Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems
The dc magnetoconductivity of the multisubband two-dimensional electron system formed on the liquid helium surface in the presence of resonant microwave irradiation is described, and a new mechanism of the negative linear response conductivity is studied using the self-consistent Born approximation....
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| Cite this: | Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems / Yu.P. Monarkha // Физика низких температур. — 2011. — Т. 37, № 8. — С. 829–841. — Бібліогр.: 21 назв. — англ. |
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Monarkha, Yu.P. 2017-05-30T18:05:32Z 2017-05-30T18:05:32Z 2011 Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems / Yu.P. Monarkha // Физика низких температур. — 2011. — Т. 37, № 8. — С. 829–841. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 73.40.–c, 73.20.–r, 73.25.+i, 78.70.Gq https://nasplib.isofts.kiev.ua/handle/123456789/118643 The dc magnetoconductivity of the multisubband two-dimensional electron system formed on the liquid helium surface in the presence of resonant microwave irradiation is described, and a new mechanism of the negative linear response conductivity is studied using the self-consistent Born approximation. Two kinds of scatterers (vapor atoms and capillary wave quanta) are considered. Besides a conductivity modulation expected near the points, where the excitation frequency for inter-subband transitions is commensurate with the cyclotron frequency, a sign-changing correction to the linear conductivity is shown to appear for usual quasi-elastic inter-subband scattering, if the collision broadening of Landau levels is much smaller than thermal energy. The decay heating of the electron system near the commensurability points leads to magnetooscillations of electron temperature, which are shown to increase the importance of the sign-changing correction. The line-shape of magnetoconductivity oscillations calculated for wide ranges of temperature and magnetic field is in a good accordance with experimental observations. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кристаллы Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems Article published earlier |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems Monarkha, Yu.P. Квантовые жидкости и квантовые кристаллы |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems |
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Monarkha, Yu.P. |
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Monarkha, Yu.P. |
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Квантовые жидкости и квантовые кристаллы |
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Квантовые жидкости и квантовые кристаллы |
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2011 |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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The dc magnetoconductivity of the multisubband two-dimensional electron system formed on the liquid helium surface in the presence of resonant microwave irradiation is described, and a new mechanism of the negative linear response conductivity is studied using the self-consistent Born approximation. Two kinds of scatterers (vapor atoms and capillary wave quanta) are considered. Besides a conductivity modulation expected near the points, where the excitation frequency for inter-subband transitions is commensurate with the cyclotron frequency, a sign-changing correction to the linear conductivity is shown to appear for usual quasi-elastic inter-subband scattering, if the collision broadening of Landau levels is much smaller than thermal energy. The decay heating of the electron system near the commensurability points leads to magnetooscillations of electron temperature, which are shown to increase the importance of the sign-changing correction. The line-shape of magnetoconductivity oscillations calculated for wide ranges of temperature and magnetic field is in a good accordance with experimental observations.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118643 |
| citation_txt |
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband two-dimensional electron systems / Yu.P. Monarkha // Физика низких температур. — 2011. — Т. 37, № 8. — С. 829–841. — Бібліогр.: 21 назв. — англ. |
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AT monarkhayup microwaveresonanceinducedmagnetooscillationsandvanishingresistancestatesinmultisubbandtwodimensionalelectronsystems |
| first_indexed |
2025-11-26T04:50:42Z |
| last_indexed |
2025-11-26T04:50:42Z |
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1850609612273745920 |
| fulltext |
© Yu.P. Monarkha, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8, p. 829–841
Microwave-resonance-induced magnetooscillations and
vanishing resistance states in multisubband
two-dimensional electron systems
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 April 5, 2011
The dc magnetoconductivity of the multisubband two-dimensional electron system formed on the liquid he-
lium surface in the presence of resonant microwave irradiation is described, and a new mechanism of the nega-
tive linear response conductivity is studied using the self-consistent Born approximation. Two kinds of scatterers
(vapor atoms and capillary wave quanta) are considered. Besides a conductivity modulation expected near the
points, where the excitation frequency for inter-subband transitions is commensurate with the cyclotron frequen-
cy, a sign-changing correction to the linear conductivity is shown to appear for usual quasi-elastic inter-subband
scattering, if the collision broadening of Landau levels is much smaller than thermal energy. The decay heating
of the electron system near the commensurability points leads to magnetooscillations of electron temperature,
which are shown to increase the importance of the sign-changing correction. The line-shape of magnetoconduc-
tivity oscillations calculated for wide ranges of temperature and magnetic field is in a good accordance with ex-
perimental observations.
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: magnetoconductivity, two-dimensional electron systems, microwave-resonance-induced magneto-
oscillations.
1. Introduction
The discovery of novel microwave-induced oscillations
of magnetoresistivity [1] as a function of the magnetic field
B and so-called zero-resistance states (ZRS) [2,3] has
sparked a large interest in quantum magnetotransport of
two-dimensional (2D) electron systems exposed to micro-
wave (MW) radiation. The 1/B-periodic oscillations were
observed for quite arbitrary MW frequencies ω larger
than the cyclotron frequency cω . The period of these os-
cillations is governed by the ratio / cω ω . ZRS appear in
ultrahigh-mobility GaAs/AlGaAs heterostructures as a
result of evolution of the minima of the oscillations with an
increase in radiation power.
Recently [4,5], MW-induced magnetooscillations and
vanishing of the magnetoconductance xxσ were observed
in the nondegenerate multisubband 2D electron system
formed on the free surface of liquid 3 He . These oscilla-
tions have many striking similarities with those observed
in semiconductor systems: they are 1/B-periodic, governed
by the ratio / cω ω , and their minima eventually evolve in
zero magnetoconductance states nearly at the same values
of / cω ω . The important distinction of these new oscil-
lations is that they are observed only for a MW frequency
fixed to the resonance condition for excitation of the
second surface subband: 2,1=ω Δ (here , =l l l lΔ Δ Δ′ ′− ,
and lΔ describes the energy spectrum of surface electron
states, = 1, 2,...l ).
The ZRS observed in semiconductor systems are shown
[6] to be understood as a direct consequence of the nega-
tive photoconductivity < 0xxσ which can appear with an
increase in the amplitude of conductivity oscillations. Re-
garding the microscopic origin of the oscillations, the most
frequently studied mechanism is based on photon-induced
impurity scattering within the ground subband, when an
electron simultaneously is scattered off impurities and ab-
sorb or emit microwave quanta [7,8]. This kind of scatter-
ing is accompanied by an electron displacement along the
Yu.P. Monarkha
830 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
applied dc-electric field whose sign depends on the sign of
( )c n nω ω− +′ (here = 0,1,...n ). Therefore, sometimes
this mechanism is termed the “displacement” mechanism.
A different mechanism, called the “inelastic” mechanism
[9], explains conductivity oscillations as a result of oscilla-
tory changes of the isotropic part of the in-plane electron
distribution function.
Both microscopic mechanisms of the negative conduc-
tivity prosed for semiconductor systems cannot be applied
for explanation of similar effects observed in the system of
surface electrons (SEs) on liquid helium, because the MW
frequency considered in these theories has no relation to
inter-subband excitation frequencies , ,/l l l lΔ ω′ ′≡ . Re-
cently, a new mechanism of negative momentum dissipa-
tion relevant to experiments with SEs on liquid helium was
briefly reported [10]. It cannot be attributed to “displace-
ment” or “inelastic” mechanisms. In this theory, the origin
of magnetooscillations and negative dissipation is an addi-
tional filling of the second surface subband induced by
MW irradiation under the resonance condition 2,1( = )ω Δ ,
which triggers quasi-elastic inter-subband electron scatter-
ing. The ordinary inter-subband scattering, which does not
involve photon quanta, is accompanied by electron dis-
placements whose sign depends on the sign of
2,1( )c n nω ω− +′ . Usually, this scattering does not lead to any
negative contribution to xxσ . A correction to xxσ proportio-
nal to 2,1( )c n nω ω− +′ was shown to appear only if
/2,1
2 1> e TeN N Δ− , where lN is the number of electrons at
the corresponding subband, and eT is the electron tem-
perature. It is important that this correction is also propor-
tional to a large parameter equal to the ratio of eT to the
collision broadening of Landau levels.
In this work, we perform a systematic theoretical study
of negative dissipation phenomena in a multisubband 2D
electron system caused by nonequilibrium filling of excited
subbands. The magnetotransport theory [10] is generalized
in order to include electron scattering by capillary wave
quanta (ripplons) which limits SE mobility in experiments
[5], where vanishing magnetoconductivity xxσ is ob-
served. In order to understand the importance of MW heat-
ing at the vicinity of commensurability points, electron
energy relaxation is analyzed. A sign-changing correction
to the energy relaxation rate similar to the sign-changing
correction to the momentum relaxation rate is found for
non-equilibrium filling of excited subbands.
