Solid StateCommunications, Vol. 9, pp. 265—269,1971. PergamonPress. Printed in GreatBrit~

RAMAN SCATTERING BY MAGNETOPLASMA OSCILLATIONS

IN MANY-VALLEY SEMICONDUCTORS*

John Sanderst

McGill University, Montreal, P.Q., Canada

(Received23 November1970 by E.F. Bertaut)

The Ramanscatteringcross sectionsof the longitudinal opticalcollective modesin a magneticfield of a polarmany-valley semi-conductorarecalculated.Numerical resultsareobtainedfor n-typePbTe. A one-bandmodel hasbeenused.

RAMAN scatteringby chargecarriers in semi- zero, anda new low frequency(LF) CDF arisconductorshasbeen actively investigatedboththeoretically and experimentallyin recentyears.’2 Calculation of the density—densityThe fact that thesecarriers move in a periodic correlationfunction shows that for not-too-hi~potential gives rise to interestingeffects which magneticfields, the chargedensity associateare not observedfor free carriers.36 Here we with this excitation is small, most of thewish to consideryet anothercollective excit- restoringforce beingdue to the circulating mcation which exists in semiconductorswith of the electrons, ratherthan to the build-up ol

ellipsoidal energysurfacesin the presenceof a charge.Hencethis modewould be difficult tostatic magneticfield.7 observedirectly throughlight scatteTing.Ho~

ever, if oneusesa polar semiconductor,theIf one applies a magneticfield in a direction longitudinal optical (LO) phononswill mix wi

orientedsymmetrically with respectto the four this mode,and should render it visible in aellipsoidal energy surfacesof, for example, Ramanexperiment.PbTe,8it is clear that the motion perpendicularand parallel to the field will couple, as the A one-bandmodel is usedto describetheeffective masstensoris then not diagonal. If semiconductor,which takesinto accountthe

onenow tries to propagatea chargedensity crystal structurein the simplest way.2 Thougfluctuation (CDF) parallel to the field, the re- it has limitations,9 it is sufficient to describ~sponseof the electron gaswill be effectedby the this mode.~existenceof the Landaulevels. In fact, whenthe frequencyof sucha fluctuation approaches The formalism is similar to that usedbythe cyclotron frequency, oneexpectsabsorption Platzman.~ Thenoninteractingsingle-particledueto single-particleexcitations betweenLandau Hamiltonian is writtenlevels. As a result, somewherebelow this

tj(l) 1 (1) 1 (~frequency, the dielectric constantgoesthrough = a . + I ILBB .

* Basedon Ph D thesis, McGill University. The nearnessof the incident frequencyto tl

band gapcould be approximate1yaccounted

t Presentaddress:Laboratoired’Electrostatique for by a90enhancementfactor in the cross

et de Physique du Metal, Cedex166, section. For modifications introducedby38. Grenoble—Gare,France. a two-bandmodel, seefor instancereferenc

265

266 RAMAN SCATTERING IN SEMICONDUCTORS Vol. 9, No.

for a particle belonging to ellipsoid ‘1’. ~~is the di~ r w2 1electronmomentum~ = p + e a the ~ ~i2 w, 1 — ~

effective masstensor, and the effectiveg-factor. We haveassumedthat the magnetic f~’Je”°~<(,~~q,t), ,t)( q,o)Il> ~[~).~[l

Refield doesnot mix the wavefunctions of thedifferent valleys appreciably.By a rescaling Here 4

9(q, 1) is the Fourier transformof theproceduretheproblem can be reducedto a chargedensity of surfaceI and spin a; r

0 is tl~sphericalonewith solutions classical electronradius; w, and0)2 are the

~nQK (x) 1 incoming and scatteredphoton frequencies,wi277 (a.~.aL)’’

