Hall field-induced resistance oscillations in tilted magnetic fields

0.18 MB·English

by  A. T. Hatke

Hallfield-induced resistanceoscillationsintiltedmagneticfields
A.T. Hatke,1 M.A. Zudov,1,∗ L.N. Pfeiffer,2 and K.W. West2
1School ofPhysicsandAstronomy, UniversityofMinnesota, Minneapolis, Minnesota 55455, USA
2Princeton University, Department of Electrical Engineering, Princeton, NJ08544, USA
(ReceivedNovember10,2010)
Wehavestudiedtheeffectofanin-planemagneticfieldonHallfield-inducedresistanceoscillationsinhigh
mobilitytwo-dimensionalelectronsystems. Wehavefoundthattheoscillationfrequencydependsonlyonthe
1 perpendicular component ofthemagneticfieldbuttheoscillationamplitudedecaysexponentiallywithanin-
1 planecomponent. Whilethesefindingscannotbeaccountedforbyexistingtheoriesofnonlineartransport,our
0 analysis suggests that the decay can be explained by an in-plane magnetic field-induced modification of the
2 quantumscatteringrate.

n
PACSnumbers:73.43.Qt,73.63.Hs,73.21.-b,73.40.-c
a
J

6 High mobility two-dimensionalelectron systems (2DESs) HIROsaredescribedby:8,10
2
exhibit an array of fascinating transport phenomena occur-

] ringinveryhighLandaulevelswheretheShubnikov-deHaas δr ≃ 16τtrλ2cos2πǫdc, (2)
l oscillations (SdHOs) are not yet resolved. Among these ρ0 π τπ
l
a are several classes of magnetoresistance oscillations, such
h as microwave-induced resistance oscillations (MIROs),1–5 whereτtristhetransportlifetimeandτπ isthetimedescribing
s- phonon induced resistance oscillations (PIROs),6 and Hall electronbackscatteringoffofimpurities.8,10
e field-induced resistance oscillations (HIROs),7,8, as well as Theeffectofanin-planemagneticfield,Bk,onMIROshas
m theircombinations.9,10Inaveryclean2DES,MIROsandHI- beenrecentlyinvestigatedintwoindependentexperiments.2,3
. ROscanleadtoexoticzero-resistance11andzero-differential In Ref.2 the magneticfield B was tilted away fromthe nor-
t
a resistance states,12 respectively, which can be explained in maltothe2DESbyanangleθ andinRef.3B wasapplied
k
m termsofinstabilitiesandformationofcurrentdomains.13 independentlyofB⊥. Whilebothexperimentsagreedthatthe
d- MIROs are observedin linear-responsemagnetoresistivity MareIRsOtrofrnegqluyesnucpypirsegsosevderunneddebryBB⊥,≃Re0f..53fTouwnhditlheaitnMRIRefO.2s
n whena2DESisirradiatedbymicrowavesandcanbeunder- k
o stoodintermsofmicrowave-inducedtransitionsbetweenthe MIROswereessentiallyunchangeduptoθ ≃80◦ (Bk <∼1.2
T).Thiscontroversyandtheveryfactthatthesuppressionof
c Landaulevelswhichleadtooscillatoryphotoresistivity:5
[ MIROsobservedinRef.3wasleftunexplainedindicatesthat

