Highly Anisotropic Transport in the Integer Quantum Hall Effect

4 Pages·0.17 MB·English

by  W. Pan

Highly Anisotropic Transport in the Integer Quantum Hall Effect
W. Pan1,2, H.L. Stormer3,4, D.C. Tsui1, L.N. Pfeiffer4, K.W. Baldwin4, and K.W. West4
1Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544
2National High Magnetic Field Laboratory, Tallahassee, Florida 32310
3Department of Physics and Department of Applied Physics, Columbia University, New York, New York 10027
1 4Bell Labs, Lucent Technologies, Murray Hill, New Jersey 07974
0
(February 6, 2008)
0
2
n At very large tilt of the magnetic (B) field with respect to the plane of a two-dimensional electron
a system the transport in the integer quantum Hall regime at ν = 4, 6, and 8 becomes strongly
J anisotropic. At these filling factors the usual deep minima in the magneto-resistance occur for the
6 current flowing perpendicular to the in-plane B field direction but develop into strong maxima for
2 the current flowing parallel to the in-plane B field. The origin of this anisotropy is unknown but
resembles therecently observed anisotropy at half-filled Landau levels.

]
l
l
a
h
-
s
e Strongly correlated electronic systems often exhibit observations.
m
stripe phases [1]. In two-dimensional electron systems Our sample consists of a 350˚A wide GaAs quantum
t. (2DES)suchastripephaseisbelievedtobeattheorigin well embedded into Al.24Ga.76As and delta-doped from
a of the recently observed electronic transport anisotropy bothsidesatadistanceof490˚A.Thespecimenhasasize
m
at half-fillings of high Landau levels [2–8]. At Lan- of 5mm × 5mm and is contacted via eight indium con-
- dau level filling factors ν = 9/2, 11/2, 13/2, etc. the tacts placed symmetrically around the perimeter. The
d
n magneto-resistance is a maximum along one current di- electrondensityisestablishedafterilluminatingthesam-
o rection, whereas it is a minimum when the current di- plewitharedlight-emittingdiodeat∼4.2Kand,within
c rection is rotated by 90◦ within the plane of the sample. limits,thedensitycanbetunedbyexposuretime. Atan
[ In a purely perpendicular magnetic field (B) the direc- electron density of n = 4.2×1011 cm−2 two electrical
1 tion of anisotropy is pinned to the crystal lattice [3,4], subbands are populated having densities n0 ∼3.1×1011
v but re-orients itself when an in-plane B field (Bip) is cm−2 andn1 ∼1.1×1011cm−2 asdeterminedbyFourier
6 added by tilting the sample. At large B the easy-axis analysis of the low-field Shubnikov-de Haas oscillations.
1 ip
of anisotropy in the plane of the sample (the direction All angular dependent measurements are carried out in
4
1 of minimum resistance) is always perpendicular to Bip a dilution refrigerator equipped with an in-situ rotator
0 [5,6]. Althoughthenatureofthisanisotropyremainsun- placed inside a 33 Tesla resistive magnet. We define the
1 certain, experimental data [2–8] and theoretical models axisofrotationasthe y-axis. Consequently,the in-plane
0 [9–21] point to the formation of a unidirectional charge field, B , extends along the x-axis when the sample is
/ ip
t density wave, often referred to as the “stripe phase”, or rotated.
a
m to astateakintoaliquidcrystalphase[11]. Averysim- We have measured Rxx and Ryy, which differ only in
ilar anisotropy is also observed at ν = 5/2 and ν = 7/2 the in-plane current direction, at more than 10 tilt an-
-
d in the second Landau level under large B [5,6]. Mod- gles(θ)between0◦ and90◦. R representsthedirection
ip xx
n eling [22] suggests that an electronic anisotropic phase, for which, under tilt, the current runs along B . Figure
ip
o notunliketheoneathalf-fillingsofhigherLandaulevels, 1 shows data at five selected angles, from θ = 81.1◦ to
c has been induced by the in-plane B field. 84.4◦. Atθ=0◦(notshown)bothR andR vanishat
: xx yy
v So far,anisotropyhas only been observedathalf-filled ν=6asexpectedforanisotropicquantumHallstate. As
Xi Landau levels. In this letter, we present data that show θ is increased towards 81.1◦, both Rxx and Ryy remain
r strong electronic transport anisotropies at fully filled vanishingly small at ν = 6, although the widths of the
a Landau levels. They are created by the very strong in- resistanceminimaandoftheHallplateaushrinkwithin-
plane B fields atvery large tilt in the regime ofthe inte- creasingθ. Verygenerally,suchanangulardependenceis
gralquantumHalleffect(IQHE)atν =4,6,and8. The readily understood for the spin-unpolarized ν = 6 state.
