Full counting statistics of crossed Andreev reflection Jan Petter Morten,1,∗ Daniel Huertas-Hernando,1 Wolfgang Belzig,2 and Arne Brataas1 1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 2University of Konstanz, Department of Physics, D-78457 Konstanz, Germany (Dated: September17, 2008) We calculate the full transport counting statistics in a three-terminal tunnel device with one superconducting source and two normal-metal or ferromagnet drains. We obtain the transport 9 probability distribution from direct Andreev reflection, crossed Andreev reflection, and electron 0 transfer which reveals how these processes’ statistics are determined by the device conductances. 0 The cross-correlation noise is a result of competing contributions from crossed Andreev reflection 2 and electron transfer, as well as antibunching due to the Pauli exclusion principle. For spin-active tunnelbarriersthatspinpolarizetheelectron flow,crossed Andreevreflectionandelectron transfer n statistics exhibit different dependencies on the magnetization configuration, and can be controlled a byrelative magnetization directions and voltage bias. J 2 PACSnumbers: 74.40.+k72.25.Mk73.23.-b74.50.+r 2 ] I. INTRODUCTION as a consequence of dominating CA. Subsequently, mea- l l surements reported in Ref. 18 on nonlocal voltages in a h In crossed Andreev reflection (CA), a Cooper pair in Au probes connectedto a wire of superconducting Al by - transparentinterfacesindicatedthattheETcontribution s a superconductor (S) is converted into an electron-hole e quasiparticlepairinnormal-metalterminals(N )orvice is larger than the contribution from CA. n m versa.1,2 Thisprocesshaspotentialapplicationsinquan- The competition between CA and ET determines the signofthenonlocalvoltageandhasbeenstudiedtheoret- . tuminformationprocessingsinceitinducesspatiallysep- at aratedentangledelectron-holepairs. Theperformanceof ically using various approaches20,21,22,23,24,25,26,27,28 in- m entanglers utilizing this effect is diminished by parasitic cluding the circuit theory of mesoscopic superconductiv- ity utilized in this paper.29,30 We will consider the linear - contributionsfromelectrontransfersbetweentheNn ter- d response nonlocal conductance G . In superconductor– minals. Thisprocesswillbereferredtoaselectrontrans- nl n normal-metal hybrid devices where transport in one fer (ET), but is also often denoted electron cotunneling o normal-metal terminal N is measured in response to or elastic cotunneling. 1 c anappliedvoltageinanothernormalmetal-terminalN , [ It has been suggested that the noise properties 2 this quantity is defined by of crossed Andreev reflection can be used to distin- 1 guish it from electron transfer between the normal- v ∂ I = G = (G G ), (1) 1 metal terminals.3 Mesoscopic transport noise also re- V2 1 − nl − ET− CA 5 veals information about charge carriers which is in- where we have introduced conductances associated with 5 accessible through average current measurements. In the charge transfer processes introduced above, G CA 3 a system with several drain terminals, i.e. cur- for crossed Andreev reflection, and G for electron ET 1. rent beam splitters, noise measurements show signa- transfer.20 The sign of the nonlocal conductance is de- 0 tures of correlations between the current flow in sep- termined by the competition between ET and CA. The- 9 arated terminals. The cross-correlation noise has at- oretical calculations based on second order perturbation 0 tained interest recently since it can be utilized to theoryinthetunnelingHamiltonianformalismpredicted v: studyentanglement4,5,6,7,8,9,9,10,11,12,13,14,15,16 andcorre- that the nonlocal conductance resulting from CA reflec- i lated transport. tion is equivalent in magnitude to the contribution from X CrossedAndreevreflectioninsuperconductor–normal- ET.20 Thus the induced