THE UNIVERSITY OF M I C H IGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Physics Technical Report SELF-INTERACTION OF ACOUSTIC WAVES IN CADMIUM SULFIDE John H. Gibson ORA Project 04941 under contract with: U. S. ATOMIC ENERGY COMMISSION CHICAGO OPERATIONS OFFICE CONTRACT NO. AT( 11-1)-1112 ARGONNE,. ILLINOIS' administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1968

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ACKNOWLEDGMENTS I wish to express my sincere gratitude and appreciation to Professor Gabriel Weinreich for his interest, guidance, and constant encouragement during the course of the experiment. I am deeply indebted to Dr. Victor E. Henrich for his help with all aspects of the problem and for many favors, without which the experiment would not have been possible. I wish also to thank Dr. William H. Wing for many interesting discussions on electronics and experimental technique. Thanks are also due Dr. Benjamin Tell for providing the cadium sulfide crystal used in the experiment. The partial support of this research by the United States Atomic Energy Commission under Contract No. AT(11-1)-1112 is gratefully acknowledged. ii

TABLE OF CONTENTS Page LIST OF FIGURES v ABSTRACT viii INTRODUCTION 1 Chapter I. ACOUSTOELECTRIC CURRENT WAVEFORMS 4 1.1 The Observed Acoustoelectric Effect 4 1.2 The Weinreich Relation 13 1.3 Current Ripple Due to the Boundary Condition 19 1.4 The Shape of the Acoustoelectric Current Trace 21 1.5 Representation of Nonelectronic Losses 26 II. THE SMALL-SIGNAL THEORY 28 2.1 Weinreich Relation in A Piezoelectric Photoconducting Crystal 31 2.2 Equivalent Circuit Model of the Interaction 44 2.3 The Attenuation Coefficient 51 2.4 The Acoustoelectric Current 56 2.5 Inclusion of the Diffusion Term 59 2.6 Acoustic Gain 66 III. THE ACOUSTOELECTRIC FEEDBACK EFFECT 70 3.1 Computer Program Outline 73 3.2 Finding the Circulating Current J(t) 75 5.3 Equivalent Circuit Model of the Crystal 78 5.4 Solving for the Electric Fields and Conductivities 80 3.5 Revised Outline 84 3.6 Machine Computation 86 3.7 The Computer Program 88 IV. THE EXPERIMENT 90 4.1 The Acoustic Assembly 90 4.2 Electronics 95 4.3 Modification for High Acoustic Power 98 iii

TABLE OF CONTENTS (Concluded) Page V. COMPARISON OF RESULTS 100 5.1 Direct Comparison of Waveforms 102 5.2 Quantitative Comparisons 121 5.3 Distortion of the Wave Packet 129 5.4 Summary 131 APPENDIX. REDUCTION TO THE ONE-DIMENSIONAL PROBLEM 135 FOOTNOTES AND REFERENCES 141 iv

LIST OF FIGURES Figure Page 1. Photograph of the acoustic assembly in the sample holder. 5 2. Diagram of the acoustic assembly. 5 3. Basic experimental arrangement. 8 4. Representative acoustoelectric current trace. 9 5. 30 Mc excitation applied to the input transducer. 9 6. Illustration of the linear relationship between acoustoelectric current and input acoustic power (at small-signal levels). 11 7. Acoustoelectric current traces produced for constant input power and changing crystal resistivity. 12 8. Elementary equivalent circuit model of the crystal. 16 9. Equivalent circuit for synthesizing acoustoelectric waveforms generated by a uniformly attenuated acoustic wave. 23 10. Illustration of the phase relationships needed for acoustic attenuation. 40 11. Equivalent circuit and phase diagram for the acoustoelectric interaction (without diffusion). 49 12. Plot of a vs. vd (without diffusion). 53 13. Plot of a vs. wcr for the case of no applied drift field (without diffusion). 56 14. Equivalent circuit and phase diagram for the acoustoelectric interaction (with diffusion). 62 15. Plot of a vs. vd including the influence of diffusion. 65 16. Plot of E vs. ar at zero drift field showing the influence of diffusion. 67 v

LIST OF FIGURES (Continued) Figure Page 17. Equivalent circuit model of the ith crystal segment. 79 18. Illustrations demonstrating the high acoustic reflectivity of the bonds between elements of the acoustic assembly. 93 19. Block diagram of the arrangement of electronic equipment used in the experiment. 96 20. Experimental data and fitted theoretical curve for attenuation as a function of crystal conductivity for the experimental sample. 101 21. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.80 x 105 ohm-cm. 104 22. Computer-generated acoustoelectric waveforms for resistivity of 1.80 x 105 ohm-cm. 105 23. Power-dependent acoustoelectric waveforms for crystal resistivity of 6.12 x 104 ohm-cm. 106 24. Computer-generated acoustoelectric waveforms for resistivity of 6.12 x 104 ohm-cm. 107 25. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.27 x 104 ohm-cm. 108 26. Computer-generated acoustoelectric waveforms for resistivity of 1.27 x 104 ohm-cm. 109 27. Power-dependent acoustoelectric waveforms for crystal resistivity of 4.46 x 103 ohm-cm. 110 28. Computer-generated acoustoelectric waveforms for resistivity of 4.46 x 103 ohm-cm. 111 29. Power-dependent acoustoelectric waveforms for crystal resistivity of 2.08 x 103 ohm-cm. 112 530 Computer-generated acoustoelectric waveforms for resistivity of 2.08 x 103 ohm-cm. 113 31. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.01 x 103 ohm-cm. 114 vi

LIST OF FIGURES (Concluded) Figure Page 32. Computer-generated acoustoelectric waveforms for resistivity of 1.01 x 103 ohm-cm. 115 33. Illustration showing how the local electric field influences the attenuation coefficient by displacing the operating 117 point. 34. Plot of the slope of the tail of the acoustoelectric current trace as a function of the input attenuator setting. 125 35. Acoustoelectric current traces generated by a 1.5 iLsec input wave. 126 36. Comparison of output signals produced for input excitation levels of 0 and -20 db. 132 37. Computer-calculated output signals for input excitation levels of 0 and -20 db. 133 vii

ABSTRACT In a piezoelectric crystal acoustic waves of certain modes and propagation directions are accompanied by electric fields. If the crystal also contains free electrons, and if there is a sink (for example, the lattice) to which they may dump their energy, then the electric fields will do work on the electrons, thereby dissipating the energy of the wave. As the wave surrenders its energy to the electrons, so must it also give up its momentum to the same absorber. The result is the production of a direct current along the wave propagation vector. This acoustoelectric current is a primary source of information on local electronic conditions encountered by the wave. For example, conservation of momentum for the interaction requires that the current be proportional to the rate of dissipation of energy from the wave (the Weinreich relation), so the acoustoelectric current pulse produced as a short acoustic wave packet traverses the crystal gives directly the acoustic dissipation rate at each instant of time during the trip, a result not obtainable by observation of the transmitted sound alone. This thesis reports a study of the acoustoelectric self-interaction of a 30 Me ultrasonic wave packet propagating in a cadmium sulfide single crystal. This self-interaction results from the fact that the acoustic attenuation rate may be modified by imparting a drift velocity to the electrons, and such a drift velocity is in turn produced by the acoustoelectric current resulting from acoustic attenuation. The effect of the self-interaction may be computed by substituting the acoustoelectric current for the drift current in the small-signal theory. We have done this in order to compute the influence of the self-interaction on the detailed shape of the acoustoelectric current pulses generated as short acoustic wave packets traverse the crystal. These predicted pulses have been compared directly with experimental results. Upon comparison it is seen that the observed effect of the self-interaction is larger than the computed effect, and the predictions are more accurate at high crystal resistivity than at low. This is in part explained as a failure of the calculation to account for the influence of the second harmonic component of the acoustoelectric current on the local attenuation rate. viii

INTRODUCTION The acoustoelectric effect is a wave-particle drag phenomenon and refers specifically to the appearance of a direct current along the direction of propagation of a decaying acoustic wave in a medium containing mobile charge carriers. The effect occurs whenever there is a possibility of energy and momentum exchange between the wave and the charges, providing there exists a sink (such as the crystal lattice) to which the charges may dump their energy. The interaction causes the charges to drain energy and momentum from the wave, and the absorbed momentum propels them in the direction of the wave propagation vector. The result is attenuation of the wave accompanied by a direct current along its direction of propagation. In 1956 Weinreich1 published a definitive theoretical treatment of the acoustoelectric effect in a semiconductor in which both holes and electrons are simultaneously involved. About a year later Weinreich and White2 detected a weak acoustoelectric effect in n-type germanium, and in 1959 Weinreich, Sanders, and White3 reported they had used the acoustoelectric effect to measure the intervalley scattering rate of electrons in arsenic-doped germanium. In 1960 cadmium sulfide was discovered to be more strongly piezoelectric then quartz. This property, in combination with the crystal's easily-controlled and wide-ranging photoconductivity, made it an ideal material for the further investigation of electronically induced acoustic attenuation, and a thorough theoretical treatment of this possibility was published in 1962 by Hutson and 1

2 White. This theory is based on a small-signal analysis of the electron-wave interaction derived from the piezoelectric equations of state for a conducting medium. Weinreich had proven in 1956 that, by application of a dc electric field along the direction of propagation, it should be possible to use the acoustoelectric interaction to achieve traveling wave amplification of sound. In 1961 this was attempted in CdS by Hutson, McFee, and White. They succeeded in 7-9 demonstrating acoustic amplification, and this phenomenon and the related 10-12 effect of current saturation have received the lion's share of attention on the subject ever since. Reference 9 contains an extensive bibliography of recent work on these effects. The study of the less glamorous basic acoustoelectric effect in CdS in the absence of an applied electric field has consequently been neglected, even though it offers much in the way of experimental simplification. Nonamplifying experiments do not require the fast rise time, high voltage apparatus needed 6, 12 for amplification work, and they avoid the instabilities and current satura6, 10-12 tion effects which have come to be associated with CdS under amplifying conditions. That there is a wealth of information to be had from nonamplifying experiments was demonstrated by Henrich, 3 when he analyzed acoustoelectric current pulse shapes to verify the attenuation predictions of the small-signal theory of Hutson and White and also to compute electron trapping parameters in several CdS samples. The work presented here is also based on an experimental study of cadmium sulfide under nonamplifying conditions. It is primarily concerned with large

signal acoustoelectric effects in CdS, the analysis of which falls outside the immediate province of the small-signal theory of Hutson and White, but which nonetheless may be treated by perturbation calculations on that theory. By this approach we shall show that under large-signal conditions the propagating acoustic wave interacts with itself, and that it is possible to make computer-calculated predictions of the acoustoelectric currents generated by acoustic waves of large (but not too large) amplitude.

CHAPTER I ACOUSTOELECTRIC CURRENT WAVEFORMS We shall begin this chapter by presenting the basic experimental evidence for the acoustoelectric effect in cadmium sulfide, placing particular emphasis on the shape of the observed acoustoelectric current waveforms. We shall then introduce the Weinreich relation, the fundamental energy-momentum conservation relation which underlies all acoustoelectric phenomena, and use this relation to explain and predict the detailed shape of the current waveforms. This approach is possible because much of what we observe in CdS is true of any acoustoelectric interaction and is unaffected by details of the electron-wave coupling for a particular material or a special experimental situation. 1.1 THE OBSERVED ACOUSTOELECTRIC EFFECT Consider the experimental arrangement pictured in Figs. 1 and 2. A cadmium sulfide crystal, cut as a cube 7 mm on a side, is clamped between two pieces of fused silica. These fused silica pieces are called "buffers" (their function will be explained in Chapter IV), and each is a cylinder 1/2 in. in diameter and 1 in. long. Quartz transducers, cut for half-wave resonance at 30 Mc, are attached to the outside circular faces of these buffers, and all of these elements are solidly cemented together to form an acoustic assembly. If we electrically excite one of these transducers with a brief burst of 30 Me rf, the acoustic wave generated will propagate through the assembly and be detected at the opposite transducer~ Shear waves for example will traverse this system in about 17,isect. 4

7,5%5 Fig.!. Photograph of the acoustic assembly i the sample holder. input transducer buffer CdS crystal wire leads buffer output transducer Fig. 2. Diagram of the acoustic assemnbly.

6 We said in the introduction that cadmium sulfide is both piezoelectric and photoconductive. The acoustoelectric effect in CdS is a consequence of the interaction between the piezoelectric fields accompanying a propagating acoustic wave and the photoconduction electrons in the crystal. We therefore choose a crystal orientation in the acoustic assembly which causes the acoustic wave to be accompanied by strong piezoelectric fields. In Chapter II we shall show that it is impossible to have strong transverse piezoelectric fields accompany a propagating acoustic plane wave. We are therefore limited to crystal orientations and acoustic modes which together generate strong longitudinal piezoelectric fields. In the system thus far described an acoustic wave packet will propagate through all elements of the assembly, suffering some attenuation or reflection at the bonded interfaces between components of the system, but experiencing negligible losses in either the buffers or the CdS crystal-providing the crystal is kept in the dark. But if we now illuminate the crystal, we will discover that it is possible to cause the propagating acoustic wave to be strongly attenuated, the severity of the attenuation being a function of the light intensity and varying from a maximum greater than 40 db for a certain optimum brightness to near zero in the dark. There is a simple explanation for this phenomenon. We know the propagating wave packet is accompanied by piezoelectric fields, and we also know that the CdS crystal is an insulator in the dark but becomes a conductor when illuminated. It therefore is reasonable that the piezoelectric fields of the

7 traveling wave do work on the free electrons produced by the illumination, and that the energy thus dissipated by the piezoelectric fields must be drawn from the propagating acoustic wave, thus producing attenuation. If this explanation is correct, then we may further expect to find a crystal conductivity for which the EdJ dissipation of the photoconduction electrons is a maximum. The argument goes as follows: It is obvious that E J losses must vanish in the dark when the crystal is an insulator, i.e., when J = 0. At the opposite extreme of very intense illumination we might look for a diminished rate of dissipation as the electric field goes to zero, short-circuited by the very high photoconductivity. In between these two extremes there must be a maximum. This prediction would be borne out by experimentation on the system we have described. We would observe a maximum acoustic attenuation in excess of 40 db at a crystal resistivity of about 6500 ohm-cm. Let us now consider an extension of the experiment. Imagine the end faces of the CdS crystal (the faces in contact with the buffers) to be coated with a metal film to allow broad, ohmic contact to the crystal, and let these contacts be made externally accessible through wire leads (see Fig. 2). These leads might have been used to measure the crystal's photoconductivity in the preceding experiment, but they will also serve to show us another interesting phenomenon. Suppose these wire leads are joined through a small resistor, and an oscilloscope is connected across the resistor to monitor any current which may flow in this circuit (see Fig. 3). If the CdS crystal is now illuminated and

repeated bursts of electrical excitation are supplied to the input transducer, then an oscilloscope trace of the type shown in Fig. 4 will be observed each time an acoustic wave packet traverses the crystal. i I I Fig. 3. Basic experimental arrangement. This phenomenon of direct current production ("direct current" in the sense that it is slowly varying compared to the 30 Mc frequency of the acoustic wave) is called the acoustoelectric effect, and the current thus generated is called the acoustoelectric current. Let us examine the oscilloscope trace in detail. The acoustic wave packet used was that of Fig. 5. The acoustoelectric current (Fig. 4) begins to rise just as the wave packet enters the crystal; it reaches a maximum just after the peak enters and then dies away. The oscillations which appear on the rising portion of the trace are generated as the traveling wave crosses the input face of the crystal. They will be discussed in Section 1.3.

Fig. 4. Representative acoustoelectric current trace produced by the input wave of Fig. 5. The current was measured in terms of the voltage drop across an external 2000 ohnm resistor. Crystal resistivity = 3.7 x 104 ohm-cm. Vertical scale =.001 v/cm. Time base =.5 isec/cm. Fig. 5. 30 Mc excitation applied to the input transducer to produce the acoustoelectric current trace of Fig. 4. Time base =.5 [~sec/cm.

10 Let us concentrate our attention upon the falling portion of the trace. The input acoustic wave had a length of 2.5 Usec, and the transit time through the 7 mm crystal is 4.0 isec. Thus that portion of the current trace recorded between 2.5 and 4.0 Utsec occurred while the wave packet was entirely within the crystal. This portion of the trace is a falling exponential. Were we to change the amplitude of the input wave (by use of a precision attenuator so as not to change the envelope shape), we would observe the height of the current trace (but not the shape) to change in direct proportion to the square of the input acoustic amplitude, i.e., to vary as the input acoustic energy. The effect is illustrated in Fig. 6. Study of oscilloscope traces recorded during experimentation at other photoresistivities (see Fig. 7) would further reveal that the time constant T describing the exponentially falling tail of the current trace is in each case the same time constant that would be needed to predict the attenuation suffered by the acoustic wave packet in traversing the crystal; i.e., if the decaying tail of the current trace is described by = e-t/Te (1.1) and if the attenuation experienced tby he acoustic wave packet is given by Wu W. e,(12)-CL out - in (1.2)

11 " I (a) Acoustoelectric current traces produced by varying the input attenuator in 1 db steps. Vertical scale =.1 v/cm. (b) Acoustoelectric current traces produced for two settings of the attenuator with compensating changes in the oscilloscope vertical sensitivity switch. The traces overlap exactly. Attenuator Vertical Scale -10 db.1 v/cm -20 db.01 v/cm Fig. 6. Illustration of the linear relationship between the acoustoelectric current and the input acoustic power (at small signal levels). Input acoustic power was adjusted with a precision attenuator controlling excitation of the input transducer. Crystal resisitivity = 1.57 x 104 ohm-cm. Time base =. 5 [sec/cm.