origin of these anisotropies is unknown, although, phe- While the ν =6state alwaysoccursatthe sameperpen-
nomenologically, they resemble the anisotropies at half- dicular magnetic field, B , the total magnetic field at
perp
filled Landau levels: the magneto-resistance is a mini- tilt angle, θ, increases as B =B /cos(θ). Since the
tot perp
mum when the current is perpendicular to B and a electron spin experiences B , the Zeeman splitting of
ip tot
maximum when the current is along B . A striped spin all Landau levels increases with increasing θ. This leads
ip
density wave phase may be at the origin of these new to a reduction of the energy gap at ν = 6 and a shrink-
1
ing width and depth (not visible on the linear scale of statesisunknown. Beforespeculatingabouttheoriginof
Fig. 1) of the R and R minima. Eventually, this thisnewphenomenonitisinstructivetoconsiderinmore
xx yy
leadsto acollapseanddisappearanceoftheν = 6IQHE detail the single particle states in this two-electric sub-
state. Indeed, at θ = 83.3◦, R has turned from a deep band specimen. Figure 4a shows the usual Landau fan
xx
minimum into a strong peak and the usual Hall plateau diagram for a density of 4.2 ×1011 cm−2. The Zeeman
has vanished. Therefore, the disappearance of R can splitting is enhanced by a factor of 10 to be visible. The
xx
be rationalized as the closing of the ν = 6 energy gap. position of the Fermi level, E , is indicated by a heavy
f
However,verysurprisingly,the electricaltransportturns line. Clearly,inthevicinityofν=4,6and8,Landaulev-
outto be strongly anisotropic. In contrastto R ,which elsfrombothelectricsubbandscontributeandE jumps
xx f
shows a strong maximum at this angle and filling fac- between levels of different origin. Using such a simple
tor, R , continues to shows a strong minimum at ν = single-particlepicture anda2DESofzerothicknesswith
yy
6. Just as in the case of half-fillings [5,6] the easy-axis densities appropriate for the data of Figs. 1 and 3, one
of this anisotropy at full-filling factors is perpendicular would expect the gaps at ν = 4, 6 and 8 to close at θ =
to B . The direction of anisotropy is not dependent on 83.4◦, 88.4◦ and 88.3◦, respectively. These values differ
ip
the orientation of the crystallographic axis with respect from experiment, especially in the case of the ν = 6 and
to the in-plane field, as we determined by performing ν = 8 states.
the same experiments on the same specimen mounted The discrepancy is largely the result of the neglect of
◦
in a configuration rotated by 90 about the sample nor- exchange and of the thickness of the wave function. In
mal. Furthermore, none of the resistance measurements the remainder we focus on the state at ν = 6, which we
showed any hysteresis as a function of the sweep direc- studied most extensively and which shows the strongest
tion of the B field. Finally, in the anisotropic regime, anisotropy in experiment. We expect similar arguments
the generallystrongHallplateauatν = 6disappearsfor toholdforν =4andν =8. Figure4bshowstheresultof
both directions of current. a self-consistent local-density-approximation calculation
This is the firsttime that suchananisotropyhas been [24] performed for a density n = 4.2×1011 cm−2 at a
observed in a state as robust as an IQHE state. To filling factor ν = 6 as a function of B . The gap at ν =
ip
learn more about this anisotropic state we perform T- 6 (shaded region) undergoes strong variations, comes al-
dependent studies of R and R . For comparisons, most to a close at B ∼2.5T (not shown), and vanishes
xx yy ip
we choose θ = 81.1◦, where the electronic transport is at B ∼ 18.5T due to level crossing. The experimental
ip
isotropic, and θ = 83.3◦, where transport is strongly value ofB for the stronganisotropyis ∼25T.However,
ip
anisotropic. in Figure 2a and Figure 2b we show three we consider the theoretical result of ∼18.5T to be suffi-
representativetraces ofR andR . At θ = 81.1◦, R ciently close to ∼25T to attribute the disappearance of
xx yy xx
and R exhibit the usual activated behavior: the value the energy gapatν = 6 in Fig.1 to the crossingofspin-
yy
of both resistances increases with increasing T. On the split Landau levels originating from different electrical
otherhand,atθ=83.3◦,R andR behaveoppositely: subbands (i = 1, 2). This provides a rationalfor the ap-
xx yy
R decreases whereas R increases with increasing T. pearanceofnovelfeaturesinthedataatthisfillingfactor
xx yy
The T-dependenciesare quantifiedin Figure 2c and Fig- and angle. However, none of such level crossing consid-
ure2d,whereR andR areshownonArrheniusplots. erations can explain the observedanisotropy, which rep-
xx yy
At θ = 81.1◦, R and R show well-behaved activated resentsthe remarkablefinding in ourdata. The originof
xx yy
behavioryieldingasingleenergygapof∆∼1Kforboth thisphenomenonmustbetheresultofcorrelatedelectron
current directions [23]. On the other hand, the data for behavior.
