R. Schuster, K. Ensslin, V. Dolgopolov*, and J. P. Kotthaus
Sektion Physik, Ludwig-Maximilians-Universität München,
Geschwister-Scholl-Platz 1,
D-80539 München, Germany
and
G. Böhm and W. Klein
Walter Schottky Institut, Technische Universität München, D-85748 Garching, Germany
PACS: 73.20.Dx, 03.65.Sq, 05.45.+b, 73.50.Jt
Abstract:
We study electron transport in an antidot superlattice with a finite number of periods. At low carrier densities and high magnetic fields the electrons travel phase coherently over the entire system in magnetic edge channels and interfere with each other under the influence of the geometry of the antidot potential landscape. The magnetoresistance displays pronounced Aharonov-Bohm-type oscillations arising from electrons that travel along the bound states which form around the antidots. The characteristic period of the oscillations is given by the area defined by the circumference of the antidots.
Antidot superlattices represent a model system to study electron transport
through a periodic potential landscape [1-4]. Starting from a high mobility
two-dimensional electron gas (2DEG) a periodic array of potential pillars can
be fabricated by various technological means. A typical potential landscape is
sketched in Fig. 1(a) where the maxima of the potential peak through the Fermi
energy. So far most experiments have been performed on systems whose extensions
are much larger than the electron mean free path
as well as the phase coherence length
.
Pronounced maxima have been observed in the magnetoresistance arising from
ballistic electron orbits around groups of antidots. The experimental
observations have successfully been described by classical dynamics neglecting
the phase of the electrons [5].
Effects related to the phase coherence of the electrons along the classical
trajectories have only recently been observed. Weiss et al. reported so-called
quantum oscillations [6] that rely on the phase coherence on the length scale
of the lattice period. Schuster et al. investigated a finite antidot lattice
whose total dimension is smaller than
at low temperatures [7]. In this case the classical commensurability
oscillations are superimposed by strong reproducible fluctuations in the
magnetoresistance that die out at higher magnetic fields where the cyclotron
diameter becomes smaller than the lattice period. The Fourier analysis reveals
an Aharonov-Bohm (AB)-type effect related to the area enclosed by the
respective classical cyclotron orbit around groups of antidots.
Usually the Fermi energy
in the antidot lattices investigated so far is relatively high [Fig. 1(a)]
in the sense that the filling factor of the Landau levels that arise at finite
magnetic field is typically larger than 10 for the magnetic fields of interest.
Here we focus on a very different regime of low carrier densities and magnetic
fields around filling factor
.
The system now schematically resembles a shallow lake (Fermi sea) between the
antidot pillars [Fig. 1(b)]. The low Fermi level and the reduced screening of
the electrons lead to relatively large antidots (see transition from Fig. 1 (a)
to (b)). Transport now occurs predominantly in edge states that also extend
into the bulk of the sample because of the presence of the antidot lattice. The
role of the classical electron trajectories as described previously is now
taken over by quantum mechanical edge states.
In order to investigate phase coherence effects in antidot systems we fabricate
an array of 9x9 antidots surrounded by a square geometry (see inset of Fig. 2).
For very low temperatures where electron-electron scattering is reduced both
and
may exceed the size of the system [8]. The electrons now carry a phase and an
amplitude and therefore can interfere with each other.
The fabrication process starts from a
heterostructure which contains a two-dimensional electrons gas 65nm below the
surface. Its electron density is
and the elastic mean free path is
at
.
A Hall bar is defined by wet etching and provided with Ohmic contacts
(AuGe/Ni). The antidots as well as the square confining geometry pattern is
produced by electron beam lithography thus providing an inherently good
alignment of the two structures. The square geometry around the finite antidot
lattice has point contact-like openings at its corners serving as contacts to
the system. The pattern is then transferred onto the sample by a carefully
tuned wet etching step. The etch rate depends sensitively on the size of the
respective features. In making the width of the bars that define the square
confining geometry larger than the diameter of the antidots it is guaranteed
that the finite lattice is decoupled from the outside 2DEG before the antidot
potential is actually formed in the 2DEG. The inset of Fig. 2 shows an atomic
force microscope image of the finite antidot lattice with a system size
and a period
.
Each antidot is well developed and the variation in size is remarkably small.
The whole structure is covered by a gate metal which allows to tune the Fermi
energy in the system. The sample is cooled in a dilution refrigerator down to
bath temperatures of
.
Typical four-terminal-measurements of the resistance
are made by passing a current
through the contacts
and
and measuring the voltage drop across the other two contacts
and 
.
