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Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field L. Wendler and V. G. Grigoryan Fachbereich Physik, Martin-Luther-Universita ¨t Halle, Friedemann-Bach-Platz 6, D-06108 Halle, Germany ~Received 12 September 1995; revised manuscript received 1 May 1996! We investigate the collective and single-particle excitations of an electron gas, quantum confined in a quasi-one-dimensional quantum-well wire, in the presence of a magnetic field. The calculations are done in the random-phase approximation with no further simplifying approximations. We derive analytical results for the dispersion relations of the intra- and intersubband magnetoplasmons. For large-magnetic fields the dispersion curves of the intersubband magnetoplasmons approach the frequencies of the principal and Bernstein modes of a two-dimensional electron gas, whereas the dispersion curve of the intrasubband magnetoplasmon tends to zero in this limit. It is shown that additional intersubband magnetoplasmons exist in comparison to a quasi- one-dimensional electron gas without a magnetic field and to a two-dimensional electron gas in the presence of a magnetic field. These magnetoplasmons have the character of confined modes with a vanishing depolariza- tion shift for zero- and infinite-magnetic fields. With these modes it is possible to explain the resonance- splitting and fine-structure effects in the recently observed far-infrared transmission spectra. We interpret the complete spectrum of the quasi-one-dimensional magnetoplasmons by the hybridization of confined principal and Bernstein modes. This spectrum is universal, i.e., independent from the concrete shape of the lateral confining potential. @S0163-1829~96!02335-1# I. INTRODUCTION Quantum-well wires ~QWW! are semiconductor nano- structures in which the carrier motion is only quasifree in one spatial direction, but quantum confined in the two other spatial directions by an artificial potential of width in the order of the wavelength of the carriers ~Fermi wavelength in a degenerate case!. The realization of such low-dimensional semiconductor structures nearly precise in atomic scale is possible with modern epitaxial layer growth techniques and nanometer technologies. The excitation spectrum of the quasi-one-dimensional electron gas ~Q1DEG! of a QWW consists of collective ex- citations and single-particle excitations, which become split in intra- and intersubband excitations due to the size quan- tization. Intrasubband excitations are connected with electron motions within one electric subband, whereas intersubband excitations are connected with electron transitions between states of two different subbands. The single-particle excita- tions ~SPE’s! have a continuous spectrum and occur in defi- nite regions of the frequency-wave vector plane. The collec- tive charge-density excitations ~CDE’s! of a Q1DEG in the presence of a magnetic field, the intra- and intersubband magnetoplasmons, have, in general, dispersion relations out- side the single-particle continua and hence, are free of Lan- dau damping. Additionally, spin-density excitations ~SDE’s! occur due to exchange and correlation interaction. This ex- citation spectrum of an electron gas depends characteristi- cally on the dimensionality of the system. Further, with the charge density many related properties of the physical sys- tem can be varied: Fermi energy, Fermi wave vector, etc. This allows a detailed investigation of the different mecha- nisms that determines the plasma resonances itself and the interaction of plasmons with different types of other collec- tive excitations, e.g., with optical phonons in polar semicon- ductors and acoustic phonons in piezoelectric semiconduc- tors. Hence, the spectrum of the plasmons contains much physical information about the low-dimensional many- particle system. Q1D plasmons have been investigated theoreti- cally 1–9,11–39 and experimentally 40–54 in isolated QWW’s and lateral multiwire superlattices. Most of the theoretical works were done using the random-phase approximation ~RPA! to calculate the linear response to an external charge neglecting retardation 1–4,8–15,19–39 and to an external current including retardation. 30 Q1D plasmons were also investigated with the semiclassical hydrodynamic model. 5–7,16 It was shown 25,34 that the intrasubband plasmon of a QWW, assuming a two-subband model from which one is occupied, has a long-wavelength dispersion relation accord- ing v p 00 5u qu a @ ( v F / a ) 2 2v s 2 ln(uqu a ) # 1/2 , where q is the one- dimensional wave vector along the wire axis. The constant a depends on the wire size and is equal to the width of a rectangular potential, the radius of a cylindrical potential or to the effective width of a parabolic potential. Further, v F is the Fermi velocity in the one occupied subband and v s 5@ v F e 2 /( p « 0 « s \ a ) # 1/2 , where « 0 is the permittivity in vacuum, « s is the static dielectric constant of the host semi- conductor, and the electron charge is 2e . In the case that more subbands are occupied it was shown 1 that the intrasub- band plasmons, which arise additionally due to the higher occupied subbands, exist in gaps between the single-particle continua and are following free of Landau damping at T 50 K. These plasmon modes have a linear dispersion for small wave vectors. The lowest-frequency intersubband plasmon is investi- gated in detail and it was shown 8 for the case of one occu- pied subband that its frequency v p 10 5(1 1a ) 1/2 V 10 is differ- PHYSICAL REVIEW B 15 SEPTEMBER 1996-II VOLUME 54, NUMBER 12 54 0163-1829/96/54~12!/8652~24!/$10.00 8652 © 1996 The American Physical Society

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Page 1: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

Collective and single-particle excitations of the quasi-one-dimensional electron gasin the presence of a magnetic field

L. Wendler and V. G. GrigoryanFachbereich Physik, Martin-Luther-Universita¨t Halle, Friedemann-Bach-Platz 6, D-06108 Halle, Germany

~Received 12 September 1995; revised manuscript received 1 May 1996!

We investigate the collective and single-particle excitations of an electron gas, quantum confined in aquasi-one-dimensional quantum-well wire, in the presence of a magnetic field. The calculations are done in therandom-phase approximation with no further simplifying approximations. We derive analytical results for thedispersion relations of the intra- and intersubband magnetoplasmons. For large-magnetic fields the dispersioncurves of the intersubband magnetoplasmons approach the frequencies of the principal and Bernstein modes ofa two-dimensional electron gas, whereas the dispersion curve of the intrasubband magnetoplasmon tends tozero in this limit. It is shown that additional intersubband magnetoplasmons exist in comparison to a quasi-one-dimensional electron gas without a magnetic field and to a two-dimensional electron gas in the presence ofa magnetic field. These magnetoplasmons have the character of confined modes with a vanishing depolariza-tion shift for zero- and infinite-magnetic fields. With these modes it is possible to explain the resonance-splitting and fine-structure effects in the recently observed far-infrared transmission spectra. We interpret thecomplete spectrum of the quasi-one-dimensional magnetoplasmons by the hybridization of confined principaland Bernstein modes. This spectrum is universal, i.e., independent from the concrete shape of the lateralconfining potential.@S0163-1829~96!02335-1#

I. INTRODUCTION

Quantum-well wires~QWW! are semiconductor nano-structuresin which the carrier motion is only quasifree inone spatial direction, butquantum confinedin the two otherspatial directions by an artificial potential of width in theorder of the wavelength of the carriers~Fermi wavelength ina degenerate case!. The realization of such low-dimensionalsemiconductor structures nearly precise in atomic scale ispossible with modern epitaxial layer growth techniques andnanometer technologies.

The excitation spectrum of thequasi-one-dimensionalelectron gas~Q1DEG! of a QWW consists ofcollective ex-citationsandsingle-particle excitations, which become splitin intra- and intersubband excitationsdue to thesize quan-tization. Intrasubband excitations are connected with electronmotions within one electric subband, whereas intersubbandexcitations are connected with electron transitions betweenstates of two different subbands. Thesingle-particle excita-tions ~SPE’s! have a continuous spectrum and occur in defi-nite regions of the frequency-wave vector plane. The collec-tive charge-density excitations~CDE’s! of a Q1DEG in thepresence of a magnetic field, theintra- and intersubbandmagnetoplasmons, have, in general, dispersion relations out-side the single-particle continua and hence, are free of Lan-dau damping. Additionally,spin-density excitations~SDE’s!occur due to exchange and correlation interaction. This ex-citation spectrum of an electron gas depends characteristi-cally on the dimensionality of the system. Further, with thecharge density many related properties of the physical sys-tem can be varied: Fermi energy, Fermi wave vector, etc.This allows a detailed investigation of the different mecha-nisms that determines the plasma resonances itself and theinteraction of plasmons with different types of other collec-

tive excitations, e.g., with optical phonons in polar semicon-ductors and acoustic phonons in piezoelectric semiconduc-tors. Hence, the spectrum of the plasmons contains muchphysical information about the low-dimensional many-particle system.

Q1D plasmons have been investigated theoreti-cally1–9,11–39and experimentally40–54in isolated QWW’s andlateral multiwire superlattices. Most of the theoretical workswere done using therandom-phase approximation~RPA! tocalculate the linear response to an external charge neglectingretardation1–4,8–15,19–39and to an external current includingretardation.30 Q1D plasmons were also investigated with thesemiclassical hydrodynamic model.5–7,16

It was shown25,34 that the intrasubband plasmon of aQWW, assuming a two-subband model from which one isoccupied, has a long-wavelength dispersion relation accord-ing vp

005uqua@(vF /a)22vs

2ln(uqua)#1/2, whereq is the one-dimensional wave vector along the wire axis. The constanta depends on the wire size and is equal to the width of arectangular potential, the radius of a cylindrical potential orto the effective width of a parabolic potential. Further,vF isthe Fermi velocity in the one occupied subband andvs5@vFe

2/(p«0«s\a)#1/2, where «0 is the permittivity in

vacuum,«s is the static dielectric constant of the host semi-conductor, and the electron charge is2e. In the case thatmore subbands are occupied it was shown1 that the intrasub-band plasmons, which arise additionally due to the higheroccupied subbands, exist ingapsbetween the single-particlecontinua and are following free of Landau damping atT50 K. These plasmon modes have a linear dispersion forsmall wave vectors.

The lowest-frequency intersubband plasmon is investi-gated in detail and it was shown8 for the case of one occu-pied subband that its frequencyvp

105(11a)1/2V10 is differ-

PHYSICAL REVIEW B 15 SEPTEMBER 1996-IIVOLUME 54, NUMBER 12

540163-1829/96/54~12!/8652~24!/$10.00 8652 © 1996 The American Physical Society

Page 2: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

ent from the corresponding frequencyV10 of the single-electron transition. This many-particle effect is caused byresonance screening and exchange-correlation effects. Thefirst effect is described in the RPA, whereas the second isbeyond this approximation. The frequency shift, which iscaused by the resonance screening (a.0), is called thede-polarization shift. It was shown in Refs. 18,20 for the case ofmore than one occupied subband thatnew additionalbranches of intersubband plasmonsexist in that regions ofthe frequency-wave vector plane, which areopened insidethe single-particle intersubband continua. Hence, these plas-mons are free of Landau damping. One of the additionalmodes was detected by Goni et al.47 performing a resonantinelastic light scattering measurement on a QWW with twooccupied subbands.

The effect of a quantizing magnetic field on intra- andintersubband plasmons has been extensively theoreticallyinvestigated.6,7,9,12,13,22,23,26,31,33,35,38,39It is shown6,7,22,35thatthe intrasubband magnetoplasmon has a negative dispersionwith increasing magnetic field, known fromedge modesof aspatially confined Q2DEG.55 The quantum-mechanical re-sponse theory of plasmons in the presence of a quantizingmagnetic field is complicated by the fact that in this case theelectron wave function depending on the coordinate of theconfinement direction, also depends on the wave-vectorcomponent of the free propagation direction via the centercoordinate of the wave function. If one would proceed in thestandard manner of linear-response theory to derive the dis-persion relation of the Q1D magnetoplasmons, one wouldhave to solve an infinite-dimensional secular equation ac-cording to this wave-vector component. To solve this prob-lem, Zhaoet al.13 used approximations in the response treat-ment which deal with the center coordinate inconsistently. Liand Das Sarma22 expanded the wave function in powers ofthe center coordinate. Unfortunately, this expansion is notvalid in the range of parameters, used inexperiments.42,46,48,49,51In Ref. 22, two different dispersioncurves of Q1D magnetoplasmons were found: the intrasub-band magnetoplasmonvmp

00 showing edge-mode behaviorand the intersubband modevmp

10 which goes tovp10 for

B→0 and approaches the cyclotron frequencyvc5eB/me(me : effective conduction-band-edge mass! for B→`. Onthe other hand Eliassonet al.6 investigated the intersubbandmagnetoplasmons in the framework of a quasiclassicalmodel neglecting thenonlocalityof the Q1DEG. The result-ing modes have the correctB50 limit, i.e., coincidence withthe intersubband plasmonsvp

10,vp20 . . . . But all the disper-

sion curves approach for large-magnetic fields, i.e., in the 2Dlimit, the cyclotron frequencyvc . This is a result of theneglection of nonlocality in the Q1DEG, i.e., the higher reso-nances of a 2D magnetoplasma occurring at multiples of thecyclotron frequency, the Bernstein modes, are absent withoutspatial dispersion. Hence, the up to now developed theoriesgive quite different results for the Q1D magnetoplasmons. InRef. 35, a different method was developed to calculate thedispersion curves of the Q1D magnetoplasmons for all rel-evant parameters: the displaced wave function is representedin terms of the closure set of undisplaced wave functions,i.e., a different basis of the corresponding Hilbert space isused. This representation has the advantage of being free ofany small parameter and hence, is valid for such physical

parameters typical for real QWW used in experiments, aslong as the RPA of the Q1D Fermi liquid model is valid.Using the full nonlocal RPA response it was shown35 that theintersubband magnetoplasmons approach for large-magneticfields multiples of the cyclotron frequency and coincidenceat B50 with the intersubband plasmons. Thus, the magne-toplasmons obtained in Ref. 35 show both limits correctly.

