Photon-assisted scattering and magnetoconductivity oscillations in a strongly correlated 2D electron system formed on the surface of liquid helium

0.2 MB·

by  Yuriy P. Monarkha

Photon-assisted scattering and magnetoconductivity oscillations in a strongly
correlated 2D electron system formed on the surface of liquid helium
Yu.P. Monarkha1
1Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkov, Ukraine
The influence of strong internal forces on photon-assisted scattering and on the displacement
mechanism of magnetoconductivity oscillations in a two-dimensional (2D) electron gas is theoreti-
cally studied. The theory is applied to the highly correlated system of surface electrons on liquid
4 heliumunderconditionsthatthemicrowavefrequencyissubstantially differentfrom inter-subband
1 resonancefrequencies. Astrongdependenceoftheamplitudeofmagnetoconductivityoscillationson
0 the electron density is established. The possibility of experimental observation of such oscillations
2 caused byphoton-assisted scattering is discussed.

n
a PACSnumbers: 73.40.-c,73.20.-r,73.25.+i,78.70.Gq
J

3
I. INTRODUCTION tant question: why theoretical mechanisms of MO and
2
negativeconductivityeffectsproposedforsemiconductor

] electrons do not display themselves in experiments with
l Radiation-induced magnetoresistivity oscillations and
l SEsonliquidhelium? The answerto this questioncould
a zero resistance states (ZRS) of a 2D electron gas sub-
shedlightalsoonthesituationinsemiconductorsystems.
h jected to a perpendicular magnetic field were discov-
The most frequently discussed mechanism of MO and
- ered in 2001-2003using high quality GaAs/AlGaAs het-
s negative conductivity effects called the displacement
e erostructures1–3. Since then, a large number of theo-
mechanism was proposed already in 1969 by Ryzhii18.
m retical mechanisms have been proposed to explain these
In this model a quasielastic scattering event of an elec-
magneto-oscillations (MO)4–11. The ZRS appeared at
.
t tron caused by an impurity potential can be accompa-
a highradiationpowerasaresultofevolutionofresistivity
nied by absorption of a photon, which leads to indirect
m minima can be caused by a negative conductivity effect
inter-Landau-level scattering (n n′). The energy con-
- (σxx <0)12, whose microscopicoriginis quite controver- →
servation of such a photon-assisted scattering event
d sial as well as the origin of MO.
n
o Inexperiments1–3,themicrowave(MW)frequencywas ~ωc(n n′)+~ω+eEk(X X′)=0 (1)
quite arbitrary: ω > ω , here ω is the cyclotron fre- − −
c c c
[ quency. The period of MO observed is controlled by the (hereE isthedcdrivingelectricfielddirectedalongthe
k
1 ratio ω/ωc. Similar 1/B-periodic oscillations of magne- x-axis) determines the displacement of the electron or-
v toconductivity σxx and ZRS were discoveredin a nonde- bit center, which can be opposite to the driving force
3 generate 2D electron system formed on the free surface (X′ > X) if ω/ωc < n′ n. This is the reason for
8 of liquid helium when the MW frequency was tuned to the negative conductivity e−ffect. It should be noted that
8 the inter-subband excitation frequency13,14: ω = ω2,1 in the intersubband displacement model15,16 the corre-
5 (∆2 ∆1)/~ (here ∆l is the energy spectrum of surfac≡e sponding energy conservation contains the intersubband
1. subb−ands, l = 1,2,...). These oscillations were theoreti- excitation energy ∆2 ∆1 instead of ~ω. The displace-
0 cally explained15 by a nonequilibrium population of the ment model18 had be−en intensively developed after dis-
4 excitedsubbandandbypeculiaritiesofquasielasticinter- covery of ZRS to include self-energy effects and multi-
1 subband scattering in the presence of the magnetic field photon processes4,5,19.
