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Optimum acoustic wave second-harmonic generation in piezoelectric semiconductors
J. Phys. Chem. Solids, 1974, Vol. 35, pp. 1133-1137,
Pergamon Press.
Printed in Great Britain
OPTIMUM ACOUSTIC WAVE SECOND-HARMONIC GENERATION IN PIEZOELECTRIC SEMICONDUCTORS* A. K. AGARWAL and S. K. SHARMA Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi- I 10029, India (Received 23 M a y 1973; in revised form 5 September 1973)
AbstractmThe phenomenologicaI approach has been used to study analytically the acoustic wave second harmonic generation in piezoelectric semiconductors in the presence of the d.c. electric and an oscillating electromagnetic field (OEF). It has been suggested that the second harmonic acoustic flux (SHAF) can be enhanced considerably by the application of an OEF polarized in the direction of the propagating acoustic wave. The SHAF exhibits a maximum at 12 = to, where 12 is the frequency of the OEF and to is the frequency of the acoustic wave. The SHAF also shows a maximum at d.c. electric fields for which the average drift velocity of the carriers is equal to the velocity of sound. It is found that for a typical case of n-type nondegenerate InSb (77°K, n = 2.5 x 10" cm -~) that the SHAF is enhanced by a factor of I0~ over its value in the absence of OEF. The present analysis is valid in the low frequency region only (i.e. ql ,~ 1).
1, INTRODUCTION In recent years a great deal of attention has been given to the phenomena of nonlinear mixing of acoustic waves in piezoelectric semiconductors, because of its importance in revealing information about the growth of acoustic d o m a i n s [ l - 6 ] . It has been recently pointed out by Pantell and Soohoo [7] and Epshtein[8], that the acoustic gain or attenuation is considerably enhanced by the application of an oscillatory electromagnetic field (OEF) in the direction of propagation of the acoustic wave. Therefore, it is of interest to study the effect of an O E F on nonlinear acousto-electric interactions. In the present paper we have used the phenomenological approach to study analytically the ultrasonic second harmonic generation in piezoelectric semiconductors due to the nonlinear e l e c t r o n - p h o n o n - p h o t o n interactions (NEPPI). The second harmonic acoustic flux (SHAF) is calculated by solving the usual piezoelectric equations of state and the Maxwell's equations along with the modified electric current densities as a result of N E P P I . It is shown that our expressions for S H A F in the absence of an OEF, reduce to those derived by Wu and Spector[5] and Mauro and Wang[2]. It is found for a typical case (77°K, n = 2 . 5 x *Work supported by CSIR (India). ]'It should be noted that equation (5) is valid in the long wavelength limit, i.e. ql <~1, where q is the wave vector of the acoustic wave and t is the mean free path of the carriers.
lO'4cm-3) of n-type nondegenerate InSb that the second harmonic yield is increased by a factor of 10-' as compared to its value in the absence of the OEF. The present treatment is valid when the acoustic wavelength is much larger than the mean free path o'f the carriers, i.e. ql ,~ 1, because of its phenomenological nature. 2. THEORY In a piezoelectric semiconductor, the equation of motion of the lattice is[9] P
= 0xj'
(1)
further we also have Tij = C,iklSkl -- ~,ikE~,
(2)
D, = eE~ + 4 ~r[3,i~sik,
(3)
where
aCq,
s, = ~ \-d-g 5Z,/
(4)
