K. L. JAT and S. GHOSH: Parametric Amplification of Acoustohelicon Waves 583

phys. stat. sol. (b) 165, 583 (1991)

Subject classification: 78.20; 78.45; S7.13

Department of Physics, Government P. G. College, Neemuch and School of Studies in Physics, Vikram University, Ujjain‘) ( b )

Parametric Amplification of Acoustohelicon Waves in Semiconductors 2,

BY K. L. JAT (a) and S. GHOSH (b)

The parametric amplification of acoustohelicon waves is analytically investigated in a longitudinally magnetised semiconductor of centrosymmetric nature. The threshold value of the pump electric field (EOth) and initial gain constants are obtained. The longitudinal magnetic field decreases the required magnitude of (EOIh) for the excitation, whereas it increases the gain constants. Numerical analyses are performed for an InSb crystal at 77 K duly irradiated by frequency-doubled pulsed 10.6 pm CO, lasers. It is found that the gain constants for both the couplings (gb) and for piezoelectricity only (g,) are nearly equal and approximately 100 times higher than that for deformation potential coupling (gd).

Die parametrische Verstarkung von Akustohelikon-Wellen wird analytisch untersucht fur einen longitudinal magnetisierten Halbleiter mit zentrosymmetrischer Struktur. Der Schwellenwert fur das elektrische Pumpfeld (EOIh) und die zugehorigen Verstarkungskonstanten werden erhalten. Das longitudinale Magnetfeld erniedrigt die fur die Erregung notige GroRe von (EOth), wahrend es die Verstiirkungskonstanten erhoht. Numerische Rechnungen werden durchgefuhrt fur einen InSb-Kristall, der bei 77 K hinreichend bestrahlt wird durch frequenzverdoppelte, gepulste 10,6 Frn C0,-Laser. Es wird gefunden, daD die Verstarkungskonstanten fur beide Kopplungen (gb) und fur nur piezoelektrische Kopplung (g,) fast gleich sind und etwa 100-ma1 groljer als fur Deformationspotentialkopplung (gd).

1. Introduction

Out of a number of basic phenomena of nonlinear nature, the phenomenon of parametric interaction has an important place in nonlinear optics, especially in generating tunable laser light at a frequency not directly available from a laser source [l]. The parametric interactions in a nonlinear medium are responsible for the construction of parametric oscillators, amplifiers, optical phase conjugators, etc. [2]. It is known that the origin of parametric interaction lies in the second-order optical susceptibility I(’) of the medium. Recently there has been considerable activity in the field of parametric excitation of acoustic waves in semiconductors [ 3 to 61.

To our knowledge no attempt has yet been made to obtain the second-order susceptibility arising due to mobile electrons in a magnetised, doped semiconductor when a dc magnetic field is applied parallel to the direction of propagation. However, Neogi and Ghosh [6] have studied the second-order susceptibility originating from the finite induced current density in a transversely magnetised n-type semiconductor, and reported dispersive as well as absorptive characteristics. Motivated by this work and the intense interest in the field of second-order susceptibility, in the present paper we have attempted to study the effect

’) Ujjain 456010, India. ,) Work partially supported by DST India under a research project

584 K. L. JAT and S. GHOSH

of a dc magnetic field applied along the direction of propagation on the parametric gain constant and second-order susceptibility in an n-type semiconductor crystal having both piezoelectricity and deformation potential couplings. We have chosen the said geometry since it helps in propagating an acoustohelicon mode through the medium, which is of substantial importance in investigating the fundamental properties of the crystal.

2. Theoretical Formulations

To study the parametric amplification process in a longitudinally magnetised doped semiconductor, arising due to nonlinear effective optical susceptibility x ( ~ ) , the hydrodynamic model of a homogeneous semiconductor plasma of infinite extent is considered. A spatially uniform pump electric field (i.e. pump wave vector (kol z 0) Eo exp ( - h o t ) and a dc magnetic field B, are applied to a semiconducting plasma, parallel to the wave vector k (along x-axis). The applied pump wave Eo gives rise to a time-varying electrostrictive strain and is thus capable of driving the acoustic waves in the medium. Assuming the acoustic wave generated internally to be a pure shear wave propagating along the cubic axis (001) of the crystal, the lattice perturbation u is then along the cubic axes (110) and (110) of the crystal. The equation of motion of the lattice in the crystal with piezoelectric as well as deformation potential couplings is then given by

a Z u , au, aE, C,F a2E, a2u e 7 + 2y,e- + p- + ~ __ = c y at at ax e ax2 ax2

and

where e is the mass density of crystal, y, the phenomenological damping parameter of the acoustohelicon mode and C the crystal elastic constant, Cd the deformation potential constant, and /3 the piezoelectric constant.

