Pram~n.a, Vol. 16, No. 1, January 1981, pp. 99-106. ~) Printed in India.
Stimulated Raman scattering from plasma modes in magnetoactive semiconductors
S G U H A a n d N A P T E *
Department of Physics, Ravishartkar University, Raipur 492 010, India
*Present Address: School of Studies in Physics, Vikram University, Ujjain 456 331, India
MS received 24 September 1979; revised 19 August 1980
Abstract. Stimulated scattering off electron plasma mode is investigated analytically for the case when the pump wave is an intense circularly polarised electromagnetic wave propagating parallel to a homogeneous dc magnetic field in an isotropic semi- conductor-plasma. The threshold electric field of the pump necessary for the stimu- lated Raman scattering and the growth rate of the parametrically unstable mode have been obtained for two cases (i) B0=0 and (ii) Bo~0. It is seen that the mag- netic field does not significantly affect the threshold electric field as well as the growth rate provided the cyclotron frequency is small compared to the frequency of the pump wave. The threshold conditions are also found to be insensitive to the electron ther- mal velocity.
Keywords. Stimulated Raman scattering; parametric instability; piezoelectric semi- conductor; plasma modes.
1. Introduction
There has been m u c h interest in the scattering o f intense electromagnetic radiation in a plasma resulting f r o m the decay o f the incident wave into a n o t h e r electromagnetic wave a n d either a L a n g m u i r wave (stimulated R a m a n scattering, SRS) or an acoustic wave (stimulated Brillouin scattering). SRS provides useful i n f o r m a t i o n a b o u t the spectrum a n d nature o f e l e m e n t a r y excitations in a p l a s m a . Originally the technique was applied to gaseous p l a s m a (Kunze et al 1964). Several theoretical studies o f stimu- lated R a m a n scattering (SRS) in a n infinite h o m o g e n e o u s isotropio gaseous p l a s m a have been r e p o r t e d recently (e.g. D r a k e et al 1974; L a s h m o r e - D a v i e s 1975; F u c h s 1976; Sodha et al 1976). The stimulated R a m a n b a c k s c a t t e r i n g o f a circularly polarized electromagnetic wave p r o p a g a t i n g along an u n i f o r m static magnetic field h a s been investigated b y M a r a g h e c h i a n d Willett (1979). T h e p h e n o m e n o n o f S R S h a s been used to s t u d y p l a s m a s in semiconductors as well ( W o l f f 1970; Patel a n d Shaw 1970). M a n y theoretical treatments considered either the p h o t o n - s y s t e m coupling with a single c o m p o n e n t o n e - b a n d electron gas, o m i t t i n g all p h o n o n effects (Wolff 1970) or considered u n d o p e d semiconductors in which no c o n d u c t i o n electron is present. F o p a n d T z o a r (1970) calculated the R a m a n scattering cross-section in the f o r w a r d direction in solid-state plasma in a magnetic field. However, the thres- hold electric field required f o r the onset o f SRS f r o m electron p l a s m a m o d e a n d the growth rate o f the unstable m o d e well a b o v e the threshold as well as their dependence on the various p a r a m e t e r s o f the system have not been studied in solid-.state plasma.
p.--7 99
100 S Guha a n d N A p t e
In this paper we report our investigations on the SRS of a circularly polarized electro- magnetic wave due to the excitation of electron plasma waves propagating along a uniform static magnetic field in a semiconductor-plasma. The electron concentra- tion in the semiconductor is chosen suitably so as to neglect the effect of optical phonons on SRS. The stimulated Raman forward as well as backscattering of a forward travelling pump wave have been studied incorporating its (pump) spatial dependence. The coupled mode theory (Lashmore-Davies 1975) which treats the pump and the excited waves on the same footing has been employed in our analysis to obtain the amplitude of the threshold electric field required for the onset of insta- bility and the growth rate of the unstable Raman mode well above the threshold has been studied. A large magnetic field is found to enhance the threshold pump ampli- tude as well as the growth rate. It can also be seen from our analysis that a right- handed circularly polarized pump wave gives rise to a left-handed circularly polarized scattered wave as a result of the scattering from the electron plasma mode.