12 where Eq. (1.2) may be considered the defining equation for the attenuation coefficient ca, then 1/Te = cvs, (1.3) where W = energy density of the acoustic wave packet L = crystal length vs = propagation velocity of the acoustic wave. Fig. 7. Acoustoelectric current traces produced for constant input power and changing crystal resistivity. Input attenuator set at -10 db. Time base =.5 sec/cm. Listed according to peak height, the traces are: Crystal Acoustic Trace Resis t i -vity, Attenuation Rate Top 9.5 x l0s o'm-cu'~ 60 db/cm CentRer 5.5 x 104 l Ohri-OX! 25 db/cca Bonto~ 1.1 x 10- onilr-cm dbj/cm

The implication of these observations is that the instantaneous value of the observed acoustoelectric current is directly proportional to the energy of the acoustic wave packet which decays exponentially as it traverses the crystal. That this is indeed the case (at least for the idealized experiment we have been describing) will be shown in the section which follows. 1.2 THE WEINREICH RELATION We have not yet studied in detail the interaction between the piezoelectric fields of the propagating acoustic wave and the photoconduction electrons in the crystal, but such information is not necessary to an understanding of the relationship between acoustic attenuation and the acoustoelectric current. This 14 was shown by Weinreich in 1957 when he proved that the energy and momentum surrendered by the decaying acoustic wave are given over entirely to driving a direct current down the crystal, with the result that the ratio of the attenuation coefficient to the current is determined entirely by wave dynamics and is independent of the detailed mechanism of the acoustoelectric interaction. (The application at that time was to the prediction of the weak acoustoelectric currents produced in multi-valley semiconductors as a consequence of the drag exerted on a traveling wave by free carriers.) The derivation goes as follows: (1) Momentum relation for the electrons: The force on the free electrons is given by dpe/dt = -nqE (1.4) where

14 n = density of electrons Pe = momentum density carried by the electrons q = absolute magnitude of the electronic charge E = an effective acoustoelectric field acting on the electron to produce the acoustoelectric current (E is defined to be positive when it points along the propagation vector of the wave). (2) Energy-momentum relation for waves: W = Vsps, (1.3) where W = energy density carried by the wave ps = momentum density carried by the wave. Therefore dW dp dp -v = S(1.6) dx s dx dt (3) Conservation of momentum for the interaction requires that dp dp s e + = 0. (1.7) dt dt Substitution of Eqs. (1.4) and (1.6) into Eq. (1.7) gives the Weinreich relation: dW nqE = - (1.8) dx If the acoustic attenuation is describable by an attenuation coefficient, that is, if

15 - dW/ dx =aW then the Weinreich relation becomes E = - l/nq OW. (1.9) We would like an expression involving the acoustoelectric current density This must be given by j = aE or j = -, wW (1. 10) where S = electron mobility in the crystal The Weinreich relation can serve as a powerful tool for the study of acoustic attenuation in CdS. Equation (1.1l.) tells -us that the local rate of acoustoelectric current generation is directly proportional to local acoustic energy density. If the traveling acoustic wave is well localized (T << Te, where T is the length of the wave packet, and -e is defined by Eq. (1.3)), then Eq. (1.10) allows us to measure the energy of the wave packet at any point, along its traverse of the crystal, assuming both 4 and a are known. For a wave packet which is not wall localized the relationship is more complicated. In Chapter III we shal. prove that th current flowing in the external circuit is given by J(t) = L j(x,t) dx, (1.11) L o where L is the length of the crystal, and j(x,t) is the rate of acoustoelectric current generation at each point on the wave. Equation (1.11) may also be

16 developed from a simple equivalent-circuit model of the crystal. Let us conceptually divide the crystal into N segments, each segment being of length L 6x = Within each segment let us represent the local rate of acoustoelectric N current generation by an ideal current generator -~iict) -aWit) (1.12) Ji(t) = -~IaWi(t) working across the local crystal resistance R = rSx as in Fig. 8. Ji-(t) Ji(t) Ji+l (t) 3(t) Jrx rx 3(t) rbx r~x r6x Fig. 8. Elementary equivalent circuit model of the crystal. We should pause here to clarify our sign convention. All flow quantities (currents and velocities) are taken as positive when they point in the direction of the acoustic propagation vector. We have just shown that for acoustic attenuation produced by mobile charge carriers, the recoil of the absorber must cause

a net drift of the carriers in the direction cf the wave. For negative carriers (as in CdS) conventional current flow is opposite in direction to the carrier drift, so j(t) becomes a negative number as in Eqo (10). Because of our sign convention, equivalent circuit representations like that of FigO 8 show generating elements which appear to produce a conventional current in the direction of the wave, but we must remember that it is the carrier velocity which is always with the wave, and the sign of the current carriers determines the sign of j(t). Flowing through all N crystal segments and in the external circuit is the circulating current J(t). The voltage drop across the ith segment is V. = R(J - ji). (1ol}) If we require as a boundary condition that the ends of the crystal be shortcircuited, then we must have N N ZV =- R(NJ- Z i j) = 0 (1.14) i=l i i=l i The solution for the circulating current is therefore J(t,) - iZE j.(t) x. (1.15) L i=l i In the limit of N becoming infinite, this passes to the integral form J(t ) = L J j(x,t) dx, (1.16) which is the same as Eq. (11). Because the integration need not be carrimd past the leading edge of the propagating wave packet, the upper limit may be changed to x = vst (for

18 v t < L). If we substitute Eq. (1.10) for j(x,t), then Eq. (1.16) becomes s 1 Vst J(t) = -~ f /o W(x,t) dx. (1_17) Within the crystal W(x,t) may be expressed in terms of the acoustic energy density at the input face. For the case of uniform acoustic attenuation we have W(x,t) = e W(O, t - s), (1.18) so that 1 vt -ax x j(t) = -4L - s e W(O, t - x ) dx. (1.19) L Vs Using Eq. (1.19) we may explicitly display the exponentially decaying character of the acoustoelectric current when the acoustic wave packet is entirely within the crystal. Let us first introduce a change of integration variables t = t - x (1.20) vs with the result that 1J(t) -pvVst t e Ovst W(O,t") dt". (1.21) This may be rewritten as 1 J(t) e Cxst St fteXVstys W(O,tt") dt". av L o If we now differentiate both sides with respect to t and then divide throughout by e s, we obtain

19 d + J(t) W(O,t) (1.22) crvs dt I22) Now the usual specification for the input energy density will be of the form W(O,t) = A2(t), t < T 0, t>T, (1.23) where T is the length of the acoustic wave packet whose amplitude is A(t). For t > T (the wave packet entirely within the crystal) the right side of Eq. (1.22) vanishes, and the remaining homogeneous equation has the solution J(t): J(T) e vs(tT) (1.24) which is the exponentially decaying result that we sought. Equation (1.24) may be compared with Eqs. (1.1) and (1.3). 1.53 CURRENT RIPPLE DUE TO THE BOUNDARY CONDITION The small oscillations superposed upon the larger acoustoelectric current trace (Figs. 4-7) are generated as the acoustic wave train crosses the input face of the crystal. Their frequency is the same as that of the traveling wave, and their envelope closely resembles the envelope of the traveling wave before it enters the crystal. They are not the result of capacitive feed-through of the excitation voltage applied to the input transducer; the 6.4 Usec transit time of the first buffer assures us of that. What they are may be understood as follows: While inside the CdS crystal the traveling acoustic wave generates longitudinal piezoelectric fields. Local intensity of the resultant electric

20 fields is approximately proportional to local mechanical strain (the exact relationship will be derived in Chapter II). The net voltage available to drive a current around the external circuit may be computed by integrating this electric field over the crystal length. This voltage divided by the total circuit res:istance is then the circulating current. The input face of the crystal is one of the limits of the integration. The integral may therefore contain a fraction of a cycle at the input face plus a number of whole cycles already in the crystal. To the extent that the traveling wave is symmetric about zero we may expect the average contribution of whole cycles already inside the crystal to be small, whereas the cycle crossing the input face contributes an amount x=O V(t) Es(x-vst) dx, (1.25) x=vst where t is the time since the leading edge (i.e., zero) of this cycle crossed the input face. Equation (1.25) may be rewritten as v(t) E(t)f X sin k(x-vst) dx = p El(t) (1 - cos Wt) (1s26) where El(t) represents the envelope of the piezoelectric field generated by the traveling wave as it crosses the input face. The current ripple is therefore given by I(t) E(t) (1 - cos It), (1.27) 2ir B

21 where R is the total circuit resistance (crystal resistance plus the external resistor). The observed amplitude of the current ripple is smaller than that predicted by Eq. (1.27). A possible explanation is that Il(t) is reduced because the acoustic wavefront does not arrive exactly parallel to the input face of the crystal. The probability of this occuring may be estimated by noting that for an acoustic wavelength X =.06 mm (3O Mc shear waves in CdS) and a crystal 7 mm wide, a deviation of O.50 could cause complete cancellation of the 30 Mc component of the circulating current. Such small deviations are unavoidable in the assembly of the acoustic system; e.g., the r-Ind faces of the buffers are only guaranteed parallel within.25~, and there is no way of measuring to such close tolerances the thickness uniformity of the bonds in the assembled acoustic system. Thus there appears little hope for quantitative use of the 30 Mc component. For this reason no attempt was made to extend the bandwidth of the measuring equipment (described in Sk ction 4.2) to accurately reproduce it. 1.4 THE SHAPE OF THE _ACi(:-STOELECTRIC C.LURRENT T}-ACE We are now ready to predict and explain til- detailed shape of the acoustoelectric current trace. We shall do thi-is wit-h the aid of Eq. (1.22), which may write as dJ + J(t) - L W(O,t), (1.28) e dt where

22 T = (129) e The behavior of J(t) in Eq. (1.23) is most easily visualized with the help of the mathematically equivalent circuit of Fig. 9(a). I: this representation the forcing function - - W(O,t) appears as an ideal corr at generator wailking L across a parallel RC circuit, J(t) i the current flowing in the resistie branch of the load, and R and C my take any value, prividing they touFrther satisfy R Ca = T = 1.30) e av Re and Ca form a low pass filtcir at the ouIput of tut ideal curre t generator, thereby restricting the speed with which the observed:.irrenrt J(t) AIs:,ble to follow changes in the generator current. The problem mf predicting the detailed sh;-pe of the acoustoelectric current trace (produced by a uniformly attenuated acoustic wave) has therefore been reduced to the problem of analyzing the responsof' a simple low pass filter to the input wave - LW(O, t). Let us first examine the effect of the filter for the extreme cases of either very large or very small values if' the attenuation coefficient. When a is very large (T << T), the influence of the filter is negligible. "Resistor" Re effectively swamps "capacitor" Ca, and the observed current is a faithful record of the acoustic energy density at the input face of tlie crystal: J(t) = - - W(O,t). (1.31) L The peak of the observed current therefore coincides with the peak of the input wave and is independent of ac Physically this is the case where the acoustic

_a (a) Equivalent circuit representation of Eq. (1.28) showing how the acoustoelectric current trace produced by a uniformly attenuated acoustic wave may be synthesized by a constant current generator working into a low pass filter. 3(t) L w(o,t) I -- R R 1 Ca I| e (b) Equivalent circuit representation of Eq. (1.55) showing the influence of nonelectronic losses. Fig. 9. Equivalent circuit for synthesizing acoustoelectric waveforms generated by a uniformly attenuated acoustic wave.

attenuation is so rapid that only that part of the wave which has just crossed the input face has enough energy to contribute significantly to the circulating current J(t), i.e., the crystal has an acoustic skin depth of 1/a. When a is very small (Te > T), the response of the filter is sluggish compared to the duration of the wave, and the output of the current generator goes wholly into charging Ca. Ca integrates the generator output, and the "leakage current" through R is e J(t) = [- W(O,t")] dt", t < T. (132) T 0 The observed current therefore rises for as long as the wave crosses the input face of the crystal. Thus we see that J(t) maximum occurs at t = T and is proportional to a. Physically this is the case where the attenuation is so slow that all portions of the wave inside the crystal are essentially undiminished and contribute accordingly to the circulating current. We therefore expect to see the observed current rise for as long as we inject acoustic energy into the system. Before discussing intermediate cases we wish to add one more observation. From the representation of Fig. 9(a), it appears that the entire output of the ideal current generator must eventually flow through Re, i.e., for a given input wave the integrated area under all possible acoustoelectric current traces must be the same, regardless of the filter time constant: J r(t) dt = Ao [-4 W(O,t)] dt (1.33)

25 This useful identity helps us maintain some perspective on the relative heights of the acoustoelectric current traces generated for different values of the attenuation coefficient. Of course the left-hand integration (over the time to infinity) can have no physical meaning for a real crystal, since the circuit representation of Fig. 9(a) ceases to be valid as soon as the wave reaches the opposite crystal face (see Eq. (1.17)), where it either propagates out of the crystal, suffers reflection, or experiences some combination of these effects. But this deficiency in no way impairs the utility of Eq. (n.z3) in comparing diff;-rent acoustoelectric current traces. Although the tails ofi ral traces L will depart from the ideal decaying exponential form after t —. tht portion o' the traces recorded up to this time is correctly given by Eq. ( a.:it) and therefore by the circuit of Fig. 9(a). What may we now say about the intermediate case? Of course we araa>, have the exact solution through Eq. (1.21). The question we wish ansv-, r here is: What will the comparison be between two acoustoelectric currm-i.t tra:- s generated by the same input wave but for different values of the att nuation coefficient? Based on our study of the low pass filter response, we conclude the following: For a very large value of a we know that the observed current is a faithful record of the input wave (Eq. (1.31)), and that the peak of the current coincides with the peaK of the wave and is indiepend':lnt of' a. By comparison with this case, diminshirig values of the attenuationi coefficient will cause the peak of the observed current to be redced in amplitude and deTlayed in time, and will slow the decay of the current fullowing the peak. These

26 phenomena are all clearly illustrated in the experimental record of Figo 7. We emphasize these relationships because they will be useful to us in Chapter V when we study the way acoustic attenuation rates can change under largesignal conditions. 1. REPRESENTATION OF NONELECTRONIC LOSSES We have not yet mentioned that there may be acoustic attenuation for reasons other than the absorption of the wave momentum by the free electrons. These nonelectronic losses may be grouped under the general heading of "transfer of momentum to the lattice" and are independent of the concentration of free carriers in the crystal. We may account for them by observing that the a of the Weinreich relation (Eq. (1.10)) is due to the electronic losses alone, whereas Eq. (1.18) for the acoustic attenuation should be written W(x,t) = e-( a+~)x W(O,t - x) (1.34) where ac is an attenuation coefficient representing the nonelectronic losses. Equation (1.22) therefore becomes 1 dJ a0 - -d + (1 + ) J(t) = L W(Ot) ~vs dt L A modified equivalent circuit representation which includes these nonelectronic losses is that shown in Fig. 9(b), where Ca is arbitrarily chosen, and Re and Ro must then satisfy T _e 1 Re = Ca O_ sC (1.36) a, CYs a

27 T 0 1 R = (1.7) o Ca 9oVsC When the wave packet is entirely inside the crystal, the homogeneous Eq. (1.35) has the solution J(t) = J(T) e-(a+~o)vs(t-T) (1 38) or, in terms of the decay time constants, J(t): J(T)-7 To) (t-T) (1.59) In Section 1.1 we argued that the acoustoelectric attenuation rate should go to zero in low light as the CdS crystal approaches an insulating condition. Because the nonelectronic losses are independent of the density of free carriers, any remaining slope to the tail of the acoustoel'ctric current trace as the crystal conductivity is r-duced to zero must be due entirely to so. This gives us an easy experimental check on;he nonelectronic losses in the crystal. We need only see whether an experimental plot of attenuation vs conductivity tends to some value ao rather than zer ir. the hig-i r.sistivity limit (WT -+ ). Such a plot will be presented at the beginning of Chapter V. For the moment we stlall merely say that the nonelectronic losses in the crystal are small enolagh (less than 1 db/cm) that they are not a cause for worry, and they will be disregarded in the theoretical dev+lopment of the next two chapters.

CHAPTER II THE SMALL-SIGNAL THEORY In this chapter we shall investigate the mechanism of the acoustoelectric interaction between the electrons and a propagating wave in a piezoelectric crystal. Our goal is to develop a small-signal theory for the prediction of acoustic attenuation and acoustoelectric current generation. The work presented here is a rederivation of the original small-signal theory of Hutson and Whi.te and its later extension by White7 to cover the case of acoustic gain. We have chosen a somewhat different approach to the problem with a consequent change in emphasis of some of the results. The theory will be developed as follows: In Section 2.1 we shall use the piezoelectric equations of state to demonstrate the Weinreich relation for the acoustoelectric interaction in a conducting piezoelectric crystal, indicating the phase relationships between electrical and mechanical components of the wave necessary to the production of acoustic attenuation and acoustoelectric current. In Section 2.2 we shall introduce a simple equivalent circuit model of the interaction and show how it gives directly these relationships along with the dissipation rate, Sections 2.3 and 2.4 will be devoted to the development of formulae for acoustic attenuation and acoustoelectric current production. In Section 2.5 we shall discuss modifications of the theory needed to accowlt for the effect of diffusion of the electrons. Finally, in Section 2.6 we shall list some of the problens associated with the experimental study of acoustic gain. 28

29 The acoustoelectric effect in CdS follows from the strong piezoelectric coupling which may exist between a propagating sound wave and the free electrons. To study this interaction for all interesting directions of polarization and propagation of the acoustic wave would appear to require that we undertake a three-dimensional analysis of the system. In fact it is a problem in only one dimension, that being along the propagation vector of the wave. Consider an acoustic plane wave of either longitudinal or shear mode propagating in an insulating piezoelectric crystal. On the basis of our understanding of the static case it would seem that the sound wave could be accompanied by longitudinal and transverse electric fields, with the strength of a particular electric field component essentially unchanged from that produced 15 by an equivalent static strain. In 1949 Kyame showed that such a description is correct for the longitudinal electric field, which is essentially electrostatic in nature, but that the transverse electric field (and consequent magnetic field) behaves like an electromagnetic wave constrained to move at the velocity of sound and is therefore reduced in amplitude compared to the expected electrostatic value by a factor (v/c)2, where v is the velocity of sound and c the velocity of light in the crystal. In 1954 Kyame expanded his theory to include conductive crystals, but there was no change in his earlier result about the magnitudes of the electric fields accompanying the wave (our Appendix contains a partial rederivation of this result for conductive piezoelectric crystals). Because of their extremely small amplitude the transverse electric fields are of no consequence to us in our study of the acoustoelectric effect in CdS, and our analysis does become a one-dimensional problem along the propagation vector of the wave.

