Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium
The dependence of radiation-induced dc magnetoconductivity oscillations on the microwave polarization is theoretically studied for a two-dimensional system of strongly interacting electrons formed on the surface of liquid helium. Two different theoretical mechanisms of magnetooscillations (the displ...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium / Yu.P. Monarkha // Физика низких температур. — 2017. — Т. 43, № 6. — С. 819-827. — Бібліогр.: 34 назв. — англ. |
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| citation_txt | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium / Yu.P. Monarkha // Физика низких температур. — 2017. — Т. 43, № 6. — С. 819-827. — Бібліогр.: 34 назв. — англ. |
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| description | The dependence of radiation-induced dc magnetoconductivity oscillations on the microwave polarization is theoretically studied for a two-dimensional system of strongly interacting electrons formed on the surface of liquid helium. Two different theoretical mechanisms of magnetooscillations (the displacement and inelastic models) are investigated. We found that both models are similarly sensitive to a change of circular polarization, but they respond differently to a change of linear polarization. Theoretical results are compared with the recent observation of a photoconductivity response at cyclotron-resonance harmonics.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6, pp. 819–827
Polarization dependence of microwave-induced
magnetoconductivity oscillations in a two-dimensional
electron gas on liquid helium
Yu.P. Monarkha
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: monarkha@ilt.kharkov.ua
Received September 21, 2016, published online April 25, 2017
The dependence of radiation-induced dc magnetoconductivity oscillations on the microwave polarization
is theoretically studied for a two-dimensional system of strongly interacting electrons formed on the surface
of liquid helium. Two different theoretical mechanisms of magnetooscillations (the displacement and inelastic mod-
els) are investigated. We found that both models are similarly sensitive to a change of circular polarization, but
they respond differently to a change of linear polarization. Theoretical results are compared with the recent ob-
servation of a photoconductivity response at cyclotron-resonance harmonics.
PACS: 73.40.–c Electronic transport in interface structures;
73.20.–r Electron states at surfaces and interfaces;
78.67.–n Optical properties of low-dimensional, mesoscopic, and nanoscale materials and structures;
73.25.+i Surface conductivity and carrier phenomena.
Keywords: 2D electron gas, microwave radiation, magnetoconductivity oscillations, zero resistance states.
1. Introduction
Microwave-induced resistance oscillations (MIRO) and
zero resistance states (ZRS) of a 2D electron gas which
were discovered in high quality GaAs/AlGaAs heterostruc-
tures [1–4] subjected to a perpendicular magnetic field have
attracted much interest. For a quite arbitrary microwave
(MW) frequency > cω ω (here cω is the cyclotron frequen-
cy), these oscillations are governed by the ratio / cω ω . In
the vicinity of / =c mω ω , where m is an integer, resistivity
[ ( )xx Bρ ] and conductivity [ ( )xx Bσ ] curves have an asym-
metrical shape. At strong enough power and low tem-
peratures, the minima of these curves positioned near
/ = 1/ 4c mω ω + evolve into zero-resistance states. A large
number of theoretical mechanisms have been proposed to
explain these magneto-oscillations [5–10] (for a review,
see [11]), but the subject is still under debate. The ZRS can
be explained [12] by an assumption that the longitudinal
linear response conductivity xxσ is negative in appropriate
ranges of the magnetic field B, regardless of details of
a microscopic mechanism.
Different MW-induced magnetoconductivity oscillations
and ZRS are observed in the nondegenerate 2D electron
system formed on the free surface of liquid helium [13,14]
if the intersubband excitation frequency 2 1 2,1( ) /∆ −∆ ≡ ω
coincides with the MW frequency ω (here, 1∆ and 2∆
are the energies of the ground and first excited surface sub-
bands, respectively). The explanation of this
phenomenon [15,16] is based on nonequilibrium popula-
tion of the second surface subband caused by the MW res-
onance which triggers quasielastic intersubband scattering
against or along the driving force, depending on the relation
between 2,1ω and the in-plane excitation energy cm ω . It
was found that these oscillations vanished if ω was sub-
stantially different from 2,1ω , which means that the origin
of these oscillations is different from that of MIRO ob-
served in semiconductor heterostructures. It is interesting
that states of surface electrons (SEs) with negative xxσ are
unstable which experimentally results in redistribution of
SEs [17]. Under these conditions, the system can form a
density-domain structure [18] caused by strong Coulomb
interaction of SEs.
Two most elaborated mechanisms of MIRO proposed
for semiconductor systems (“displacement” [5,6] and “in-
elastic” [7,8]) are based on photon-assisted scattering. In
the displacement model, a displacement of the electron
© Yu.P. Monarkha, 2017
Yu.P. Monarkha
orbit center X' X− which follows from energy conserva-
tion for photon-assisted scattering by impurities depends
strongly on the relation between ω and cm ω . In the ine-
lastic mechanism, photon-assisted scattering to high Lan-
dau levels ( =n' n m− ) changes the electron distribution
function ( )f ε near n'ε ≈ ε [here ( )= 1/ 2n c nε ω + is the
Landau spectrum] which affects usual impurity scattering.
A theoretical analysis [19] indicates that both these mech-
anisms potentially can reveal themselves in the nondege-
nerate system of SEs on liquid helium. Still, observation of
these effects in this system requires significantly higher
MW power, because the mass of free electrons M is much
larger than the effective mass of semiconductor electrons
M ∗ ( / 0.06M M∗
).