v: B. Predictions of this intersubband displacement model Another popular mechanism of MO and negative con-
i concerning the influence of internal Coulomb forces on ductivity effects called the inelastic model9 originates
X
positions of σxx extrema16 were recently supported by from oscillations of the electron distribution function
ar experimental observations17. f(ε) induced by MW radiation. An oscillatory behav-
For mechanisms of MO proposed4,5,9, electron gas de- ior of f(ε) means that at certain ranges of ε there is a
generacy is not a crucial point. Therefore, they can population inversion, ∂f(ε)/∂ε > 0, and the magneto-
be applied also to a nondegenerate 2D electron system conductivity σ could be negative. These ideas were
xx
like surface electrons (SEs) on liquid helium. Electrons also discussed in Ref. 20.
bound to the free surface ofliquid helium representa re- In this work, we consider mostly the displacement
markable model 2D system which is quite simple and mechanismof MO applied to the system of SEs in liquid
clean. SEs are scattered quasi-elastically by capillary helium: photon-assisted scattering by ripplons. The in-
wave quanta (ripplons) and by vapor atoms. Their in- elastic mechanismwill be shortly discussedas well,since
teraction parameters are well established. Experiments we see its certain relation to photon-assisted scattering.
on SEs13,14 employed approximately the same MW fre- As compared to the initial version of photon-assisted
quencies and power as those used for the 2D electron scattering18,21, the theory is extended to include strong
gas in GaAs/AlGaAs1–3. Therefore, there is an impor- Coulombforcesactingbetweenelectronsandthecollision
2
broadeningof Landaulevels (LLs). We found the reason where b and b† are creationanddestructionoperators
q −q
whyMOofσxx andZRScausedbyphoton-assistedscat- of ripplons, Vr,q is the electron-ripplon coupling22, Qq =
teringwerenotseeninexperimentsonSEs13,14forchosen ~q/2ρω D, ω α/ρq3/2, αandρ arethe surface
r,q r,q
ranges of the MW frequency and amplitude, and formu- ≃
tension and mass density of liquid helium respectively,
lated conditions under which they could be potentially p p
and D is the surface area.
observed.
Probabilities of photon-assisted scattering are calcu-
lated according to the Golden Rule with the following
matrix elements21
II. PROBABILITIES OF PHOTON-ASSISTED
SCATTERING i V v v V f
i V˜ f = h | r| ih | mw| ±i+
±
h | | i E E
i v
v −
Consider a 2D electron gas on the free surface of liq- X
uid helium in the presence of a static magnetic field B
directed normally to the interface. The corresponding i V v v V f
mw r ±
vectorpotentialofthemagneticfieldA =(0,Bx,0). In + h | | ih | | i, (8)
0 E E
i v
the presence of the driving electric field Ek = (Ek,0,0), Xv −
eigenfunctions and the energy spectrum of SEs are char-
where E is the energy of the initial state, the final
i
acterized by the surface subband number l, the electron
state f = n′,X′,n(r) 1 ,n(mw) 1 , corre-
orbit center | ±i q ± ±q k − −k
sponds to proc(cid:12)esses of destruction of a photon Eand cre-
cp eE (cid:12)
X = y +ζ, ζ = k , (2) ation(signplus(cid:12))ordestruction(signminus)ofaripplon.
−eB −meωc2 Then(r) = n(r) andn(mw) = n(mw) arethevectors
q q k k
and by the LL number n: describing onccupaotion numbers onf ripploons and photons.
Further evaluations are based on the relationship
Ψl,n,X =ψl(z)√21π~e−i(X−ζ)y/l2Bψn(os)(x−X) , eiqre n,X;n′,X′ = δX,X′−l2BqyeiqxXMn,n′ which is valid
for the chosen gauge. Here we use the following nota-
(3) (cid:0) (cid:1)
tions
El,n,X =∆l+~ωc(n+1/2)+eEkX + 2em2Eωk22 , (4) Mn,n′(xq,ϕ)=i|n′−n|Jn,n′(xq)×
e c
Here ψn(os)(x) are oscillator eigenfunctions, ψl(z) are ×exp[ixqcos(ϕ)sin(ϕ)+i(n′−n)ϕ], (9)
wavefunctions of vertical motion, and l = ~c/eB is
B
the magnetic length.
At low temperatures T < 1 K, electronps predomi- min(n,n′)! |n′−n|
nantly occupy the ground surface subband (l =1), since Jn,n′(xq)=smax(n,n′)!xq 2 ×
∆ ∆ is about 6K (liquid 4He) or 3.2K (liquid 3He).
