Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface
We measured the resonance spectra of edge magnetoplasmon (EMP) oscillations in a two-dimensional (2D) electron system located on a liquid-helium surface below 1.1 K. Systematic measurements of the resonance fre-quency and the damping rate as a function of the lateral confinement electric field stren...
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irk-123456789-1188122017-06-01T03:03:14Z Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface Yamanaka, Shuji Arai, Toshikazu Sawada, Anju Fukuda, Akira Yayama, Hideki Электроны и ионы над/в сверхтекучем гелии We measured the resonance spectra of edge magnetoplasmon (EMP) oscillations in a two-dimensional (2D) electron system located on a liquid-helium surface below 1.1 K. Systematic measurements of the resonance fre-quency and the damping rate as a function of the lateral confinement electric field strength shows clear evidence of the oscillation mode transformation. A pronounced change corresponding to the mode transformation was ob-served in the damping rate. When 2D electrons are confined in a strong lateral electric field, the damping is weak. As the lateral confinement electric field is reduced below a certain threshold value, an abrupt enhancement of the damping rate is observed. We hypothesize that the weak damping mode in the strong lateral confinement electric field is the compressive density oscillation of the electrons near the edge (conventional EMP) and the strong damping mode in the weak confinement field is the coupled mode of conventional EMP and the boundary displacement wave (BDW). The observation of the strong damping in the BDW–EMP coupled mode is a manifestation of the nearly incompressible feature of strongly interacting classical electrons, which agrees with earlier theoretical predictions. 2013 Article Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface / Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, Hideki Yayama // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1086–1095. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS: 73.20.–r, 73.20.Mf http://dspace.nbuv.gov.ua/handle/123456789/118812 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электроны и ионы над/в сверхтекучем гелии Электроны и ионы над/в сверхтекучем гелии |
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Электроны и ионы над/в сверхтекучем гелии Электроны и ионы над/в сверхтекучем гелии Yamanaka, Shuji Arai, Toshikazu Sawada, Anju Fukuda, Akira Yayama, Hideki Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface Физика низких температур |
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We measured the resonance spectra of edge magnetoplasmon (EMP) oscillations in a two-dimensional (2D) electron system located on a liquid-helium surface below 1.1 K. Systematic measurements of the resonance fre-quency and the damping rate as a function of the lateral confinement electric field strength shows clear evidence of the oscillation mode transformation. A pronounced change corresponding to the mode transformation was ob-served in the damping rate. When 2D electrons are confined in a strong lateral electric field, the damping is weak. As the lateral confinement electric field is reduced below a certain threshold value, an abrupt enhancement of the damping rate is observed. We hypothesize that the weak damping mode in the strong lateral confinement electric field is the compressive density oscillation of the electrons near the edge (conventional EMP) and the strong damping mode in the weak confinement field is the coupled mode of conventional EMP and the boundary displacement wave (BDW). The observation of the strong damping in the BDW–EMP coupled mode is a manifestation of the nearly incompressible feature of strongly interacting classical electrons, which agrees with earlier theoretical predictions. |
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Article |
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Yamanaka, Shuji Arai, Toshikazu Sawada, Anju Fukuda, Akira Yayama, Hideki |
| author_facet |
Yamanaka, Shuji Arai, Toshikazu Sawada, Anju Fukuda, Akira Yayama, Hideki |
| author_sort |
Yamanaka, Shuji |
| title |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
| title_short |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
| title_full |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
| title_fullStr |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
| title_full_unstemmed |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
| title_sort |
oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
| topic_facet |
Электроны и ионы над/в сверхтекучем гелии |
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http://dspace.nbuv.gov.ua/handle/123456789/118812 |
| citation_txt |
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface / Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, Hideki Yayama // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1086–1095. — Бібліогр.: 33 назв. — англ. |
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Физика низких температур |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10, pp. 1086–1095
Oscillation mode transformation of edge
magnetoplasmons in two-dimensional electron system
on liquid-helium surface
Shuji Yamanaka1, Toshikazu Arai2*, Anju Sawada2, Akira Fukuda3, and Hideki Yayama1
1Department of Physics, Graduate School of Sciences, Kyushu University, Fukuoka 812-8581, Japan
2Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8502, Japan
E-mail: toshikaz@tohtech.ac.jp
3Department of Physics, Hyogo College of Medicine, Nishinomiya, Hyogo 663-8501, Japan
Received March 12, 2013
We measured the resonance spectra of edge magnetoplasmon (EMP) oscillations in a two-dimensional (2D)
electron system located on a liquid-helium surface below 1.1 K. Systematic measurements of the resonance fre-
quency and the damping rate as a function of the lateral confinement electric field strength shows clear evidence
of the oscillation mode transformation. A pronounced change corresponding to the mode transformation was ob-
served in the damping rate. When 2D electrons are confined in a strong lateral electric field, the damping is
weak. As the lateral confinement electric field is reduced below a certain threshold value, an abrupt enhancement
of the damping rate is observed. We hypothesize that the weak damping mode in the strong lateral confinement
electric field is the compressive density oscillation of the electrons near the edge (conventional EMP) and the
strong damping mode in the weak confinement field is the coupled mode of conventional EMP and the boundary
displacement wave (BDW). The observation of the strong damping in the BDW–EMP coupled mode is a ma-
nifestation of the nearly incompressible feature of strongly interacting classical electrons, which agrees with ear-
lier theoretical predictions.