In recent experiments [20], the amplitude of the MW
electric field acE was significantly increased (up to
10 V/cmacE ≈ ) and / cω ω -periodic dc magnetoconductiv-
ity oscillations induced by the MW were observed in the
nondegenerate 2D electron system on the surface of liquid
4He. This proved the universality of the effect of MIRO.
A preliminary theoretical analysis given there have shown
that the observation can be explained by an oscillatory cor-
rection to the electron distribution function caused by pho-
ton-assisted scattering (the inelastic model) affected by
strong internal forces.
For semiconductor experiments on MIRO, the import-
ant point is the polarization sensitivity or immunity of the
microwave magnetoresistance response [21]. Theoretical
models discussed in the literature usually predict different
microwave polarization sensitivity in the radiation-induced
oscillations. Therefore, an appropriate experimental study
can be a test for the theory. For SEs on liquid helium,
MW-induced magnetoconductivity oscillations were inves-
tigated theoretically [19] only for the linear polarization
fixed parallel to the dc-electric field. Therefore, additional
theoretical investigation on polarization dependence of
MW-induced oscillations is required.
In this work, we report the theory of dc magnetocond-
uctivity oscillations of SEs on liquid helium induced by the
MW field of an arbitrary polarization (linear and circular).
For both displacement and inelastic mechanisms of MIRO,
photon-assisted scattering of SEs by ripplons (capillary
wave quanta) is considered using Landau–Floquet states
which include the MW field in an exact way. This allows
us to find the polarization dependence of MIRO in a sim-
ple analytical form. Strong Coulomb interaction of SEs is
taken into account considering an ensemble of electrons
moving fast in the electric field fE of the fluctuational
origin [22,23]. A comparison of our results with observa-
tions [20] indicates that, for a given amplitude of the MW
field ( 10 V/cmacE ≈ ), the inelastic model results in suffi-
ciently large magnetoconductivity oscillations similar to
those reported experimentally.
2. Landau–Floquet states for an arbitrary
MW polarization
SEs on liquid helium are bound in a 1D potential well
formed by the liquid repulsion barrier 0 1 eVU ≈ , image
attraction potential ( ) = /U z z−Λ , and the potential of a
pressing electric field eE z⊥ . The image potential is rather
weak because 2= e ( 1) / 4( 1)Λ − − and the liquid helium
dielectric constant is very close to unity (for liquid 4He,
1 0.057− ). Therefore, in the ground subband electrons
are gliding above the surface at a height of about 100 Å. At
low temperatures ( 0.5 KT ), the population of higher
subbands can be neglected. Here we concentrate on the in-
plane motion of SEs, assuming that the wave function of
vertical motion ( )h z is well known, and the intersubband
excitation frequency 2,1ω is substantially higher than the
MW frequency ω.
We consider the magnetic field B directed perpendicu-
lar to the electron layer and choose the Landau gauge for
the vector potential. Then, in the presence of a MW elec-
tric field ( )mw tE and a dc electric field dcE directed along
the x axis, the Hamiltonian of an electron can be written as
2
21 ˆ ˆ=
2 x y dc
eBH p p x eE x
M c
+ + + +
( ) ( ) ( ) ( ) .x y
mw mweE t x eE t y+ + (1)
For this Hamiltonian, the wave equation can be solved
in an exact way using a generalization of the well-known
nonperturbative method (for a recent example, see Ref. 24).
The wave equation can be satisfied by
( )
( )
= , e
i p yyx X t
−ζ
ψ ϕ − − ξ ×
( ) ( ) ( )exp ,i iM x X y t × ξ − − ξ + ζ − ζ + η
(2)
where
2= y dc
c
cp eE
X
eB M
− −
ω
,
and the surface area is set to unity. The functions ( )tξ ,
( )tζ and ( )tη are found from conditions chosen to reduce
the wave equation to the conventional oscillator equation.
After lengthy algebra we arrive at
( ) ( )2 = 0,x
c c mwM M M eE tξ + ω ξ + ω ζ + (3)
( ) ( ) = 0.y
mwM eE tζ + (4)
The function ( )tη can be written as ( ) ( ) ( )0= Xt t tη η +η ,
where
( ) ( )2 2 2 2
0 = ,
2 2 2
x y
c mw mw c
M M M eE eE Mη ξ + ζ − ω ξ − ξ − ζ − ω ξζ
820 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Polarization dependence of microwave-induced magnetoconductivity oscillations
and
( ) ( ) ( ) ( )
0
= 0 .
t
x
X c mwM X t X eE t dt'′η − ω ζ − ζ − ∫ (5)
The ( )tξ represents the classical motion of an electron in
a magnetic field forced by an ac electric field, while ( )tζ
represents a 1D electron motion in the field ( ) ( )y
mwE t .
Thus, instead of ( ),x tϕ entering Eq. (2), we can use
usual eigenfunctions ( )n xϕ of an oscillator Hamiltonian
( ) ( ) ,, = exp ,n X
n nx t x i t
ε
ϕ ϕ −
2 2
, 2
e
=
2
dc
n X n dc
c
E
eE X
M
ε ε + +
ω
. (6)
Therefore, the Landau–Floquet eigenfunctions of the
Hamiltonian given in Eq. (1) can be represented as
( ) ( ) ( )2/ , ,
, = , e e e ,iXy l F t i x y tB Xn X n x X t − θψ ϕ − − ξ (7)
where
( ) ( ) ( ) ( )
0
= 0 ,
t
x
X ac c
i i iF t MX X eE t' dt M X′− ξ − + ω ζ∫
(8)
and the exact form of the function ( ), ,x y tθ independent of
n and X is not important for the following consideration.