2. General definitions
Consider a multisubband 2D electron system under
magnetic field applied perpendicular. The electron energy
spectrum is described by l nΔ ε+ , where ( )= 1n c nε ω +
represents Landau levels ( = 0,1,2...n ). For SEs on liquid
helium (for review see Ref. 11) under a weak holding elec-
tric field ( 0E⊥ → ), 2/l RE lΔ − , where RE is the ef-
fective Rydberg energy of SE states,
( )
( )
22 2
2
1
= , = , = ,
4 12
R B
ee B
e
E a
mm a
ε
Λ
Λ ε
−
+
(1)
Ba is the effective Bohr radius, em is the electron mass,
and ε is the dielectric constant of liquid helium. The exci-
tation energy 2,1Δ is about 6 K (liquid 4 He ) or 3.2 K
(liquid 3 He ). It increases with the holding electric field
E⊥ , which allows also to tune 2,1Δ in resonance with the
MW frequency.
Under typical experimental conditions, the electron-
electron collision rate e eν − of SEs is much higher than the
energy and momentum relaxation rates. Therefore, the
electron distribution as a function of the in-plane energy ε
can be characterized by the effective electron temperature,
( )
2
/2
= e ,TB el l
l
f N
AZ
επ
ε − (2)
where 2 = /Bl c eB , and A is the surface area. According
to the normalization condition ( ) ( ) =l l lf D d Nε ε ε∫ (here
( )lD ε is the density-of-state function for the correspond-
ing subband), /= e Tn e
nZ ε−∑ .
The approach reported here will be formulated in a
quite general way to be applicable for any weak quasi-
elastic scattering. As important examples, we shall consid-
er interactions which are well established for SEs on liquid
helium. Vapor atoms are described by a free-particle ener-
gy spectrum ( ) 2 2= / 2a K MεK with eM m . For elec-
tron interaction with vapor atoms, it is conventional to
adopt the effective potential approximation
( ) ( ) ( )int = ,a a
e a
e a
H V δ −∑∑ R R (3)
where ( )aV is proportional to the electron-atom scattering
length [12]. Ripplons represent a sort of 2D phonons, and
the electron-ripplon interaction Hamiltonian is usually
written as
( )( ) †
int
1= ( ) e ,ir e
q e q
e
H U z Q b b
A
⋅
−+∑∑ q r
q q
q
(4)
where = / 2q qQ q ρω , 3/2/q qω α ρ is the ripplon
spectrum, { }= ,e e ezR r , q is the ripplon momentum,
†b−q and bq are the creation and destruction operators, and
( )q eU z is the electron-ripplon coupling [11] which has a
complicated dependence on q .
For both kinds of SE scattering, the energy exchange at
a collision is extremely small, which allows to consider
scattering events as quasi-elastic processes. In the case of
vapor atoms, it is so because eM m . One-ripplon scat-
tering processes are quasi-elastic ( q Tω ) because the
wave-vector of a ripplon involved is usually restricted by
the condition 1Bql .
For quasi-elastic processes in a 2D electron system un-
der magnetic field, probabilities of electron scattering are
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 831
usually found in the self-consistent Born approximation
(SCBA) [13]. Following Ref. 14, we shall express the scat-
tering probabilities in terms of the level densities at the
initial and the final states. Then, Landau level densities
will be broadened according to the SCBA [13] or to the
cumulant expansion method [15],
( ) ( ),2 2= Im ,
2
l l n
nB
AD G
l
ε ε
π
− ∑ (5)
where ( ),l nG ε is the single-electron Green's function. The
later method is a bit more convenient for analytical evalua-
tions because it results in a Gaussian shape of level densi-
ties
( ) ( )2
, 2
, ,
22Im = exp .n
l n
l n l n
G
ε επ
ε
Γ Γ
⎡ ⎤−⎢ ⎥− −
⎢ ⎥⎣ ⎦
(6)
Here ,l nΓ coincides with the broadening of Landau levels
given in the SCBA. For different scattering regimes of
SEs, equations for ,l nΓ are given in Ref. 11. We shall also
take into account an additional increase in 2,nΓ due to
inter-subband scattering.
Effects considered in this work are important only un-
der the condition ,l n TΓ which is fulfilled for SEs on
liquid helium. Therefore, we shall disregard small correc-
tions to Z caused by collision broadening because they
are proportional to 2 2
, / 8l n eTΓ . In other equations, some-
times we shall keep terms proportional to , /l n eTΓ , if they
provide important physical properties.
Average scattering probabilities of SEs on liquid helium
and even the effective collision frequency effν can be ex-
pressed in terms of the dynamical structure factor (DSF) of
the 2D electron liquid [11] ( ),S q ω . This procedure some-
how reminds the theory of thermal neutron (or X-ray) scat-
tering by solids, where the scattering cross-section is ex-
pressed as an integral form of a DSF. Without MW
irradiation, most of unusual properties of the quantum
magnetotransport of SEs on liquid helium are well de-
scribed by the equilibrium DSF of the 2D electron liquid
[11,16]. A multisubband electron system is actually a set of
2D electron systems. Therefore, the single factor ( ),S q ω
is not appropriate for description of inter-subband electron
scattering. Luckily, for non-interacting electrons, we can
easily find an extension of ( ),S q ω which could be used in
expressions for average scattering probabilities of a multi-
subband system:
( ) / 2
, ,
,
2, = e ( )Te
l l n n q
n n
S q d J x
Z
εω ε
π
−
′ ′
′
×∑ ∫
( ) ( ), ,Im Im ,l n l nG Gε ε ω′ ′× + (7)
where
2| |2 | |
, min ( , )
[min ( , )]!( ) = e ( ) ,
[max ( , )]!
n nn n x
n n n n
n nJ x x L x
n n
− ′− −′
′ ′
′ ⎡ ⎤
⎣ ⎦′
2 2= / 2q Bx q l , and ( )m
nL x are the associated Laguerre po-
lynomials. The factor ( ), ,l lS q ω′ contains the level densi-
ties at the initial and the final states, and it includes averag-
ing over initial in-plane states. At =l l ′ , this factor
coincides with the DSF of a nondegenerate 2D system of
non-interacting electrons. Generally, ( ), ,l lS q ω′ is not the
dynamical structure factor of the whole system, neverthe-
less this function is very useful for description of dissipa-
tive processes in presence of MW irradiation.
As a useful example, consider the average inter-
subband scattering rate l lν → ′ caused by quasi-elastic scat-
tering, which is important for obtaining subband occupan-
cies = /l l en N N under the MW resonance [17]. Using the
damping theoretical formulation [14] and the SCBA [13],
l lν → ′ can be represented in the following form
( ) ( ), , ,= , ,l l l l l l l l
e
q S q
m A
ν χ ω→ ′ ′ ′ ′∑
q
(8)
where ,l lχ ′ ( ,= l lχ ′ ) describes electron coupling with scat-
terers. For SEs on liquid helium, we have two kinds of
scatterers: ripplons and vapor atoms. Therefore,
( ) ( )
, , ,= r a
l l l l l lχ χ χ′ ′ ′+ . Electron–ripplon scattering gives
( ) ( ) ( ) ( )
2 2
2
, 3 3 2, ,
= 2 ,r e e
q q q ql l l l l l
m m T
q Q N U U
q
χ
α′ ′ ′
(9)
where
1/
= e 1 1
Tq
qN
ω −⎛ ⎞−⎝ ⎠ , and ,( ) ( ) .q l l q eU l U z l′ ≡ ′
For electron scattering at vapor atoms,
( ) ( )
( ) ( )23 ( )
( ) ( )
,0 0, 3
1,1
= , =
D a
e aa a a
l ll l
m n V
q p
B
χ ν ν′′ (10)
where
( )
2
1,1 1 1
, ,
,,
= , = e ,iK zz el l l l z
l ll l Kz
B
p B L
B
− −
′ ′
′′
∑
zL is the height above the liquid surface, ( )3D
an is the den-
sity of vapor atoms, and zK is the projection of the vapor
atom wave-vector. The ( )
0
aν represents the SE collision
frequency at vapor atoms for = 0B .
