Memory Effects in Electron Transport in Si Inversion Layers in the Dilute Regime: Individuality versus Universality

4 Pages·0.3 MB·English

by  V. M. Pudalov

Memory Effects in Electron Transport in Si Inversion Layers in the Dilute Regime:
Individuality versus Universality
V. M. Pudalova,b, M. E. Gershensona, and H. Kojimaa
a Department of Physics and Astronomy, Rutgers University, New Jersey 08854, USA
b P. N. Lebedev Physics Institute, 119991 Moscow, Russia
Inordertoseparatetheuniversalandsample-specificeffectsintheconductivityofhigh-mobility
2 Si inversion layers, we studied the electron transport in the same device after cooling it down to
0 4K at different fixed values of the gate voltage Vcool. Different Vcool did not modify significantly
0 eitherthemomentumrelaxationrateorthestrengthofelectron-electroninteractions. However,the
2
temperature dependences of the resistance and the magnetoresistance in parallel magnetic fields,

n measured in thevicinityof themetal-insulator transition in2D,carry astrongimprint ofindividu-
a alityofthequencheddisorderdeterminedbyVcool. Thisdemonstratesthattheobservedtransition
J between “metallic” and insulating regimes involves both universal effects of electron-electron inter-

1 actionandsample-specificeffects. Farawayfromthetransition,atlowercarrierdensitiesandlower
resistivities ρ<0.1h/e2, thetransport and magnetotransport become nearly universal.

]
l
e Afteralmostadecadeofintensiveresearch,theappar- the range n = (0.7−3)×1011cm−2 in a system with a
-
r ent metal-insulator transition (MIT) in two-dimensional snapshot disorder pattern. Two key features of the 2D
t
s (2D) systems remains a rapidly evolving field [1]. The MIT, strong dependences of the resistivity on the tem-
.
t central problem in this field is whether the anomalous perature and parallel magnetic field, have been studied.
a
low-temperature behavior of the conductivity, observed Wehaveobservedtwodistinctregimesasafunctionof
m
in high-mobility structures in the dilute limit, signifies a the electron density. At relatively high densities (resis-
-
d novel quantum ground state in strongly-correlated sys- tivity ρ≤0.1h/e2), the dependences R(T)and R(Bk) in
n tems, or this is a semiclassical effect of disorder on elec- weak parallel magnetic fields B are similar for different
k
o tron transport. Indeed, a great body of experimental cooldowns; the similarity indicates ‘universal’ behavior.
c data demonstrates that, at leastat sufficiently large car- In contrast, at low densities (ρ ∼ (0.1−1)h/e2), or in
[
rier densities (consequently, weak interactions), the low- moderateandstrongparallelfieldsgµ B ∼E ≫k T,
B k F B
1 temperature behavior of disordered systems is governed the cooling conditions affect dramatically the electron
v
by the universal quantum corrections to the conductiv- transport. This observation provides direct experimen-
1
ity [2]. On the other hand, there are also observations talevidence thatthe behaviorofdilute systems becomes
0
0 that near the apparent 2D MIT, the behavior of dilute sample-specificneartheapparent2DMITnomatterhow
1 systems is very rich, and does not necessarily follow the one approaches the transition (either by decreasing the
0 same pattern (e. g., even for the same system, as Si electron density, or by increasing the parallel magnetic
2
MOSFETs, the ”critical” resistance and the high-field field).
0
magnetoresistancevarysignificantlyfordifferentsamples Theresistivitymeasurementswereperformedonahigh
/
at [3,4]). This duality (universality versus individuality) is mobility Si-MOSFET sample [17] at the bath tempera-
m reflectedintwoapproachestothe theoreticaldescription tures 0.05−1.2K. The crossedmagnetic field system al-
of the ”metallic” regime: some models, based on strong lowed to accurately align the magnetic field parallel to
-
d electron-electron interactions, treat the transition as a the plane of the 2DEG [18]. The carrier density, found
n
universalphenomenon[5–10],whereastheothersempha- fromtheperiodofShubnikov-deHaas(SdH)oscillations,
o
sizetheroleofasample-specificdisorder(traps,localized varies linearly with V : n ≈ C ×(V − V ), where C
c g g th
: spins, potential fluctuations, etc.) [11–15]. (= 1.103×1011/Vcm2 for the studied sample) is deter-
v
In order to find a definite experimental answer to mined by the oxide thickness. The ‘threshold’ voltage
i
X thisproblemandtoseparateuniversalandnon-universal Vth varied little (within 0.15V) for different cool-downs
r (“individuality”)effectsinthevicinityofthe2DMIT,we andremainedfixedaslongasthesamplewasmaintained
a
have studied the electron transport in the same Si MOS at low temperatures (up to a few months).
structure,whichwasslowlycooleddownfromroomtem- Figure 1 shows the mobility µ versus V for five differ-
g
perature to T =4K at different fixed values of the gate ent cool-downs with Vcool = 0,5,10,18, and 25V. The
voltageV =Vcool [16]. Changingthe coolingconditions peakmobilityfordifferentcool-downsvariesbylessthan
g
allowedustovarytheconfiningpotentialandthedensity ∼7%);thisdemonstratesthatthemomentumrelaxation
of quenched localized states without affecting the main time τ is not strongly affected by the cooling conditions.
parameters which control electron-electron interactions We also observed that the amplitudes of the SdH oscil-
in the system of mobile electrons: the momentum relax- lations are similar for different cool-downs, as shown in
ationrateandtheinteractionconstants. BytuningV at theinsettoFig.1. Thesetwoobservationsareconsistent
g
low temperatures, we varied the electron density n over with each other, since the quantum lifetime τ is nearly
q
1
equal to τ for Si-MOSFETs. criticaldependences ρ (T) suggeststhat a) saturationof
c
thetemperaturedependencesatn=n ,reportedin[21],
c
is notauniversaleffect, andb) the criticaldensityis not
necessarily associated with the sign change of dρ/dT(n)
[19].
2
0.5
10
Vcool= 5V
= 25V
s)
V
2m/ 5
m ( 1 2h/e)0.4 VVccooooll== 05VV 2e) 3 nc5 21
(xx VVccooooll==1108VV (p/ 3
r Vcool=25V r 4
0.3 n25 5
c
0.0 0.5 B (T) 1.0 6
0
0 2 4 6 8 10 12 1 7
V (V)
g

