Giant negative magnetoresistance in high-mobility 2D electron systems

0.36 MB·English

by  A. T. Hatke

Giantnegativemagnetoresistance inhigh-mobility 2D electron systems
A.T. Hatke,1 M.A. Zudov,1,∗ J.L. Reno,2 L.N. Pfeiffer,3 and K.W. West3
1School ofPhysicsandAstronomy, UniversityofMinnesota, Minneapolis, Minnesota 55455, USA
2Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
3Department of ElectricalEngineering, PrincetonUniversity, Princeton, NewJersey08544, USA
Wereport onagiant negativemagnetoresistance inveryhigh mobilityGaAs/AlGaAsheterostructures and
quantum wells. Theeffectisthestrongest atB ≃ 1 kG,where themagnetoresistivity develops aminimum
2 emergingatT .2K.Unlikethezero-fieldresistivitywhichsaturatesatT ≃2K,theresistivityatthisminimum
1 continuestodropatanacceleratedratetomuchlowertemperaturesandbecomesseveraltimessmallerthanthe
0 zero-fieldresistivity. Unexpectedly,wealsofindthattheeffectisdestroyednotonlybyincreasingtemperature
2 butalsobymodestin-planemagneticfields. Theanalysisshowsthatgiantnegativemagnetoresistancecannot
beexplainedbyexistingtheoriesconsideringinteraction-inducedordisorder-inducedcorrections.
n
a
PACSnumbers:73.43.Qt,73.63.Hs,73.40.-c
J

7
2 Over the past decade, low field magnetotransport in high turesandlowin-planefields,theresistivityatthisminimumis

mobility two-dimensionalelectron systems (2DESs) became asmallfractionofthezero-fieldresistivity. Remarkably,the
]
l a subject of considerable interest, in part, owing to the dis- GNMRisquicklysuppressednotonlybytemperaturebutalso
l
a covery of many unexpected phenomena.1–14 While the char- by modest (a few kG) in-plane magnetic fields. Our analy-
h acteristicfeaturesofthemajorityofthesephenomenaarenow sisofthelow-fieldmagnetoresistivityshowsthattheobserved
- understoodreasonablywell,15–27therearestillexistmanyun- GNMRcannotbeexplainedbyexistingtheoriesconsidering
s
e solvedpuzzles.Onesuchpuzzleistherecentlyreportedgiant either interaction-inducedor disorder-induced corrections to
m
microwavephotoresistivitypeakwhichemergesinthevicin- theDruderesistivity.
. ityofthesecondharmonicofthecyclotronresonance.13,14,28 Oursamples(A,B,andC)arelithographicallydefinedHall
t
a While its origin remains unclear, this peak so far has been bars(widthswA = 50µm, wB = 150µm, wC = 100µm).
m observed only in 2DESs which also exhibit giant negative Sample A is fabricated from a GaAs/AlGaAs Sandia-grown
- magnetoresistance(GNMR).13,14 Therefore,investigatingthe heterostructurewith densitynA 1.6 1011 cm−2 andmo-
d GNMR effect29 is not only interesting and important in its bility µA 5.4 106 cm2/Vs. S≈ample·B (C) is made from
n ≈ ·
ownrightbutmayalsoprovidenecessarycluestoaccountfor a Princeton-grown 24(30) nm-wide GaAs/AlGaAs quantum
o
c otherphenomena. well with density nB 4.3 1011 cm−2 (nC 3.4 1011)
[ The magnetoresistance can be characterized by the ratio andmobilityµB 1.0≈107cm· 2/Vs(µC 1.2 ≈107cm·2/Vs).
ρ(B)/ρ0,whereρ(B)andρ0arethelongitudinalresistivities Magnetoresistivit≈y ρ(B·) was measured ≈in a 3H· e cryostat at
1
measured with and without perpendicular magnetic field B, temperaturesuptoT =6.0Kusingastandardlowfrequency
v
respectively. Inthepresentstudy,we focusontheregimeof lock-intechnique.
