Volume142, number1 PHYSICSLETTERSA 27November1989

COLLECTIVE EXCITATIONS OF EDGE STATE ELECTRONSIN QUASI-ONE-DIMENSIONAL QUANTUM WELL WIRES

Hui Lin ZHAO, Yun ZHU, Lihong WANG and ShechaoFENG’DepartmentofPhysicsand theSolidStateScienceCenter,UniversityofCalifornia, LosAngeles,CA 90024, USA

Received12 September1989; acceptedfor publication27September1989Communicatedby A.A. Maradudin

Collectiveexcitationsof edgestateelectronsin a quasi-one-dimensionalquantumwell wire undera largetransversemagneticfield atzerotemperatureareinvestigatedtheoretically.Theplasmonmodesfor both intrasubbandandintersubbandexcitationsarefoundatdifferentmagneticfield strengthsandquasi-IDelectrondensity,in thelongwavelengthlimit. Theplasmondisper-sionrelations,whichareall acoustic-like,alsoshowananisotropywhichcanbeattributedto theappliedfield andthefinite widthofthewire. Thesepredictionsareamenableto experimentalverifications.

Therehavebeenmany studiesof the low dimen- Thetransversemagneticfield configurationfor quasi-sional electroniccollective excitationsin semicon- 1 D wiresis particularlyimportantbecauseit is in theductordevicessincethe first paperby Stern [11 on quantumHall effect(QHE) regime,which hasgiventhe plasmoncollectivemodesin a two-dimensional riseto oneof themostexcitingdevelopmentsin solidelectron gas (2DEG). Until very recently, these statephysicsin this decade.Many recentstudiesofstudiesweremainlyfocusedon 2DEG systemssuch the quasi-1 D QHE were focusedon the edgestates,as a metal-oxide-semiconductor(MOS) or hetero- a termfirst introducedby Halperin [71in a film ofstructureslike GaAs/Al~Ga,_~Asheterojunctions. annulargeometryand later studiedby MacDonaldWith the advancesin lithographic and molecular- andStfedain a one-dimensionalquantumchannelbeam-epitaxytechnologies,peopleare now able to [8]. In the quasi-1 D QHE, the edgestateelectronsfabricate microstructures with quasi-one-dimen- are localizedat the two edgesof the quantumwiresional (quasi-1 D) transportbehaviorfrom GaAs! but are extendedin the directionof the wire. Clas-Al~Ga,_~Asheterojunctionsas narrow as 0.1 ~.tm sically, the edgestateelectronsskipalongtheedgeof[2,31.Thesequasi-1 D quantumwireshaveattracted the wire due to the transverseconfinementandtheconsiderableattentionbecauseof their highmobil- magneticfield. It has beenshown that theseelec-ity. Severaltheoreticalpapers [4] haveappearedin tronscarrytheHall currentandplay the key role inthe literature dealing with various aspectsof elec- the quasi-1 D QHE [7—9].It is thereforeimportanttronic transportin thesequasi-1 D systemsin theab- to studythe collectiveexcitationsof theseelectronssenceof a magneticfield. The singleparticlestates in orderto gain abetterunderstandingof quasi-1 Din a longitudinal magneticfield andthe associated electronicsystemsand especiallythe role of edgeinterestingphenomenahaverecently beenconsid- statesin quasi-1 D QHE.ered [5,61, butasfar asweknow no studyhasbeen In this paperwe studya quasi-1 D quantumwireperformedon the collectiveexcitationsof suchde- with parabolic transverseconfinement. The elec-vices in the presenceof a transversemagneticfield. trons areconfinedin the z=0 planeandare free in

thex-direction.Thetransverseconfiningpotential is

Also atUniversitätzu Köln, II. PhysikalischesInstitut, Insti.. assumedto be V(y) = ~mw~y2,anda magneticfieldtut für TheoretischePhysik, Zülpicher Strasse77, 5000 B is appliedalongthe z-axis.The singleparticleei-Cologne41,FRG. genstatecanbe written as [10]

36 0375-9601/89/S03.50© ElsevierSciencePublishersB.V. (North-Holland)

Volume 142,number1 PHYSICSLETTERSA 27November1989

wheref is the Fermi—Diracdistribution function.

