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A theory of parametric resonance in semiconductors
0038-1098/83/080697-05503.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 45, No. 8, pp. 697-701, 1983. Printed in Great Britain.
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS I.E. Aronov and O. N. Baranetz Institute for Radiophysics and Electronics, Academy of Sciences of the Ukrainian SSR, Kharkov 310085, U.S.S.R.
(Received 26 August
1982
by E.A. Kaner)
A theory of parameteric resonance (PR) in semiconductors is constructed with an account of the interaction between electrons and volume scatterers. A new phenomenon is predicted, namely appearance of a rigid PR excitation regime. Mechanisms stabilizing the parametric instability have been analyzed. The shape of the electromagnetic wave absorption line at the PR has been found. The absorption is shown to become in some cases negative. Numerical estimates show that observation of this effect is quite feasible for modem experiment. time varying field component (1). If the field E~ (r, t) in the specimen is weakly nonuniform, and the latter is at a peak of the magnetic field in the resonantor, the vector E~ (r, t) has the form
1. In [1 ] a parametric resonance (PR) was predicted for semiconductors and investigated theoretically. The resonance should be observed in a spatially uniform magnetic field H modulated in time with a frequency 3':
H(t)
= He(1 + acos3't), a < 1, H(/)IIOZ.
E~(z, t) =
(1)
In such a field the cyclotron electron frequency is also a periodic function of time, i.e. ~2(t) = I2o(1 + a cos 3'0, I2o = eHo/mc. This means that a resonance parametric instability is possible in the system of charge carriers, of the type typical of the mechanical PR [2]. An appreciably nonequilibrium and anisotropic distribution of electrons is formed in the semiconductor when the parametric resonance conditions ([2 o ~3'/2) are approached. A semiconductor with such a distribution function can support different unstable electromagnetic excitations. In [1 ] conditions were established for the absorption of an external power (averaged over time) to become negative, i.e. the external signal to be enhanced. In [3 ] it was shown that under the PR conditions longitudinal plasma oscillations (plasmons) become absolutely unstable. In [4] a resonance parametric instablility of longitudinal lowfrequency sound was investigated (cos ,~ 3'/2, where cos is the sound frequency connected with the plasmon instability), i.e. enhancement of the sound signal occurs. As shown in [5 ], in a semiconductor under the PR conditions a high-frequency longitudinal sound of frequency cos ~ 3'/2 is generated. A crucial point in the considered problem is the presence of a vortex electric field E ~ (r, t) which appears in the specimen owing to the modulation of the magnetic field H(t) of equation (1). The space variation of E_(r, t) is determined by the excitation condition of 697
(l?I/cXdy, - / I x ,
0), a +/3 = 1
(2)
and is a linear function of the coordinates. The values of the parameters tx and/3 depend on the form of the resonator and determine the polarization of the vortex field E~ (r, t) in the resonator. The magnetic field of equation (1) can be assumed uniform while the electric field of equation (2) a given function of coordinates, provided both the skin depth and the electromagnetic wavelength at the modulation frequency 3' are much larger than the specimen size, i.e.
L ~ c (27ro3') -'/2,
c/3",
(3)
where a is the static conductivity of the semiconductor. For a linearly polarized electric field of equation (2) (t~ = i,/3 = 0, or a = 0,/3 = 1)the equations of motion of the electrons in the fields (1) and (2) can possess resonance solutions if the modulation frequency satisfies the condition 3" = 2~2[n (n = 1 , 2 , . . . ) . The width of the PR zone decreases with an increase in n, the number of the resonance, as t/a [t~ ,~ 1 is the modulation amplitude of the magnetic field (1)]. The fundamental resonance which has the instability zone width of the order a appears at the frequency 7 ~ 2[2o. With a circular polarization (a =/3 = 1/2) the fundamental resonance occurs at 3' ~ [20.* With an elliptically polarized field (2) when a and/3 are arbitrary numbers it turns out that *The existence of a parametric instability in the equations of motion of electrons in such fields was first noted in [6].
698
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
the fundamental resonance occurs at two frequencies, viz. I2o "" 3'[2 and ~2o = 3'. The PR zone width at I2o = 3'[2 is determined by the inequality 1~20- T/21 < (a~2o2/T)I1 -- 2otl
(4)
and at ~20 ~- 3' by
I~o - 3"1 < ( a ~ o ~ / 3 ' ) V ~ .
(5)
At I2o = 3' the PR can only occur if ot/~> 0. Obviously, equations (4) and (5), include the limiting cases of circular and linear polarizations. With a circular polarization the resonance zone width at the frequency T ~- 212o is zero, hence the fundamental resonance occurs only at 9' = ~2o. With a linear polarization the resonance at 3' = I2o disappears. Further on, we shall consider the electric field E~(r, t), equation (2), linearly polarized along the axis 0X (t~ = 1,/~ = 0). As is known, a weak linear friction does not stablize the parametric instability, and stable regimes of steady oscillation modes occur only if one takes into account nonlinear processes. A natural nonlinear mechanism in semiconductors is the nonparabolic energy-momentum relationship for electrons. In [1 ], the energy e(p) was chosen to be of the form e(p) = p2/2m +/i(Px 2 + py2), where/~ is the parameter describing deviation from the quadratic law;* the non-quadratic part of the energy being independent of the longitudinal momentum Pz; in this case the longitudinal and transverse motions of electrons separate. It has been shown [1 ] that such a nonlinearity under PR conditions leads to stable values of the amplitude and phase of steady-state oscillations, which do not depend on initial conditions. Hence, in the system of charge carriers distributed initially according to the Boltzmann law with temperature T, a nonequilibrium anisotropic distribution function is formed such that along the magnetic field (the Pz axis) it remains Boltzmann-like, but in the transverse plane it becomes a delta-function. When looking for the distribution function F(z, p, t) it is highly important to taken into consideration the interaction of electrons with volume scatterers. In [ 1], the scattering was taken into account by introducing into the equations of motion of electrons a weak linear friction. With the aid of this model it has been shown that the amplitude and phase of steady-state oscillations will be stable. In semiconductors the electron momentum relaxation frequences are not constant but depend on the total electron energy. Let us consider the situation when small-angle scattering prevails in the semiconductors and the main term in the collision integral is the outgoing one. Here we shall consider the collision integral with * For the Kein's zone I/~I-~ "" 2meg, e# is the forbidden zone width.
