Volume149, number4 PHYSICSLETTERSA 24 September1990

Discreteperimeterplasmonexcitationsof electrongasesin a quasi-one-dimensionalwire: ring geometry

FengyiHuangSchool ofElectricalEngineering,UniversityofBath, BathBA2 7AY, UK

Received26March1990; revisedmanuscriptreceived21 June1990; acceptedfor publication11 July 1990Communicatedby R.C. Davidson

A self-consistent-fieldtheory is appliedto electrongasesconfinedto a quasi-one-dimensionalwire with a ring geometry.Ananalyticalexpressionof thedynamicdielectricfunction is obtained.Collectiveexcitationsof theelectrongasesarestudied.Dif-ferentfrom theusualtwo-dimensionalorquasi-one-dimensionalplasmonmodes,novelperimeterplasmonmodesofthediscretedispersionrelationduetoangularsubbandtransitionsarefound.

Thedevelopmentof molecular-beamepitaxyasa called the angularinter-subbanddiscreteperimetertechniquefor the growth of high-quality semicon- plasmon.ductorcrystalshas causeda new dimensionin the The structureof the electrongaseswith a ring ge-experimentalstudyofthepropertiesof narrow-chan- ometry is basedon therecentexperimentfor study-nel microstructures.Quantumwell wire (QWW) and ing Aharonov—Bohmoscillationsof a heterostruc-quantumdotstructureshavebeenrealizedandstud- ture ring [7]. Using a novel split-gate technique,ied extensively.Recentresearchinterestin thesesys- high-mobility two-dimensionalelectrongas (2DEG)temsinclude the investigationof magnetotransport dericeshavebeenfabricatedon GaAs—A1GaAshet-properties[1] andplasmonexcitations [2—6].Us- erostructureswith a ring structure.A layerof metaling a self-consistent-fieldtheory, the plasmonexci- on the top of the AlAs actsas a Schottkygate, de-tationsin a singleQWW andinter-subbandmodes pletingthe 2DEGbeneathit whena negativebiasisof quasi-one-dimensionalQWWswith a superlattice applied.The 2DEGcanbeconstrainedto a narrowstructurehavebeenstudied [2—4]. Plasmonexci- channelwith variablewidth controlledby changingtationsin a specialkind of QWW with electronscon- the bias voltage.If the typical length of the ring isfinedto a cylindrical surfacehavealsobeenstudied comparableto the phasecoherencelength of the[5]. Different from the quantum approach,a hy- electrons,the motion of electronsin this structureisdrodynamicmodel hasbeenproposedto explainthe quantized,which allows us to considerthe subbandmagneto-edgeplasmonmodesof electrongasescon- transitionof electronsin this structure.finedto a restrictedtwo-dimensionalplane,suchas A schematicexplanationof the quasi-one-dimen-a disk structure[6]. In thispaperwestudytheplas- sional cylindrical QWW with a ring geometry ismonmodesofelectrongasesconfinedto aquasi-one- shownin fig. 1. Electronsare confinedin a ringwithdimensionalQWW with a ringgeometry.It is found insideradiusa1 andoutsideradiusa2. Electronsarethatwhen the angularinter-subbandtransitionsare only allowedto movein the planeperpendiculartotaken into consideration,the collective modesbe- the ring axis,i.e., the z direction, in our discussion,comediscretein the plasmondispersionwith an- as a result a sequenceof electronicsubbandsisgularquantumnumbers,which is differentfrom any formed.kind of two-dimensionalor quasi-one-dimensional The effective-massHamiltonian describing anplasmonmode.Sincethis novel plasmonmode is electronconfinedto a ring structurein the presenceconfinedto theperimeterof thering structure,it is of an effectivepotential V~ff(r)canbe written as

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Volume149, number4 PHYSICSLETTERSA 24 September1990

eV

2OH=__öflô(z). (5)Cs

After a Fourier transformationof 0 and a Hankeltransformaboutr, eq. (5) becomes

(02/8z2—p2)Ø(p, atm, z)Fig. 1. Schematicexplanationofthequasi-one-dimensionalelec-trongasin acylindrical QWWwith aring geometry.Themotion = — ~n(p) ~(z) , (6)ofelectronsis confinedto theplaneperpendicularto thering axis.

