VOLUME 72, NUMBER 10 PHYSICAL REVIEW LETTERS

Absence of Spin-Density Excitations in Quasi-Two-Dixne~sional Electron Systems

R. Decca, ~ A. Pinczuk, ~ S. Das Sarma, 2 B. S. Dennis, ~ L. N. Pfeifkr, ~ and K. W. West'ATST Bell Laboratories, Murray Hill, ¹mJersey 0797$

Department of Physics, University of Maryland, College Park, Maryland MES(Received 9 October 1993)

Inelastic light scattering spectra reveal a new behavior of electron-electron interactions in tvro-dimensiona1 double quantum vre11 semiconductor structures. It manifests in intersubband excitationsa&hen both subbauds are densely populated. In such high density 1imit, spin excitations, we11 de6nedwhen only one subband is populated, cannot be distinguished from single-particle transitions. Aquantitative analysis showa a collapse of vertex corrections (exciton shift), which we attribute tothe cancellation between contributions from electrons in the bvo subbands.

PACS numbers: 71.45.Gm, 73.20.Dx, 78.30.Fs

The study of the electron gas in semiconductor nano-structures has revealed remarkable new phenomena asso-ciated with electron-electron interactions under reduced

imensions [1]. In collective excitation spectra the shiftsfrom single-particle transition energies are a measure ofthe strength of Coulomb interactions [2]. The split-ting between spin-density collective excitations (SDE)and single-particle transitions (SPE) is due to exchange-correlation vertex-corrections (or exciton shift), while theenergy spacings between charge-density modes (CDE)and SDE follow from depolarization shifts due to directterms [2,3]. The need to understand fundamental inter-actions stimulates theoretical and experimental researchof the elementary excitation spectra. These studies seekthe mode energies and character with the aim of under-standing electron-electron interactions.

Resonant inelastic light scattering experiments giveunique insights into Coulomb interactions in the in-

tersubband excitations of the two-dimensional electrongas [3,4]. Spectra of sharp CDE and SDE, togetherwith broader SPE, enable quantitative determinationsof Coulomb interactions. SDE observed at energies well

below the SPE demonstrate that exchange vertex cor-rections are significant [3]. Different theoretical approxi-mations that explicitly take into account exchange inter-actions were used to describe such experimental results.The local-density approximation [2,5], variational tech-niques [6], and the solution of the Bethe-Sslpeter equa-tion [7] were used to calculate the irreducible polarizabil-ity of the 2D electron gas. These calculations show anexcellent agreement with light scattering results. In in-

frared optical absorption (FIR) experiments light coupleswith electric dipole excitations, SDE are not measuredand vertex corrections are inferred from energies of CDEmodes [8]. Surprisingly, RPA calculations [9], where ex-change is not considered, show agreement with FIR ab-sorption experiments in quasi-one-dimensional systemswith many occupied subbands [10].

Here we report on a new regime of intersubband ex-citations of the 2D electron gas, which reveals an unex-pected behavior of electron-electron interactions in low-

Antisym

CDE

Symm

Polari

Depol

4 I I I I

0 1 2 3 4Energy (meV)

FIG. 1. Inelastic light scattering spectra of Intersubbandexcitations of the DQW. The peaks of spin-derLsity excitations(SDE), charge-density exeitations (CDE), and si~&le-particleexeitations (SPE) are shown. The sample has only one sub-band occupied, g = 1.04. Inset: The syaunetric and. antisym-metric states of the DQW.

dimensional systems. At high densities, defined as therange showing substantial occupation of the two sub-bands, the energy spacing between SDE and SPE modesdecreases with increasing density. These determinationsreveal the existence of a "critical" density where SDE andSPE merge. A quantitative analysis indicates the abruptcollapse of vertex corrections when the critical density isapproached. Our results show that the evolution froma purely 2D system towards a more classical one is notcontinuous at all.

