 11). So the spectrum and the growth rates depend on more general features of the boundary conditions than their numerical values ( f (s) does depend explicitly on TI, T,) .

The first expression in (23) is the formula published in [a] and the second one is our correction.

600 J. SLECHTA

We did our calculation for two materials CdS and ZnO with characteristics given in [3]. ZnO: wD = 7.17 x 1O1O Hz ,

W, = 3.13 x 10' Hz , vs = 2.73 x lo3 m/s , 1 = 20 pm .

The basis frequency of the platelet WB = 4.2 x 10' Hz CdS: w U = 3 . 6 x 109Hz, ~ ~ = 1 . 3 7 ~ 1 0 ' H z ,

W, = 1.87 x 10" H z , us = 1.75 x lo3 m/s .

1 = 4 0 p m ,

It is easy to show the conditions under which the self-oscillations were derived are fulfilled in both cases.

For 1 2.5 t o 3 times larger than those used the correction (24) disappears, as the conditions under which it was derived do not hold place.

The results for our choice of the boundary conditions are plotted in Fig. 1 and 2. Fig. 1 shows the real part (growth rate) of the Maines-Paige part of (23), the Fig. 2 depicts the real part of the complete formulae (23). In the case of the sim- pler theory the growth rate of the first four harmonics is plotted, in the case of our theory the growth rate of the first three odd frequencies is plotted. In both cases the maximum growth is for the fifth harmonic. The results are given for CdS samples, the results for ZnO are very similar.

6. Discussion and Conclusion We see that the correction due to the electrical boundary conditions pro-

foundly changes the oscillation spectrum. The first harmonic is completely suppressed having negative growth rate for all applied voltages. For V > 1 only the odd harmonics starting from the third starts to grow. The even frequencies have positive growth rate lo2 times weaker than the odd ones with

0

-1

-2 Fig. 1. The dependence of the growth rates of the first four harmonics of the natural basic frequency of the platelet on V = p E,/v, in the case of the Maines and Paige

Fig. 2. The dependence of the growth rates of the first three odd harmonics of the natural basic frequency of the platelet on V = p E,/v, in the case of our choice of the

theory effective boundary conditions


a positive growth rate and they are not included in Fig. 2. The boundary conditions modify the frequencies by an amount of lo7 Hz which is still 10 times smaller than the basic frequency of the platelet and so the method of successive approximation used is valid.

At this point we would like to discuss in more detail our choice of the last two boundary conditions.

The first of them we can get out of the assumption that the acoustoelectric oscillator is in fact an active capacitor on which plates are charges Q ( z ) and - Q ( z ) respectively. From i t and the general electrical boundary conditions for D(x, t ) which state D-(O, t ) - D+(O, t ) = 4 II. &(t) and D$, t ) - D+(Z, t ) = = - 4 n Q ( t ) , where D+ means the value of D(x, t ) on a boundary from the right, D- from the left. Further D-(O, t ) = D+(Z, t ) = 0 and so D+(O, t ) = = D-(Z, t ) which is our boundary condition. The treatment is precise in the quasistatic limit of the electrodynamic and so it is consistent with the basic set of equations, which are valid in the same limit. From that point that condition is precise, but because it may raise some questions in connection with the contact we included it among effective boundary conditions.

The last boundary is valid precisely under assumptions: there is no electric current, both contacts are identical. It is applicable to the situation of a uni- form sample with identical metal electrodes on opposite faces. It may be violat- ed by the current flow. To the test sensitivity of the self oscillations to the choice of t,hat condition we tried an(0, t)/ax = an(Z, t)/ax and found that the results for the odd harmonics were qualitatively similar to those obtained for the choice of the boundary conditions discussed above. The growth rate of the even harmonics was found lo6 smaller than for the odd ones.

With the boundary condition replaced by n(0, t ) = f n(Z, t ) we obtained a spectrum very different with features not resembling the behaviour of the oscillator.