The generalized factor of a multi-subband 2D electron
system ( ), ,l lS q ω′ will be used throughout this work be-
cause its basic property
( ) ( )/
, ,, = e ,Tel l l lS q S qωω ω−
′ ′− (11)
allows us straightforwardly to obtain terms responsible
for negative dissipation. This property follows from the
detailed balancing for quasi-elastic processes, =l lν →′
/,= e Tl l e
l l
Δ ν− ′
→ ′ , and also directly from the definition of
Eq. (7). Using Gaussian level shapes of Eq. (6), one can
find
Yu.P. Monarkha
832 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
( ) ( )
21/2 , /
, , ; ,
, ; ,,
( )2, = e ,n n q Tn el l l n l n
l n l nn n
J x
S q I
Z
επ
ω ω
Γ
′ −
′ ′ ′
′ ′′
∑ (12)
where 2 2 2
, ; , , ,2 =l n l n l n l nΓ Γ Γ′ ′ ′ ′+ , and
( ), ; , =l n l n mI ω+′
22 2
, ,
2
, ; ,
/ 4
= exp .
8
c l n e l n
l n l n m e
m T
T
ω ω Γ Γ
Γ +′
⎡ ⎤⎛ ⎞− −⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
(13)
The ( ), ,l lS q ω′ , as a function of frequency, has sharp max-
ima when ω equals the in-plane excitation energy
( ) cn n ω−′ . The parameter 2
, ; ,l n l nΓ ′ ′ describes broadening
of these maxima. Equations (12) and (13) satisfy the condi-
tion of Eq. (11). Terms of the order of ( )2, /l n eTΓ entering
the argument of Eq. (13) could be omitted, as it was done
for Z , because even the linear in , /l n eTΓ term provides
us the necessary condition of Eq. (11). Anyway, our final
results will be represented in forms which allow to disre-
gard even the linear in , /l n eTΓ term entering ( ), ; ,l n l nI ω′ ′ .
Consider the decay rate of the first excited subband
2 1ν → . Under typical experimental condition, , ; ,l n l nΓ ′ ′ is
much smaller than cω . Therefore, most of terms entering
( ), 2,1,l lS q ω′ are exponentially small and can be disre-
garded. The exceptional terms satisfy the condition
( )=n n m B∗−′ , where ( ) ( )2,1round / cm B ω ω∗ ≡ is an in-
teger nearest to 2,1 / cω ω . In this notation,
/
2, ;1, *
2 1 1/2
=0 2, ;1, *
e=
Tn e c n n m
n n n mZ
ε ω β
ν
π Γ
−∞
+
→
+
×∑
( )22
2,1
2
2, ;1, *
exp ,
c
n n m
mω ω
Γ
∗
+
⎧ ⎫−⎪ ⎪× −⎨ ⎬
⎪ ⎪
⎩ ⎭
(14)
where
2
, ; , , ,
0
= ( ) .l n l n m l l n n m q qJ x dxβ χ
∞
+ +′ ′∫
For electron scattering at vapor atoms, ( )
, ; , ,
a
l n l n m l lβ β+′ ′≡ =
( )
,0= a
l lpν ′ which coincides with ( )
,
a
l lχ ′ . In the case of elec-
tron–ripplon scattering, 2, ;1,n n mβ + has a more complicated
expression due to a particular form of ( )q eU z entering the
definition of ( )
,
r
l lχ ′ . The 2 1( )Bν → is a 1/B-periodic func-
tion. It has sharp maxima when 2,1 / cω ω equals an integ-
er. In the argument of the exponential function of Eq. (14),
we have disregarded terms which are small for
, / 1l n eTΓ .
Transition rates l lν → ′ determine subband occupancies
ln under the MW resonance. At low electron tempera-
tures, the two-subband model is applicable, and the rate
equation gives
/2,1
2 2 1
1 2 1
e
= ,
Ten r
n r
Δ
ν
ν
−
→
→
+
+
(15)
where r is the stimulated absorption (emission) rate due to
the MW field, and 1 2 = 1n n+ . Thus, under the MW reson-
ance, magnetooscillations of 2 1ν → lead to magnetooscilla-
tions of subband occupancies 1n and 2n . For further anal-
ysis, it is important that MW excitation provides the
condition /2,1
2 1> e Ten n Δ− , which is the main cause of
negative momentum dissipation.
3. Magnetoconductivity under resonance MW
irradiation
Consider now an infinite isotropic multisubband 2D
electron system under an in-plane dc-electric field, assum-
ing arbitrary occupancies of surface subbands ln induced
by the MW resonance. In the linear transport regime, the
average friction force acting on electrons due to interaction
with scatterers scatF is proportional to the average electron
velocity av = eV v . This relationship can be conveniently
written as scat eff av= e eN m ν−F V , where the proportionali-
ty factor effν represents an effective collision frequency
which depends on B and, generally, on electron density.
The scatF is balanced by the average Lorentz force
fieldF , which yields the usual Drude form for the elec-
tron conductivity tensor ,i kσ , where the quasi-classical
collision frequency 0ν is substituted for effν [11,16].
The effective collision frequency effν can be obtained
directly from the expression for the average momentum
gained by scatterers per unite time. Usually, to describe
momentum relaxation, one have to obtain deviations of the
in-plane electron distribution function from the simple
form of Eq. (2) induced by the dc-electric field. For the
highly correlated 2D system of SEs on liquid helium under
magnetic field, this problem was solved in a general way,
assuming that in the center-of-mass reference frame the
electron DSF has its equilibrium form ( ) ( )0 ,S q ω . In the
laboratory frame, its frequency argument acquires the
Doppler shift ( ) ( ) ( )0
av, = ,S S qω ω − ⋅q q V due to Gali-
lean invariance [11,16]. This approach is similar to the
description of electron transport by a velocity shifted Fer-
mi-function of the kinetic equation method, where avV is
found from the momentum balance equation. The same
properties can be ascribed to the generalized factor [10]
( ), ,l lS ω′ q .
Here, we consider a different way, taking into account
that scatF , as well as the momentum gained by scatterers,
can be evaluated in any inertial reference frame. We
choose the reference frame fixed to the electron liquid cen-
ter-of-mass, because in it the in-plane distribution function
of highly correlated electrons ( effeeν ν ) has its simplest
form of Eq. (2), and the generalized factor ( ), ,l lS ω′ q has
it equilibrium form of Eq. (7). Then, scatF can be consi-
dered as the drag due to moving scatterers. At the same
time, distribution functions of scatterers which are not af-
fected by external fields can be easily found according to
well-known rules.
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 833
In the electron liquid center-of-mass reference frame, the
in-plane spectrum of electrons is close to the Landau spect-
rum, because the driving electric field av= (1/ )' c− ×E E B V
is nearly zero, at least for effcω ν . At the same time, in
this frame the ripplon excitation energy is changed to
( )
av=rE ω −q q qV , because the gas of ripplons moves as
a whole with the drift velocity equal to av−V . The same
Doppler shift correction av− qV appears for the energy
exchange in the case of electron scattering at vapor atoms,
even for the limiting case M →∞ (impurities which are
motionless in the laboratory reference frame). In the elec-
tron center-of-mass reference frame, vapor atoms move
with the velocity av−V and hit electrons which results in
the energy exchange av− qV .
Describing electron-ripplon scattering probabilities in
terms of the equilibrium factor ( ), ,l lS q ω′ , as discussed
above, contributions to the frictional force from creation
and destruction processes can be found as
( )
2
2
scat 2 ,
,
= e
q l q l l
l l
N
Q n U
A ′
′
− ×∑ ∑
q
F q
( ) ( )
, , , ,( 1) ( , / ) ( , / ) .r r
l l l l l l l lN S q E N S q Eω ω′ ′ ′ ′
⎡ ⎤× + − − +⎢ ⎥⎣ ⎦q q q q
(16)
It is clear that disregarding the Doppler-shift correction
av− qV in this equation yields zero result. This correction
enters the ripplon distribution function Nq and the fre-
quency argument of the factor ,l lS ′ . In the linear transport
regime, the Doppler-shift correction entering the ripplon
distribution function is unimportant. This can be seen di-
rectly from Eq. (16): setting ( ) 0rE →q in the frequency
argument of ,l lS ′ gives zero result for scatF . Therefore, in
this equation one can substitute Nq for qN defined in
Eq. (9). We can also disregard qω in the frequency ar-
gument of ,l lS ′ . Then, interchanging the running indices of
the second term in the square brackets, and using the basic
property of ( ), ,l lS q ω′ given in Eq. (11), Eq. (16) can be
represented as
( ) ( )scat , , , av
,
= ,
2
e
l l l l l l
e l l
N
q S q
m A
χ ω′ ′ ′
′
− + ⋅ ×∑∑
q
F q q V
( )/ /, ave e ,
T Tl l e el ln n
Δ− − ⋅′
′× − q V (17)
where ( ) ( ) ( ), ,= r
l l l lq qχ χ′ ′ was defined in Eq. (9). This equ-
ation has the most convenient form for expansion in
av⋅q V .