9
7 weak magnetic fields where Shubnikov-deHaas oscillations InFig.1(a)[(b)]wepresentthemagnetoresistivityρ(B)in
6 arenotyetdeveloped.Inthisregime,thecharacteristicfeature sample A [sample B] measured at T from 0.5 K to 1.75 K
5 ofρ(B)isa broadminimumoccurringatB0 1kG.Quite [from 0.4 K to 1.6 K], in a step of 0.25 K [0.2 K]. In addi-
1. remarkably, the resistivity at this minimum, ρ≃(B0) ρmin, tiontoShubnikov-deHaasoscillations,bothsamplesreveala
≡
0 canbesignificantlylowerthanρ0,i.e. ρmin/ρ0 1,invery GNMR effectmarked by a pronouncedminimumwhich oc-
2 high mobility samples.13,14 In what follows we≪will use the cursatB0 1kGandbecomesprogressivelydeeperwithde-
1 valueofρmin/ρ0toquantitativelydescribetheGNMR. creasingT≃; incontrasttothezero-fieldresistivity, ρ0, which
:
v While negative magnetoresistance effect has been known remainsnearlytemperature-independent,theresistanceatthis
i for nearly three decades,30–32 systematic experimental stud- minimum, ρmin, decays rapidly and becomes a small frac-
X
iesinveryhighmobility(µ 107 cm2/Vs)2DESshaveap- tion of the zero-field resistivity. For example, in sample A,
ar pearedonlyrecently. Moresp∼ecifically,Bockhornetal.33 re- ρmin/ρ0 0.2atT =0.5K.
≈
portedthattheeffectquicklydisappearswithincreasingden- To examine the MR effect at higher T, we present in
sity; ρmin/ρ0 increased from 0.3 to 0.7 as the carrier Fig.1(c)themagnetoresistivityρ(B)insampleAattempera-
densitychangedfrom 2to ≈3 1011≈cm−2.34 Inaddition, turesfrom2Kto6K,inastepof0.5K.Here,wenoticethat
itwasfound33 (forthe≈carrier≈den·sityof 2.3 1011 cm−2) atT <4K,ρ(B)exhibitsphonon-inducedresistanceoscilla-
≈ ·
that the minimum resistivity roughly doubleswhen the tem- tions, owing to resonantelectron scattering on thermallyex-
peratureisraisedfrom0.1to0.8K. cited2k -acousticphonons.2,25,26,35–37Thesecondordermax-
F
InthisRapidCommunicationwesystematicallyinvestigate imaoftheseoscillationsoccuratB 1.3kG,asmarkedby
the roles of temperature and in-plane magnetic field on the nextto thetrace atT = 3.0K in F≈ig.1(c).38 AtT & 4K,
↓
GNMR effect observed in high mobility GaAs/AlGaAs het- the position of the resistivity minimumis shifted to a higher
erostructures and quantum wells. In all of our samples, the field( 1.5kG)andbothρ0andρmingrowataboutthesame
≈
effectmanifestsitself asa welldefinedminimumin the lon- rate,asevidencedbyroughlyparalleltracesinFig.1(c). The
gitudinalresistivityemergingatB0 1kG.Atlowtempera- spacingbetweenadjacenttracesremainsroughlyconstantin-
≃
2
a c a
8 1.75 K T = 6.0 K
14 12
6 )(cid:58) Sample A
()(cid:85)(cid:58)4 ((cid:85)min 8
12 ,0
(cid:85)
4
2
Sample A T = 0.5 K
0 0
b b
2 1.6 K 10 1.0
Sample B
(cid:70) (cid:70)
0.8
0
)(cid:58) (cid:18)(cid:85) 0.6
((cid:85) 1 8 (cid:85)min0.4
Sample A
0.2
Sample B T = 0.4 K Sample A T = 2.0 K
0 6 0.0
-2 -1 0 1 2 -2 -1 0 1 2 0 1 2 3 4 5 6
B (kG) B (kG) T (K)
FIG.2. (Coloronline)(a)ρ (opencircles)andρ (solidcircles)
0 min
FIG.1.(Coloronline)(a)ρ(B)ofsampleAat0.5K≤T ≤1.75K, versus T in sample A. Vertical line“separates” high and low tem-
withastep∆T =0.25K.(b)ρ(B)ofsampleBat0.4K≤T ≤1.6 perature regimes in sample A. (b) ρ /ρ versus T in sample A
min 0
K,∆T = 0.2K.(c)ρ(B)ofsampleAat2K≤ T ≤ 6K,∆T = (circles)andinsampleB(squares).