R and R at θ = 83.3◦ show no longer activated be- Previously,largeelectricalanisotropieshaveonlybeen
xx yy
havior. R and R appear to start from similar values observed at half-filled Landau levels [2–8]. It is believed
xx yy
at high temperature but then diverge from each other thatthere theelectronsystemspontaneouslybreaksinto
roughly exponentially with exponents of similar magni- stripeddomainsofalternatingfillingfactorssuchasν=4
tude but opposite sign. At the lowesttemperatures both andν =5aroundν =9/2[9–21]. Giventhesimilarityof
resistances assume an approximately T-independent be- the observedproperties ofthe anisotropicphases around
havior. This dependence is qualitatively the same as the ν = 9/2 and ν = 6 one might speculate on a similar
T-dependence of the anisotropic state at ν = 9/2 [4,17]. underlying striped geometry. The driving force behind
The remarkable anisotropy found in the IQHE is not the phase separation in the ν = 9/2 case is exchange.
limited to the ν = 6 state. Similar anisotropies are ob- The energetic gain from breaking into domains of ν = 4
servedatfilling factors ν = 4 and ν = 8. Figure 3 shows and ν = 5 is counteracted by a strong electrostatic cost
the ν = 8 and ν = 4 anisotropy in the same sample for creating an inhomogeneous charge distribution. This
atslightlydifferentdensities,tunedbyapplyingdifferent is the reason for the formation of very narrow stripes of
dosesoflight. Wehavenotperformedasystematicstudy ν = 4 and ν = 5 states, which are only a few magnetic
of these states. lengths wide. A phase, consisting of stripes around ν =
The cause of the anisotropy at integral quantum Hall 6, would carry a much smaller, electrostatic burden.
2
At the point of collapse of the ν = 6 energy gap in the State of Florida. D.C.T. and W.P. are supported by
Fig. 4b two electronic configurations are degenerate. At the DOE and the NSF.
B smallerthanthelevelcrossinginFig.4btheelectrons
ip
occupy three spin-unpolarizedlevels emanatingfromthe
lowestthree Landaulevels(N = 0,1,and2)ofthe lower
electronic subband, i = 1 [25]. (Note, an earlier anti-
crossing at B ∼ 2.5T exchanges states i = 1, N = 2
ip
and i = 2, N = 0). The total system is spin-unpolarized
[1] V.J.Emery,E.Fradkin,S.A.Kivelson,andT.C.Luben-
(3 spin-up, 3 spin-down). At Bip larger than the level sky,Phys. Rev.Lett. 85, 2160 (2000).
crossing in Fig. 4b the electrons occupy only two spin-
[2] H.L. Stormer, R.R. Du, D.C. Tsui, L.N. Pfeiffer, and
unpolarizedlevelsemanatingfromthelowesttwoLandau K.W. West, Bull. Amer.Phys. Soc. 38, 235 (1993).
levels (N = 0, 1)of the lowerelectronic subband (i = 1). [3] M. P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeiffer,
In addition, they occupy the spin-up states (solid lines) and K.W. West,Phys. Rev.Lett. 82, 394 (1999).
oftwolevelsemanatingfromthei=1,N=2andthei= [4] R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfeiffer, K.W.
2,N=0states. There,thetotalsystemispartially spin- Baldwin,andK.W.West,SolidStateCommun.109,389
polarized (4 spin-up, 2 spin-down). In the vicinity of the (1999).