W~k(x)= e ø~k(Y), (1) The electronsareassumedto respondto the totalself-consistentpotential i(x, cv, t), which implies

whereØ~k(y)is the shifted simpleharmonicoscil- that our approachis a RPA which is a goodapprox-lator wave function given by imationfor a highdensityelectrongas. The reason

1/2 thattheRPA canbeusedin theinversionlayerprob-

ø~k(Y)=(~~~I~)Hfl(..f~(y—y~)) lem where the densityis only moderatelies in thelow effectivemassof GaAs,’Al~Ga,_~As(thereforexexp[—1(y—y~)2!2], (2) a largeeffectiveBohr radius).

In orderto obtain the dispersionof thecollectivewhere H

5(x) is a Hermite polynomial andthe re- excitations,we mustrelate~nto b. SolvingPoison’slated parameters are defined as follows: equationin a two-dimensionalgeometry,we obtain)~=m&!h, y~=kl~(w~!c2)

2,1~=hc/eB, w~=eB,’mcand&2=cv~+a~.Theeffectivemassof electronsin (1)(q, y, z—0)GaAs!Al~Ga

1_~Asis m~0.07me.If we introducea = 2e f K0(qly—y’ I)~n(q,y’)dy’ , (7)neweffectivemassin thepresenceof magneticfield,m*=m(öj/wo)

2, theeigenenergiescanbewritten asEflk=(n+~)hi+h2k2!2m* . (3) wheree~is the dielectricconstantfor the medium,

K0(x) is a modifiedBesselfunction.Takingthe ma-

We thenassumethatN electronsarefilled into the trix elementof c1(q, y), we havelowest eigenstates.Following Berggrenet al. [10],we canrelateto0 now with thewidth of the quantum ~ [5~?c~2Hflfl~(q, w)V~~(q)]nn kchannelas W= 27t(N/L)~

3(2h/37tmw0)2~/3, where

L is the samplelength. The Fermi wave numbers X <ønk I I øn’k±q> = 0, (8)~ for different subbandsn andthe Fermi energycan be simply determinedby ~nO ~ = ~N!2L, whereö~’ is a Kroneckersymbolwhich equals 1where only whenn=l, n’ =1’, k=k’ andis zero otherwise.

The potential V is givenbykfP~={2m*[EF_(n+~)hth]!h

2}l~’2. (4)

Next we considerthe responseof the systemto an V~~(q)~$ Jdydy’ ølk~(Y’)Øi’~+q(Y’)externalperturbationof frequencycv. Thetotal per-turbation, including the externalperturbationand xK

0(qIy—y’ I )ønk(Y)ønk+q(Y) . (9)the inducedHartreeandexchange—correlationterms,will also be of the form ~(x, cv, t)=P(x, y)e_~(~t. Eq. (8) determinesthe plasmonmodes in theDefining the Fourier component‘I~(q,y)=JcP(x, presenceof magneticfield. It is a difficult equationy)e~ dx, and using the Ehrenreich—Cohen[11] to solve exactlybecauseit givesa determinantof in-self-consistentfield prescription,we obtain for the finite dimensions,due to the presenceof k in the ckinducedelectrondensity term. In a strongfield, however,thiskind of “cou-

pling” disappearsbecause6n(q,y,w)= L fl?fk cv) <ØnkI~IØn’k+q> <øn I~Iøn’q>(l+k1~!w).

