transverse wave interaction and acoustic amplification in piezoelectric semiconductors

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Physica 132B (1985) 323-336 North-Holland, Amsterdam TRANSVERSE WAVE INTERACTION AND ACOUSTIC AMPLIFICATION IN PIEZOELECTRIC SEMICONDUCTORS Manvir S. KUSHWAHA Department of Physics, Banaras Hindu University, Varanasi-221iXI.5, India Received 17 January 1985 The mechanism of the acoustic wave amplification in semiconductors involving piezoelectric coupling and deformation potential coupling is investigated using small signal approximation theory. The effect of the ratio of the effective masses of the charge carriers on the transverse phonon-helicon interaction is incorporated. The dispersion relation for the coupled mode propagation is used to derive expressions for the gain constants, complex effective elastic constant, and the relative change in the velocity of the acoustic wave. The dispersion relation for the coupled mode propagation is computerized to obtain the wave frequency as a function of the wave vector. The results are discussed in the light of earlier results reported by various authors. It is shown that the amplification in any drifted plasma with a drift velocity less than the acoustic wave velocity does not exist. The possibility of resonant (collisionless interaction) amplification is explored. 1. Introduction The discovery of acoustic wave amplification in CdS by Hutson et al. [l], originated a growing interest in the study and development of solid state amplifiers. The early work in this area was reviewed by Solymar [2]. It is generally believed that electrons amplify the acoustic wave if their drift velocity was made to exceed the velocity of phonons (acoustic wave). White [3] had shown that the necessary condition for amplification is that the ordered motion of the carriers should be in the direction of the electric field in the semiconductor. In accordance with this criterion, the electrons drifting as a result of the applied electric field can not interact with the acoustic waves which give rise to transverse electric fields. However, Browne [4] suggested that such interactions would become possible in the presence of a sufficiently large axial magnetic field which causes the ordered motion of the electrons to become helical. Following this work a number of theoretical studies have been carried out on the transverse wave interaction in piezoelectric semiconductors [5-121. The mathematically derived expressions and the physical interpretations given by these authors have aroused a considerable controversy regarding the optimum values of the drift velocity and the existence of resonant (collisionless) modes leading to the growing waves. The transverse wave interaction giving rise to the acoustic wave amplification was for the first time studied by Browne [4]. He showed that resonant modes (interaction effects in the absence of collisions) could lead to growing waves in the transverse interaction and that amplification in any drifted plasma could be obtained even with a drift velocity appreciably below the acoustic velocity. Solymar and Lashmore-Davies [5] made a theoretical analysis of this interaction in the piezoelectric semiconductors but obtained results in disagreement with the above-mentioned results of Browne [4]. Subsequently, Singh [7] extended the interaction mechanism of Solymar and Lashmore-Davies [5] by including the effect of charge density perturbations and showed that resonant modes may lead to a growing wave as shown by Browne [4]. It is, however, noteworthy that the expression for the gain constant obtained by Singh [7] (his eq. (6)) agreed with that of Solymar and Lashmore-Davies [5]. In yet another paper Singh [9], using the effective elastic constant approach, has derived an expression for the gain constant (his eq. (13)) which clearly reveals that amplification is possible even when the drift velocity is smaller 0378-4363/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Page 1: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

Physica 132B (1985) 323-336

North-Holland, Amsterdam

TRANSVERSE WAVE INTERACTION AND ACOUSTIC AMPLIFICATION IN PIEZOELECTRIC SEMICONDUCTORS

Manvir S. KUSHWAHA

Department of Physics, Banaras Hindu University, Varanasi-221iXI.5, India

Received 17 January 1985

The mechanism of the acoustic wave amplification in semiconductors involving piezoelectric coupling and deformation

potential coupling is investigated using small signal approximation theory. The effect of the ratio of the effective masses of

the charge carriers on the transverse phonon-helicon interaction is incorporated. The dispersion relation for the coupled

mode propagation is used to derive expressions for the gain constants, complex effective elastic constant, and the relative change in the velocity of the acoustic wave. The dispersion relation for the coupled mode propagation is computerized to

obtain the wave frequency as a function of the wave vector. The results are discussed in the light of earlier results reported by various authors. It is shown that the amplification in any drifted plasma with a drift velocity less than the acoustic wave

velocity does not exist. The possibility of resonant (collisionless interaction) amplification is explored.