Contrary to the theoretical work,far-infrared ~FIR! opti-cal transmission experimentson QWW’s with applied mag-netic field show a more complex structure of resonances. Incomparison to the 2D case, two unexpected effects are ob-served:~i! a resonance splitting~anticrossing! of one inter-subband mode at the line 2vc ~Ref. @48#! and ~ii ! for largerelectron densities a very pronounced splitting of the lowest-frequency intersubband mode into several branches in thetransition region from size to magnetic quantization.48,51Thefirst experimental finding gives a mode which is for small-magnetic fields a (320) intersubband magnetoplasmonvmp30 but approaches for large-magnetic fields 2vc , i.e., has

the character of the first 2D Bernstein mode. The authorsinterpret this resonance phenomena as the anticrossing be-havior of the (320) intersubband magnetoplasmonvmp

30 ,with the second harmonic of the cyclotron resonance 2vc .But this picture is not consistent, because the modevmp

30 is anormal mode of the Q1DEG, whereas the Bernstein mode at2vc is a normal mode of the 2DEG and not of the Q1DEG.For the second phenomenon mentioned above the authors ofRefs. 48,51 assumed that it could be explained by interwireCoulomb interaction~Ref. 48! and the effect of nonparabo-licities of the lateral confining potential~Ref. 51! ~in theexperiments not a single wire but many QWW’s in lateraldirection are used!. It becomes obvious that the up to nowdeveloped theories6,22,26 fail to explain these two detectedphenomena. In Ref. 35 the existence of additional modes ismentioned, showing a quite different behaviour than thatmodes which are to be expected and which are discussed inRef. 35. Consequently, the question arises whether these ad-ditional modes are responsible for the unexpected experi-mental findings and whether they complete the mode spec-trum of Q1D magnetoplasmons. A simple gedankenexperiment shows that the modes discussed in detail in Ref.35 cannot form the complete mode spectrum. This is truebecause the Q1D modes have to approach the 2D limit forlarge magnetic fields and thus, they have to approach mul-tiples of the cyclotron frequency. In other words, in this limitthe Q1D magnetoplasmon modes have to go over into theprincipal mode and the Bernstein modes of a 2DEG. Whenreducing the magnetic field each of these modes should splitinto several branches, due the quantizing action of the lateralconfining potential. Therefore, it is necessary to investigateall appearing modes in more detail.

The aim of this paper is to give a detailed investigation ofthe magnetoplasmons and the single-particle excitations of aQWW for the case ofoneandtwooccupied subbands includ-ing nonlocality and size quantization. The case of two occu-pied subbands is important, because the QWW’s used in themost experiments performed up to now have for zero-magnetic field a few, typically between 1 and 10, occupiedsubbands, which become depopulated with increasing mag-netic field. The theory is based on the RPA of the Fermiliquid theory of a Q1DEG. It is well known that a~strict!

54 8653COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 3: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

1DEG forms a Luttinger liquid,56,57 because the 1D Fermisystem becomes in this case a singular, strongly correlatedmany-particle system. But the up to now performedexperiments40–54 on real QWW’s and theoreticalinvestigations25 show that the Q1DEG is quantitatively welldescribed by the RPA, i.e., in the lowest order of Feynman-Dyson perturbation series of a Q1D Fermi liquid. In thispaper, we calculate the dispersion relation without any fur-ther approximation on the RPA. Hence, our theory is validfor all wave vectors, magnetic-field strengths, and electrondensities as long as the underlying many-particle concept,the RPA of the Q1D Fermi liquid model, is valid. The elec-tronic ground state is considered in Sec. II. The density re-sponse of a Q1DEG in the presence of a quantizing magneticfield is developed in Sec. III, using the representation of thedisplaced wave function by the undisplaced one. The intra-and intersubband SPE’s and magnetoplasmons are discussedin detail in Secs. IV and V, respectively. In Sec. VI, weapply the obtained results to discuss the recent far-infraredtransmission experiments on QWW’s and give our conclu-sions in Sec. VII.

II. GROUND STATE

The model of the QWW used in this paper is the follow-ing. We assume that the electrons are confined by aneffec-tive potential Veff(x)5Veff(y)1Veff(z), whereVeff(z) con-fines the electron motion along thez direction in a zero-thicknessx-y plane atz50. In they direction the electronmotion is quantum confined by an effective potential, as-sumed to be parabolic:Veff(y)5meV

2y2/2, whereV is theconfining frequency. The unperturbed single-particle Hamil-tonian in the effective-mass approximation for the electronsof the Q1DEG in the presence of a homogeneous quantizingmagnetic fieldB5(0,0,B) is given by

H051

2me~pe1eA!21Veff~x!, ~1!

where we ignore the Zeeman spin splitting, i.e., we shallomit the spin quantum number and coordinate, but assumethat the spin summation is included when necessary withoutany explicit indication. Here,pe5(\/ i )¹ is the momentumoperator of the electron with charge2e andA is the vectorpotential of the magnetic fieldB5¹3A. It is noticeable thatthe use of the effective potential,Veff(x) in the Hamiltonianof Eq. ~1!, which is for an interacting many-particle systemthe sum of a bare potentialV0(x) and the Hartree potentialVH(x) if we ignore exchange and correlation effects, makesH0 effectively to the Hartree Hamiltonian, assuming that theHartree potential is always calculated and known. With theuse of the Landau gaugeA5(2yB,0,0) for the vector po-tential, the single-particle Schro¨dinger equation of theHamiltonian given in Eq.~1! reads

F pe22me2vcypex1

me

2vc2y21Veff~z!GCNkx

~x!

5EN~kx!CNkx~x!, ~2!

where pex is the x component of the electron momentum

operator andvc5(vc21V2)1/2 is the hybrid frequency. The

eigenvalue problem becomes equivalent to two separateequations, one for the size quantization inz direction and onefor the mixed size and magnetic quantization iny direction.It is well known58 that the single-particle Schro¨dinger equa-tion for the electron motion iny direction is exactly solvablewith the shifted harmonic-oscillator wave function

^xuNkx&5CNkx~x!5

1

ALxeikxxFN~y2Ykx

!w~z!, ~3!

where

FN~y2Ykx!5

1

A2NN!p1/2l 0expF2

1

2l 02 ~y2Ykx

!2G3HNF 1l 0 ~y2Ykx

!G . ~4!

The associated energy eigenvalues are given by the subbands~hybrid levels!,

EN~kx!5EN1\2kx

2

2me, ~5!

where the subband bottoms are

EN5\vc~N1 12 !, N50,1,2, . . . . ~6!

In the equations above,Lx is the length of the QWW in thex direction ~we assume Born–von Ka´rman periodic bound-ary conditions!, kx is the electron wave-vector component inthe x direction, andYkx

5g l 02kx is the center coordinate,

where l 05@\/(mevc)#1/2 is the typical width of the wave

function in y direction. For vanishing confining potential(V50), l 0 results in the magnetic lengthl 05(\/eB)1/2. Fur-ther, we have definedg5vc /vc , HN(y) is Hermite’s poly-nomial,me5me(vc /V)2 is the renormalized magnetic-field-dependent mass, anduw(z)u25d(z). The subbandsdegenerate forB→` to Landau levelsEN5\vc(N11/2)and for B→0 the electric subbands EN(kx)5\V(N11/2)1\2kx

2/(2me) result. It is seen from Eq.~4!that in the presence of a quantizing magnetic field, applied inz direction, the wave function describing the electron motionin y direction depends on the wave-vector componentkx ,representing the fact that the magnetic field couples the elec-tron motion in thex2y plane.

Because forBÞ0 the time-reversal symmetry is broken,generally the density of states%(E) is different for 6kx :%1(E)Þ%2(E). However, for the parabolic confining poten-tial EN(kx)5EN(2kx) is valid with the result that%(E)52%1(E)52%2(E), where

%~E!5Lxp

A2me

\ (N

Q~E2EN!

~E2EN!1/2. ~7!

In Eq. ~7!, Q(x) is the Heaviside unit step function,Q(x)51 for x.0 andQ(x)50 for x,0. The Fermi energyEF follows from Eq.~7! and is obtained atT50 K from thecondition

8654 54L. WENDLER AND V. G. GRIGORYAN

Page 4: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

n1DEG54

p\Ame

2 (N

AEF2ENQ~EF2EN!, ~8!

wheren1DEG is the 1D electron number density~number ofelectrons per unit length!. This equation defines the Fermiwave vectors,

kF~N!5HA2me~EF2EN!/\2 if EF.EN

0 if EF<EN~9!

and the Fermi velocitiesvF(N)5\kF

(N)/me of the different sub-bandsEN(kx).

III. DENSITY RESPONSE OF A Q1DEG

In this section, we calculate the linear response of aQ1DEG to an external potential on a quantum-mechanicallevel within the RPA. There are different methods to developthe RPA. Here, we use theself-consistent field~SCF! methodof Ehrenreich und Cohen.59 The single-particle Hamiltonianof the electrons of the Q1DEG in the presence of the pertur-bation is written asH(x,t)5H0(x)1H1(x,t), whereH0 isthe unperturbed Hamiltonian of a single-electron, quantumconfined in the QWW, satisfyingH0uNkx&5EN(kx)uNkx&andH15Vsc(x,t) is theself-consistent potential, which is asum of theexternal potential Vext(x,t) and theinduced po-tential Vind(x,t). The single-particle von Neumann equationi\(]/]t)rG5@H,rG# for the statistical operatorrG of thegrand-canonical ensemble, describing the response of thesystem to the self-consistent potential, can belinearizedwithrG5rG

(0)1rG(1) , whererG

(0) is the single-particle statisticaloperator of the unperturbed system, andrG

(1) is the correctionto the statistical operator to the first order in the perturbation.The external potential is adiabatically switched on att52`, giving Vext(x,t)5Vext(x,v)exp@2i(v1id)t# withd→01. This time dependence is assumed for all the poten-tials. The steady-state properties of the system are evaluatedat t50.

The total electron number densityn(x,v)5n0(x)1nind(x,v) of the Q1DEG, wheren0(x)5Tr$rG

(0)d(x2xe)% is the ground-state electron number density in equi-librium, is given by n(x,v)5Tr$rGd(x2xe)%, and thetrace~Tr! is evaluated in the grand-canonical ensemble. Forthe induced electron number density,nind(x,v)5Tr$rG

(1)d(x2xe)%, it follows

nind~x,v!5E d3x8P~1!~x,x8uv!Vsc~x8,v!, ~10!

whereP(1)(x,x8uv) is the irreducible (proper)RPA polar-ization functionof the Q1D magnetoplasma:

P~1!~x,x8uv!51

Lx(qx

(N,N8

(kx

eiqx~x2x8!

3PNN8~1!

~qx ,kx ;x' ,x'8 uv!. ~11!

Herein, we have

PNN8~1!

~qx ,kx ;x' ,x'8 uv!

5PNN8~1!

~qx ,kxuv!FN~y2Ykx1qx!

3FN* ~y82Ykx1qx!FN8~y82Ykx

!

3FN8* ~y2Ykx

!d~z!d~z8!, ~12!

with thematrix polarization function

PNN8~1!

~qx ,kxuv!52

Lx

nF„EN8~kx!…2nF„EN~kx1qx!…

\~v1 id!1EN8~kx!2EN~kx1qx!,

~13!

where nF(En)5Q(EF2En) is the Fermi distribution func-tion atT50 K and we have definedx'5(0,y,z).

In the RPA and neglecting retardation effects the induceddensity is related to the induced potential by Poisson’s equa-tion:

@¹x'•«s~x'!¹x'

2«s~x'!qx2#Vind~qx ;x'uv!

52e2

«0nind~qx ;x'uv!. ~14!

In this equation«s(x') is the static dielectric function of thesemiconductor background, arising from the high-energyelectronic excitations across the band gap and the opticalphonons. Because QWW’s are usually realized from semi-conductors which are polar and piezoelectric, optical andacoustic phonons are present. For typical QWW’s, realizedup to now, the frequencies of the acoustic phonons are muchsmaller than the typical magnetoplasmon frequencies, butthose of the optical phonons are much higher than the typicalfrequencies of the lowest-lying magnetoplasma modes.Hence, the acoustic phonons have a vanishing influence onthe magnetoplasmons, but the optical phonons screen the fre-quencies of the magnetoplasmons. On grounds of this physi-cal situation it is possible to neglect the influence of theacoustic phonons but use the«s approximation for thescreening of the dielectric background, which includes theoptical phonons to respond instantaneously. Because theself-consistent potentialVsc(qx ;x'uv) is composed of theexternal potentialVext(qx ;x'uv) and the induced potentialVind(qx ;x'uv), it follows from Eq. ~14!,

Vsc~qx ;x'uv!5Vext~qx ;x'uv!1e2

2p«0«sE d2x'8 E d2x'9

3K0@ uqx~x'2x'8 !u#

3P~1!~qx ;x'8 ,x'9 uv!Vsc~qx ;x'9 uv!, ~15!

where we have neglectedimage effectsby solving Poisson’sequation. In Eq.~15!, K0(y) is the modified Bessel functionof zeroth order. Performing matrix elements of this equationwith the wave function of Eq.~2! and assuming that thecollective excitations of the Q1D magnetoplasma exist underthe condition thatVscÞ0, while Vext50, the existence con-dition for collective excitations reads

54 8655COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 5: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

(N,N8

(kx

F dN1NdN2N8dkxkx12qx2V

sN1

kx1

N2

kx12qx

N8kx

Nkx1qx

~qx!P

~1!

Nkx1qx

N8kx

~qx,v!GV SCN

kx1qx

N8kx

~qx,v!50, ~16!

where

V

sN1

kx1

N2

kx12qx

N8kx

Nkx1qx

~qx!5e2

2p«0«sE

2`

`

dyE2`

`

dy8FN1* ~y2Ykx1

!FN2~y2Ykx12qx

!K0@ uqx~y2y8!u#

3FN8* ~y82Ykx

!FN~y82Ykx1qx! ~17!

and

V

scN

kx1qx

N8kx

~qx ,v!5E2`

`

dyFN* ~y2Ykx1qx!Vsc~qx ;yuv!FN8~y2Ykx

!. ~18!

