Coulomb Oscillations in Antidots in the Integer and Fractional Quantum Hall Regimes

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by  A. Kou

Coulomb Oscillations in Antidots in the Integer and Fractional Quantum Hall Regimes
A. Kou,1 C. M. Marcus,1 L. N. Pfeiffer,2 and K. W. West2
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
2Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Dated: January 10, 2012)
We report measurements of resistance oscillations in micron-scale antidots in both the integer
and fractional quantum Hall regimes. In the integer regime, we conclude that oscillations are of
theCoulombtypefromthescalingofmagneticfieldperiodwiththenumberofedgesboundtothe
antidot. Based on both gate-voltage and field periods, we find at filling factor ν = 2 a tunneling
chargeofeandtwochargededges. Generalizingthispicturetothefractionalregime,wefind(again,
2 basedonfieldandgate-voltageperiods)atν =2/3atunnelingchargeof(2/3)eandasinglecharged
1 edge.
0
2
ThefractionalquantumHalleffectoccurswhenahigh- tive interpretation in terms of Coulomb charging of two
n
a mobility two-dimensional electronic gas (2DEG) is sub- isolated edge states [12]. Coulomb charging of an anti-
J ject to a perpendicular applied magnetic field. At low dotwasobserveddirectlybyKataokaetal.atbothν =1

8 temperature, electrons in the bulk of the 2DEG con- and ν = 2 using a QPC charge sensor [16]. Goldman et
denseintoincompressiblestates[1],withextendedstates al. foundthefieldperiodofoscillationstodependonthe

]
l at the sample edge carrying charge, spin, and energy. filling factor through the constriction, interpreting this
l
a It was theoretically shown that for Laughlin states such dependence as a signature of Coulomb charging of mul-
h as filling factor ν = 1/3, a single chiral edge state car- tiple isolated edges surrounding the antidot [20]. Recent
- ries excitations with fractional charge [2]. A more com- theory by Ihnatsenka et al. captures many of the effects
s
e plicated structure is predicted for hole-conjugate frac- observed experimentally in antidots in the integer quan-
m
tions such as ν = 2/3, where counter-propagating edge tum Hall regime [28]. Much less experimental work on
. states hybridize in the presence of edge disorder, lead- antidots has been reported in the fractional regime de-
t
a ingtoaforward-propagatingchargemodeandareverse- spite numerous theoretical proposals related to this sys-
m
propagating neutral mode [3–6]. Contrasts between the tem [23–27]. A key experiment was the measurement
- two counter-propagating modes at ν =2/3 and the sim- of fractional tunneling charge, e∗ = e/3, for an antidot
d
pler situation at ν = 2, where two forward-propagating withν =1/3intheconstrictions,basedontheback-gate
n
o modesremainseparatedandindependent,havebeendis- voltageandmagneticfieldperiodsofobservedresistance
c cussed theoretically [4, 6]. oscillations [13].
[
Ashoori et al. investigated the ν = 2/3 edge exper- InthisLetter,wereportmeasurementsofresistanceos-

1 imentally using edge magnetoplasmon propagation and
v found only a single charged edge mode [7]. Recent mea-
0 surementsofcurrentnoisethroughaquantumpointcon- (a) (b)
0 tact(QPC)atbulkfillingν =2/3foundfiniteshotnoise (a) (b) gate depletion gate
6 athalftransmissionoftheQPC,whichwasinterpretedas B Vq1 sHcrfeOe2n screening gate
1 HfO2
1. indicating a single composite 2/3 edge [8]. In addition, a Vs GaAs 2DEG
tunnelingchargeofe∗ =(2/3)ewasobservedatlowtem-
0 Vg (c)
2 perature, decreasing to e/3 above 0.1 K [8]. Subsequent (c)
1 theory addressed these surprising results by considering I VHVD
: the agglomeration of e/3 quasiparticles at low tempera-
v 1 (cid:43)m (cid:105)
i tures [9]. c
X Antidots have been used to study tunneling and con- νc
ar finement effects in both the integer and fractional quan- (cid:105)b νb
tum Hall regimes [10–20]. Early experiments by Hwang Vq2
et al. [10] found resistance oscillations as a function of
FIG. 1: (a) False color SEM of the larger antidot device with
magnetic field and channel gate voltage for a density
slightly below ν = 1 in the constrictions between the 2 µm diameter. Diagonal voltage, VD, across the device, as well
as bulk Hall voltage, VH, are measured as shown. Depleted gates
antidot and sample edges. The observed field period, areindicatedinyellow,groundedgatesareindicatedinblue. The
corresponding to one flux quantum through the antidot, QPC constriction sizes range from 500 nm to 1.5 µm. (b) Cross
sectionofthedevicetakenalongthedashedlinein(a). Theantidot
was interpreted in terms of a single-particle Aharonov-
gate is separated from the screening gate by a 30 nm HfO2 layer.