In high magnetic fields the electrons in a homogeneous 2DEG travel along the
boundary of the confining geometry in magnetic edge channels [9]. The
suppression of backscattering leads to a quantized resistance in the Hall
effect [10]. In our antidot samples edge states also develop along the
circumference of the antidots. The electrons in these states are sensitive to
the flux through the antidots if the phase coherence is maintained along the
circumference. The edge states which form along the boundary of the square
confining geometry are not sensitive to the flux through the antidots. However,
if the Fermi Energy
is lowered by a very negative gate voltage the constrictions between the walls
and the antidots and between two neighboring antidots become very narrow. This
leads to a coupling of the edge states which form at the edge of the confining
geometry and the localized states around the circumference of the antidots.
Figure 2 presents the magnetoresistance
for different negative gate voltages. At low magnetic fields
the resistance maximum corresponding to a classical orbit around a single
antidot can be seen (see vertical arrow). Superimposed on this classical
commensurability maximum there are oscillations in the magnetoresistance that
reveal an Aharonov-Bohm effect or, equivalently, a modifiedShubnikov-de Haas
effect related to the area enclosed by the classical cyclotron orbit that fits
around a single antidot [6, 7]. At higher magnetic fields a wide minimum occurs
at about
which is related to the minimum of the Shubnikov-de Haas oscillations at
filling factor
. The filling factor defined as
specifies the number of occupied Landau levels below the Fermi energy. A series
of highly periodic AB-oscillations with a period of
arises in the magnetoresistance traces just in this
minimum in the magnetic field regime
.
The characteristic area leading to the AB-oscillations is defined by the
circumference of the antidots. The role of the classically pinned trajectories
is now taken over by quantum mechanical edge states also in the sense of
commensurability effects. In single quantum dots such AB-oscillations have been
observed where the electrons are confined to edge states close to the perimeter
of the dot [11,12]. Similar results were obtained by Kirczenow et al. who
inserted a single antidot inside a narrow wire [13].
At high magnetic fields the energy of an edge channel is determined by its
guiding center energy
where
is the cyclotron frequency,
is the Landau level index and the last term accounts for spin splitting. The
electrons follow the equipotential lines
around the antidots. During one revolution they accumulate a phase
where
is the elementary flux quantum. The resistance oscillations are periodic in B
corresponding to the addition of one flux quantum through the enclosed area
.
In order to get a quantitative understanding of the oscillations we Fourier
transform the magnetoresistance as displayed in Fig. 3. If the window for the
transformation is chosen around the minimum corresponding to
pronounced maxima in the Fourier transforms are found. The maxima shift to
higher frequencies
for more negative gate voltages. This is in agreement with the simple picture
that with decreasing carrier density the antidots size increases. Consequently,
the area given by the circumference of the antidots and the corresponding
frequency of the AB-oscillations increases. The inset in Fig. 3 shows the
normalized frequency as a function of the applied gate voltage. The frequencies
approach a value
which is determined by a circular area
with the diameter equal to the lattice period. If the gate voltage is lowered
further the edge channels are reflected at the barriers between the antidots
and the resistance diverges.
We have studied the behavior of the minimum in the magnetoresistance related to
filling factor
in more detail for a different sample whose antidots are almost as big as the
lattice period and are therefore larger than in sample one [Fig. 4]. On this
second sample the main features observed are very similar to what was described
before. In particular the gate voltage dependence of the AB-period or, in other
words, the area dependence follows the same trend. In this second sample the
coupling between the edge channels can be achieved be applying moderate
negative gate voltages. Therefore the mobility in the unpatterned regions
remains comparatively high and the edge channels are spin resolved. The barrier
height between the antidots can be adjusted by means of the gate voltage. At
there are no oscillations present (lowest trace in Fig. 4(c)). As the gate
voltage becomes more negative the coupling between edge states becomes
apparent. The period of the oscillations in this case is only
.
At very strong depletion, 4(a), we observe a doubling of the period of the
oscillations. In between there is a transition regime where every second
minimum of the oscillations is only weakly pronounced [Fig. 4(b)].
In order to explain the occurrence of the period doubling we discuss several
possibilities. In general a factor of two may arise by a transition from
to
oscillations. The latter originating from the interference of time reversed
trajectories. However, it is not clear why the
oscillations are only observed in a certain gate voltage regime. Since all
scattering lengths (elastic and inelastic) decrease with increasing depletion
the
oscillations should persist for negative gate voltages while the
effect could be averaged out more easily. This is in contrast to our
experimental observations. Another possibility is the contribution of edge
channels which form in the region between four antidots. These may, in
principle, provide an additional frequency. But since they will enclose a
similar area than the one defined by the circumference of the antidots they can
only accidentally provide a double frequency signal corresponding to an area
twice as large.
On this basis we propose that the additional period is related to resonant
processes between two spin resolved edge channels. Quantum mechanically bound
states around the antidots exist if the enclosed flux is a multiple integer of
the flux quantum
which corresponds to a phase change
.
This spatial quantization condition determines the energy
of the bound states around the antidots. Resonant transmission between bound
states occurs if the Fermi energy
coincides with energy of the bound states
.
This situation is the basis of the following discussion. In Fig. 5 a schematic
representation of the edge states at two different gate voltages is presented.