A similar equation for scatF can be found considering
electron scattering at vapor atoms. Evaluating momentum
relaxation rate, one can disregard ( ) ( )= a aε ε′ −′ ′−K,K K K K
which represents the energy exchange at a collision in the
laboratory reference frame. In the center-of-mass reference
frame, Doppler-shift corrections enter the vapor atom dis-
tribution function ( )aN ′K and the frequency argument of the
factor ,l lS ′ due to the new energy exchange at a collision
av− qV . The correction entering ( )aN ′K is unimportant
because of the normalization condition: ( ) =aN ′′∑ KK
(3 )D
a zn L A= . Therefore, we have
( )
0
scat , , , av
,
= ( , ).
a
e
l l l l l l l
e l l
N
n p S q
m A
ν
ω′ ′ ′
′
− + ⋅∑ ∑
q
F q q V (18)
In order to obtain the form of Eq. (17), we represent the
right side of Eq. (18) as a sum of two identical halves and
change the running indices in the second half: → −q q and
l l ′ . Then, the basic property of ( ), ,l lS q ω′ yields
Eq. (17) with ( )
, ,= a
l l l lχ χ′ ′ , where ( )
,
a
l lχ ′ is from Eq. (10).
Thus, Eq. (17) is applicable for both scattering mechan-
isms. In the general case, ( ) ( )
, , ,= r a
l l l l l lχ χ χ′ ′ ′+ . The effective
collision frequency under magnetic field effν can be found
expanding Eq. (17) in av⋅q V up to linear terms. We shall
represent effν as a sum of two different contributions:
eff N A=ν ν ν+ . The normal contribution Nν originates
from the expansion of the exponential factor
( )avexp / eT− ⋅q V . In turn, Nν can be represented as a
sum of contributions from intra-subband and inter-subband
scattering N N,intra N,inter=ν ν ν+ . The sums of N,interν
take account of all ,l l ′ . It is useful to rearrange terms with
<l l ′ ( , < 0l lΔ ′ ) by interchanging the running indices
l l ′ , and using the basic property of ( ), ,l lS q ω′ . Then,
we have
( ) ( )
2
N,intra , ,
0
= ,0 ,
4
c
l q l l l l q
e l
n x q S q dx
T
ω
ν χ
π
∞
∑ ∫ (19)
( )2 /,
N,inter
>
= e
4
Tc l l e
l l
e l l
n n
T
Δω
ν
π
− ′
′
′
+ ×∑
, , ,
0
( ) ( , ) .q l l l l l l qx q S q dxχ ω
∞
′ ′ ′× ∫ (20)
The ( )N Bν is always positive. In the limiting case of a
one-subband 2D electron system ( ,1=l ln δ ), Eq. (19) re-
produces the known relationship between the effective
collision frequency and the electron DSF [11]. In the pa-
rentheses of Eq. (20), the first term is due to scattering
from l to l ′ , while the second term describes the contri-
bution of scattering back from l ′ to l . It should be noted
that the forms of Eqs. (19) and (20) allow to simplify
( ), ,,l l l lS q ω′ ′ of Eq. (12) by disregarding small corrections
proportional to , /l n eTΓ and 2
,( / )l n eTΓ entering
( ), ; ,l n l nI ω′ ′ defined by Eq. (13).
The anomalous contribution to the effective colli-
sion frequency ( )A Bν can be found from Eq. (17) ex-
panding , , av( , )l l l lS q ω′ ′ + ⋅q V in av⋅q V , and setting
avexp ( / ) 1eT− ⋅ →q V in the parentheses. In this case, to
rearrange terms with <l l ′ ( , < 0l lΔ ′ ), we shall use the
property
Yu.P. Monarkha
834 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
( ) ( ) ( )/ /
, , ,, = e , e ,T Te el l l l l l
e
' 'S q S q S q
T
ω ωω ω ω− −
′ ′ ′− − +
( )/
,e , .Te
l l
'S qω ω−
′− (21)
Here ( ) ( ), ,, , /l l l l
'S q S qω ω ω′ ′≡ ∂ ∂ , and the last transfor-
mation assumes that ,l n eTΓ . Interchanging the running
indices l l ′ of terms with , < 0l lΔ ′ and using Eq. (21),
( )A Bν can be found as
( ) ( ) ( )
2 /,
A , , ,
> 0
= e , .
2
Tc l l e
l l q l l l l l l q
l l
'n n x q S q dx
Δω
ν χ ω
π
∞
− ′
′ ′ ′ ′
′
−∑ ∫
(22)
As compared to N,interν of the normal contribution, here
the second term in parentheses has the opposite sign.
Therefore, for usual Boltzmann distribution of subband
occupancies, ( )A = 0Bν . The anomalous contribution
appears only when
/,e
Tl l e
l ln n
Δ− ′
′≠ , which occurs under
the MW resonance condition ,= l lω ω ′ .
In the form of Eq. (22), it is possible to use a simplified
expression
( )
( ) ( )
21/2 , /
, ,
, ; ,,
2 2
, ,
2 2
, ; , , ; ,
( )2, e
2
exp ,
n n q T' n e
l l l l
l n l nn n
l l c l l c
l n l n l n l n
J x
S q
Z
n n n n
επ
ω
Γ
Δ ω ω ω
Γ Γ
′ −
′ ′
′ ′′
′ ′
′ ′ ′ ′
− ×
⎧ ⎫⎡ ⎤ ⎡ ⎤− − − −′ ′⎪ ⎪⎣ ⎦ ⎣ ⎦× −⎨ ⎬
⎪ ⎪⎩ ⎭
∑
(23)
which disregards terms proportional to , /l n eTΓ and
2
,( / )l n eTΓ . From Eqs. (22) and (23) one can see that at
/,> e Tl l e
l ln n Δ− ′
′ , the sign of ( )A Bν is opposite to the
sign of ( ),l l cn nω ω′ − −′ . Therefore, ( )A < 0Bν when the
magnetic field B is slightly lower the commensurability
condition 2,1 / =c mΔ ω (here m is an integer), which
agrees with the experimental observation for minima of
xxσ .
For further analysis, it is convenient to introduce
( ) ( )2
, ; , , ,
0
= .l n l n q l l n n q qx q J x dxλ χ
∞
′ ′ ′ ′∫ (24)
When referring to a particular scattering mechanism, we
shall use a superscript, ( ) ( )
, ; , , ; , , ; ,= r a
l n l n l n l n l n l nλ λ λ′ ′ ′ ′ ′ ′+ . Con-
sider a two-subband model which is valid at low enough
electron temperatures. Using the new definitions given
above, the normal contribution to the effective collision
frequency can be represented as
/ 2
1, ;1, 2, ;2,
N,intra 1 2
1, 2,=0
e ( )
= ,
2
Tn e n n n nc
n nen
n n
T Z
ε λ λω
ν
Γ Γπ
−∞ ⎡ ⎤
+⎢ ⎥
⎢ ⎥⎣ ⎦
∑ (25)
( ) ( )2 /
/2,1
N,inter 2 1
=0
e= e
2
Tn eT ce
e n
n n
ZT
ε
Δ ω
ν
π
−∞−
+ ×∑
( )22
2,12, ;1, *
2
2, ;1, * 2, ;1, *
*
exp ,cn n m
n n m n n m
mω ωλ
Γ Γ
+
+ +
⎡ ⎤−⎢ ⎥× −⎢ ⎥
⎢ ⎥⎣ ⎦
(26)
where ( )2,1* round / cm ω ω≡ is the function of B defined
in the previous Section. The N,intraν and N,interν have
magnetooscillations of two kinds. Oscillations of N,interν
are quite obvious, because quasi-elastic inter-subband scat-
tering increases sharply at the commensurability condition:
2,1 = cmω ω . The shape of these peaks is symmetrical with
respect to the point 2,1 / =c mΔ ω . It is formed by the
interplay of the exponential factor, having 2, ;1, *n n mΓ + for
the broadening parameter, and the line-shapes of the sub-
band occupancies. It should be noted that at low electron
temperatures, N,interν is exponentially small. The intra-
subband scattering contribution N,intraν oscillates with
1/ B in an indirect way because of oscillations in level
occupancies 2n and 1n induced by oscillations in the de-
cay rate 2 1ν → , according to Eqs. (14) and (15). These os-
cillations have also a symmetrical shape whose broadening
is affected by the relation between r and 2 1ν → .
