Positive noise cross-correlations in superconducting hybrids: Role of interface transparencies R. M´lin1 , C. Benjamin2,3 and T. Martin2,4 e arXiv:0708.0566v3 [cond-mat.mes-hall] 3 Jan 2018 1 Institut NEEL, CNRS and Universit´ Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France e 2 Centre de Physique Th´orique, Case 907 Luminy, 13288 Marseille cedex 9, France e 3 Quantum Information Group, School of Physics and Astronomy, University of Leeds, Woodhouse Lane, Leeds, LS29JT, UK and 4 Universit´ de la M´dit´rann´e, 13288 Marseille Cedex 9, France e e e e (Dated: June 1, 2018) Shot noise cross-correlations in normal metal-superconductor-normal metal structures are discussed at arbitrary interface transparencies using both the scattering approach of Blonder, Tinkham and Klapwik and a microscopic Green’s function approach. Surprisingly, negative crossed conductance in such set-ups [R. M´lin and D. Feinberg, Phys. Rev. B 70, 174509 (2004)] does e not preclude the possibility of positive noise cross-correlations for almost transparent contacts. We conclude with a phenomenological discussion of interactions in the one dimensional leads connected to the superconductor, which induce sign changes in the noise cross-correlations. I. INTRODUCTION Among the challenges in condensed matter systems at the nanoscale is the manipulation of electronic entanglement in connection with quantum information processing. Some theoretical proposals1–4 involving superconductivity have been implemented by three groups5–7 , aiming at the observation of non-local effects in transport, and in the long run at the realization of a source of entangled pairs of electrons. The devices realized up to now5–7 consist of ferromagnet-superconductor-ferromagnet (Fa SFb ) and normal metal-superconductor-normal metal (Na SNb ) threeterminal hybrids, designed in such a way that Cooper pairs from the superconductor have the opportunity to split in the two different normal (Na,b ) or ferromagnetic (Fa,b ) electrodes. Noise measurements, on the other hand, have focused mostly on two-terminal devices such as a two dimensional electron gas-superconductor junction8 and a junction between a diffusive normal metal and superconductor9 . The possibility of separating pairs of electrons into different electrodes10,11 has aroused a considerable theoretical interest recently, and calls for a new look at fundamental issues related to non local transport through a superconductor, namely tunneling in three-terminal configurations in the presence of a condensate12–18,20–27 . In most of the above studies of three-terminal hybrids, the leads are separately connected to the source of Cooper pairs (the superconductor). Some early proposals considered a normal metal “fork” with two leads each connected to voltage probes, and with the third lead connected to a superconductor. There, the motivation was to compute the current noise cross-correlations between the two normal leads, and to see whether the sign of this signal would be reversed compared to the negative noise cross-correlations of a normal metal three-terminal device (the fermionic analog of the Hanbury Brown and Twiss experiment). Ref.[28] considered a ballistic fork with an electron beam splitter, and found that positive noise correlations could arise upon reducing the transparency of the superconductor interface. Ref.[13] considered a chaotic cavity connected to the superconductor and to the leads, and found positive noise correlations which are enhanced by the backscattering of the contacts, and are robust when the proximity effect in the dot is destroyed by an external magnetic field. Positive correlations may also occur in the absence of correlated injection29 . It was shown already2,3 that noise cross-correlations between the currents in different electrodes can lead in the long term to Einstein, Podolsky, Rosen experiments with electrons, being massive particles. In the short term, current-current correlations among electrodes Na and Nb in Na SNb three-terminal structures may lead to detailed informations about the microscopic processes mediating non local transport. A spin-up electron from electrode Nb may be transmitted as a spin-up electron in electrode Na across the superconductor (a channel