For gate voltages which lead to low barriers between the antidots both edge
channel are fully transmitting. The bound states are spin resolved and are
denoted
and
.
Since spin is conserved in the resonant tunneling process the current flows
through states
and
.
The subsequent minima in the resistance can be attributed to resonant tunneling
of electrons with opposite spin. The observed phase between the two
oscillations could be explained by a possible Coulomb interaction between the
two edge states as suggested by Sachrajda et al. for resonant reflection
processes in a single quantum dot geometry [12]. For higher potential barriers
the edge channel with
is partly reflected while the
channel is fully transmitted. In this regime every second minimum is only
weakly pronounced. Eventually the
channel is completely reflected and only the
channel contributes to the conductivity [Fig. 5(b)]. Resonant tunneling should
in principle be independent of the thickness of the barrier. Our explanation
relies on the assumption that the tunneling probabilities among spin-up and
spin-down edge channels are comparable. For steep potential walls this could
indeed be the case since the spatial separation between edge states could be
small. However, for a more refined analysis one has to consider that in an
antidot lattice each bound state interacts with the bound states of the four
neighboring antidots and that both resonant tunneling and resonant reflection
can occur.
At higher magnetic fields a series of B-periodic oscillations arises in the
minimum where only a single edge state is occupied. The oscillations in this
regime have only a single period. These observations support the idea that spin
related processes are responsible for the occurrence of the additional period.
Further experiments may tell us which type of interpretation can lead to a
physical picture.
Recently Lenssen et al. [14] reported on the observation of quantum interference in a two-dimensional lateral superlattice. They explained their observations by the phase-coherent coupling of localized classical orbits. Our experimental results as presented in this paper unambiguously demonstrate that AB-oscillations occur in the edge state regime and that their frequency is related to the lattice period. In the sample as presented in Fig. 3 the Fermi wavelength is about 1/8 of the circumference of an antidot. In order to be able to observe Aharonov-Bohm oscillations the sample has to be homogeneous at least on that scale. For higher carrier densities the improved screening behavior will lead to an even more homogeneous sample. This fact as well as the observation of pronounced fluctuations in the high-density regime [7] imply that the phase coherence length in our system is of the order of the lattice dimension.
In summary, we have reported on edge state transport through an antidot
lattices with a finite number of periods (9x9). In the regime of small antidots
and large carrier densities commensurability oscillations are known to dominate
the low-field magnetoresistance. In finite antidot lattices pronounced
fluctuations are superimposed reflecting the dominant role of chaotic
trajectories in phase space. [7] In the present publication, however, where we
concentrate on the quantum Hall regime the survival of regular trajectories is
thought to lead to the transition to quantum mechanical edge states. This is
achieved with the application of a negative gate voltage which lowers the Fermi
energy such that the Fermi sea ends up to be a shallow lake that just covers
the minima of the potential. In this regime where transport is dominated by
edge states we observe AB-type oscillations superimposed on the SdH-minimum
corresponding to filling factor
.
The characteristic area for the AB-oscillations is given by the circumference
of the antidots. By carefully adjusting the barrier height between the antidots
the contribution of spin resolved edge states can be observed.
It is a pleasure to thank T. Schlösser, M. Suhrke and D. Wharam for most valuable discussions and M. Wendel for help with the atomic force microscope. We acknowledge financial support by the Deutsche Forschungsgemeinschaft (SFB 348) and the Volkswagen Stiftung (V. D. ).
*Permanent address: Institute of Solid State Physics, Chernogolovka, 142432 Moscow District, Russia
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FIGURE CAPTIONS
Figure 1:
Antidot potential with a high Fermi energy
(a) and very low Fermi Energy (b). The position of the Fermi energy is depicted
by the white planes.
Figure 2:
Magnetoresistance traces of a finite antidot lattice at T= 30 mK at very low carrier densities. The arrow indicates the classical commensurability maximum. The inset shows an image taken with an atomic force microscope of a wet etched surface of a GaAs/AlGaAs- heterostructure with the characteristic dimensions as indicated. Ohmic contacts are made to the corners of the square indicated by i,j,k and l.
Figure 3:
Fourier transform calculated from the experimentally obtained magnetoresistance traces [Fig. 2]. The right inset shows the dependence of the AB-frequency on the applied gate voltage. The frequencies approach a value which is given by the circular area as indicated in the left inset.
Figure 4:
Minimum of the magnetoresistance related to filling factor
.
At very strong depletion (c) every second minimum of the oscillations is
suppressed which leads to doubling of the period.
Figure 5:
A schematic representation of the edge channels for two different barrier
heights between the antidots. The barrier height is adjusted the gate voltage (
).
In (a) both spin resolved edge channels are fully transmitted which corresponds
to the situation in Fig. 4 (c). In (b) the second edge channel is reflected at
the barriers between the antidots corresponding to the experimental trace in
Fig. 4 (a).