2 1
−
The Hamiltonian of electron interaction with MWs has
the usual form xq n′−n
e e ×exp − 2 Lm| in(n,|n′)(xq), (10)
Vmw = p+ A0 A˜, (5) (cid:16) (cid:17)
cm c
e where x = q2l2/2 is a dimensionless parameter, q =
(cid:16) (cid:17) q B x
where A˜ is the vector potential of the MW field, qcosϕ, and Lmn(x) are the associated Laguerre polyno-
mials. Our expression for Mn,n′(xq,ϕ) differs from the
2π~c2 1/2 similar equation of Ref. 21 because we have chosen the
A˜ (r)= e eik·r a +a† , different gauge. The result of Ref. 21 can be restored
α ω K α,λ,k k,λ −k,λ
Xλ,k(cid:18) k v(cid:19) (cid:16) (cid:17) from Eq. (9) using the substitution ϕ→ϕ−π/2.
(6) Introducing
a and a† are creation and destruction operators of
k,λ k,λ
pizhaotitoonnsv,eKctvori.s tIhteisvocluonmvee,naienndt etoλ,kcoinsstihdeerutnhite plionleaarr- (ω)= 4πn(mw)(ω)~ω (11)
polarization with A˜ =0. E s Kv
y
The Hamiltonian of the electron-ripplon interaction,
1/2
which dominates under conditions of the experiments and Cq(±) =Vr,qQq n±(rq) + 21 ± 21 , and following the
(T =0.2K)14, can be written as procedure describedhin Ref. 21, onie can find
V = V Q b +b† eiq·r, (7) e (ω)
r Xq r,q q(cid:16) q −q(cid:17) hi|V˜ |f±i =Cq(±)2mEeω2lBδX,X′−l2Bqy×
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
ωMn,n′′pn′′,n′ ωpn,n′′Mn′′,n′ III. THE CONDUCTIVITY OF STRONGLY
+ ,
ω (n n′′) q V ω ω (n n′′)+ω INTERACTING ELECTRONS
n′′ (cid:20) c − − y H ∓ r,q c − (cid:21)
X
(12)
Generally, the structure of Eq. (15) is similar to the
where pn,n′ = i √nδn′,n−1 √n′δn′,n+1 are dimen-
− structure of the corresponding probability of the usual
sionless matrix el(cid:16)ements of the electron m(cid:17)omentum op- electron-ripplon scattering. Therefore, considering the
erator,and V is the absolute value of the Hall velocity:
H contribution of photon-assisted scattering into the mo-
eE (X X′)= ~q V .
k y H mentum relaxation rate, we can use advantages of the
− −
Keeping in mind the energy conservation delta- approach22–24, which allows to express average scatter-
function, in the first term of Eq. (12) we can use the ing probabilities and the momentum relaxation rate ν
eff
replacementωc(n n′′) qyVH ωr,q ωc(n′ n′′) ω. in terms of the dynamic structure factor (DSF) of the
− − ∓ → − −
For slightly broadened LLs, this is an approximate pro- 2D electron system. Since w(±) (q) does not
cedure which has the same accuracy as the replacement X′ n,X→n′,X′
depend on X, when averaging over the initial electron
1 f(ε′) 1 f(ε) in usual conductivity equations. P
− → − stateswecanconsiderelectrondistributionoverLLsonly
Then, using properties of the associated Laguerre poly-
nomials, one can find fn ≃ Zk−1e−εn/Te (here Zk = ne−εn/Te). Then, the
average probability of electron scattering with the mo-
mentum exchange ~q caused byPdestruction of a photon
2 2 e (ω) 2
i V˜ f = C(±) E and creation or destruction of a ripplon can be written
(cid:12)h | | ±i(cid:12) (cid:16) q (cid:17) (cid:20)2meω2lB(cid:21) × in the following form
(cid:12) (cid:12)
(cid:12) (cid:12) 1 2
w¯(±) = C(±) λ χ (ω)
×δX,X′−l2Bqyχq(ω)xqJn2,n′(xq), (13) mw,q ~2 (cid:16) r,q (cid:17) mw q ×
where x S(q, q V ω +ω), (17)
q y H r,q
× − ∓
eiϕω e−iϕω 2 where C(±) is obtained from C(±) replacing the ripplon
χ (ω)= . (14) r,q q
q (cid:12)ωc+ω − ωc−ω(cid:12) occupationnumbern±(rq) withthe Bosedistributionfunc-
(cid:12) (cid:12) tion N , the function
(cid:12) (cid:12) r,±q
The dependence of χq(cid:12)on ϕ is important(cid:12) for calculation
of the momentum relaxationrate, which contains an ad- 2
ditional factor q2 =q2sin2ϕ. S(q,Ω)= π~Z Jn2,n′(xq)×
y k n,n′
After summation over k the probabilities of photon- X
assisted scattering from n,X to n′,X′ accompanied by
the momentum exchange ~q can be found as
dεe−ε/Tegn(ε)gn′(ε+~Ω) (18)
×
2π 2 Z
w(±) (q)= C(±) λ χ (ω)x
n,X→n′,X′ ~ q mw q q× is the DSF of a nondegenerate 2D electron gas, gn(ε)=
(cid:16) (cid:17) ImG (ε), and G (ε) is the single-electron Green’s
n n
−
function.
