Resonance spectroscopy of solids and plasmas

Journal of Magnetism and Magnetic Materials 11 (1979) 1-15 © North-Holland Publishing Company

RESONANCE SPECTROSCOPY OF SOLIDS AND PLASMAS B. LAX Francis Bitter National Magnet Laboratory * and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 10 November 1978

A brief review of the history of cyclotron resonance in plasmas and solids over a 30 year span at MIT is presented together with the interband resonant spectroscopy of semiconductors and semimetals which was a consequence. Magnetoplasma phenomena at microwave and infrared frequencies is also treated. Finally, the role of lasers is weaved into the fabric of the more recent extensions of these experimental and theoretical studies.

1. Historical review Cyclotron resonance was first demonstrated by ionospheric physicists and later in the laboratory in gas discharges at UHF. The first microwave experiment was a demonstration for a thesis project by the author in 1948. In essence microwave resonance spectroscopy of electrons at MIT began with this project on gaseous plasmas [1] which was further explored in subsequent years [2]. Using similar techniques in semiconductors, cyclotron resonance was observed in germanium by the Berkeley [3] and Lincoln groups [4] following a theoretical proposal by Shockley [5]. The Berkeley group observed an apparently isotropic resonance, while the Lincoln group showed structure and the existance of a large anisotropy. The latter were the first to determine the mass parameters of electrons and holes [6]. They also introduced light excitation of carriers which allowed the observation of anisotropy of both holes and electrons in silicon as well [7]. The next significant advance was made with the introduction of infrared techniques and high magnetic fields by the Naval Research Laboratory (NRL) [8] and Lincoln groups [9]. These experiments demonstrated the quantum nature of cyclotron resonance in InSb and the existence of the non-parabolic nature * Supported by the National Science Foundation.

of the conduction band [10]. However, the optimum system for such experiments was achieved with the introduction of submillimeter lasers into the high field water cooled magnet at the National Magnet Laboratory [ 11 ]. This ideal combination of high frequencies and high fields has been exploited to determine the mass parameters of many I I I - V [12], I I - V I [13] compounds, alkali halides [t4] as well as a number of ternary compounds [15]. It also allowed the observation of combination resonances involving spins and phonons and harmonic combinations as well [16]. One of the most novel experiments was the demonstration of the fundamental nature of the polaron cyclotron resonances in CdTe [17]. Microwave and millimeter resonances have also demonstrated the magnetoplasma phenomena [18] and the observation of helicon waves in semiconductors and semimetals. The cyclotron resonance of metals was also begun at Bell Telephone Laboratories [19] and Lincoln Laboratory [20]. The proper theoretical analysis of Azbel' and Kaner [21 ] elucidated the true nature of cyclotron resonance in metals. The resonance experiments which were observed in the quantum limit as intra-band transitions led to the discovery of the analogous quantum phenomena of intraband transitions in semiconductors by the NRL [22] and Lincoln [23] groups. The latter also discovered this effect in magneto-reflection in semimetals as well [24]. The intraband cyclotron resonance

2

B. Lax /Resonance spectroscopy of solids and plasmas

and the interband magneto-optical experiments combined proved to be powerful tools for the quantitative study of the band properties of both semiconductors and semimetals. The interband magneto-absorption experiments were the first to determine the effective mass of a higher band in Ge [23] and the first to measure the anomalous g-factor of electrons in InSb [25]. These interband transitions were later observed as stimulated emission in laser diodes in a magnetic field [26]. The magneto-plasma experiments were also extended to the infrared in analogy to the ionospheric experiments [27]. These also served to explore the plasmon-phonon coupling in the presence of ~t magnetic field [28] and to demonstrate the many-body nature of these coupled systems [29]. Resonance experiments in ionized plasmas here at MIT have also received attention. Bekefi and co-workers [30] showed the nature of the cyclotron harmonic emission from a hot plasma. These now serve as important diagnostics for the tokamak plasmas such as Alcator [31 ]. Another extension has been the use of high power submillimeter lasers for the breakdown of gases and for heating them resonantly in a magnetic field [32]. With the advent of gyrotrons such resonant heating may be important supplementary energy sources for tokamaks. In fact, lower hybrid and ion cyclotron heating at microwave and UHF respectively are being explored for heating Alcator.

~.~0.

5

P

To

4

0

2

3

cO Fig. 1. The power absorbed by an electron from a microwave field as a function of magnetic field (after Lax et al. [6]).

demonstrated most easily in a gas where the pressure is varied, hence car, as indicated in the breakdown threshold for He gas in fig. 2 in a large volume cavity where diffusion losses are minimal. In a flat plate cav-

IO00

BREAKDOWNO F HELIUMINTRANSVERSE ELECTRICANDMAGNETICFIELDS D I A M . = 7. 32c m HEIGHT

= 4.60cm

E

X 03 I---I 0 >

2. Cyclotron resonance in plasmas

Z a

The classical concept of cyclotron resonance at microwaves is a simple one. The electrons execute helical motion along a magnetic field with a cyclotron frequency cac = eB/m. In the presence of a transverse microwave electric field in a cavity, the absorption is given by P Po

1 + (ca2 + we2) r 2 [1 + (cae2 - 6o2) r2] 2 + 4ca2r 2 '

LI-

I00 ~

30mm

Z

3

0

Llul rr rn

(1)

where Po is the power absorbed at dc electric field, i.e., co = 0, cae = 0. When the expression in eq. (1) is plotted as a function of magnetic field for different values of car, the curves clearly demonstrate the required criterion for cyclotron resonances as shown in fig. 1. This resonance behavior is experimentally

to

~m 0

I000 2000 B, MAGNETIC FIELD IN GAUSS

3000

Fig. 2. Breakdown of helium at microwaves as a function of magnetic field (after Lax et al. [1 ]).