is the strain tensor, c,j~ are the elastic constants,/3~jk is the piezoelectric tensor, E is the electric vector, T~j is the stress tensor, D is the electric displacement vector, ~ is the displacement, p is the density of the material, and e is the dielectric constant of the medium. The electronic current density J is given b y t
1133
J = netzE-
e~Vn
(5)
A. K, AGARWAL a n d S. K . SHARMA
1134
harmonics and sum and difference frequencies are not retained, as their amplitudes are small as compared to those retained. In the presence of the electric field given by equation (6a) the carrier concentration can be written as
where e is the electronic charge, n is the electronic concentration,/z is the mobility of the carriers, and is the diffusion coefficient given b y the Einstein's relation ( ~ = McBTle, k~ is the Boltzmann's constant, T is the temperature of the lattice). E is the total electric field[10]: E = E0 + E~ exp i (tot - q~z~) + E~.~.exp i ~ t + E2 exp i (2tot - q2zO +E3 exp 2 i l ~ t + E+ exp i ((to + l~)t - q+zO +E_ exp i ((to - 12)t - q - z O ,
n = no + nl exp i (tot - qlzl) + n2 exp i (2tot - q2zO 4 n+ exp i((to + f / ) t - q+z~) + n- exp i((to - f l ) t - q-zO. (6b)
(6a)
where E0 is the external d.c. electric field, E~.~.is the O E F of frequency f~; E~ is the a.c. electric field induced b y the acoustic w a v e of frequency to; E~, E~, E_- are the a.c. electric fields generated due to the nonlinearity of the medium, and q~, q~, q-_ are the w a v e vectors associated with the fundamental, second harmonic, sum and difference frequency waves, respectively. In equation (6a), the higher
Considering a dispersionless medium (i.e. taking q ~ = 2 q , , q - = ( l + I ~ l t o ) q O and making use of the continuity equation, one obtains = ff_L ill
J2
- ' el)s ' r12 = -evs
n± = - -~.
el)s
F r o m equations (5) and (6) one obtains the following expressions for the time d e p e n d e n t and timeindependent components of the electronic current densities (7a)
Jo = OroEo; O'o = rloe/~
" [,+ =
t (,+'°o?) ',~'~.E,l"
o'0E2
. ~ + D.A/ COD /
rl ~ E . _ _
_
d
/
(7b)
....
1 too/\
~D
(7c)
oo,: I1 too
+ i~o - t'l~
_
(,+,°:?; _
(6c)
mo
(7d)
.I I
where 3' = 1 - (Jo/noev,) = 1 - (va tv,), and too = v,2/~, is the diffusion frequency. It is obvious from equation (7), that the fundamental and the second harmonic current densities d e p e n d not only on their own fields i.e. El, E2, but also on the applied O E F . Using equation (7) in the Maxwell's equations, one obtains the following expressions for the longitudinal electric fields associated with the fundamental, second harmonic, sum and difference frequency waves, respectively
Eix
E2x
[~+i(-~ +~tl ' \con to t J
too~l[~,+~/~+~'/Y [.,/+ i(~+ to~'~l [~,+/2,o+ \too 2 t o / J \g"g'o 2d/_l ',~,o to,,~
(8a)
1135
Optimum acoustic wave second-harmonic generation
7
+-,(.v ~:.°.
E+
~(~,)(;)~"°"' +'°
,,,0 ,., (.~)(y+i~o2Sl)s
. +
D]
(8b)
t~
~sJ
m__~v,)(~_~) [ # z I~14:/3i '17 ....s,
E+_~ =
to -t-1"~
',"""~o
E_
'+
(8c)
t,O~
~--~--fi-+~iJ
\o,o
~tJ
and X = f tim. Here o~. = 4~rcr0/e, is the dielectric relaxation frequency. In equations (8) s~, s2, s_~are the strains associated with the fundamental, second, harmonic, and the sum and difference frequency components respectively and/3 stands for the appropriate piezoelectric constant• Making use of equation (8) in equations (1)-(4), one obtains the following equations for the sound-wave amplitudes for the fundamental and second harmonic: c
~'~,
P-a~" = a'~.
[4w[32~[
+" t o k \
t _-~_7W*%--;) last, ÷ ~ + '. ~co+ ~oJ. // ax 2' 4~r/3')(y + "2i~!~
(
(~
(9a) 4w z 3(~)(~,)(, to~ /~ '•(-D' " '~"°'- 0'" 0"~.x
a'f.,
p-~7~-= .+ ~+i2
•
~
(-.tic 2
/tO/
+A
r.2 -~-r a2~¢i~-. a~, .....