Further, we have definedPNkx1qx

N8kx

(1)(qx ,v)[PNN8

(1) (qx ,kxuv). The corresponding secular equation

detF dN1NdN2N8dkxkx12qx2V

sN1

kx1

N2

kx12qx

N8kx

Nkx1qx

~qx!P

~1!

Nkx1qx

N8kx

~qx ,v!G50 ~19!

is the dispersion relationof the Q1D magnetoplasmons if

ImPNkx1qx

N8kx

(1)(qx ,v)50. Analyzing Eqs.~16! to ~19!, it fol-

lowsv(qx)52v(2qx), independent from the symmetry ofthe energy dispersionEN(kx). But, here, for the valid case ofEN(kx)5EN(2kx), it follows uv(qx)u5uv(2qx)u and hence,we can restrict to the quadrantqx.0 and v.0 of thev2qx plane. It is seen that the algebraic system of equa-tions, Eq.~16!, consists of an infinite number of equations,resulting from the different wave vectorskx52pnx /Lx ;nx50,61,62, . . . and an infinite number of equations re-sulting from N and N8, which are ~formally! infinite formodel confining potentials with infinite high barriers. Fortu-nately, the number of equations resulting from the subbandquantum numbers is restricted to a small numberN,N850,1,2, . . . ,M21. First, the matrix polarization func-

tion has the propertyPNkx1qx

N8kx

(1)(qx ,v)→0 for uN2N8u→`.

Second, in the typical QWW’s there are only a few subbands

occupied and becausePNkx1qx

N8kx

(1)(qx ,v)50 if both inequali-

ties EF<EN andEF<EN8 are valid, it is possible to restrictthe calculation to a finite numberM of subbands, i.e., to useamultisubband model, respectively, forM52,3 the two- orthree-subband model with a high accuracy of the results. Butthe remaining infinite number of equations, Eq.~16!, accord-ing to the number of differentkx values results in an infinite-dimensional secular equation~19!, which is impossible to

solve. Only atB50 and forB→`, VNkx1qx

N8kx

sc(qx ,v) is inde-

pendent ofkx and VN1kx1

N2kx12qx

N8kx

Nkx1qx

s(qx) is independent of

kx andkx1. This allows us to perform the sum overkx in Eq.

~16!: (kxPNkx1qx

N8kx

(1)(qx ,v)5PNN8

(1) (qx ,v). Thus, only in this

case it is possible to solve the secular equation~19!. Weobtain an infinite number of identical algebraic equations,i.e., for B50 andB→` the problem becomes degenerateaccordingkx .

To solve this problem for nonvanishing magnetic fieldsZhao et al.13 used approximations, which finally results inthe neglection of the influence of the center coordinate andhence, the results for the dispersion relations of theQ1D magnetoplasmons seems to be incorrect. Going further,Li and Das Sarma22 proposed a perturbation scheme,the a expansion FN(y2Ykx

)5FN(y)2F8(y)akx1FN9(y)a

2kx2/21O(a3), wherea5Ykx

/kx . This expan-

sion converges ifakx / l 0!1. Because the largestkx of anelectron in the Q1D Fermi sea with one occupied subband iskx5kF

(0) the small physical parameter isakF(0)/ l 0. Thus, this

expansion converges only ifakF(0)/ l 0!1 and following, the

a-expansion scheme is restricted to three different ranges:~i!to very small electron densities, or~ii ! to weak magneticfields, or ~iii ! to strong magnetic fields. Unfortunately, thisexpansion is not valid in the range of parameters, used inexperiments, e.g., Refs. 42,46,48,49, and 51, it breaks downfor typical QWW’s, e.g., for \V52 meV andn1DEG513106 cm21 betweenB50.3 T andB520 T. Fur-ther, the use of the power expansion forFN(y2Ykx

) in Eq.~16! strictly restricts the results for the dispersion relation ofthe Q1D magnetoplasmons on the used order of this expan-sion. In Ref. 22, three terms are used. To include for exampleone further term in the expansion or one further subband in

8656 54L. WENDLER AND V. G. GRIGORYAN

Page 6: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

the consideration, it becomes necessary to perform all thecalculations from the very beginning.

A different approach was developed in our recent paper.35

In this paper, the shifted harmonic-oscillator wave functionis represented by the closure set of undisplaced wave func-tions $FN(y)%:

FN~y2Ykx!5 (

L50

`

CLN~Ykx!FL~y!, ~20!

where

CLN~Ykx!5AN!

L!2~N2L !/2e2~jkx /2!2~jkx!

L2NLNL2NF ~jkx!

2

2 G~21!

for L>N and CLN(Ykx)5@CNL(2Ykx

)#* for L,N with

j5g l 0, andLNN8(x) is the Laguerre polynomial.

Performing matrix elements of Eq.~15! with the wavefunctionFN(y2Ykx

) in the above representation, we obtaininstead of Eq.~16!

(L,L8

$dLL1dL8L22LL1L2LL8~qx ,v!%VLL8sc

~qx ,v!50

~22!

for the existence condition of collective excitations. Herein,we have defined

VLL8sc

~qx ,v!5E2`

`

dyFL* ~y!Vsc~qx ,yuv!FL8~y!,

~23!

LL1L2LL8~qx ,v!5 (L3L4

VL1L2L3L4s ~qx!PL3L4LL8

~1!~qx ,v!,

~24!

where

PL3L4LL8~1!

~qx ,v!5 (NN8

(kx

PNN8~1!

~qx ,kxuv!

3CL3N8* ~Ykx

!CL4N~Ykx1qx

!

3CLN* ~Ykx1qx!CL8N8~Ykx

! ~25!

and

VL1L2L3L4s ~qx!5

e2

2p«0«sE

2`

`

dyE2`

`

dy8FL1* ~y!FL2

~y!

3K0@ uqx~y2y8!u#FL3* ~y8!FL4

~y8!.

~26!

The matrix elements of the Coulomb interaction potentialVL1L2L3L4s (qx) are explicitly given in Appendix A. It is seen

that the symmetry relations PL1L2L3L4(1) (qx ,v)

5PL4L2L3L1(1) (qx ,v)5PL1L3L2L4

(1) (qx ,v), PL1L2L3L4(1) (2qx ,v)

5(21)L11L21L31L4PL1L2L3L4(1) (qx ,v), and PL1L2L3L4

(1) (qx ,

2v)5(21)L11L21L31L4@PL2L1L4L3(1) (qx ,v)#* are valid. In

the case thatVeff(y)5Veff(2y), the parity of the single-particle states FL(y) results in VL1L2L3L4

s (qx)50 if

(L11L21L31L4) is an odd number. Further, for boundelectrons under consideration the envelope wave functionsare real and hence, we haveVL1L2L3L4

s (qx)

5VL2L1L3L4s (qx)5VL1L2L4L3

s (qx)5VL3L4L1L2s (qx) and VLL8

sc

5VL8Lsc (qx,v). Using the resulting symmetry relations

LL1L2LL8(qx ,v)5LL2L1LL8(qx ,v) and VLL8sc (qx ,v)

5VL8Lsc (qx ,v) , we finally obtain thedispersion relation of

the Q1D magnetoplasmonsin the form

det@dLL1dL8L22JL1L2LL8~qx ,v!#50, ~27!

where

JL1L2LL8~qx ,v!5LL1L2LL8~qx ,v!1LL2L1L8L~qx ,v!

11dLL8~28!

and thus, in Eq.~27! L8<L is valid. With the symmetrypropertyJL1L2L3L4

(qx ,v)5JL1L2L4L3(qx ,v) it follows the

caseL8.L, which gives the identical results.Let us consider two special cases. For thevanishing mag-

netic field(B50, anyqx) we haveCLN(0)5dLN in Eq. ~20!and hence, we can changeL by N with the result

LN1N2NN8~qx ,v!5VN1N2N8Ns

~qx!PNN8~1!

~qx ,v!, ~29!

whereVN1N2NN8s (qx) is given by Eq.~26! andPNN8

(1) (qx ,v)

5(kxPNN8(1) (qx ,kxuv) 5RePNN8

(1) (qx ,v) 1 i ImPNN8(1) (qx ,v)

reads20

RePNN8~1!

~qx ,v!52me

p\2qxH lnU kF~N8!1qx2

2me

\qx~v2VNN8!

kF~N8!2

qx2

1me

\qx~v2VNN8!

U1 lnU kF~N!1qx2

1me

\qx~v2VNN8!

kF~N!2

qx2

2me

\qx~v2VNN8!

UJ ~30!

and

ImPNN8~1!

~qx ,v!52me

\2uqxuH QFkF~N8!1

qx2

2me

\qx~v2VNN8!GQFkF~N8!2

qx2

1me

\qx~v2VNN8!G

2QFkF~N!1qx2

1me

\qx~v2VNN8!GQFkF~N!2

qx2

2me

\qx~v2VNN8!G J , ~31!

54 8657COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 7: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

where VNN85(EN2EN8 )/\ is the subband separation fre-quency. Because of the symmetries given above, the systemof algebraic equations, Eq.~22! with Eq. ~29!, splits in twoseparate systems with the corresponding secular equations

det@dN1N12ndN2N2VN1N2N12nNs ~qx!xN12nN

~1! ~qx ,v!#50

~32!

and

det@dN1N1~2n11!dN2N2VN1N2N1~2n11!Ns ~qx!

3xN1~2n11!N~1! ~qx ,v!#50, ~33!

wheren50,1,2, . . . . Equation~32! is the dispersion rela-tion of the symmetricQ1D plasmons~even parity modes!connected with collective electron transitions between stateswith the same parity and Eq.~33! is the dispersion relation ofthe antisymmetric plasmons~odd parity modes! connectedwith collective electron transitions between states with oppo-site parity. Herein, we have defined

xNN8~1!

~qx ,v!5PNN8

~1!~qx ,v!1PN8N

~1!~qx ,v!

11dNN8. ~34!

The second special case concernsvanishing wave vector(qx50, any B). From Eqs.~21! and ~25!, it follows thatPL1L2L3L4(0,v)50 if (L11L21L31L4) is an odd number.

As mentioned above, the same is true forVL1L2L3L4s (qx). Us-

ing these properties in Eqs.~24! and ~28!, we obtain thatJL1L2L3L4

(0,v)Þ0 if (L11L21L31L4) is an even number

and JL1L2L3L4(0,v)50 if this number is odd. Hence, the

determinant in Eq.~27! becomes a determinant of a block-diagonal matrix, i.e., two independent determinantal equa-tions result:

det@dL1L812ndL2L82JL1L2L812nL8~qx ,v!#50, ~35!

and

det@dL1L81~2n11!dL2L82JL1L2L81~2n11!L8~qx ,v!#50.~36!

Performing the substitutiony→2y in Eq. ~23!, we obtainVsc(0,yuv)5Vsc(0,2yuv) if ( L1L8) is an even number.Because the modes are determined by the self-consistent po-tential in this case, Eq.~35! is valid and thus determinessymmetric excitation. On the other hand,Vsc(0,yuv)52Vsc(0,2yuv) if ( L1L8) is an odd number and follow-ing, Eq.~36! describesantisymmetric excitation. It is impor-tant to note that this symmetry results from the fact that forqx50 in the y direction forwards and backwards travelingwaves are equivalent. But this is not the case forqxÞ0.

IV. SINGLE-PARTICLE EXCITATIONS

The excitation spectrum of the Q1DEG in the presence ofa magnetic field consists ofcollective excitationsandsingle-particle excitations. Within the RPA, the collective excita-tions are magnetoplasmons arising from collectivecharge-densityoscillations.Spin-densityexcitations are beyond theRPA, because this scheme is restricted to the Hartree part ofthe electron-electron interaction~time-dependent Hartree ap-

proximation! neglecting exchange and correlation contribu-tions. The SPE’s are single-electron transitions from occu-pied states below the Fermi surface to empty states above it.Caused by the size quantization of the confining potential,present in semiconductor nanostructures, both collective ex-citations as well as SPE’s become split inintra- and inter-subbandexcitations. An intrasubband excitation is connectedwith electron transitions from occupied states below theFermi surface to empty states above it in the same subband,while for an intersubband excitation these transitions are be-tween different subbands. In general, the collective transi-tions are not independent from each other because the Cou-lomb interaction between the electrons couples thetransitions. Here, for the considered case of aparabolicQWW, all collective transitions become coupled for non-vanishing magnetic fields. Only in two cases,~i! for a van-ishing magnetic field and~ii ! for a vanishing wave vector,the spatial symmetry of the confining potential causes thesymmetricelectron transitionsN↔N85N12n, i.e., betweenstates with the same parity, to become independent from theantisymmetricones,N↔N85N1(2n11), i.e., from suchtransitions between states with opposite parity. Because theRPA consideres the response of noninteracting electrons onthe self-consistent potential, the single-particle electron tran-sitions are always independent, when described in the RPA,while the collective electron transitions are coupled withinthis approach through the Coulomb potential. The SPE’shave a continuous excitation spectrum and exist in regions ofthe v2qx plane, where ImPNN8

(1) (qx ,kxuv)Þ0 for all kx .These regions are plotted for a three-subband model(N,N850,1,2) in Figs. 1 and 2. The boundaries of thesecontinua are given by

v1,2NN85U6

\kF~N8!qx

me

1\qx

2

2me1vc~N2N8!U ~37!

and

v3,4NN85U6

\kF~N!qx

me

2\qx

2

2me1vc~N2N8!U . ~38!

For numerical work throughout this paper we have chosen aGaAs-Ga12xAl xAs QWW ~GaAs: «s512.87 andme50.066 24m0; m0: bare electron mass!, with a confiningenergy\V52 meV. In the case of one occupied subband,the three possible electron transitions 0↔0, 0↔1, and0↔2 form the continua shown in Fig. 1. It is important tonote that for a Q1D plasma (B50) and a Q1D magneto-plasma, no single-particle (020) intrasubband excitationsexist in the region 0,v,v2

00 in contrast to a Q2D plasma,where in this region also single-particle (020) intrasubbandexcitations occur.60 This nonexistance of low-energyelectron-hole pairs for 0,qx,2kF

(0) is the districtive featureof Q1D systems. It is seen from Fig. 1~b! that with increasingmagnetic field the widths of the continua decrease and de-generate for large-magnetic fields to single lines at multiples

of the cyclotron frequency:v1,2NN8(B→`)→(N2N8)vc .