2. Theoretical formulation
Let us consider an infinite and isotropic semiconductor-plasma to which an uniform and static magnetic field is applied such that B0 : B0 x. The plasma is subjected to a high frequency electromagnetic wave which is circularly polarized in a plane per- pendicular to 13o and propagates in the x-direction. The scattering is completely characterized by the following wave vector and frequency selection rules:
kTo = kT1 -P kl, (1)
and o, TO = ¢.OT1 -~- ¢o t. (2)
We have assumed the spatial uniformity and the perfect frequency matching and hence we take equations (1) and (2) to be satisfied exactly. Here kT0(kTl) and COT0 (COT1) are, respectively, the incident pump (scattered) wave vector and frequency, k~ and ~o~ are the wave vector and frequency of the electron plasma wave that is excited in the semiconductor. The subscripts To, T 1 and 1 stand for the pump, scattered wave and Langmuir wave respectively. The electric field of the large amplitude circularly polarized pump wave is described by
ETO 4- = ETO exp [i (kTO x - - toTO t)].
The first step towards studying the parametric interaction of the two electromag- netic and an electron plasma wave is to obtain the normal modes for these waves.
For this the starting equations are:
_ _ + e _ Or±
Ov~ _]_ vv± E± :t= iWcV + = ~: i e v x B ± - - Vx ,
Ot m m Ox
(3)
i S E ± 4- 8 B ± = O,
(4)
8x 8t
Stimulated Raman scattering in semiconductors 101 eoE± q: ioB±
- - - - no e v± = n l e v±, ( 5 )
Ot ~o Ox
ant + no i i (~l VX)' (6)
Ot OX
Ovx eEx + v ~ anl
+ vv~ +
O t m no Ox
.r.<.+._ i7] vihnlOnl VxOVx
m L 2 -- " + no Ox -- ~ Ox (7)
and e OE~ _ no e v~ = n 1 e v~. (8)
Ot
Equations (3), (4) and (5) are used to obtain the normal modes of the pump wave and the scattered electromagnetic wave while (6), (7) and (8) yield the normal mode o f Langmuir wave (electron plasma wave). Equations (3) and (7) are the momentum transfer equations while (4), (5) and (8) are the Maxwell's equations. All o f them are written in the component form corresponding to the electromagnetic and Lang- muir modes, v± and vx are the perturbed fluid velocities, v is the electron collision frequency and coc ( : [ e [ Bo/m) denotes the electron cyclotron frequency. E~, Ex and B± are the perturbed electric field components and perturbed magnetic induction respectively. Equation (6) is the continuity equation, n o and n 1 denote the unper- turbed and perturbed electron density respectively. VTh=(2KBTe[m)½ represents the thermal velocity of the electrons. The terms on the right side of (3) to (8) denote the nonlinear contributions. To obtain the normal mode equations we neglect the quadratic terms in the wave variables and take the linear combination o f (3) to (5) and (6) to (8) such that
~t (VT+ :J: aTBT+ + E~TET~ ) + v VT+ + eET+ T io% VT±
m fiT k r BT~
- - aT kT ET+ + - - f i t eno VT4- = 0 , ( 9 )
IZo
and 0 (n I + at v~ + El ~E~) + ikt n o v~ + a l v v~
Ot
eE,, + iai V~h ki n x
+al -- #l enovx=O, (10)
r n n o
where we have assumed O/Ox = ikT, l. The normal mode equations are obtained by choosing aT, t and #T, t such that equations (9) and (10) have the form
OAT++ i~oTaT+ -t- ~TaT+ =0, ( l l a )
Ot
102 S Guha and N Apte
and Oa_j _~ itot at + 7g a l : O . (1 lb)
c3t
aT± and at are the normal mode amplitudes represented as
aT+ = OT± 4- a r Bra: + af T E r ± , (12a)
at = nl q- at vx -~- efi~ Ex, (12b)
where we assume
aT+, l ~'~ exp [i(kT, I x -- toT, I t)] exp [-- 7T, l t].