PACS: 73.20.–r Electron states at surfaces and interfaces;
73.20.Mf Collective excitations (including excitons, polarons, plasmons and other charge-density exci-
tations).
Keywords: surface, liquid helium, magnetoplasmons.
1. Introduction
In bound two-dimensional electron systems (2DESs)
placed in a strong perpendicular magnetic field, electrons in
the vicinity of the system edges play an important role in
influencing the system's transport properties. For example,
edge electronic states significantly contribute to the magne-
totransport of a degenerate 2DES in the quantum Hall re-
gime [1,2] where the entire current is carried by edge elec-
trons. An interesting application for spintronics devices
proposes the use of edge electrons in graphene with zigzag
edges, which support spin-dependent transport [3].
In this light, the important discovery of collective exci-
tation modes was made in a classical (non-degenerate)
2DES formed on a liquid-helium surface [4,5]. The collec-
tive excitation is called an edge magnetoplasmons (EMPs),
in which an oscillation of charge fluctuation localizes in a
very narrow strip near the edge of a 2DES and the charge
oscillation propagates along the perimeter. EMPs appear as
the lower frequency branch of 2D magnetoplasmons, and
the frequency decreases as the external perpendicular mag-
netic field is increased. Subsequently, EMPs were ob-
served in a wide variety of 2DESs including degenerate
2DESs in semiconductor heterostructures [6], 2DESs in
nanostructures [7], and in 2D ion systems trapped under a
liquid-helium surface [8]. Taking advantage of the EMP
property that the charge oscillation localizes near the edge,
researchers have attempt to employ EMPs as sensitive
probes to study the structure of edge electronic states and
electronic transport along edge channels [9,10].
* Present address: Center of General Education, Tohoku Institute of Technology, Sendai 982-8577, Japan.
© Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, and Hideki Yayama, 2013
mailto:toshikaz@tohtech.ac.jp
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system
In EMPs, a deviation from the equilibrium electron dis-
tribution causes accumulation of charge in a narrow strip
near the edge due to the Lorentz force acting on the elec-
trons, and consequently, the EMP wave propagates along
the perimeter [11,12]. The conventional EMP theory as-
sumes that the EMP is a compression wave of the electron
density near the edge, while the boundary position is fixed.
The conventional EMP theory fairly accurately describes
the basic features of EMPs. It is understood that the EMP
spectrum is gapless, and thus, an EMP is a candidate for
the lowest energy excitation from the ground state. One of
the specific characteristics of an EMP is that its frequency
is inversely proportional to the applied magnetic field.
Therefore, at a sufficiently large value of magnetic field,
EMP resonance can be observed at frequencies of the order
10 kHz or less. Thus far, several types of EMP modes have
been studied theoretically and experimentally [13]. Among
the types of modes, there is one that propagates along the
edges of an incompressible 2DES, and it is particularly
observed in 2DESs in the quantum Hall states. In principle,
an incompressible 2DES does not support the compression
density wave. In such a case, the charge strip is induced by
the displacement of the boundary from the equilibrium
position while the uniform density distribution is main-
tained [14]. We call this mode as the boundary displace-
ment wave (BDW). The 2DES formed on a liquid-helium
surface is a classical non-degenerate 2DES, since the Fer-
mi temperature 10FT mK corresponding to the typical
areal electron density 1210n m–2 is considerably lower
than the working temperature of standard experimental
conditions ( > 100T mK).
In general, a classical 2DES is not incompressible and
the BDW is not responsible for the collective edge mode;
however, it is predicted that the classical 2DES at low
temperature can be nearly incompressible when the elec-
tron–electron Coulomb interaction energy dominates over
the thermal kinetic energy of the electrons [13,15]. In the
present paper, we report our observation of a novel BDW
mode in a classical 2DES formed on a liquid-helium sur-
face. We show that our novel BDW mode is an in-phase
coupled oscillation of the conventional EMP and the BDW
(BDW–EMP coupled mode, ),+ω theoretically predicted
by Monarkha [13,15]. Experimental observation of BDW–
EMP coupled mode has been reported as a small resonance
peak whose frequency is slightly downshifted from the
large resonance peak of a conventional EMP. The amount
of the frequency shift in Ref. 16 was in reasonable agree-
ment with theoretical calculation, and thus, the observed
mode was identified as the out-of-phase coupled mode of
the BDW–EMP ( ).−ω In order to confirm that the BDW–
EMP coupled mode reflects the nearly incompressible fea-
ture of a classical 2DES, it is crucial to measure the damp-
ing rate of the resonance. The energy dissipation of the
electron motion is determined by electron collision with
the quanta of the capillary wave (ripplon) in the superfluid-
helium surface. If the 2DES is nearly incompressible, elec-
trons located in the deep interiors from the edge are in mo-
tion, and ripplon scattering of these electrons contributes to
the dissipation. On the other hand, in the case of the com-
pression wave of a conventional EMP, only the electrons
within the narrow strip near the edge are responsible for
the damping [13,15]. In the present work, we were able to
transform the conventional EMP into the BDW–EMP
coupled mode by lowering the strength of the electric field
for lateral confinement. A pronounced increase was seen in
the damping rate. We measured the enhanced damping
rates of BDW–EMP coupled mode. Our results provide
clear evidence of the nearly incompressible feature of the
classical 2D Coulomb system.