Magnetooscillations of ( )A Bν have a completely dif-
ferent shape:
( ) ( )2 /
/ 2, ;1, *2,1
A 2 1 2
2, ;1, *=0
e= e
Tn eT n n mce
n n mn
n n
Z
ε
Δ λω
ν
π Γ
−∞− +
+
− − ×∑
( ) ( )22
2,1 2,1
2
2, ;1, *2, ;1, *
* 2 *
exp .c c
n n mn n m
m mω ω ω ω
ΓΓ ++
⎡ ⎤− −⎢ ⎥× −⎢ ⎥
⎢ ⎥⎣ ⎦
(27)
In the ultra-quantum limit c eTω , terms with > 0n
entering Eq. (27) can be omitted, which allows to describe
magneto-oscillations of ( )A Bν in an analytical form. In
contrast with oscillations of the normal contribution Nν ,
in the vicinity of the commensurability condition, Aν is an
odd function of 2,1 / *.c mω ω −
Thus, the effective collision frequency eff N A=ν ν ν+
and magnetoconductivity xxσ of SEs are found for any
given electron temperature. In order to obtain eT as a func-
tion of the magnetic field, it is necessary to describe energy
relaxation of SEs for arbitrary subband occupancies.
4. Energy dissipation
It is instructive to analyze another important example of
negative dissipation which can be induced by the MW re-
sonance. Consider the energy loss rate of a multisubband
2D electron system due to quasi-elastic scattering
processes discussed in the previous Section. In this case,
there are no complications with the dc-driving electric field
or with the Doppler shifts which can be set to zero. This
analysis will be important also for description of electron
heating due to decay of the SE state excited by the MW.
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 835
It should be noted that for SEs above superfluid 4 He ,
there are inelastic inter-subband scattering processes ac-
companying by simultaneous emission of two short wave-
length ripplons [11,17]. These processes cause strong addi-
tional energy relaxation. Experiments of Refs. 4 and 5
were performed for SEs on the free surface of Fermi-liquid
3He . For such a substrate, short wavelength capillary
waves with 710q are so heavily damped that even the
existence of ripplons with such wave-numbers is doubtful.
Therefore, here we shall confine ourselves to one-ripplon
scattering processes.
The energy loss rate per an electron due to one-ripplon
creation and destruction processes can be represented in
terms of ( ), ,l lS q ω′ quite straightforwardly:
( ) ( )
2
2
2 ,
,
1= 1q q q l l
l l
W Q N U
A
ω
′
′
− + ×∑ ∑q
q
/
, , , ,( , ) e ( , ) .
Tq
l l l l l q l l l l l qn S q n S q
ω
ω ω ω ω
−
′ ′ ′ ′
⎡ ⎤× − − +⎢ ⎥⎣ ⎦
(28)
Interchanging the running indices ( ,l l ′ ) in the second
term, and using the basic property of ( ), ,l lS q ω′ given in
Eq. (11), the terms entering the square brackets can be
rearranged as
( ) ( )1/ 1//,
, ,, e e .
T TT q el l e
l l l l q l lS q n n
ωΔ
ω ω
− −− ′
′ ′ ′
⎡ ⎤− −⎢ ⎥⎣ ⎦
(29)
Since the processes considered here are quasi-elastic, we
can expand this equation in qω and represent W as a
sum of two different contributions: N A=W W W+ . The
normal energy loss rate NW is proportional to eT T− ,
which is a measure of deviation from the equilibrium,
( ) ( ) ( )/,
N , ,,
,
= e , .
T re l l e
l l l l ll l
e l l
T T
W n S q
m A
Δ
χ ω
− ′
′ ′ ′′
′
−
− ∑∑
q
(30)
Here
( ) 2
,, = ( ) .
2
r e
q l ll l
e
m q
U
T
χ
ρ ′′
This contribution originates from expansion of the expo-
nential function in ( )1/ 1/q eT Tω − .
It is conventional to represent the energy loss as
( ) ( )
N N= r
eW T T ν− − , where ( )
N
rν is the energy relaxation
rate of an electron. Rearranging terms with <l l ′
,( < 0)l lΔ ′ ), as described in the previous Section, one can
find
( ) ( )( )
,N ,= ,0r r
l l ll l
e l
n S q
m A
ν χ +∑∑
q
( ) ( )/ ( ),
, ,,
>
e , .
T rl l e
l l l l l ll l
e l l
n n S q
m A
Δ
χ ω
− ′
′ ′ ′′
′
+ +∑∑
q
(31)
The normal contribution ( )
N
rν is always positive, which
means positive dissipation ( N < 0W ) regardless of sub-
band occupancies ln .
An anomalous contribution AW appears when expand-
ing ( ), ,,l l l l qS q ω ω′ ′ − of Eq. (29) in qω and setting
exp [ (1/ 1/ )]q eT Tω− − to unity. The rearrangement of
terms with <l l ′ ( , < 0l lΔ ′ ) based on the property of
Eq. (21) yields
( ) ( ) ( )/,
A , ,,
>
2
= e , .
T re l l e
l l l l l ll l
e l l
TT 'W n n S q
m A
Δ
χ ω
− ′
′ ′ ′′
′
−∑ ∑
q
(32)
Here
/,e
Tl l e
l ln n
Δ− ′
′− represents an additional measure of
deviation from the equilibrium induced by the MW. For
equilibrium distribution of fractional occupancies ln , the
anomalous term equals zero, but for occupancies
/2,1
2 1> e
Ten n
Δ−
induced by the MW resonance, AW can
lead to negative energy dissipation of the electron system.
In Eq. (32), one can use the approximate expression for
( ), ,,l l l l
'S q ω′ ′ given in Eq. (23). According to Eqs. (23)
and (32), the sign of AW− coincides with the sign of
( )2,1 cn nω ω− −′ which can be negative or positive depen-
ding on the magnetic field. Since , ; , 2,1<l n l n cΓ ω Δ′ ′ ,
the contribution AW is mostly exponentially small with
the exception of magnetic fields where ,l lΔ ′ −
( ) , ; ,c l n l nn n ω Γ ′ ′− −′ .
The appearance of negative corrections to energy dissi-
pation under the condition
/2,1
2 1 e > 0
Ten n
Δ−
− can be
explained quite easily. The negative anomalous contribu-
tion ( A > 0W ) corresponds to ( ) 2,1>cn n ω Δ−′ . For
narrow Landau levels, this means that scattering from the
excited subband ( = 2l ) to the ground subband ( = 1l ′ ) is
accompanied by destruction of a ripplon, while the corres-
ponding scattering back from the ground subband to the
excited subband is accompanied by creation of a ripplon.
When
/2,1
2 1= e
Ten n
Δ−
, these two processes compensate
each other in the expression for AW . If /2,1
2 1> e Ten n Δ− ,
destruction of ripplons dominates, which leads to negative
dissipation. In the opposite case, when ( ) 2,1<cn n ω Δ−′ ,
creation of ripplons dominates, which results in additional
positive dissipation. It should be noted that the negative
contribution to energy dissipation and negative momentum
dissipation occur at the opposite sides of the point
2,1= /c mω Δ . Comparing Eq. (17) with Eqs. (28) and
(29) one can conclude that the origin of this difference is
the negative sign of the Doppler-shift correction in the rip-
plon excitation spectrum considered in the center-of-mass
reference frame: ( )
av=rE ω −q q qV .
Consider now the energy loss rate of SEs due to elec-
tron scattering at vapor atoms. In this case, the interaction
Hamiltonian is proportional to the density fluctuation oper-
ator of vapor atoms †= a aρ ′−′′∑K KK KK , where
{ },zKK = q represents the momentum exchange between
an electron and a scatterer. In terms of ( )(0)
, ,l lS q ω′ , the
energy loss rate per an electron can be obtained as
Yu.P. Monarkha
836 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
( )
2( ) 2
2 2 ,, ,
( )= e
a
iK zz e
l
l ll lz
VW n
A L
′
′′ ′
− ×∑ ∑ K,K
K K
( ) ( ), ,, ,a
l l l lN S q ω′ ′ ′′× − K,KK (33)
where ( ) ( )= a aε ε′ −′ ′−K,K K K K is the energy exchange at a
collision. In order to obtain NW and AW , we shall firstly
rewrite Eq. (33) trivially as a sum of two identical halves.
Then, in the second half, the running indices ,′K K will be
substituted as − →′ ′K K K , and → −K K , which
changes the sign of the energy exchange,
′ ′→ −K,K K,K . The next steps are the same as those
resulting in Eq. (29). Interchanging the running indices
l l ′ in the second half, and using the basic property of
( )(0)
, ,l lS q ω′ one can find
( ) ( )
2( ) 2
2 2 ,, ,
( )= e
2
a
aiK zz e
l ll lz
VW N
A L
′ ′′′ ′
− ×∑ ∑ K,K K
K K
( ), ,,l l l lS q ω′ ′ ′× − ×K,K
( )/ 1/ 1/,e e .
T T Tl l e e
l ln n
Δ− − −′ ′
′
⎡ ⎤× −⎢ ⎥⎣ ⎦
K,K (34)
This equation is more convenient for expansion in ′K,K
than Eq. (33).
