PHYSICAL REVIEW B 70, 153305 (2004)

Intersubband electronic Raman scattering in narrow GaAs single quantum wells dominatedby single-particle excitations
Takeya Unuma,1,* Kensuke Kobayashi,1 Aishi Yamamoto,2 Masahiro Yoshita,1 Yoshiaki Hashimoto,1 Shingo Katsumoto,1Yasuhiro Iye,1 Yoshihiko Kanemitsu,2,† and Hidefumi Akiyama1
1Institute for Solid State Physics, University of Tokyo, andCREST, JST, Kashiwa, Chiba 277-8581, Japan
2Graduate School of Materials Science, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan(Received 8 June 2004; published 25 October 2004)
We measured resonant Raman scattering by intersubband electronic excitations in GaAs/AlAs single quantum wells (QWs) with well widths ranging from 8.5 to 18 nm. In narrow (less than ⬃10 nm) QWs with
sufficiently high electron concentrations, only single-particle excitations (SPEs) were observed in intersubband
Raman scattering, which was confirmed by the well-width dependence of Raman spectra. We found characteristic variations in Raman shift and line shape for SPEs with incident photon energy in the narrow QWs.
DOI: 10.1103/PhysRevB.70.153305

PACS number(s): 78.67.De, 78.30.Fs, 73.21.Fg

Intersubband electronic excitations in quantum wells(QWs) are strongly affected by many-body Coulombinteractions.1 In electronic Raman scattering in doped QWs,
two types of intersubband collective excitations have been
confirmed by many researchers2 since the first reports by
Abstreiter and Ploog3 and by Pinczuk et al.4 in 1979: a collective charge-density wave (CDW) appears if the polarizations of incident and scattered lights are parallel 共储兲, whereas
a collective spin-density wave (SDW) appears if they arecrossed 共⬜兲.
The CDW and SDW excitation energies are given by1,5ECD ⬇ E10 + 共␣ − ␤兲NS ,共1兲

ESD ⬇ E10 − ␤NS ,共2兲

based on the local-density-functional theory,6–8 where NS is
the sheet electron concentration in a QW, E10 is the intersubband energy separation that includes static many-body corrections, and ␣NS and ␤NS are dynamical many-body corrections (called the depolarization shift and excitonic shift) due
to the direct and exchange-correlation intersubband Coulombinteractions, respectively. In the crudest approximation,8,9 thedepolarization shift ␣NS is proportional to NSdeff and the excitonic shift ␤NS is proportional to 共NS / deff兲1/3, where deff isthe effective thickness of the two-dimensional electron gas.1
Experimentally, ␣NS can be determined from the measured
value of ECD − ESD; ␤NS can be estimated only when there isinformation about E10.
Later, in 1989, Pinczuk et al. reported10 that not only
CDW and SDW but also single-particle excitations (SPEs)
are observed in intersubband electronic Raman scatteringand their transition energy is E10, for modulation-doped
25-nm GaAs/ Al0.3Ga0.7As single QWs with NS = 共1.5–3兲
⫻ 1011 cm−2. SPEs are observed for both the parallel andcrossed polarizations and seem to become stronger10,11 at
higher values of NS.
There are many other reports on intersubband electronic
Raman scattering for wide (more than ⬃20 nm) GaAs QWs.
It will be interesting to measure Raman scattering also in
1098-0121/2004/70(15)/153305(4)/$22.50

2 −2/3narrower QWs, considering ␤ / ␣ ⬃ 共NSdeff
兲 . However,there are only a few reports for narrower GaAs QWs12,13
because interface roughness makes Raman peaks broader
and more difficult to observe. Such an experiment has not yet
been reported for a single QW.
In this paper, we report resonant Raman scattering by thelowest 共0 → 1兲 intersubband electronic excitations in a set ofmodulation-doped GaAs/AlAs single QWs with well widths
ranging from 8.5 to 18 nm. We found unexpected results thatparallel- and crossed-polarization Raman spectra of narrow
QWs had only a single peak at almost the same energy,
though electron concentrations in the QWs were high enoughthat the theoretical values of the depolarization shift weremore than 10 meV. On the other hand, spectra of a wide QW
had a typical feature in that CDW, SDW, and SPE peaks all
appeared clearly. In an intermediate-width QW we found no
SDW peak, a very small CDW peak, and large SPE peaks,
which led us to conclude that the peaks observed in narrow
QWs correspond to the SPEs. Moreover, characteristic variations in Raman shift and line shape with incident photon
energy were observed commonly in the narrow QWs. They
were found to have a correlation with resonance strength andcan be qualitatively understood by considering relaxation of
the in-plane wave-vector conservation rule.
The samples used in this study were n-type modulationdoped GaAs/AlAs single QWs grown by molecular-beam
epitaxy on (001) GaAs substrates: N1, N2, W, and M. Their
well widths L ranged from 8.5 to 18 nm, as shown in Table I.
The QW structure of sample N1, for example, consisted of aSi-doped Al0.33Ga0.67As layer, a 4.0-nm undoped
Al0.33Ga0.67As spacer layer, a 6.0-nm undoped AlAs barrier,a 10-nm undoped GaAs QW, a 6.0-nm undoped AlAs barrier,and a Si-doped Al0.33Ga0.67As layer. The other samples hadsimilar QW structures. The samples were designed such that
calculated values of the depolarization shifts1 were morethan 10 meV, which is larger than the linewidths of intersubband transition14 and large enough to see in Raman spectra.
The electron concentrations were controlled mainly by thethickness of spacer layers. Exceptionally, In0.1Ga0.9As alloy
was used for the QW layer of sample N2 in order to raise its

