Low-frequency peak in the magnetoconductivity of a non-degenerate 2D electron liquid

4 Pages·0.15 MB·English

by  Frank Kuehnel

Low-frequency peak in the magnetoconductivity of a
non-degenerate 2D electron liquid
Frank Kuehnel1, Leonid P. Pryadko2, and M.I. Dykman1
1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823
2School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540
0
(February 1, 2008)
0
0
Westudythefrequency-dependentmagnetoconductivityofastronglycorrelatednondegenerate2D
2
electronsysteminaquantizingmagneticfield. Wefirstrestorethesingle-electronconductivityfrom

n calculated 14 spectral moments. It has a maximum for ω ∼ γ (h¯γ is the disorder-induced width
a of the Landau level), and scales as a power of ω for ω → 0, with a universal exponent. Even for
J
strongcouplingtoscatterers,theelectron-electroninteractionmodifiestheconductivityforlowand

0 high frequencies, andgives rise toa nonzerostatic conductivity. Weanalyzethefull many-electron
3 conductivity,and discuss the experiment.

] PACS numbers: 73.23.-b, 73.50.-h, 73.40.Hm
l
l
a
Oneofthemostinterestingproblemsinphysicsoflow- sal exponent µ 0.215. Whereas the onset of the peak
h ≈
- dimensional systems is the effect of the electron-electron is a single-electron effect, the nonzero value of the static
s interaction (EEI) on electron transport. In many cases conductivity and the form of σ (ω) for big ω/γ are de-
e xx
m the EEI is the major factor, fractional quantum Hall termined entirely by the EEI. We obtain an estimate for
effect (QHE) being an example. At the same time, σ (0)andanalyzetheoverallshapeofσ (ω)inthepa-
. xx xx
t single-electron picture is often also used for interpreting rameter range where exp(¯hω /k T) 1 and k T ¯hγ
a c B ≫ B ≫
m transport, as in the integer QHE. Another closely re- (ωc is the cyclotron frequency), the conditions usually
lated example is magnetotransportof a low-density two- met in strong-field experiments on electrons on helium.
-
d dimensional electron system (2DES) on helium surface
n [1]. For strong quantizing magnetic fields, experimental 4
6.5
o k=7
data on electron transport in this system are reasonably
c
[ well described [2–4] by the single-electron theory based σ G k=2
on the self-consistent Born approximation (SCBA) [5].
2 k=1
v This theory does not take into account the interference 2 5.5
effects that lead to electron localization in the random 0 0.2 ω/γ 0.4
7
2 potential of scatterers. Such a description appears to
4 contradictthephenomenologyoftheintegerQHE,where
1 all but a finite number of single-particle states in the
0
random potential are localized [6,7]. The static single- 0
0
electron magnetoconductivity σ (0) must vanish, as il- 0 1 ω/γ 2
0 xx
/ lustrated in Fig. 1, since the statistical weight of the ex- FIG.1. Reducedmicrowaveconductivity(1)ofanon-inter-
t
a tended states is equal to zero. acting2DESinashort-rangedisorderpotentialforkBT ≫¯hγ
m In this paper we discuss the case where the EEI is (solid line). The electron-electron interaction results in flat-
- strong and the electrons are correlated, as for 2DES on tening of the conductivity for ω <∼ ωl (13), and in a much
d heliumandinfractionalQHE.Yetthecharacteristicforce slower decay of σ˜ for moderately big ω ≫ γ, as shown by
n
on an electron from the short-range random potential dashedlines. Inset: convergenceoftheinterpolationfactorG
o
c mayexceedtheforcefromotherelectrons. Theinterrela- (10) with theincreasing numberof moments 2k.
