Unusual Landau Level Pinning and Correlated $\nu=1$ Quantum Hall Effect in Hole Systems Confined to Wide GaAs Quantum Wells

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by  Yang Liu

Unusual Landau Level Pinning and Correlated ν = 1 Quantum Hall Effect in Hole
Systems Confined to Wide GaAs Quantum Wells
Yang Liu, S. Hasdemir, M. Shayegan, L.N. Pfeiffer, K.W. West, and K.W. Baldwin
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544
(Dated: January 19, 2016)
In two-dimensional hole systems confined to wide GaAs quantum wells, where the heavy- and
light- hole states are close in energy, we observe a very unusual crossing of the lowest two Landau
levels as the sample is tilted in magnetic field. At a magic tilt angle θ (cid:39)34◦, which surprisingly is
independent of the well-width or hole density, in a large filling factor range near ν = 1 the lowest
twolevelsarenearlydegenerateasevincedbythepresenceoftwo-componentquantumHallstates.
Remarkably, a quantum Hall state is seen at ν =1, consistent with a correlated Ψ state.
111
6
1
0 I. INTRODUCTION (a) θ = 0° (b) θ ≈ 34° (c) θ ≈ 50°
2 y
g
n Amongthemostfascinatingphasesoftwo-dimensional ner
a E
J electronsystems(2DESs)inastrongperpendicularmag- ν = 1 1/ν ν = 1 1/ν ν = 1 1/ν
netic field (B ) are the quantum Hall states (QHSs).
6 ⊥
These are incompressible phases signaled by vanishing
1 FIG.1. SchematicdiagramofthelowesttwoLLsatdifferent
longitudinal resistance (Rxx) and quantized Hall resis- tilt angles (θ).

] tance(Rxy), andareobserved atintegralorcertain frac-
el tional Landau level (LL) filling factors (ν)1–3. Adding
- a layer (or subband) degree of freedom leads to exciting
tr twists. A bilayer electron system with nearly degenerate observation indicates that the two lowest-energy LLs are
.s LLs from different subbands and comparable inter- and nearlydegenerateandisconsistentwithaB||-inducedLL
at intra-layer interaction can support new, two-component crossing16. Remarkably, as schematically shown in Fig.
m (2C) QHSs that have no counterpart in standard single- 1(b), this near degeneracy persists in a large magnetic
layer (or one-component, 1C) 2DESs. An example is the fieldrange nearν =1whenθ (cid:39)34◦,amagicanglewhich
-
d Ψ state, a QHS formed at the even-denominator fill- doesnotdependonW orp. Moreover,whenthetwoLLs
331
n ing ν = 1/24–7. The correlated Ψ QHS, stabilized at are degenerate, the 2DHS is compressible at ν = 1 if p
111
o ν = 1, is another example5,8–12. This state is generally and W are large so that d/lB >∼1.3, but exhibits a QHS
[c considered to be an excitonic superfluid which can sup- when d/lB <∼ 1.3, consistent with the development of a
port Josephson-like interlayer tunneling and superfluid correlated 2C (Ψ111) state (d is the interlayer separation
1 transport. and lB is the magnetic length)8,12,17.
v
Recent experimental studies of 2D hole systems
5
(2DHSs) confined to wide GaAs quantum wells (QWs)
1
2 haveunraveleduniquephenomena,arisingfromthenon-
4 trivial spin-orbit coupling of the heavy- and light-holes. II. METHOD
0 Graninger et al. reported a reentrant behavior of the
1. ν =1 QHS as a function of parallel magnetic field B|| in Our samples, grown by molecular beam epitaxy on
0 symmetric, wide QWs13. Later, Liu et al. observed an GaAs (001) wafers, consist of GaAs QWs flanked by un-
16 unusual crossing of the two lowest-energy LLs at B|| =0 doped Al0.3Ga0.7As spacer and carbon δ-doped layers.
