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Zero-Bias Anomaly in a Nanowire Quantum Dot Coupled to
Superconductors
Eduardo J. H. Lee,1 Xiaocheng Jiang,2 Ramón Aguado,3 Georgios
Katsaros,1,*
Charles M. Lieber,2 and Silvano De Franceschi1,†
1SPSMS, CEA-INAC/UJF-Grenoble 1, 17 rue des Martyrs, 38054
Grenoble Cedex 9, France2Harvard University, Department of
Chemistry and Chemical Biology, Cambridge, Massachusetts 02138,
USA
3Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC
Cantoblanco, 28049 Madrid, Spain(Received 5 July 2012; published 31
October 2012)
We studied the low-energy states of spin-1=2 quantum dots
defined in InAs=InP nanowires and coupled
to aluminum superconducting leads. By varying the
superconducting gap � with a magnetic field B we
investigated the transition from strong coupling � � TK to
weak-coupling � � TK , where TK is theKondo temperature. Below the
critical field, we observe a persisting zero-bias Kondo resonance
that
vanishes only for low B or higher temperatures, leaving the room
to more robust subgap structures at bias
voltages between � and 2�. For strong and approximately
symmetric tunnel couplings, a Josephson
supercurrent is observed in addition to the Kondo peak. We
ascribe the coexistence of a Kondo resonance
and a superconducting gap to a significant density of intragap
quasiparticle states, and the finite-bias
subgap structures to tunneling through Shiba states. Our
results, supported by numerical calculations, own
relevance also in relation to tunnel-spectroscopy experiments
aiming at the observation of Majorana
fermions in hybrid nanostructures.
DOI: 10.1103/PhysRevLett.109.186802 PACS numbers: 73.21.La,
72.15.Qm, 73.63.Kv, 74.45.+c
Hybrid devices which couple superconducting (S)electrical leads
to low-dimensional semiconductors havereceived great attention due
to their fascinating underlyingphysics [1]. Further interest in
this field has been generatedby recent theoretical predictions on
the existence ofMajorana fermions at the edges of one-dimensional
semi-conductor nanowires (NWs) with strong spin-orbit interac-tion
connected to S electrodes [2]. Zero-bias conductancepeaks meeting
some of the expected characteristic signa-tures of Majorana physics
were recently reported in hybriddevices based on InSb [3,4] and
InAs NWs [5]. In the pastyears, quantum dots (QDs) coupled to
superconductingleads have been widely explored as tunable
Josephsonjunctions [6,7], or as building blocks of Cooper-pair
split-ters [8–10]. Hybrid superconductor-QD devices also
con-stitute versatile platforms for studying fundamental
issues,such as the physics of the Andreev bound states (ABS)[11–13]
or the interplay between the Kondo effect and thesuperconducting
proximity effect [14–16,16–24].
The Kondo effect usually stems from the antiferromag-netic
coupling of a localized electron spin and a Fermi seaof conduction
electrons. Below a characteristic tempera-ture TK, the so-called
Kondo temperature, a many-bodyspin-singlet state is formed, leading
to the partial orcomplete screening of the local magnetic moment.