[5] W.Pan,R.R.Du,H.L.Stormer,D.C.Tsui,L.N.Pfeiffer,
level crossing in Fig. 4b, a phase separation of the elec-
K.W.Baldwin,andK.W.West,Phys.Rev.Lett.83,820
tronic system [26,27] into spin-unpolarized and partially
(1999).
spin-polarizeddomainsmayoccurdrivenbyexchange. A
[6] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeiffer,
verysmallgaininexchangeenergymaysuffice, sincethe and K.W. West,Phys. Rev.Lett. 83, 824 (1999).
charge density in both configurations is identical and, to
[7] M.Shayegan,H.C.Manoharan,S.J.Papadakis,andE.P.
first order, there is no associated electrostatic cost. De Poortere, Physica E, 6, 40 (2000).
Such a pattern resemblesthe pattern of a spin-density [8] W. Pan, T. Jungwirth, H.L. Stormer, D.C. Tsui, A.H.
wave, SDW. The existence of an in-plane magnetic field MacDonald,S.M.Girvin,L.Smrˇcka,L.N.Pfeiffer,K.W.
and the so-induced coupling of spin and orbital mo- Baldwin, and K.W. West, Phys. Rev. Lett. 85, 3257
tion will energetically favor a given orientation of the (2000).
stripes with respect to B . The resulting stripe phase [9] A.A.Koulakov,M.M.Fogler,andB.I.Shklovskii,Phys.
ip Rev. Lett. 76, 499 (1996); M.M. Fogler, A.A. Koulakov,
of alternating IQHE configurations is bound to have
andB.I.Shklovskii,Phys.Rev.B54,1853(1996); M.M.
one-dimensional edge-states along its interface between
FoglerandA.A.Koulakov,Phys.Rev.B55,9326(1997).
neighboringdomains,whichcarrytheelectriccurrentina
[10] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006
highlyanisotropicfashion. Thistransportpatternwould
(1996).
be analogous to the pattern invoked in the stripe phases [11] E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065
thatarebelievedtoformathalf-fillingsofLandaulevels, (1999).
such as ν = 9/2 and 13/2and believed to be responsible [12] H.A. Fertig, Phys. Rev.Lett. 82, 3593 (1999).
for the anisotropic electronic behavior. However, with- [13] E.H.Rezayi,F.D.M.Haldane,andKunYang,Phys.Rev.
outtheapplicationofotherexperimentaltechniquesand Lett. 83, 1219 (1999).
without a detailed theoretical investigation this picture [14] S.H. Simon, Phys.Rev. Lett. 83, 4223 (1999).
remains speculative. [15] T. Jungwirth, A.H. MacDonald, L. Smrˇcka, and S.M.
Girvin, Phys.Rev.B 60, 15574 (1999).
In summary, we have observed strongly anisotropic
[16] T. Stanescu, I.Martin, and P.Phillips, Phys.Rev.Lett.
transport under high in-plane magnetic field in the
84, 1288 (2000).
regime of the IQHE in a quantum well sample with
[17] E. Fradkin, S.A. Kivelson, E.Manousakis, and K. Nho,
two occupied electrical subbands. Phenomenologically, Phys. Rev.Lett. 84, 1982 (2000).
the data have much in common with the previously dis- [18] N. Maeda, Phys.Rev. B 61, 4766 (2000).
covered anisotropy at half-fillings of high Landau levels. [19] A.H. MacDonald and M.P.A. Fisher, Phys. Rev. B 61,
From a simple level crossing picture we conjecture that 5724 (2000).
a novelstriped spin-densitywavemaybe atthe originof [20] F. von Oppen, B.I. Halperin, and A. Stern, Phys. Rev.
this phenomenon. Lett. 84, 2937 (2000).