This paper is organized as follows: In Sec. 2, a quick
overview of EMPs is provided. In Sec. 3, we present our
experimental techniques of EMP resonance and signal
analysis. In Sec. 4, the experimental results concerning the
enhanced damping rate are presented. In Sec. 5, we present
our hypothesis that the oscillation mode with the enhanced
damping rate is the BDW–EMP coupled ω+ mode, and
that the incompressible motion of the 2DES is responsible
for the damping rate enhancement.
2. Overview of edge excitation modes
In this section, we provide a quick overview of EMPs.
Owing to the characteristic properties listed below, EMPs
have been extensively studied for various 2DESs both ex-
perimentally and theoretically. In the course of such stu-
dies several types of EMPs have been observed, such as
the acoustic mode, boundary displacement wave (BDW),
and the BDW–EMP coupled mode. These modes differ in
their manner of charge oscillation; however, they have the
following main characteristic properties in common: EMPs
are gapless, i.e., ln | |,EMP q qω where EMPω denotes
the frequency, and q denotes the wave number. Hence,
EMPs is considered as the lowest energy excitation from
the ground state of the 2DES. The EMP frequency is pro-
portional to the 2DES density n and inversely propor-
tional to the applied normal magnetic field B, i.e.,
/EMP n Bω ∝ . In contrast to the bulk 2D magnetoplas-
mons [5], the frequency of the EMPs decreases with in-
creasing magnetic field. For sufficiently large value of B,
EMPω is several orders of magnitude smaller than the cyc-
lotron frequency = / ,c eeB mω where e and em denote the
charge and the mass of an electron, respectively. The EMP
wave propagates along the perimeter of a bounded 2DES
in a single direction determined by the sign of the Hall
conductivity .xyσ Accordingly, a resonance occurs when
the perimeter length P is an integer multiple of the wa-
velength, i.e., = 2 /q m Pπ ( = 1, 2, ...).m These common
properties can be understood within the conventional
EMP model.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1087
Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, and Hideki Yayama
2.1. Conventional EMP
Let us suppose that a charge fluctuation occurs in a
2DES. In applying a strong normal magnetic field, due to
the Lorentz force, electrons move perpendicular to the
electric field. In a bounded system, the charge fluctuation
accumulates in a narrow strip near the edge and move only
within the strip. A conventional EMP is a collective oscil-
lation of electron density that propagates along the edges
of a 2DES [4,5]. The conventional EMP theories of Vol-
kov and Mikhailov [11,17] and Fetter [12,18,19] assume a
fixed boundary position, and the charge fluctuation occurs
as a density fluctuation. Therefore, the conventional EMP
is a compression wave. Volkov and Mikhailov arrived at
the expressions for the frequency EMPω and the damping
rate EMPγ as below:
= 2 = [ln (| | ) ],
2
xy
EMP EMP f
q
f q b C
σ
ω π − +
πε
(1)
( )
1 .
4 ln | |
xx
EMP b q b
σ
γ −
πε
(2)
We denote the frequency of conventional EMP as EMPω
hereafter. Here, ε denotes the dielectric constant of the
substrate; for the case of 2DES on liquid helium,
0= 1.057ε ε is the dielectric constant of liquid helium 0(ε
is the permittivity of vacuum). The parameter b denotes
the characteristic length scale of the charge strip where the
EMPs localize. The term xxσ denotes the magnetocon-
ductivity, and 0= /xy n e Bσ is the Hall conductivity of the
2DES 0(n is the electron density at the center of the sam-
ple). The parameter fC denotes a geometrical constant.
2.2. Acoustic modes
In the theories of conventional EMPs, the equilibrium
density profile near the edge is regarded as the step-
function-shaped. In actual systems, the electron density
falls off from 0n to 0 at the edge with a characteristic
length .w The step-function approximation holds as long
as .b w For the case of a 2DES on a liquid-helium sur-
face, standard experiments involve a pair of parallel metal-
plate pressing electrodes above and below the surface. The
pressing electrodes screen the electric field that is generat-
ed in the 2DES. The screening length d is approximately
the same as the distance between the 2DES and the press-
ing electrode, typically 1 mm. Therefore, the oscillating
EMP electric field does not penetrate deep within the
2DES over the distance d so that the length b may be of
the same order as d. Likewise, the length scale w is of the
same order as d. Hence, for a 2DES on a liquid-helium
surface, .b w d In such a situation, the step-function
approximation becomes inaccurate and a more realistic
equilibrium density profile is needed to account for the
resulting profile. Nazin and Shikin [20] considered EMPs
in a 2DES with a smooth equilibrium density profile near
the edge, and they discovered a number of possible novel
acoustic modes. In the acoustic mode, the charge density
oscillates in the direction perpendicular to the edge, as well
as in the direction along the edge. The spectrum of the
acoustic mode changes according to the applied magnetic
field. In weak magnetic fields, the frequency of the acous-
tic mode is calculated [20] to be proportional to ,B and in
strong magnetic fields, it is proportional to 1/B as in the
case of a conventional EMP [21]. The acoustic mode has
been experimentally identified by Kirichek et al. [22].
Compared with the conventional EMP, a considerably
smaller amplitude is expected to be observed in the acous-
tic mode, because of the screening effect by the alternating
charge perpendicular to the edge [21].