0.5 K. Arrows mark the second order maxima of phonon-induced
resistanceoscillations(seetext). which is introduced by tilting the sample normal by angle θ
with respect to the magnet axis. Figure 3(a) shows magne-
dicatinglineartemperaturedependenceoftheresistivityover toresistivity ρ(B) at selected θ from 0◦ to 89◦ measured in
theentirerangeofmagneticfields. sampleCatT 0.3K.Atθ = 0◦ weagainobserveGNMR
≃
For a quantitative analysis of the GNMR we present in characterizedbyρmin/ρ0 0.14. Withincreasingθ thedata
≈
Fig.2(a) the zero-field resistivity, ρ0 (open circles), and the revealrathercomplexbehavior;ρmin increaseswhileB0 be-
resistivity at the minimum, ρmin (solid circles), measured in comes smaller, decreasing roughlyby a factor of four at the
sample A for each T studied. The data clearly show that at highestangle.
T & 2.5K(totherightofthedashedverticalline),theresis- Toestimatethecharacteristicin-planefieldrequiredtosup-
tivitiesareclosetoeachother,ρ0 ρmin,bothfeaturingvery pressGNMRweextractρmin/ρ0fromthedatainFig.3(a)and
similar,approximatelylinear,tem≃peraturedependence. Such presenttheresultinFig.3(b)asafunctionof1/cosθ.Wefind
behaviorisconsistentwiththeelectronscatteringonthermal that ρmin/ρ0 doublesat 1/cosθ 5 which gives the scale
≃
acousticphonons.36,39 ofthein-planefield,Bk = B0/cosθ 5kG.Wenotethat
≃
similarin-planefieldvalueswerefoundnecessarytosuppress
At lower temperatures, T . 2.5 K (to the left of the
microwave-induced42 and Hall field-induced43 resistance os-
vertical line), the T dependences of ρ0 and ρmin become
cillationsoccurringinasimilarperpendicularfieldrange. At
markedly different. The decrease of ρ0 gets considerably
slowerastheacousticphononcontributionbecomesirrelevant highertiltanglesρmin/ρ0appearstosaturateat 0.8.
≈
andtheresistivitysaturatesatavaluedeterminedbyimpurity At first glance observed increase of ρmin with increasing
scattering.36,39,40Quiteremarkably,incontrasttoρ0,ρminnot tiltanglemightoriginatefromthein-planefield-inducedpos-
itivemagnetoresistanceeffect,recentlyreportedinveryhigh
onlycontinuestodropatlowertemperaturesbutalsodoesso
mobility2DEG.44 However,accordingtoRef.44anorderof
ata muchfasterrate. Sucha suddenchangeofthetempera-
magntitude higher B is needed to double the resistance in
turedependenceofρministotallyunexpected.Quantitatively, k
a 30 nm-wide quantum well. Therefore, further studies are
oncethe temperatureis loweredfrom2.5K to 0.5K, ρ0 de-
neededtoclarifytheoriginoftheB -inducedsuppressionof
creasesonlyby about20%while ρmin dropsbymorethana k
factoroffive.41 theGNMReffect.
IntheremainderofthisRapidCommunication,wefocuson
Usingρ0andρminshowninFig.2(a),wecalculateρmin/ρ0
thetemperaturedependenceofthelow-fieldmagnetoresistiv-
and present the result (circles) in Fig.2(b) as a function of
itypreceedingtheformationofthedeepminimumatB =B0.