B. Lax /Resonance spectroscopy o f solids and plasmas

rotating circulary polarized modes of a circular cavity as perturbed by the decaying plasma. From perturbation theory one can show that

ity where diffusion dominates when this is taken into account using the proper dependence of the diffusion tensor on magnetic field then the result for the breakdown microwave electric field E m as a function of magnetic field normalized by the breakdown field Eo at B = 0 is given by Em _ / Eo

[1 + (we2

-

w 2) r2] 2 + 4w% 2

Aw_ 1 0.94Wc/~ A6o~- - 1 + 0.94Wc/CU " -

/ 1/2

/ [1 + (~¢~ +~-~)-~2-~- + ~ 2 ~ - ~ ~--¢o¢2r2)j

3

•

(2) The diffusion has the effect of further reducing the breakdown field at increased values of the magnetic field. The single electron result given in eq. (2) can be rigorously justified in He for a flat cavity from the Boltzmann theory. Experimental results are in good agreement with these theoretical predictions as shown in fig. 3. A second experiment of electron cyclotron resonance at microwaves was carried out in the after-glow o f a discharge in which a variable frequency probe measures the frequency shift of the two counter-

(3)

Experiment agreed well [2] with the above expression and were useful in determining the variation of the diffusion coefficient with magnetic field. The experiments analogous to the microwave breakdown were considered for the submillimeter lasers [33] which have achieved considerable power. Indeed both cyclotron breakdown and heating of electrons were studied quantitatively [32]. The simple extension of the theory of the microwave analysis was inadequate because the plasma produced by laser was unbounded and the absorption of the laser resulted in an inhomogeneous electric field. The result was a broadened line as a function of power which has been analyzed by Hacker et al. [34] to explain the anomalous absorption. Recently Biron et al. [35] have fitted Hacker's theory very well at lower pressures to the

' II ' BREAKDOWNOF HELIUMIN TRANSVERSE ELECTRICAND M4GI~TICFIELDS THEORETICAL e----.-e EXPERfMENTAL

p • 2.0 MM

k

e [

p,l.O MM

z_ ° .J

_w

b.

I

I

~.m. . ' ' ' ~ ' ' l r ~ "e

I

I

1

Fig. 3. Comparison of theory and experiment [eq.

I

I

I

t

t

E

i

i

J

80O 12OO 1600 2OOO 24OO B, M ~ T I C FIELDIN GAUSS (2)] for microwave breakdown of helium vs. magnetic field (after Lax et al.

[1]).

B. Lax /Resonancespectroscopy of solids and plasmas

4

experimental data with the following expression for the absorption

A =eUeff[ 7ra2 In I1 + [ e ° N ° WL [ o L ~ ira 2

~

I

l) e °

CRYSTAL ISOLATOR~

7ra~

(4) where o=

e2

•

Pc

.

VAR,ABLE

~r

To observe resonance especially with a multicomponent carrier present, it is necessary to have 6or > 1. To achieve this we examined pure Ge and Si during 1953 and 1954 at microwave frequencies at X-band and K-band 9 GHz and 24 GHz, respectively, immersing the samples at liquid He. The experiments were carried out with a microwave paramagnetic resonance spectrometer with the sample at the center of the microwave cavity. Our innovation was to freeze out the carriers to avoid magnetoplasma effects and then reionize them in a controlled way with microwaves or with light. The latter proved to be superior. A schematic of the experimental system is shown in fig. 4.

rr

fl

/"~ . R E ~

Ho [no(

CAVITY 1i

f l

L,OU,

PRECI SION~1IIF CALI BRATED II][ ATTENUATOR!.~.1CRYSTAL?l . ~ ~

SAMPLE IN~R)A~LAMP

/ NETWORK = --

I DETECTION

Fig. 4. Experimental system for microwave cyclotron resonance of semiconductors (after Lax et al. [6]).

With infrared light using a hollow tube we excited either electrons or holes in n-type or p-type materials, respectively. With white light through a quartz light pipe both holes and electrons appeared. The :;pectra appeared as shown in fig. 5 for electrons and holes. Both were anisotropic. The effective masses, ~a function of orientation were found for electrons i~l Ge and hole~ in Si. From these experiments, the mass parameters cf electrons were first determined as follows [6,7]:

3. Cyclotron resonance in semiconductors

3.1. Microwave resonance

L

TTENOATO.

2mc (6o -- 6o¢)2 + Vc2 '

and ueff is an effective ionization energy; WL is the laser energy; a is the focal radius; vc is the collision frequency and No is the electron density. Another pioneering effort at MIT has b~en the study of cyclotron emission from hot plasmas. Bekefi and co-workers [30] observed these effects and developed the theoretical models which account for the phenomenon. With the development of hot plasmas in tokamaks it soon became apparent that the cyclotron emission including its harmonics offers another important diagnostic tool for measuring the properties of a plasma in a magnetic field. Using a fast scan Michelson or a Fabry Perot interferometer, the cyclotron harmonics were clearly observed in the Alcator tokamak. Hutchinson and Komm [31] have used this data to determine relative electron temperatures of the plasma.

Emo.

STA~/Lc!IATT'0N~ " - -

Ge Si

mt/m

ml/m

0.0819 -+0.003 0.192 -+0.01

1.64 -+0.03 0.98 -+0.04

More accurate values are available today. However, for the initial values these were remarkably good. For holes, the band structure was theoretically derived by Dresselhaus et al. [35] using the k • p perturbation as follows: h2

g(k)=_T~m[A/c:+X/B2k,

+ C~ (kxky 2 2 + kxkz 2 2 + k~kz)] 2 2 • (5)

Using this model we were able to fit the data for the light and heavy holes in Ge and Si by using Boltzmann theory or Shockley's integral for warped sur-

B. Lax /Resonance spectroscopy of solids and plasmas

0

CO I.~ Z 0

0 ~

0

-

0

1§;

m

~_ _ o O 0 W

Z(3= 01--

Z 0

--

W

~b.I ~ ~. I u-

~

of an effective mass equation analogous to the KleinGordon equation for relativistic electrons. The data fit the solutions given by [38]/x~ = ~ n - / ~ n - I where

W ._1

o

"r >-

(6)

&n = l~/gg2 + 4&g [(n + }) hwc + g/aBH],

giving approximate expressions for the effective masses and g-factors as follows

-

1000

2000

5000

MAGNETIC FIELD (oe)

Fig. 5. Cyclotron resonance trace in germanium at 4 K and 23 000 MHz (after Dexter et al. [7]).

faces. The parameters A, B and C were then determined [37].

Ge Si

5

A

B

C

13.1 -+0.4 4.0 -+0.1

8.3 -+0.6 1.1 -+0.6

12.5 • 015 4.1 -+0.4

Again, greater accuracy has been achieved by more recent experiments for these band parameters but the essence of the early results was definitive.

m* (H) ..~ m*~/1 + 4nhwc/gg . g(H) =g(0)/x/1 + 2(2n + 1) ~COC/~g .