: tion between the forces determines the effective strength
v
i of the coupling to scatterers. The analysis allows us to The single-electron conductivity at low frequen-
X understand the strong coupling limit and the crossover cies is determined by the correlation function of the ve-
r to weak coupling, and to resolvethe apparentcontradic- locity of the guiding center R of the electron cyclotron
a
tionbetweenlocalizationofsingle-electronstatesandthe orbit in the potential of scatterers. For ω kBT/¯h
≪
experimental data for electrons on helium. and exp(h¯ωc/kBT) 1, it can be written as σxx(ω) =
≫
We show that, for strong coupling to scatterers, the (ne2l2γ/8kBT)σ˜(ω), where n is the electron density, l =
low-frequency magnetoconductivity σxx(ω) of a nonde- (h¯/mωc)1/2 is the magnetic length, and σ˜ is the reduced
generate 2DES becomes nonmonotonic: it has a maxi- conductivity,
mum at a finite frequency ω 0.3γ, where ¯hγ is the ∞
max 2h¯γ
SCBA level broadening [Fig. 1].≈For small but not too σ˜(ω)= dteiωt (qq′)
small ω/γ, the conductivity scales as ωµ with a univer- −mωc Z−∞ Xq,q′
V˜qV˜q′exp[iqR(t)]exp[iq′R(0)] . (1)
×
(cid:10) (cid:11)
1
3 443 25003 13608949709
Here, stands for thermal averaging followed by the
M =1; ; ; ; ;
h·i 2k
averaging over realizations of the random potential of 8 1152 38400 8941363200
defects V(r), and V˜ = (V /¯hγ) exp( l2q2/4) are pro- 4.47809;15.7244;63.7499 (7)
q q
−
portionalto the Fouriercomponents V ofV(r). We will
q (we give approximate values of M for k 5).
assume that V(r) is Gaussian and delta-correlated, 2k ≥
To restore the conductivity σ˜(ω) from the calculated
V(r)V(r′) =v2δ(r r′), (2) finite number of moments, we need its asymptotic form
h i − for ω γ. It can be found from the method of optimal
≫
in which case ¯hγ =(2/π)1/2v/l [5]. fluctuation [11], by calculating the thermal average in
Time evolution of the guiding center R (X,Y) in Eq.(1)ontheexacteigenstates n ofthe lowestLandau
≡ | i
Eq. (1) is determined by the dynamics of a 1D quantum band ofthe disorderedsystem. All states n are equally
| i
particle with the generalized momentum and coordinate populated for kBT ¯hγ. Their energies En are sym-
≫
X and Y, and with the Hamiltonian metrically distributed around the band center (E = 0),
with the density of states ρ(E) exp( 4E2/¯h2γ2) [12].
∝ −
H =h¯γ V˜ exp(iqR), [X,Y]= il2. (3) For large ω/γ, the conductivity is formed by transitions
q
Xq − between states n , m with large and opposite in sign
| i | i
BecauseoftheLandauleveldegeneracyintheabsence energies En,m (En Em = h¯ω). The major contribu-
| − |
of random potential, the problem of dissipative conduc- tion comes from En = Em. Only those configurations
−
tivity is to some extent similar to the problem of the ofV(r) aresignificant,wherethe states n , m arespa-
| i | i
absorption spectra of Jahn-Teller centers in solids [8], tially close. However, the overlap matrix elements affect
which are often analyzed using the method of spectral onlythe prefactorinσ˜ [10],andtologarithmicaccuracy,
moments. This method can be applied to the conduc-
σ˜(ω) [ρ(h¯ω/2)]2 exp( 2ω2/γ2). (8)
tivity (1) as well [9]. It allows, at least in principle, to ∝ ∝ −
restore σxx(ω). In addition, the moments Since the tail of the conductivity is Gaussian, one is
1 ∞ tempted to restore σ˜(ω) from the moments Mn using
M = dω(ω/γ)kσ˜(ω) (4) a standard expansion in Hermite polynomials, σ˜(γx) =
k 2πγ Z−∞ c H (√2x)exp( 2x2). From(4),the coefficientsc
n n n − n
aPre recursively related to the moments M with k n.
can be directly found from measured σxx(ω), and there- However,for the moments values (7), suchkan expan≤sion
fore are of interest by themselves.
