Table Of ContentCyclotron-resonance-induced negative dc conductivity in a two-dimensional electron
system on liquid helium
Yu.P. Monarkha
Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkov, Ukraine
We theoretically predict instability of a zero-dc-current state of the two-dimensional electron
system formed on the surface of liquid helium induced by the cyclotron resonance (CR). This
conclusion follows from the theoretical analysis of the dc magnetoconductivity which takes into
5
account the contribution from radiation in an exact way. A many-electron model of the dynamic
1
structurefactor of the2D Coulomb liquid isused to describe theinfluenceof stronginternal forces
0
2 actingbetweenelectrons. Forlowelectrondensitiesandhighamplitudesofthemicrowavefield,the
dc magnetoconductivity is shown to become negative in the vicinity of the CR which causes the
n
instability. This effect is strongly suppressed by Coulomb forces in theregion of high densities.
a
J
PACSnumbers: 73.40.-c,73.20.-r,73.25.+i,78.70.Gq
0
2
Microwave-induced resistance oscillations and zero- on liquid helium was recently found in studies of the
] resistance states (ZRS) observed in a two-dimensional Coulombic effect onpositions ofconductivity extrema14.
l
al (2D) electron gas subjected to a transverse magnetic The most frequently discussed mechanism of negative
h field1–3 represent a surprising discovery in condensed conductivity effects called the displacement mechanism
- matter physics. A number of theoretical mechanisms wasproposedalreadyin1969byRyzhii15. Inthismodel,
s
e have been proposed to explain these oscillations and anelectronscatteringbyanimpuritypotentialaccompa-
m ZRS4–8. Still, by now the origin of the phenomenon is nyingbyabsorptionofaphotonistheoriginofthenega-
. controversial. A crucial result9, independent of the de- tivelinearresponseconductivity. Inrecentdevelopments
at tails of the microscopic mechanism, is that the ZRS can ofthemodel16,17,thecontributionfromradiationistaken
m be explainedby anassumptionthat the longitudinallin- into account exactly which is important for high MW
ear response conductivity σ is negative in appropri- powers. Even though, in semiconductor systems, other
- xx
d ate ranges of the magnetic field B. In this case, a zero- mechanisms reportedly18,19 could give a stronger effect,
n current state is unstable and the system spontaneously it is very attractive to study the displacement model for
o
develops a nonvanishing local current density. the 2D electron system on liquid helium.
c
[ The above noted phenomena were observed in high In the displacement model, the strength of the effect
quality GaAs/AlGaAs heterostructures. There is an- dependsontheproductoftwodimensionlessparameters:
1
v other extremely clean 2D electron system formed on
7 the surface of liquid helium which exhibits remarkable λ= eEa(0c)lB, κ = ωc2 , (1)
2 quantum magnetotransport phenomena10 although it is ~ω ω2 ω2
7 anondegeneratesystemofstronglyinteractingelectrons. | − c|
4 Since electron gas degeneracy is not a crucial point where E(0) is the amplitude of the MW field, and l =
0 ac B
. for some of the theoretical mechanisms explaining ZRS ~c/eB is the magnetic length. For a fixed ratio ω/ωc,
1 in semiconductor systems, these mechanisms potentially the value of κ is the same in both semiconductor and
0 canbeappliedtosurfaceelectrons(SEs)onliquidhelium SpE systems. The effective mass of SEs is very close to
5
as well. Complementary studies of these phenomena in the free electron mass m , while the effective mass of
1 e
v: tthifiecasytisotnemofotfhSeEosriogninlioqfuZidRhSeolibusmervceodulidnhGealpAws/iAthlGidaeAns- sTehmeirceofonrdeu,cattorfiexleedctEro(n0)siasnmduωc,hisnmeaxlpleerr:imme∗en≃ts0w.0it6h4SmEes.