(7)

A comparison between theory and experiment for the effective mass is shown in fig. 6. The problem with the infrared experiments at very high fields and relatively short wavelengths is that the technique is limited to materials with small effective masses. Hence, it became evident that the optimum combination for cyclotron resonance to explore a larger class of materials was the submillimeter frequencies and continuous magnetic fields in the range of 100 to 200 kG. With the advent of the molecular lasers and the development of high field water-cooled and superconducting magnets this situation became a reality. The first such experiments were carried out at the MIT National Magnet Laboratory using an HCN laser developed by Gebbie and co-workers [ 11 ]. For a demonstration we selected

3.2. Infrared cyclotron resonance 0.030

The next significant step in cyclotron resonance in semiconductors was to extend the microwave technique to other materials. A review of these and at millimeters is given in detail [38]. Although limited success was made in observing electron cyclotron resonance in InSb at microwaves, the holes were not resolved to provide the band parameters [39]. Two groups - one at NRL [8], the other at Lincoln [9] carried out experiments in the far infrared region on electrons in InSb. The Lincoln results used pulsed fields up to 300 kG and wavelengths from 10 to 20 /am, while NRL used dc fields up to 60 kG at 40/am. The very high field data illustrated the nonparabolic nature of the conduction band and the interpretation has to be made in terms of resonance in the quantum limit between Landau states of low quantum number. The resultant expressions can be expressed in terms

o

E

_

In Sb _

-

Z

~

0.025

0.020

UNCOLN LAB DATA

t0.015 l,i

/

h-

z ne

:} T.A.SM,SS,O.

0.010 THEORY

Q.

< 0.005

o

[ 50

I

I

l

I

I

100

t50

200

250

300

MAGNETIC FIELD (kilogouss)

Fig. 6. Variation of apparent effective mass with magnetic field in InSb at room temperature (after Lax et al. [ 10]).

6

B. Lax /Resonance spectroscopy o f solids and plasmas

p-InSb (IOO)

T=35 K x= 33,m~

-rlj3 / ~

~I~If

2

]

\

A

~

A = H(O, x, 0 ) , af~n = [PZx+ (Py - m*wex) 2 + P~I ~n/2m'.

(8)

The solution to the quantum mechanical problem is

~n = (n + ~) booe + p2z/Zm * .

(9)

In a semiconductor the full wave function is a product of the envelope function, which is the.solution of the effective mass Hamiltonian and extends through the crystal, and a band edge function which is characteristic of the unit cell, i.e. L

0

50

I00

150

MAGNETIC FIELD (kilo-Oersteds) Fig. 7. Cyclotron resonanceof holesin InSb at 337/~m,~howing quantum effects of light and heavy holes. Inset is millimeter resonance (after Button et al. [13 ]).

p-type InSb among the first candidates. The spectrometer was analogous to the microwave case with the exception of the cavity, instead we used simple transmission through the sample which was located at the center of a Bitter magnet in a variable temperature dewar. By varying the magnetic field, the absorption spectra was observed from the output of a Golay cell or bolometer. The spectra showed a great deal of structure [40] which is absent even with a millimeter spectrometer as shown in the inset of fig. 7. The important result is that at relatively modest temperatures in a moderately impure crystal one can see the quantum effects of holes in InSb which were observed in "pure" p-type Ge at 2 ° with microwaves [41]. In recent experiments using megagauss fields and molecular lasers Miura and co-workers [42] have extended the high field measurements to a number of semiconductors.

4. Quantum aspects of resonance spectroscopy 4.1. Theory It is known from the work of Landau that the Hamiltonian of an electron in a magnetic field gives rise to harmonic oscillator-like levels. In a semiconductor one simple writes the Hamiltonian in the effective mass approximation as follows in the Landau gauge

~in ~ ~kn(r) ui ,

(10)

where i refers to a particular band. Hence, when we consider the selection rules for transitions between Landau levels there are two classes, namely, intraband and interband. From the matric element which can be represented as

f t~in(P " e) ~lm = f Uiui dr f ~,.O, " e) ~m dr cell

+ fu,(p.e)u, crystal

dr

crystal

fq',,~m

dr.

(11)

cell

The selection for the first is the intraband or cyclotron resonance transition with the selection rule An = +1 and the second is the interband transition with An = 0. These two phenomena are complementary and together have proved to be very powerful tools for studying the band properties of semiconductors.

4.2. Polaron cyclotron resonance One of the most dramatic applications of cyclotron resonance at submillimeter wavelengths is the polaron cyclotron resonance in CdTe [17]. It was first shown by Larsen [43] that an electron in a magnetic field in a polar semiconductor introduces additional terms which alter the transition energies between quantum levels. He used a variational treatment. However, one can show from perturbation theory that the energy levels can be represented by an expression of the form

a( 6°L)3/2 = (" + ½)

+ x/(. + ½)

c

sin -11/(n +½) v (12)

B. Lax / Resonance spectroscopy of solids and plasmas HZ0

phonon energy are taken into account. The diagram also shows various submillimeters lines for different lasers that have been used to observe the cyclotron transitions. These experiments have been extended by Litton and co-workers [44] and the mass as a function of magnetic field is shown for the transition n = 0 to n = 1 in fig. 9. The solid curve is a theoretical one using the theoretical variation of the effective mass with magnetic field which is given approximately by an expression

n -CdTe T"

Z

50

K

o t-O. re

Z I"

0

I

~

0.060--

I

z

zo

(J :I:

(-} N O(:~

///

o N "r

0.040 --

7

/ /n=2

mp m* ~

/

1 +--+ 6 20 col

....

(13)

Z

oo

- ~ ~0.~ ~ L

~: 0 . 0 2 0 L~J -1

0

50 IO0 q50 MAGNETIC FIELD, N (k0e)

200

Fig. 8. Polaron effect on cyclotron resonance in CdTe. The plot shows energy vs. magnetic field for the three lowest levels. Arrows indicate transitions which were observed (after Waldman et al. [ 17 ]). One can expand this expression to higher order in terms of the cyclotron frequency to show that the electron phonon interaction as represented by the coefficient a introduces nonparabolic terms. A plot of the quantum levels as a function of magnetic field for CdTe is shown in fig. 8 where effects due to pinning near degeneracies of the quantum levels including the

0.112 --

f

experiments --

*E 0.108

HzO=;~ laser /

theoretical, a =0.40

D z O' ~ "

ffl

FTS-,~ j

~ 0.104

02O

bJ

_>

HeN

0.I00

loser

w h

~ ~ D C 1~p.wove

h

'" 0.096

I

O

.