×δX,X′−l2BqyJn2,n′(xq)δ(εn−εn′ −~qyVH ∓~ωq+~ω), weTsahkailnlgusinettoheacrceosuunlttotfhtehecoclulimsiounlanbtroaapdpernoiancgh2o5f LLs,
(15)
where ε =~ω (n+1/2),
n c √2π~ 2(ε ε )2
n
g (ε)= exp − , (19)
λ = e2Em2w , E2 =4πNmw(ω)~ω, (16) n Γn "− Γ2n #
mw 4m2ω4l2 mw K
e B v where Γ is the collision broadening of LLs26. This ap-
n
proximation, being quite accurate for low LLs, greatly
N (ω) is the number ofphotonswith the frequency ω,
mw
simplifies further evaluations. The equilibrium DSF has
and E ( ergs/cm3) is the amplitude of the electric
mw an important property
field in the MW. As compared to usual electron-ripplon
scattering,Epq.(15)containsthephotonenergy~ω inthe S(q, Ω)=e−~Ω/TeS(q,Ω) (20)
argumentofthedelta-functionandadditionaldimension- −
less proportionality factors λmw, χq(ω), and xq. The which allows to shorten evaluations. According to
χq(ω) is of the order of unity (here ω is substantially Eqs.(18)and(19),theS(q,Ω)asafunctionoffrequency
larger than ωc), while xq is of the order of n′ n due has sharp maxima when Ω approachesthe LL excitation
−
to Jn2,n′(xq). The λmw depends only on the MW field frequency (n′ n)ωc.
−
parameters (E and ω) and on basic properties of the The momentum relaxation rate can be found by eval-
mw
electron gas under magnetic field. uating the totalmomentum gainedby scatterers. Foran
4
infiniteisotropicsystem,thekineticfrictionactingonthe w¯ , otherwise scattering probabilities in the direction
r,q
electron gas, of the driving force and in the opposite direction would
be the same leading to j =0.
x
F = N ~q w¯ +w¯(+) +w¯(−) , (21) SEsonliquidhelium formahighly correlated2Delec-
fric − eXq (cid:16) r,q mw,q mw,q(cid:17) tron system. Even for small densities ne ∼ 106cm−2,
theaverageenergyofCoulombinteractionperparticleis
should be antiparallel to the current. Here w¯r,q ≡ muchlargerthan the averagekinetic energy(Te). Under
w¯(+)+w¯(−) isthecorrespondingprobabilityobtainedfor such conditions, the 2D electron system subjected to a
r,q r,q
usual electron-ripplon scattering in the absence of MW magneticfieldcanbe welldescribedusingthe conceptof
radiation,andw¯(±) canbefoundfromEq.(17)usingthe thefluctuationalelectricfield27. Atafinite temperature,
r,q
substitution λmwχq(ω)xq 1 and setting ω 0 in the for each electron there is an electric field Ef of other
frequency argument of the→DSF. → electrons causedby fluctuations and directedto an equi-
Thus, the momentum relaxation rate ν can be de- librium position of the electron. Within the cyclotron
eff
VfinedbyVth.eUresliantgioenlashstiipc(aFpfprirco)xyim=a−tiNonem(~eωνeffVy ,0w),htehree aorubnitifloernmgthfi,eltdh,eiEf fBciasnsbtreonapgpernooxuimgha.telTyhcuosn,stidheeremdaags-
y H r,q
≃− → netic fieldandE causefastrotationofthe electronobit
correction into the effective collision frequency induced f
center around the equilibrium position. Therefore, such
by MW radiation is found as
astronglyinteractingsystemcanbeconsideredasanen-
2λ semble of noninteracting electrons, whose orbit centers
ν = mw q2χ (ω)x C2 S′(q,ω), (22)
mw m ~ y q q r,q movefastinthe fluctuationalfield22,23. The distribution
e Xq of Ef is known from numerical calculations28.