i ac
X heterostructures. onliquidheliumλissmallerthanitis forsemiconductor
ar It should be noted that another kind of magnetocon- electronsbythefactor m∗e/me ≃1/4. Thereductionof
ductivity oscillations and ZRS (zero-σ states to be ex- λcanbe wellcompensatedbyapproachingthecyclotron
act) was alreadyobservedin the systemxxof SEs on liquid resonance (CR) conditpion ω ω, which increases κ.
c
→
helium11,12 when the energy of excitation of the second Thus, for SEs onliquid helium, negativedc conductivity
surface subband (∆ ∆ ) was tuned to the resonance (similar to that of the semiconductor model) could be
2 1
−
with the microwave (MW) frequency. These phenom- expected in the vicinity of the CR. Since the average
ena were explained13 by nonequilibrium population of Coulomb interaction energy per SE U is usually much
C
the second surface subband which triggers quasi-elastic larger than the average kinetic energy, a many-electron
intersubband decay processes accompanying by electron treatment of the displacement model is required.
scattering against or along the driving force, depending In this work, we report results of a theory of the dis-
on the ratio (∆ ∆ )/~ω (here ω is the cyclotron placementmechanismofnegativedcconductivityapplied
2 1 c c
−
frequency). An important evidence for identification of to SEs on liquid helium interacting with capillary-wave
the mechanism of oscillations and ZRS observed for SEs quanta (ripplons). The ac electric field is taken into ac-
2
countinanexactwaysimilartoRefs.16and17. Scatter- dεe−ε/Tegn(ε)gn′(ε+~Ω), (5)
ing with ripplons is described using a perturbation the-
Z
ory. Strong Coulomb interaction is taken into account
where g (ε) = ImG (ε) represents the Landau level
employing a model based on the dynamic structure fac- n − n
density of states, G (ε) is the single-electron Green’s
tor (DSF) of a 2D electronliquid. We found that results n
function, x =q2l2/2,
ofthemany-electrontreatmentdrasticallydependonSE q B
density n , which allows us to predict the range of n
s s
whWereewnielglautsievethceoLnadnudctaiuvigtyauegffeeicntswchainchbaemobosmerevnetdu.min In,n′(x)=smmainx((nn,,nn′′))!!x|n′2−n|e−x2Lm|ni′n−(nn,|n′)(x), (6)
y direction (p ) is a good quantum number. The dc and
y
ac electric fields (E and E ) are taken to be parallel and Lm(x) are the associated Laguerre polynomials.
dc ac n
to the x-axis. The ac field E (t) = E(0)cosωt. In the This representation is similar to that of the theory of
ac ac
absence of scatterers, the exact solution of the single- thermal neutron (or X-ray) scattering by solids, where
electron Hamiltonian can be written as16,20 the scattering crosssectionofa particle flux is expressed
in terms of the DSF of the target.
ψ (x,y,t)=eiϑ(x,y,t)exp iXβsinωt Comparing with the case Ea(0c) = 0, matrix elements,
n,X
(cid:26) lB (cid:27) × describingelectronscattering,containtheadditionalfac-
tor exp( iq l βsinωt). Using the expansion eizsinφ =
y B
J (−z)eimφ [here J (z) is the Bessel function], the
exp iXy/l2 ϕ (x X ξ(t),t), (2) m m m
× − B n − − procedure of finding scattering probabilities can be re-
P
where (cid:0) (cid:1) duced to a quite usual treatment. Then, for ripplon
creation (index +) and destruction (index ) processes,
−
w¯(±)(V ) can be found as
cp eE ω2 q H
X = y dc , β =λ c , (3)
−eB − meωc2 (ω2−ωc2) w¯(±)(V )= C¯q± 2 ∞ J2 (βq l )
q H ~2 m y B ×
ϕn(x,t)isthewell-knownsolutionfortheunforcedquan- (cid:12)(cid:12) (cid:12)(cid:12) mX=−∞
