Spin-dependent resistivity at transitions between integer quantum Hall states

0.37 MB·English

by  K. Vakili

Spin-dependent resistivity at transitions between integer quantum Hall states
K. Vakili, Y. P. Shkolnikov, E. Tutuc, N. C. Bishop, E. P. De Poortere, and M. Shayegan
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544
(Dated: February 2, 2008)
ThelongitudinalresistivityattransitionsbetweenintegerquantumHallstatesintwo-dimensional
5
electronsconfinedtoAlAsquantumwellsisfoundtodependonthespinorientationofthepartially-
0
filled Landau level in which the Fermi energy resides. The resistivity can be enhanced by an order
0
of magnitude as the spin orientation of this energy level is aligned with the majority spin. We
2
discuss possible causes and suggest a new explanation for spike-like features observed at the edges

n of quantumHall minima near Landau level crossings.
a
J PACSnumbers: 71.70.-d,72.25.Dc,72.60.+g,73.43.-f,73.50.-h

7

] The integer quantum Hall (QH) effect is a rela- mobilities are 5.0m2/Vs inthe narrowQW samplesand
l 2
l tively well-understoodphenomenonin condensedmatter 17 m /Vs for the low-mass direction of the wide QW.
a
physicsthatoccurswhenatwo-dimensionalelectronsys- We performed measurements down to T = 20 mK us-
h
- tem (2DES) is subjected to a large perpendicular mag- ing low-frequency lock-in techniques. The samples were
s
netic field [1]. Associated with the effect are oscillations mountedon a single-axistilting stageto allowthe angle,
e
m inthelongitudinalmagnetoresistivity(MR),ρxx,includ- θ, between the normal to the sample and the magnetic
ing wide regions of zero resistivity, that occur as the field to be varied from 0o to 90o in situ. We define B
. tot
t Fermilevelpassesthroughsuccessivedisorder-broadened as the total magnetic field, and B and B as the com-
a k ⊥
m Landau levels (LLs) containing bands of localized and ponents parallel or perpendicular to the 2DES plane.
- extended states [2]. At transitions between integer QH BecausetheZeemanenergy,EZ,dependsonBtotwhile
d states, there are peaks in ρxx that have been extensively thecyclotronenergy,Ec,dependsonB⊥,theratioEZ/Ec
n
studied in the context of universal scaling phenomena can be adjusted by changing θ [4]. At the so-called co-
o
[3]. ThemagneticfieldresolvestheLLsintoupanddown incidence angles, E /E takes integer values as different
c Z c
[ spinbranchessplitbytheZeemanenergy,causingtheρxx spin branches of the various LLs are brought into ener-

maximatosplitaswell,butthespindegreeoffreedomis getic coincidence. These level crossings can be labelled
1
v generallythoughttoplayaminorroleindeterminingthe by a crossing index given by the difference between the
1 valueofthe ρxx maximainthe spin-resolvedQHregime. LL indices for the crossing up and down spin levels, i =
5 We observeunexpectedly largechangesofρxx attran- N↓1 - N↑2. By tilting the sample through coincidence an-
1
sitions between integer QH states as the 2DES is tilted gles, the spin for a LL associated with a given ρxx max-
1
with respect to an applied magnetic field such that the imum can be reversed [see Fig. 2(a)]. This argument is
0
5 LL spins arepolarized(Fig. 1). Specifically, the value of basedona single-particlepictureand, asdiscussedlater,
0 ρ at a transition is higher when the spin of the energy many-bodyeffectscanalterthissimplebehaviornearthe
xx
/ level in which the Fermi energy (E ) resides is aligned coincidence angles.
t F
a with the majority, lowest LL spin, and lower for those AlAs QWs areverywellsuited to tilted-field measure-
m
levels with the opposite spin orientation. The resistivity ments. Their relatively large value of g*m* [5, 6] (g*
- difference can be as large as an order of magnitude. andm*aretherenormalizedLand´eg-factorandelectron
d
n We performed measurements on electrons confined to mass respectively) allows LL coincidences to be reached
o a narrow (45˚A wide) AlAs quantum well (QW) with atmodest tilt angles as compared,for example, to GaAs
:c Al0.4Ga0.6As barriers,grownona GaAs (100)substrate. 2DESs which have an order of magnitude smaller g*m*.