30 We may effect a further simplification by restricting the directions of propagation and displacement of the acoustic wave to lie along the crystallographic axes, thereby avoiding the task of recomputing the crystal's anisotropic properties in a rotated coordinate frame. Cadmium sulfide is a hexagonal crystal belonging to crystal class C6v. The existence of a sixfold symmetry axis imposes a high degree of degeneracy upon the crystal's anisotropic physical properties.7 The nonzero elements of the dielectric constant are ell 22' 33 and those of the piezoelectric coefficient are e e, e e = e 11 322 333' 13 223' where the 3-axis is the axis of hexagonal symmetry. As a consequence of this degeneracy only certain electric field pclarizations may be associated with a given acoustic plane wave. For acoustic propagation along the hexagonal axis longitudinal waves (e ) are accompanied only by longitudinal electric fields, but shear waves (e e ) have only transverse field components. For propagation at right angles to the hexagonal axis, shear waves with displacement along the hexagonal axis (e = e ) are accompanied 113 223 only by longitudinal electric fields, but those with displacement at right angles to the hexagonal axis see no piezoelectric effect. Longitudinal waves propagating at right angles to the hexagonal axis (e = e ) are accompanied only by 311 322 transverse electric fields.

31 Therefore, the only acoustic propagation modes of interest to us are (i) the shear mode with propagation vector at right angles to the hexagonal axis and displacement along the hexagonal axis (e = e ), and (2) the longitudinal 113 223 mode with propagation along the hexagonal axis (e 33). 2.1 WEINREICH RELATION IN A PIEZOELECTRIC PHOTOCONDUCTING CRYSTAL Consider an acoustic wave propagating in the x-direction in a piezoelectric photoconducting medium and define a strain S, a stress T. and a displacement u such that au aT a'u S - and - = m ax ax t2 where m is the mass density. Further, assume that the medium is characterized by a piezoelectric coefficient e such that S produces an electric field in the x-direction. Under adiabatic conditions the piezoelectric equations of state corresponding to the one-dimensional problem are T = cS - eE (2.1) D = eS + cE, (2.2) where T = stress S = strain E = electric field D = electric displacement

aTSE = adiabatic short-circuit elastic constant = aE = adiabatic clamped dielectric constant e = = - = adiabatic piezoelectric coupling coefficient. E S In the absence of the piezoelectric coupling we recognize these equations as being simply: (1) Hooke's Law (2) The usual relation for D and E. The piezoelectric equations of state tell us that, in the presence of piezoelectric coupling, the propagating pattern of mechanical stresses and strains which characterize the traveling acoustic wave will be accompanied by proportional E and D fields. The fact that the piezoelectric coefficients e in the two equations of state are identical may be proven by a thermodynamic 18 argument that the electric enthalpy of the system must be a total differential A propagating wave packet may be mathematically represented as the product of an envelope function and an oscillating function. In particular, for a dispersionless medium we may represent the strain associated with an acoust;ic plane wave of frequency w by: S(x,t) = Re[S1(- - t) ei(k-t) (2,3) where k = v (2.4)

33 and v is the propagation velocity. The envelope function Sl(x-vst) then represents the local amplitude of the propagating wave and can be said to modulate the oscillatory component. But how can such a description have any application to a medium where there is attenuation and dispersion? In the preceding chapter we stated that the maximum acoustic attenuation we shall encounter with our particular experimental sample is about 65 db/cm. Although this would seem to be a very large rate of attenuation, it is still small compared to a wavelength of sound; e.g., for a 30 Mc wave the attenuation is less than 0.1 db/radian. Therefore, although the attenuation should be explicitly displayed in a complex propagation constant as W a vs 2(2.5) our approach will be to treat the attenuation over one wavelength of the sound as being so small that for the purpose of computing local relationships among the components T, S, D, and E of the traveling wave we are justified in taking the propagation vector k as being entirely real. To this approximation, the strain associated with the traveling acoustic wave may be represented by x t) eim(, - t) S(x,t) = Re[Sl( - t) ei( - t) (2.6) If we assume similar representations for T, E, and D and substitute these expressions into the piezoelectric equations of state, then the oscillatory components will cancel, leaving the following relationships among the amplitudes: T1 = cS1 - eEl (2.7)

34 D1 = eS1 + eE1 ~ (2.8) These amplitudes may be complex in order that they properly represent possible differences in phase between the components of the traveling wave. However, the physical constants c, e, and C will all be real, because the ultrasonic frequency (30 Mc) is small compared to a typical lattice vibration frequency 12 -l (a- 10 sec ). If the acoustic wave is attenuated in its passage through the crystal, then the local rate of acoustic energy dissipation is dW 1 2 T dS dt 2- o Td (2.9) where the phase = W( - t). (2.10) If we assume the envelope of the wave is slowly varying compared to the frequency ~), then we may replace the derivative by dS dt - _ iwS, (2.11) so that Eq. (2.9) becomes d-W =1 2 T (-iaS) dm (2.12) dt 2T o 2 Re[Ti*(-in Si)] (2-13) 2 where the asterisk indicates complex conjugation. Thus the acoustic energy dissipation is given by

35 dW = 2 Im[T1*Sl]. (2.14) dt 2 The local rate of acoustic energy dissipation is therefore different from zero only when there is a phase difference between the amplitudes Ti and S1. This we expect, since the condition that the stress and strain of a propagating wave be in phase requires that they be linked only by an effective elastic constant which is entirely real and therefore dissipationless. From Eq. (2.'7) we see that any phase difference between T1 and S1 must come from El being out of phase with S1. By substituting Eq. (2.7) into Eq. (2.14), we may display directly this relationship between the acoustic dissipation rate and the relative phase of the electric field and the strain. We obtain: dW dW = ~ Im[-eEl*S1]. (2.15) dt 2 We normally think of the piezoelectric effect as describing a linear dependence of electric field upon mechanical strain. On this basis we should expect the electric field to be exactly proportional to the strain everywhere on the traveling wave, thus making E1 in phase with S1. But it is apparent, from Eq. (2.15) that there must be a phase difference between these quantities for acoustic dissipation. The second equation of state (Eq. (2.8)) reveals that this phase difference is directly traceable to D1. This is where the photoconduction electrons enter the picture. It is instructive first to study the case of the insulating crystal. In the absence of illumination the crystal is an insulator, a condition of charge dD neutrality exists throughout, and dis therefore everywhere zero, Thus we must have D1 = 0, and Eq. (2.8) becomes:

36 El - - S- (2.16) C giving E1 in phase with S1. If we substitute this expression into Eq. (2.7), we get e2 T1 = cS1 + - S1 (2.17) 2 = c(l +) S1. (2.18) cc We may define a new elastic constant 2 ST \D_ e cD ==c(l +-), (2.19) cc so that T1 = cDS1 ~ (2.20) We shall later have use for the electromechanical coupling coefficient K2, which is usually defined by 2 __D Ec K (2- 21) cD e2 1 + - cc Equa,;io ns (2.18) and (2.20) may therefore be written as T1 S- 1 (2.22) Thus, in ar insulating crystal, the effect of the piezoelectric coupling is to stiffen the elastic constant. Typically K2 has a value of about.036 in CdS9 so the effect is small.

37 Let us now return to the general case of the piezoelectric crystal with free conduction electrons. In the quiescent crystal there will be a condition of charge neutrality with d = 0 and therefore D1 = 0. But in the presence of dx the wave we may expect the piezoelectric field accompanying the wave to upset local charge equilibrium conditons. In particular, since the piezoelectric field reverses sign every half wavelength along the wave, we may expect periodic bunching of the electrons by the wave. If we let ns represent the local density of electrons in excess of that required for charge neutrality, then D must reflect the bunching through Gauss' law dD d = -qn, (2 23) where q is the absolute magnitude of the electronic charge. The amplitude of' D is assumed to be slowly varying compared to the frequency of the wave, so we may write Eq. (2.23) as il) D -qn (2.24) The charge bunching ns has the periodicity of the wave and may also be represented as the product of an amplitude and an oscillatory component: x iai(k - t) ns(x,t) = Re[nl(ss - t) e iv (2.25) so that Eq. (2.24) may be written in terms of the amplitudes alone: - D1 = -qnl (2.26)

38 In discussing Eq. (2.15) we observed that there can be attenuation only if there is a relative phase difference between E1 and S1 and that this phase difference is directly traceable to D1. This dependence of the acoustic dissipation on D1 may be explicitly shown by using Eq. (2.8) to eliminate E1 in Eq. (2.15), with the result that dW w e d-~t= 2 Im - I Dl-S1] dt 2 m - em Im [D1*(- S1)] (2~27) We recognize - e S1 as just the piezoelectric field which would accompany the acoustic wave in the absence of the conduction electrons (see Eq. (2.16)). Let us replace this quantity by an equivalent piezoelectromotive fieldL given by 2jx,t) = - e S(x,t); (2.28) or, in terms of the amplitudes e( x t) = -CSl(X- t). (2.29) 5 s This is tantamount to writing the second piezoelectric equation of state (Eq. (2-8)) as El = ~1 +-D1 X (2.30) If we substitute Eq. (2.29) into Eq. (2.27), the expression for the rate of acoustic energy dissipation becomes

39 dW = w dt - Im[D1* 1 ] (2.31) But we know D1 in terms of the charge bunching amplitude nl. We therefore substitute Eq. (2.26) into Eq. (2.31) and obtain after some manipulation dW Vs _dW- -2- Re[(-qnl)* 1]. (2.32) We shall later wish to compute ar attenuation coefficient. For this we need dW dW - rather than - - The relation between the two is simply dx dt dW 1 dW V5~ ~ dt' ~~~(2. 55) so that dW 1 dx - (-qnl)* 1] (2.34) dx 2 Thus we see, as we expected, that the energy of the acoustic wrave is dissipated through work done on the conduction electrons by the piezoelectric field which accompanies the wave. What perhaps was not intuitively apparent was that the piezoelectric field would cause periodic bunching of the electrons, and that the attenuaticn suffered by the wave would be strongly dependent on the relative phase between the charge bunching and the piezoelectric field. Had we been looking for the charge bunching, we might have expected the distribution of electric fields and conduction electrons illustrated in Fig. 10(a); iLe., we might have looked for symmetric bunching around electric field minima. But this symmetric condition unfortunately gives

n n S S kx-wt n n iis s S Fig. 10(a). Illustration of the phase relationships needed for acoustic attenuation. Phase condition for the static case. The electrons are symmetrically bunched around the zeros of the piezoelectric field. This phase condition cannot produce acoustic attenuation.

~ ~ ns ns / / kx-wt n n Fig. 1O(b). Illustration of the phase relationships needed for acoustic attenuation. Phase condition needed for acoustic attenuation. The electron bunching lags the zeros of the piezoelectric field.

Re[(-qnlj-] 0 ~ and therefore no attenuation. We shall later show that this condition also delivers no net direct current down the crystal. It is'the relative phase lac of the charge bunching which produces acoustic attenuation and acoustoelectri& current generation. We may guess that a phase lag occurs because the electrons' finite mobility makes them sluggish in response to the rapidly oscillating local piezotlec-ric field induced by the passing wave. It' this is the case, then we see th,t; there are two ways we might expect to influence the rate of acoustic attenuation: (1) By adjustment of the illumination we may change the rate of generation of free electrons, thus regulating th_ density of carriers available i'or bunching by the acoustic wave. This technique is effective, as will be shown in Section 235. (2) A more interesting possibility is that by application of an external dc fi.-ld E to the crystal we may be able to adjust the relative phase by vihich the e — ctron bunching lags the acoustic wave. An external dc field would impart an average drift, velocity vd = -BE0 to the free -_lictrons, and that particular value of E. which gives vd - vs should produce no phase lag and therefore no attenuation~ The logical extension of th:s speculation is that the application of even larger drif't fields may cause: the electron bunching to lead in phas-. thereby reversing the direction of momernt-um exchange between the electrons and t he wave, i.e., we should be able to acheive acoustic gain. This point will be considered further in Secti..n 2.3, and some of the problems associated withL the experimental study of acoustic gain will be discussed in Section 2K60

43 To finish our proof of the Weinreich relation in a piezoelectric crystal we need an expression for the acoustoelectric current to compare with Eq. (2.34) for the acoustic attenuation. If there is a net flow of direct current (i.e., current slowly varying compared to the ultrasonic frequency of the acoustic wave) down the crystal, then the local production of the current must be given by 1 2ii J = 2 f (qn)(4E)d1 (2. 5) 2 - Re[(qnl)* HE1]. (2.34) To eliminate E1 from Eq. (2.36) we first combine Eq. (2.30) with Eq. (2.26), obtaining E1 = + -i- qn nl (2.37) This expression may then be substituted into Eq. (2.36), with the result 1 j = 4 2 Re [(qnl)*1l]. (2.38) Our proof is finished. Upon comparing Eq. (2.38) with Eq. (2.34) we see that we have indeed demonstrated the Weinreich relation for a piezoelectric conducting crystal: 1 dW 1. =d(2.59) dx

44 2.2 EQUIVALENT CIRCUIT MODEL OF THE INTERACTION Working from the piezoelectric equations of state and Gauss' law we have succeeded in demonstrating the Weinreich relation in a piezoelectric photoconducting crystal, and we have also obtained some insight into the conditions necessary to the production of acoustic attenuation. So far we have not independently predicted either the attenuation or the acoustoelectric current; we have only proven that each could be computed in terms of the other through the Weinreich relation. To be able to predict the attenuation and acoustoelectric current at a given crystal conductivity for a particular input acoustic wave packet, we must first understand the mechanism which causes the charge bunching to lag in phase behind the zeros of the piezoelectric field. We begin by writing the expression for the electron current in the crystal. Under illumination cadmium sulfide behaves liKe a n-type extrinsic semiconductor. At the maximum available light intensity our experimental sample has a photoconductivity of 10-3 (ohm-cm). For an electron mobility of 515 V/cm, we may compute the maximum density of free electrons at 2 x 10 3/cm. V/cm At such low concentrations the electrons may be regarded as classical particles whose behavior is described by Maxwell-Boltzman statistics. For a short mean free path the electron current may be written as the sum of drift and diffusion currents: drift diffusion dnc J = qncE + 4kT dxc, (2.40) where nc = density of electrons in the conduction band. The coefficient of the diffusion term follows from the Einstein relation. In most circumstances the contribution of the diffusion term is small, and presentation of the theory is clearer if this term is omitted. In the theoretical development which follows we shall first derive results without this term; then in Section 2.5 we shall show how the theoretical results must be modified to include the influence of diffusion.