temperature. ResultsforsampleBobtainedinthesameway
More specifically, we analyze the low B part of the data in
using the data in Fig.1(b) are represented by squares. Both
termsof
samples show a rapid increase of ρmin/ρ0 with increasing
temperatureandeventualsaturationatρmin/ρ0 1. ρ(B)
≃ =1 βB2, (1)
We next examine the effect of an in-plane magnetic field ρ0 −
3
a a
z (cid:84) Btot 1.0 2.0 K Sample A
2.0
y 0.8 1.5 K
x o
89
(cid:70) 87o (cid:85)00.6
1.5 (cid:70) o /(cid:85) 1.0 K
86
(cid:70) 0.4
)(cid:58) 0.5 K
((cid:85) 83o 0.2
1.0 (cid:70)
0.0
80o -2 -1 0 1 2
(cid:70) B (kG)
0.5
7(cid:84)4 =o 0o b c
1
Sample C T = 0.3 K
0.0 )
-2 -1 0 1 2 -2G0.1
B (kG) k
(
1.0 (cid:69)
b
0.8 0.01
(cid:85)00.6 4 6 8 1 2 4 60 1 2 3 4 5 6
(cid:18)
(cid:85)min0.4 T (K) T (K)
FIG.4.(Coloronline)(a)Solidcurvesrepresentρ(B)/ρ measured
0
0.2 insampleAat T from0.5 Kto2.0 K,asmarked. Dashedcurves
arefitstothedata,ρ(B)/ρ = 1−βB2,at|B| ≤ 0.5kG.(b,c)β
0
0.0 versusT. Solidlinesarefitstothedata(seetext)anddashedlines
1 5 10 15 20 areβsmcalculatedusingEq.(3).
i
1/cos(cid:84)
FIG.3.(Coloronline)(a)ρ(B)ofsampleCatT =0.3Katdifferent
tivemagnetoresistance,seeEq.(1),withβ givenby
tiltanglesθ(asmarked).(b)ρ /ρ versus1/cosθ(circles).Solid
min 0
curveisaguidetoaneye.
e2 τ 1/2
β = L , 0<τ−1 τ−1. (2)
andthenexamineβ asafunctionoftemperature. InFig.4(a) d 2πn p2 (cid:18)2τ (cid:19) L ≪ S
S F S
weplotnormalizedmagnetoresistivity,ρ(B)/ρ0,measuredin
sample A atT from0.5 K to 2.0 K, in a step of 0.5K.45 To
Here,τ−1andτ−1arelong-andshort-rangedisordermomen-
extractβwefitthedatausingEq.(1)overtherange B 0.5 L S
| |≤ tumrelaxationrates,τ−1 =τ−1+τ−1,47n isthearealden-
kG(cf.dashedlines)andobservethatthecurvatureofthelow L S S
sityofshort-rangescatterers,andp istheFermimomentum.
fieldresistivityβ decreaseswithincreasingtemperature. F
Equation(2)isvalidforβ B2 1andathigherB theresis-
iedA,fwteer rperpeeseantitngextthreacfitetdtinβg ipnroFciegd.u4r(eb)foarnadllFoigth.e4r(cT) usstuindg- tivityisexpectedtosaturatdeatρ≪min ≃ρ0·(τS/τL)≪ρ0.46,48
log-log and log-linear scale, respectively. First, we notice Whilethedisordermodelcan,inprinciple,leadtoGNMR,
that at T . 1 K, β shows a sign of saturation and can be it clearlyfails to explainourexperimentalfindings. First, as
well described by β 1.45 T2/T02, T0 1.7 K [cf. shown above, β exhibits strong dependence on temperature
solid curve in Fig.4(b≈)]. At h−igher T the da≈ta can be de- whichdoesnotenterEq.(2). Second,webelievethattheas-
scribed by either β T−2.6, T & 2.5 K [cf. solid line in sumptionof τ−1 τ−1 is notsatisfied in our samples. In-
F1[c.i0fg..Ks4o(lb.i)d]Tloinr.ebsy3in.β5FKi∝g∝.a4en(xdcp)T(]2.−I≈Tti/s1Tc.91l,e2Ka)r,ftwohraht3et.rh5eeKTte1m.≈pTe1ra..0tu6Kre.0dfoeK-r ndinaelesladym,ntpholeteicaAen4at9hlyLassutigswg≪ohefislHetsSaRollepffip.eo1ls3dit-ceinordneuclalcuteidodenrd,eτsthLi−sat1tan≃thcee5MτoSs−Rc1i.lilnWattiehoefinis-r
pendenceofβisrathercomplexwhichislikelyaresultofone samplescanbeconsistentlydescribedbyEq.(2),50neitherthe
orseveralcrossoversbetweendifferentregimes. Inwhatfol- temperaturedependencenorthevalidityofτ−1 τ−1 con-
L ≪ S
lowsweexamineβ(T)intermsofexistingtheoreticalmodels ditionhasbeenexamined.
