Because the valleys in the band structure of graphene are related by time-reversal symmetry, electrons from one valley are reflected as holes from the other valley at the junction with a superconductor. We show how this Andreev reflection can be used to de

Detectionofvalleypolarizationingraphenebyasuperconductingcontact

A.R.AkhmerovandC.W.J.Beenakker

Instituut-Lorentz,UniversiteitLeiden,P.O.Box9506,2300RALeiden,TheNetherlands

(Dated:December2006)

Dec 2006

Becausethevalleysinthebandstructureofgraphenearerelatedbytime-reversalsymmetry,electronsfromonevalleyarere ectedasholesfromtheothervalleyatthejunctionwithasuper-conductor.WeshowhowthisAndreevre ectioncanbeusedtodetectthevalleypolarizationofedgestatesproducedbyamagnetic eld.Intheabsenceofintervalleyrelaxation,theconductanceGNS=(2e2/h)(1 cosΘ)ofthejunctiononthelowestquantumHallplateauisentirelydeter-minedbytheangleΘbetweenthevalleyisospinsoftheedgestatesapproachingandleavingthesuperconductor.Ifthesuperconductorcoversasingleedge,Θ=0andnocurrentcanenterthesuperconductor.AmeasurementofGNSthendeterminestheintervalleyrelaxationtime.

92 PACSnumbers:74.45.+c,73.23.-b,73.43.-f,73.50.Jt

]llaThequantizedHallconductanceingrapheneexhibitshthehalf-integerquantizationGH=(n+1

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(1 cosΘ),(1)

whentheHallconductanceGH=2e2/hisonthelowestplateau[15].HerecosΘ=ν1·ν2isthecosineoftheanglebetweenthevalleyisospinsν1,ν2ofthestatesalong

FIG.1:Threediagramsofagraphenesheetcontactedbyonenormal-metal(N)andonesuperconducting(S)electrode.Edgestatesapproachingandleavingthesuperconductorareindicatedbyarrows.Thesolidlinerepresentsanelectronstate(green:isospinν1;blue:isospinν2),andthedashedlinerepresentsaholestate(red:isospin ν2).

thetwographeneedgesconnectedbythesuperconductor(seeFig.1).Ifthesuperconductorcoversasingleedge(Fig.thenΘ=0 GNS=0—nocurrentcanenterintothesuperconductorwithoutintervalleyrelaxation.Ifthesuperconductorconnectsdi erentedges(Figs.1b,c)thenGNScanvaryfrom0to4e2/h—dependingontherelativeorientationofthevalleyisospinsalongthetwoedges.

WestartouranalysisfromtheDirac-Bogoliubov-DeGennes(DBdG)equation

H µ

µ THT 1 Ψ=εΨ,(2)

withHtheDiracHamiltonian, thesuperconductingpairpotential,andTthetimereversaloperator.The