Multicomponent fractional quantum Hall states with subband and spin degrees of freedom

0.72 MB·English

by  Yang Liu

Multicomponent fractional quantum Hall states with
subband and spin degrees of freedom
Yang Liu, S. Hasdemir, J. Shabani, M. Shayegan, L.N. Pfeiffer, K.W. West, and K.W. Baldwin
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544
(Dated: November4, 2015)
In two-dimensional electron systemsconfined towide GaAsquantumwells weobservea remark-
ablesequenceoftransitionsofthefractionalquantumHallstatesinthefillingfactorrange1<ν <3
5
nearthecrossingoftwoN =0Landaulevelsfromdifferentsubbandsandwithoppositespins. The
1
transitionsattest totheinterplaybetweenthespin andsubbanddegrees offreedom andcan beex-
0
plainedaspseudospinpolarizationtransitionswherethepseudospinsarethetwocrossinglevels. The
2
magneticfieldpositionsofthetransitionsyieldanewandquantitativemeasureofthetransitionen-

v ergiesor,equivalently,thecompositeFermions’discreteenergylevelseparations. Surprisingly,these
o energies are much larger than those reported for other systems with SU(2) symmetry; moreover,
N theyare larger when theelectron system is less spin polarized.

3
I. INTRODUCTION FQHSs in the filling range 1 < ν < 3, revealing the for-
]
l mationofFQHSswithbothspinandsubbanddegreesof
al freedom simultaneously.19 We can qualitatively describe
h Fractional quantum Hall states (FQHSs) are the hall- the number of observed transitions and their positions
- marks of an interacting two-dimensionalelectron system
in a simplified two-component picture with SU(2) sym-
s
(2DES)atlargeperpendicularmagneticfield(B )when
e ⊥ metry, where the two pseudospins are the S0↓ and A0↑
m a Landau level (LL) is partially occupied.1,2 Adding ex- levels. However,ourquantitativeanalysisrevealsseveral
tra electronic (pseudospin) degrees of freedom leads to
. puzzling features. We find that at a fixed ν, the tran-
t additional sets of LLs which are separated by, e.g., the
a sition energies are much larger than reported previously
m Zeemanenergy(EZ),subbandseparation(∆SAS),orval- in systems with SU(2) symmetry where the two pseu-
- leysplittingenergy(EV),insystemswithspin,subband, dospins are either spin or valley levels. Also, the tran-
d orvalleydegreeoffreedom,respectively. Whentwosuch sition energies are about twice larger when the 2DES is
n LLs cross at a particular LL filling factor (ν), the QHS
less spin-polarized. We discuss possible origins of these
o at that filling typically weakens or disappears.3 In very
unexpected features.
c high quality samples, however, if two N = 0 LLs cross,
[
the 2DES can form two-component FQHSs where the

2 components are the crossing LLs (the pseudospins), and
v exhibit pseudospin polarization transitions as one tunes
8 the pseudospin energy splitting, or the separation be-
5
tween the two LLs. Such transitions have been reported
9 II. METHOD
for2DESsconfinedtoAlAsquantumwells(QWs)where
6
0 theenergyseparationbetweenthetwooccupiedvalleysis
. tunedviathe applicationofstrainsothatthetwolowest
1 Our sample, grown by molecular beam epitaxy, is a
(N =0)LLsofthetwovalleyscross.4–7Multi-component
0 65-nm-wideGaAsQWboundedoneachsidebyundoped
FQHSs and transitions between their pseudospin config-
15 urations have also been studied in other 2DESs where Ausle0.2a44G×a0.476mAms s2psaqcuearrleapyeiercseawndithSialδlo-dyoedpeIdnSlanyceorns.taWctes
: two or more LLs with different spin or valley indices are
v atfourcorners,andanevaporatedTi/Aufront-gateand
close in energy.8–17
i an In back-gate to change the 2DES density (n) while
X
Here we present a study of 2DESs confined to wide keeping the charge distribution symmetric [Fig. 1(b)].