Following, the Q1D magnetoplasma behaves like a 2D mag-netoplasma for large-magnetic fields because in this case themagnetic quantization of the electron motion in thex2y

8658 54L. WENDLER AND V. G. GRIGORYAN

Page 8: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

plane is dominant in comparison to the size quantization iny direction. In this limit the subbandsEN(kx) become Landaulevels EN , eacheBLxLy /2p\ - fold degenerate. The fre-quencies of the single-particle (020) intrasubband transi-tions go to zero for large-magnetic fields. This is a result ofthe vanishing dispersion of the subbands in this limit. Thecollective excitations discussed in detail in the next sectionexist on definite curves in thev2qx plane, the dispersionrelations. Outside the single-particle continua the magneto-plasmons are undamped within the used model~collisionlessQ1D magnetoplasma!, but become damped inside the single-particle continua by the resonant and collisionless mecha-nism of theLandau damping, i.e., the Q1D magnetoplas-mons are able to decay into single electron-hole pairs.

If two subbands are occupied for the three-subband modelthe transitions 0↔0, 0↔1, 0↔2, and additionally 1↔1,1↔2 are possible. The corresponding continua, where the

SPE’s exist, are plotted in Fig. 2. The regions of the SPE’sare compared for the cases of one occupied subband and twooccupied subbands in Figs. 2~a! and 2~b!. It is seen from Fig.2~b! that below the parabolav2

00 again regions between thesingle-particle (020) and (121) intrasubband continua inthev2qx plane exist, where no SPE’s occur and hence, noLandau damping of the collective excitations is possible.Looking to the single-particle transitions plotted in Fig. 2~a!and 2~b! it is seen that in the case of two occupied subbandsa region inside the single particle (120) intersubband con-tinuum isopenedin which SPE’s are forbidden and follow-ing this region is free of Landau damping. The dependenceof the single-particle continua from the magnetic field isplotted in Fig. 2~c!. It is seen that with increasing magneticfield the subbandE1(kx) becomes depopulated slightly above1.5 T with the result that the (121) and (221) continua areabsent above this magnetic-field strength. In this case alsothe gap region free of Landau damping within the single-particle (120) intersubband continuum disappears.

V. COLLECTIVE EXCITATIONS:Q1D MAGNETOPLASMONS

The dispersion relations of the collective excitations ofthe Q1D magnetoplasma are determined by Eq.~27! with theresulting dispersion curvesv5vmp(qx ;B) of the differentmodes, the Q1Dmagnetoplasmons. If the electron gas oscil-lates in such a mode (qxÞ0), in principle, all possible elec-tron transitionsN8↔N contribute for nonvanishing magneticfields. Only in adiagonal approximation, i.e., by neglectingthe coupling between the different transitions in thedispersion relation, each mode is connected with one typeof collective electron transitionN8↔N. The dispersionrelations of the Q1D plasmons (B50) in the diagonalapproximation follows from Eqs.~32! and ~33! byretaining only the elements VN1N2N8N

s (qx)

5dN1N8dN2NVN1N2N1N2s (qx) @which corresponds toxNN8

(1)

3(qx ,v)5dN1N8dN2NxN1N2(1) (qx ,v)#, with the result

12VNN8NN8s

~qx!xNN8~1!

~qx ,v!50. ~39!

Here, the denotation diagonal approximation becomes obvi-ous if one introduces combined indicesl5$N1N2% andk5$N N8%: 12Vkk

s xk(1)50. In the case of nonvanishing

magnetic fields, the diagonal approximation followsfrom Eq. ~27! if we consider in Eq.~25! each summandof the sum over N,N8 separately. This is achievedif one uses PNN8

(1) (qx ,v) 5@dN1NdN2N8PN1N2(1) (qx ,v)

1dN1N8dN2NPN2N1(1) (qx ,v)]/(11dN1N2) in Eq. ~25!. The in-

teraction between the different electron transitions, i.e., theintersubband coupling~ISC! beyond the diagonal approxi-mation, results in ahybrid-typespectrum of the modes.

As stated in Sec. III, the number of subbandsN,N8needed to solve the dispersion relation~27! is restricted to afinite number. In the following, we use a three-subbandmodel:N,N850, 1, 2. But in general, the order of the deter-minant det@d i j2J i j #50 with i5$L1L2%, j5$L L8% in Eq.~27! is infinite according to the combined indicesi and j . Tosolve this dispersion relation, it is necessary to restrict the

FIG. 1. Single-particle continua of a parabolic QWW~hatchedareas! in the v2qx ~a! and v2B ~b! plane for a three-subbandmodel if one subband is occupied:~a! n1DEG533105 cm21,B51 T; ~b! n1DEG533105 cm21, qx523105 cm21.

54 8659COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 9: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

representation of the displaced wave functionsFN(y2Ykx)

to a finite numberL1 ,L2 ,L,L850,1, . . . ,Lmax of terms. Wetake into account so many terms in Eq.~20! that the accuracyof theobtained resultsis better than 1%. Because the sum inEq. ~20! converges rapidly, the number of needed terms issmall; e.g., at n1DEG533105cm21: Lmax53, atn1DEG583105cm21: Lmax56, and at n1DEG51.23106cm21: Lmax59. The calculation of the determi-nantal equation~27! is very complicated due to the four-index structure of the elements. One has to transform thisstructure to the usual two-index structure by the followingrelation to the combined indicesi5$L1L2% and j5$L L8%:for Lmax51: 15$0 0%, 25$1 0%, 35$11%; for Lmax52: 1 5$0 0%, 2 5$10%, 3 5$2 0%, 4 5$11%, 55$21%, 65$22%, and thus, generally, it followsi5L2@2(Lmax11)2(L221)]/21(L12L2)11 and j5L8@2(Lmax11)2(L821)]/21(L2L8)11. The order of theresulting determinantal dispersion relation is (Lmax11)3(Lmax12)/2.

A. One-subband occupied

1. Numerical calculations

In the three-subband model assuming at first that only onesubband is occupied, i.e., in theelectric quantum limit~EQL!, the dispersion relationsvmp(qx ;B) of the Q1D mag-netoplasmons are given by Eq.~27!. The full RPA dispersionrelations of the magnetoplasmons including ISC in depen-dence on the wave vector are plotted for a parabolic QWWand for different magnetic-field strengths and electron densi-ties in Figs. 3~a!–3~c!. In the case of vanishing magneticfield @Fig. 3~a!#, three different branches of dispersion curvesresult from Eqs.~32! and ~33!. There are two symmetricplasmon modes, denoted byvp

00 andvp20. Both modes are

accompanied with the collective electron transitions 0↔0and 0↔2 and are independent from the transition 0↔1.This collective electron transition results in the antisymmet-ric plasmon modevp

10. The plasmon modevp00 is not

damped within the single-particle (120) intersubband con-tinuum andvp

10 not within the (220) SPE continuum. Thedifference between the single-particle intersubband transitionfrequencyNV and the collective one,vp

N0(qx50), is in theframework of the RPA the well-knowndepolarization shift.The depolarization shift results from the change of the Har-tree potential due to the nonzero induced electron densitywhen a collective excitation is excited and this shift is apolarization effect~resonance screening!. In addition to thesolid lines, the dashed lines represent the dispersion curvesof the (020) intra-, (120), and (220) intersubband plas-mons calculated within the diagonal approximation. In thisapproximation the electron motions oscillating in the(N20) mode are pure collective transitions between thelowest subband and theNth subband. The coupling betweenthe different transitions results in the hybrid spectrum ofplasmons, e.g., themixed(020)2(220) plasmons, whichhave two differentbranchesof dispersion curves. One is the(020) intrasubband plasmonlike branchvp

00 and the otherone is the (220) intersubband plasmonlike branchvp

20. Itis profitable atB50 to denote the different branches startingfrom the picture of independent electron transitions

FIG. 2. Single-particle continua of a parabolic QWW~hatchedareas! in thev2qx plane calculated for a three-subband model.~a!one subband is occupied:n1DEG533105 cm21, B51T; ~b! twosubbands are occupied:n1DEG583105 cm21, B51T. The corre-sponding regions of~b! in the v2B plane are plotted in~c! forqx523105 cm21.

8660 54L. WENDLER AND V. G. GRIGORYAN

Page 10: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

(N8↔N) in the diagonal approximation:vpNN8, i.e., counting

the main contributing electron transition to this hybrid modein the name of the branch and omitting the word ‘‘like’’ forsimplicity. In general, such a denotation makes sense if theISC is weak. It is important to note that the word mixed~hybrid or coupled! modes only results from the point ofview of the diagonal approximation. Each branch of the Q1Dplasmon and magnetoplasmon spectrum represents an inde-pendent excited state of the system. It is seen that the cou-pling between the transitions results in arepulsing of thedispersion curvesvp

00 andvp20.

The case of a nonvanishing magnetic field is plotted inFig. 3~b!. It is obvious from comparing the dispersion curvesof the hybrid modes including ISC with that calculated in thediagonal approximation, that the coupling strength betweenthe electron transitions becomes larger for nonvanishingmagnetic fields, and hence, the mode mixing effect increasesin magnitude. Now the (120) mode hybridizes with theother modes, i.e., the solid and dashed line~denoted byvmp10;1) become quite different forBÞ0. Because for non-

vanishing magnetic fields all electron transitions betweendifferent subbands become coupled, the dispersion curvevmp00 inside the (120) continuum andvmp

10;1 inside the(220) continuum become Landau damped. In this case themodes lose there meaning in the strict sense of the definitionof a mode as the resonant response to the perturbation in theform of a coherent motion of all electrons. Following, theirdispersion curves are not plotted inside the SPE continua.They arise as broad peaks in a suitable response function,e.g., in the dynamic structure factorS(q,v) of the Q1DEG.61

Only such an investigation gives the answer to the questionwhether the collective mode is better interpreted:~i! as sepa-rate collective excitation of the system or~ii ! as a weak reso-nance in the SPE’s spectrum.61 It is seen that forB51 T wehave one (020) intrasubband magnetoplasmon branchvmp00 , but three (120) intersubband magnetoplasmon

branchesvmp10;1, vmp

10;2, vmp10;3, and also three (220) intersub-

band magnetoplasmon branchesvmp20;1, vmp

20;2, and vmp20;3.

Only the modesvmp00 , vmp

10;1, and vmp20;1 survive the limit

B→0.We call the modesvmp

N0;1 of a Q1DEG with one occupiedsubband, which exist forB50, the fundamental modes~FM’s! and the modesvmp

N0; j.1 theadditional modes~AM’s !.We denote each group of modesvmp

N0; j ~FM’s and AM’s! bythe pairN0, corresponding to the electron transition 0↔N ofthat SPE continuum to which these modes tend towards withincreasing wave vector and magnetic field.

We denote the different AM’s byj : at larger magneticfields the AM’s have with increasingj a decreasing depo-larization shift. These modes are characterized by dispersioncurves which have for a given magnetic field astop pointqsN0; j5qs

N0; j (B) at the upper boundaryv1N0 of the corre-

sponding single-particle continuum. Hence, forqx.qs

N0; j (B) these modes become Landau damped. Furtherstop points may occur at the boundaries of other single-particle continua. It is seen from Fig. 3~c! that for a largerelectron density also the number of ‘‘resolvable’’ AM’s in-creases. Thus, dependent on electron concentration,magnetic-field strength, and wave vector, adifferentnumberof dispersion curves of intersubband magnetoplasmons ex-

FIG. 3. Dispersion relation of the magnetoplasmons in depen-dence on the wave-vector componentqx calculated in a RPA~in-cluding ISC: heavy solid lines; diagonal approximation: dashedlines! of a parabolic QWW for a three-subband model, where onesubband is occupied:~a! B50 T, n1DEG533105 cm21; ~b! B51T, n1DEG533105 cm21; ~c! B51T, n1DEG55.53105 cm21. Thehatched areas correspond to the single-particle intra- and intersub-band continua.

54 8661COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 11: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

ists, but always only one dispersion curve of the intrasub-band magnetoplasmon.

The full RPA dispersion curves of the mixed(020)2(120)2(220) magnetoplasmons in thev2Bplane are plotted in Figs. 4~a! and 4~b! for two different wavevectors. It is seen that for the used physical parameters thespectrum consists of the modesvmp

00 , vmp10;j , andvmp

20;j withj51,2, respectively. The intrasubband magnetoplasmonmodevmp

00 has a decreasing frequency with increasing mag-netic field. Such a behavior is known from the edge modes ofthe spatially confined Q2DEG.55 Hence, the Q1D (020)intrasubband magnetoplasmon behaves like anedge mode,while the intersubband branches approach for large magneticfields multiples of the cyclotron frequency@see inset of Fig.4~a!#. What follows is that in this limit the (120) intersub-band magnetoplasmon branchesvmp

10;j approach for large-magnetic fields the dispersion curve of theprincipal mode

vmp2D;p and the (220) intersubband magnetoplasmon

branchesvmp20;j approach the dispersion relation of the first

Bernstein modevmp2D;B1 of a 2DEG, i.e., the (N20) intersub-

band magnetoplasmon branches approach the dispersioncurve of the (N21)th Bernstein mode. The principal modeand the first Bernstein mode propagating in thex direction ofa 2DEG have the long-wavelength@qx!6me«0«svc

2/(n2DEGe

2)# dispersion relations:62

vmp2D;p5H ~vp

2D!21vc21

3

2n2DEGp l 0

4

3@~vp

2D!21vc2#vc

2

~vp2D!21vc

22~2vc!2qxJ 1/2 ~40!

and

vmp2D;B15H ~2vc!