Expanding Eq. (34) in ′K,K , one can find again that
N A=W W W+ , where NW and AW have the same forms
as that given in Eqs. (30) and (32), where ( )
,
r
l lχ ′ should be
substituted for
( ) ( ) ( ) ( )
2( ) 2 2
, 2 ,,
( )
= e .
2
a
a a iK ze z
l l l lKz e z
m V
N
AL TT
χ ′′ ′ ′′
∑ K,K K
K
(35)
Using the condition K K′ , this equation can be simpli-
fied as
( ) ( )
, ,0, = ,a a e cR
l l q l ll l
e e
m E
u x p
M T T
ω
χ ν ′ ′′
⎛ ⎞
+⎜ ⎟⎝ ⎠
(36)
where
2 21 211
,
1
= , = (e ) .iK zB z e
l l l l z l l
l l z Kz
a B
u C K
C L
−
′ ′ ′
′
∑
Expressions for 1
l lC−
′ and 1
,l lB−
′ convenient for numerical
evaluations were given in Refs. 12 and 18. Some useful
expressions for the SE energy relaxation rate obtained for
arbitrary subband occupancies are given in the Appendix.
The energy loss W transferred to vapor atoms and rip-
plons is balanced by the energy taken from the MW field:
( )1 2 2,1=W n n rΔ− , where r is the MW excitation rate
defined by
( )
2
2 2
2,1
1= ,
2
Rr
Ω γ
ω ω γ− +
(37)
where γ is the half-width of the MW resonance, and RΩ
is the Rabi frequency proportional to the amplitude of
the MW field. It is clear that negative contribution of AW
will be compensated by an increase in NW due to electron
heating.
5. Results and discussions
5.1. Vapor atom scattering regime
Electron scattering at vapor atoms represents the most
simple case for the magnetotransport theory, because the
collision broadening of Landau levels of the same subband
( lΓ ) is independent of the level number n . The same is
obviously valid for the broadening of the generalized fac-
tor ( ), ,l lS q ω′ , which now can be denoted as ;l lΓ . Addi-
tionally, the parameter defined in Eq. (24) has a very sim-
ple form ( ) ( )( )
,0, ; , = 2 1a a
l ll n l n m p n mλ ν ′+′ + + which greatly
simplifies evaluations.
Consider electron temperature as a function of the mag-
netic field. It is defined by the energy balance equation
which contains the MW excitation rate r given in
Eq. (37). In turn, r depends on the half-width of the MW
resonance γ , which was studied theoretically with no
magnetic field and under a parallel magnetic field [19]. If
B is applied perpendicular to the surface, γ should also
have 1/B-oscillating terms, because inter-subband scatter-
ing increases when 2,1 / c mΔ ω → . In our numerical
evaluation, we shall use a qualitative extension of the re-
sult obtained for = 0B . According to this result, γ con-
tains the contribution from intra-subband scattering 22 11γ −
and the contribution from inter-subband scattering
2,1 2 1= / 2γ ν → . Under the magnetic field applied normal-
ly, electron scattering is enhanced by the factor
/c lω πΓ [13]. Therefore, we can use an approximation
( )
0
22 11 2,2 1,1 2,1
2,1
2 ,
2
a
c p p p
ν ω
γ
πΓ− ⎡ ⎤≈ + −⎣ ⎦
where 2,1Γ numerically is rather close to 1 =Γ
( )
02 /a
cω ν π= . As for the oscillatory part 2 1ν → entering
2,1γ , we shall use the exact form of Eq. (14).
It should be noted that the oscillatory part of γ is not
large because 2,1 0.14p . Still, it leads to some important
consequences for electron temperature as a function of the
magnetic field shown in Fig. 1. Solid curves represent re-
sults of numerical evaluations for the two-subband model
taking into account oscillatory corrections to the MW re-
sonance half-width γ , as described above. In this case, the
electron temperature has small local minima at
2,1 / c mΔ ω → due to oscillatory decrease in 1/r γ∝ .
Three typical values of the Rabi frequency are chosen to
provide MW excitation rate levels of 5 110 s− , 5 13 10 s−⋅
and 5 15 10 s−⋅ at = 1 TB . For a model with a constant
MW excitation rate r , which is applicable when inhomo-
geneous broadening dominates, the corresponding results
are shown by dashed curves. At 2,1 / c mΔ ω ≈ these
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 837
curves are nearly straight lines (without minima). For both
models, the shape of curves describing the oscillatory in-
crease of electron temperature has asymmetry with regard
to the point 2,1 / =c mΔ ω . This asymmetry is due to the
negative correction of the anomalous term AW leading to
additional heating of the electron system at
2,1> / *c mω Δ . The asymmetry increases strongly with
the MW excitation rate r and with *( )m B .
Electron heating increases with *m (lowering B ), and,
for the excitation rate 5 1= 5 10 sr −⋅ at = 1 TB , the two-
subband model fails at * > 11m . The applicability range of
the two-subband model can be extended by using a strong-
er holding electric field which increases 2,1Δ . In Fig. 2,
electron temperature is shown as a function of the parame-
ter 2,1 / 1/c BΔ ω ∝ for a substantially higher MW fre-
quency used in experiments on SEs [20]. For the solid
curve, the two-subband model is applicable up to * = 15m .
MW heating affects strongly the shape of conductivity
oscillations because effν depends on electron temperature.
For example, the normal contribution to the effective colli-
sion frequency can be represented as
( ) 2 2
,( ) 0
N,intra
;
= coth ,
22
a
l l lca c
l l ee l
n p
TT
ν ω ω
ν
Γπ
⎛ ⎞
⎜ ⎟⎝ ⎠∑ (38)
( )( ) 2 2
/ ,( ) 0 ,
N,inter
;>
= e
2
a
T l lca l l e
l l
l le l l
p
n n
T
Δν ω
ν
Γπ
− ′′
′
′′
+ ×∑
( ) ( ), ,coth ,
2
c
l l c l l c
e
F H
T
ω
ω ω′ ′
⎡ ⎤⎛ ⎞
× +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
(39)
where the new functions
( ) ( )22
,
, 2
=1 ;
= exp ,l l c
l l c
m l l
m
F
ω ω
ω
Γ
∞ ′
′
′
⎡ ⎤−⎢ ⎥−⎢ ⎥
⎢ ⎥⎣ ⎦
∑ (40)
( ) ( )22
,
, 2
=1 ;
= exp l l c
l l c
m l l
m
H m
ω ω
ω
Γ
∞ ′
′
′
⎡ ⎤−⎢ ⎥−⎢ ⎥
⎢ ⎥⎣ ⎦
∑ (41)
defined for >l l ′ are independent of eT . For narrow Lan-
dau levels ( ,l l cΓ ω′ ), the series defining ,l lF ′ or ,l lH ′
can be approximated by a single term with = *m m , where
*m depends on the magnetic field according to the above
noted rule: ( ),* = round /l l cm ω ω′ .
The anomalous contribution to the effective collision
frequency has a different form
( )( ) 2 2
/ ,( ) 0 ,
A 1/2 2
> ;
= e
a
T l lca l l e
l l
l l l l
p
n n
Δν ω
ν
π Γ
− ′′
′
′ ′
− − ×∑
( ) ( ), ,coth ,
2
c
l l c l l c
eT
ω
Φ ω Θ ω′ ′
⎡ ⎤⎛ ⎞
× +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
(42)
where functions ( ),l l cΦ ω′ and ( ),l l cΘ ω′ are defined
similar to ( ),l l cF ω′ and ( ),l l cH ω′ of Eqs. (40) and (41)
respectively, with the exception that their right sides con-
tain the additional factor ( ), ;2 /l l c l lmω ω Γ′ ′− originated
from Eq. (23). Similar equations for SE energy relaxation
rate are given in the Appendix.
Comparing eT -dependencies of ( )
N
aν and ( )
A
aν given in
Eqs. (38), (39) and (42), we conclude that heating of the
electron system reduces the normal contribution to the ef-
fective collision frequency. In contrast with this, the ano-
malous sign-changing correction ( )
A
aν , can be even en-
hanced to a some extent with heating of SEs due to the
factor coth ( /2 )c eTω . Typical magnetoconductivity os-
cillations of SEs calculated for the conditions of the expe-
riment of Ref. 5 are shown in Fig. 3. Electron temperature
Fig. 1. Electron temperature vs the magnetic field for three levels
of MW irradiation estimated at = 1 TB : 5 1= 10 sr − (1),
5 13 10 s−⋅ (2), and 5 15 10 s−⋅ (3). The solid curve was calcu-
lated for the model of ( )Bγ discussed in the text, the dashed
curve represents the case ( ) = constr B .
