Bistability and Hysteresis of Intersubband Absorption in Strongly Interacting Electrons on Liquid Helium

0.53 MB·English

by  Denis Konstantinov

Bistability and Hysteresis of Intersubband Absorption in Strongly Interacting
Electrons on Liquid Helium
Denis Konstantinov,1,2 M. I. Dykman,3 M. J. Lea,4 Yu. P. Monarkha,5 and K. Kono2
1Okinawa Institute of Science and Technology, Okinawa 904-0495, Japan
2 Low Temperature Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
3 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
4Department of Physics, Royal Holloway, University of London, TW20 0EX, United Kingdom
5 Institute for Low Temperature Physics and Engineering, Kharkov 61103, Ukraine
2
(Dated: January 31, 2012)
1
0 We study nonlinear inter-subband microwave absorption of electrons bound to the liquid helium
2 surface. Alreadyforacomparativelylow radiation intensity,resonant absorption duetotransitions
n between the two lowest subbands is accompanied by electron overheating. The overheating results
a in a significant population of higher subbands. The Coulomb interaction between electrons causes
J a shift of the resonant frequency, which depends on the population of the excited states and thus
7 on the electron temperature Te. The latter is determined experimentally from the electron pho-
2 toconductivity. The experimentally established relationship between the frequency shift and Te is
in reasonable agreement with the theory. The dependence of the shift on the radiation intensity

] introducesnonlinearityintotherateoftheinter-subbandabsorptionresultinginbistabilityandhys-
ll teresis of theresonant response. The hysteresisof theresponse explainsthebehavior in theregime
a of frequency modulation, which we observe for electrons on liquid 3He and which was previously
h seen for electrons on liquid 4He.
-
s
e PACSnumbers: 67.90.+z,73.20.-r,78.70.Gq,42.65.Pc
m
.
t I. INTRODUCTION modes, ripplons.11–14
a
m The recent interest in resonant absorption was stimu-
- Free electrons outside liquid helium can be trapped at lated by the proposal to use laterally confined electrons
d the vapor-liquid interface owing to the attractive polar- on the helium surface as qubits for quantum computing,
n izationpotentialabovethesurfaceandarepulsivebarrier with the qubit states being the out-of-plane states with
o at the surface, which prevents electrons from entering α = 1 and 2.15,16 Resonant interaction with an exter-
c
[ the liquid.1–3 As a result, the electron motion perpen- nally applied microwave field provides a natural way to
dicular to the helium surface is quantized, with energies manipulatethequbitstate. Couplingbetweenthequbits
1 ǫ =−R/α2 (α=1,2,..),whereRistheeffectiveRydberg comes from the Coulomb interaction between the elec-
v α
energy,R≈7.6Kforliquid4HeandR≈4.2Kforliquid trons. The interaction energy depends on the electron
0
1 3He. An externalelectric field E⊥ applied perpendicular quantum numbers α, because the electron distance from
9 to the surface Stark-shifts the energy levels. The shift the surface depends on α. Therefore, the change of the
5 depends on the quantum number α. Consequently, the quantum state of one electron results in the shifts of the
. level spacing can be varied by changing E . If there are quantum levels of other electrons.
1 ⊥
0 no other confining fields the electrons are free to move In microwave absorption experiments with electrons
2 along the surface and their overallenergy spectrum con- that are free to move along the surface, one would ex-
1 sists of two-dimensional (2D) subbands. pect to see a shift in the resonance frequency for the
v: Resonantinter-subbandopticalabsorptionduetotran- transition from the ground to the first excited sub-
i sitions between the subbands has been directly observed band(andalsoinothertransitionfrequencies)depending
X
for surface electrons (SEs) above liquid 4He using ra- on the population of excited subbands. The subband-
ar diation in the millimeter4,5 and far-infrared6 ranges. population dependent shifts of the energy levels due to
Recently resonant absorption of millimeter-range mi- many-electron effects were predicted earlier for semicon-
crowaves was also observed for electrons on liquid 3He.7 ductorheterostructures.17,18Insuchsystemstheelectron
In the experiments, SEs were tuned in resonance with correlationeffectsareaperturbation,becausethesystem
radiation by adjusting the value of E and optical is close to metallic.
⊥
absorption was detected as the variation in radiation A distinctive feature of the electron system on he-
power passing through the system. Inter-subband ab- lium is that it is strongly correlated.2,3,19–22 Therefore
sorption has been also probed indirectly by observing a the frequency shifts turn out to be much stronger than
changeintheconductivityofirradiatedSEsonboth4He in semiconductors. Signs of the dependence of the fre-
and 3He.8,9 This change is mainly caused by radiation- quency shift on the state of the system were seen earlier
induced overheating of the electron system.10 The over- in a complicated response to radiation with modulated
heating occurs both where electrons are scattered pri- frequency23 andthe power-dependentshift of the photo-
marily by helium vapor atoms or by surface vibrational conductivity signal.12 However, no systematic studies of
2
these effects have been carried out. II. THEORETICAL BACKGROUND
The state-dependent shift of resonance frequency in- A. Qualitative picture of electron dynamics
troduces nonlinearity in the absorption rate. Such non-
linearity can cause bistability and hysteresis of resonant
Intheneglectoftheelectron-electroninteraction,elec-
response. Hysteretic effects related to nonlinear absorp-
tronmotionnormaltotheheliumsurfaceisdescribedby
tion are familiar already from the early work on ferro-
the Schr¨odinger equation
magnetic resonance in ferrite crystal disks.24–26 For 2D
systems, magnetization bistability was observed in spin- ~2 d2
polarizedatomichydrogengasabsorbedonthesuperfluid − 2mdz2ψα(z)+V(z)ψα(z)=ǫαψα(z), (1)
helium film.27. For semiconductor heterostructures, the
nonlinearity of inter-subband absorption was predicted where z is the out-of-plane coordinate, ψ (z) and ǫ
α α
andstudiedsincelate80s28–33 andabsorptionbistability are the eigenfunctions and eigenvalues (α = 1,2,...),
has been long sought.34–37 m is the electron mass, and V(z) is the electron po-
tential energy, V(z) = −Λ/z +eE z for z > 0. Here,
⊥
In this work, we present theoretical and experimen- Λ=e2(ε−1)/4(ε+1),−eisthe electroncharge,εis the
tal studies of nonlinear absorption effects for a strongly dielectric constant of liquid helium, and E is the elec-
⊥
correlated electron system on liquid helium surface. A tric field applied normal to the surface. On the surface,
short account of the results appeared earlier.38 We pro- z =0,thepotentialV(z)hasahighstep∼1eV.Neglect-
vide a theory of the frequency shift associatedwith elec- ingpenetrationoftheelectronwavefunctionintotheliq-
tron correlations and show that this shift causes bista- uid,Eq.(1)canbesolvednumericallywiththeboundary
bility and hysteresis of resonant response. We describe condition ψ (0) = 0. For the field E < 102 V/cm, the
α ⊥
experimentalresultsonthefrequencyshiftandnonlinear typicallocalizationlengthofthelow-lyingelectronstates
absorptioneffects for electrons on liquid 3He in the tem- is r =~2/Λm≈0.8×10−6 cm−1 for 4He.