with

h2 ~l ô ô 1 ô2

H=— + ~ + Ve~(r)). (1) Sn(p)= J~rn(Pr)8n(r, i~m)rdr. (7)Thesolutiontoeq. (6),with theboundarycondition

Weassumethat the completeset of quantumnum- that the potential is zero at infinity, isbers representingthe quantizedmotion of electronsin thissystemis v= (m, n), wherem is the angular ØH(p /~m,z)quantumnumberwith m=0, ±1, ±2,..., anddif-

=(—e/t5)~n(p)(—l/2p)exp(—pIzI). (8)

ferent subbandswithin the sameangularquantumlevel aredenotedby n with n=0, 1, 2, 3 Thecor- At z=0 we getrespondingeigenenergyandeigenfunctionare given eby mn and~mn(r),with theangularpartof thewave- ØH(p, ~m)= — ~n(p) . (9)function of the form exp(imØ). 2pe.

Let usestablishthedynamicresponsetheoryof an The inverseHankeltransformgiveselectrongasin a QWW in theabsenceof an externalmagneticfield. An externalperturbationof the form OH(r ~m)= JpdpJ~m(pr)OH(p,sm). (10)

~ r, 0; t) = Ø~( r, L~m;U)) exp( iwt— thm0) (2) In all theexpressionsthetimedependenceof thepo-

tentialsis of the form exp(iwt).will inducea perturbedelectrondensity,which in Theexchange—correlationpotentialØ’”~canbe ob-turn inducesperturbedHartreeandexchange—cor- tamedby following thetreatmentof Kohn andShamrelationpotentials.The total perturbation [9]. In the following treatmentthe exchange—cor-

relationpart is omitted.= Øex

t + ~ +Ø~ (3) After substitutingön obtainedfrom eq. (4) intoeq. (10), the perturbedHartreepotentialcanbe re-

isalsoof theform of (2). FollowingtheEhrenreich— lated to the total potentialdefinedby eq. (3). TheCohenself-consistent-fieldprescription[8], the lin- propertiesof theelectrongascanbedescribedby theear-responseapproximationleads to the induced responseof the electrongas to theexternalpertur-electrondensity bations.In thefollowing discussion,wetaketheelec-

tron profile in theradialdirectionasa 5 function,i.e.,~5~ (,,—C,,—ha) ~n(r,Ø;t)=8n(Ø,t)d(r—a) , (11)

X <vIH1 Iv’> <ii’ Iô(x’ —xflv> , (4) with a the ring radius.This correspondsto a very

narrow ring width suchthat ~—~a2= a. In this casewhere f0 is the Fermi—Dirac distribution function, we need only consider the angular subbandandH1 = — eØ. transitions.

The perturbedHartreepotentialcanbe obtained The matrix elementsfor the Hartreepotentialbe-from the Poissonequation, tweenthe initial stateand the final stateis

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Volume 149, number4 PHYSICSLETTERSA 24 September1990

<OIOH(a,Am;w) I Am>

V(Am)TI(Am,w)<0IO(a,Am;w)IAm>.5 xlØ

3(12) ~2.8

i04Theparametersare definedas follows,

a

V(Eim)=f-JJ~m(pa)dP~ (13)

mo / 1 1.4~H(Am,w)=2 ~

m=—mo

+ 1 (14)Cm Cm+tsm +hwj

Thesummationis over all the occupiedstates,and 0 ~m

0 is thehighestangularlevel occupied,which is re-latedto thenumberdensityof theelectrongas in thering, n, by Fig. 2. Plasmonexcitationsof electrongasesin a QWW with a

ring geometry.The dispersionof themodeshasdiscretevalues= ~ (rtan — I) . (15) for differentangulartransitionquantumnumbers~m, thesolid

linesaresimplyguidesto theeyes.The numbersrepresentingtwo

Thedynamicdielectnc functionis definedas differentstructuresdenotethehighestquantumlevelsoccupied.<oext>

= <0> (16) ture hasbeenneglected,which shouldbeconsideredfor a realsystem.The effectof afinite ring width re-

From the expressionsabovefollows sults in additionalquantizedmotion for a givenan-

gular quantum numberm, as mentionedbefore,�~m(w)lV(Am)17~ (w). (17) .