FIG. 1. The mobility versus the gate voltage for dif-
0.3
cool
ferent cool-downs. The V values for both the main
panel and the inset are shown in the figure. Examples 2)
e
of the SdH oscillations, shown in the inset for the same p/ 8
Vg = 1.15V, T = 0.1K, Bk = 0.03T, demonstrate that (
r
the quantum time τq is not very sensitive to the cooling 0.1 9

conditions. Thecarrierdensitiesare(fromtoptobottom)
n=1.081, 1.092, 1.070 in units1011cm−2. 10
11
0.5
Figure 2a shows the dependences ρ(T) for two cool- 12
downs in the vicinity of the 2D MIT. Far from the 0.0 0.4 0.8 1.2
transition, where ρ ≪ h/e2, the dependences ρ(T) are T (K)
cooldown-independent. However, in the vicinity of the
FIG.2. Temperature dependencesof theresistivity for
transition (ρ ∼ h/e2), a dramatically different behav-
twodifferentcool-downs. Thedensities,whichcorrespond
ior is observed. The irreproducibility of ρ(T) for differ-
tocurves1to12,areasfollow: 0.783, 0.827,0.882, 0.942,
ent cool-downs is clearly seen for the curves in Fig. 2 0.972, 1.001, 1.021, 1.31, 1.53,1.87, 2.29,2.58inunitesof
which correspond to the same ρ at the lowest T: these 1011cm−2.
curves, being different at higher temperatures, converge
with decreasing T. The sample-specific memory effects We now turn to the magnetoresistance (MR) in par-
vanishalsoatsufficientlylowtemperatures: thissuggests allel fields; the data are shown in Figs. 3 and 4. This
that the underlying mechanism is related to the finite- MR is usually associated with the spin effects [1,4]. In
temperatureeffectsinasystemwhichretainsaquenched the theoretical models of the parallel-field MR, based
disorder. These results suggest that, in addition to uni- on electron-electron interactions, the MR is controlled
versal effects, a finite-temperature and sample-specific by the effective g∗-factor and the momentum relaxation
mechanism, which strongly affects the resistivity, comes time τ [5,10,22]. Animportantadvantageofour method
into play. isthatcoolingofthe samesampleatdifferentVcool does
The ‘critical’ density n , which corresponds to the not affect these parameters. Thus, one might expect to
c
transition, was found from a linear extrapolationto zero observeasample-independentbehavioriftheMRiscon-
of the density dependence of the activation energy ∆(n) trolled solely by the universal interaction effects.
in the insulating regime ρ(T) ∝ exp(∆/T) [3,19]. The Firstly,let us considerthe rangeoffields muchweaker
dependencesρ(T),whichcorrespondedton=nc fortwo than the field of complete spin polarization (g∗µBBk ≪
cool-downsshowninFig.2a,arehighlightedinbold. Itis EF). The insets to Figs. 3a and 3b show that the MR
clearthat(i) the ‘critical’densitiesand‘critical’resistiv- is proportional to Bk2 at g∗µBBk/kBT ≤ 1. We found
ities depend on the cooling conditions, and (ii) the ‘crit- that the slope dρ/dB2 is nearly cooldown-independent
ical’ dependences ρ(T,n = n ) are non-monotonic (see (i.e. universal)only for the densities n>1.3×1011cm−2
c
also Refs. [19,20]). Observation of the non-monotonic (which are by 30% greater than the critical density n ),
c
2
or for the resistivities ρ < 0.16h/e2 (compare insets to the intercept of the tangents at fields below and above
Figs.3aand3b). Withapproachingn ,thisuniversality MR saturation [4].
c
vanishes: Figure 3a shows that even when the zero-field Figure 4c shows the dependences B (n) for differ-
sat
resistivity is as small as 0.22h/e2, the slope varies by a ent cool-downs. We also plotted here the density depen-
factor of 1.3 for different Vcool. dence of the field B = 2E∗/g∗µ = nπ¯h2/m∗g∗µ ,
pol F B B
For the intermediate fields, kBT < g∗µBBk < EF, for the complete spin polarization of mobile electrons
the ρ(Bk) behavior is not universal over the whole den- (m∗ istherenormalizedeffectivemass). Thedependence
sity range n = (1−3)×1011cm−2 (Figs. 3a,b). As n B (n) was calculated using sample-independent (uni-
pol
decreases and approaches nc, the cooldown-dependent versal) g∗m∗ values [18]. Comparison between Bsat and
variations of R(Bk) increase progressively. Observation Bpol shows that Bsat does not necessarily manifest the
of large (∼ 50%) non-universal variations of the MR at complete spin polarization, and that the coincidence of
intermediate fields makes the scaling analysis [23] of the B with B reported in Ref. [24] may be rather acci-
sat pol
MR in this field range dubious. dental. This non-universal, sample-dependent behavior
ofB agreeswithearlierobservationsmadeondifferent
sat
5
samples [4]. We emphasize that the curves for different
Vcool (in eachofFigs.4a andb) correspondto the same
0.4
4 2e) density, as follows from Hall voltage and/or SdH oscilla-
2 (h/e) 3 00..23r (h/ tdpieaoprnaeslnlmedleefinaetsludprsaermisaemnnoetttse.sro,TleshluyegrgfeaelcsatttsetdhthattaotBsptshainte-ipMsoaRlarciioznoaltsditoornwonnog-f
r 2 0.00 0.02B2 (T2)0.04 mobile electrons; we speculate that it also reflects the
spinpolarizationofthesample-specificlocalizedelectron
10V
states, which might have the effective g-factor quite dif-
18V
1 25V ferent from that for the mobile electrons [4].

