Hot electrons and phonons under high intensity photoexcitation of semiconductors

Solid-%lt

Ekctmnies, 1918,Vol. 21. pp. 43-93. Pwgtdnon Press.

Printed in Great Llrkain

HOT ELECTRONS AND PHONONS UNDER HIGH INTENSITY PHOTOEXCITATION OF SEMICONDUCTORS JAGDEEPSHAH Bell TelephoneLaboratories,Holmdel,NJ 07733,U.S.A. Ahstrati-It has become well establishedduringthe last few years that intense photoexcitationof a semiconductor leadsto the heatingof the carriersandthe generationof nonequilibrium phonons.These phenomenawhichresultfrom the relaxationof photoexcitedcarriersto the band extremaby interactionwith other carriersand by emission of phonons,are reviewed in this paper.At relativelylow intensities(40’ W/cm*for GaAs) the photoexcitedcarrier distributionis Maxwellianwith a carriertemperatureT, differentfrom the latticetemperature.T. as high as ISOK and effective phonon temperaturesas hi as 3700Khave been observedin GaAs. The observed variationof Z’, with excitationintensityleads to the conclusionthat in semiconductorslike CiaAsthe polar opticalmodescattering is the dominantenergyloss mechanismfromthe electrongas to the lattice.Similarresultsare obtainedin CdSe and CdS. At higherintensities(> Id W/cm’for GaAs), the carrierdistributionbecomes non-Maxwellianfor reasonsnot well understoodat present. We will also discuss some recent measurementsof variationof T, with excitation wavelengthand of the transmissionspectraof photoexcitedGaAs.

1. lNTRoDucTIoN

When a crystal of semiconductor is excited with photons of energy (hv,,) larger than the energy bandgap (E,) of the semiconductor, electrons and holes are created with excess kinetic energy of the order of (hv, - E,). These photoexcited carriers lose this excess energy and relax to the band minimum or maximum by interaction with phonons and other carriers in the system. If the photoexcitation is sufficiently intense, this relaxation process results in the generation of hot electrons[ 11 and phonons[2] in the system. A study of these hot electron and phonon distributions provides detailed information about electron-electron and electron-phonon interactions in a semiconductor. The purpose of this paper is to review the field of hot electrons and phonons in semiconductors at high ph&oexcitation intensities. Hot electron phenomena at low intensities are reviewed by Ulbrich[3]. We will begin with a discussion of theoretical considerations in Section 2. We will then discuss various experiments on hot electrons (see Section 3) and phonons (see Section 4) in GaAs. In Section 5, we will discuss hot electron results in other semiconductors such as CdSe and CdS; these results show that the concepts developed to explain the results in GaAs are equally applicable to other similar semiconductors. In Section 6 we discuss how intense photoexcitation might lead to non-equilibrium acoustic phonons and a possible technique for detecting them. We conclude in Section 7 with a summary and a brief discussion of future directions.

CONDUCTION BAND

ELECTRON

GAS

VALENCE

BAND

Fig. I. Schematicrepresentationof various relaxationprocesses of photoexcited electrons. I and 3 correspondto emission of optical phonons with different wavevectors; 2 correspondsto collision of the photoexcitedelectronwith the electrongas.

electron is created with an excess energy AE, given by AE, = (hu, - E,) X (1 + mJmJ’

(1)

where m, and m,, are electron and hole effective masses respectively. The excess energy of the photoexcited hole is AE,, =(hv,-EJ-AE, We will consider in detail what happens to the photoexcited electrons; similar considerations apply to the holes. The distribution function of hot electrons is determined by the following considerations: (1) What are the mechanisms by which the photoexcited electron loses its excess energy AE, and relaxes to the bottom of the conduction band? What are the rates of loss of energy by these mechanisms? (2) Once the electrons have relaxed close to the bot-

2.-CAL CONSIDERATIONS Photoexcitation can generate electron-hole pairs in a semiconductor either by interband absorption mechanism (hv,,>E,) or by some nonlinear effects (hv,,c E,) such as two photon absorption or second harmonic generation. We consider in Fig. 1 the case of hv,,> Eff If we assume parabolic bands, the photoexcited 43

