......
Surface Science NorthHolland
......................._,_,:j:.:.:,,,,,; ii:.~::::::.:::::::::::::jgi:jj::::::::::~:~:
:::‘:‘:” ‘.‘.‘. ..‘..i. . . .../ ,,,, :,,,,, ,,,,, . . . .A’:‘:‘: : : :,::::.:.:.:.y i......... ............ . . . . .. . . . . .,.,.,,.,.,,, ,,,,:,:;,.,;,, :::j:::,:,:,::: :.:,....... +.....
263 (1992) 451456
surface science ::.,.>... ..,.(. $:j Z.) ‘:‘;.:.:‘~.r:‘::.:.:....: ...../.... ~,.:.~~......~..:+y::::j::
:.:.:.::.:.:.y ““““““.~.‘.~.):.:.,:.:.~ .:.,,.,.,,,,,/,,,,,, .......A.. _,:,,,,,,(,,,,,,(,_ ,,,,:.,:jj::.:.. ‘:‘=‘~~~~~~ ‘.‘.‘........ .................:.:.:. ,,.,.,.,~,~.:,~.,.,,,,,; ““:~~‘.:.::~.::::i:::~jjijj::::.:.::::::::: ~,,,./,...
Magnetorotons
in wide parabolic quantum wells
K. Karrai a, X. Ying ‘, H.D. Drew a, M. Santos b, M. Shayegan b, S.R.E. Yang ’ and A.H. MacDonald d a Joint Program for Adcanced Materials, Department of Physics and Astronomy, University of Maryland, College Park,
MD 20742, USA
and Laboratory for Physical Sciences, College Park, MD 20740, USA ’ Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA ’ National Research Council of Canada, Ottawa, Ontario, Canada KIA OR6 d Department of Physics and Material Research Institute, Indiana Unir~ersity,Bloomington, IN 47405, USA Received
4 June
1991; accepted
for publication
26 August
1991
The magnetoplasmon dispersion relation of a quasithreedimensional electron gas confined in a wide AI,Ga, _,As parabolic quantum well is investigated in the magnetic quantum limit. Magnetoplasma excitations polarized perpendicular to the magnetic field and with wave vector 4 of the order of the inverse magnetic length l/la are measured in far infrared magnetotransmission experiments on 100 nm thick electron slabs. A minimum is found in the magnetoplasma mode frequency at qlo = 2 suggesting the existence of a magnetoroton excitation. The collective excitation spectrum is calculated in analogy with Feynman’s theory for rotons in helium, and is found to be in good qualitative agreement with the measurement.
The dispersion of plasma modes in interacting electron systems is sensitive to the manybody correlations. Consequently, the mode dispersion w(q) versus wave vector q of plasma excitations in a degenerate threedimensional electron system has been extensively studied both experimentally and theoretically [l]. The case of an electron gas in a magnetic field B sufficiently high that all the electrons are in the lowest magnetic level (the extreme quantum limit) is particularly interesting since the system becomes susceptible to novel correlated ground states. So far the magnetoplasmon dispersion relation for this situation has been little studied. Measurements of w(q) in this context can be used to monitor possible phase transitions in the interacting electron system. Although this regime has been treated theoretically using a magnetohydrodynamic model [3,4] and the randomphase approximation (RPA) [5], the existing theoretical treatments of this problem are not expected to give a valid description of the magnetoplasma mode dispersion when q becomes larger than the inverse average interelec00396028/92/$05.00
0 1992  Elsevier
Science
Publishers
tron separation. So far experiments in the interesting range of parameters have not been feasible. In bulk semiconductors, in which the quantum limit can be reached, the electrons interact dominantly with the ionized donors making the carrier lifetimes short and the magnetoplasma properties uninformative [2]. In remotely doped twodimensional electron systems this difficulty is avoided. Theoretically, a collective excitation, which is the analog of the roton excitation in superfluid 4He was predicted for twodimensional electron ’ systems (2DES) [6,7]. Recently, Raman spectroscopy measurements on 2DES in the extreme quantum limit, indicated the existence of extrema in the magnetoplasma mode dispersion, supporting the existence of a roton minimum [Sl. These measurements alone, however, could not lead to the q dependence of the magnetoplasma modes. Recently, quasithreedimensional electron systems (Q3DES) have been realized in remotely doped wide parabolic Al,Ga,_,As quantum wells [9171. They have attracted much attention since it is now possible
B.V. and Yamada
Science
Foundation.