B
perature range 0.2−0.4 K that covers both vapor-atom Of central interest for this paper is electron response
andripplonscatteringregimes. Theresponseisobserved to a microwavefield which is resonantwith the α=1→
as a change of the longitudinal conductivity σxx of SEs α = 2 transition. This response depends on electron re-
in a weak perpendicular magnetic fields. The frequency laxation. To describe the relaxation we note that, for
shift is measured as a function of electron temperature electron densities n and temperatures T studied in ex-
s
Te, which is determined from the relative change of σxx periment,theelectron-electroninteractionisstrong. The
at resonance. The shift is compared with the theoretical interaction is characterized by the plasma parameter Γ
estimate and a reasonable agreement is found. and the characteristic plasma frequency ω ,
p
At high values of input microwave power, the absorp- Γ=e2(πn )1/2/k T, ω =(2πe2n3/2/m)1/2. (2)
s B p s
tion cross-sectionshows a jump as a function of E and
⊥
depends on the direction of sweeping E⊥, that is, dis- Mostofthe existing experimentalresultsonelectronson
plays hysteresis. This behavior agrees well with our the- helium refer to the range Γ ≫ 1. This means that the
oreticallypredictedbistabilityofnonlinearresponse. We electron system is strongly correlated, SEs form a non-
also describe the results of the experiment, in which res- degenerate liquid or, for Γ > 130, a Wigner solid; the
onance is observed by direct measurements of the mi- experiment described in this paper refers to the range
crowavepowerabsorbedbytheelectronsystem. Weshow where the system is a liquid. The reciprocal plasma fre-
that the complicated behavior of the response to radia- quencyω−1 givesthecharacteristictimeofinter-electron
p
tion with slowly modulated frequency can be explained in-plane momentum and energy exchange. In our exper-
bytakingintoaccountthenonlinearityoftheabsorption imentω >6×109 s−1. Thiswasthe shortestrelaxation
p
rate. time in the range of temperatures studied in our experi-
ment. Also, ω exceeded the microwave absorption rate,
p
In Sec. II we derive a many-body quantum kinetic see below.
equationforastronglycorrelatedsystemanduseittode- Thefastinter-electronenergyexchangeleadstoather-
scribenonlinearresonantabsorptionandtopredictbista- mal distribution over the in-plane electron energy, with
bility of resonant response. In Sec. III we describe the an effective electron temperature T . We note that the
e
experimentaltechniques. InSec.IVthe experimentalre- time it takes to form such distribution in a correlated
sults are presented. In Section V we discuss the results electron system can be longer than ω−1. For example,
p
ofthe measurementsandmakecomparisonwiththethe- if one thinks of electrons forming a Wigner crystal, this
ory. The experimentally observed nonlinear effects, in is the thermalization time of the phonons of the Wigner
particular the bistability and hysteresis of inter-subband crystal. Still this time is much shorter than the char-
absorption,are also discussed in this section. Section VI acteristicenergyrelaxationtimedue toinelasticelectron
provides a summary of the main results and the conclu- scatteringbyheliumvaporatomsorripplons/phononsin
sions. helium. Of major interest to us is the parameter range
3
where k T > ~ω . In this range electron in-plane mo- As we will see, this interaction leads to an ana-
B e p
tion is semi-classical and the electron in-plane state can log of the depolarization effect in semiconductor
be describedby well-defined momentum and coordinate. heterostructires.17,18,29–33 However, because SEs on he-
The temperature T should be the same in all sub- lium are strongly correlated the effect is much stronger
e
bands, even though the distribution over the subbands and is described in a qualitatively different way.
does not necessarily has to be thermal. This is a con- The electron coupling to a resonant microwave field is
sequence of the strong difference between the character- described by the Hamiltonian
istic in-plane and out-of-plane inter-electron distance in
the correlatedsystem, ns−1/2 and rB. Since ns−1/2 ≫rB, HF =−~ΩRcosωFt σn12+σn21 . (6)
the in-plane momentum exchange∆p is essentially the Xn (cid:0) (cid:1)
ee
samewhethertheelectronsareinthesameorindifferent
Here, Ω = eE z /~ is the Rabi frequency; E
subbands,leadingtothesamein-planemomentumdistri- R MW 12 MW
andω arethe amplitude andfrequencyoftheradiation
butionirrespectiveofthesubband. Atthesametime,the F
field, respectively, and z = hα|z|βi. Frequency ω is
electron-electron interaction practically does not lead to αβ F
assumed to be close to the transition frequency of the
inter-subbandtransitions,becauseitrequiresshort-range
electron system, ω =(ǫ −ǫ )/~, |ω −ω |≪ω .
collisions which do not occur in the correlated system. 21 2 1 21 F 21
Thenonlinearresponseofastronglycorrelatedsystem
Other scattering mechanisms come from the interac-
canbe analyzedusingthe many-electrondensity matrix.
tion of SEs with the surrounding, primarily with he-
Itdependsonthestatesofin-planeandout-of-planemo-
lium vapor atoms, for comparatively high temperatures,
tionofallelectrons. Itisconvenienttowrite itinthe in-
T > 0.3 K for electrons on 3He, and with surface
teraction representation using canonical transformation
waves and bulk excitations in liquid helium, for lower
U(t)=exp(−iH t), where
temperatures.3,15 Scattering by vapor atoms and by sur- t
face capillary waves, ripplons, is mostly quasi-elastic.