Bohm phase [10]. Subsequent experiments near ν = 2
(c) Schematic layout of the edge states through the device when
intheconstrictionsfoundafieldperiodcorrespondingto νc = νc−. The bulk filling factor, νb, and the constriction filling
h/2e (φ /2) through the antidot, motivating an alterna- factor,νc,areindicated.
0
2
cillationsingate-definedantidotsoftwosizes,comparing
integer and fractional filling factors in the constrictions
betweentheantidotandadjacentgatesthatextendtothe
sample edge. Oscillations as a function of perpendicular
magnetic field and antidot gate voltage were measured ν = 2/3
c
in 2D sweeps, and dominant frequencies extracted from
2D Fourier spectra. For integer filling in the constric-
R
D
tions,magneticfieldoscillationfrequencieswerefoundto
be proportional to the filling factor in the constrictions,
consistent with a Coulomb charging model. At ν =2/3,
the magnetic field oscillation frequency was found to be RH
consistent with a single charged edge within a general-
ν = 1
ized Coulomb charging picture, with the charge-carrying b
edge state located slightly closer to the antidot than the ν b = 4/3
singleedgefoundatν =1. Gate-voltageoscillationspro-
vide a direct measurement of the tunneling quasiparticle ν = 5/3
b
charge. Normalizing to a tunneling charge of e at ν =1,
ν = 2
which determines the gate voltage lever arm, we find a b
tunneling charge consistent with e for all measured inte-
gerfillingfactors,andatunnelingchargeconsistentwith
e∗ =(2/3)e at ν =2/3.
Antidot devices with 1 µm and 2 µm diameter an-
tidots were fabricated on a symmetrically Si-doped FIG. 2: Diagonal resistance, RD (red), and bulk Hall resistance,
GaAs/AlGaAs 30 nm quantum well structure located RH (blue),asafunctionofperpendicularmagneticfield,B,inthe
1 µm diameter antidot with the 500 nm QPCs activated. Insets
230 nm below the wafer surface, with density n = 1.6× showregionsofoscillationsatνc=2−,νc=1−,andνc=2/3−.
1015 m−2 and mobility µ = 1,200 m2/V·s measured in
the dark. A Ti/Au (8 nm/42 nm) screening gate was
firstpatternedusingelectron-beamlithographyonawet- well-quantized plateaus. This filling, measured on the
etched mesa [purple gate in Fig. 1(a)]. The sample was high-field side of the ν plateau in R , is denoted ν−.