All rights reserved
to obtain highmobility quasithreedimensional electron slabs (100 to 300 nm thick) with densities in the range of 5 x 101’ to 5 X 10lh cm‘, that remain metallic in magnetic fields as high as 17 T at temperatures as low as 25 mK. In this paper, we report an investigation of the magnetoplasma mode dispersion of Q3DES up to very large wave vectors with 0 I 41,) 2 6 where I,, is the magnetic length. The magnetofarinfrared properties of Q3 DES confined in wide parabolic wells were recently reported with B applied in the plane of the electron gas [lo13,161. In this geometry, a single resonance is observed at a frequency w= (0: + wf,)‘/* where w, is the cyclotron frequency and wC, s the harmonic oscillator frequency characteristic of the parabolic well. When the electron slab is thick compared to I,,, this excitation is the plasma shifted cyclotron resonance [10,11,13], and the plasma frequency is w,, = w(,. In particular, it has been shown [11,13,14] that the only mode of the parabolic well that couples to a homogeneous (i.e., JJ = 0) electromagnetic excitation is the plasma shifted cyclotron resonance mode which corresponds to the uniform oscillation of the center of mass of the electron system. This result, is known as the generalized Kohn theorem [14]. It is possible, however, to achieve a controlled breakdown of the generalized Kohn theorem by confining a quasithreedimensional electron system in a wide parabolic well which has small controlled deviations from the parabolic shape [13171 grown into its potential profile. In this paper we report measurements on wide parabolic wells with a set of planar 6 potential perturbations grown in a periodic array (period u = 20 and 30 nm> along the growth axis of the sample. For an infinitely thick electron slab, the periodic perturbation leads to zone folding of the plasmon dispersion relation at 9 = r/a and r/a allowing, therefore, the electromagnetic radiation to couple to the new modes at the center of the artificial Brillouin zone. If the perturbations created by the array of 6 potentials are small enough, it is possible to reconstruct the bulk dispersion relation at q = 0, 2r/a, 4r/a,. .. of the magnetoplasmons with polarizations normal to the electron slab.
We have studied four samples grown by molecular beam epitaxy. The first sample is a pure parabolic well grown with o,, = 48 cm ‘. Two other samples were grown with a = 20 nm. One of them was grown with an array of positil~ 6 planar potentials obtained by periodically superimposing a 0.5 nm thick layer of Ga, , Al, As with _Y= 0.005 on a 300 nm wide Ga, ~, Al, As parabolic well. Each Ga, ._,.A1 ,.As layer is a 6 perturbing potential with an integrated strength I = +2.S meV nm. In the other a = 20 nm sample we periodically inserted a GaAs layer in the wide parabolic well. The inserted GaAs layers constitute a periodic array of negutil~ planar 6 perturbations for the electrons in the conduction band. The width of the GaAs layers are such that the 6 perturbations have an integrated strength of I = ~ 5 meV nm. The fourth sample was grown with u = 30 nm and a periodic array of planar 6 perturbations with I = +2.5 meV nm. This latter sample was grown in order to achieve ql,, = 2 at 7 T. In all samples the thicknesses of the inplane perturbations are kept small compared to I,, so that the perturbations can be considered as ii potentials. The Al profile of all samples wcrc characterized by secondary ion mass spectroscopy in order to measure the thickness of the wide parabolic well, and therefore the period n. During the magnetooptical measurements it was possiblc to reduce persistently the electron arcal density N, using a red light emitting diode placed near the sample. A’, ranges typically from 10”’ to 2.5 x 10” cm‘. At the largest electron densities, the effective thickness of the electron slab, given approximately as W = N,/n, , is W = 100 nm, where II, is the threedimensional electron density determined optically from wr, [lo.1 1,131. The number of 6 planar perturbations experienced by the electrons depends on W (hence on y,) and varies from 1 to 6 in the present work. Magnetoplasma excitations were measured in the quantum limit (i.e., only one spin degenerate Landau level occupied) with B (O9 T) applied in the plane of the electron slab. In this (Voigt) geometry the electromagnetic radiation, which is polarized perpendicular to B, propagates perpendicular to the electron slab. The samples were immersed in liquid helium and maintained at a
