A. Wixforth, M. Kaloudis, C. Rocke, K. Ensslin

Sektion Physik, Universität München, D-8000 München 22, Germany

and

M. Sundaram, J.H. English, and A.C. Gossard

Materials Department, University of California, Santa Barbara, California 93106

The far - infrared response of electron systems confined in a parabolic potential is investigated experimentally. We present recent studies on cyclotron resonance, intersubband resonance and hybrid modes using various experimental techniques and geometries and compare our experimental findings with theoretical results based on a generalization of Kohn's Theorem. We also focus on the far-infrared spectrum of so-called imperfect parabolic wells with intentionally induced deviations from ideal parabolicity. Here, new symmetry - forbidden resonances are observed which yield a sensitive test for a theoretical description. At finite wave vector intra-subband plasmonic excitations are also possible. We study their dispersion and investigate their interaction with the intersubband modes. If subjected to an in-plane magnetic field, the intra-subband plasmon exhibits a strongly anisotropic dispersion which can be directly related to so-called one dimensional plasmons. Due to the simplicity of the confining potentials most of our experimental results can be explained in straightforward ways. Many of our results as well as the theoretical descriptions directly apply to recent investigations of lateral nanostructures and thus may serve for a better understanding also of this rapidly developing field.

PACS numbers : 73.20.Dx, 73.20.Mf, 73.40.Kp

**I. INTRODUCTION **

The interaction between electron systems in semiconductor quantum well structures and optical fields has been studied intensively during the last two decades [1]. These studies include intersubband-absorption, cyclotron resonance in high magnetic fields as well as plasmon emission and absorption. The collective excitation spectrum of an electronic system contains valuable information as it is one of its most fundamental properties. For quasi two-dimensional electron systems (Q2DES, quantum films) as realized in space charge layers in semiconductors the study of plasmonic (intrasubband-) excitations as well as intersubband-transitions have proven invaluable in the characterization and understanding of these systems [2]. More recently [3], also the collective excitations in quasi-one-dimensional (Q1DES, quantum wires) [4,5] and quasi-zero-dimensional electron systems [6] (Q0DES, quantum dots) have attracted very much attention. This is because in the last few years the realization of lateral nanostructures has become possible which yielded a rapidly growing field of interest in semiconductor physics. On the other hand tremendous improvements of semiconductor growth techniques like molecular-beam-epitaxy (MBE) nowadays offer the possibility to engineer practically every desired kind of bandstructure for semiconductor structures.

Here, we like to review some of our recent experimental results on the
far-infrared response of electron systems confined in parabolic potentials
[7..12, and references therein]. The systems investigated are realized in
so-called parabolic quantum wells (PQW). Originally, they have been proposed as
an attempt to realize the theoretical construct of *jellium *by creating a
wide, highly mobile quasi three-dimensional electron system (Q3DES) [13]. Such
a system in the presence of strong magnetic fields is expected to show
interesting and exotic properties: Depending on the strength of the field,
different kinds of broken-symmetry ground states like spin-density-, and
charge-density waves or crystallization of the electron system in form of the
Wigner crystal have been proposed [14,15].

Another very attractive feature of an electron system in a PQW is the striking similarity of many of its properties to the ones of a Q1DES or even a Q0DES. Many experimental results obtained in such nanostructures have in the recent past been successfully described in terms of a parabolic confining potential [3]. As a matter of fact, early experiments on the FIR response of a PQW [16,17] have resulted in the generalization of Kohn's theorem [18,19,20], which subsequently has been used to explain and understand many interesting properties of laterally confined low-dimensional systems. For these reasons, throughout this review article, we shall point out the similarities and the applicability of our experimental and theoretical results to the case of lateral nanostructures and give representative examples.

In Sec. II of this report, we first give a brief review of some main aspects and features of parabolic quantum wells and the electron systems under investigation. We shortly discuss the fabrication and the characterization of these structures and present some fundamentals of the resulting subband spectrum. It will be followed by Sec. III which is devoted to a short description of the generalized Kohn theorem and further existing theories that have been used to model our and others experimental results. In Sec. IV we present some experimental details and describe the sample-geometries used in our experiments. The next section (V) focuses on experimental results and their discussion. We first present some results as obtained in 'ideal' PQW to demonstrate the connection with Kohn's theorem and some other fundamental theoretical models that have been developed in the recent past. We then switch to so-called 'imperfect' PQW, i.e. structures where we intentionally introduce some degree of controlled non-parabolicity to study its influence on the FIR spectrum. We show that we are able to electrically alter specific samples between the 'perfect' and the 'imperfect' state and thus study the influence on, e.g., non-local interactions which are also very important in the interpretation of the FIR response of lateral nanostructures. The last paragraph of Sec. V is devoted to experiments at finite wave vector in the FIR. Here, we use a grating coupler technique to couple to intra-subband plasmonic excitations. Using this technique, we gain access to the complete collective excitation spectrum at small wave-vector of an electron slab of finite width, resulting in bulk- and surface-like modes with characteristic dispersion. We study their interaction with the inter-subband resonances, and by using an in-plane magnetic field we investigate the field-induced anisotropic bandstructure of a semiconductor space charge layer. These experiments are very closely related to the recent observation of one-dimensional plasmons in mesa-etched quantum wires [5].

**II. ELECTRON SYSTEMS IN PARABOLIC QUANTUM WELLS**

Besides their application in graded-index semiconductor (GRINSCH)
lasers parabolic quantum wells have first been successfully studied in optical
experiments by Miller, Gossard, Kleinman, and Munteanu [21]. The typical,
equally spaced energy spectrum of such parabolic structures that has been
observed in photoluminescence excitation experiments demonstrated the
possibility of a very precise control of molecular beam growth resulting in
graded-bandgap structures. The parabolic profiles of both the conduction as
well as the valence bands have been obtained by properly grading the aluminum
content of the ternary AlxGa1-xAs alloy . In the range 0 <= x <= 0.3 the
band gap of AlxGa1-xAs varies nearly linearly with the Al mole fraction such
that a controlled variation of x directly leads to the desired structure.
Subsequently, remotely doped PQW have been proposed to result in very wide (W
<= 0.6um) and highly mobile (u <=
30m^{2}V^{-1}sec^{-1}) quasi three-dimensional electron
systems [13] . The basic idea of these structures is to create a conduction
band profile EC(z) in the growth direction such that it mimics the parabolic
potential of a uniformly distributed slab of positive charge.

In Fig. 1 we depict the basic ideas that led to the realization of such
structures [22]. In (a), we schematically plot the conduction band profile of a
conventional narrow rectangular quantum well in a remotely doped semiconductor
heterostructure. Remote doping spatially separates the dopants form the free
carriers, thus increasing the electron mobility by a considerable amount. The
width of this confining potential is of the order of the de Broglie wavelength
of the electrons, and the system forms a Q2DES with quantized levels in the
growth-direction and a free dispersion along the heterointerfaces. Part (b)
depicts the situation for a wide rectangular quantum well. The resulting
selfconsistent electron distribution is far from being a uniform and wide
electron slab. Instead, electrostatics leads to a band bending such that charge
accumulates near the edges of the well and in general two independent Q2DES are
formed. Such systems, however, are of special interest themselves since here a
possibility is given to investigate the properties of coupled electron systems
in great detail [23]. Fig. 1(c) shows a sketch of the conduction band profile
of a graded PQW. Once this structure is remotely doped, the donors release
electrons into the well which in turn will screen the man-made parabolic
potential, thus forming a wide and nearly homogeneous electron layer (d). An
undoped layer between the dopants and the well reduces ionized impurity
scattering and thus enhances the electron mobility in the well like for
conventional heterostrucures. The bare man-made parabolic potential is
equivalent to the potential of a positive charge density n^{+}. This
fictitious charge is related to the curvature of the grown parabola by
Poisson's equation, namely

**
**(1)

Here, [[epsilon]] denotes the mean dielectric constant of AlxGa1-xAs, [[Delta]] is the energy height of the parabola from its bottom to the edges, e the electronic charge, and W the width of the grown PQW.

The curvature of the parabola and thus the fictitious charge n^{+} can
be varied over a wide range by proper control of the growth process. Due to the
similarity to a real existing positive space charge this concept has been
referred to as 'quasi-doping' in the past [13]. In Fig. 2, we show the two
basic attempts that have been employed to grow PQW by means of MBE. The goal is
a graded aluminum profile that can be controlled very precisely. In (a) we show
the technique of the so-called digital alloy. A short period (2nm) superlattice
of GaAs and Al0.3Ga0.7As is grown such that the duty cycle between the two
species varies quadratically during the growth. The average Al content is then
given by the solid line. A prerequisite for this procedure is a very short
period of the superlattice with thin wells and barriers to allow for nearly
perfect tunnelling between the wells. In (b) a different, more straightforward
technique is shown. By varying the crucible temperature of the aluminum furnace
from, e.g., T=1000^{o}C to T=1300^{o}C one can control the Al
flux towards the substrate in a quite controlled manner. The problem, however,
is the large thermal inertia of the crucibles which makes it difficult to avoid
feed-back-oscillations of the growth rate. All the samples presented in this
report are grown using the digital technique, no essential differences have
been observed due to the superlattice structure as compared to 'analog'
samples. The actual profile of the Al mole fraction is not easily measured
directly. Its general shape is sometimes deduced from optical and electrical
measurements on the resulting structures, followed by a fitting of the data to
the calculations for the desired energy bandgap profile. However, it is very
useful, at any rate, to be able to calibrate the deposited Al mole fraction
versus depth in a direct way. A usual way is to employ analytical methods like
secondary ion mass spectroscopy (SIMS) which is, however, quite time consuming
and not straightforward. Recently, we presented a technique [24] which is
simple and reproducible having reasonable resolution and accuracy. This
technique is based on a calibration run directly before growth of the actual
PQW and monitoring the Al flux directly using a fast ion gauge scheme. Using
this method we were able to calibrate our grown PQW to within [[Delta]]xAl
<= 0.01 and to show that our digital alloy approach produces more accurate
parabolic profiles than we have been able to produce by analog alloying.

The parameters of the four different samples investigated in this report are
listed in Table 1 where they are identified by the wafer number. Two different
types of parabolic quantum wells are used : (a) Three PQW with different
curvatures and additional vertical sidewalls (PB25, PB26, and PB31), and (b)
for comparison one PQW (PB48) without vertical sidewalls but with the same
curvature as PB25. The difference between PB26 and PB31 is a higher electron
mobility of the latter, leading to narrower linewidths in the FIR spectra.
Using a conduction band offset between Al0.3Ga0.7As and GaAs of 65% we find
from the relation [[Delta]]Ec=750meV*XAl and eq. (1) the values for the
quasi-charge density n^{+} for the different wells as given in the last
row of table 1. Of course, many more PQW with other parameters have been
investigated but the scope of this report is perfectly covered by the choice of
data and samples as presented here.

We now turn to a short description of the resulting density distribution and
subband spectrum of an electron system in such a PQW. Mobile carriers that are
introduced by remote doping will tend to screen the external potential of the
positive quasi-charge n^{+} leading to a wide and nearly homogeneous
electron slab. This behavior is already intuitively clear from Fig. 1(c) and
(d). A more detailed description, however, requires a self consistent
calculation [25]. There have been many different investigations of the self
consistency for the case of a PQW and we wish here to review only the basic
results. For the case of an empty PQW the subband spectrum is the one of a
harmonic oscillator with a characteristic energy

(2)

which is solely determined by the design and growth of the structure under
consideration. As the well is filled, the electrons distribute themselves to
form a wide layer and the resulting conduction band profile is essentially flat
over the occupied width of the PQW. Then, the single-particle subband spectrum
resembles more the one of a rectangular quantum well of width W, leading to
non-equally spaced subband energies. In first approximation these are given by
Eo*n^{2}, with Eo of the order of 0.1 to 1 meV, depending on the actual
sample. In other words, with decreasing carrier density and related to this
with decreasing width of the electron layer the subband energies increase
tending towards the equal spacing of the empty well given by eq. (2).