c D c
then coated with 30 nm of HfO using atomic layer de- Figure 2 shows R (B) and R (B) at fixed gate volt-
2 H D
position. A circular antidot [Ti/Au (8 nm/42 nm)] was ages in the 1 µm diameter antidot with the 500 nm
next patterned on top of the HfO , positioned to extend QPCs activated. Zoom-ins reveal periodic oscillations
2
beyond the edge of the screening gate, with connection in R (B), with periods ∆B = 2.1 mT for ν = 1−,
D c
to a remote bonding pad via a “pan handle” depletion ∆B = 1.0 mT for ν = 2−, and ∆B = 2.9 mT for
c
gate that runs on top of the screening layer, so that only ν = 2/3−. Aperiodic fluctuations in R (B) were also
c D
the2DEGundertheantidotisdepletedwhenthegateis observed on the low-field sides of the plateaus. The pe-
activated [Fig. 1(a, b)]. riod of 2.1 mT at ν = 1− corresponds to φ = h/e
c 0
Transport measurements were made using a current through an area of 2.0 µm2, larger than the lithographic
bias I of 0.3 nA at 101 Hz, with magnetic field, B, ap- areaoftheantidot,A=0.8µm2. Weattributethelarger
plied perpendicular to the plane of the 2DEG, in a dilu- areatothefinitedepletionlengthoftheantidottopgate,
tion refrigerator with base temperature ∼10 mK. QPC as discussed quantitatively below. The period of 1.0 mT
gatevoltages(V ,V ), weretrimmedaround−1.0V to at ν = 2− corresponds to φ /2 going through a device
q1 q2 c 0
symmetrize the device, i.e., to give the same filling fac- ofaboutthesamearea. Continuingthistrendforinteger
tors in the two constrictions. The antidot gate voltage, states,oscillationsobservedatν =3− hadafieldperiod
c
V , was then swept around −0.9 V, yielding periodic re- correspondingtoφ /3;oscillationsatν =4− hadafield
g 0 c
sistance oscillations. The screening gate and all unused periodcorrespondingtoφ /4(notshown). Theobserved
0
QPC gates were grounded. The diagonal voltage, V , scalingoffieldperiodswithconstrictionfillingfactor,i.e.,
D
was measured between incoming edge states on opposite the field period at ν corresponds to φ /ν , is consistent
c 0 c
sides of the device [Fig. 1(c)], and the diagonal resis- with previous experiments in the integer quantum Hall
tance, R ≡ dV /dI, was used to determine the filling regime in antidots [12, 20].
D D
factor in the QPCs, ν = h/(R e2). The bulk Hall re- Figure 3 shows R with a smooth background sub-
c D D
sistance, R ≡ dV /dI, was simultaneously measured tracted, denoted δR , as a function of both B and V
H H D g
using contacts away from the antidot [Fig. 1(a)]. Figure in the 2 µm diameter antidot with 500 nm QPCs acti-
1(c) schematically shows the bulk filling factor, ν , along vated at ν = 1−, ν = 2−, and ν = 2/3− along with
b c c c
with the filling factor in the constrictions, ν , in the con- correspondingtwo-dimensional(2D)Fourierpowerspec-
c
dition where R is slightly larger than the resistance of tra. The 2D plots of δR reveal positively sloped stripes
D D
3
D [µm] ν d [nm] ∆B [mT] N
(a) c
νc = 1 - δR νc = 1 - 1 320 0.74 1
D A 2 2 310 0.38 2
[xh1/0e- 23 ] -1V ] [a.u.] 2/3 190 0.92 1
1 m 1 320 2.1 1
V [g 1.0 1 2 310 1.0 2
0 Δ1/ 0.5 2/3 190 2.9 1
-1 0
TABLE I: Summary of magnetic field oscillations data, for
antidotswithlithographicdiameterD andfillingfactorν in
c
theconstrictionsconnectingtheantidottothesampleedges.
(b)
ν = 2- ν = 2- Oscillation periods, ∆B, used to determine depletion length,
c δRD c A d,takentobethesameforthetwoantidotsatthesamefilling
[xh1/0e- 23 ] -1V ] [a.u.] factor, and the number of edge channels, N.
