Supercurrent Quantization in Narrow Channel SNS Junctions

We determine the quasi-particle excitation spectrum in the normal region of a narrow ballistic superconductor--normal-metal--superconductor (SNS) Josephson contact. Increasing the effective chemical potential in the contact converts the electronic levels i

SupercurrentQuantizationinNarrowChannelSNSJunctions

NikolaiM.Chtchelkatcheva,GordeyB.LesovikaandGianniBlatterb

ab

L.D.LandauInstituteforTheoreticalPhysics,117940Moscow,RussiaTheoretischePhysik,ETH-H¨onggerberg,CH-8093Z¨urich,Switzerland

(February1,2008)

arXiv:cond-mat/9907460v1 [cond-mat.mes-hall] 29 Jul 1999

Wedeterminethequasi-particleexcitationspectruminthenormalregionofanarrowballisticsuperconductor–normal-metal–superconductor(SNS)Josephsoncontact.Increasingthee ectivechemicalpotentialinthecontactconvertstheelectroniclevelsintoAndreev-levelscarryingsuper-current.Theopeningofthesesuperchannelsleadstoasupercurrentquantizationwhichexhibitsanon-universalbehavioringeneralandwediscussitsdependenceonthejunctionparameters.

TheJosephsone ect[1],ahallmarkofsuperconduc-tivity,isoffundamentalinterestandbearsconsiderablepotentialforapplicationsinsuperconductingelectronicsToday,theminiaturizationofelectronicstructureshasreachedtheregimewherethetransportproceedsviafeworevenasingleconductingchannelUs-inggatedstructures,junctionscanbetransformedfrominsulatingSIStosuperconductingSNSstateswiththequasi-particlespectrumevolvingfromphase-insensitiveelectronic-tophase-sensitiveAndreevstatescarryinglargesupercurrents.Theonsetofsuper owproceedsinstepsassociatedwiththeopeningoftransversechannels,similartotheconductancequantizationinametalliccon-tact.WhileinthelattertheconductanceG=I/Visquantizedinunitsof2e2/¯hitisthemaximalsuper-currentIcwhichisquantizedinunitsofe /¯hinthecaseofashortJosephsonlink( isthesuperconductinggapinthebanks).However,contrarytotheuniversalityofthequantizationinanormalcontact,thequantizationofthecriticalsupercurrentisperfectonlyinthelimitofshortjunctionsL ξ0(ξ0=h¯vF/π isthesupercon-ductingcoherencelength)butisnon-universalingeneralIndeed,whileexperimentsonsuperconductingquan-tumpointcontactsdoshowstepsinthecriticalcurrentIc,thesearemuchlessprominentthanthecorrespondingstepsintheconductanceGInthisletter,westudytheopeningofsuperconductingchannelsinthemetalliclinkofanarrowballisticSNSJosephsoncontactandde-terminetheevolutionofthequasi-particlespectrumandthedependenceofthesupercurrentquantizationonthejunctionparameters.

WhilethebehaviorofmacroscopicSNSJosephsonjunctionsiswellunderstood[8],thepresentinterestcon-centratesongatedstructuresofmesoscopicsize.Suchquantumpointcontactsarerealizedinheterostructures[3,7]orviamanipulationswithascanningtunnelingmi-croscope[9].Theoretically,thesupercurrent-phasere-lationinmesoscopicSNSjunctionswithaδ-scattererhasbeenanalyzedandthephenomenonofsuper-currentquantizationhasbeenstudiedinshortjunctions[5].Non-universalfeaturesofsupercurrentquantizationhave rstbeenobservedbyFurusakietal.[6]—unfor-tunately,thesenumericalresultsprovidelimitedinsightintothephysicaloriginandtheparametricdependenceof

thesee ects.Here,wepresentadetaileddiscussionoftheopeningofsuperchannelsinmesoscopicSNSjunctionsus-ingquasi-classicalandscatteringmatrixtechniques.Wediscussthenon-trivialevolutionoftheexcitationspec-trumasthechemicalpotentialdropsbelowthesuper-conductinggapandanalyzethetransformationoftheballisticSNSstructureintoaSIStunneljunction.Thephase-dependenceofthequasi-particlespectrumallowsusto ndthesupercurrentquantizationinshortandlongjunctions,wherethecontributionfromthecontinuousspectrumcanbeignored.FIG.1.NarrowchannelSNScontact:(a)geometricalsetupshowingtheadiabaticjoiningofthewiretothesu-perconductor,(b)potentiallandscapewitha atpotentialinthewirecenterandsmoothlydroppingtothebandbot-tominthesuperconductingbanks.Whileelectronsandholeswithsmallexcitationenergiesε<µxformcurrentcarryingAndreevstates,theholepropagationisquenchedatlargeen-ergies >µxandtheAndreevlevelstransformintolocalizedelectronicstates.

Consideranarrowmetalliclead(withfewtransversechannels)connectingtwosuperconductingcontactsassketchedinFig.1(a)[weassumepiecewiseconstantgapparameters (x< L/2)= exp(i L), (|x|<L/2)=0,and (L/2<x)= exp(i R)].Foreachtransversechannelinthemetallicwirethequasi-particlespectrumενisdeterminedthroughthe1DBogoliubov-deGennesequation(wechoosestateswithεν≥0) 22 h¯ x uν(x)uν(x)

,=εν

v(x)v(x)νν+µ(x)x2mwhereuνandvνdenotetheelectron-andhole-likecom-ponentsofthewavefunctionΨν.Thespectrumsplits

intocontinuousanddiscretepartsandweconcentrate

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