Continuing with our small-signal approach to the problem, let us separate the variables on the right hand side of Eq. (2.40) into their do and first order oscillatory components: nc = no + ns (2.41) E = E + E (2,42) where no = steady-state (quiescent) density of electrons in the conduction bind. The crystal's dc photoconductivity is given by a = noqgo ns = Re [n e i Vs ] represents the acoustic charge bunching. Eo = any externally applied dc drift field. x iw(, - t) Es Re [El e (s )] is the oscillatory component of the local electric field. After this separation (and dropping the diffusion term) Eq. (2.40) becomes J = qnoEo + qnspEs + qns4EO + qnES. (2.43) We may effect a similar separation of J into dc and first order parts: J = Jo +Js' (2. 44) If we then collect first order (oscillatory) terms only, we will have Js = Es + qns4Eo. (2.45)

46 We recognize that the intermodulation term qns5Es in Eq. (2.43) has dc and second order oscillatory components only. Hence it will contribute to J~ but not to Js. In terms of amplitudes Eq. (2.45) is J1 = aE1 + qnlEo. (2.46) We may use the continuity equation dJ d dx - ( d n (.qnc) 47j to eliminate nl from Eq. (2.46). If we make our usual assumption about amplitudes varying slowly compared to the frequency of the wave, then the continuity equation may be written as i V Js = -iw qns (2.48) or, in terms of the amplitudes 1 qn - J1 (2X49) Substitution of Eq. (2.49) into Eq. (2.45) then gives + +- 0OJ o = aE1 (2.50) or E1 = y r J1, (2.51) where r- =, and

47 0E = 1 + v. (2. 52) vs The externally applied drift field imparts a drift velocity vd = -.E~ to the conduction electrons. Thus the factor y may also be written vd -=1-v. V(2. 3) We demonstrated earlier (Eq. (2.32)) that the traveling acoustic wave gives up its energy by doing work on the conduction electrons. Thus the rate of dissipation of energy frcm the wave must be given by dW 1 _d - Re [E1* J1] (2.54) (this expression may also be derived by manipulation of Eq. (2.32)). From Eq. (2.51) we see that the product E1*J1 is entirely real and is positive for y positive (vd < vs), negative for'y negative (vd > vs), and zero when the drift velocity of the electrons is matched to the propagation veloci —y of the wave, Equation (2.54) clearly shows how the direction of energy transfer between *the wave and the electrons may be reversed to produce acoustic gain. Knowing E1 in terms of J1 is not particularly useful. We would rather know b.th E1 and J1 in terms ofZ-, to which we have access through Eq. (253';): El = ~ + qn, (2.55) We may combine this with the continuity Eq. (2.49) to obtain

48 = E1 + J1. (2.56) - icl If we further substitute Eq. (2. 1) to eliminate El, we have 1 = (7r + il ) J1. (2.57) The relationships among es, Es, and Js may be neatly summarized in an equivalent circuit diagram (Fig. 11). The corresponding equations for the dependence of the amplitudes E1 and J1 upon E1 are J - = F 1 1 r (2.58) 1 1 r E1 1 (2.59) yr + -- 7+ - icuG - iAT where T = re is the dielectric relaxation time. These relationships are also displayed in the phase diagram of Fig. 11(b). If we define @ as the relative phase angle by which the electron bunching nj lags the zeros of the piezoelectric field l (our time convention makes a phase lag appear as a counterclockwise rotation), then zr 0 = arctan = arctan w(eT l/n c The angle 0 may take any value betweern - JT/2 and + 1T/2, with the negative values (phase lead) representing acoustic gain. Equation (2.51) requires that El be colinear with J1, and Eqs. (2.30) and (2.99) are satisfied only if the tip of E1 is constrained to lie on the circle shown in the diagram. The lower semicircle is the region of acoustic attenuation, and the apper semi-circle is the

49 r E S s = e = d = 1 + (a) Equivalent circuit. 1 0 = arctan ywT wt 1 J1 (b) Phase diagrarm. Fig. 11. Equivalent circuit and phase diagram for the acoustoelectric interaction (without diffusion).

o50 region of acoustic gain. E1 reverses sigh when G goes through zero, so E1 and J1 are paralll in the region of acoustic attenuation and anti-parallel in the region of acoustic gain. Maximum acoustic gain or attenuation occurs at - = + We make the following observations: (1) Ihe electric field E1 vanishes when y = 0, that is, when the drift velocity of the electrons is matchled to the propagation velocity of the traveling wave. For' this case the wave is stationary in a referencQ frame "fixed with respect to the average drift velocity of the electrons. Thus the electrons have "enough titre" to completely neutralize the piezoelectric f'eild of the acoustic wave. (2) The cLrrent J1 and therefore eharge bunching n, are largest at 0. This is also cInsistent with the concept of "enough time." Also notice that J1 and nl are exactly 901' out of phase with k- at y = O, this bi:ing the phase condition which we earlier showed could give no attenuation and nc r.et direct current. (3) The equivalent c> suit model clearly displays the conditions needed for maximum attenuation, i. -., maximum dissipation in yr. Maximum transfer o' power to "load resistor" yr occurs when it is matched so the equivalen- generator impedance 1/-ieo. For a given <rystal resistivity this condition occurs wh,-. 1 ( -.60) with the minus sign representing the condition of acoustic gain. Aitern<a ively if'? = 1 (no external drift field available) then maximum disslpstion wil ccr for

51 r = AC (2. 6i) 2.3 THE ATTENUATION COEFFICIENT We may now compute an attenuation coefficient. From Eq. (2.34) the decay rate of the wave is given by dW = Re [(-qnl)*~1] (2.62) dx 2 To eliminate -qnl from Eq. (2.62), we combine the continuity Eq. (2. 49) with Eq. (2. 59) -qnl = v J1 = v r (2.63) and substitute the result into Eq. (2.62), obtaining dW _ 1 11 1 1 dW 1| <Re (2.64) dx -Vs 2r Re 1 - iwCT' vs 2r. 2(2.65) Vs 2r + 2 IY + -ieO (Our rtason for writing Re[y] instead of just y will become apparent when we discuss the influence of diffusion in Section 2.5. ) Alternatively, Eq. (2.65) may also be derived directly from the equivalent circuit model. The dissipation in "resistorr" yr is dW 1 dW = 1 Re[El* J1]. (2.66) We may use Eqs. (2. 8) and (2. 9) to replace E1 and J1. obtaining:

dW 2 Re[7] dt 2r 1 -il WT Multiplication throughout by 1/vs will again give Eq. (2.65). To compute an attenuation coefficient we would like - dW/dx in terms of the strain S1 rather than the equivalent piezoelectromotive field _l-. From Eq. (2.29) we may write 1% 22 2 2 =s 12 e 2= 1 e cS2 1 _1e2 SS (2.68) r 2r c rc cc 2 We may combine this with Eqs. (2.19) and (2.21), with the result 2 22 2 2 _ 1 1 K cS K T 2 CDS1 1 (2.69) 2r T1. K 2 T Thus dW K2 1 2 Re[y] vd =VKT 2 cDS1 1 2 (2.70) + -iWT The attenuation coefficient a is defined by 1 dW 1 dW - - - *(2. 71) W dx If we approximate W by the energy density of a traveling acoustic wave in an insulating crystal W - S1 (2. 72) 2 D then our final expression for the attenuation coefficient is K-2 Re[] 2' (2. 73) - +1(1

53 Figure 12 is a plot of a vs. vd. The plot is antisymmetric about y = 0 (vd = vs), and the peaks of maximum attenuation and maximum gain are displaced from 7 = 0 by an amount 7= + 1 — (2. 74) Vd. \I y= O (>E = -Wv) Fig. 12. Plot of CX vs. vd (without diffusion). These maxima therefore occur at 1 yr = I (2. 7t) and correspond to values of externally applied drift field and crystal resistance for which the charge bunching nl lags or leads the zeros of -1 by a relative phase of 45~. These maximum and minimum values of a are

54 K2 WT - + 2 - VeT 2 _ K2 X' (2.76) For 30 Me shear waves propagating in CdS the length of a radian is X_ Vs 1.75 x 105 cm/sec -3 ~x= = -0xl0sec.95x 10 cm 21 2t x 30 x 10 6/sec In our particular experimental sample the electromechanical coupling coefficient has the value 3 K2 =.0284 The maximum value of a is therefore -i = 15.0 cm = 65.2 db/cm This is attenuation so large that it nearly deserves to be called annihilation. One might fairly ask whether we have violated our criterion that c/2 be small co:r.par;-d with W/vs (see Eq. (2.5)), but t-his is not the case, since XY 2 (15.0 cm )(.93 x 10 cm) W/Vs 2 If we wish to observe this much attenuation with no externally applied drift field, then we shall have to set y = 1 and choose the crystal resistivity:9 to satisfy Eq. (2.75). For the shear mode the crystal capacitivity 22 is -I -1222 c =.8 x 10- farads/cm, 22

55 so the required crystal resistivity is r = 6650 ohm-cm This is the particular crystal resistivity for which the separation between peaks of maximum and minimum a (Eq. (2.74)) is given by y = ~i, thus centering the positive peak on the vertical axis. Values of crystal resistivity different from this amount must therefore produce diminished acoustic attenuation at zero drift field. This last statement is illustrat d in Fig. 13, where we see a plot of ca as a function of crystal resistivity for the case of no applied drift field. We have taken ST (assuming c constant) rather than the resistivity as the independent variable, and we have chosen to make the horizontal axis a logarithmic scale. When plotted this way the curve is symmetric about its maximum at XT 1. The simple analytic function describing the shape of the curve may be derived as follows: If there is no externally applied drift field, then 7 - 1, and Eq. (2.73) becomes K2 1 Kx - (l'2' (2.77) 1+ \+ V which reduces to K23 2 a( = x 21 (2.78) WJT + ()T Here the symmetry about T 1 on a logarithmic scale is clearly apparent. We may find the exact shape of the curve by defining

X W.i!W 1111111111111111 111 1111111111 111111111111111111111!11111111i 1 1 1 I1 100 10 1.1 1I. ig. 11 1i111111111111111111 for the case of no applied l II a = log T, (.79) - 1" 2111 (2.80)!I1 11111II!11111111111111llllllllllllilll I i TrIr I I,.I I I Itll]lli 1 illI11111111111111IIII!111NI!11111111111 1 xi I I I I I 1f I t. F I 1 1.. WI.l. iT 1111111111111111111 11111 NJ 1.1. i sech a I (2.81)II nn0 n 1 0 w Ir r 1 IIi!11I The shape of the curve is therefore simply th e hyperbolic secant: a = ch(log (1 T) (2.82) 2. 4 THE ACOUSTOELECTRIC CURRENT In the last section we computed an attenuation coefficient by collecting the first order oscillatory terms from Eq. (2. 43) and using these terms to

57 calculate the rate of dissipation of energy from the wave. We wish now to compute the acoustoelectric current. If we return to Eq. (2.43) and collect the do terms, we will have 1 2~T Jo = qn04E 0+-f qnos4E d+ (2.83) = qnoiEo + Re [(qnl)*.E1]. (2.84) We identify the acoustoelectric current j as j = - Re [(qnl)* E1] (2.85) 2 The product qnsiEs also has an oscillating component whose frequency is 2w and whose amplitude is equal to that of the dc term. The relationship between these intermodulation products and the initial interaction term is simply that of the trigonometric identity 2 1 cos2, = - (1 + cos 2P). (2.86) We are -unable to observe this oscillatory component of the acoustoelectric current. There are two reasons: (1) The observed current is physically integrated over the entire crystal length, so any ripples are "smoothed out" by the action of the "low pass filter" discussed in Section 1.4; (2) The requirement of parallelism between the acoustic wavefront and the crystal end faces is twice as severe as that for observation of the ripple due to the boundary condition (Section 153). For these reasons we have made no attempt to extend the bandwidth of our measuring equipment to 60 Me where these oscillations occur.

58 Following a procedure similar to that used in developing Eq. (2.73) for the attenuation coefficient, we use Eqs. (2.59) and (2.63) to eliminate qnl and pE1 from Eq. (2.85), obtaining j - vs 2r 1 2 Substitution of Eq. (2.69) and further manipulation then gives K2 Re[y] J = 1WVT 1 2 (2.87) I + -iTIComparison of Eq. ("2.87) with Eq. (2.73) f:r the attenuation coefficient again illustrates the Weinro-ich relation i = - (2.88) At the beginning of the introduction we characterized the acoustoelectric effect as a wave-particle drag phenomenon. We think of the wave and the elec;rons as exerting a drag on each other, the drag being the result of a momentum exchange which attempts to equalize their velocities. In the absence of an externally applied drift field the electrons are initially "at rest" with respect to the crystal lattice, and the momentum exchange results in attenuation of the wave and the production of a net acoustoelectric direct current. From the Weinreich relation and from the work of this section we know that the acoust"oelectric current is proportional to the acoustic attenuation and that the current must reverse direction when the attenuation coefficient changes sign. An amplifying drift field propels the electrons at a velocity higher

59 than that of the acoustic wave, and the retarding drag on these carriers by the wave causes a reduction of the drift current, the amount of this reduction then being the reverse acoustoelectric current associated with acoustic gain. 2.5 INCLUSION OF THE DIFFUSION TERM Let us now return to Eq. (2.40) and make the corrections needed to account for the influence of diffusion. The expression for the current was drift diffusion dn, J = qncE + 4kT dx The gradint in nc must be th:- result of charge bunching; i.e., dn_ dns ix dx; ax = v ns (2~89) = - -. (2.89) We see that the diffusion term will contribute only to J1 (no dc c ntribution). Equation (2.46) for th-: amplitudes therefor- becomes i kT J: = El ~ + (4Eo + Vs - ) qnl (290) If we use the continuity Eq. (2.49) to eliminate qn1 from Eq. (2.90), we obtain [iEo ie. kT (1 + s + s2 q ) J = oEl (2.91) HE or, after multiplying throughi by r and substituting y = +Vs r + r) J - E (2.92) Ir+V q rv)J- q II' w-: now introduc th-: difusi.n frequency %:

60 i 1 p 4kT WD Vs2 q 29) then Eq. (2.92) may be reduced to I w ( + ~i )) l - Ez. (2 91 ) By comparisun with Eq. (2.51) we see that all of tht f~rmulae we derived in Sections 2.2 - 2.4 will be preserved if we everywhere replace 7y by the new quantity F, where 1i w r - r + i -D. (209s) For 30 Mc shear waves propagating in CdS (at 3000K) - has an approximate numeri.al value of _o 1'D- 2 -o we see that in the absence of+ an amplifying drift field (i.e., for y = 1) the influence of the diffusion term if indeed small. Equations (2. 58) and (2. 5>) now b —com-',U' J = - 1 (2. 96) fr + ic i E1 - - 1 (2.97) 7+ We may includ- the influence of tie ditf-'usion term in the equivalent circuit rmodel. The factor -i in the dencminator of Eqo (2.94) fixes the phase and dictates that proper representfti n must be as a capacitive impedance20

61 D -i (D r (2.98) At the risk of concealing its dependence upon the frequency and the conductivity, we can make the diffusion term appear like a capacitor. We do this by returning to Eq. (2.92) and substituting D kT ikT c q r ~ (2.99) noq2 where LD is the Debye screening length. After some manipulation Eq. (2.92) becomes 1 (r + ) J1 = El, (2.100) where the effective dielectric permitivity due to diffusion is defined by c (r>) = ( ), (2.101) and the equivalent circuit representation of the interaction takes the form of Fig. 14(a). If we substitute Eq. (2.95) into Eqs. (2.96) and (2.97), the expressions for local current density and electric field intensity become J = 1 1 (2,102) yr + - (CU- + D r) 1 W 7 + - - (T + )3) -i W

62 II (a) Equivalent circuit. ~ o wt E 1 E:D1J (b) Phase diagram. Fig. 14. Equivalent circuit and phase diagram for the acoustoelectric interaction (with diffusion).

Notice that J1 and therefore nl are still a maximum at vd = vs (7 = 0), but that E1 no longer vanishes at this point. This is because diffusion prevents charge bunching from completely neutralizingl at vd = vs. Figure 14(b) is a phase diagram of the interaction. Here 0 is the relat iJve phase angle by which the electron bunching n1 lags the zeros ot the piezoelctric field1l and is given by = arctan 1 W ()T CpD In this diagram the tip oi tihe electric field amplitude E1 is constrain d t(o li- on the smaller circle, and its position on this circle is determined by the point of intersection of J1 with the larger circle. The diiferenc — in diameters between the circles is ~$1~, where A(roy) o 1'i + 1 D +-U- 1+LD, T We may now recompute the attenuation coefficient and the expression -,or the acoustoelectric current. The new expressions replacing Eqs. (2.73) and (2.87) are K2 R~ [?] K2 R ] 2 (2.104) sT 1 1 - icoT K2 Re[ r] -' K Re[ -. (2. 10o) Ir + 1WT:- T f After substitution of Eq, (2.95) these expr~ ssi ns take t!he form

64 1 K r (2~10)2 W j K2 1 (2.106) sJs 1 1 Y _i (-1 + l ) Figure 15 is a plot of a vs. vd from Eq. (2.106). The cross-over voltage (value of E for y = O) is unchanged, but the displacement of the peaks of maximum and minimum a from this point is increased to r = ~ (M7 + T), (2.007) while the height of these peaks is reduced to K2To 1 + KiT l + LT (2.10v28) 1 + WT SD If there is no externally applied drift field, then y = 1 and Eq. (2o106) becomes K2 1 VsT I 2 (2.109) 1 +!,T + T) \r K2. 2 ___~ 2 ~(2.110 [1 + () T + 2 + T It is clear from Eq. (2.110) that the influence of diffusion is strongest, at high resistivity and becomes negligible in the limit; of very low resist:iv14,y. If we further define 1 b, (2.111) L;hen Eq. (2.110) may be written as

1 o 1 o / \ X ~ ~ ~ ~ WI I K 2 D D y= 0 (>E0 = -v5) Fig. 15. Plot of a vs. vd including the influence of' diffusion. K 2 2b 7 + 2b D + (2112) a is therefore symmetric on a logarithmic scale about CUT = b. If we now let a= loge CT (2.115a) ao = loge b (2.113b) then the analytic form of Eq. (2.112) will be explicitly displayed as

66 K2Tt b X cosh (a-ao) + b (2114) Figure 16 is a plot of cy vs. eT (assuming X constant) as given by Eq. (2.112). The numerical value taken for /D is that given by Hutson and White. Equation (2.78) for c vs. LT in the absence of diffusion is also plotted for comparison. 2.6 ACOUSTIC GAIN The possibility of using the acoustoelectric interaction to acheive traveling wave amplification was proposed by Weinreich in 1956. He proved that the application of a longitudinal electric field large enough to cause the electrons to drift faster than the propagation velocity of the acoustic wave could result in a negative attenuation coefficient. The experiments of interest at that time3 were concerned with the acoustoelectric effect in n-type germanium, an interaction so weak as to make experimental observation of acoustic attenuation extremely difficult, and studies of acoustic amplification were not attempted. Experimental observation of acoustic gain in cadmium sulfide was first 6 reported by Hutson, McFee, and White. Except for the additional apparatus required to impress a drift field across the crystal, the experimental arrangement they used was essentially the same as that outlined in Chapter I and discussed in detail in Chapter IV. These investigators observed maximum acoustic gains of about 18 db at 15 Mc and 38 db at 45 Mc for shear waves in a 7 mm crystal. These maxima were not sharply defined and occured at about 1050 v/cm for both frequencies. The agreement with theory was only qualitatively correct, and the experimentally determined plots of c vs. E did not have the nice inverse symmetry about the point y = O predicted by theory.