X (ønk(Y )I~(q,Y’)Iønk....q(Y’)> (10)

X Ø~k(Y)ØnJ~_q(Y), (5) Thusaslongask1~.~ Wwe cangetasecularequation

wherethefactor2 is due to spinand11 is definedas

f(Enk_q) —f(Eflk) det(ö~—2 ~ H~~k(q,cv) V~r~(q))=0. (11)En~k_q_Enk+hW (6)

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Volume 142, number1 PHYSICSLETTERSA 27 November1989

Before we performany meaningfulnumericalcal- The rest of relatedparametersare assumedto haveculations,we must first evaluatethe summationin the values B= 6 T, k~°~3.68x 106 cm~,EFeq. (11). It turns outthat this canbe easilydonein 0.72h&~7.2meY.Thetwo branchesoftheplasmonthe longwavelengthlimit (small q) at zero temper- modes,computednumerically,are shown in fig. 1.ature, andwe obtain The branch with lower frequencycorrespondsto

k’ = — k~°~which in turn correspondsto the lower~ H,,,,k(q,w)V~y~(q) edge (y<O side) of the quantumwire. The otherk

branchcorrespondsto the upperedge(y>’O side).e2

= ~“~(k~?~ —k~+q) Thisanisotropycanbe understoodphysically as fol-lows: in the presenceof transverseconfinementandmagneticfield, the edgestateelectronsare involved

0(k~~—q)+0(k~—k}/~—q) not only in collectivemotionbut also skipalongthex

cv — (hk,~! m* ) q + (n’ — fl ) tO edges.The directionsof the skippingmotion areop-

+A~”~(kfJ’~—k~P~—q) positefor the two edges,the upperonebeing in thesamedirection as the collective excitation motion

0(k_k~)+q)+0(k~_k~—q))

while the lower onebeing oppositeto it. Thus theX ~ (~k~/m*)q±(n’ —n)& ‘ upperedgeexcitationhasa higher frequency,as fig.

1 indicates.(12)

2. Two subbandsare filled with electrons. We

where kF= (k~!°+kfJ’~)/2,0(x) is the step func- choosein this casethefollowing parametersfor ourtion, and numerical calculation: B=2 T, kf~°~~2.24x 106

cm’, k~’~l.36xl06cm’, EF~2.O9hth~7.8~ = J Jdydy’ K

0(qIy—y’ meV. We needto discussthe intrasubbandexcita-

tions in bothsubbandsandthe intersubbandexci-+(WC1B/&)2( ±kF—k’) I)

XØi(Y’)Øtq(Y’)Øn(Y)Øn’q(Y). (13) 5

In derivingeq. (11), we haveassumedthat the nthandn’ th subbandsarefilled with electrons,andthatq<< (k~?~,k~P~).

Now we are ready to calculatethe plasmondis- I 3persionrelations.For a typical quasi-lD quantum ‘~

wire cut from GaAs,/Al~Ga,_~As,W—~0.lI.Lm, N!

© 4’.L’-~2.3Xl06cm’ande~’-~13. Itiseasytocheckthat ‘-~ 2at B=6 T only the n=0 subbandis filled with elec-trons while at B=2 T both the n=0 andn= 1 sub-bandsarefilled. Forclarity of presentation,weshallconfineourselvesonly in thesetwo casesin the sub-

sequentdiscussion. o1. Only onesubbandis filled with electrons.In this 0 1 2

case,we only have intrasubbandexcitations.The q(105cm1)plasmonmodesare determinedby

Fig. 1. Intrasubbandcollective excitations.The curvesmarked1—2 ~ HØØk(q,cv) V~~.(q)=0. (14) by 0 belongtoplasmonmodeofn=Osubbandforonesubband

koccupation.Thecurvesmarkedby A and• respectivelycorre-

Sinceonly thoseelectronsneartheFermi surface(a spondto plasmonmodesof n = 0 andn= 1 subbandsfor two sub-bandsoccupation.Foreachmode,thelowerbranchcorresponds

point in lD system),i.e.,the edgestates,contribute totheloweredgeexcitation(y<Oside)ofthequantumwirewhile

to the collectiveexcitations,we canset k’ = ±k,c/”. theupperbranchcorrespondstotheupperedge.