K. Karrai’et al. / Magnetorotons in wide parabolic quantum wells
temperature of 1.4 K. The excitations were measured by sweeping B at fixed laser frequencies. The farinfrared radiation, which was generated by an optically pumped moleculargas laser, was propagated to the sample by light pipe optics, and was linearly polarized a few millimeters before the sample. We detected the transmitted light using a composite Ge bolometer located outside the magnetic field at 4.2 K. The sample substrate was wedged 3’ in order to avoid lineshape distortion due to interference fringes. Typical transmission spectra for the three samples grown with arrays of 6 planar perturbations are shown in fig. 1. At high enough magnetic fields (i.e., when the quantum limit is reached), a satellite resonance develops on the high magnetic field side of the plasma shifted cyclotron resonance. As shown in fig. 2, this resonance is not present in the transmission spectrum of the non
l.05 piLzzLz
_
0.95
3
0.90

0.85

0.80

0.75

T=1.45
r
1.00
K
0 z \
E z
6
5 B(T)
6
5
B(T)
Fig. 2. Left panel: magnetotransmission spectra of a pure parabolic well (w, = 48 cm‘) taken for increasing N, are shown. From top to bottom the traces correspond to increasing W from 20 to 100 nm. Right panel: similar data are shown for a parabolic well with a periodic array of 6 planar potential perturbations (period a = 30 nm). As W is increased, a first satellite resonance whose frequency reduces with increasing W appears. We identify this resonance as a dimensional mode. As W further increases, another resonance appears which we identify as the magnetoplasma mode w(q = 27r/a).
T=1.4 VOIGT
K GEOMETRY
I
O
1
2
3
4
5
6
7
5
B(T) Fig. 1. Magnetotransmission spectra are shown for three samples. (A): sample with negative 6 planar periodic potential, period a = 20 nm, wP = 51 cm‘; (B): sample with positive 6 planar periodic perturbations, a = 20 nm, or = 64 cm ‘; (C): same as (B) with a= 30 nm and w,,= 48 cm‘. The numbers above the spectra indicate the farinfrared radiation wavelength in brn. The spectra are displaced vertically for clarity.
perturbed wide parabolic quantum well. It appears in the transmission spectra of the samples with positive as well as negative S periodic perturbations. In fig. 3 we show the position of both the main and the satellite mode in a w2 versus B2 plot. A similar dependence is observed for all samples containing a periodic 6 planar perturbation. We attribute this new resonant mode to the excitation of the magnetoplasma mode with a periodic charge distribution of period a. The coupling to this new resonant mode can be understood as follows. In the vicinity of the plasma shifted cyclotron resonance field, the radiation polarizes the electron slab along the growth axis. The resulting charge distribution in the periodically perturbed parabolic confining potential leads to a selfconsistent field with Fourier
components q = 27rN/u (N = I, 2. 3, . . ). In order to reconstruct the dispersion of the magnetoplasma mode, we have plotted the normalized satellite frequency (w2  w~)‘/‘/w, against the effective wave vector q1,, = 2rl,,/a in fig. 4. The corresponding data points are shown for 5 5 B I 7 T and a = 20 and 30 nm. On the same plot the position of the main resonance was also plotted in a similar way (open symbols). As expected, the main mode shows no dependence on ql,, since it represents the uniform plasma mode at q = 0. As W is decreased, an additional satellite resonance appears closer to the plasma shifted cyclotron resonance. The mode shifts away from the main mode as W decreases. We identify this new mode as a dimensional resonance. Since the slab has a finite effective width W, standing plasma waves with displacements normal to the electron slab (i.e., the dimensional resonances)
0.6
1
2
a
a
~20rlrn,
30nm/ a 2w
l
I.,’ 0
.
