Volume70A, number2 PHYSICSLETTERS 19 February1979

QUANTUM THEORY OF THE CYCLOTRON RESONANCE LINESHAPEFOR A TWO-DIMENSIONAL ELECTRON-PHONON SYSTEM

MahendraPRASAD1Departmentof Physicsand Astronomy,StateUniversityofNew Yorkat Buffalo, Amherst,NY14260, USA

Received12 July 1978Revisedmanuscriptreceived22 November1978

Basedon the superresolventoperatorrepresentationof Kubo’s currentcorrelationformula [1] and its properconnecteddiagramexpansion[2] a theoryof cyclotronresonancelineshapeis presented.Effectsof theaveragephononfield are takeninto accountwhentreatingthe scatteringof a givenelectronwith thereferencephonon.Thisgives rise to a self-consistentequationfor the irreduciblecollision operator,which possessesa built in struc-ture of the “gain” and “loss” like termsoccurringin thekinetic descriptionof therelaxationprocess.Equationsfor thecyclotronresonancelinewidthsrN andthe frequencyshifts ~N associatedwith theelectronictransitionsbetweenthe LandausubbandsN andN+ 1 arederivedfor the casewhentheelectron—phononinteractioninducesthescatteringof the electronin thesameLandausubband.Temperaturedependentcyclotronresonancelinewidthdataof KueblbeckandKotthaus[3] compareverywell with thetheoreticalpredictionmadehereandone param-eterfitting yieldsthe deformationpotentialin reasonableagreementwith that found for silicon.

Kubo’s formulafor the dynamicmagnetoconductivitytensoris givenby

à~_(w)= i lim urn UmL2 tr [TR{(a~i/au) [‘to~‘ph + 2’-11e—ph — z]1J~}] (1)a-.O u—~O

where“tr” and“TR” standfor the singleelectronandmanyphonontraces,respectively,h0, singleelectronhamil-

tonian undera magneticfield perpendicularto thesurfacein the Landaugauge,HPh,manyphononhamiltonianandHe_phstandsfor theelectron—phononcouplinghamiltonian.A “hat” () on the lettersdenotesthe Liouvilleoperatorcorrespondingto them.

= {1 + exp [I3(12~+ HPh + ~‘1e—ph— j~~u— ~)]}—1 (2)

1+ andj_ arethe currentcomponents;~ is the Fermienergy,L2 thenormalizationarea,u isac-numberandzstandsfor — w + ia. The cyclotronresonancelineshapearisesmainly from the superresolventoperatoroccurringin eq.(1) andtheinteractionterm from ffcanbe dropped,then ,i-~ii~.

Properconnecteddiagramanalysisof eq.(1) thenyields the following result:

o÷_(w)= i lim lim LimL2 tr [(a/au) ~O>ph~’I‘ (3)a-GO u—*O

‘I’ [1z0~‘ph — ~_z]~hj~, b A2<I~e_ph[IO+~ph— b_Z]

1He_ph)~~, (4,5)

wherethe superscript(P, C) meansthat only properconnecteddiagramsare to be selectedandsubscript(ph)meansmanyphononaverage.

Finding thetracesoccurringin eq.(3) andtheu-derivativewe obtain thefollowingresult:

Presentaddress:ClarendonLaboratoryandDepartmentof TheoreticalPhysics,Universityof Oxford, OxfordOX1 3NP, UK.

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Volume 70A, number2 PHYSICSLETTERS 19 February1979

e2 i E[f(~)_f(EN÷1)1(N+l)N (wo—w+z~N)+iFN (6)

whereEN= (N+~)hw0,f(E), the Fermi distribution functionand rN, ~N satisfy thefollowing equations:

1_2fd2q(l+nq)IcqI2[{K1(N+l,N+l;q)_K2(N,N;q)}{wq2+r~}_1

+ {K1(N, N; q) — K2(N, N; q)}{w~+

+h_2fd2qnqIcqI2[{K1(N,N;q)_K2(N,N;q)}{w~+F~}1

+{K1(N+ 1,N+ 1; q) —K2(N,N;q)}{o.~~~+ F~}—

1] , (7)

= ~ fd2q(1 +flq)ICqI2[{Ki(N+ 1,N+ 1; q)—K2(N,N;q)}(wq ~~N)’((~’q ~~N)

2 +

+ {K1(N,N; q) — K2(N,N; q)}(’~N— Wq)/((wq— ~N)

2 + F~)]

+ 7j_2 fd2qnqlCql2[{K1(N, N; q) — K2(N,N;q)}(wq + L\N)/((Wq+ ~N)

2 + F~)

+ {K1(N+ 1, N+ 1; q) — K2(N, N, q)} (~N— “q)’((~N — (.~.)q)

2+ I’~)] . (8)

The termsoccurringin theaboveequationscanbe physicallyinterpretedas arisingfrom the processesinvolvingemission(1 + flq terms)andabsorption(flq terms)of phonons.The functionsK

1 andK2 aregivenby

~ (9)

~ ~ (10)

J~,~’(X,q~,X’)- f dx~(XX)exp(iq~x)~N(~X). (11)

ØN(x) are the Landauwave functions,r0 = (h/eB)112 is the radiusof thegroundLandauorbit andX thecentre

coordinateof the cyclotronorbit. flq standsfor the occupancyof thephononsin the modeq, givenby

flq = [exp(i371wq) — 1]_l. (12)

Eq. (8) implies that the frequencyshift no longervanishesas opposedto the electronimpurity case [4] andisrathercomplicated.