voltagein N in response to the 1 r a metal systems has been experimentally studied in Refs. bias on N2 should vanish since CA and ET give currents 17,18,19. In Ref. 17 the nonlocal voltage was mea- with opposite sign, in contrast to the measurements re- sured in a multilayer Al/Nb structure with tunnel con- ported in Refs. 17,18,19. The tunneling limit was also tacts between the superconducting Nb and the normal- considered in Refs. 26,29,31, and it has been found that metal Al layers. Current was injected through one of the nonlocalconductanceis infact offourthorderin the the normal-metal–superconductor contacts, and a non- tunneling and favours ET. local voltage measured between the superconductor and Experimental investigations of crossed Andreev re- the other normal-metal. At injection bias voltage below flection in superconductor-ferromagnet (S-F) structures the Thouless energy E = ℏD/d2 associated with the havebeenreportedinRefs. 32,33. Themeasurementsin Th separation d between the normal-metals, positive non- Ref. 32 were modeled using the theory of Ref. 20. local voltage was measured and this was interpreted as Experimental studies of the CA and ET noise proper- the resultof dominating ET.For voltageseV aboveE ties can be used to determine the relative contributions Th thenonlocalvoltagechangedsign,whichwasinterpreted of these processes to the nonlocal conductance. It was 2 shown theoretically in that CA contributes positively to N N the noise crosscorrelations,whereas ET givesa negative 1 2 contribution.3 Calculations ofhigher ordernoise correla- torsorthenoisedependenceonspin-polarizinginterfaces g g canrevealfurtherinformationabouttheCAandETpro- g1MR1 c g2MR2 cesses. We will consider the full counting statistics (FCS) gS whichencompassesallstatisticalmomentsofthecurrent flow.34,35,36 The noise properties of ET and CA reflec- S tion thus obtained can be used to study the competition between these processes and reveal information that is notaccessibleinthe meancurrents. Ourcalculationalso FIG.1: Circuittheoryrepresentationoftheconsideredbeam- determinesthecontributiontothenoisecomingfromthe splitter where supercurrent flows from a superconducting fermion statistics (Pauli exclusion principle). Moreover, reservoir (S) into normal metal drains (Nn). Tunnel barriers the charge transfer probability distribution provided by betweencavity(c)andthedrainscanbespin-active,andare FCSrevealsinformationabouttheprobabilityofelemen- characterized by the conductances gn and spin polarizations tary processes in the circuit.37 gMRn/gn. In this paper we calculate the FCS of multitermi- nal superconductor-normal metal and superconductor- gap of the superconducting terminal. In addition to the ferromagnet proximity structures, and study the cur- ET and CA processes described above,there can also be rents, noise and cross correlations associated with the directAndreev(DA)reflectionbetweenthesuperconduc- various transport processes. We obtain the probabil- tor andone normal-metalterminal, where both particles ity distribution for transportat one normal-metaldrain, of the Andreev reflected pair are transferred into N . n and show that the probability associated with ET is Semiclassical probability arguments show that the sub- largerthantheprobabilityassociatedwithCA.Forspin- gapchargecurrentinthe connectorbetweenN andc in 1 active interfaces we show how spin filtering can be uti- the three terminal network in Fig. 1 has the following lized to control the relative magnitude of the CA and structure,20,29 ET contributions to the transport. Finally, we consider crossed Andreev reflection for spin triplet superconduc- I (E)= G (V +V ) G (V V )+2G V , (2) 1 CA 1 2 ET 2 1 DA1 1 − − tors. This paper goes beyond our previous publications Refs. 29,30,37 in that we consider different bias volt- where we have