i

N ~ys. Re~ e

13 .5592 ( 1 9 7 6 )

"

I

i

The best fit to the curve is for the value of a = 0.4 which differs from the optical value. However, the latter depends on the difference of the measurements of e0 and e** which is subject to error. Hence, not only does the cyclotron resonance experimentally confirm the Fr6hlich theory of the polaron quantitatively, it also gives the best value of the electron phonon coupling a. 4.3. Spin resonance

Spin resonance of electrons was first observed in semiconductors by Fletcher et al. [45] in silicon in a bound donor state. This work and that in germanium was extended by Feher [46] who used the electron nuclear double resonance technique called ENDOR. With this technique Feher not only determined the g-factor but showed that with the hyperfine interaction of the electron with Si nuclei, he was able to establish that the electron minimum was inside the zone edge at ko/kmax = 0.85. Further experiments in silicon and germanium shallow donors showed that the g-factors were anisotropic [47] and, in particular, in germanium where gll = 0.87 andg± = 1.92 was in good agreement with the theory of Roth and Lax [48]. Spin resonance in other materials has also been studied. The one of particular interest has been in that of the electron in InSb where the g-factor is anomalous as first shown from interband magnetoabsorption a value g ~- 50 in agreement with the theory of Roth [49]

J

20

40 60 80 IOO 120 FREQUE NCY, D- (cm -~ ) Fig. 9. Variation o f polaron effective mass in CdTe with frequency (magnetic field) (after Litton et al. [44]).

g = 2 I1 + (1 - ~ * * )

A

3~g + 2A

-1

"

(14)

The first spin resonance was actually observed at mi-

B. Lax / Resonance spectroscopy of solids and plasmas

8

crowaves by Bell [50] which confirmed the earlier magnetoabsorption value.

sion using the following form A~ ~

4.4. Combination and harmonic resonances

The possibility of combination resonance in which an electric dipole transition is accompanied by a spin flip transition simultaneously, i.e., An = +1 and Am s = 1 was first predicted theoretically by Rashba and Sheka [51 ]. The first observation of this phenomenon was made by McCombe and co-workers [15] in InSb with an infrared spectrometer. Subsequently, even more dramatic results were obtained by the NRL group in Hgo.alCd0J9Te alloys with a much smaller gap using a submillimeter laser spectrometer [52]. The data as a function of magnetic field in InSb shows multiple resonances, namely: spin, cyclotron resonance and combination resonance involving spin and harmonics, as shown in fig. 10. The data can be explained by using approximate expressions similar to those given in eqs. (6) and (7). For the combination resonance the energy difference can be fitted by an expres-

26o c

2.0-

(a) Ep ± H 1.5-

--

,,

~ II till] ,Sample # 16

]I

.... ~,,[,,o],somp,e# 17 ---- HII[IO0], Sample # 16

i,,i

2Wc+W e//~

Magneto-plasma resonances have been encountered in conjunction with microwave cyclotron resonance [18] and in the far infrared region [27]. These exhibit themselves as a shift of the cyclotron frequency which, in the language of plasma p h ~ c a l l e d the upper hybrid resonance, i.e., co = ~/co~ + co~. There are other resonances which occur at long wavelengths when one assumes k ~ 0 and results in expressions for single carriers of the form

oo.o

20

?

(b) Epll H

-r 0.5

40

60

t

3~C

- - H II[lll], Sample #16

i80

-

I00

120

,

I

2We+We l'~ 2Wc

/ i...>- " 0.1-

20

40 60 MAGNETIC

80 IO0 F I E L D (kOe)

--J 120 140

Fig. 10. Magnetoabsorption vs. magnetic field in InSb with ~- ~. X 106 cm -3 at 10.6 urn showing anisotropy o f harmonies and combination resonances (after Favrot et al. [55]).

ne

+ co )/coc,

co = co2/coc

(17)

140

- - - R ,[,c]. Semp~ w' t7 o.2 . . . . ~ ,[lo0],Sample # 1 9

0.0

(16)

These also give rise to plasma shifted cyclotron resonance or a low frequency plasma

I

.//.Y'

(15)

4.5. Magneto-plasma resonances

co = (co -

gol.tBH

Other types of combination resonances have involved phonons as well as spins, i.e., phonon assisted cyclotron resonance. In fact, in such materials as InSb a whole host of harmonic resonances with phonon assisted and spin flip transitions were first observed by Enck et al. [53] and the harmonics including phonons were observed by Johnson and Dickey [54]. A more complete study of these resonances and their anisotropies have been examined at high magnetic field by Favrot and co-workers [55]. The theoretical analysis of these transitions with suitable polarization has been well accounted for by Weiler et al. [56].

co-+= ½(coo +x/co2 + 4co2) •

i i

, .o -

~"

hcoc

-+ X/1 + 4hcoc/P~g X/1 + 6hcoc/~g

resonance when 6% > > cop which has been observed by Dresselhaus and co-workers [18] in InSb. However, one can also show that in the limit of cop > > coc CO + ~COp -+ ICe c .

(18)

This splitting of the plasma resonance has been observed at infrared frequencies as shown in fig. 11 and was first used to determine the effective mass of electron in doped HgSe, which normally is a zero gap semiconductor [57]. The infrared experiment was extended to longer wavelengths to measure the optical phonon coupled magneto-plasma system in InAs [28].

B. Lax /Resonance spectroscopy of solids and plasmas ¶00

I

I

I

I

I

I

I

90

I

I

I

for COp< < we and COp > > COcas follows

I

o 38.5 kO • 25kG

80

~,

\

7O z

,~

CO~ COc+ ~2p/(COc+ COo ,]

$2.1kG

a

CO~ CO

=

COh + ~/(COc + COh),/COp<<

I

COp > > coc

(21)

where 2

2

~ p = 4COpe q- COph ;

2O

I

CO~

,.Q2/(COe .iI ( . O h ) ,

CO = ~ p + I(COe -- COh),

i. t .055 .057 .059 .06t

9

I

I

~

'

T

.OG3 .065 .067 ,069 .07t PHOTON

ENERGY

"~

I

I

.073 .(~5 .0"/7

COc

--

eH mcC

;

COh

_

eH mhC

In a similar manner we can obtain results for the coupled hybrid frequencies for the electron hole droplets where the secular equation then becomes

(e.v.)

Fig. 11. Magneto-plasmaeffect in n-type InSb, n e = 1.8 × 1018 cm-3 (after Lax and Wright [57]).