44

JAGDEEP

tom of the band what are the mechanisms (and rates) by which this electron gas loses energy? (3) Do the electrons within the electron gas interact with each other sufficiently strongly so that we may characterize the distribution of the electron gas by a temperature T,? (4) If the electron gas can be characterized by T,. which factors determine the value of this temperature? The answer to the last question is clear: under steady state conditions the electron gas temperature T, must be such that the rate at which the electron gas receives energy from the photoexcited electrons is equal to the rate at which the electron gas loses energy to phonons, impurities, phonons, etc. This forms the basis of the analy-

sis of all photoexcitation induced heating phenomena in semiconductors. These considerations for photoexcited carriers are similar to those applied to the case of electric field induced hot carrier effects which have been extensively investigated[4]. We can apply much of the formalism developed for this case to the case of photoexcitation induced heating. From the considerations discussed above it is clear that rates of exchange of energy between different excitations (electrons, holes, phonons, impurities, etc.) are the important parameters. The expressions for these rates are well known[4] so one can, in principle, analyze any semiconductor of interest. However, for the purpose of this review, we will restrict ourselves to semiconductors like GaAs, for which it has been shown by studies of transport properties[4,5] and of photoexcited heating phenomena[l] that the dominant electron-phonon interaction mechanism is polar optical scattering except at very low temperatures. For such a semiconductor the photoexcited electron can relax to the band minimum by emitting successive longitudinal optical (LO) phonons or by interacting directly with the electron gas. The relevant rates for energy loss are [4,6] (in MKS units)

de 2eEo y, Z P-0 = (2mp)

0

[N, sinh-’ (&)I”

-(IV, + 1) sinh-’ (%)“*]

(4)

Here e-e and p-o stand for electron-electron and polar optical mode scattering, n is the density of electrons in the electron gas, r is the energy of the photoexcited electron, K is an average dielectric constant, lo is the free space permittivity, boo is the LO phonon energy, N, is the occupation number of phonon of wavevector q interacting with the electron, and E. is an effective electric field given by

SHAH

where K, and KS are optical and static dielectric constants of the semiconductor. Let n: be the electron density at which the two rates are equal:

.

(6)

For n 4 nr, the photoexcited electron will relax primarily by successive LO phonon emissions until it doesn’t have sufficient energy to emit an LO phonon, and then give the remaining energy SE to the electron gas, provided e-e collisions are more frequent than collisions with acoustic phonons. 6E = (BE, - mhwo)

(7)

where m is the largest integer for which SE > 0. For the case when n - n$, the photoexcited electron loses a fraction F of its energy AE, to the electron gas even before it has relaxed to energy
(8)

The power per electron given to the electron gas from the photoexcited electrons is P. = (I/d) ( W/hvo)(l/n)

(9)

where I is the laser power absorbed per unit area, and d is the larger of the absorption length, or the diffusion length in the sample for thick samples, and sample thickness for thin samples. For semiconductors like GaAs, it has been shown[4,5] that the dominant interaction and energy loss mechanism for the electron gas is electron-polar optical phonon scattering, except at very low temperatures when only an insignificant fraction of the electron population has energy> IUVJ~For this case a characteristic electron temperature T. can be defined if electron-electron collisions are more frequent than electron-LO phonon scattering. For an electron gas with Maxwell-Boltzmann distribution at temperature T, the rate of energy loss per electron is obtained[4] from eqn 4

sz

(2)1/zeE, (s) (xJ2)“’ e’“i” K,,(xJ2) V(?r/2)

1 (10)

where x0 = (&J&T,), TL = crystal lattice temperature x, = (k&kT,) and K. is the mod&d Bessel function of order zero [7].

High intensityphotoexcitationof semiconductors

The condition for defining an electron temperature T. collisions more frequent than electron-phonon collisions) is satisfied if n > n, where

(electron-electron

n, = (8d(?r)/e’) ha0K’e,’ eE,

x

($t&-!) (exJ2Ko(xJ2))

(11)