v The 2DES in this system occupies a single, out-of-plane In narrowAlAs QWs, g*m* is smallenoughto allow the
Xi valley with an isotropic Fermi contour, a band effective i = 1 coincidence to be observed, while wide AlAs QWs
mass of 0.21m , and a band g-factor of 2. We also made are past this coincidence at θ = 0o for most accessible
r e
a measurements on 2D electrons in a strained, wide (110˚A densities [6, 7]. Because of their relatively narrow well
wide) AlAs QW, where a single, in-plane valley with an widths, AlAs QWs have the further advantage of being
anisotropicFermicontourwithtransverseandlongitudi- largely free of finite-thickness effects associated with the
nal band effective masses of 1.1me and 0.21me is occu- coupling of Bk to the orbital degree of freedom.
pied. Westudiedatotaloffoursamplesfromthreediffer- InFigs. 1(a)and(b), we showρ inanarrowQWat
xx
ent wafers, all of which were lithographically patterned several different θ. The value of g*m* is about 1.5 at n
in a Hall bar configuration with Ohmic contacts made = 2.1 x 1011 cm−2, and previous measurements have in-
by depositing AuGeNi and alloying in a reducing envi- dicated no strong dependence of g*m* on magnetic field
ronment. Metallic front and back gates were deposited or filling factor (ν) in this system [5]. This means that
for in situ control of the charge density, n. The peak E < E at θ = 0o, so that the ρ maximum near B
Z c xx ⊥
2
B (T) (a)
^ 2
2 4 6 2 1
0
1 2
3 (a) n = 2.1 x 1011 cm-2 1 0 2
E
44.8o c E 0 1 1
(cid:1)) 45.4o Z 0 0 0
/ 63.8o n = 2 i=1 i=2
Wk
( (b) (d)
xx
r
10
0 (cid:1))
1125 (b) EF 10flfl flflflflflfl n = 2.1 x 1 0651351.. 86cmoo -2 Wr (k/xx i=3 i=4 n = 1.3 x 1 qn0 ==11 93c0/m2o-2
(cid:1)/) 9 EF 00›fl ››flfl›fl 544297...561ooo 1 i=1 ni ==2 1. 3 x 1011 cm-2 nn == 57//22
Wk
(xx 6 45.9o n = 2.1 x 1011 cm-2 (c) (e) n = 2.1 x 1011 cm-2
r 0o
n = 3 2
1 10
3
(cid:1)) i=2
/
0
0 2 4 6 8 10 W (kxx
B (T) r
i=3
^
1
i=1
FIG.1: (coloronline)ρxx vs. B⊥ atT=300mKinanarrow
AlAs QW as the sample is tilted through the first (i = 1) 1 2 3 4 5 0 4 8 12 16
coincidence angle (θ ≃ 45o). (a) A resistance spike is visible 1/cos(q ) BTotal (T)
near the ν = 2 minimum, and (b) subsequently merges with
theρxxmaximumbetweenν =2and1,leadingtoadramatic FIG. 2: (color online) (a) Energy level configurations for a
enhancementofρxxatthatmaximum. Thetwolowestenergy 2DESinanarrowAlAsQWasitistiltedinfield. Many-body
levels, labelled by their LL index and spin, and the location effects near the coincidence angles are not shown. (b, c) ρxx
ofEF areshownschematically fortheindicatedρxx maxima. vs. 1/cos(θ)atT=20mKforhalf-integerν attwodensities,
with the coincidence angles marked by vertical dotted lines.
(d,e)ThesamedataplottedagainstBtot alongwithparallel-
= 6 T corresponds to EF passing through the 0↑ energy fieldMR(redtraces). Thekeyforthesymbolsinpanels(b-e)
level (transition between ν = 2 and 1 QH states) [see is in panel(d).
Fig. 1(b)]. As the sample is tilted to angles near the i
= 1 coincidence, a spike develops in the ν = 2 minimum
[Fig. 1(a)]. Similar spikes were observed previously in attains finite values at lower B⊥ when the second lowest
wide AlAs QWs [7] and were associated with scattering LLis1↓thanwhenitis0↑. Thiscausesanapparentshift
at boundaries between different spin domains concomi- in the position of the ν = 2 minimum to lower B⊥ [9].
tant with a first-order,many-body spin reversal[8]. The The Hall resistance plateaus (not shown) are truncated
propertiesofthespikesobservedinnarrowQWsaresim- in step with the ρxx minima.