The other basic equations used in this analysis are

a#, e

at m - + V U ~ = - - (Eo + UO x Bo),

ax1 an, ax at '

uo ~ + no div u1 = - - (4)

ao at

V X H = J + - - ,

D = & E + P

Parametric Amplification of Acoustohelicon Waves in Semiconductors 585

These equations and notations are explained in the paper of Neogi and Ghosh [6] except for the additional terms on the left-hand side of (1) and (5 ) representing the contribution due to deformation potential coupling.

In a doped semiconductor, the low-frequency phonon mode (w) as well as the pump electromagnetic mode (0,) produce a density perturbation (nl) at the respective frequencies in the medium which can be obtained by using the standard approach [7] on assuming that the low-frequency perturbations are proportional to exp (i(kx - wt)), while the high-frequency perturbations vary as exp (--coot). Using (l), (2), (4) in the collision dominated regime (v % w ; k . u,), we get

where l7 = - (i) E , and 8 = w - k . u,; up [= (noe/rn~,~,)”2] is the plasma frequency, . .

w, [ = eB,/rn] the electron cyclotron frequency, and cL [ = ( , u , E , E ~ ) ~ / ~ ] the electromagnetic wave velocity in the crystal with lattice dielectric constant E ~ . In obtaining (9) we have taken u+ = uy t iuz, v + = v, -t iv,, B , = B, f iB,, and E , = E, iE,, where plus and minus signs correspond to the right- and left-handed circular polarizations. Here both vy and v, represent the sum of the slow and the fast components of u along y- and z-axes. The perturbed electron concentration n, also has components known as slow and fast; the slow component (n,) being associated with the phonon mode and the fast component (n,) with the high-frequency electromagnetic wave, arising due to the three-wave parametric interac- tion, will propagate at generated frequencies w and wo w, respectively. For these modes, the phase matching conditions wo = w , + w and k , = k , + k, known as the energy and momentum conservation relations, should be fulfilled. From (9) we obtain

here w , = wo - w and 6; z w;O/w (assuming w, % v, 6; which is one of the conditions for helicon wave propagation). In deriving (10) we have restricted our analysis to the Stoke’s component (w, - w ) only of the scattered electromagnetic wave. It may be inferred from (10) that components of the density perturbation are coupled to each other via the pump electric field E , ( B = -eE,/m). Using (1) and (9) we get

i e k 3 n , ~ w v ~ ( K 2 + L2k2) E , m(k2cZ - w2) (w2 - v:k2 + 2iy,w)

n, =

_ _ - wft82 w(k2cZ - w’) (6‘ + iv) ( S 2 + ivw,)

586 K. L. JAT and S. GHOSH

where

v, being the sound velocity in the lattice. It is evident from (11) that n, depends on the pump intensity (I) which is given by

= t v w , IEOl2 > (12)

where q is the background refractive index of the crystal, E, the absolute permittivity, and co the velocity of light in free space.

In this present investigation, in order to study the effect of nonlinear current density on the induced polarization in a longitudinally magnetised doped semiconductor, the effect of the transition dipole moment is neglected while analysing parametric interaction in the crystal. The Stoke’s component of the induced current density is given by

J , = -n,*eu,. (13)

Thus in quasi-static approximation, (11) and (13) yield the components of J , as

oio J i = ieswikv:0EoE$(K2 + L’k’) 1 - - - k’cZ(6’ - iv) O;(S2 - ivo,)

(14)

P , = J J i d t . (15)

Treating the induced polarization P , as the time integral of current density J we have

Now the second-order optical susceptibility can be obtained by defining the induced

(16)

polarization at frequency (0,) as

P,(wi) = COXEN o * . (2)E E*

The effective nonlinear optical susceptibility in the coupled mode scheme obtained by using (14) to (16), neglecting the induced polarization due to transition dipoles, becomes

, where

e e , k 3 v ~ o ~ G ( K 2 + L’k’) A =

2mYsWoW1 9

& j1 = 6’, pz = v , y 1 = 6’, y 2 = vo, , and E , = ~

80

The above formulation reveals that the total crystal susceptibility is influenced by the carrier concentration through 0, + 0 and by the longitudinal magnetostatic field through w, 4= 0, which is one of the preconditions to evoke a helicon mode in the crystal.