2.3. Boundary-displacement-wave (BDW) modes
In the theories of the conventional EMP and the acous-
tic modes, it is assumed that the 2DES is a compressible
Coulomb liquid. But in fact, EMP waves have been expe-
rimentally observed in incompressible 2DESs under the
conditions of the quantum Hall effect [23]. By nature, the
electron liquids in the quantum Hall regime are incompres-
sible because of the presence of the excitation gap from the
ground state. In the EMPs of incompressible 2DESs,
charge fluctuation occurs as the boundary position dis-
placement instead of the charge density fluctuation [14].
Hence, the edge wave that propagates in this manner is
termed the boundary displacement wave. Although the
origin of the charge fluctuation is different in the BDW
from the conventional EMP, the restoring force for both
modes is the Lorentz force acting on the electrons. There-
fore, the BDW provides practically the same spectrum as
the conventional EMP. Hence, from an experimental point
of view, it is very difficult to distinguish a BDW from a
conventional EMP by frequency measurement.
Intuitively, since classical electron liquids are in prin-
ciple compressive, a BDW is not related in any way to the
edge wave in a classical 2DES formed on a liquid-helium
surface. However, it has been reported that the 2DES on
liquid helium can be nearly incompressible at low tempera-
tures, and the occurrence of a type of EMP mode along
with boundary position motion is possible [15,16,24]. The
theory predicts that when the Coulomb interaction energy
between electrons dominates over the thermal kinetic ener-
gy, the strong repulsion force keeps the electrons apart and
the 2DES can be nearly incompressible. In such a case,
since the frequency of a BDW is close to that of a conven-
tional EMP, these two modes can couple, thereby resulting
in two novel modes: the in-phase and the out-of-phase
BDW–EMP coupled modes, ω+ and –ω modes, respec-
tively. The spectra of the BDW–EMP coupled modes are
calculated as
= ,xyqω γ σ± ± (3)
where γ ± denote dimensionless geometrical factors. The
frequency ω+ ( )−ω appears at a slightly higher (lower)
1088 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system
frequency value than that of the conventional EMP. In fact,
the BDW–EMP coupled mode was experimentally identi-
fied by Kirichek et al. [16]. In the experiment, the authors of
Ref. 16 observed small resonance peaks that were slightly
downshifted to the lower frequency region apart from the
main resonance peaks corresponding to conventional EMPs.
The measured frequencies were in good agreement with the
−ω values given by Eq. (3). The resonance amplitude of the
out-of-phase BDW–EMP coupled oscillation should be
small compared with that of the conventional EMP for the
same reason that the amplitude of the acoustic mode is small
[24]. The small observed amplitude of −ω also agrees with
this qualitative picture.
A more significant difference between BDWs (includ-
ing the BDW–EMP coupled modes) and conventional
EMPs is expected to be observed in terms of the damping
rate than in the frequency [15,24]. The number of electrons
involved in the oscillation motion is large in BDWs be-
cause the incompressible motion of electrons penetrates
deep towards the interior from the edge over the distance
where the electric field fluctuation penetrates (screening
length). The penetration depth of the incompressible elec-
tron motion is 1,q− in a similar manner to the fluid motion
in gravity waves of water. On the other hand, in conven-
tional EMPs, the electrons in motion are confined to the
narrow charge strip in the vicinity of the edge. Therefore,
the energy dissipation via the electron–ripplon collision
and consequently the damping rate must be enhanced. The
damping rate enhancement arising from the incompressible
motion of a 2DES has not thus far been observed.
2.4. Discussion on incompressible motion
Among the two BDW–EMP coupled modes, the ω+
mode has not been experimentally identified. There is a
suggestion that in the EMP resonance experiment of 2D
4He+ ion pool trapped below a liquid-helium surface
[25,26], the observed “extra satellites of unknown origin”
may be the ω+ mode [24]. In this respect, there is an ar-
gument about whether or not the BDW–EMP coupled
mode with incompressible motion exists in classical 2D
Coulomb systems. The author of Ref. 24 suggests that the
higher shifted frequency of the satellites from the conven-
tional EMP frequency may correspond to .+ω Against this
suggestion, an argument was presented [26,27] based on an
analysis comparing the amplitudes of electric field pertur-
bation arising from boundary displacement and electron
density compression. However, this analysis is questiona-
ble [24] in terms of the treatment of electron pressure.
The origin of the satellite mode is unclear, and the con-
troversy over the incompressible motion is still unre-
solved. In order to clarify the nature of incompressible
2DES motion in classical 2D Coulomb systems, mea-
surement of the enhanced damping rate in the BDW–EMP
coupled mode is required.
3. Experiment
3.1. Sample cell
In order to study the nearly incompressible feature of a
classical 2DES formed on liquid helium arising from
strong electron–electron interaction, we designed our expe-
riment in a manner that allows us to transform the conven-
tional EMP into the BDW–EMP coupled mode and meas-
ure the difference in the damping rate.
Our sample cell is made of ordinary copper, and it is at-
tached to the mixing chamber of a dilution refrigerator. We
ensured that the cell was perfectly horizontally mounted.
High purity (>
99.99995 %) 4He gas is condensed in the
sample cell. The liquid surface level in the sample cell is
carefully aligned with the center of a 6 T superconducting
solenoid. The sample gas is filled from a room-temperature
gas cylinder through a 0.7-mm-inner-diameter Cu–Ni fill-
ing tube. During the filling process, the sample cell tem-
perature is maintained below 1 K, for which temperature
the vapor pressure of 4He is sufficiently low, in order to
avoid shift in the liquid level by residual gas condensation
on further cooling. Figure 1 shows the structure of the
sample cell. The liquid-helium depth in the sample cell is
precisely measured from the capacitance between the top
and bottom electrodes. The top and bottom electrodes are
circular in shape, 25 mm in diameter, and 3 mm apart. All
data presented in this paper were measured under the con-
dition that the depth d from the bottom electrode was
constant at 1.6 mm.