does not show convergence. This indicates possible non-
For ω,γ k T/¯h, the states within the broadened
≪ B analyticity of the conductivity at ω =0.
lowest Landau level are equally populated and the re-
Forω 0,the conductivitycanbe foundfromscaling
duced conductivity is symmetric, σ˜(ω) = σ˜( ω). Then →
− arguments [13,14]by noticing that it is formed by states
oddmoments vanish, M =0. For evenmoments, we
2k+1 within a narrow energy band E ¯hγ. The spatial ex-
obtain from Eqs. (1), (4) | |≪
tentoflow-energystatesisoftheorderofthelocalization
M2k =−2l2 (q1q2k+2) V˜q1...V˜q2k+2 (5) liesntghtehloξc∼alilza|εt|i−oνn,ewxpheorneenεt=[7E]./T¯hhγeafnredquνe=ncy2.3ω3,±on0t.h03e
X (cid:10) (cid:11)
... eiq1R, eiq2R ,... , eiq2k+1R eiq2k+2R, other hand, sets a “transport” length L l(γ/ω)1/2.
ω
× ∼
(cid:2)(cid:2) (cid:2) (cid:3) (cid:3) (cid:3) It is the distance overwhich an electron woulddiffuse in
where the sum is taken over all q1,...,q2k+2. The com- therandomfieldV(r)overtime1/ω,withacharacteristic
mutators (5) can be evaluated recursively using diffusioncoefficientD =l2γ,iftherewerenointerference
effects. Forlargeξ,L l,thescalingparametercanbe
eiqR, eiq′R =2isin 1l2q q′ ei(q+q′)R. (6) chosen as g = (L /ξω)1≫/ν ε (ω/γ)−1/2ν [14,15]. The
ω
(cid:2) (cid:3) (cid:16)2 ∧ (cid:17) conductivity σ˜(ω) is determ∼in|ed| by the states within the
From Eq. (2), hV˜qV˜q′i = (πl2/2S)exp(−l2q2/2)δq+q′, ceonnertgriybubtaenndeawrhlyereeqgua<∼lly,1.anFdor high T, all these states
whereS isthe area. Theevaluationofthe 2kthmoment
comes then to choosing pairs (qi,−qi) and integrating σ˜(ω) ωµ (ω 0), µ=(2ν)−1 0.215. (9)
overk+1independentq . FromEq.(6),theintegrandis ∝ → ≈
i
a (weighted with q1q2k+2) exponential of the quadratic With Eqs. (8), (9), the conductivity can be written as
form (l2/2) q Aˆ q , where i,j = 1,...,k + 1. The
i ij j
matrixelemePntsAˆ arethemselves2 2matrices,Aˆ = σ˜(ω)=xµG(x)exp( 2x2), x= ω /γ. (10)
ij ij − | |
×
Iˆδ +a σˆ , where σˆ is the Pauli matrix, and a =
− ij ij y y ij The function G(x) (x 0) can be expanded in Laguerre
−Aˆ,ajtihe=m0o,m±e1n.tsBMecauasereogfivthene bstyruracttiuorneaolfntuhmebmerast.riFcoesr polynomials L(nµ−1)/2(2≥x2), which are orthogonal for the
2k weighting factor in Eq. (10). We have restored the cor-
k =0,1,...,7 we obtain [10]
responding expansioncoefficients from the moments (7).