Specifying ( )mw tE , we shall consider
( ) ( )= cos , = sin ,x y
mw ac mw acE aE t E bE tω ω (9)
where acE is the amplitude of the MW field, and, general-
ly, a and b are arbitrary parameters. For particular cases,
we assume that a can be 0 or 1, and b can be 0 or 1± . In
this way, we can describe two linear polarizations [parallel
( = 1a , = 0b ) and perpendicular ( = 0a , = 1b ) to the dc
electric field] and two circular polarizations ( = 1a , = 1).b ±
Respectively, we define the polarization index = , , , ,s ⊥ + −
where the first two symbols ( and ⊥ ) correspond to linear
polarizations, and the last two symbols (+ and −) corre-
spond to circular polarizations. Now we have
2 2 2
( )
= sin , = cos ,
( )
ac c ac
c
beE a b eE
t t
MM
ω+ ω
ζ ω ξ ω
ωω ω −ω
(10)
2 2
( )
( ) = sin ,
( )
c c
X
B c
a bXF t i t
l
ω ω + ω
λ ω
ω −ω
(11)
where 2 = /Bl c eB , and = /ac BeE lλ ω . Using these defi-
nitions, one can determine the matrix elements for electron
scattering probabilities.
3. Scattering probabilities
At low temperatures, SEs on liquid helium interact with
capillary-wave excitations (ripplons). The electron-ripplon
interaction Hamiltonian (ripplons represent a sort of 2D
phonons) is usually described as
( )†= e ,i
I q qV U Q b b − ⋅
− +∑ q r
q q
q
(12)
where †bq and bq are creation and destruction operators of
ripplons, qU is the electron-ripplon coupling [23],
2
,= / 2 ,q r qQ q ρω 3/2
, /r q qω α ρ is the ripplon spec-
trum, α and ρ are the surface tension and mass density of
liquid helium, respectively.
Scattering probabilities depend on matrix elements
, , ,(e )i
n' X' n X
− ⋅q r which now acquire additional time-depend-
ent factors: one factor comes directly from ( )exp XF t of
Eq. (7), and another factor ( )exp xiq− ξ appears because of
the change of the integration variable 1=x x X+ + ξ .
Gathering these two factors, one can find
( )sin,
, , ,(e ) = e
i ti s
n' X n X
− β ω +γ− ⋅
′ ×q r q
( )0
2 , ; ,,
(e )iq xx
n' X' n XX X' l qB y
−
−
× δ , (13)
where
( ) ( )2 22 2
, 2 2= ,
( )
c B
s y c x c
c
l
q a b q a b
λω
β ω + ω + ω+ ω
ω −ω
q (14)
( )
( )
tan = ,x c
y c
q a b
q a b
ω+ ω
γ
ω + ω
and ( )0
, ; ,(e )iq xx
n' X' n X
− are matrix elements in the absence of
the MW field. Here the first term under the sign of the
square root of Eq. (14) originates from the factor
exp [ ( )]XF t , and the second is from ( )exp xiq− ξ . Then,
using the Jacobi–Anger expansion sine = ( ) eiz ik
kkJ zϕ ϕ∑
[here ( )kJ z is the Bessel function], the procedure of find-
ing scattering probabilities can be reduced to a quite usual
treatment.
The probability of electron scattering , , 'n X n' X→
with the momentum exchange q due to ripplon creation (+)
and destruction (−) can be found as
( ) ( ) ( )
2 2
, 2, , ,
2= n n' qn X n' X' X X' l qB y
w C I x± ±
→ −
π
δ ×qq,
( ) ( )2
, , , ,
=
,k s n' X' n X r q
k
J k
∞
−∞
× β δ ε − ε − ω± ω∑ q (15)
where ( ) 1/2( )= 1/ 2 1/ 2r
q qC U Q n±
±
+ ± q q , ( )rn±q is the number
of ripplons with the wave vector ±q, ,n Xε is from Eq. (6),
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 821
Yu.P. Monarkha
( )
( )
2
,
min , !
( ) =
max , !
n n
n n' q q
n n
I x x
n n'
′−′
×
( )
2
min ,exp ( ) ( ) ,n' n
q qn n'x L x− × −
(16)
( )m
n qL x are the associated Laguerre polynomials, and
2 2= / 2q Bx q l is a dimensionless variable. We note that for
Landau–Floquet states, matrix elements of Eq. (13) fix the
displacement of the orbit center to the same value
2='
y BX X q l− as that of usual scattering. It is instructive
also that, considering the MW field in a classical way, we
still found that the energy exchange in Eq. (15) occurs by
MW quanta k ω .
Energies of ripplons ,r qω involved in one-ripplon scat-
tering processes are usually much smaller than typical
electron energies, and we can use a quasi-elastic treatment
of electron-ripplon scattering. Therefore, in the energy
conservation delta-function, we shall set ,r qω to zero.
This allows us to write the whole probability as a sum
( ) ( )=w w w+ −+q q q neglecting a small difference between
( )C +
q and ( )C −
q .