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TABLE I. Main parameters of the samples. L is the well width, NS is the sheet electron concentration, and␮ is the mobility at 4.2 K. Theoretical values of the depolarization shift are calculated after Ref. 1.

SampleWell/barrier

L共nm兲

NS
共1012 cm−2兲

␮
共104 cm2 / Vs兲

Calculated depolarization shift(meV)

N1N2W
M

GaAs/AlAs
In0.1Ga0.9As/ AlAs
GaAs/AlAs
GaAs/AlAs

108.5
18
13.5

1.2
2.0
0.63
1.1

4.3
2.8
44
5.6

1114912electron concentration sufficiently. The sheet electron concentrations NS and and mobilities ␮ at 4.2 K are also shownin Table I. The values of NS were confirmed by
Shubnikov-de Haas measurements for samples N1, W, andM, and by Hall measurements for sample N2.
Raman spectra were measured at 12 K in a geometry ofbackscattering normal to QW layers such that in-plane wavevector transfer from an incident photon to an electron wasless than ⬃104 cm−1. Typical values of incident laser powerand spot size were 30 mW and 0.3 mm2, respectively. Finding a suitable resonance condition is crucial for electronic
Raman experiments, particularly for those on narrow single
QWs, in order to make Raman signals intensified and avoid
their overlap with the strong photoluminescence (PL) fromthe GaAs QW or AlGaAs layers. In our experiments, theincident photon energies Einc were probably resonant withthe energy gap between the electron first-excited 共E1兲 subband and the heavy-hole second-excited 共H2兲 subband in aGaAs QW.15 This was the only resonance condition thatmade intersubband Raman scattering observable in the energy range between the PL peak positions for the QW andAlGaAs layers in samples N1, N2, and M.
Figure 1(a) shows typical intersubband Raman spectra ofsample N1. The backgrounds of the spectra were due to PL
from the QW. The parallel- and crossed-polarization spectra
have only a single peak at almost the same energies of 134.2
and 134.0 meV, respectively, while the theoretical value ofthe depolarization shift calculated after Ref. 1 was about 11
meV for the QW with NS = 1.2⫻ 1012 cm−2. A similar polarization dependence was observed in sample N2, as shown inFig. 1(b), where NS = 2.0⫻ 1012 cm−2 and the calculated depolarization shift was 14 meV. Apparently, these experimental results do not correspond to the conventional interpretation for wide QWs that CDW and SDW peaks are in theparallel- and crossed-polarization spectra, respectively, andtheir energy difference is the depolarization shift. We assigned the Raman peaks in Fig. 1 to SPEs, which were confirmed by the well-width dependence of Raman spectra as
discussed in later paragraphs.
The resonance condition for sample N1 had an allowable
range of about 70 meV in the vicinity of Einc = 1.80 eV, due
to the electron Fermi distribution and the relaxation of thein-plane wave-vector conservation rule caused by interface
roughness or other scatterers. In this range, we measured
Raman spectra of sample N1 and found peculiar variations inRaman shift and line shape with incident photon energy.
Similar characteristics were also found in sample N2.
Figure 2 shows the spectra of sample N1 at various inci-dent photon energies Einc: (a) 1.790, (b) 1.807, (c) 1.817, and(d) 1.835 eV. In Fig. 2(a), the parallel- and crossedpolarization spectra have almost the same peak positions. InFig. 2(b), however, the crossed-polarization spectrum has aslightly higher peak position than the parallel-polarization
spectrum. In Fig. 2(c), on the other hand, the parallelpolarization spectrum has a slightly higher peak position
than the crossed-polarization spectrum. In Fig. 2(d), theparallel-polarization spectrum again has a slightly higher
peak position than the crossed-polarization spectrum, andtheir line shapes are very asymmetric. Note, on the whole inFig. 2, that the polarization dependence of the peak positionat each Einc is small compared with the theoretically expected depolarization shift of 11 meV, and that the Raman

FIG. 1. Intersubband electronic Raman spectra of (a) 10-nm
GaAs/AlAs (sample N1) and (b) 8.5-nm In0.1Ga0.9As/ AlAs (sampleN2) single QWs at 12 K. The 储 and ⬜ marks mean that incidentand scattered-light polarizations were parallel and crossed,respectively.

