Photoresonance and conductivity of surface electrons on liquid ³He
Resonance variations of the in-plane conductivity of surface electrons (SEs) over liquid ³He induced by microwave (MW) radiation of a fixed frequency are experimentally and theoretically studied for low temperature scattering regimes (T < 0.5 K). The system was tuned to resonance by varying the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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| Цитувати: | Photoresonance and conductivity of surface electrons on liquid ³He / D. Konstantinov, Y. Monarkha, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 470–479. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1169232017-05-19T03:03:03Z Photoresonance and conductivity of surface electrons on liquid ³He Konstantinov, D. Monarkha, Y. Kono, K. Электроны над жидким гелием Resonance variations of the in-plane conductivity of surface electrons (SEs) over liquid ³He induced by microwave (MW) radiation of a fixed frequency are experimentally and theoretically studied for low temperature scattering regimes (T < 0.5 K). The system was tuned to resonance by varying the amplitude of the vertical electric field which shifts the positions of SE Rydberg levels. The line-shape change and reversing of the sign of the effect are found to be opposite to that reported previously for weak vertical electric fields. A theoretical analysis of conductivity of the SE system heated due to decay of electrons excited to the second Rydberg level by the MW explains well the line-shape variations observed. It shows also that shifting the MW resonance into the range of weak vertical fields leads to important qualitative changes in the line-shape of SE conductivity which are in agreement with observations reported previously. 2008 Article Photoresonance and conductivity of surface electrons on liquid ³He / D. Konstantinov, Y. Monarkha, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 470–479. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 67.90.+z;73.20.–r;73.25.+i;78.70.Gq http://dspace.nbuv.gov.ua/handle/123456789/116923 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Электроны над жидким гелием Электроны над жидким гелием |
| spellingShingle |
Электроны над жидким гелием Электроны над жидким гелием Konstantinov, D. Monarkha, Y. Kono, K. Photoresonance and conductivity of surface electrons on liquid ³He Физика низких температур |
| description |
Resonance variations of the in-plane conductivity of surface electrons (SEs) over liquid ³He induced by
microwave (MW) radiation of a fixed frequency are experimentally and theoretically studied for low temperature
scattering regimes (T < 0.5 K). The system was tuned to resonance by varying the amplitude of the
vertical electric field which shifts the positions of SE Rydberg levels. The line-shape change and reversing
of the sign of the effect are found to be opposite to that reported previously for weak vertical electric fields.
A theoretical analysis of conductivity of the SE system heated due to decay of electrons excited to the second
Rydberg level by the MW explains well the line-shape variations observed. It shows also that shifting
the MW resonance into the range of weak vertical fields leads to important qualitative changes in the
line-shape of SE conductivity which are in agreement with observations reported previously. |
| format |
Article |
| author |
Konstantinov, D. Monarkha, Y. Kono, K. |
| author_facet |
Konstantinov, D. Monarkha, Y. Kono, K. |
| author_sort |
Konstantinov, D. |
| title |
Photoresonance and conductivity of surface electrons on liquid ³He |
| title_short |
Photoresonance and conductivity of surface electrons on liquid ³He |
| title_full |
Photoresonance and conductivity of surface electrons on liquid ³He |
| title_fullStr |
Photoresonance and conductivity of surface electrons on liquid ³He |
| title_full_unstemmed |
Photoresonance and conductivity of surface electrons on liquid ³He |
| title_sort |
photoresonance and conductivity of surface electrons on liquid ³he |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Электроны над жидким гелием |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116923 |
| citation_txt |
Photoresonance and conductivity of surface electrons
on liquid ³He / D. Konstantinov, Y. Monarkha, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 470–479. — Бібліогр.: 24 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT konstantinovd photoresonanceandconductivityofsurfaceelectronsonliquid3he AT monarkhay photoresonanceandconductivityofsurfaceelectronsonliquid3he AT konok photoresonanceandconductivityofsurfaceelectronsonliquid3he |
| first_indexed |
2025-07-08T11:19:30Z |
| last_indexed |
2025-07-08T11:19:30Z |
| _version_ |
1837077439197478912 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5, p. 470–479
Photoresonance and conductivity of surface electrons
on liquid 3He
Denis Konstantinov1, Yuriy Monarkha1,2, and Kimitoshi Kono1
1
Low Temperature Physics Laboratory, RIKEN, Hirosawa 2-1, Wako 351-0198, Japan
2
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: konstantinov@riken.jp
Received October 26, 2007
Resonance variations of the in-plane conductivity of surface electrons (SEs) over liquid
3
He induced by
microwave (MW) radiation of a fixed frequency are experimentally and theoretically studied for low tem-
perature scattering regimes (T � 0.5 K). The system was tuned to resonance by varying the amplitude of the
vertical electric field which shifts the positions of SE Rydberg levels. The line-shape change and reversing
of the sign of the effect are found to be opposite to that reported previously for weak vertical electric fields.
A theoretical analysis of conductivity of the SE system heated due to decay of electrons excited to the sec-
ond Rydberg level by the MW explains well the line-shape variations observed. It shows also that shifting
the MW resonance into the range of weak vertical fields leads to important qualitative changes in the
line-shape of SE conductivity which are in agreement with observations reported previously.
PACS: 67.90.+z Other topics in quantum fluids and solids; liquid and solid helium;
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: liquid helium, surface electrons, microwave, resonance, conductivity.