introduced the conductance GDA1 associ- ages on the normal-metal/ferromagnetic drains and dis- ated with direct Andreev reflectionbetween terminal N1 cuss the effect of triplet superconductivity. andS.Eq. (2)leadstothedefinitionofthenonlocalcon- Thepaperisorganizedinthefollowingway: InSec. II ductance in(1)whichshowsthat whenV2 >V1, ETand CA give competing negative and positive contributions we describe the electroniccircuitandoutline the formal- ismutilizedtocalculatethecumulantgeneratingfunction respectively to the current. The conductances in (2) will be determined in the calculation below. oftheprobabilitydistributions. InSec. IIIwediscussthe results in the case of normal metals, and in Sec. IV we consider the spin-active connectors. Finally, our conclu- A. Circuit theory sions are given in Sec. V. The circuit theoryof mesoscopictransportis reviewed inRef. 38andisasuitableformalismtostudyproximity II. MODEL effects insuperconducting nanostructures. The theoryis developed from a discretization of the quasiclassicalthe- The systems we have in mind can be represented by oryofsuperconductivity,39 incombinationwith atheory the circuit theory diagram (see Sec. IIA) shown in Fig. of boundary conditions based on scattering theory. 1. A superconducting source terminal (S) and normal The circuit theory is formulated in terms of the qua- metaldrainterminals (N ) areconnectedby tunnel bar- n siclassicalGreen functions of the terminals and nodes in riers with conductance g to a common scattering re- n the system. Nodes can represent small islands or lat- gion which is modeled as a chaotic cavity (c). The as- tice points of diffusive parts of the system. Under the sumptions on the cavity is that the Green function is assumptions described above, the Green function of the isotropic due to diffusion or chaotic scattering at the in- spin-singletS terminalin Fig. 1 is Gˇ =τˆ where τˆ is a S 1 k terfaces, and that charging effects and dephasing can be Pauli matrix in Nambu space. The Green functions Gˇ disregarded. The tunnel barrierscan be spin-active with n of normal-metal terminals N are given by n spin polarization g /g . We consider elastic trans- MRn n portatzerotemperature. Thesuperconductingterminal τˆ τˇ +(τˇ +iτˇ ) E <eV , itsergmrionuanlsd.edW, eanadssubmiaseesthVant Vanre ap∆pl0i,edwhtoereth∆e0noisrmthael Gˇc(E)=(τˆ33τˇ33+sg1n(E)τˆ23(τˇ1+iτˇ2) ||E||≥eVnn, (3) ≪ 3 where τˇ are Pauli matrices in Keldysh space. B. Full counting statistics k Matrix currents Iˇdescribe the flow of charge, energy andcoherencebetweenterminalsandnodesthroughcon- Full counting statistics is a useful tool to compute nectors, and conservation of these currents are imposed currents and noise in a multiterminal structure,43 and at each node. This generalized Kirchhofflaw determines also provides the higher statistical moments that may the Green function of the cavityGˇc. The balance of ma- become experimentally accessible in these systems. Ad- trixcurrentsIˇn flowingbetweeneachterminaln=1,2,S ditionally,one canobtain informationabout the elemen- and the cavity, including the effect of superconducting tary charge transport processes by studying the proba- pairing in c, can be written bility distributions.37 The cumulant generating function (CGF) ( χ )oftheprobabilitydistributionisdirectly n Iˇ e2ν0Vc∆ τˆ , Gˇ =0. (4) accessibSle{by t}he Green function method, and is defined n− ℏ c 1 c by n (cid:20) (cid:21) X e−S({χn}) = P( N ;t )e−iPnχnNn Here, ∆ is the superconducting order parameter on the { n} 0 c cavity,ν isthedensityofstatesand thevolumeofthe XNn ceaqvuiattyi.o3n00,3a9bTovhee isnedcouncdesteelremctroonnt-hhoelelVecfptahirainngd.siSdiencoef tthhee P({Nn};t0)=(2π1)M Z−ππdMχe−S({χn})+iPnNnχn. pairingtermhasthesamestructureasthecouplingtothe (7) superconducting terminal (see (5)), it gives quantitative Here, P( N ;t ) is the probability to transfer geffectsgthaantdarweilclabpetuormeditbteydriennothrmeafolilzloinwgineg2.