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Volume142,numberI PHYSICSLETTERSA 27 November1989

tationbetweenthetwo subbandsseparately.Strictly i 2

speaking,we haveto solve a 4x4 determinant.Buttogeta qualitativepicture,we canfirst simply ignore 10

the off-diagonalmatrix elements,and the resultingmodescanbe discussedas follows. ~ 8

(a) Intrasubbandexcitationin the first subband(n=0). The equationwhich determinesthe plas- 6 __________________

monmode is the sameas eq. (16) apartfrom the ‘~ u-.-u-._._.~~~

numerical differences in B and kf~°~.The two 4

branchesof excitationsarethenshownalsoin fig. 1.(b) Intrasubbandexcitation in the secondsub- 2

band (n= 1). The plasmonmode is determinedby

1—2 ~ H,,k(q, cv) Vft~.(q)=0. (15) 0 2k

Its two branchescorrespondingtok’ = ±kf~°~arealso q(105an~)

givenin fig. 1. Comparingthethreemodesin fig. 1, Fig. 2. Intersubbandcollectiveexcitationsfor two subbandsoc-it is easyto seethat the two branchesof eachmode cupation.Thecurvesmarkedby A and~arerespectivelythe

aresqueezedcloseras the subbandnumbern is in- plasmonmodesof n=0 to n=1 andn =1 ton = 0. Foreachmode,

creased,or asthe appliedfield B is decreased.This thelowerbranchcorrespondsto thelower edgeexcitation(y< 0

is reasonablephysicallybecausein eithercase,the side) of thequantumwire while theupperbranchcorresponds

correspondingedgestategoescloserto the centerof to theupperedge.Theshadedareais theintersubbandsinglepar-ticleexcitationregion.

the quantumchannelandthe edgeeffect (thus theconfiningpotential) becomesless important.In the explainedearlier. We also note that the dispersionlimit of free electrons(no transverseconfinement),

curvescorrespondingto n = 0 to n = 1 havepositivethetwo branchesshoulddegenerateinto one, which slopeswhile thecurvescorrespondingto n = 1 to n = 0is what we expectfor a 2DEG. havenegativeslopes.Thiscould againbe explained(c) Intersubbandcollective excitationsbetween

n =0 andn = 1 subbands.Thereare two modesfor as an edgeeffect.In considerationof measurementsof theseplas-intersubbandcollectiveexcitations.Whenwe ignore

mon modesexperimentally,we mustalso computetheoff-diagonalelements,the two modesaresimply

the singleparticleexcitationregion, which is deter-thecollectiveexcitationscorrespondingto n =0 sub-

mined bybandto n = 1 subbandand n = 1 subbandto n = 0subband.Thesetwo plasmonmodesare given by lim Im ~ H~~.k(q,w+ii

1) V~}’~(q)~0. (18),~—.O k

1—2 > HO,k(q, cv) V00~.(q) =0

k Whenthe collectivemodesfall into the singlepar-

(n= to n= 1) , (16) tide excitation region, they will decay into singleparticleexcitations(Landaudamping).Thusonecan

and only expect to detectthose stableplasmonmodes

1 —2 ~ H,Ok(q, cv) V13~.(q) =0 outsidethe regionof singleparticleexcitations.Fig.k 2 shows that onebranchof the intersubbandplas-

monmodesis remarkablydampedwhile the others(n=lton=0), (17)

are unaffected.Fig. 3 gives the intrasubbandsinglewherek’ = kF = ±(k~°~+ k~’~) !2. Thedispersion particle excitations relative to the three lowestcurvesfor thesetwo modesare shownin fig. 2. Again branches of intrasubbandcollective excitations.we seethat the branchcorrespondingto the upper Thesefigures indicatethat the singleparticleexci-side of the quantumwire (k’ = +k,~)hasa higher tation regionsaremuch narrowerin thesequasi-1Dfrequency,in agreementwith our physicalintuition systemsthan thosein the corresponding2D system

39

Volume142, number1 PHYSICSLETTERSA 27 November1989

_______________________________ moredifficult) treatmentis needed.(2) In the case1.5 wheretwo subbandsare occupied,thereshouldbe

coupling betweenthe intrasubbandand intersub-/ bandexcitationsandwe hadto include the off-di-agonalelementswhenwesolvedthesecularequation