ilar, including hysteretic behavior and the movement of To further understand the enhancement, we plot ρ
xx
the spike to higher B as θ is increased. In contrast at half-integer ν in Fig. 2 as a function of 1/cos(θ) [(b)
⊥
to previous studies, however, when the spike eventually and (c)] and B [(d) and (e)] for two different n in a
tot
mergeswiththeρ maximumbetweenν =2and1[Fig. narrowQW.Wealsoshowschematicallytheevolutionof
xx
1(b)], the spike’s height and width increase continuously the energy levels with increasing θ for the narrow QW
untilitisnolongerdistinctfromthemaximumitself,and in Fig. 2(a), although we have omitted peculiarities of
we are left with a single, enhanced ρ maximum. Even the level configurations associated with many-body ef-
xx
after the spike is unresolvable,the ρ maximumcontin- fects near the coincidence angles. Prior to the i = 1
xx
ues to increase in height as the sample is tilted until a coincidence, ρ at ν = 3/2 and 7/2 is locally minimal
xx
saturationangleisreached. Theapparentpositionofthe as a function of 1/cos(θ) while at ν = 5/2 it is maximal.
QHminimumalsodependsonthespinorientationofthe From the energy level diagram in Fig. 2(a), we see that
adjacent LLs. It is evident in Fig. 1 that ρxx at ν >∼ 2 3/2 and 7/2 correspond to EF passing through spin-up
3

branches while 5/2 corresponds to a down branch. As 9
2 (a)
the i = 1 coincidence is traversed, the spin for the en- n = 76 5 4 3
ergylevelscorrespondingtothese ν flip, andaccordingly 1
ρxx increasesordecreasesas the spinflips todownorup (cid:1)) 6 0
respectively. With further increase of θ, this correspon- 0 2 4 6 8
/
adtenacgeivbeentwνeseantρuxrxataensdatspthinecaonngtlienbueeys,oannddwehviecnhtuthaellryeρaxrxe W (kxx n = 4.5 x I1011 cm-2
3
no further spin reversals. Beyond this saturation, there r 56.4o n = 3
sometimes appears a slow decrease in ρ (e.g. ν = 3/2 47.8o
xx
atn = 1.3 x 1011 cm−2), howeverthis is small compared 0o
0
to the initial increase and does not appear to depend on E 2fl flflfl (b)
the spins ofthe nearby(higher)LLs (i.e.,itis unaffected I F 1fl flflfl
by passage through subsequent coincidence angles). 0fl flflfl
Plottingρxx atthe varioushalf-integerν asa function (cid:1))
of Btot [Figs. 2(d,e)] reveals that the enhancement sat- Wk/ 1 EF 10fl› flfl››fl› n = 3
tuhreatseastuatraatiroonufighelldy,νB-in,dfeoprenthdeenptaBratloltelw-fiheilcdhMisRclowsheetroe (xx 0fl flflfl
P r
θ = 90o and B = B . B has been shown experimen-
tot k P
tally [10, 11] and theoretically [12] to correspond to the
total spin polarization of the charges. Thus, the corre-
0
spondence between the saturation fields, which is con- 0 3 6 9
B (T)
sistent with previous observations of isotropy and field- ^
independence ofg*m*in narrowAlAs QWs [5], corrobo-
ratestheassertionthattheenhancementoftheρxx max- FIG. 3: (color online) ρxx vs. B⊥ at T = 300 mK in a wide
ima are associated with spin polarization. The analogy AlAsQWfordifferentθwiththecurrent(I)directed(a)along
with parallel-field MR implies a possible explanation for and (b) transverse to the long-axis of the occupied valley, as
indicatedschematically in eachpanel. Theinset of (a)shows
the occurrence of spin-dependent ρ in the QH regime.
xx a closeup of the θ = 0o trace, and the LL indices and spins
In the absence of orbital effects, the parallel-field MR
for the lowest three energy levels are shown in (b) for the
can be attributed largely to the spin-polarizationdepen- indicated traces.
dence of screening in a 2DES [12]. This is essentially
a consequence of the Pauli exclusion principle, whereby
like-spin charges screen disorder from each other less ef- calculated directly instead of the resistivities, as is usu-
fectively than opposite spin charges. This might seem ally the case, then some additional mechanism must be
irrelevant to the QH case where the density of states is postulated to account for the spin-dependence. For ρ↑
xx
quantizedintodiscrete(albeitdisorder-broadened),spin- = ρ↓ ≪ ρ↓,↑, this model gives a factor of 9 change in
xx xy
resolved, and macroscopically degenerate LLs. However, ρ at ν = 3/2 when tilting past the i = 1 coincidence,
xx
screeningassociatedwithinter-LLexcitationscanindeed which is in reasonable agreement with the narrow QW
play a significant role in this regime [13]. In particu- results. As we show below, however, measurements in
lar, considering that the relatively large band mass in wide AlAs QWs give results that are not accounted for
AlAs causes the electron-electron interaction energy, E by this simple model for spin-dependent resistivity.