Introduction
Surface electron (SE) states on liquid helium [1,2] are
formed owing to the interplay between an attractive im-
age potential acting above the surface, V z zimg ( ) /� �� ,
and a very high repulsion barrier V0 1� eV appearing at
the interface (z � 0 ). The image potential is very weak be-
cause � � � �e 2 1 4 1( ) / ( )� � and the dielectric constant
of liquid helium � is uniquely close to 1 (� �1 0 057� . for
liquid 4He, and � �1 0 043� . for liquid 3He). In an approxi-
mation V0 � , SE energy levels are spaced similarly to
Rydberg levels of a Hydrogen atom:
l R l� � / 2, where
l �12, , ..., and
R is the corresponding Rydberg energy
which are about 7.6 K for liquid 4He and about 4.2 K for
liquid 3He. The effective Bohr radius of SE Rydberg
states a mB e� �
2 / � is about two orders of magnitude
larger than the conventional Bohr radius, which makes
these states insensitive to small surface distortions.
Therefore, at low temperatures, liquid helium provides a
remarkable opportunity to study a very clean two-dimen-
sional system of highly correlated electrons. The SE
Rydberg states and a Stark effect in the vertical electric
filed E� were first observed by Grimes and Brown [3] in
a microwave (MW) resonance absorption experiment.
If Coulomb interaction is disregarded, SE motion
along the interface can be described by the free electron
spectrum � k ek m� �
2 2 2/ , where k is a two-dimensional
wave-vector. At typical helium temperatures, electron in-
teractions with vapor atoms and capillary wave quanta
(ripplons) are weak enough to be treated in terms of elec-
tron scattering which limits SEs conductivity along the
interface. For liquid 4He (3He), scattering by vapor atoms
dominates at T � 1K (0.5 K), while at lower temperatures
T � 0 5. K (0.3 K) SEs are mostly scattered by ripplons
whose wave vectors q k
2 are much smaller than typical
wave-vectors of thermal ripplons. The ripplon-limited
conductivity was observed by measuring the power ab-
sorption in a parallel AC electric field [4], the plasmon
resonance [5] and cyclotron resonance (CR) [6]
broadening.
© Denis Konstantinov, Yuriy Monarkha, and Kimitoshi Kono, 2008
Combined photoresonance and mobility measure-
ments [7] were performed as an interesting indirect probe
of SE Rydberg states. These measurements had shown
that in the case of liquid 3He at T � 0 35. K, electron mobil-
ity decreases at the MW resonance for low powers, but the
effect changes the sign for higher powers. The reduction
in electron conductivity was attributed to electron-vapor
atom scattering, even though this explanation was in con-
tradiction with results of similar quantum CR measure-
ments [6] which display an increase in SE conductivity.
Of course, the comparison with quantum CR data could
be misleading because in the presence of a strong mag-
netic field SE conductivity is affected by the many-elec-
tron effect which can lead to a substantial narrowing of
the CR [8,9].
Recent conductivity measurements for SEs exposed to
resonant MW radiation above liquid 3He in the vapor
atom scattering regime [10] indicate that heating of the
electron system is the key occurrence of a MW resonance
experiment with SEs. This heating occurs because of
electron decay from the excited Rydberg level to the
ground level induced by interaction with vapor atoms.
Because the mass of a helium atom M is much larger than
the free electron mass me , the energy difference between
the two levels is not given out but is transferred to the ki-
netic energy of electron motion along the interface. Under
the resonance condition the effective electron tempera-
ture Te can be much higher than the ambient temperature
T . Such decay heating of SEs appears already at very low
excitation rates affecting electron conductivity [10] and
the MW resonance linewidth [11]. The important point is
that in the vapor atom scattering regime, SE conductivity
at the MW resonance steadily decreases with power, and
there is no the sign change of this effect at high excitation
powers similar to that reported previously [7]. At high
powers, there is only saturation of the conductivity de-
crease caused by the quantum saturation of the fractional
occupancy of the first excited level n2 approaching that of
the ground level n1. Therefore new studies of SE conduc-
tivity affected by photoresonance are necessary for un-
derstanding these interesting phenomena. An additional
interest in such studies is evoked by a possible use of elec-
trons occupying the two lowest SE Rydberg levels as
electronic qubits controlled by MW radiation (for a recent
review of this problem, see Ref. 12).
In this work we report the results of experimental and
theoretical investigations of SE conductivity changes in-
duced by MW resonance excitation for liquid 3He in a low
temperature range (T � 0 5. K) covering both vapor atom
and ripplon scattering regimes. At T � 0.35 K and low
MW powers, we observed an increase in mobility of SEs
which is opposite to the result reported previously [7].
This discrepancy is explained by our theoretical analysis,
which indicates that experimental results obtained at
T � 0 35. K can be understood only if electron–ripplon in-
teraction is taken into account. We found that the sign of
the conductivity change induced by the photoresonance
crucially depends on the range of the vertical (holding)
electric fields E� used for tuning of the resonant fre-
quency. For weak holding fields, E� � 3 V/cm, corre-
sponding to the conditions of Ref. 7, conductivity indeed
decreases at the MW resonance, if the excitation power is
low. For holding fields used in our measurements,
E� � 93 V/cm, in the same power range, decay heating
leads to the opposite effect — SE conductivity limited by
ripplons increases at the photoresonance. Because the
conductivity changes induced by MW radiation occur un-
der conditions of the linear transport regime, such hot
electrons could be used for probing the electron coupling
with surface excitations of Fermi-liquid 3He.