1. Introduction

The discovery of acoustic wave amplification in CdS by Hutson et al. [l], originated a growing

interest in the study and development of solid state amplifiers. The early work in this area was reviewed by Solymar [2]. It is generally believed that electrons amplify the acoustic wave if their drift velocity was made to exceed the velocity of phonons (acoustic wave). White [3] had shown that the necessary condition for amplification is that the ordered motion of the carriers should be in the direction of the electric field in the semiconductor. In accordance with this criterion, the electrons drifting as a result of the applied electric field can not interact with the acoustic waves which give rise to transverse electric fields. However, Browne [4] suggested that such interactions would become possible in the presence of a sufficiently large axial magnetic field which causes the ordered motion of the electrons to become helical. Following this work a number of theoretical studies have been carried out on the transverse wave interaction in piezoelectric semiconductors [5-121. The mathematically derived expressions and the physical interpretations given by these authors have aroused a considerable controversy regarding the optimum values of the drift velocity and the existence of resonant (collisionless) modes leading to the growing waves.

The transverse wave interaction giving rise to the acoustic wave amplification was for the first time

studied by Browne [4]. He showed that resonant modes (interaction effects in the absence of collisions) could lead to growing waves in the transverse interaction and that amplification in any drifted plasma could be obtained even with a drift velocity appreciably below the acoustic velocity. Solymar and Lashmore-Davies [5] made a theoretical analysis of this interaction in the piezoelectric semiconductors but obtained results in disagreement with the above-mentioned results of Browne [4]. Subsequently, Singh [7] extended the interaction mechanism of Solymar and Lashmore-Davies [5] by including the effect of charge density perturbations and showed that resonant modes may lead to a growing wave as shown by Browne [4]. It is, however, noteworthy that the expression for the gain constant obtained by Singh [7] (his eq. (6)) agreed with that of Solymar and Lashmore-Davies [5]. In yet another paper Singh [9], using the effective elastic constant approach, has derived an expression for the gain constant (his eq. (13)) which clearly reveals that amplification is possible even when the drift velocity is smaller

0378-4363/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Page 2: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

324 M.S. Kush waha / Acoustics in piezoelectric semiconductors

than the sound velocity. Effective transverse phonon-helicon interaction in piezoelectric semiconductors was also studied by Hsieh [lo] but the author’s emphasis was on the amplification of the helicons. Chandra and Verma [ll] have also studied the transverse phonon-helicon interaction and derived an expression for the gain constant (their eq. (18)) which again leads to the conclusion that a drift velocity smaller than the sound velocity (V, < s) can also give rise to the acoustic amplification. These authors [ll] have, however, reversed the aforesaid statement while explaining their gain mechanism. Recently, Guha et al. [12] have studied the wave interaction phenomenon in an inhomogeneous piezoelectric semiconductor with a different (crossed electric and magnetic fields) configuration and have shown the possibility, though with a number of crude approximations, of acoustic wave amplification even when the electrons drift with a velocity less than the velocity of sound in the medium.