The excitations studied here by inelastic light scatter-ing are associated with transitions between the two low-

est subbands of double quantum wells (DQW). Free elec-trons in DQW are characterized by the parameter rl =~sos/EF, where bsAs is the symmetric-antisymmetricsplitting (see the inset of Fig. 1) and E~ is the Fermi en-

ergy. In these systems the critical density for the collapseof vertex corrections occurs at rl 0.12. We attribute

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VOLUME 72, NUMBER 10 PHYSICAL REVIEW LETTERS 7 MARcH 1994

this efFect to the cancellation between contributions tovertex corrections from particles in difFerent subbands.The collapse is not described by any of the current ap-proximations used to calculate the many-body electron-electron interactions, and we suggest second order Feyn-man diagrams that could explain our observations. Whilethe critical density may vary for other systems, the col-lapse of vertex corrections could be a behavior commonto electron gases with dense population of several sub-bands. A similar cancellation of vertex-corrections would

explain the success of RPA in dense 1D electron systems.The DQW were grown by molecular beam epitsxy.

They consist of two nominally identical GaAs quantumwells of width d = 139 A. separated by an Al Gap Asbarrier. The electrons are provided by remote Si-6-dopeddonor layers, set back f'rom each side of the DQW byAl Gaq As spacer layers. The total density is in the(0.75-6.35)x10~~ cm z range and the low temperaturemobility is in the range (0.5—2.0)xl0 . EsAs is easilytuned by changing the thickness and Al content of thebarrier [ll]. We selected symmetric DQW because highdensities, where rI & 1, can be easily reached. rI wasdetermined from values of b,sos and EF obtained in

spectra of inelastic light scattering and luminescence, re-spectively. Spectra were obtained at 1.7 K using a dyelaser tuned to excitonic optical transitions. A conven-tional backscattering geometry allowed in-plane wave-vector transfer by rotating the sample relative to the in-

cident light beam [3,12]. In this configuration the wavevector is q = ~&~ sin 8, where A is the wavelength of the in-cident light and 8 is the angle between the incident beamand the normal to the sample.

Figure 1 shows spectra obtained in a sample whereonly the lowest symmetric subband is occupied and

q & 0.5 x 104 cm ~. The sharp peaks with well definedpolarization selection rules arise from SDE and CDE[3,4,12]. SDE are active in depolarized spectra, where

incident and scattered light polarizations are perpendic-ular and CDE occur in polarized spectra measured withparallel polarizations .The collective modes have FWHM

0.2 meV, comparable to those in the best single quan-tum wells (SQW) [3]. The sharpness of the collectiveexcitations is consistent with the high electron mobility.

The spectra of Fig. 1 also show a broader peak at en-

ergies in between those of SDE and CDE. As in previouswork [3,4], this feature is assigned to SPE. For q « K~,where K~ is the Fermi wave vector, the intersubbandSPE occur within the continuum of energies bounded byb, sp, s 6 hqv~ [13], where v~ is the Fermi velocity. Re-sults for a sample with two densely populated subbandsare shown in Fig. 2. We see that the SPE broaden withincreasing q, showing the expected width of about 2hqv~,and, more importantly, two well-defined maxima emerge.This is the fingerprint of SPE in a 2D system with twooccupied subbands [13]. In the high density limit understudy the phase space for the transition is severely re-duced by the Pauli exclusion principle, and the only par-ticles that can be excited are those with v vF. In com-parison to q = 0, the efFect of phase-space reduction isnegligible for particles with k either parallel or antiparal-lel to q (whose transition energies are E b,s~s +hqv~),but it is very important when the particles have k per-pendicular to q, giving rise to the observed characteristicminimum. Figure 2 shows that the imaginary part of thepolarizability function of a noninteracting 2D electrongas with two occupied subbands, which represents thedensity of states for the SPE, gives a very good fit to thespectra [13].

Figure 3(a) shows depolarized spectra of samples withdifFerent rl. We normalized the energy shift by the b,sAsof each sample. The figure shows that the SDE move

Cm 1]

a.- Depolarized

SDE(~SA

EF

b.- Polarized

CDE

2.7

2.2

tl

COI 0.56

1.4

0.80.0

I I

1 2Energy (meV)

FIG. 2. VVave-vector dependence of the intersubband SPEof the sample arith g 0.12. The dashed line is a fit tothe experimeaial data using the polarizability function forthe intersubband transition of the noninteracting gas. The fitincorporates a phenomenological broade~~~g of 0.08 meV.

0.22

0.12I I I ~ ~ ~ I

05 10 15 1 2Normalized Energy (E/AsAs)

FIG. 3. (a) Depolarized spectra as a function of the nor-malized energy shift. For each spectrum the energy shiR hasbeen normalized by the As~s of the corresponding sample.The spectra are labeled with their values of q = bsxs/&s.(b) Same as (a), but for polarized spectra.

