Short Notes K131

phys. stat. sol. (b)103 Kl31 (1981) Subject classification: 13 and 20; 22.2.3 Department of Physics, Ravishankar University, Raipur (a) and Department of Physics, Heriot-Watt University, Riccarton, Currie, Edinburgh 1,

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Parametric Decay of Alfvgn Waves in Piezoelectric Semiconductors

BY S. GUHA (a) and P. S E g ) (b)

The nonlinear behaviour of Alfv6n waves has recently received considerable attention. The nonlinear decay of an Alfvgn wave into fast and slow acoustic waves has been studied by Karplyuk and Oraevskii /l/. Lehane and Paoloni /2/ have observed experimentally the parametric excitation of an Alfvkn wave by a pump magnetic field and found that the maximum amplification of the wave oc- curs at pump frequency twice the excited wave frequency. Hung /3/ has in- vestigate# the decay of an Alfvh wave into second Alfvkn and an ion acoustic wave. It is found that the instability occurs with standing a s well a s the propa- gating pump wave. Most of the earlier work has been done in gaseous plasma in which there is a severe limitation on the frequency of the Alfvh waves a s it should be low compared with the ion cyclotron frequency. The cyclotron fre- quency of holes in solids is appreciably greater than that of ions in gaseous plasma. Hence, a larger frequency range of Alfvbn waves stimulates interest in the study of the decay of anAlfv6n wave in solid state plasmas. Guha et al. /4/ have studied theoretically the modulational instability of Alfvgn waves in bismuth.

In the present paper we have analysed the possibility of the decay of an Alfvbn wave into an acoustic and another Alfvh wave in a piezoelectric semi- conductor. The instabilities have been observed only with the propagating pump wave under the perfect frequency matching condition. The analysis yields an interesting result that at ve = vh, the instability disappears.

piezoelectric semiconductor plasma subjected to an externally driven Alfvkn wave propagatlng along the direction of the applied magnetostatic field. The

We consider the fluid model of a compensated, homogeneous, magnetised

1) Edinburgh EH 144AS, Great Britain. 2) To whom all the correspondence should be sent.

K132 physica status solidi (b) 103

excited acoustic wave as well as the scattered Alfven wave are assumed to prop- agate along the direction of the magnetostatic field. The nonlinearity arising due to the second order te rms in the basic equations has been taken into account. The fluid model used here restricts the validity of our analysis to the limit kl < 1 where k' is the wave vector and 1 is the electron mean free path. The basic equations used are

Equations (1) and (2) a re momentum transfer equations for electrons and holes, respectively. so is the applied magnetostatic field, -e(te) and m,(mh) a re the charge and effective mass of electron (hole). In the continuity equation (3) no and n represent the unperturbed and the perturbed car r ie r densities. Equations (4 )

and (5) a re Maxwell's equations in which E and po describe the dielectric con- stant and the absolute Permeability. The electric displacement 6 in piezoelec- tric semiconductor is defined by (6). We consider that the motion of lattice is governed by (7) in which u, e ) c, and @ represent the lattice displacement, density, elastic constant, and piezoelectric coefficient of the crystal, respec- tively. Equations (1) to (5) a r e arranged in such a way that the terms on the right

Short Notes Kl33

hand side of the equation represent the nonlinear contribution.

O), Efo = (0, Efo, O),and Bfo = (0, 0, B 1. Applying the coupled mode theory and using the linearised forms of basic equations, the normal mode equations a r e obtained a s

.c The pump wave is assumed to have the polarisation defined as kfo = (kfo, 0, * -b

fo

and

+ + a a: a t - s s s s - + i w a - + a a - = O ,

where the subscripts fo, fl , and s stand for the quantities describing the pump

A l f v h wave, excited Alfvh wave, and acoustic wave, respectively. afo, fl, and

of the wives. In deriving these normal mode equations we have assumed the low collisional damping of waves. The space and time dependence of the mode ampli- tudes has been takenas

represent the normal mode amplitude and the collisional damping rate ffo, fl , s

+

+ being the wave vector of the corresponding mode. The superscript + and

- represent the forward and the backward propagating mode, respectively. It is

worth mentioning that the combination of modes la+l = la-1 denotes a standing wave whereas the absence of backward propagating wave ( la-1 = 0) represents a propagating wave.

) and also assumed that w s v s w In the case 2 2 mation (k C wfo,fl,s fo 1 fo,fl eh’ s’ of non-piezoelectric semiconductor, we found that the damping coefficient of acoustic waves becomes of the order of hole collision frequency ( v ~ , ~ ) . Thus according to the approximation taken ( v > us), we get a highly damped acous- tic wave in the medium. Hence the analysis used here is strictly valid for piezo- electric semiconductors only.

kfo, f l , s

In deriving the normal mode equations we have used the quasi-static approxi-

e, h

W e now address ourselves to the nonlinear interaction by retaining the non- linear terms in the basic set of equations. We take into account only those non- linear terms which have the same space and time dependences as the normal

K134 physica status solidi (b) 103

mode concerned on the right hand sides of equations - by using the following forms of energy and momentum conservation rules:

* w = w + ~ and k = ~ + ~ s . fo f l s fo f l

As a result we get a set of coupled equations

2 io k (1 - ve/vh)Cfo ce s [at (a t )*- aio(a,)*] (9) a ail t i -

fo s a t 2 iwflafl - a a- = Of1 % f l f l

and

12iwcekf0(l 2 - ve/v )C C i + h fo f 1 k (a+)*- a+- (a-)*],

a a: - 2 fo fl fo f l fo

a t + i w a- + 7 a- = - s s s s w

o dium, and w

above set of equations we have neglected kfo/ofo in comparison with k /O o as

being the electron plasma frequency, C1 is the velocity of light in the me- Pe

represents the electron cyclotron frequency. In deriving the 2 ce

s f l s kfo-ks and o <c o s fo'

Equations (9) and (10) contain the phase variations of the normal modes. In order to obtain coupled mode equations for slowly varying amplitudes, we write a = A t (where the subscript f = fo, f l 1 Af,

being the time dependent mode amplitude and is mathematically represented by exp i [kf, sx - w f , s f, s f, s 1

A = a exp i(- wi t) , f, s f , s

where w. represents the magnitude of imaginary part of frequency. Using the ap- proximation of constant pump power and introducing the combination

1

A> = (Afo) + * + Afl and A; = (Aio)*Afl . W e obtain the following Set of coupled equations:

io2 ce k s (1 - ve/vh)Cfo [& + a,)+ =

w w f l s

Short Notes K135

The purpose of the above choice of combination of amplitudes, is to solve the set of four coupled equations. However, the resulting equations (11) and (12) also give an insight of the effect of propagating and stationary pump waves on the present excitation. The explicit explanation of this effect has been explained later.

The equations (11) and (12) are solved by taking an exp -i(wit) dependence of mode amplitudes. A straightforward mathematical analysis yields the dis- persion relation as

4(1 - ve/vh’2u~e k k s fo C 2 C fo f l

2 fo f l s

( - iwi + r ) ( - iw. + y ) = w w w f 1 s

The above dispersion relation contains entirely the imaginary parts of frequen- cies as the real parts have been already removed while obtaining (11) and (12). It can be,observed clearly that the excited waves are coupled via the pump wave amplitude I Aio I . In case of a standing pump wave ( I A’ 1 = IAio I 1, the coupling term vanishes whereas it remains finite for a propagating pump wave ( [A- I =O) .

This result can be compared with that of Hung /3/, if perfect frequency matching condition is applied in his case. The other interesting result observable from the dispersion relation is that at a car r ie r temperature, where the electron collision frequency equals the hole collision frequency ( ve = v,), the coupling term vanishes which consequently shows the absence of instability.

fo fo

The dispersion relation is now employed to obtain the threshold value of the

pump field required for the onset of instability by equating wi to zero, substi- tuting the values of rfo, fl and rs and writing the mode amplitudes in terms of electric field amplitude of the waves and one gets

where Eth is the threshold value of the electric field amplitude of pump wave re- quired for the onset of’instability. In obtaining the above expression we have as= sumed that the pump wave is propagating in nature. In order to achieve the in- stability at much low values of electric field amplitude of the pump wave, one can use the following conditions which a re evident from (I 4): The medium should be

of moderate piezoelectric characteristics and it should be immersed in moder-

Kl36 physica status solidi (b) 103

ab ly low magnetostatic fie1d.A larger car r ie r concentration will further help in achieving low values of threshold electric field amplitude of waves. The medium with low values of collision frequency will be always favourable.

The growth rate of the unstable mode well above the threshold can be ob- tained from the dispersion relation (13). We consider the fact that at very high values of electric field amplitude (higher than the threshold), the damping is completely overcome and hence, neglecting the damping term, we get the growth rate

In obtaining the growth rate, the propagating nature of the pump wave has

been taken into account. The best results can be achieved by using moderately low values of applied magnetostatic field and by choosing large car r ie r concen- tration in the medium.

The numerical estimates have been made for InSb at 77 K. The physical con- - .

stants chosen a re me-= 0.013 mo, m = 0 . 4 m do = 4 . 4 ~ 1 0 " syand the pump frequency is taken as ofo 0 10l2 s-'. The thresh-

2 = 1 . 3 ~ 1 0 - ~ , w = 1 . 4 ~ -1 h O' -1 ce 13 -1

s , w = l . l 8 d 0 l 4 s-', vs = 4 . 0 ~ 1 0 3 m s , ve = 3 . 5 ~ 1 0 1 1 s , vh =

old electric field is found to be of the order of 1 , 9x104 Vm-', and the growth rate at an electric field % 1 0 Vm 5 -1 10 s-l is found to be w 2 . 5 ~ 1 0

This work is partially supported by U.G.C., India, under the research pro- ject "Instabilities in Solid State Plasmas". One of the authors (P. S. ) is grateful to CSIR (India) for the financial support during the progress of the work. She is also grateful for Prof. P.G. Harper, Department of Physics, Heriot-Watt Uni- versity, for kind hospitality and encouragement. The discussions with Dr. P. K. Sen and Miss N. Apte are also gratefully acknowledged.

References /1/ K. KARPLYUK and V. ORAEWSKII, Soviet Phys. - J. exper. theor. Phys.,

/2/ J.A. LEHANE, and F.J. PAOLONI, Plasma Phys. -7 12 461 (1972).

/3/ N. T. HUNG, J. Plasma Phys. l.4, 445 (1974). /4/ S. GUHA, P.K. SEN, and S. GHOSH, phys. stat. sol. (b) 91, Kl35 (1979).

Letters - 5, 365 (1968).

(Received November 26, 1980)