H =H +~(ω −ω ) σ22+H . (7)
This is a consequence, respectively, of the large ratio of t 0 F 21 n He
X
theheliumatommasstotheelectronmassandtheslow- n
ness of ripplons: even for the ripplon wave number as The term H in Eq. (7) describes excitations in liquid
largeasrB−1theripplonfrequencyisverysmall,∼108Hz, helium and tHhee almostideal gas formed by helium vapor
and the ripplon energy is much smaller than the average atoms. The transformed Hamiltonian of coupling to the
energy of in-plane electron motion. fieldU†H U isindependent oftime inthe rotatingwave
F
approximation. We emphasize that the electron-electron
interaction H(0) is incorporated into H , this is not a
B. Hamiltonian of the many-electron system ee t
perturbation.
We write the density matrix in the interaction rep-
Using the condition ns−1/2 ≫ rB, the Hamiltonian of resentation with Hamiltonian Ht using the Wigner
theisolatedelectronsystemcanbewrittenasH +H(1), representation with respect to in-plane motion as
0 ee
where the leading term is ρ (t;{α ,β ,r ,p }). Forweakcouplingtoripplonsand
0 n n n n
helium vapor atoms, one can obtain a many-electron ki-
1
H =H +H(0)+H , H = p2, netic equation for ρ . This equation is Markovian in
0 K ee z K 2mXn n time slow comparedt0o ~/kBTe,ω2−11 and presents an im-
e2 mediate extension to the multi-subband case of the ki-
He(e0) = 2 rn−n1′, Hz = ǫασnαα. (3) netic equation discussed previously in the one-subband
nX6=n′ Xn,α approximation.39
Because the Coulomb interaction is strong, to zeroth
Here, r and p ate lateral coordinate and momentum
n n
order in the interaction with excitations in helium and
of an nth electron, respectively, rnn′ ≡ |rn−rn′| is the in H(1) function ρ depends on the coordinates and mo-
in-plane inter-electron distance, and ee 0
menta of individual electrons in terms of the total in-
σαβ =|αi hβ|, (4) plane momentum P= p and the total in-plane en-
n nn n n
ergyH +H(0). ThedPistributionoverthetotalin-plane
where |αi ≡ [ψ ] is the αth out-of-plane state of the K ee
n α n energyisoftheBoltzmannform. Moreover,intheclassi-
nth electron.
cal range k T >~ω the distribution over momenta p
The part of the electron-electron interaction incorpo- B e p n
is also of the Maxwell form.
rated into H corresponds to the approximation of the
0 Coupling to helium vapor atoms and ripplons causes
electrons being in a plane. The part of the interaction
mixing of states with different P and also of different
related to the out-of-plane electron displacement is de-
out-of-plane states |αi . The coupling Hamiltonian can
scribed by He(e1). To the leading order in rB/rnn′ be written as n
e2
He(e1) =− 4 (zn−zn′)2/rn3n′. (5) Hi = Vˆqαβeiqrnσnαβ. (8)
nX6=n′ Xq,nXα,β
4
Here, Vˆ is an operator with respect to the variables of To find the smooth terms H(1)(t) we write in Eq. (5)
ee
ripplons or phonons in helium and the positions of the z = z σαβ (and similarly for z2) and notice that
n αβ αβ n n
helium vapor atoms, while ~q is the lateral momentum U†(t)σPαβU(t) ∝ exp(iω t). Here z = hα|z |βi ; the
n αβ αβ n n n
transferred from these excitations to an electron. Terms matrix elements of z2 are defined similarly. The inter-
n
with α = β describe intrasubband coupling, whereas subbandtransitionfrequenciesω largelyexceedthein-
αβ
terms with α 6= β describe mixing of states in different plane vibration frequencies, which are ∼ ω and charac-
p
subbands. terizetimevariationofthefactorsr−3 inH(1)(t). There-
Thecharacteristicvaluesofthetransferredmomentum nn′ ee
fore the first type of slowly varying matrix elements in
are comparatively large, q ≫n−1/2, because the density
s H(1)(t) are those containing only diagonal components
of states of excitations in helium and often the interac- ee
z σαα and (z2) σαα. The second type comes from
tion strengthincrease with q. Thus scatteringby helium αα n αα n
excitations is short-range. As a result, in the electron the terms ∝ znzn′ with n 6= n′. The corresponding
slowly varying terms are proportional to |z |2σαβσβα
liquid each electron is scattered individually, processes αβ n n′
with α6=β.
where two electrons are scattered by the same ripplon,
phonon, or a vapor atom can be disregarded.19,39 Since We will analyze the interaction He(e1) in the mean-field
between the scattering events the electrons have time to approximation. Thisapproximationisjustifiedwhenthe
exchangelateralmomentumandenergywitheachother, number of nearest neighbors for each electron is large,
the scatteringratescanbe averagedoverlateralelectron sothatcorrelationsbetweenout-of-planestatesofneigh-
motion with the Boltzmann factor. boring electrons andtheir positions are averagedout. In
other words, tracing over the positions of coupled elec-
Resonantresponseoftheelectronsystemtomicrowave
trons and over their out-of-plane states is done inde-
field is independent of the electron lateral motion. Of
pendently. In a strongly correlated 2D electron liquid
interestforthestudyofthisresponseisthesingle-particle
the number of nearest neighbors is 6, on average, which
density matrix
makes the approximation reasonable. For typical times
ontheorderofthedurationofacollisionwithheliumex-
ρ (t)=Tr σβαρˆ (t) =N−1 Tr σβαρˆ (t) . (9)
αβ n 0 n 0 citations or a radiation-inducedinter-subbandtransition
(cid:2) (cid:3) Xn (cid:2) (cid:3)
therelativeshiftofthepositionsofneighboringelectrons
is small. It is a good approximationthen to describe the
The traceistakenoverthe in-planecoordinatesandmo-
mentaandovertheout-of-planestatesofallelectrons,N distribution of inter-electron distances rnn′ by the static
is the total number of electrons. In a spatially uniform pair correlationfunction g(rnn′).