The motivation behind the present investigation is two fold. Firstly, we present here details of a systematic analysis of the transverse phonon-helicon interaction in a homogeneous and isotropic semiconductor magnetoplasma when both piezoelectric coupling and deformation potential coupling are present. The effective masses of the charge carriers in a high mobility semiconductor are comparable and hence their ratio (d = m,/m,) has been accounted for in the present study. These inclusions make the dispersion relation for the coupled mode propagation sufficiently general. The dispersion relation thus obtained has been used to derive the expressions for the gain constant, complex effective elastic constant, and relative change in the velocity of sound. Secondly, if we put d = 0 and neglect either of the two couplings, the resulting simplified expressions offer us an opportunity to resolve the contradic- tion raised in some of the earlier work. The general dispersion relation has been computerized, without imposing any stringent simplifying conditions to obtain the dispersion characteristics of coupled modes in semiconductor magnetoplasma. The detailed analysis and computed results lead us to draw various important conclusions as discussed in the following sections.

2. Theory

The geometrical configuration of the wave propagation in a semiconductor plasma assumed to be homogeneous, isotropic and infinite in extent is as shown in fig. 1. The CGS system of units is used.

The expressions for strain S and stress T, are defined in terms of lattice displacement u as

S=$ and z=p$.

The pertinent relations connecting S, T, D and E are

Cd&L as D=QE+S---

e a2.

The general equation of motion for charge carriers is written as [13]

(2)

(3)

The subscript cx refers either to electrons ((Y = e) or holes (LY = h) and the - and + signs in eq. (4)

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M.S. Kushwaha / Acoustics in piezoelectric semiconductors 325

Lattice d.c. Magnetic field -To

3 z

Wave propogation

direction -3

d. c. Electric field --To

X

Electron drift -- vo

Fig. 1. The geometrical configuration of wave propagation.

respectively are to be taken for electrons and holes. The quantities with a subscript zero are d.c. parts. The coupling between helicons and phonons is provided by the Maxwell field equations

1 6’B VxE=----,

c at

Vx,=fkJ+i!? C c at ’

J = J, + Jh = e(n,, V,, - II, V,) .

The dispersion relation for the coupled mode propagation has assumptions:

i) Only the first order perturbations of the form -ei(kz-o’)

proximation);

(5)

(6)

(7)

been derived with the following

are considered (small signal ap-

ii) the drift is imparted only to the electrons V, = V,( = - eE,/mv) and hence V,, = 0. iii) the collision frequencies for electrons and holes are equal i.e. ve = I+, = v; iv) the electrons are the majority carriers (in n-type) II, > n,,, the effective mass ratio (d) satisfies

O<d<l and Ve> V,,. Hence n,Ve~nn,Vh. With these assumptions the dispersion relation for the coupled mode propagation has been derived,

using the complete set of equations

(w’- k's') k*c; - w* + w;(aJ - kVJ(lT~)

u*d w-kV,+iv-tw,(l-d)-A

w+iv > I = - K,o=k*s* , (8)

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326 MS. Kushwaha ! Acoustics in piezoelectric semiconductors

where

c, = c& ) co; = 4wze2/E,m,, 1

w, = eB,lm,c, K, = ____ 47r&,C, i (9)

In view of the coming discussion of the properties of various wave modes, eq. (8) is rewritten in the form

o-kVO+iv?w.(l-d)---$&)(k2ci-u2)(k2-$)

= KPk2w2 w2d

w-kV,+iv?w,(l-d)----C w+iv )

s)(k’-;).

The t signs are a consequence of the fact that the only permissible solutions are the left- and right-handed circularly polarized ones, i.e.

E,=E,?iE,, B,=B,-+iB,, V++ V,kiV,, U,=ux*iuy. (11)

2.1. Solution of the dispersion relation

We obtain the dispersion relations for the individual waves from the dispersion relation, eq. (lo), for the coupled mode propagation. In what follows we study the various cases:

Case I If wP = 0 and K,, = 0, the dispersion relation (10) is decomposed into three sets of uncoupled

equations,

(w+-w,(l-d)-g)

k= v,

, kc+& k=+w. (12) clil s

These waves are called respectively the (slow and fast) cyclotron wave [14], the (forward and backward) electromagnetic wave, and the (forward and backward) acoustic wave.

