Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium
It is shown that a nonequilibrium filling of an upper surface subband induced by the microwave resonance can be the origin of the absolute negative conductivity and zero-resistance states for the two-dimensional electron system on liquid helium under magnetic field applied normally. Contrary to the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1184422025-02-23T20:00:50Z Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium Monarkha, Yu.P. Письма редактору It is shown that a nonequilibrium filling of an upper surface subband induced by the microwave resonance can be the origin of the absolute negative conductivity and zero-resistance states for the two-dimensional electron system on liquid helium under magnetic field applied normally. Contrary to the similar effect reported for semiconductor systems, an oscillating sign-changing correction to the dc-magnetoconductivity appears due to quasi-elastic inter-subband scattering which does not involve photons. The analysis given explains remarkable magnetooscillations and zero-resistance states recently observed for electrons on liquid helium. The author is grateful to K. Kono and D. Konstantinov for acquainting with experimental data before publication and for helpful discussions. 2011 Article Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium / Yu.P. Monarkha // Физика низких температур. — 2011. — Т. 37, № 1. — С. 108–111. — Бібліогр.: 9 назв. — англ. 0132-6414 PACS: 73.40.–c, 73.20.–r, 73.25.+i, 78.70.Gq https://nasplib.isofts.kiev.ua/handle/123456789/118442 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Письма редактору Письма редактору |
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Письма редактору Письма редактору Monarkha, Yu.P. Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium Физика низких температур |
| description |
It is shown that a nonequilibrium filling of an upper surface subband induced by the microwave resonance can be the origin of the absolute negative conductivity and zero-resistance states for the two-dimensional electron system on liquid helium under magnetic field applied normally. Contrary to the similar effect reported for semiconductor systems, an oscillating sign-changing correction to the dc-magnetoconductivity appears due to quasi-elastic inter-subband scattering which does not involve photons. The analysis given explains remarkable magnetooscillations and zero-resistance states recently observed for electrons on liquid helium. |
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Monarkha, Yu.P. |
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Monarkha, Yu.P. |
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Monarkha, Yu.P. |
| title |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| title_short |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| title_full |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| title_fullStr |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| title_full_unstemmed |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| title_sort |
microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2011 |
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Письма редактору |
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https://nasplib.isofts.kiev.ua/handle/123456789/118442 |
| citation_txt |
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband twodimensional electron system on liquid helium / Yu.P. Monarkha // Физика низких температур. — 2011. — Т. 37, № 1. — С. 108–111. — Бібліогр.: 9 назв. — англ. |
| series |
Физика низких температур |
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2025-11-24T20:17:57Z |
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2025-11-24T20:17:57Z |
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1849704292939726848 |
| fulltext |
© Yu.P. Monarkha, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1, p. 108–111
Письма редактору
Microwave-induced magnetooscillations and absolute
negative conductivity in the multisubband two-
dimensional electron system on liquid helium
Yu.P. Monarkha
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: monarkha@ilt.kharkov.ua
Received October 7, 2010
It is shown that a nonequilibrium filling of an upper surface subband induced by the microwave resonance
can be the origin of the absolute negative conductivity and zero-resistance states for the two-dimensional elec-
tron system on liquid helium under magnetic field applied normally. Contrary to the similar effect reported for
semiconductor systems, an oscillating sign-changing correction to the dc-magnetoconductivity appears due to
quasi-elastic inter-subband scattering which does not involve photons. The analysis given explains remarkable
magnetooscillations and zero-resistance states recently observed for electrons on liquid helium.
PACS: 73.40.–c Electronic transport in interface structures;
73.20.–r Electron states at surfaces and interfaces;
73.25.+i Surface conductivity and carrier phenomena;
78.70.Gq Microwave and radio-frequency interactions.
Keywords: magnetooscillations, surface electrons, negative conductivity, microwave resonance.
Magnetooscillations and absolute negative conductivity
induced by microwave (MW) radiation under magnetic field
applied perpendicular to a thin semiconductor film were
predicted theoretically by Ryzhii already in 1969 [1]. For a
long while, these effects were not observed experimentally.
Quite recently, magnetooscillations and the effect of vanish-
ing resistivity (zero-resistance states) induced by MW radia-
tion were observed in semiconductor two-dimensional (2D)
electron systems of ultrahigh mobility [2]. It was shown that
the negative conductivity ( < 0xxσ ) by itself suffices to
explain the zero-dc-resistance state (ZRS) observed [3].
There are also alternative explanations which refer to an
excitonic mechanism for electron pairing [2].