The many-electron frequency shift ∆ω of the
system ρ is independent of the subscript n in Eq. (9). βα
αβ
|αi → |βi transition is determined by the ratio
~−1Tr σαβ H(1),ρˆ /ρ . Fromtheabovearguments
n ee 0 βα
n (cid:2) (cid:3)o
C. Many-electron shift of the transition frequency
′
Tr σαβ r−3ρˆ(0) ≈Fn3/2ρ ,
n n Xn′ nn′ o s βα
onQrueasolintaatnitvealyb,sothrpetimonajoisr tehffeecltinoefatrhSetianrtkersahcitfitonofHte(he1e) TrnσnαβX′n′rn−n3′σnα′′β′ρˆ(0)o≈Fn3s/2ρβαρβ′α′, (10)
frequency of the inter-subband|1i→|2i transitionof an
where the prime over the sum indicates that n′ 6=n and
electron due to the out-of-plane component of the elec-
tric field created by other electrons. This field for an ′
F =n−3/2 r−3 . (11)
nth electron is given by e−1∂He(e1)/∂zn. As seen from s DXn′ nn′E
Eq. (5), the field depends on the relative distance be-
Numerically, in a strongly correlated liquid F ≈8.9140.
tween the electrons normal to the surface and therefore
From Eqs. (5) and (10) with account taken
onthestatesofout-of-planemotionofdifferentelectrons,
aswellasonthe inter-electrondistanceintheplane. We of the commutation relation σnαβ,σnα′′β′ =
h i
etrmopnh,ansoitzeinththatewaveeararegeinfiteelrdesitnedthienetlhecetrfioenldlaoynera,nwehliecch- δnn′ σnαβ′δβ′,α′ −σnα′βδα,β′ , the frequency shift of
(cid:16) (cid:17)
is significantly different. the |1i → |2i transition due to the electron-electron
Formally, the term H(1) couples a single-particle den- interaction is
ee
sitymatrix,inparticularthatgivenbyEq.(9), toatwo- Fe2n3/2
particleone. Thiscouplingisdescribedbytheterminthe ∆ω = s (z2) −(z2) −2 z −z
kinetic equation (∂ ρˆ ) = i~−1 ρˆ (t),H(1)(t) , where 21 2~ h 11 22 (cid:0) 11 22(cid:1)
t 0 ee h 0 ee i × zααραα+2|z12|2 ρ11−ρ22 . (12)
He(e1)(t) = U†(t)He(e1)U(t). The density matrix in the in- Xα (cid:0) (cid:1)i
teraction representationslowly varies in time. Therefore From Eq. (12), the frequency shift ∆ω is determined
21
the main contribution to ρˆ0 comes from slowly varying by the populations of all subbands ραα. In Eq. (12) we
terms in H(1)(t). dropped terms proportional to ρ ρ with α 6= 1,2.
ee 1α α2
5
These terms would be important if radiation excited
transitions other than |1i → |2i. However, such transi-
tionsarenonresonant,andforacomparativelysmallmi-
= 4 23
crowavepowerthatweconsidertheycanbedisregarded.
The effect of the electron correlations on the position
= 3
of the absorptionline was considered earlier by Lambert
and Richards6. Their results referred to the weak-power
limit, ραα ∝ δα,1, and the expression they were using = 2 21
differed from Eq. (12); in particular, the last term was
missing all together.
11
The physics of the terms proportional to the diago-
p
nal and off-diagonalmatrix elements of z, z2 in Eq. (12)
is different. The diagonal terms come from the fact = 1
that SEs in different out-of-plane states |αi have differ- FIG.1: (color online)Sketchoftheenergysubbandsandthe
ent static dipole moments in the out-of-plane direction. electron transitions. Vertical (red) arrows indicate photon-
These static moments lead to the static Stark shift of induced transitions, while horizontal (blue) arrows indicate
thetransitionfrequencyω ofotherelectrons,whichde- transitions duetoelastic scattering by excitations in helium.
21
pends onthe occupanciesρ . Thetermproportionalto
αα
|z |2 in Eq. (12) is dynamic in nature: it describes res-
12
onant excitation transfer between different electrons.15 is independent of the electron number n. In deriving it
we used the aforementioned picture in which electrons
are scattered by a short-range potential independently,
D. Kinetic equation that is one electron collides with a short-range scatterer
at a time. However, during a collision the electron is in
the in-plane potential created by other electrons, which
An equation for the single-electron density matrix for
out-of-planemotionραβ canbeobtainedfromthefullki- is determined by He(e0). This potential is smooth on the
netic equation for ρˆ0 in the interaction representation,41 electron wavelength λT and for kBTe ≫ ~ωp weakly af-
fects short-rangescattering in the absence of a magnetic
field.39
ρ˙ = (γ ρ −γ ρ )+Ω (δ −δ )Imρ ,
αα βα ββ αβ αα R α,1 α,2 12
X The electron-electron interaction plays a critical role
β
if the electron system is in a strong magnetic field B
1
ρ˙12 =−(iδω+γ0)ρ12+ iΩR(ρ22−ρ11), (13) normal to the surface. Here, in the single-electron ap-
2
proximationtheelectronenergyspectrumisdiscrete(the
where Landauquantization),andthe Bornapproximationused
to derive Eq. (15) would not apply. However, because
δω =ω −ω −∆ω . (14) an electron is in an a fluctuational electric field of other
F 21 21
electrons E , its potential (and thus kinetic) energy is
f
InEq.(13)wedonotincorporateexplicitly theelectron- uncertain by eE λ ∼ ~ω , and if this uncertainty ex-
f T p
electron interaction, which leads to establishing thermal ceeds~ω ,whereω =eB/mcisthecyclotronfrequency,
c c
distribution over lateral motion, with effective tempera- the effect of the Landauquantizationon scatteringis es-
ture Te. We also do not incorporate the relaxation pro- sentially eliminated.20 This condition was met in the ex-
cessesthatleadtoenergyexchangebetweentheelectrons periment, even though the magnetic field was classically
and the helium excitations. strong.
