Vo H o ~ a ANH and NQUYEN NHU DAT: Parametric Excitation in Semiconductors 585

phys. stat. sol. (b) 86, 585 (1978)

Subject classification: 6 and 13.5.2; 20; 22

Laboratory of Thoretieal Physics, Institute of Physica, Centre for Scientific Research5 of Vietnam, Hanoi

Parametric Excitation in Kane's Semiconductors BY

Vo HONG ANH and NGUYEN NHU DAT

The parametric excitation of electromagnetic vibrations in narrow-gap semiconductors with non. parabolic energy dispersion defined by Kane's model is investigated using the quantum formalism proposed previously. The numerical estimates performed for the typical sample of n-InSb crystals in different cases show that the deviation from parabolicity lowers the instability threshold field values by about one order of magnitude.

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1. Introduction

As it is well known, the parametric excitation in solid state plasmas is intensively studied in the last decade.

I n this phenomenon the energy of external electromagnetic wave field is transferred to the system under consideration by a resonant mechanism that takes place when the field amplitude is large enough to cause the vibration (with the external field frequency) of certain physical parameters of the system. In the case of relativistic systems such a vibrating parameter is the mass of particles that depends upon their velocity in the radiation field. This effect was first considered by Tsintsadze [2]. Later it was pointed out [3] that in the plasma of narrow-gap semiconductors in which the electron energy dispersion is expressed in Kane's model [4] by the pseudorelativistic formula

E ( p ) = I/cmc*",2 - p 2 ~ * 2 , (1)

where m and p are the particle effective mass and canonical momentum, respectively, c * = with E, being the energy gap value, there realizes the same excitation mechanism. The ,authors of [2, 31 have investigated this case for a single electron system in the hydrodynamical approximation without taking into account the thermal motion effects.

The purpose of this paper is to make an account of the effect of non-parabolicity of the type (1) in the parametric excitation process in interacting electron-phonon sys- tems. The calculations will be carried out in the framework of the quantum formalism proposed in [l]. Analytical results for instability growth rates will be presented for interacting electron-phonon (Section 3) as well as for single electron (Section 4) systems. For comparison with the results presented in [l, 5,3], the numerical estimates for the threshold field values have been performed for a sample of n-InSb crystal. 78.

586 Vo HONQ ANII and NUIJYEN NHU DAT

2. Equations for Potentials of Electromagnetic Perturbation Field

Let us consider an interacting electron-phonon system under the action of a strong radiation field that can be presented in the well-known dipole approximation by an oscillatorv electric field

c For arbitrary electron energy dispersion i5 and in the presence of the

external field E,(t), the Hamiltonian of such a system introduced in [ 13 takes the form

Here ~ ( p ) = aS(p)/ap is the electron velocity. All other notations are the same as in [l].

Introducing the “quantum distribution function” in the form

(a&,+,) = np(t) L o + f ( ~ + q, P , t ) ; np(t) = (a,+apit , (4)

one can obtain by using the Hamiltonian (3) the linearized equation of motion for f ( P + Q, P, t ) as

( 5 )

e e 2c

Following the work [3], we shall consider np(t) as an equilibrium Fermi distribution function.

Now we must solve the complete system of Maxwell equations with charge density e and current density j expressed through the function f as follows:

Parametric Excitation in Kane's Semiconductors 587

Introducing the transformation - X +. X = X exp { - i (A sin mot + 9 sin 3w0t) ,

where

for the quantities f , A, (p, j, e, one obtains the system of equations for the Fourier components of d and @:

x {&(a)) wj.) + Loo) - & L ( O ) A"(@ + Lwo) + B " S ( 4 [A"@ + (L + 2) 0 0 ) + L

for q I Eo, where the following notations have been introduced :

588 Vo HONG ANH and NGUYEN NHU DAT

pF = mvF is the electron Fernii niomentum, J1 the Bessel function of the first kind; n, n’, 1, l ’ , 11, 1; are integers. In obtaining (7a), (7b) we have set p < 1, p2 < 1 and also taken into account the

fact that the quantities lRLl < 1, < 1 when i + j as they represent the emall eontribution of non-parabolicity effects. In agreement with this only the terms linear with respect to the mentioned quantities have been retained.

3. Instability Analysis for Coupled Electron-Phonon Systems

I n thissection, proceeding from (7a), (7 b), we obtain the dispersion relations for elec- trostatic and electromagnetic waves in an electron-phonon system and solve for pos- sible instability growth rates of these waves in different cases.

We shall work in the rectangular system of coordinates with z-axis parallel to the wave vector q and the electric field vector E, lying in the xz plane. As the parameters 2, rl, ql,q2involved in (7a), (7 b) are much smaller than unity, infurther calculationsonly terms linear in them will be retained and in accordance with this one will have J,(I) = = J,(V) = 1.

The analysis will be done in the high-frequency long-wavelength limit (qoF < w, .q < qFT = wp/oF) and in the so-called “two-mode approximation” (see [ I ] ) .