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FIG. 4. Intersubband electronic Raman spectra of (a) 18-nm(sample W) and (b) 13.5-nm (sample M) GaAs/AlAs single QWs at
12 K. 储: parallel polarizations, ⬜: crossed polarizations.

FIG. 2. Intersubband electronic Raman spectra of a 10-nm
GaAs/AlAs QW (sample N1) at various incident photon energies
Einc: (a) 1.790, (b) 1.807, (c) 1.817, and (d) 1.835 eV. 储: parallel
polarizations, ⬜: crossed polarizations. (e) and (f) are schematic
diagrams of intersubband electronic excitations with the relaxationof the in-plane wave-vector conservation rule at low and high Einc,respectively.

shift of the peak position for the parallel polarizations becomes larger as Einc increases from 1.790 to 1.835 eV.
Figure 3 shows the Raman shifts of peak position (circles)
and the Raman integrated intensities (squares) versus incident photon energies in sample N1. Open and closed symbols correspond to the values for the parallel and crossed
polarizations, respectively. In Fig. 3, there is a correlation
between the Raman shift and the integrated intensity: theRaman shift becomes smaller as the resonance gets stronger.
Near Einc = 1.81 eV, where the Raman shift is larger for theFIG. 3. Raman shift of peak position and Raman integrated
intensity vs incident photon energy in a 10-nm GaAs/AlAs QW(sample N1). Open and closed symbols show the values for theparallel- and crossed-polarization spectra, respectively.

crossed polarizations than for the parallel polarizations, theresonance curve for the crossed polarizations has a dip. At
Einc = 1.83–1.85 eV, where spectra have larger Raman shifts
and very asymmetric line shapes as shown in Fig. 2(d), theresonance is weaker for both the parallel and crossed polarizations.
Next, to clarify the origin of the intersubband Ramanpeaks observed in samples N1 and N2, we also measured
Raman spectra in wider QWs with well widths of 18 (sampleW) and 13.5 nm (sample M), and investigated how spectralfeatures changed with well width. Figure 4(a) shows Raman
spectra of sample W. The CDW peak at 56.1 meV and theSDW peak at 45.1 meV are clearly seen in parallel- andcrossed-polarization spectra, respectively, as well as the SPEpeaks at 48.3 meV in both the spectra. The energy differenceof 11.0 meV between the CDW and SDW is in good agreement with the theoretical calculation of the depolarization
shift. This is a conventional result that follows Refs. 10 and11, and other reports.
Figure 4(b) shows Raman spectra of sample M, which had
an intermediate well width: the large SPE peaks at 69.5 meVfor both polarizations and the small CDW peak at 81.8 meVfor the parallel polarizations. (Sharp peaks at 72.6 and 69.8
meV for the parallel polarizations are from 2-LO-phonon
excitations in GaAs and AlGaAs layers, respectively.) The
SDW peak is not seen, probably because it is too small oroverlapped by the SPE peak. The change in spectral featureswith well width shows that the peaks observed in samples N1
and N2 should be assigned to the SPEs.
The electronic Raman process for intersubband SPEs ismicroscopically described as follows: An electron is excitedfrom a valence-band state to a conduction first-excited subband state 关k1 , E10 + ␧共k1兲兴 by an incident photon with theresonant energy Einc, where ␧共k兲 = ប2k2 / 2mⴱ with k being
the electron in-plane wave vector and mⴱ being the effective
mass. Then, an electron in a conduction ground subband
state 关k0 , ␧共k0兲兴 recombines with the virtual hole remaining
in the valence-band state, which produces a scattered photonwith energy Escatt. As a result, an intersubband SPE
关k0 , ␧共k0兲兴 → 关k1 , E10 + ␧共k1兲兴 occurs.
In a geometry of backscattering normal to QW layers,
basically, the change in the in-plane wave vector q = k1 − k0 is
0 and the Raman shift Einc − Escatt for the intersubband SPE is153305-3