called as “elastic cotunneling”), or it may be transmitted as a spin-down hole in electrode Na with a pair left in the superconductor (a channel called as “crossed Andreev reflection”) (see Fig. 1). However, in Na SNb structures with tunnel contacts, the elastic cotunneling current of spin-up electrons in electrode Na turns out to be opposite to the crossed Andreev reflection current of spin-down holes in electrode Na : as shown by Falci et al.12 the lowest order contribution to the crossed conductance is vanishingly small in a Na ISINb structure, where an insulating layer I is supposed to be inserted in between the normal and superconducting electrodes. A finite crossed conductance is however restored to next orders in the tunnel amplitudes20,23,24 , if the normal electrodes Na,b are replaced by ferromagnets12 Fa,b , or if interactions in the superconductor are taken into account22 . We are motivated by the fact that contacts with moderate to large interface transparency may be used in future experiments in order to maximize the measured current noise signal. A relevant question is then: do basic properties 2 Vb (a) Superconductor e Va Vb (b) Normal metal "b" Ib Va Ia e Normal metal "a" Ib Normal metal "b" Ia Superconductor e h e e Normal metal "a" FIG. 1: Schematic representation of elastic cotunneling (a) in a Na SNb structure, in which a spin-up electron (e↑) tunnels from electrode Nb to electrode Na across the superconductor. (b) shows crossed Andreev reflection in which a spin-up electron (e↑) coming from the normal electrode Nb is converted as a hole in the spin-down band (h↓) in electrode Na while a pair is transfered in the superconductor. The balance between (a) and (b) can be controlled by the relative spin orientation if the normal metals Na,b are replaced by ferromagnets Fa,b . The voltage Va is set to zero in the available crossed conductance experiments5–7 . We evaluate in the article cross-correlations between the currents Ia and Ib in the presence of arbitrary voltages Va and Vb in the normal or ferromagnetic electrodes30 . Ia n R Ib Vb Normal metal "b" R Superconductor Sites a Va Normal metal "a" N S N Ib Ia Va Vb Sites b n (a) (b) FIG. 2: Schematic representation of the tight-binding model of Na SNb structure with lateral contacts used in Green’s function calculations (a), and of the one dimensional (1D) Na SNb structure used in the BTK approach (b). such as the sign of current cross-correlations depend or not on interface transparency, as compared to lowest order perturbation theory in the tunnel amplitudes discussed by Bignon et al.30 ? Interestingly, we obtain below a positive answer to this question, using two complementary approaches: the scattering approach of Blonder Tinkham and Klapwijk as well as a microscopic tight binding scheme using the Keldysh method. The two approaches differ in the way interfaces are treated. The scattering approach assumes that the Na SNb system has a single mode connected throughout, while in the Green’s function approach the local tunneling term connects “many” modes in the leads to “many” other modes in the superconductor. The study of these two limits is necessary to establish the robustness of the positive noise cross-correlation signal discussed in this paper. As we shall see, we find qualitative agreement between the two approaches. Concerning the non local conductance calculation, the situation is summarized rapidly. It was shown already20,23,24 that the non local conductance of a Na SNb structure is negative for transparent interfaces. This was interpreted in terms of the next higher order process in the tunnel transparencies23 corresponding to the co-tunneling of pairs from electrode Nb to electrode Na . Such processes is not expected to result in entanglement in between the two electrodes Na and Nb because no Cooper pair from the superconductor is split between the normal electrodes Na and Nb . However, noise crossed correlations constitutes a quantity which is distinct from the non local conductance, and, surprisingly, we show that the corresponding noise cross-correlations of highly transparent Na SNb structures are positive in spite of negative non local conductance. This is confirmed by keeping track of the processes contributing