Recently, microwave-induced magnetooscillations [4]
and even vanishing resistivity [4,5] were observed in the
quasi-2D electron system on liquid helium. Despite strik-
ing similarity of these results to those obtained for semi-
conductor systems, there are important differences in expe-
rimental conditions. First, it should be emphasized that
surface electrons (SEs) on helium is a nondegenerate sys-
tem with strong Coulomb interaction. Therefore, it is
doubtful whether electron paring can be realized in such a
system. The second important difference is that for semi-
conductor 2D systems magnetooscillations and ZRS were
predicted and observed for a quite arbitrary MW frequency
ω without excitation of higher subbands, while for SEs on
liquid helium the effect is observed only when the MW
frequency is under the resonance condition for exciting
electrons to the second surface subband: 2 1=ω Δ −Δ
(here lΔ describes energy spectrum of surface levels, and
= 1, 2, ...l ). This means that the theory of photon-
stimulated impurity scattering [1] which leads to magne-
tooscillations and negative conductivity in semiconductor
2D systems is not sufficient for explaining ZRS of SEs on
liquid helium, at least under conditions of the experiment.
In this Letter we show that additional electron popula-
tion of the first excited subband induced by the MW reson-
ance provides a possibility for ordinary impurity inter-
subband scattering to result in an oscillating sign-changing
correction to .xxσ At high enough MW powers and at cer-
tain magnetic fields, this correction leads to absolute nega-
tive conductivity and, according to Ref. 3, to the zero-
resistance state observed for SEs on liquid helium [4,5].
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband two-dimensional electron
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 109
In order to understand the physics of appearing of the
sign-changing correction to the dc-magnetoconductivity
under the MW resonance condition, it is instructive to re-
mind the explanation given for photon-induced impurity
scattering [1]. Consider a multisubband 2D electron system
under perpendicular magnetic field B , and assume that its
energy spectrum is described by , = ( 1/ 2),l n l c nε Δ + ω +
where cω is the cyclotron frequency / eeB m c and
= 0,1, 2,...n For photon-induced impurity scattering with-
in the ground subband, electron current xj is proportional
to 2( )y c B yq n n eEl q⎡ ⎤′δ ω − + ω+⎣ ⎦ [1], where q is the
momentum exchange, 2 = /Bl c eB , E is the in-plane
driving electric field, ω is the photon energy. The delta-
function represents the energy conservation which takes
into account Landau level tilting in the driving field;
2 ( )B y y HeEl q eE X X q v′≡ − ≡ (here = /Hv cE B ). De-
pending on the sign of ( )c n n′ω − + ω , the yq could be
positive or negative which was the explanation for chang-
ing the direction of the current.
Under the MW resonance condition 2 1= ,ω Δ −Δ a
nonequilibrium filling of the first excited subband gives rise
to usual inter-subband impurity scattering which is restricted
by a similar delta-function 2
2,1( ) ,c B yn n eEl q⎡ ⎤′δ ω − + Δ +⎣ ⎦
where , = .l l l l′ ′Δ Δ −Δ Therefore, the sign of yq depends
on 2,1( ) ,c n n′ω − + Δ and, if for ordinary impurity scatter-
ing there is a current correction linear in ,yq it will be an
oscillating sign-changing correction. We shall show that a
similar correction to electron conductivity appears when an
additional filling of the second surface subband exceeds
that given by the Boltzmann factor: /2,1
2 1e > 0TeN N −Δ
−
(Te being electron temperature). In this case, the MW af-
fects electron transport in an indirect way, just changing
subband occupation numbers .lN
SEs on liquid helium are scattered quasi-elastically by
helium vapor atoms and by capillary wave quanta (rip-
plons). In this work we consider only scattering at vapor
atoms which is similar to usual impurity scattering in semi-
conductor systems. It is conventionally described by the
effective potential ( ) = ( )e a a e aV V− δ −R R R R and by
the interaction Hamiltonian
( ) †
int = e ,ia e
aV V a a ′′−
′
∑ KR
KK K
K,K
(1)
where { , }zK≡K q is the 3D momentum of a vapor
atom, and †aK ( aK ) is the creation (destruction) operator
of vapor atoms. The 2D vector q represents momentum
exchange at a collision.