Thefluctuationalelectricfieldintroducesanadditional
where Cr,q ≡ Cr(,−q) ≃ Cr(,+q), and S′(q,Ω) = broadening of maxima of the DSF16,22:
∂S(q,Ω)/∂Ω.
aintgiIontnhtrehaerteeapbνlasr(e0c)nemcceaenonftbMeλmfWowrmχraqadl(liωya)toixobqnta,→itnhe1edamfnrodommseetEnttqiun.mg(2ωr2e)→lauxs0-- S(q,Ω)= 2Z√kπ nX,n′ Jγnn2,,nn′′ exp(cid:20)−εTne −Pn,n′(Ω)(cid:21) , (27)
in the frequency argument of S′:
where
2
νr(0) = me~Xq qy2Cr2,qS′(q,0). (23) Pn,n′ = [Ω−(n′−γ2n)ωc−φn]2, φn = Γ2n4+Tx~qΓ2C,
n,n′ e
The derivative of the Eq. (20) gives the relationship (28)
and
~
S′(q,0)= S(q,0)>0, (24)
2Te ~γn,n′ = Γ2n+2Γ2n′ +xqΓ2C. (29)
r
(0)
which transforms ν into the result of the SCBA the-
r
ory26 applied to the system of SEs on liquid helium22. In Eq. (28), defining Pn,n′(Ω), we have neglected terms
The total momentum relaxation rate ν =ν(0)+ν . of the order of Γ2n/8Te2 which are very small for the
eff r mw
considered system. The Coulomb broadening parame-
The results of Eqs. (22) and (23) can be obtained also
using the direct definition of the electron current ter ΓC = √2eEf(0)lB increases with ne and Te because
E(0) 3√T n3/4.
jx =−ens (X′−X)q w¯r,q+w¯m(+w),q+w¯m(−w),q . fTh≃e flucteuaetional field introduces also an additional
Xq (cid:16) (cid:17) shiftinpositionsofmaximaoftheDSFφC =xqΓ2C/4Te~
(25) which enters the definition of φ . This Coulomb shift
n
Taking into account (X′−X)q =qylB2, one can find is restored from the expression for the DSF of the 2D
Wigner solid under a strong magnetic field. It preserves
σ = enslB2 q w¯ +w¯(+) +w¯(−) the equilibrium property of Eq. (20). The influence of
xx E y r,q mw,q mw,q ≃ this shift on the intersubband displacement mechanism
k Xq (cid:16) (cid:17) ofMO16 wasrecentlyconfirmedinexperiments17 onSEs
above liquid 3He.