Due to the smallness of the subband energies usually more than one subband is occupied for typical carrier densities in the wells. This is demonstrated in Fig. 3, where we plot the results of a fully self consistent calculation for the carrier density and the subband energies of one of the samples (PB31) investigated. In (a) the calculated carrier densities in the different electrical subbands are depicted as a function of the applied bias Vg between a metal gate electrode on top of the sample surface and the electron system in the well. The inset shows schematically the sample geometry used in all our experiments. Four indium pellets are alloyed to the electron system to form an Ohmic contact. The gate electrode consists of a thin (5nm) NiCr layer also serving as a semitransparent electrode compatible with FIR spectroscopy. On some samples in addition a highly conducting Ag grating coupler is deposited on top of the gate electrode. This grating coupler is used to couple FIR to both inter- and intrasubband collective modes and is described in detail in Sec. IV. One sees how with increasing negative bias Vg the total carrier density Ntot decreases and at some particular gate voltages the electrical subbands become depopulated. At the same time the selfconsistent Hartree potential adjusts itself such that the corresponding subband energies Ei increase as a function of decreasing carrier density as expected from the above simple arguments. This behavior is shown in Fig. 3b, where we plot the calculated subband energies Ei-Ef as a function of the gate bias. Here, Ef denotes the Fermi level which is held to be Ef =0 in the figure.

In Fig. 4 we show the self consistent potential (thick solid line) together
with the calculated wavefunctions weighted by the carrier densities
in the subbands as a function of depth in growth direction z. The total
carrier density in this case is Ns =
2.4^{.}10^{11}cm^{-2}, corresponding to a gate bias of
Vg=0V. The total width We of the electron slab in this case is close to
We=115nm as expected from the simple relation

(3)

For this particular sample and the given carrier density three electrical subbands are occupied and the resulting total density distribution as well as the conduction band profile are nearly flat. Experimentally, the decreasing width of the slab as well as the shift of the potential minimum and thus the center of mass of the total wavefunction can be determined by, for example, a measurement of the gate capacitance as a function of the gate bias [25]. Also, the subsequent depopulation of electrical subbands can be observed using this technique.

**III. LONG-WAVELENGTH SPECTROSCOPY ON PARABOLICALLY CONFINED ELECTRON
SYSTEMS**

Shortly after the first successful realization of remotely doped PQW some very enlightening initial experiments of Karaii and co-workers [16,17] stimulated a lot of further experimental as well as theoretical work on this subject. Subsequently many more interesting features of parabolic quantum wells have been investigated both experimentally as well as theoretically. Both (magneto-)transport [26, 27] as well as FIR investigations [28] uncovered a large amount of new and interesting results which shed some light onto the understanding of many fundamental properties of low-dimensional electron systems. Here, we restrict ourselves to a short description of those experiments related to the subject of the present report and refer the reader to recent review articles on, for example, the investigation of the single particle spectrum on similar structure as obtained in magnetotransport experiments [27].

**(a) The Generalized Kohn Theorem**

In a tilted magnetic field experiment similar to the one of Schlesinger and
co-workers and Wieck and co-workers [29] to study the resonant interaction of
subband- and Landau-levels in a Q2DES the authors of ref. [16] observed that
the FIR response of the electron system in a PQW is governed by a single, well
defined frequency very close to the plasma frequency of a three dimensional
electron system of density n^{+} which *by construction* is given
by [[omega]]o in Eq. (1). Initially, this behavior was interpreted in terms of
a successful realization of a highly mobile Q3DES. In a celebrated theoretical
article, however, Brey, Johnson, and Halperin [19] explained this result by a
generalization of Kohn's theorem [18]. It states that in a purely parabolically
confined electron system long-wavelength radiation only couples to the center
of mass coordinates and its motion. The reason is the decoupling of the
center-of-mass modes of the interacting electron gas from its internal modes.
Relative coordinates and thus particularly electron-electron interactions in
such systems do not affect the resonance frequency of the observed transitions.
We here shortly review the basic idea of the generalized Kohn theorem which has
been formulated in the recent past by numerous authors. We directly follow the
work of Yip [20] and use the same formalism: Suppose an electron system in a
three-dimensional parabolic confining potential subjected to a magnetic field.
Putting
,
the Hamiltonian in this case in its most general form can be written as:

**
**(4)

Here, the three-dimensional parabolic potential is parametrized in terms of the characteristic frequencies [[omega]]i, the influence of the magnetic field is included via [[pi]]j, and electron-electron interaction is included via an interaction potential U which depends only on relative coordinates (rj-rk). N represents the total number of carriers in the well. Yip now introduces center of mass and relative coordinates of the form

(5)

and similar for Y^{(2)}, ...,Y^{(N)}, Z^{(2)}, ...,
Z^{(N)}, and [[Pi]]^{(2)}, ..., [[Pi]]^{([[Nu]])}. If
the following relations are used for the coordinates xj,

(6)

and equivalently for yj,zj, and [[pi]]j, it can be shown that now the total Hamiltonian can be separated in two terms H = HC.M. + Hrel., where HC.M. is given by

(7)

and the term Hrel. being more complicated but only involving the relative coordinates and momenta. Since the external magnetic field is assumed to be uniform across the sample, A(ri) is a linear functional and thus . It follows that [HC.M., Hrel.]=0 and .

The center of mass and the relative motion thus separate and the eigenfrequencies are identical to those of a single electron in the bare parabolic confining potential. The interaction of the electron system with the optical field of the long-wavelength FIR is expressed via a uniform, time-dependent perturbation

(8)

which leads to

(9)

In other words, in a purely (one-dimensional) parabolic confining external potential only the frequency [[omega]]o given by Eq. (2) is observed in a FIR experiment, independent of the choice of the electron-electron interaction. This mode is of inter-subband type and represents a sloshing of the whole electron system, represented by its center of mass (CM) in the external parabolic potential. Meanwhile, many different experiments including those on Q1DES and Q0DES have proven the validity of this statement [3]. It was shown that the generalization of Kohn's theorem indeed holds and that the observed intersubband-like resonance frequency of an electron system in a parabolic confining potential is independent of electron-electron interactions and thus the actual number of carriers in the well.

**(b) Imperfect Parabolic Quantum Wells**

** **So far we have been dealing with electrons in ideal parabolic confining
potentials. Here, the generalized Kohn theorem is directly applicable. No real
system, however, can be regarded as to be completely ideal. First, no
artificial potential is infinitely deep, in other words, at some finite
energies there must be a cutoff where parabolicity is no longer maintained.
Secondly, there might be some unintentionally induced non-parabolic terms in
the potential which are related to some imperfection during the growth of the
structures. Even more serious is the situation for quantum wires and -dots
where the parabolic model is certainly only valid in first order. Thus, it is
highly desirable to gain some insight into the behavior and FIR response of
electron systems which are confined in 'imperfect' parabolic potentials. As far
as MBE grown quantum wells are concerned we have the possibility to tailor any
kind of desired bandstructure artificially and thus are able to intentionally
induce a certain degree of nonparabolicity [9,30]. These structures then may
serve to a better understanding of many properties also of quantum dots and
wires where the confining potential is not exactly known a priori. Early
experimental results on such samples which will be addressed in Sec. V of this
report have indeed triggered a lot of valuable theoretical work on the subject
of imperfect PQW such that we now are in the pleasant situation to also have
some knowledge about the properties of nearly parabolic confining potentials,
where Kohn's theorem is no longer strictly applicable. Most interesting is the
fact that in such systems symmetry-forbidden transitions can be observed which
occur due to nonlocal interactions in the electron system. Such calculations
have been performed either by using a hydrodynamic approach [31] or a fully
selfconsistent calculation in a random-phase-approximation (RPA) framework
[32..35]. In Sec. V of this report we shall directly apply the results of the
existing theories to our experimental findings and also give more details on
the models that have been developed and used in the past.

**(c) Finite Wavevector**

** **A very interesting subject of investigation is the collective
excitation spectrum of electron systems at finite wave vector q. Here, the
coupling of FIR to intra- subband plasmons also becomes possible [11,12,36,37].
These are collective modes of charge density oscillation with surface
excitation character. For Q2DES there exists a whole library of literature on
those modes which has proven invaluable to the understanding of the collective
spectrum of low-dimensional systems [2]. More recently, the collective
excitation spectrum of Q1DES has also attracted very much attention. Demel and
co-workers [5] succeeded in the observation of so-called one-dimensional
plasmons which represent a charge density oscillation along the free direction
of a quantum wire. A very characteristic magnetic field dispersion of these
modes has been observed which indicates the edge-excitation character of such
collective modes. On the other hand, the resonant coupling between intra- and
inter-subband collective excitation is also of great interest since here much
information on the dynamic behavior of low-dimensional electron systems can be
obtained. Such coupling has been first observed by Oelting and co-workers [38]
for a 2DES in an elegant experiment after it has been theoretically predicted
by Das Sarma [39]. More recently Li and Das Sarma [40] as well as Gold [41]
focused on the interaction of 1D intersubband resonances with 1D intra-subband
plasmons. Here a resonant interaction is more likely as compared to the 2D
case since for typical sample parameters, both modes are energetically of
approximately the same size.

For a Q3DES of finite width a strict separation of surface- and bulk-character of the plasma modes is no longer possible once the thickness of the electron slab becomes comparable to the inverse of the wave-vector of the excitation. Then, strong mode coupling occurs leading to a characteristic dispersion of the modes. In a PQW one has the possibility to change both the carrier density in the well as well as the effective width of the electron slab which makes it a favorable subject for very detailed investigation of these interesting facts.

**IV. EXPERIMENTAL REMARKS**

** **All our experiments are performed in transmission using a rapid scan
Fourier transform spectrometer (BRUKER IFS 113) connected to a low temperature
(T >= 2K) and high magnetic field (B <= 15T) system. The samples under
investigation are mounted on a sample stage centered in the super conducting
solenoid and providing the possibility to tilt the sample with respect to the
magnetic field direction ( 0 <= [[Theta]] <= [[pi]]/2). Usually
unpolarized FIR is normally incident so that we can investigate all
configurations between Faraday ( [[Theta]] = 0) and Voigt-geometry ( [[Theta]]
= [[pi]]/2 ). A thin NiCr layer on top of the sample serves as a
semitransparent gate electrode and Indium-pellets are alloyed to the electron
system to provide Ohmic contacts. Application of a negative bias VG between the
gate electrode and the electron system tends to deplete the well as already
mentioned in Sec. II. Experimentally, we determine the relative change in
transmission -[[Delta]]T/T = [T(0)-T(NS)]/T(0) which is proportional to the
real part of the effective conductivity
of the system, depending on the actual polarization used in the experiment
[42]. T(0) is the transmission of the sample with the well being completely
depleted, T(NS) the one at finite carrier densities. A silicon composite
bolometer held at T=2K is used to detect the transmitted radiation. On some
samples also silver grating couplers have been prepared either using standard
optical lithography or holographic techniques for the shorter period gratings.
The periodicities of the grating couplers used in our experiments varies
between a = 6 um and a = 0.8 um to provide both the necessary z-component of
the FIR to couple directly to inter-subband-like excitations as well as the
finite q= 2[[pi]]/a for the surface (intra-subband) plasmon experiments [2].
The sample substrate is wedged to about 3 degrees to avoid disturbing
interference effects on the transmission spectra.

**V. EXPERIMENTAL RESULTS AND DISCUSSION**

**(a) Ideal Parabolic Quantum Wells**

In this section we shall concentrate on a review of the FIR response of what
we call 'ideal' PQW, i.e., quantum wells with no intentionally induced
deviation from parabolicity. Such samples are represented by PB26, a 200 nm
wide PQW, where the Al mole fraction was varied during the growth between x = 0
in the center to x = 0.2 at the edges and with additional vertical sidewalls
up to x = 0.3, and a 130 nm wide PQW (PB48) with x going from zero to x =0.3 at
the edges, respectively. For PB 26 this corresponds to a curvature of the
parabola which simulates a positive background charge of n^{+ }=
2.1^{.}10^{16}cm^{-3}, leading to an expected
characteristic energy of
**=
**5.8 meV
47cm^{-1}. For PB 48 we expect from growth
**=
**10.7 meV
86cm^{-1}. The sheet carrier density of both samples at Vg = 0V is
about Ns = 3^{.}10^{11} cm^{-2}, the electron mobility
for both samples is of the order of u = 15
m^{2}V^{-1}sec^{-1}.