Considering a multisubband 2D system under the MW
resonance condition, we cannot use the conventional li-
near-response theory. As we shall see, even the effective
electron temperature approximation will not help to ob-
tain a sign-changing correction to .xxσ For arbitrary po-
pulation of surface subbands = /l l en N N , it is conve-
nient to use the force-balance method (for details, see
Ref. 6). For an isotropic system, this method assumes that
the average kinetic friction scattF acting on electrons is
proportional to the electron current, which can be written
as scatt eff av= ,e eN m− νF v where avv is the average elec-
tron velocity, eN is the total electron number, em is the
electron mass. The proportionality factor eff ( )Bν rep-
resents the effective collision frequency which generally
depends on magnetic field B (quantum limit) and even on
the areal electron density sn when many-electron effects
are considered. Once eff ( )Bν is found, the conductivity
tensor can be obtained from equations similar to the con-
ventional Drude equations.
For noninteracting electrons, scattF can be found by
evaluating the momentum gained by scatterers. Consider
an arbitrary electron distribution ( ) = ( ),l lf N fε ε where ε
is the in-plane energy. We shall use the condition
( ) ( ) = ,l l lf D d Nε ε ε∫ where ( )lD ε is the density-of-state
function 2 2 1
,( ) = (2 ) Im ( ),l B l nnD l G−ε − π ε∑ and , ( )l nG ε
is the single-electron Green’s function for the correspond-
ing subband. In the Born approximation, we obtain
( )2
0
scatt , , ,
,
= ( , ),
a
e l l l l l l l
e l l
N n s S
m ′ ′ ′
′
ν
ω∑ ∑
q
F q q (2)
where = / ,l l en N N ,l ls ′ represents
2
,(e )iK zz l lKz
′∑
normalized according to Ref. 7, , ,= /l l l l′ ′ω Δ ,
, 2 2
1( , ) = ( )l l
B
S d f
l
′ ω ε ε ×
π ∫q
2
, , ,
,
( )Im ( )Im ( ),n n q l n l n
n n
J x G G′ ′ ′
′
× ε ε + ω∑
( )
22
,
!( ) = e ( ) ,
!
m x m
n n m n
nJ x x L x
n m
−
+ ⎡ ⎤
⎣ ⎦+
( )m
nL x is the associated Laguerre polynomials,
2 2= / 2q Bx q l , and ( )
0
aν is the collision frequency in the
absence of the magnetic field. If the collision broadening
of Landau levels does not depend on ,l the , ( , )l lS ′ ωq
coincides with the dynamical structure factor ( , )S ωq of a
nondegenerate 2D system of noninteracting electrons.
For SEs on liquid helium, electron–electron collision fre-
quency is much higher than eff .ν In this case, , ( , )l lS ′ ωq
coincides with its equilibrium form (0)
, ( , )l lS q′ ω and ( )f ε
equals to /e TeA −ε only in the center-of-mass reference
frame moving with the drift velocity avv (here A is a nor-
malization factor). Because of the Galilean invariance along
the interface, in the laboratory frame, , ( , )l lS ′ ωq (as well as
the dynamical structure factor [6]) acquires the Doppler
shift: ( ) (0)
, av,, = ( , ).l l l lS S q′ ′ω ω− ⋅q q v The av⋅q v repre-
sents an additional energy exchange at a collision which
appears in the center-of-mass reference frame.
In the cumulant approach [8], ,Im ( )l nG ε has a Gaus-
sian shape with the Landau level broadening lΓ indepen-
Yu.P. Monarkha
110 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1
dent of n (for the lowest surface subband,
( )2
1 0
2= a
cΓ ω ν
π
). In this case,
21/2 , /(0)
, ;,
,,
( )2( , ) = e ( ),n n q n Tc e l l n nl l
l ln n
J x
S q I
Z
′ − ω
′ ′−′
′′
π
ω ω
Γ∑
(3)
where /1 = 1 e Tc eZ − ω− − ,
( )
22 2
, ; 2
,
/ 4
= exp ,
8
c l e l
l l m
l l e
m T
I
T
′
′
⎡ ⎤⎛ ⎞ω− ω −Γ Γ⎢ ⎥ω − +⎜ ⎟⎜ ⎟⎢ ⎥Γ⎝ ⎠⎢ ⎥⎣ ⎦
and 2 2 2
, = ( ) / 2l l l l′ ′Γ Γ + Γ . We assume that .l TΓ
Using /(0) (0)
, ,( , ) = e ( , )Te
l l l lS q S q− ω
′ ′−ω ω , Eq. (2) can be
represented as
( )2
(0)0
scatt , , av,
,
= ( , )
2
a
e l l l ll l
e l l
N s S q
m ′ ′′
′
ν
ω − ⋅ ×∑ ∑
q
F q q v
/ /, ave e .