2. Experiment
Since DC current measurements are practically impos-
sible for SEs on liquid helium, the capacitive detection
method is frequently used for measuring conductivity
[13,14]. In magnetoconductivity studies, this method em-
ploys an electrode array in the form of a Corbino disk usu-
ally placed below the helium surface. In our case, the
Corbino electrodes were placed above the surface, as in-
dicated in Fig. 1. A positive voltage VB was applied to the
bottom electrode to hold electrons. The parallel metal
electrodes were separated by 2 6 01. .� mm. To confine
electrons laterally, each electrode was surrounded by a
guard ring charged negatively (not shown in Fig. 1). To
improve the heat contact between liquid 3He and the cell
body, the bottom part of the cell contained a sintered sil-
ver heat exchanger. The cell was placed inside of the
superconducting magnet to create magnetic field B di-
rected perpendicular to the helium surface. One electrode
of the Corbino array was driven with an AC voltage in-
Photoresonance and conductivity of surface electrons on liquid 3He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 471
d
e3
He
VIN
IOUT
a
b
c
VB
Fig. 1. Schematic drawing of the experimental setup: micro-
wave waveguides (a), Corbino electrodes (b), surface electrons
(c), bottom electrode (d), heat exchanger (e).
ducing a current in the electron layer. The measured cur-
rent was analyzed by means of the transmission line mo-
del, which allows to extract � xx data using a conventional
relationship [15].
In the cell, electrons were accumulated on the va-
por-liquid 3He interface placed in the middle between the
parallel metal electrodes. At typical helium temperatures,
practically all electrons occupy the ground surface level
( l �1). In our measurements the areal electron density
ns � � �17 10 7 2. cm . To excite electrons to the second sur-
face level (l � 2), MW radiation of a fixed frequency
( /� �2 130� GHz) was arranged to be passing through the
cell, as shown in Fig. 1 . Owing to the Stark effect, the SE
system was tuned in resonance with MW radiation by
varying the holding electric field E� . A constant mag-
netic field was applied, and the Corbino signal was re-
corded as E� was swept through the resonance. In order
to avoid complications with many electron effects affect-
ing quantum magnetotransport of SEs on liquid helium,
we employed only a weak magnetic field for which the
magnetoconductivity can be described by the classical
Drude equation
�
�
� �
xx
s
e c
e n
m
�
�
2
2 2
, (1)
where �c is the cyclotron frequency, and � is the collision
frequency describing momentum relaxation of SEs. For
example, at T � 0.35 K, the magnetic field B was about
233 G which is within the semi-classical transport regime
(��c T�� ). Of course, the frequency � of the AC voltage
VIN is too much lower than �c and � to be taken into ac-
count in the Drude eqbation. Extracting � from the
magnetoconductivity data we can judge of the DC con-
ductivity of SEs in the absence of the magnetic field
� �� e n ms e
2 / and compare its variations induced by MW
with the old results of Ref. 7.
The MW resonance itself can be described by the usual
Lorentzian form determining the stimulated absorption
(emission) rate [16]
r
E
12
2
2
21
2
0 5
�
� � �
.
[ ( )]
� �
� � �
, (2)
where �21 2 1� �( ) /
� is the resonant frequency de-
pending on the holding electric field, � is the half-width
calculated previously in Ref. 17, � � � �eE zMW 1 2| | / � is
the Rabi frequency, EMW is the MW field amplitude,
� �1 2| |z is the electric dipole length for the transition. For
low excitation, the energy absorbed from the MW field is
proportional to ��21 12r . Since � ��� 21, the excitation
rate and energy absorption as functions of E� have a
sharp resonance structure when �21( )E� is close to the
MW frequency �.
Variations in electron collision frequency �( )E� ob-
tained from our magnetoconductivity data, as described
above, are shown in Fig. 2. At T � 0 48. K, the electron col-
lision frequency has a typical resonance structure with a
maximum positioned at E� � 93 V/cm. Thus, under the
condition, the MW resonance excitation leads to a de-
crease in SE conductivity, as expected for the vapor atom
scattering regime [10]. In this case, the increase in the col-
lision rate is caused by decay heating of SEs and their oc-
cupation of higher surface levels, where the inter-level
collision rate is high.
The recording signal greatly changes, if the ambient
temperature T � 0 35. K which corresponds to the condi-
tions and results of Ref. 7 discussed in the Introduction.
The most important conclusion which comes out from
Fig. 2 for this case is that MW radiation leads to an in-
crease in conductivity (� �� e n ms
2 / ) at low powers. This
is opposite to the result of Ref. 7. The resonance line in
472 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Denis Konstantinov, Yuriy Monarkha, and Kimitoshi Kono
18
17
16
15
88 90 92 94 96 98
88 90 92 94 96 98
88 90 92 94 96 98
T = 0.48 K
T = 0.35 K
T = 0.3 K
2.45
2.40
2.35
2.30
2.25
2.20
1.40
1.35
1.30
1.25
1.20
1.15
1.10
E , V/cm�
�,
1
0
s
8
–
1
�,
1
0
s
8
–
1
�,
1
0
s
8
–
1
Fig. 2. Variations of the effective collision frequency with E�
induced by MW radiation and measured at three typical ambi-
ent temperatures.
not of the Lorentzian shape and the effect changes the
sigh as E� approaching the resonance value. It is clear
that the sign change of the effect is due to the contribution
from vapor atom scattering. Therefore, the conductivity
increase in the region of resonance tails is, obviously, due
to electron–ripplon scattering. This conclusion is con-
firmed also by the data obtained at T � 0.3 K, when vapor
atom scattering is substantially reduced. For such condi-
tions, the dependence �( )E� has a usual resonance shape,
but the sign of its variations with E� is opposite to that
obtained in the vapor atom scattering regime (T � 0 48. K).
Scattering by vapor atoms only slightly affects the reso-
nance curve near its minimum. Since our new low temper-
ature data conflict with the data reported in Ref. 7, the
situation should be clarified by an appropriate theoretical
analysis.