.
I 3
4
5
6
Znl,/a Fig. 1. Normnlizcd
dispetsion of the magnetoplasma
The dashed lines are the calculated mode
approximation
for
w,, = 51
disperAon cm
‘.
mode.
in the Gngle
Filled
symholx:
squares are data taken on samples with (I = 20 nm. triangle\: N = 30 nm, circles are the dimensional plasma shifted for the satellite
cyclotron
mode plotted
resonance.
at ql,, = r! is identified
modes. Open symbols: vt’rsu~ (/I,, as defined
The deep minimum as the “roton
that appear\
minimum”
10000
5000
2
wP
Main resonance Satellite resonance
l
1
n=2.5x101s/cm3
o ,.I.’ 0
,
I
10
/
20
I
30
J
I
/
40
50
60
‘70
B2 (T2) Fig.
3.
w’
versus
B’
dependence
cyclotron
resonance
(squares)
are shown. The
dashed
magnetoplasma
mode
calculated nm
and
(circles)
wP = 51 cm ‘.
The
of
the
plasma
and of the satellite
line is the position
of the
w(q = 2r/u).
u = 20
dotted
resonance.
shifted
resonance
line
with
is the
cyclotron
[4,16 181 are also expected. The array of H potcntial perturbations within the slab should allow the farinfrared radiation to couple to such magnetoplasma excitations at cffectivc y = NT/F+ (N = 1, 2. 3, . . .I. Larger effective r~ arc favored for small W. hence smaller N,. Transmission spectra taken for different values of W are shown in fig. 2. It is not yet clear which standing modes can be excited in the present arrangement. Dipole active modes cannot be reliably identified by symmetry arguments using this present experimental configuration. A finite static internal electric field would shift the electron slab along the growth direction with respect to the array of 6 potentials breaking down the reflection symmetry of the system. In fig. 4, we have plotted the dispersion of this new mode assuming q = NT/IV with N = 1 or 2 (filled circles). The uncertainty N = 1 or N = 2 is represented in the horizontal error bars.
K. Karrai’et al. / Magnetorotons in wide parabolic quantum wells
The overall reconstructed mode dispersion indicates the existence of a minimum at gI, = 2. In order to relate these experimental observations to theory we have calculated the dispersion relation of threedimensional magnetoplasmons propagating perpendicular to B. We used the singlemode approximation (SMA) in a calculation very similar to that presented in ref. 171.The calculations were carried out using a HartreeFock ground state. The results are presented in fig. 4 for two different magnetic fields. The pronounced minimum found at ql, = 2 can be understood in analogy with the theory of ref. [7] as a roton minimum. In fig. 2 the calculated w* versus B2 dependence is shown for q = 2rr/a (a = 20 nm). It should be noted that in the SMA, the mode dispersion is entirely characterized by the ground state of the electron system. Our calculations show that, at higher magnetic fields and ground state is not lower wp, the HartreeFock the lowest energy state. Therefore, mapping the mode dispersion at higher fields and lower or, would provide, within the SMA, a direct signature of any electronic phase transitions. Several experimental aspects remain to be clarified. The higher harmonic q = 4r/a has not been observed. One difficulty may be due the to finite number of 6 perturbations experienced by the electron gas (up to 6 6 perturbations). Also, the frequency region where the higher harmonics of the plasma mode are expected to occur has a strong absorption background due to the plasma shifted cyclotron resonance and the first harmonic mode. The effect of the cornmensuration of the q = 2n/a mode to the dimensional mode is not yet clearly understood. Finally, we remark that a more quantitative theory of the mode coupling is needed. In conclusion, using molecular beam epitaxial growth techniques we have realized a quasithreedimensional system with an imbedded grating coupler which permits measurements of the magnetoplasma mode dispersion up to very large effective wave vectors. The measured dispersion relation is in a good qualitative agreement with our SMA calculations for a threedimensional electron gas in the quantum limit. Our results suggest the existence of a roton minimum in the
455
magnetoplasma mode dispersion of threedimensional electron systems.