ar GaAs QWs where EZ and ∆SAS are much smaller than We measure the transport coefficients in a dilution re-
the Coulomb energy at large B⊥. Thus the lowest four frigeratorwith base temperature T ≈30 mK, using low-
LLs–the S0↑,S0↓,A0↑andA0↓levels–arecloseinen- frequency (< 30 Hz) lock-in technique and a rotatable
ergy so that, in principle, a full description of FQHSs sample platform to induce B . At B = 0, the Fourier
|| ||
would require the inclusion of all four LLs and using transformof the Shubnikov-de Haas oscillations exhibits
SU(4) symmetry (S and A refer to symmetric and anti- two peaks that correspond to the electron densities in
symmetricsubbands,↑and↓refertoup-anddown-spin; the two subbands. The difference between these densi-
see Fig. 1(a)). Via applying either a parallel magnetic tiesyields∆SAS whichisinexcellentagreementwiththe
field component (B||, see Fig. 1(b)), or changing the results of our B = 0 self-consistent calculations. To cal-
electron density in the QW,18 we reduce ∆SAS and in- culate ∆SAS at finite B|| and B⊥ [Fig. 1(a)], we employ
crease EZ, so that the S0↓ and A0↑ levels cross when a perturbative simulation introduced in Ref. [20], where
∆SAS = EZ; see Fig. 1(a). Near the crossing, we ob- we assume B|| only mixes LLs from different subbands
servearemarkablepatternofappearinganddisappearing but does not change the QW potential.
2
ν = 2 (a) (b) T ≈ 65 mK (e)
A0↓ B
A0↑ 10 K B|| 5/3 7/5
B θ 10/7 4/3
Δ ┴ W = 65 nm ν = 2 8/5
SAS 11/7
S0↓ E 0.4 0.6
S0↑ Z T ≈ 65 mK (d)
0 2 4 6
B (T) θ
||
0.4 8/3 7/3 41.1˚
(c) T ≈ 135 mK 12/5 ν = 2
0.3 3
7/3 2 13/5 40.5˚
8/3 ÷ 2 θ 0.4
40˚
0.3
ν = 4/3 48.9˚
46˚ 5/3 39.4˚
θ 46.1˚
45˚ 0.2 38.1˚
41˚
44.9˚ 37.5˚
0.2
40˚ 43.6˚ 36.2˚
)
kΩ 42.4˚ 0.2 35˚
( 37˚
R xx 0.1 40.5˚ 32.5˚
0.1 30˚ 38.1˚ 30˚
25˚ 36.9˚ 27.5˚
35˚ 25˚
0 0˚ 0 0˚ 0 0˚
3 4 5 6 7 3 4 4 5 6 7
B (T) B (T) B (T)
┴ ┴ ┴
FIG. 1. (color online) (a) Relevant energy levels for n=2.12×1011 cm−2 electrons confined to a 65-nm-wide GaAs QW near
filling factor ν =2, calculated as a function of parallel magnetic field (B ). (b) Charge distribution (red curve) and potential
||
(black curve) calculated viasolving Poisson-Schr¨odinger equations self-consistently at B=0. The experimentalgeometry and
the tilt angle θ are also shown. (c)-(e) Waterfall plots of Rxx vs B⊥, measured at the indicated temperatures and different θ.
Each trace is shifted vertically. The observed FQHS transitions are marked by theblack arrows.
III. EXPERIMENTAL RESULTS and has also been reported in previous studies.3,17,21,22
More interestingly, near θ ≃ 37◦, the FQHSs on both
Figure 1 (c) shows the longitudinal magnetoresistance sidesofν =2showarichseriesoftransitions,asmarked
(Rxx) traces for 1<ν <3, measured at n=2.12×1011 with arrows in Figs. 1(c)-(e).26 The ν = 4/3 FQHS is
cm−2, T ≃ 135 mK, and different tilt angles (θ). As strong at both small and large θ, but becomes weak at
we increase θ, the ν = 2 Rxx plateau narrows near θ ≃ 30◦ [Figs. 1(c) and (e)]. The ν = 8/3 FQHS also
θ ≃ 37◦ and then widens again at larger θ. The re- experiences one transition, at θ ≃ 45◦ [Figs. 1(c) and
duction of the Rxx plateau width signals a weaken- (d)]. Meanwhile, the ν = 5/3 and 7/3 FQHSs become
ing of the integer QHS, which is widely used to iden- weak twice: at θ ≃ 30◦ and 40◦ for ν = 5/3, and at θ ≃
tify LL crossings in high quality 2DESs and at low 37◦and45◦forν =7/3. DatatakenatlowerT ≃65mK,
temperatures.3,21,22 This weakening, but not disappear- shown in Figs. 1(d) and (e), reveal a more remarkable
ing, is similar to what is seen at certain LL crossings at pattern of higher-order FQHS transitions. On the left
other integral ν when interaction preserves the energy side of ν = 2 [Fig. 1(d)], the ν = 13/5 FQHS weakens
gap through QHS ferromagnetism.3,23,24 In our samples, twice at θ ≃ 40.5◦ and 46.1◦. The ν = 12/5 FQHS, on
∆SAS is larger than EZ at B|| = 0. As we increase B||, the other hand, becomes weak three times, at θ ≃36.9◦,
∆SAS decreases and EZ increases, so that the S0↓ and 42.4◦ and 46.1◦. On the high-field side of ν = 2, as
A0↑ levels cross when ∆SAS ≃ EZ.25 In Fig. 1(a), we seen in Fig. 1(e), the ν = 7/5 FQHS weakens twice,
show the calculated energies of the four N =0 LLs rela- at θ ≃ 27.5◦ and 35◦, and the ν = 8/5 FQHS thrice,
tive to the S0↑ level as a function of B at ν = 2.20 We at θ ≃ 27.5◦, 36.2◦ and 39.4◦. We also observe in Fig.