223

2n2DEGp l 0

4

3@~vp

2D!21vc2#vc

2

~vp2D!21vc

22~2vc!2qxJ 1/2, ~41!

where vp2D5@n2DEGe

2qx /(2«0«sme)#1/2 is the long-

wavelength dispersion relation of a 2D plasmon, and for aparabolic QWW n2DEG'@3pmeVn1DEG/(2\)#2/3/(2p) isvalid.63 For very large-magnetic fields all modes of theQ1DEG are free of Landau damping, because of the com-plete quantized situation, i.e., the single-particle continua de-generate to single lines at multiples of the cyclotron fre-quency. If one compares Figs. 4~a! with 4~b! it is seen thatincreasing wave vector results in adecreasing phase spacefor the collective excitations free of Landau damping. It be-comes obvious from Fig. 4~b! that the branchesvmp

00 ,vmp10;1, vmp

10;2, and vmp20;2 only exist for magnetic fields

B.0.7 T and have stop points at the boundaries of the SPEcontinua.

The depolarization shifts of the FM’s and the AM’s havea different dependence on the magnetic field. As shown inRef. 38, the FM’s have for all magnetic fields a finite depo-larization shift, whereas the AM’s have for zero- and large-magnetic fields a vanishing depolarization shift. This depo-larization shift has a maximum at those magnetic fieldswhere the magnetic quantization becomes comparable withsize quantization. Thus, the FM’s are the modes which existat B50 andB5`. For qx50, the SPE’s only exist at thelines vc , 2vc , . . . , theintrasubband magnetoplasmon van-ishes, and the intersubband modes correspond tocollectiveintersubband resonances~collective intersubband transi-tions, dimensional resonances!. As shown above forqx50,in general, the mode spectrum consists ofsymmetric modes:vmp10;2, vmp

10;4, . . . , vmp20;1, vmp

20;3, . . . and of antisymmetricmodes: vmp

10;1, vmp10;3, . . . , vmp

20;2, vmp20;4, . . . even forBÞ0. In

Fig. 5, the frequencies of the collective (120) and (220)intersubband resonances are plotted versus the electron den-sity. It is seen from this figure that for larger electron densi-ties more AM’s become observable and the depolarizationshift Dmp

N0; j5vmpN0; j (qx50)2Nvc of the AM’s increases with

increasing electron density. Furthermore, the number of ob-servable AM’s increases also with increasing electron den-

FIG. 4. Dispersion relation of the magnetoplasmons as a func-tion of the magnetic-field calculated in RPA~heavy solid lines! of aparabolic QWW for a three-subband model, where one subband isoccupied: ~a! qx513105 cm21, n1DEG533105 cm21; ~b!qx523105 cm21, n1DEG533105 cm21. The hatched areas corre-spond to the single-particle intra- and intersubband continua.

8662 54L. WENDLER AND V. G. GRIGORYAN

Page 12: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

sity, e.g., forn1DEG513105 cm21 only the FM’s are ob-servable but forn1DEG513106 cm21 the six modesvmp

10;j

( j51,2, . . . , 6) areresolvable from the single-particle fre-quenciesvc and 2vc at which the other~higher! AM’s be-come degenerate. There are crossings between the antisym-metric and symmetric modesvmp

10;1, vmp10;2, vmp

20;2, vmp20;1, and

vmp20;4, andvmp

20;3, respectively, which correspond to system-atic degeneracies. Thus, at intermediate magnetic fields thefirst AM of each group may have a larger depolarization shiftthan that of the corresponding FM. Only for large- and forsmall-magnetic fields the FM has the largest depolarizationshift of each group of modes~see also Fig. 4!. ForqxÞ0 theISC lifts the systematic degeneracy and gaps between thedispersion curves become open. Atn1DEG51.073106

cm21, the second subband becomes occupied and the curveschange their behavior.

The influence of the electron density on the spectrum ofthe Q1D magnetoplasmons is depicted in Figs. 6~a! and 6~b!for the vanishing wave vector in dependence on the magneticfield. The spectrum consists under the given condition of themodesvmp

10;j andvmp20;j with j51,2, @Fig. 6~a!# and j51,2,3

@Fig. 6~b!#. It is seen from Fig. 6~a! that forn1DEG523105 cm21 the antisymmetric modevmp

10;1 is forB50 T below 2V, but for n1DEG533 105 cm21 above2V atB50 T @see Fig. 6~b!# and hence, crosses the antisym-metric modevmp

20;2 and the symmetric modevmp20;3, when cal-

culated within the diagonal approximation~dashed lines!.The critical density, wherevmp

10;1(qx50;B50)52V isn1DEG56p«0«s\V/e2 @see Eqs.~45! and ~46! below#. Be-cause the accompanied electron transitions 0↔1 and 0↔2are not independent, the interaction results intwo antisym-metric resonance split hybrid modes(vmp

10;1 andvmp20;2), when

calculated with ISC~heavy solid lines!. The symmetric modevmp20;3 is independent from the antisymmetric branches and

thus, not influenced by these modes. It is obvious from theinset of Fig. 6~b! that for small magnetic fields the branchvmp20;2 of the mixed modes behaves like the uncoupled mode

vmp10;1 and the branchvmp

10;1 like the uncoupled modevmp20;2.

For decreasing electron densityn1DEG the depolarizationshift of the modevmp

10;1 decreases and at a certain densityvmp10;1 it is below 2V at B50 and no anticrossing effect

occurs. Hence, the existence of the AM’s gives the possibil-ity of resonance splitting~anticrossing! of dispersion curvesin the spectrum of the magnetoplasmons.

On this point the important question arises about thephysical origin of the AM’s (j.1) in comparision to thecaseB50 and why in the opposite limit (B5`), i.e., in the2D limit these modes vanish as well as the intrasubbandmagnetoplasmon mode. In a semiclassical picture the occur-rence of the AM’svmp

N0; j can be explained in a very simplephysical picture. Let us start from the limit of a 2DEG real-ized if B→`, i.e., where the size quantization is unimportantfor the single-electron motion. In the absence of the confin-ing potential,Vef f(y)50, a nonlocal theory of 2D magneto-

FIG. 5. Frequencies of the (120) and (220) intersubbandresonances~FM’s and AM’s; solid lines, asymmetric modes;dashed lines, symmetric modes! at qx50 as a function of the 1Delectron densityn1DEG.

FIG. 6. Dispersion relation of the intersubband magnetoplas-mons as a function of the magnetic field atqx50 ~collective inter-subband resonances! calculated in a RPA~including ISC: heavysolid lines; diagonal approximation: dashed lines! of a parabolicQWW for a three-subband model, where one subband is occupied:~a! n1DEG523105 cm21; ~b! n1DEG533105 cm21.

54 8663COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 13: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

plasmons results in the principal modevmp2D;p andN Bern-

stein modesvmp2D;BN . If Veff(y)50, the collective excitations,

i.e., the principal modevmp2D;p and the Bernstein modes

vmp2D;BN can propagate freely in the homogeneous and isotro-

pic x-y plane and hence, these modes are degenerate withrespect to the propagation direction. Introducing the confin-ing potential this reduces the symmetry with the result thatthe principal mode as well as the Bernstein modes becomeconfined modes, i.e., behaves like standing waves. The addi-tional confining energy ifVeff(y)Þ0 lifts the degeneracy.Thus, thefine structure( j51,2,3, . . . ) of the intersubbandmagnetoplasmon dispersion relations occurs. With decreas-ing magnetic field the contribution of the size quantizationincreases. Hence, the separation between the different con-fined modes, i.e., between the FM and the AM’s of onegroup (N0), increases. In the other limit, i.e., forB50, andin the absence of the lateral confining potentialVeff(y) onlythe 2D plasmonvp

2D is present. This dispersion curve splits

in the Q1D branchesvpNN8 in the presence of the confining

potential. ForB→0 the Bernstein modes lose their collectivecharacter and thus, the dispersion curves of the confinedBernstein modes approach atB50 the frequencies of theSPE’s, e.g., atqx50: V105V, V2052V, . . . . For smallmagnetic fields (B→0) the FM’s result from the confinedprincipal mode which becomes the Q1D plasmon forB50,and the AM’s result from the confined Bernstein modes.Thus, the AM’s may be called the Q1D analogous modes tothe Bernstein modes. At large but finite magnetic fields wehave the confined principal modes~FM and AM’s: vmp

10;j )and the confined Bernstein modes@FM and AM’s: vmp

N0; j toeach pair (N0) with N.1# showing fine-structure effects.Hence, atB50 andB5` we have the simple picture ofQ1D plasmons and 2D magnetoplasmons, respectively.

As mentioned above the different electron transitions arenot independent. Considering the coupling between the dif-ferent transitions the confined principal and Bernstein modeshybridize and the spectrum of the Q1D magnetoplasmonsresults, which show the fine-structure effects. Hence, theQ1D magnetoplasmons may be considered from the point ofview of a 2DEG as hybrid-type modes. But to keep the 2Dpicture makes sense only for large-magnetic fields and thus,it is only possible to speak about principal and Bernsteinmodes atB5`. The strong hybridization at intermediatemagnetic fields, in general, makes no sense to identify theQ1D magnetoplasmons in the 2D picture. But nevertheless,for B→0 the origin of the Q1D magnetoplasmons from theprincipal and Bernstein modes of a 2DEG becomes obvious~see discussion above!. To support these ideas we havedrawn a sketch of the origin of the Q1D magnetoplasmons in

Fig. 7. ForB50 the modes start atvpNN8 andNV ~to the left

of the line denoted by ‘‘1’’! and for large-magnetic fields thedispersion curves approach multiples of the cyclotron fre-quency~to the right of the line denoted by ‘‘2’’!. Betweenthese two limits the hybridization results in the Q1D magne-toplasmons~the range between the lines denoted by ‘‘1’’ and‘‘2’’ !.

The intrasubband magnetoplasmon dispersion curve goesto zero frequency for infinite-magnetic fields, because in thislimit the subbands degenerate to nondispersive Landau lev-

els, i.e., the intrasubband magnetoplasmon does not exist inthis limit. The occurrence of a dispersive energy-momentumrelationEN(kx) allows for finite-magnetic fields the electrontransition with finite energy difference within one subbandwith the resulting intrasubband magnetoplasmon. Hence,only one branch of the intrasubband magnetoplasmon re-sults. Because in the limitB→0 the eigenenergiesEN(kx)remain a dispersion, the (020) intrasubband magnetoplas-mon exists in this limit.

There are several possibilities for the denotation of theQ1D magnetoplasmons. For this, there are two ‘‘fix points’’in the spectrum:~i! at B50 we have the Q1D plasmons

vpNN8(qx) and ~ii ! at B→` we have the 2D magnetoplas-

mons, i.e., the principal and Bernstein modes. It seems thatthe case of larger-magnetic fields, where the intersubbandmodes approach multiples of the cyclotron frequency, givesthe best choice for it. This we will do throughout this paper.

The quantum-mechanical picture of the Q1D magneto-plasmons supplements the consideration made above. In the2D limit, the single-particle ground state is characterized bythe discrete quantum numberN and the quasicontinuousquantum numberkx ~or Ykx

) and the eigenenergiesEN are

degenerate accordingkx . There areeBLxLy/2p\ kx valuesbelonging to the same Landau levelEN . Decreasing the mag-netic field, the size quantization of the electrons caused bythe confining potential becomes more and more important.The size-quantization results in a lifting of the degeneracyaccordingkx with the occurrence of the subbandsEN(kx).This has the consequence that the collective intersubbandtransitions split in different modes. This becomes obviousfrom the dispersion relation, Eq.~19!. At B50 andB5`this equation becomes independent fromkx and kx1, i.e.,gives for allkx andkx1 the same dispersion curves. But for

FIG. 7. Schematical representation of the physical origin of theQ1D magnetoplasmons. The solid and dashed lines correspond tothe ‘‘unhybridized’’ modes. For large-magnetic fields the confinedprincipal modes~solid lines! and the confined Bernstein modes~dashed lines! are plotted. The confined principal modes become theQ1D plasmons atB50, whereas the confined Bernstein modes goto NV. The hybridization of these modes gives the Q1D magneto-plasmon dispersion relations.

8664 54L. WENDLER AND V. G. GRIGORYAN

Page 14: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

finite-magnetic fields the dispersion relation depends onkxand kx1. Hence, different frequencies result and the degen-eracy is lifted. The occurrence of the AM’s results from thereduced symmetry atBÞ0 in comparison toB50 andB5`.

Now the improvements of the present theory to the earlierresults on Q1D magnetoplasmons of Eliassonet al.6 and Liand Das Sarma22 become obvious. Within a local approach,

as done by Eliassonet al.6 all the branchesvpNN8 of confined

modes tend tovc at large-magnetic fields, because in a localtheory higher resonances atNvc , i.e., Bernstein modes, areabsent. Our model includes nonlocal effects with the resultthat for nonvanishing magnetic fields both the confined prin-cipal modes and the Bernstein modes are present. The ap-proach of Li and Das Sarma22 results in the two dispersioncurvesvmp

00 andvmp10;1 Thus, this model neglects the AM’s

~for B→`) because it considers only two subbands andgives no fine-structure effects.