Case II If K,, = 0 but wP # 0 then eq. (10) reduces to

w;(w - kVJ(lTs)

w2 w2d ’

w-kV,+iv?o,(l-d)-A w+iv > 1 (13)

Eq. (13) is a dispersion relation for helicons in a drifted semiconductor magnetoplasma Since the plasma frequency (w,) is considerably larger in n-type semiconductors, with electrons as the

Page 5: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

MS. Kushwaha / Acoustics in piezoelecttic semiconductors 327

majority carriers, we avoid using the terms cyclotron and/or electromagnetic for the waves interacting with transverse phonons and would rather prefer to name them simply as helicons. Thus in what follows we shall study the transverse phonon-helicon interaction with a finite coupling provided both by piezoelectric field and deformation-potential. The theoretical analysis of the dispersion relation (eq. (8)) is made under specific assumptions, whereas the general solution of the same is computerized without imposing stringent conditions to obtain the complete w vs. k plot as discussed in the subsequent section.

2.2. Collision induced instability

The mathematical analysis of eq. (8) is made assuming that the forward acoustic wave vector is perturbed in the system. Eq. (8) is rewritten in the form

k-%K k2ti2

‘(k +0/s) (k 2c2 -

m 02)+

o&J - kV,) (1 T -$)

s (w-kl/,+iv?o,(l-d)-*)

w +iv

(14)

Using the perturbation for the wave vector

k=%cx, aslw41, s (15)

and substituting in the right-hand side (rhs) the unperturbed value of k and separating the real and imaginary part of (Y we obtain

Ima =~(~)‘($)~[lk~(I+ y)],

where

D=

and

If eq. (14) is solved under the collision dominated limit, i.e. u 4 V, kV,4 v, then we obtain

(16)

(17)

(19)

where

Page 6: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

328 M.S. Kushwaha I Acoustics in piezoelectric semiconductors

(20)

In eqs. (16) and (19), it is the upper sign of polarization which is of interest to us since it pertains to the negative energy carrying drifted helicons which interact with the positive energy carrying phonons (acoustic wave). The square brackets in eqs. (16) and (19) lead one to conclude that the second term in

the brackets is either zero (eq. (19)) or it is negligible (in the collision dominated limit) (eq. (16)). Thus the quantity in the square brackets in eq. (16) and eq. (19) is always positive. To make the picture clearer we, however, simplify eq. (16) in the collision-dominated limit and assume in both eqs. (16) and (19) that w,e v. In this situation both from eqs. (16) and (19) we obtain an identical expression written

as

(21)

In view of the assumed nature of perturbations in the present treatment, the acoustic wave is growing when Im (Y < 0. It thus follows from eq. (21) that for y < 0, there is a possible growing (convective) instability. The condition y < 0 requires that V, > s. In other words, the acoustic amplification occurs when electrons drift with a velocity greater than the velocity of phonons. Further, the growth rate disappears when Y+O and/or when V, = s, i.e. when electrons drift with the velocity of phonons. It seems worthwhile to note that the condition for amplification obtained by Chandra and Verma [ll] disagrees with the well-established result that in the transverse interaction the acoustic am-

plification occurs only when the condition V, > s is satisfied. Their conclusion seems to be arising due to

inadvertent neglect of a minus sign in their expression (their eq. (18)).

2.3. Resonant modes

In order to ascertain whether the ‘resonant mode’ (wave interaction effect in the absence of collisions) leads to the growing waves, we study the dispersion relation (lo), under two different

approximations:

Case I wp = 0: In this case the dispersion relation, eq. (lo), is written as

(22)

where C(w, k) determines the strength of the effective coupling. Assuming that the maximum inter- action occurs at resonance, one finds

k=% w -+ w,(l - d) - g d)

w

v, >

s (23)

Page 7: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

MS. Kushwaha I Acoustics in piezoelectric semiconductors 329

k = - dl- 4 l-c I t dr l/2

+ 2sy (l-d)2+l * I I (24)

It is obvious that the propagation constant (k) has no complex roots and hence the resonant mode in the existing conditions does not lead to growing waves. An error in the dispersion expression for the coupled modes led Singh [7] to demonstrate the growth rate of waves.