We first present the results of tilted magnetic field experiments, i.e., where coupling to the inter-subband like mode is achieved by a resonant coupling of the cyclotron resonance to the sloshing mode of the center of mass [[omega]]o. Since for an ideal PQW due to Kohn's theorem only this mode couples to the CR in a tilted magnetic field [19]. In this case the theoretical description of the interaction is particularly simple and can be treated like the coupling of two harmonic oscillators of frequencies [[omega]]o and [[omega]]c.The tilt angle enters the problem via the normal and parallel projections of the CR, namely and . This problem has been treated originally by Maan [43] and independently by Merlin [44]. The result is the occurrence of an mode-anticrossing between both resonances if the magnetic field is varied such that [[omega]]o and [[omega]]c energetically degenerate. The resonance positions in this case are given by the simple expression

(10)

For harmonic oscillators the resulting gap due to the anticrossing is solely determined by the tilt angle, whereas for more complicated potential profiles in general the appropriate matrix elements are involved. The oscillator strengths of both CM-modes are also very easily calculated:

For low magnetic fields, [[omega]]- has the character of the cyclotron resonance, whereas [[omega]]+ represents an intersubband-like mode. The situation is reversed for high magnetic fields and in the magnetic field regime where both modes are energetically degenerate, they exchange oscillator strength. The results of eqs. (10) and (11) are depicted in Fig. 5, where we plot the resonance positions of as a function of the total magnetic field in units of [[omega]]o and tilt angles of [[Theta]] = 0, 20, and 45 degrees. For [[Theta]] != 0, the degeneracy between both modes is lifted, resulting in a characteristic anticrossing of the two lines.

A typical example for the mode-anticrossing at a very small tilt angle is
given in Fig. 6, where we show a series of observed cyclotron-resonance lines
for sample PB26 at different magnetic fields between B=2.5 T and B= 4.0 T using
small magnetic field steps of [[Delta]]B=0.05 T. Clearly one observes the
region of interaction resulting in a strong modulation of the CR lineshape. At
B = Bo ~ 3.5T and [[omega]] = [[omega]]o ~ 46 cm^{-1}, there is a sharp
dip in the envelope of the spectra, reflecting the exchange of oscillator
strength and the anticrossing of both lines involved. The small tilt angle in
this case is only given by the wedge angle of the sample, namely [[Theta]]~3
degrees. Even here, clearly an interaction between the CR and the sloshing mode
of the center of mass of the electron system is detectable although both modes
are not yet strictly separated.

In Fig. 7, we depict the result of a tilted field experiment on sample PB48 for three different tilt angles. The symbols represent the extracted resonance positions and the solid lines are the result of a calculation according to eq. (10). For all three experiments the same parameters have been used as given in the inset of the figure. The agreement between our experimental results and the simple model calculation is nearly perfect for all three angles investigated. The small but reproducible deviations between theory and experiment at higher magnetic fields are due to band-nonparabolicity, which has not been included in the calculation.

In Fig. 8 we show the result of the tilted field experiment for PB48 at a tilt
angle of [[Theta]]=23^{o} together with the amplitude of the observed
resonances which has been normalized to the one of the CR in perpendicular
magnetic fields. Taking the width of the resonance fixed, which is justified
here, this quantity is proportional to the oscillator strength. The dashed
lines represents the prediction in the simple harmonic oscillator picture as
given in eq. (11). Especially at low magnetic fields there exists a quite large
scatter around the expected value for the oscillator strength. This we believe
is caused by effects arising from the population and depopulation of different
subbands in the system: The different spatial extend of the wavefunctions in
different subbands leads to the occurrence of an 'artificial nonparabolicity'
[7], resulting in a superposition of different CR lines which then may
complicate the proper evaluation of the oscillator strength. Also, the
effective electron mass m* turns out to be a function of position in the well.
These effects as well as those of subband depopulation and intersubband
scattering, however, are beyond the scope of the present report and will be
discussed elsewhere.

Under the extreme condition of an in-plane magnetic field
([[Theta]]=90^{o} , Voigt-geometry), only [[omega]]+ is observable. The
electrical and the magnetic confinement act in the same direction and the CR
hybridizes with the oscillation in the bare potential as represented by
[[omega]]o. For small magnetic fields, electrical quantization is stronger than
the magnetic one and the observed resonance frequency is close to [[omega]]o.
For higher magnetic fields the magnetic quantization becomes dominant and the
observed resonance approaches [[omega]]c. In a well with perfect parabolicity
this resonance is a discrete excitation that is split off from the continuum of
inter-Landau level transitions by collective polarization effects. The
resonance position and the oscillator strength of the hybrid mode are then
given by

and (12)

Here, it is interesting to note that exactly the same magnetic field dispersion is expected and in fact observed for quantum wires (Q1DES) with parabolic confinement in a perpendicular magnetic field [3], so that our PQW in Voigt geometry may serve as a perfect model system for the investigations of such Q1DES. The result of such an experiment on a PQW is shown in Fig. 9, where we plot the extracted resonance position together with the amplitude of the line for sample PB26. Again, it has been normalized to the one of the CR in Faraday-geometry. The symbols represent the experimental results and the lines are the result of the calculation according to Eq. (12). The origin of the large deviation of the measured oscillator strength from the expected value above B= 12T is not yet known. For the geometry of this experiment and the related FIR polarization the oscillator strength vanishes for B 0. Using a grating coupler technique, however, we are able to follow the resonance down to B=0.

An example for such a direct observation of the intersubband-like sloshing
mode of a PQW [9] is given in Fig. 10. Here, we plot the result of a grating
coupler induced resonance for sample PB26 for three different gate voltages Vg
and correspondingly three different carrier densities in the well. To induce
longitudinal electric components in the FIR radiation that are transmitted
through the sample parallel to the growth direction we use a 70 nm thick Ag
grating coupler of periodicity d= 6 um that is deposited in addition to the
semitransparent NiCr gate electrode on top of the sample. Although the carrier
density in the well is changed from Ns =
2.5^{.}10^{11}cm^{-2} at Vg = 0V down to Ns =
1.6^{.}10^{11}cm^{-2} at Vg=-0.3V, the position of the
grating coupler induced resonance has not changed. The decrease of the carrier
density, however, is reflected in a reduced oscillator strength as can be seen
in Fig. 10. The exact amplitude of the observed resonance depends not only on
the dynamic conductivity [[sigma]]zz([[omega]]) and thus on the carrier density
Ns but also on the efficiency of the grating coupler used in the experiment.
For this reason, we cannot directly compare different spectra obtained for
different samples and grating couplers.

The resonance positions of the inter-subband like sloshing mode in sample PB26
as obtained from all the different experimental techniques are summarized in
Fig. 11 as a function of the gate bias Vg. Different symbols represent the
results of different experiments as given in the inset. Part (b) of Fig. 11
sketches the change in width of the electron layer confined in the parabolic
well as a function of gate bias. Although this width is changed by
approximately a factor of four no significant change of the resonance position
is observed. However, there is a small but reproducible scatter of the data
around the expected value of [[omega]]o
46.9 cm^{-1} which is indicated by a thin horizontal line in the
figure. Within the resolution of the experiments this scatter is independent of
the way that it is obtained, indicating an intrinsic origin. We believe that it
is the result of a sensitive test of the local curvature of the PQW around the
potential minimum. Such small changes in the curvature of a MBE-grown PQW
cannot be completely excluded and have in fact been observed in different
experiments and direct measurements [45]. Application of a negative gate bias
shifts the potential minimum deeper into the sample such that a specific gate
bias is related to a possible slightly different local curvature. This
assumption appears to be justified by the fact that the scatter becomes larger
once the width of the electron slab is reduced: The wider the electron system,
the less important short range fluctuations of the actual curvature of the PQW
become and vice versa. Since the positions of the resonances crucially depend
on the actual shape of the bare potential, small deviations from a constant
curvature have a large impact on the resonance positions.

In a theoretical analysis of the spectrum of 'imperfect parabolic quantum wells', Brey and coworkers [32] studied the influence of small additional quartic terms [[Delta]]q in the potential as a measure for deviations from parabolicity. They found, that such a correction is quite effective in shifting the absorption line. For example, a positive [[Delta]]q causes a convex parabolic correction to the nearly uniform charge distribution produced by the parabolic potential alone. This leads to an additional confinement of the carriers and thus to an increase of the resonance position. A negative [[Delta]]q, however, induces an additional concave parabolic component, leading to a spreading of the charge as compared to the ideal parabola and thus to a reduction of the resonance position. A rough estimate yields that a shift of the absorption line by 5% can already be caused by quartic contributions to the potential of the order of [[Delta]]q/[[Delta]]p ~ 0.1 where [[Delta]]p denotes the parabolic coefficient of the potential. In other words, although we cannot state definitely that the observed resonance position does not depend on the gate bias Vg, we have strong indication that it does not depend on the actual carrier density in the well in accordance with Kohn's theorem.

**(b) Imperfect Parabolic Quantum Wells**

** **In the last section we saw that for electron systems confined in purely
parabolic potentials the generalized Kohn theorem is applicable. Purely
parabolic confining potentials, however, do not exist in reality. There are two
major sources for deviations from parabolicity:

(i) No real potential is infinitely deep. For the special case of PQW this is equivalent with the finite height of the AlGaAs barriers enclosing the quantum well. In other words, the finite size of the system under consideration in principle violates the theorem. We will refer this source of deviation to as finite size effects.

(ii) There might be some unintentional small deviations in the curvature of the MBE grown parabola. In first order approximation those deviations may be considered in a series expansion of the confining potential. These effects will be referred to as higher order effects.

Both (i), and (ii) are not only restricted to the case of a MBE grown PQW. Finite size effects certainly occur in all kind of man-made potential profiles, the question only reduces to a comparison of relevant energy scales, like, e.g., the ratio of the Fermi-energy to the height of the potential. Effects arising from higher orders in the confining potential (ii) are the most universal approach to model deviations from parabolicity. Self consistent calculations for quantum wires [46] show that, independent of the way of preparing such wires (electrostatically or by deep mesa etching) the resulting confining potential is somewhere in between a parabola and a rectangular quantum well. Such a potential then can be modelled by a parabola plus additional higher order terms.

To study the influence of deviations from perfect parabolicity and thus from
the validity range of the generalized Kohn theorem parabolic quantum wells seem
to be a nearly perfect tool. Unlike the case of quantum wires or dots the
external confining potential can be tailored in a very precise and controlled
way during the growth of the structure. Moreover, optical experiments on
imperfect parabolic quantum wells can yield information not only about the
extent to which the confining potential deviates from perfect parabolicity, but
also - for small deviations - about the forbidden excitations of an ideal
system [31]. Here, we like to present the experimental results obtained in a
structure (PB25) where we intentionally induced a certain degree of
nonparabolicity to study its influence on the FIR spectrum. Moreover, the
sample has been designed such that we can electrically tune the degree of
deviations and switch between different limiting cases. The sample is a
nominally 75 nm wide PQW with [[Delta]]=75 meV having vertical sidewalls which
are 150 meV high. The curvature of PB25 is chosen such that we expect the bare
harmonic oscillator frequency to be [[omega]]o
86 cm^{-1}. The result of a self consistent potential and subband
calculation for this sample is shown in Fig. 12, where we plot the self
consistent potential together with the total wavefunction as a function of
depth for three different gate voltages or carrier densities in the well,
respectively. The gate electrode is situated at the left-hand side, the
substrate at the right-hand side. In (a), the well is nearly completely filled
at a carrier density close to Ns = 5^{.}10^{11}cm^{-2}
at Vg = 0V. Here, the vertical sidewalls certainly will influence the finite
size effects as mentioned above. For intermediate carrier densities or well
fillings the wavefunction is located in the purely parabolic part of the
confining potential and we expect the sample to behave like a 'normal' PQW.
This situation is shown in Fig. 12(b) for a carrier density of Ns =
3^{.}10^{11}cm^{-2} at Vg = -0.4V. For very low carrier
density or very negative gate bias (Fig. 12c) close to total depletion of the
well we expect that the resulting selfconsistent potential looks more like a
half-parabola, thus containing strong higher-order coefficients in the
curvature. In this case only one electrical subband remains occupied leaving
the system in the electrical quantum limit. In Fig. 13, we depict the single
particle subband energies and the resulting subband population for sample PB25
as obtained from selfconsistent calculations. This calculated single particle
bandstructure and subband filling has been shown before to be very close to
reality, since in various (magneto-) transport experiments we can directly
compare it to our experimental results [27].

An experimental spectrum for this 'imperfect' sample as obtained using the
grating coupler technique is shown in Fig. 14. Here, we plot the relative
change in transmission vs. FIR frequency for different gate voltages Vg.
Capacitance-voltage measurements on the same sample reveal a threshold voltage
of VTh = -1.2V for complete depletion. Trace A has been recorded after
illuminating the sample at Vg = 0V for some seconds with an IR light emitting
diode, thus increasing the carrier concentration via the persistent
photoeffect. Trace B is taken at a gate bias of Vg = +0.6V and after
illumination, which further increases Ns. The exact values for the densities at
A and B, however, have not been determined in the experiment. From an analysis
of the oscillator strength of the resonance NS(B) is estimated to be about
6^{.}10^{11}cm^{-2}. As can be seen from the
figure, in this special sample the electron system not only absorbs at the
frequency of the bare harmonic potential, but side lines appear in the
spectrum. The main line, however, remains quite unaffected throughout the range
of carrier densities shown, indicating the rigidity of Kohn's theorem even for
this highly perturbed sample.