If this condition is fulfilled then T. is obtained by balancing power flow into the electron gas against the power flow out of the gas, i.e. by setting

where P. and P(f,) are given by eqns (9) and (10). 2PEOTClEX~HOTCAING6AS

The first experiment demonstrating the heating produced by intense photoexcitation was performed in GaAs [ 11.The crystal, immersed in superfluid He at =2K, was excited by a cw Ar’ laser (hi = 2.41 eV) and its luminescence spectrum was measured using a double spectrometer. Figure 2 shows typical spectraPI obtained in this manner; the characteristic features are that the high energy luminescence tail broadens with increasing pump powers, and that the luminescence intensity falls off exponentially with increasing photon energy. For non-degenerate carriers at temperature T, the intensity of luminescence is expected to be[8] Z(hv a (hv)2e:‘h’-Ee~. Therefore, these observations imply that the carriers are thermalized and that T. in-

10r------- r-- ----

1

45

creases with pump power. 7’. as large as 76K was measured in thii first experiment while at the same time a variety of arguments were used to show that the rise in lattice temperature was insignificant. In the analysis of these results, the following parameters were used111 to characterize &As: m, = 0.07 m,,mh =O.Sm,,K,= 10.62, KS= 12.53, K = 11.7 and ho= 36.8 meu (=427K). Values of R, = 3 x 10” crnm3 for electrons and 2 x 10” crne3 for holes were obtained. The photoexcited carrier density in the sample was estimated111 to be 15 x 10’6cm-3 at the highest pump power so that one expects the thermalization of carriers observed experimentally. As we discussed in Section 2, the variation of T. with pump intensity (I) provides information about how the electron gas acquires energy as well as how it loses energy. In Fig. 3 we show a plot of (l/T,) vs Z for a wide range of intensities[ 1,9]. T, varies rapidly with Z at low intensities but rather slowly with I at intermediate and high intensities. This behavior was explained[ 1,9] as follows: In GaAs nz = 6 x 10” cm-’ so it was assumed that W = (AI?, x n/n@ for I< lO’W/cm’ so that P,= (Z/d) x (AEJhv,,)x (l/n*,) from eqns (8) and (12). For the experimental case of x, B 1 and (x0- x.) * 1, the square bracket in eqn (9) is = 1[7] so that eqn (12) reduces to

U/d)Wdhv) (l/n*,) =

(2)

eE,xexp (-&,/kT,) (13)

or Z a exp (-g&kT,). Thus a semilog plot of (l/T.) vs I (see Fig. 3) should be linear with a slope of (-g&k). The solid line in Fig. 3 is drawn with this slope using b,, = 36.8 meV; the agreement at intermediate Z is quite good. Equation 13 also predicts (for d = 0.3 pm) that T. i I 0.06

-

0.05

-

-

\ 1 \ I \

GoAs Tboth = 2K \\

-20

b \ I T

15

0.04 -

\

\

- 25 h

-30

z 3

- 35 -40

0.01

ENEROY OF EMITTED

PHOTONS (a’il

2

The high energy luminescencetail of photoexcited@As for various excitationpowers.POis the maximumpower absorbed by the sample. The straight lines through the data points define car&r temperatures T, indicated in the @ure. (From Shah and L-site,Ref.[l]). SSE Vol. 21, No. I-D

I

0’

1

“““”

10

‘.“““I

102

l’,,,,‘a ‘..““” 103 104

PUMP INTENSITY

(W/cm*

100

1

Fii. 3. (l/Z’,) vs Excitation Intensity I for GaAs at 2K. The pointsare experimentaldata from Refs. [ l] and [9]. The solid lime corresponds to a slope of (&,/k) =427K where kvs = 36.8meV = LO phononenergy.

46

should be 76K at I = lo’ W/cm*, in good agreement with experiments. From the good agreement between theory and experiments it was concluded that polar optical mode scattering is the dominant energy loss mechanism for the electron gas at these Z’, The good agreement also shows the validity of the assumption that the electron gas acquires energy from the photoexcited electron before it has relaxed close to the bottom of the band by successive LO phonon emissions. More evidence for this second point has been obtained by varying (hv,) and will be discussed later. The agreement between theory and experiments is clearly not good at low I. The reason for this is that different energy loss mechanisms dominate at low temperatures[4,10]. The reason for the disagreement at high Z is that n is no longer negligible compared to nt[9]. Under these conditions the left hand side of eqn 13 should be multiplied by l/(1 + n/n:) as can be seen from eqn 8. Thus the power per electron transferred to the electron gas and hence T, increases less rapidly with I than at lower I. One can in fact estimate n/n? from the observed variation of T, with I, using the modified eqn (13); the results are shown in Fig. 4. The estimated n implies a carrier lifetime r = 2 x lo-” set, which is not unreasonable. The data in Fig. 3 is restricted to I s 2 x ld W/cm*; the reason for this is that the carrier distribution can no longer be characterized by a temperature at larger intensities, as shown in Fig. 5. In fact, some very weak emissions extend to the red region of the spectrum as shown in Fig. 4 of Ref.[9]. The reason for this nonMaxwellian behavior is not clear at this time. Until now we have discussed the’ high energy luminescence tail as the probe for carrier heating. The carrier distribution function can also be obtained by analysing the shape of the single particle light scattering spectrum from photoexcited carriers[ll]. This technique has been used by Mooradian to obtain carrier dis-