ν0HVecr∆e,c/gℏ+is N1, N2, ...,{Nnn}ele0ctrons into terminal N1,N2, ...,Nn in thSe→supSerconductor tunnel conductance. S time t0, and M is the total number of terminals in the circuit. The CGF is a function of the set of count- Spin dependent transmission and reflection is de- ing fields χ which are embedded in the Green func- scribed by tunneling amplitudes tnk,σ and rkn,σ for elec- tion at ea{chn}terminal by the transformation Gˇn trons with spin σ incident from the cavity side on the eiχnτˇK/2Gˇne−iχnτˇK/2 where τˇK = τˆ3τˇ1. The CGF w→ill interface between the cavity and terminal n in channel be determined by the following relation,43 k. The matrix currentIˇ throughsuch spin-activeinter- n faces is40 ie∂ ( χn ) S { } = dE I ( χ ), (8) n n t ∂χ { } Iˇ =gn Gˇ ,Gˇ + gMRn m σ¯ τ¯ ,Gˇ ,Gˇ . (5) 0 n Z n 2 n c 4 n· 3 n c whereIn( χn )istheparticle(counting)currentthrough { } (cid:2) (cid:3) (cid:2)(cid:8) (cid:9) (cid:3) connector n in presence of the counting fields. Our task Here, g = g tn 2 is the tunnel conductance is now to integrate this equation and obtain the CGF n Q k,σ| k,σ| wghere(gQtn=2e2/thPnis2t)hiescthonedcuocntdaunccteanqcueapnotulamri,zagtMioRnn,=σ¯ rSen(t{χcnon})s.erUvastiniognt(h6e),gietnewraasl sfoouluntdioinntRoetfh.e1m1,atthriaxtctuhris- isQthekve|ckt,o↑r|o−fP| ak,u↓l|imatricesinspinspace,andtheunit is possible by rewriting the counting current in terms of vecPtor m points in the direction of the magnetization a derivative of Mˇ with respect to the counting fields. n of the spin polarizing contact. In (5) we have neglected Explicit derivation shows that anadditionaltermrelatedtospindependentphaseshifts 1 1 from reflection at the interface which can be suppressed In({χn})= 8eTr τˇKIˇn({χn}) = 4ei∂χnTr Mˇ2 . by a thin, non-magnetic oxide layer.41,42 The effects of (cid:8) (cid:9) np o(9) spin filtering contained in the polarization g , which MRn This result is valid also in the presence of spin-active can be obtained experimentally using ferromagnetic ter- contacts (5). Combining (9) with (8) yields ( χ ) minals, will be studied in Sec. IV. n S { } straightforwardly. Insystemswherealltheconnectorsaretunnelbarriers Practicalcalculations of CGFs are performedby diag- describedbythematrixcurrent(5),itispossibletosolve onalizing the matrix Mˇ, which allows us to express the (4) analytically and obtain the cavity Green function in CGF in terms of the eigenvalues of Mˇ, terms of the terminal Green functions and the tunneling parameters. To this end, we note that it is possible to t write (4) as [Mˇ,Gˇc] = 0. Employing the normalization S =−4e02 dE λ2k. (10) conditionGˇ2 =1,the solutioncanbe expressedinterms Z Xk q c of the matrix Mˇ as11 In this equation, λ is the set of eigenvalues of Mˇ. k { } We can obtain the cumulants of the transport proba- Gˇ =Mˇ/ Mˇ2. (6) bilitydistributionbysuccessivederivativesoftheCGF.36 c Specifically, we obtain the mean current from p This result facilitates calculation of the cumulant gener- ie ∂ ( χ ) atingfunctionofthechargetransferprobabilitydistribu- I = S { } . (11) n −t ∂χ tion in tunnel barrier multiterminal circuits. 0 n (cid:12){χ=0} (cid:12) (cid:12) (cid:12) 4 The current noise power is given by case V = V , the total CGF following from (10) has 1 2 6 S one contribution from the energy range E < eV ,eV , 1 2 e2 ∂2 ( χ ) and if the voltages are different, another contribution in C =2 S { } , (12) m,n t ∂χ ∂χ the energy range eV E <eV (we assume V >V ), 0 m n (cid:12){χ=0} 1 ≤ 2 2 1 (cid:12) (cid:12) whereinthemultiterminalstructure(cid:12),theautocorrelation t −eV1 eV1 eV2 noise at terminal n is given by C . When m=n, (12) = 0 + + E λ2(E) gives the noise cross-correlations.n,n 6 S −4e2 Xk Z−eV2 Z−eV1 ZeV1 ! q k = (V )+ (V V ). (13) a 1 b 2 1 S