which yields the eigenstatesdenotedby two quan-Mostof theimportantpropertiesof electrongases, turn numbers,mand n. Since bothmandn repre-

which respondto an externalelectric field, will be sentquantizedmotionsthe discretecharacterof thereflectedby the dynamicaldielectric function. The plasmamodeswould notbechangedasthequantumconditionfor the collectiveexcitation (plasmon)of numbern is considered.It is worth noting that, inthe systemis that self-sustainingoscillationsin the order to realizethe quantizedmotion alongthe an-electrondensityoccur.As is well knownthe disper- gular direction, the typical length of the ring is re-sionrelationofthecollectivemodesisgivenby Cs,,,(q, quired to be comparableto the phasecoherencew) = 0. From eq. (17) weget length of the electrons,so that electronswill move

/ ~ / ,~ / ~~ many timesalongthe ring beforethey are scatteredl=V~Am,iJ~m~U)j. ~ . ,

by impuntiesand lose their phasecoherence.TheNumericalcalculationswherecarriedoutandthe discreteplasmonmodeswill bebroadenedasa result

resultsareshownin fig. 2. Theelectronnumberden- of the scattering.Thewell width fluctuation,whichsity is takenasn = 108 cm’. Theplasmonfrequen- is expectedto be importantas the ring radius be-ciesas a function of theangulartransitionquantum comes very small, is anothermechanisminducingnumberare calculatedfor two conditionswith dif- plasmamodebroadening.All theseeffectsresult inferent highest level occupied, m0= 5 x10~,and a changein the plasmonmodesso that the originalm0= 1 0~.Thedispersionhasdiscretevaluesfor dif- discretemodefor a givenquantumnumberwill beferentangularquantumnumbersAm, andthe solid extendedto a finite width, but the modesfor differ-lines are simply guidesto the eyes. entquantumnumbersremaindiscrete.The electron

In ourdiscussionthefinite width ofthe ring struc- gassystemconsideredherehasfewer electronscorn-

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Volume 149,number4 PHYSICSLETTERSA 24 September1990

paredto theusual two-dimensionalor quasi-one-di- ring. Thisdiscreteplasmonmodeshouldbe observ-mensionalelectrongasesof a superlatticestructure, able in experiment.Theproof thatcollectivemodesshouldexist in thissystem is the recentexperimentin quantumdots[10], fractionalquantizationhasbeenobservedin Referencesa structurewith the numberof electronsevenlessthanhundred.Thereare manymethodsto observe [I ] G. Timp, A.M. Chang,P. Mankiewich,R. Behringer,J.E.Cunningham,T.Y. ChangandR.E.Howard,Phys.Rev.Lett.the low energy acousticplasmonexcitationsof elec- 59 (1987)732.tron gases,suchaselectronenergylossandinfrared [21P.F.Williams andA.N. Bloch,Phys.Rev.B 10(1974)1097.

absorption.In orderto observethisangularplasmon [3] W.I. FriesenandB. Bergersen,J.Phys.C 13 (1980)6627.

mode;whenusingthe light absorptionspectroscopy [4] W.M. QueandKirezenow,Phys.Rev.B 37 (1988)7153.technique,the incident light should have the re- [5] Y. Zhu,F.Y. Huang,X.M. Xiong andS.X.Zhou,Phys.Rev.B37 (1988)8992.quiredangularcomponentto excitethe correspond- [6] A.L. Fetter,Phys.Rev.B 33 (1986) 5221.ing discretecollective modes. [7] C.J.B.Ford, T.J. Thornton,R. Newbury, M. Pepper,H.

In summary,a novel type of plasmonmodeof an Ahmed,D.C. Peacock,D.A. Ritchie,J.G.F.FrostandG.A.C.

electrongasconfinedto a QWW in a ring geometry Jones,AppI. Phys.lett. 54 (1989)21.

hasbeenfound.Thecollectivemodesarediscretefor [8] H. EhrenreichandM. Cohen,Phys.Rev. 115 (1959)786.[9]W. KohnandL.J.Sham,Pliys. Rev. 140 (1965)A1133.different angularquantumnumbersdue to the an- [10] W. Hansen,T.P. Smith III, K.Y. Lee,J.M. HongandC.M.

gular quantizationof the electronmovementin the Knoedler,AppI. Phys.Lett. 56 (1990)168.

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