tumharmonicoscillator,nistheLandaulevelindex,and
ξ(t) is the classical solution of the forced harmonic os-
S(q, q V +mω ω ), (7)
cillator, ξ =eEa(0c)cosωt/m ω2−ωc2 . The resonant de- − y H ∓ r,q
nominator of β originates from ξ˙. The exact expression 1/2
(cid:0) (cid:1) where C¯± = V Q N(r)+1/2 1/2 , N(r) is the
forϑ(x,y,t)isnotimportantinthe followingtreatment. q r,q q q ± q
The interaction with ripplons causes electron scatter- ripplon distribution fuhnction, Vr,q is theielectron-ripplon
ing between different states given in Eq. (2). The dc coupling10,Q = ~q/2ρω ,ω α/ρq3/2,αandρ
q r,q r,q
≃
conductivity σ of SEs can be found from the equation arethe surfacetensionandmassdensity ofliquidhelium
xx
for current density j = en l2 q w¯ , where w¯ is respectively. p p
x − s B q y q q
the average probability of electron scattering with the Generally, the structure of Eq. (7) is similar to that
momentum exchange ~q, and l2qPrepresents a change found for other scattering mechanisms important for
B y
of the orbit center number X for such a process. The semiconductor electrons16,17,21. The main advantage of
effective collision frequency ν , entering the usual con- the form of Eq. (7) is that we can employ the properties
eff
ductivity form, is found as ofthe equilibrium DSF andmodel the effect of Coulomb
interaction using the DSF of strongly interacting SEs.
1
νeff =−meVH q ~qyw¯q(VH), (4) S(huecrheaaipsoasstiybpiliictaylealpepcteraorns supnadceinrgt)hwehciocnhdaitllioownslBto≪cona-
X sider a fluctuational electric field E , acting on a partic-
f
where VH = cEdc/B is the absolute value of the Hall ular electron, as a quasi-uniform field22. This reduces
velocity. The w¯q depends on Edc because, in addition the many-electron problem to a single-electron dynam-
to εn = ~ωc(n+1/2), we have a term eEdcX. For ics. Even in this limit, the situation remains to be very
nondegenerate electrons, the probability wq is indepen- complicated; still anaccurateform ofthe DSF ofthe 2D
dent of the quantum number X due to eEdc(X′ X)= Coulomb liquid in a magnetic field can be found10:
~q V . In this case, w can be averaged over −Landau
y H q
lZekv−e1lenxupm(−beεrns/oTnel)y,(haessruemZikngisatnheeqpuairlitbitriiounmfudniscttriiobnu)t.ion S(q,Ω)= 2Z√kπ n,n′ γInn2,,nn′′ exp(cid:20)−Tεne −Pn,n′(Ω)(cid:21) , (8)
In a single-electron treatment, w¯ can be found in X
q
terms of the DSF of a nondegenerate 2D electron gas where
S(q,Ω)= π~2Zk nX,n′In2,n′(xq)× Pn,n′ = [Ω−(n′−γn2n,n)′ωc−φn]2, φn = Γ2n4+Txe~qΓ2C, (9)
3
~γn,n′ = Γ2n+2Γ2n′ +xqΓ2C, (10)
r
10
Γn is the collision broadening of Landau levels, ΓC = mmax
√2eE(0)l , and E(0) 3√T n3/4 is the typical fluctua-
tionalfeleBctric field2f3 u≃nder theescondition U /T >10.
C
Remarkably, the DSF of the Coulomb liquid with )
0
strong interaction given in Eq. (8) is similar to the DSF B, 5
(
ofnon-interactingelectrons(the latercorrespondsto the eff
rreeflgiemctesΓthCe≪singΓunl)a.r nTahteurperoopfotrhteiomnaalgitnyetfoatcrtaonrsp1/oγrtn,inn′ (0) )/Eac
e2nDhasnycsetemmesn.t fFacotrorΓC~ωc=/Γ0n, oefvethnetuSaCllyB,Aitthleeaodrsy,twohtihche (B,eff 0 mmax
describes the effect of multiple electron scattering. The
fluctuational electric filed drives anelectronfrom a scat-
terer which reduces multiple scattering by increasing
γn,n′ given in Eq. (10). As the function of frequency, -5
the DSF has maxima near Landau excitation energies.