The tungsten filament attached to the upper electrode is
an electron source. The electrons emitted from the filament
accumulate on the liquid-helium surface. The filament is
made from a tiny light bulb from which the glass cover is
removed. The typical operating condition to charge the
surface is 0.6 V/10 mA with a duration of 1 s. To charge
the surface, it is necessary to slow down the energetic
thermoelectrons on the path from the filament to the sur-
Fig. 1. (Color online) Electrode arrangement of the sample cell:
(a) piezo actuator, (b) top electrode, (c) guard ring, (d) electrons,
(e) bottom electrode, (f) mechanical connection supporting (a)
and (b), (g) tungsten filament, (h) liquid 4He.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1089
Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, and Hideki Yayama
face. Electrons possessing larger kinetic energy than the
surface potential barrier (~ 1 eV) penetrate the liquid. Re-
peated collisions of electrons with cold helium vapor
atoms during the flight can reduce the kinetic energy of
electrons. To ensure repeated collision, we perform elec-
tron emission at a temperature close to 1.5 K, at which
temperature helium vapor is abundant.
The bottom electrode is biased by a positive dc source
dcV = 10–200 V, while the top electrode is always gro-
unded. The generated vertical electric field presses elec-
trons onto the helium surface. The top electrode is mechan-
ically connected to a piezo actuator. The application of an
ac voltage to the piezo actuator causes vertical vibration of
the top electrode. The estimated amplitude of vibration of
~10–5 m is small compared with the distance between the
top electrode and the surface (~10–3 m). Combined with
the positively biased lower electrode, this assembly serves
as a vibrating capacitor electrometer (VCE), which meas-
ures the total charge on the surface [28].
The electric field created by the guard ring electrode
(26 mm inner diameter) surrounding the 2DES laterally
confines the 2DES. By varying the applied potential ,GV
which is normally negative, the strength of the lateral con-
finement electric field is controlled. A large negative value
of GV corresponds to strong confinement. The value of
GV changes the electron density distribution near the edge.
The curves shown in Fig. 2 are the numerically calculated
electron density distributions ( ),n r where r denotes the
radial coordinate, for our sample cell geometry and applied
electrode potentials. The electron density distribution is
uniform in the interior and falls off near the edge. We de-
fine eR as the 2DES radius at which ( )n r drops to zero.
The density distribution curves are calculated under the
boundary conditions of realistic experimental parameters:
= 10GV − V (small radius) and 0 V (large radius), respec-
tively, for a given dcV = 74 V, and the total number of
electrons eN = 91.16 10 ,⋅ and these are conditions corres-
ponding to the resonance spectra shown in Fig. 4. As the
lateral confinement field is decreased, the 2DES expands,
and, therefore, eR increases and the central density 0n
becomes small. For a given surface electron density ,n
Gauss’s law determines the potential eV of the 2DES. In
this work, all measurements were carried out under unsatu-
rated electron densities, i.e., > 0eV V. The electrons are
confined without loss as long as > .e GV V
3.2. EMP resonance technique
We employed the standard frequency sweep conti-
nuous-wave EMP resonance technique in our experiment
[29]. The EMP signals are measured via the capacitive
coupling between the 2DES and the bottom electrode [30].
The bottom electrode is divided into five segments by us-
ing 0.1 mm spacings, as illustrated in Fig. 3. The electrode
array was photo-etched on a glass-epoxy printed circuit
board. The area of the central disk E is half of that of the
entire disk, as are the areas of the segments A–D. As shown
in Fig. 3, all the segments of the bottom electrode are equal-
ly biased by dcV through identical resistors of = 10R MΩ.
The input ac voltage to the segment A drives charge oscilla-
tions near the edge. The charge oscillations propagate as
EMP waves along the 2DES perimeter and are detected as
an ac current in segment C. The inserted blocking capacitor
BC isolates the ac signals from .dcV The time constant
= 1BRC s is sufficiently large compared with the period of
ac signals (< 10–4 s), thereby making the signal current lea-
kage in the circuit of dcV negligible. We measured the sig-
nal current amplitude as a function of the excitation frequen-
cy .f Resonance occurs when the 2DES perimeter is an
integer multiple of the EMP wavelength. Too high a driving
voltage sets off a nonlinear transport of 2DES, as reported
by Monarkha et al. [31]. We varied the driving voltage and
determined a value such that the EMP signals behaved well
in the linear transport regime.
Fig. 2. (Color online) Calculated electron density profiles in the
(1) — strong (red) and (2) — weak (green) confinement regimes
for the immersed-guard-ring case. Inset: enlarged view at the edge.
Fig. 3. (Color online) Schematic explanation of the edge magne-
toplasmon (EMP) resonance experiment electronics and the di-
vided bottom electrode.
1090 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system
We systematically investigated the influence of the GV
variation on the EMP spectra. As mentioned previously,
GV determines the strength of the lateral confinement
electric field. When comparing a conventional EMP and a
BDW (or the BDW–EMP coupled mode), it is easy to as-
sume that the BDW is favored in weak lateral confinement
conditions because the energy cost for the boundary dis-
placement against the confining electric field can be low.