S − III. NORMAL METAL DRAINS Thereisnocontributiontotransportat E >eV . Here, 2 | | we have defined two separate contributions to the CGF In this section, we will consider the FCS of the super- that govern transport in the regime E < eV ,eV ( ) 1 2 a S conducting beamsplitter in Fig. 1 when the connectors where only Andreev reflections (CA and DA) can occur, arenotspinpolarizing,andgeneralizepreviousworksby andthe regime eV E <eV ( ) where in additionto 1 2 b ≤ S takingintoaccountadifferenceindrainterminalvoltages Andreevreflections,ETcantakeplace. Thecontribution V =V . was calculated in Ref. 11, see (14a), where we have 1 2 a 6 S IntheregimeE <eV ,eV theonlycontributiontothe defined g =[g2+g2]1/2 and g =g +g . The counting 1 2 Σ S 1 2 nonlocal conductance comes from CA since we consider factors e2iχS−iχm−iχn describe processes where two par- zerotemperature. TheresultingCGFwasstudiedinRef. ticles are transferredfrom S, and one particle is counted 11 where it was assumed that V = V . In the general at terminal N and at terminal N (m,n=1,2). 1 2 m n t V Sa =−√02e1vgΣ2 + gΣ4 +4gS2 gmgn(e2iχS−iχm−iχn −1), (14a) u s Xm,n u t (Vt V ) 0 2 1 S = − b − √2e ×vgΣ2 +2g1g2(eiχ1−iχ2 −1)+ gΣ4 +4gS2 gng2(e2iχS−iχn−iχ2 −1)+4g1g2g2(eiχ1−iχ2 −1) (14b) u s n u X t m m 2 1 m m 2 1 EF EF eV eV −| 2| ∆ −| 2| N S N N c N 2 1 2 1 (a) (b) FIG.2: Transport processes inthethreeterminaldevicewheneV1 =0: (a) Crossed Andreevreflection: ACooperpairfrom S is convertedinto an electron-hole pair in c byAndreevreflection, and theelectron with energy +E is transferred into N1,and theholewithenergy−E istransferredintoN2. Tunnelbarriersbetweenthereservoirsmaybespin-activeandaredescribedby magnetization vectorsmthatinthispaperareconsideredcollinear. (b)Electron transfer: AparticlefromN1 tunnelsthrough the cavity c into N2. The density of states in the cavity c is suppressed due to the proximity effect from the superconducting terminal. In (14b) we show the calculated which has con- tributions from electron transfer. The counting factor b S 5 eiχ1−iχ2 describeseventswhereanelectronistransferred currentswith oppositesignsinN1 andN2 resulting from from N to N . Compared to , we see that DA events thisprocess. Thisdemonstratesthatitispossibletotune 1 2 a S between S and N that would be described by counting the signof cross-correlationsby the voltagesV andV .3 1 1 2 factors e2iχS−2iχ1 no longer occur. This can be under- The contribution to C1,2 in (16c) is proportional to the stood from the electron-hole–natureof Cooper pairs, see voltage difference V V , and vanishes in the limit of 1 2 − Fig. 2. Two quasiparticles,with energy+E for the elec- asymmetric conductances g g or g g ,g . S 1(2) 2(1) S ≫ ≫ tronand EfortheholeconstitutetheAndreevreflected It is interesting to compare with the corresponding b − S quasiparticle pairs in c. In the energy range considered CGF when S is in the normal state, here, eV E <eV , the states in N are occupied, pre- cludingD1A≤reflectio2nintoN . Asimi1larargumentshows = t0(V2−V1) (17) 1 b S − 2e × that in CA processes, the electron must be transferred into N1 and the hole into N2. (g1+gS+g2)2+4g2g1eiχ1−iχ2 +4g2gSeiχS−iχ2. ThenonlocalconductanceG =G G following nl ET− CA Here pwe see a contribution due to transport between from (14b) is in agreement with Ref. 29 where N and N that is similar to the one outside the dou- 1 2 ble square root in (14b). Superconductivity leads to the g g 2g2+g2 g g g2 G = 1 2 S , G = 1 2 S . double square root in (14b) that takes into account the ET 2 [g2+gS2]3/2 CA 2 [g2+gS2]3/2 correlationoftransportthroughcbyAndreevreflections (15) and ET. The complicated dependence on the counting fields in (14b) precludes a simple interpretation of in b The nonlocal conductance is dominated by ET and is S terms of the probabilities of elementary charge transfer dofucotradnecreOva(ngni4s)h.2e6s,2d9uWe