4 w = — &~ �~is the optical dielectricconstantwhich takescareof interbandscreeni

exp (iQX2 + iKX3) ~. (x, + Q) andthe A’s are the polarizationfactors

= e~•d~ e~where e1 (e2) is thepolar-C ization vector of the incoming (scattered)phot

with

= ~ ~,,J[a2~(a~ 2a~) eB Theretardedcommutator,in the random3 1 mC____________ = — (2) phaseapproximation,is

~ i~~-’~r (2)

= ~ 1 J di e <tp (q, t), ~)(_q~)]> =

77 ~/(2’~n!) 0

exp (— ~ [~‘(mc~~)x,] ~P~(q,w) ~477e P(q,w)P~(q,a~q,w)

and!~I~arethe Hermite polynomials. The corre- the dielectric constantbeinggiven by1’

spondingenergiesare 2 — ~ ~ 5~p~)(q,

~(q,0))= E~ 2

W—WT q 1O~EflK=(n+~)wC+ ~ (3)In thecalculation only the Fröhlich interactioiwastaken into account,and thedeformation

The coordinatesX~are related to x, by thepotential interaction wasneglectedas the latt

transformationM hasa muchsmaller intrabandmatrix element.2= XkMk, (4)

The responseof the electrongasis

We will only needthe last column characterizedby’2

(2’\ aL — aT (q, w) = - _______ ~ Y,~’(M~q~,~ q)M1~=j~) \(aT+

2°L)\ fin

= f EnK -i~ q —dKf~

(EflK~/YQ —E~~)2--(a + i5)2I_3a~a~-M

33=j~ + 2at) (5) with f~~theFermi function, and the7’s givei

by

for eachof the ellipsoids. a~andaL are the 1~eigenvaluesof the matrix a, with aT>

0-LW ~ = J e~Z L1.,,(x) (-~~

The crosssection can be written3 If we now specializeto q = (0, 0, q), P becorrindependentof I as eachsurfaceentersthe

Vol. 9, No.4 RAMAN SCATTERING IN SEMICONDUCTORS

Table1. Numerical results for the longitudinal collectivemodesin PbTe propagatingparallel to thestatic magneticfield for different valuesof this field. A(w) is the relative crosssection,the fullexpressionbeing given by equatIon(11).

B Modefrequency A (w) Linewidth(Kgauss) (10’~see’ ) (10’~sec’) (cgsunits) (1010 sec’)

5.85 1390 1.65 0.23 0.47 1680 0.18

0.14 9.4 3.4

5.86 1390 1.610 0.47 — — —

0.27 118 3.2

5.86 1390 1.6

12 0.56 — — —

0.31 249 2.9

5.87 1390 1.6

15 0.70 0.54 1040 1.30.36 616 2.3

5.89 1390 1.620 0.93 0.64 455 2.4

0.40 1170 1.2

6.13 1350 1.750 2.33 1.44 15 3.0

0.43 1570 0.46

7.06 1150 2.3100 4.67 2.50 63 2.5

0.43 1600 0.44

10.50 657 3.0200 9.34 3.35 388 . 1.8

0.43 1600 0.43

problemin the sameway. For valuesof q ~ i0~cm’ and magneticfields B ~ 5KG it is usually sufficient to

The collective modesaregiven by the zeros work to lowest order in q•* The dielectric13of E (q, w), and are undampedin regionsin which constantbecomes

lm E (q, w) = 0. For such a modeat w’ the1 2 2 2 2 2 2~

cross section is �(0, a)) =0)L — ~. w ~ a~

2 I ~ ~2 a)2a)~2jd al 22 2= A(w’) q r

0 e2~e~ 8 (w — w’)(ll) withdQdwL,’ - -

with ~2 M~a3

A(w’) =:~x_(2a 2 47Te2 P~(q 12 + M~

3q

4 w’)j 477Ne2 2g�3/2 T+aL) —

(M,3÷M:3)

m E,1

d � (q, w)~ (12) Numerical resultshavebeenobtained(indoi long-wavelengthlimit) for A (w) for PbTewitF

the following parameters

268 RAMAN SCATTERING IN SEMICONDUCTORS Vol. 9, No

a~=42

= 1.8 x 1D’~sec’ = 4.9 x 10’2sec’18 —3N = 10 cm V = 10~cd E,~=30 -~

They are tabulatedin Table 1.

