Coherent tunnelling across a quantumpoint contact in the quantum Hall regime
F. Martins
1
, S. Faniel
1,2
, B. Rosenow
3
, H. Sellier
4
, S. Huant
4
, M. G. Pala
5
, L. Desplanque
6
, X. Wallart
6
, V. Bayot
1,4
& B. Hackens
1
1
IMCN/NAPS, Universite´ catholique de Louvain, LouvainlaNeuve B1348, Belgium,
2
ICTEAM/ELEN, Universite´ catholique deLouvain, LouvainlaNeuve B1348, Belgium,
3
Institute for Theoretical Physics, Leipzig University, Leipzig D04009, Germany,
4
Institut Ne´el, CNRS and Universite´ Joseph Fourier, Grenoble F38042, France,
5
IMEPLAHC, Grenoble INP, Minatec, Grenoble F 38016, France,
6
IEMN, Cite´ scientifique, Villeneuve d’Ascq F59652, France.
Theuniquepropertiesofquantumhalldevicesarisefromtheidealonedimensionaledgestatesthatforminatwodimensionalelectronsystemathighmagneticfield.Tunnellingbetweenedgestatesacrossaquantumpoint contact (QPC) has already revealed rich physics, like fractionally charged excitations, or chiralLuttinger liquid. Thanks to scanning gate microscopy, we show that a single QPC can turn into aninterferometerforspecificpotentiallandscapes.Spectroscopy,magneticfieldandtemperaturedependencesof electron transport reveal a quantitatively consistent interferometric behavior of the studied QPC. Toexplain this unexpected behavior, we put forward a new model which relies on the presence of a quantumHall island at the centre of the constriction as well as on different tunnelling paths surrounding the island,thereby creating a new type of interferometer. This work sets the ground for new device concepts based oncoherent tunnelling.
E
lectron phase coherence is the cornerstone of quantum devices and computation
1,2
. In that perspective,Quantum Hall (QH) devices are particularly attractive in view of their large coherence times
3
. QuantumHall edge states (ES) formed by Landau levels (LL) crossing the Fermi energy near sample borders are idealonedimensional (1D) systems in which scattering vanishes exponentially at low temperature
T
1,4
. Edge stateloops surrounding potential hills or wells, referred to as localized states or quantum Hall islands (QHIs), thenform unique zerodimensional (0D) systems
5
. The last few years witnessed great progresses in the transportspectroscopy of model QH localized states created by patterning quantum dots
6
or antidots
7–9
in a twodimensional electron system (2DES).Inparallel,newtoolsweredevelopedtoprobethemicroscopicstructureofconfinedelectronsystemsintheQHregime. Inparticular, scanning gate microscopy
10–16
(SGM) makes use of a movable metallic tip, which is voltagebiased, to finely tune the electrons’ confining potential in its vicinity. This way, the geometry of propagating edgestates and localized states can be modified at will
17
. Very recently, SGM allowed us to locate active QHIs in a QHinterferometer
18
. Importantly, it appeared that QHIs do not only form around antidots, but potential inhomogeneities also induce QHIs in the arms or near the constrictions connecting a quantum ring to source and drainreservoirs
18
. Therefore, lateral confinement, e.g. in quantum point contacts (QPC), offers the possibility toconnect a QHI to ES through tunnel junctions, and thus form a new class of 1D0D1D QH devices (Fig. 1).Inthiscase,the0Dislandischaracterizedbyaweakcoupling(
s
=
e
2
/
h
)andalargechargingenergy(
E
c
5
e
2
/
C
?
k
B
T
)(
C
istheislandcapacitance),whichinduceCoulombblockade(CB)
1
.Insuchdevices,AharonovBohm(AB)likeoscillations oftheresistance canbeexplained byCoulombcoupling betweenfullyoccupied LLsandconfinedstates in the QHI
7,18–22
. It was also suggested that AB oscillations reported on a QPC
23
could be attributed totunnelling paths around the saddle point
24
. In contrast, transport through QH devices, but in the strong coupling limit (
s
?
e
2
/
h
), revealed coherent effects analog to those observed in optical MachZehnder
3,25,26
or FabryPe´rot
22,27–32
interferometers.Here, we examine an unexplored regime of transport across a QPC where QH edge states are weakly coupled,but phase coherence is preserved. The SGM tip is used as a nanogate to tune the potential landscape and henceedge states’ pattern and coupling. At first sight, one expects that transport should be driven by tunnelling, andpossibly by Coulomb blockade if a quantum Hall island were mediating transport between edge states (Fig. 1)
18
.Indeed, SGM and magnetoresistance data corroborate with Coulomb blockade across a QHI located near thesaddle point of the QPC. However, temperature dependence and scanning gate spectroscopy show clear
SUBJECT AREAS:
ELECTRONIC ANDSPINTRONIC DEVICESELECTRONIC DEVICESQUANTUM HALLELECTRONIC PROPERTIES ANDMATERIALS
Received21 December 2012 Accepted22 February 2013Published11 March 2013
Correspondence andrequests for materialsshouldbeaddressedtoF.M. (frederico.rodrigues@uclouvain.be) or B.H. (benoit.hackens@uclouvain.be)
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signatures of quantum interferences. Since, up to now, such interferences were exclusively observed in open QH devices, this observationsetsthe stage for a new electrontransportscenario. Wepropose a new model that provides a quantitative interpretation of the data.
Results
Our sample is a QPC etched in an InGaAs/InAlAs heterostructureholding a 2DES 25 nm below the surface. The QPC lithographicwidth is 300 nm. All the experiments were performed at
T
between 4.2 K and 100 mK, in a dilution refrigerator. Here, theperpendicular magnetic field
B
,
9.5 T, which corresponds to aLL filling factor
n
,
6 in the 2DES. The SGM experiment isschematically depicted in Fig. 1. It consists in scanning a metallicatomic force microscope tip, polarized at voltage
V
tip
, along aplane parallel to the 2DES at a tip2DES distance of 50 nm whilerecording a map of the device resistance
R
13,14
. The QPC resistanceis defined as
R
5
dV
/
dI
, where
V
and
I
are the voltage and thecurrent through the device, respectively.The 2DES being on a quantized Hall plateau, whenever somecurrent tunnels between opposite edge channels,
R
deviates fromthe zero value expected in QH systems at very low
T
4,18,33
. In ourcase, the SGM resistance map recorded at
B
5
9.5 T,
V
tip
52
4 Vand
T
5
4.2 K and presented in Fig. 2(a) reveals concentric fringessuperimposed on a slowly varying background. The srcin of thebackground, related to reflection of ES at the QPC, is discussed inthe supplementary information. The fringe pattern can easily beunderstood in the presence of a QHI surrounding a potential hill,close to the saddle point of the QPC and tunnelcoupled to thepropagatingES(Fig.1).Indeed,approachingthepolarizedtipgraduallychangesthepotentialoftheQHI,andhenceitsarea
A
,definedasthesurfaceenclosed bythe‘‘looping’’ES.Theenclosedmagnetic flux
w
varies accordingly and the tip generates iso
w
lines when circling around the QHI. Since adding one flux quantum
w
0
corresponds totrapping one electron per populated LL in the island, CB oscillationsare generated whenever
B
or
A
are varied
21
, thereby producing ABlike oscillations
7,18–20,22
. Isoresistance lines visible on Fig. 2(a) are,therefore, iso
w
lines that are crossed as the tipisland distance is varied
18
. Consequently, the center of concentric fringes in Fig. 2(a)indicates the position of the active QHI, which connects oppositepropagating edge channels through tunnel junctions (Fig. 1).In the framework of this model, the area of the QHI can be determined thanks to the
B
dependence of ABlike oscillations
21
:
D
B
~
w
0
=
A
ð Þ
=
N
ð
1
Þ
where
N
is the number of completely filled LL in the bulk (here
N
5
6). The combined effect of moving the tip along the dashed line inFig. 2(a) and changing
B
is illustrated in Fig. 2(b) for
V
tip
52
6 V.Along the
B
axis, ABlike oscillations are highlighted with the whitedashed lines. The negatively polarized tip approaching the QHIraises its potential, which increases its area
A
, and hence reducesthe magnetic field that separates two resistance peaks
D
B
. This isillustrated in Fig. 2(c), where we assume that the QHI has a surfaceequivalenttothatofadiskwithdiameter
d
obtainedfromEq.(1):
d
isfoundtoincreasefrom
,
65 nmto
,
95 nmasthetipislanddistance
d
x
decreases from 1300 nm to 300 nm, respectively. Noteworthy, asexpected for Coulomb dominated transport in a QH interferometer,increasing
B
isequivalent toapplying a morenegative
V
tip
, yielding apositive
dV
tip
/
dB
for isoresistance stripes
22,31,32
. Since approaching the negatively charged tip has the same effect as decreasing
V
tip
,Fig. 2(b) seems consistent with the Coulomb dominated transport.But, surprisingly, the temperature dependence of fringes amplitude(
d
R
, measured on SGM maps), shown on Fig. 3, reveals a peculiarbehaviour: it clearly does not follow the
T
2
1
dependence expected inthe quantum regime of CB
18,34,35
(data from ref. 18 are presented forcomparison in Fig. 3). Instead,
d
R
deceases very slowly from 100 mKto 4.2 K. Indeed, for coherent transport through a FabryPe´rot geometry, thermal smearing of interference gives rise to a temperaturedependence
d
R
(
T
)
,
exp(
2
T
/
T
0
) in the low temperature regime. Incontrast,for transport processes involving a weaklycoupled Coulombisland, this form for
d
R
(
T
) is expected only for temperatures largerthan the charging energy
32
. In the Fabry–Pe´rot situation,
T
0
is linkedto the excited states level spacing
D
E
Ex
according to the relation
T
0
5
D
E
Ex
/
k
B
5
2
v
ES
/(
dk
B
) where
v
ES
is the local edge state velocity,related to the gradient of the confining potential. From experimentaldata measured in a GaAs QH FabryPe´rot interferometer
29
, one caninfer that, in our sample, 5
3
10
4
m/s
,
v
ES
,
10
5
m/s. Given thisrange for
v
ES
, and
d
,
80 nm (from Fig. 2(c), taking into account thatthe
T
dependence data were measured at
d
x
,
630 nm), we obtainthe range of
T
dependence represented as a gray region in Fig. 3,which reproduces quite well the behaviour observed experimentally.The corresponding range of 9.5 K
,
T
0
,
19.1 K is consistent withthe low temperature limit and hence with a FabryPe´rot behaviour.
Figure 1

Schematic representation of our model and experimentalsetup.
Tunnelling paths (dotted lines) connect opposite ES throughaquantumHallisland(circle).Currentcarryingcontacts(1–2)andvoltageprobes (3–4) allow resistance measurements. (only one edge state isrepresented for the sake of clarity).
Figure 2

Imaging tunnelling across a QPC.
(a) SGM map at
B
5
9.5 T,
T
5
4.2 K, and
V
tip
52
4 V. Continuous lines correspond to the edges of the QPC. The black bar represents 1
m
m. (b)
B
dependence of
R
profilesover the region marked with a dashed line in (a), with
V
tip
52
6 V. UsingEq. (1) for the two consecutive fringes highlighted with the white dashedlines in (b), we calculate in (c) the diameter of the QHI as the tipQHIdistance
d
x
is varied.
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Earlier experiments already evidenced such an exponential decay with temperature, but only in MachZehnder and ballistic devices,which are known to be coherent
30,36,37
.However,ourmainobservationconfirmingthepreservedelectronphase coherence emerges from the analysis of nonlinear transportthroughtheQPC.Scanninggatespectroscopyisrealizedbypositioning the tip right above the QHI, and sweeping both
V
tip
and the dccurrent
I
through the QPC. The voltage across our tunnel device, i.e.between propagating edge states, is the Hall voltage:
V
bias
5
h
/(
e
2
N
*
)
I
33
, where
N
*
is the (integer) number of transmitted ES at theQPC (Fig. S1  supplementary material). The measurement configuration is indeed identical to that used to perform conventionalelectrical spectroscopy on isolated quantum dots. If the QHI wereweakly tunnelcoupled to the propagating edge states, one wouldexpect to observe a ‘‘Coulomb diamond’’ pattern
38
. Fig. 4(a) shows
dR
/
dV
tip
as a function of the dc component of
V
tip
and
V
bias
. Insteadof Coulomb diamonds, the spectroscopy displays a checkerboardpattern of maxima and minima, indicating that both
V
tip
and
V
bias
tunetheinterferenceoftransitingelectrons.Eachbiasindependently adds a phase shift between interfering paths, so that the transresistance is modulated by a product of cosines and an exponential termaccounting for a voltagedependent dephasing induced by electronsinjected at an energy
e
j
V
bias
j
29,30,36,39
:
dRdV
tip
~
D
cos 2
p
V
bias
D
V
bias
cos 2
p
V
tip
D
V
tip
z
Q
exp
{
2
p
c
V
bias
D
V
bias
n
,
ð
2
Þ
where
D
is the zerobias visibility of the oscillations,
D
V
tip
is theoscillation period induced by
V
tip
,
Q
is a constant phase factor,
D
V
bias
5
4
v
ES
/(
ed
) is the oscillation period along the
V
bias
axis,and
c
is directly related to the
V
bias
dependent dephasing rate:
t
{
1
Q
~
c
e V
bias
j jð Þ
=
2
29
.
n
varies from 1 to 2 according to Ref. 29, 30,36, 39 and was set to 1 as we could not discriminate from fitting thedata.AsshowninFigs.4(b–d),weobtainanexcellentfitofthedatainFig. 4(a) using Eq. (2) with a transist time
t
t
5
d
/
v
ES
5
1.7
3
10
2
12
s,and a parameter
c
5
0.2 in the range found in Ref. 29. Note that insuch a small QHI,
t
t
turns out to be smaller by at least one order of magnitude than the intrinsic
t
Q
in the same 2DES
40
. This renderscoherentresonanttunnellingthroughthewholeQHIdevicepossible.
Discussion
Tointerpret
D
V
tip
obtainedfromthefit,onefirstnotesthat
R
evolves very similarly when changing either
V
tip
or
B
in the vicinity of
B
5
9 T (Fig. S2  supplementary material). Therefore, one can convert
D
V
tip
into an equivalent
D
B
, through a lever arm
D
B
/
D
V
tip
5
0.108 T/V. Hence,
D
V
tip
5
0.46 V corresponds to
D
B
5
50 mTfor the ABlike oscillations. In that range of
V
tip
,
N
*
5
5(Fig. S1(d)). This means that
d
~
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
w
0
=
p
N
D
B
ð Þ
p
~
145 nm, consistent with data in Fig. 2(c) since
d
is at a maximum when the tip isabove the QHI (
d
x
5
0). Moreover, given the value of
t
t
5
1.7
3
10
2
12
s found in fitting the spectroscopy data, one obtains
v
ES
5
8.5
3
10
4
m/s, within the range of values that was expected from data inref. 29, and in agreement with the exponential temperature dependence in Fig. 3. We therefore have a fully consistent picture thatexplains all magnetoresistance, temperature dependence and spectroscopy data, and shows that tunnelling across the QHI is indeedcoherent.One fundamental question remains: why do we observe two distinctbehavioursoftransportthroughapparentlysimilarQHdevices,Coulomb blockaded transport in our previous work
18
, and coherenttransport in this one? The qualitative difference cannot be explainedby the fact that
d
is smaller than previously examined QHIs.
T
0
and
D
V
bias
would be reduced proportionally, but not enough to explainthe observed
T
dependence and spectroscopy. On the other hand,signs of coherent transport through CB quantum dots were only
Figure 4

Evidence for coherent transport in spectroscopy.
(a)
dR
/
dV
tip
as a function of the dc component of
V
tip
and
V
bias
at
B
5
9.5 T and
T
5
100 mK. Voltage modulation of
V
tip
was set to 50 mV. (b) 2D fit of
dR
/
dV
tip
using Eq. (2). (c–d) Transresistance vs
V
bias
taken along the red(c) and blue (d) dashed lines in (a–b). The circles correspond to theexperimental data and the continuous lines to the fit.
Figure 3

Temperature dependence : Coulomb blockade vs coherenttransport.
d
R
vs
T
obtained from SGM maps with
V
tip
52
1 V (circles)and from data in ref. 18 (squares). The dashed line corresponds to a
T
2
1
dependence. The gray region corresponds to an exponential dependenceexp(
2
T
/
T
0
) with 9.5 K
,
T
0
,
19.1 K, consistent with magnetoresistancedata and edge state velocity estimate along Ref. 29. The solid linecorresponds to
T
0
5
16.2 K, consistent with the spectroscopy data (seetext).
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obtained for symmetric tunnel junctions
34
that allow resonanttunnelling instead of sequential tunnelling. In that framework, onemight thus ascribe the loss of electron coherence in other QHIs to anasymmetry of tunnel junctions. However, a difference in the transmission coefficients
T
c
of the tunnel barriers may point towards analternative explanation. In the coherent regime, we find a ratherstrong coupling between the QHI and propagating ES (0.27
,
T
c
,
0.43), which contrasts with the Coulomb blockade regime where
T
c
=
1
41
.Asimilartrendisobservedintransportexperiments at
B
5
0 T on carbon nanotubes
42
: phase coherence is maintained whenelectrons tunnel through barriers with a large transmission coefficient, so that interference effects can be observed.Up to this point, our analysis is based on the presence of a QHInear the QPC, connected to propagating ES on both sides throughtunnellingpaths(Fig.1).However,onecouldimaginethepresenceof additionaltunnellingpathsbetweenpropagatingES,inthevicinityof the QPC saddle point. The resulting model is presented in Fig. 5(a–b).Whilethe‘‘green’’pathsoccurnaturallywhenpropagatingESandtheQHIarecloseenough,the‘‘red’’pathsmayoriginatefrompotentialanharmonicities(
i.e.
nonparabolicity)onbothsidesofthesaddlepoint, similar to the fast potential variations suggested in Ref. 24. Inthis model, transport depends in principle on the various tunnelling probabilities,denoted
T
1,2
and
T
3,4
inFig.5.However,thepresenceof the QHI should always induce oscillations in the magnetoresistanceand spectroscopy of the QPC, either because it is enclosed in aninterferometer, created by the ‘‘red’’ paths and propagating ES, when
T
1,2
,
T
3,4
, or because tunnelling occurs directly through it (
T
1,2
.
T
3,4
) as discussed above (Fig. 1). Therefore, whichever
T
1,2
or
T
3,4
dominates,transportisstillcontrolledbythefluxtrappedintheQHIand hence its Coulomb charging, so that the analysis developedabove to extract parameters from the magnetoresistance and spectroscopy are still valid. In that case, i. e.
T
3,4
.
T
1,2
, the amplitude of the fringes leads us to 0.043
,
T
3,4
,
0.078 (for details seeSupplementary Information).In summary, we report first evidence for preserved electron phasecoherence in tunnelling across a quantum point contact in thequantum Hall regime. We propose a framework that explains allmagnetoresistance, temperature dependence and spectroscopy data.Thisscenarioreliesonthepresenceofapotentialhillthatgeneratesaquantum Hall island near the saddle point of the QPC. Our datatherefore provide new signatures of coherent tunnelling in anultrasmall QH device.