Vol. 45, No. 8
volume scatterers in the kinetic equation of the following model form: J(F) = ravtk (e)v k aF/aP~, where m, v, p and e are the mass, velocity, momentum and total energy of electron, respectively, vik(e ) is the electron scatterer collision frequency. As a result of such a choice of the collision integral the nonlinear friction force mv(e)v appears in the equations of motion of electron which accounts for the isotropization of the electron states due to the electron-scatterer interaction. The energy dependence of the momentum relaxation frequency v~k(e), according to [7], has the form
v~k(e) = vfiik (e/T) 2, (i, k = x, y, z)
(6)
where 8tk is the Kroneker symbol, T is the temperature, q is the parameter which, depending on the kind of scatterers, takes the values 1/2, 0, - 1 / 2 , - 1 , - 3 / 2 [7]. In this paper we shall consider an isotropic nonquadratic electron energy, viz. e(p) of the form
e(p) = p2/2m + 8p 4.
(7)
The nonlinear friction force in the equations of motion determined by the friction coefficient of equation (6) and the specific energy-momentum law of equation (7) do not allow the longitudinal and transverse motions of electron to separate. As will be seen below, for q > 0 the nonlinear mechanism due to the friction coefficient equation (6) can stabilize the parametric instability by itself. 2. In the PR range (I20 ~ 7/2) the equations of motion for the Y the coordinate are _ Qx { - A ( t ) c o s [ ~ + 0 ( t ) ] ; Y -mI2o = h lal slngo - v[ ~
(8)
+ 2mT]
;
0 = l a 2 + l a l c o s 2 0 + l a 3 (A 2 ~. - ~p2| ; lit
(9) ~ / ~ 0 ¢,
la x = aI2~13",la2 = I2o -- 3'/2, la3 = 8maI2~13", = v x + vy
(10)
Qx and Pz are integrals of motion. The parameters lal, #2, la3 and 9 are assumed to be small as long as the amplitude a, the off-resonance shift 112o -- 3'/21 and the nonparabolic parameter 8 are small. Equation (9) describe the dynamics of averaged electron motion in the slow time scale [8]. Integration of the system (9) is generally impossible. However, it is possible to investigate the behavior of solutions to the system (9) for large time t >> r, where r ~ [(p~ -- la22)1/2 + J] -1 is the inverse increment of solutions with lea = 0 and q = 0. The nonlinearity determined by the parameters la3 and
Vol. 45, No. 8
699
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
q stabilizes the parametric instability leading to steadystate solutions of the system (9) (A = 0, d = 0). Nonzero steady solutions of the system are determined by the following equations [K(r/ + z) + /32sin173] 2 = 1 - - ~ l ( r l + z ) 2 q ;
(11)
~2
~
"rrr "r
I
~2o
0~t) = ½ arcos [--/32 -- K(r/k + z) sin #31, aP
Oi2) = Oi') + lr,
0
s
(12)
J
,8, I
-I
where ~/= m~2~A2/2T is the transverse electron energy to temperature ratio, r/k are the positive roots of equation (10). The following notations were introduced above ~1 = J/lal, ~2 = ([2o -- 7/2)//ax, K = 2la3T/m~lal, z = p2z/2mT.
(13)
The existence and stability of steady solutions of the system (9) depend on the master parameters/31 and /~2. Analysis of equation (i0) and the system equation (9) shows that the plane of the master parameters can be divided into three regions (Fig. I). In region I the zero solution of the system (9) is unstable, and there a single stable steady transverse energy 17exists, i.e. in this range equation (2) has one solution and it is stable. In region 2 of the master parameters the zero solution of equation (9) is stable, however there are also two steady values of the transverse energy r/0h > r/2), the larger one Oh) being stable and the smaller one (r/2) unstable. In other words, in region 2 under PR a rigid steady excitation mode is realized. This means that the electrons with initial transverse energies smaller than #2 T do not participate in the PR, the remaining electrons are in resonance, and for them a stable steady oscillation mode is realized with the transverse energy rh T. In region 2 there is not PR - the zero solution is stable and there are no nonzero steady solutions of the system (9). The lines demarcating the regions 1,2 and 3 on the plane of the master parameters/3~ and #2 are given by the following equations [3~z2q + (f12 + Kz sin#3) 2 = I; /31 = [2K(q-- 1)] ~
(o) q" I/2
(~) q
(b) q - O
Fig. 1. Plane of the master parameters for r/a < 0. The common point of the three regions is determined by the expression:/32o sin/a a = -- Kz/2q + sin q(1 +
~ z2 /4q2) 1'2.