0.495 0.500 0.505
Thefluctuationalfieldintroducesanadditionalbroaden-
ing of these maxima √xqΓC and the shift in their posi- B (T)
tions φ = x Γ2/4T ~. This form of the DSF describes
welltheCmagnqetCotranesportpropertiesofSEs10eventhose FIG. 1. Contributions from partial sums Pmmm=a0x νm to νeff
induced by the intersubband MW resonance14. normalized vs the magnetic field B for a sequence of mmax:
In most cases, the ripplon energy can be disregarded from mmax = 1 to mmax = 7 (solid). The conditions are
in the frequency argument of the DSF which allows the following: T = 0.2K (liquid 4He), ns = 106cm−2, and
to consider electron scattering as quasi-elastic (in this Ea(0c) =0.05V/cm.
limit C¯± C¯ = V Q T/~ω ). Then, us-
q → q r,q q r,q
ing the property of the equilibrium DSF, S(q, Ω) =
exp( ~Ω/T )S(q,Ω), the effepctive collision fre−quency TheEq.(11)indicatesthatν ,asthefunctionofω ω ,
e 0 c
can b−e represented as the sum ν = ∞ ν , where has a symmetricalminimum atω =ω. To the cont−rary,
eff m=0 m c
the next termν andthe followingterms ν with m>1
1 m
ν = 1 q2 C¯ 2J2(βq l P)S(q,0), (11) have an asymmetrical shape of a derivative of a max-
0 meTe q y q 0 y B imum affected by the symmetrical factor Jm2 (λκqylB).
X (cid:12) (cid:12) Fornoninteractingelectrons,S′(q,ω)hasanegativemin-
(cid:12) (cid:12)
and imumatω ω =γ /√2. Atthispoint, thewholesum
c 0,1
ν become−snegativealreadyatλ=1.4 10−3 duetothe
ν = 2 q2 C¯ 2J2 (βq l ) perffoximity of the CR condition and the·extreme sharp-
m m ~ y q m y B ×
e q ness of Landau levels. This estimate is very promising
X (cid:12) (cid:12)
(cid:12) (cid:12) for experimental studies of negative conductivity effects
in the system of SEs on liquid helium.
~
1 e−mT~eω S′(q,mω)+ e−mT~eωS(q,mω) (12) Atthechosenvalueofω ωc,thesumovermconverges
− T −
(cid:20)(cid:16) (cid:17) e (cid:21) quite rapidly. Still, each next term in the sum of νeff
has its own minimum which is closer to the point ω =
for m > 0. The derivative S′ ∂S/∂Ω appears be- c
≡ ω. This follows from Eqs. (8) and (9): for each energy
cause of the linear expansion of the function w¯ (V )
q H exchange ~Ω = ~mω there is a term with n′ n = m
in Eq. (4). The Eq. (11) follows from the relationship −
S′(q,0) = (~/2T )S(q,0). Usually, the second term in having a sharp maximum near ωc = ω (the sharpness
e of the maximum increases with m). The derivative of
squarebracketsofEq.(12)isverysmall. Inthefollowing,
such a term contributes to ν of Eq. (12). Since the
weshallconsidertheregimeofsharpmaximaofS(q,Ω), m
effect of E eventually comes from ξ˙(t), an approach
realized at γn,n′ ωc. ac
Since λ, enter≪ing the definition of β = λκ, is usu- to the resonance condition is equivalent to an effective
| | increase in E which explains the importance of multi-
ally very small, we shall concentrate on effects induced ac
photon terms. Thus, in the vicinity of the resonance, a
by the CR, when ω is quite close to ω. Sometimes we
c
substantial number of ν should be taken into account.
shall use a damping form (ω2−ωc2)2+4γ∗2ω2 instead This situation is illustramted in Fig. 1 where partial sums
of ω2 ω2 in the denomqinator of κ given in Eq. (1). mmax ν withdifferentm areshownasfunctionsof
− c m=0 m max
This denominator originates from the classical equation; B. Here,thedampingparameterγ =ν ,andthemany-
∗ cl
the(cid:12)(cid:12)refore, it(cid:12)(cid:12)is reasonable to set the damping parameter Pelectron DSF is taken into account for ns =1 106cm−2.