Parametric Amplification of Acoustohelicon Waves in Semiconductors 587

Transforming (1 7) into

X&3 = -’”: + m ( Y : + Y 3 - b(YI + Y I ) (PI - $ 2 ) - c(BI + B:) (rl - 0211

x [{a(PlYl - P 2 Y 2 ) - by1 - C P I Y + ( 4 P 2 Y I + P l Y J - b y 2 - c P 2 ) ’ I r ’

(18) the second-order susceptibility appears in a form, split into real and imaginary parts,

(19) ( 2 ) - ( 2 ) ( 2 ) XEN - XEN, + ~ X E N ~ .

The real part describes the parametric dispersive characteristics of the scattered acous- tohelicon wave in the crystal. It reveals that in the case of a laser driven acoustohelicon mode excitation only negative nonlinear dispersion characteristic occurs which leads to self-defocusing of the signal in the material. The above expression also reveals that the dispersion characteristics is heavily influenced by the applied magnetic field.

To obtain the parametric excitation of the acoustohelicon wave in the presence of the high-frequency oscillating electric field, the imaginary part of (1 8) is substituted in the expression for the parametric gain constant given by

where gEN is the effective nonlinear absorption coefficient. The non-linear growth of the signal (ol = wo - w ) as well as the idler (w) modes are possible only if aEN obtained from (20) is negative.

It is necessary to determine the threshold value of the pump amplitude for the onset of the parametric process in order to study the parametric amplification. This is obtained by setting x&$, = 0, which yields the expression for the threshold electric field (EOth) as

mw,6, (.sf - 0:)’ + v2w: v2 + 0: (EOth)

~ ek2c, [: { This reveals that application of a longitudinal magnetic field reduces the value of EOth,

in contradiction to the result obtained by Guha et al. [8]. The parametric growth rate well above the threshold electric field can be obtained from

(20) in the presence of both piezoelectric and deformation potentials (gb ) , for deformation (gd) only, and for piezoelectricity (g , ) only; they are given by

g b = [ A B { b P Z ( y ? + ”?$) + c Y 2 ( P : + 8:))1 x [{4PlYl - P 2 Y 2 ) -by , - C P J 2 + {a(P,y,+ B 1 Y 2 ) - by2 - cP2}21-1 3 (22)

(23) g , = L2k2gb(K2 + L 2 k 2 ) - ’ ,

588 K. L. JAT and S. GHOSH

\ Z

-x- - r -x- *- - -1- - - + --x .-_ -x-

It can be inferred from (22) to (24) that the parametric gain constants are dependent not only on the dc magnetic field but also on the magnitude of the applied laser field. In the absence of the laser field (i.e. E,) the gain constants become zero. Thus the presence of a laser field is the precondition to achieve parametric amplification.

Fig. 1. Threshold field Eorh vs. mag-

3. Results and Discussion

A numerical analysis has been performed to study the parametric growth of the acous- tohelicon wave in n-InSb at 77 K with the help of IBM PC/AT. The physical constants used are no = 2 x loz4 mP3, m = 0.014m0, us = 4 x lo3 m s-', v = 4 x 10" s-l , p = 0.054 CmP2, Cd = 7.209 x = 18.0, y, = lO-"s, and 4 = 3.9; m, is the free electron mass.

We have studied the effect of the longitudinal magnetic field (in terms of w,) and the wave vector k on the threshold electric field in the piezoelectric and deformation potential semiconducting crystal. Fig. 1 shows the variation of Eoth with the magnetic field (in terms of w,) at a constant k = 2 x lo6 m-l. It is seen that Eoth decreases rapidly with w, up to w, 2 2 x l O I 3 s-l, and becomes nearly constant for w, > 2 x 1013 sP1. The variation of Eoth with wave vector k is shown in Fig. 2 at w, = l O I 3 s-'; Eoth decreases rapidly with k up to k & 5 x lo6 m-', and becomes independent of it for k > lo7 m-'.