||
use a three-foldenhancedEZ to matchthe experimental 2(e)three transitionsatν =10/7andfourtransitionsat
observation that the ν = 2 crossing occurs near θ ≃ 37◦ ν =11/7.18 We summarize in Fig. 2(a) the values of B
||
(B|| ≃3.5 T). This enhancement of EZ is not surprising and B⊥ for all the observed transitions.
3
5 (a) -4 In this model, the FQHS at ν = 4/3 has only one oc-
Δ < 0 νCF = -2-3 32 cupied Λ level (νCF =1).29 It therefore has two possible
Δ < E configurations with the Λ-level filling factors of the up-
SAS Z 1
and down-pseudospin CFs being (ν ,ν )=(1,0) or
-3 CF↑ CF↓
4 -2 -1 (0,1), and should exhibit one transition at ∆ = 0 [see
2
) Fig. 2(b)], consistentwith Fig. 2(a) data. The FQHS at
T
B (|| 1 2 4/3 ν =7/5,ontheotherhand,hastwofilledΛ-levels(νCF =
2),threepseudospinconfigurations[(ν ,ν )=(2,0),
CF↓ CF↓
3 ν = 5/3 10/7 (1,1) and (0,2)], and two transitions when ∆ = ±¯hωCF
8/3 7/5 [Fig. 2(b)], alsoin agreementwithFig. 2(a)data. Using
8/5
13/5 Δ > 0 similar logic, we can explain the three transitions ob-
12/5 11/7
7/3 ΔSAS > EZ served for the ν = 10/7 FQHS which has νCF = 3 and
2
3 4 5 6 7 four configurations [Fig. 2(b)]. By invoking particle-
B (T) hole symmetry, whichlinks the FQHSs at ν to (4−ν) in
(b) ┴
Energy our system (e.g., 4/3 to 8/3), and also utilizing negative
CF fillings, e.g., ν = −2 for the ν = 5/3 state, we
CF
can explain all the transitions summarized in Fig. 2(a).
ℏω
CF In general, a FQHS with νCF has |νCF|+1 pseudospin
(4,0)(3,0) (3,1) (2,1) (2,2) (1,2) (1,3) (0,3)(0,4)±4 configurations,and |νCF| pseudospin polarizationtransi-
|Δ| (2,0)(1,0) (1,1) (0,1) (0,2) ±3 tionswhichoccurwhenever|∆| equalsamultiple integer
±2 of ¯hω [see Fig. 2(b)].
νCF = ±1 CF
Δ > 0 Δ < 0 NextweproceedtoaquantitativeanalysisofFig. 2(a)
B
|| data. Figure 2(b) implies that, when νCF is odd, one
transition occurs exactly at ∆ = 0. In Fig. 2(a), we
FIG. 2. (color online) (a) Summary of B|| and B⊥ where first fit the dashed line through these transition points.
pseudospin transitions are seen in Fig. 1. The dashed We then focus on a particular ν, e.g. 8/5, and calculate
line represenst zero pseudospin splitting ∆ = 0 (see text).
∆=∆SAS−EZ ispositiveinthegreen region, andbecomes ∆3(SaA).SUassinagfutnhcetivoanluoefBof||Battahtiswfihlilcinhgt,haes∆sho=w0nitnraFnisgi-.
negativeintheyellowregion. Typicalerrorbarsareshownfor ||
tion for ν = 8/5 occurs (B = 3.96 T), we determine a
theq/3FQHStransitions. (b)SchematicdepictionoftheCF ||
Λ-level energies in a simplified two-pseudospin picture. The value for g∗ (≃−1.34)so that ∆SAS =EZ at B|| =3.96
dashed black and solid red lines represent the CF Λ-levels T.We thenplotEZ (=g∗µBB)atν =8/5asa function
with up- and down-pseudospins (the electron S0↓ and A0↑ of B|| in Fig. 3(a), and determine ∆ for the other two
levels). Each configuration is labeled with its pseudospin Λ- 8/5 transitions. (See Fig. 3(b) for the four pseudospin
level filling factors (νCF↑, νCF↓), where ↑ and ↓ stand for the configurationsofthe ν =8/5 FQHS.) Since these transi-
up- and down-pseudospins. tions are expected to occur when ∆=±2h¯ω [see Fig.