e2n ν(0)+ν In Eq. (22), describing νmw, the function χq(ω)
s r mw
, (26) is averaged over directions of the ripplon vector
≃ m ω2
e c χ (ω)sin2ϕ χ (ω). Simple integration yields
q ϕ ≡ tr
which proves that F is antiparallel to the current. It
fric (cid:10) (cid:11)
should be noted that one cannot disregard the driving 1 3ω2/ω2+1
χ (ω)= c . (30)
field correction −qyVH in the expressions for w¯m(±w),q and tr 2(ωc2/ω2−1)2
5
For ω2 ≫ 3ωc2, the χtr(ω) → 1/2 which coincides with
sin2ϕ ϕ entering νr(0). Using this notation, the photon- 6 6 -2
assisted scattering correction ν can be represented in ne /10 cm =1
(cid:10) (cid:11) mw
an analytical form
2m TΛ2 3
e
ν =λ χ (ω) F (B), (31)
mw mw tr √πα~3l2 ω 2.5
B 4 0 10
1
where )/ 0
B
1 ∞ (
F (B)= e−εn/Te dx x V2 (x )J2 F 5
ω −Z q q 1,1 q n,n+m×
k n,m Z0
X -3
2
ω/ω m φ /ω ω/ω m φ /ω
c n c c n c
− − exp − − ,
γ˜n3,n+m "−(cid:18) γ˜n,n+m (cid:19) # -6
(32) 0.6 0.7
γ˜ =γ /ω ,andm=n′ n. Thedimensionless
n,n+m n,n+m c
electron-ripplon coupling22 − B , T
V1,1(xq)=xqwc xq/2g2lB2 +eE⊥lB2/Λ (33) 0F.I2GK. 1a.ndFfωou(Br)eleccatlcrounlatdeednsniteiaers:ωn/eωc==1·410a6ncdm5−2fo(rsoTlid=),
is defined by the function(cid:0) (cid:1) 2.5·106cm−2 (dashed), 5·106cm−2 (dotted), 10·106cm−2
(dash-dotted).
1 1 1+√1 x
w (x)= + ln − . (34)
c −1 x (1 x)3/2 √x
− − (cid:18) (cid:19) φ /ω and F (B) > 0. In the case of photon-assisted
n c 0
−
The V1,1(x) depends also on the pressing elec- scattering with ω > ωc, the extrema of Fω(B) oc-
etr2ic(ǫ fie1ld)/4E(⊥ǫ,+t1h)e(himeraegǫe ispottheentdiaiellepcatrriacmceotnesrtaΛnt o=f cFuωr(Bat)/|Fω0/(ωBc)−m|4T∼e/~γ˜γnn,,nn+,ma.nd Twheecraefnorreo,ugthhelyreasttiio-
liquid−helium), and on the localization parameter (g) of mate ≈
the SE wave function: ψ (z)=2g3/2zexp( gz).
EqT.h(3e2a)bisnoclruetaesevsalsuheaorpf1ltyhewhfuennctωio/nωcFbωe(cBo−m) edseficnloesdebtyo ννmr(0w) ∼λmw Γ42nT+e Γ2C = mee~2ωE4m2wΓω2ncT+eΓ2C. (35)
an integer m = n′ n > 0. In the vicinity of this
point, Fω(B) chang−es its sign because of the factor The νmw/νr(0) inpcreases with the MpW field amplitude
ω/ωc m φn/ωc. The typical dependence Fω(B) is Emw and decreases with the MW frequency.
shown−in F−ig. 1 for four different electron densities. We A more accurate comparison of ν and ν(0) can be
mw r
have chosen the MW frequency ω/2π = 79GHz which given using numerical evaluation of Eq. (32) shown in
is typical for SEs above liquid 3He. It is obvious that Fig. 1. Consider typical conditions of the experiment14
an increase in n suppresses the amplitude of MO and with SEs on liquid 3He: ω/2π 79GHz, T = 0.2K
e
broadens their shape. As expected, negative values of and n = 106cm−2. Under these≃conditions, the inter-
e
F (B) and ν mostly appear at ω/ω (B)>m. subband displacement mechanism16 leads to giant MO
ω mw c
and σ < 0 already at a MW field amplitude which
xx
corresponds to the Rabi frequency eE 1 z 2 /~ =
mw
IV. DISCUSSION AND CONCLUSIONS 108s−1. For such a MW power, E |h | 0|.1iV| /cm,
mw
≃
and the calculation based on Eqs. (31) and (32) gives
Negative conductivity effects and ZRS appear when δν /ν(0) 0.65 10−5, if B is close to 0.7T (m = 4).