For zero external field (E, = 0 ) one can obtain from the system (7 a), (7 b) the expres- sions for the frequencies w1,2 of the two longitudinal (electrostatic) and wl,2(q) of the two transverse (electromagnetic) eigenmodes involving the terms due to the non-para- bolicity effect. Thus, one has

Parametric Excitation in Kane’s Semiconductors 589

where wL is the longitudinal optical phonon frequency and

The expression for w?,z(q) has the same form as (8) with

For E, + 0, q I I E, the system (7) leads to uncoupled dispersion equations describing longitudinal and transverse waves. The equation for longitudinal modes is

~ ( w ) ~ ( w - w , ) = {Jl(A) [ E ( o - ~ ~ ) - E ( ~ ) ] + P ~ ~ ( w - ~ ~ ~ ) - P ~ ~ ( w - w,)}~, €(a)) 3 E(W) - Po,:!(w),

(E(w) is the lattice dielectric function) that yields the growth rate y (the resonant couling modes are: w = w2, w, - w = w1 and so wo = w1 + w,, w, > w,) in the form

y = - eE,q 4m eW(w1 + wd2

For transverse modes (w = wl(q) , w, - w = w,(q)) we have the dispersion relation

A*(o) A*(o - w,) = {Jl(A) [A(o) - A(o - o,)] +

where

I.

Q(w) = &8:8(w)

(note that QT = Q%u 3 Q L for q 1 1 E,). the growth rate in this case is expressed by the formula

As it i s seen, in the case of parabolic energy dispersion the growth rates (10) and (12) coincide with those calculated in [l].

We now turn to the case of q 1 E,. Equations (7) show that there exist purely electromagnetic (transverse) perturbations polarized perpendicularly to Z0(E 1 E,) and coupled electrostatic (with E 1 E,) -electromagnetic (with E 1 1 E,) ones with “mixed” polarization: The usual procedure leads to the following growth rate for-

590 Vo HONQ ANH end NQUYEN NHU DAT

mula for the purely transverse modes (with 0 = 02(q), 2w, - o = wl(q)) :

Equation (13) shows that the excitation in this case is caused essentially by the non- parabolicity effect. For the coupled plasmon-polariton modes (o = 02, coo - w = ol(q)) the growth rate y has the form

(0: - b H - 4 q ) l )1’2 x y = - eEoq w$

4m - 5 c 4 [ 0 2 + ~l(Cf) l wf l l (q ) (WE - 0;) [WE(@ - w4(q)]

The numerical values of the threshold fields are estimated in the same way as shown in [l]. For this purpose a sample of n-InSb crystal was taken. This is a typical narrow-gap semiconductor that may be described by the electron energy dispersion formula (1). The results are

1. for the longitudinal modes with q 1 1 E,:

Eoth k: 10’ V/cm for q = 5 X lo6 cm-1 , ( 15)

EOth k: 1W V/cm for q = 5 x 1oS cm-1. ( 16)

2. for the coupled electrostatic-electromagnetic modes with q 1- E,:

Here the following data have been employed (see [l, 51): n = l0le cm-8, m = O.O1me, Eg = 0.234 eV, = 9, c0 = 27, wL = 1013 s-l, cop/&2 = 2 x 101s s-1, the damping constants: r p h = 7 X lolo s-l for phonons and rp = 1.7 X

Comparison of (16) to (16) with the results obtained in [ l , 51 shows that the non- parabolicity effect makes the threshold field values lowered by several times in all cases.

s-l for plasmons.

4. Single Electron Systems

The problem of parametric excitation in single electron plasma can be treated by

The plasmon frequency oP+ in the absence of any external influence is then obtained using equation (7) with E ( W ) = 1.

from ~ ( o ) = 0 and has the form

while the frequencies of electromagnetic modes are determined by A * ( o ) = 0 and are

&(q) = m y + q w . (17b) The instability analysis in the presence of the radiation field E,(t) is carried out

analogously to the case of electron-phonon systems. The results for instability growth rates and threshold fields are (with TI, = lo1’ cm-s, up = 2.7 x 1014 s-l and the effec- tive collision frequency of electrons y e ~ = 6 x 10lls-l [3]):

1. For the plasmon mode w = COP+ with Q 1 1 E, when w, = oQ:

Parametric Excitation in Kane’s Semiconductors 591

2. for the purely electromagnetic mode ol (q ) with q 1 1 E, when w, = w l ( q ) :

3. for the electromagnetic mode w l ( q ) with q 1 E,, E I Eo (w, = ol(q)):

4. for the coupled electrostatic-electromagnetic mode w$ with q J- E, (0, = u?), the growth rate y and the threshold field Eo th are determined by (18).

As i t is seen from (18) to (20), the results for the growth rate expressions coincide with those calculated in [3] if the thermal motion effect is neglected. So it is clear that this effect has made the threshold field values somewhat lower than those obtained in [3].

References [l] Vo H o ~ o ANH and NQUYEN VAN TRONQ, phys. stat. sol. (b) 83, 395 (1977). [2] N. L. TSINTSADZE, Zh. eksper. teor. Fiz. 69, 1251 (1970). [3] N. L. TSINTSADZE and V. S. PAVERMAN, Fiz. tverd. Tela 14,3427 (1972). [a] 0. E. KANE, J. Phys. Chem. Solids 1, 249 (1957). [5] N. Tmm, Phys. Rev. 164,518 (1967).

(Received January 27, 1978)