Thisphenomenon, discovered in metals containing diluted mag-netic
impurities, is now routinely found in individual QDswith a
spin-degenerate ground state, e.g., QDs hosting anodd number of
electrons. The Kondo effect manifests itselfas a zero-bias
conductance peak whose width is propor-tional to TK. In S-QD-S
devices, the quasiparticle densityof states (DOS) around the Fermi
level (EF) of the leads
vanishes due to the opening of the superconductinggap (�). This
lack of quasiparticles precludes Kondoscreening.The competition
between the Kondo effect and super-
conductivity is governed by the corresponding energyscales, kBTK
and �. While no Kondo screening occursfor kBTK � � (weak coupling),
a Kondo singlet is ex-pected to form for kBTK � � (strong coupling)
at theexpense of the breaking of Cooper pairs at the Fermi
level[18,25]. A quantum phase transition is predicted to takeplace
at kBTK � � [15–17]. Experimental signatures ofthis exotic
crossover have been investigated both in theJosephson supercurrent
regime [21–23] and in the dissipa-tive subgap transport regime
[18–20]. Yet a full under-standing of these experimental findings
is still lacking. Inthis Letter, we report an experimental study on
S-QD-Sdevices where the relative strength between Kondo
andsuperconducting pairing correlations is tuned by means ofa
magnetic field B acting on �. The transition from strongto weak
coupling is continuously achieved by sweeping Bfrom above the
critical field Bc, where� ¼ 0, to zero field,where � attains its
maximum value �0 exceeding kBTK.The S-QD-S devices were fabricated
from individual
InAs=InP core/shell NWs grown by thermal evapora-tion (total
diameter �30 nm). The InP shell (thickness� 2 nm) acts as a
confinement barrier resulting in anenhanced mobility of the
one-dimensional electron gas inthe InAs core [26]. After growth,
the NWs were depositedonto a degenerately doped, p-type Si
substrate (used as aback gate), covered by a 300-nm-thick thermal
oxide.Device fabrication was accomplished by e-beam lithogra-phy,
Arþ bombardment (to remove native oxides), metal
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evaporation, and lift-off. Source and drain contacts con-sisted
of Ti ð2:5 nmÞ=Al (45 nm) bilayers with a lateralseparation of
approximately 200 nm, and a superconduct-ing critical temperature
of�1 K. Transport measurementswere performed in a He3-He4 dilution
refrigerator with abase temperature of 15 mK.
At low temperature, electron transport is dominated byCoulomb
blockade with the NW channel behaving as asingle QD. Charge
stability measurements (i.e., differentialconductance dI=dV as a
function of source-drain bias Vsdand back-gate voltage VG) were
performed to identifyKondo resonances in Coulomb diamonds with a
spin-1=2ground state. These measurements were taken at 15 mKwith
the leads in the normal state (superconductivity wassuppressed by
means of a magnetic field B? ¼ 70 mTperpendicular to the substrate
and exceeding the perpen-dicular critical field B?c ). In each
Kondo diamond, TK wasmeasured from the half width at half maximum
of the zero-bias dI=dV peak, while the tunnel coupling asymmetrywas
extracted from the peak height, i.e., the linearconductance G,
according to the relation: G=G0 ¼4�L�R=ð�L þ �LÞ2, where G0 ¼ 2e2=h
and �LðRÞ is thetunnel coupling to the left (right) lead. Here we
presentdata corresponding to three Kondo diamonds labeled as �,�,
and �, where TK;� � 0:56 K, ð�L=�RÞ� � 6:6� 10�3,TK;� � 1 K,
ð�L=�RÞ� � 0:44, and TK;� � 0:71 K,ð�L=�RÞ� � 7:6� 10�3.
Figure 1(a) shows a dI=dVðB?; VsdÞ measurementtaken at the
center of Kondo diamond �. The zero-biasKondo peak is apparent
above B?c � 23 mT (we notethat at such low fields the Zeeman
splitting is muchsmaller than kBTK;�, explaining the absence of a
splitKondo peak). Surprisingly, reducing the field belowB?c does
not lead to an abrupt suppression of theKondo peak. Instead, the
peak becomes progressivelynarrower and smaller, vanishing
completely only belowB? � 9 mT.
We argue that the observed zero-bias peak is a mani-festation of
Kondo screening due to intragap quasiparticlestates. To support
this interpretation, we show in Fig. 1(b)a data set taken in an
adjacent diamond with even occu-pation (i.e., with no Kondo
effect). The dI=dVðVsdÞ tracesshown correspond to different
in-plane fields, Bk, rangingfrom zero to just above the in-plane
critical field Bkc �200 mT. When lowering the fields from above to
below
Bkc, the subgap dI=dV does not drop abruptly, supportingour
hypothesis of a sizable quasiparticle DOS at EF [wenote that,
although the measurement of Fig. 1(b) refers toin-plane fields, a
qualitatively similar behavior can beexpected for perpendicular
fields, see SupplementalMaterial [27] ]. The development of a
‘‘soft’’ gap just
below Bkc is additionally marked by the absence ofdI=dV peaks
characteristic of the BCS DOS singularities(a discussion of the
origin of the soft gap is included inthe Supplemental Material
[27]). Such peaks develop only
at fields well below Bkc, becoming most pronouncedat B ¼ 0. In
this low-field limit, the subgap conductancesimultaneously vanishes
and first-order multiple-Andreev-reflection resonances emerge at
eVsd � ��.To reproduce the observed subgap features and the
coex-
istence of a Kondo peak and a superconducting gap, wecalculated
the dI=dV of a QD modeled by an AndersonHamiltonian including
coupling to BCS-type supercon-ducting reservoirs. We used the
so-called noncrossing ap-proximation, a fully nonperturbative
theory that includesboth thermal and quantumfluctuations,
complementedwiththe Keldysh-Green’s function method to take into
accountnonequilibrium effects at finite Vsd (see
SupplementalMaterial [27]). In order to fit the experimental data,
theDOS of the leads was modeled by a Dynes function:
FIG. 1 (color online). (a) Left panel: Color plot of dI=dV
vs(B?, VSD) measured at the center of diamond �. The dashedlines
highlight the emergence of finite-bias peaks related to theopening
of a superconducting gap. The superimposed line traceshows the B?