e
∼ e2/4πǫlB (lB is the magnetic length), to be greater Our measurements of strained, wide AlAs QWs with
than Ec for all accessible fields, it becomes evident that a single anisotropic valley occupied [14] give results that
mixingbetweendifferentLLsmustbetakenintoaccount. are similar to those in the narrow QWs but with some
Disordermaythenbescreenedmosteffectivelywhenthe notable differences. Figure 3 displays MR data for cur-
spin of the LL in which EF resides is different than the rent(I) oriented(a)along and(b) transverseto the long
majority spin of the low-lying LLs. axis of the occupied valley. At n = 4.5 x 1011 cm−2,
Treating different spin levels as parallel conduction the value of g*m* in this system is about 3 [6], so the
channels could also point to a cause of spin-dependent two lowest energy levels are spin-down and the highest
ρ . The spin-channel Hall resistivities (ρs , s ∈ {↑,↓}) fully occupied levels are spin-up for odd integer ν > 1
xx xy
are different by virtue of the different filling factors for at θ = 0o. Like narrow QWs, the ρ maxima in the
xx
the two spin channels. Thus, if the spin-channellongitu- wideQWarelargerforE passingthroughthe majority
F
dinal resistivities (ρs ) are close, then the spin-channel spinlevels than throughthe minority spinlevels. This is
xx
longitudinalconductivities,σs =ρs /(ρs 2+ρs 2),are clearly evident in wide AlAs QWs even at θ = 0o [Fig.
xx xx xx xy
different and, when added and inverted, give a spin de- 3(a) inset], where the ρ maxima to the left of odd ν
xx
pendent total ρ . Of course, ρ↑ and ρ↓ are not nec- minima (spin-down levels) are large while those to the
xx xx xx
essarily close, and if the spin-channel conductivities are right(spin-up levels)are small. In tilted field, resistivity
4
spikes form near the coincidence angles (the i = 2 co- enhanced(spin-down)ρ maxima(Figs. 1and3),which
xx
incidence at n = 4.5 x 1011 cm−2 in this system occurs is consistent with them originating from spin-dependent
at θ ≃ 46o) and large ρ changes occur when these an- ρ . Therefore, the relative sizes of the spikes as they
xx xx
gles are passed [Figs. 3(a,b)]. In contrast to the narrow passacrossthelandscapeofρ areneatlyaccountedfor,
xx
QWs, however, the enhancement at ν = 5/2 in the wide as is the fact that the largest spikes are observed when
QW is anisotropic, with a remarkable factor of ∼19 en- the spin-dependent enhancement is strongest [e.g. Fig.
hancement in (a) but only ∼5 in (b). By straining the 3(a)]. Itisimportanttopointout,however,thatresistiv-
sample to depopulate one valley and populate the other ity spikes are also observed deep within the QH minima
and repeating the measurement, we have confirmed that (e.g. θ = 44.8o trace in Fig. 1(a)), i.e. in the local-
theenhancementanisotropyarisesfromthevalley(mass) ized states, that do not fall within the envelopes defined
anisotropy and not from the orientation of B with re- by the enhanced ρ maxima. Domain-wall scattering
|| xx
spect to I or the valley. The narrow wells, which have may account for these, but we reiterate that large spike-
an isotropic in-plane Fermi contour, show no enhance- like features at the edges of QH states are an inevitable
mentanisotropy. Suchananisotropicenhancementisnot consequenceofaspin-dependentρ withasuddenspin-
xx
explained by the aforementioned two-channel model for reversal transition. Spikes will occur even for second-
spin-dependent ρ . We also note that the strong spin- order(continuous)transitionsaslongasthe transitionis
xx
dependence is suppressedwhen the sample is strainedto sufficientlyrapidinmagneticfieldonthescaleoftheQH
occupy a second valley. This might be expected in the features. Infact,thesharpnessofthetransitionsappears
context of an inter-LL screening picture, since the ad- todiminishastheymovedeeperintotheextendedstates
ditional valley degree of freedom would allow like-spin (i.e. deeperintotheflanksoftheQHminima)[Fig. 1(b)],
charges to screen disorder from each other as effectively possibly signaling a crossover from a first to a second-
as opposite-spin. That spin-dependent ρ is strongest order transition. Finally, for spikes arising from spin-
xx
insingle-valleysystemswithlargeelectroneffectivemass dependent ρ , the spike maximum may not correspond
xx
mayexplainwhythiseffecthasnotbeenreportedforSi- to the center of the phase transition as was previously
MOSFETs (metal-oxide-semiconductor field-effect tran- thought to be the case, nor does the full-width-at-half-
sistors) or GaAs heterostructures [15]. maximum give the width of the transition. These facts
Whatever the cause, it is clear that ρ at QH transi- place restrictions on deriving information about the ex-
xx
tionsisspin-dependent. Thisobservationpromptsasim- change energy or Curie temperature associated with the
ple explanation for ρ spikes that exist on the flanks of ferromagnetictransitionforspikesoccurringattheedges
xx
QHstatesthatisindependent ofdomain-wallscattering. of QH states.