67 K21r ~II~~~~~~~~~~~~~~fillIIIIIII ~ ~ ~ i l I iil.1IT l IIIIIII I I1H HI1 1 1 z ~~ 1 111111111~~~~~1111111 I I IE M 11 I lllillllll11111I i1111 I I I I I HINIIIiii llll 1111 il l IIII I ilk!; lfil llIl i I IHT l I i l I l 1 II I I I [ EdI I ll 4i -iIi -i~I I I I; I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l I A l 111111 111111 IIIIIII I IL.,IIII IH fill I IIII II Hill Jill I llil i OIIOI I H iI iT I Oi I I i. i I.... Il IIll II ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iIr ~ c....~. wihot ifuso the, in f u eceo d iffsin W ill 11111 111 11 1~ I Il I 1 1111 1 II I l Mill I lill I III W. HIT I I Jill I ITH i i i II 1 11 I film Il [Jil I IN Il I II I H ll I AI I I life,111 I f il i 1i Ill 4 It 11111 I I I I I I V!Ili U1111111 I v~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ill IH I I II v 1 11I I ilI il l 1 11 0I IU H11 ia l 1 HU T Il T I,,ul 111 IlI llI II I, - IL III ifilI' IM I I T 1HI I 11111 il [ Il U lli l I III I HL 111 I a l 111 A a l I 111' i C I d llM II I I al i1 I -I ls. 111 11 TU II I i Il! l fim l Fi ll 1 6. I I I I I I I H 1 - ze II I L' A I I I Iin the influence of diffusi I U od IM nF.1111111.II I I 11 II 1 II HIII II

68 These authors also detected a buildup of noise at the output transducer each time an amplifying drift field (a drift field large enough for the production of acoustic gain) was applied across the crystal in the absence of ar! input signal. This buildup occurred over a period equal to several acoustic transit times within the crystal. They concluded that they were seeing an ultrasonic flux of amplified background noise. The round trip gain, including reflection losses, was greater than unity, so this flux buildup was able to occur over several passes as the noise bounced back and forth within the cryst-alo This phenomenon was further investigated by McFee, who was pursuing ~the 6 noise seen by Hutson, McFee, and White and also seeking an explanation for the current satura;ion reported by Smith. Smith had observed that the crystal current produced by an amplifying drift field always decayed to a steady-state saturation value which could be calculated by assuming that the carrier drift velocity was the same as the sound velocity in the crystal. McFee's results supported the theory of Hutson that the acoustoelectric current generated by the flux was responsible for the apparent current saturation. At the end of Section 20 4 we showed that the acoustoelectric current generated under amplifying conditions must be experimentally observed as a reduction in the drift current, produced by *the applied electric field. This is what finially l-imits the growth of ultrasonic flux amplified from the background noise in the crystalo The reverse acoustoelectric current grows in intensity with the ultrasonic flux until the carrier drift velocity matches the propagatirion velocity of the wave~ This is the saturation value of the crystal currernt, and at this point there is no further amplification of the ultrasonic flux

69 This reverse acoustoelectric current makes accurate experimental study of acoustic amplification extremely difficult, and it probably contributed to the asymmetry of the experimental c vs. E plots published by Hutson, McFee, and White. It might seem possible to avoid this complication by transmitting a relatively weak acoustic wave packet down the crystal during the first few microseconds that the drift field is turned on. This unfortunately does not entirely solve the problem. Rapid application of the drift field results in the generation of shock-excited noise within the crystal, and gradual application leaves too much time for current saturation to begin. This is not to say that such studies are impossible; they are merely beset with vexing experimental problems. It has been the practical experience of this investigator that the difficulties associated with the experimental study of the acoustoelectric effect in CdS seem to increase in direct proportion to the magnitude of the externally applied drift field. Fortunately the acoustoelectric interaction in CdS contains a wealth of experimental information for the case of no externally applied drift field. For example, under large input signal conditions the acoustoelectric current accompanying a propagating wave may be large enough to effectively modify rates of acoustic attenuation and acoustoelectric current production. It is this phenomenon which will occupy our attention in the remaining chapters, where we shall show that it is possible to predict the results of this interaction of the wave with itself and to verify these predictions by experiment.

CHAPTER III THE ACOUSTOELECTRIC FEEDBACK EFFECT We have proven (a) that acoustic attenuation will produce a direct current in an external wire joining the end faces of the crystal, and (b) that an externally applied drift field will influence acoustic attenuation within the crystal. We shall now show that these cannot be independent effects. In Section 1.2 we devised a simple equivalent circuit model (Fig. 8) which showed every region of the crystal crossed by the total acoustoelectric current J(t). According to this model the direct current crossing the ith crystal segment is carried in part by the local acoustoelectric interaction ji(t), and the remaining current J(t)-ji(t) is carried by the conduction electrons. The conduction electrons therefore have a local average drift velocity (vd)i = 4Ei, where Eiox is the potential drop across the ith segment. But this conduction electron drift is locally indistinguishable from the drift produced by an external electric field applied across the crystal to control acoustic attenuation. The acoustoelectric current must therefore influence acoustic attenuation. We have an acoustoelectric feedback effect. The concept of an acoustoelectric feedback effect (although not so named) was first proposed by Carleton, Kroger, and Prohofsky.9 These authors concluded that the acoustoelectric current would have no observable influence on 70

71 the overall acoustic attenuation, but their calculation was based on an oversimplified model of the acoustoelectric interaction and also assumed the special case of a cw (continuous wave) input wave train. In a later paper Prohofsky22 once more mentioned the influence of the acoustoelectric current on the attenuation, this time including dielectric relaxation effects. However, his calculations were again for the cw case, and he was interested only in using the local acoustoelectric field as an explanation for nonlinear mixing and the production of collective waves of second sound. The drift current density will vary from one region to another in the crystal, depending upon local rates of acoustoelectric current production. From +he continuity equation we know that linear divergences in the drift current must produce local accumulations or depletions of the free electrons. This must also affect acoustic attenuation. Thus there are two ways in which the acoustoelectric current may contribute to the acoustoelectric feedback effect; they are (1) through the local carrier drifts which transport the circulating current, and (2) through local changes in the conductivity resulting from divergences in these drifts, These are essentially large-signal effects. The drift currents and divergences depend upon local current densities which in turn are proportional to power dissipation from the acoustic wave. These electronic effects are therefore unobservable for acoustic waves of small amplitude. A complete large-signal theory for the acoustoelectric effect in CdS does not exist at present. Such a theory was attempted in 1964 by Beale,23 but he was forced to neglect the very important influence of space charge in order to

72 reach a solution, thus rendering his theory useless for application to the work described here. Lacking an adequate large-signal theory, it might be hoped that an extension of the small-signal theory to include the acoustoelectric feedback effect would be useful for predicting acoustoelectric effects at high acoustic energy densities. This is -the approach we shall take. In this chapter we shall treat the acoustoelectric feedback effect (including both of the electronic effects described above) as a perturbation -upon the small-signal theory of Chapter II, and we shall use the perturbed theory to compute the acoustoelectric currents generated'by acoustic waves of large amplitude. We recognize that there eventually comes a limit beyond which it must be hopeless to expect a small-signal -theory to make accurate predictions of largesignal effects. For example, our physical intuition tells us that at sufficiently large sound amplitudes all of the conduction electrons must be trapped within Athe piezoelectric potential wells of -the traveling wave and thus be constrained to move at the propagation velocity of'the wave. To the extent that *the total number of conduction electrons is conserved at such amplitudes, there is then an upper limit to the acoustoelectric current density, this limit being max J = qnovs Proper accountcing of t~his effect and the intermediate cases leading -to it must await -the development of a complete large-signal theory. In the absence of such a theory we shall show that. extensioon of tXhe small-signal theory too include the acou-isit;oelectric feedback effect does yield meaningful predictions which may be -verified by experimento

35l COMPUTTER PROGRAM OUTLINE We shall set as our goal the development of a computer program for the prediction of the total circulating current J(t) generated by an acoustic wave of large amplitude as it traverses the crystal~ With -this aim we see:cr task to be the derivation of an iterated series of computational steps which may eventually be assembled into a complete program for machine calculation of acoustoelectric current traces for direct comparison with experimental oscilloscope photographs~ In this section we present an outline of such a programr We schematically divide the crystal into N segments, each of length tx- The transit time across one such segment is 6t = Fx/vs. We then bring the i;raveling wave into the crystal, successively advancing it by increments tx at time intervals St. During each time interval 5t we do the following: a. Using local values of conductivity and electric field, compute an attenuation coefficient a in each segment of the crystal where the wave is passing. bo From the results of (a) compute the rate of acoustoelectric current; generation at each point on the waveo c. Compute the circulating current J(t)o Plot t his poin-t on a graph. d. From the results of (b) and (c) calculate the electric field intensit.y and local conductivity in each crystal segment for use in the comp-utation of step (a) -the next time through the program. e. Advance the wave by an amount bx while attenuating each portion of in according to the local atternuation coeff:icients computed in step (a)j

74 f. Return to step (a) and repeat the cycle. Certain practical considerations govern the selection of _t and bx. The experimental crystal is 7 mm long, resulting in a transit time of 4)0 *.sec fozr shear waves. The program makes N points available for plotting J(t) before the wave reaches the end of the crystal, so a choice of N = 80 will give a plot, of' 80 points at intervals ct =.05 lsec. These are certainly enough to accurately fix the shape of J(t). Although it might seem that greater theoretical accuracy should result; from a choice of even finer sputial division, this is nct the case. *The period of -the 30 Mc acoustic wave is.033 4sec, and it is this interval whJich effectively limits the experimental resolution and makes finer theoretioai resolution devoid of physical meaning. Except for the computations of steps (c) ard (d) we already have all of t'he formulae necessary for -the writing of this program. For step (a) we shall use the results of the small-signal theory, and for step (b) we need only;he Weinreich relation~ Step (c) req'uires that; we derive an expression for'the rciriulat ing currentl which includes the influence of the acoustoelectrric feedback effet.; we shall do this in Section 3.20 Several formulae are needed for step (d). These are most easily developed w:-i th I'he aid of an equivalent circuit model. Ir Chapter I we assumed a simple cryst.al model of fixed resistors and constant current generators to help -us L-nderstarnd the t ime-dependent, behavior of the circulating current produced by a uniformly attenuat;ed acoustic wave. This model is not adequate for our present purposes. In Sectiorn 3.3 we shall use the macroscopic electrcnic equations

75 of the crystal to derive an equivalent circui-t; mode l which is more acc;arat,e arid complete, and in Section 3.4 we shall use t-hls model to analyze the time-denerid eoLerit behavior of the elect,ric fields and curren'ts i-n the crystaLo 3.2 FINDING THE CIRCULATING CURRENT J(t) The electric fields and currents within,he crystal are locally relat;ed by the following set of four equations D = cE (7, 6D = q (nc rio) (5L) a64(c + jo at (qrj = -i (j + ) ( 5).C j cE qn cE, (5.4) where nc instantraneous local density of free electrons no quiescent; densi.ty of free electrons (for charge neural.ity) j -' local rate+ of acoust4oelectric current. generati.on j local drift (conduct'ion) current densi ty. We may co mbine Eqso (t1l) and (3o2) to eliminat-e D: aE e ae= qn' (5 5) where n n - n. (53 7i

76 is the local density of electrons in excess of that required for charge neutrality. Equation (3.3) is therefore (qn') = (jc + j) (37) We may substitute Eq. (3.5) into Eq. (3.7) to eliminate qn': ~2E _ c F a'xt a- -(jC + j) (58) or a[C E(x,t) + jc (x,t) + j(x,t)]: 0. (3.9) Satisfaction of Eq. (3.9) requires that the bracketed quantity be a function of the time only, that is C a E(x,t) + j c(x,t) + j(x,t) = i(t). (3,10) We identify J(t) as the circulating current. We may show for a short-circuit boundary condition that Eq. (3.10) gives exact;ly the same expression for the circulating current as did Eq. (1o16) of Chapter I. W7- prove this by integrating Eq. (3.10) over the length of the crystal: 1 L 1 L c. 1 L -(t):= L j(x,t) dx + L o j (xt) dx + C L t E(x,t) dx. 3.11) The short-circuit boundary conditicn requires that

77 f E(x,t) dx O, (012) so the last term of Eq. (3.11) must vanish~ We may show that the term in j also vanishes. From the constitutive Eq. (3.4) we have j - aE -aE + 4qn'E. (313) We may substitute Eq. (3.5) into Eq. (3.13), obtaining.c 1 E2 J E co- c (E2) (3 1i) so that r'L c 1 E L jE dx E dx 2 [E2(L) - E2(0) (3,1 The first right-hand term of Eq. (3.15) vanishes because of the short-circuit boundary condition. The remaining term will also vanish if we require that the number of free electrons in the crystal (at a given illumination) be conserved, ioe,: if L qnc dx qnO ( ) so that 1 L L /o qn' dx = (0351) Substitution of Eq. (3.5) into Eq. (3.16) gives 1 f L dEx = e [E(L) - E(0) ] = 0 (3 18) L ox L

Thus Eq. (3.15) vanishes, and Eq. (3o11) finally becomes 1 L J(t) = L fL j(x,t) dx, (519) which is identical with Eq. (1.16) of Chapter I. Therefore, to do step (c) of the computer program outline, we need only apply Eq. (3.19). 3-3 THE EQUIVALENT CIRCUIT MODEL OF THE CRYSTAL Equation (3.10) has a simple equivalent circuit representation. If we conceptually divide the crystal into N segments and within each segment repr-esent the local conductivity, capacitivity, and rate of acoustoelectric current generation by idealized lumped elements, then the equivalent circuit for the ith segment is just that of Fig. 17. That this is the correct representation may be easily shown. At either node we must have i c + j 5i = J 3') If each se-gment is of length 6x -N( 2) then the voltage drop Vi across the ith segment is Vi- Ex, (3 22) and the capacitance C. is Ci o - (3,23)

79 Thus the charge Qi stored on capacitance Ci is just Qi = CivC = E Ei (3.24) so Eq. (3.20) may be written as C i + jSc f + = J (3.25) which is just the iterated form of Eq. (3.10). The circuit model is indeed a valid representation of Eq. (3.10). If we use the constitutive Eq. (7.4) to eliminate the j ic, we have finally C Ei + iEi = j i ( 26) Ji 3 r I dQt dt Ci. 1 Fig. 17. Equivalent circuit model of the ith crystal segment.

80 3.4 SOLVING FOR THE ELECTRIC FIELDS AND CONDUCTIVITIES Having derived the equivalent circuit model and found an expression for the circulating current J(t), we are in principle now prepared to do step (d) of the computer program outline. It is now apparent that this computational step involves finding a solution to the set of simultaneous differential Eqs. (3.26) within each successive interval 6t. Unfortunately the calculation is more difficult; than is perhaps indicated by the deceptively simple appearance of the circuit model, since the "resistor" representing the local conluctivity ai of Eq. (3526) is also a variable. We may display the true complexity of the problem by using Eqs. (3.4), (3.5), and (3.6) to eliminate jc in Eq. (3.10), obtaining i + (o - 4C 6-) E = J(t) - j(x,t) Neither this nonlinear differential equation for the continuous crystal nor the equivalent set of N simultaneous nonlinear differential equations for the circuit model has an easy solution in closed form. But we are urier no obligation to seek a pFurely analytic solution. We are attacking the entire problem of wave propagation and current generation in the crystal by iterative calculation, and there is no reason why this particular iterative step cannot be further divided into easily handled computational parts. Le.t us do this. W- furth-r divide time interval Et into subintervals dt. with dt tSaken small enough that all of the variables of Eq. (3.26) change only slightly within each sub-interval. This will be the case if dt satisfies

81 sidt << 1. (3.28) for all i = 1,...N, where the local dielectric relaxation rate si is given by 1 i Si = - = (329) Within dt let us try changing only one variable at a time while holding all other variables constant. We may start by holding Ji and Gi constant and studying the variation in Ei. Working under this assumption, we multiply both sit' sides of Eq. (3.26) by e i, where t' is the time measured from the beginning of the subinterval dt. We obtain: ~ ~sit' 1 siit' (Ei + siEi) es = (J i) es The left side of this equation is a total differential, so the expression may be rewritten as d sit 1 st' (E e - Bi) (J - ji) e - (-j30) where Bi is the constant of integration. The solution for Ei(t') is therefore -s-tt' 1 -sit' t? sit If E.(t) BBi e m -- ji) e dterval we have t' = 0, and Eq. (35.31) reduces to Ei(O) = Bi ~ (3.32) We have assumed that all of the Ji are constant during sub-interval dt. From Eq. (3.19) we know that we may also take J(t) as constant. The factor (J - ji) may therefore be taken outside the integral, and we have

82 -sit -sit' 1! e Ei(t') = Ei(O) e + C (J - i i) si (33) Let us assume that the set Ei(O) satisfies the short-circuit boundary condition. We would like the new set Ei(dt) also to satisfy this condition, but Eq. (3.533) offers us no assurance that this will be the case. In general we may expect that the new set Ei(dt) will not meet the boundary condition, i.e., that N E(dt) = N i=l Ei(dt) O. (3.34) However, if we have been careful to take dt small enough, then each of the Ei(dt) will not be greatly different from the Ei(O), and as a result E(dt) will be smalls 1 We may therefore specify a new set of electric field values Ei(dt) = Ei(dt) - E(dt) (335) which will satisfy the boundary condition. It is this set that we shall take as representing the electric field distribution in the crystal at the end of the interval dt. Our calculation of the new electric field values is therefore a two step processo We first compute those changes due to local relaxation processes (fit st term of Eq. (3.33)) and due to the charge rfedistribution caused by divergences in the current (second term of Eq. (3.33)); then we adjust the new values of the field to conform to the short-circuit boundary condition6 We may now compute the local onductivity changes caused by the charge redistribution. Combining Eqs. (3.4), (355), and (3.6) we have

83 a(x) = oo - x (3.6) To use Eq. (3-36) we must know how to express the electric field gradient in terms applicable to the equivalent cir.uit model. One possibility is to write the symmetric expression - E aE.' =,1 i-l xi/. 2-x but tnis approach invites trouble. It divides the crystal into the two subsets of odd and even segments with the sub-sets coupled to each other only through the expression for the electric fi-ld gradient. The computer is in effect asked to solve the problem of two coupled large systems, a situation which of course does not exist in the real crystal. We may avoid this pitfall by writing the gradient as E'\ E E -8.x/i. x ( 3 v 37 ) The expression for the conductivity therefore becomes E. - E. i = ao E S - (3538) Notice that Eq. (3.38) may be evaluate using either the set Ei(dt) or the set E'(dt). The computed values for the local conductivity are unaffected by the adjustment of electric tl'ield terms necessitated by the short-circuit b undary condition.