The influence of nonparabolic contributions to the well potential as listed
above has recently been discussed in considerable detail by Brey and coworkers
[32]. They calculate the IR absorption of 'imperfect' parabolic wells using the
Local Density Approximation for a PQW similar to the ones we use in our
experiments. For comparison, in Fig.15 we replot the result of Brey et al. for
a completely filled well, where finite size effects become important, together
with trace B from Fig.14. The amplitude of the main line of the calculation has
been normalized to the experimental one. Clearly all essential features of the
experiment are reproduced in the calculation, although the linewidths and the
exact position of the satellite structure differ slightly. This is believed to
be caused by the fact that the parameters of the sample assumed in the
calculation are not exactly identical with ours and that a phenomenological
scattering time [[tau]] has been used in the calculation. According to Brey et
al. for a well-filling [[eta]] = NS/(n^{+}W) ~ 1 the perturbation has
two main effects on the IR absorption of a PQW. First, the additional
confinement leads to a slight shift of the main absorption line to a higher
frequency and secondly, small satellites begin to appear, reflecting the
coupling of light to the internal oscillations of the electron system. As can
be seen in Fig.14, this crossover is clearly discernible from our data. At high
[[eta]] clearly two lines are observed. The one of smaller oscillator strength
and of higher frequency disappears with decreasing NS. At intermediate
densities only one resonance is observed until at low NS the system begins to
feel the strong deformation of the potential associated with the vertical
confining side wall (cf. Fig.12c). Here, the model of the harmonic oscillator
is certainly not valid any more and as a consequence, additional resonances
appear.

Experiments in a tilted magnetic field confirm this observation. Here, we make
use of the effect of a certain 'contrast enhancement' in such kind of
measurements: Although the oscillator strengths of the additional lines may be
quite small as compared to the main line, their existence may lead to a
resonant interaction between these modes and the cyclotron resonance similar to
the one for a perfect well. Typical examples of such tilted field experiments
for sample PB25 at different gate bias Vg and carrier density Ns, respectively,
are given in Figs. 16 to 19 for a series of magnetic fields. Here, we depict a
series of FIR spectra obtained for a tilt angle of [[Theta]]=23^{o}
between the sample normal and the magnetic field direction. The spectra shown
cover the upper magnetic field range between B=6T and B=15T. The most
remarkable feature here is the occurrence of additional small lines in the gap
between the CR-like and the ISR-like modes, which are not present in ideal
parabolic wells (cf. Figs. 7,8,9). The more the well is disturbed from ideal
parabolicity, the more those additional lines gain in importance. In Fig. 17
three distinct lines appear and the onset of a new anticrossing around B=12T is
observed, being reflected in a strong broadening of the ISR-like line
[[omega]]+ at around 160 cm^{-1}. This anticrossing is clearly resolved
in the spectra of Fig. 18 and 19, taken at a quite low carrier density and a
strongly deformed potential profile, respectively. Here, the major anticrossing
between the CR and the ISR-like sloshing mode is shifted upwards in frequency,
such that the center of the gap occurs at considerably higher energies as
compared to the unperturbed case. In Fig. 20, we summarize all the resonance
positions as obtained under different experimental conditions and different
well fillings or carrier densities. The harmonic oscillator-coupling as derived
in eq.(10) is still rather applicable to Fig. 20a, of course leaving the
additional lines within the gap unconsidered. This can be regarded as an
indication that with decreasing carrier density or equivalently more deformed
potential profile, the mode character significantly changes and self
consistency as well as the appropriate matrix elements of the problem have to
be considered to describe the interaction.

A first approach to explain the extra modes occurring in our experiments has been recently carried out by Dempsey and coworkers [31] using a classical hydrodynamic model to describe the magneto plasma excitations in a PQW. The electron system confined in the well in this case is treated as a classical charge fluid with an internal pressure p proportional to a density (N-Nc), where N is the density of the electrons and Nc a density at which the pressure vanishes. Linearizing the dynamical equations and neglecting retardation effects, they calculate the dispersion of the magnetoplasmon frequencies as a function of an in-plane wave vector q. Applied to our experiment, i.e. for small in-plane q they show that at least for the case of a completely filled PQW ([[eta]] ~ 1) where finite size effects become important, this model gives an impression of the character of the internal modes of the electron system. Those excitations may be regarded as 'standing-wave'-like bulk magneto plasmons with an integral number of wavelengths within the electron slab of width We. Thus no longer only two harmonic oscillators are coupled in a tilted field experiment but now one has to take into account the pressure-driven internal oscillations of the electron slab. Such non-local interaction effects are well known for homogeneous 3DES and 2DES. They arise from the inherent finite compressibility of the Fermi gas, the 'Fermi-pressure', and lead to corrections for the squared plasma frequency. For an ideal three-dimensional system, the frequencies of these oscillations are then given by

(13)

where vF denotes the Fermi-velocity of the electrons of the 'natural' density
n^{+} in the well. For a strictly two-dimensional system, the prefactor
3/5 in eq. (13) has to be replaced by 3/4 [47]. Such non-local effects become
important at small plasmon wavelengths or correspondingly high values of q. A
well-known and famous manifestation of it is observed if photons are incident
on a boundary of a semi-infinite 3DES. Here, it is manifested in a coupling of
the incident transverse photons with longitudinal plasmons [48]. Also related
to this is the problem of the widely discussed 'additional boundary conditions'
(ABC) [49]. A similar non-local interaction has recently also been observed in
0DES [2,6]. For the case of our PQW, the different branches of the coupled
system now can be written as

(14)

Here, [[omega]]cz again denotes the z-component of the cyclotron frequency whereas the other symbols have the meaning as defined above. It is important to note that only the modes with n = 0 (CM-modes) exhibit a non-zero dipole-moment. For symmetry reasons, only these modes should be observable in a FIR transmission experiment. The presence of a gate bias, however, may break this symmetry such that symmetry-forbidden resonances also are detectable in our experiments.

The result of the calculation is shown in Fig. 21, where we plot the first five internal modes together with the experimental data as extracted from Fig. 20(a). The thin lines represent the interaction of the Fermi-pressure-driven internal oscillations and the cyclotron resonance, whereas the thick lines are the center-of-mass modes as discussed above. The only parameters necessary to make the comparison are [[omega]]o and We, which are known independently. Given the simplicity of the model, the agreement between the calculation and the experiment is remarkable. It suggests quite strongly that the actual modes excited in the experiment are closely related to the magnetoplasmon modes as obtained in the simple hydrodynamic approach. In principle, the hydrodynamic calculations should give good results for plasmon-like modes in systems with large, slowly varying electron density, as long as the wavelength of the excitation is large as compared to the inter particle spacing [31].

There are, however, some differences between the results of this model and the
real experimental data which cannot be explained within this framework. For
instance, the model does not predict resonances above the CR-like mode below
the point of anticrossing, whereas the experiment exhibits a quite strong line
that slowly levels off with decreasing magnetic field. Furthermore, the
agreement between the result of the hydrodynamic model and the experiment
becomes worse if the carrier density in the well is considerably reduced as
compared to the Vg=0V case. Also for different tilt angles and especially for
[[Theta]]=90^{o}, the agreement between model and experiment is
unsatisfactory [31].

The collective excitations in a PQW can be regarded as representing something in between the true plasmonic collective excitation of a pure 3D system and the intersubband-transitions of a pure 2DES, where self-consistent screening leads to an upward shift of the already large transition energies. In a parabolic well the smallness of the selfconsistent single-particle energies as compared to the characteristic energy of the bare potential strongly mixes the intersubband transitions forming collective excitations that occur at much higher energies and all the plasmon energy arises from the coherent polarization of the electronic system. This strong mixing together with the self-consistent screening requires a fully self consistent calculation of the mode-spectrum taking into account not only the center-of-mass modes. Such calculations have been recently carried out also by Dempsey and coworkers [33..35] using a selfconsistent field approach in Local Density Approximation (LDA-SCF) and, for comparison also in Random Phase Approximation (RPA). The calculated spectra using both methods, however, only differ by a few percent in the resonance positions. This indicates that in a PQW exciton - like effects (at least those calculated using a zero-field local exchange potential) are small in comparison with depolarization effects [33,34]. This is very different from the situation in quasi 2D systems, where depolarization- and exciton-like effects are known to be of the same order of magnitude. Experiments that are sensitive to these contributions like Raman-scattering using polarized and depolarized geometries certainly promise to be very interesting for a further understanding of the underlying mechanisms.

In their calculations, Dempsey and coworkers chose well parameters to match our real experiments for the sister-samples PB25 (hard vertical sidewalls) and PB48 (same curvature but no vertical sidewalls) and find excellent quantitative agreement over a wide range of magnetic field strengths and carrier densities in the wells. Referring the reader to a detailed description of the calculation method in the original articles [33..35], we here restrict ourselves to give only the essential results.

We start with a description of the spectra of PB25 at a gate bias of Vg = 0V as shown in Figs. 16, 20(a), and 21. The most prominent feature as compared to the spectrum of an 'ideal' PQW is the occurrence of two extra peaks between the center of mass modes, which are only little changed with respect to the 'ideal' well PB48. A direct comparison of both samples is shown in Fig. 22, where we plot the measured resonance positions for both samples and identical experimental conditions. Filled symbols represent the result for the 'imperfect' well PB25 and the open circles those for the 'ideal' well PB48. As can be seen, the major differences are the extra peaks and a slight energy decrease of the CM-mode [[omega]]- for the perturbed sample. The upper CM-modes [[omega]]+ are nearly indistinguishable for both samples.

Dempsey and coworkers accurately reproduced the extra peaks that are observed in the experiment as well as some additional lines and anticrossings by just taking into account the finite size of the well. However, since self consistent screening strongly mixes the intersubband transitions, it is in general very demanding to associate those peaks in the spectrum with particular transitions. One special case is the upper extra peak in the gap between the CM-like modes. To account for the occurrence of this second extra peak especially at high magnetic fields the authors of ref. [34] had to include effects that arise from the finite temperature of the experiment. In a T=0 calculation this peak which is associated with the 1 4 intersubband transition disappears above B=10.5T due to a magnetic depopulation of the E1-subband. Since the single particle subband separations, however, are so small in a PQW there is a reasonable occupation of the E1-subband even at low temperatures. This occupation has a strong effect on the absorption spectrum leading, for example, to the occurrence of the upper extra peak in the spectrum.

The result of Dempsey's RPA-calculation together with our experimental data is
shown in Fig. 23. The dots are the calculated resonance positions with their
size being proportional to the calculated oscillator strength. The boxes are
our experimental data. As can be seen, the agreement is close to being perfect
as far as the resonance positions are concerned. Also the behavior of the
oscillator strength is calculated quite perfectly as can be seen from Fig. 16.
Given the results of the calculations, the authors also calculated the spectra
for a strongly biased PQW, where the electric-field-induced asymmetry becomes
the dominant effect on the absorption spectrum. This is shown in Fig. 24 where
again both the calculated and the experimental resonance positions are shown
together for a gate bias of Vg=-0.7V (cf. Fig.18), leading to a linear
potential gradient of 205 meV across the well. The inset shows the calculated
ground-state density and self-consistent potential under these conditions. It
is interesting to note that charge accumulation at the edge of the well leads
to a maximum electron density that is more than 1.5 times larger than the
'natural' density n^{+ }given by eq. (1).