T 1.0 (11). Thesecouplingswill eliminatethe interceptsof differentdispersionbranchesandbreakthem intoseparatecurveswith mixed natureof intrasubbandand intersubbandexcitations.A similar coupling

0.5 effect in the collective excitations in GaAs!Al~Ga,_~Assuperlatticesin theabsenceof magneticfield hasbeenrecentlystudiedby Jam andDasSarma

___________________________ [12].0.0

0 1 2We wouldlike to thankProfessorFan-AnZengfor

q(105~) valuableassistancein numericalcalculations.SF also

Fig. 3. Intrasubbandsingle particleexcitation regions(shaded acknowledgestheUniversity of Colognefor its hos-areas)relativeto thethreelowestbranchesofintrasubbandcol- pitality during his visit. Thiswork was supportedinlectiveexcitations(seefig. 1). Thethreeshadedareas,fromlow part by a grant from the DOE undergrantnumberfrequencyto highfrequency,respectivelycorrespondto thosein DE-FGO3-88ER45378.n =0 subbandwhenonly onesubbandis occupied;in n = 1 sub-bandwhen twosubbandsareoccupiedandin n = 0 subbandwhentwosubbandsareoccupied.

References[12], at leastfor the frequencyrangeweconsidered.Therefore,we expectthat it will be easierto detect [1] F. Stern,Phys.Rev. Lett. 18 (1967) 546.

the plasmonmodesin a 1 D systemthan in a 2D 121 W. Hansen,M. Horst,J.P.Kotthaus,U.Merkt, Ch. SikorskiandK. Ploog,Phys.Rev. Lett.58 (1987)2586.

system. [3] G. Timp, A.M. Chang,P. Mankiewich,R. Behringer,J.E.The energyscalesof the collectiveexcitationswe Cunningham,T.Y. ChangandR.E. Howard,Phys.Rev.Lett.

computed(-..~1—10 meY) suggestthat they be ob- 59 (1987) 732.servedby Ramanscatteringor infraredspectroscopy [4] V.K. Arora,Phys.Rev.B 23 (1981) 5611;

experimentally.To increasethespectralintensity,the P.A. Lee,Phys.Rev.Lett. 53 (1984) 2042;S. Das SarmaandWu-yuanLai, Phys.Rev. B 32 (1985)

samplecanbe preparedasanarrayof parallelquasi- 1401.1 D channels.Theinteractionbetweenthe electrons [5] T.P. SmithIII, J.A. Hong,C.M. Knoedler,H. ArnotandH.

in the adjacentchannelswill modify the dispersion Schmid,Phys.Rev.Lett. 59 (1987)2802;

curvesinto bands.But this doesnot changequali- T.P. Smith III, J.A. Brum, J.A. Hong,C.M. Knoedler,H.

tatively thegeneralfeaturesof thedispersionspectra ArnotandL. Esaki,Phys.Rev.Lettt 61(1988)585.[6] Hui Lin Zhao,Yun Zhu andShechaoFeng, Phys.Rev.B,presentedabove, tobepublished.

Finally, we would like to elaboratea bit moreon [7]B.I.Halperin,Phys.Rev.B25 (1982)2185.

two pointswementionedbefore: (1) Thestrongfield [8] A.H. MacDonaldandP. Stteda,Phys.Rev. B 29 (1984)

condition (from eq. (10)) is satisfiedfor the two 1616.

cases(B= 6 T, B= 2 T) we consideredparticularly. [9] M. BUttiker, Phys.Rev.B 38 (1988) 9375.[10] K.F. Berggren,G. RoosandH. van Houten,Phys.Rev.B

Thesefields correspondto the systembeing in the 37 (1988) 10118.

quantumHall regime. Forweakerfield, we cannot [11] H. EhrenreichandM. Cohen,Phys.Rev. 115 (1959) 786.

obtaineq. (11) from eq. (8). A morecareful (and [12] J.K.Jam andS. DasSarma,Phys.Rev.B 36 (1987)5949.

40