Here 7T a n d 7t represent the collisional damping rates. The subscript T denotes electromagnetic waves and subscripts i represent the nature o f polarization. F r o m (10) and (11), the constants aT, I, fiT, l and ~'T, l are obtained as
a T - - . ~
to~ i~T I tozrtoT 1 t
1 4 - - - + 1 2toc ~
kT oj T (2toT(to T ± toc)2 qz ,op !,(13a)
e.o 1 -- (2 r (tot 4- T tog tot}
fiT = - - e ~ o L 4- - -
tot
1 --"[-2toT(toT
4- to*)~ ~ to~ tot) , (13b)vto~ toT
7T -- 2toT(toT 4- to,)2 qz to~v,Oc ' (13c)
fit -- ieno (13d)
~mv~, h kt '
-- n°to~t (1 iv i, (13e)
a n d ~t = v/2, (13f)
where we have used the dispersion relations
to'~ = " kT cl q- "2 2
to~ ,o r (to T -4- toc)
2 2
2 + kt VTh.
and w~ = top
C~ is the velocity o f light in the crystal. We take the same linear combinations o f equations (3)-(5) and (6)-(8) as done for the normal modes, and retain only those
Stimulated Raman scattering in semiconductors 103 nonlinear terms on the right side which have the space and time-dependence of the mode concerned. For simplicity of calculation, we have assumed that the incident pump wave is right-handed circularly polarized so that the scattered electromagnetic mode is a left-handed circularly polarized wave. Thus using (3)-(11), we obtain the coupled mode equations as
Oa~l- + i°~T1 a~l- + 7T1 a~'l- = fl~:l- enx V~'o+ ~ - - -- v~
Ot m Ox
(14a)
e l(iv~,l - BTO+). (14b)
and 8a...j + io~tal + Ylai = a t _
Ot m
These equations have been considered for a~, and a + only, since these modes do not couple with a f and a T in view of the fact that the modes are o f high frequencies and the coupling is weak. The equations for a~- and a T are of the same form as (14) and do not give any additional information. Thus (14) covers both the cases of forward and backward propagating modes respectively. It is assumed that the mode ampli- tudes are small and the dependence of original variables (VT+, nl, etc.) on the mode amplitudes aT_ 4_ and a~ is obtained using linearized forms of equations (3) to (8) and the definition of normal modes equation (12). The effect of collisions on the non- linear terms is neglected since we have considered only weakly damped waves.
The normal mode amplitudes are taken as a product of a slowly varying amplitude and a rapidly varying phase as
aT+, l = AT+, l (t) exp [i (kT, l x -- Re tOT, I t)],
where the imaginary part of toT, l is contained in the mode amplitudes AT, I" Using the above form in (14a, b) and dividing by the corresponding phase factors (so that Re tOT, 1 are completely eliminated, and what remains is the imaginary part denoted as o~) one gets the following dispersion relation
( - - ito + 7T1) (-- ito q- 7t) = Co, CoiI ATO + I s,
(15)
where it is assumed that
ATO + --- constant, l ATO+ [ >> l ATl _[, [ ATo+ { >> ] A, [.