So far we have considered the experiments at tilted magnetic fields where the coupling of the CR to the center-of-mass and the internal oscillations of the PQW led to the occurrence of a quite complicated absorption spectrum. For in-plane magnetic fields also, there are strong deviations from the 'ideal' case. This is a very interesting result, since it can directly be compared to FIR investigations of quantum wires, where the confining potential is not exactly known. An example for the in-plane spectrum of the 'imperfect' PQW PB25 is given in Fig. 25. The corresponding squared resonance positions vs. the squared magnetic field for two different carrier densities in the well are shown in Figs. 26(a) and (b), respectively. The most striking fact is that now two resonances are observed which are well separated at low magnetic fields and merge at high magnetic fields at the position of the 'unperturbed' hybrid resonance of a PQW in Voigt-geometry (see, e.g., Fig. 9). The oscillator strengths of the two resonances vanish for B 0 with the low-energy resonance disappearing earlier. It is interesting to note that for low magnetic fields, the observed transitions as a function of the magnetic field can be described by . A similar behavior has very recently been also observed in FIR-studies of 1D quantum wires [50]. There, the magneto-electric hybrid mode exhibited a splitting at around , which was attributed to a nonlocal effect arising form nonparabolic terms in the confining potential. It is interesting to note that this higher frequency as well as the apparent mode - crossing at corresponds to the situation where [[omega]]o=[[omega]]c. This is exactly the condition, at which the density of states in a PQW in parallel magnetic field changes from quasi 2D behavior at low magnetic fields towards quasi 1D behavior in high magnetic fields [60].

The essential result of Dempsey's calculation using a self-consistent-field
formalism introduced by Ando [1,51] is the occurrence of a band of transitions
instead of the sharp line (cf. eq. 12) predicted for an 'ideal' well. The
interesting fact is that the oscillator strengths of the single transitions
forming a continuum are not distributed symmetrically among these lines. At low
magnetic fields, where the electrical quantization is governing the system, the
oscillator strength is peaked at the upper edge of the continuum, then moving
towards the lower edge until for high magnetic fields both maxima merge. For
very large magnetic fields a single line representing the cyclotron resonance
is recovered. The point, at which the oscillator strength is equal for both
lines, again corresponds to a magnetic field, at which
or [[omega]]o=[[omega]]c, respectively. This behavior is shown in Fig. 27(a)
taken from ref. [35], where the calculated resonance positions and oscillator
strengths (dots) together with our experimental data (boxes) at a gate bias of
Vg=0V are shown. In Fig. 27(b), the calculated spectrum assuming a
phenomenological scattering time [[tau]] [35] is plotted for comparison.
Although the calculated resonance positions match the data of our experiment,
we do not really obion according to eq. (10). For all three experiments the
same parameters have been used as given in the inset of the figure. The
agreement between our experimental results and the simple model calculation is
nearly perfect for all three angles investigated. The small but reproducible
deviations between theory and experiment at higher magnetic fields [[alpha]]re
due to band-nonparabolicity, which has not been included** in the calculation.
**

** In Fig. 8 we s**how the result of the tilted field experiment for PB48 at
a tilt angle of Q=23o together with the amplitude of the observed resonances
which has been normalized to the one of the CR in perpendicular magnetic
fields. Taking the width of the resonance fixed, which is justified here, this
quantity is proportional to the oscillator strength. The dashed lines
represents the prediction in the simple harmonic oscillator pic^{tur}e
as given in eq. (11). Especially at low magnetic fields there exists a quite
large scatter around the expected value for the oscillator strength. This we
believe is caused by effects arising from the population and depopulation of
different subbands in the system: The different spatial extend of the
wavefunctions in different subbands leads to the occurrence of an 'artificial
nonparabolicity' [7], resulting in a superposition of different CR lines which
then may complicate the proper evaluation of the oscillator strength. Also, the
effective electron mass m* turns out to be a function of position in the well.
These effects as well as those of subband depopulation and intersubband
scattering, however, are beyond the scope of the present report and will be
discussed elsewhere.

Under the extreme condition of an in-plane magnetic field (Q=90o ,
Voigt-geometry), only w+ is observable. The electrical and the magnetic
confinement act in the same direction and the CR hybridizes with the
oscillation in the bare potential as represented by wo. For small magnetic
fields, electrical quantization is stronger than the magnetic one and the
observed resonance frequency is close to wo. For higher magnetic fields the
magnetic quantization becomes dominant and the observed resonance approaches
wc. In a well with perfect parabolicity this resonance is a discrete
excitatio[[nu]] that is split off from the continuum of inter-Landau
l** e**vel transitions by collective polarization effects. The
resonance position and the oscillator strength of the hybrid mode are then
given by

and (12)

Here, it is interesting to note that exactly the same magnetic field dispersion is expected and in fact observed for quantum wires (Q1DES) with parabolic confinement in a perpendicular magnetic field [3], so that our PQW in Voigt geometry may serve as a perfect model system for the investigations of such Q1DES. The result of such an experiment on a PQW is shown in Fig. 9, where we plot the extracted resonance position together with the amplitude of the line for sample PB26. Again, it has been normalized to the one of the CR in Faraday-geometry. The symbols represent the experimental results and the lines are the result of the calculation according to Eq. (12). The origin of the large deviation of the measured oscillator strength from the expected value above B= 12T is not yet known. For the geometry of this experiment and the related FIR polarization the oscillator strength vanishes for B 0. Using a grating coupler technique, however, we are able to follow the resonance down to B=0.

An example for such a direct observation of the intersubband-like sloshing
mode of a PQW [9] is given in Fig. 10. Here, we plot the result of a grating
coupler induced resonance for sample PB26 for three different gate voltages Vg
and correspondingly three different carrier densities in the well. To induce
longitudinal electric components in the FIR radiation that are transmitted
through the sample parallel to the growth direction we use a 70 nm thick Ag
grating coupler of periodicity d= 6 mm that is deposited in addition to the
semitransparent NiCr gate electrode on top of the sample. Although the carrier
density in the well is changed from Ns = 2.5.1011cm-2 at Vg = 0V down to Ns =
1.6.1011cm-2 at Vg=-0.3V, the position of the grating coupler induced resonance
has not changed. The decrease of the carrier density, however, is reflected in
a reduced oscillator strength as can be seen in Fig. 10. The exact amplitude of
the observed resonance depends not only on the dynamic conductivity szz(w) and
thus on the carrier density Ns but also on the efficiency of the grating
coupler used in the experiment. For this reason, we cannot directly compare
different spectra obtained for different sam^{p}les and grating
couplers.

The resonance positions of the inter-subband like sloshing mode in sam[[pi]]le
[[Pi]]B26 as obtained from all the different experimental techniques are
summarized in Fig. 11 as a function of the gate bias Vg. Different symbols
represent the results of different experiments as given in the inset. Part (b)
of Fig. 11 sketches the change in width of the e[[lambda]]ect[[rho]]on layer
confined in the parabolic well as a function of gate bias. Although this width
is changed by approximately a factor of four no significant change of the
resonance position is observed. However, there is a small but reproducible
scatter of the data around the expected value of wo 46.9 cm-1 which is
indica^{t}ed by a thin horizontal line in the figure. Within the
resolution of the experiments this scatter is independent of the way that it is
obtained, indicating an intrinsic origin. We believe that it is the result of a
sensit[[iota]]ve tes^{t }of the local curvature of the PQW around the
potential minimum. Such small changes in the curvature of a
M^{B}E-^{gr}ow^{n} ^{PQ}W c^{an}not be
completely excluded and have in fact been observed in differ^{e}nt^{
e}xp^{er}iments and direct measurements [45]. Application of a
negative gate bias shifts the potential minimum deeper into the sample such
that a specific gate bias is related to a possible slightly different local
curvature. This ass^{u}mption appears to be justified by the fact that
the scatter becomes larger once the width of the electron slab is reduced: The
wider the electron system, the less important short range fluctuations of the
actual curvature of the PQW become and vice versa. Since the positions of the
resonances crucially depend on the actual shape of the bare potential, small
deviations from a constant curvature have a large impact on the resonance
positions.

In a theoretical analysis of the spectrum of 'imperfect parabolic quantum
wells', Brey and coworkers [32] studied the influence of small additional
quartic terms Dq in the potential as a measure for deviations from
parabolicity. They found, th[[alpha]][[tau]] such^{ a} correction is
quite effective in shifting the absorption line. For example, a positive Dq
causes a convex parabolic correction to the nearly uniform charge distribution
produced by the parabolic potential alone. This leads to an additional
confinement of the carriers and thus to an increase of the resonance position.
A negative Dq, however, induces an additional concave parabolic component,
leading to a spreading of the charge as compared to the ideal parabola and thus
to a reduction of the resonance position. A rough estimate yields that a shift
of the absorption line by 5% can already be caused by quartic contributions to
the potential of the order of Dq/Dp ~ 0.1 where Dp denotes the parabolic
coefficient of the potential. In other words, although we cannot state
definitely that the observed resonance position does not depend on the gate
bias Vg, we have strong indication that it does not depend on the actual
carrier density in the well in accordance with Kohn's theorem.

(b) Imperfect Parabolic Quantum Wells

In the last section we saw that for electron systems confined in purely parabolic potentials the generalized Kohn theorem is applicable. Purely parabolic confining potentials, however, do not exist in reality. There are two major sources for deviations from parabolicity:

(i) No real potential is infinitely deep. For the special case of PQW this is equivalent with the finite height of the AlGaAs barriers enclosing the quantum well. In other words, the finite size of the system under consideration in principle violates the theorem. We will refer this source of deviation to as finite size effects.

(ii) There might be some unintentional small deviations in the curvature of the MBE grown parabola. In first order approximation those deviations may be considered in a series expansion of the confining potential. These effects will be referred to as higher order effects.

Both (i), and (ii) are not only restricted to the case of a MBE grown PQW.
Finite size effects certainly occur in all kind of man-made potential profiles,
the question only reduces to a comparison of relevant energy scales, like,
e.g., the ratio of the Fermi-energy to the height of the potential.
Ef^{fe}cts arising from higher orders in the confining potential
(i[[iota]]) are the most universal approach to model deviations from
parabolicity. Self consistent calculations for quantum wires [46] [[sigma]]how
that, independent of the way of preparing such wires (electrostatically or by
deep mesa etching) the resulting confining potential is somewh^{ere} in
be^{tw}een a parabola and a rectangular quantum well. Such a potential
then can be modelled by a parabola plus additional higher order terms.

To study the influence of deviations from perfect parabolicity and thus from
the validity range of the generalized Kohn theorem parabolic quantum wells seem
to be a nearly perfect tool. Unlike the case of quantum wires or dots the
external confining potential can be tailored in a very precise and controlled
way during the growth of the structure. Moreover, optical experiments on
imperfect parabolic quantum wells can yield information not only about the
extent to which the confining potential deviates from perfect parabolicity, but
also - for small deviations - about the forbidden excitations of an ideal
system [31]. Here, we like to present the experimental results obtained in a
structure (PB25) where we intentionally [[iota]]nduced a certain degree of
nonparabolicity to study its influence on the FIR spectrum. Moreover, the
sample has been designed such that we can electrically tune the degree of
deviations and switch between different limiting cases. The sample is a
nominally 75 nm wide PQW with D=75 meV having vertical sidewalls which are 150
meV high. The curvature of PB25 is chosen such that we expect the bare harmonic
oscillator frequency to be wo 86 cm-1. The result of a self consistent
potential and subband calculation for this sample is shown in Fig. 12, where we
plot the self consistent potential together with the total wavefunction as a
function of depth for three different gate voltages or carrier densities in the
well, respectively. The gate electrode is situated at the left-hand side, the
substrate at the right-hand side. In (a), the well is nearly completely filled
at a carrier density close to Ns = 5.1011cm-2 at Vg = 0V. Here, the vertical
sidewalls certainly will influence the finite size effects as mentioned above.
For intermediate carrier densities or well fillings the wavefunction is located
in the purely parabolic part of the confining potential and we expect the
sample to behave like a 'normal' PQW. This situation is shown in Fig. 12(b) for
a carrier density of Ns = 3.1011cm-2 at Vg = -0.4V. For very low carrie[[rho]]
density or very negative gate bias (Fig. 12c) close to total depletion of the
well we expect that the resulting s[[epsilon]]lfconsistent potential looks more
like a half-parabola, thus containing strong higher-order coefficients in the
curvature. In this case only one electrical subband remains occupied leaving
the system in the electri[[chi]]al quantum limit. In Fig. 13, we depict the
single particle subband energies and the resulting s^{ubba}nd
population for sample PB25 as obtained from selfconsistent calculations. This
calculated single particle bandstructure and subband filling has been shown
before to be very close to reality, since in various (magneto-) transport
experiments we can directly compare it to our experimental results [27].