Introducing Landau level densities of states, the average
probability of electron scattering wq with the momentum
exchange q, can be found in terms of the dynamic structure
factor (DSF) ( ),S q Ω of a nondegenerate 2D electron gas
2
2
,2
=
2
= ( ) ( , ),q
k s y H
k
C
w J S q k q V
∞
−∞
β ω−∑q q
(17)
where 1/2
,q q q r qC U Q N , ,r qN is the ripplon distribution
function, = /H dcV cE B is the absolute value of the Hall
velocity which enters the frequency argument due to
( ) =dc y HeE X' X q V− ,
( ) ( ) ( )/2
,
,
2, = ( ) e ,Ten n' q n n'
n n'
S q I x d g g
Z
−εΩ ε ε ε + Ω
π ∑ ∫
(18)
eT is the electron temperature, Z
is the partition function
for the Landau spectrum nε , and ( )ng− ε is the imaginary
part of the single-electron Green’s function. For the Gauss-
ian shape of level densities [25],
( ) ( )2
2
22= exp ,n
n
n n
g
ε − επ ε −
Γ Γ
where nΓ is the Landau level broadening. For SEs on liq-
uid helium, at low temperatures nΓ is usually much smaller
than T and cω .
4. The displacement model
Using Eq. (17), the dc magnetoconductivity xxσ of SEs
under MW radiation can be found directly considering the
current density
( ) 2= =x e e B yj en X' X w en l q w− − −∑ ∑q qqq q .
Therefore, the effective collision frequency
( )eff
1= y H
H
q w V
MV
ν − ∑ q
q
(19)
defines 2 2
effe /xx e cn Mσ ν ω . Expanding Eq. (19) in HV
yields the linear dc magnetoconductivity in terms of the
derivative of the DSF = /'S S∂ ∂Ω. The property
( ) ( ) ( ), = exp / ,eS q T S q−Ω − Ω Ω allows us to represent
the effective collision frequency as a sum eff =0= kk
∞
ν ν∑ ,
where
( ) ( )2 2 2
0 0 ,
1= ,0 ,y q s
e
q C J S q
MT
ν β∑ q
q
(20)
and
( )
( )
/
2 2 2
,
2 1 e
= ( ) ,
k Te
k y k sq C J S' q k
M
− ω−
ν β ω∑ q q
q
(21)
for > 0k . Here we neglected a small term
/( / ) e ( , )k TeeT S q k− ω ω
as compared to ( ),S' q kω . In Eq.
(20), the derivative ( ),0S' q was transformed employing
the relationship ( ) ( ) ( ),0 = / 2 ,0eS' q T S q . It is instructive
that in the nonperturbative method, the MW field affects
also the usual contribution to the effective collision fre-
quency 0ν due to the factor 2
0 ,( )sJ β q .
SEs on liquid helium represent a highly correlated elec-
tron system because the average Coulomb interaction ener-
gy per an electron 2e enπ (here en is the electron density) is
usually much larger than the average kinetic energy T . In
this case, each electron is affected by a strong internal elec-
tric field fE of fluctuational origin [26]. The typical value
of the fluctuational field depends strongly on the electron
temperature and density: ( )0 3/43 e efE T n . Self-energy
effects (collision broadening of Landau levels) can be
combined with the Coulomb effect by considering the DSF
of an ensemble of electrons moving fast in the fluctuational
field [22,23]. Therefore, to model the effect of Coulomb
interaction on MW-induced conductivity oscillations we
shall use the DSF obtained previously [23] for strongly
interactions electrons
( )
2
/ ,
,,
2, e T n n'n e
n n'n n'
I
S q
Z
−επ
Ω ×
γ∑
( ) 2
2
,
exp ,c n
n n'
n' n Ω − − ω −φ × −
γ
(22)
where
2 2
2 2
, ,= , = ,
4
n q C
n n' n n' q C n
e
x
x
T
Γ + Γ
γ Γ + Γ φ
(23)
822 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Polarization dependence of microwave-induced magnetoconductivity oscillations
2 2 2
,2 =n n' n n'Γ Γ +Γ , and (0)= 2C BfeE lΓ . In most cases, the
parameter nφ can be neglected. Thus, the strong Coulomb
interaction of SEs results in a broadening of DSF maxima
which occur at ( ) cn' nΩ→ − ω . In the presence of MW
radiation of a circular polarization, the employment of
Eq. (22) gives absolutely correct description of the many-
electron effect on magnetooscillations. For a linear MW
polarization, this procedure is a reasonable approximation.
With the exception of the cyclotron resonance (CR)
condition ( cω →ω), the parameter ,sβ q defined in Eq. (14)
is small and one can expand the Bessel functions in ,sβ q. In
this case, the main term of Eq. (20) transforms into the
well known result of the self-consistent Born approxima-
tion [27]. In spite of small values of ,sβ q, the contribution
from 1ν can be substantial because it contains the derivative
of the function ( ),S q ω having sharp maxima. Using dimen-
sionless coupling function 2( ) = /c q q BU x U l Λ (its exact
form is given in Refs. 16, 19) and Eq. (22), we arrive at
( ) ( )/2
1, = 1 e ,TR es s
c
T F B− ωπ ν
ν λ − χ
ω
(24)
( )
/
2 2
,
, 0
e= ( )
Tn e
q c q q n n'
n n'
F B dx U x x I
Z
∞−ε
− ×∑ ∫
( )2
3 2
, ,
/ /
exp
( / )
c nc n c
n n' c n n'
mm ω− ω −φω ω − −φ ω × −
γ ω γ
, (25)
where 2 4= / 8R Blν Λ π α is a characteristic collision fre-
quency, =m n' n− , and sχ is a dimensionless polarization
factor with the polarization index = , , ,s ⊥ + − . For two
linear polarizations,
2 2 2 2 2 2
2 2 2 2 2 2
(3 ) (3 )
= , = ,
( ) ( )
c c c c
c c
⊥
ω ω +ω ω ω +ω
χ χ
ω −ω ω −ω
(26)
and for two circular polarizations,
2 2
2 2 2
4 ( )
= .