Parameters γαβ in Eq. (13) are the rates of inter- The processes described by Eq. (15) are shown
subband transitions |αi → |βi, which are essentially schematically in Fig. 1. The calculation of the rates γ
αβ
quasi-elastic. The electron momentum goes to helium is similar to that of the in-plane relaxationrate in a cor-
excitations whereas the energy of motion normal to the related electron system.39 Even where T 6= T, since the
e
heliumsurfacegoesintothe kinetic energyoflateralmo- scattering described by Eq. (15) is elastic and the elec-
tion and the energy of the electron-electron interaction, tron distribution is thermal, the transition rates satisfy
the detailed balance condition
∞
γ = ~−2 |V |2 dtexp[i(ǫ −ǫ )t/~]
αβ qαβ Z α β
Xq −∞ γαβ =γβαexp[(ǫα−ǫβ)/kBTe]. (16)
×hexp[iqr (t)]exp[−iqr (0)]i. (15)
n n
The decay rate γ of the off-diagonal matrix element
0
Here, the overline indicates averaging over the thermal ρ in Eq. (13) in the single-electron approximation for
12
distribution(with temperatureT)ofthe relevantexcita- T =T ≪(ǫ −ǫ )~wasconsideredinRef.42. Usingthe
e 2 1
tionsinhelium; h...idenotesaveragingoverthe in-plane approach of Ref. 39 the results can be extended to the
electron motion with temperature T . Expression (15) many-electron case and made applicable in the presence
e
6
of a magnetic field normal to the helium surface, 1, again resonantly absorb a photon, and ne scattered
into subbands 4, 5, etc.
1 1
γ = ( γ + γ )+ |V −V |2 The cascade of photon-induced transitions and inter-
0 1α 2α q11 q22
2 X X 2X subband scattering leads to population of higher and
α6=1 α6=2 q
∞ higher subbands and to a decrease of the population of
×~−2 dthexp[iqr (t)]exp[−iqr (0)]i.(17) subband 1. Such decrease, along with the fact that the
Z n n
−∞ populations of subbands 1 and 2 become closer with in-
creasing T , lead to an overall decrease of resonant ab-
The frequency detuning (14) depends on the distribu- e
sorption.
tion of SEs, cf. Eq. (12), which makes the overall quan-
tum kinetic equation (13) nonlinear. As shown below,
this nonlinearity may result in the bistability of the re-
F. Bistability of resonant response
sponse of the electron system to resonant radiation.
A complete analysis of nonlinear response requires in-
E. Absorption bleaching corporating inelastic processes that lead to energy ex-
change between the electrons and excitations in helium.
By now several types of inelastic processes have been
To find the stationary distribution of the system,
identified,includingtheweakinelasticityofscatteringby
Eq. (13) must be complemented with the equation that
helium vapor atoms, the inelastic two-ripplon scattering
describes electron energy relaxation. However, an im-
and the inelastic scattering by phonons in liquid helium.
portantconclusionaboutthedistributioncanbereached
For all of them, the energy relaxation rate depends on
evenwithouttheanalysisofenergyrelaxationinthecase
the electron state and helium temperature.3,14,16,43
ofweaktomoderatelystrongmicrowaveradiation,where
The description of energy relaxation is simplified by
Ω2 ≪γ γ . (18) the fact that it is slow. Between inelastic scattering
R 0 21
events electrons have time to exchange energy with each
Condition (18) is sufficient to suppress the conven- other, and the state of the electron system is described
tional absorption saturation. Such saturation is well- by electron temperature. Therefore one can introduce a
knownfor two levels systems andrequires that radiation single energy relaxation rate ν , which is independent
E
makes the populations of the excited and ground states of the electron state and depends only on T ,T, and
e
ρ and ρ close to each other, so that the probabilities n .3,10–12,14,41. This leads to an implicit equation for T .
22 11 s e
of radiation-induced transitions up and down in energy It has the form of a simple energy balance equation
become close, too. In the saturation regime the popu-
lation ratio ρ22/ρ11 significantly differs from the Boltz- νEkB(Te−T)=~ωFrZ−1[e−ǫ1/kBTe −e−ǫ2/kBTe],
mann factor exp[(ǫ −ǫ )/k T ]. In our case instead of 1
1 2 B e r = Ω2γ /[γ2+(ω −ω −∆ω )2]. (20)
the levels we have subbands of electron motion. It is 2 R 0 0 F 21 21
seen from Eq. (13) that in the range (18) the deviation
We have introduced here absorption coefficient r, which
of ρ /ρ from the Boltzmann factor is small even at
22 11 is determinedby Eqs.(13)and(14). The left- andright-
resonance,δω =0,and evenif one disregardstransitions
hand sides of Eq. (20) are, respectively, the energies dis-
between subbands 1, 2 and other subbands. This shows
sipated into helium and absorbed by an electron from
thatabsorptionsaturationdoes notoccurinour system.
radiation per unit time. For all energy relaxation mech-
FromEq.(13)andfromthedetailedbalancecondition
anisms considered so far, the left-hand side of Eq. (20)
(16)itfollowsthatintherange(18)theoverallstationary
is an increasing function of T −T, at least for not too
distribution of the electron system both over the energy e
large T −T.
of lateral motion and over the subbands of out-of-plane e
Parameterr isanonlinearfunctionofT ,asseenfrom
motion is characterizedby the same temperature T , e
e Eqs. (12) and (14). Therefore Eq. (20) is fairly compli-
ρ ≈Z−1e−ǫα/kBTe, Z = e−ǫα/kBTe. (19) cated. A convenient way of understanding the possible
αα types of its solution is based on the graphical solution,
X
α
an example of which is shown in Fig. 2. The plot shows
Even though there is no absorption saturation in the the rates of energy absorption and relaxation calculated
range (18), the absorptionsignificantly changeswith the for T = 0.4 K where the scattering is dominated by in-
increasing radiation intensity. The mechanism of this teraction with helium vapor atoms. In this regime, the
changeis absorption bleaching.12,41 Itcanbe understood T -dependence ofν , γ and∆ω are calculatedfor the
e E 0 21
from Fig. 1. As a result of inter-subband scattering an typical experimental parameters with account taken of
electron that resonantly absorbed a photon in subband 200 subbands of electron motion normal to the helium
1 and made a transition to subband 2 can go back to surface.
subband 1, but with higher energy. It can then again The left-hand side of Eq. (20) is shown in Fig. 2 by a
absorbaphoton. Nowitcanbeelasticallyscatteredfrom dashed line, while the right-hand side (rhs) is shown by
subband 2 to subband 3. It can also go back to subband thesolidlinesforω −ω =0.1and0.4GHz(linesaand
F 21
7
b,respectively). ThemajorpartofT -dependenceinthe onsetofbistabilityisageneralconsequenceofabsorption
e
rhscomesfromthefrequencyshift∆ω . FromEq.(12), bleaching.