andcomparetheresultsofouranalysistootherexperimental Electon-electron interaction model,32,51,52 on the other
studies.13,33 hand,predictsatemperature-dependentmagnetoresistance.In
Quasiclassicaldisordermodel,46predictsaparabolicnega- theballisticregime,~/τ k T,andforsmoothdisorderpo-
B
≪
4
tentialthismodelalsoleadstoEq.(1),withβ givenby52 In summary, a giant negative magnetoresistance effect in
high-mobility GaAs/AlGaAs heterostructures and quantum
βsm =µ2 ρ0 c0 ~/τ 1/2, τ−1 =0. (3) wells is marked by a pronounced minimum of the longitu-
i R π (cid:18)k T(cid:19) S dinal resistivity appearing at B 1 kG. The temperature
K B ≃
dependence clearly reveals a crossover between two distinct
Here, RK = h/e2 is the von Klitzing constant and c0 = regimes. In the high temperature regime, the zero-field re-
3ζ(3/2)/16√π 0.276. However, Eq.(3) also fails to de- sistivityandtheminimumresistivitybothexhibitlineartem-
scribe our findin≃gs. Indeed, takingT = 1 K as an example, perature dependence, due to scattering on thermal acoustic
ourexperimentgivesβ 1.1kG−2 whichis nearlytwo or- phonons. Inthelowtemperatureregime,however,zero-field
ders of magnitude larger≈than βsm 0.014 kG−2 obtained resistivityquicklysaturatesbuttheminimumresistivity con-
fromEq.(3). Comparisonofβsmi ob≈tainedusingEq.(3)[cf. tinuestodecreaseatanevenfaster rateeventuallybecoming
i
dashed line in Figs. 4(b) and 4(c)] with our data shows that asmallfractionofthezero-fieldresitivity. Unexpectedly,we
thediscrepancyremainssignificantoverthewholerangeofT also find that the GNMR is destroyed not only by tempera-
studied.Moreover,itthisclearthattheinteractionmodelfails ture but also by very modest (a few kG) in-plane magnetic
toexplainourdataevenonaqualitativelevel. Wealsonotice fields. Finally, ouranalysisof the low-fieldmagnetoresistiv-
thatsignificantdisagreementwithEq.(3)wasfoundinRef.33 itydemonstratesthattheGNMReffectcannotbeunderstood
reportinglow-temperatureβwhichisroughly30(n 2 1011 byexistingtheoreticalmodelsconsideringeitherinteraction-
cm−2)to150(n 3 1011cm−2)timeslargerthan≈βsm·.53 inducedordisorder-inducedcorrections,evenonaqualitative
We next cons≈ider· several scenarios for the obiserved level. Takentogether,these findingsprovideimportantclues
discrepancy. First, in a realistic high-mobility 2DEG, foremergingtheoriesandshouldhelptoelucidatetheorigin
sharp disorder, which is not present in Eq.(3), plays a oftheGNMRinveryhighmobility2DES.
crucial role in many of the low-field magnetotransport
WethankM.Dyakonov,R.Houg,M.Khodas,D.Polyakov,
phenomena.3,9–11,22–24,35,46,48,54–58 For the case of mixed dis-
M.Raikh,andB.ShklovskiifordiscussionsandG.Jones,T.