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VOLUME 72, NUMBER 10 PHYSICAL REVIEW LETTERS 7 MARCH 1994

closer to the SPE with decreasing g and disappears atthe critical density rl 0.12. On the other hand, theCDE remains always well defined, as can be seen in Fig.3(b). The disappearance of the intersubband SDE intothe intersubband SPE at high densities is the main resultof the present work. We show below that this behavioris due to a collapse of vertex corrections.

In order to obtain quantitative values for the depolar-ization shift n and exciton shift due to vertex correctionsP we use the usual parametrization [2—4]

and

4P —4)Q = t(ldCD)

2&SAS(ns nAs)

2 2+SAS ~SD

2&SAS(ns nAS)

where ucD (~sD) is the energy of the CDE (SDE), ns(nAs) is the electron density of the symmetric (antisym-metric) band, and s(u) is the dielectric function of thepolar lattice [5]. The factor (ns —nAs) takes into ac-count the reduction of available phase space when two

subbands are populated. From the measured values oftdcD wsD +sAs ns, and nAs, and using Eqs. (1) and(2), we obtain the values of o. and P displayed in Fig. 4.The figure shows that at q 0.25, P begins a relativelyabrupt decrease which ends in a full collapse at g 0.12.

In Fig. 4 the strength of vertex corrections when rl ) 1is the same as that obtained in 250 A asymmetric SQWwith similar electron densities. This is not unexpectedsince the matrix element for P is determined by thespread of the wave function, that is similar in the caseof the SQW and in our DQW of total width 300 A..However, here o. is substantially larger because the de-polarization shift is proportional to the overlap integralbetween the wave functions [2), which is larger in thesymmetric DQW. These considerations also explain thesmaller ratio & = 0.15 obtained in symmetric DQW.

The collapse of P is not caused by screening of theexchange interaction. Screening is related to the totalnumber of particles in the system and SQW with similartotal electron densities do not show the collapse. Further,in the system under study, the samples with g = 0.12,0.13, 0.15, 0.22, and 0.30 have a total density of 6.35,2.82, 5.05, 2.68, and 1.28 x10~~ cm 2, respectively. Thisshows that there is no correlation between P and the totalcarrier density.

The shift between SDE and SPE is dictated by theexchange interaction [2,6,7]. Given that the strengthof exchange interactions in DQW and in 250 A SQWare comparable, the collapse of vertex corrections shouldarise from a cancellation of terms when the two subbandsare occupied. In this framework, the key point of our ex-periment is that it reveals the importance of exchange in-

teractions with electrons in the antisymmetric subband,which must overwhelm the contribution of particles inthe lowest symmetric subband.

However, from the theoretical point of view the answeris far from trivial. The simplest contribution to vertexcorrections arises from the ladder diagrams, which arethe only vertex corrections included in any of the exist-ing response calculations using the generalized RPA. Theladder diagrams can be explicitly calculated in the caseof two subband occupancy. The calculation of P, definedin Eq. (2), with the only contribution of ladder diagramsdoes not show the observed vertex corrections collapse.For this reason the usual ladder diagrmns (diagrams 1.Iand 2.I in Fig. 5, together with all higher order ones) can-not be the explanation for the observed behavior. Theexperimental efFect is, therefore, beyond the TDLDA andladder diagrams. The lowest order vertex corrections be-yond the ladder diagrams are those shown as diagrams2.11 and 2.III in Fig. 5. The diagram 2.III does not par-ticipate in SDE vertex corrections because the Coulombinteraction is spin conserving [14]. A crude estimate ofdiagram 2.II shows that the vertex correction is indeed re-duced by this higher-order diagram when both subbandsare occupied. An explicit calculation is not possible to

7.0-6.0-5.0-4.0-

C)3.0-2.0-

Eo 1.0-)tD

E~ 0.5- O-I

0 0

0

2 —II 2 —III

0.00.0 0.2 0.4 0.6 0.8 1.0

~ = ~SA.«F

FIG. 4. Depolarization shift n and vertex corrections P asa function of g for the investigated samples. The line is aguide to the eye.

FIG. 5. Some topologicaQy distinct irreducible polarimabil-

ity diagrams in the first fern orders of the many-body pertur-bation theory. Diagram 2.11 (and its counterparts in aQ thehigher orders) is responsible for the observed collapse of ver-tex corrections. The indices i, j have the values 1 or 2 vrhenboth subbands are occupied (1 corresponds to the symmetricand 2 to the antisymmetric sllbbaxlds).