htoeneqgu/aglSp≪rob1a,bthileitynofnorloEcaTl caonnd- processes. However, when gS ≫g or g1(2) ≫g2(1),gS we can expand the square roots in and obtain the CGF b CA as we will explicitly show by inspection of the prob- S t (V V ) ability distribution below. In the opposite limit that = 0 2− 1 g g (g2+2g2)eiχ1−iχ2 the coupling to normal terminals dominates, g /g 1, Sb − 2g3e 1 2 S S Σ ≪ h the conductance for ET is to lowest order given by +g2g g e2iχS−iχ1−iχ2 g g (1 δN)/g. Here δN = (g /g)2/2 is the lowest or- S 1 2 1 2 S der cor−rection to the low-energy density of states due to +g2g2e2iχS−2iχ2 . (18) S 2 superconducting correlations. The CA conductance be- i In this limit the CGF describes independent CA, ET, comes g g δN/g in this limit. 1 2 andDAPoissonprocesses. The prefactorsdetermine the Let us now consider cross-correlations. Ingeneral,CA averagenumberofchargestransferredbyeachprocessin leads to a positive contribution and ET leads to a nega- time t . tive contributionto the cross-correlation. An additional, 0 To illustrate the physics described by in the limit negative contribution is induced by the Pauli exclusion Sb introduced above, let us examine the probability distri- principle. The cross-correlation between N and N fol- 1 2 bution obtained by the definition in (7). If we consider lowing from (13) is the current response in N to a voltage in N , we can 1 2 C1,2 =2e(V1+V2)GCA 2e(V2 V1)GET (16a) consider that V1 = 0 and the only contribution to the − − totalCGF comesfrom . Thenormalizedprobability 10eV1(G1 Gnl)(G2 Gnl) (16b) distributionSfor the transSpbort at terminal N1 following − g − − Σ from (18) then becomes 4e(V V ) + 2g− 1 [GCA(G2+2GDA2)−GDA2Gnl], (16c) P(N1;t0)=e−N¯1gΣ2/g2 Σ k≥X|N1| wherewehavedefinedthelocaldifferentialconductances k+N1even Gn = ∂nIn and the conductance for direct Andreev re- g2 k+2N1 g2 k−2N1 flection into terminal n is G = g2g2/2g .29 We now N¯ S N¯ S +1 DAn S n Σ × 12g2 1 2g2 focusonthecompetitionbetweenCAandET.Whenthe (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) −1 twonormal-metalterminalsareatequalvoltageV =V k+N k N 1 2 1 1 ! − ! . (19) thecontributionfromETandalsothetermin(16c)van- × 2 2 (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21) ishes and we are left with a positive contribution to the Herewehavedefinedthemeannumberofparticlestrans- cross-correlations from CA due to the correlated parti- ferred in time t , cle transfer into N and N . An additional, negative 0 1 2 contribution due to the Pauli principle in (16b) van- I t V t g g N¯ = | 1| 0 = 2 0 1 2 . (20) ishes in the limit of asymmetric conductances gS ≫g or 1 e e g(1+g2/g2)3/2 g g ,g due to the noisy (Poissonian) statistics S 1(2) 2(1) S ≫ of the incoming supercurrent. A negative contribution Eq. (19)describesajointprobabilitydistributionforCA from ET in (16a) is induced when there is a voltage dif- and ET processes with Poissonian statistics, and is con- ference between the normal-metal terminals due to the strained such that the number of CA events described 6 by the weight g2/2g2 subtracted by the number of ET IV. SPIN-ACTIVE CONNECTORS S events described by the weight g2/2g2+1, is N as re- S 1 quired. When g /g 1, the mean number of particles S ≫ transferred vanishes according to (20) and the probabil- ity distribution (19) is symmetric around N = 0. This 1 means that the averagecurrent vanishes, since the prob- abilitiesforETandCAareequal. Ingeneral,the proba- bility distribution has its maximum for negative N , i.e. 1 ET is more probable than CA reflection. In Fig. 3 we Qualitatively, the effect of spin polarizing interfaces have plotted the probability distribution (19) for differ- onthecompetitionbetweenCAandETprocessesinS-F ent values of g /g. For small ratios g /g, ET dominates systems can be understood as follows.20 The ET process S S and the