We would like to thank E. Palm and T. Murphy for [21] R. Cˆot´e and H.A.Fertig, Phys.Rev.B 62, 1993 (2000).
experimentalassistance,E.P.De Poortere,S.P.Shukla, [22] E.H. Rezayi and F.D.M. Haldane, Phys. Rev. Lett. 84,
4685 (2000).
and E. Tutuc for the help in numerical calculation, and
N. Bonesteel, R. R. Du, A.H. MacDonald, N. Read, and [23] Rxx is about three times larger than Ryy at the same
temperature. This level of initial anisotropy is not un-
K. Yang for useful discussion. We are indebted to T.
commonin2DES,whethertheyarepatternedorcleaved
JungwirthforprovidingtheenergylevelschemeofFigure
into squares. It is probably due to small density vari-
4b. AportionofthisworkwasperformedattheNational
ations leading to current inhomogeneities. However, it
High Magnetic Field Laboratory, which is supported by cannot be the origin of the strong anisotropy since the
NSF Cooperative Agreement No. DMR-9527035 and by
3
strong anisotropy observed at ν = 4, 6, and 8 is pegged FIG. 2. Panels (a) and (b): Temperature dependence
to Bip and not to any particular sample direction. of Rxx◦(solid lines) a◦nd Ryy (dashed lines) at tilt angles θ
= 81.1 (a) and 83.3 (b) and three different temperatures
[24] T. Jungwirth, unpublished.
each. Traces are shifted vertically for clarity. The position
[25] Strictlyspeaking,theseparationintosubbandlevelsand
of the ν = 6 filling factor is indicated. Panels (c) and (d):
Landau levels only holds at zero tilt angle. At non-zero
tilt they are mixed and one can use this nomenclature Corresponding Arrhenius plots for Rxx and Ryy at ν = 6 at
thetiltanglesofthepanelsabove. Thestraightlinesinpanel
only approximately.
[26] G.F. Giuliani and J.J. Quinn, Phys. Rev. B 31, 6228 (c) are a linear fit to the data [23]. The energy gap is ∼ 1K
(1985). for Rxx and Ryy.
[27] SudhakarYalagadda, Phys.Rev.B 44, 13101 (1991).
Pan et al, Figure 3

Pan et al, Figure 1
ν= 6 0.5 RRxx ((II /⊥/ BBip))
81.1° T ~ 50 mK n=4.8×1y0y11cm-2ip ν=8
θ = 84.0°
b. units) 82.0° Ωd R (k)yy 0.0 22 24 26
R(aryy 82.6° R anxx 0.5 RRxyxy ((II /⊥/ BBiipp)) ν=4
d n=3.7×1011cm-2
n θ = 82.5°
R axx 83.3°
500Ω 0.0 24 28 32
84.4° B (T)
2.6 2.8 3.0 3.2 FIG. 3. (a) Anisotropic trantostport around ν = 8 at θ =
B (T) 84.0◦. (b) Anisotropic transport around ν = 4 at θ = 82.5◦.
=F6IaGt.T1.∼R5x0xm(sKol,idIl=ine1s0)annAdpearRpnydyf(odrafishveedtillitneasn)galerso,ufnrdomν Thahveesbaemenpleaddjuenstseitdiebsyadreiffselriegnhttllyigdhtiffeexrepnotsufrreo.mFForigR. x1xatnhde
θ = 81.1◦ to θ = 84.4◦. For Rxx the current runs along the current runsalong thein-plane magnetic field.
in-planemagnetic field. n=4.2×1011 cm−2.
Pan et al, Figure 4
Pan et al, Figure 2 8 7 ν=6 5 4
1.0 (a)
θ = 81.1° (a) (b) θ = 83.3° V)10
ν=6 RRxyyx gy (me
1 ν=6 0.07K er
n
E
0.08K 0
0.5 0 2 Bperp (T) 4
0.25K
40 (b) level-crossing
0.22K V)
Ω) me 39 i=2,N=0
k 0.40K y ( spin-up
d R (yy0.016 18 200.30K22 0 22 24 26 28 30 Energ 3378 i=1,N=2 si=pi1n,-Ndo=w1n
n B (T)
a tot 15 16 17 18 19 20
R xx0.1 (c) 1 (d) FIG. 4. Panel (a): SimBpipl e(T)Landau fan diagram for
θ = 81.1° the two-electric subband sample of density n = 4.2×1011
θ = 83.3° cm−2. The Zeeman splitting is enhanced by a factor of
0.01 0.1 R 10 to be visible. The position of the Fermi level is indi-
Rxx cated by heavy line. Panel (b): Result of self-consistent lo-
Rxx yy cal-density-approximation calculation [24] at ν = 6 and as a
R
1E-3 yy function of in-plane magnetic field, Bip, at the same density
5 10 5 10 15 as panel (a). The electric subband index, i, and the Landau
1/T (K-1) levelindex,N,are indicated. Theshaded area representsthe
gap at ν = 6.
4