It is important to note that the spectrum, i.e., the occur-rence of FM’s, AM’s with fine structure, isuniversalandthus independent from the concrete shape of the lateral con-fining potential. Our gedanken experiment makes no specialattribution for this potential. The numerical calculations, per-formed here, are done for aneffective perfect parabolic po-tential. This model has the advantage of being simple and itgives quite general results. It was shown in Refs. 28,37 byperforming self-consistent calculations of the ground stateand the response for QWW’s withbare parabolic and non-parabolic potentials that the qualitative features of the plas-mon spectrum~number and types of modes, etc.! are inde-pendent from the shape of the lateral confining potential.Varying the shape of the potential more or less only scalesthe frequencies of the modes. Thus, for a bare parabolic, abare nonparabolic, or an effective parabolic potential thesame mode spectrum arises, i.e., the mode spectrum is uni-versal. But in difference to the universal nature of the spec-trum of the collective modes their observation, e.g., in a FIRtransmission experiment or in a Raman scattering experimentstrongly depends on the lateral confining potential. The po-tential shape influences the selection rules and the oscillatorstrengths for the excitation of certain modes. It is well knownthat for abare perfect parabolic potential, the generalizedKohn’s theorem64,65 predicts that in a FIR transmission ex-periment the Q1DEG absorbs in the absence of a magnetic-field radiation only at the bare harmonic-oscillator fre-quency, independent of the electron-electron interaction andthe number of electrons in the QWW. Usually this intersub-band resonance (vmp

10;1) is called Kohn’s mode. Thus, in thisspecial situation this dispersion curve is pinned atvmp10;1(qx50)5(vc

21V02)1/2 for BÞ0, with the bare

harmonic-oscillator frequencyV0. But the mode still has afinitedepolarization shiftDmp

10;1. The manifestation of Kohn’stheorem in the spectra of Q1D plasmons is investigated indetail in Ref. 37. In the case of magnetoplasmons, it mani-fests in such a manner that the renormalization of the hybridfrequencyv05(vc

21V02) of the bare perfect parabolic po-

tentialV0(y)5meV02y2/2 to v0*5v02Dv0 of the effective

potential due to many-particle effects~tadpole self-energydiagrams! is compensated by the depolarization shiftDmp10;15vmp

10;1(qx50)2v0* : vmp10;1(qx50)5v0*1Dmp

10;15v0

only if Dv05Dmp10;1 is valid. In this very special situation of

a bare perfect parabolic potential the collective motion of theelectrons in the Kohn mode is connected by the center-of-mass motion of the Q1DEG. Because the center-of-mass mo-tion becomes independent from the relative motions, the dis-persion relation of the Kohn modevmp

10;1 may cross otherdispersion curves. In the quantum-mechanical picture of thesubband space this means an accidental degeneracy of thedispersion relations. Nonparobolicity of the lateral confiningpotential, as well as a finite wave-vector componentqx , willbreak the validity of Kohn’s theorem. In this case, the acci-dental degeneracy becomes lifted, i.e., anticrossing behaviorof the dispersion curvevmp

10;1 with other dispersion curvesarises. Thus, for an effective perfect parabolic potential,where the associated bare potential is nonparabolic, the dis-persion curvevmp

10;1 does not cross any other dispersion curveat finite qx . Further, this induces a redistribution of the os-cillator strength for FIR absorption and thus, in principle, allmodes become observable. But the mode spectrum itself isonly shifted to lower or higher frequencies depending on thetype of nonparabolicity, i.e., nonparabolicity of the bare lat-eral confining potential does not induce any new modes.37

Hence, the use of an effective parabolic confining potential,which may be considered as the final result of a self-consistent calculation of the ground state, is without loss ofgenerality concerning the universal features of the Q1D mag-netoplasmon dispersion relations.

2. Analytical calculations

For the quantitative discussion of the physical propertiesof the Q1D magnetoplasmons it is necessary to derive ana-lytical expressions for the dispersion relations. In these cal-culations, we will use a two-subband model:N,N850,1. Ifthe inequalityjkF

(0)!1 is fulfilled it is possible to restrict therepresentation of the displaced center wave function on asmall number of terms. Here, we use three terms:L50,1,2.The determinantal dispersion relation, given by Eq.~27!,det@ai j #50, has the elementsai j5d i j2J i j (qx ,v), wherei5$L1L2% and j5$L L8%, with 15$00%, 25$10%, 35$20%, 45$11%, 55$21%, and 65$22%.

Let us start with the caseqx→0. In this case, we cancalculate analytically the functionJL1L2LL8(qx ,v), given inEq. ~28! with the result

JL1L2LL8~qx ,v!51

AL!L8! ~11dLL8!

2vc

p\~v22vc2!

3 (L350

`

(L450

L3

VL1L2L3L4s ~qx!AL3L4LL8~v!,

~42!

where

AL3L4LL8~v!52@12 l /2#

AL3!L4! ~11dL3L4!j@~L1L8!

3~L31L4!J~ l22!2 lJ ~ l !1J~ l12!# ~43!

and

54 8665COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 15: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

J~n!5E2jkF

~0!

jkF~0!

dx xne2x252 dn,2m(k50

`~21!k

k!

~jkF~0!!2k1n11

2k1n11,

~44!

and wherel5L1L81L31L4 andm50,1,2,3, . . . . The ma-trix elements of the Coulomb potential for the caseqx→0are given in Appendix A. Using these results in Eq.~27!, weobtain up to powers of (jkF

(0))5 the following solutions:

vmp10;j5~11a10;j !

1/2vc1O@~jkF~0!!5# ~45!

with

a10;15n1DEGe

2

2p«0«s\vcS 12

pn1DEG2 \vc

2

8mevc3 D , ~46!

a10;25n1DEGe

2

2p«0«s\vcS p2n1DEG

2 \vc2

6mevc3 D . ~47!

The two dispersion relations are valid for all magnetic fields.We note that in the next order ofjkF

(0) the next modevmp10;3

would appear as a solution of the dispersion relation. Thus, itbecomes clear that the additional Q1D magnetoplasmamodes arise in higher order injkF

(0). The extra term}a10;j

in Eq. ~45! is due to the depolarization effect.Let us discuss the limiting cases of these two dispersion

relations. In the case ofB→0, it results from Eqs.~45! and~46!,

vmp10;15S 11

n1DEGe2

2p«0«s\V D 1/2V1O~B2!, ~48!

which is the dispersion relation of the well-known (120)intersubband plasmon. In this limit, Eq.~45! with Eq. ~47!reads

vmp10;25F11

vc2

V2S 11pn1DEG

3 e2

12«0«smeV2D G1/2V1O~B4!.

~49!

Hence, forB50 the mode has the frequencyvmp10;25V and

thus, has a vanishing depolarization shift, i.e., approaches thesingle-particle continuum. But a collective intersubband tran-sition with no depolarization shift means that the densityinduced if the Q1D plasma oscillates in this mode vanishes,i.e., the modevmp

10;2 disappears forB50. So, we see that themodes resulting from our theory have the correct zero-magnetic-field limit. Let us now discuss the opposite limit,i.e.,B→`. In this case, Eq.~45! with Eq. ~46! reads

vmp10;15S 11

n1DEGe2

2p«0«s\vcD 1/2vc1OS 1BD ~50!

and it follows from Eqs.~45! and ~47!

vmp10;25S 11

V2

vc2 1

pe2n1DEG3

12«0«smevc2D 1/2vc1OS 1

B2D . ~51!

It is seen from the two equations that for large-magneticfields, vmp

10;j (B→`)→vc is valid at qx50. Hence, thesemodes behave like the principal mode of a 2DEG@comparewith Eq. ~40!# and our analytical results have the correctlarge-magnetic-field limit.

To derive analytical expressions of the dispersion rela-tions of the Bernstein modesvmp

20;j , it is necessary to use thethree-subband model, i.e.,N50,1,2 with at least three termsL50,1,2 in the representation of the displaced center wavefunction. However, this model results in very tedious andcumbersome calculations. If the frequencies of the (120)intersubband magnetoplasmons are well separated from thatof the (220) intersubband magnetoplasmons the ISC is neg-ligible small and thus, the diagonal approximation givesgood quantitative results for the dispersion curves. Perform-ing the diagonal approximation of Eq. ~27!, usingN,N8 5 0,2 and three termsL,L85 0,1,2 in Eq. ~20!, weobtain for the functionJL1L2LL8(qx ,v),

JL1L2LL8~qx ,v!51

AL!L8! ~11dLL8!

2vc

p\~v224vc2! (

L350

`

(L450

L3

VL1L2L3L4s ~qx!AL3L4LL8~v!, ~52!

with

AL3L4LL8522 l /2

AL3!L4! ~11dL3L4!j„4$@L~L21!1L8~L821!#@L3~L321!1L4~L421!#%

3J~ l24!24$~L1L8!@L3~L321!1L4~L421!#1~L31L4!@L~L21!1L8~L821!#%J~ l22!

12$~L81L3!~L81L321!1~L1L4!~L1L421!12~LL31L8L4!%J~ l !22lJ ~ l12!1J~ l14!

…. ~53!

8666 54L. WENDLER AND V. G. GRIGORYAN

Page 16: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

Substituting, this result in the dispersion relation, Eq.~27!,we obtain

vmp20;j5~11a20;j !

1/2~2vc!1O@~jkF~0!!5# ~54!

with

a20;15n1DEGe

2

8p«0«s\vcS 12

p2n1DEG2 \vc

2

3mevc3 D ~55!

and

a20;25n1DEGe

2

8p«0«s\vcS p2n1DEG

2 \vc2

12mevc4 D . ~56!

Notice, that in the next order ofjkF(0) the modevmp

20;3 wouldarise as a result of Eq.~27!. In the limit of vanishing mag-netic fieldB→0, Eq. ~54! gives for j51

vmp20;15S 11

n1DEGe2

8p«0«s\V D 1/2~2V!1O~B2!, ~57!

which is the dispersion relation of the well-known (220)intersubband plasmon. Comparing Eq.~48! for vmp

10;1 withEq. ~57! for vmp

20;1, it is obvious that the depolarization shiftis identical for both modes in this approximation. Equation~54! with j52 results forB→0 in

vmp20;25S 11

vc2

V2 1pn1DEG

3 e2vc2

96«0«smeV4D 1/2~2V!1O~B4! ~58!

and hence, atB50: vmp20;252V. Again, the first (220) AM

has a vanishing depolarization shift with the result that thismode @and all higher-order (220) AM’s# disappears forvanishing magnetic field. In the opposite limit, i.e.,B→`,we obtain

vmp20;15S 11

n1DEGe2

8p«0«s\vcD 1/2~2vc!1OS 1BD ~59!

and

vmp20;25S 11

V2

vc21

pn1DEG3 e2

96«0«smevc2D 1/2~2vc!1OS 1

B2D .~60!

Both dispersion curves approach 2vc for B→` and thus,result in the first Bernstein mode of a 2DEG@compare withEq. ~41!#. Also, the modesvmp

20;j show the correct zero- andinfinite-magnetic-field limit.

Now we consider the caseqxÞ0. The analytical calcula-tions become very complicated in this case. Hence, we useagain the two-subband model:N,N850,1, but restrict therepresentation of the displaced center wave function only ontwo terms:L50,1. In this case the determinantal dispersionrelation det@ai j #50 considered up to the order (jkF

(0))3 hasthe elements@we omit the argumentqx in VL1L2L3L4

s (qx) and

the argumentsqx ,kx , andv in PNN8(1) (qx ,kxuv)#

a11512(kx

SV0000s P00

~1!2j2

2„V0000

s $P00~1!@kx

21~kx1qx!2#2P10

~1!~kx1qx!22P01

~1!kx2%2V0011

s @P00~1!2P10

~1!2P01~1!#kx~kx1qx!…D ,

a125j

A2(kx$V0000

s @2P00~1!~2kx1qx!1P10

~1!~kx1qx!1P01~1!kx#1V0011

s @P10~1!kx1P01

~1!~kx1qx!#%,

a1352j2

2(kx

$V0000s @~P00

~1!2P10~1!2P01

~1!!kx~kx1qx!#1V0011s @P10

~1!kx21P01

~1!~kx1qx!2#%,

a2152j

A2V1010s (

kx@~P00

~1!2P10~1!!~kx1qx!1~P00

~1!2P01~1!!kx#,

a22512V1010s (

kxS P10

~1!1P01~1!1

j2

2@P00

~1!~2kx1qx!22P10

~1!~6kx218kxqx13qx

2!2P01~1!~6kx

214kxqx1qx2!# D , ~61!

a2352j

A2V1010s (

kx@P10

~1!kx1P01~1!~kx1qx!#,

a315(kx

S 2V1100s P00

~1!1j2

2„V1100

s $P00~1!@kx

21~kx1qx!2#2P10

~1!~kx1qx!22P01

~1!kx2%2V1111

s ~P00~1!2P10

~1!2P01~1!!kx~kx1qx!…D ,

a3252j

A2(kx$V1100

s @P00~1!~2kx1qx!2P10

~1!~kx1qx!2P01~1!kx#1V1111

s @P10~1!kx1P01

~1!~kx1qx!#%,

54 8667COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 17: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

a33512j2

2(kx

$V1100s ~P00

~1!2P10~1!2P01

~1!!kx~kx1qx!1V1111s @P10

~1!kx21P01

~1!~kx1qx!2#%.

To calculate explicitly the elementsai j , we use the long-wavelength expressions of the Coulomb matrix elements~upto the order qx

2) given in Appendix A and the long-wavelength expressions of the different sumsSNN8

(n)

5(kxkxnPNN8

(1) (qx ,kxuv), n50,1,2 ~up to the orderqx2),

which are given in Appendix B. The analytical calculationgives two dispersion relations. One describes the (020) in-trasubband magnetoplasmon35

vmp005qxl 0F S vF~0!

l 0D 22 n1DEGe

2

2p«0«smel 02ln~ uqxu l 0!G 1/2

3H 11n1DEGe

2vc2

2V2~2p«0«s\vc1n1DEGe2!

J . ~62!

For vanishing magnetic field the expression~62! gives theknown result for the (020) intrasubband plasmon. The sec-ond dispersion relation is

vmp10;15~11a10;1!

1/2vc , ~63!

where in this approximationa10;1 is given by Eq.~46! with-out the last term in the parentheses.