x ci, 6000

7600

7600

1.55

1.60

1.64

hv (rV1

Fig. 5. High EnergyPhotoluminescencetail in GaAs showingthe non-Maxwelliancarrierdistributionsfor I > 2 x lo’ W/cm’ (from Shah, Ref. [9]).

tribution in the presence of electric field[l2]. We show in Fig. 6 the single particle scattering spectra obtained by Turtelli et aL[l3] at various photoexcitation intensities. They also find that Z’, increases with excitation intensity. The results are in good agreement with the luminescence results and will be discussed in more detail by Leite [14]. Another technique for measuring the carrier distribution is to measure the near bandgap transmission spectrum of a probe beam while the sample is simultaneously excited by an intense pump beam[l5]. Typical transmission spectra obtained at different pump intensities are shown in Fig. 7. They have been analysed[ IS] in terms of the band filling by optically excited carriers at finite temperatures: carriers temperatures obtained in this manner are also in agreement with luminescence measurements. This technique will also be discussed in more detail by

lo5

10'

PUMP INTENSITY

(WIcm2)

4. Fkctron dcasity II ve I for Gala at 2K as deducedfrom the hi intensity r&on of Fig. 3 where deviations occur from Fi.

the solii line.

ENERGY

DIFFERENCE

110’ i’,

6. Spectra of electron sin&e particle scatteringfrom GaAs (3OOK)at various excitation powers. T, =S6OK was deduced from the curve at 1 kW. (FromTmtelli et al, Ref.[ 19. IQ.

47

High intensityphotoexcitationof semiconductors AE,

L,,__

WI

c, , , , , , , , , , , 4

0.5

-

L.I. XxXxXa05 PO . ..” ____ WEoRy (NO EWTON)

GoAs 2-K

p0

(0) ”

---------I

--._ I

t

I

I

1.53

.

*.I

I

1.52

1.51

PHOTON

ENERGY

(eV)

PHOTON

ENERGY (eV)

1.5

Fii. 7. (a) Optical density of GaAs determinedby transmission measurements.The low intensity (L.I.) was measured with a tungstenlamp,while the remainingspectrawere measuredwith a dye laser. The dashed curve illustratesthe absorptioncoefficient expected for parabolic bands with no Coulombinteraction. (b) Data for high excitation intensity. POcorrespondsto incident intensity of ==5x Id W/cm’. The solid lines are calculated T, = 60K was deduced for 20 POcurve (from Shah et al., Ref.1151).

Leheny and Shah[lfj. While these techniques provide alternative probes for carrier distributions, luminescence studies provide the simplest and most straightforward probe for carrier distributions. We now consider the question of how the carrier temperature varies with the photon energy Iv,, of the excitation source. The first experiment[l7] was performed using a tunable dye laser pumped by a pulsed N2 laser. The photon energy was varied from 2.83 to 1.54eV corresponding to a factor of 50 change in AE. (eqn (1)). At each pump wavelength, the laser power was increased until the high energy luminescence tail indicated a carrier temperature of 45K. The photon flux (F = Z//M,)required to obtain T. = 45K increased with decreasing hv,, as shown in Fig. 8. These results can be understood as follows: assume that n is such that 8E in eqn 8 can be neglected and W -(AZ&/n:) so that eqn (9) may be written as P. = (f/d) x (A&/n!). Since diffusion length of carriers exceeds absorption length, d is independent of hv, and the flux required to heat carriers to a given T. is expected to vary as (n:/AE,). This last quantity, as well as n: and AE,, are plotted in Fig. 8 as a function of hv,,. We see that the experimental points agree well with this analysis, supporting the assumption that for the intensities used in this experiment the electron gas acquires energy directly from the photoexcited carrier before the photoexcited carrier relaxes to the bottom of the band by successive LO phonon emission. Variation of T, with /IV, for hv, close to E, of GaAs is recently reported by Goebel and Hildebrandtl81. Their results are reproduced in Fi. 9. They find that T, varies linearly with AE, for AE, < Ilo, and somewhat slower for larger AE, There is a break in the curve at BE, = &I,,, but no oscillations in T. with AE, are observed.