This means that in this region the electrons with any longitudinal energy z are off-resonance. In the region /3] z2. The lower part of the ellipse (14) up to the point of intersection with the curve (15) provides the larger root of equation (14), viz. the line z2 = z2 fill,/32). This curve separates the PR region 1 from the no-resonance region 3. The upper part of the ellipse down to the intersection point with the curve of equation (15) provides the smaller root of equation (14), viz. the curve z2 = z2(#1, ~2). This curve separates region 2, the region of rigid PR excitation, from the resonance region. We note here that the steady transverse energy values rh and r/2 are determined by the relations rh = g I - - Z, 'r] 2 -~--g 2 - - Z .
It is noteworthy that for q = -- 3/2 (the Coulomb scattering) [7] the stability of steady solutions depends on the relation between the transverse and longitudinal energies of electron. If ~ / z < 2, the steady solutions are stable, if rh/z > 2 the steady solutions become unstable in all the regions of the master parameters. For stable steady oscillation modes to appear in the system, it is necessary in this case that the parameter K (see equation (13)) describing the non-quadratic term of the energy-momentum relation would satisfy the following condition:
(14)
1 -- [ 4 ( q - 1)] -2 La2 sin#3 + x//1] + 4q2(] - l/q] 2 = M~2). [ ( 2 q - 1 ~ sin/as + X//~ + 4q2(1 -- l/q)] ~
The curve (14) and the line/~ = 0 confine region 1 and determine the PR zone width in which the zero solution of the system (9) is unstable. The curve (1 5) separates regions 2 and 3 (Fig. 1). In the following we shall need of the transverse energy values zl and z~ which are the positive roots of equation (1 4). In the region of the master parameters which is determined by the inequality /~ > Ma (fl~) (15), equation (14) has no positive roots.
(15)[
~ sinu~ + ~ ./~ ~ 3z V " 27z s " 3z The physical meaning of the above described nonlinear electron dynamics in semiconductors under the PR conditions can be understood with the help of the following comprehensive considerations. With non. quadratic electron energy, the effective mass and,
700
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
therefore, the electron cyclotron frequency depend on the energy e and the momentum projection on the magnetic field Pz: lbS
I2" = ell~m* (e, pz)C, m*
21r ae'
where S is the cross-section area of the isoenergetic surface e(p) = e cut by the plane Pz = const. In the chosen model energy-momentum relation of equation (7) the effective cyclotron frequency I2" with an accuracy to the first order in the parameter 6 is ~2" = I2o +/23 (,4 2 + Pz2 ~ m~]
(16) "
It is well known [2] that the PR instability zone in linear systems is determined by the condition p~ p] = ~ . Let us write down the equation for the boundary of the linear PR instability zone where the cyclotron frequency ~2o will be substituted by I2*(e, Pz) and the linear friction coefficient ~ by ~(e) (6) (Pl ~ g~, ta~ -* p~, v ~ g(e)). Then we obtain ,2
/.t~' --/a 2 = v(e).
(17)
This equation coincides with equation (1 1) which determines the amplitudes of steady oscillations in the nonlinear mode under PR. In other words, the steady oscillation electron energy in the nonlinear mode under PR is the energy corresponding to the resonance zone boundary (1 7).
n
The first term in the braces of equation (18) is due to the resonance electrons. Their longitudinal energies correspond to the master parameter I and II (Fig. 1). The transverse energy for electrons from region 1 may be arbitrary. In region 2, those electrons are resonance whose transverse energy 7/> 7/1. The remaining terms in the distribution function of equation (18) are formed by the electrons from region 3 and the off-resonance electrons from region 2 with initial transverse energies 77< ~1. The functions O(I), (9(I1) and O(III) are the Heaviside theta-functions which determine the distribution function in the respective regions of the master parameters 31 and 32. For/31 > M a (f12), equation (1 5) O (I) = O, (II) = 0, O (III) = 1; for/31 < Mq(fl2), ® (I) = O (z -- z2)O(zl -- z), O (II) = O (z2 -- z), O (III) = O(z -- zl), where zl > z2 are the positive roots of equation (14). Thus, in semiconductors under the PR conditions when taking into account the electronvolume scatterer interaction equation (6), the anisotropic nonequilibrium distribution function (1 8 ) o f charge carders is formed. The distribution function (1 8) allow to obtain the limiting result of [1] for q = 0 provided that the non-quadratic part of the electron energy (7) depends on the transverse momentum projections Px and py only. 4. We present the final expression for the time-averaged specific power Q = Q1 + Q2 absorbed by the semiconductor due to resonance electrons (18) in the weak signal field E(t) = El + E2 cos 60t, E(t)IIOX under the PR conditions:
3. Of considerable significance is the fact that the electron amplitude A and phase 0 reach the steady values irrespective of initial conditions. Another feature of the steady mode is the existence of two equiprobable values of the phase 0t O and v(2) (12) with the same amplitude A. = A 1. Owing to this, the system of charge carders, distributed initially according to the Boltzmann law, for t >>r under PR acquires the following distribution function F(p, t) - (2ranT)3/2 e -z~
01
Q2 _ E---~
602
602
(2--f)~q¢q (zl, z2), [
X- zl q ] (20)
X 1/ z , ( e - X , +
{ lrmT [O(I) + e -n~ (~)O(II)
2
(19)
87r~/--~ (60~- 7/2)21- d ~zl~)i] q ~ka(zl'z2)"
]-~2 e_Z2i
'%/ Z 1
]
(q - - 3~12)
)
e-t r ( x , ½) =
d+
X =
0
x ~, 6 [Px -- m~2ogt(t, z)] x 8 [Px-- m~l(t, z)] |ffil
+ [O [r/2(z)--
Vol. 45, No. 8
2mT]O(II)+O(III)]e-°~''~/2mr}• (189
Here n is the electron density, ~i(t, z) designates the second term in the right-hand part of equation (8) where the amplitude A(t) should be substituted by A 1 = A® (II) and the phase 0 by 1910from equation (12).
Here QI is the time-averaged specific power absorbed in the static field El, Q2 is the power absorbed in the oscillating field E2; 60j, = (4nne2[m) u2 is the semiconductor plasma frequency. The absorption Q is determined by zl and z2 which are the positive roots of the equation (14), and, besides, by the parameter q from equation (6). If the exponent of the energy dependence of the relaxation frequency (equation 6) q < 0, the d.c. absorption QI is negative,
Vol. 45, No. 8
-i -1+,,02
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
1
P __
I
S
absorption IQ21 for X < 1 is greater by a factor ot-2 as compared to IQ21 for ), ~" 1, It should be noted that the resonance lineshape depends on the sign of the parameter q, which is determined by the nature of electron scattering in semiconductors (see Figs. 2 and 3).
_.__~ -~ {o) J>>l
o
,-~2,
~2
(3) J<
Fig. 2. Absorption of the oscillating part of the signal field 0s = 81r~4r~'ffQ2]E]oop2) as a function of the off-resonance shift/32 for the case q = -- 1/2, gt3 < 0. The absorption line for/a3 > 0 can be obtained from the line shown in the figure by reflection in the axis /32 ~ 0. (a) For ), ~, 1, Q2 < 0 the minimum is at /32 = - 1 + ~ ; (b~ for X < 1, Q2 > 0 the maximum is at ~ = - 1 - ~1.
p(A)2
j -I
701
,P
5. The possibility of existence of the parametric instability in the above considered situation is determined by the form of the steady distribution, (equation 18). PR can manifest itself not only in the electromagnetic wave absorption but in many other phenomena (see [3-5] ). As far as we know, the above considered phenomena have not been observed experimentally. It is likely to be observed in pure indium antimouide samples at low temperatures. For typical values of the parameters for In Sb (n = (1 + 30) x 1014 cm -a, m = 0.0134mo) ([9]) for the collision frequency of~ ~ 101° s -1, PR should be observed at the frequency 3,/2 ~- ~o = (0, 9 + 1,3) x 1012 s-I in the fields H0 of the order of several KOe at modulation amplitudes aHo ~- (20 + 100)Oe. Under these conditions the required cyclotron parametric pumping is evaluated to be of order 10 -1 to 1 kW which would probably necessitate pulsed technique.
Acknowledgements - We are very grateful to E.A.
0 (a) ,~>>1
(~) ~A<
Kaner and A.A. Slutskin for stimulating discussions and their interest to the work and R.I. Shekhter for useful discussions.
8
Fig. 3. Absorption of the oscillating part of the signal field (p = 81r~/r~Q2/E~ coy 2) as a function of the offresonance shift/32 for q = 1/2. The resonance lines are symmetric with respect to the axis/32 = 0. (a) For ), >> 1, Q2 > 0 the maximum is at/32 = 0, (b) for ~, < 1 the minimum is at/32 = 0, the maximum at/32 = +x/1 -- 0 , / 3 ) ~ .
REFERENCES 1. 2. 3. 4.
if q > 0, the absorption of the oscillating signal field E2 at co "" 3,[2, ~ < 1 will be negative. If ~ ~, 1, the expression (20) for Q2 coincides with the expression (19) for Q1. Since Q depends on zl and z2 (equation 14), the absorption line is always of the resonance type, i.e. the absorption is nonzero only near PR. The expressions (19) and (20) describe the shape of the PR line. In figs. 2 and 3, the absorption of the varying part of the signal field Q2 (equation 20) is plotted as a function of the off-resonance shift/32 (equation 12) for low temperature, when K = T[8Qeg < 1, and small relaxation frequencies ~(/31 ~ 1). In the case o f q = -- 1[2 (Fig. 2), the absolute value of the absorption iQ2I for X < 1 is greater by a factor (3,2/~2K)1/2 as compared to IQ21 for ), ~. 1. In the case o f q = 1/2, the
5. 6. 7.
8.
I.E. Aronov, E.A. Kaner & A.A. Slutskin, Solid State Commun. 38,245 (1981). L.D. Landau & E.M. Lifshitz, Mechanics, p. 103. Nauka Publ. Co., Moscow, (1965) (in Russian). I.E. Aronov & E.A. Kaner, ZhETF Pisma, 34, N6, 341 (1981). I.E. Aronov & E.A. Kaner, FTT, 25, N1,200 (1982). I.E. Aronov & E.A. Kaner, FTT, 25, 2529 (1982). L.I. Rudakov & R.Z. Sagdeev, Physics of Plasma
and the Problems of the Controlled Thermonuclear Reactions, p. 153.3 Moscow (1958). F.G. Bass & Yu.G. Gurewich, Hot Electrons and Powerful Electromagnetic Waves in Semiconductor and Gas Discharge Plasmas, p. 36. Nauak Publ. Co., Moscow (1975) (in Russian). N.N. Bogolyubov & Yu.A. Mitropolsky, Asymp-
totic Methods in the Theory of Nonlinear Oscillations, p. 250. Nauka Publ. Co., Moscow (1974) (in Russian).
9.
Optical Properties of Semiconductors (Arab vtype Semiconductor Compounds), (Edited by R. Willardson & A. Beer), Mir Publ. Co., Moscow (1970) (Russian translation).
Solid State Communications, Vol. 45, No. 8, pp. 697-701, 1983. Printed in Great Britain.
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS I.E. Aronov and O. N. Baranetz Institute for Radiophysics and Electronics, Academy of Sciences of the Ukrainian SSR, Kharkov 310085, U.S.S.R.
(Received 26 August
1982
by E.A. Kaner)
A theory of parameteric resonance (PR) in semiconductors is constructed with an account of the interaction between electrons and volume scatterers. A new phenomenon is predicted, namely appearance of a rigid PR excitation regime. Mechanisms stabilizing the parametric instability have been analyzed. The shape of the electromagnetic wave absorption line at the PR has been found. The absorption is shown to become in some cases negative. Numerical estimates show that observation of this effect is quite feasible for modem experiment. time varying field component (1). If the field E~ (r, t) in the specimen is weakly nonuniform, and the latter is at a peak of the magnetic field in the resonantor, the vector E~ (r, t) has the form
1. In [1 ] a parametric resonance (PR) was predicted for semiconductors and investigated theoretically. The resonance should be observed in a spatially uniform magnetic field H modulated in time with a frequency 3':
H(t)
= He(1 + acos3't), a < 1, H(/)IIOZ.
E~(z, t) =
(1)
In such a field the cyclotron electron frequency is also a periodic function of time, i.e. ~2(t) = I2o(1 + a cos 3'0, I2o = eHo/mc. This means that a resonance parametric instability is possible in the system of charge carriers, of the type typical of the mechanical PR [2]. An appreciably nonequilibrium and anisotropic distribution of electrons is formed in the semiconductor when the parametric resonance conditions ([2 o ~3'/2) are approached. A semiconductor with such a distribution function can support different unstable electromagnetic excitations. In [1 ] conditions were established for the absorption of an external power (averaged over time) to become negative, i.e. the external signal to be enhanced. In [3 ] it was shown that under the PR conditions longitudinal plasma oscillations (plasmons) become absolutely unstable. In [4] a resonance parametric instablility of longitudinal lowfrequency sound was investigated (cos ,~ 3'/2, where cos is the sound frequency connected with the plasmon instability), i.e. enhancement of the sound signal occurs. As shown in [5 ], in a semiconductor under the PR conditions a high-frequency longitudinal sound of frequency cos ~ 3'/2 is generated. A crucial point in the considered problem is the presence of a vortex electric field E ~ (r, t) which appears in the specimen owing to the modulation of the magnetic field H(t) of equation (1). The space variation of E_(r, t) is determined by the excitation condition of 697
(l?I/cXdy, - / I x ,
0), a +/3 = 1
(2)
and is a linear function of the coordinates. The values of the parameters tx and/3 depend on the form of the resonator and determine the polarization of the vortex field E~ (r, t) in the resonator. The magnetic field of equation (1) can be assumed uniform while the electric field of equation (2) a given function of coordinates, provided both the skin depth and the electromagnetic wavelength at the modulation frequency 3' are much larger than the specimen size, i.e.
L ~ c (27ro3') -'/2,
c/3",
(3)
where a is the static conductivity of the semiconductor. For a linearly polarized electric field of equation (2) (t~ = i,/3 = 0, or a = 0,/3 = 1)the equations of motion of the electrons in the fields (1) and (2) can possess resonance solutions if the modulation frequency satisfies the condition 3" = 2~2[n (n = 1 , 2 , . . . ) . The width of the PR zone decreases with an increase in n, the number of the resonance, as t/a [t~ ,~ 1 is the modulation amplitude of the magnetic field (1)]. The fundamental resonance which has the instability zone width of the order a appears at the frequency 7 ~ 2[2o. With a circular polarization (a =/3 = 1/2) the fundamental resonance occurs at 3' ~ [20.* With an elliptically polarized field (2) when a and/3 are arbitrary numbers it turns out that *The existence of a parametric instability in the equations of motion of electrons in such fields was first noted in [6].
698
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
the fundamental resonance occurs at two frequencies, viz. I2o "" 3'[2 and ~2o = 3'. The PR zone width at I2o = 3'[2 is determined by the inequality 1~20- T/21 < (a~2o2/T)I1 -- 2otl
(4)
and at ~20 ~- 3' by
I~o - 3"1 < ( a ~ o ~ / 3 ' ) V ~ .
(5)
At I2o = 3' the PR can only occur if ot/~> 0. Obviously, equations (4) and (5), include the limiting cases of circular and linear polarizations. With a circular polarization the resonance zone width at the frequency T ~- 212o is zero, hence the fundamental resonance occurs only at 9' = ~2o. With a linear polarization the resonance at 3' = I2o disappears. Further on, we shall consider the electric field E~(r, t), equation (2), linearly polarized along the axis 0X (t~ = 1,/~ = 0). As is known, a weak linear friction does not stablize the parametric instability, and stable regimes of steady oscillation modes occur only if one takes into account nonlinear processes. A natural nonlinear mechanism in semiconductors is the nonparabolic energy-momentum relationship for electrons. In [1 ], the energy e(p) was chosen to be of the form e(p) = p2/2m +/i(Px 2 + py2), where/~ is the parameter describing deviation from the quadratic law;* the non-quadratic part of the energy being independent of the longitudinal momentum Pz; in this case the longitudinal and transverse motions of electrons separate. It has been shown [1 ] that such a nonlinearity under PR conditions leads to stable values of the amplitude and phase of steady-state oscillations, which do not depend on initial conditions. Hence, in the system of charge carriers distributed initially according to the Boltzmann law with temperature T, a nonequilibrium anisotropic distribution function is formed such that along the magnetic field (the Pz axis) it remains Boltzmann-like, but in the transverse plane it becomes a delta-function. When looking for the distribution function F(z, p, t) it is highly important to taken into consideration the interaction of electrons with volume scatterers. In [ 1], the scattering was taken into account by introducing into the equations of motion of electrons a weak linear friction. With the aid of this model it has been shown that the amplitude and phase of steady-state oscillations will be stable. In semiconductors the electron momentum relaxation frequences are not constant but depend on the total electron energy. Let us consider the situation when small-angle scattering prevails in the semiconductors and the main term in the collision integral is the outgoing one. Here we shall consider the collision integral with * For the Kein's zone I/~I-~ "" 2meg, e# is the forbidden zone width.
Vol. 45, No. 8
volume scatterers in the kinetic equation of the following model form: J(F) = ravtk (e)v k aF/aP~, where m, v, p and e are the mass, velocity, momentum and total energy of electron, respectively, vik(e ) is the electron scatterer collision frequency. As a result of such a choice of the collision integral the nonlinear friction force mv(e)v appears in the equations of motion of electron which accounts for the isotropization of the electron states due to the electron-scatterer interaction. The energy dependence of the momentum relaxation frequency v~k(e), according to [7], has the form
v~k(e) = vfiik (e/T) 2, (i, k = x, y, z)
(6)
where 8tk is the Kroneker symbol, T is the temperature, q is the parameter which, depending on the kind of scatterers, takes the values 1/2, 0, - 1 / 2 , - 1 , - 3 / 2 [7]. In this paper we shall consider an isotropic nonquadratic electron energy, viz. e(p) of the form
e(p) = p2/2m + 8p 4.
(7)
The nonlinear friction force in the equations of motion determined by the friction coefficient of equation (6) and the specific energy-momentum law of equation (7) do not allow the longitudinal and transverse motions of electron to separate. As will be seen below, for q > 0 the nonlinear mechanism due to the friction coefficient equation (6) can stabilize the parametric instability by itself. 2. In the PR range (I20 ~ 7/2) the equations of motion for the Y the coordinate are _ Qx { - A ( t ) c o s [ ~ + 0 ( t ) ] ; Y -mI2o = h lal slngo - v[ ~
(8)
+ 2mT]
;
0 = l a 2 + l a l c o s 2 0 + l a 3 (A 2 ~. - ~p2| ; lit
(9) ~ / ~ 0 ¢,
la x = aI2~13",la2 = I2o -- 3'/2, la3 = 8maI2~13", = v x + vy
(10)
Qx and Pz are integrals of motion. The parameters lal, #2, la3 and 9 are assumed to be small as long as the amplitude a, the off-resonance shift 112o -- 3'/21 and the nonparabolic parameter 8 are small. Equation (9) describe the dynamics of averaged electron motion in the slow time scale [8]. Integration of the system (9) is generally impossible. However, it is possible to investigate the behavior of solutions to the system (9) for large time t >> r, where r ~ [(p~ -- la22)1/2 + J] -1 is the inverse increment of solutions with lea = 0 and q = 0. The nonlinearity determined by the parameters la3 and
Vol. 45, No. 8
699
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
q stabilizes the parametric instability leading to steadystate solutions of the system (9) (A = 0, d = 0). Nonzero steady solutions of the system are determined by the following equations [K(r/ + z) + /32sin173] 2 = 1 - - ~ l ( r l + z ) 2 q ;
(11)
~2
~
"rrr "r
I
~2o
0~t) = ½ arcos [--/32 -- K(r/k + z) sin #31, aP
Oi2) = Oi') + lr,
0
s
(12)
J
,8, I
-I
where ~/= m~2~A2/2T is the transverse electron energy to temperature ratio, r/k are the positive roots of equation (10). The following notations were introduced above ~1 = J/lal, ~2 = ([2o -- 7/2)//ax, K = 2la3T/m~lal, z = p2z/2mT.
(13)
The existence and stability of steady solutions of the system (9) depend on the master parameters/31 and /~2. Analysis of equation (i0) and the system equation (9) shows that the plane of the master parameters can be divided into three regions (Fig. I). In region I the zero solution of the system (9) is unstable, and there a single stable steady transverse energy 17exists, i.e. in this range equation (2) has one solution and it is stable. In region 2 of the master parameters the zero solution of equation (9) is stable, however there are also two steady values of the transverse energy r/0h > r/2), the larger one Oh) being stable and the smaller one (r/2) unstable. In other words, in region 2 under PR a rigid steady excitation mode is realized. This means that the electrons with initial transverse energies smaller than #2 T do not participate in the PR, the remaining electrons are in resonance, and for them a stable steady oscillation mode is realized with the transverse energy rh T. In region 2 there is not PR - the zero solution is stable and there are no nonzero steady solutions of the system (9). The lines demarcating the regions 1,2 and 3 on the plane of the master parameters/3~ and #2 are given by the following equations [3~z2q + (f12 + Kz sin#3) 2 = I; /31 = [2K(q-- 1)] ~
(o) q" I/2
(~) q
(b) q - O
Fig. 1. Plane of the master parameters for r/a < 0. The common point of the three regions is determined by the expression:/32o sin/a a = -- Kz/2q + sin q(1 +
~ z2 /4q2) 1'2.
This means that in this region the electrons with any longitudinal energy z are off-resonance. In the region /3]
It is noteworthy that for q = -- 3/2 (the Coulomb scattering) [7] the stability of steady solutions depends on the relation between the transverse and longitudinal energies of electron. If ~ / z < 2, the steady solutions are stable, if rh/z > 2 the steady solutions become unstable in all the regions of the master parameters. For stable steady oscillation modes to appear in the system, it is necessary in this case that the parameter K (see equation (13)) describing the non-quadratic term of the energy-momentum relation would satisfy the following condition:
(14)
1 -- [ 4 ( q - 1)] -2 La2 sin#3 + x//1] + 4q2(] - l/q] 2 = M~2). [ ( 2 q - 1 ~ sin/as + X//~ + 4q2(1 -- l/q)] ~
The curve (14) and the line/~ = 0 confine region 1 and determine the PR zone width in which the zero solution of the system (9) is unstable. The curve (1 5) separates regions 2 and 3 (Fig. 1). In the following we shall need of the transverse energy values zl and z~ which are the positive roots of equation (1 4). In the region of the master parameters which is determined by the inequality /~ > Ma (fl~) (15), equation (14) has no positive roots.
(15)[
~ sinu~ + ~ ./~ ~ 3z V " 27z s " 3z The physical meaning of the above described nonlinear electron dynamics in semiconductors under the PR conditions can be understood with the help of the following comprehensive considerations. With non. quadratic electron energy, the effective mass and,
700
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
therefore, the electron cyclotron frequency depend on the energy e and the momentum projection on the magnetic field Pz: lbS
I2" = ell~m* (e, pz)C, m*
21r ae'
where S is the cross-section area of the isoenergetic surface e(p) = e cut by the plane Pz = const. In the chosen model energy-momentum relation of equation (7) the effective cyclotron frequency I2" with an accuracy to the first order in the parameter 6 is ~2" = I2o +/23 (,4 2 + Pz2 ~ m~]
(16) "
It is well known [2] that the PR instability zone in linear systems is determined by the condition p~ p] = ~ . Let us write down the equation for the boundary of the linear PR instability zone where the cyclotron frequency ~2o will be substituted by I2*(e, Pz) and the linear friction coefficient ~ by ~(e) (6) (Pl ~ g~, ta~ -* p~, v ~ g(e)). Then we obtain ,2
/.t~' --/a 2 = v(e).
(17)
This equation coincides with equation (1 1) which determines the amplitudes of steady oscillations in the nonlinear mode under PR. In other words, the steady oscillation electron energy in the nonlinear mode under PR is the energy corresponding to the resonance zone boundary (1 7).
n
The first term in the braces of equation (18) is due to the resonance electrons. Their longitudinal energies correspond to the master parameter I and II (Fig. 1). The transverse energy for electrons from region 1 may be arbitrary. In region 2, those electrons are resonance whose transverse energy 7/> 7/1. The remaining terms in the distribution function of equation (18) are formed by the electrons from region 3 and the off-resonance electrons from region 2 with initial transverse energies 77< ~1. The functions O(I), (9(I1) and O(III) are the Heaviside theta-functions which determine the distribution function in the respective regions of the master parameters 31 and 32. For/31 > M a (f12), equation (1 5) O (I) = O, (II) = 0, O (III) = 1; for/31 < Mq(fl2), ® (I) = O (z -- z2)O(zl -- z), O (II) = O (z2 -- z), O (III) = O(z -- zl), where zl > z2 are the positive roots of equation (14). Thus, in semiconductors under the PR conditions when taking into account the electronvolume scatterer interaction equation (6), the anisotropic nonequilibrium distribution function (1 8 ) o f charge carders is formed. The distribution function (1 8) allow to obtain the limiting result of [1] for q = 0 provided that the non-quadratic part of the electron energy (7) depends on the transverse momentum projections Px and py only. 4. We present the final expression for the time-averaged specific power Q = Q1 + Q2 absorbed by the semiconductor due to resonance electrons (18) in the weak signal field E(t) = El + E2 cos 60t, E(t)IIOX under the PR conditions:
3. Of considerable significance is the fact that the electron amplitude A and phase 0 reach the steady values irrespective of initial conditions. Another feature of the steady mode is the existence of two equiprobable values of the phase 0t O and v(2) (12) with the same amplitude A. = A 1. Owing to this, the system of charge carders, distributed initially according to the Boltzmann law, for t >>r under PR acquires the following distribution function F(p, t) - (2ranT)3/2 e -z~
01
Q2 _ E---~
602
602
(2--f)~q¢q (zl, z2), [
X- zl q ] (20)
X 1/ z , ( e - X , +
{ lrmT [O(I) + e -n~ (~)O(II)
2
(19)
87r~/--~ (60~- 7/2)21- d ~zl~)i] q ~ka(zl'z2)"
]-~2 e_Z2i
'%/ Z 1
]
(q - - 3~12)
)
e-t r ( x , ½) =
d+
X =
0
x ~, 6 [Px -- m~2ogt(t, z)] x 8 [Px-- m~l(t, z)] |ffil
+ [O [r/2(z)--
Vol. 45, No. 8
2mT]O(II)+O(III)]e-°~''~/2mr}• (189
Here n is the electron density, ~i(t, z) designates the second term in the right-hand part of equation (8) where the amplitude A(t) should be substituted by A 1 = A® (II) and the phase 0 by 1910from equation (12).
Here QI is the time-averaged specific power absorbed in the static field El, Q2 is the power absorbed in the oscillating field E2; 60j, = (4nne2[m) u2 is the semiconductor plasma frequency. The absorption Q is determined by zl and z2 which are the positive roots of the equation (14), and, besides, by the parameter q from equation (6). If the exponent of the energy dependence of the relaxation frequency (equation 6) q < 0, the d.c. absorption QI is negative,
Vol. 45, No. 8
-i -1+,,02
A THEORY OF PARAMETRIC RESONANCE IN SEMICONDUCTORS
1
P __
I
S
absorption IQ21 for X < 1 is greater by a factor ot-2 as compared to IQ21 for ), ~" 1, It should be noted that the resonance lineshape depends on the sign of the parameter q, which is determined by the nature of electron scattering in semiconductors (see Figs. 2 and 3).
_.__~ -~ {o) J>>l
o
,-~2,
~2
(3) J<
Fig. 2. Absorption of the oscillating part of the signal field 0s = 81r~4r~'ffQ2]E]oop2) as a function of the off-resonance shift/32 for the case q = -- 1/2, gt3 < 0. The absorption line for/a3 > 0 can be obtained from the line shown in the figure by reflection in the axis /32 ~ 0. (a) For ), ~, 1, Q2 < 0 the minimum is at /32 = - 1 + ~ ; (b~ for X < 1, Q2 > 0 the maximum is at ~ = - 1 - ~1.
p(A)2
j -I
701
,P
5. The possibility of existence of the parametric instability in the above considered situation is determined by the form of the steady distribution, (equation 18). PR can manifest itself not only in the electromagnetic wave absorption but in many other phenomena (see [3-5] ). As far as we know, the above considered phenomena have not been observed experimentally. It is likely to be observed in pure indium antimouide samples at low temperatures. For typical values of the parameters for In Sb (n = (1 + 30) x 1014 cm -a, m = 0.0134mo) ([9]) for the collision frequency of~ ~ 101° s -1, PR should be observed at the frequency 3,/2 ~- ~o = (0, 9 + 1,3) x 1012 s-I in the fields H0 of the order of several KOe at modulation amplitudes aHo ~- (20 + 100)Oe. Under these conditions the required cyclotron parametric pumping is evaluated to be of order 10 -1 to 1 kW which would probably necessitate pulsed technique.
Acknowledgements - We are very grateful to E.A.
0 (a) ,~>>1
(~) ~A<
Kaner and A.A. Slutskin for stimulating discussions and their interest to the work and R.I. Shekhter for useful discussions.
8
Fig. 3. Absorption of the oscillating part of the signal field (p = 81r~/r~Q2/E~ coy 2) as a function of the offresonance shift/32 for q = 1/2. The resonance lines are symmetric with respect to the axis/32 = 0. (a) For ), >> 1, Q2 > 0 the maximum is at/32 = 0, (b) for ~, < 1 the minimum is at/32 = 0, the maximum at/32 = +x/1 -- 0 , / 3 ) ~ .
REFERENCES 1. 2. 3. 4.
if q > 0, the absorption of the oscillating signal field E2 at co "" 3,[2, ~ < 1 will be negative. If ~ ~, 1, the expression (20) for Q2 coincides with the expression (19) for Q1. Since Q depends on zl and z2 (equation 14), the absorption line is always of the resonance type, i.e. the absorption is nonzero only near PR. The expressions (19) and (20) describe the shape of the PR line. In figs. 2 and 3, the absorption of the varying part of the signal field Q2 (equation 20) is plotted as a function of the off-resonance shift/32 (equation 12) for low temperature, when K = T[8Qeg < 1, and small relaxation frequencies ~(/31 ~ 1). In the case o f q = -- 1[2 (Fig. 2), the absolute value of the absorption iQ2I for X < 1 is greater by a factor (3,2/~2K)1/2 as compared to IQ21 for ), ~. 1. In the case o f q = 1/2, the
5. 6. 7.
8.
I.E. Aronov, E.A. Kaner & A.A. Slutskin, Solid State Commun. 38,245 (1981). L.D. Landau & E.M. Lifshitz, Mechanics, p. 103. Nauka Publ. Co., Moscow, (1965) (in Russian). I.E. Aronov & E.A. Kaner, ZhETF Pisma, 34, N6, 341 (1981). I.E. Aronov & E.A. Kaner, FTT, 25, N1,200 (1982). I.E. Aronov & E.A. Kaner, FTT, 25, 2529 (1982). L.I. Rudakov & R.Z. Sagdeev, Physics of Plasma
and the Problems of the Controlled Thermonuclear Reactions, p. 153.3 Moscow (1958). F.G. Bass & Yu.G. Gurewich, Hot Electrons and Powerful Electromagnetic Waves in Semiconductor and Gas Discharge Plasmas, p. 36. Nauak Publ. Co., Moscow (1975) (in Russian). N.N. Bogolyubov & Yu.A. Mitropolsky, Asymp-
totic Methods in the Theory of Nonlinear Oscillations, p. 250. Nauka Publ. Co., Moscow (1974) (in Russian).
9.
Optical Properties of Semiconductors (Arab vtype Semiconductor Compounds), (Edited by R. Willardson & A. Beer), Mir Publ. Co., Moscow (1970) (Russian translation).