ν &ν(0). It should be notes that χ (ω) is about 1/2, Theiffs exeffpla≃ins why· MO caused by photon-assisted scat-
mw eff tr
if ω is substantially larger than ω . As compared to the tering were not observed together with MO caused by
c
result of usual electron-ripplon scattering, the momen- the intersubband displacement mechanism.
tumrelaxationratecausedbyphoton-assistedscattering The correctionto ν(0) causedby the intersubbanddis-
r
[Eq. (31)] has an important proportionality factor λmw, placement mechanism16 does not have the proportion-
which originates from V2 /(E E )2 V2 /~2ω2. ality factor λ , because this mechanism does not in-
mw i− v ∼ mw mw
We note also the presence of the MW frequency ω in volve photons. Instead, there is the proportionality fac-
expression for Fω(B) given in Eq. (32). For usual scat- tor[n¯2 e−(∆2−∆1)/Ten¯1],wheren¯l =Nl/Nall isthefrac-
−
tering, the main contribution into F (B) comes from tionaloccupancyofthel-thsurfacesubband. Thisfactor
0
terms withm=0,whenthe factorω/ω m φ /ω isnottoosmall,typicallyabout0.1orevenlarger. There-
c n c
− − →
6
fore,theenhancementfactor4Te/~γn,n′ notedabovecan function f(ε)[1 f(ε)]= Te∂f/∂ε, which is not valid
− −
make the amplitude of oscillations very large. for an arbitrary function [moreover f(1 f) > 0]. The
−
Thus, the main reason why photon-assistedscattering properrelationshipbetweenν(0) and∂f/∂εcanbefound
r
is small in the system of SEs on liquid helium is the consideringthenonequilibriumDSF ofa2Delectrongas
parameter λ , which actually does not depend on the
mw
D
onnaltyuroenopfasrcaamtteetreerrsso.fFtohreaMfiWxefidelrdat(iEomωw/ωacn,ditω)deapnednodns S(q,Ω)= Neπ2lB2~n,n′Jn2,n′(xq)Z dεf(ε)×
basic parameters of charge carriers (the charge and the X
effective mass m∗). According to recent treatments of
photon-assisted secattering in semiconductor 2D electron ×[1−f(ε+~Ω)]gn(ε)gn′(ε+~Ω), (37)
systems19,the effectofMWonthe in-planecurrentjx is wheref(ε)isanarbitrarydistributionfunctionandD is
characterized by the factor JM2 2xqEmw/E˜ω , where othneescuarnfnacoet agrueaar.anNtoeewtthhaetSS′′((qq,,00))6=en~teSri(nqg,0E)q/.2T(2e3a)nids
JM(z)istheBesselfunction,M(cid:16)ispthenumberof(cid:17)photons positive.
assisted in a scattering event, and ˜ is a characteristic
Eω Generally, S′(q,Ω 0) consist of the term with the
MW field →
derivative of the factor 1 f(ε) and the term with the
−
√2m∗ω ω2 ω2 ~1/2 derivative of gn′(ε). The secondterm can be rearranged
E˜ω = e eωc2(cid:12)(cid:12) +c −ω2ωc1(cid:12)(cid:12)/2 (36) uJsni,nng′ =inJteng′,rnatoinoencbayn fipnadrts. Then, using the property
Fothfoerthlωiem2iB≫teosfωsewc2l,efatuhkneMcEt˜iWωon≃fie√pld22sxmEq∗eEωm2wlB//E˜eω˜, a→,ntdh2etheffexeqacλrtmgouwfm.oenInent- S′(q,0)= Ne2Dπ2lB2 nX,n′Jn2,n′(xq)×
p mw ≪Eω p
photon assisted scattering is characterized by the small
∂f(ε)
proportionality factor J12 2 xqλmw ≃ xqλmw, which × dε − ∂ε gn(ε)gn′(ε). (38)
agrees with our calculations. Z (cid:20) (cid:21)
(cid:0) p (cid:1)
The main peculiarity of the electron gas in ThisequationtogetherwithEq.(23)establishesthenec-
GaAs/AlGaAs, as compared to SEs on liquid helium, essary relationship between the momentum relaxation
is the small effective mass m∗e ≃ 0.067me. There- rate and ∂f(ε)/∂ε. For interaction with short-range
fore, for the fixed ratio ω/ωc = 2, the Eq. (36) yields: scattererslikevaporatoms,thecouplingparameterCq2 =
E˜ω ≃ 74V/cm if ω/2π = 79GHz, and E˜ω ≃ 22V/cm if ~3ν0/2meD,whereν0 isthecollisionfrequencyatB =0.