i.7

1.9

2.1

PHOTON

2.3

ENERGY

2.5

2.7

2.9

WI

Fig. 8. Photon Eux F requiredto obtain T, = 45K vs photon energy(hv&.The data are normalizedto unity at hv, = 2.834eV. The othercurves show the calculatedvariationsof n$ I/A& and n:/AE, as a function of photon energy (from Shah et al., Ref.[l’l]).

10

100

AE [meV]

IO

20

30 40 50 AE [meV] Fig. 9. Variationof T, with AE, in GaAs at 2K (a) log. log plot, (b) linear plot (from E. Goebel and 0. Hilderbrand,to be published).

These oscillations would be expected if successive emission of LO phonons was faster than any other relaxation process. The absence of oscillations is not surprising in view of the fact that n = n? in their experiments. However, the rapid variation of T. with AE, is surprising and more work will be required to understand this interesting behavior. We have discussed a variety of experiments designed

48

JAGDEEP SHAH

to understand the relaxation mechanisms relevant to the problem of photoexcited hot carriers. In concluding this section I would like to mention that in recent years a different kind of photoexcited non-equilibrium phenomenon has also been investigated: this involves the use of appropriately polarized exciting light to preferentially create electrons in a given spin state. The study of luminescence from such photoexcited electrons provides information about spin-spin and spin-lattice relaxation mechanisms and ratesll91. A PHOTOEXCITED HOT PHONONS IN GlAs The results on hot electrons discussed in Section 3 indicate that a large number of LO phonons are emitted when photoexcited carriers relax to the bottom of the band and when the heated electron gas loses energy to the lattice. Therefore, we now turn our attention to what happens to phonons in GaAs under intense photoexcitation. It is well known that the ratio of the Stokes to antiStokes Raman scattering intensities provides an accurate probe for the occupation number of the phonon which participates in Raman scattering. Shah et a[.[21 used this fact to probe photoexcited nonequilibrium phonous in GaAs. A cw Ar’ laser was used to excite the crystal which was maintained at room temperature. The same laser was used to study the Raman scattering spectrum of LO and TO phonons in the backscattering geometry so that only phonons of wavevector k- 2k, were

IOO-

probed. Here k. = 2n$rv,,/c, where q is the refractive index and c is the velocity of light in vacuum. It was observed that with increasing pump power the ratio of anti-Stokes to Stokes intensity (A/S) increased for the LO phonon but not for the TO phonon indicating that a nonequilibrium population of LO phonons was generated as a result of intense photoexcitation. The Raman spectra at two different intensities are plotted in Fig. 10 to demonstrate these effects. The phonon occupation number E = l/(S/A - 1) vs incident pump power is plotted in Fig. 11. From a simple analysis, it was shown121 that the slope of this curve is related to the phonon lifetime r. A

GoAs s

1.4

300’K

$ EXPERIMENTAL i.2

A CORRECTED

t

P

FOR To

0.2 i 0

9

3

j

1 0.5

0

’

c 10

P/PO

Fig. 11.LO phonon occupation number A= l/@/A - 1) vs normalized pump power. The largest 1 corresponds to an effective temperature of WOKfor LO phonons (from Shah et al., Ref.121).

STOKES

ANTI

GoAs

- STOKES

300’ +

00-

K

NOISE

P=Po

-I 60-

I-

RESOLUTION

LO ;

40-

c Ll ; q

zo-

: t y

o-

3 n

IO-

5-

ORAMAN

SHIFTS

(cn?)

Fig. 10.Typical Raman spectra of TO and LO phonons in GaAs at 3OOK at two different pump powers (from Shah et al., Ref.[2]).