0.8 1.0
0.4
0.6
B, T
T = 0.4 K
f = 79 GHz
1
2
3
T
e
,
K
Fig. 2. Electron temperature vs 2,1 / cΔ ω for the MW of higher
resonant frequency, = 130 GHzf . Three levels of MW irradia-
tion are the same as those described in the caption of Fig. 1. The
all curves were calculated for the model of ( )Bγ discussed in the
text.
10 11 12 13 14 15
0.5
1.0
1.5
2.0
Δ ω2,1/ћ c
T = 0.4 K
f = 130 GHz
T
e
,
K
Yu.P. Monarkha
838 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
calculated for these curves was shown in Fig. 1 by solid
curves. At low MW excitation ( 5 1= 10 sr − at = 1 TB ),
magnetooscillations of xxσ are just simple maxima cen-
tered at 2,1 / =c mΔ ω , which agrees with experimental
observations. Between the commensurability conditions,
xxσ is suppressed, as compared to the dash-dot-dot line
calculated for zero MW power. This suppression is due to
2n → 1 1/ 2n → and weaker scattering at the excited sub-
band. The increase in the decay rate 2 1ν → at
2,1 / c mΔ ω → leads to a sharp decrease in 2n , which
restores xxσ values obtained without the MW field. This
restoration is not complete if >eT T , because ( )
N,intra
aν
decreases with heating, as discussed above. The ( )
N,inter
aν is
very small under these conditions.
At higher MW excitation ( 5 1= 3 10 sr −⋅ and 5 15 10 s−⋅
at = 1 TB ), the shape of conductivity oscillations is af-
fected strongly by the anomalous term ( )
A
aν leading to
local minima at 2,1 / >c mΔ ω . The ( )
A
aν increases with
r because of two reasons. The first reason is the increase
in /2,1
2 1= e Ten n n ΔΔ −− at higher MW excitation shown
in Fig. 4. The second reason is electron heating due to de-
cay of the excited SE state which increases
( )coth / 2c eTω of Eq. (42). Further shape evolution is
shown in Fig. 5 for lager *m and = 130 GHzf , where,
according to Fig. 2, electron temperature can take a value
of about 2 K . As expected, under these conditions the
anomalous contribution strongly increases. Near the com-
mensurability conditions 2,1 / c mΔ ω → , the shape of
conductivity oscillations becomes similar to that observed
for the electron-ripplon scattering regime [5] at = 0.2 KT .
It is important that the results given in Fig. 5 are still ob-
tained in the validity range of the two-subband model.
It is instructive to compare the peak broadening of dif-
ferent quantities shown in Fig. 6. The broadening of the
decay rate 2 1ν → coincides with 2,1Γ , which is an average
of 2Γ and 1Γ . In contrast, such quantities as Aν ,
1 1/ 2n − , and eT T− rise in a much broader magnetic
field range having nearly the same width which does not
represent the broadening of Landau levels directly. We
shall use this similarity in the line widths of eT T− and
1 1/ 2n − later, considering electron heating for the elec-
tron-ripplon scattering regime.
5.2. Electron-ripplon scattering regime
For electron-ripplon scattering, the anomalous (sign-
changing) contribution to the effective collision frequency
is induced by the MW resonance absolutely in the same
way, as it is for electron scattering at vapor atoms. In the
Fig. 3. Magnetoconductivity vs 2,1 / 1 /c BΔ ω ∝ for the MW of
= 79 GHzf . The dashed-dot-dot line was calculated with no
MW irradiation. Dotted, dashed and solid curves represent three
levels of MW irradiation ( r ) given in the caption of Fig. 1.
4 5 6
10
15
σ
Ω
xx
·1
0
,
1
1
–
1
T = 0.4 K
f = 79 GHz
Δ ω2,1/ћ c
Fig. 4. Deviation of subband occupancies from the equilibrium dis-
tribution, /2,12 1= e Ten n n ΔΔ −− , vs the parameter 2,1/ 1/c BΔ ω ∝ .
Dotted, dashed and solid curves represent three levels of MW
irradiation ( r ) given in the caption of Fig. 1.
4.8 4.9 5.0 5.1 5.2
0
0.1
0.2
0.3
0.4
0.5
Δn
Δ ω2,1/ћ c
Fig. 5. Magnetoconductivity vs 2,1 / cΔ ω for the MW field of
= 130 GHzf . The dashed-dot-dot line was calculated with no
MW irradiation. Dotted, dashed and solid curves represent three
levels of MW irradiation ( r ) the same as those in Fig. 2.
10 11 12 13 14 15
20
30
T = 0.4 K
f = 130 GHz
σ
Ω
xx
·1
0
,
1
1
–
1
Δ ω2,1/ћ c
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 839
case of liquid 3 He , electron-ripplon scattering dominates
at low temperatures 0.2T ≤ , where the half-width of the
MW resonance γ is substantially reduced. According to
Eq. (37), at the same amplitude of the MW field, this de-
crease in γ leads to a strong increase in the MW excitation
rate r at the resonance 2,1=ω ω , which greatly magnifies
Aν .
Unfortunately, the electron-ripplon scattering regime is
much more difficult for the analysis of the effect of elec-
tron heating than the vapor atom scattering regime because
of different reasons. First, the electron-ripplon coupling
qU has a very complicated form [11]:
( ) ( )
(0)
1
1= ,e
q
e
VqU z K qz eE
z qz z
Λ
⊥
∂⎡ ⎤
− + −⎢ ⎥ ∂⎣ ⎦
where ( )1K x is the modified Bessel function of the second
kind, and ( )(0)
eV z is the electron potential energy over a
flat surface. Therefore, it is impossible to obtain simple
analytical equations for the energy loss function ( )eW T .
Moreover, if 3 He is used as the liquid substrate, there
might be contributions from other mechanisms of energy
relaxation, which by now have no strict theoretical descrip-
tions.
As indicated above, heating of SEs only increases the
importance of the anomalous contribution to the effective
collision frequency. For electron-ripplon scattering, it fol-
lows directly from Eqs. (25)–(27). Therefore, in order to
prove the possibility of existence of zero resistance states
due to non-equilibrium filling of the excited subband, it is
sufficient to show that negative xxσ can appear even with-
out electron heating. At = = 0.2 KeT T , the MW field
amplitude, which gave 5 1= 10 sr − (at = 1 TB ) for the
vapor atom scattering regime shown in Fig. 3 (dotted
curve), now gives 6 1= 2 10 sr −⋅ , because the MW reson-
ance line width 2 0.3 GHzγ due to inhomogeneous
broadening [5]. This excitation rate is very high, because it
leads to < 0xxσ already at * = 4m . For presentation of
Fig. 7, we had chosen a two-times lower excitation rate
6 1= 10 sr − independent of the magnetic field. This figure
shows the evolution of the line shape of conductivity oscil-
lations with the gradual increase in the integer parameter
*m . It is quite convincing that even without heating of
SEs the anomalous contribution to the effective collision
frequency increases strongly with *m , and the conductivi-
ty curve corresponding to * = 6m enters the negative con-
ductivity regime in the vicinity of the minimum. This is in
accordance with experimental observations reported for the
high magnetic field range ( * < 10)m .
Maxima and minima of A ( )Bν have the same ampli-
tude. Without heating of SEs, amplitudes of conductivity
maxima obtained here are larger than amplitudes of mini-
ma, because the normal contribution Nν increases at
2,1 / c mΔ ω → due to oscillations of subband occupan-
cies. Experimental curves [5] show that at strong MW
power and large *m amplitudes of minima are larger. This
could be an indication of electron heating, because
N 1/ eTν ∼ . To analyze the effect of heating of SEs on
conductivity oscillations, we shall model electron tempera-
ture oscillations using similarities in the line shapes of
eT T− and 1 1/ 2n − shown in Fig. 6. In particular, we
assume that an electron temperature peak is described by
( ) ( ) [ ]1max= 2 ( , ) 1/ 2e eT B T T n B TΔ+ − , where the maxi-
mum elevation ( )maxeTΔ depends of *m . We disregard
the asymmetry of the peak induced by AW because it does
not lead to a substantial change in final results. The results
of such a model treatment of the heating effect are shown
in Fig. 8. They indicate that even moderate heating of SEs
affects strongly the shape of magnetooscillations, making
amplitudes of minima larger than amplitudes of maxima
(dotted curve) in accordance with experimental data.
Fig. 6. Line shapes of 2 1ν → (dashed), Aν (solid), eT T−
(short-dotted), and 1 1 / 2n − (short-dashed) as functions of B
near the commensurability point with = 5m , under the condi-
tions: = 0.4 KT , and 5 1= 3 10 sr −⋅ .