During a series of measurements, we carefully maintained
constant the values of ,T ,B and the total number of elec-
trons .eN Since maintaining eN constant is of prime im-
portance in this work, we carefully checked the reproduci-
bility of the spectra.
3.3. Signal analysis
Figure 4 shows our resonance spectra for = 10GV −
and 0 V, measured at = 0.58T K and = 3.2B T. The re-
sonance condition for the mth harmonic is given as
2 = ,eR mπ λ where λ denotes the wavelength. Clear re-
sonances of the first few harmonics are visible for both
values of .GV
We assume simple damped harmonic oscillations for
the EMP resonance and fit the first two resonance peaks
( = 1, 2)m to the following function ( ):F ω
1 2
2 2
( ) ( )
( ) =
( 1/ )
A AF
R L C
ω + ω
ω
+ ω − ω
(4)
with 2 2 2 2 2( ) = / ( ) ,m m m mA aω ω −ω + γ ω = 2 fω π denotes
the driving angular frequency, mω denotes the resonant an-
gular frequency, mγ denotes the damping rate, ma denotes
the amplitude parameter, and ,R ,L and C represent the
external circuit resistance, inductance, and capacitance, re-
spectively. Since the quantity being measured is the electric
current, it is necessary to divide the voltage 1 2( ) ( )A Aω ω+
by the external circuit impedance. We adjusted the values of
,mω ,mγ and ma as fitting parameters, while ,R ,L and
C were fixed. As shown by the solid curves in Fig. 4, the
fitting results reproduce the spectra well.
A frequency shift to the lower side is observed in the
weak confinement spectrum (green squares in Fig. 4). This
shift can be intuitively understood as the consequence of
the enlarged 2DES edge perimeter, corresponding to an
increase in the wavelength in the weak lateral confinement
electric field. The resonance peak amplitudes of the weak
confinement spectrum in Fig. 4 appear to be suppressed
compared with those of the strong confinement spectrum.
However, this suppression is due to the external circuit
impedance factor in Eq. (4). The impedance increases with
increasing ω within our frequency range of interest. It
should be emphasized that there is no significant difference
in the intrinsic amplitude of the numerator of Eq. (4), re-
gardless of variation in the values of GV .
4. Results and analysis
Figure 5 shows the GV dependence of (a) the resonance
frequency, (b) the damping rate ,γ and (c) the Q-value
Fig. 4. (Color online) EMP spectra for strong ( =GV –10 V, red
circles) and weak ( =GV 0 V, green squares) lateral confine-
ments. The black solid lines indicate the fitting results of ( ).F ω
The conditions for acquiring the spectra are following: T = 0.55 K,
= 3.2B T, 9= 1.16 10 ,eN ⋅ and the distance from the bottom elec-
trode and the liquid surface = 1.6d mm. The guard ring is im-
mersed in the liquid helium. Electrons are emitted at =dcV 70 V
and the measurements are performed at = 74dcV V.
C
ur
re
nt
, n
A
Frequency, kHz
m = 1
m = 2
m = 3
60
40
20
0 50 100 150
Fig. 5. (Color online) Relation between GV and frequency (a),
linewidth (b), and Q-value (c). The red open circles indicate the
experimental results (T = 0.55 K, B = 3.2 T, 9= 1.16 10 ,eN ⋅ and
= 74.0dcV V). The blue solid lines indicate the numerical con-
ventional EMP frequency and linewidth from the Volkov and
Mikhailov theory [17].
Fr
eq
ue
nc
y,
k
H
z
Li
ne
w
id
th
, k
H
z
Q
-v
al
ue
(a)
(b)
(c)
VG , V
60
50
40
8
6
4
10
8
6
I: conv. EMP II III: BDW ( )ω+
–10 –8 –6 –4 –2 0
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1091
Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, and Hideki Yayama
1 1( / )f γ for the first resonance peaks. Since the behavior of
the second peak is essentially the same as that of the first
peak, only the first-peak results are shown.
The most striking behavior is observed in the damping
rate shown in Fig. 5(b). In the course of decreasing | |,GV
the damping rate reaches a minimum value near =GV –6 V,
and it increases in the intermediate region II while finally
reaching the limiting value at = 4GV − V (region III). In a
similar manner, the Q-value changes from about 10 in re-
gion I to 6 in region III. The behavior of the damping rate
as well as the Q-value apparently indicates that the damp-
ing mechanisms for region I and III are different. The re-
sonance frequency monotonically reduces with decreasing
| |,GV but a kink is observed at around = 4GV − V.
We define the region boundaries as the point at which
the damping rate is minimum (I–II boundary) and the point
at which the frequency curve shows the kink (II–III boun-
dary). We hypothesize that an oscillation mode transforma-
tion occurs from region I to III across the intermediate re-
gion II. Decrease in the lateral confinement electric field
results in an expansion of the 2DES and thus, the 2DES
edge approaches the guard ring electrode. If the top of the
guard ring electrode is positioned upright, as shown in
Fig. 1, above the surface, the influence of the surface de-
formation of the meniscus will also need to be considered. In
order to examine the influence of the surface deformation,
we compared the EMP spectra using two types of guard
ring: one was upright (half-immersed in the liquid) and the
other was completely immersed in the liquid helium, thereby
ensuring the liquid surface was flat. The oscillation mode
transformation was observed in both configurations, and
therefore, the meniscus is not a factor in our experiment.