3. Theory
3.1. Decay heating
For the vapor atom scattering regime, it is proven that
even a very small fraction of SE excited by the MW from
the first to the second surface level can substantially heat
the entire electron system [10,18]. This happens because
the decay rate of electrons occupying the second surface
level � 21
1� is determined mostly by electron interaction
with heavy vapor atoms. Therefore, a return of an elec-
tron back to the ground level is accompanied by a very
small energy exchange between an electron and a vapor
atom. Thus, practically all the excitation energy
21 (here
and below
ll l l� �� � ) is transferred to the kinetic en-
ergy of electron motion along the interface which is even-
tually redistributed between other electrons because of
electron–electron collisions. At the same time, the energy
relaxation rate ~� for electron–atom scattering is much
lower than the decay rate of the first excited level � 21
1� .
This leads to a strong increase in electron temperature
even at very low excitation rates r12 21
310� � � .
Determination of electron temperature requires the
knowledge of the electron energy relaxation rate due to
interaction with scatterers for arbitrary level occupancies.
Decay heating leads to electron occupation of higher sur-
face levels, therefore outer levels (l � 2) cannot be disre-
garded in the expression for the energy relaxation rate
[18]
~ exp [ (| | ) / ]( )
,
� �a
e
a l
l l
l l l l e
m
M
n T� � � �
�
� ��0 2
� � �
�
�
��
�
!!
"
#
$
%
&
'�
�
�
R
e
l l
l l
e
l l
T
u
T
s
| |
2 , (3)
where
R e Bm a� �
2 22/ , and n N Nl l e� / are fractional
occupancies of surface levels. We used
�a
e a am n V
B
( )0
2
3
11
�
�
,
s
B
B
B dz f z f zl l
l l
l l l l�
�
�
�
�� � (11 1
0
2, [ ( ) ( )] , (4)
u
a B
C
C dz
d
dz
f z f zl l
B
l l
l l l l�
�
�
�
�� � )
*
+
,
-
.(
2
11 1
0
, [ ( ) ( )]
2
, (5)
where na is the density of vapor atoms, f zl ( ) are electron
wavefunctions describing surface states, Va is the ampli-
tude of the pseudo-potential Va e a/( )R R� frequently
used for description of the electron-atom interaction, and
�a
( )0 is the momentum relaxation rate of SEs for scattering
within the ground surface level (l l� � �1). In terms of ~�a
the power transfer can be written as ( )~ ( )T T Te a e� � .
For electron–ripplon scattering, energy relaxation is
much more complicated. Scattering processes which in-
volve only one ripplon also can be considered as
quasi-elastic, because the energy exchange ��q is very
small for q k
2 . Energy relaxation is much more effec-
tive, when electrons emit couples of shortwave ripplons
with small total wave vectors (| |q + q� �� q) [19]. In this
case, the energy exchange 2��q at a scattering event is of
the order of Te and the energy transfer rate to the environ-
ment increases by about two orders of magnitude. Be-
cause of the strong repulsion barrier existing at the inter-
face V0, the largest contribution to the energy relaxation
rate is given from the nonlinear interaction term [2]
V z
V
z
e
int
( )
( )
( , ) ( )
2
2 0
2
21
2
r r�
0
0
1 , (6)
where Ve
( )0 is the electron potential for the flat interface,
and 1( )r is the surface displacement operator. The matrix
elements of V
int
( )2
describe two-ripplon scattering proba-
bilities already in the second order of the perturbation
theory.
The matrix elements g V zll e ll� �2 0 00 5 2 0 2. ( / )( ) [here
( )� ll� means � � �l l| |� ] can be expressed in terms of the
derivative of the electronic wave functions �f zl ( ) at the
helium surface (z � 0) calculated in the approximation
V0 � . This results in
g V f f
z z
ll l l
ll l
�
�
�� � �
0
0
�
�
�
�
!
0
0
�
�
�
�
!� 0 0
1
00 03 3( ) ( )
v v
� �l
, (7)
where 30
1
02� � � / m Ve is the penetration depth of the
electron wavefunction into liquid, and v( ) /z z� � ��
� �eE z. Since ( / )0 0v z ll is finite in the limiting case
V0 � , the matrix elements g Vll� 4 0 .
Following the previously described procedure [20],
the energy loss of SEs per unit time can be presented in
Photoresonance and conductivity of surface electrons on liquid 3He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 473
the conventional form � ~ ( )( )� � � �N T T Te r e e�2 with the
energy relaxation rate defined as
~
( )
| | ( )�
�5 �2 2
2
0
3
2
2
1r
e l l
ll l
q
q
m
T T
g n dq
q
N�
�
� �
�
�
� (
� � � �
"
#
$
%
&
'
)
*
6
+6
,
-
6
.6
�� �1
2n
n T T T
T Tl
l
l l
e
q
e
eexp ( )
��
� �
� � �"
#
$
%
&
'
� �
exp
| |
l l q l l q
eT
2 2
2
� �� �
, (8)
where N q is the ripplon distribution function and 5 is the
liquid helium mass density. A detailed analysis of Eq. (8)
indicates that this expression is positive. For the
Boltzmann distribution of level occupancies nl and
l l� � �1, this expression transforms into the result ob-
tained in Ref. 20.