T Tl l e el ln n
−Δ ⋅′
′
⎡ ⎤× −⎢ ⎥⎣ ⎦
q v (4)
For slow drift velocities, av ,l l′⋅ Γq v and av ,eT⋅q v
linear in avv terms of scattF and the effective collision
frequency effν can be easily obtained from Eq. (4). The
linear term obtained expanding /ave Te⋅q v represents the
usual result of which the effective collision frequency is
always positive.
There is another term originated from the expansion of
/ (0),
, av,( e ) ( , )
Tl l e
l l l ll ln n S q
−Δ ′
′ ′′− ω − ⋅q q v in av ,/ l l′⋅ Γq v .
This is the linear in q term we were searching for. It is simi-
lar to the corresponding term of Ref. 1 obtained for photon-
induced impurity scattering because (0)
, av, ( , )l ll lS q ′′ ω − ⋅q v
initially contained the δ-function representing energy con-
servation. Expanding , ; , av( )l l m l lI ′ ′ω − ⋅q v of Eq. (4), we
obtain the factor 2
, ( ) / 4l l c l en n T′ ′Δ − − ω −Γ which has
different signs at the opposite sides of the point where it
equals zero.
If heating of the electron system is small ( 2,1eT Δ ),
we can use the two-subband model ( = 1, 2l ) and find the
sign-changing correction to effν
( ) 2 2
/1,20 2,1
eff 2 12
2,1
= e
a
Tc es
n n
−Δν ω ⎛ ⎞δν − − ×⎜ ⎟
⎝ ⎠π Γ
coth ( ) ( ) ,
2
c
c c
eT
⎡ ⎤⎛ ⎞ω
× Φ ω +Θ ω⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
(5)
where
/2,1
2,1 1,2( ) = ( ) e ( ),
Te
c c cF F
Δ
Φ ω δ ω − δ ω
/2,1
2,1 1,2( ) = ( ) e ( ).
Te
c c cH H
Δ
Θ ω δ ω − δ ω
For , > 0l l′Δ ,
( )2
,
, , ; ,
,=0
/ 4
( ) = ( ) .
l l c l e
l l c l l m l l
l lm
m T
F I
∞ ′
′ ′ ′
′
Δ − ω −Γ
δ ω ω
Γ∑
In the opposite case ( , < 0l l′Δ ),
/
, , ; ,
=0
( ) = e ( )m Tc e
l l c l l m l l
m
F I
∞
− ω
′ ′ ′−δ ω ω ×∑
( )2
,
,
/ 4
.
l l c l e
l l
m T′
′
Δ + ω −Γ
×
Γ
The corresponding form of , ( )l l cH ′δ ω differs from that
of , ( )l l cF ′δ ω only by the additional factor m inside the
sum. One can see that 2,1Fδ and 1,2Fδ entering ( )cΦ ω (as
well as 2,1Hδ and 1,2Hδ in ( )cΘ ω ) have opposite signs.
This means that the contribution of scattering from = 2l
to = 1l is not cancelled by the contribution of scattering
from = 1l to = 2.l
The conventional contribution to effν originated from
the expansion of /ave Te⋅q v can be written as
( ) 2 2
/,0 ,
1/2
,,
( ) = e
2
a
Tl lc l l e
l
l ll le
s
B n
T
−Δ′ ′
′
′′
ν ω
ν ×
Γπ
∑
, ,coth ( ) ( ) ,
2
c
l l c l l c
e
F H
T ′ ′
⎡ ⎤⎛ ⎞ω
× ω + ω⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
(6)
where ,l lF ′ and ,l lH ′ are defined similar to ,l lF ′δ and
,l lH ′δ with the exception that their equations do not con-
tain the factor 2
, ,( / 4 ) / .l l c l e l lm T′ ′Δ ω −Γ Γ∓ For SEs on
liquid helium, Landau levels are extremely narrow:
.l TΓ In this case, the collision frequency term of
Eq. (5) originated from the expansion in av ,/ l l′⋅ Γq v
acquires an additional large factor ,/ 1e l lT ′Γ as com-
pared to the conventional term of Eq. (6). Therefore, even
a small nonequilibrium filling of the excited subband
/2,1
2 1e > 0Ten n −Δ− can lead to giant oscillations in eff .ν
The results of numerical evaluations of ,xxσ taking in-
to account both the usual contribution to effν defined by
Eq. (6) and the sign-changing correction of Eq. (5), are
shown in Fig. 1 for two levels of MW power defined by
the ratio of the Rabi frequency RΩ to ( )
0
aν . Fractional
occupancies ln are found from the rate equation of the
two-subband model, while electron temperature eT is ob-
tained from the energy balance equation. The MW reson-
ance frequency equals 95 GHz. Other parameters corres-
pond to experiments with liquid 3He at T = 0.4 K. One can
see that for noninteracting electrons, the amplitude of the
sign-changing correction to xxσ becomes larger in the low
magnetic field range, and, at high MW powers, xxσ be-
comes negative. The shape of xxσ oscillations obtained
here is in striking accordance with experimental data ob-
served for SEs on liquid helium [5]. Conductivity minima
Microwave-induced magnetooscillations and absolute negative conductivity in the multisubband two-dimensional electron
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 111
Fig. 2. The fractional occupancy of the second subband for two
MW power levels: ( )
0/ = 0.2a
RΩ ν (solid) and ( )
0/ = 0.4a
RΩ ν
(dashed).