Case II oP # 0 and k2ci * w2: With these approximations eq. (8) is written as

Further, setting d = 0 the solution of eq. (25) at resonance (w = ks) is

2

k’f&k+$=O.

The solution for k is

(25)

(26)

(27)

It is quite clear from eq. (27) that k can have complex roots, i.e. resonant modes may lead to growing waves in the transverse interaction, if y2 > (o,c,Rwps)‘. This condition may be easily satisfied in the presence of low magnetic field in a high carrier density semiconductor when the drift velocity attains a value of the order of - 10’ cm/s’.

2.4. Diagnostic importance of interaction studies

The interaction between transverse phonons and helicons is not only interesting in its own right, but it also provides a very useful tool for many diagnostic purposes [15]. For example, this interaction provides a mechanism for amplifying and generating high frequency phonons. The dependence of the strength of the interaction on the electric and magnetic fields is an interesting piece of information that can be obtained from the acoustic absorption. The absorption coefficient and any change in the velocity of phonons (sound) may be used to obtain valuable information about the interaction itself. We, therefore, in the following subsections derive expressions for the absorption coefficient and the relative change in the velocity of the acoustic wave.

2.4.1. Complex effective elastic constant The dispersion relation, eq. (8), governing the acoustic signal in semiconductors with piezoelectric-

* It is my belief that Browne [4] who showed that the resonant modes might lead to the growing waves has used approximations

similar to the ones made above. Since the original paper of Browne [4] is not available to the author no further comments can be

made on this aspect.

Page 8: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

330 M.S. Kushwaha I Acoustics in piezoelectric semiconductors

and deformation potential coupling may also be written as

pw= = c*k2,

where

(28)

(29)

Here c* is the complex effective elastic constant and includes the electronic contribution to the elastic properties. The second term in the square bracket in eq. (29) provides a small correction to the observed elastic constant (C,). In terms of the complex elastic constant c*, the absorption coefficient

(4 is

ff ab

where s = qCc,/p. When KP is treated as a small perturbation, we find

(30)

(31)

where D is already defined in eq. (17). It is thus quite clear that under above mentioned approximations the expression for the absorption coefficient calculated using wave interaction method and using the equations of motion in the theory of the elasticity is the same [15]. If eq. (31) is further simplified by assuming o 4 V, w, 4 v and 0 < d < 1, one obtains the expression for (Y,~ which is the same as eq. (21). If d = 0, Eq. (21) further reduces to

provided that c, I- S. White [3] obtained the following expression for the gain constant in longitudinal wave interaction (in the absence of bunching and diffusion):

(32)

(33)

where K2 is the piezoelectric coupling constant. It is obvious that eq. (33) is derivable from the

Page 9: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

M.S. Kushwaha / Acoustics in piezoelectric semiconductors 331

generalized equation (32) by neglecting the deformation potential coupling in Kp, replacing wP by w,/y, and substituting O~/V = wC, the so-called dielectric relaxation frequency. These substitutions lead one to believe that under special conditions (c, = s) the behaviour of transverse wave interaction is identical to that of longitudinal wave interaction. Under such conditions interaction becomes cumulative and can be observed experimentally [5]. Using wrong sign (his eq. (13)) Singh [9] argued for (Y > 0 which implied p > 0 or V, < Vso, a conclusion which is in contradiction with the fact that acoustic amplification in the transverse interaction occurs only when the drift velocity is greater than the sound velocity. On the other hand, if the condition Vs,< V,, is fulfilled, then his (Singh [9]) (Y is to be identified as an attenuation constant and not the gain constant. The use of electron cyclotron frequency as an imaginary quantity in the dispersion relation by Singh [7-91 is erroneous. This has not been commented upon earlier probably because the results were mostly derived in the limit w,r + 1 and w, generally did not explicitly appear in his final results.