Equation (8) shows that for two-ripplon emission
within the surface levels (l l� � ), typical ripplons contrib-
uted to ~�2r belong to the short wavelength range
2��q eT� . In the case of liquid 3He, the behavior of the
surface excitation spectrum in this range is unknown. It is
not known also how the strong damping of ripplons ex-
pected at low temperatures affects the scattering probabil-
ities. Anyway, experimental data indicate that the strong
damping of capillary waves does not affect much
one-ripplon scattering probabilities which determine the
momentum relaxation of SEs. Therefore, in our numerical
evaluations of ~�2r we shall disregard damping effects and
assume that even in the short wavelength range the
ripplon spectrum coincides with the capillary wave
asymptote � 7 5q q� / /3 2.
The electron temperature is determined by the energy
balance equation
( ) ( )~( )n n r T T Te e1 2 21 12� � �
� , (9)
where ~( )� Te is the total energy relaxation rate due to va-
por atoms and ripplons. At the same time, fractional occu-
pancies nl should be obtained from the rate equations
dn dtl / � 0 which ensure the balance between electron
transition to and from surface levels caused by scatterers
and the MW field. In the case of electron-vapor atom in-
teraction, scattering frequencies which enter the rate
equations for level occupancies have a very simple form
w s T
l l
a
a ll l l l l e� � � � �� � �( ) ( ) [ (| | ) / ]� 0 2exp
. (10)
For transitions down the surface levels (l l� �and
l l� � 0),
the scattering rate w
l l
a
� �
( )
does not depend on Te , while for
scattering up (l l� �,
l l� � 0), the scattering rate w
l l
a
� �
( )
ac-
quires an additional exponential factor e
� �
l l eT/
.
If electron–ripplon scattering dominates, at T � 01. K,
the scattering frequencies are determined mostly by
one-ripplon processes. Collecting corresponding proba-
bilities found in the framework of the usual perturbation
treatment, the scattering rates can be presented in the
following form
w
T
T
d
l U l
l l
r
e
q
q q
q� �
� � � � �(( )
| | | |
1
0
2
4 �7
�
� ��
� � � �exp [ ( ) / ( )]� �q l l q eT
2 4 , (11)
where � q eq m� �
2 2 2/ . The matrix elements of the elect-
ron–ripplon couplingU q can be written as [2] � � � �l U z lq| ( )|
� �� �( )eE Fq ll ll , where
eE z q
qz
K qz
qz
q ( )
( )
( )
� �
"
#
$
$
%
&
'
'
� 2
2
11
, (12)
F eEll ll ll� � � �� �/ 8 ,
8 ll
e
ll
e
l l
V
z m
f f
z
�
�
�� �
0
0
�
�
�
�
�
!
!
� � � �
0
0
�( )
( ) ( )
0 2
2
0 0
� v
�
�
�
!
�ll
, (13)
and K x1( ) is the modified Bessel function of the second
kind. It is obvious that w w T
l l
r
l l
r
ll e�� � � �� �( ) ( )
exp ( / )
1 1
.
Since 8 ll � 0, diagonal matrix elements of U q are deter-
mined only by the holding field term eE� and the term
eEq which originates from the polarization interaction
with oscillating liquid. For off-diagonal terms, 8 ll� sub-
stitutes the holding field term which turns to zero.
At T � 0.1 K, the decay rate of excited SE states is de-
termined mostly by quasielastic one-ripplon scattering,
while the energy relaxation is mostly due to inelastic
two-ripplon scattering discussed above. Electron scatter-
ing induced by one-ripplon processes limits the lifetime
of the first excited Rydberg level and transfers the energy
difference
21 into the kinetic energy of electron motion
along the interface. Then electron–electron collisions
which have the highest rate redistribute it among other
electrons forming a nondegenerate distribution of the
in-plane momentum with an effective electron tempera-
ture T Te � . With much lower rate (about 10 6 s–1) all the
electrons transfer the energy to the environment emitting
couples of short wavelength ripplons. Therefore, decay
heating of SEs at the MW resonance is expected in the
ripplon scattering regime as well. At much lower temper-
atures, the one-ripplon contribution to the decay rate of
excited surface levels freezes out and one have to take
into account two-ripplon emission processes [21] which
start to limit the lifetime of excited surface levels. Still,
this limiting case, which requires a separate examination,
is beyond the conditions of the experiments discussed
above.
To obtain the electron temperature at the MW reso-
nance (� �� 21 ) as a function of the Rabi frequency, we
474 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Denis Konstantinov, Yuriy Monarkha, and Kimitoshi Kono
found numerical solutions of the energy balance equation
and the rate equations for nl under different experimental
conditions. Typical dependence Te ( )� at the resonance
maximum is shown in Fig. 3. In these calculations, the
wave functions of the first 10 levels were obtained by
means of numerical solution of the corresponding
Schrodinger equation, while the next 190 levels were ap-
proximately described by Airy functions. At the begin-
ning, with an increase in the Rabi frequency, Te increases
fast. Then, at high excitations (r12 21 1� � ), electron tem-
perature saturates due to the saturation of the level occu-
pancy of the first excited level n n2 1� . Figure 3 indi-
cates that electron temperatures of about 10 K can be
reached at the saturation condition. For a fixed �, elec-
t ron tempera ture decreases wi th | ( )|� �� �21 E in
accordance with Eq. (2), because it reduces the excitation
rate r12.
3.2. Conductivity of hot electrons
It is clear that decay heating of SEs is a very important
effect induced by the MW resonance. In the vicinity of the
resonance one can have ultra-hot electrons covering the
cold surface of liquid helium (T Te �� ). For weak driving
electric fields, conductivity of such hot electrons can be
described by the linear transport theory. Additionally, in
this system, electron–electron collisions are usually much
higher than other relaxation rates which allows us to sim-
plify the conductivity treatment. To obtain the effective
collision frequency �which enters the Drude conductivity
equation we can just evaluate the kinetic friction acting
on the whole electron system, assuming that in the mov-
ing reference frame, the electron liquid can be described
by the equilibrium dynamical structure factor [2]. Gener-
a l ly, the kinet ic f r ic t ion can be represented as
F Vfric � �N me e a�
v
, where Vav
is the average velocity.