CF
2(b)],wefind¯hω =5.5Kfor∆>0andh¯ω =2.5K
CF CF
for ∆<0. Using this procedurewe candeduce ¯hω for
CF
IV. DISCUSSION FQHSs at ν = 8/5, 12/5, and 10/7 which have a ∆ = 0
transitiononthedashedlineinFig. 3(a). FortheFQHSs
at ν = 13/5, 7/3, 5/3 and 7/5, we assume ∆= 0 at the
The fact that all these transtions occur near the
intersectionofthe dashedline andthe verticallines that
crossing of the S0↓ and A0↑ levels when ∆SAS ≃ EZ
mark these fillings in Fig. 3(a) and, following a similar
suggests that these are FQHS pseudospin polarization
transitions.19 Suchtransitionscanbe readilyunderstood procedure, find ¯hωCF from the transitions’ B|| values.
in the framework of composite Fermions (CFs), quasi- In Fig. 3(c) we plot as a function of 1/(2νCF+1) all
particles formed by attaching two flux quanta to each the deduced values of ¯hωCF, normalized to the Coulomb
electron.2,27,28 The CFs form their own discrete en- energy(VC =e2/4πǫlB,wherelB =p¯h/eB⊥isthemag-
ergy levels, the so-called Λ-levels which are separated neticlengthandǫisthedielectricconstant),fordifferent
by the CF cyclotron energy ¯hω , and the FQHS at FQHSs. Figure 3(c) data allow us to make a quanti-
CF
ν =νCF/(2νCF+1) is the integer QHSofCFs with inte- tative comparison of the deduced ¯hωCF/VC to previous
gralΛ-levelfillingfactorν . Sincetheothertwoenergy experimental reports. The dashed lines in Fig. 3(c) rep-
CF
levels,S0↑andA0↓,arereasonablyfarinenergyandare resent h¯ωCF/VC, measured near ν = 1/2 and 3/2, from
alwaysfullorempty,19 weneglectthemandinterpretour FQHSspin transitionsinQWswithsamewell-width(65
data in a simple (SU(2)) picture where the two crossing nm) but much lower density where the S0↑ and S0↓ lev-
levels,S0↓andA0↑,arethetwopseudospins.29Thepseu- els are closer in energy and the A0↑ level is well above
dospin splitting energy ∆ = ∆SAS −EZ is then tuned S0↓.13,18,19 Figure 3(c) plot clearly shows that our mea-
frompositivetonegativeviaapplyingB|| [Figs. 1(a)and sured h¯ωCF/VC is much larger than the dashed lines.
2(b)]. Asdiscussedbelow,thissimplifiedtwo-pseudospin Itispossiblethatthetransitionenergieswededuceare
model qualitatively explains our data. somewhat exaggerated because of the inaccuracy of the
4
(a) (b) ν = -3 (ν = 8/5) toroftwolargerthanthedashedlinesinFig. 3(c).18 We
CF
thereforeconcludethat¯hω inour2DESwithbothspin
CF
Δ andsubbanddegreesoffreedomislargerthanina2DES
10 ℏω
Δ CF with only spin degree of freedom. The reason for this
K) Δ SAS E is not entirely clear. Perhaps the nature of pseudospins
( Z
y plays a role in determining the polarizationenergies and
g
er ¯hωCF. It is also possible that in our system the close
n
E ν = 8/5 proximity of the other two LLs (S0↑ and A0↓; see Fig.
0 1(a))is important. Bothofthese interestingpossibilities
3 4 5 deserve further theoretical examination.
B (T)
|| We highlight two additional noteworthy features of
2 11/7 ν = 10/7 (c) Fig. 3(c). First, the results for 1<ν <2 approximately
matchthosefor2<ν <3,suggestingthat¯hω doesnot
) CF
% Δ < 0 dependonwhichLL(S0↓orA0↑)hoststheCFs. Second,
(0
C Δ > 0 inour2DESwetunethe crossingoftwoN =0LLswith
V opposite spins while a lowerlevel(S0↑)is fully occupied.