84 355 REVISED OUTLINE We now have all of the formulae needed to do step (d) of the computer program outline. We must first divide ht into sub-intervals dt small enough to s- isfy Eq. (3p28)~ Let P be the number of these sub-intervals within 5t;. Then e-.cI time we encounter step (d). Pe must per-orm P times the computational sequence dI scribed by Eqs. (3.33), (3.3, (4) (3.3'5), and (3.,8), where each equation (except (3034)) is evaluated for all segments i = 1,...N before going on to the next. But; t~his program does not really abide by the rules we laid down in developing th iterated solution to Eq. (3.26): that within the sub-int- rval dt we woud cQeparately adust each variable of the equation while holding the remaining variables constant. The rate of acoustoelectric current generation ji(t) and t;he circulating curr-nt J t) are also variables of Eq. (3.26) and should therefore be recomputed as many times as are the electric field and thL. conductivity. W should therefore.nc-ude steps (a), (b), and (c) witshin the loop of com!;utai lons repeated P times over each in-tlerval St. This is not:; the.ame as incr- asing the number of spat5ial divisions N of the crystal, an idea we rejected in Section 31 because it produced a finer spatial resolution than was physically meaningful for the 30 Mc wave. Instead we have chosen to su -divide the transit time Ft of the segment 6x withc-ut, sudividing the segment itself, the sub-division of the time being necessitated by — he it5erated soluticn tz Eq. (3.26). There is another vemy practi al reasonb for sub-dividing only Tt and not Tx, if P is thLe number of sub- ihntervals dt wit;hin Tt, then we are increasing

85 by a factor P the number of computations we must do in a practical computer program, thereby multiplying by P the running time of such a program. For N = 80 and P = 1 a practical program will take about 10 sec of machine time. Thus N = 80 and P - 10 gives a running time of about 100 sec. If, however, we had taken N = 800 (and P = 1), then we would have increased by 102 the number of calculations, thereby raising the running time to about 1000 sec or 18 min of computer time for each set of data. We now present a revised outline for the computer program based on our new technique for sub-division of the interval 5t. I. Do parts A through E for p = 1,...P, where P is the number of subintervals into which bt is divided. A. Using local values of conductivity and electric field, compute an attenuation coefficient a in Mach segment of the crystal where the wave is passing. B. From the results or (A) comp:ute the rate of acoustoelectric current generation at each point on the wave. C. Compute the instantaneous circulating current J p(t). Do From the results of (B) and (C) calculate the electric field intensity and local conductivity in each crystal segment. E. Locally attenuate each portion of the wave by multiplying by the -revsdt factor e. where a is the local attenuation coefficient computed in (A)o II Do each of th- following steps in sequence. F. Compute the average circulating current over the inverval St.

P l P Piot the m:oint J(t). G. Advance th- wave b>- an amount bx. H. Increase the tim'- by- an 2a-:ount bt and transfer to I. 3.6 MACHINE COMPUTATIQ N The computer used was an IBM 7090 on The University of Michigan campus. The program was presentsd to th- machine in MAD (Michigan Algorithm Decoder), a compile.r language written at the University. It is not necessary to sift through all of the step-by-step details -of the machin calculation in'rder to be able to evaluate the computer program in its final form. It is important, however, that w- define and explain a few basic rul -s of machine computation. Th comuter treats names of variablb —s as 1 abeLs for locations in its memory, Stubscripteo` var able are labels for individual locations wit-,hin an array c: locations~ Thu. E,3 is a label "'or a _articular location in the memor;,, and ith- num- rical value of F2 is the number currently stored in that location. 25 Th-e computer memory is characterized by destructive read-in and nondestru t;ive read-out. By the Latter we mean that.ie may at any time obtain i'rom any memcry location the numbe. contained -within, and this "read-out" operationr doACs nrt change t-.e number stores or in any way jeopardize our ability to obtair. tat same number again when the occasion demands. By destructive read-in w icean that whenever we wish to nter a new nu-:ber at a given memory location,

the "read-in" operation causes the number previously stored at that memory location to be erased and therefore no longer accessible; the location is ineffect "set" to the new number. For example, consider the following machine instruction: E4 + C4 - E4 + 6 Here the machine is instructed to take the number stored in location C4, sulbtract from it the number stored in E4, add to the difference the number 6, and finally to store the result of the computation in location E4. If before the computation E4 had contained the number 3, and C4 had contained the number 4. then after the computation E4 would contain the number 7, but C4 would still contain the number 4. The computer can perform repeated operations over a running index, i.e., it can be instructed to successively do the computations E. Eo + 6 for all values of i ranging from i =, 2,...N In Section 357 we show the final form of the program as it is presented to the computer. In each step the computation to the right of the arrow is performed using numbers obtained from memory locations labeled by the variable names shown, and the result of the computation is then stored in the memory location indicated to the left of the arrow, the new number displacing (and therefore erasing) whatever number that location previously contained.

88 A succession of steps involving an iterated subscript is performed in its entirety before advancing the running index, for example in part A all str-ps I through 4 are performed first for i = t, then again for i = t-l, etc. The quantities contained in round brackets are calculated before the pr: gram is begun, and each round-bracketed quantity is then treated as a single constant within the program. For our purpose the Conductivity ai, the dielectric relaxation rate si, and tKe dielectric relaxation time Ti are merely different ways of writing a single physical quantity, and it would be redundant to calculate more than onof them. We shall therefore compute and use only si. 3o7 THE COMFIUTER PROGRAM I. Do parts A through E for p = I.,.. P, where P is the number of sub-intervals dt into which the interval St is divided.. Do steps 1 trough 4 for i = t, t -l,..t-T, where T is the length of the wave inside the crystal S i ( 1 eil + 1 + (s-2) Ei + - K2 Re F.:] 2. ~ i +) si ( s. 2 2. ai Si i -2 iri + - I i 35. ji - pCiWi 4. W. Wi e-Ci(6x/P) B. Compute the instantaneous circulating current JP N l. Up+ C? J.l

89 C. Do the following computation for all i = 1,.N -s.dt 1 1 - e sidt E. -E.e 1 + [j _ji] 1 1 C p 1 5i D. Compute the average electric field 1 N' +N il Ei E. Correct for the short-circuit boundary condition and find the dielectric relaxation rates by doing the following steps for all i = 1,,ooNo 1. E +. Ei - E 1'1 20 Si, SO ( ) [E ^ E - IIo Do each of the following steps F. Compute the average circulating current over the interval St. J(t) -+ p Jp, and plot this point on a graph. p=l P Go Advance the wave by doing the following steps l If the wave is not entirely within the crystal, then admit the next portion of the input; wave by setting Wo + A2(t), where A(t) is the amplitude of the input wave~ 2. Do the following for i - t, t-l,.. t-T W.+ Ww 1 i-1 H. Set t + + ~ bt, return to the beginning of the program, and repeat the entire sequence~

CHAPTER IV THE EXPERIMENT The experimental arrangement was similar to that used by Hutson, McFee, and White and also by Henricn with whom this author shared equipment, crystals, and an experimental setup. There was one important difference which will be discussed in Section 4.5. 4.1 THE ACOUSTIC ASSEVIBLY The basic experimental arrangement was outlined in Chapter I. The acoustic system (see Fig. 2) consisted of (a) a transmitting transducer, (b) a fused silica buffer, (c) the photoconducting 2dS crystal, (d) another fused silica buffer, and finally (e) a receiving transducer. The transducers ~ere 24 5/8 in. diameter quartz platelets y-cut for half-wave resonance at 27.5 Mc. The fused silica buffers were cylinders 1/2 in. in diameter and 1 in. long. The CdS crystal 5 was a cube 7 mm on a side oriented with the c-axis in the direction of the polarization vector of the propagating wave. There are two reasons for using the buffers: (1) They provide electrical insulation. This is absolutely necessary for those experiments requiring the application of a large drift field (around 1000 v) across the crystal for the purpose of studying ultrasonic gain. In other experiments (such as this one) not requiring a drift field it is still imperative that at least one end of the crystal not be grounded. But the inboard faces of the transducers must be grounded, thus the need for the insulating property. 90

91 (2) The buffers permit a convenient separation in time between the excitation of the input transducer and the occurrence of acoustoelectric phenomena in the CdS crystal, and again between these phenomena and the appearance of the acoustic wave at the receiving transducer. Capacitively-coupled electrical feedthrough effects are thus prevented from interfering with observations. This may be clarified by a study of the times involved. Typically the input transducer was excited by a 1.5-2.5 usec burst of 30 Mc rf. For shear waves and a 7 mm CdS crystal the relationships are: Length Velocity Transit Time CdS 7 mm 1.75 x 105 cm/sec 4.0 usec Fused silica 1 in. 5.8 x 105 cm/sec 6.4-1 Cisec All four end faces of the buffers were coated with a brushed-on layer of Hanovia liquid bright platinum. Individual wire leads were wrapped tightly around the ends of the buffers and brought into electrical contact with the end faces by painting on a slender ring of silver paste to bridge the space between them. The two outside faces of the buffers provided electrical contact to the inside (grounded) surfaces of the transmitting and receiving transducers (the transducers were not plated). The platinum layers on the insidle faces of the buffers enabled electrical contact to the ends of the CdS crystal which were coated with an evaporated layer of indium. This was necessary in order that the contacts be ohmic, that is possess a linear volt-amrlpere chara r 13, 26-28 acteristic.

92 The illumination source was a General Electric H-100-A4T 100 watt mercury discharge lamp. A Kodak No. 8 Wratten filter was used to pass only the 577-9 and 546 my lines of Hg. These lines are weakly absorbed in cadmium sulfide, thus guaranteeing homogeneous generation of photoelectrons throughout the crystal volume. Illumination intensity was adjusted by placing Wratten neutral density gelatin filters in front of the lamp. All elements of the acoustic system were bonded together with poly-alpha methyl styrene (Dow resin V-276-9), an extremely viscous liquid (4800 poise at 25~C) which may be readily softened with a heat gun. This substance was the best available bonding material which also afforded relative ease of repeated disassembly and reassembly of the components of the acoustic system. Unfortunately it provided a poor acoustic impedance match to the elements of the system; reflection coefficients were in the neighborhood of.8 at the interfaces between elements. This condition is illustrated in Fig. 18(a), where we see an oscilloscope trace recorded at the receiving transducer of the transmitted sound pulse and its many echoes. Figure 18(b) shows the acoustoelectric current produced over an extended period of time (50 Usec) by the transmitted sound and its echoes. The echoes were in themselves no problem, since observations of the acoustoelectric current were usually confined to that brief time interval during which the [directly transmitted] sound pulse (but none of its echoes) traversed the crystal, enough time being allowed between excitations of the input transducer to permit the echoes to die away. However, the poor transmission efficiency of the acoustic bonds was a cause for worry; there were two reasons: (1) The bonds were erratic in the

93 (a) Transmitted sound pulse and its echoes as detected at the output transducer. Time base = 10 4sec/cm. The crystal is in the dark. (b) Acoustoelectric current trace for a crystal resistivity of 1.35 megohm-cm. Time base = 5 isec/cm. Fig. 18. Illustrations demonstrating the high acoustic reflectivity of the bonds between elements of the acoustic assembly. In each photograph the sweep starts as excitation is applied to the input transducer.

94 sense that there was variation in their transmission efficiency from one assembly of the acoustic system to the next. Furthermore, since the mercury vapor lamp generated considerable heat, it was necessary to keep a stream of air continuously flowing across the acoustic assembly in order that the transmission efficiency of the bonds be reasonably stable during a single experimental run. (2) This poor transmission efficiency at first crippled chances for the observation of power-dependent acoustoelectric effects, since we were unable to deliver enough acoustic energy to the crystal to allow these effects to be seen. The solution to this problem is discussed in Section 4.3. The acoustic assembly was mounted in a sample holder (see Fig. 1), a copper box partitioned into three chambers. The CdS crystal occupied the central chamber. Electric contacts to the end faces of the crystal were brought out through ceramic insulators. The end chambers housed the transducers and the terminating impedances of the rf feed lines. The outboard faces of the transducers were electrically driven through spring-loaded polished brass buttons. The grounded partitions between chambers provided electrical screening and also offered some mechanical support to the acoustic assembly. The tunable inductors (visible in the end chambers in Fig. 1) were connected across the transducers to null their capacitance. In practice these adjustments had little influence on transducer conversion efficiency. A series of lenses and mirrors served to deliver the light to the crystal through two rectangular openings machined into opposite faces of the sample holder. Uniformity of the illumination over the two crystal faces receiving

95 the light was mapped with a Texas Instruments H-ll photo device. The sensitive surface of this device was approximately 1.5 mm in diameter. By adjusting the lenses, mirrors, and the position of the light source it was possible to make the illumination uniform to within 5% across each crystal face. The problems of illumination uniformity and the optical arrangement used in this experiment are treated extensively in Ref. 13. 4.2 ELECTRONICS In Fig. 19 there is presented a block diagram of the arrangement of electronic equipment used in the experiment. An Arenberg PG-650-C pulsed oscillator generated the excitation voltage for the input transducer. Excitation amplitude was controlled and adjusted by a pair of Hewlett-Pakcard model 55 attenuators inserted into the 50 Mc rf feed line. A Tektronix model 547 oscilloscope was used for display of acoustoelectric current waveforms. The oscilloscope triggered sweep was internally locked to the 60 cps power line frequency, and a gating signal from the oscilloscope was fed through a divider circuit to trigger the pulsed oscillator. The divider put out one sync pulse for each two it received, so the pulsed oscillator was triggered only on alternate sweeps of the electron beam across the oscilloscope face, thus providing a base line for the acoustoelectric current waveforms. The acoustoelectric current was usually measured in terms of the voltage drop across a 200 ohm resistor shunting the current output terminals of the sample holder. This particular resistance represented the best compromise between the need to approximate a short-circuit boundary condition and the re

96 rf power pulsed rf power AA amplifier power n~v~~V amoscillator _h 0-0 I 30~ input Isync attenuator sync divider w A_60~ sync ij0 i attenuator Fig. 19. Block diagram of the arrangement of electronic equipment used in the experiment.

97 quirement that there be enough output signal at most crystal resistivities to permit observation of the acoustoelectric current over a wide range of input acoustic energy densities. At very high crystal resistivity (r > 5 x 104 ohm-cm) it was necessary to substitute a 2000 ohm resistor in order that there be enough voltage drop for acoustoelectric current measurements in the small-signal region. When this resistor was first installed, capacitive loading by the oscilloscope and the connecting cable caused appreciable distortion of the trace. In Section 1.4 we showed that the rate of change of the observed current J(t) was limited by the "low pass filter" time constant T 1 e avs In our experimental crystal the maximum value of a was 65 db/cm. Thus Te was never less than.62 isec. Therefore, for accurate reproduction of acoustoelectric current waveforms (we neglect the 50 Mc ripple component), the current sampling circuit had to have a time constant short compared to 620 nanoseconds. Capacitive loading of the 2000 ohm resistor was limited by placing a cathode follower immediately adjacent to the current output terminals of the sample holder. The cathode follower was a single 6AK5 vacuum tube connected with the screen grid driven by the cathode. This arrangement shunted the crystal with a measured capacitance less then 7 pf, giving a time constant smaller than 14 nanoseconds. It is thus apparent that the measuring circuit should not distort the acoustoelectric current pulse. That it did not was

98 experimentally verified by substituting much smaller resistors and carefully comparing the observed current traces. All observations of acoustoelectric waveforms were taken directly from the oscilloscope screen as recorded by a Tektronix C-12 camera with a Polaroid roll film back. This permitted convenient recording on a single photograph (with appropriate adjustments of the oscilloscope vertical sensitivity switch) of acoustoelectric current traces taken at two or more different acoustic input power levels. The usefulness of this technique in displaying and studying power-dependent effects will be shown in Section 5.1. There were times during the experiment when it was necessary to view the transmitted acoustic wave as it appeared at the output transducer. At those times the output transducer was coupled through a Telonic TG-950 attenuator to a RHG model E5010 30 Mc broadband tuned amplifier, and the amplified signal was then available for display on the oscilloscope screen. 4.3 MODIFICATION FOR HIGH ACOUSTIC POWER At 30 Me the Arenberg pulsed oscillator was capable of a maximum output of about 90 v peak across 50 ohm. Trial experimentation revealed this level of excitation to be insufficient for observation of power-dependent acoustcelectric effects. This was in large part due to the poor transmission efficiency of the poly-alpha methyl styrene acoustic bonds (Section 4.1). Comparison of observed acoustoelectric current levels with results generated by machine computation verified that additional excitation was needed for the production of power-dependent effects. This was achieved by feeding the

99 pulsed oscillator output through a Heathkit HA-10 linear rf power amplifier (a 1 kw grounded-grid amplifier designed primarily for class-B amplification of radio amateur single sideband signals) to boost the excitation voltage to an amplitude of 350 v peak across 50 ohms, a gain of 12 db. Power-dependent effects were then clearly evident. The power amplifier did distort the envelope of the rf excitation pulse. This in no way upset the accuracy of the experiment, as it was only necessary to record the shape of the excitation envelope as it appeared at the input transducer in order to properly include it in the computer program.