Methods
Device fabrication and 2DES parameters
.
Our device is fabricated from a InGaAs/InAlAs heterostructure grown by molecular beam epitaxy where a 2DES is confined25 nm below the surface (the layer sequence of this heterostructure is detailed in
13,14
).The QPC was patterned using ebeam lithography followed by wet etching. The QPCresistance
R
is measured in a fourprobe configuration: a lowfrequency (typically 10to 20 Hz) oscillating current
I
is driven between contacts 1 and 2 on Fig. 1, and
V
ismeasured between contacts 3 and 4 using a lockin technique, with
V
across the QPCalwayslessthan
k
B
T/e
.NexttotheQPC,wepatternedaHallbarwherewemeasuredalow
T
electron density and mobility of 1.4
3
10
16
m
2
2
and 4 m
2
/Vs, respectively.
SGM and SGS techniques
.
All the experiments are carried out inside a
3
He/
4
Hedilution refrigerator where a homemade cryogenic atomic force microscope (AFM)was integrated
18
. The AFM is based on a quartz tuning fork to which a commercialmetallized Si cantilever (model CSC17 from MikroMasch) is glued by means of aconductive silver epoxy. We image the sample topography by imposing a feedback loop on the shift in the tuning fork resonant frequency and using standard dynamicAFM mode of operation. After locating the QPC we perform SGM. It consists of scanningthetipalongaplaneparalleltothe2DESatconstantdistanceof25 nmfromthe surface, i.e. 50 nm from the 2DES, with a bias voltage
V
tip
applied to the tip andrecording simultaneously the device resistance
R
. At the end of a set of SGMexperiments, we image the topography of the QPC to ensure that, during that period,the position of the QPC did not change. The SGS is performed by positioning theAFM tip at a fixed position in the vicinity of the QHI and by adding a dc current
I
tothelockinacsignalbetweencontacts1and2(Fig.1).Thevoltagebetweenedgestates
V
bias
is obtained by multiplying the dc current
I
by
h
/(
e
2
N
*
). The transresistance
dR
/
dV
tip
is measured with a second lockin using an ac modulation of
V
tip
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Figure 5

Potential landscape and tunnelling paths across the QPC.
(a) Schematic representation of the electrostatic potential in the vicinity of the QPC (in brown), with the ES in yellow and the tunnelling pathsconnectingoppositeES(redandgreen).Onlyoneedgestateisrepresented,for the sake of clarity. (b) top view of the threedimensional figure in(a), with the various tunnelling probabilities
T
i
between edge states.
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Acknowledgements
TheauthorsaregratefultoM.TreffkornandT.Hyartforhelpfuldiscussions.F.M.andB.H.are postdoctoral and associate researchers with the Belgian FRSFNRS, respectively. Thiswork has been supported by FRFC grants no. 2.4.546.08.F and 2.4503.12, FNRS grant no.1.5.044.07.F, by the FSR and ARC program ‘‘Stresstronics’’, by BELSPO (Interuniversity Attraction Pole IAP6/42), and by the PNANO 2007 program of the ANR (MICATECproject). V.B. acknowledges the award of a Chaire d’excellence by the NanoscienceFoundation in Grenoble.
Authorcontributions
F.M., B.H. and S.F. performed the lowtemperature SGM experiment; F.M., B.H., V.B. andB.R. analysed the experimental data; L.D. and X.W. grew the InGaAs heterostructure; B.H.processed the sample; B.H., S.F. and F.M. built the low temperature scanning gatemicroscope; B.H., F.M., S.F., H.S., S.H., M.P. and V.B. contributed to the conception of theexperiment; F.M., B.H. and V.B. wrote the paper and all authors discussed the results andcommented on the manuscript.
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How to cite this article:
Martins, F.
et al
. Coherent tunnelling across a quantum pointcontact in the quantum Hall regime.
Sci. Rep.
3
, 1416; DOI:10.1038/srep01416 (2013).
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