probability distribution is centered at a nega- is favoured when magnetizations of ferromagnetic leads tive value for N . As expected, we see that the center are parallel since the same spin must traverse both the 1 of the probability distribution (mean number of parti- interfacesbetweenc andthe ferromagnets. Onthe other cles transferred)is shifted froma negative value towards hand, CA reflection is favouredin an antiparallelconfig- zerowithincreasingg /g. Thewidthofthedistribution, urationsince two particles with opposite spins must tra- S described by the autocorrelation noise C (see (12)), verse the interfaces. This behaviour was experimentally 1,1 decreases with increasing g /g. observed in Refs. 32,33 where ferromagnetic Fe probes S were contacted to a superconducting Al wire. g /g=5.00 S 0.05 0.2 )0 The FCS of a beam splitter with spin-active contacts ;t1 and V = V was considered in our previous paper Ref. N 1 2 (P 0.1 37 and constitutes a, see (21a). In this case the only S transport processes are DA and CA reflection, and we found that CA is enhanced in an antiparallel alignment of the magnetizations as expected. 0 -30 -20 -10 0 10 N 1 FIG. 3: (Color online) Probability distribution for transport When V2 > V1, there can also be ET, and an addi- of N1 electrons into terminal N1, P(N1;t0) Eq. (19). We tional effect is spin accumulation at the node. With show distributions for two different values of the parameter collinear magnetizations (sign of g describes mag- MRn gS/g = 5.00 (red [gray] solid impulses), and 0.05 (blue [dark netization directions up (positive) or down (negative) gray] dotted impulses). We have chosen the parameter α = alongthez quantizationaxis),wefindthatintheregime g1g2V2t0/eg = 20, which gives the mean value N¯1 for small eV < E < eV , = + where is given 1 2 b b+ b− bσ gS/g, see Eq. (20). below (21b). S S S S 7 t V 4g2 4g2 = 0 1+ 1 MR(g2+g2)+ S (g g g g )(e2iχ−iχm−iχn 1), (21a) Sa − √2evuuu vuu − gΣ4 S gΣ4 Xm,n m n− MRm MRn − t t t g (V V ) 2 = 0 Σ 2− 1 1+ (g +σg )(g +σg ) eiχ1−iχ2 1 Sbσ − 2√2e ( gΣ2 1 MR1 2 MR2 − (cid:0) (cid:1) 4g2 4g2 + 1 MR(g2+g2)+ S (g σg )(g +σg ) e2iχS−iχn−iχ2 1 " − gΣ4 S gΣ4 n n− MRn 2 MR2 − X (cid:0) (cid:1) 4(g g )2 1/2 1/2 + − MR (g +σg )(g +σg ) eiχ1−iχ2 1 . (21b) gΣ4 1 MR1 2 MR2 − (cid:21) ) (cid:0) (cid:1) Herewehaveredefinedg =(g2+g2+g2 )1/2andintro- The cross-correlationfollowing from (22) is Σ S MR ducedg =g +g . TheexpressionforS reduces MR MR1 MR2 b totheresultfornonpolarizingcontacts,Eq. (14b),inthe C1,2 =2e(V2+V1)GCA 2e(V2 V1)GET. (24) − − limit that g 0. The two terms correspond to MRn bσ the two possible→directions of the spin(Ss) involved in the The sign of C1,2 can now be tuned by two experimental charge transfer processes. The spin-dependent conduc- control parameters: The bias voltages through the pref- tance for a spin up (down) is g +( )g . In ET, one actors in (24), and the relative magnetization direction n MRn spinmusttraversethetwospin-activ−einterfaces,thusthe that determines the magnitudes of GET and GCA. counting factor for spin σ is proportional to the weight In the energy range V1 < E <V2 we are in this setup (g +σg )(g +σg ). The two spin channels are measuring the energy of the quasiparticles involved in 1 MR1 2 MR2 independent. The two opposite spins of an Andreev re- crossedAndreevreflection,seeFig. 2. Sincetheelectron- flected quasiparticlepair canbe CA reflectedinto termi- like quasiparticle with energy +E must flow into N1, nals with different polarizations gMR1 and gMR2 accord- and the −E hole-like quasiparticle must flow into N2, ing to the prefactor (g σg )(g +σg ), and each this means that the entanglement in the energy degree 1 MR1 2 MR2 possibility for the direc−tions of the two spins gives an of freedom of Andreev reflected quasiparticle pairs has independent contribution to the CGF . collapsed. b S In the limit that g g or g g ,g (g S 1(2) 2(1) S MRn ≫ ≫ ≤ g by definition) we can expand the double square roots n and perform the summation over which yields A. Triplet superconductivity bσ S t (V V ) 0 2 1 Sb =− 2eg−3 Superconductingcorrelationswithtripletpairingsym- Σ metry in the spin space can be induced by magnetic × gΣ2 +(g−gMR)2 (g1g2+gMR1gMR2)eiχ1−iχ2 exchange fields in singlet superconductor heterostruc- (cid:8)+(cid:2) gS2(g1g2−gMR1(cid:3)gMR2)e2iχS−iχ1−iχ2 thuarserse.cTenhtilsyebffeeecntehxapseartimtaeinnteadllcyodnesmidoernasbtrleatiendte(rseeset,Raenfsd. +g2(g2 g2 )e2iχS−2iχ2 . (22) S 2 − MR2 44,45,46andreferenceswithin). The differentspin-space Thenonlocalconductancefollowing(cid:9)from(22)isgivenby symmetryoftheinducedelectron-holecorrelationsopens interesting experimental applications where e.g. super- G =(g g +g g ) gΣ2 +(g−gMR)2 , (23a) conducting correlations can propagate through a strong ET 1 2 MR1 MR2 2g3 ferromagnetic material.47,48 We have studied the FCS (cid:2) Σ (cid:3) g2 whenSis asourceofspintripletquasiparticlepairs,and GCA =(g1g2−gMR1gMR2)2gS3. (23b) in this subsection we summarize our results for collinear Σ magnetizations when V =V . 1 2 This immediately demonstrates that ET is favoured The CGF for the spin triplet Cooper pairs with in a parallel configuration of the magnetizations S Ψ = 0, where S is the spin operator along the z- z z | i (g g > 0) as the same spin in this case tun- axis and Ψ is the spin part of the Cooper pair wave MR1 MR2 | i nels through both interfaces. On the other hand, function, is identical to the CGF for conventional spin CA reflection is favoured by antiparallel magnetizations singlet superconductors. We showed in Ref. 37 that the (g g <0)sincetheoppositespinsofasinglettun- CGF (21a) reveals the entangled nature of the quasipar- MR1 MR2 nel through different interfaces. These qualitative fea- ticle pairs. The S Ψ = 0 spin triplet states are also z | i tures are in agreement with Ref. 20. oneofthe maximallyentangledBellstateswhichimplies 8 that the magnetization dependence for CA is the same The tripletstateswhereS Ψ = ℏ Ψ giveriseto a z | i ± | i for singlet and triplet in the collinear case. This result differentdependenceonthemagnetizationconfigurations can be shown also straightforwardly by computing the in the CGF since the quasiparticle pairs are not in spin two-electron tunneling probability p which is propor- entangledstates,butratherinproductstates. Weobtain 1,2 tional to Ψ (g +g σ¯ ) (g +g σ¯ ) Ψ +1 2. the CGF 1 MR1 3 2 MR2 3 h | ⊗ | i ↔ t V 4g2 4g2 = 0 1+ 1 MR(g2+g2)+ S (g +σg )(g +σg )(e2iχ−iχm−iχn 1), (25) Sk −Xσ 2√2evuu s − gΣ4 S gΣ4 1 MR1 2 MR2 − t Compared to the singlet case (21a), we see that the cross-correlations from electron transport between nor- CA counting prefactor factorizes as a result of the non- mal terminals and crossed Andreev reflection. Finally, entangled product state of the quasiparticle pairs. This we have shownhow spin-active contacts act as filters for magnetization dependence can be recovered also by cal- spin, and calculated the cumulant generating function. culatingthetwo-particletunnelingprobabilityp asdis- The sign of the cross-correlation due to the competing 1,2 cussed above. contributions from electron transportbetween drain ter- minals and crossed Andreev reflection can in this case be determined by two external control parameters, i.e. V. CONCLUSION bias voltages and the relative magnetizationorientation. Finally,wemakesomeremarksaboutthecountingstatis- We have calculated the full counting statistics of mul- tics in the case of spin triplet superconductors. titerminaltunnel-junctionsuperconductor–normal-metal and superconductor–ferromagnet beam splitter devices, and studied the resulting currents and cross-correlation. 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