2.4.2. Change in the velocity of sound The oscillatory and resonant effects of the transverse wave interaction appear in both the absorption

coefficient and the velocity of sound. Usually it is much easier to measure these effects in the absorption coefficient than in the sound velocity [15]. This is because all of the absorption arises from the interaction, whereas the change in the velocity of sound resulting from the interaction is small compared to the velocity of sound itself. Since the presence of the d.c. fields has such a strong effect on phonon-helicon interaction, it is of interest to see whether d.c. fields will also have measurable effect on the velocity of sound. For this purpose, we rewrite eq. (14) in the form

(34)

The second term on the right-hand side is obviously the perturbation in the wave vector k due to the interaction. Assuming cik* * W* (i.e. neglecting the fast electromagnetic branch), using the unperturbed value of k in the right-hand side and separating real and imaginary parts after using

+., k real

we find the following expression for the relative change in the velocity of sound:

(35)

LL WV k,/ ’ Y J I v WV \c,I ’ J J

This solution is subject to the assumption of high v (i.e., w, e V) and small perturbation Kp. For d = 0

eq. (36) reduces to

j$f=- !%I 2

s *

( > -

Ctll

[1+y(y+$($)]

[l+[~+$(~)2Y}2] . (37)

Page 10: Transverse wave interaction and acoustic amplification in piezoelectric semiconductors

332 M.S. Kushwaha / Acoustics in piezoelecfric semiconductors

If (w&JV)(s/c,)” e 1, we obtain from eq. (37)

M

On the other hand, if u + V, then eq. (37) gives

M -_- !G 2

s 2 ( > - %I

[1+$($2] [1+ pp)‘Y)*]~

(38)

(39)

The results obtained in eqs. (38) and (39) lead us to believe that the relative change in the velocity of sound does not vanish even when the coupling is small provided that the latter is not entirely zero. The relative change in the velocity of sound has its maximum or minimum value when y = 0, or V,, = s. Such a relative change in velocity occurs because the interaction between the helicons and phonons maximises when the electrons drift with the velocity of the acoustic wave in the system. Singh’s [7] results compared to our results are different and erroneous since his equations are dimensionally incorred.

2.5. Relative strength of the piezoelectric coupling and the deformation potential coupling

In the preceding sections the coupling constant K, is expressed as

1 &=---- ___ 47r&r& (

e2 + k2C&; > e2 ’

(40)

where the first term refers to the piezoelectric coupling and the second one the deformation potential coupling. If we fix the parameters of the entire system, we can compare the relative contribution of the two effects contributing to the total coupling. The ratio of the deformation potential coupling constant (K,) to the piezoelectric coupling constant (K,,) is

$ci _ k’C:d. PP

e2tT2

Substituting the unperturbed value of k(= w/s) in eq. (41) yields

2

K@= w ,

&P ( > WC,

(41)

(42)

where w,,= (see/C,&,). (For InSb, we have w,, = 3.45 X 10” Hz.) Therefore, the contribution of the deformation potential coupling will be greater than that of the piezoelectric coupling when w > w,. Furthermore, when the deformation potential coupling is much larger than the piezoelectric coupling the gain (section 2.2), as well as, the relative change in the velocity of sound (section 2.4.2) depend on the signal frequency and increase as w increases.

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MS. Kushwaha / Acoustics in piezoelectric semiconductors 333

3. Numerical computation

The computation of w as a function of real k in the presence of collisions is analogous to an

approximate solution of eq. (8) used for studying the absolute instability. The procedure is to write w = ks + S, where S/ks 4 1. Im S > 0 leads to the possibility of the growing (absolute) instability. The dispersion relation, eq. (8), has been numerically solved without making any assumption to restrict the frequency and/or the wavelength regime. The collision frequency (v) is, however, assumed to be much less than the electron cyclotron frequency (w,) so that O,T %= 1. The data used for the computation are listed in table I.