orderpotentialEq.(3)isgeneralizedto52
Murphy,andD.Smirnovfortechnicalassistance.Aportionof
thisworkwasperformedattheNHMFL,whichissupported
3τ τ
βmix = 4 Lβsm. (4) by NSF CooperativeAgreementNo. DMR-0654118,by the
i (cid:18) − τL(cid:19)r τ i State of Florida, and by the DOE and at the Center for Inte-
grated Nanotechnologies, a U.S. Department of Energy, Of-
oI4fu(τrτLL−s/a1τm)p≪1l/e2Aτ≫S−,1a,s1tmwheehrniectihoanpleepadedaasrbstooavβep,imaτrix−am1≫etrβi5cisaτml−l.y1Hlfaroorwgmeevwfeahrc,itcoihnr fifeorcgreyI,onOfteBfgfiarcasetiecodEfNBneaarnsgiocyteEScnchieneronglcyoegSsiceuises,enracefUasc.uSils.ietyrDfeaapncadilritatmyt.ethTnetheCofwenEotrnekr-
L ≃ S
weestimate(4 3τ/τ ) τ /τ 1.5. Suchasmallfactor at Minnesota was supported by the NSF Grant No. DMR-
L L
− ≈
isclearlynotsufficienttopexplainthediscrepancy. 0548014(measurementsat Minnesota on SamplesA and B)
Anotherpossiblecause forlargeβ isthedisorder-induced andbytheDOEGrantNo. DE-SC002567(tilt-fieldmeasure-
T-independent correction, similar to that given by Eq.(2). ments at NHMFL on sample C). The work at Princeton was
Assuming that the contributions are additive, one has β = partiallyfundedbytheGordonandBetty MooreFoundation
β +β , where β (β ) T0(T−1/2). It is clear, however, and the NSF MRSEC Program through the Princeton Cen-
d i d i
∝
thattheexperimentallyobtainedβ(T)cannotbedescribedby ter for Complex Materials (DMR-0819860)and the work at
suchdependence.59 SandiawassupportedbytheSandiaCorporationunderCon-
Finally, theory should consider a possibility that the low- tract No. DE-AC04-94AL85000. Sandia National Labora-
temperaturemagnetoresistanceoriginatesprimarily from the tories is a multi-program laboratory managed and operated
quasiclassicaldisordermechanismwhich,however,issignif- by Sandia Corporation, a wholly owned subsidiary of Lock-
icantly altered by the electron-electron interactions with in- heedMartinCorporation,fortheU.S.DepartmentofEnergy’s
creasing temperature.60 However, such a theory remains a NationalNuclearSecurityAdministrationundercontractDE-
subjectoffuturework. AC04-94AL85000.
∗ Correspondingauthor:[email protected] Lett.90,046807(2003).
1 M.A.Zudov,R.R.Du,J.A.Simmons,andJ.L.Reno,Phys.Rev. 6 C.L.Yang,M.A.Zudov,T.A.Knuuttila,R.R.Du,L.N.Pfeiffer,
B64,201311(R)(2001). etal.,Phys.Rev.Lett.91,096803(2003).
2 M.A.Zudov,I.V.Ponomarev,A.L.Efros,R.R.Du,J.A.Sim- 7 I. V. Kukushkin, M. Y. Akimov, J. H. Smet, S. A. Mikhailov,
mons,etal.,Phys.Rev.Lett.86,3614(2001). K.vonKlitzing,etal.,Phys.Rev.Lett.92,236803(2004).
3 C.L.Yang,J.Zhang, R.R.Du,J.A.Simmons,andJ.L.Reno, 8 A.A.Bykov,J.-Q.Zhang,S.Vitkalov,A.K.Kalagin,andA.K.
Phys.Rev.Lett.89,076801(2002). Bakarov,Phys.Rev.Lett.99,116801(2007).
4 R.G.Mani,J.H.Smet,K.vonKlitzing,V.Narayanamurti,W.B. 9 W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys.
Johnson,etal.,Nature(London)420,646(2002). Rev.Lett.98,106804(2007).