For most magneticfields threecollective ________________________

modesexist in theplasma;they havebeen 2

plotted againstmagneticfield in Fig. 1. The __________________________________

graph shows the usual ‘repulsion’ o interacting o so ióomodes.For fields near 10 Kg, whenw~.=ak,, ${KGA~.5S)

the centralmode is damped,and the one-band FIG. 1. The threelongitudinal collectivemodmodel doesnot provide accuratevaluesfor the frequenciesfor PbTeareplotted againstsingle-particlescatteringcross section~~14 magneticfield (for q /1 B).

15The tableshows that the chargedensities Bobayashiet al. and incorporatedin the con

associatedwith the LF CDF are small, and ductivity accordingto 0) -= C~)+ i/7 Sincethrthat it is the mixing with the LO phononswhich phononsareassumedto have infinite lifetime~may enableone to detectits presenceat low they are damped(i.e. show a linewidth) only t

magneticfields, the extent that they are mixed with theelectrc

modes.In a dopedsemiconductorperhapsoneought

to worry aboutelectronlifetimes, and their Acknowledgements— The authorwould like tceffect on the normal modes.In the last column thankProf. P.R. Wallacewho suggestedtheof the table the linewidth hasbeenestimatedfor problem, andunderwhoseguidancethe workeachof the modes.A lifetime for the electrons was carriedout, andthe National Research

Council of Canadafor the financial support itwasassumedon the basisof the mobility dataof hasprovided.

REFERENCES

1. WOLFF P,A., Phys. Rev. Bi, 950 (1970).

2. PLATZMAN P.M. and TZOAR N., Phys.Rev. 182, 510 (1969).

3. PLATZMAN P.M., Phys. Rev. 139, A379 (1965).

4. WOLFF P.A , Phys. Rev. 171, 436 (1968).

5. WOLFF P.A., Phys. Rev.BI, 164 (1970).

6. FOO E-NI andTZOAR N., Phys.Rev. 1b~,644 (1969).

7. WALLACE P.R., Can. J. Phys.43, 2162 (1965).

8. PERKOWITZ S., Phys. Rev. ~82,828 (1969).

9. McWHORTER A.L. and ARGYRESP.N., in Light scatteringspectraof solids,Proc. mt. Conf. or~Light Scattering,p. 325 NewYork 1968, Springer(1969).

10. BLUM F.A., Phys. Rev. Bi, 1125 (1970).

11. VARGA B.B., Phys.Rev.137, A1896 (1965).

12. MERMIN N.D. and CANEL E., Ann. Phys.(USA) 26, 247 (1964).

*This is not true for modescloseto nc~ for n = 0, 1,

Vol. 9, No.4 RAMAN SCATTERING IN SEMICONDUCTORS 269

13. WALLACE P.R., in Physicsof Solids in IntenseMagneticFields, p.82. (editedbyHAIDEMENAKIS E.D., Plenum,New York (1969).

14. JHA S.S.,Phys.Rev. 179, 764 (1969).

15. BOBAYASHI A., SATO Y. and FUJIMOTO M., in Physicsof SemicondactorsProceedings,Paris

Conference,p. 1258, 1964, Dunod, Paris (1964).

Les sectionsefficacesde diffusion Ramanpour desmodesoptiqueslongitudinauxen presenced’un champmagnétiquesontcalculeesdans le casd’un semiconducteurpolaireI surfacesde Fermiellipsoidales.Des résultatsnumériquessontobtenuspour du PbTetype n. On a utilisé an modéleI unebande.