theroleofB isnotunderstood.Itisthereforeinterestingand
2 k
v δρω ≃−ηπǫacPωλ2sin2πǫac. (1) timelytoexaminehowotherclassesofresistanceoscillations,
1 ρ0 e.g.HIROs,respondtoBk.
7 InthisRapidCommunicationwereportontheeffectofan
8 in-plane magnetic field on Hall field-induced resistance os-
2 Here, ρ0 is the resistivity at B = 0, ǫac = ω/ωc, ω = 2πf cillations in a high mobility 2DES. We employ a tilted-field
1. is the microwavefrequency,ωc = eB⊥/m∗ is the cyclotron setup and observe that, similar to MIROs,2,3 the HIRO fre-
frequency,B isthemagneticfieldnormaltothe2DES,m∗is
0 ⊥ quencydependsonlyon the perpendicularcomponentof the
1 theeffectivemass,λ=exp(−π/ωcτq)istheDinglefactor,τq magnetic field, B . We further find that with increasing tilt
⊥
1 is the quantum lifetime, Pω is the dimensionlessmicrowave angle θ, HIROs are strongly suppressed by modest B ∼ 1
: power, and η is the scattering parameter which depends on k
v T. The observed suppression is nonuniform and depends on
temperatureandtypeofdisorderinthe2DES.5
i theoscillationorder;thelowerorderoscillationsdecayfaster
X
HIROs are observed in differential resistivity r ≡ dV/dI thanthehigherorderoscillationswithincreasingθ. Whilethe
r
a when a direct current I is passed through a 2DES and the suppressionof the HIRO amplitudeby Bk cannotbe readily
perpendicular magnetic field, B , is varied. HIROs origi- explainedbyexistingtheories,wediscussourfindingsinthe
⊥
natefromshort-rangeimpurity-mediatedtransitionsbetween context of Eq.(2) and show that the suppression can be un-
Landaulevelstilted bytheHallelectricfield, Edc = ρHI/w derstoodin termsof a Bk-inducedreductionof the quantum
(ρH is the Hallresistivity, w is the sample width).8,10 Inthis lifetime.However,identifyingtheoriginofsuchmodification
scenario,acharacteristicscatteringeventinvolvesanelectron remainsthesubjectoffuturestudies.
which is backscattered off of an impurity. The guidingcen- Whilesimilarresultshavebeenobtainedfromavarietyof
ter of such an electron is displaced by the cyclotron diame- samples,thedatapresentedherewereobtainedfromaHallbar
ter, 2Rc, and when 2Rc matches an integral multiple of the (w = 100µm)cleavedfromaGaAs/Al0.24Ga0.76As300A˚-
real-spaceLandaulevelseparation,~ωc/eEdc,theprobability widequantumwellgrownbymolecularbeamepitaxy. After
of such events is enhanced. This enhancement gives rise to abrieflowtemperatureilluminationwitharedlightemitting
a maximuminthedifferentialresistivityoccurringwhenever diode,thissamplehadelectrondensityne ≃3.6×1011cm−2
ǫdc ≡ eEdc(2Rc)/~ωc isequaltoaninteger. At2πǫdc ≫ 1, and mobility µ ≃ 1.0×107 cm2/Vs. The experiment was
2
isno longera monotonicfunctionofB andthuscannotbe
(cid:72) =1 ⊥
z dc describedbyanexponentialdependenceofEq.(2).
B
2.5 (cid:84) Since Eq.(2) dictates that the HIRO amplitude A ≡
I = 100 (cid:80)A (16/π)·(τtr/τπ)·exp(−2π/ωcτq), onehasto examinepos-
T = 1.0 K y sible effects of B on the variousscattering parameters, i.e.,
k
x onτtr,τπ,andτq. Inour2DESatweakB⊥,ρ ≃ ρ0 ∝ 1/τtr,
andone can estimate the effectof B on1/τ by investigat-
k tr
2 ing the evolution of the background part of r with increas-
2.0 ing θ. This is done in Fig.2(a) showing r/ρ0 as a function
(cid:84) = 0o 4 3 of B⊥ for different θ (as marked) without a vertical offset.
Plotted in such a way, the data clearly show thatr oscillates
o about a smooth, slowly varying background which does not
0 62.4
(cid:85) change with θ. This conclusion is further supported by the
/r
existence of common crossing points (cf.↓,↑) which occur
o
1.5 75.0 at ǫdc ≃ n ± 1/4(n = 1,2,3,...) and where δρ ≃ 0, as
prescribed by Eq.(2). We thus conclude that 1/τ does not
tr
79.5o changesignificantlyinour2DESunderBk <∼1T.
The backscattering rate 1/τπ appearing in Eq.(2) is es-
sentially a scattering rate from the sharp disorder potential,
o
81.9 e.g., residual background impurities and/or interface rough-