: as a function of B⊥14. For a given density (p) and well- The2DHSshaveas-growndensitiesrangingfrom0.98to
v width(W),thecrossingoccursataparticularfilling(Fig. 2.12, inunitsof1011 cm−2 whichweusethroughoutthis
Xi 1(a)); it destroys or weakens the odd-denominator QHSs report,andveryhighlow-temperaturemobilitiesµ≥100
nearthisfilling,andstabilizesauniqueeven-denominator m2/Vs. We made samples in a van der Pauw geometry,
r
a QHS when it happens at ν =1/214. 4 × 4 mm2, and alloyed In:Zn contacts at their four cor-
Here we present low-temperature transport data for ners. Each sample is fitted with an evaporated Ti/Au
2DHSs confined to symmetric, wide GaAs QWs, as we front-gate and an In back-gate to control the 2DHS den-
change the tilt angle (θ) between the sample normal and sity and QW symmetry. The data presented here were
themagneticfielddirection. Wefindthatatlowandhigh taken in symmetric QWs. The transport measurements
θ,ifW andparesufficientlylarge,LLsfromdifferentsub- were carried out in a dilution refrigerator with a base
bandsarewellseparatedfromeachotherandthe2DHSs temperature of T ≈ 30 mK and a superconducting mag-
exhibit normal QHSs at the standard fillings ν =2/3, 1, netupto18T.Wechangedθwithanin-siturotator,and
4/3, 7/5, 8/5 and 5/3. But near an intermediate θ, the usedlow-frequency(∼30Hz)lock-intechnique. Herewe
2DHSs exhibit 2C QHSs similar to those reported in bi- focus primarily on R traces; the R data corroborate
xx xy
layer 2DESs with vanishing subband separation15. This R and show corresponding plateaus.
xx
2
Energy the z-direction, the heavy-hole (HH) subband is lower in
W = 40 nm 4/5
(d) HH-S0 energy than the light-hole (LH) subband. But the HHs
have a smaller effective mass in the xy-plane than the
3 ν = 1 2/3
LHs, so the ground-state (N = 0) LL of the HH sym-
4/3 LH-S0
5/3 metric subband, which we refer to as HH-S0 for simplic-
7/5
8/5 p ν > 2 ity, increases faster in energy than the LH-S0 LL as we
(1011 cm-2) sufficiently increase B , leading to a LL crossing. In a
(c) ⊥
3/2 1.59(d) more quantitative picture, the spin-orbit coupling mixes
2 theHHandLHsubbandsandLLs,andresultsinamore
1.31 (c) complex,non-linearLLfandiagram. However,thecross-
Ωk) 19/15 ν = 5/3 ing between the two lowest-energy LLs is preserved in
R (xx 1.20 symmetric QWs14. In our wide QW samples, the HH
(b)
and LH subbands are close in energy, so the two levels
cross at moderate B .
1.01 ⊥
1 (b)
4/3 ÷4 ν = 1
5/3
7/5
8/5 (a)
0.82(a)
IV. EXPERIMENTAL RESULTS OF FINITE θ
0.76
0
ν = 2/3
0.5 1.0 1.5 Data presented in Fig. 3 reveal that QHS transitions
1/ν 1/ν can also be induced at a fixed density by varying θ, but
the behavior is dramatically different. In Fig. 3(c), we
FIG. 2. R traces measured in a 2DHS confined to a 40-
xx show R vs B traces measured at p=2.05 and differ-
nm-QW at θ = 0 and different densities. Right panels show xx ⊥
ent θ. The density is high so that the LH-S0 LL is well
schematically the two lowest-energy LLs’ energy vs 1/ν at
below the HH-S0 LL at θ = 0◦ in the range ν < 2 (see
different densities, corresponding to the traces as marked.