(9b)
where [ \
E2.e
-~| OjD } t ~/ "t" i ('O(O~D~'~) (4,n-i,)',' (8~2) (~,)'
1
A-'=F
L
/to _+f~ to~ \ y+i t O)D + gO-t-a) ~
.[ {[~ .o+. + (o°
÷ [~/+ .~+__I~II"
.<
o_+D][
"
i a~
•
'o,. JL''+ ( ~ + ~
Assuming the sound wave amplitudes associated with the fundamental and second harmonic, respectively ~ = ~:loexp i(tot -qtz~);
~:2= ~2oexp i(2tot - q2zt); 5+ = ~÷oexp i((oJ + ~ ) t - q+z,); ~-= ~-o exp i((to - f ~ ) t - q - z , ) ,
(10)
1136
A. K. AGARWALand S. K. SHAR~
and on solving equations (8)-(10), one obtains the following expressions for the second harmonic sound wave amplitude, 2 ,
°°'L
('+'%")('+
oo,,,,,,,,j
.......
( ~ I v , ) 3 ~ o q ( ~ ' ~ l ~ % ) ( A . + A_)ELo.~, ~
The acoustic flux associated with a wave of frequency o~j is[1]
Pi
I
2
=
v,l ,l 2 •
(I I)
It is seen from equation (11) and (13) that the second harmonic acoustic flux depends upon the amplitude and the frequency of the OEF.
(12) 3. N U M E R I C A L R E S U L T S AND D I S C U S S I O N
Therefore, the ratio of the acoustic flux in the second harmonic to the initial flux in the fundamental is
___L_8
2 (13)
p--~ - pv, to ~ [ ~ 2 •
Equation (13) has been used to obtain some results for the typical case of n - t y p e InSb at 77°K. The following parameters were used to calculate P 2 / P , 2, n = 2-5 x 10 ~4 cm -~, t~ = l0 s c m 2 / V s e c , ~ = 18, m = 1.17x 10-29 g, 0 = 5 . 8 g/cm 3, v, = 4 x l0 S cm/sec. Fig. 1 shows the variation of the relative 10-9
F-2 10-10
IO-H cq 0.
10-12
0
0"5
I
1"5
2
£I,I~ Fig. 1. Variation of the second-harmonic acoustic flux P2tPi 2 with Q/co for ql = 0.1, F = 1 and for various values
of-,/.
I0-15 0
0-5
I
I-5
J 2
I-y Fig. 2. Variation of the second-harmonic acoustic flux P21P12 with 7 for ql = 0.1, [l/to = I and for various values of F. The vertical scale is to be multiplied by 10-' for F = 0 curve,
Optimum acoustic wave second-harmonic generation flux (P2/P~2) as a function of the frequency It of the OEF. It can be seen from Fig. 1 that the relative acoustic flux is maximum when the frequency of the OEF is equal to the frequency of the acoustic wave, viz. It = co. This maximum in the relative flux may be attributed to the resonant transfer of electrical energy to the acoustic energy. Figure 2 shows the dependence of the relative acoustic flux on the strength of the applied d.c. electric field/3o for It = co. The values of other parameters are indicated in the figure. It is seen from Fig. 2 that the second harmonic yield, in the presence of OEF first increases as the d.c. electric field is increased, till the carrier drift velocity becomes equal to the sound velocity, and then decreases with the further increase in the strength of the d.c. electric field va > vs. In the absence of the OEF, our expressions for the relative flux in the second harmonic reduce to those of Wu and Spector [5]. The harmonic yield in the absence of OEF, first decreases slightly in the range 0 < v~/vs < 1 and then shows a slight increase for higher values of vd/v~ (i.e. vd > v~). Further the second harmonic power is enhanced by a factor of 103 over its value in the absence of an OEF, for an OEF having an amplitude of twice the value of d.c. field, and when the frequencies of the OEF and acoustic wave are equal (F = 2, A = I). 4. CONCLUSIONS
From the numerical results presented in Section 3, the following conclusions can be drawn: (i) The second harmonic acoustic flux can be enhanced considerably by the application of an oscillating electric fields, (ii) The second harmonic acoustic flux exhibits a maximum when the frequency of the applied oscil-
1137
lating electric field is equal t o the frequency of the acoustic wave, (iii) The second harmonic yield also shows a maximum when the carrier drift velocity is equal to the velocity of sound in the presence of the oscillating electric field. It shows a minimum in the absence of the OEF for the same value of the drift velocity. Therefore, the present study suggests that the second harmonic yield can be optimized by the suitable choice of various parameters e.g. amplitude and frequency of the oscillating electric field, the strength of the externally applied d.c. electric field. Because of the non-availability of the relevant experimental data, the present results cannot be verified experimentally.