( )
c c
c
±
ω ω±ω
χ
ω −ω
(27)
Besides the small factor 2λ , the quantum form of Eq. (24)
contains two large factors / cT ω and 2
,( / )c n n'ω γ . It
should be noted that in the ultra-quantum limit eTω ,
Eq. (24) with the linear MW polarization =s transforms
into the result found previously [19] using the conventional
perturbation theory, if we set
( )2 2= 4 / 2,mw
ac
N
E
V
ω
π ω→E
where ( )mwN ω is the number of photons with the frequency
ω in the volume V . This confirms the validity of the ap-
proach to description of probabilities of photon-assisted
scattering using Landau–Floquet states.
The MW-induced correction to the effective collision
frequency of noninteracting electrons can be obtained from
Eqs. (24) and (25) by setting the Coulomb broadening CΓ
to zero. In this case, ,n n'γ is independent of qx , and, there-
fore, the derivative of the Gaussian [which can be formed
in the lowest line of Eq. (25)] can be moved out from the
integrand. Thus, the shape of MW-induced oscillations near
/ =c mω ω represents a simple derivative of a Gaussian.
Formally, in the limit of strong broadening of the Gaussi-
ans the positions of minima approach the condition
/ ( ) 1/ 4c B mω ω → + , which agrees with observations. The
Coulomb interaction broadens the derivative of Gaussians
and involves it into averaging over qx because ,n n'γ of
Eq. (23) becomes dependent on qx .
The Eq. (24) indicates that, for two different polariza-
tions described by indexes s and s' , the ratio 1, 1,/ =s s'ν ν
= /s s'χ χ . When changing the parameter / cω ω the ratio
/⊥χ χ
varies from 1.86 ( / = 2cω ω ) to 3 ( / cω ω →∞).
For circular polarizations, the ratio /+ −χ χ decreases with m:
from 9 ( / = 2cω ω ) to 1 ( / cω ω →∞). Results of numeri-
cal evaluations of 1,sν obtained for two linear polarizations
are shown in Fig. 1. Here the main parameters of the SE
system are the same as those in the experiment [20]. The
many-electron treatment of the displacement model results
in the broadening of MW-induced oscillations which agrees
with experimental data. Still, the amplitude of oscillations
caused by the MW field directed parallel to dcE is approx-
imately an order of magnitude smaller than the amplitude
observed. As expected, the MW field with the perpendicu-
lar polarization ( =s ⊥) results in dc magnetoconductivity
oscillations having a substantially larger amplitude.
Fig. 1. An oscillatory contribution to the effective collisions fre-
quency calculated for two different MW polarizations: =s
(dashed) and =s ⊥ (solid). The conditions are the following:
= 10 V/cmacE , 6 2= 17 10 cmen −⋅ , / 2 = 88.34 GHzω π , and
= 0.56 KT (liquid 4He).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 823
Yu.P. Monarkha
For the CR condition ( cω ω ), / 1⊥χ χ
and the
photoconductivity is immune to a change of linear polari-
zation. In this case, the parameter ,β q entering the Bessel
function of Eq. (24) can be simplified as
2
, 2 2( )
c
B
c
ql
ω
β λ
ω −ω
q . (28)
As noted previously [28] [in the brief report, there is
a small misprint in the expression similar to Eq. (28)],
the parameter ,β q increases much when cω →ω and, gen-
erally, it is impossible to expand the Bessel functions.
Moreover, according to the expression for the DSF
( ),S q kω , sharp Gaussian maxima appear for the all k at
= ( ) ck n n′ω − ω , and, if cω →ω, it is necessary to take
account of the all terms in the sum =0 kk
∞
ν∑ . A formal
inclusion of a damping parameter in the classical equations
for ξ and ζ can reduce the number of kν to be taken into
account.
Results of numerical calculations are shown in Fig. 2.
As a damping parameter, here we chosen the classical col-
lision frequency of SEs. This figure illustrates that at a
fixed value of 0cω−ω ≠ , the sum over k converges quite
rapidly. Still, each next term in the sum =0 kk
∞ ν∑ has its
own extrema which are closer to the point =cω ω. There-
fore, in the vicinity of the resonance, a substantial number
of mν should be taken into account.
As mentioned above, these calculations does not de-
pend on the orientation of the linear polarization of the
microwave field. On the contrary, for the circular polariza-
tion with =s − , the strong enhancement of the parameter
,sβ q caused by the CR vanishes and one can restrict calcu-
lations to 1ν given by Eqs. (24) and (25). For the chosen
amplitude of the MW field, corresponding oscillatory fea-
tures could not be seen on the scale of Fig. 2.
Results shown in Fig. 2 are obtained assuming =eT T .