B. Two subbands occupied

The theory of the Q1D magnetoplasmons developedabove gives the possibility of calculating their dispersion re-lations for aM -subband model from whichM 8 may be oc-cupied. In the three-subband model assuming that two sub-bands are occupied, the dispersion curves of the Q1Dmagnetoplasmons are determined by Eq.~27!. The full RPAdispersion curves of the Q1D magnetoplasmons includingISC, i.e., the mixed (020)2(121)2(120)2(220)2(221) magnetoplasmons are plotted, in dependence onthe wave vector for different magnetic-field strengths andelectron densities in Figs. 8~a!–8~c!. The case of zero-magnetic field is plotted in Fig. 8~a!. It is seen that forB50 the hybrid modesvp

11, vp102, andvp

21 occur in additionto the modes of a QWW with one occupied subband@cp. Fig.3~a!#. The (121) intrasubband plasmon branch arises due tothe intrasubband electron transitions within the second occu-pied subband and the (221) intersubband plasmon branchdue to the intersubband electron transitions 1↔2. The modevp11 occurs in the region of thev2qx plane which is free of

Landau damping and is between the single-particle (020)and the (121) intrasubband continuum. Further, it is seenthat the modevp

10, is split in the two branchesvp106. The

lower frequency mode occurs inside the gap of the single-particle (120) intersubband continuum.20 It was shownrecently36 that the electron densities induced in subbandsE0(kx) and E1(kx) oscillate in phase if the branchvp

00 isexcited, but in antiphase for the branchesvp

11, andvp106.

Thesymmetricmodesvp00, vp

11, andvp20 form one set, which

is decoupled from the set of theantisymmetricmodesvp101, vp

102, and vp21. For nonvanishing magnetic fields

@Figs. 8~b! and 8~c!# it is seen that the intersubband magne-toplasmon dispersion relations show a fine structure, i.e.,again FM’s and AM’s occur for the same reasons as in thecase of one occupied subband.

For the unique denotation of the modes in the case of twooccupied subbands one additional problem arises. ForB50, we have the intersubband modesvp

21 andvp102 which

vanish if with increasing magnetic field the second subbandbecomes depopulated. Because these modes do not exist forlarge-magnetic fields they do not fit the classification schemedeveloped above. The best way to extend this scheme to thecase of few occupied subbands is as follows. At first themodes should be classified byvmp

N0; j in FM’s with j51 andAM’s with j.0 at large-magnetic field according to theirasymptotic behavior. Second, only at zero and small-magnetic fields the denotationvmp

21 and vmp10;12 should be

introduced if necessary. The modesvmp21 andvmp

10;12 are ad-ditional modes with respect to case of one occupied subbandand hence, we call themoccupational modes~OM’s!. Thesemodes exist in a finite range of magnetic fields. For the con-ditions used in calculation of the dispersion curves of Fig.8~b! we have two FM’svmp

10;11, vmp20;1, and five AM’s:

vmp10;2, vmp

10;3, vmp20;3, vmp

10;4. But there is only one branch of theOM vmp

10;12. It is seen from Fig. 8~c! that increasing mag-netic field strengths and increasing electron density result inmore AM’s in the regions free of Landau damping.

The frequencies of the Q1D magnetoplasmons are plottedin dependence on the magnetic field for different wave vec-tors in Figs. 9~a! and 9~b!. For the QWW used in the calcu-lations, two subbands are occupied forB<1.5 T, but forlarger strengths of the magnetic field the second subbandbecomes depopulated. Because of the strong hybrid type ofthe modes at intermediate magnetic fields we use for thedenotation of the modes, as mentioned above, the limit oflarge-magnetic fields. In this case we have onlyvmp

N0; j

branches, i.e., we do not denote a branch byvmp21 in the figure

to avoid confusion in this very complicated spectrum. Themodevmp

21 exists for small-magnetic fields, but hybridizesstrongly with the other modes for finite-magnetic fields,where it is senseless to speak from a (221) intersubbandmagnetoplasmon mode. The case of zero wave vector is plot-ted in Fig. 9~a!. In this case, we have five collective(120) intersubband resonancesvmp

10;j ; j51, 2, 3, 4, 5 andfive collective (220) intersubband resonancesvmp

20;j ; j51,2, 3, 4, 5. The dispersion curves of the Q1D magnetoplas-mons are plotted for finite wave vectors in Fig. 9~b!. In thiscase the two branches of the intrasubband magnetoplasmons(020) and (121) arise. Both dispersion curves have de-creasing frequencies with increasing magnetic field, i.e.,show edge mode behavior. Further, in comparison to Fig.9~a!, the branchesvmp

10;5 andvmp20;5 disappear from the spec-

trum. From Fig. 9~b! it becomes obvious that the modes

8668 54L. WENDLER AND V. G. GRIGORYAN

Page 18: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

vmp10;3 andvmp

10;4 only exist in a finite interval of the magneticfield, i.e., havetwo stop points for finite magnetic fields. Ingeneral, all the AM’svmp

NN8; j.1 start and stop at the boundary

v1NN8 of the corresponding single-particle (N2N8) con-

tinuum at certain values of the magnetic field in dependenceon the physical parameters characterizing the QWW, e.g.,confining potential, electron density. The OMvmp

21 shows anegative dispersion at small-magnetic fields@see Fig. 9~b!#.In general, the Q1D magnetoplasmon spectrum becomesvery complex if more subbands become occupied. Whichmodes are present strongly depends on the physical param-eters of the QWW, the magnetic field, electron density, andwave vector.

VI. COMPARISON WITH EXPERIMENTAL RESULTS

Because the theory developed above is valid for allmagnetic-field strengths and electron densities as long as the

FIG. 8. Dispersion relation of the magnetoplasmons in depen-dence on the wave-vector componentqx calculated in a RPA~heavysolid lines! of a parabolic QWW for a three-subband model, wheretwo subbands are occupied:~a! B50 T, n1DEG583105 cm21; ~b!B51 T, n1DEG583105 cm21; ~c! B52 T, n1DEG51.23106

cm21. The hatched areas correspond to the single-particle intra-and intersubband continua.

FIG. 9. Dispersion relation of the magnetoplasmons as a func-tion of the magnetic field calculated in a RPA~heavy solid lines! ofa parabolic QWW for a three-subband model, where two subbandsare occupied:~a! qx50, n1DEG583105 cm21; ~b! qx513105

cm21, n1DEG583105 cm21; The hatched areas correspond to thesingle-particle intra- and intersubband continua.

54 8669COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 19: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

RPA of the Q1D Fermi liquid model is valid, it should bepossible to give a qualitative explanation of the experimentson Q1D magnetoplasmons in QWW’s. It is obvious from theresults of the last section that in a FIR experiment as well asin a Raman scattering experiment for nonvanishing magneticfields and wave vectors a very complex spectrum appears.However, in such experiments one has to couple an externalelectromagnetic field with the Q1D magnetoplasmons. An-swering the question of what modes couple with the incidentlight needs to know the selection rules of the correspondingprocess is outside the scope of the present paper. But never-theless, a qualitative explanation of the experiments ongrounds of the complete spectrum given here of the Q1Dmagnetoplasmons should be possible. Here, we focus on thephysical explanation of the two ‘‘unexpected’’ findings inFIR transmission spectroscopy. In difference to the 2D case,Drexler et al.48 found a resonance splitting~anticrossing! ofthe (320) intersubband mode at the line 2vc , whereas the(120) intersubband mode does not show anticrossing be-havior with this line. The observation of the (320) intersub-band mode in the FIR transmission indicates that the barelateral confining potential of the QWW has a nonparabolicshape and thus, the condition for the validity of the general-ized Kohn’s theorem is not given. This led to the question,why does the (320) intersubband mode show anticrossingbehavior and why does the (120) intersubband mode not.The authors of Ref. 48 interpret their experimental results asthe anticrossing of the (320) intersubband mode with the‘‘2 vc mode,’’ i.e., with the second harmonic of the cyclo-tron resonance. From the discussion of the Q1D magneto-plasmons it becomes clear that this picture seems to be nottrue, because the second harmonic of the cyclotron resonanceis not a normal mode of the Q1D magnetoplasma. Accordingto the complicated mode structure many resonance-splittingeffects with respect to a diagonal approximation are possiblefor bare nonparabolic lateral confining potentials. One ex-ample is the discussed resonance splitting of the FMvmp

10;1

with the AM vmp20;2, demonstrated in Fig. 6~b!. But this reso-

nance splitting is very small in magnitude and thus, it shouldnot be observable~i.e., resolvable! in FIR experiments. Theexperiments46,48 justify this assumption, because in these ex-periments the FMvmp

10;1 shows no anticrossing effects. It isimportant to note that this anticrossing behavior shown inFig. 6~b! only results if thebare confining potential is non-parabolic. For a bare perfect parabolic potential the Kohnmode vmp

10;1 would cross the dispersion curvesvmp20;2 and

vmp20;3. But deviations from the parabolic shape would result

in an anticrossing behavior of the dispersion curves. On theother hand, the upper resonance-split mode observed in Ref.48 is for small-magnetic fields the (320) intersubbandmodevmp

30;1, but approaches for large-magnetic fields the line2vc . Quite general, for all QWW’s a mode with such abehavior does not exist. Because in the experiment48 the caseqx50 is investigated, only the antisymmetric modesvmp

10;1,vmp10;3, . . . ,vmp

20;2,vmp20;4, . . . ,vmp

30;1,vmp30;3, . . . aredipole ac-

tive and thus, are observable. Hence, the only explanation isthat in the FIR experiment for small-magnetic fields themodevmp

30;1 is detected and for larger-magnetic fields onedetects the modevmp

20;2. This is only possible if the modevmp30;1 has for small-magnetic fields a largeoscillator

strength, which is the measure of the coupling strength of theexternal field with the mode, and this oscillator strength de-creases with increasing magnetic field, whereas the oppositeis valid for the modevmp

20;2. Thus, the upper resonance-split‘‘mode’’ detected in the experiment is in reality the measure-ment of theexchangeof oscillator strength between themodesvmp

30;1 and vmp20;2. The lower resonance-split mode,

measured in the experiment, we interpret as one of the modesvmp10;j . Hence, the detected phenomenon at 2vc is in our opin-

ion explained by an exchange of oscillator strength betweentwo modes. In Fig. 10, we have plotted the dispersion curvesof the Q1D magnetoplasmons for the situation typical in theexperiment. The symbols (d) indicate the expected posi-tions of the absorption peaks~at integer values of the mag-netic fields!.

Similar anticrossing behavior at 2vc is reported by Demelet al.46 It becomes obvious that the lateral potential of theQWW used in Ref. 46 has a bare parabolic potential becausein the FIR transmission atqx50 only the modevmp

10;1 isobserved as a result of the validity of the generalized Kohn’stheorem in difference to the experiments of Drexleret al.48

In the caseqx50, Demelet al.46 found no anticrossing be-havior. But forqxÞ0, realized in the experiment with a grat-ing coupler on top of the sample,46 Kohn’s theorem is brokenand hence more modes become observable. In the experi-ment of Demelet al.,46 the observed magnetoplasmon spec-tra seem to show the following. The lowest-frequency inter-subband mode crosses forqx50 the line 2vc , but showsanticrossing with this line ifqxÞ0. Further, the (020) in-trasubband magnetoplasmonvmp

00 was detected showingedge-mode behavior and crosses the line 2vc . In this paper,the question remains open whyvmp

00 shows no resonancesplitting with 2vc . The complete spectrum of Q1D magne-toplasmons allows the correct classification of the modes ob-served in this experiment. As mentioned above, the line2vc has no physical meaning in a Q1DEG, because it is not

FIG. 10. Dispersion relation of the intersubband magnetoplas-mons as a function of the magnetic field atqx50 ~collective inter-subband resonances!. The full RPA dispersion relations are calcu-lated for n1DEG53.73105 cm21 and are given by solid lines~antisymmetric modes! and dashed lines~symmetric modes!. Thesymbols (d) represent the peaks which should be measured in aFIR transmission experiment.

8670 54L. WENDLER AND V. G. GRIGORYAN

Page 20: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

connected with any normal mode and hence, interaction withsuch a fictivious line cannot take place. The modes detectedin this experiment we classify as follows. The upper mode isthe AM vmp

20;2 which approaches 2vc for large-magneticfields. The second-highest mode we interpret asvmp

10;1(qxÞ0), which has a stop point at the lower boundary of thesingle-particle (220) intersubband continuum forqxÞ0@see Figs. 4~a! and 4~b!#. The third-highest resonance is thecollective (120) intersubband transitionvmp

10;1(qx50).Thus, in our opinion there is no resonance splitting at 2vc inthe spectrum.

In the experiments of Drexleret al.48 and Hertelet al.51 asecond phenomenon was detected: for larger electron densi-ties the modevmp

10;1 shows a very pronounced splitting intoseveral branches in the transition region from size to mag-netic quantization. Originally, the authors of Refs. 48,51 sug-gested that this splitting could be interpreted as an effect ofthe interwire Coulomb interaction~Ref. 48! and of the non-parabolicity of the lateral confing potential~Ref. 51!. But, inour opinion this observation looks like the detection of thefine structure of the modevmp

10;j . This conclusion is supportedby the fact that the fine structure is observed only for largerelectron densities and only in a finite range of the magneticfield, because only in this range the (120) AM’s have afinite depolarization shift. As discussed above, the depolar-ization shift of the AM’s has its maximum just at those mag-netic fields where the size quantization and the magneticquantization are of the same order~see Ref. 38!, and theabsolute value of the depolarization shift increases with in-creasing electron density~see Fig. 5!. These are the exactconditions realized in experiments48,51 to resolve more andmore AM’s. Thus, comparing our results with the experi-mental findings, we come to the conclusion that in the ex-periment the observed splitting of the (120) intersubbandmode in several branches is thefirst detectionof the finestructure of the Q1D (120) intersubband magnetoplasmondispersion relation. We show the Q1D magnetoplasmon dis-persion relationsvmp

10;j for the experimental situation in Fig.11. The symbols (d) indicate the largest peaks which shouldbe measured in a FIR transmission experiment. In Fig. 11~a!,we have plotted the case of low carrier concentration, whereonly the modevmp

10;1 should be observable, but for the largerelectron density@Fig. 11~b!# the modesvmp

10;1 and vmp10;3

should be detected, and in a narrow range of the magneticfield also the modevmp

10;5. Becausevmp10;3 is in the very near

vicinity of the SPE continuum its observation is very difficultand thus, we have not indicated this mode in Fig. 11~a! ~indifference to the global analysis given in Fig. 10!. Very re-cently, the fine-structure of the~1-0!, ~2-0!, and ~3-0! inter-subband magnetoplasmous is observed in inelastic llightscattering experiments.66 These experiments very weill agreewith our results and confirm the theory developed here ofQ1D magnetoplasmous.