Acknowledgements--The authors are thankful to Professor M. S. Sodha and Dr. S. K. Sharma for their helpful suggestions. REFERENCES
1. Tell B., Phys. Rev. 136, A772 (1964). 2. Mauro R. and Wang W. C., Phys. Rev. B1, 683 (1970). 3. Zemon S. and Zucker J., IBM Z Res. Dev. 13, 494
(I~). 4. Palik E. D. and Bray R., Phys. Rev. B3, 3302 (1971). 5. Wu C. C. and Spector H. N., J. appl. Phys. 43, 2937
(1972). 6. Spector i-I.N., Phys. Rev. B6, 2309 (1972). 7. Pantell R. H. and Sochoo J., f. appl. Phys. 41, 441 (1970). 8. Epshtein E. M., Soviet Phys. Solid State I0, 2325 (1969). 9. Spector H. N., Solid State Phys. 19, 291 (1966) (see more references in this article). 10. The expansion of the electric field is done under a perturbation approach, i.e.
Eo~E,.o.; Et >E.., E,, E = > - • -
Pergamon Press.
Printed in Great Britain
OPTIMUM ACOUSTIC WAVE SECOND-HARMONIC GENERATION IN PIEZOELECTRIC SEMICONDUCTORS* A. K. AGARWAL and S. K. SHARMA Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi- I 10029, India (Received 23 M a y 1973; in revised form 5 September 1973)
AbstractmThe phenomenologicaI approach has been used to study analytically the acoustic wave second harmonic generation in piezoelectric semiconductors in the presence of the d.c. electric and an oscillating electromagnetic field (OEF). It has been suggested that the second harmonic acoustic flux (SHAF) can be enhanced considerably by the application of an OEF polarized in the direction of the propagating acoustic wave. The SHAF exhibits a maximum at 12 = to, where 12 is the frequency of the OEF and to is the frequency of the acoustic wave. The SHAF also shows a maximum at d.c. electric fields for which the average drift velocity of the carriers is equal to the velocity of sound. It is found that for a typical case of n-type nondegenerate InSb (77°K, n = 2.5 x 10" cm -~) that the SHAF is enhanced by a factor of I0~ over its value in the absence of OEF. The present analysis is valid in the low frequency region only (i.e. ql ,~ 1).
1, INTRODUCTION In recent years a great deal of attention has been given to the phenomena of nonlinear mixing of acoustic waves in piezoelectric semiconductors, because of its importance in revealing information about the growth of acoustic d o m a i n s [ l - 6 ] . It has been recently pointed out by Pantell and Soohoo [7] and Epshtein[8], that the acoustic gain or attenuation is considerably enhanced by the application of an oscillatory electromagnetic field (OEF) in the direction of propagation of the acoustic wave. Therefore, it is of interest to study the effect of an O E F on nonlinear acousto-electric interactions. In the present paper we have used the phenomenological approach to study analytically the ultrasonic second harmonic generation in piezoelectric semiconductors due to the nonlinear e l e c t r o n - p h o n o n - p h o t o n interactions (NEPPI). The second harmonic acoustic flux (SHAF) is calculated by solving the usual piezoelectric equations of state and the Maxwell's equations along with the modified electric current densities as a result of N E P P I . It is shown that our expressions for S H A F in the absence of an OEF, reduce to those derived by Wu and Spector[5] and Mauro and Wang[2]. It is found for a typical case (77°K, n = 2 . 5 x *Work supported by CSIR (India). ]'It should be noted that equation (5) is valid in the long wavelength limit, i.e. ql <~1, where q is the wave vector of the acoustic wave and t is the mean free path of the carriers.
lO'4cm-3) of n-type nondegenerate InSb that the second harmonic yield is increased by a factor of 10-' as compared to its value in the absence of the OEF. The present treatment is valid when the acoustic wavelength is much larger than the mean free path o'f the carriers, i.e. ql ,~ 1, because of its phenomenological nature. 2. THEORY In a piezoelectric semiconductor, the equation of motion of the lattice is[9] P
= 0xj'
(1)
further we also have Tij = C,iklSkl -- ~,ikE~,
(2)
D, = eE~ + 4 ~r[3,i~sik,
(3)
where
aCq,
s, = ~ \-d-g 5Z,/
(4)
is the strain tensor, c,j~ are the elastic constants,/3~jk is the piezoelectric tensor, E is the electric vector, T~j is the stress tensor, D is the electric displacement vector, ~ is the displacement, p is the density of the material, and e is the dielectric constant of the medium. The electronic current density J is given b y t
1133
J = netzE-
e~Vn
(5)
A. K, AGARWAL a n d S. K . SHARMA
1134
harmonics and sum and difference frequencies are not retained, as their amplitudes are small as compared to those retained. In the presence of the electric field given by equation (6a) the carrier concentration can be written as
where e is the electronic charge, n is the electronic concentration,/z is the mobility of the carriers, and is the diffusion coefficient given b y the Einstein's relation ( ~ = McBTle, k~ is the Boltzmann's constant, T is the temperature of the lattice). E is the total electric field[10]: E = E0 + E~ exp i (tot - q~z~) + E~.~.exp i ~ t + E2 exp i (2tot - q2zO +E3 exp 2 i l ~ t + E+ exp i ((to + l~)t - q+zO +E_ exp i ((to - 12)t - q - z O ,
n = no + nl exp i (tot - qlzl) + n2 exp i (2tot - q2zO 4 n+ exp i((to + f / ) t - q+z~) + n- exp i((to - f l ) t - q-zO. (6b)
(6a)
where E0 is the external d.c. electric field, E~.~.is the O E F of frequency f~; E~ is the a.c. electric field induced b y the acoustic w a v e of frequency to; E~, E~, E_- are the a.c. electric fields generated due to the nonlinearity of the medium, and q~, q~, q-_ are the w a v e vectors associated with the fundamental, second harmonic, sum and difference frequency waves, respectively. In equation (6a), the higher
Considering a dispersionless medium (i.e. taking q ~ = 2 q , , q - = ( l + I ~ l t o ) q O and making use of the continuity equation, one obtains = ff_L ill
J2
- ' el)s ' r12 = -evs
n± = - -~.
el)s
F r o m equations (5) and (6) one obtains the following expressions for the time d e p e n d e n t and timeindependent components of the electronic current densities (7a)
Jo = OroEo; O'o = rloe/~
" [,+ =
t (,+'°o?) ',~'~.E,l"
o'0E2
. ~ + D.A/ COD /
rl ~ E . _ _
_
d
/
(7b)
....