(16)
In order to obtain the coupling terms Co, and Cox, we substitute the values of the wave variable (v, TO + nl, etc.) in (14) and we get
v~hk~,o ~
Co, = 2n ° to~ (2OJT0 (°JT0 + oJ¢) 2 _ wv ~ wc~. [wTO(~Tl--w~)k,--wt ~c kTo]
and
Col = 2O h (2 r ° ( ro + (2 rl ( rl +
104 S Guha and N Apte
Equation (15) is solved analytically to obtain the threshold condition for growing Langmuir and scattered electromagnetic wave. The threshold electric field for the onset of instability is obtained by putting oJ = 0 in (15) as a result
m y inTO
F 2 (coTO +
coc) cu t] 1/°-
(eth)B0
#o
- 7Lkr0
{o, r0 (,~rl-o,c) k,--~c o,t kr0}J " (t7) To obtain the growth rate well above the threshold, we neglect the damping terms (YTI- and Yt) in (15) and is solved for oJ, then_e of l
(°J)Bo # o 2m ,or0 L(o, T0 + o~) ¢.O/ { 2~OTl (mTl --Wc) ~ -~- w~
¢Oc} j
08)
In (17) and (18) we have assumed that
~°T0, 1 ~ ~°l ( ~ co/,) ~ co c.
If we approximate
coTO ,,~ OJTi, kTo ~ kt ~ k equations (17) and (18) give
m v oJTO F (°~TO "j- me) COp 11/z
(Eth)B°~0 -- ek" " L{COT0/(oj--~0:o%)-'--"~ ¢OpyJ ' (19a)
and (°J)Bo # 0 = ekETo ~ oJp
l
2m "L~°T0 (°JT; T~_" ojc--- ~ ~2co-'--~0 (OJT0_OJ"-'--~)"S _~_ oJ~ coc}.] "
(19b)
m y
(Eth)Bo = 0 = ~-~ [2°JTo C°p] 1/a' (20a)
ek ETO
a n d (°~)Bo = 0 -- - - [°Jp/2°~ro ]1/~' (20b)
2m coTO
where it is assumed that cot ~ COp. For a magnetoactive plasma (i.e. Bo # 0), we get from (19 a, b) and (20 a, b)
(Eth)Bo =/= 0
[ O~r0 (~,r0
+ o~) ]1/.o(21a) Equations (19a) and (19b) give the threshold pump amplitude required for the onset o f instability and the growth rate for the ease when the pressure term ( ~ h in (7)) has been included in the analysis.
F o r an isotropic plasma (i.e. B0 = 0), (19a) and (19b) yield
Stimulated Raman scattering in semiconductors 105
and (~°)Bo ~- 0
_ _ - - tOT0
L
(°~)ao = o (°~ro + (21b)
Neglecting the pressure term in (7) and following a similar procedure if the expressions for the threshold pump electric field and the growth rate of the unstable scattered electromagnetic and the Langmuir modes are obtained, it is found that they are the same as equations (19a) and (19b). Thus it can be con- cluded that the threshold conditions are insensitive to the electron thermal velocity VTh in isotropic as well as magnetoactive plasmas.
3. R e s u l t s a n d d i s c u s s i o n s
The analytical results obtained have been invoked to study the dependence o f the threshold electric field and the growth rate on different parameters such as the pump wave frequency, carrier concentration, cyclotron frequency, thermal velocity, etc.