5. Discussion
Let us consider the oscillation mode in region I. Since the
electrons are strongly confined in the lateral direction, it is
reasonable to assume the occurrence of a conventional EMP.
The curves in Figs. 5(a) and (b) indicate the calculated val-
ues of the conventional EMP. We used the Eqs. (1) and (2)
for our calculations. We see that the Volkov and Mikhailov
[11,17] equations well reproduce the experimental results
for region I. In the calculation, we arbitrarily assumed b as
the distance from eR to the radius where ( )n r rises up to
50% of the central density 0.n This assumption is justified
for the following reason: The top and bottom pressing elec-
trodes screen the oscillating electric field of the charge strip
with a screening length ,d which is the distance between
the 2DES and the bottom electrode, and the width of the
charge strip b is estimated to be of the same order as .d
The characteristic length scale of the ( )n r change near
the edge is also of the order of .d The fC and xxσ values
were adjusted to fit the data. For the purpose of confir-
mation, the obtained xxσ values were compared with
the result of an earlier experiment [31] measured at
11
0 = 3.15 10n ⋅ m–2 and = 1.84B T. We measured the
xxσ values of Fig. 6 under conditions of 12
0 = 2.5 10n ⋅ m2
and = 3.19B T. In order to compare the xxσ data ob-
tained for different values of 0n and ,B we assume the
result of Drude model of magnetoconductivity in the limit
of strong magnetic field, 2
0 / .xx n Bσ ∝ According to the
Drude model, the xxσ values of Ref. 31 are required to be
scaled by multiplying a factor of 2.4. As shown in Fig. 6,
the behavior of our obtained xxσ curve is in good agree-
ment with the result of an earlier experiment [31]. There-
fore, the oscillation mode of region I can be regarded as
corresponding to the conventional EMP. We note that the
xxσ of 2DES on liquid helium at low temperatures is
known to be not fully described within the simple Drude
model and it requires more sophisticated theory [31]. Tak-
ing that into account, the coincidence of the scaling factor
of 2.4 could be accidental.
The oscillation mode of the weak-lateral-confinement
region III is expected to be the BDW. The enhanced damp-
ing rate agrees with the qualitative prediction in the study
by Monarkha [13], in which it is pointed out that the in-
compressible electron motion of the BDW penetrates deep
within the interior of the 2DES beyond the screening
length where electric field perturbation is absent; this pene-
tration results in a strong damping, while the electron mo-
tion in the conventional EMP is confined within the
screening length. Since the 2DES on helium in practice
remains compressible, the BDW couples with the conven-
tional EMP in strong magnetic fields [15,16]. The wave
that we considered as the BDW would consequently cor-
respond to the BDW–EMP coupled mode.
Fig. 6. (Color online) Comparison of our results (red circles) with
the results in Ref. 31 (blue pluses) as regards the relation between
temperature and .xxσ In order to compare the xxσ values ob-
tained for different electron density and magnetic field, we as-
sumed Drude model of magnetoconductivity and the xxσ data of
Ref. 31 inelastic are multiplied by a factor of 2.4.
1092 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10
Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system
The BDW–EMP coupled mode was first observed by
Kirichek et al. [16]. Upon comparison of our results with
the resonance signals obtained in Ref. 16, there are both
similarities and differences. The appearance of the BDW–
EMP coupled mode in Ref. 16 is characterized by separate
small resonance peaks in the lower frequency domain
away from the conventional EMP main peaks. The fre-
quency is proportional to 1/ ,B in the same manner as that
of the conventional EMP. The coupled mode was observed
only at < 0.9T K, where the damping is sufficiently
small; however, the damping rate was not measured quan-
titatively. The above-mentioned observations were suc-
cessfully explained by the out-of-phase BDW–EMP coupl-
ing (the −ω mode) [15,16]. Similarly, our resonance
frequency in region III varies as 1/B (Fig. 7(a)) and the
oscillation mode transformation is clear at < 0.9T K
(Fig. 8). At higher temperatures, it becomes difficult to
recognize the change in the damping rate.
Our resonances in region III shows some differences.
We observed the BDW as a single resonance peak, in con-
trast to the separate peaks observed in Ref. 16. This indi-
cates that the oscillation mode in our case transforms from
the conventional EMP of region I into the BDW–EMP
coupled mode of region III depending on the lateral confin-
ing electric field. When compared with the conventional
EMP, in our case, the frequency is higher and the ampli-
tude is of the same order. The comparison of our results
and the reported behavior of −ω [16] is summarized in
Table 1.
Table 1. Comparison of other EMP modes observed in our
study and Ref. 16
Parameter This work Ref. 16
Appearance single peak separate peak
Amplitude identical considerably smaller
Temperature 0.9T K 0.9T K
Frequency 0 /n B∝ 0 /n B∝
Damping strong no data
Identification ω+ −ω
Upon evaluating these observations together, along with
the enhanced damping rate, we conclude that the oscilla-
tion mode in region III is consistent with the in-phase
BDW–EMP coupled mode (the ω+ mode) [15]. Our iden-
tification of the ω+ mode is further confirmed from the
dependence of the frequency on the magnetic field
(Fig. 7(a)), central density (Fig. 7(b)), and temperature
(Fig. 7(c)). All our observations from Fig. 7 are consistent
with the behavior expressed by Eq. (3); ω+ is proportional
to 0 /n B and independent of temperature. The observation
of the ω+ mode has not yet been confirmed. There has
been a speculation that the “satellite mode” in Ref. 15 ob-
served in a 2D ion pool trapped below a liquid-helium sur-
face may be the ω+ mode [13,15]; however, the specula-
tion is still controversial [26,27] and the origin of the
satellite mode is still unclear.