/F
ωC2 5/3 This allows us to observe FQHS pseudospin transitions
ℏ 7/3 1 < ν < 2 at a given ν for both ∆ < 0 and ∆ > 0, when the net
8/5 2 < ν < 3 7/5 spinorientationoftheCFsisalignedoranti-alignedwith
4 12/5 13/5 the spin (↑) of majority electrons, respectively. Data of
-0.4 -0.2 0 0.2 0.4 Fig. 3(c)indicatethat¯hωCF/VC isabouttwicelargerfor
1/(2ν +1) the case ∆ > 0 compared to the ∆ < 0 case. This is an
CF important observation as it sheds light on the mysteri-
ous asymmetry between h¯ωCF/VC for CFs near ν =3/2
FIG.3. (coloronline)(a)Calculated∆SAS andEZ asafunc- and 1/2, deduced from CF spin-polarization transitions
tion of B at ν =8/5. (b) Λ-level diagram showing the four
|| inGaAs2DESs,13orCFvalley-polarizationtransitionsin
different pseudospin configurations for ν = 8/5 (νCF = −3) AlAs2DESs.6 Inthose2DESsthepolarizationtransition
in our two-component picture. The pseudospin polarization
energiesofFQHSsnear3/2,whereelectronsarepartially
transitionsareexpectedwhen∆=0,±2¯hωCF [seeFig. 2(b)].
(c) The Λ-level separation in units of VC = e2/(4πǫlB) as a spin- (or valley-) polarized, are much larger than those
functionof1/(2νCF+1),deducedfrom∆atwhichtheFQHS near1/2wheretheyarefully polarized.6,13 Ourdata,af-
transitionsareobservedinFig. 2(a). Dataareshownforboth forded by the tunability of the subband/spin degrees of
1<ν <2(black)and2<ν <3(red). Thedashedlinesrep- freedom,provideclearevidencethatthespinpolarization
resent data for 2DESs with only spin degree of freedom13; ofthe totalelectronsystemcanaffectthe CFproperties.
results from transitions near ν = 1/2 and 3/2 are shown in
the∆<0 and ∆>0 regions, respectively.
ACKNOWLEDGMENTS
We acknowledge support through the NSF (DMR-
perturbative calculations we use to determine the B - 1305691), and DOE BES (DE-FG02-00-ER45841) for
||
dependence of ∆SAS. To test this possibility, we made measurements, and the NSF (Grant MRSEC DMR-
measurementsonanotherQWwithslightlynarrowerwell 1420541), the Gordon and Betty Moore Foundation
width (55 nm).18 In this sample, we induce the crossing (Grant GBMF4420), and Keck Foundation for sample
of the S0↓ and A0↑ levels via applying B and also by fabrication and characterization. A portion of this work
||
tuning n at B = 0. The ¯hω deduced from the tilt- was performed at the NHMFL, which is supported by
|| CF
ing data matches Fig. 3 data, but the n-tuning results theNSFCooperativeAgreementNo. DMR-1157490,the
aresomewhatsmaller,18 suggestingtheinaccuracyofour State of Florida, and the DOE. We thank S. Hannahs,
perturbative calculations.30 However,we emphasize that G. E. Jones, T. P. Murphy, E. Palm, A. Suslov, and J.
eventhe n-tuning datawhichdo notrelyoncalculations H. Park for technical assistance, and J. K. Jain for illu-
of ∆SAS, yield ¯hωCF/VC values that are about a fac- minating discussions.
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||
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Nat. Phys. 11, 347 (2015). occupied in the range of interest here, a FQHS at ν is
18 See SupplementalMaterial.. equivalenttotheν′ =ν−1FQHSinasystem suchas2D
19 As reported in Refs. [8-13], spin transitions of FQHSs electronsinanAlAsQWwherethecrossing levelsarethe
around ν = 3/2 and 1/2 are seen in GaAs 2DESs as a lowest two (valley) S0 LLs; see, e.g., Refs. [4–7].
functionofdensityortiltangle.Ina65-nm-wideQWsam- 30 We indeed hope that our data will stimulate more precise
ple, wealso see suchspin transitions at low densities near futurecalculations; see, e.g., Refs. [31]and [32].
n=1.2×1011 cm−2(seeRef.[13]fordetails).Inthiswork, 31 T. S. Lay, T. Jungwirth, L. Smrˇcka, and M. Shayegan,
ataveryhighdensity(n=2.12×1011 cm−2)andwithad- Phys. Rev.B 56, R7092 (1997).
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befully occupiedand nospin transitionsareexpected,we Tsui, A. H. MacDonald, S. M. Girvin, L. Smrc˘ka,
observe additional transitions which, in the simplest and L. N. Pfeiffer, K. W. Baldwin, and K. W. West,
most plausible explanation,are inducedbythecrossing of Phys. Rev.Lett. 85, 3257 (2000).