21
∆ω rather quickly increases with the increasing T . In The occurrence of two stable regimes can be under-
21 e
a certain range of ω −ω , the change of T can bring stood in the following way. In one regime, which cor-
F 21 e
∆ω inresonancewithω −ω . Thisleadstoaresonant responds to lower T , the overall detuning of the elec-
21 F 21 e
peak of r and thus of the rhs of Eq. (20) as function of tron transition frequency ω +∆ω from the radiation
21 21
T seen in Fig . 2. frequency ω is large compared to γ . Therefore, the
e F 0
The heights of the resonantpeaks in Fig. 2 depend on absorption rate is comparatively small and Te is self-
ωF −ω21. From Eq. (20), the coefficient in front of r consistently low. On the other hand, if Te is sufficiently
decreases with increasing Te. The maximal value of r at high, ω21 +∆ω21 is close to ωF, absorption is compar-
resonance, rmax ∝ 1/γ0, also decreases. Therefore the atively strong and Te is self-consistently high. We note
higher is the value of T where ∆ω = ω −ω , the that, since ∆ω is positive, such bistability can occur
e 21 F 21 21
smaller is the height of the peak of the rhs of Eq. (20). only for ωF −ω21 >0.
The bistability leads to the hysteresis of electron ab-
sorptionwithvaryingω −ω . Ifwestartfromlargede-
F 21
0.2 tuningω −ω (curveAinFig.2)andthendecreaseit,
F 21
the system will stay in the small-absorptionstate, which
T=0.4 K
corresponds to left circle in Fig. 2. On the other hand,
)
s if we start from small ω −ω , the system will be on
g/ a F 21
r the decreasing with increasing T part of the resonant
e e
9 0 solid line in Fig. 2. As ωF −ω21 increases, the system
1 0.1 will get into the region of strong absorption, but ulti-
x
E ( matelywillswitchtothelow-absorptionbranchforlarge
T b ω −ω . Belowwepresenttheexperimentalobservation
A F 21
R of frequency shift, bistability, and hysteresis and make a
comparison with the theory.
0.0
0 2 4
III. EXPERIMENT
T -T (K)
e
The details of the experimental apparatus have been
FIG. 2: (color online) Graphical solution of the energy bal-
ance equation (20) for T = 0.4 K and ns = 4.0×107 cm2. describedelsewhere14. SEswereaccumulatedonthesur-
The dashed line is the energy loss rate [the left hand side of face of liquid 3He placed about half way between two
Eq. (20)] due to inelastic scattering by helium vapor atoms. round electrodes of diameter 26 mm separated by ap-
The solid lines show the rate of the microwave power ab- proximately 2.6 mm and forming a parallel-plate capac-
sorption [the right hand side of Eq. (20)] calculated for itor. A positive voltage V was applied to the bottom
B
ωF = 104.5 GHz, ΩR = 2.0 MHz, and ωF −ω21 = 0.1 GHz electrode to create an electric field E perpendicular to
⊥
(a) and 0.4 GHz (b). For the case (a), the stable solutions the helium surface. In addition, a magnetic field B was
of Eq. (20), which are given by the intersections of solid and
applied perpendicular to the surface using a supercon-
dashed lines, are marked by(red) circles.
ductingmagnetplacedaroundtheexperimentalcell. Mi-
crowaveradiationatfixedfrequencyω /2π=104.5GHz
F
The solutions of Eq. (20) are determined by the cross- was passed through the experimental cell, and the inter-
ings ofthe dashedand solidline in Fig.2. They givethe subband splitting ǫ −ǫ was brought in resonance with
2 1
effective temperature Te in the stationary state. Where ~ωF by varying E⊥.
the microwavefrequencyisrelativelyfarfromresonance, The top plate of the capacitor consisted of two
i.e., ωF −ω21 is relatively large,there exists only one so- concentric-ring electrodes (the Corbino disk geometry),
lutionforTe,withTe closetotheheliumtemperatureT. which were used to measure the longitudinal magneto-
Closertoresonance,ontheotherhand,Eq.(20)canhave conductivityoftheelectronsystemσ . TheCorbinosig-
xx
two stable solutions with comparatively large and small nal proportional to σ−1 was recorded while slowly vary-
xx
Te−T. TheyaremarkedbycirclesinFig.2. Inthiscase ingVB atafixedmicrowavepowerandatafixedvalueof
Eq. (20) also has a stationary solution for intermediate the magneticfieldB. The variationofσ causedbythe
xx
Te, as seen from Fig. 2. By writing the energy change radiation-induced heating of SEs allowed us to observe
perunittime asthedifferenceofthe right-andleft-hand resonant response from inter-subband absorption. Sub-
sides of Eq. (20), one can see that this intermediate so- tracting the value of the signal recorded in the absence
lution is unstable. of radiation, the relative change ∆σ−1/σ−1 due to the
xx xx
The left and right hand sides of Eq. (20) display a heating was determined and plotted vs V . The mea-
B
qualitatively similar behavior for other parameters and surements reported here were done in the temperature
for otherrelaxationmechanisms. This indicates thatthe range from 0.2 to 0.4 K.
8
For each curve ∆σ−1/σ−1 vs V , the electron tem- previously,10,14 in this case radiation-induced heating
xx xx B
peratureT wascalculatedfromthepreviouslydiscussed leads to a decrease of the relaxation time τ. Therefore
e
dependence on T of the quasi-elastic electron transport σ−1 should decrease at resonance.
e xx
timeτ.10,14FormagneticfieldsBusedintheexperiment,
the many-electron conductivity σ follows the Drude
xx
law,20
0.15 T=0.4 K
7 -2
σx−x1 =(1+µ2B2)/σ0, (21) k ns=4.0x10 cm
-1 xx g
where σ = n eµ is the conductivity in zero magnetic 0.10
0 s 1 / f
fieldandµ=eτ/mistheelectronmobility. ForµB ≫1, - xx
σ−1 is proportional to τ. To find the proportional- e
xx -
ity coefficient, which depends on the electron density 0.05 d
n , the Corbino signal was recorded without radiation c
s b
while SEs were slowly cooled down until they formed a a
Wigner crystal. The crystallization could be easily de- 0.00
tected as an abrupt change in σ .44 The value of n
xx s
was found from the crystallization temperature and the 11.2 11.6 12.0 12.4 12.8
well established critical value of the plasma parameter V (V)
B
Γ=e2(πn )1/2/k T ≈130.