Our observation of the enhanced damping rate supports
the hypothesis by Monarkha [13,15]. The electron correla-
tion of classical electron liquids becomes strong for high
densities, in contrast to quantum electron liquids that tend
Fig. 7. (Color online) Measured first-peak frequencies vs 1/ ,B the central electron density, and temperature in the weak confinement
regime ( = 0GV V). The blue solid lines indicate the values obtained using the least mean square method. The guard ring is im-
mersed ( = 1.6d mm) and = 74dcV V. The other conditions are (a): = 0.55T K and 9= 1.16 10eN ⋅ , (b): = 3.2B T and = 0.55T K,
(c): = 3.2B T and 9= 1.16 10 .eN ⋅
(a)
100 100
50 50
0 00.2 0.2 0.4 0.4 0.6 0.6
Fr
eq
ue
nc
y,
k
H
z
Fr
eq
ue
nc
y,
k
H
z
Fr
eq
ue
nc
y,
k
H
z
1 2 3 4 5
(b) (c)
80
60
40
20
0 0.8 1.0 1.2
T, K1/ , TB
–1 Central electron density, 10 m
12 –2
Fig. 8. (Color online) GV vs Q-value at various temperature val-
ues ( = 3.2B T and 9= 1.16 10 ).eN ⋅ The temperature values are
0.28 (red circles), 0.58 (green triangles) and 0.88 K (blue
squares). The guard ring is immersed ( = 1.6d mm).
12
10
8
6
4
2
–10 –8 –6 –4 –2 0
VG , V
Q
-v
al
ue
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1093
Shuji Yamanaka, Toshikazu Arai, Anju Sawada, Akira Fukuda, and Hideki Yayama
to behave as an ideal Fermi gas in the high density limit.
To examine the relationship between the incompressibility
and electron correlation, we measured the region bounda-
ries of the oscillation modes at various electron densities.
From the GV values for the measured region boundaries,
we calculated the central electron density 0n and the aver-
age lateral confining electric field GE at the outer proximi-
ty of the 2DES for the equilibrium conditions.
The field GE was calculated as follows [32]. For a giv-
en GV value, we solved the electrostatic equation for the
potential ( )rφ of the helium surface level under appropri-
ate boundary conditions. For a small distance r∆ outside
the edge, we define GE as the average gradient of ( ):rφ
( ) ( )
= .e e
G
R r R
E
r
φ + ∆ −φ
−
∆
(5)
We assumed = 20r∆ µm for the sake of numerical preci-
sion. At equilibrium, the expanding force of Coulomb re-
pulsion within the 2DES and the confining force caused by
the guard electrode field acting on the electrons at the edge
are balanced so that = 0GE for 0r∆ → [13]. The GE
value obtained for a finite r∆ value provides a measure of
the strength of lateral confinement.
Figure 9 shows the GE values at the mode transforma-
tions as a function of 0.n The filled circles and triangles
represent the experimentally obtained boundaries of re-
gions I–II and II–III, respectively, and the lines show the
linear fit of these values. These values were measured us-
ing the half-immersed guard ring configuration. The results
for the I–II boundary region for the completely immersed
guard ring configuration are also plotted (open circles). It
can be observed that the lines dividing the oscillation-
mode regions are independent of the sample cell geometry,
thereby indicating that the curves in Fig. 9 indicate the
universality of the oscillation mode for given values of 0n
and .GE The appearance of the BDW in the strong con-
finement regime at high densities suggests that the strong
electron correlation gives rise to the incompressibility of
the classical 2DES.
Our results can be explained by the following simple
analysis. The compressibility κ is related to the density n
and the chemical potential ( )nµ by the relation
1 2= .dn
dn
− µ
κ (6)
Using Seitz’s theorem, ( ) = ( / )G Gn n d dnµ ε + ε and the
ground-state energy [33] 1/2 ,G nε − we have 3/2n−κ − .
Therefore, the compressibility would be small for large
values of .n
6. Conclusion
In summary, we obtained clear evidence of an oscilla-
tion mode transformation from the conventional EMP into
the BDW–EMP coupled mode in a classical 2DES formed
on the surface of liquid helium. The latter mode appears
when the strength of the lateral confinement electric field
is weak. The behavior the BDW–EMP coupled mode is
consistent with the theoretical prediction of the in-phase
coupled mode ( +ω mode) [15]. The coupling of the EMP
with the BDW enhances the damping of the oscillation,
thereby indicating the incompressibility arising from the
strong correlation in the classical Coulomb liquid.
Acknowledgments
We would like to thank K. Kono for his attention to our
work and for helpful discussions. We would like to express
our gratitude to R. Nishinakagawa and S. Isoyama for their
assistance with our experiments. This work was supported
by a Grant-in-Aid for Challenging Exploratory Research
from the Ministry of Education, Culture, Sports, Science
and Technology (MEXT), Japan (Research No. 16654058)
and by a Grant-in-Aid for Scientific Research (B) from
Japan Society for the Promotion of Science (JSPS)
(Nos. 16340108, 19540337, and 21540319).
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