Table I

The data for n-InSb used for the computation

Quantity Value Units

L n e

me mh

d s VO BO

P

Cd

P G4 % UC ”

EL

6.48 x 1O-8

2.0 x 10’4

4.802 x lo-”

1.822 x 1O-29

1.822 x 10-a

0.1

2.287 x lo5

2s

200

2.262 x 16

7.2 x lo-l2

5.773

3.05 x 10”

8.976 x 10”

2.796 x lOLo

0,/200

16.0

cm

cme3

esu

g

g

cm/s

G esulcm

erg gm/cm3

dyne/cm*

Hz

Hz

Hz

In order to calculate the frequency (w) as a function of the wave vector (k), we use the dispersion equation (8) and fix the values of the wave vector k by analogy with the standard theory of crystal dynamics [16]. The wave vector k in the reciprocal space is, in general, expressed as

k = s (k,, k,, k,) ,

where N is the cube-root of the total number of subdivisions in the first Brillouin zone, L is the lattice constant, and k,, k,, k, are the integers which satisfy specific conditions for a particular crystal structure [16]. Assuming k2ci Q w2, which allows us to neglect the fast electromagnetic branch, we are left with the slow-wave branches of the dispersion relation. Introduction of collisions (V f 0) and effective mass ratio (d) gives us the dispersion curves displayed in fig. 2. Since we are interested in interaction between positive energy carrying acoustic waves and negative energy carrying drifted helicons, we focus our attention on the k, w > 0 quadrant.

We have to mention here that if eq. (8) is solved for the maximum interaction when the phase

velocity of helicons is equal to the velocity of sound (see eq. (25), for d # 0 and C’(w, k) = 0), we obtain

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334 MS. Kushwaha I Acoustics in piezoelechic semiconductors

Fig. 2. The dispersion diagram for coupled phonon-helicon interaction. H- (solid line) and H+ (dotted line) respectively refer to the negative energy carrying and positive energy carrying drifted helicons. S+ and S- respectively denote the positive energy carrying and the negative energy carrying acoustic modes.

a third degree polynomial in k which when solved gives three finite values of k:

k, = 0.732 x 10m2 cm-‘, k, = 0.159 x lo5 cm-‘, k, = 0.942 X lo5 cm-’ .

Here k, is very low frequency (W < lo5 Hz) long wavelength cross-over point. Since this frequency range is relatively uninteresting, we restricted our computation of w as a function of real k to the range covering the points k, and k,. The cross-over points are shown as C, and C, which correspond to relatively high frequency-short wavelength resonance between the interacting drifted helicons and phonons. The crossing of the two branches clearly indicates the vanishingly small coupling at higher frequencies. The cross-over region results because the high frequency transverse electromagnetic waves which would normally propagate at the velocity of light (c,) in the medium if caused to propagate at the velocity of sound reduce the effective coupling considerably [5].

A comparison of the present dispersion curves (fig. 2) with those reported by Steele and Vural [6] reveals two major differences. Firstly, the drifted helicons show a non-zero frequency at wave vector k = 0. This is attributed to a consequence of including the terms involving effective mass ratio (d) of the

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MS. Kushwaha / Acoustics in piezoelectric semiconductors 335

charge carriers. The frequencies of the drifted helicons at wave vector k = 0 are given, using the dispersion relation (eq. (S)), as o = _tw,d. This gives w = +2.8 x lOlo Hz and -2.8 x 10” Hz, respectively, for the negative energy carrying and positive energy carrying drifted helicons. This effect of including terms with d # 0 is analogous to the presence of non-zero frequency optic modes at k = 0 in the phonon dispersion relations in the diatomic cubic crystals having different ionic masses. Therefore, the presence of non-zero frequencies for the drifted helicons at k = 0 may be termed as ‘helico-optic’ effect. Secondly, the frequencies in the lower half right quadrant (k < 0, w > 0) rise gradually and continuously into the upper half right quadrant (k > 0, w > 0). The shift of the frequencies may again, reasonably, be ascribed to the effect of the effective mass ratio (d). As a result of this shift of frequencies of the interacting drifted helicons, the interaction between the latter and the positive energy carrying acoustic wave is expected to give rise to growing instabilities at relatively higher frequencies for d = 0.