s B
FIG. 3: Relative change of the reciprocal conductivity
Theexperimentalprocedurefordirectabsorptionmea- −∆σx−x1/σx−x1 vs the voltage applied to the bottom electrode
surements was similar to that described in Ref. 7. The VB obtained for SEs on liquid 3He at T = 0.4 K, ns =
powerofthemicrowaveradiationthatpassedthroughthe 4×107 cm−2 and several values of the input power of ra-
cell was measured with an InSb bolometer mounted in- diation. The lines a to k correspond to -17.5, -14.0, -11.4,
sidethecryostat. Resonantabsorptioncouldbeobserved -8.8, -6.2, -4.2, -2.3and -0.6 dBof theinputpowerexpressed
as the variation of the bolometer signal at T =0.4 K as as a fraction (in decibel) of themaximum power ∼1 mW.
electronsweredriventhroughresonancebysweepingthe
bottom plate voltage V . Simultaneously, this voltage
B TheresistivitycurvesinFig.3showapronouncedres-
wasmodulatedwithasmallsinusoidalsignalatfrequency
onant peak. The height of the peak monotonically in-
10 kHz, and the in-phase and quadrature components of
creases with the input power. At small power levels, the
the demodulated bolometer signalwereobtained using a
resistivity shows a slight increase on the sides of the res-
dual-phaselock-inamplifier. Withsuchanarrangement,
onance (respectively, −∆σ−1 becomes negative). This
in the limit of small modulation amplitude the in-phase xx
effect is attributed to the contribution of ripplon scat-
component of the obtained signal is proportional to the
tering to the momentum relaxationrate.14 As the power
derivative of the power absorbed by SEs with respect to
increases,the resonanceshiftstowardlowervaluesofV ,
B
V . Normally, the quadrature component of the signal
B which corresponds to the increase in the transition fre-
should be zero. The absorption lineshape was obtained
quency. Also, the lineshape becomes asymmetric. We
bynumericalintegrationofthesignalwithrespecttoV .
B believe the shift is due to the many-electron effect de-
scribed by Eq. (12) and is associatedwith thermal pop-
ulation of the excited subbands as the effective electron
IV. RESULTS
temperature increases.
At high power levels, when the voltage is swept
A. Resistivity measurements through the resonance from the low-V side, the re-
B
sistivity jumps abruptly to a lower value; respectively,
In Fig. 3 we show the relative radiation-induced con- −∆σ−1/σ−1 jumpsup. Thisunusualbehaviorisdemon-
xx xx
ductivitychange−∆σ−1/σ−1asafunctionofthevoltage strated in Fig. 4. The data used for this plot were taken
xx xx
that presses the electrons to the helium surface V . All at similar conditions to those in Fig. 3, except that the
B
plotswereobtainedfromthedatatakenatT =0.4Kand sweeping rate was slower by a factor of two. The jump
electrondensityn =4×107cm−2. Theradiationpower always occurs on the low-field side of the resonance and
s
was measured at the output of the microwave source; is well reproducible. When the sweeping direction is
showninFig.3isthe ratioofthe appliedpower(indeci- reversed, the resistivity curve first follows the forward-
bel) to the maximum power, which was approximately sweepingcurveas itpassesthroughthe resonance. How-
1 mW. Experimental traces were obtained by increasing ever,onthe low-fieldsideit doesnotshowanyjump but
VB slowly to drive SEs through resonance. A conversion slowly goes to zero with decreasing VB. This behavior
of the increase of V to the increase of the transition is also well reproducible. Therefore, overall the system
B
frequency ω by ≈25 MHz is discussed in Sec. V A. displays a well-pronounced hysteresis.
21
AtT =0.4K,electronscatteringismostlydetermined Theresistivityjumpbecomes moreprominentandthe
by collisions with helium vapor atoms. As we showed width of the hysteresis loop increases at lower tempera-
9

0.15 T=0.4 K T=0.2 K
7 -2 7 -2
ns=4.0x10 cm 0.2 ns=4.2x10 cm
-1 xx 0.10 -1 xx cb c
-1 /xx 0 dB -1 /xx a ba
0.1
- 0.05
"satellite"
0.00
0.0
10.5 11.0 11.5 12.0 12.5 13.0 10.5 11.0 11.5 12.0 12.5 13.0
VB (V) VB (V)
FIG. 4: Relative change of the reciprocal conductivity FIG. 5: Relative change of the reciprocal conductivity
−∆σx−x1/σx−x1 vsVB obtainedatT =0.4K,ns =4×107 cm−2 ∆σx−x1/σx−x1 vsVB obtainedatT =0.2K,ns =4.2×107 cm−2
andatthemaximuminputpower(0dB).Thedashedandsolid and three different values of the input power: -6.2, -5.4 and
lines correspond to sweeping through theresonance from the -4.2dB(linesa, bandc,respectively). Thedashedandsolid
low and high side of VB, respectively, as indicated by the ar- lines correspond tothe sweeping through theresonance from
rows. thelow-andhigh-VBsides,respectively. Asatellitepeakthat
appears on thelow-field sideof theresonance is indicated by
thefull triangle.
tures. In Fig. 5 we plot ∆σ−1/σ−1 vs V for the data
xx xx B
taken at T = 0.2 K, n = 4.2×107 cm−2, and at three
s
different input power levels. At this temperature, the
relaxation rate is determined by the interaction of the
electrons with ripplons, and electron heating leads to an
itanetcrrereessiasosneoabonfsceterhv.e1e4rdeIlnaatxaadTtdioi=tniot0ni.m4teoKτt,h.etThaheberlriuenfpeotsrhejuaσpmx−exp1siahnnocdwreshaysaesns- 0.3 Tns==08..20 xK106 cm-2 -1-1 /xxxx0.2
1 0.1
interesting structure. It seems to be a superposition of - xx 0.2
a broad peak centered at VB ≈ 11.9 V and a high peak -1 /x 0.0
locatedonthe low-V side ofthe broadpeak. While this x 11.3 11.4 11.5
B
latter peak grows and shifts with the increasing input VB (V)
0.1
power, the position of the broader peak does not change
andthe heightofthe peak slightlydecreaseswith thein-
creasing power. We identify the high peak as being due
to resonanttransitionsfromthe groundstate to the first 0.0
excited state. The broader peak remains unexplained at 10.5 11.0 11.5 12.0 12.5 13.0
this time.