5 M.A.Zudov,R.R.Du,L.N.Pfeiffer,andK.W.West,Phys.Rev. 10 A.T.Hatke,H.-S.Chiang,M.A.Zudov,L.N.Pfeiffer,andK.W.
5
West,Phys.Rev.Lett.101,246811(2008). whenbothtypesofdisorderarepresent,theresistivityshouldre-
11 M.Khodas,H.S.Chiang,A.T.Hatke,M.A.Zudov,M.G.Vav- mainfinite46andcanevendivergeathighB.48Closeexamination
ilov,etal.,Phys.Rev.Lett.104,206801(2010). ofFig.2(a-b)showsasignofsaturationofρ atlowtempera-
min
12 A.T.Hatke,H.-S.Chiang,M.A.Zudov,L.N.Pfeiffer,andK.W. tures.
West,Phys.Rev.B82,041304(R)(2010). 42 C.L.Yang,R.R.Du,L.N.Pfeiffer,andK.W.West,Phys.Rev.
13 Y.Dai,R.R.Du,L.N.Pfeiffer,andK.W.West,Phys.Rev.Lett. B74,045315(2006).
105,246802(2010). 43 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys.
14 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys. Rev.B83,081301(R)(2011).
Rev.B83,121301(R)(2011). 44 X.Zhou,B.A.Piot,M.Bonin,L.W.Engel,S.DasSarma,etal.,
15 V.I.Ryzhii,Sov.Phys.SolidState11,2078(1970). Phys.Rev.Lett.104,216801(2010).
16 A.C.Durst, S.Sachdev, N.Read, andS.M. Girvin, Phys.Rev. 45 Inorder toimprove thefittingprocedure, aslight asymmetry of
Lett.91,086803(2003). therawdatainFig.1(a)waseliminatedbysubtractingalinearin
17 X.L.LeiandS.Y.Liu,Phys.Rev.Lett.91,226805(2003). Bcomponenttoensureρ(B )=ρ(−B ).
0 0
18 I.A.Dmitriev,A.D.Mirlin,andD.G.Polyakov,Phys.Rev.Lett. 46 A.D.Mirlin,D.G.Polyakov,F.Evers,andP.Wo¨lfle,Phys.Rev.
91,226802(2003). Lett.87,126805(2001).
19 M.G.VavilovandI.L.Aleiner,Phys.Rev.B69,035303(2004). 47 The limiting cases of τ−1 = 0, known as Lorentz gas, and of
20 I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and τ−1 = 0 have been stuLdied in Refs.64,65,66 and Refs.67,68,
S
D.G.Polyakov,Phys.Rev.B71,115316(2005). respectively.
21 I.A.Dmitriev,A.D.Mirlin,andD.G.Polyakov,Phys.Rev.B75, 48 D.G.Polyakov,F.Evers,A.D.Mirlin,andP.Wo¨lfle,Phys.Rev.
245320(2007). B64,205306(2001).
22 M.G.Vavilov,I.L.Aleiner,andL.I.Glazman,Phys.Rev.B76, 49 TheapproachofRef.57yieldsτ /τ ≈6,whereτ−1istheelec-
π π
115331(2007). tron backscattering rateoff short-range disorder. Withτ ≃ τ
S π
23 M.KhodasandM.G.Vavilov,Phys.Rev.B78,245319(2008). oneobtainsτ−1 ≈ (1/6)τ−1 andτ−1 ≈ (5/6)τ−1.Thisanal-
S L
24 I. A. Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov, and ysisassumed isotropic scatteringoff short-range disorder which
M.G.Vavilov,Phys.Rev.B80,165327(2009). mightnotbeexactlythecase.
25 O.E.Raichev,Phys.Rev.B80,075318(2009). 50 Ref.13assumedτ /τ ≃ ρ /ρ andusingEq.(2)foundthat
S L min 0
26 I. Dmitriev, S. Dorozhkin, and A. Mirlin, Physica E 42, 1159 intheirsamples(withµrangingfrom0.9·107cm2/Vsto3.0·107
(2010). cm2/Vs)n variesbetween(2.6µm)−2 and(8.0µm)−2.These
S
27 A.T.Hatke,M.Khodas,M.A.Zudov,L.N.Pfeiffer,andK.W. values correspond ton(3D) ∼ n /a ∼ 1013 cm−3 and 1012
West,Phys.Rev.B84,241302(R)(2011). cm−3 (a ≃ 10 nm isSa Bohr raSdiuBs in GaAs). The same pro-
B
28 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys. cedureappliedtothedatafromsampleAatT = 0.5Kleadsto
Rev.B83,201301(R)(2011). n ≃(1.6µm)−2.