An experimental spectrum for this 'imperfect' sample a[[sigma]] obtained using the grating coupler technique is shown in Fig. 14. Here, we plot the relative change in transmission vs. FIR frequency for different gate voltages Vg. Capacitance-voltage measurements on the same sample reveal a threshold voltage of VTh = -1.2V for complete depletion. Trace A has been recorded after illuminating the sample at Vg = 0V for some seconds with an IR light emitting diode, thus increasing the carrier concentration via the persistent photoeffect. Trace B is taken at a gate bias of Vg = +0.6V and after illumination, which further increases Ns. The exact values for the densities at A and B, however, have not been determined in the experiment. From an analysis of the oscillator strength of the resonance NS(B) is estimated to be about 6.1011cm-2. As can be seen from the figure, i[[nu]] this special sample the electron sy[[sigma]]tem not only absorbs at the frequency of the bare harmonic potential, but side lines appear in the spectrum. The main line, however, remains quite unaffected throughout the range of carrier densities shown, indicating the rigidity of Kohn's theorem even for this highly perturbed sample.

The influence of nonparabolic contributions to the well potential as listed
above has recently been discussed in considerable detail by Brey and coworkers
[32]. They calculate the IR absorption of 'imperfect' parabolic wells using the
Local Density Approximation for a PQW similar to the ones we use in our
experiments. For comparison, in Fig.15 we replot the result of Brey et al. for
a completely filled well, where finite size effects become important, together
with trace B from Fig.14. The amplitude of the main line of the calculation has
been normalized to the experimental one. Clearly all essential features of the
experiment are reproduced in the calculation, although the linewidths and the
exact position of the satellite structure differ slightly. This is believed to
be caused by [[tau]]he fact th^{at} the parameters of the sample
assumed in the calculation are not exactly identical with ours and that a
phenomenological scattering time t has been used in the calculation. According
to Brey et al. for a well-filling h = NS/(n+W) ~ 1 the perturbation has two
main effects on the IR absorption of a PQW. First, the additional confinement
leads to a slight shift of the main absorption line to a higher frequency and
secondly, small satellites begin to appear, reflecting the coupling of light to
the internal oscillations of the electron system. As can be seen in Fig.14,
this crossover is clearly discernible from our data. At high h clearly
t^{wo} lines are observed. The one of smaller oscillator strength and
of higher frequency disappears with decreasing NS. At intermediate densities
only one resonance is observed until at low NS the system begins to feel the
strong deformation of the potential associated with the vertical confining side
wall (cf. Fig.12c). Here, the model of the harmonic oscillator is certainly not
valid any more and as a consequence, additional resonances appear.

Experiments in a tilted magnetic field confirm this observation. Here, we make use of the effect of a certain 'contrast enhancement' in such kind of measurements: Although the oscillator strengths of the additional lines may be quite small as compared to the main line, their existence may lead to a resonant interaction between these modes and the cyclotron resonance similar to the one for a pe[[rho]]fect well. Typical examples of such tilted field experiments for sample PB25 at different gate bias Vg and carrier density Ns, respectively, are given in Figs. 16 to 19 for a series of magnetic fields. Here, we depict a series of FIR spectra obtained for a tilt angle of Q=23o between the sample normal and the magnetic field direction. The spectra shown cover the upper magnetic field range between B=6T and B=15T. The most remarkable feature here is the occurrence of additional small lines in the gap between the CR-like and the ISR-l[[iota]]ke modes, which are not present in ideal parabolic wells (cf. Figs. 7,8,9). The more the well is disturbed from ideal parabolicity, the more those additional lines gain in importance. In Fig. 17 three distinct lines appear and the onset of a new anticrossing around B=12T is observed, being reflected in a strong broadening of the ISR-like line w+ at around 160 cm-1. This anticrossing is clearly resolved in the spectra of Fig. 18 and 19, taken at a quite low carrier density and a strongly deformed potential profile, respectively. Here, the major anticrossing between the CR and the ISR-like sloshing mode is shifted upwards in frequency, such that the center of the gap occurs at considerably higher energies as compared to the unperturbed case. In Fig. 20, we summarize all the resonance positions as obtained under different experimental conditions and different well fillings or carrier densities. The harmonic oscillator-coupling as derived in eq.(10) is still rather applicable to Fig. 20a, of course leaving the additional lines within the gap unconsidered. This can be regarded as an indication that with decreasing carrier density or equivalently more deformed potential profile, the mode character significantly changes and self consistency as well as the appropriate matrix elements of the problem have to be considered to describe the interaction.

A first approach to explain the extra modes occurring in our experiments has
been recently carried out by Dempsey and coworkers [31] using a classical
hydrodynamic model to describe the magneto plasma excitations in a PQW. The
electron system confined in the well in this case is treated as a classical
charge fluid with an internal pressure p proportional to a density (N-Nc),
where N is the density of the electrons and Nc a density at which the pressure
vanishes. Linearizing the dynamical equations and neglecting retardation
effects, they calculate the dispersion of the magnetoplasmon frequencies as a
fun[[chi]]tion of an in-plane wave vector q. Applied to our experiment, i.e.
for small in-plane q they show that at least for the case of a completely
filled PQW (h ~ 1) where finite size effects become important, this model
gives an impression of the character of the internal modes of the electron
system. Those excitations may be regarded as 'standing-wave'-like bulk magneto
plasmons with an integral number of wavelengths within the electron slab of
width We. Thus no longer only two harmonic oscillators are coupled in a tilted
field experiment but now one has to take into account the pressure-driven
internal oscillations of the electron slab. Such non-local interaction effects
are well known for homogeneous 3DES ^{an}d 2DES. They arise from the
inherent finite compressibility of the Fermi gas, the 'Fermi-pressure', and
lead to corrections for the squared plasma frequency. For an ideal
three-dimensional system, the frequencies of these oscillations are then given
by

(13)

where vF denotes the Fermi-velocity of the electrons of the 'natural' density n+ in the well. For a strictly two-dimensional system, the pr[[epsilon]]factor 3/5 in eq. (13) has to be replaced by 3/4 [47]. Such non-local effects become important at small plasmon wavelengths or correspondingly high values of q. A well-known and famous manifestation of it is observed if photons are incident on a boundary of a semi-infinite 3DES. Here, it is manifested in a coupling of the incident transverse photons wit[[eta]] longitudinal plasmons [48]. Also related to this is the problem of the widely discussed 'additional [[beta]]oundary conditions' (ABC) [49]. A similar non-local interaction has recently also been observed in 0DES [2,6]. For the c[[alpha]]se of our PQW, the different branches of the coupled system now can be written as

(14)

Here, wcz again denotes the z-component of the cyclotron frequency whereas the other symbols have the meaning as defined above. It is important to note that on[[lambda]]y the modes with n = 0 (CM-modes) exhibit a non-zero dipole-moment. For symmetry reasons, only these modes should be observable in a FIR transmission experiment. The presence of a gate bias, however, may break this symmetry such that symmetry-forbidden resonances also are detectable in our experiments.

The result of the calculation is shown in Fig. 21, where we plot the first
five internal modes together with the experimental data as extracted from Fig.
20(a). The thin lines represent the interaction of the Fermi-pressure-driven
internal oscillations and the cyclotron resonance, whereas the thick lines are
the center-of-mass modes as discussed above. The only parameters necessary to
make the^{ }comparison are wo and We, which are known independently.
Given the simplicity of the model, the agreement betwe[[epsilon]]n the
calculation and the experiment is remarkable. It suggests quite strongly that
the actual modes exci^{t}ed in the experiment are closely related
t^{o} the magnetoplasmon modes as obtained in the simple hydrodynamic
approach. In principle, the hydrodynamic calculations should give good results
for plasmon-like modes in systems with large, slowly varying electron density,
as long as the wavelength of the excitation is large as compared to the inter
particle spacing [31].

There are, however, some differences between the results of this model and the real experimental data which cannot be explained within this framework. For instance, the model does not predict resonances above the CR-like mode below the point of anticrossing, whereas the experiment exhibits a quite strong line that slowly levels off with decreasing magnetic field. Furthermore, the agreement between the result of the hydrodynamic model and the experiment becomes worse if the carrier density in the well is considerably reduced as compared to the Vg=0V case. Also for different tilt angles and especially for Q=90o, the agreement between model and experiment is unsatisfactory [31].

The collective excitations in a PQW can^{ }be regarded as representing
something in between the true plasmonic collective excitation of a pure 3D
system and the intersubband-transitions of a pure 2DES, where self-consistent
screening leads to an upward shift of the already large transition energies. In
a parabolic well the smallness of the selfconsistent single-pa^{r}ticle
energies as compared to the characteristic energy of the bare potential
strongly mixes the in^{t}ersubband transitions forming collective
excitations that occur at much higher energies and all the plasmon energy
arises from the coherent polarization of the electronic system. This strong
mixing together with the self-consistent screening requires a fully self
consistent calculation of the mode-spectrum taking into account not only the
center-of-mass modes. Such calculations have been recently carried out also by
Dempsey and coworkers [33..35] using a selfconsistent field approach in Local
Density Approximation (LDA-SCF) and, for comparison also in Random Phase
Approximation (RPA). The calculated spectra using both methods, however, only
differ by a few percent in the resonance positions. This indicates that in a
PQW exciton - like effects (at least those calculated using a zero-field local
exchange potential) are small in comparison with depolarization effects
[33,34]. This is very different from the situation in quasi 2D systems, where
depol^{ar}ization- and exciton-like effects are known to be of the same
order of magnitude. Experiments that are sensitive to these contributions like
Raman-scattering using polarized and depolarized geometries certainly promise
to be very interesting for a further understanding of the underlying
mechanisms.

I**n their calcu**lations, Dempsey and coworkers chose well parameters to
match our real experiments for the sister-samples PB25 (hard vertical
sidewalls) and PB48 (same curvature but no vertical sidewalls) and find
excellent quantitative agreement over a wide range of magnetic field strengths
and carrier densities in the wells. Referring the reader to a detailed
description of the calculation method in the original articles [33..35], we
here restrict ourselves to give only the essential results.

We start with a description of the spectra of PB25 at a gate bias of Vg = 0V as shown in Figs. 16, 20(a), and 21. The most prominent feature as compared to the spectrum of an 'ideal' PQW is the occurrence of two extra peaks between the center of mass modes, which are only little changed with respect to the 'ideal' well PB48. A direct comparison of both samples is shown in Fig. 22, where we plot the measured resonance positions for both samples and identical experimental conditions. Filled symbols represent the result for the 'imperfect' well PB25 and the open circles those for the 'ideal' well PB48. As can be seen, the major differences are the extra peaks and a slight energy decrease of the CM-mode w- for the perturbed sample. The upper CM-modes w+ are nearly indistinguishable for both samples.

Dempsey and coworkers accurately reproduced the extra peaks that are observed in the experiment as well as some additional lines and anticrossings by just taking into account the finite size of the well. However, since self consistent screening strongly mixes the intersubband transitions, it is in general very demanding to associate those peaks in the spectrum with particular transitions. One special case is the upper extra peak in the gap between the CM-like modes. To account for the occurrence of this second extra peak especially at high magnetic fields the authors of ref. [34] had to include effects that arise from the finite temperature of the experiment. In a T=0 calculation this peak which is associated with the 14 intersubband transition disappears above B=10.5T due to a magnetic depopulation of the E1-subband. Since the single particle subband separations, however, are so small in a PQW there is a reasonable occupation of the E1-subband even at low temperatures. This occupation has a strong effect on the absorption spectrum leading, for example, to the occurrence of the upper extra peak in the spectrum.

The result of Dempsey's RPA-calculation together with our experimental data is
shown* in Fig*. 23. The dots are the calculated resonance positions with
their size being proportional to the calculated oscillator strength. The boxes
are our experimental data. As can be seen, the agreement is close to being
perfect as far as the resonance positions are concerned. Also the behavior of
the oscillator strength is calculated quite perfectly as can be seen from Fig.
16. Given the results of the calculations, the authors also calculated the
spectra for a strongly biased PQW, where the electric-field-induced asymmetry
becomes the dominant effect on the absorption spectrum. This is shown in Fig.
24 where again both the calculated and the experimental resonance positions are
shown together for a gate bias of Vg=-0.7V (cf. Fig.18), leading to a linear
potential gradient of 205 meV across the well. The inset shows the calculated
ground-state density and self-consistent potential under these conditions. It
is interesting to note that charge accumulation at the edge of the well leads
to a maximum electron density that is more than 1.5 times larger than the
'natural' density n+ given by eq. (1).