((02 _ (.,32c _ 632)(032 _ CO~h -- 022) = COpcCOph2 2

These experiments suggested a more general quantum treatment of magneto-plasma phenomena which considers coupled systems of multiple carriers and can include phonons as well [29]. It can also be used to treat the plasma resonance of finite plasmoids as in the case of electron hole droplets where depolarizing factors enter into the coupling terms as first shown by the classical treatment. Since we assume relatively cold plasmas in semiconductors, we neglect the dispersion and represent the Hamiltonian in the form for electron hole pairs as

CO ~ N/r'~cCOh ;

= a+a[22 + b+b~22 + a+bcopccoph ,

(22)

where CO~ X/J22 + COc 2 + co~ is the upper hybrid analog for the gaseous plasmas case including depolarizing factors and

(19)

where the eigenvalues ~2c and ~h are those given for each of the carriers by eq. (16). In this c a s e COp2 _>LCO2 where L = ~- is the depolarizing factor for a sphere which is assumed for the droplets. If the droplet is deformed then the result is rewritten in terms of a tensor. In any event, the secular equation results in an expression of the form 2 2 2 2 ~"2c~'2h = COphCOpc , or ( 6o2 -+ COCOc -- CO2c)(CO2 -+ COCOh -- CO2ph) = COpcCOph2 2

(20) for the long wavelength limit with resultant solutions

~'2p ~

COc, COh

(23)

is the lower hybrid. Other solutions can be obtained numerically for such cases as germanium and silicon where there are two holes and multi-valley electrons in an analogous manner. The eigenvalues for the electron plasma resonances have been treated by Lax and Roth for Si and Ge [58]. The equivalent results were rederived by Kononenko [59]. The important result is that these plasma resonances are observed at submillimeter waves with present lasers [60]. The large number of components can be reduced to the equivalent of two carrier analogies for Ge and Si by orienting the magnetic field along the high symmetry axes.

5. Magnetospectroscopy in semiconductors 5.1. Interband magneto-optical effects

As indicated previously the quantum nature of cyclotron resonance led to the discovery of the analogous interband transition in semiconductors. It was observed in germanium [23] and indium antimonide [22] for the direct transition in thin samples by magnetoabsorption. It was shown theoretically that the magnetoabsorption coefficient for the direct transition

10

B. Lax / R esonance spectroscopy o f solids and plasmas

has the form [61 ]

(72-0

el(B) =Aoo c ~ (co - COn)- ' ' 2 ,

t '

(24)

n

where Wn = 6% + (n + -~) 6% and coc = ell/pc where p is the reduced effective mass of the holes and electrons. The initial bonus for germanium was that the effective mass of the electron in the higher P2 band was measured for the first time and the direct gap measured with accuracy at low temperature as well as room temperature. In InSb the determination and first observation of the anomalous g-factor [49] was made where it was shown to be - 5 0 . Subsequently, the indirect transition magnetoabsorption in germanium was also made and was shown to be a step function and with broadening [62] as shown in fig. 12.

a(H) =Bwe]OOc2

tan -] (co - (,On) r

'

o

'

GERMANIUM(IO0)

771.0 --

'//Z o

/ o//

~ T?O.O

/

(25)

using this technique not only the electrons were studied at L point but the structure of the indirect exciton was detected with and without magnetic field [63]. This work was extended to higher fields [64] and the spectra is shown in fig. 13. The data show the splitting of the indirect exciton of 0.0010 eV. The later development of magnetoreflection became more suitable for the study of the I I I - V and other low gap materials [57]. With the introduction of the modulation techniques a more careful study of the conduction band in germanium [65] explored the nonparabolicity of this band to reasonably high energies as indicated by the closer spacings of the transitions in the spectra shown in fig. 14. A confirmation of these results on a quantitative basis was made by the cyclotron results

76e-0

1

I

20

I

r

I

40 H ( KilogCNJIS}

I

1

60

Fig. 13. The Zeeman spectrum of the indirect exciton in Ge vs. magnetic field (after Halpern and Lax [64]).

of Miura et al. [40] using megagauss fields. The nonparabolicity of the mass variation as a function of energy above the bottom of the band was the same as that of the interband studies at lower fields.

5.C 4.0 -

~ i r ~.-~ GREASE

3.0 8 o

--J~ v Z _o Ul U) Z OE I--

6

~ZERO~X, \ FIELD , ~ - 3S.gl ; ; %

",:, 2.0

B ALONG[lO0] 1.5IK

:<

LANDAU

[ EiCTG/Ni ['~~~TRANSITIONS ~ 4

0.768

0.772 0.776 0.780 PHOTON ENERGY(electronvolts)

Fig. 12. Magneto-absorption of the indirect transition in Ge at low temperature (after Zwerdling et al. [62]).

! .0

~~

0.0

Ls

-I .0

[

[

L'I

I

]

L7

[

Ls

.750 .765 .780 :?95 .810 .825 .840 •855 PHOTONENERGY(eV) Fig.]4. Magneto-piezo-transmissispectrum on of theindirect transition in germanium (after Aggarwal e t a1.[65]).

B. Lax / Resonance spectroscopy of solids and plasmas I

go.o~6 tO

I

I

I

i

H g l _ x CdR Te

o.o 4

l

+g ~" :

:/FI~

T=24K

:

bJ

/

--> 0.012 t-'-

/

+g

/ /

Ld

o0 c W

/

0.008 t,i I

/ O00E

nn

z 0004 0

//+c

+d •

PRESENT

WORK

L

¢ /

0.002

/ 0

+f +e

/÷g

I-.-

0

+g

/

/

<

r~ Z

/

/

/ IV

+ PREVIOUS

WORK

;

I I [ I 1 018 0 2 0 0.22 0 2 4 0.26 A L L O Y COMPOSITION x

I 0 28

Fig. 15. Effective mass of electron in Hg I _xCdxTe as a function of alloy composition (after Weiler et al. [66 ]).