We have found a system -of effective boundary conditions, which explains some features of the thin oscillator self-oscillations spectrum, especially the missing first and second harmonics. When we look at the Table in [6] in all interactions not including the first and second harmonics the third is strongest. This fact still further supports our conclusion ([S]), that the rest of the basic properties of the saturated state of the thin A-E oscillator, are due to the non- linearities.

The amplitudes pl, tp2 in (18) correspond to the “acoustic” modes, p3 and tpa to the “space charge” modes. All four modes will oscillate in time as

exp ( 8 , t ) = exp (a, t ) exp [i(a?) + b,) tl , where s, is given by (22) but from (18) and (8) is evident that in the “acoustical” modes the even harmonics in the total current are almost completely suppressed due to the factor (1 - exp [k, I ] ) (in agreement with [a]. When the “space charge” modes are weak then practically only odd harmonics may be observed in the total current. To analyse the strength of different modes quantitatively in more detail needs the solving of equation (10) in tpl and to take more realistic classical facts of the driving forces (nonhomogeneous E(x, t ) in [ l l ] ) . The relative strength of different modes in the saturated state in the nonlinear region may be also effective by an equation similar to (15).

The numerical values of the growth rate in the present calculation seem to be large compared with the experimental evidence. This may be caused partly 39 physica (a) 16/2

602 J. SLECHTA: Study of the Thin Acoustoelectric Oscillator

by the fact that our effective boundary conditions are not physically compre- hensive and partly due to the above open question about the relative values of different yi, but i t clearly shows the role of the elect,rical boundary conditions in the observed features of the spectrum of the thin oscillator is not negligible. We think our thcory is complementary to that in [ll]. The authors allow for nonhomogeneous electrical field inside the sample but their formalism does not allow them to separate clearly the influences of the electrical boundary conditions and the external driving forces on the spectrum. We believe a more complete theory should be a synthesis of both points of view.

To find a solution of (20) which is valid also in the vicinity of Z , given by (22) is extremely complicated, but it follows out of (21) and (22), a t least qualitatively, that some chosen values of Z may change the standard spectrum severely (e.g. selective killing modes observed by [12]).

We believe for making more quantitative theory it is desirable to investigate the role of the contact in controlling of the self-oscillation of the linearized equations.

From the flow of the calculation we may draw an important conclusion, which is not dependent on a particular choice of the boundary conditions:

The structure of B depends only on how the system is controlled, that means what characteristics of the system are controlled and what are the relations between them. However is does not depend on the detailed numerical values of them.

This feature of the boundary conditions is very general but we preferred to demonstrate i t on a concrete choice of boundary conditions, though it does not depend on it. To realise it is of crucial importance when tackling the problem by means of a computer.

Acknotmledgemeats

The author would like to thank Prof. P. N. Butcher for his critical comments concerning the problem and Drs. F. G. Marshall, H. M. Janus and E. G. S. Paige for valuable discussion.

Refercnees [I] F. G. MARSHALL, Electronic Letters 6, 382 (1970). 121 H. M. JANUS and N. I. MEYER, Solid State Commun. 8, 417 (1970). [3] H. M. JANUS and F. G. MARSHALL, private communication (1970). [4] J . D. MAINES and E. G. S. PAIGE, J. Phys. C 2, 175 (1969). [5] V. L. GUREVICH and B. D. LAIKHTMAN, Soviet Phys. - Solid State 7, 2603 (1966). [6] J. SLECHTA, J. Phys. C 5, 582 (1972). [7] V. L. GUREVICH and B. D. LAIRHTMAN, Sh. eksper. i teor. Fiz. 46, 598 (1964). [8] P. K. TIEN, Phys. Rev. 151, 970 (1968). [9] A. R. FORSYTH, A Treatise on Differential Equations, MacMillan.& Co. Ltd., London

1961. -101 D. L. WHITE, J. appl. Phys. 33, 2547 (1962). 1113 P. N. BUTCHER and H. M. JANUS, J. Phys. C 6 ,567 (1972). 1121 F. G. MARSHALL, Electronic Letters 5, 581 (1969).

(Received February 19, 1973)

on boundary problem study of the thin acoustoelectric oscillator

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