dependence of the linear conductance. Rightpanel: dI=dVðVSDÞ traces
taken at different B? values. The insetshows a scanning electron
micrograph of a typical device (scalebar: 200 nm). (b) dI=dVðVSDÞ
traces measured in an evendiamond revealing the Dynes-like DOS of
leads. (c) Numericalcalculations. The smaller solid dots denote the
position of the �and 2� peaks, whereas the open dots highlight the
position of theShiba bound state peaks. (d) Linear conductance (G0)
normal-ized to the normal-state value (GN0 ) and plotted as a
function of�=kBTK for diamonds � (red dots) and � (black
squares).
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NsðE; �; BÞ ¼ Re" jEj þ
i�ðBÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðjEj
þ i�ðBÞÞ2 ��ðBÞ2p
#; (1)
where �ðBÞ is a phenomenological broadening term [28].For small
�ðBÞ, the �ðBÞ term is particularly importantleading to a finite
quasiparticle DOS at EF. By contrast,as �ðBÞ increases (with
decreasing B), the DOS of thesuperconducting leads approaches the
ideal BCS profile.
Figure 1(c) shows a set of calculated dI=dVðVsdÞ tracesat
different B. By adjusting the DOS parameters �ðBÞ and�ðBÞ, these
calculations clearly show a zero-bias peakpersisting below the Bc,
in agreement with the experimen-tal data of Fig. 1(a). Since this
peak emerges only in thecase of a finite �ðBÞ, we conclude that the
finite DOS at theFermi level is at the origin of the experimentally
observedzero-bias anomaly. The narrowing of this Kondo anomalywith
increasing � can be interpreted as a decreasing TKdue to the
shrinking quasiparticle DOS around the Fermilevel (this aspect is
more quantitatively discussed in theSupplemental Material [27]). As
TK approaches the elec-tronic temperature, the peak height gets
smaller leading tothe disappearance of the zero-bias peak.
Figure 2 shows a second data set taken in Kondo dia-mond �. In
this case, the stronger and more symmetriccoupling to the leads
results in a higher TK, and a largerpeak conductance. Nevertheless,
the field dependence[Fig. 2(a)] shows substantially the same
behavior as inFig. 1(a), i.e., a zero-bias Kondo peak persisting
belowB?c , becoming progressively narrower with decreasing B?,and
vanishing below B? � 10 mT. Interestingly, a sharpdI=dV resonance
is found around Vsd ¼ 0 superimposed tothe (wider) zero-bias Kondo
peak. This resonance persiststhroughout the entire field range in
which the leads aresuperconducting. By performing current-bias
measure-ments [inset of Fig. 2(b)], we were able to ascribe
thissharp resonance to a Josephson supercurrent as high as0.9 nA at
B ¼ 0. This finding reveals the possibility of acoexistence between
the Josephson effect, linked to thesuperconducting nature of the
leads, and a Kondo effect
arising from the exchange coupling between the localizedelectron
and intragap quasiparticle states.The peak heights of Kondo
resonances � and � appear
to follow approximately the same dependence on �=kBTK[Fig.
1(d)], where TK refers to the Kondo temperature inthe normal state.
A similar scaling was reported earlier byBuizert et al. [18], for a
S-QD-S device fabricated from anInAs self-assembled QD using Ti=Al
contacts. In thatLetter, it was speculated that when kBTK � �, it
becomesenergetically favorable for Cooper pairs to split in order
toscreen the local spin and create a Kondo resonance at theFermi
level. The results presented here point at a
differentinterpretation based on the presence of the already
dis-cussed intragap quasiparticle states, which become
particu-larly important when B approaches Bc. According to
thisinterpretation, the apparent scaling in Fig. 1(d) is
inti-mately related to a quasiparticle ‘‘poisoning’’ of the
super-conducting gap.Well below Bc, as the Kondo anomaly
disappears, the
dI=dVðVsdÞ is dominated by a pair of peaks
symmetricallypositioned with respect to Vsd ¼ 0 [Fig. 1(a)]. These
peaksbecome most pronounced at B ¼ 0. Similar types of sub-gap
structures (SGS) have been reported in earlier worksand were given
different interpretations: a Kondo enhance-ment of the first order
Andreev reflection process [18–20],or, in the case of
asymmetrically coupled S-QD-S devices,a persisting Kondo resonance,
created by the stronglycoupled lead, which is probed by the BCS DOS
of thesecond, weakly coupled lead [19]. Some of the
aboveinterpretations [18,19] invoke Kondo correlations to ex-plain
the observed subgap structure, even though, as wehave pointed out,
these correlations get suppressed as �reaches its largest value at
B ¼ 0.More recently, finite-bias SGS were explained in terms
of tunneling through Yu-Shiba-Rusinov states [29,30].These
intragap bound states, often referred to as Shibastates, were
originally discussed in the case of magneticimpurities embedded in
a superconductor [31–34]. Theycan be seen as ABS emerging as a
result of the exchangecoupling J between the impurity and the
superconductor.As later confirmed by experiments based on
scanningtunneling spectroscopy [35], Shiba states emerge as pairsof
peaks in the local DOS symmetrically positioned atenergies �EB
relative to the Fermi level, where EB de-pends on J and jEBj<
�.The zero-field dI=dVðVsdÞ trace in the right panel of
Fig. 1(a) exhibits a pair of dI=dV peaks at eVsd �
�1:4�,followed by negative dI=dV regions. Taking into accountthe
strong asymmetry in the tunnel couplings, these fea-tures can be
well explained in terms of a pair of Shibalevels with EB ¼ 0:4�,
created by the strongly coupled Slead, and tunnel probed by the
weakly coupled S lead [seeFig. 3(a)]. The observed dI=dV peaks
result from thealignment of these Shiba levels with the BCS
gap-edgesingularities. Precisely, one dI=dV peak is due to the
onset
FIG. 2 (color online). (a) B? dependence measured in dia-mond �.
(b) dI=dV line profiles taken at different B. The inset isa
voltage-current measurement carried out at B ¼ 0, whichreveals
transport of a dissipationless supercurrent in the device.
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of electron tunneling from the Shiba level below EF tothe empty
quasiparticle band of the S probe. The other peakis due to the
onset of electron tunneling from the occupiedquasiparticle band of
the S probe to the Shiba level aboveEF. Increasing jVsdj beyond
these resonance conditionsleads to a reduced tunneling probability
and hence anegative dI=dV. The Shiba-related features observed
inFig. 1(a) are very well reproduced by the numerical resultsin
Fig. 1(c). In the case of relatively low contact asymmetry[lower
panel of Fig. 2(b)], both S leads interact with the QDspin
resulting in a stronger J. We find a pair of dI=dVpeaks at eVsd �
��, which implies EB � 0. In addition,we observe a rather complex
set of smaller peaks mostlikely due to multiple-Andreev-reflection
processes.
In the weak-coupling limit, the energy of the Shiba statesis
related to � through EB ¼ �ð1� xÞ=ð1þ xÞ, wherex ¼ 3ð��FJ=4Þ2 and
�F is the Fermi velocity [30]. SinceJ is presumably independent of
B, the energy of the Shibastates should evolve proportionally to
�ðBÞ as B is varied.We have verified this dependence with a
measurementperformed in diamond �, where, as in diamond �,
tunnelcouplings are strongly asymmetric. Differently from dia-mond
�, however, the low-energy transport is character-ized by two pairs
of intragap dI=dV peaks, as shown by thezero-field curve in Fig.