2
The resulting conductivity is shown in Fig. 1 with solid ofallotherelectrons. TheoverallchangeoftheCoulomb
line. Theexpansionforσ˜convergesrapidlyforµbetween energyoftheelectronsystemoverasmalltimeintervalis
0.19 and 0.28 (as illustrated in Fig. 1 for µ = 0.215), givenby e(E δr ),whereδr isthedisplacementof
n n n n
whereasoutsidethis regiontheconvergencedeteriorates. thenthelPectronduetothepotentialofdefects,andE is
n
The electron-electron interaction (EEI) can theelectricfieldonthenthelectronfromotherelectrons.
strongly affect the magnetoconductivity even for low Clearly, E and δr are statistically independent. This
n n
electron densities, where the 2DES is nondegenerate. allows us to relate the coefficient of energy diffusion of
Of particular interest for both theory and experiment an electron D to the characteristic coefficient D = γl2
ǫ
are many-electron effects for densities and temperatures of spatial diffusion in the potential V(r),
where Γ e2(πn)1/2/k T 1. The 2DES is then
≡ B ≫ D =(e2/2) E2 D γ(h¯/t )2. (12)
strongly correlated and forms a nondegenerate electron ǫ h fli ∼ e
liquid or, for Γ > 130 [16], a Wigner crystal. The mo-
Energydiffusioneliminateselectronlocalizationwhich
tion of an electron is mostly thermal vibrations about
caused vanishing of the single-electron static conductiv-
the (quasi)equilibrium position inside the “cell” formed
ity. The low-frequency boundary ω of the range of ap-
l
by other electrons. For strong B, the characteristic vi-
plicability of the single-electron approximation can be
bration frequency is Ω = 2πe2n3/2/mω , Ω ω (for
p c p ≪ c estimated from the condition that the diffusion over the
a Wigner crystal, Ω is the zone-boundary frequency
p energy layer of width δε = (ω /γ)µ [which forms the
l l
of the lower phonon branch [1]). We will assume that ∼
single-electronconductivity (9) at frequency ω γ] oc-
l
k T ¯hΩ . Thenthevibrationsarequasiclassical,with ≪
amBpli≫tude δp (k T/e2n3/2)1/2 l. curred over the time 1/ωl. For µ=1/(2ν), this gives
fl B
Therestorin∼gforceonanelectro≫nisdeterminedbythe ω /γ =C (γt )−2ν/(ν+1), C 1. (13)
ℓ l e l
∼
electric field E from other electrons. The distribution
fl
of this field is Gaussian, except for far tails, and E2 = All states with energies ε < δǫl contribute to the con-
F(Γ)n3/2kBT, with F(Γ) varying only slightly, frhomfli8.9 ductivity for frequencies|ω| ∼< ωl. Therefore the many-
to10.5,inthewholerangeΓ>20[17]. Sinceδ l,the electron conductivity may only weakly depend on ω for
fl
field Efl is uniform over the ∼electron wavelength≫l. The ω <ωl, as shown in Fig. 1, and the static conductivity
electronmotioncanbethoughtofasasemiclassicaldrift σ (0) σ (ω ) (ne2γl2/k T)(γt )−1/(ν+1). (14)
of an electron wave packet in the crossed fields Efl and xx ≈ xx l ∼ B e
B, with velocity cEfl/B. We note that there is a similarity between the EEI-
Inthepresenceofdefects,movingelectronswillcollide induced energy diffusion, which we could quantitatively
with them. If the density of defects is small and their characterize for a correlated nondegenerate system, and
potential V(r) is short-range [cf. Eq. (2)], the duration the EEI-induced phase breaking in QHE [20,21]. The
of a collision is cutoff frequency ω can be loosely associated with the
ℓ
reciprocal phase breaking time.
t =l(B/c) E−1 (h¯/el)n−3/4(k T)−1/2, (11)
e h fl i∼ B TheEEIalsochangesthehigh-frequency tail ofσxx(ω)
in the range ω ω . In the many-electron system, the
and the scattering cross-section is ∝ γ2. For γte ≪ 1, tail is formed b≪y proccesses in which a guiding center of
electron-defect collisions occur independently and suc-
theelectroncyclotronorbitshiftsinthefieldE (byδR).