Still, under the CR condition, the system of SEs on liquid
helium is strongly heated [29] and the electron temperature
as a function of cω−ω usually has a sharp maximum [30].
Because 0 1/ eTν ∝ , the wavy variations of ( )xx Bσ induced
by photon-assisted scattering are expected to appear at the
bottom of a broad minimum caused by electron heating.
5. The inelastic model
In the displacement model discussed above, it is as-
sumed that the electron distribution function ( )f ε , enter-
ing the average probability of the momentum exchange wq
and the effective collision frequency effν of Eq. (19), coin-
cides with the equilibrium distribution function /1e TeZ −ε−
.
Still, photon-assisted scattering from a level n to a higher
level n' selected by the condition n n′ε − ε ω might in-
crease the population of the level n' . Moreover, if the en-
ergy exchange with the medium (in our case, the ripplon
energy) can be neglected in the energy conservation delta-
function, the sharp structure of the level density ( )ng ε
could make a sharp shape of ( )f ε at n'ε ε which even-
tually results in MW-induced magnetoconductivity oscilla-
tions. This is the inelastic model, introduced in Ref. 7.
The photon energy ω and the magnetic field B usually
select two Landau levels with n' nε − ε ω . In the ultra-
quantum limit important for SEs on liquid helium, this
allows us to consider a simple two-level model for obtain-
ing a correction to the distribution function. When analyzing
the average transition rate up (n n'→ ) and the all transition
rates down (n n′ → ), we can represent them as integral
forms ( ...d d 'ε ε∫ ∫ ) similar to that of Eqs. (17) and (18)
using the Landau level density of states. Then, disregard-
ing the Coulomb interaction, for an energy n''ε ε , it is
possible to obtain the rate-balance condition
( )
,
2
,
( ) ( )
( ) = ,
( )
n n n
n' R
n n'n'
f ' r '
f
r '
′ε − ω ε
′ε
ν + ε
(29)
where
( )
2
, ,( ) =
2n n' s R n n' n
Tr ' P g 'λ
ε χ ν ε − ω
is the excitation rate,
2 2
, ,
0
= ( ) ( ) ,n n' c q n n' q qP U x I x dx
∞
∫
2 2 2
2 2 2
2 ( )
= = , = ,
( )
c c
c
⊥ ± ±
ω ω +ω
χ χ χ χ
ω −ω
(30)
( )2R
n'ν is the inelastic transition rate from n' to the all
<n n' caused by two-ripplon emission processes [31]. In
the distribution function ( )nf '′ ε , the subscript n' indicates
Fig. 2. Contributions from partial sums max
=0
k
kk ν∑ to effν nor-
malized vs the magnetic field for a sequence of maxk : from
max = 1k to max = 7k (solid). The conditions are the following:
= 0.2 KT (liquid 4He), 6 2= 2 10 cmen −⋅ , and = 0.05 V/cmacE .
824 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Polarization dependence of microwave-induced magnetoconductivity oscillations
that its argument 'ε is close to n'ε . Dimensionless integrals
,n n'P describe the strength of the electron-ripplon coupling
in the presence of a strong magnetic field.
Here the polarization immunity =⊥χ χ
appears be-
cause the transition rate ,n n'r is finite in the limit 0dcE →
and integration over the angle of the vector q results in
equal averaging of 2
xq and 2
yq entering ,sβ q of Eq. (14)
(note that the probability of scattering is proportional to
2
,sβ q). In the displacement model, one have to expand wq in
y Hq V up to a linear term which results in unequal averag-
ing of 2
xq and 2
yq . This leads to the linear MW polarization
sensitivity of the displacement model in radiation-induced
oscillations.
Direct electron transitions between different Landau
levels accompanied by one-ripplon creation or destruction
are practically impossible because the typical value of q
which follows from energy conservation is much larger
than 1
Bl
− (the dimensionless parameter 1qx ). Still, there
are transitions from a level n' to lower levels accompanied
by emission of pairs of short wavelength ripplons with
1 ,Bl q q'−′+q q . According to Ref. 31, ( )2R
n'ν can be
written as
1
(2 ) 2 4 2
,2
=0
2= ( 1) ( 2 ),
n'
R
q q r q n' n qn'
nB
W Q N
l
−
ν + δ ε − ε − ω∑∑
q
(31)
where qW is the coupling function for two-ripplon scatter-
ing. In terms of continuous variables ′ε and ε, the ( )2R
n'ν
varies slow within the width of a Landau level since
n cΓ ω . This allows us to use the form of Eq. (31) in
Eq. (29).
The Eq. (29) reminds the solution of the rate equation
of a two-level model usually obtained in quantum optics.
The second term in the denominator of this equation is
caused by backward electron transitions accompanied by
emission of a photon. Firstly, we note that in the absence
of the inelastic decay rate ( )2R
n'ν , the solution satisfies the
saturation condition ( ) ( )=n' nf ' f 'ε ε − ω which is quite
obvious. Secondly, a sharp shape of ( )n'f 'ε appears when
the inelastic scattering rate is stronger then the excitation
rate: ( )2
,
R
n n'n' rν . In this case, the single-electron treat-
ment yields ( ) ( )n' nf ' g 'ε ∝ ε − ω because ( )nf ε can be
approximately set to the equilibrium function. This is the
reason why the mechanism discussed here is called the
inelastic model.