For the vapor atom scattering regime, the effective col-
lision frequency which determines DC conductivity
( / )� �� e n ms e
2 has a simple form [18]
� �( ) ( )
,
exp
| |a a
l l
l l
l
l l l l
e
s n
T
� �
��
�
��
�
!!�
�
� ��0 2
1
2
�
�"
#
$
%
&
'
� �| |
l l l l
eT
.
(14)
There are two important factors which cause changes in
� ( )a , if SE are heated. The first factor is the electron occu-
pation of high surface levels which increases inter-level
scattering and the momentum relaxation rate � ( )a . The
second factor is a decrease in matrix elements sl l� with
level numbers which acts in the opposite way. It is re-
markable that the outcome of the competition of these two
factors strongly depends on the holding electric field.
For typical holding fields used in our experiments
(E� �100 V/cm), the first factor dominates and the eff-
ective collision frequency � ( )a increases with � with-
out an observable sign of a decrease [10]. Nevertheless,
our new calculations conducted for weak holding fields
( .E� � 315 V/cm) related to the experiment of Ref. 7 re-
sults in a more complicated behavior of � ( )( )a � shown in
Fig. 4. At low powers, � ( )a decreases with the MW exci-
tation by about 4%, and then we have the sign change of
the effect. This decrease is induced by the reduction in
matrix elements sl l� whose magnitude depends strongly
on E� . Thus, for weak holding fields scattering by vapor
atoms should be reduced by MW radiation at low powers,
which is opposite to the mobility decrease at the pho-
toresonance reported previously [7]. This means that va-
por atom scattering cannot be the origin of that mobility
decrease.
The ripplon contribution to the momentum collision
frequency of SEs can be obtained similarly to Eq. (11)
Photoresonance and conductivity of surface electrons on liquid 3He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 475
10
1
T
,
K
e
T = 0.35 K
10
–5
10
–4
10
–3
10
–2
10
–1
10
0
�, GHz
Fig. 3. Electron temperature at the MW resonance vs the Rabi
frequency calculated for T � 035. K and E� � 93 V/cm.
2.4
2.2
2.0
1.8
1.6
10
–3
10
–2
10
–1
10
1
10
0
�, GHz
v,
1
0
s
9
–
1
T = 0.5 K , E = 3.15 V/cm�
Fig. 4. Collision frequency at the MW resonance vs the Rabi
frequency calculated for the vapor atom scattering regime.
�
�7
�
�
( ) | | ( )| |r
e
l
l
q
q l
q e
T
T
n
d
l U z l� � � � �� ( �
�4
0
2
�
�
�
�
�"
#
$
$
%
&
'
'
� �( )
exp
( )�
�
�
�
q l l
q e
q l l
q eT T
4 4
2
. (15)
The additional factor of the integrand containing
( )� q l l� �
originates from the factor qk which usually en-
ters the equation for the momentum relaxation rate. In our
evaluations of matrix elements � � �l U z lq e| ( )| , we shall use
the variational form of SE wave functions f zl ( ) �
� �A zP z b zl l l( ) exp [ / ]2 , where Al is the normalization
constant, P zl ( ) is a polynomial of power l �1. We shall use
also the following notations b b bll l l� �� �( ) / 2.
To understand the origin of strange variations in � ( )r
induced by decay heating, it is convenient to rearrange the
collision rate of electrons using the specific form of elec-
tron-ripplon coupling and the Boltzmann approximation
for fractional occupancies n nl l
B� ( )
. The later works well
for low and medium excitations and fails under the satura-
tion condition [18]. In this approximation, using � ( )r �
� � l l
B
l
r
n
( ) ( )� , we introduce the collision frequency of
electrons occupying a level l,
�
7l
r
e
ll
l
l l l l
e
T
T
F
T
( )
exp
| |
� �
�"
#
$
%
&
' ��
�
� ��4 2
2
�
� �
�"
#
$
$
%
&
'
'
�
�
�(�4 4
2
1 2
0
2TT
a
x
x T
x
e
B l
l l e
�7�
/ exp
( / )
� �
"
#
$
$
%
&
'�
�
�
� �
xW x
T F a
T
W x
T
ll
e
b
ll B
e
ll
e
bll ll
2 4
2
4
( ) ( )
� � '
dx, (16)
where functions W yll� ( ) are defined by
W y
A A
b
x P x b P x bll
l l
ll
l ll l ll�
�
�
� � �� �(( ) exp ( ) ( / ) ( / )
2
3
0
�
� �
"
#
$
$
%
&
'
'
1 1
y
x
K x y
y
dx
( )
, (17)
and � bll �
is defined similar to � q with bll� standing for q.
Usually, low temperature asymptotes of W yll� ( ) valid for
y �� 1 are used for specific evaluations, such as mobility
calculations [22] and the nonlinear conductivity studies
[23]. For example, in this approximation, W y11( ) �
� 0 5 4 1. ln ( / )y � . In our treatment of hot electrons, we
cannot restrict ourselves to this approximation because
4Te can be of the order of � bll �
.