0.54 0.55 0.56 0.57 0.58 0.59
–0.2
0.2
0.4
0.6
0
νA
T – Te
n1 – 1/2
ν2 1→
B, T
ν/
1
0
,
s
,
K
;
–
1
/2
9
–
1
;
T
–
T
e
n
1
Fig. 7. Evolution of the ( )xx Bσ line shape near commensurabili-
ty points with the gradual increase in *m at = 0.2 KT and
= 79 GHzf .
–0.05 0 0.05
0
4
8
12
m* = 3
4
5
6
σ
Ω
xx
·1
0
,
1
1
–
1
Δ ω2,1/ћ c – m*
Yu.P. Monarkha
840 Fizika Nizkikh Temperatur, 2011, v. 37, No. 8
In Fig. 8 we had chosen the excitation rate 6 1= 10 sr − ,
so that the initial curve (solid) calculated for =eT T have
a small minima with > 0xxσ . Then, we found that heating
with ( )max = 0.1 KTΔ strongly reduces conductivity ex-
tremes due to N 1/ eTν ∼ , and moderate heating with
( )max = 0.5 KTΔ leads to a minimum with < 0xxσ .
Therefore, decay heating of electrons, which occurs in the
vicinity of the commensurability conditions, helps to ob-
tain zero resistance states. For example, within the validity
range of the two-subband model, electron temperature
peaks of about 2 K can reduce Nν by an order of magni-
tude. Still, heating alone cannot make 0xxσ ≤ . It is only
the anomalous contribution Aν which eventually leads to
negative conductivity and zero-resistance states. Without
Aν , a conductivity dip would be an even function of the
parameter 2,1 * cmω ω− with > 0xxσ . The existence of a
magnetoconductivity maxima at the opposite side of the
point 2,1 * = 0cmω ω− in experimental curves, which
demonstrate vanishing magnetoconductivity [5], is an addi-
tional evidence for a sign-changing correction convincing
that ZRS are realized at the vanishing points.
It should be noted that at = 0.2 KT , one-ripplon scat-
tering processes are not sufficient to prevent strong heating
of the electron system at the commensurability conditions.
In particular, for 5 1= 5 10 sr −⋅ , estimation gives
( )max 3 KeT ∼ at * = 4m . The model treatment of the
heating effect discussed here allows to draw conclusions
about actual role of the electron heating in experiments
with SEs [5]. For example, the firm conductivity maximum
(without a minima) observed for radiation power P of –
25 dB at * = 4m surely indicates that electron heating is
small or moderate under these conditions, and there is an
additional mechanism of energy relaxation at low ambient
temperatures. We speculate, that the magnetopolaronic
effect and electron coupling with bulk quasi-particles, giv-
ing a very small correction to the momentum relaxation
rate under experimental conditions, can contribute substan-
tially to the energy relaxation rate reducing electron tem-
perature.
Experiments [4,5] are conducted for low surface elec-
tron densities sn of about 6 210 cm− . Nevertheless, elec-
tron-electron interaction affects noticeably experimental
data. According to Ref. 21, under magnetic field an elec-
tron moves in a quasi-uniform electric field of other elec-
trons fE of fluctuational origin. Its average value
(0) 3/43 e sfE T n increases strongly with electron tempera-
ture and density. The fluctuational electric field increases
the broadening of the DSF [16,11]
2 2
, ; , , ; ,l n l n l n l n q CxΓ Γ Γ′ ′ ′ ′→ + ,
where (0)= 2 1/C BfeE l BΓ ∝ . Thus, at = 0.2 KeT ,
and 6 2= 0.9 10 cmsn −⋅ , the Coulombic correction increas-
es , ; ,l n l nΓ ′ ′ by about 1.3, if we assume 1qx . If we take
into account that the integrand of Eq. (24) has a maximum
at * 2qx m +∼ , the broadening increases approximately
two times. Therefore, a qualitative analysis indicates that
the many-electron effect becomes more important in the
low magnetic field range where it increases the width of
conductivity oscillations and reduces amplitudes of max-
ima and minima, which also agrees with experimental ob-
servations. Decay heating increases the Coulombic cor-
rection to the broadening of magnetooscillations. Still, a
strict description of Coulombic effects on magnetoconduc-
tivity oscillations requires a more careful study.
6. Conclusion
In summary, we have developed the theory of magneto-
conductivity oscillations in a multi-subband 2D electron
system under MW irradiation of a resonant frequency. We
have shown that besides the quite obvious 1/B-modulation
of conductivity, the non-equilibrium filling of the excited
subband induced by the MW resonance leads also to sign-
changing corrections to the effective collision frequency
due to usual inter-subband scattering. As the MW power
goes up, the corresponding increase in the amplitude of
these sign-changing corrections can result in the negative
linear response conductivity and zero-resistance states.
Our theory is based on the self-consistent Born approx-
imations, and it is presented in a general way applicable for
any quasi-elastic scattering mechanism. As particular ex-
amples, we have considered two kinds of scatterers which
are typical for the electron system formed on the free sur-
face of liquid helium: helium vapor atoms and capillary
wave quanta (ripplons). In the vapor atom scattering re-
gime, we found a strong 1/B-modulation of the electron
temperature, which increases sharply in the vicinity of
commensurability conditions. This decay heating is shown
to enhance the effect of the sign-changing terms in the lon-
Fig. 8. Evolution of the ( )xx Bσ line shape near the commensu-
rability point * = 4m with the increase in ( )maxeTΔ at
= 0.2 KT and = 79 GHzf : ( )max = 0eTΔ (solid), 0.1 K
(dashed), and 0.5 K (dotted).
0,695 0,700 0,705 0,710 0,715
0
2
4
6
8
B, T
( ) = 0.3 KTe max
( ) = 0.7 KTe max
m* = 4
σ
Ω
xx
·1
0
,
1
1
–
1
Microwave-resonance-induced magnetooscillations and vanishing resistance states in multisubband 2D electron systems
Fizika Nizkikh Temperatur, 2011, v. 37, No. 8 841
gitudinal conductivity xxσ . The evolution of the line-
shape of conductivity oscillations with an increase of the
MW field amplitude is studied, taking into account heating
of surface electrons.
For the electron-ripplon scattering regime, we have
shown that magnetooscillations of large amplitude and the
negative linear response conductivity of SEs can easily
appear under moderate MW excitation even for cold SEs.
The evolution of the line-shape of xxσ extremes caused by
an increase in the electron temperature is studied using a
model treatment. We believe that theoretical results pre-
sented in this work explain all major features of MW-
resonance-induced magnetooscillations observed in the
system of SEs on liquid helium, and support the suggestion
[4,5] that novel zero-resistance states are realized in such a
system.
Appendix A: Energy relaxation rate
Here we give final expressions for the energy relaxation
rate of SEs due to scattering with vapor atoms. The nor-
mal ( )
N
aν and anomalous ( )
A
aν energy relaxation rates
are defined by the following relationships: N =W ( )
N( ) a
eT T ν= − − , and ( )
A A= aW Tν− . In turn, ( ) ( )
N N,intra=a aν ν +
( )
N,inter
aν+ , where
( )
( ) 0
, ,N,intra 1/2
;
= coth ,
2
a
e c Ra l c c
l l l l
l l R ele
m E n
u p
E TMT
ν ω ω ω
ν
Γπ
⎡ ⎤⎛ ⎞
+⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
∑
(A.1)
( ) ( ) ( )
/( ) ,
( ) 0
N,inter 1/2
;>
, , , , ,
e
=
coth ,
2
Ta l l e
e c Ra l l
l ll le
c c
l l l l c l l l l c l l c
R e
m E n n
MT
u F p F H
E T
Δν ω
ν
Γπ
ω ω
ω ω ω
− ′
′
′′
′ ′ ′ ′ ′
+
×
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪× + +⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∑
(A.2)
functions ,l lF ′ and ,l lH ′ were given in Eqs. (40) and (41).
The anomalous energy relaxation rate can be repre-
sented as
( )
( ) ( ) ( )
/,( )
( ) 0
A 1/2 2
> ;
, , , , ,
e2
=
coth ,
2
Tl l ea l le c Ra
l l l l
c c
l l l l c l l l l c l l c
R e
n nm E
M
u p
E T
Δ
ν ω
ν
π Γ
ω ω
Φ ω Φ ω Θ ω
− ′
′
′ ′
′ ′ ′ ′ ′
−
×
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪× + +⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
∑
(A.3)
where ,l lΦ ′ and ,l lΘ ′ are the same as those of Eq. (42).
For equilibrium subband occupancies, A = 0ν . These equ-
ations were used for obtaining electron temperature as a
function of the magnetic field under the MW resonance.
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