It has been shown theoretically [8, 16] that, when oppo-
WethanktheNSFforsupport,andA.H.MacDonald,
site spin LLs are broughtnear energetic coincidence, B
⊥
R. N. Bhatt, S. L. Sondhi, F. D. M. Haldane, and R.
can induce a discontinuous transition where the nearly
Winkler fordiscussions. Partofourworkwasperformed
coincident LLs suddenly trade places in energy. For ex-
at the Florida NHMFL, also supported by the NSF; we
ample,at low B the energy levels may have the con-
⊥
thank E. Palm, T. Murphy, and G. Jones for assistance.
figuration of the middle panel of Fig. 2(a), but as B is
⊥
increasedenoughtobegindepopulatingthelowlyingLLs
a rapid transition to the configuration of the left panel
ofFig. 2(a)canoccur. Spikesinρ havebeenshownto
xx
accompanysuchtransitions [7], andhavebeen explained [1] K.v. Klitzing et al.,Phys.Rev.Lett. 45, 494 (1980).
asarisingfromscatteringatboundariesbetweendomains [2] See, e.g., The Quantum Hall Effect, edited by R.E.
of the two competing energy level configurations [8]. In Prange and S.M. Girvin (Springer, New York,1990).
the case of a spin-dependent ρ , however, such spikes [3] B. Huckestein,Rev.Mod.Phys.67, 357(1995), andref-
xx
would accompany a collective spin-reversal transition at erences therein.
the edges of QH states even in the absence of domain- [4] F.F. Fang and P.J. Stiles, Phys. Rev. 174, 823 (1968).
[5] K. Vakili et al.,Phys. Rev.Lett. 92, 226401 (2004).
wallscattering. Asanexample,foranappropriateθ,the
[6] Y.P. Shkolnikov et al., Phys. Rev. Lett. 92, 246804
system may have the configuration of the middle panel
(2004).
of Fig. 2(a) as EF enters the extended states of the 1↓ [7] E.P. DePoortere et al., Science290, 1546 (2000).
energy level, and because the spin of that LL is down, [8] T. Jungwirth and A. H. MacDonald, Phys. Rev. Lett.
ρ would be large. As B is increased and a transi- 87, 216801 (2001).
xx ⊥
tion to the configuration in the left panel of Fig. 2(a) is [9] We determined θ from Hall resistance and by matching
made, the partially occupiedLL wouldchangeto 0↑ and low-field,tilted-fieldρxx minima(thosenotexperiencing
a LL crossing) to thesame minima at θ = 0o.
ρ would rapidly drop to the un-enhanced values. This
xx [10] T. Okamotoet al.,Phys.Rev. Lett.82, 3875 (1999).
would register in ρxx as a spike-like feature. [11] E. Tutucet al.,Phys. Rev.Lett. 86, 2858 (2001).
The spikes occurring at the edges of QH states are [12] V.T.DolgopolovandA.Gold,JETPLett.71,27(2000);
observed to follow the same rising curve defined by the I.F. Herbut,Phys. Rev.B 63, 113102 (2001).
5
[13] A.P.Smith et al.,Phys. Rev.B 45, 8829 (1992). in that measurement.
[14] Y.P.Shkolnikovetal.,Appl.Phys.Lett.85,3766(2004). [16] G.F. Giuliani and J.J. Quinn, Phys. Rev. B 31, 6228
[15] Aspin-dependentρxx isevident,thoughnothighlighted, (1985).
in [11]. Dueto thelarge 2DES width,there islikely also
anorbitalcomponenttothetilted-fieldMRenhancement