VII. SUMMARY

In this paper, we have studied the collective and single-particle excitations of a Q1D magnetoplasma, quantum con-fined in a QWW, within the RPA. Here, we have used themethod of representing the single-particle wave function in

the basis of undisplaced harmonic-oscillator wave functions.This method have two advantages. The first is that the theorydeveloped here is free of any small parameter and hence,valid for all magnetic-field strengths, electron densities, andwave vectors as long as the RPA of the Q1D Fermi liquidmodel is valid. Further, the results are given in closed form,i.e., the form of the dispersion relation is independent fromthe number of the undisplaced wave functions used in prac-tical calculations. This theory is valid for many subbandsfrom which any number may be occupied. With this theory itis possible to calculate the complete spectrum of Q1D mag-netoplasmons. This method provides the calculation of theresponse of a Q1DEG in the presence of a magnetic field fordifferent lateral confining potentials. It is also profitable anduseful if in a first step the ground-state problem must besolved self-consistently. Further, this method is also appli-cable for a Q2DEG in the presence of an in-plane magneticfield.

The collective excitations of the Q1DEG in the presenceof a magnetic field can be classified into three groups. Thefirst type of modes are called fundamental modes. These

FIG. 11. Dispersion relation of the intersubband magnetoplas-mons as a function of the magnetic field atqx50 ~collective inter-subband resonances!. The calculated full RPA dispersion relationsare given by solid lines~antisymmetric modes! and dashed lines~symmetric modes!: ~a! n1DEG563105 cm21; ~b! n1DEG513106

cm21; the symbols (d) represent the main peaks which should bemeasured in a FIR transmission experiment.

54 8671COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 21: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

modes are defined as such, which exist atB50 ~and also atB50!. For B50 the fundamental modes are the Q1D plas-mons and atB5` these modes correspond to the 2D mag-netoplasmons~principal mode and Bernstein modes!. For BÞ0 andBÞ` additional modes arise due to the reducedsymmetry of the system, i.e., the spectrum of Q1D magne-toplasmons shows fine-structure effects. At large-magneticfields these modes can be interpreted as confined principaland Bernstein modes which strongly hybridize at smaller-magnetic fields. Thus, the confining energy results in the finestructure of the former 2D principal and Bernstein modes.ForB→0, the additional modes approach the frequencies ofthe single-particle excitations and hence, lose their collectivecharacter analogous as the Bernstein modes of higher-dimensional electron gases. Thus, at small-magnetic fieldsthe additional modes may be interpreted as the Q1D analogyto the Bernstein modes. If more than one subband is occu-pied due to the occupation effects, further modes appear.These modes, which are present atB50, we classify as oc-cupational modes.

The complete mode spectrum of the Q1DEG is indepen-dent from the concrete shape of the lateral confining poten-tial. It is the same for parabolic and nonparabolic bare andeffective potentials, and thus is universal. According to thesymmetry the modes can show systematic degeneracies~symmetric and antisymmetric modes! and for a bare perfectparabolic potential accidental degeneracies~Kohn’s mode!.But which of the Q1D magnetoplasmons are excited in anexperiment, e.g., FIR transmission spectroscopy, Ramanspectroscopy, depends on the experiment and on the shape ofthe lateral confining potential. With this knowledge it is pos-sible to explain the recent findings in FIR experiments ofanticrossing at 2vc as an exchange phenomenon of the os-cillator strength of two modes and the splitting of the(120) intersubband mode at larger electron densities andintermediate-magnetic fields as the first detection of the finestructure of the Q1D magnetoplasmon spectrum. Our theoryovercomes the problems and inconsistencies of the earliertheoretical works on Q1D magnetoplasmons.6,13,22

The calculations are done for an effective parabolic con-fining potential using the full RPA dispersion relations. It isshown that, in general, in the presence of a magnetic field allelectron transitions become coupled with the result that theQ1D magnetoplasmons are hybrid modes concerning the dif-ferent transitions. The spectrum consists of differentbranches of intra- and intersubband magnetoplasmon disper-sion curves. It is shown that the intrasubband branches arecharacterized by decreasing frequencies for increasing mag-netic fields, i.e., show edge-mode behavior. The intersub-

band branches approach for large-magnetic fields multiplesof the cyclotron frequency. For vanishing wave vector ormagnetic field the modes split into two groups:~i! the sym-metric modes and~ii ! the antisymmetric modes.

We note that the results obtained here are also valid forQWW’s, in which the lateral confining potential is field-effect induced by a gratinglike gate on top of the sample aslong as the Q1D quantization occurs. This is true because thescreening of the metallic gates induces in the 2DEG alternat-ing compressible and incompressible stripes only if the dis-tance between the gates is so large that the electron gas be-tween the gates can be considered as a spatially confined2DEG.67,68

ACKNOWLEDGMENTS

The authors wish to thank W. Hansen and A. Wixforth forvaluable discussions. We gratefully acknowledge financialsupport by the Deutsche Forschungsgemeinschaft~DFG!,Project No. We 1532/3-2.

APPENDIX A: MATRIX ELEMENTSOF THE COULOMB POTENTIAL

In this appendix we calculate the matrix elements of theCoulomb potential defined in Eq.~27! neglecting image ef-fects:

VL1L2L3L4s ~qx!5

e2

2p«0«sf L1L2L3L4C ~qx!, ~A1!

where the form factor is

f L1L2L3L4C ~qx!5E

2`

`

dyE2`

`

dy8FL1* ~y!FL2

~y!

3K0@ uqx~y2y8!u#FL3* ~y8!FL4

~y8!.

~A2!

Using the integral representation for the modified Besselfunction in Eq.~A1! ~69a!, we obtain for the form factor ofthe Coulomb potential

f L1L2L3L4C ~qx!5E

0

`

dt1

At21qx2E

2`

`

dyE2`

`

dy8FL1* ~y!

3FL2~y!FL3

* ~y8!FL4~y8!cos@ t~y2y8!#.

~A3!

We integrate Eq.~A3! over y and y8 with the followingresult ~69b!:

f L1L2L3L4C ~qx!5dL11L21L31L31L4 , 2n

~21!~N/2!1L11L2 22~N/2!AL2!L4!

L1!L3!

3E0

`

dt1

At21qx2 ~ l 0t !

Ne2@ l 0t !2/2]LL2

L12L2F ~ l 0t !2

2 GLL4L32L4F ~ l 0t !2

2 G , ~A4!

where n50,1,2, . . . , N5L12L21L32L4 is an even number,L1>L2, L3>L4, and LNN8(x) is the Laguerre polynomial.

Because of the spatial symmetry of the confining potential, we havef L1L2L3L4C (qx)5 f L2L1L3L4

C (qx)5 f L1L2L4L3C (qx) from which

8672 54L. WENDLER AND V. G. GRIGORYAN

Page 22: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

f L1L2L3L4C (qx) follows in the caseL1,L2 andL3,L4. Using the definition of the Laguerre polynomial

Lna~x!5 (

n50

n~21!m~n1a!!

~a1m!! ~n2m!!

xm

m!~A5!

and ~69c!

E0

`

dx xn21~x1b!2remx5b~n2r21!/2m~r2n21!/2ebm/2G~n!W~12n2r!/2 ,~n2r!/2~bm!, ~A6!

we obtain form51, r51/2, b5(qxl 0)2/2, n5(m21m41N/211/2), l5m21m41N/2:

f L1L2L3L4C ~qx!5~21!~N/2!1L11L222@N/211#AL1!L2!L3!L4!pe~qxl0 /2!2

3 (m250

L2

(m450

L4 ~22!2~m21m4!~2l21!!! @~qxl 0!2/2#~l21!/2

m2!m4! ~L22m2!! ~L42m4!! ~L12L21m2!! ~L32L41m4!!W2l/2,l/2F ~qxl 0!

2

2 G , ~A7!

where

W2l/2,l/2~x!5 (n50

l

CnlW0,n~x! ~A8!

is Whittaker’s function and

Cnl5H 2~21!l2nxl/2Y @~l2n!! ~l11! . . . ~l1n!! # if nÞ0

~21!lxl/2/l! if n50.~A9!

Using the relationW0,l(x)5Ax/pKn(x/2), it follows that

f L1L2L3L4C ~qx!5~21!L11L2dL11L21L31L4 ,2n

AL1!L2!L3!L4! (m250

L2

(m450

L4

(n50

l

ea~21!n

3alCn

l~2l21!!!Kn~a!

m2!m4! ~L22m2!! ~L42m4!! ~L12L21m2!! ~L32L41m4!! ~l2n!!, ~A10!

with n50,1,2, . . . , a5(qxl 0/2)2, l5m21m41(L12L21L32L4)/2, (2l21)!!51•3• . . . •(2l21), L1>L2 ,L3>L4, and

Cnl5H 1/@~l11! . . . ~l1n!# if nÞ0

1/2 if n50.~A11!

In the limit qxl 0!1, we use the long-wavelength relations for the modified Bessel functions~69d! in Eq.~A10!:K0(x)' ln22ln(x)2C andKn(x)'G(n)(2/x)n/2, whereC50.577 215 665 is Euler’s constant. Then we have up to the order(qxl 0)

2:

f L1L2L3L4C ~qx!5

1

2 H @22lnuqxl 0u13ln22C#dL1L2dL3L41~21!L11L2AL1!L2!L3!L4!

3 (m250

L2

(m450

L4 ~21!l~12d0l!~l21!!

m2!m4! ~L22m2!! ~L42m4!! ~L12L21m2!! ~L32L41m4!!J . ~A12!

APPENDIX B: MATRIX POLARIZATION FUNCTION

For the calculation of the matrix elementsai j , given in Eq.~61!, we need the follow sums:

SNN8~0!

~qx ,v![(kx

RePNN8~1!

~qx ,kxuv!, ~B1!

SNN8~1!

~qx ,v![(kx

kxRePNN8~1!

~qx ,kxuv!, ~B2!

54 8673COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .

Page 23: Collective and single-particle excitations of the quasi-one-dimensional electron gas in the presence of a magnetic field

SNN8~2!

~qx ,v![(kx

kx2RePNN8

~1!~qx ,kxuv!. ~B3!

These sums can be performed analytically with the following results:

SNN8~0!

~qx ,v!52me

p\2qx H lnU kF~N8!1qx2

2me

\qx@v2~N2N8!vc#

kF~N8!2

qx2

1me

\qx@v2~N2N8!vc#

U1 lnU kF~N!1qx2

1me

\qx@v2~N2N8!vc#

kF~N!2

qx2

2me

\qx@v2~N2N8!vc#

UJ ,~B4!

SNN8~1!

~qx ,v!5me

p\2qxH 2@kF

~N!2kF~N8!#2

p\2qxme

Fqx2 2me

\qx

3@v2~N2N8!vc#GSNN8~0!

~qx ,v!J , ~B5!

and

SNN8~2!

~qx ,v!52me

p\2qxH qx@3kF~N!2kF

~N8!#2me

\qx

3@v2~N2N8!vc#@kF~N8!2kF

~N!#2p\2qxme

3Fqx2 2me

\qx@v2~N2N8!vc#G2

3SNN8~0!

~qx ,v!J . ~B6!

In the limit of small wave vectors, i.e., ifaqx6

5uqxu/kF(0)6mev/\uqxu!1 is valid, it follows for N5N8

50 up to the order@aqx6 #2 ~Refs. 34 and 36!

S00~0!~qx ,v!52

2mekF~0!

p\2

1

~kF~0!!22S mev

\qxD 2 , ~B7!

S00~1!~qx ,v!5

2mekF~0!

p\2

qx2

2mev

\qx

~kF~0!!22S mev

\qxD 2 , ~B8!

and

SNN8~2!

~qx ,v!522mekF

~0!

p\2

~kF~0!!22

mev

\

~kF~0!!22S mev

\qxD 2 . ~B9!

If NÞN8 and for \kF(0)uqxu/2me , \qx

2/2me!v2uN2N8uvc,v1uN2N8uvc , we have

SNN8~0!

~qx ,v!52me

p\2qxH 2\qx~kF

~N!2kF~N8!!

me@v2~N2N8!vc#

2\2qx

3~kF~N!1kF

~N8!!

me2@v2~N2N8!vc#

2

12\3qx

3@~kF~N!!32~kF

~N8!!3#

3me3@v2~N2N8!vc#

3 J 1O~qx4!

~B10!

and from Eq.~B10! it follows for N51, N850 andN50,N851 up to the orderqx

2 ,

S10~0!~qx ,v!5

2kF~0!

p\~v2vc!, ~B11!

S10~1!~qx ,v!5

2~kF~0!!3qx

3pme~v2vc!2, ~B12!

S10~2!~qx ,v!5

2~kF~0!!3

3p\~v2vc!, ~B13!

S01~0!~qx ,v!52

2kF~0!

p\~v1vc!, ~B14!

S01~1!~qx ,v!5

2kF~0!qx

p\~v1vc!F12

\~kF~0!!2

3me~v1vc!G , ~B15!

and

S01~2!~qx ,v!52

2~kF~0!!3

3p\~v1vc!. ~B16!

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54 8675COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS . . .