Magnetically-driven colossal supercurrent enhancement in InAs nanowire Josephson junctions

1.6 MB·

by  J. Paajaste

Magnetically-drivencolossalsupercurrentenhancementinInAsnanowireJosephsonjunctions
J. Paajaste,1 E. Strambini,1,∗ M. Amado,1,2 S. Roddaro,1 P. San-Jose,3
R. Aguado,3 F. S. Bergeret,4,5 D. Ercolani,1 L. Sorba,1 and F. Giazotto1,†
1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
2Materials Science and Metallurgy, University of Cambridge CB3 OFS, United Kingdom
3Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones
Cient´ıficas (ICMM-CSIC), Sor Juana Ine´s de la Cruz 3, 28049 Madrid, Spain
4CentrodeFisicadeMateriales(CFM-MPC),CentroMixtoCSIC-UPV/EHU,E-20018SanSebastian,Spain
5Donostia International Physics Center (DIPC), E-20018 San Sebastian, Spain
6 TheJosephsoneffectisafundamentalquantumphenomenonconsistingintheappearanceofadissipa-
1 tionlesssupercurrentinaweaklinkbetweentwosuperconducting(S)electrodes. Whilethemechanism
0 leading to the Josephson effect is quite general, i.e., Andreev reflections at the interface between the S
2 electrodesandtheweaklink,theprecisephysicaldetailsandtopologyofthejunctiondrasticallymodify
thepropertiesofthesupercurrent. Specifically,astrongenhancementofthecriticalsupercurrentI is
n C
a expectedtooccurwhenthetopologyofthejunctionallowstheemergenceofMajoranaboundstates.Here
J wereportchargetransportmeasurementsinmesoscopicJosephsonjunctionsformedbyInAsnanowires
andTi/Alsuperconductingleads. Ourmainobservationisacolossalenhancementofthecriticalsuper-
2
1 currentinducedbyanexternalmagneticfieldappliedperpendiculartothesubstrate. Thisstrikingand
anomaloussupercurrentenhancementcannotbeascribedtoanyknownconventionalphenomenonex-
] istinginJosephson junctionsincluding, forinstance, Fraunhofer-like diffractionoraπ-statebehavior.
l WealsoinvestigateanunconventionalmodelrelatedtoinhomogenousZeemanfieldcausedbymagnetic
l
a focusing,andnotethatitcannotaccountfortheobservedbehaviour. Finally,weconsidertheseresults
h inthecontextoftopologicalsuperconductivity,andshowthattheobservedI enhancementiscompatible
C
-
withamagneticfield-inducedtopologicaltransitionofthejunction.
s
e
m
Couplingaconventionals-wavesuperconductortomateri- spect to the junctions substrate and the NW. This is in stark
.
t als based on helical electrons such as topological insulators contrast with what has been observed so far in conventional
a
m orsemiconductorswithstrongspin-orbit(SO)interactionlike weaklinks,wheretheJosephsoncouplingturnsouttobesup-
InAs or InSb nanowires (NWs) leads to an unconventional pressed by any applied magnetic field. Notably, this abrupt
-
d p-wave superconductor. The latter may undergo a topologi- switchingofIC alwaysoccursaroundthesamemagneticfield
n caltransition,andbecomeatopologicalsuperconductor(TS) (B )injunctionsmadeofnominallyidenticalNWsbutchar-
sw
o hostingexoticedgestateswithMajorana-likecharacter[1,2]. acterizedbydifferentlengthsofthenormal(N)region,there-
c
Most of the early experimental efforts to demonstrate these foresuggestingthattheoriginoftheobservedcriticalcurrent
[
modes have been focused on normal metal-superconductor enhancement is intrinsic, i.e., it is not related to geometrical

1 junctions realized with strong-SO NWs [3–5] with the aim resonances existing in the junction or linked to the Thouless
v todetectsignaturesofMajoranaboundstates(MBSs)emerg- energy of the system [21]. In addition, the temperature de-
5
ingforincreasingZeemanfields. Soonafter,experimentson pendenceofB followsaBCS-likebehaviorthusindicating
sw
5
Josephsonjunctionsbasedonhelicalmaterialshavebeenper- astronglinktotheproximityeffectpresentinthejunction.
9
2 formed as well to detect peculiar hallmarks of MBSs in the Our results are discussed on the basis of the most com-
0 phaseevolutionoftheJosephsoneffect,bothinthecontextof mon and understood phenomena that may affect the Joseph-
. topologicalinsulators[6–10]andNWs[11].Yet,aconclusive son supercurrent of the junction. Although no final conclu-
1
evidence of MBSs emerging from these hybrid systems still sions can be drawn from such an analysis we stress that a
0
6 remainsanoutstandingexperimentalgoal.Thisisparticularly topological transition developing at Bsw appears to be fully
1 trueforNW-basedJosephsonweaklinkswherenosignatures compatiblefromaqualitativepointofviewwiththeobserved
: ofTShavebeenreportedinthesupercurrent[12–19]. phenomenology: our proposed physical scenario consists of
v
In this work, we investigate the Josephson coupling in amagnetically-drivenzero-energyparitycrossingofAndreev
i
X Al/InAs-NW/Alhybridjunctionsinthepresenceofanexter- levelsinthejunction[20]whichisexpectedtooccurformag-
r nalmagneticfield. Inparticular,wefocusontheamplitudeof netic fields applied perpendicularly to the wire SO axis, ex-
a
theJosephsoncriticalcurrentIC whichisexpectedtostrongly actlyasobservedintheexperiment.
increasewhenthetopologyofthejunctionenablestheemer-
gence of MBSs [20]. Our key result is the observation of a
colossal enhancement of I manifesting just above a thresh- Experimental results Figure 1a shows a scanning elec-
C
oldmagneticfieldappliedinperpendiculardirectionwithre- tron micrograph of a typical n-InAs-NW-based Josephson
junction. Thejunction’sinterelectrodespacingLrangesfrom
∼40 nm to ∼113 nm. The growth and physical details of
the n-doped InAs NWs are reported in the Methods section.