ness. Scattering off of backgroundimpurities can hardly be
1.0
83.3o affected by Bk because of their 3D character. It is also un-
likely that B can reduce interfaceroughnessscattering. Fi-
k
nally,asignificantdecreasein1/τπshouldleadtoanoticeable
decreasein1/τ whichisnotobserved.Wethusconcludethat
tr
-2 -1 0 1 2 thedecayoriginatesfromanincreaseofthequantumscatter-
B [kG] ingrate1/τ underappliedB .
(cid:65) q k
ToexaminetheeffectofB onthequantumscatteringrate
k
FIG. 1: [Color online] Normalized differential resistivity r/ρ0 vs 1/τq, we first extract the oscillatory part δr/ρ0 by subtract-
B⊥ for tilt angles θ ≃ 0◦,62.4◦,75.0◦,79.5◦,81.9◦,83.3◦. The ingtheBk-independentbackgroundfromthedatainFig.2(a).
data arevertically offset for clarityin step of 0.2 startingfrom the TheresultispresentedinFig.2(b)asafunctionofǫdcshow-
highest θ. HIRO maxima are marked by ǫdc = 1,2,3,4. Inset: ingtheexpectedperiodandphasewiththemaximaoccurring
directionofBwithrespecttothe2DES. nearintegerǫdcforallθ. Inaddition,Fig.2(b)allowseasyex-
tractionoftheoscillationamplitudeAandshows,again,that
the rate at which the oscillations disappearstrongly depends
performedin a 3He cryostatatT ≃ 1.0 K andI = 100µA onǫdc,withthehigher-orderHIROspersistingtohigherθ.
usingastandardlow-frequencylock-indetectionscheme. Weproceedwiththeanalysisbyadoptinganempiricalre-
In Fig.1 we present the normalized differential resistivity lation, 1/τq = 1/τq0 +∆, where 1/τq0 is the scattering rate
r/ρ0vsB⊥fordifferentθ,asmarked.Thedataarevertically at Bk = 0 and ∆ is the Bk-induced correction. Then it
offsetforclarityinstepof0.2startingfromthehighestθ. At follows that to account for a faster decay at higher B⊥, ∆
thetopcurve(θ =0)thefirstfourHIROmaximaaremarked should increase faster than Bk. Indeed, if ∆ ∝ Bk, then
the argument of the Dingle factor would acquire a correc-
by ǫdc = 1,2,3,4. Examinationof the data revealsthat, re-
gardless of θ, the positions of the same-order maxima and tion −π∆/ωc ∝ −Bk/B⊥ = −tanθ. As a result, oscil-
minima roughly coincide with each other. This finding pro- lations of all orders would decay with θ at the same rate,
vides firm experimental evidence that, similar to MIROs,2,3 in contradiction with our findings. If, however, ∆ ∝ Bk2,
the frequency of HIROs is controlled by B⊥. At the same then −π∆/ωc ∝ −Bk2/B⊥ ∝ −tan2θ/ǫdc. This correc-
time,itisevidentthattheoscillationamplitudedecreaseswith tionincreaseswithθ (foragivenǫdc)anddecreaseswithǫdc
increasingθ. (foragivenθ),consistentwithourexperimentalobservations.
ThetoptraceinFig.1obtainedatB =0showsthattheos- Thisresultimpliesthattheamplitudeshoulddecaywithθ as
cillationamplitudegrowsmonotonicakllywithincreasingB A = A0exp(−αtan2θ/ǫdc), where A0 is the amplitude at
⊥
duetotheincreaseoftheDinglefactorλenteringEq.(2).Ex- θ =0andαisaBk-independentconstant.
aminationoftheotherdatainFig.1revealsthatHIROsgrad- InFig.3(a)wepresentnormalizedHIROamplitudeA/A0
ually decay with increasing θ and that the decay is nonuni- vs 1/ǫdc on a semi-log scale for three different θ (as
form;thestrongest,fundamentalHIROpeak(cf.“ǫdc =1”)is marked). We observe that the data are described by A =
muchmoresensitivetoθ thanthehigherorderpeaksappear- A0exp(−β/ǫdc) reasonably well and that β increases with
ingatlowerB . Indeed,itvirtuallydisappearsatθ ≃ 81.9◦ θ. In Fig.3(b) we show that β scales roughly linearly with
⊥
(B ≃ 1.1 T) while the higher-order peaks are still clearly tan2θ, the dependence which follows from ∆ ∝ B2. We
k k
observed. Thisfindingindicatesthatatfiniteθ theamplitude thus conclude that the observed decay of HIROs can be ex-
3
a a
1.6 2
o
(cid:84) = 62.4 1
1.4
6
4
1.2 75.0o (cid:84) = 79.5o
(cid:85)0 A0 2
/r 1.0 79.5o /A0.1
o
(cid:70) (cid:70) 81.9o 6 81.9
0.8 4
(cid:67)
o
2 83.3
0.6
0.01
0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0
B [kG] 1/(cid:72)
(cid:65) dc
b b
0.4
8
o
0.2 81.9
o 6
79.5
0 o
(cid:85) 75.0
/r(cid:71) 62.4o (cid:69) 4
0.0
2
-0.2
0
1 2 3 4 0 20 40 60 80 100 120
(cid:72)dc tan2(cid:84)
FIG. 2: [Color online] (a) Normalized differential resistivity r/ρ0 FIG. 3: [Color online] (a) Normalized HIRO amplitude A/A0 vs
vs B⊥ for θ = 62.4◦,75.0◦,79.5◦, and 81.9◦. The data are not 1/ǫdc for θ = 79.5◦,81.9◦, and 83.3◦ (cirles) and fits to a ·
verticallyoffset. (b) Normalized oscillatory part of thedifferential exp(−β/ǫdc) (lines). (b) Extractedvalues ofβ vstan2θ (circles)
resistivityδr/ρ0vsǫdcforthesametiltangles. andafit,β≃0.064·tan2θ(line).
plainedbyaBk-inducedcorrectiontothequantumscattering which indicates the same increase in the quantum scattering
ratewhichscalesroughlyasB2. rate 1/τ′ which enters the SdHO amplitude. Such a modest
k q
It is interesting to examine the possibility that the B - increaseofquantumscatteringrate,indeed,isnotsufficientto
k
induced suppression of MIROs observed in Ref.3 can also explaincompletequenchingofMIROs. However,1/τq′ isof-
beexplainedwithinthesamepicture.First,wenoticethatthe tenconsiderablylargerthan1/τq enteringEqs.(1),(2),which
DinglefactorentersequallyinthedescriptionofbothMIROs is also the case for the 2DESused in Ref.3 since the MIRO
[Eq.(1)] and HIROs [Eq.(2)] and that both MIROs and HI- onset is much smaller than the SdHO onset. The overesti-
ROsaredestroyedbysimilarBk ∼1T.Second,ifBkisused mated1/τq′ isusuallyattributedtothefactthattheamplitude
asaparameter(insteadofθ),thecorrectiontotheargumentof of the SdHOs (∝ λ1) is very sensitive to macroscopic den-
theDinglefactorcanbewrittenas−π∆/ωc ∝−ǫac·∆. This sityinhomogeneitieswhich,however,donotsignificantlyaf-
correctionscales with ǫac (and notwith 1/ǫac if θ = const) fect the amplitude of “induced” oscillations (∝ λ2). Since
and therefore higher order oscillations should be more sen- inhomogeneitiescan hardly be affected by B , a modest in-
k
sitive to B . Indeed, direct examination of Fig.1 in Ref.3 creasein1/τ′ observedinRef.3likelysignalsamuchlarger
k q
reveals that (i) the number of oscillations quickly decreases increasein1/τ which,inturn,isresponsibleforthedecayof
q
with increasing B , (ii) this decrease occurs at the expense MIROsobservedinRef.3.
k
ofhigherorders,(iii)thelowerorderoscillationspersisttoa Insummary,wehavestudiedtheeffectofanin-planemag-
muchhigherBk thanthehigherorders, and(iv)the onsetof netic field on Hall field-induced resistance oscillations in a
theoscillationsincreaseswithB . Alltheseobservationsim- highmobility2DES.Wehavefoundthatwhiletheoscillation
k
ply that 1/τq increases with Bk. We believe that a standard frequencyremainsunchanged,the amplitude quicklydecays
Dingle plot analysis (which is not feasible on our data ob- as the magnetic field is tilted away from the normal to the
tained at fixed θ) would readily revealthe exactdependence 2DES.Thedecayisverysensitivetotheoscillationorderand
of1/τqonBk. cannotbe readilyexplainedbyexistingtheoriesof nonlinear
Finally, we mention that Ref.3 observed that the onset of transport. Our analysis shows that the decay can be under-
the SdHOs increased by ≃ 50% under applied B ≃ 1 T, stoodintermsoftheB -inducedincreaseofasingleparticle
k k
4
scatteringrate. However,theexactmechanismofsuchanin- Laboratory, which is supported by NFS Cooperative Agree-
creaseremainsasubjectoffuturetheoreticalandexperimental ment No. DMR-0654118, by the State of Florida, and the
studies. DOE.TheworkatMinnesotawassupportedbytheNSFGrant
We thank I. Dmitriev, A. Kamenev, M. Khodas, I. No. DMR-0548014 and by the DOE Grant de-sc0002567.
Kukushkin,andB. I. Shklovskiifordiscussionsand S. Han- A.T.H. acknowledges support by the Thesis Research Grant
nas, G.Jones, J. Krzystek,T.Murphy,E.Palm, J.Park, and, andtheDoctoralDissertationFellowshipoftheUniversityof
especially,D.Smirnovfortechnicalassistance. A portionof Minnesota.
thisworkwasperformedattheNationalHighMagneticField
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