In the inelastic model, the correction to xxσ is usually
obtained from the conductivity equation which contains the
derivative /f∂ ∂ε . This picture is instructive to see how the
derivative of maxima appears in the final result. Still, ac-
cording to Eq. (19), the initial form of the effective colli-
sion frequency contains the derivative ( ) = /n' ng g ′′ ε ∂ ∂ε.
Therefore, in the ultra-quantum limit ( 0n → ), the correc-
tion to the effective collision frequency induced by addi-
tional population of higher Landau levels can be written as
( ) ( ) ( ),2
=1
4
= R c
mw n' n' n' n' n'
n'
T
P d f g g
∞ν ω
′ν ε ε ε ε
π
∑ ∫
. (32)
Here we neglected the overlapping of different Landau
levels. The Eq. (32) is obtained in the same way as Eqs.
(20), (21). The form of mwν with the derivative /f∂ ∂ε can
be found from Eq. (32) using integration by parts [19].
Assuming low exitation regime ( ( )2
,
R
n n' nr ′ν ) and in-
serting ( )n'f ε in Eq. (32), we find
( )
, 0,2 2 2
2
=1 0,
= n' n n'
mw s R c R
n' n' n'n'
P P
T
G
∞
′ν πλ χ ν ω ×
ν Γ
∑
( ) ( )2
2 2
0, 0,
2
exp ,c c
n' n'
n' n'
G G
ω − ω ω − ω
× −
(33)
where 2 2 2
, ,= / 4n n' n n' n'G Γ −Γ . The Eq. (33) indicates that,
in the inelastic model, the shape of MW-induced magne-
toconductivity oscillations represents the derivative of a
Gaussian similar to that of the displacement model [Eq. (24)].
In contrast with the displacement model, additional large
parameters ( )2/ R
R n'ν ν and 0,/ nT G appear in the expression
for mwν which makes MIRO more pronounced.
For experimental conditions [20], the Eq. (33) results in
conductivity variations near / = ='
c n mω ω which are
substantially sharper than those observed. Respectively,
the amplitude of oscillations is much higher than in the
experiment. This discrepancy can be attributed to the effect
of strong Coulomb interaction which broadens the Gaussi-
ans and accordingly reduces the amplitude of oscillations.
Unfortunately, a strict description of this effect is very dif-
ficult because now the fluctuational electric field affects
the both ( )n'f ε and ( )n'g ′ ε . In order to describe the many-
electron effect qualitatively, we can introduce a correction
to the energy exchange caused by a fluctuational electric
field ( )f feE X X′ − ⋅q u (here fu is a drift velocity)
and average the both quantities n'f and n'g ′ over fE , inde-
pendently. This yields an additional broadening of the pa-
rameter
4
2 2 2
0, 0, 2 24( )
n'
n' n' q C
n' q C
G x
x
′
Γ
→ Γ + Γ −
Γ + Γ
(34)
which becomes dependent on qx (the integration variable
of ,n' n'P ) and q'x (the integration variable of 0,n'P ). The
later means that the derivative of Gaussians now enters the
integrand of the double-integral ...q q'dx dx∫ ∫ , and calcula-
tions become more complicated. Moreover, the averaging
over the fluctuational field changes the factor
2
2 2 3/2
1
( )
n'
n' n' q Cx
Γ
→
Γ Γ + Γ
(35)
reducing electron scattering. As a result, conductivity vari-
ations near / =c mω ω aquire the broadening which agrees
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 825
Yu.P. Monarkha
with experimental data. Still, the new factor of Eq. (35)
leads to a more rapid decrease of the amplitude of oscilla-
tions with m than it is observed. Thus, Eq. (34) gives a
reasonable approximation describing the effect of Cou-
lomb broadening of conductivity oscillations induced by
the MW, while the replacement of Eq. (35) is expected to
be improved in a more accurate treatment. For example,
the procedure of averaging of mwν , taken in the form con-
taining the derivative /f∂ ∂ε , results in a different re-
placement: 2 2 1/2( )n' n q Cx′Γ → Γ + Γ .
In the inelastic model discussed here, the amplitude of
oscillations does not depend on the orientation of the linear
polarization of the microwave field because =⊥χ χ
. This
is in contrast with the results found for the displacement
model and illustrated in Fig. 1. For semiconductor systems,
the same conclusion was drown previously [8]. Regarding
the circular polarization, the both mechanisms of MIRO
give the same dependence on the polarization index =s ±
described by the factors =± ±χ χ . This dependence on the
polarization index follows directly from the classical pa-
rameter ( )tξ entering the Landau–Floquet wave function.
According to Eq. (10), the amplitude of ( )tξ depends
strongly on the sign of b . The most obvious example is
that the classical CR vanishes if = 1b − ( =s − ). The differ-
ence between +χ and −χ is strong for a few lowest
= 2, 3, ...m , and it decreases with / cω ω .
Typical magnetoconductivity oscillations, obtained for
the inelastic model including the Coulomb interaction as
described in Eqs. (34) and (35), are shown in Fig. 3. Here
we considered two circular polarizations. The amplitude of
oscillations is significantly larger if the polarization vector
rotates in phase with respect to the cyclotron rotation (in
our notations =s + ). A similar dependence of MW in-
duced oscillations on a choice of the circular polarization
were reported for different versions of the displacement
model applied to a degenerate 2D electron system [32,33].