Fig. 2(d)), and the 2DHS exhibits 1C QHSs at standard
fillings. At θ (cid:39) 34◦, the 2DHS becomes 2C in a large
range of fillings 2/3 < ν < 2. This is evinced by the
III. EXPERIMENTAL RESULTS AT θ=0
development of insulating phases around ν = 2/3 (i.e.,
around ν = 1/3 for each component24), the complete
We first describe data taken in a 2DHS confined to disappearance of the QHSs at ν = 5/3 and 1, as well as
a 40-nm-wide QW as a function of density at θ = 0◦. the stabilization of QHSs at twice the standard fillings
In Fig. 2, the QHS transitions (marked by solid circles) ν = 4/3, 6/5, 6/7, 2/3, and at unusual fillings such as
which appear when two LLs are nearly degenerate, can ν =19/15and29/3515,23. Atlargerθ,theν =1and5/3
be seen moving from low to high ν as we increase p. QHSs reappear while many 2C QHSs remain, suggesting
At p = 0.76, we observe QHSs at the standard fillings, the two lowest-energy LLs are separated by a small but
similar to what is seen in systems where LLs from dif- finite energy25.
ferent subbands are well-separated3. The ν = 2/3 QHS
Figure 3(d) data taken at p = 1.59 exhibit a more
becomes weak at p = 0.82 but is restored at higher p.
complete and revealing evolution. The system is essen-
The weakening of the ν = 1 QHS at p (cid:39) 1.01 is evi-
denced by a profound narrowing of its R plateau, and tially 1C for θ <∼20◦ and θ >∼44◦, showing strong QHSs
xx at standard fillings25. It becomes 2C for ν < 2 when
serves as direct evidence that the two lowest-energy LLs
are crossing at ν = 118–22. At p = 1.20, a strong ν = 1 25◦ <∼ θ <∼ 44◦, exhibiting insulating phases flanking
ν = 2/3 and 2C QHSs at ν = 19/5, 6/5, 29/35, etc.,
QHS is restored, and a 2C QHS develops at an unusual
filling ν = 19/1515,23. The transition continues moving while QHSs at ν = 1 and 5/3 become weak and essen-
tially disappear as θ approaches 34◦.
to higher ν at p = 1.31. The ν = 5/3 QHS disappears
and another 2C QHS develops at ν = 3/2, which is the Figure 3(e) shows traces taken at p = 1.28 where, at
particle-holecounterpartofthe2Cν =1/2(Ψ331)QHS6. θ =0, the LL crossing occurs near ν =3/2, as evidenced
In the top trace (p = 1.59), the 2DHS reverts back to bythestabilizationofthecorrelated,2CQHSatν =3/2,
1C for ν < 2, exhibiting QHSs at standard fillings. The and the absence of a QHS at ν = 5/3. Similar to the
aboveevolutionoftheQHSs,whichimpliesaLLcrossing data of Figs. 3(c) and (d), the system becomes 2C near
that moves from low ν to high ν as density is increased, θ (cid:39) 34◦ and 1C when θ >∼ 49◦. However, in contrast to
is consistent with previous observations and theoretical Figs. 3(c)and(d)data,theν =1QHSbecomesweakat
calculations14. θ = 34◦ but never disappears. The fact that the system
TheaboveLLcrossingcanbequalitativelyunderstood is 2C near ν = 1 suggests that the ν = 1 QHS seen at
in a simplified picture (see the right panels in Fig. (2)). θ (cid:39)34◦ in Fig. 3(d) is also a 2C QHS; we will return to
When confined to QWs, because of their heavier mass in this later.
3
(a) d B (b) (d) p = 1.59 x 1011 cm-2 (e) p = 1.28 x 1011 cm-2
B 4/3
┴
B 4/3
θ || 5/3 7/5 ν = 1 2/3
W = 40 nm S a m ple 5/3 ν = 1 4/5 2/33/5 2 8/5 3/5
2 4/5
(c) p = 2.05 x 1011 cm-2 66° 59°
6 19/15 ν = 1 49° 49°
4/3
5/3 6/5
8/5
2 44° 44°29/35
59° 4/5
39° 19/15 6/7
4 4/5 5/3 4/3 1 2/3
Ω) 54° 29/35 2/3 4/319/15 1 29/345/5 8/5 6/534°
R (kxx 19/15 44° 6/5 34°
4/36/5 34° 6/7 30° 4/5
2 27°
25°
25° 25°
3/2
4/3 2/3 3/5
θ = 0° θ = 0° θ = 0°
0
4 6 8 10 12 4 6 8 10 4 6 8
B (T) B (T) B (T)
┴ ┴ ┴
FIG.3. (a)Self-consistentlycalculatedchargedistributionofthe2DHSconfinedtothe40-nm-wideQWatdensitiesp=2.05,
1.59 and 1.28. (b) Experimental geometry. (c)-(e) R vs B traces measured at different θ. In all panels, the QHS at ν =1
xx ⊥
is strong at θ=0, disappears or weakens at θ(cid:39)34◦, and becomes strong again at larger θ.