1 too/\
~D
(7c)
oo,: I1 too
+ i~o - t'l~
_
(,+,°:?; _
(6c)
mo
(7d)
.I I
where 3' = 1 - (Jo/noev,) = 1 - (va tv,), and too = v,2/~, is the diffusion frequency. It is obvious from equation (7), that the fundamental and the second harmonic current densities d e p e n d not only on their own fields i.e. El, E2, but also on the applied O E F . Using equation (7) in the Maxwell's equations, one obtains the following expressions for the longitudinal electric fields associated with the fundamental, second harmonic, sum and difference frequency waves, respectively
Eix
E2x
[~+i(-~ +~tl ' \con to t J
too~l[~,+~/~+~'/Y [.,/+ i(~+ to~'~l [~,+/2,o+ \too 2 t o / J \g"g'o 2d/_l ',~,o to,,~
(8a)
1135
Optimum acoustic wave second-harmonic generation
7
+-,(.v ~:.°.
E+
~(~,)(;)~"°"' +'°
,,,0 ,., (.~)(y+i~o2Sl)s
. +
D]
(8b)
t~
~sJ
m__~v,)(~_~) [ # z I~14:/3i '17 ....s,
E+_~ =
to -t-1"~
',"""~o
E_
'+
(8c)
t,O~
~--~--fi-+~iJ
\o,o
~tJ
and X = f tim. Here o~. = 4~rcr0/e, is the dielectric relaxation frequency. In equations (8) s~, s2, s_~are the strains associated with the fundamental, second, harmonic, and the sum and difference frequency components respectively and/3 stands for the appropriate piezoelectric constant• Making use of equation (8) in equations (1)-(4), one obtains the following equations for the sound-wave amplitudes for the fundamental and second harmonic: c
~'~,
P-a~" = a'~.
[4w[32~[
+" t o k \
t _-~_7W*%--;) last, ÷ ~ + '. ~co+ ~oJ. // ax 2' 4~r/3')(y + "2i~!~
(
(~
(9a) 4w z 3(~)(~,)(, to~ /~ '•(-D' " '~"°'- 0'" 0"~.x
a'f.,
p-~7~-= .+ ~+i2
•
~
(-.tic 2
/tO/
+A
r.2 -~-r a2~¢i~-. a~, .....
(9b)
where [ \
E2.e
-~| OjD } t ~/ "t" i ('O(O~D~'~) (4,n-i,)',' (8~2) (~,)'
1
A-'=F
L
/to _+f~ to~ \ y+i t O)D + gO-t-a) ~
.[ {[~ .o+. + (o°
÷ [~/+ .~+__I~II"
.<
o_+D][
"
i a~
•
'o,. JL''+ ( ~ + ~
Assuming the sound wave amplitudes associated with the fundamental and second harmonic, respectively ~ = ~:loexp i(tot -qtz~);
~:2= ~2oexp i(2tot - q2zt); 5+ = ~÷oexp i((oJ + ~ ) t - q+z,); ~-= ~-o exp i((to - f ~ ) t - q - z , ) ,
(10)
1136
A. K. AGARWALand S. K. SHAR~
and on solving equations (8)-(10), one obtains the following expressions for the second harmonic sound wave amplitude, 2 ,
°°'L
('+'%")('+
oo,,,,,,,,j
.......
( ~ I v , ) 3 ~ o q ( ~ ' ~ l ~ % ) ( A . + A_)ELo.~, ~
The acoustic flux associated with a wave of frequency o~j is[1]
Pi
I
2
=
v,l ,l 2 •
(I I)
It is seen from equation (11) and (13) that the second harmonic acoustic flux depends upon the amplitude and the frequency of the OEF.
(12) 3. N U M E R I C A L R E S U L T S AND D I S C U S S I O N
Therefore, the ratio of the acoustic flux in the second harmonic to the initial flux in the fundamental is
___L_8
2 (13)
p--~ - pv, to ~ [ ~ 2 •
Equation (13) has been used to obtain some results for the typical case of n - t y p e InSb at 77°K. The following parameters were used to calculate P 2 / P , 2, n = 2-5 x 10 ~4 cm -~, t~ = l0 s c m 2 / V s e c , ~ = 18, m = 1.17x 10-29 g, 0 = 5 . 8 g/cm 3, v, = 4 x l0 S cm/sec. Fig. 1 shows the variation of the relative 10-9
F-2 10-10
IO-H cq 0.