So far we have considered the experiments at tilted magnetic fields where the
coupling of the CR to the center-of-mass and the internal oscillations of the
PQW led to the occurrence of a quite complicated absorption spectrum. For
in-plane magnetic fields also, there are strong deviations from the 'ideal'
case. This is a very interesting result, since it can directly be compared to
FIR investigations of quantum wires, where the confining potential is not
exactly known. An example for the in-plane spectrum of the 'imperfe**ct' PQW
PB25 is given in **Fig. 25. The corresponding squared resonance positions vs.
the squared magnetic field for two different carrier densities in the well are
shown in Figs. 26(a) and (b), respectively. The most striking fact is that now
two resonances are observed which are well separated at low magnetic fields and
merge at high magnetic fields at the position of the 'unperturbed' hybrid
resonance of a PQW in Voigt-geometry (see, e.g., Fig. 9). The oscillator
strengths of the two resonances vanish for B0 with the low-energy resonance
disappearing earlier. It (QUEST) at the University of California, Santa
Barbara for the hospitality during a sabbatical stay. This work has been
sponsored in part by the Deutsche Forschungsgemeinschaft, the Volkswagen
Stiftung, and in part by U.S. Air Force Office of Scientific Research under
Contract No. AFOSR-88-099.

** REFERENCES :**

[1] for an excellent review on the electronic properties of low-dimensional
systems, see, e.g., T. Ando, A. B. Fowler, and F. Stern,

Rev. Mod. Phys. **54**, 437 (1982)

[2] see, e.g., D. Heitmann, *Two-Dimensional Systems : Physics and New
Devices *(ed. G. Bauer, F. Kuchar, and H.Heinrich, Springer, Berlin) p. 285
(1986)

[3] for a recent review, see, e.g., : W. Hansen, U. Merkt, and J. P. Kotthaus,
in *Semiconductors and Semimetals*, (ed. R. K. Williardson, A. C. Beer,
E. R. Weber), Vol. **35**, (Academic Press, San Diego 1992) p. 279.

[4] W. Hansen, M. Horst, J.P. Kotthaus, U. Merkt, Ch. Sikorski, and K. Ploog,

Phys. Rev. Lett.* ***58**, 2586 (1987)

[5] T. Demel, D. Heitmann, P. Grambow, and K. Ploog,

Phys. Rev. Lett.* ***64**, 788 (1990)

[6] Ch. Sikorski, and U. Merkt,

Phys. Rev. Lett. **62**, 2164 (1989)

[7] Due to the varying Al content in a PQW the wavefunctions of different
subbands extend into regions of a different average effective mass, which
influences the position of the cyclotron resonance. See, e.g.,

A. Wixforth, M. Sundaram, D. Donnelly, J.H. English, and A.C. Gossard,

Surf. Sci. **228**, 489 (1990), and

K. Karrai, X. Ying, H.D. Drew, and M. Shayegan

Phys. Rev. **B42**, 9732 (1990)

[8] A. Wixforth, M. Sundaram, J.H. English, and A.C. Gossard, in

*The Physics of Semiconductors*, eds. E.M. Anastassakis, and J.D.
Joannopoulos, **Vol. II**, 1705 (World Scientific, 1990)

[9] A. Wixforth, M. Sundaram, K. Ensslin, J.H. English, and A.C. Gossard,

Phys. Rev. **B43**, 10000 (1991)

[10] A. Wixforth, M. Sundaram, K. Ensslin, J.H. English, and A.C. Gossard,

Surf. Sci. **267**, 523 (1992)

[11] M. Kaloudis, K. Ensslin, A. Wixforth, M. Sundaram, J.H. English, and A.C.
Gossard, Phys. Rev. **B46**, 12469 (1992)

[12] A. Wixforth, M. Kaloudis, M. Sundaram, and A. C. Gossard,

Sol.St. Comm.**84**, 861 (1992)

[13] M. Sundaram, A.C. Gossard, J.H. English, and R.M. Westervelt

Superlatt. Microstruct. **4,** 683 (1988), see also

M. Shayegan, T. Sajoto, M. Santos, and C. Silvestre,

Appl. Phys. Lett. **53**, 791 (1988)

[14] B.I. Halperin, Jap. Journ. of Appl. Phys. **26**, 1913 (1987)

[15] V. Celli, and N.D. Mermin, Phys. Rev. **140,** A839 (1965)

[16] K. Karrai, H.D. Drew, H.W. Lee, and M. Shayegan,

Phys. Rev. **B39**, 1426 (1989)

[17] K. Karrai, X. Ying, H.D. Drew, and M. Shayegan,

Phys. Rev. **B40**, 12020 (1989)

[18] W. Kohn, Phys. Rev. **123**, 1242 (1961)

[19] L. Brey, N.F. Johnson, and B.I. Halperin, Phys. Rev. **B40**, 647
(1989)

[20] S. K. Yip, Phys. Rev. **B43**, 1707 (1991)

[21] R. C. Miller, A. C. Gossard, D.A. Kleinmann, and O. Muntaneau,

Phys. Rev. **B29**, 3740 (1984)

[22] M. Sundaram, S. A. Chalmers, P.F. Hopkins, and A. C. Gossard

Science Vol. **254**, pp. 1326-1335 (1991)

[23] J.P. Eisenstein, Superlatt. Microstruct. **12**, 107 (1992)

[24] M. Sundaram, A. Wixforth, R.S. Geels, A.C. Gossard, and J.H. English,

Journ. Vac. Sci. Technol. **B9**, 1524 (1991)

[25] see, e.g., A. Wixforth, M. Sundaram, K. Ensslin, J.H. English, and A.C.
Gossard, Appl. Phys. Lett. **56**, 454 (1990)

[26] E.G. Gwinn, R. M. Westervelt, P.F. Hopkins, A.J. Rimberg, M. Sundaram,
and A.C. Gossard, Phys. Rev. **B39**, 6260 (1989)

[27] K. Ensslin, A, Wixforth, M. Sundaram, P.F. Hopkins, J.H. English, and A.C.
Gossard, Phys. Rev. **B47**, 1366 (1993)

[28] K. Karrai, X. Ying, H. D. Drew, M. Santos, M. Shayegan, S. R. E. Yang, and
A. H. MacDonald, Phys. Rev. Lett. **67**, 3428 (1991)

[29] Z. Schlesinger, J.C.M. Hwang, and S.J. Allen, Jr.

Phys. Rev. Lett. **50**, 2098 (1983), see also

A. D. Wieck, J.C. Maan, U. Merkt, J.P. Kotthaus, K. Ploog, and G. Weimann,

Phys. Rev. **B35**, 4145 (1987)

[30] M. Sundaram, A. Wixforth, P.F. Hopkins, and A.C. Gossard,

Journ. of Appl. Physics **72**, 1460 (1992)

[31] J. Dempsey, and B. I. Halperin, Phys. Rev. **B45**, 1719 (1992)

[32] L. Brey, J. Dempsey, N. F. Johnson, and B. I. Halperin

Phys. Rev. **B42**, 1240 (1990)

[33] J. Dempsey, and B.I. Halperin, Phys. Rev. **B45**, 3902 (1992)

[34] J. Dempsey, and B.I. Halperin, Phys. Rev. **B47**, 4662 (1993)

[35] J. Dempsey, and B.I. Halperin, Phys. Rev. **B47**, 4674 (1993)

[36] N. G. Asmar, and E.G. Gwinn, Phys. Rev. **B46**, 4752 (1992)

[37] P.R. Pinsukanjana, E.G. Gwinn, John F. Dobson, E.L. Yuh, N. G. Asmar,

M. Sundaram, and A. C. Gossard, Phys. Rev. **B46**, 7284 (1992)

[38] S. Oelting, D. Heitmann, and J.P. Kotthaus, Phys. Rev. Lett. **56**,
1846 (1986)

[39] S. Das Sarma, Phys. Rev. **B29**, 2334 (1984)

[40] Q. Li, and S. Das Sarma, Phys. Rev. **B40**, 5680 (1989)

[41] A. Gold, and A. Ghazali, Phys. Rev. **B41**, 8318 (1990)

[42] E. Batke, and D. Heitmann, Infrared Phys.** 24**, 189 (1984)

[43] J.C. Maan, in *Two Dimensional Systems: Heterostructures and
Superlattices*

edited by G. Bauer, F. Kuchar, and H.Heinrich, Springer, Berlin) p.183
(1984)

[44] R. Merlin, Sol. St. Comm. **84**, 99 (1987)

[45] E.G. Gwinn, P.F. Hopkins, A.J. Rimberg, R.M. Westervelt, M. Sundaram, and
A.C. Gossard, Phys. Rev. **B41**, 10700 (1990)

[46] S. E. Laux, D.J. Frank, and F. Stern, Surf. Sci. **196**, 101 (1988)

[47] F. Stern, Phys. Rev. Lett. **18**, 546 (1967)

[48] M. Anderegg, B. Feuerbach, and B. Filton, Phys. Rev. Lett. **27**, 1565
(1971)

[49] F. Sauter, Z. Phys. **203**, 488 (1967), A. R. Melnyh, and M.J.
Harrison,

Phys. Rev. Lett. **21**, 85 (1967)

[50] H. Drexler, W. Hansen, J.P. Kotthaus, M. Holland, and S. P.
Beaumont

Phys. Rev. **B46**, 12849 (1992).

[51] T. Ando, J. Phys. Soc. Japan **39**, 411 (1975), and

T. Ando, J. Phys. Soc. Japan **44**, 475 (1978)

[52] In principle, any periodic modulation of the dielectric properties in the vicinity of the electron system, e.g. a modulation of the system itself, causes a grating coupler effect.

[53] D. Heitmann, in *Physics and Applications of Quantum Wells and
Superlattices*, NATO ASI Series B: Physics , (eds. E.E. Mendez, and K. von
Klitzing),

Vol. **170**, pp.317-346 (1987)

[54] F. Stern, Phys. Rev. Lett. **18**, 546 (1967), for a general review see
also

A. V. Chaplik, Surface Science Reports** 5**, 289-336 (1985)

[55] E. Batke, D. Heitmann, and C.W. Tu, Phys. Rev. **B34**, 6951 (1986)

[56] for a review see, e.g., H. Raether : *Surface Plasmons *,

(Springer Verlag, Berlin, 1988)

[57] A.V. Chaplik, Soviet Phys. JETP **35**, 395 (1972)

[58] G. Eliasson, J.W. Wu, P. Hawrylak, and J.J. Quinn

Sol. St. Comm. **60**, 41 (1986)

[59] D.B. Mast, A.J. Dahm, and A.L. Fetter,

Phys. Rev. Lett. **54**, 1706 (1985)

[60] W. Zawadzki, in : *High Magnetic Fields in Semiconductor Physics II
*(ed. G. Landwehr, Springer Verlag, Berlin), p. 220 (1989)

[61] E. Batke, and C.W. Tu, Phys. Rev. **B34**, 3027 (1986)

[62] F. Stern, Phys. Rev. Lett. **21**, 1687 (1968)

[63] V. Fock, Z. Phys. **47,** 446 (1928)

**FIGURE CAPTIONS :**

**Fig. 1:**

Schematic of different potential energy wells for four different heterostructures, and the corresponding electron distributions. The barriers surrounding the wells are doped in order to supply carriers to the wells. For a narrow square well (a), the electrons are concentrated at the well center. For a wide square well (b), they repel each other and accumulate at the two interfaces. To obtain an electron gas with uniform density, the well must be graded parabolically (c) to compensate for this electrostatic repulsion. From Poisson's equation, electrons introduced into a parabolic well by doping the barriers (d) distribute themselves uniformly at a density given by the curvature of the parabola [taken from ref. 22].

**Fig. 2:**

Realization of a PQW by synthesis of a graded alloy of AlxGa1-xAs with parabolically varying Al mole fraction x. Computer-controlled molecular beam epitaxy is used to create either a digital alloy (top) or an analog alloy (bottom). The digital alloy is a superlattice consisting of GaAs and Al0.3Ga0.7As with a 2nm period as explained in the text [taken from ref.22].

**Fig. 3:**

Result of a selfconsistent calculation (symbols) for the electron densities in the different electrical subbands (a) as well as for the single particle (Hartree-) energy levels (b) of a 200nm wide PQW as a function of the applied gate bias Vg. The interpolating lines are guides to the eye. With increasing negative bias the well becomes depleted while the upper subbands depopulate. At the same time the subband levels increase in energy. The inset shows a typical sample geometry used in our experiments.

**Fig. 4:**

Self consistently obtained Hartree-potential (thick solid line) together with the calculated wavefunctions of a PQW at high well filling. The total wavefunction represents the nearly uniform electron distribution over in this case nearly 100 nm width. At the same time the potential profile in this region is nearly flat.