4. Conclusion

The analysis of general dispersion equation under various approximations leads us to make the following categorical concluding remarks:

i) In the present configuration of wave propagation the acoustic amplification occurs only when V, > s, i.e. when the drift velocity of the electrons is greater than the velocity of the acoustic wave.

ii) If wp = 0, the resonant mode does not lead to any growing waves. On the other hand, if wp f 0, the resonant mode gives rise to a growing instability in high carrier density mobile semiconductors in the presence of small magnetic field when the drift velocity attains a value of the order of -lO*cm/s.

iii) The absorption coefficient calculated by the wave interaction method and by using the equation of motion for the acoustic wave system is found to be the same. Under special condition (viz. c, = s), the behaviour of transverse wave interaction and longitudinal wave interaction is identical. In such conditions the transverse interaction is experimentally observable.

iv) The relative change in the velocity of sound has its maximum/minimum value when y = 0, (or V, = s) and does not vanish until and unless the coupling constant is entirely zero. The calculations due to Spector [17] also confirm the finite change in the velocity of sound in a chosen system.

The inclusion of the effective mass ratio of the charge carriers in the dispersion equation for the coupled mode propagation is believed to have a significant effect particularly when the investigation is restrained within the helical regime: w 4 w,, w,r % 1. In general, the existence of non-zero frequencies of the drifted helicons at k = 0 and the shift of the frequencies of the interacting helicons from k ~0, w > 0 quadrant to k > 0, w > 0 quadrant are due to the inclusion of the effective mass ratio of the charge carriers.

Acknowledgements

I would like to thank Prof. R.N. Singh for stimulating discussions and Prof. D.K. Rai for critical reading of the manuscript. The financial assistance from U.G.C., New Delhi, is gratefully ac- knowledged.

References

[l] A.R. Hutson, J.M. HcFee and D.L. White, Phys. Rev. Lett. 2 (1961) 239. [2] L. Solymar, Solid State Electr. 9 (1966) 879.

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336 MS. Kushwaha I Acoustics in piezoelecttic semiconductors

[3] D.L. White, J. Appl. Phys. 33 (1%2) 2547.

[4] M.E. Browne, Helv. Phys. Acta 37 (1964) 545.

(51 L. Solymar and C.N. Lashmore-Davies, Int. J. Electr. 22 (1967) 549.

[6] M.C. Steele and B. Vural, Wave Interactions in Solid State Plasmas, (McGraw-Hill, New York, 1969).

[7] A. Singh, Int. J. Electr. 25 (1968) 495.

[8] A. Singh, Int. J. Electr. 27 (1969) 193.

[9] A. Singh, Int. J. Electr. 34 (1973) 127.

[lo] H.C. Hsieh, J. Appl. Phys. 45 (1974) 482.

[ll] R. Chandra and J.S. Verma, Ind. J. Phys. 49 (1975) 944.

[12] S. Guha, S. Ghosh and N. Apte, Phys. Lett. A 71 (1979) 382.

[13] G.F. Friere, Proc. IEEE 58 (1970) 482.

[ 141 W.H. LouiselI, Coupled Mode and Parametric Electronics (Wiley, New York, 1980).

1151 H.N. Spector, Solid State Phys. 19 (1966) 291.

[16] M.S. Kushwaha, Ph.D. Thesis, Banaras Hindu University (1980).

[17] H.N. Spector, Phys. Rev. 134 (1964) A507.