a)
0.0 0.1 0.2 0.3 a) b)
B2 (T2)
10 n=1.092 n=1.330
)
25V 0.25 2e) T=0.3K T=0.3K 12/e
2 15V8V 0.202 (h/e) r (h/ 1 VVccooooll==11 08VV r (h
r Vcool= 5V
2)
(h/e 10.00 0.03 B2 (T20).06 0.15 0 2 4 B (T) 6 0 2 B (T4) 6 0.1
r
25V c)
5V ) 6
18V b) (T B
00.0 0.2 0.4 0.6 pol pol
B2 (T2) 2 B, Bsat 4 Vcool=5V
FIG. 3. Examples of the dependences ρ(Bk) at Vcool=10V
T =0.3Kforthecarrierdensity(a)1.20×1011cm−2 and Vcool=18V
(b) 1.34×1011cm−2. The insets blow up the low-field Vcool=25V
region of the quadratic behavior. The values of Vcool are 2
1.0 1.2 1.4
indicated for each curve.
density (1011cm-2)
The influence of cooldown conditions becomes even
more dramatic in strong fields, B >∼ EF/g∗µB. De- cooFlI-Gdo.w4n.sRaetsisttwivoitdyevnssitinie-sp:laan)e nma=gn1et.0ic92fie×ld1f0o1r1ctmhr−e2e
spite the fact that the dependences µ(n) for different and b) n = 1.33 × 1011cm−2. c) The saturation field
cool-downs are very similar (Fig. 1), we observed very
Bsat versusnforfourdifferentcool-downs. Solidlinesare
large variations in the strong-field MR. Figures 4a and
guidestotheeye. Dashedlineshowsthefieldofcomplete
4b show R(Bk) for different cool-downs at two values of spin polarization calculated on the basis of direct mea-
n. The cooldown conditions cause factor-of-five changes surements of the spin susceptibility for mobile electrons
inρ(B)inhighfieldsandfactor-of-twochangesintheval- [18].
ues of B =B at which the MR ‘saturates’ at a given
k sat
carrierdensity. Thelatterquantitywasdeterminedfrom It is worth mentioning that the influence of vari-
3
able disorder on transport and magnetotransport in Si- [3] V. M. Pudalov, G. Brunthaler, A. Prinz, and G. Bauer,
MOSFETshasbeenobservedearlier. Boththetempera- JETP Lett., 68, 442 (1998).
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k
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from the tails of upper subbands. At low temperatures,
insulator transition in 2D involves both universal and
whenever gate voltage is varied, the potential well re-
sample-specific effects. These results also help to estab-
mains almost unchanged and memorizes an imprint of
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[19] Thetwomethodsoffindingnc,suggestedasequivalentin
TheworkwaspartiallysupportedbytheNSF,INTAS,
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NATO, NWO, and Russian Programs RFBR, ‘Statisti-
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4