V. DISCUSSION (d)). Finite B introduces additional confinement of the
||
2DHS in the z-direction, raises the LH-S0 LL relative to
the HH-S0 LL, and causes a crossing of these levels at
The transition from 1C to 2C as a function of increas-
intermediate θ (see Fig. 4(d)).
ing B has been reported previously for electrons con-
|| ThemostremarkablefeatureofFig. 3data,however,is
fined to wide GaAs QWs15,26. In such systems, the cou-
nottheLLcrossingatanintermediateθ. Rather,itisthe
pling of B to the orbital (out-of-plane) motion of elec-
|| behaviorofthe2DHSnearthecrossingangle,suggesting
trons renders the system progressively more bilayer-like
a very unusual ”pinning” or near-pinning of the LLs in
athigherB andquenchestheenergyseparationbetween
|| a very large range of ν (Fig. 1(b)). Note in Fig. 3 that
the N =0LLsofthe symmetricandantisymmetric sub-
atagivendensitythesystemexhibits2Cbehaviorinthe
bands, making them essentially degenerate15,26. Further
entire range of ν <4/3 at θ (cid:39)34◦. This is very different
increasing B does not lift this degeneracy and the sys-
|| from the θ = 0 data of Fig. 2 where the LL crossing
tem remains 2C at the highest B . This is very different
|| features for any given density appear near a specific ν
from our data shown in Figs. 3(d) and (e), where the
which moves from low to high values as the density is
2DHS near ν = 1 becomes 2C only near θ (cid:39) 34◦, but is
increased. Moreover, in Fig. 3 the angle θ (cid:39) 34◦ at
1C at smaller and higher B .
|| which the 2DHS becomes 2C appears to be independent
We attribute the evolution in Fig. 3 data to a B - ofthe2DHSdensity. Inother2DHSsamples,confinedto
||
induced LL crossing13,27,28. Unfortunately, no accurate QWswithW rangingfrom35to50nm,wehaveobserved
calculationsofLLsinthepresenceofbothB andB are similarphenomenaasinFig. 3atthesameθ (cid:39)34◦. This
⊥ ||
available, particularly for 2DHSs with multiband struc- independence of the 2C behavior on ν, p, and W at this
ture. The tilted-field geometry implies complicated cou- critical angle is astonishing, and demands a theoretical
plings between Landau harmonic oscillators from dif- explanation.
ferent subbands, and makes numerical calculations ex- The evolution of the QHS at ν = 1 is also very in-
tremely demanding. Qualitatively, we can explain the triguing. As seen in Fig. 3, it disappears completely
crossing as follows. The densities of Fig. 3 data are at θ (cid:39) 34◦ when p = 2.05 but only becomes weak at
sufficiently large so that the LH-S0 level is lower than p = 1.28. In Fig. 4(a) we summarize our results for
the HH-S0 level near ν = 1 at B = 0 (Figs. 2(c) and many 2DHSs, illustrating the conditions for the stability
||
4
d/l QHSatthecrossing,andthefactthatthe2DHSis2Cat
d B (c)
(b) nearby fillings, suggest that it is a 2C QHS with strong
No QHS
interlayer correlations, likely the Ψ state reported in
111
GaAsbilayerelectron8–10,12 orhole11,17 systemsconfined
QHS
Δ to double QWs. In those systems, when the lowest LLs
y ≈ 34° (d) from different subbands are degenerate, the ν = 1 QHS
W = 45 nm g LH-S0
p = 1.39 × 1011 cm-2 ner Δ HH-S0 isstableatd/lB <∼2,andturnsintoacompressiblestate
E ν = 1 θ if d/l becomes large8,12. Also note that in our exper-
B
1.8 iments the energy separation between the two crossing
p (1011 cm-2) No QHS (a) LLs increases as θ deviates from (cid:39) 34◦ (see Fig. 4(d)).