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VOLUME 72, NUMBER 10 PH YSICAL REVI EW LETTERS 7 MARCH 1994

carry out because one must sum an infinite series of diagrams (of the type 2-II) to obtain P. At this stage, all wecan say is that while we believe that these higher-order di-agrams are responsible for the observed vertex correctioncollapse, considerable theoretical work will be requiredbefore the situation can be clarified quantitatively [15].No cancellation is expected for the depolarization shift.

The observations of SPE in light scattering spectraare critical to our study. Figure 2 shows that the SPEhave the wave vector defined by the scattering geome-try. This demonstrates that their presence in spectraof high quality systems is intrinsic to resonant inelasticlight scattering, and rules out the disorder-assisted pro-cesses proposed in previous work [3]. The results indicatethat the resonant light scattering cross sections do nothave a simple proportional dependence with the dynam-ical structure factor, which would predict only collectivemodes, as in the nonresonant cases. At this time we donot have a full explanation. We note, however, that res-onant enhancements were previously invoked to explainthe unexpected observation of SPE in polarized spectraof bulk n-GaAs [16].

In conclusion, we have uncovered a new aspect ofelectron-electron interactions in the 2D electron gas. Itappears as a collapse of the exciton shift due to vertexcorrections in dense systems. We attribute the collapseto a cancellation of contributions when the two subbandsare densely populated. A similar collapse could also ex-plain the success of quantitative interpretations of FIRspectra that ignore exchange interactions in dense 1Delectron systems [9,10]. The theoretical interpretationis beyond the reach of commonly used approximations, afeature that is critically manifested when the occupanciesof two subbands are similar.

We had the benefit of discussions with G. S. Boebinger,J. P Eisens. tein, and Y. Garini. The work of S.D.S. atMaryland is supported by the U.S.-ARO, U.S.-ONR, andNSF. We want to acknowledge P. M. Platzman and P. B.Littlewood for a critical reading of the manuscript.

[1] For recent contributions, see Proceedings of the ¹nth

International Conference on Electronic Properties of 8-

Dimensional Systems (ENDS), Nara, Japan, 1991 [Surf.Sci. 263 (1992)]; Proceedings of the Tenth InternationalConference on Electronic Properties of 2-DimensionalSystems (EP2DS), Newport, USA, 1993 (to be pub-lished).

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[3] A. Pinczuk, S. Schmitt-Rink, G. Danan, J. P. Valladares,L. N. PfeifFer, and K. W. West, Phys. Rev. Lett. 63, 1633(1989).

[4] D. Gammon et al. , Phys. Rev. Lett. 68, 1884 (1992); D.Gammon et al. , Phys. Rev. B 41, 12311 (1991).

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[6] J. C. Ryan, Phys. Rev. B 43, 12406 (1991).[7] S. L. Chuang et al. , Phys. Rev. B 46, 1897 (1992).[8] M. O. Manasreh et aL, Phys. Rev. B 43, 9996 (1991).[9] Wei-ming Que and George Kirczenow, Phys. Rev. B 43,

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[10] W. Hansen et al. , Phys. Rev. Lett. 58, 2586 (1987); T.Demel et al. , Phys. Rev. B 38, 12732 (1988).

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[12] G. Abstreiter, R. Merlin, and A. Pinczuk, IEEE J. Quan-tum Electron. 22, 1771 (1986).

[13] J. K. Jain and S. Das Sarma, Phys. Rev. B 36, 5949(1987).

[14] It is interesting to note that not all irreducible polar-izability diagrams contribute to SDE as has commonlybeen assumed (e.g. , the diagram 2.III is an irreducible po-iarizability diagram which does not contribute to SDE).Only singlet-exciton diagrams where the electron andhole have opposite spins contribute.

[15] Explicit time-dependent local-density-approximation cal-culations [P. I. Tamborenea and S. Das Sarma (unpub-lished)] for the structures used in these experiments givegood agreement with the data only for g ) 0.3, but doesnot lead to the collapse of vertex corrections.

[16] A. Pinczuk, L. Brilison, E. Burstein, and E. Anastassakis,Phys. Rev. Lett. 27, 317 (1971).

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