2 m 6.0
0 Δ1/V [g 24..00 D [µm] νc ∆Vg[mV] ∆V∆gνVcg=1
-2 0 1 0.28 ≡ 1
2 2 0.28 1.0
2/3 0.18 0.64
(c) 1 0.62 ≡ 1
- -
νc = 2/3 νc = 2/3 1 2 0.62 1.0
δRD A 2/3 0.42 0.67
[xh1/0e- 24 ] -1V ] [a.u.]
m
5 V [g 0.2 TtiAdoBtLsEwiItIh:lSituhmomgraarpyhiocfdgiaatme-evtoelrtaDgeanosdcifilllalitnigonfascdtaotraν,foinrtahne-
0 Δ c
1/ 0.1 constrictions. Oscillation periods, ∆V , and periods relative
g
-5 to the period at ν =1− determine the tunneling charge, e∗,
0 c
as discussed in the text.
the same depletion length, d, for two antidot sizes mea-
FIG.3: Diagonalresistancewithbackgroundsubtracted,δRD,as sured at the same antidot gate voltage. To calculate the
a function of B and Vg in 2 µm diameter antidot at (a) νc =1−, depletion length from the data at ν = 2−, we take one
(b) νc = 2−, and (c) νc = 2/3−, with corresponding 2D Fourier c
flux quantum to correspond to twice the measured mag-
power spectra. Dominant peaks in the power spectra are marked
by dashed lines, located at (1.4 mT−1, 3.6 mV−1) for νc = 1−, neticfieldperiod. Notethatthedepletionlengthdeduced
(2.7 mT−1, 3.6 mV−1) for νc = 2−, and (1.1 mT−1, 5.6 mV−1) in this way is ∼ 100 nm smaller at νc = 2/3− than at
forνc=2/3−. ν =1− and ν =2−.
c c
Thenumberofcharge-carryingedges,N,ateachfilling
factorisdeducedfromthemagneticfieldperiodwithina
in all cases. The dominant peaks in the Fourier spec- Coulomb charging picture, where each edge contributes
trashowtheexpectedscalingofthefieldperiodbetween one oscillation in resistance per flux quantum. In the
νc =1− andνc =2−,withpeaksat1.4mT−1 atνc =1− integer regime, this interpretation is consistent with pre-
and 2.7 mT−1 at νc = 2− differing roughly by the same vious experiments and data in both antidots and quan-
factoroftwoasdiscussedabove. Atνc =2/3−,thedom- tumdots [12,20, 29,30], andreconciles theunphysically
inant Fourier peak is at 1.1 mT−1, which is close to, but large effective area that would be inferred from the field
somewhat smaller than the magnetic field frequency at period at ν = 2− from a single-edge picture. We find,
c
νc =1−. This scaling of magnetic field periods is similar therefore, N = 2 separate charge-carrying edges bound
to what was observed in the 1 µm device above. The to the antidot at ν =2− . At ν =2/3−, the measured
c c
position of the dominant spectral peak as a function of period is too large to correspond to two charged edges
gate-voltage frequency, 1/∆Vg, is the same for νc = 1− within an analogous Coulomb picture, as the large field
and νc = 2−, in both cases 3.6 mV−1, but increases to period would imply d < 0. We thus find N = 1 charged
5.6 mV−1 at νc =2/3−. edge at νc =2/3−.
TableIsummarizestheobservedoscillationsasafunc- TableIIsummarizesobservedoscillationsasafunction
tion of magnetic field. Magnetic field periods at each of antidot gate voltage. Gate-voltage periods at ν =1−
c
filling factor were used to find a best-fit effective area and ν = 2− are the same within measurement uncer-
c
enclosed by the charged edge state, constrained to have tainty. The gate-voltage period at ν = 2/3−, is ∼ 2/3
c
4
of this value. No features are visible in the ν = 2/3− B. Rosenow, and A. Yacoby for useful discussions, and
c
Fourierspectrumat1/3timesthegate-voltageperiodat P. Gallagher for experimental contributions. Research
ν =1− ineitherdevice. Wealsodonotfindanychange supported by Microsoft Project Q, the National Science
c
in the gate-voltage period above 100 mK, in contrast to Foundation(DMR-0501796),andtheDepartmentofEn-
the behavior observed in Ref. [8]. ergy,OfficeofScience. DevicefabricationusedHarvard’s
In summary, based on magnetic field and gate-voltage Center for Nanoscale Systems.
periods, we infer a Coulomb charging mechanism for the
observed resistance oscillations. In the case of single-
particleAharonov-Bohmoscillations,onewouldexpecta
constantfieldperiodandagate-voltageperiodthatscales
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