High intensityphotoexcitationof semiconductors

value of 7 = 5 x lo-” set obtained in this manner agrees with the estimates of phonon lifetime based on I&man linewidths. Similar measurements were also made at lower lattice temperatures[21. These Raman scattering results on photoexcited phonons corroborate the picture of relaxation processes that emerged from the study of photoexcited hot carrier processes. Much more work on photoexcited optical phonons has been done at University of Campinas in Brazil[20-231. A detailed discussion of these results is given in a recent review [241. 5.PEOIY)EXCFFEDEorCAIIP[FBSMOTBPIlsEMICoNDumOIIS

Most of the work on photoexcited hot carriers and phonons has been done on GaAs but the detailed considerations of Section 2 should be applicable to other semiconductors in which polar optical mode scattering dominates. Experiments have been performed on CdS[25,26] as well as CdSe[27l both of which are II-VI compound semiconductors with more polar character than GaAs. We discuss here the results on CdSe[27]. A thin platelet of CdSe was immersed in superfluid He and excited by 15nsec wide probes from a cavity dumped Ar’ laser (hv,,=2.41 eV). From the exponentially decaying high energy luminescence tail, carrier temperature T. was determined in the same way as in GaAs. Fiie 12 shows a plot of (l/T.) vs pump intensity. The solid line in the figure has a slope corresponding to 26.5 meV which is the LO phonon energy in CdSe. Thus the results show that polar optical scattering is the dominant energy loss mechanism for the electron gas in CdSe. The deviation at large Z is similar to that observed in GaAs (Fig. 3) and has a similar explanation. At Z = 2 x 10’W/cm’, one would expect that T. = 74K, whereas T, = NJK was observed experimentally. It should be noted that n* = 1.1 x lo’* cmm3,and AZ?, is smaller than in GaAs for fhe same pump photon energy (hv,,) so that

o’03~~_

49

one expects to find lower temperatures than in GaAs for the same pump intensity. This is indeed observed experimentally. The results in CdSe give us confidence in our understanding of the relaxation processes in semiconductors in which polar optical scattering dominates. 6.

ACOUSl’IC

Np

PEONONS

We have seen above that intense photoexcitation can create large nonequilibrium populations of optical phonons. These phonons primarilydecay by emitting two or more acoustic phonons. The process is schematically illustrated in Fig. 13. The large wavevector acoustic phonons so created will decay into small wave vector phonons which will eventually lose energy to the helium bath. While these ideas seem to be generally accepted, no experiments studying photoexcited nonequilibrium acoustic phonons have been reported. The use of the I, bound exciton luminescence feature in CdS as an acoustic phonon spectrometer was demonstrated a few years ago[28]. The observation of nonequilibrium acoustic phonons and its relaxation to equilibriumfollowing pulsed photoexcitation would be very interesting because it would provide important information regarding acoustic phonon processes. 7. CONCUJSIONS

idea that intense photoexcitation leads to the generation of hot carriers and phonons has become well established during the past several years. We have reThe

NON-EQUILIBRIUM

ACOUSTIC

PHONONS

hw

-‘R/a

0

k-

‘17/a

Fig. 13. A schematicillustrationof relaxationprocesses of optical and acoustic phononsin &As.

ELECTRON EXCESS

4 h&K = UnK (or & = 26.5meV = LO phonon energy). The dashed curve at hi intmsities is calculatedby assumingthat dn: = 2 at I/&= 0.5 and II01I. 4 = 4 x ld W/cm*(from Shah, Ref. [271).

ACOUSTIC PHONONS Fii.

14. A block dim illusuatiaa the flow of energy in photoexcitationstudies of semiconductors.

50

JAGD~~EP SHAH

viewed here the experimental results and discussed the information about relaxation processes that one can obtain from such measurements. In Fig. 14 we show a block diagram of how the energy of the absorbed photon is ultimately given to the helium bath in which the crystal is immersed. The experiments described here, and many others on transport properties, have provided information about various relaxation rates. The missing link in

II. See,for example, A. Mooradian, Light Scartting Spectra of So/ids, (Edited by G. B. Wright), p. 285. Springer-Verlag, New York(1968). 12. A. Mooradian and A. L. McWhorter, In Proc. of the XIrh Int. Conf. on Physics of Semiconductors, (Ed&ed by S. P. Keller, J. C. Hensel and F. Stern), p. 380. U.S.A.E.C. Div. of Tech. Information, Oak Ridge, Tennessee (1970).