S
29 Significant low-field negative magnetoresistance has been ob- 51 I.V.GornyiandA.D.Mirlin,Phys.Rev.Lett.90,076801(2003).
servedinseveralearlyexperiments.2,3,61. 52 I.V.GornyiandA.D.Mirlin,Phys.Rev.B69,045313(2004).
30 A.M.Paalanen,D.C.Tsui,andJ.C.M.Hwang,Phys.Rev.Lett. 53 Thedisagreement withEq.(3)isnotonlyinthemagnitudeofβ
51,2226(1983). butalsoinitsmuchstrongerdependenceondensity.
31 K. K. Choi, D. C. Tsui, and S. C. Palmateer, Phys. Rev. B 33, 54 W.Zhang,H.-S.Chiang,M.A.Zudov,L.N.Pfeiffer,andK.W.
8216(1986). West,Phys.Rev.B75,041304(R)(2007).
32 L.Li,Y.Y.Proskuryakov, A.K.Savchenko, E.H.Linfield,and 55 A.T.Hatke,H.-S.Chiang,M.A.Zudov,L.N.Pfeiffer,andK.W.
D.A.Ritchie,Phys.Rev.Lett.90,076802(2003). West,Phys.Rev.B77,201304(R)(2008).
33 L.Bockhorn,P.Barthhold,D.Schuh,W.Wegscheider,andR.J. 56 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys.
Haug,Phys.Rev.B83,113301(2011). Rev.Lett.102,066804(2009).
34 The corresponding mobility variation was from µ ≃ 0.9·107 57 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys.
cm2/Vstoµ≃1.1·107cm2/Vs. Rev.B79,161308(R)(2009).
35 W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. 58 A.Hatke,M.Zudov,L.Pfeiffer,andK.West,PhysicaE42,1081
Rev.Lett.100,036805(2008). (2010).
36 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys. 59 Fittingthedatawithβ =β +b·T−1/2leadstoβ < 0atany
0 0
Rev.Lett.102,086808(2009). temperature.
37 A.T.Hatke,M.A.Zudov,L.N.Pfeiffer,andK.W.West,Phys. 60 D.G.Polyakov,privatecommunication.
Rev.B84,121301(2011). 61 V.Umansky, R.dePicciotto,andM.Heiblum,Appl.Phys.Lett.
38 Themaximaoftheseoscillationsoccurclosetointegervaluesof 71,683(1997).
theratio, 2kFs/ωc (Ref.2), wheres ≃ 3.4km/sisthevelocity 62 A.GoldandV.T.Dolgopolov,Phys.Rev.B33,1076(1986).
of the transverse acoustic mode in GaAs. In sample A, we find 63 A.Gold,Phys.Rev.B41,8537(1990).
2~kFs/kB ≈2.6K. 64 E.M.Baskin,L.N.Magarill,andM.V.Entin,Sov.Phys.JETP
39 H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, 48,365(1978).
Phys.Rev.B41,1278(1990). 65 A.V. Bobylev, F.A.Maaø, A.Hansen, and E.H. Hauge, Phys.
40 Effectsoftemperature-dependentscreening(see,e.g.Refs.62,63) Rev.Lett.75,197(1995).
onthelow-temperatureelectronmobilityinoursamplesareneg- 66 A. Dmitriev, M. Dyakonov, and R. Jullien, Phys. Rev. B 64,
ligible. 233321(2001).
41 Itmightappearthatρmin(T) → 0asT → 0,abehaviorwhich 67 M.M.Fogler,A.Y.Dobin,V.I.Perel,andB.I.Shklovskii,Phys.
wouldsignalaB-inducedmetal-insulatortransitionpredictedin Rev.B56,6823(1997).
2DESwithonlysharp64–66oronlysmooth67,68disorder.However, 68 A.D.Mirlin, J.Wilke, F.Evers, D.G.Polyakov, andP.Wo¨lfle,
Phys.Rev.Lett.83,2801(1999).