cessively in time. This corresponds to weak coupling to fl
The energy ¯hω goes into the change of the potential en-
the defects, and allows one to use a single-electron type
ergy of the electron system eE δR, whereas the recoil
transport theory, with the collision rate τ−1 calculated fl
momentum¯hδR/l2 goestodefects. Forlargeω,itisnec-
for the electron velocity cE /B determined by the EEI,
τ−1 γ2t [18]. The manfly-electron weak-coupling re- essary to find optimal δR and Efl. For weak coupling to
sults∼have beeen fully confirmed by experiments [4,19]. defects, γte ≪ 1, the correlator (1) can be evaluated to
the lowest order in γ, which gives
For γt 1, collisions with defects “overlap” in time,
e
≫
which corresponds to the strong coupling limit. In this σ˜(ω)=γωt2exp (2/π)1/2ωt . (15)
case, from Eqs. (2), (11), the characteristic force on an e h− ei
electron from the random field of defects F = h¯γ/l
rf The exponential tail (15) is determined by the charac-
≫
eE . One might expect therefore that the EEI does not
fl teristic many-electron time (11), and the exponent is
affecttheconductivity,andthesingle-electrontheorydis-
just linear in ω. For larger ω, the decay of σ˜ slows
cussed above would apply. It turns out, however, that down to lnσ˜(ω) (ωt )2/3/[ln(ω/γ)]1/3, provided
e
this is not the case for the low- and high-frequency con- n1/2δ (ωt|)1/3 |1 [∝10]. This asymptotics results from
fl e
ductivity. ≪
anomalous tunneling [22] due to multiple scattering by
AsaresultoftheEEI,theenergyofanelectroninthe
defects. It also applies for strong coupling to defects,
potentialofdefectsV(r)isnolongerconserved. Themo-
γt 1, and replaces the much steeper single-electron
e
tion of each electron gives rise to modulation of energies ≫
Gaussian asymptotics (8).
3
We note that the overall frequency dependence of imation, which provides an insight into numerous exper-
σ (ω)isqualitativelydifferentforstrongandweakcou- imental observations for electrons on helium surface.
xx
pling toscatterers. Inthe latter case,σ is maximal for We are grateful to M. M. Fogler and S. L. Sondhi for
xx
ω = 0 and decreases monotonously with the increasing useful discussions. Work at MSU was supported in part
ω, in contrast to the behavior of σ (ω) in the strong- by the Center for Fundamental Materials Research and
xx
coupling case shown in Fig. 1. Both γ and t increase by the NSF through Grant no. PHY-972205. L.P. was
e
with the magnetic field, and by varying magnetic field, supported in part by DOE grant DE-FG02-90ER40542
electron density, and temperature one can explore the
crossoverbetweenthelimitsofstrongandweakcoupling.
It is interesting that both the static and the high-
frequency conductivities are many-electron even for
γt 1, where the coupling to defects is strong. It
e
≫
follows from Eq. (14) (see also Fig. 1) that the many-
[1] Two-dimensional electron systems on helium and other
electronσ (0)isoftheorderofthesingle-electronSCBA
xx cryogenic substrates, ed. by E. Y. Andrei (Kluwer,
conductivity σSCBA(0) = (4/3π)ne2γl2 [3] for not ex-
xx Boston, 1997).
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frequency dependence of the full single-electron conduc-
(1988).
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→
It follows from the above arguments that the random [4] M. J. Lea et al., Phys. Rev. B 55, 16280 (1997) and
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and S.M. Girvin (Springer, New York,1990), p. 117.
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from helium vapor atoms [1]. Ripplons, althoug∼h they
structure of defects in insulators and semiconductors
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(Clarendon Press, Oxford,1975).
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∝
Randompotentialofvaporatomsistime-dependent(and [10] F. Kuehnelet al.,in preparation.
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T
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v
then if γt 1, coupling to the vapor atoms≫is strong, [14] Z.-Q. Wang et al.,cond-mat/9906454.
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≫ [15] S. L. Sondhi,unpublished.
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ℓ
order.∼In a∝certain range of magnetic fields and electron
densities, the value ofσ (0) (14)is reasonablyclose nu-
xx
mericallytotheresultoftheself-consistentBornapprox-
4