Considering the various ranges of pump intensities well above the threshold EOth, the gain constants for all the three types of couplings (i.e. piezoelectric g,, deformation g,, and both g,) can be estimated, and their variation with other parameters can be studied. One may infer from Fig. 3 that the gain constants g, and g, rapidly decrease with k, gd decreases exponentially. But at higher values of k all the three gain constants become independent of k. The values of g, and g, are nearly equal, but always found to be greater than gd. It is also clear from Fig. 3 that the effect of the deformation potential is comparable with the piezoelectric coupling at higher frequency. Fig. 4 shows the variation of gain constants with magnetic field (w,); they increase monotonously with w,. It is also inferred that at higher values of w, (> they increase more rapidly, but we cannot increase w, infinitely because of the restriction w t < w i imposed in our analysis.

J, = 15.8,

t 5 1

I I 1 I 1 I I -I

589 Parametric Amplification of Acoustohelicon Waves in Semiconductors

1 - 4 1

S I

3 1 s u030-

1°-

10-

0

I I I I

- 40-7

- 1

I - I I

- I I I I I

- 1 I \

-

- \ \

- ‘x

‘x y- -x-x- -*-- - - -,- - - - -4 . 5 I0 15 20

Fig. 4

Fig. 4. Parametric gain constants (gbr g,, and gd) vs. magnetic field o, at E , = 10 V/m and k = lo7 m - Fig. 5. Parametric gain constants (gb, gp, and gd) vs. pump amplitude E , at k = lo7 and o, = 1014~-1

590 K. L. JAT and S. GHOSH: Parametric Amplification of Acoustohelicon Waves

Fig. 5 shows the variation of gain constants with pump amplitude E , (E , > EOth); all three decrease exponentially with E,. It is also found that g, and g, are nearly equal and approximately lo2 times higher than g,.

From the above discussion it is clear that the required threshold electric field Eoth = 1.708 x lo9 V/m corresponds to an excitation intensity I = 5 x 1 O I 2 W/cm2. Such a pump intensity is very easily obtained by using double cw 10.6 pm CO, lasers. The most important conclusions obtained in the above study are as follows:

1. The applied magnetic field linearly increases the values of all three gain constants, while it reduces the required pump intensity for excitation of the acoustohelicon wave.

2. The gains with piezoelectricity g, and with both the couplings g, are nearly equal, but these gain constants are higher than the gain constant with deformation coupling only

3. Eoth and all the three gain constants rapidly decrease with k at lower values of k, but become constant for higher k.

Thus one may conclude from the above discussion that magnetoactive, highly doped semiconductors, having both coupling mechanisms, are the best choice for the fabrication of parametric backward acoustohelicon wave amplifiers.

k d ) .

Acknowledgement

The author (KLJ) acknowledges the encouragement received from the principal, Gov- ernment P. G. College, Neemuch-458441 (MP).

References

[ l ] V. VASILYAUSKAS, A. PISKARSKAS, V. SIRUTKAITIS, A. STRABINIS, and A. YANKAUSKAS, Izv. Akad.

[2] A. PISKARASKAS, A. STABINIS, and A. YANKAUSKAS, Soviet Phys. - Uspekhi 29, 869 (1986). [3] R. BUKOWSKI and Z. KLESZCZEWSKI, Acustica 56, 48 (1984). [4] S. N. GURBATOV and N. V. PRONCHATOV-RUBTSOV, Soviet Phys. - Acoust. 31, 281 (1985). [S] P. AGHAMKAR, P. SEN, and P. K. SEN, phys. stat. sol. (b) 145, 343 (1988). [6] A. NEOCI and S. GHOSH, phys. stat. sol. (b) 152, 691 (1989). [7] S. GUHA, P. K. SEN, and S. GHOSH, phys. stat. sol. (a) 52, 407 (1979). [8] S. GUHA, P. K. SEN, and S. GHOSH, Phys. Letters A 69, 442 (1979).

Nauk SSSR, Ser. fiz. 50, 32 (1986).

(Received June 11, 1990; in revised form December 7 , 1990)