ω/2π = 35.5GHz. It should be noted that these values To find corrections to the equilibrium distribution
of ˜ are much larger than the estimate of the MW field function induced by MW radiation, a sort of kinetic
ω
E
E 0.5V/cm given in Ref. 29 for typical experimen- equation was used9 where the effect of microwaves
mw
talcon≈ditionsrealizedinsemiconductorsystems. ForSEs St f(ε) wasproportionaltof(ε ~ω) f(ε). Since
mw
{ } ± −
on liquid helium the estimate of the characteristic MW theMWitselfcannotcauseelectrontransitionsfromεto
field ˜ increases only by the factor 1/√0.067 3.9. ε ~ω when ω/ω 2, we conclude that the main con-
ω c
E ≃ ± ≥
This factor could be compensated by coordinatedreduc- tribution into St f(ε) comes from photon-assisted
mw
{ }
tions in the MW frequency and the cyclotron frequency scattering. Though, inelastic mechanism applied to SEs
provided the ratio ω /ω and the MW power are fixed. onliquidheliumrequiresaseparateinvestigation,weex-
c
Therefore,ifphoton-assistedscatteringisthemainorigin pectthatoscillatorycorrectionstotheequilibriumdistri-
ofMOinGaAs/AlGaAs,itcouldbepotentiallyobserved butionfunctionwillcontainthesmallparameterλ in-
mw
alsointhesystemofSEsonliquidheliuminthelowMW troduced above. Additionally, electron-electron interac-
frequency range and at highest radiation power. tion, which is extremely strongfor SEs onliquid helium,
At this point it is instructive to discuss briefly the in- shouldincreasesubstantiallytheinelasticrelaxationrate
elastic mechanism of MO and ZRS, which is expected and reduce the amplitude of MO caused by the inelastic
to produce larger MO amplitudes than those of the dis- mechanism.
placement mechanism. In the inelastic model9, MO and Concluding, for observation of MO of σ caused by
xx
thenegativeconductivityeffectoriginatefromoscillatory both of the mechanisms (displacement and inelastic) in
behavioroftheelectrondistributionfunctionf(ε)enter- the 2D electron system formed on the free surface of liq-
ing the conductivity equation for usual scattering which uid helium, it is necessary to use low electron densities
does not involve photons. Oscillatory corrections δf to andtheparametersoftheMWfieldgivingalargestvalue
the equilibrium distribution function f (ε) are caused of λ defined by Eq. (15).
F mw
by the MW field. The important point of the model is In summary, we have investigated the influence of
the relationship between σ and ∂f(ε)/∂ε. Sometimes strong Coulomb forces acting between SEs in liquid he-
xx
this relationship is referred to Kubo conductivity equa- lium on photon-assisted scattering and on the displace-
tions. Itshouldbe notedthat there the factor∂f(ε)/∂ε mentmechanismofMW-inducedoscillationsofmagneto-
appearsfromthepropertyoftheequilibriumdistribution conductivity. Coulomb interaction is shown to suppress
7
strongly the amplitude of oscillations and affect their mulateconditionsunderwhichphoton-assistedscattering
shape. Under conditions of the experiment with SE on could be observed. In order to obtain the same effect as
liquid helium14, the amplitude of MOcausedby photon- in heterostructures, one need to increase the MW field
assisted scattering is shown to be very small, which ex- amplitude by a factor of about 3.9. Therefore, the sys-
plains why oscillations were not detected for MW fre- temofSEs onliquidhelium canserveas amodelsystem
quencies substantially different from the inter-subband for testing mechanisms of MO and ZRS proposed for a
resonance frequency. The relationship between the am- 2D electron gas in heterostructures.
plitude of MO and parameters characterized the MW TheworkwaspartiallysupportedbyaGrantofSFFR
fieldandthe electronsystemobtainedhereallowstofor- and JSPS (F52.2/005).
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