In spite of the cumbersome form of Eq. (16), the main
features of �
l
r( )
can be seen quite easily. In contrast with
�
0
( )a
, the electron–ripplon collision frequency for elec-
tron scattering within the ground level depends strongly
on Te . For l l� � �1, the first term of Eq. (16), representing
the holding field term eE� of U q , decreases with elect-
ron temperature as Te
�1. The second term of Eq. (16) has a
more complicated dependence on Te . Nevertheless, it is
quite clear that for warm SEs the term whose integrand
containsWll�
2 increases with electron temperature approxi-
mately as T Te b eln ( / )2
1
� , because the low temperature
asymptote ofW y11( ) given above has only logarithmic de-
pendence on y. At higher Te , when 4 1
1
� b eT/ � , we can
simplify W y y11 1 3( ) /9 and the second term becomes
approximately independent of Te . With further increase
in Te , when the argument ofW11 becomes large, even this
term start decreasing with electron temperature. Thus, we
conclude that the integral effect of decay heating on � ( )r
depends strongly on the magnitude of the holding electric
field E� . For zero or a weak value of E� , the collision
rate starts to increase with warming of SEs, then attains a
maximum and starts to decrease. If the holding field is
sufficiently strong, the collision rate decreases with heat-
ing even for warm SEs (Te � T).
4. Results and discussions
The momentum collision rate � ( )r given by Eqs. (15)
and (16) has a very complicated form, which is not conve-
nient for evaluations when many surface levels should be
taken into account. In this case, there are two options: one
can simplify the electron–ripplon coupling, or take into
account only a very restricted number of surface levels. In
this work, we consider only 1 and 3-level models with the
exact form of U q . The wave functions of surface levels
are found according to the conventional variational
principle.
For electron–vapor atom scattering, the momentum re-
laxation rate given in Eq. (14) is much more simple, and,
therefore, it can be evaluated for a large number of sur-
face levels sufficient for the convergence of the result, if
we approximate higher levels (l � 3) by the corresponding
Airy functions. This allows us to take into account up to
400 surface levels. The chosen approximation is expected
to describe well the conductivity of hot SEs for strong
holding fields (E� � 100 V/cm). For weak holding fields,
it can describe well only the conductivity of warm and not
very much heated electrons (Te � 2 K).
The numerical evaluations of � ( )( )a
eT and � ( )( )r
eT
support the qualitative analysis of the momentum relax-
ation rate given at the end of preceding Section. Since the
MW power used in our experiments is much lower than
that providing the electron temperature saturation, we ex-
pect that the Boltzmann distribution of level occupancies
is a reasonable approximation for evaluation of � ( )( )r
eT .
For the conditions of the experiment of Ref. 7 (E� �
= 3.15 V/cm, T � 0.35 K), the effective collision fre-
quency calculated employing different models is shown
476 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Denis Konstantinov, Yuriy Monarkha, and Kimitoshi Kono
in Fig. 5. Though, at T Te � , the electron–ripplon colli-
sion rate is substantially lower than that induced by vapor
atoms, it depends on electron temperature even at low
MW powers, where � ( )a is approximately constant. Cal-
culations conducted for the 1-level model result in a rapid
increase in � ( )r with Te (dotted line), which turns into a
slow decrease at Te � 13. K.
Taking into account more surface levels affects mostly
the decreasing part of the dependence � ( )( )r
eT which be-
comes sharper as shown by the dash-dotted line. The total
collision frequency � � �� �( ) ( )a r of the 3-level model as
a function of Te preserves the conductivity maximum
(dash-dot-dotted line) and decreases strongly at Te � 1 K.
Even the inclusion of 400 levels in calculations of � ( )a
(solid line) does not affect the maximum of the total colli-
sion frequency (here the contribution � ( )( )r
eT is still cal-
culated for the 3-level model).
The above given theoretical results explain the mobil-
ity decrease at the photoresonance previously observed
for low powers [7]. Even numerically, the change in the
collision frequency at the maximum which is about 3%
(Fig. 5) agrees well with typical changes of SE mobility
observed in the experiment. Regarding the reversing of
the sign of the effect observed at high powers, it agrees
with the 3-level model considered here (dash-dot-dotted
line), though this model cannot be applied for Te � 12. K.
Therefore, this property should be verified by an all-level
treatment which is very difficult for such a weak holding
electric field. The existence of the collision frequency
maximum at Te � 0 88. K leads to a complication of the
shape of the conductivity resonance (�( )E� ) induced by
MW radiation of intermediate powers, which also agrees
with observations [7].
In our experiments, the holding electric field is much
stronger than that used in Ref. 7: E� � 93 V/cm. For such
a field, at Te � 0 48. K, scattering by vapor atoms is domi-
nant, and, as shown previously [10,18], the correspond-
ing collision frequency of SEs increases steadily with
heating. The experimental data relevant to this tempera-
ture and shown by the first data line of Fig. 2 are consis-
tent with this conclusion. The photoresonance decreases
the conductivity of SEs.
For the ambient temperature T � 0 35. K, the depend-
ence �( )Te becomes completely different, as indicated in
Fig. 6. At first, consider the contribution from the
electron–ripplon scattering only (lines R). For the models
of the SE system analyzed here, the momentum collision
frequency � ( )r mostly decreases with heating. The pla-
teau feature which is seen for the 3-level model at T2 2�
K remains remarkably even in models taking into account
larger numbers of surface levels which we do not discuss
in this publication. At the same time, the contribution
from vapor atoms (line A) calculated for the all-level
treatment increases steadily with Te .
Photoresonance and conductivity of surface electrons on liquid 3He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 477
1.1
1.0
0.9
0.8
0.7
0.4
0.3
0.2
0.1
1 2 3 4 5
T , Ke
Total
3LA
R
1L
3L
T = 0.35 K , E = 3.15 V/cm��
�
(T
)/
(T
)
e
Fig. 5. Momentum collision frequency vs Te normalized by
�( ) .T � 152 108� s
–1
: contribution from scattering by ripplons
(R) for 1-level (1L dotted line) and 3-level (3L dash-dotted
line) models, contribution from vapor atoms (A) obtained in an
all-level treatment, total �( )Te for the 3-level model (dash-dot-
dotted line) and for the all-level treatment of vapor–atom scat-
tering as described in the text (solid line).