V (V)
B
Inadditiontothepeaksdescribedabove,asmallsatel-
lite peak appears on the low-field side of the resonance. FIG.6: ∆σx−x1/σx−x1 vsVB forT =0.2K,ns =8.0×106 cm−2,
and -2.3 dB of the input power. The dashed and solid lines
This peak becomes visible only at very high power lev-
correspond to the sweeping through the resonance from the
els. It was reported in our previous work9 and was at-
low and high VB side, respectively, as indicated by arrows.
tributed to the excitation of collective in-plane electron
Inset: plots obtained under the same conditions, but with
vibrations.45 At extremely high radiation intensities, we
theslower bya factor of two rate of sweeping from thelower
observe a number of satellite peaks on the both sides (opened squares) and the higher(solid squares) VB side.
of the resonance. Some of them can be due to reso-
nant transitions between higher excited subbands and
emerge through self-sustained absorption, which was re-
cently discovered in our experiments.46 The origin of
other peaks is not clear at the moment. A detailed de- peak is significantly smaller, while the higher peak is
scription of the satellite structure of the resonance will much narrower. Interestingly, for lower electron density
be given elsewhere. the signal shows abrupt jumps for both sweeping direc-
The signal strongly changes at lower electron den- tions. Ithasacharacteristicsawtoothshapeexpectedfor
sity. In Fig. 6 we plot ∆σ−1/σ−1 vs V for the data switching between two stable states.The abrupt switch-
xx xx B
taken at the same temperature as in Fig. 5 but at n = ing for sweeping the voltage up and down is highlighted
s
8.0×106 cm−2. Here, the intensity (area) of the broad in the inset of Fig. 6.
10

B. Absorption measurements
A
We have also studied absorption directly by measur- 1.0
)
ingthepowerofthe microwaveradiationpassedthrough V
the electron system. The experiment was done by si- (
tnhuesocidenatllryalmpoodinutlaVt¯iBngabtohuetpwohteicnhtiatlheVBmoadnudlastwioenepwinags I 0.0 bac
performed. The measured transmitted power was then T=0.4 K d
7 -2
Fourier-transformedusingalock-inamplifier. Unexpect- -1.0 ns=4.0x10 cm e
edly we found that, in the absorbed power, along with

the standard in-phase component I(V¯ ), for high radia-
B )
tion power the signal displayed a quadrature component V B
Q(V¯B). The data were obtained for the modulation fre- Q ( 0.0 ab
quency ω /2π=10 kHz and the modulation amplitude
m c
Vm = 10 mV, which corresponds to the modulation am- -0.5 ed
plitude ≈ 25 MHz of the transition frequency ω (see
21
11.0 11.5 12.0 12.5 13.0
below). For modulation ∝ cosω t we define, following
m
the standard notations, the quadrature component as a VB (V)
coefficient at −sinω t.
m FIG. 7: Integrated lineshape of the absorption signal mea-
In Fig. 7 we show functions suredwithSEsonliquid3HeforT =0.4K,ns =4×107cm−2
atdifferentpowerlevels. Thegraphshowsthein-phase(panel
VB VB
I(V )= dV′I(V′), Q(V )= dV′Q(V′), A)andthequadrature(panelB)componentsofthemeasured
B Z B B B Z B B signal. Thelinesatoecorrespondto-27.7,-20.6,-17.5,-14.4,
(22) and -11.4 dBof theinput power attenuation.
sometimes called integrated lineshapes. They were ob-
tained by sweeping V¯ upward for T = 0.4 K and
B
ns = 4×107 cm−2 at different power levels. The lower frequency shift, ωF −ω21 = ∆ω21 with ∆ω21(Te) given
limitsoftheintegralsinEq.(22)werechosenatavalueof by Eq. 12. The position of the resonance depends on
V¯B wellbelowthe resonance;the resultweaklydepended the electron temperature Te. In turn, Te depends on the
onthislimit,asitisclearfromFig.7. ForT =0.4K the absorbed power and for fixed incident power is expected
intrinsic linewidth of the absorption spectrum is rather to be maximal at resonance. In the parameter ranges
smallandtheobservedlineshapeisdeterminedbythein- usedinourexperimentσ−1variesmonotonicallywithT .
xx e
homogeneousbroadeningduetothenonuniformityofthe Therefore we assume that the maxima of the resistivity
electric field at the helium surface as well as the nonuni- curves plotted in Fig. 3 correspond to exact resonance.
formity of the microwavefield7,23. Then, for eachpower level, the frequency shift ∆ω can
21
As expected, for low radiation power the quadrature be found from the shift of the maximum of the corre-
component of the absorption signal is zero. However, at sponding curve with respect to the resonant absorption
higher powers hysteresis occurs within each modulation curvemeasuredatverylowpowerusingtheInSbbolome-
cycle itself. This gives a distinctive quadrature compo- ter. For this curve one can assume T = T, and then in
e
nent in the modulated output, see Sec. V. The effective Eq.(12) for ∆ω one can set ρ =δ . Generally, the
21 αα α,1
lineshape which is obtained from the integrated modu- parameters z in Eq. (12) depend on the pressing field
αβ
lation signal then has a quadrature component and an E andthusonV ;however,thisdependence issmooth,
⊥ B
offset in both components. Note that the quadrature and in the narrow range of V that we studied it could
B
component itself is not hysteretic - it is the same sweep- be disregarded.
ing the voltage up or down. But this component gives a To find the shift in frequency units, the frequency of
sensitiveindicationoflocalhysteresisineachmodulation the microwave source was changed by a known amount
cycle. The features of the integrated lineshape become and the corresponding shift in V was recorded. Such
B
more prominent with the increasing power. For lower T, measurementsweredoneatlowpowersothatthemany-
the quadraturecomponentis seenfor lowerpowerlevels. electronshiftremainedconstant. Thisprocedureallowed
ustoestablishtherelationshipbetweenω andV . The
21 B
conversion factor, i.e., the slope of ω /2π vs V , was
21 B
V. DISCUSSION found to be 2.5±0.1 GHz/V.
InFig.8weplotthemany-electronfrequencyshiftand
A. The many-electron frequency shift theelectrontemperaturedeterminedforthemaximumof
eachcurveinFig.3. WeremindthatT wasobtainedby
e
The resonance condition for exciting an electron from comparing the measured σ−1 with its theoretical value
xx
the ground to the first excited level of motion normal to for given n and T. The frequency shift is counted off
s
the helium surface is determined by the many-electron from its value for T = T. It monotonically increases
e