∗[email protected] ForeachjunctionaneighbouringTi/Alsuperconductingpair
†[email protected] is used for transport measurements. The current vs voltage
2
a c 800 mK. Furthermore, for temperatures below 250 mK a
remarkable hysteretic behaviour between the switching (I )
L C
InAs I and retrapping (I ) critical currents is observed, as shown
Cr
b in Fig.1d,e. This hysteresis stems from quasiparticle heating
Ti/Al 150 nm
within the NW region whilst switching from the resistive to
200 nm
V Ti/TAi/lAl the dissipationless regime [22], and has been often observed
+ - SiOSxiOx inhybridJosephsonjunctionsindependentlyofthegeometry
Si n++ Si n++
orcompositionoftheweaklink[17,23–27].

15 mK 800 mK ThemonotonicdecayofI (T)andthesaturationofI (T)
d e C Cr
T(mK) = 15 Ballistic at low T is displayed in Fig. 1(e) [28] together with the two
200 I Diffusive 150 best fits for IC(T) obtained by modelling the junction as an
C ideal diffusive or ballistic NW (red and blue dashed line, re-
400
150 spectively). The details of the theoretical model for each fit
600
aredescribedintheMethodssection. Noneofthetwofitting
A) 800 A) curvescanaccuratelydescribethemonotonicdecayofIC(T)
I (n 0 I (nC wdihaitcehreisgicmoensbisettewnetewnitthheatjwuoncatiboonvebelliomnigtsin[g18to],ain.e.i,ntLer∼mel-
e
I 75 where l ∼60 nm is the electron mean free path estimated
Cr e
in our InAs-NWs [19, 29]. Moreover, the diffusive fit sug-
700 gests an effective junction length of the order of ∼300 nm
-150 500 which largely exceeds the interelectrode spacing. The actual
300
geometry of the junction supports this observation since the
100
superconducting electrodes cover a considerable portion of
100 V
the NW. In addition, the same fit yieds an estimate for the
0
0 400 800 junction Thouless energy Eth =h¯D/L2 ≈160µeV (D is the
V(V) T (mK) diffusion constant in the InAs NW), and for the induced su-
perconductingminigap[30]intheNW,∆(cid:39)80µeV.
FIG. 1. | Sample layout and zero magnetic field characteri- As long as the external magnetic field is absent, the be-
zation. a, Pseudo-color scanning electron micrograph of a typical havior of our NW-based Josephson junctions is fully consis-
n-InAs nanowire-based Josephson junction along with a sketch of tentwithwhathasbeenobservedforsimilarAl/InAs-NW/Al
the four-wire measurement setup. The Josephson junction is cur-
weak links [15, 16, 18, 19]. A strikingly different and un-
rentbiased(I)whereasthevoltagedrop(V)ismeasuredthrougha
expected phenomenology develops when the magnetic field
room-temperature differential preamplifier. The junction length is
is applied perpendicular to the substrate (B ): the amplitude
denotedwithL,andthewidthoftheTi/Alelectrodesis∼150nm. ⊥
of I drastically changes in a way that has never been ob-
b,Sideviewofthejunctionshowingthedifferentmaterialsforming C
thestructure.c,Colored45◦tiltedscanningelectronmicrographofa served so far in similar devices [15, 18] and, to the best of
typicaldistributionofInAsnanowiresaftertheirgrowth.d,Backand our knowledge, in any other kind of Josephson weak links.
forth current vs voltage characteristics of an Ti/Al-InAs Josephson As shown in Fig. 2a, where the IV characteristics of one of
junctionwithL=100nmmeasuredatdifferentbathtemperaturesT. thejunctionswithL=100nmisdisplayedfordifferentval-
Thecurvesarehorizontallyoffsetforclarity. e,Switching(IC,dots) uesofB⊥,thecriticalcurrentICremainsalmostconstantupto
and retrapping (ICr, triangles) supercurrents vs temperature. Two ∼15mT,thenquicklydoublesitsamplitudeata“switching”
distincttheoreticalmodelsforthecriticalcurrentICholdingeitherin fieldBsw(cid:39)23mT,anddecaysatlargermagneticfields. Fur-
theballistic(dash-dottedline)ordiffusive(dashedline)regimeare
themore,for|B |>B ,theIV characteristicsdevelopadis-
⊥ sw
shown.
sipativebehavior(i.e.,theyshowafiniteslopearoundV ≈0,
seeFig.2b)whichcorrespondstoaresistanceof∼60Ω. The
colossalenhancementofI (whichexceeds100%)isveryro-
C
(IV)characteristicsoftheJosephsonweaklinksareobtained bustandreproducibleoverdifferentcoolingcycles,andmea-
byapplyingabiascurrentI,andmeasuringtheresultingvolt- suredjunctions.
age drop across the NW via a room-temperature differential
Figure 2c shows the behavior of I (B ) for three junc-
C ⊥
preamplifier,asshowninFig.1a.Ashematicsideviewshow-
tions of different lengths. Apart from sample specific fluc-
ingthematerialsformingthejunctionsaswellasa45◦ tilted
tuationsofI (0),thesupercurrentenhancementoccursatthe
C
scanning electron micrograph of a typical distribution of n-
same magnetic field for all the junctions. This suggests that
InAs NWs after the growth are displayed in Fig. 1b and c,
the origin of the effect is intrinsic to the materials combi-
respectively.
nation, and cannot be attributed to geometrical resonances
Figure 1d shows the temperature dependence of the IV in the junction [21]. A nontrivial behaviour characterizes
characteristics of a typical Josephson junction with L = alsothetemperatureevolutionoftherelativeI enhancement,
C
100nm. AcriticalcurrentI exceeding∼150nAisobserved ∆I = max[I (B )]/I (0)−1, shown in Fig. 2d,e for two
C C C ⊥ C
atthebasetemperatureofadilutionrefrigerator(15mK)[18]. junctions with L=40 nm and 100 nm belonging to differ-
Without any applied magnetic field I persists up to ∼ ent NWs. In contrast to the usual temperature-driven weak-
C
3
a b c
B (mT)

0 16 32 48 64 80 96 112 B (mT) L(nm) 100
100  40
27 100 A)
26 113 I (nC
I (nA) 0 V (V) 24 50
22
21
-100
V 19
8 24 40 56 72 88 104 120 128 0
5 0
L = 100 nm V (V) 100 V -100 0 100 -100 -50 0 50 100
I (nA) B (mT)
d L = 40 nm T(mK) e  f
25 120 L = 100 nm B L = 100 nm
25 mK C
100 0.10
200
T)
A) 300 %)80 B (
I (nC 400 I (C 0.05
500 700 mK 40 40 nm
B
600 SW
Stack
20 nA
700 -30 nA
-50 -25 0 25 50 0 0.00
B (mT) 0 300 600 0 400 800
 T (mK) T (mK)

FIG.2. |Enhancementofthecriticalcurrentatfiniteout-of-planemagneticfield. a,BackandforthIV characteristicsofaJosephson
junctionwithL=100nmmeasuredfordifferentvaluesoftheout-ofplanemagneticfield(B )at15mK.Thecurvesarehorizontallyoffsetfor
⊥
clarity.AstrongenhancementofI occursatB (cid:39)23mT.b,Blow-upofselectedIV characteristicsofthesamejunctionofpanelashowing
C ⊥
the dissipative behaviour of the weak link after the transition to the enhanced I state. c, Comparison of the I vs B behavior for three
C C ⊥
junctionsofdifferentlengthLat15mK.d,CriticalcurrentI vsB measuredatdifferentbathtemperaturesforajunctionwithL=40nm.
C ⊥
Thecurvesareverticallyoffsetforclarity. EachdatapointrepresentsthecriticalcurrentobtainedfromasingleIV measurementatconstant
magneticfieldandtemperature. e,Temperaturedependenceofthecriticalcurrentrelativeenhancement,∆IC=max[IC(B⊥)]/IC(0)−1,for
twodifferentjunctionswithL=100nmandL=40nm. NotethatthejunctionwithL=100nmisdifferentfromtheoneshowninpanel
c. f,TemperaturedependenceofthecriticalfieldBC forthedisappearanceoftheJosephsoneffect,andoftheswitchingfieldBswforIC fora
junctionwithL=100nm. DottedlinesaretheBCSfittingofthetwodatasetsusingtheequationBx(T)=Bx(0)[1−(T/TC)2]obtainedfor
thesamesuperconductingcriticaltemperatureT =900mK.
C
eningoftheJosephsoneffect,∆I showsanonmonotonicbe- thepossibleoriginsofthiseffect,isthestrongdependenceof
C
haviourwithamaximumatT ∼300mKforthejunctionwith I (B) on the orientation of the external magnetic field. This
C
L=100 nm whereas in the shorter junction (L=40 nm) the clearlyappearsinFig.3a,wherewecomparethecriticalcur-
behaviour is almost monotonic, and simply decays with the rentsoftwojunctionswithdifferentlengths(L=40nmand
temperature. Severalcharacterizationsperformedondifferent 113 nm) measured for three different orientations of B. The
samples seem to indicate that the temperature behaviour of maximum I enhancement is observed when the field is ap-
C
∆I is not related to the length of the junction but rather de- pliedperpendiculartothesubstrate(andthusalsototheNW
C
pendsonthespecificNW.Moreover,thebehaviorofB (T) axis) as shown in the left panel, and occurs at B (cid:39)23 mT.
sw sw
showninFig.2fforajunctionwithL=100nm,followsthe Differently,incantedfieldconfigurations(B30◦,centralpanel)
sametemperaturedependenceofthecriticalfieldforthedis- B shifts to higher fields according to the amplitude of the
sw
appearance of the Josephson effect B (also displayed in the projection B . When B is applied in-plane (B , right panel)
C ⊥ ||
same plot) therefore suggesting a common origin of the two theeffectisalmostabsentapartatinysupercurrentenhance-
phenomena,i.e.,theproximityeffect. mentaround∼160mTwhichcanbeascribedtoasmallmis-
alignment present in our setup as well as to an incomplete
AcrucialfeatureobservedintheI enhancementofallthe magneticfieldscreeninginthejunctionregion,aswillbedis-
C
junctions,andwhichisessentialinordertodiscriminateover
4
a
2 B B L(nm) B
 30° ||
)=0 113
B
C( 40
/IC
I
1
0
0 40 80 80 160 80 160 240
B (mT) B (mT) B (mT)
b c
200
B B
SO B ┴
B ||
┴
150
%)
B (ext100 BSO
/Beff
50
B
||
0
0 50 100 150 200
0% 100% 200%
Position (nm)
FIG.3. |Angledependenceofthecriticalcurrentenhancement. a,ComparisonbetweentheI behaviourasafunctionofthemagnetic
C
fieldappliedinthreedifferentorientationsfortwodifferentjunctionlengths:out-ofplane(B ),30◦fromtheplane(B ),andin-plane(B ).
⊥ 30° ||
b,COMSOLfinite-elementsimulationofthenonuniformdistributionoftheamplitudeofBduetotheMeissnereffectinthesuperconducting
leads, evaluatedforamagneticfieldappliedalongthethreemainorthogonalaxesofthejunction: out-ofplane(B , leftpannel), in-plane
⊥
orthogonaltotheNWandparalleltotheSOvector(B ,middlepannel),andalongtheNW(B ,rightpannel).Verticalgrayregionsindicate
SO ||
thesuperconductingelectrodeswhereasthelight-greenhorizontalonestheNW.c, Amplitudeprofileoftheeffectivemagneticfield(B )
eff
normalizedbytheappliedmagneticfield(Bext)alongtheportionoftheNWindicatedbythedashedlinesshowninb. Notethatwhileinthe
out-of-planedirection(B⊥,redline)Beff isalmostuniformlydoubledwithrespecttoBext owingtothefocusingeffect,alongtheNW(B||,
blueline)B itisdrasticallysuppressedbyMeissnerscreening,andreachesatmost∼25%oftheintensityoftheexternalfieldatthecenter
eff
ofthejunction.Inthein-planedirectionorthogonaltotheNW(BSO,blackline)Beff almostcoincideswithBext.
cussedbelow. Notably, thein-planecomponentsofthemag- forourjunctionsgeometryforthethreerelevantdirectionsof
netic field (see right panel) seem to have a marginal role in theexternalmagneticfield(B ). Theresultsaresummarized
ext
determiningtheactualvalueofB ,thisconclusionfollowing in Fig. 3b where the space distribution of B is calculated as-
sw
fromthesimilarbehaviordisplayedbythetwoaboveJoseph- suminganidealMeissnereffectinthesuperconductingleads.
sonweaklinkswhichwerefabricatedwithNWshavingvery In particular, we obtain that when B is orthogonal to the
ext
differentorientationsinthesubstrateplane. substrate(B ,leftpanel)themagneticfieldB inthejunc-
⊥ eff
tion region is strongly amplified, and its intensity is almost
ThepeculiardependenceoftheI enhancementasafunc-
C
doubledwithrespecttotheexternalfieldduetomagneticfo-
tion of magnetic field direction imposes an important con-
cusing,asrecentlyreportedforPb-basedInAsNWJosephson
straintoverpossiblemodelsthatcanexplaintheeffect. These
junctions [17]. When B is applied in-plane along the SO
observationsjoinedwiththein-planepinningofthespin-orbit ext
vector (B , central panel) there is complete penetration of
(SO) vector in InAs NWs laying on top of a SiO /Si sub- SO
2 themagneticfieldwithinthejunctionregion,i.e.,B ∼B .
strate [3], seem to lead to the intriguing conclusion that the eff SO
By contrast, when B is applied parallel to the NW axis (B ,
I enhancement requires the magnetic field to be perpendic- ||
C
rightpanel)theweaklinkareaisalmostscreenedbythemag-
ular to the SO vector. The lack of I enhancement observed
C neticfieldthankstoMeissnerexpulsionintheleads,andB
for field configurations orthogonal to the SO vector but par- eff
obtains values up to ∼25% of the external field at the cen-
alleltotheNWcanbeattributedtothestrongmagneticfield
ter of the junction. Figure 3c summarizes the above results
expulsionintheweaklinkregionduetotheMeissnerscreen-
fortheamplitudeprofileofB alongtheportionoftheNW
ing of the superconducting electrodes forming the junction. eff
indicatedbythedashedlinesinpanelb.
The magnitude of this effect has been numerically estimated
5
a c
Theoretical interpretations The scenario drawn above
bytheseexperimentalevidencesisclearbutitsinterpretation B B
ispuzzlingduetothecomplexityofthesystemandthelarge S S S S S S
amount of non-trivial phenomenologies possible for Joseph- S’ S’ S’ S’ S’ S’ W S’ VS VN VS S’
Z Z Z
son junctions. In the first approximation it is known that a
magnetic field destroys superconductivity via the orbital and b L LS d
paramagneticeffect,andthereforeonewouldexpectamono-
2.0
tonic decay of the critical current by the application of an 0.03 V ZS/V ZN
external field. An experimental exception to this behavior is 0.10 100
represented by field-enhanced superconductivity observed in TN = 0.2 0.00 5303
25
Josephson junctions made with metallic NWs covered with 0.08 20
magnetic impurities. In that case, the I (B) enhancement is -0.03 17
inducedbythefieldpolarizationoflocalCmoments,andbythe    1.5 1142
rirneitliaCetsoivoienpqeourueprnacNihrsWin[gs3,1o]af.nthdDeuoeewxcitnohgatnhtgeoeatchboesuepsnplcieencgoififwcmitmhagathngeenteeiltceiccimtfiropenulds- I ()eC00..0046 0.07 I /()e0C
orientationleadingtotheI enhancementwecanexcludethis
C
pictureastheexplanationofourobservations. 1.0
0.02
Two further mechanisms are known that yield an I in-
C
crease as a function of field: Fraunofer-like diffraction [32], 0.009 0.0004
andtheπ-junctionbehaviour[33]. Theformerisexpectedto 0.00
0 2 4 6 0.0 0.3 0.6 0.9
occur whenever the magnetic flux enclosed by the junction, B/B V S (eV)
crit Z
Φ=LWB (W istheNWdiameter),equalsanintegermulti-
⊥
pleofthefluxquantum,Φ0=h/2e. Notethatneithertheab- FIG. 4. | Two possible models for the IC enhancement. a, A
senceofanIC maximumaroundB⊥=0northeevidencethat sketch of the four Majorana bound states (red circles) formed in
thecriticalcurrentenhancementisindependentofthejunction thetwoproximizedregionsS’ofa(ballistic)NWundertheaction
lengthcanbeexplainedwithinthispicture.Ontheotherhand, of an out-of-plane magnetic field B . b, Theoretical critical cur-
⊥
aπ-junctionbehaviorinducedbytheexternalfieldistypically rent IC calculated for increasing magnetic field values in an InAs
characterized by oscillations of I (B) with periodicity of the Josephsonjunctionsimilartotheonesinvestigatedintheexperiment
C
order∼gµ B/E ,i.e.,theratiobetweentheZeemanandthe (L=50 nm, LS =500 nm, andW =50 nm). The NW chemical
B T
potential µ is set to have six spinful subbands at half filling. Rest
Thoulessenergyofthejunction,wheregistheNWgyromag-
neticfactorandµ istheBohrmagneton.Suchoscillatorybe- of the parameters: spin-orbit coupling αSO=0.13eVA˚, gyromag-
B netic factor g=15, and zero-temperature superconducting energy
haviour,whencombinedwithSOcouplinganddisorder[33],
gap∆=150µeV.Theredcurveisqualitativelyrepresentativeofthe
might explain enhancement of the critical current for some
IC enhancementobservedintheexperiment. TN denotesthecontact
specificvaluesofthemagneticfield. However,neitherthees- transmissivity. Inset: correspondingAndreevlevelsε vsphasedif-
timatedvaluefortheratiogµBBsw/ET ∼0.25northeabsence ferenceφ atB=6.2Bcrit,whichexhibitanavoidedcrossingatphase
ofanydependenceonthejunctionlength(throughtheThou- differenceφ =π duetothehybridizationofinner(blue)andouter
lessenergyET)canbeaccommodatedwithinthisexplanation. (red)MajoranaboundstatesacrossLS.c,Sketchofa(diffusive)NW
Josephsonjunctioninthepresenceofaninhomogeneousmagnetic
fieldduetothefocusingeffectinducedbythetwosuperconducting
DespitebeingaratherrarephenomenoninJosephsonjunc-
leads. ThelocalZeemanexchangeenergyinthesuperconductorsS’
tions we then consider two less obvious alternatives which andintheNWisdenotedbyVSandVN,respectively,withVS>VN.
account, at least from a qualitative point of view, for the Z Z Z Z
d, Theoreticalcritical current I calculated asa function ofVS for
observed colossal critical current enhancement induced by a C Z
differentdegreesofinhomogeneity(VS/VN).
magneticfield. Toembracebothtransportregimeswhichare Z Z
relevant for our junctions, we will present: (i) a first model
basedontopologicaltransitionsintheballisticapproximation;
(ii)asecondmodelbasedoninhomogeneousZeemanfieldin pling,liketheonesstudiedhere,canbestronglyenhancedrel-
thediffusivelimit. ativetothetrivialphaseforsmalljunctiontransmissivity(T ).
N
(i) Ballistic scenario -Topological transition and MBSs- This happens by virtue of the additional supercurrent con-
The enhancement of the critical current in a NW- tributedbyMajoranazeromodesinthejunctionastheexter-
based superconductor-normal metal-superconductor Joseph- nalZeemanfieldexceedsacriticalvalue(Bcrit).Inaquasi-one
sonjunctionwaspredictedtooccuraftertheproximizedsec- dimensional geometry, this topological transition occurs as a
tionsofthenanowire(theS’regionsshownintheschemeof givensubbandnundergoesabandinversionatZeemanenergy
Fig. 4a) are driven into a topologically non-trivial phase by V(n) = (cid:112)∆2+µ2, where µ is the Fermi energy measured
Z n n
anexternalZeemanfield. Inparticular,itwasshown[20]that fromthebottomofthesubband,and∆isthesuperconducting
inthetopologicallynon-trivialphasesthecriticalsupercurrent minigapinducedintheNW.Afterthetransition,andforlong
ofaJosephsonjunctionrealizedwithamultimodequasi-one enough proximized regions, i.e., for L (cid:29)ξ ≡h¯v /∆ where
S F
dimensionalsemiconductorNWwithSO(Rashba-type)cou- ξ istheMajoranastatelocalizationlength,eachS’sectionof
6
theNWbecomesatopologicalsuperconductorwithemergent region(VS(N))havetobeverydifferentinamplitude; specif-
Z
Majorana zero modes at its ends (the red circles in Fig. 4a). ically, itis requiredVS (cid:29)VN to achievethe observed I en-
Z Z C
The topological transition can be directly seen as a closing hancement.(iii)TheinterfacebetweentheSelectrodeandthe
andreopeningoftheBoboliubov-DeGennes(BdG)spectrum Nwireshouldexhibitatunnel-liketransmissivity. Condition
nearzeroenergy. Afterthetransition,theBdGspectrumcon- (i)isplausiblefortheregionsoftheNWclosetotheS/Ncon-
tainsemergentMajoranazeromodeswithprotectedcrossings tacts,thereforeonecanidentifythestructureasaSF-I-N-I-SF,
atasuperconductingphasedifferenceφ =π whichgiverise wherenowSFdenotesaBCSsuperconductorwithavariable
toa4π-period√icJosephsoneffect,andtoanenhancedcritical spin-splittingproportionaltotheappliedmagneticfield. Con-
current (IC ∼ TN) relative to that of conventional Andreev dition (ii), namely an inhomogeneously distributed Zeeman
levels(IC∼TN)thereforeonlyobservableforreducedcontact field, is possible as well owing to the strong magnetic field
transmissivity TN (cid:28)1. For shorter wires (with LS (cid:28)ξ) the focusing originating from the S electrodes (see Figs. 3b,c)
Majoranamodesarealwaysstronglyoverlapping,andmerge so one can safely assume thatVS (cid:54)=VNS (see the scheme in
Z Z
intostandardfinite-energyAndreevlevels. Fig. 4c). Condition (iii), on the other hand, is more unlikely
Figure.4billustratesthisphenomenondisplayingthecalcu- butstillpossiblebyconsideringlocaldefectsintheNW.
latedspectrumofaNWJosephsonjunctionwithL =500nm
S Having in mind these limitations, we have computed the
correspondingtotheexperimentalsamples(notethatwetake
critical current of a diffusive SF-I-N-I-SF junction with the
intoaccountalltheSfingersoneachsideofthejunction). In
help of the quasiclassical Green’s function formalism [36]
this limit the junction should be always topologically trivial
(seeMethodsformoredetails),andusingtheparametersob-
since the protected φ =π crossing is lifted, as shown in the
tainedfromtheI (T)fit(seeFig.1e).Theresultsaresumma-
inset of Fig. 4b. Despite this, the I enhancement predicted C
C rizedinFig.4d. Theoverallbehaviourisqualitativelyconsis-
tooccuratthetopologicaltransitionforidealjunctionlengths
tentwiththeexperimentalobservationalthoughaquantitative
(L (cid:29)ξ), remains visible even in our short-contact geome-
S comparison is difficult due to the complexity of the junction
try (L (cid:28)ξ) as a reminiscence of the Majorana zero modes
S geometry. Despite the good qualitative description of I (B),
which,evenifstronglyoverlapping,areeffectivelycontribut- C
the maindiscrepancy between themodel andthe experiment
ing to I . This is shown in Fig. 4b where a remnant of a
C liesinthelargevalueoffieldinhomogeneity,VS/VN(cid:38)10,re-
topological transition for short L leads to a critical current Z Z
S quiredtoobtainamagnetically-drivenI enhancement. This
enhancementaroundacrossovermagneticfieldVZ(n)(cid:38)∆(for last condition is indeed much stronger tChan the maximal ex-
µn∼0)asthejunctiontransmissivityisreduced.Forthetypi- pectation obtained for our junctions geometry in the out-of-
calenergygapdeducedfromtheexperiment,∆∼80µeV,and planemagneticfieldconfiguration,i.e.,VS/VN (cid:46)1.2,asdis-
Z Z
assumingg=15forInAsNWs,thisyieldsacriticalmagnetic
playedinFig.3c.
fieldB =∆/(1gµ )(cid:39)184mT.Thislattervalueisinrough
crit 2 B
agreementwiththeexperimentalmagneticfieldsatwhichthe
I enhancement is observed, B≈23 mT, if we consider the
C
field focusing factor of 2 (as shown in Fig. 3c) and a large g
Conclusions We have reported a colossal enhancement
factor,g∼60.
ofthecriticalsupercurrentatfinitemagneticfieldsinn-InAs
(ii) Diffusive scenario -Inhomogeneous Zeeman field- If
NW-based weak links which is in total contrast with the be-
we exclude the above topological transition scenario as the
havior observed so far in conventional Josephson junctions.
originoftheobservedeffectanalternativemodeltodescribe
The effect manifests itself only for very specific experimen-
the colossal I enhancement is based on the increase of the
C talconditions: themagneticfieldneedstobeappliedperpen-
criticalcurrentpredictedtooccurinSF-I-SFJosephsonjunc-
dicular to the NW axis and to the substrate, i.e., in the only
tions,whereSFdenotesasuperconductorwithaZeeman-split
configurationwhichisexpectedtobeperpendiculartotheSO
density of states, and I is an insulator. If the effective spin-
vector in the weak link. Despite this clear but puzzling ex-
splittingZeemanfieldsintheleftandrightelectrodesaredif-
perimentalevidence,aconclusivetheorycapabletofullyde-
l(r)
ferent, V , one expects to observe an enhancement of the scribeourresultsisstillmissing. Wepresentedtwopossible
Z
criticalcurrentwithamaximumatVl−Vr=∆ +∆ [34,35] models that allow to qualitatively grasp the observed effect,
Z Z l r
asaconsequenceofarobustspintripletcomponentinthesu- one holding in the ballistic limit, the other in the diffusive
perconductingcondensate,where∆ isthepairingpotential regime. Theavailableevidencesfallinacomplexintermedi-
l(r)
in the left(right) superconductor. It is worthwhile to empha- ateregimethatpreventsadetailedagreementbetweentheory
sizethattheI enhancementonlytakesplaceifthetransmis- and experiment. Yet, the realization of fully ballistic junc-
C
sivityofthebarrierislow[35]toavoidthecouplingbetween tionsisexpectedtoprovideanimprovedunderstandingofthe
thetwospin-polarizedsuperconductingcondensates. Inaddi- present phenomenology. In addition, extension of the theory
tion,thecriticalcurrentenhancementisalsopossibleinSF-I- oftopologicallynon-trivialpairingtothecaseofdiffusiveNW
N-I-SF[36]andeveninS-I-N-I-Sjunctions,thelatterbeinga Josephson weak links will offer a refined description of our
geometrysomewhatclosertotheexperimentalone,ifthefol- system, and further clarify the role of MBSs in the experi-
lowingconditionsarefulfilled: (i)Thedensityofthestatesof ment. Webelievethatthesenewsworthyresultswillstimulate
theSelectrodesisBCSlikewiththecoherentpeaksZeeman the development of novel theoretical models, and will con-
split thanks to the presence of the magnetic field (the orbital tributetotheprogressoftheinvestigationandunderstanding
effectsaredisregarded);(ii)TheZeemanfieldsintheSandN ofMBSsincondensedmatterphysics.