Forelectronacousticphononinteractionswe have

ICqI2Aq and wq—sq, (13,14)

sbeingthe soundwavevelocity.A is constantandcontainsthemassdensityandthedeformationpotentialcorre-spondingto the Si (100)surface.Whenthephononenergiesare smallcomparedto thethermalenergy( kBT= 1 /i3) we canwrite flq kBT/hqsandthen the solutionsof eq.(7) for a few lowest Landautransitionsare sum-marizedbelow.

(i) N 0 * N 1 transition.

1 r[1 —7~+7~exp(’y~)Ei(’y~)]. (15)

This resulthasbeenreportedearlier [5].

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Volume 70A, number2 PHYSICS LETTERS 19 February1979

(ii) N=1 * N=2 transition.

1 =~r[2_2’y~ —3’y~—’y~~ . (16)

(iii) N 2* N=3 transition.

1 = ~ r [4 — {4’y~+ 14’y~+ ~ + i~1~ + ~ ~4O} + {~‘y~+4~4+ 164 + 24y~+ 1 2} ~ exp(~) Ei(ij)]. (17)

(iv) N=3* N 4 transition.

~ (18)

—{36y~+ 2104 + 318~4+q2.y~+2ji~y~O+~ 7~2+~‘y~4}]

where r = 2IrAkBT/s3h3,“N = rNrO/s.~/~(N= 0, 1, 2, ... ), andEi(x) standsfor an exponentialintegralgivenby

Ei(x)f~-_—du. (19)

Reduced resonance linewidths 7N givenby eqs.(15), (16), (17)and(18) are plotted against the reduced tempera-ture r in fig. 1. Someinterestingfeaturesof thesecurvesareenumeratedbelow.(1) Reducedwidths ‘IN monoto-nouslygrow with the increaseof the temperature.(2) At any fixed temperaturedifferencesamongthe linewidthsdue to varioustransitionsarealmostconstant.Also this constancyis maintainedthroughoutthetemperaturere-gion of interest.(3) Thelinewidth arisingdueto theN= 0* N= 1 transitionis maximumandreducesin magni-tude for the transitionsof higherLandauindexN. Thisreductionof theresonancewidth for higherLandaulevelscanbe explainedasbelow.

The de Broglie wavelengthof any Landaulevel is of the order of r0/(2N+ 1)1/2,thusdecreasesasN increases

andtheelectron—phononinteractionweakensasN increasestherebyreducingthe linewidthsdueto thetransi-tionsbetweenhigherLandaulevels.

Temperaturedependentcyclotronresonancelinewidthdataof KueblbeckandKotthausare fitted to the elec-

- s-5.9~jlO5cmAec. I i / /

r0100A / / / /

— Theoreticalcurvee Exp~riment~l

an- pointsFromreF3 •

2.5

2.0

N-0--N- 115 N-i ~N-2

N-2—-N-3I -3-=N-4

to ________________________________0 2.0 4 6.0 8.0 10.0 12.0

—T---.

Fig. 1. Reducedwidths “N = rNro/s%J~areplotted againstthereducedtemperaturer = 2irAk8T/s31t3.

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Volume70A, number2 PHYSICSLETTERS 19 February1979

tronic transitionsbetweentwo lowest Landaulevels,characterizedby eq.(15). Oneexperimentalpoint correspond-

ing to T 65 K is fitted on the theoreticalcurve andthustheconstantA is found (A = 2.8X l0~50cgs). This inturn gives rise to a deformation potential 10 eV. This valueis in reasonableagreementwith that foundfor theinversionlayersin Si. Otherexperimentalpointsare close to the theoreticalcurve and thusa goodagreementisfound to result.

It shouldbe remarkedthat the treatmentbasedon theextensionof Adams andHolstein’s [6] theoryleadsto adivergenceat w = w0, which is circumventedin the presentpaper.The two-dimensionalphononmodelusedhereis alsojustifiedbasedon the momentumbalancein theemissionandabsorptionscatteringrates.More detailedre-sultswill be reportedlater.

I am gratefulto ProfessorR.J.Elliott andDr. R.A. Stradlingfor encouragementandfruitful discussions.Initialguidanceof ProfessorS. Fujita is gratefullyacknowledged.

References

[1] R. Kubo,J.Phys.Soc.Japan12(1957)570.121 S. Fujita andC.C.Chen,Intern.J. Theor. Phys. 2(1969)59;

J.R. Barker,J. Phys.C 6 (1973) 2663;A. LodderandS. Fujita, J. Phys.Soc. Japan25 (1968)775.

[3] H. KueblbeckandJ.P.Kotthaus,Phys.Rev. Lett. 35(1975)1019.[4] M. PrasadandS. Fujita, SolidStateCommun.23 (1977) 551;

S. Fujita andM. Prasad,J. Phys.Chem. Solids, to be published.

[51 M. Prasad,T.K. SrinivasandS. Fujita, Solid StateCommun.24 (1977)439.

[6] E.N. AdamsandT.D. Holstein,J. Phys. (Them. Solids 10 (1959) 254.

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