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E10, since photon in-plane wave vectors are negligible. Infact, there are some dephasing mechanisms that cause broadening of the in-plane wave vectors of the electron and hole
states involved, or the relaxation of the in-plane wave-vector
conservation rule; so q ⫽ 0 intersubband SPE processes arealso allowed.
Figure 2(e) schematically shows a standard case of兩k1 兩 ⬍ kF for intersubband SPEs, where kF is the Fermi wave
vector. In this case, various q ⫽ 0 excitation processes only
cause a symmetric width of Raman spectra around E10,
which we think corresponds to the case of Einc ⬍ 1.83 eV in
our experiments. Near the higher limit of Einc for intersubband Raman scattering, however, 兩k1兩 is more than ⬃kF as
shown in Fig. 2(f). In this case, particular q ⫽ 0 excitation
processes such that k0 · q ⬎ 0 are dominant and Einc− Escatt ⬎ E10. Thus, Raman spectra have a larger Raman shift
and more asymmetric line shape (with a high-energy tail) at
Einc = 1.83–1.85 eV than at Einc ⬍ 1.83 eV. It is not clear atpresent which carrier state was most deeply involved in thein-plane wave-vector conservation relaxation: the electron
ground state, the electron first-excited state, or the virtualhole state.
In the above discussions, we have assumed for simplicity
that electron subbands have the same dispersion ␧共k兲. Here,
we add a note on the case of their different dispersions due toband nonparabolicity. In general, nonparabolicity is expected
to make the Raman shifts of intersubband SPE peaks smallerwith increasing incident photon energy, but to have little influence on their line shapes in resonant Raman scattering,
because only carriers around k1, where resonance occurs, areinvolved. With regard to our results for sample N1, the Raman shifts of the intersubband SPE peaks become larger as
incident photon energy increases (Fig. 3), which suggeststhat nonparabolicity has a smaller effect on Raman scatteringin narrow GaAs QWs than the relaxation of the in-planewave-vector conservation rule.
In summary, we have measured intersubband electronic
Raman scattering in GaAs/AlAs single QWs with wellwidths ranging from 8.5 to 18 nm. The change in Raman
spectral features with well width shows that only SPEs havebeen observed in narrow (8.5 and 10 nm) QWs for bothparallel and crossed polarizations. Variations in Raman shift
and line shape with incident photon energy have been foundin the narrow QWs and qualitatively explained by considering q ⫽ 0 intersubband SPEs inside and across kF.

*Electronic address: unuma@issp.u-tokyo.ac.jp

radius in GaAs, the exchange-correlation potential Vxc关n共z兲兴 has
an approximate dependence of −1 / rs, leading to ␤NS⬃ 共NS / deff兲1/3.
10
A. Pinczuk, S. Schmitt-Rink, G. Danan, J. P. Valladares, L. N.
Pfeiffer, and K. W. West, Phys. Rev. Lett. 63, 1633 (1989).
11 H. Peric, B. Jusserand, D. Richards, and B. Etienne, Phys. Rev. B
47, 12 722 (1993).
12 G. Abstreiter, T. Egeler, S. Beeck, A. Seilmeier, H. J. Hübner, G.
Weimann, and W. Schlapp, Surf. Sci. 196, 613 (1988).
13 M. Ramsteiner, J. D. Ralston, P. Koidl, B. Dischler, H. Biebl, J.
Wagner, and H. Ennen, J. Appl. Phys. 67, 3900 (1990).
14 T. Unuma, M. Yoshita, T. Noda, H. Sakaki, and H. Akiyama, J.
Appl. Phys. 93, 1586 (2003); T. Unuma, T. Takahashi, T. Noda,M. Yoshita, H. Sakaki, M. Baba, and H. Akiyama, Appl. Phys.
Lett. 78, 3448 (2001).
15 G. Danan, A. Pinczuk, J. P. Valladares, L. N. Pfeiffer, K. W. West,and C. W. Tu, Phys. Rev. B 39, 5512 (1989).

†Also at: Institute for Chemical Research, Kyoto University, Uji,Kyoto 611-0011, Japan.
T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437(1982).
2 For a review, see A. Pinczuk and G. Abstreiter, Light Scatteringin Solids V (Springer-Verlag, Berlin, 1988), pp. 153–211.
3 G. Abstreiter and K. Ploog, Phys. Rev. Lett. 42, 1308 (1979).
4 A. Pinczuk, H. L. Störmer, R. Dingle, J. M. Worlock, W. Wiegmann, and A. C. Gossard, Solid State Commun. 32, 1001
(1979).
5 We define ␣ and ␤ following the convention of Ref. 10. They arerelated to Ando’s convention by ␣ = ␣AndoE10 / 共2NS兲 and ␤= ␤AndoE10 / 共2NS兲.
6 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
7 L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).
8 W. L. Bloss, J. Appl. Phys. 66, 3639 (1989).
9 When r = 关4␲a 3n共z兲 / 3兴−1/3 is much less than unity, where n共z兲 iss0
the three-dimensional electron concentration and a0 is the Bohr1

We are grateful to T. Ando for his helpful discussions
about the theory of intersubband electronic Raman scattering. This work was partly supported by a Grant-in-Aid from
the Ministry of Education, Culture, Sports, Science, andTechnology, Japan. One of us (T.U.) also thanks the Japan
Society for the Promotion of Science for partial financial
support.

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