It can be inferred from (19)-(21) that (i) the threshold electric field amplitude can be lowered by reducing the Langmuir wave frequency oJz(,~, oJp). SRS from electron plasma waves in a semiconductor occurs only when %, is far away from the optical phonon frequency o~pn. Eth can be reduced further by increasing the wa.ve number kz such that it satisfied the condition k~L < 1 (where L is the electron mean free path) for the validity of hydrodynamic model of a semiconductor-plasma. (ii) Increasing longitudinal magnetostatic field increases the threshold pump amplitude as well as the growth rate. But at very high magnetic fields which yield toTO(tOTO-- tO c)
<coco, p in (21a) and (21b), the instability does not exist. (iii) The growth rate can be increased at higher values of wave number k and can be enhanced further by taking a pump wave with lower frequency such that it satisfies the frequency match- ing condition (equation (2)) and by using a semiconductor with smaller effective electronic mass and higher carrier concentrations. (iv) When the magnetic field is such that
0 < °~c < 0 . 1
°~T0
the threshold electric field amplitude and the growth rate differ very little from those at the vanishing magnetic field, (v) Since k 1 is always greater for the backscattering (i.e. k~=kTo+kT1), it has a much lower threshold than that for the forward scatter-
ing. (vi) Assuming tOT0 ' l > t % and using (21a) and (21b) one obtains
(Eth) Bo~O/(Eth) Bo=0 = [(oJTO+OJc)l(OJTO--Co~)]l/~ (22a)
and (co) ao~o/(OJ) ao=o = [°JTO/(°J~'O--C°~)xl2] (22b)
Comparison of our results (equations (22a) and (22b)) with Maraghechi and Willett (1979) yields
(Eth) A Bo~O I (Eth)MB0 ¢ 0 = 2%/2 (23)
106 S Guha and N Apte
E q u a t i o n (23) shows that the threshold electric field a m p l i t u d e o b t a i n e d in o u r ease is 2 x / 2 times t h a t obtained b y M a r a g h e c h i a n d Willett (1979) f o r a n isotropic semi- c o n d u c t o r p l a s m a . The difference arises due to the fact t h a t we have considered r i g h t - h a n d e d polarized p u m p wave giving rise to a left-handed polarized scattered wave b o t h travelling in the f o r w a r d direction whereas M a r a g h e c h i a n d Willett (1979) h a v e t a k e n a left-handed polarized p u m p wave a n d only a backscattered wave.
(vii) F u r t h e r it c a n be seen f r o m (22a, b) t h a t at o~ _ ~OT0 the threshold as well as g r o w t h rate b e c o m e infinite. (viii) At oJp _~ O~ph electron p l a s m a as well as the optical p h o n o n s contribute to the stimulated R a m a n scattering a n d then one c a n n o t consider the case o f SRS f r o m electron p l a s m a m o d e only.
A numerical estimation o f the threshold electric field required f o r the onset o f the S R S a n d o f the growth rate o f the unstable m o d e s a t a n electric field larger t h a n the t h r e s h o l d electric field has been m a d e b y using the analytical results (20) a n d (22) f o r isotropic a n d the magnetoactive plasmas. W e t a k e a n n-type I n S b crystal at 77°K a n d pulsed C O 2 laser o f 10"6/zm as a high frequency electromagnetic p u m p . W h e n k : 5 × 105 m -1, cop : 0"2× 10 is sec -1 (i.e. n o : 3"18× 1030 m -3) a n d (colli- sion frequency related to m o m e n t u m transfer which is a s s u m e d to be constant) : 2 " 5 2 × 1 0 1 1 sec -1 (obtained f r o m the value o f electron mobility / ~ : 5 × 105 c m ~ v -1 sec -x at 77°K used by G e r s h e n z o n et al 1974), (Eth)B= 0 is equal to 3"4 × 105 V m -1.
I t c a n be applied to n-InSb without a n y damage. T h e d a m a g e threshold c a n be increased by reducing the pulse duration. T h e g r o w t h rate (oJ)B0= 0 is equal to 10 a sec -1 f o r a n electric field o f 108 V m -1. I n a m a g n e t o p l a s m a it is f o u n d that when a p p l i e d de magnetic field is increased f r o m 1-8 tesla (Eth)Bo~ 0 /(Eth)B0= 0 increases f r o m 1 to 1.85 a n d (O~)Bo~ 0 / (~O)B0=0 increases f r o m 1"0 to 1"2.
Acknowledgements
One o f the a u t h o r s (NA) is thankful to her colleagues D r P K Sen, D r S K G h o s h a n d Miss P N a m j o s h i for m a n y helpful discussions. She also gratefully acknowledges the financial s u p p o r t f r o m the Council o f Scientific a n d Industrial Research, N e w Delhi. T h e a u t h o r s thank a referee f o r critical c o m m e n t s .
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