4 5 6 7 8 9 10
0
0.2
0.4
N
N
2
e
/
� �2,1 c/�
occur when the parameter 2,1 / cΔ ω is slightly above an
integer, which also agrees with the observation.
The fractional occupancy of the second subband 2n
as a function of the parameter 2,1 / cΔ ω is shown in
Fig. 2. It is reduced in the vicinity of the condition
2,1 / c mΔ ω → (here m is an integer) due to elastic inter-
subband scattering.
The many-electron effect can be taken into account con-
sidering an electron moving in a quasi-uniform electric field
of other electrons fE due to thermal fluctuations [9]. In this
case, according to Ref. 6, ,l l′Γ entering (0)
, ( , )l lS q′ ω should
be replaced by 2 2
, ,l l q Cx′Γ + Γ where (0)= 2C BfeE lΓ , and
(0) 3/43 .e sfE T n Since 1 / ,Bl B∝ the many-electron
effect broadens magnetooscillations and reduces their ampli-
tudes in the low magnetic field range, which is also in ac-
cordance with experimental observations.
It should be noted that ( )xx Bσ curves do not change
much even if we set = .eT T Nevertheless, electron tem-
perature as a function of the parameter 2,1 / cΔ ω oscil-
lates and rises when 2,1 / .c mΔ ω → In this case, quasi-
elastic decay transfers 2,1Δ to the kinetic energy of the in-
plane motion. At ( )
0/ 0.2,a
RΩ ν ∼ the maximum increase
in eT is less than 2 K which allows to use the two-subband
approximation. For high MW powers, under the condition
2,1 / c mΔ ω → electron population of higher subbands
should be taken into account. Electron heating can be sup-
pressed by increasing the holding electric field E⊥ or by
lowering the ambient temperature .T In the later case, elec-
trons are predominately scattered by ripplons. Nevertheless,
qualitatively magnetooscillations of xxσ should be the same
because typical energy of ripplons involved is extremely
small. There is also a possibility to use a higher MW reson-
ance condition 1= lω Δ −Δ with > 2.l
Concluding, we have shown that remarkable MW-
induced oscillations of magnetoconductivity of SEs on liq-
uid helium and zero-resistance states observed in Refs. 4, 5
can be understood as a consequence of inter-subband
electron scattering by vapor atoms or ripplons under the
condition that the fractional occupancy of the excited
subband exceeds that given by the conventional
Boltzmann factor. This explains why the MW resonance
condition 2 1=ω Δ −Δ is crucial for observation of mag-
netooscillations and ZRS in this system. Electron–electron
interaction, increasing with electron density, is shown to
suppress the amplitude of magnetooscillations and to de-
stroy the ZRS.
The author is grateful to K. Kono and D. Konstantinov
for acquainting with experimental data before publication
and for helpful discussions.
1. V.I. Ryzhii, Fiz. Tverd. Tela 11, 2577 (1969) [Sov. Phys.
Solid State 11, 2078 (1970)].
2. R. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B.
Johnson, and V. Umansky, Nature 420, 646 (2002).
3. A.V. Andreev, I.L. Aleiner, and A.J. Millis, Phys. Rev. Lett.
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Fig. 1. Magnetoconductivity vs magnetic field for two MW pow-
er levels: ( )
0/ = 0.2a
RΩ ν (solid) and ( )
0/ = 0.4a
RΩ ν (dashed).
0.3 0.4 0.5 0.6 0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
B, T
T = 0.4 K
�
�
x
x
,
1
0
–
9
–
1
|