Dependencies of the amplitude of magnetoconductivity
oscillations on the MW polarization obtained here are valid
only for small values of the parameter ,sβ q given in
Eq. (14), when we can neglect multi-photon processes and
expand the Bessel function 1 ,( )sJ β q . For the system of SEs
on liquid helium, these conditions are fulfilled if > 1m .
It should be noted that in a semiconductor 2D electron
system, the microwave induced resistance oscillations and
the zero resistance regions are notably immune to the sense
of circular polarization [34]. This can be regarded as a cru-
cial test for theories. Obviously, the circular polarization
immunity of the photoconductivity can not be reconciled
with the displacement and inelastic mechanisms of MIRO.
On the other hand, for SEs on liquid helium, the many-elec-
tron treatment of the inelastic model results in an oscillation-
amplitude consistent with experimental observations [20].
One may assume that magnetoconductivity oscillations ob-
served for SEs on liquid helium and MIRO reported for
GaAs/AlGaAs heterostructures are caused by different me-
chanisms. Therefore, circular-polarization-dependent meas-
urements of photoconductivity response of SEs at CR har-
monics are of great interest.
6. Conclusions
Summarizing, we developed a theoretical approach which
enabled us to obtain the dependence of MW-induced dc
magnetoconductivity oscillations on the MW polarization
in a nondegenerate 2D electron system bound to the free
surface of liquid helium. Probabilities of photon-assisted
scattering of surface electrons by capillary-wave quanta
(ripplons) are shown to be well described using Landau–
Floquet states which include the microwave field and the
dc electric field in a nonperturbative exact way. The pho-
toconductivity is found expanding the average probability
of a momentum exchange in the dc electric field. Photon-
assisted scattering affects the magnetoconductivity in two
different ways by changing directly the momentum ex-
change between SEs and ripplons (the displacement model),
and changing the distribution function of excited Landau
levels (the inelastic model). The effect of strong Coulomb
interaction is described considering an ensemble of elec-
trons moving fast in a quasi-uniform internal electric field
of the fluctuational origin.
We found that the ratio of oscillation-amplitudes ob-
tained for MW fields of different polarizations is described
by a simple analytical form. Contributions of the both the-
oretical models to the photoconductivity of SEs are shown
to be very sensitive to a change of circular polarization, if
the parameter / cω ω is not large. For different circular po-
larizations, the ratio of oscillation-amplitudes is the same
in the both models. This is caused by the nature of photon-
assisted scattering whose strength depends on the ampli-
Fig. 3. Magnetoconductivity of SEs on liquid helium exposed to
the MW field with = 10 V/cmacE calculated for two circular
polarizations: s = − (dashed) and s = + (solid). Other conditions
are the same as in Fig. 1
826 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Polarization dependence of microwave-induced magnetoconductivity oscillations
tude of classical motion sensitive to a circular polarization.
On the contrary, the displacement and inelastic models
respond differently to a change of the direction of the MW
field having a linear polarization. In the displacement mo-
del, the oscillation-amplitude depends strongly on whether
the MW electric field is parallel or perpendicular to the dc
electric field. At the same time, the inelastic model is in-
sensitive to a change of linear polarization. Therefore, an
experiment with MW fields of different circular polariza-
tions could be a test for photon-assisted scattering, while
an experiment with different linear polarizations could be
a test for particular mechanisms of magnetooscillations.
Numerical calculations performed for experimental condi-
tions [20] indicate that MW-induced dc magnetoconduct-
ivity oscillations observed can be explained by an oscilla-
tory correction to the electron distribution function caused
by photon-assisted scattering affected by strong Coulomb
forces.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 827
1. Introduction
2. Landau–Floquet states for an arbitrary MW polarization
3. Scattering probabilities
4. The displacement model
5. The inelastic model
6. Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-129513 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T15:20:45Z |
| publishDate | 2017 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Monarkha, Yu.P. 2018-01-19T20:34:10Z 2018-01-19T20:34:10Z 2017 Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium / Yu.P. Monarkha // Физика низких температур. — 2017. — Т. 43, № 6. — С. 819-827. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 73.40.–c, 73.20.–r, 78.67.–n, 73.25.+i https://nasplib.isofts.kiev.ua/handle/123456789/129513 The dependence of radiation-induced dc magnetoconductivity oscillations on the microwave polarization is theoretically studied for a two-dimensional system of strongly interacting electrons formed on the surface of liquid helium. Two different theoretical mechanisms of magnetooscillations (the displacement and inelastic models) are investigated. We found that both models are similarly sensitive to a change of circular polarization, but they respond differently to a change of linear polarization. Theoretical results are compared with the recent observation of a photoconductivity response at cyclotron-resonance harmonics. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кpисталлы Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium Article published earlier |
| spellingShingle | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium Monarkha, Yu.P. Квантовые жидкости и квантовые кpисталлы |
| title | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| title_full | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| title_fullStr | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| title_full_unstemmed | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| title_short | Polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| title_sort | polarization dependence of microwave-induced magnetoconductivity oscillations in a two-dimensional electron gas on liquid helium |
| topic | Квантовые жидкости и квантовые кpисталлы |
| topic_facet | Квантовые жидкости и квантовые кpисталлы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129513 |
| work_keys_str_mv | AT monarkhayup polarizationdependenceofmicrowaveinducedmagnetoconductivityoscillationsinatwodimensionalelectrongasonliquidhelium |