3(c). As in Fig. 1(a), the most
prominent peaks (at eVsd ��1:4�) correspond to thealignment of
the Shiba levels with the BCS coherencepeaks of the weakly coupled
contact, and they are consis-tently followed by negative dI=dV
dips. The weaker peaksat eVsd � �0:53� (in fact only one of them is
clearlyvisible) can be interpreted as ‘‘replicas’’ of the
Shibapeaks. Such replicas are expected when the Shiba levelsline up
with the Fermi level of the weakly coupled S lead(for eVsd ¼ �EB),
provided a non negligible density ofquasiparticles is present
throughout its superconductinggap. This is apparent from the
calculated dI=dVðVsdÞtrace in Fig. 3(d), which agrees fairly well
with the experi-mental one.The Shiba peaks and their replicas shift
towards Vsd ¼ 0
as Bk is increased. Their positions are plotted in the inset
ofFig. 3(e) as solid and open dots, respectively. For compari-
son, the �ðBkÞ dependence measured in a non-Kondo dia-mond is
also plotted. Up to Bk � 120 mT, the Shiba peaksand their replicas
evolve proportionally to �ðBkÞ in agree-ment with the theoretical
prediction. Above Bk � 120 mT,this behavior begins to be affected
by the increasingZeeman splitting (� 0:3 meV=T) of the QD spin
doublet.The main Shiba peaks get strongly suppressed and theyseem
to eventually merge into the normal-state, Zeeman-split Kondo peaks
through a nontrivial transition region(roughly between 120 and 180
mT). This large Zeemansplitting prevents the observation of a
zero-bias Kondo peak
when approaching Bkc.In conclusion, we have studied the
transport properties
of a spin-1=2 QD coupled to S contacts. The ability
tocontinuously tune�with an externalmagnetic field enabledus to
investigate the transition from a normal-state Kondo toa
superconducting-state Shiba ground state.We showed thatthe presence
of a finite quasiparticle DOS within � canpromote the formation of
a zero-biasKondo peak coexistingwith superconductivity. The origin
of this quasiparticleDOS remains to be clarified, also through a
better under-standing of the superconducting proximity effect in
semi-conductor NW structures (e.g., the role of
disorder-inducedpair breaking [36]). Finally, we should like to
emphasizethat our results bear clear implications in the
interpretationof subgap transport features in hybrid
superconductor-semiconductor systems, especially in the presence of
rela-tively high B that can cause a significant suppression of
�.This is precisely the regime where Majorana fermions areexpected
to arise as zero-energy quasiparticle states. Weargue that intragap
quasiparticle states,manifesting througha sizable background
conductance (as seen in Refs. [3–5]),could result in the screening
of a local spin (or orbital)degeneracy leading to zero-bias
anomalies that are notrelated to Majorana physics.This work was
supported by the EU Marie Curie pro-
gram and by the Agence Nationale de la Recherche. R.
A.acknowledges support from the Spanish Ministry ofScience and
Innovation through Grant No. FIS2009-08744.
FIG. 3 (color online). (a) Schematics of the formation of
Yu-Shiba-Rusinov bound states resulting from the interaction of
theQD with the strongly coupled lead. (b) Bk dependence of
dI=dVmeasured in diamond �. (c) The solid line depicts the
dI=dVtaken at B ¼ 0. Two pairs of peaks are observed at jeVsdj ¼�þ
EB, EB. The dashed line shows the equivalent dI=dV takenin an even
diamond, for comparison. (d) Numerical calculationof the dI=dV for
B ¼ 0. (e) Position of the intragap peaks as afunction of B. The
inset shows the data before performing therescaling in units of �.
The dashed line displays thefield dependence of �.
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*Present address: Johannes Kepler University, Institute
ofSemiconductor and Solid State Physics, Altenbergstr. 69,A-4040
Linz, Austria.†[email protected]
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detailsof the theoretical model, a discussion of the origin of
thesoft gaps, and temperature dependence data of the zero-bias
anomaly.
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