7
ACKNOWLEDGEMENTS ferentNWs,andmeasuredatlowtemperature.
The magneto-electric characterization of the InAs-NW
PartialfinancialsupportfromtheMarieCurieInitialTrain- Josephson junctions was performed in a filtered dilution re-
ingAction(ITN)Q-NET264034,fromtheMIUR-FIRB2013 frigeratordownto15mKusingastandard4-wiretechnique.
ProjectCoca(GrantNo. RBFR1379UX),fromCNRthrough The current-voltage characteristics of the junctions were ob-
thebilateralCNR-RFBRproject2015-2017,andfromtheEu- tained by applying a low-noise biasing current, with volt-
ropean Research Council under the European Union’s Sev- age across the NW being measured by a room-temperature
enth Framework Programme (FP7/2007-2013)/ERC Grant battery-powereddifferentialpreamplifier.
615187-COMANCHE is acknowledged. The work of E.S.
is funded by the Marie Curie Individual Fellowship MSCA- FittingdetailsofI (T). TostudythedecayofI (T)pre-
C C
IFEF-STNo.660532-SupeMag.M.A.acknowledgesthesup- sentedinFig.1eandidentifythemaintransportregimehold-
portoftheTuscanyregionthroughtheproject’TERASQUID’ ing in the NW we have modelled the Josephson junction in
andthePeopleProgramme(MarieCurieActions)oftheEu- two opposite limits: diffusive and ballistic. In the diffusive
ropeanUnionH2020ProgrammeunderREAgrantagreement regime,I (T)isfittedwiththeexpressionofthecriticalcur-
C
n [656485]. The work by P.S-J. is supported by the Span- rentofasuperconductor-normalmetal-superconductor(SNS)
ish Ministry of Economy and Innovation through Grant No. junction obtained by solving the linearized Usadel equation
FIS2011-23713andtheRamo´nyCajalprogramme.Thework [39]
of R.A. is funded by the Spanish Ministerio de Econom´ıa y
Competitividad through grant FIS2012-33521. The work of I = πkBT ∑ (κωL)∆2 , (1)
F.S.B is supported by the Spanish Ministerio de Econom´ıa C eRNW ω (ω2+∆2)[αsinh(κωL)+βcosh(κωL)]
y Competitividad through the Project No. FIS2014-55987-
P and Grupos Consolidados UPV/EHU del Gobierno Vasco where the sum is over the Matsubara frequencies ω =
(GrantNo. IT-756-13). πkBT(2n+1), n = 0,±1,±2,..., kB is the Boltzmann con-
(cid:112)
stant, κ = 2|ω|/(h¯D), D isthe NWdiffusioncoefficient,
ω
h¯ is the reduced Planck constant, R is the resistance of
NW
METHODS the NW of length L, α =1+r2(κωL)2, β =2r(κωL), and
r=R /R withR beingtheresistanceoftheSNinterface.
b NW b
ThebestfitwiththediffusivemodelpresentedinFig.1e(pink
Devicefabrication TheJosephsonjunctionspresentedin
dashed line) is obtained from Eq. 1 by using L = 300 nm,
thisworkarebasedonheavilyn-dopedInAsNWsgrownby
D=0.0416m2/s,R =4Ω,andR =741Ω.
metal-assisted chemical beam epitaxy [37]. NWs are grown b NW
Ontheotherhand,forthefitintheballisticregimeweuse
inaRiberC-21reactorbyusingmetallicseedsobtainedfrom
the expression of the Josephson current I valid for a multi-
thermal dewetting of a thin Au film layer evaporated on a J
channeljunction[40]
InAs substrate [17, 38]. Trimethylindium (TMIn) and ter-
tiarybutylarsine (TBAs)(cracked at 1000°C) are usedin the e∆(T)2 N D (cid:20)E (T)(cid:21)
growth as metallorganic precursors while ditertiarybutyl se- I (φ)= sinφ ∑ ntanh n , (2)
J
lenide(DtBSe)isusedasaseleniumsourceforn-typedoping. 2h¯ n=1En 2kBT
Based on previous experiments performed on similar NWs
[17,29]weestimateacarrierdensityn ∼3×1018cm−3and where Dn are the eigenvalues of the transmission matrix de-
s
anelectronmobilityµ∼2000cm2/Vsfromwhichwededuce scribingthejunction,Nisthenumberofconductingchannels,
(cid:113)
a Fermi velocity vF ∼2.2×106m/s, an electron mean free En(T) = ∆(T) 1−Dnsin2φ/2, ∆(T) is the temperature-
pathle∼60nmandtheeffectiveelectronmassforInAsNWs dependent BCS energy gap, and φ is the macroscopic quan-
m∗=0.023me,wheremeisthefree-electronmass. tumphasedifferenceoverthejunction. Thecriticalcurrentat
After the growth, the NWs are transferred mechanically each temperature is then obtained by maximizing I (φ) with
J
onto a SiO2/n-Si commercial substrate pre-patterned with respect to φ, IC =max|φ[IJ(φ)]. The best fit with the ballis-
Ti/Au pads and alignment markers which are defined by op- tic model shown in Fig. 1e (blue dashed line) is obtained by
tical and electron beam lithography, and deposited by ther- setting∆ =120µeV,D =1,andN=5.