1.2
1.0
0.8
0.6
0.4
0.2
T , Ke
Total 3L
A
R
1L
3L
T = 0.35 K , E = 93 V/cm�
2 4 6 8 10
�
�
(T
)/
(T
)
e
Fig. 6. Momentum collision frequency vs Te calculated at
E� � 93 V/cm and T � 035. K: contribution from scattering by
ripplons (R) for 1-level (1L dotted line), and 3-level (3L dash-
dotted line) models, contribution from vapor atoms (A) obtain-
ed in an all-level treatment, total � � �� �( ) ( )a r for the 3-level
model (dash-dot-dotted line) and for the all-level treatment of
vapor-atom scattering (solid line) as described in the text.
For the total collision frequency shown by the solid
line, the contribution � ( )a was calculated using the
all-level treatment, while � ( )r was taken from the 3-level
analysis, as described above. Thus, instead of the maxi-
mum found above for E� � 315. V/cm, we have a mini-
mum of the momentum relaxation rate. This minimum af-
fects the shape of the conductivity resonance induced by
the MW in the way which agrees with the corresponding
data shown in Fig. 2. Even numerically, the decrease in �
at the minimum is about 12% which is close to that ob-
tained in our measurements. At high electron tempera-
tures, we have a reversing of the sign of the effect which
is opposite to that previously observed for the weak hold-
ing field: at low MW excitation powers, SE mobility in-
creases with Te , while at high powers it decreases.
At even lower T � 0 3. K, the contribution from vapor
atom scattering is substantial for high MW excitations as
shown in Fig. 7. The total collision frequency normalized
decreases stronger than at T � 0 35. K, attains a minimum
of about 0.76, and then very slowly increases reaching
only 0.8 at Te �10 K. In this case, we conclude that the
shape of the conductivity resonance at medium powers
should be similar to the Lorentzian shape, slightly af-
fected at the maximum (the minimum of �), as it is for the
third data line in Fig. 2. According to the solid line of
Fig. 7, vapor atoms restrict the total reduction in � and it
should not exceed 24%. The observed reduction in � as
shown in Fig. 2, is about 17% which is consistent with our
calculations.
It should be noted that the absolute value of the mo-
mentum relaxation rate of SEs on liquid 3He previously
measured employing different experimental setups [7,24]
used to differ from the theoretical value by a factor of 2.
Our new data of � obtained by measuring � xx under a
weak magnetic field providing the semi-classical trans-
port regime are in much better agreement with the theory.
For example, the out-of-resonance data shown in Fig. 2
give: � � 15 3 10 8. � s–1 (T � 0 48. K), 2 4 10 8. � s–1 (T �
= 0.35 K), and135 10 8. � s–1 (T � 0 3. K). The corresponding
theoretical values for the total collision frequency are rea-
sonably close: � � 16 10 8� s–1 (T � 0 48. K), 2 6 10 8. � s–1
(T � 0.35 K), and 146 10 8. � s–1 (T � 0 3. K). Thus, our new
equilibrium conductivity data obtained for SEs on liquid
3He and the theory agree even numerically with the accu-
racy of about 7.5% in both the ripplon and vapor atom
scattering regimes.
5. Conclusions
SEs on liquid helium exposed to resonance MW radia-
tion represent an interesting system of hot electrons
whose temperature can be much higher than the ambient
temperature. The transport properties of such hot elec-
trons along the interface can be described by the linear
transport theory which allows to use them for probing the
electron coupling with surface excitations of quantum liq-
uids. At low ambient temperatures, the energy relaxation
time of hot electrons is expected to be limited by emission
of couples of short wavelength ripplons (��q eT� / 2),
which potentially could be used for experimental study of
the spectrum of surface excitations of liquid 3He in a high
energy range (up to about 5 K).
Our measurements of the conductivity resonance in-
duced by MW radiation show that decay heating of SEs
previously reported for scattering by vapor atoms remains
to be an important factor in the ripplon scattering regime
as well. It changes the momentum collision rate for elec-
tron scattering within the ground Rydberg level and leads
to electron occupation of higher levels. At certain condi-
tions, the interplay between scattering by ripplons and
scattering by vapor atoms affects strongly the shape of the
conductivity resonance and leads to a reversing of the
sign of the effect induced by MW radiation. Surprisingly,
this reversing of the sign of the effect is opposite to that
reported previously [7]. The theoretical analysis of SE re-
laxation rates in the presence of MW radiation given here
indicates that conductivity changes induced by MW radi-
ation are very sensitive to the magnitude of the MW fre-
quency which determines the range of holding electric
fields used for tuning to the resonance. This analysis ex-
plains well the conductivity variations observed, and
eliminates the discrepancy between data obtained for
different experimental setups.
Acknowledgments
The work is partly supported by the Grant-in-Aids for
Scientific Research from Monka-sho.
478 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Denis Konstantinov, Yuriy Monarkha, and Kimitoshi Kono
1.0
0.8
0.6
0.4
0.2
T , Ke
�
�
(T
)/
(T
)
e
Total
3L
A
R
1L
3L
T = 0.3 K , E = 93 V/cm�
2 4 6 8 10
Fig. 7. Momentum collision frequency vs Te calculated at
E� � 93 V/cm and T � 03. K. Notations are the same as in Fig. 6.
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