13. R. S. Turtelli,A. R. B. de CastroandR C. C. Leite,Solid-St.

Comm. 16 %9 (1975). 14. R. C. C. Leite, Solid-St.Electron. 21, I77 (1978). the picture is how fast the acoustic phonons generated 15. Jagdeep Shah, R. F. Lebeny and W. Wiegmann, Phys. Reu. B15, August 1977. by the decay of optical phonons relax within the acoustic branch. Experiments capable of determining the im- 16. R. F. Leheny and Jagdeep Shah, Solid-St. He&on 21, 167 (1978). portant rates involved in these processes would be very 17. Jagdeep Shah, C. Lin, R. F. Leheny and A. E. DiGiovanni, valuable. Solid Stare Commm. 1,487 (1976). 18. E. Goebel and 0. Hilde brand, to bepublished. 19. See the section on “Spin Dependent Properties and Spin Acknowledgemenls-1 would like to acknowledge many useful Flip” in Proc. of the XI&h Ink Conf. on Physics of Semidiscussions with R. C. C. L&e and R. F. Leheny and to thank E. conductors, (Edited by M. H. Pilkuhn), pp. 743-779. B. G. Goebel and 0. Hildebrand for the permission to present their Teubner, Stutm (1974). unpublished data. 20. E. Lluesma. G. Mendes, C. A. Arguello and R. C. C. Leite, Solid Srare Comm. 14, 1195(1974). REFERENCES 21. J. C. V. Mattos, J. A. Freitas, E. Lluesma and R. C. C. Leite, 1. Jagdeep Shah and R. C. C. Leite, Phys. Reu. Ldt 22, 1304 Solid State Comm. 14, 1199(1974). 22. J. A. Freitas, J. C. V. Mattos, E. Lluesma and R. C. C. Leite, (1964). 2. Jagdeep Shah, R. C. C. Leite and J. F. Scott, So/id Stale Proc. of the XIIh Int Conference on Physics of SemiconCommun. 8, 1089(1970). ducrors (Edited by M. H. Pilkuhn), p. 463. B. G. Teubner, 3. R. G. Ulbrich, Solid-Sf. Electron. 21.51 (1978). Stuttgart (1974). 4. E. M. Conwell Solid Stare Physics, Supplement 9, (Edited by 23. A. R. Vasconcellos, R. S. Turtelli and A. R. B. decastro, F. Seitz, D. Turnbull and H. Ehrenreich). Academic Press, Solid Stare Comm. 22,97 (1977). New York (1%7). 24. Jagdeep Shah and J. C. V. Mattos, Proc. of the 3rd Int. 5. H. Ehrenreich, Phys. Rev. lu) 1951(1960)also E. Haga and Conference (Edited by M. Balkan&i, R. C. C. Leite and S. H. Kimura, I. Phys. Sot., Japan 19,658 (1964). P. S. Porto), p. 145.“Flammarion,Paris (1975). 6. R. Stratton, P~oc. Roy. Soc.,(London) A246,4% (1958)also 25. E. A. Meneses, N. Jannuzi and R. C. C. Leite, Solid Store C. 1. Heam, Proc. Phys. Sot. 86. 881 (1965). Comm. 13 245 (1973);see also E. A. Meneses, N. Jannuzzi, J. (Edited by M. 7. Handbook of Malhemarical knclions G. P. Ramos, R. Luzzi and R. C. C. Leite, Phys. Rev. Bll, Abramowitz and I. A. Stegun), pp. 374-378 and p. 417. 2213(1975). Dover, New York (1%5). 26. P. Motisuke, C. A. Arguello and R. C. C. Leite, Solid Stare 8. See, for example, A. Mooradian and H. Y. Fan, Phys. Rev. Comm. 16 763 (1975). 148,873 (1966). 27. Jagdeep Shah, Phys. Rev. B9,562 (1974). 9. Jagdeep Shah, Phys. Reo. BlO, 3697(1974). 28. Jagdeep Shah, R. F. Leheny and A. H. Dayem. Phys. Reu. 10. R. G. Ulbricb, Phys. Rev. BE, 5719(1973). IA. 33 818 (1974).