11

Most recently Weiler et al. [66] have made a systematic study of the interband magnetorefiectance of a series of semiconducting Hg l_xCdxTe alloys. Not only were the band parameters of the holes and electrons determined from these measurements but the theory was extended to include additional parameters contributed by higher bands. The power of this technique is best summarized by two sets of curves shown in figs. 15 and 16. The first shows the variation of the effective mass of electrons from mc/m = 0.002 to 0.015 for cadmium concentrations of 18 to 27%, respectively. The second figure shows the corresponding g-factor of the electron which changes radically from about 500 to 50 with change in composition. This change is consistent with the model of g-factor given by eq. (7) when the values of the energy gap of ~20 meV and 200 meV and the spin orbit splitting A ~ 1 eV are inserted. 5.2. Zeeman effect o f shallow impurities

I

I

I

I

500 Hgl_ x Cd= Te T = 24K

u

"~'~40C

•

0 I-

+ PREVIOUS WORK

PRESENT

WORK

I 0

tu 3 0 C (.9

I

I.i,.I I

I I

Z

I

< 20C

+d +e

Z

_o

I.-

:

(J :::3

~ o

~

+:f \ x.

+g

IOC

U

0

I 0.16

t 0.18

I I [ 0.20 0.22 0.24 ALLOY COMPOSITION

I 026 x

I 0.28

Fig. 16. The g-factor of electrons in Hg 1 _xCdxTe as a function of alloy composition (after Weiler et al. [66 ]).

The Zeeman effect of impurities is silicon [67] and germanium [68] were examined in the far infrared. For donors the effective mass model was straightforward and confirmed the mass parameters obtained from cyclotron resonance. However, for acceptors the structure was more complicated due to the degeneracies of the valence band. Nevertheless, a relatively simple spectrum was obtained for the spin split valence band in silicon and yielded the first accurate measurement of the spin orbit splitting in this material [69]. With the development of Fourier transform interferometers and submillimeter lasers a renewed interest in the study of Zeeman spectroscopy of shallow impurities in semiconductors was shown. In particular, the study of donors in GaAs [70] and CdTe [71 ] which have simple conduction bands has made it possible to investigate the Zeeman structure to relatively high fields. The spectra for GaAs is shown in fig. 17 with comparison of the theory of Larsen [72]. He introduced a variational function which is the product of the hydrogenic and Landau functions. Hence, in the limit of low or high magnetic fields this function is asymptotic to the former and latter, respectively. At the same time the wave function is well behaved for all values of magnetic fields as can be seen from the excellent fit of theory and experiment. A similar treatment was applied to the spectra of Cd-

B. Lax I Resonance spectroscopy of solids and plasmas

12 tZO

-o--

THEORETICAL /

/

d

o/ ( t s ~ 3 p ,

U

o~ Z

/o / //('ls--3p,

/

80

I

~

mo ro

/

.o

i/o / i 60

o/

/ //

,io

50

I.s

_o i

i

°/

/

io

(ts~3p,

f

/

/"

o/

210

li

m =-t)

Y 190

el

/

/

m = O)

o/

//

i

o'"

.//~

/ °J/ ~

(4s--2p,

m = +t)

j - -

i >_ 170 (J

"

o/://

. , ::-2p, m =o)

~ -.~- ~

u_

150 ,

3oL

~

0

fl

i

230

,o/

II

90

I

m = +tl

pI

t00

I

250

//

4t0

IE

I

/ o/

EXPERIMENTAL

~ 10

P ~5

20

MAGNETIC

25 FIELD

~ 50

t

I

35

40

=

130

(kG)

Fig. 17. Comparison of experiment and theory of impurity of Zeeman effect in n-type GaAs (after Stillman et al. ref. [70]).

I10

90

Te [71]. Here, however, the polaron effect produced a splitting of the Zeeman levels. Hence, when the polaron correction is applied the theory then agrees well with the experiments. Unfortunately, the latter have not been carried to higher magnetic fields and corresponding wavelengths to check the theory completely but with the large number of lines from optically pumped molecular lasers and fields in excess of 200 kG these can now be experimentally investigated. The comparison of theory with existing data as shown in fig. 18 is, however, sufficient to confirm the value of c~ = 0.4 to give the best fit.

6. Resonance spectroscopy in semimetals The resonance study of metals at microwaves will be presented by Azbel' as a review at this Conference. Nevertheless, it is only appropriate to say a few words about resonance spectroscopy of semimetals at microwaves and infrared, since together with Bell Laboratories, we pioneered in developing the techniques for these studies [73]. The first observation was carried out in bismuth using an apparatus similar to that for the semiconductors. The principal difference was that we investigated the derivative of the absorption by

"

(Is ~2p,m:-I)

7o

I 0

50

I

I

[

I O0

150

200

MAGNETIC

FIELD

250

(kOe)

Fig. 18. Polaron effect on the impurity Zeeman levels in n-type CdTe. Solid fine is theory and solid state are experiment (after Cohn et al. [71]). using magnetic field modulation. The microwave apparatus is shown in fig. 19. The groups at Bell Labs., i.e. Gait et al. [74], actually improved considerably on our results and also obtained ra+her spectacular data in single crystals of graphite [75] with the derivative technique showing a rich harmonic content in the spectra. This work inspired considerable theoretical effort to analyze the Bi data and more sophisticated work on graphite. Another attempt to observe cyclotron resonance in bismuth was made with the high pulse field infrared apparatus so successfully used for InSb [9]. In the analysis of the data, the author developed a HamiltonJan for bismuth based on the two band anisotropic model [27], namely +

=

½,,"

+

g'/4, 2

&n(1 + £ni&g) = (n + 1) h ~ c + 2 ~ . + ~ g e f f H ' g

(26)

B. Lax / Resonance spectroscopy of solids and plasmas

,

SAMPLE TRIGONAL AXIS BINARY AXIS RIX

BI

CYCLOTRON ABSORPTION APPARATUS

13

parabolic character of the electrons as well as the anisotropy, thereby complementing the data of the microwave cyclotron resonance experiments. The technique was extended to Bi-Sb alloys, Sb, As and to graphite. The latter which were extensively studied by M.S. Dresselhaus, Mavroides and co-workers [78] have elucidated the Fermi surface of graphite in great detail. This work has also served to properly account for the transport properties in these materials.

7. Semiconductor lasers

Fig. 19. Microwavesystem for the study of cyclotron resonance in semimetals (after Dexter and Lax [73]).

where geff ~ 2 m / m ° and 0% = eB/m*, the anisotropic cyclotron frequency for bismuth, which was worked out in detail [76]. Fortunately, the analysis did not explain the infrared data in the 10-20 tam region. The author recognized that what was being observed was the interband magneto-reflection transitions. This analysis immediately suggested a straightforward experiment with dc fields in magnetoreflection [24]. The experiments [77] were rather successful and started a new class of studies in semimetals. The data appeared as shown in fig. 20 and showed many peaks characteristic of the interband transition between levels of different quantum numbers. The analysis of the data by the traditional fan chart showed the highly non-

Bi

With the development of the semiconductor lasers it was only natural to speculate on the properties of these devices in high magnetic fields [79]. There were two considerations of importance, namely, the effect of the threshold and the tunability of the lasers with magnetic field. As expected the threshold was considerably reduced [26] as a function of magnetic field as expected. However, the theory based on the simple singular density of states did not explain this phenomenon at all. In order to do this Sachs and the author [80] found it necessary to use the Kubo density of states to account for this behavior. The magneto-optical laser also demonstrated interesting tuning properties which were relatively large for such low gap materials as InSb, PbTe and PbGe. Fig. 21 shows the tuning of a PbSe laser with two sets of stimulated transitions at lower fields [81 ]. This splitting is amea-

0.150

0.148

ZO 0,146 ttl o n

0.144

o

I 9

I

I

18 27 MAGNETIC

I 36 FIELD

I

I

45 54 (kllogaus$)

I

63

72

Fig. 20. Interband magneto-reflectance in Bi (after Brown et al. [77]).

0.142

O

I

J , 20

I

I I I 40 60 MAGNETIC FIELD

I I 80 (kgouss)

J

I I00

Fig. 21. The emission spectrum of a PbSe laser vs. magnetic field (after Butler et al. [81]).

14

B. Lax /Resonance spectroscopy o f solids and plasmas

sure of the g-factor of the carriers and the tuning using the proper band model has provided the best measurements of the mass parameter for this material.

8. Conclusions This review is much too brief to do justice to a great deal of the extensive work that has been performed in the study of resonance spectroscopy of solids and plasmas. The author apologizes for any omissions and must confess to the normal subjective view of the topic. Nevertheless, the conclusion to be drawn is that much has been learned from resonance spectroscopy from the microwave region well into the infrared. Fundamental quantitative data about the band structure of many semiconductors and semimetals such as the effective mass parameters, degeneracies, position of bands in the Brillouin zone, g-factors, non-parabolicity of bands, polarons, electronphonon interactions, plasma effects, etc. constitute an impressive record of accomplishments over two and a half decades. The study of microwaves and laser resonance in plasmas has important implications for the supplementary heating of tokamak fusion machines. With the development of superior optically pumped submillimeter laser and magnets with higher fields well into the megagauss region, the exploration into the study of new and old materials will continue and much needed basic information about solids and plasmas of practical and fundamental interest will be discovered.

References [1] B. Lax, W.P. Allis and S.C. Brown, J. Appl. Phys. 21 (1950) 1297. [2] B. Lax, Phys. Rev. 84 (1951) 1074; B. Lax and A.D. Berk, I.R.E. Natl. Cony. Record 1, Pt 10 (1953) p. 70. [3] G. Dresselhaus, A.F. Kip and C. Kittel, Phys. Rev. 92 (1953) 827. [4] B. Lax, H.J. Zeiger, R.N. Dexter and E.S. Rosenblum, Phys. Rev. 93 (1954) 1418. [5] W. Shockley, Phys. Rev. 90 (1953) 491. [6] B. Lax, H.J. Zeiger and R.N. Dexter, Physica XX (1954) 818. [7] R.N. Dexter, H.J. Zeiger and B. Lax, Phys. Rev. 95 (1954) 557; R.N. Dexter, B. Lax, A.F. Kip and G. Dresselhaus, Phys. Rev. 96 (1954) 222; R.N. Dexter and B. Lax, Phys. Rev. 96 (1954) 223.

[8] E. Burstein, G.S. Picus, J.A. Gebbie, Phys. Rev. 103 (1956) 825. [9] R.J. Keyes, S. Zwerdling, S. Foner, H.H. Kolm and B. Lax, Phys. Rev. 104 (1956) 1804. [10] B. Lax, J.G. Mavroides, H.J. Zeiger and R.J. Keyes, Phys. Rev. 122 (1961) 31. [11] K.J. Button, H.A. Gebbie and B. Lax, IEEE J. Quantum Electron. 2 (1966) 202. [12] R.A. Stradling, Proc. Int. Conf. the Generation of High Magnetic Fields and Applications in Solid State Physics, Wurzburg (1972) and (1974) p. 434. [13] K.J. Button, Proc. Int. Conf. Applications of High Magnetic Fields in Semiconductors Physics, Wurzburg (1976) p. 170. [141 J.W. Hodby, J. Phys. C4 (1971) LG; J.W. Hodby, J.G. Crowder and C.C. Bradley, J. Phys. C7 (1974) 303. [15] B.D. McCombe, S.G. Bishop and R. Kaplan, Phys. Rev. Lett. 18 (1967) 740. [16] G. Favrot, R.L. Aggarwal and B. Lax, Solid State Commun. 18 (1970) 557. [17] J. Waldman, D.M. Larsen, P.E. Tannenwald, C.C. Bradley, D.R. Cohn and B. Lax, Phys. Rev. 106 (1967) 51. [18] G. Dresselhaus, A.F. Kip and C. Kittel, Phys. Rev. 100 (1955) 618. [19] J.K. Gait, W..A. Yager, F.R. Merritt, B.B. Cetlin, H.W. Daft, Jr., Phys. Rev. 100 (1955) 748. [20] R.N. Dexter and B. Lax, Phys. Rev. 100 (1955) 1216. [21] M.Ya. Azbel and E.A. Kaner, Soviet Phys. JETP 3 (1956) 772; 5 (1957) 730. [22] E. Burstein and G.S. Picus, Phys. Rev. 105 (1957) 1123. [23] S. Zwerdling and B. Lax, Phys. Rev. 106 (1957) 51. [24] B. Lax, J.G. Mavroides, H.J. Zeiger and R.J. Keyes, Phys. Rev. Lett. 5 (1960) 241; R.N. Brown, J.G. Mavroides, M.S. Dresselhaus and B. Lax, Phys. Rev. Lett. 5 (1960) 243. [25] B. Lax, L.M. Roth and S. Zwerdling, J. Phys. Chem. Solids 8 (1959) 311. [26] R. Phelan, A. Calawa, R.H. Rediker, R.J. Keyes and B. Lax, Appl. Phys. Lett. 3 (1963) 143. [27] B. Lax and G.B. Wright, Phys. Rev. Lett. 4 (1960) 16. [28] R.W. Stimets and B. Lax, Phys. Rev. B1 (1970) 4720. [29] R. Rabitz and B. Lax, J. Phys. Chem. Solids 32 (1971) 359. [30] G. Bekefi, J.L. Hirshfield and S.C. Brown, Phys. Rev. 122 (1961a) 1037. [31] I.H. Hutchinson and D.S. Komm, Nucl. Fusion 17 (1977) 1077. [32] M.P. Hacker, R.J. Temkin and B. Lax, Appl. Phys. Lett. 29 (1976) 146. [33] B. Lax and D.R. Cohn, Appl. Phys. Lett. 23 (1973) 363. [34] M.P. Hacker, R.J. Temkin and B. Lax, J. Appl. Phys., to be published. [35] D. Biron, R.J. Temkin and B. Lax, J. Magn. Magn. Mat. 11 (1979) 000. [36] G. Dresselhaus, A.F. Kip and C. Kittel, Phys. Rev. 95 (1954) 568. [37] R.N. Dexter, H.J. Zeiger and B. Lax, Phys. Rev. 104 (1956) 637.

B. Lax / Resonance spectroscopy of solids and plasmas [38] B. Lax and J.G. Mavroides, Solid State Physics, eds. Seitz and Turnbull (Academic Press, New York, 1960) p. 261. [39] G. Dresselhaus, A.F. Kip, C. Kittel and G. Wagoner, Phys. Rev. 98 (1955) 556. [40] K.J. Button, B. Lax and C.C. Bradley, Phys. Rev. Lett. 21 (1968) 350. [41] R.C. Fletcher, W.A. Yager and F.R. Merritt, Phys. Rev. 100 (1955) 747. [42] N. Miura, G. Kido, K. Suzuki and S. Chikazumi, Proc. Int. Conf. Application of High Magnetic Fields in Semiconductor Physics, Wurzburg (1976) p. 441. [43] D.M. Larsen, Phys. Rev. 135A (1964) 419. [44 ] C.W. Litton, K.J. Button, J. Waldman, D.R. Cohn and B. Lax, Phys. Rev. 13B (1976) 5392. [45] R.C. Fletcher, W.A. Yager, G.L. Pearson, F:R. Merritt, Phys. Rev. 95 (1954) 844. [46] G. Feher, J. Phys. Chem. Solids 8 (1958) 486. [47] G. Feher, Proc. Int. Conf. Semicon. Phys., Prague (1960) p. 579. [48] L.M. Roth and B. Lax, Phys. Rev. Lett. 3 (1959) 217. [49] L.M. Roth, B. Lax and S. Zwerdling, Phys. Rev. 114 (1959) 80. [50] R.L. Bell, Phys. Rev. Lett. 9 (1962) 52. [51 ] E.I. Rashba and V.I. Sheka, Sov. Phys., Solid State 3 (1961) 1256. [52] B.D. McCombe, R.J. Wagner and G.A. Prinz, Phys. Rev. Lett. 25 (1970) 87. [53] R.C. Enck, A.L. Saleh and H.Y. Fan, Phys. Rev. 182 (1969) 790. [54] E.J. Johnson and D.H. Dickey, Phys. Rev. B1 (1970) 2675. [55] G. Favrot, R.L. Aggarwal and B. Lax, Solid State Commun. 18 (1970) 577. [56] M.H. Weiler, R.L. Aggarwal and B. Lax, Phys. Rev. B17 (1978) 3269; M.H. Weiler, J. Magn. Magn. Mat. 11 (1979) 000. [57] G.B. Wright and B. Lax, J. Appl. Phys. 32 (1961) 2113. [58] B. Lax and L.M. Roth, Phys. Rev. 98 (1955) 548. [59] V.L. Kononenko, Soy. Phys. Solid State 17 (1976) 2146. [60] K. Fujii and E. Otsuka, J. Phys. Soc. Japan 38 (1975) 742; Muff Y. Nisida, J. Phys. Soc. Japan 40 (1976) 1069.

15

[61] L.M. Roth, B. Lax and S. Zwerdling, Phys. Rev. 114 (1959) 90. [62] B. Lax and S. Zwerdling, Progress in Semiconductors, ed. Gibson (Wiley, New York, 1960) p. 221. [63] S.Zwerdling, L.M. Roth and B. Lax, Phys. Rev. 109 (1958) 2707. [64] J. Halpern and B. Lax, J. Phys. Chem. Solids 27 (1966) 111. [65] R.L. Aggarwal, M.D. Zuteck and B. Lax, Phys. Rev. 180 (1969) 800. [66] M.H. Weiler, R.L. Aggarwal and B. Lax, Phys. Rev. B16 (1977) 3603. [67] S. Zwerdling, K.J. Button and B. Lax, Phys. Rev. 118 (1960) 975. [68] H.Y. Fan and P. Fisher, J. Phys. Chem. Solids 8 (1959) 270. [69] S. Zwerdling, K.J. Button, B. Lax and L.M. Roth, Phys. Rev. Lett. 4 (1960) 1539. [70] G.E. Stillman, C.W. Wolfe and J.O. Dimmock, Solid State Commun. 7 (1969) 921. [71] D.R. Cohn, D.M. Larsen and B. Lax, Solid State Commun. 8 (1970) 1707. [72] D.M. Larsen, unpublished. [73] R.N. Dexter and B. Lax, Phys. Rev. 99 (1955A) 635. [74] J.K. Gait, W.A. Yager, F.R. Merritt, B.B. Cetlin and A.D. Bradsford, Phys. Rev. 114 (1959) 1396. [75] J.K. Gait, W.A. Yager and H.W. Dail, Jr., Phys. Rev. 103 (1956) 1586. [76] B. Lax, K.J. Button, H.J. Zeiger and L.M. Roth, Phys. Rev. 102 (1956) 715. [77] R.N. Brown, J.G. Mavroides and B. Lax, Phys. Rev. 129 (1963) 2055. [78] M.S. Dresselhaus, and J.G. Mavroides, IBM J. 8 (1964) 262; G. Dresselhaus and M.S. Dresselhaus, Proe. Int. School of Physics Optical Properties of Solids (Academic Press, New York, 1966) p. 198. [79] B. Lax, 7th Int. Conf. Phys. Semicon. (Dunod, Paris, 1964) p. 253. [80] B. Sacks and B. Lax, IEEE J. Quantum Electron. QE2 (1966) 607. [81] L.F. Butler and A.R. Calawa, Proc. Int. Conf. Quantum Electronics, Phoenix, Arizona (1966).