·
γ to its classical value ν . Itisclearthataninclusionofhighertermsonlyenhances
∗ cl
4
minimum positioned between the maxima at ω ωc. It
−
shouldbenotedthatasimilarshapeofthedcconductiv-
5 ity affected by the CR was experimentally observed for
the vapor atom scattering regime26. An important con-
4 clusionwhich follows fromFig. 2 is that electrondensity
should be rather small to obtain the negative conductiv-
0) ity regime induced be the CR.
B, 3 Recent experiments27 indicate an unusually large ex-
(
xx pansion of the electron system in a lateral direction,
(0) E)/ac 2 ns cm-2 30 wgehniecrhalclyanancocetpbteeduenffdecetrisvteooeldecitnrotnhteemfrpaemraetwuorrekapopfrtohxe-
B, imation. Electrondensities used in this experiment were
(xx 1 2.5 ratherhighns >40·106cm−2 whichdoesnotallowusto
usedirectlyanexplanationbasedonthenegativedccon-
0 ductivity. Still, for electron temperatures Te estimated
2
there, the system enters the regime U /T < 1, where
1 c e
the fluctuational field model fails and the single-electron
-1
treatmentcouldbeamuchbetterapproximation. Inthis
0.495 0.500 0.505
regime, the negative dc conductivity could appear even
B (T) forahighn ,becauseν decreaseswithT strongerthan
s 0 e
thesign-changingtermsν withm 1. InaCorbinoge-
m
≥
FIG. 2. The magnetoconductivity σxx normalized vs B ometry,the formationofa steadyring currentj, making
for different electron densities: ns/106cm−2 = 1 (dash-dot- σxx(j)=0, should be accompanied by a lateral redistri-
dotted), 2 (dash-dotted), 2.5 (dashed), and 30 (solid). Here bution of SEs. Predictions on properties of the ring cur-
mmax=7. Otherconditions are thesame as in Fig. 1 . rent are possible only in a nonlinear (in Edc) treatment
which requires separate investigations. Nevertheless, the
present theory allows us to formulate experimental con-
ditions,wherenegativedcconductivityeffectscanbeob-
the effect of negative conductivity making the minimum
served.
deeperandshiftingitspositionclosertothepointω =ω.
c
In summary,we haveinvestigatedtheoreticallythe in-
The Fig. 2 illustrates how Coulomb forces affect
fluence of cyclotron resonant excitation on the dc mag-
σ (B). Here it was instructive to set γ = 0. One
xx ∗ netoconductivity of the highly correlated 2D electron
can see that an increase in n strongly suppresses the
s system formed on the surface of liquid helium. In the
conductivityminimum andmaximumnearthe CRwith- low electron density region (n . 106cm−2), where the
s
out substantial changes in their positions and broaden-
Coulombinteractionisweakenough,thedcmagnetocon-
ing. This is contrary to the Coulombic effect reported
ductivityisshowntoreachnegativevaluesforquiteusual
previously for intersubband displacement mechanism24
amplitudes of the MW field which causes instability of
and for conductivity oscillations with ω/ω 2 caused
c the zero-dc-current state. The Coulomb interaction, in-
≥
by one-photon assisted scattering25.
creasingwith electrondensity, is shownto eliminate this
For n & 2 106cm−2, an additional maximum is effect, and at high densities (above 107cm−2) the the-
s
·
formed at ω ω >0. Both the minimum and the max- ory presented here yields the dc conductivity behavior
c
−
imum of the region ω ω > 0 are moving up when which is very similar to that observed in experiments.
c
−
n increases. Eventually, the curve σ (B) obtains the We found also that in the high density range, instability
s xx
shapewithtwomaximasettledinthe regionsω ω >0 couldbe triggeredbyelectronheatingwhichrestoresthe
c
−
andω ω <0(thelateroneishigher)andwithastrong applicability of the single-electron theory.
c
−
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