Acknowledgements
We would like to thank L. Brey, J. Dempsey, H. Fertig, and J.P. Kotthaus for fruitful discussions. This work was supported, in part, by NSF grant nos. DMR9000553 and DMR8921073.
References [l] See, e.g., D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963). [2] B.I. Halperin, Jpn. J. Appl. Phys. 24, Suppl. 263 (1987) 1913. [3] A.L. Fetter, Phys. Rev. B 32 (1985) 7676. [4] J. Dempsey and B.I. Halperin, Bull. Am. Phys. Sot. 3 (1991) 1046. [5] N.J. Horing, Ann. Phys. 31 (1965) 1. 161 C. Kallin and B.I. Halperin, Phys. Rev. B 30 (1984) 5655; 31 (19851 3635. [7] A.H MacDonald, H.C.A. Oji and S.M. Girvin, Phys. Rev. Lett. 55 (1985) 2208; S.M. Girvin, A.H. MacDonald and P.M. Platzman, Phys. Rev. Lett. 54 (1985) 581; Phys. Rev. B 33 (1986) 2481. [S] A. Pinczuk, J.P. Valladares, D. Heinman, A.C. Gossard, J.H. English, C.W. Tu, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 61 (1988) 2701. [91 M. Shayegan, J. Jo, Y.W. Suen, M. Santos and V.J. Goldman, Phys. Rev. Lett. 65 (1990) 2916, and references therein. [lOI K. Karrai, H.D. Drew, M.W. Lee and M. Shayegan, Phys. Rev. B 39 (1989) 1426. [ill K. Karrai, X. Ying, H.D. Drew and M. Shayegan, Phys. Rev. B 40 (1989) 12020; Surf. Sci. 229 (1990) 515. [=I K. Karrai’, M. Stopa, X. Ying, H.D. Drew, S. Das Sarma and M. Shayegan, Phys. Rev. B 42 (1990) 9732. [131 K. Karrai, X. Ying, H.D. Drew, M. Santos and M. Shayegan, in: 20th Int. Conf. on the Physics of Semiconductors, Eds. E.M. Anastassakis and J.D. Joannopoulos (World Scientific, Singapore, 19901 p. 1278. [141 L. Brey, N.F. Johnson and B.I. Halperin, Phys. Rev. B 40 (1989) 10647. [I51 L. Brey, J. Dempsey, N.F. Johnson and B.I. Halperin, Phys. Rev. B 42 (1990) 1240. [161 A. Wixforth, M. Sundaram, J.H. English and A.C. Gossard, in: 20th Int. Conf. on the Physics of Semiconductors, Eds. E.M. Anastassakis and J.D. Joannopoulos (World Scientific, Singapore, 19901 p. 1705.
456
K. Karrai et al. / Magnetorotors
[17] A. Wixforth, M. Sundaram, K. Ensslin, J.H. English and A.C. Gossard, Phys. Rev. B 43 (1991) 10000. [IS] J. Alsmeier, E. Batke and J.P. Kotthaus. Phys. Rev. B 49 (1989) 12574;
in wide parabolic quantum
wells
J. Alsmeier, J.P. Kotthaus, T.M. Klapwijk and S. Bakker. in: 20th Int. Conf. on the Physics of Semiconductors, Eds. E.M. Anastassakis and J.D. Joannapoulos (World Scientific. Singapore. 1990) p. 2355.