CHAPTER V COMPARISON OF RESULTS As has been mentioned in other chapters, the experimental data were collected on a 7 mm CdS crystal oriented for use with shear waves. The shear mode was chosen because the relatively longer transit time (4.0 psec vs. 1.63 isec for the longitudinal mode) produces longer acoustoelectric current traces for study. Certain physical constants of the crystal are needed as input data for the computer program. They are = 315 cm/sec v/cm K2 =.0284 vs = 1.75 x 105 cm/sec Wc/oWD =.0393 Numerical values for the mobility and the electromechanical coupling coefficient in this experimental sample are due to Henrich. 3 The mobility was determined from Hall measurements on the crystal, and K2 was chosen by Henrich to give the best fit of a theoretical curve (computed from the small-signal theory of Hutson and White) to an experimental plot of attenuation as a function of' conductivity. These curves are reproduced in Fig. 20. Incidentally, the fact that the fitted curve tends to zero in the high resistivity limit indicates that nonelectronic losses in the crystal (see Section 1.5) are small enough to be neglected. 100

60 - 7mm SHEAR *=SOUND IN TOP o=SOUND IN BOTTOM + K2 40 -.01ol42, co o ~ = 15 (CM2/VOLT- SEC.) ++S I 20 0 - I(T? I(56 I(T5 I(4 I3 107 lo-6 10-5 104 10 0- (S-CM)' Fig. 20. Experimental data and fitted theoretical curve for attenuation as a function of crystal conductivity for our experimental sample. The two sets of data points are for the two directions of sound propagation in the crystal. The data and the fitted curve are the work of Henrich.13

102 Numerical values for vs and AD are those given by Hutson and White. No attempt has been made to adjust caD for the slight differences in mobility or ambient temperature in our particular experiment, since'such corrections would be too small to influence significantly the final form of the computergenerated acoustoelectric current traces. This crystal was one of several reported on by Henrich and was the only one of that group which had a mobility independent of photoconouctivity and at the same time did not exhibit a large first-order (linear) conductivity gradient in the direction of acoustic propagation. None of the crystals examined by Henrich was homogeneous in its photoconductivity (presumably due to a nonuniform distribution of impurities), but this one was at least a little better than the others. As we shall see later in the chapter, the inhomogeneity ultimately limits our ability to make quantitative comparisons between the predictions of the acoustoelectric feedback theory and the results of experiment. 5.1 DIRECT COMPARISON OF WAVEFORMS The most striking verification of the acoustoelectric feedback theory comes from a comparison of oscilloscope traces of the acoustoelectric current with computer-generated predictions of those traces. For this comparison the experimental data were recorded by a method especially suited to the display of power-dependent effects. At a given crystal resistivity acoustoelectric current traces produced for several different settings of the input attenuator were superimposed (by multiple exposure) on a single oscilloscope

103 photograph. Each of these traces was normalized with respect to the acoustic input power by compensating every adjustment of the input attenuator with an opposite adjustment of the oscilloscope vertical sensitivity switch. Had there been only a simple linear relationship between acoustoelectric current and input acoustic power (as is predicted by the basic small-signal theory), then the traces for the different levels of input acoustic energy would have coincided exactly, as in Fig. 6(b). The fact that they did not is clear evidence that power-dependent effects were indeed present. The 30 Mc ripple component (due to the boundary condition; Section 1.3) permits us to easily distinguish which trace is which on an oscilloscope photograph. The ripple is proportional to the input acoustic amplitude, whereas the acoustoelectric current is proportional to the input acoustic energy. Since all traces on a photograph are normalized with respect to the acoustic input power, the trace with the smaller ripple is the one produced by the higher energy input wave. (The amplitude of the ripple component is also proportional to the crystal conductivity, so traces generated at low crystal resistivity have a larger ripple than those generated at high.) Application of the full amplified rf excitation (described in Section 4.3) to the input transducer produced a maximum mechanical strain amplitude (at the peak of the wave) of about 6.4 x 10-5 at the input face of the CdS crystal. Th'is corresponds to a local acoustic energy aensity of 150 ergs/cm3 (the measurement of which will be discussed in Section 5.2), and all settings of the input attenuator are referred to this level. These settings serve as a convenient way of specifying the acoustic input energy used to produce the oscilloscope traces.

10k4 (a) Sound in "A" direction. (b) Sound in "B" direction. Fig. 21. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.80 x 105 ohm-cmr. Time base -.5 4sec/cm. Ext. resistor = 2000 ohm. Listed according to peak height, the traces on each photograph are: Trace Attenuator Vertical Scale Top -10 db.02 v/cm Center -j db.1 v/cm Bottom 0 db.2 v/cm

105 Fig. 22. Computer-generated acoustoelectric waveforms for resistivity of 1.80 x 105 ohm-cm. Time base =.5 psec/division. External resistor 2000 ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale - - - -10 db.02 v/division O db.2 v/division

106 (a) Sound in "A" direction (b) Sound in "B" direction. Fig. 23. Power-dependent acoustoelectric waveforms for crystal resistivity of 6.12 x 104 ohm-cm. Time base =.5 4sec/cm. External resistor = 2000 ohm. Listed according to peak height, the traces on each photograph are: Trace Attenuator Vertical Scale Top -10 db.05 v/cm Center - 4 db.2 v/cm Bottom 0 db.5 v/cm

107 -. / -10 db.05 v/division..0 db. 5 v/division ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale - - - -10 db.05 v/division 0 db.5 v/division

1o8 (a) Sound in "A" direction. (b) Sound in "B" direction Fig. 25. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.27 x 104 ohm-cm. Time base =.5 Ctsec/cm. Ext. resistor = 200 ohm. Listed according to peak height, the traces on each photograph are: Trace Attenuator Vertical Scale Top -l0 db.01 v/cm Center -3 db.05 v/cm Bottom 0 db.1 v/cm

109 ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale - - - -10 db.01 v/division 0 db.1 v/division ~~~-0d 0 livIs O d.1v/ivii\

110 (a) Sound in "A" direction. (b) Sound in "B" direction. Fig. 27. Power-dependent acoustoelectric waveforms for crystal resistivity of 4.46 x 103 ohm-cm. Time base =.5 [isec/cm. Ext. resistor = 200 ohm. Listed according to peak height, the traces on each photograph are: Trace Attenuator Vertical Scale Top -10 db.02 v/cm Bottom 0 db.2 v/cm

111 Fig. 28. Computer-generated acoustoelectric waveforms for resistivity of 4.46 x 103 ohm-cm. Time base =.5 psec/division. External resistor = 200 ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale - - - -10 db.02 v/division 0 db.2 v/division

112 (a) Sound in "A" direction. (b) Sound in "B" direction. Fig. 29. Power-dependent acoustoelectric waveforms for crystal resistivity of 2.08 x 103 ohm-cm. Time base =.5 isec/cm. Ext. resistor = 200 ohm. Listed according to peak height, the traces on each photograph are: TracM Attenuator Vertical Scale Top -10 db..01 v/cm Bottom 0 db.1 v/cm

113 Fig. 30. Computer-generated acoustoelectric waveforms for resistivity of 2.08 x 103 ohm-cm. Time base =.5,isec/division. External resistor = 200 ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale -10 db.01 v/division 0 db.1 v/division

114 (a) Souind in "A" direction. (b) Sound in "B" direction. Fig. 51. Power-dependent acoustoelectric waveforms for crystal resistivity of 1.01 x 100 ohm-cm. Time base =.5 psec/cm. Ext. resistor = 200 ohm. Listed according to peak height, the traces on each photograph are: Trace Atte nuator Vertical Scale Top -10 db.01 v/cm Bottom 0 db.1 v/cm

115 Fig. 32. Computer-generated acoustoelectric waveforms for resistivity of 1.01 x 103 ohm-cm. Time base =.5 isec/division. External resistor = 200 ohm. The traces shown are: Acoustic Trace Input Energy Vertical Scale -10 db.01 v/division 0 db.1 v/division

116 The computer-generated plots of the acoustoelectric current may be compared directly with the oscilloscope traces (Figs. 21-32). To aid the comparison, acoustic input energies for the computed traces are also given in db referred to 150 ergs/cm3 at the peak of the wave. For the high resistivity cases ( < 1) we immediately notice three areas of agreement between theory and experiment (Figs. 21-26). With increasing acoustic input energy we observe (1) a relative decrease in the peak height of the trace, (2) a slight delay in time of the occurrence of this peak, and (3) a slowed rate of decay of the current following the peak. For the low resistivity cases (- > 1) there is much less high power disUYT tortion of the acoustoelectric current trace (Figs. 27-32), at least with the acoustic power level achievable in this experiment. This is apparent in both the experimental and the computer-generated traces (the 4.5 x 103 ohm-cm experimental case of Fig. 27 needs special consideration and will be discussed later). In comparing other details of this generally diminished power-dependent effect, we see that both theory and experiment show a reduced relative peak height at high acoustic input power, but there otherwise is not the satisfying overall agreement evident in the high resistivity case. We must ask whether these are the results we expected. We know that the attenuation coefficient can be modified either through local changes in the crystal conductivity or by local electric fields. These two influences are best understood by studying their behavior on a plot of a vs. -HE (Fig. 33). On such a plot the conductivity determines the displacement of the peaks of maximum and minimum a from the point y = 0:

117 1 x 1 + displaced operating point, - y= 0 (a) The high resistivity case ( < 1). iX __W i __+_ displaced operating point Y= 0 (b) The low resistivity case (a > 1). Fig. 33. Illustration showing how the local electric field influences the attenuation coefficient by displacing the operating point.

118 ymax = +1 (5.1) min WT whereas the local electric field fixes the position of the operating point on this curve: = 1 + _. (5.2) VS The influence of the local electric field is relatively easy to predict. In general, the electric field will be largest in the region of highest acoustic energy density, and its sign will be positive (see Eq. (5.26)). It therefore displaces the operating point to the left on a plot of a vs. -HE. To estimate the influence of local conductivity changes we notice that, because the electric field is a maximum near the peak of the wave, the average value of its slope is zero in the immediate neighborhood of this maximum. From Eq. (3-5) it follows that the average conductivity change near the peak of the wave should be small, since qn' _ - e (5 3) ax Therefore, to first approximation any changes in the acoustic attenuation rate at the peak of the wave may be attributed to the influence of the electric field alone. At high crystal resistivity (1 < 1) the peak of maximum a falls to the CUT right of the vertical axis, and the local electric field produces a reduced attenuation coefficient in regiin: of high acoustic energy density (Fig. 533(a)). The effect is to some degree self-sustaining, since the high acoustic

119 energy density causes a local reduction in the acoustic attenuation rate which in turn helps maintain the high acoustic energy density. In Section 1.4 we developed an equivalent circuit model for predicting the detailed shape of the acoustoelectric current trace produced by a uniformly attenuated wave. According to the model a decrease in the attenuation rate will cause the peak of the circulating current to be reduced in amplitude and delayed in time and be followed by a more slowly decaying tail. For the case of high crystal resistivity we have shown that a propagating acoustic wave of large amplitude encounters a (nonuniformly) reduced attenuation rate. These generalized predictions for the changed shape of the observed current trace should therefore be valid. That they do apply is verified both by the output of the computer program and by the results of experiment. At lcw crystal resistivity (H4 > 1) the peak of maximum a falls to the left of the vertical axis (see Fig. 33(b)). Again the trend is for the local electric field to move the operating point to the left, but the operating point is now being pushed toward the peak. If it does not pass the peak (this is something about which we cannot accurately guess), then large amplitude regionrs of the wave should see an enhanced acoustic attenuation rate. According to the model of Section 1.4, this should cause the peak of the observed current to;ccur slightly earlier, be a little higher, and be followed by a more rapidly decaying tail. However, the increased dissipation will also more rapidly consume the acoustic energy, thereby diminishing the power-dependent effect and making changes in the acoustoelectric current trace less pronounced than they were for the high resistivity case.

120 The observable effect is much milder, as we have already noted. The computer-generated traces (Figs. 28, 30, and 32) also show a slightly earlier peak and an almost infinitesimally enhanced decay rate following the peak, but they predict a reduced rather than an increased peak height. Since our informal estimate was based on only a perfunctory consideration of electric field and conductivity behavior which is treated in detail by the computer program, the computer-generated results must of course take precedence. The computer program also accounts for the possibility that the peak and the operating point may cross. The important question is whether or not the computer-generated traces agree with the results of experiment. There is little agreement where there is a significant power-dependent effect, as may be seen by comparing the experimental and theoretical results of Figs. 27-32. In fact, the experimentally recorded traces for the two directions of propagation do not even look alike. These experimental discrepencies are at least partially explained by a peculiarity of our CdS crystal. In Section 2.3 we observed that the point of maximum height on a plot of a vs. WA (Fig. 13) occurs at the conductivity value for which the peak of maximum attenuation falls exactly on the vertical axis on a plot of a vs. Eo (Fig. 12). But the experimental plots of Fig. 20 are split at high conductivity (because of inhomogeneous crystal conductivity, as we shall explain shortly). It is therefore not surprising that the low resistivity experimental data recorded for the two directions of propagation agree neither with the theory nor with each other. This explains the serious discrepancy between theory and experiment for the 4.5 x 103 ohm-cm case (Figs. 27 and 28),

121 since this particular value of the conductivity (2.2 x 10-4 ohm/cm) falls between the two experimental peaks. 5.2 QUANTITATIVE COMPARISONS We have not yet said how we experimentally determine the input acoustic energy so that we may have this data for use in the computer program. Ideally we should know exactly the acoustic strain amplitude S(O,t) at the input face of the CdS crystal. The envelope shape of the acoustic wave packet is easily measured by placing a low-capacitance oscilloscope probe at the input transducer, but, since the transmission efficiency of the bonds is unknown, this is of value as a relative measurement only. For absolute determination of the acoustic input power we must resort to the following indirect technique: We choose a crystal resistivity for which there is good agreement between experimental and theoretical values of the attenuation coefficient as determined by the crystal conductivity. For our particular experimental sample (see Fig. 20) it is clear that this will have to be a resistivity for which 1/WT < 1. Then, by using the measured shape of the acoustic wave as input to the computer program and scaling it up and down over a wide range of closely spaced input power levels, we generate a family of theoretical acoustoelectric current plots. These may be compared with an experimentally-generated family of oscilloscope traces produced at the same crystal resistivity. The computer plots may then be paired with the oscilloscope traces by equating peak heights. The pairing must be done in the smallsignal region where there is a linear relationship between peak current and

122 acoustic input power. Then, if all goes well, the matching should be consistent into the large-signal region where power-dependent effects occur. The matching is not exact, as may be seen by careful study of experimental and computer-generated traces for the high-resistivity case (Figs. 2126). In general the observed power-dependent effect is stronger than that predicted by the computer program, that is the difference in peak heights is greater- for the observed than for the computed traces. There is therefore a discrepancy between the predictions of the acoustoelectric feedback theory and the results of experiment. Let us first criticize the theory. Perhaps the most serious deficiency of the acoustoelectric feedback theory is a failure to account for the influence of the second harmonic component of the acoustoelectric current (see Section 2.4) on the local charge densities and velocities which control acoustic attenuation. Because of its large amplitude (equal to the local dc value of the acoustoelectzric current) this influence must be considerable. But it would have been extremely difficult to include this effect in the uccustoelectric feedback theory, since it lies outside the scope of the smallsignal theory of Hutson and White on which it is based. The small-signal theory treats the influence of'!dc" electric fields and conductivities on the local attenuation experienced by a propagating acoustic wave and in its present form cannot account for the interaction of the wave with oscillatory currents of frequency 2cL. We would have been surprised indeed had our perturbation extension of the small-signal theory been capable of accounting entirely for the changed

123 shape of the acoustoelectric current trace generated under large-signal conditions. At the beginning of Chapter III we claimed that, in the absence of an adequate large-signal theory, the acoustoelectric feedback theory would make meaningful predictions of large-signal effects. This much we have certainly done, as is shown by the very gratifying qualitative agreement between the experimental photographs and the computer-generated traces for the high resistivity case (the only case where we can also get low power agreement). But it would be foolish to insist that all large-signal distortions of the observed current trace could be explained in terms of the influence of local "dcT" electric fields and charge densities on the small-signal interaction. What we can claim is that this influence most assuredly contributes heavily to the changed shape of the observed current trace, as we have demonstrated blp the experimental and theoretical results of Figs. 21-26. Now let us criticize the experiment. For the low resistivity case we have seen that the observed current traces recorded for opposite directions of propagation at a given crystal iesistivity not only do not match the computer-generated plots; they also do not match each other. For the high resistivity case we already know that. there is some disagreement between computed and measured heights of the trace under large-signal conditions. What about other cetails of the changed shape of acoustoelectric current traces generated by acoustic waves of large amplitude? The most meaningful comparison of shapes is between slopes of that part of the acoustoelectric current trace produced while the wave is entirely within the crystal. It is this portion of the trace which should be a perfect

124 decaying exponential under the ideal small-signal condition of uniform acoustic attenuation. The method of comparing these slopes need not be elaborate or sophisticated, providing we consistently use the same technique for both the observed and the computed waveforms, e.g., we can simply measure the height of each trace at the same two points in time and then compute an'average" attenuation coefficient from the formula adb - 1 10 logl j(tl) (5.4) (t2-tl)v5 j(t2) The results of such a comparison are plotted in Fig. 34 for two different crystal resistivities. It is apparent that the experimental and theoretical plots show the proper qualitative behavior, i.e., they all tend toward a reduced acoustic attenuation rate with increasing input acoustic power, but it is disturbing to observe the poor quantitative agreement. For neither resistivity do the experimental plots (for the two directions of propagation) coincide with the theoretical curve or with each other. This discrepancy extends even into the low acoustic power (small-signal)region where power-dependent effects must vanish. In fact, the two cases shown were chosen for illustration b1ecause their low power agreement was better than that attainable at other resistivities. This low power disagreement is clear evidence that the attenuation is not uniform throughout the crystal and is a direct consequence of inhomogeneity in thi- crystal conductivity. This inhomogeneity may be "mapped" by studying tie acoustoelectric current traces generated <- short, low energy acoustic wave packets. Figure ~5 shows traces generated by a low power wave packet prcoagating in either direc

A, III i ii C H —I ADO O m - titi~ m m X 110 I( 11t1 11111111111114t1110 14t14X1Xtg t:1 1 1l! X ~1 Iif I I -.1tilm, itI il A0 I(I' i ~~GD oO O - 0 -,c~~~~~ ~I T TITT

126 _ — mmlil!roll (a) Crystal resistivity 4.4 x 106 ohm-cm. (b) Crystal resistivity = 1.0 x 106 ohm-cm. Fig. 35. Acoustoelectric current traces generated by a 1.5 4sec input wave. Traces shown are for both directions of propagation. Vertical scale =.002 v/cm. Time base =.5 4sec/cm. Ext. resistor = 2000 ohm. The input attenuator was adjusted at each crystal resistivity to keep the traces on scale.

127 (c) Crystal resistivity = 2.2 x 105 ohm-cm. (d) Crystal resistivity = 7.0 x 104 ohm-cm.

128 tion in the crystal. These rough and irregular traces clearly illustrate hiow much our experimental crystal deviates from the ideal homogeneous model we implicitly assumed while writing the computer program. Computer-predicted acoustoelectric current traces based on a single external measurement of the crystal resistance cannot hope to agree with this data. Under small-signal conditions the computer program simulating a homogeneous crystal will generate an acoustoelectric current trace identical to that predicted by the model of Section 1.4, but traces experimentally recorded from an inhomogeneous crystal will differ in both height and shape from that ideal result. Quantitative comparisons of height and shape for high acoustic input power therefore become impossibly difficult, particularly since the calibration of acoustic input power levels by pairing of experimental and theoretical traces at low power levels cannot be trusted. Quantitative comparison of the slopes of the tails of these traces is also hindered, since it is not possible to ludge how much of the slipe is truly a measure of the attenuation coefficient and how much merely reflects a conductivity gradient. How can we minimize these difficulties? We can tbegin by selecting an experimental sample which is as nearly homogeneous as is possible to obtain. This selection was discussea at the beginning of the chapter. We can also smooth out the current traces by using longer acoustic wave trains, thereby physically integrating out some of the roughness due to the irregular conductivity. This has been done for data presented here in support of the acoustoelectric feedback theory. Such a technique certainly gives better

129 looking experimental results but unfortunately still does not allow quantitative comparison between a computer program which assumes homogeneity and a real crystal which is not so simple. We may ask whether it is possible to develop a program for treating the inhomogeneous crystal. One approach would be to enter local conductivity values point-by-point into our present one-dimensional program, providing it were possible to collect such information from photographs of the acoustoelectric current traces generated by very short acoustic wave packets. This scheme would be successful if the conductivity variation occurred only in the direction of the wave's propagation vector, but the irregularity of the traces displayed in Fig. 35 leads us to suspect that this is not the case, and we at present have neither the theory nor the experimental technique to treat an irregular, three-dimensional conductivity variation. Furthermore, the fact that the experimental plots of a vs. a in Fig. 20 agree at low conductivity but split at high conductivity indicates that the crystal's inhomogeneity pattern may shift with changing photoconductivity, so that a "map" made at one conductivity would not be correct for another. In any event, rather than burden this computer program with the corrections and adjustments needed to compensate for an unsatisfactory experimental situation, the better approach would be to seek future improvement in the quality of experimental samples available for study. b.3 DISTORTION OF THE WAVE PACKET All of our discussion so far has been concerned with the detailed shape

130 of the observed current trace with little attention being given to the acoustic wave packet itself. What observable changes take place in the wave packet as a result of the acoustoelectric feedback effect? Is it possible to make predictions about these changes which may be verified by experiment? In Section 5.1 we showed that qualitative interpretation of the altered shape of the observed current trace was relatively straightforward in the case of high resistivity, that is, that the local electric field produces a locally diminished acoustic attenuation coefficient in regions of high acoustic energy density. Different regions of the wave packet will therefore experience different acoustic attenuation rates. The sign of the effect is unambiguous for the high resistivity case; regions of higher acoustic energy density will suffer less attenuation than those of lower. The observable result will be distortion of the wave packet envelope, with irregularities in the envelope shape being emphasized so that a high region becomes relatively higher still. The effect is dramatically illustrated in the experimental photographs of Fig. 36. Here we see signals taken from the output transducer which resulted from input signals of identical shape but differing in amplitude by 20 db. The output signals have been normalized with respect to the input acoustic amplitudes, and the larger signal is that produced by the high power input signal. We can clearly see that the irregularities in the wave packet envelope have been greatly emphasized for the high power case. For the photograph taken at a crystal resistivity of 6.21 x 104 ohm-cm, that region of the wave packet which initially had the largest amplitude experiences about 6 db

131 less total attenuation for the high power case than for the low power case, whereas a region of initially small amplitude suffers about the same attenuation in both cases. The distortion of the wave packet is correctly predicted by the computer program, the results o f which are shown in Fig. 37 for a crystal resistivity of 6.21 x 104 ohm-cm. The wave packet unfortunately suffers some distortion in traversing the acoustic system, even when the crystal is in the dark (condition of no acoustoelectric interaction), so accurate quantitative comparison of output wave shapes is not possible, but the strong qualitative agreement gives additional support to the validity of our acoustoelectric feedback theory. -. 4 SUMMARY We have shown that it is possible to make meaningful predictions of large-signal distortion -f the acoustoelectric current waveform by extending the small-signal theory of Hutson and White to account for the self-interaczion of a decaying acoustic wave. This self-interaction occurs because the local charge velocities and changes in charge density brought about by the acoustoelectric interaction must also influence the attenuation, just as if these conditions were imposed upon the carriers from outside the crystal. The extended small-signal theory does not completely describe the interaction of a large-amplitude acoustic wave with the electrons (as was discussed in Section 5.2), so we do not expect that such a theoretical treatment can completely explain the changed shape of the acotstoelectric current trace generated under this condition. Accordingly, we have not attempted a detailed

132 (a) Crystal resistivity = 6.21 x 104 ohm-cm. (b) Crystal resistivity = 1.80 x 105 ohm-cm. Fig. 36. Comparison of output signals produced for input excitation levels of 0 and -20 db. In each case the larger amplitude output signal corresponds to the higher power input wave. On both photographs the output signals have been normalized with respect to the input wave, that is the vertical scale for the high power case is 10x that for the low power case.

I * I ** I I * * I I * I I * I. 4. 1 I * I I ** I I I I * I I I I * I I *.... I 1~~~~~~~~~~~... I I.. I! I I I I *.. * I I *., 9 I I. I I *. I I * * I I * * * * *.I I *.. I I *44...4.. I I... I I * I 1 4'C I I I H I I 1 * 1 I * I I I I I I * 1 I I.. I I I 1 * 1 I * I I I I * * I 1 * 1 I * I. I I I. * I I I I * I I I I *. I I * 1 * * * I 0 —- ~-4-1-4' - 2 -+IIIIIIII10 ~ i ~ 2...+ TIME IN MICROSECONDS Fig. 37. Computer-calculated output signal produced for input excitation levels of 0 and -20 db. The vertical scale for the high power case is 10x that for the lac power case. Crystal resistivity = 6.21 x 104 ohm-cm.

134 quantitative comparison. However, the very satisfying qualitative agreement obtained at high crystal resistivity (- < 1) indicates that this self-interaction phenomenon must contribute heavily to the changed attenuation, distorted wave packet envelope, and altered acoustoelectric current trace which are experimentally observed under large-signal conditions.

APPENDIX REDUCTION TO THE ONE-DIMENSIONAL PROBLEM We shall sketch here the derivation of expressions for the electric fields which may accompany a propagating acoustic plane wave of any mode in a threedimensional piezoelectric conducting medium. Our approach is basically that used by Hutson and White when they re-derived with some corrections and simplifying assumptions the results of Kyame.5' In his attack upon the problem Kyame derivedthefifth-order secular determinant for the phase velocities of acoustoelectromagnetic waves propagating in a conductive piezoelectric crystal. The solutions of this determinant correspond to two transverse electromagnetic waves traveling at nearly the speed of light in the medium and three acoustic waves traveling at approximately the speed of sound, where, depending upon the direction of propagation and the piezoelectric tensor, the acoustic waves could be accompanied by longitudinal electric fields which effectively modify the elastic constants of the medium. Kyame then showed that to a good approximation the secular determinant may be separated into a third-order acoustic wave determinant and a second-order electromagnetic wave determinant, with the two sub-determinants weakly coupled to each other through matrix elements of order v/c e; where v is the velocity of sound and c is the velocity of light in the crystal, and e is the appropriate component of the piezoelectric tensor. He thereby demonstrated that by ignoring the influence of the electromagnetic waves in his solution of the acoustic wave determinant, he was neglecting corrections to the acoustic phase velocity of order v/c e. 135

136 Our goal is less ambitious. We wish merely to show that of all possible polarizations of the electric field which may accompany a propagating acoustic wave, only the longitudinal component can be large enough to significantly influence acoustic attenuation. We shall derive this result under the same assumption used for the small-signal theory development of Chapter II, namely that the attenuation over one wavelength of the sound is so small, that for the purpose of computing local relationships among the components of the traveling wave we are justified in taking the propagation vector as being entirely real. The derivation then goes as follows: Let xli, x2, and x3 be orthogonal axes arbitrarily oriented with respect to the crystal axes and consider the propagation of plane waves in the xl direction. Under adiabatic conditions the piezoelectric equations of state for the medium are T C. S e E (A.1) Tij =ijkmSkm - ekijEk k = ekijSij +i ikEi' (A.2) where repeated indicies indicate summation. Here cijkm = short-circuit adiabatic elastic tensor e adiabatic piezoelectric tensor kij elik = clamped adiabatic dielectric permitivity tensor The constitutive equations for the medium are B = oHi (A.3)

137 k = CikEi (A.4) The electromagnetic field quantities must also satisfy the Maxwell equations. The equations of particular interest to this derivation are aDk (curl H)k = at k (A.) aB (curl Ek - at (Ao) We shall also define the piezoelectromotive field vector e k = Sij'/k approximates the kth component of the total piezoelectric driving field produced by all directions and polarizations of acoustic plane waves propagating in the medium (for k = 1 and i or j = 1 the relationship is exact). The second piezoelectric equation of state (Eq. (A.2)) )ray therefore be written as D C." + E (A,8) Dk -11'-k + ikEi. If we substitute Eqs. (A.4) and (A.8) into Eq. (A.5) we obtain 6Sk a {curl H}k = -ell a + (elk a- + ik) Ei' (A.9) We are interested only in the propagation of plane waves sinusoidal in the time. We may therefore everywhere replace 6/dt with -iou, with the result that (curlH k H -il (- etI k ik ) (A+ CD )E)

138 where the relaxed dielectric permitivity tensor ek is defined by 1k ik (A ik ik -i (All) Equation (A.10) may be solved for two cases. For the first case we take k = 1; we are interested in the component along the propagation vector of the plane wave. The left side of Eq. (A.10) vanishes, leaving: ElEl= c11 - eEp, (A.12) where p = 2, 3 labels the two transverse modes of the electric field polarization. If we solve for El, the result is clll - c' E El = ~ p1 p (A.l3) For the second case we take k = p - 2, 3. The left side of Eq. (A.10) does not vanish. Following the usual program for the treatment of electromagnetic plane wave propagation we substitute Eq. (A.3) into Eq. (A.6) and take the curl: (curl curl E} {curl H} (A.14) We then substitute Eq. (Ao.10) into Eq. (A.14): (cip E.) (A.l5) (curl curl E = (-l + ipEi) (A5 Now

139 (curl curl E)i = (grad(div E))i - V2E (A.16) 1 axi ax:. ax, ax' axk m (A.17) For a plane wave propagating in the xl direction, only d/6xl gives a result different from zero. (curl curl E)p therefore collapses to a2E [curl curl E)p = - = ()2 E, (A.18) where v is the phase velocity of the acoustic wave (in the xl direction). Thus Eq. (Aol15) becomes Ep= - v2( flip + ipEi) (A19) This may be rewritten as p (c)2 (- Ei ), (A.20) p p 111 where c (A. 21) is the velocity of an electromagnetic wave propagating in the same direction as the acoustic plane wave. If the electric fields Ek accompanying the wave are to produce observable acoustic attenuation, they must be nearly as large as the piezoelectromotive fields which are generated in the piezoelectrically active modes of propagation. Although Eq. (A.20) is not truly a solution for Ep (it also appears on

140 the right as one component of Ei), it is clear that the magnitude of Ep is down by a factor of (v/c)2 from values of the transverse piezoelectromotive field componentp and the longitudinal electric field El. Thus it is apparent that the transverse electric fields are far too weak to produce observable acoustic attenuation. By contrast Eq. (A.13) tells us that the longitudinal electric field component E1 is approximately equal to the longitudinal piezoelectromotive field component l. (We shall drop the insignificant transverse fields Ep from Eq. (A.l3)). If the direction and mode of acoustic wave propagation are such as to generate a significant longitudinal piezoelectromotive field, then Eq. (A.13) may be written E =' (A,22) 1 + 1+- iWT where the dielectric relaxation time T is defined by T = (A 23) Equation (A.22) may be compared with Eq. (2.59) of Chapter II. Thus we see that the three-dimensional problem is essentially reduced to a problem in one-dimension, that being along the propagation vector of the acoustic wave.

FOOTNOTES AND REFERENCES 1. G. Weinreich, Phys. Rev. 104, 321 (1956). 2. G. Weinreich and H. G. White, Phys. Rev. 106, 1104 (1957). 3. G. Weinreich, T. M. Sanders, and H. G. White, Phys. Rev. 114, 33 (1959). 4. A. R. Hutson, Phys. Rev. Letters 4, 505 (1960). 5. A. R. Hutson and D. L. White, J. Appl. Phys. 33, 40 (1962). 6. A. R. Hutson, J. H. McFee, and D. L. White, Phys. Rev. Letters ], 237 (1961). 7. D. L. White, J. Appl. Phys. 33, 2547 (1962). 8. I. Uchida, T. Ishiguro, Y. Sasaki, and T. Suzuki, J. Phys. Soc. Japan 19, 674 (1964). 9. H. R. Carleton, H. Kroger, and E. W. Prohofsky, Proc. IEEE 53, 1452 (1965). This reference contains an extensive bibliography of recent work on acoustoelectric ultrasonic amplification and current saturation. 10. R. W. Smith, Phys. Rev. Letters 9, 87 (1962). 11. A. R. Hutson, Phys. Rev. Letters 9, 296 (1962). 12o J. H. McFee, J. Appl. Phys. 35, 465 (1964). 131 V. E. Henrich, Ph.D. thesis, Univ. of Mich. (1967). This reference contains an extensive discussion of experimental technique with cadmium sulfide and the acoustoelectric effect. 14. G. Weinreich, Phys. Rev. 107, 317 (1957). 15. Jo J. Kyame, J. Acoust. Soc. Am. 21, 159 (1949). 16. J. J. Kyame, J. Acoust. Soc. Am. 26, 990 (1954). 17. See for example W. P. Mason, Physical Acoustics and the Properties of Solids (D. van Nostrand Co., Inc., Princeton, New Jersey 1958), Appendix Ao3 o 18. Ibid., Appendix A.2. 141

142 19. D. Berlincourt, H. Jaffe, and L. R. Shiozawa, Phys. Rev. 129, 1009 (1963). 20. Our time convention is e; hence the minus sign attached to i. 21. B. Tell, Phys. Rev. 136, A772 (1964). 22. E W. Prohofsky, J. Appl. Phys. 37, 4729 (1966). 23. J.R.A. Beale, Phys. Rev. 135, A1761 (1964). 24. For use at an operating frequency of 30 Mc the transducers are actually cut for half-wave resonance at 29.2 Mc, this being the condition for maximum acoustic radiation when they are lightly loaded. For a further explanation see W. P. Mason, Physical Acoustics and the Properties of Solids, page 99. 25. This crystal was given to us by Dr. B. Tell. 26. R. W. Smith, Physo Rev. 2_, 1525 (1955). 27'. A. M. Goodman, J. Appl. Phys. 35, 573 (1964). 28. F. A. Kroger, G. Diemer, and H. A. Klasens, Phys. Rev. 103, 279 (1956).

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