**Fig. 5:**

Calculated resonance positions of the modes [[omega]]+ and [[omega]]- according to eq. (10) for a parabolic well subjected to a tilted magnetic field. Due to mode coupling the degeneracy between both modes is lifted as soon as the tilt angle [[Theta]] between the sample normal and the magnetic field direction is non-zero. For two coupled harmonic oscillators, the resulting gap between both lines is solely determined by the tilt angle.

**Fig. 6:**

Experimentally obtained spectra of the cyclotron resonance in a PQW in a tilted
magnetic field. The tilt angle in this case is very small
([[Theta]]=3^{[[omicron]]}) such that a splitting of both lines
according to eq. (10) is not yet achieved. Nevertheless, a sharp dip in the
envelope of the spectrum indicates the region of anticrossing between both
lines and allows for an exact determination of the resonance condition.

**Fig. 7:**

Experimentally obtained resonance positions for three different tilted field experiments as a function of the total magnetic field strength B. With increasing tilt angle the mode anticrossing becomes more pronounced leading to a well defined separation of both lines. The solid lines in the figure are the result of a calculation acc. to eq. (10) using m*=0.07 mo, and for all three measurements. Thus, the experimental data are perfectly described within the simple model of two coupled harmonic oscillators.

**Fig. 8:**

Experimentally obtained resonance positions (filled symbols) of a tilted
field experiment at [[Theta]]=23^{o} together with the result of a
calculation (solid line) according to eq. (10). Also shown are the extracted
intensities (open symbols) normalized to the one obtained in a normal magnetic
field, which are proportional to the oscillator strength if the line width does
not change significantly over the magnetic filed range shown. The dotted line
is the result of eq. (11), assuming unpolarized far-infrared radiation and
normal incidence.

**Fig. 9:**

The analogue experiment of Fig. 8 in Voigt-geometry. Here, the magnetic field is directing along the plane of the PQW which leads to a complete hybridization of both the cyclotron resonance and the sloshing intersubband-like mode of the PQW that obeys Kohn's theorem. Filled symbols and the solid line represent the extracted and calculated resonance positions as a function of the in-plane magnetic field. Open symbols and the dashed line depict the intensity (oscillator-strength) as extracted from our experiment and calculated using eq. (12). The origin of the discrepancies between the oscillator strength as obtained experimentally and theoretically is not known, to date.

**Fig. 10:**

Grating coupler induced intersubband-like resonance (sloshing mode) of a 200 nm
wide PQW (PB26) for three different gate voltages Vg or carrier densities Ns,
respectively. Although the density has been changed from Ns =
2.5^{.}10^{11}cm^{-2} down to Ns =
1.6^{.}10^{11}cm^{-2 }the resonance position does not
change in agreement with Kohn's theorem. The spectra have been taken at zero
magnetic field.

**Fig. 11:**

Resonance energy of the absorption of a PQW as obtained using three different techniques as indicated in the inset vs. the gate bias or the carrier density, respectively (top). The thin horizontal line indicates the natural frequency of the respective well (PB26) as expected from the growth alone. There is a small but reproducible scatter of the experimentally obtained positions around the expected value, which we attribute to a sensitive test to the local curvature of the well. The bottom panel sketches the change in carrier density or equivalently the width of the electron slab with gate bias [22].

**Fig. 12:**

Results of a self-consistent calculation for the sample PB25 (W=75 nm, [[Delta]]=75meV). Both the self-consistent potential and the resulting charge distribution are depicted as functions of depth z. The center of the PQW is located at z=0. Note the considerable shift of the electron distribution with increasing negative gate bias. In (a) we show the situation for the completely filled well, (b) depicts intermediate filling, whereas in (c) the well is nearly depleted and strongly asymmetric [9].

**Fig. 13:**

Self consistent subband energies (a) and carrier densities (b) for the sample PB25. At high well-filling up to three electrical subbands are occupied.

**Fig. 14:**

Experimentally obtained grating coupler induced spectra for sample PB25. The relative change in transmission is shown for different gate voltages Vg and carrier densities Ns. Traces A and B have been taken after a short illumination of the sample [Vg(A)=0V, Vg(B)=+0.6V], which still increases Ns in the well. In both limits of high and low electron concentration deviations from the harmonic oscillator picture are observed, manifesting themselves in the occurrence of additional lines [9].

**Fig. 15:**

Comparison of the experimentally obtained resonances of trace B in Fig.14 with those calculated by Brey et al. for a completely filled PQW. The sidelines are due to internal oscillations of the electron system caused by the finite width of the well. All essential features of the experiment are reproduced in the calculation of the far infrared absorption. The deviations are believed to occur due to differences in the sample parameters in both cases and the uncertainty of the exact filling factor [[eta]] in the experiment [9].

**Fig. 16:**

Experimentally obtained spectra for sample PB25 at high well filling in tilted
magnetic fields 6T<= B<=15T. The tilt angle between the sample normal and
the direction of the magnetic field is [[Theta]]=23^{o}. Due to the
intentionally induced deviations from 'ideal parabolicity' new lines which are
not present in an 'ideal' PQW are observed within the gap between both
center-of-mass modes [[omega]]+ and [[omega]]- .

**Fig. 17:**

Experimentally obtained spectra for sample PB25 at intermediate well filling in tilted magnetic fields 6T<= B<=15T. Note the occurrence of a strong additional line between both center-of-mass modes, indicating strong deviations from ideal parabolicity. Also note the broadening and decrease in intensity of the upper CM-modes around B=12T, indicating the onset of an additional anticrossing.

**Fig. 18:**

Experimentally obtained spectra for sample PB25 at even lower well filling in tilted magnetic fields 6T<= B<=15T. The anticrossing as already indicated in Fig. 17 now has become stronger such that a new line splitting can be clearly resolved.

**Fig. 19:**

Experimentally obtained spectra for sample PB25 at very low well filling in tilted magnetic fields 6T<= B<=15T. The strong asymmetry of the confining potential (cf. Fig. 12) completely changes the observed spectra with respect to those of Fig. 16. The simple harmonic oscillator picture is by no means applicable under these experimental conditions.

**Fig. 20:**

Summary of all the experimentally obtained resonance positions from the tilted-field experiment on sample PB25 for different well fillings or gate voltages, respectively. With decreasing carrier density the spectra deviate more and more from the prediction of the simple harmonic oscillator picture as represented by eq. (10). Additional lines besides the CM modes appear and the whole spectrum is shifted towards higher energies, indicating a 'stiffening' of the confining potential.

**Fig. 21:**

Comparison of the theoretical results (lines) obtained employing a classical hydrodynamic approach [31] and our experimental data corresponding to Fig. 16 (symbols). The additional lines in the calculation are caused by non-local interactions driven by the Fermi-pressure and resulting in the occurence of internal oscillations of the electron slab.

**Fig. 22:**

Direct comparison of the extracted resonance positions obtained in a tilted field experiment for two different PQW with the same curvature. In PB25, however, the parabola is truncated by additional vertical sidewalls, leading to the occurence of additional lines and a slight energetical increase of the low-energy CM mode. The experimental conditions for both samples are the same.

**Fig. 23:**

Comparison of the result of a fully self consistent calculation (dots) as described in the text [33,34] and our experimentally obtained resonance positions (boxes). The area of the dots is proportional to the oscillator strength of the respective lines. The inset depicts the corresponding ground state electron density (solid line, left scale) and the self consistent potential (dashed line, right scale) at B=12T. Taken with permission from ref. [34].

**Fig. 24:**

Comparison between the calculation of ref. [34] (dots) and our experimental results for the partially depopulated well PB25 at a gate-bias of Vg=-0.7V (cf. Figs. 18, 20d). Even for this strongly symmetry-broken case the agreement between the calculation and our experimental result is excellent. The inset depicts the corresponding ground state electron density (solid line, left scale) and the self-consistent potential (dashed line, right scale) at B=12T. Taken with permission from ref. [34].

**Fig. 25:**

Experimental spectra for sample PB25 for Vg = 0V and Ns =
5^{.}10^{11}cm^{-2} in Voigt-geometry (cf. Fig.9). The
magnetic field B is directing along the plane of the PQW, using normally
incident unpolarized FIR. The most striking fact is the occurrence of two lines
instead of the single mode of the plasma-shifted CR in an 'ideal PQW'.

**Fig. 26:**

Squared resonance positions for PB25 at two different gate voltages versus the square of the in-plane magnetic field B. The simple harmonic oscillator picture predicts the solid line, representing the plasma-shifted CR as observed in an 'ideal' PQW. Note that in (b) the upper mode intersects the energy-axis with a slope approximately given by 2 as described in the text.

**Fig. 27:**

(a) Comparison of the result of a fully self-consistent calculation (dots) as described in the text [33,34] and our experimentally obtained resonance positions (boxes) in Voigt-geometry. As compared to an 'ideal' system the single line of the plasma-shifted CR is now broadened into a 'continuum' with asymmetrically distributed oscillator strength. The area of the dots is proportional to the oscillator strength of the respective lines.

(b) Calculated spectra for the resonances of (a). A phenomenological scattering time has been used to model the absorption lines. Taken with permission from ref. [35].

**Fig. 28:**

Calculated B=0 uniform-slab dispersion of the collective modes for a PQW as a
function of q.We using a classical hydrodynamic model [31]. Both bulk-like
([[omega]]>[[omega]]o) and surface like excitations
([[omega]]<[[omega]]o) can be excited. The surface-like modes degenerate at
for large q. If the slab width is comparable to q^{-1}, a strict
separation of the mode characters is no longer possible.

**Fig. 29:**

Experimentally obtained resonance positions of the collective mode spectrum for sample PB31 as a function of the carrier density Ns or slab width, respectively. A 2um-periodicity grating coupler has been deposited on top of the sample surface to provide both the necessary z-component of the FIR to couple to the inter-subband-like mode as well as to provide a wave vector q to couple to the intra-subband plasmon. The lines are the result of the classical hydrodynamic model [31] whereas the dots represent our experimental data.

**Fig. 30:**

Typical set of spectra as obtained on a 200 nm wide PQW (PB31) with a 2um grating coupler. We plot the relative change in transmission as a function of the FIR energy for different magnetic fields. The spectra have been vertically offset for clarity. At low as well as for high magnetic fields three lines are observed which are interpreted in terms of cyclotron resonance (CR), intra-subband-magnetoplasmon (MP), and intersubband -plasmon (ISR). Around B ~ 2.8 T the intra- and the intersubband mode interact resonantly, whereas the CR remains unaffected.

**Fig. 31:**

(a) Experimentally obtained resonance positions of an experiment as shown in Fig. 31, but for lower carrier density Ns. In this case presumably only the lowest electrical subband is occupied and the confining potential is symmetric. Within the resolution of our experiment no resonant interaction between the inter- and intra-subband plasmon is observed which is in agreement with eq. (19).

(b) Resonance positions as obtained from the spectra of Fig. 31. The dots mark the experimental result whereas the lines are the result of the calculation involving Das Sarma's two band model [39] for all three modes. Clearly a resonant coupling of the intra - and inter-subband like mode is observed and results in an anticrossing of the two modes. To obtain best agreement between theory and experiment the parameters as listed in the inset have been used.

(c) For comparison the same result for a more positive gate bias corresponding
to a higher electron concentration with at least three occupied subbands. In
this case the mode coupling is strongly enhanced as compared to (a) and (b) and
the two-band model does not hold any longer.

**Fig. 32:**

(a) Typical spectra as obtained in transmission and Voigt geometry for different magnetic fields B. Both cases of q being perpendicular (a) and parallel (b) to B are depicted. In both experiments we observe three resonances, which are related to the intra-subband plasmon at finite wave vector (p,q), the inter-subband plasmon at finite wave vector (0,q), and the inter-subband plasmon at zero wave vector (0,0). The latter is also called plasma shifted cyclotron resonance. In (a) the resonance (p,q) has a characteristic negative magnetic field dispersion, whereas (0,q) and (0,0) follow the dispersion for a magneto-electric hybrid excitation as described in the text.

(b) In contrast to (a), for parallel configuration the intra-subband plasmon (p,q) exhibits no magnetic field dependence as expected from the simple model given by eq. (26).

**Fig. 33:**

Resonance positions as extracted from Fig. 33 for both orientations of the intra-subband plasmon wave vector q with respect to B. They reflect the magnetic field induced anisotropy of the plasmon mass. Symbols represent the experimental data, whereas the solid lines are the result of the simple model as described in the text. The parameters used which are in good agreement with the ones expected are given in the inset.