10-12
0
0"5
I
1"5
2
£I,I~ Fig. 1. Variation of the second-harmonic acoustic flux P2tPi 2 with Q/co for ql = 0.1, F = 1 and for various values
of-,/.
I0-15 0
0-5
I
I-5
J 2
I-y Fig. 2. Variation of the second-harmonic acoustic flux P21P12 with 7 for ql = 0.1, [l/to = I and for various values of F. The vertical scale is to be multiplied by 10-' for F = 0 curve,
Optimum acoustic wave second-harmonic generation flux (P2/P~2) as a function of the frequency It of the OEF. It can be seen from Fig. 1 that the relative acoustic flux is maximum when the frequency of the OEF is equal to the frequency of the acoustic wave, viz. It = co. This maximum in the relative flux may be attributed to the resonant transfer of electrical energy to the acoustic energy. Figure 2 shows the dependence of the relative acoustic flux on the strength of the applied d.c. electric field/3o for It = co. The values of other parameters are indicated in the figure. It is seen from Fig. 2 that the second harmonic yield, in the presence of OEF first increases as the d.c. electric field is increased, till the carrier drift velocity becomes equal to the sound velocity, and then decreases with the further increase in the strength of the d.c. electric field va > vs. In the absence of the OEF, our expressions for the relative flux in the second harmonic reduce to those of Wu and Spector [5]. The harmonic yield in the absence of OEF, first decreases slightly in the range 0 < v~/vs < 1 and then shows a slight increase for higher values of vd/v~ (i.e. vd > v~). Further the second harmonic power is enhanced by a factor of 103 over its value in the absence of an OEF, for an OEF having an amplitude of twice the value of d.c. field, and when the frequencies of the OEF and acoustic wave are equal (F = 2, A = I). 4. CONCLUSIONS
From the numerical results presented in Section 3, the following conclusions can be drawn: (i) The second harmonic acoustic flux can be enhanced considerably by the application of an oscillating electric fields, (ii) The second harmonic acoustic flux exhibits a maximum when the frequency of the applied oscil-
1137
lating electric field is equal t o the frequency of the acoustic wave, (iii) The second harmonic yield also shows a maximum when the carrier drift velocity is equal to the velocity of sound in the presence of the oscillating electric field. It shows a minimum in the absence of the OEF for the same value of the drift velocity. Therefore, the present study suggests that the second harmonic yield can be optimized by the suitable choice of various parameters e.g. amplitude and frequency of the oscillating electric field, the strength of the externally applied d.c. electric field. Because of the non-availability of the relevant experimental data, the present results cannot be verified experimentally.
Acknowledgements--The authors are thankful to Professor M. S. Sodha and Dr. S. K. Sharma for their helpful suggestions. REFERENCES
1. Tell B., Phys. Rev. 136, A772 (1964). 2. Mauro R. and Wang W. C., Phys. Rev. B1, 683 (1970). 3. Zemon S. and Zucker J., IBM Z Res. Dev. 13, 494
(I~). 4. Palik E. D. and Bray R., Phys. Rev. B3, 3302 (1971). 5. Wu C. C. and Spector H. N., J. appl. Phys. 43, 2937
(1972). 6. Spector i-I.N., Phys. Rev. B6, 2309 (1972). 7. Pantell R. H. and Sochoo J., f. appl. Phys. 41, 441 (1970). 8. Epshtein E. M., Soviet Phys. Solid State I0, 2325 (1969). 9. Spector H. N., Solid State Phys. 19, 291 (1966) (see more references in this article). 10. The expansion of the electric field is done under a perturbation approach, i.e.
Eo~E,.o.; Et >E.., E,, E = > - • -