2.05
1.6 We show in Fig. 4(c) a schematic ”phase diagram” for
1.10
1.39 the stability of the ν = 1 QHS as functions of ∆ and
1.59 d/l . TheresemblanceofFig. 4(c)tothephasediagram
1.4 B
d/lB 002...971582 QHS osifzeνt=ha1tQinHoSurinexdpoeurbimleeQntWs,sw8,1e2airsesetsrsikenintgia.llWyetuenminpgh∆a-
1.2 1.28 throughzeroaswetiltthesamplenearθ (cid:39)34◦;seeFigs.
1.71 35 nm
4(d).
40 nm
1.0
45 nm Inconclusion,2DHSsconfinedtowideGaAsQWsand
No crossing
50 nm
with sufficiently high density, reveal an unusual crossing
0.98
0.8 of the two lowest-energy LLs near ν = 1 as we tilt the
0 10 20 30 40 50 60
sample in magnetic field. It appears at a magic angle
θ (degrees)
θ (cid:39) 34◦, essentially independent of the QW width, den-
sity, orB (filling), suggestingapinningoftheLLsnear
FIG. 4. (a) Phase diagram for the stability of QHS at ν = ⊥
the crossing. The crossing and the pinning likely stem
1 as a function of the tilting angle θ and d/l . The solid
B
(open)symbolsmarkthepresence(absence)ofaQHSatν = from the complex interplay of the heavy- and light-hole
1. In narrow QWs and at low density, no crossing is seen LLsinB||,andshouldstimulatefurthertheoreticalinves-
as a function of θ, shown as the blue region. Once d/lB >∼ tigation. Nearthisangle,the2DHSbecomes2Catν <2
1.0, a crossing occurs near θ (cid:39) 34◦. At the crossing, a QHS and, if d/l is small, exhibits a ν = 1 QHS, consistent
B
appears at ν = 1 if d/lB <∼ 1.3, consistent with the Ψ111 with a correlated, 2C, Ψ111 state.
state. (b) Calculated charge distribution for a 45-nm-wide
QW with p=1.39 showing the interlayer distance d. (c)-(d)
Schematic phase and LL diagrams at ν =1 showing how the
LL separation ∆ increases as θ deviates from (cid:39)34◦. ACKNOWLEDGMENTS
We acknowledge the DOE BES (DE-FG02-00-
of the ν = 1 QHS. Data are shown as a function of θ ER45841)grantformeasurements,andtheNSF(Grants
and d/l , which compares the interlayer (e2/4π(cid:15)d) and DMR-1305691 and MRSEC DMR-1420541), the Gordon
B
intra-layer (e2/4π(cid:15)l ) correlations and is widely used to and Betty Moore Foundation (Grant GBMF4420), and
B
characterize bilayer QHSs8–12,17,29,30. Figure 4(a) shows Keck Foundation for sample fabrication and character-
thatnoLLcrossingatν =1canbeinducedviatiltingif ization. A portion of this work was performed at the
d/lB <∼ 1.0, and the ν = 1 QHS is always strong. When NHMFL, which is supported by the NSF Cooperative
d/lB >∼ 1.0, at ν = 1, the LH-S0 level is lower than the Agreement No. DMR-1157490, the State of Florida, and
HH-S0levelatθ =0,andthetwolevelscrossatθ (cid:39)34◦; theDOE.WethankS.Hannahs,G.E.Jones,T.P.Mur-
seeFig. 4(d). Atthecrossing,weobserveaQHSatν =1 phy, E. Palm, A. Suslov, and J. H. Park for technical as-
ifd/lB <∼1.3,andthegroundstatebecomescompressible sistance. WeareindebtedtoR.Winklerforilluminating
if d/lB >∼1.3. discussionsandprovidingtheself-consistentlycalculated
The d/lB <∼1.3 condition for the stability of the ν =1 charge distribution.
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xx
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|| ||
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||
increases.