0 n
mal evaporation. The position of the NWs on the substrate
ismappedwithascanningelectronmicroscope,andusedfor
ModelfortheI (B)enhancementbyatopologicaltran-
C
thealignedelectronbeamlithographyoftheJosephsonjunc-
sition. TheBogoliubov-deGennesHamiltonianofaproxi-
tions. Prior to the deposition of the Ti/Al superconducting
mizedtwo-dimensionalRashbasemiconductorreads
electrodes the NWs are etched with a highly-diluted ammo-
niumpolysulfide(NH4)2Sx solutiontoremovethenativeox- H=(cid:18) p2 −µ(cid:19)τ +αSO(σ p τ −σ p )
ide layer present on the semiconductor surface. This proce- 2m∗ z h¯ y x z x y
dureimprovesthequalityoftheohmiccontact,limitingunde- +∆σ τ +V σ τ , (3)
y y Z x z
siredsurfacescatteringprocesses. ThedepositionoftheTi/Al
(12/78 nm) leads is performed at room temperature in ultra- where p2= p2+p2,m∗ istheeffectivemass,σ arethespin
x y i
high vacuum conditions by electron beam evaporation [17]. Paulimatrices, τ arePaulimatricesintheelectron-holesec-
i
Morethan10junctionswerefabricatedstartingfromfivedif- tor, and α is the Rashba spin-orbit coupling. The last two
SO
8
termsdescribeaninducedsuperconductingpairingofstrength dimensional semiconducting nanowires, as demonstrated in
∆ and the Zeeman energyV produced by an external mag- Ref.[20].
Z
neticfield. Ithasbeenshown(fordetailssee,e. g.,[1,2])that
thes-wavepairinginEq.(3)inducesbothaneffectivep ±ip
x y
intrabandpairing,andaninterbands-wavepairingwhenreex- ModelfortheI (B)enhancementbyaninhomogeneous
C
pressed in terms of the ± eigenbasis of the helical Rashba + Zeemanfield. WehavecalculatedthecriticalcurrentI ofa
C
Zeemannormalproblem. Intermsoftheseeffectivepairings, diffusiveSF-I-F’-I-SFjunctionbysolvingtheUsadelequation
the system is topologically non-trivial when only one of the within the quasiclassical Green’s functions formalism [36].
two p-wavepairingsdevelops. ThisoccurswhentheZeeman Above,SFdenotesasuperconductor(S’)Zeemansplitbyan
energyVZ exceeds a critical value. In a quasi-1D geometry exchange field h1, F’ is a N wire possessing an internal ex-
(withdiscreteconfinementsubbands),thiscriticalvaluereads changeenergyh ,andIrepresentaninsulatingtunnelbarrier
2
(seethesketchshowninFig.4c). I canbeexpressedas
C
(cid:113)
VZ(n)= ∆2+µn2, (4) I = πAkBT ∑(cid:20) (f+)2 + (f−)2 (cid:21), (5)
C e sinh(χ+L)χ+ sinh(χ−L)χ−
with µ the Fermi energy measured from the bottom of the ω
n
subband. If the subbands are coupled, through the trans-
verseSOcoupling, anevennumberofMajoranazeromodes where χ± = (cid:112)2(|ω|±isign[ωh2])/D, f± =
(cid:112)
per edge hybridise in pairs and form (cid:98)N/2(cid:99) full fermions ∆(h )/ (ω±ih )2+∆2(h ), the sum is over the Mat-
1 1 1
(standardAndreevlevels)atnon-zeroenergiesε(n) (withn= subara frequencies ω already introduced, and ∆(h1) is
0
1,...(cid:98)N/2(cid:99))localisedateachedge. TheremainingN mod2 the effective superconducting order parameter calculated
(0) self-consistentlyfromtheBCSgapequation[41]. Moreover,
endstatesstayasMajoranazeromodes,atε =0.
0 A= RNWANW,whereR istheresistanceoftheNWoflength
This spectral even-odd effect is therefore a manifestation 2R2A2L NW
C C
of changes in the topological order in the multiband system, L, A is the NW crossection, R is the contact resistance,
NW C
and is reflected in the Josephson effect. In particular, the and A is the contact area. For the calculation of the curves
C
additional supercurrent contributed by Majorana zero modes displayed in Fig. 4d we have used the junction geometrical
in the junction in the non-trivial regime directly maps the dimensions,theparametersobtainedfromthediffusiveI (T)
C
topologicalphasediagramofproximizedmultibandquasi-one fitshowninFig.1e,andT =60mK.
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