to the noise cross-correlations in the Anantram and Datta18 formula for the noise in multiterminal superconducting 3 hybrids. Finally, we mention that for experimental and practical reasons, the calculations presented in this paper are performed with the assumption that the distance which separates the two normal metal/superconductor interfaces is comparable or larger than the superconducting coherence length. Indeed, all experiments carried out so far for the non local conductance5–7 have been performed within this range of parameters. For the BTK approach, this condition is necessary in order to avoid the meaningless situation where all dimensions of the superconductor are smaller than the coherence length. In the Green’s function approach, lateral contacts can be at a distance smaller than the coherence length while all dimensions of the superconductor are much larger than the coherence length. We verified carefully that our results are also valid in this situation. The article is organized as follows. Preliminaries for the Green’s function and for the BTK approach are presented in Sec. II. The results of the two methods are discussed in Sec. III. Interactions in the normal metal leads are addressed in Sec. IV. II. A. PRELIMINARIES Noise cross-correlations The noise cross-correlation is given by the current fluctuations Sa,b (t, t′ ) = δIa (t + t′ )δIb (t) , (1) where Ia (t + t′ ) and Ib (t) are the currents through the normal electrodes Na,b (or ferromagnetic electrodes Fa,b ) (see the notations “a” and “b” on Figs. 1 and 2) at times t + t′ and t respectively. The current operator at time t is defined as c+n ,σ (t)can ,σ (t) + h.c., α Ia (t) = ta,α σ (2) n where h.c. is the Hermitian conjugate, ... denotes a quantum mechanical expectation value, can ,σ (t) destroys a spin-σ electron at time t and site “an ”, and c+ (t) creates a spin-σ electron at time t and site “αn ” (see Fig. 2 for the set α,σ of tight-binding sites labeled by an , and for their counterpart αn in the superconducting electrode). For DC bias applied to the electron reservoirs, time translational invariance imposes that the real time crosscorrelation Sa,b (t2 − t1 ) depends only on the difference of times of the two current operators. This correlator is related ˆ ˆ in a standard fashion to the Green’s functions31,32 G+ (t2 − t1 ) and G− (t2 − t1 ) connecting the two branches of the Keldysh contour: Sa,b = e 2 ˆ ˆ ˆ ˆ Tr −G+ ⊗ tα,a ⊗ G− ⊗ tβ,b b,α a,β h ¯ ˆ ˆ ˆ ˆ +G+ ⊗ tα,a ⊗ G− ⊗ tb,β β,α (3) a,b ˆ ˆ ˆ ˆ +tβ,b ⊗ G+ ⊗ ta,α ⊗ G− b,a α,β ˆ ˆ ˆ ˆ − G+ ⊗ ta,α ⊗ G− ⊗ tb,β , β,a α,b where the trace is a sum over the channels in real space (see Fig. 2) and over electron and hole components of the Nambu representation. The symbol ⊗ denotes convolution over time variables and the dependence on time variables is implicit in Eq. (3). The notations αn , βn are used for the counterparts in the superconductor of the tight-binding sites an and bn at the interfaces of electrodes Na,b (see Fig. 2). The notation gi,j corresponds to the Nambu Green’s ˆ ˆ + and G− are used for electrodes connected to ˆ function of electrodes isolated from each other, while the notations Gi,j i,j each other by arbitrary interface transparencies. For instance one has the following for the Keldysh Green’s function G+ (t1 , t2 ): i,j ˆ G+ (t1 , t2 ) = i i,j c+ (t2 )ci,↑ (t1 ) j,↑ c+ (t2 )c+ (t1 ) j,↑ i,↓ cj,↓ (t2 )ci,↑ (t1 ) cj,↓ (t2 )c+ (t1 ) i,↓ . (4) ˆ In Eq. (3), ti,j denotes the Nambu hopping amplitude from i to j: ˆ ti,j = |ti,j | 0 0 −|ti,j | . (5) 4 Based on a standard description20 , we suppose that propagation in the normal electrodes Na,b is local, as for a vanishingly small phase coherence length. Some processes that delocalize in the direction parallel to the interfaces21 are not accounted for. Then the multichannel contact on Fig. 2 can be replaced by a single channel contact, and this is why we use in Eq. (3) the notations a, α, β, b instead of an , αn , βn and bn (see Fig. 2a), and average over the Fermi phase factor kF R entering non local propagation in the superconductor. To discuss both highly transparent interfaces and an arbitrary distance between the contacts, we solve the Dyson equation of the form ˆ ˆˆ ˆ AGα,β = gα,β + Xα,β Gβ,β ˆ ˆ ˆˆ ˆ B Gβ,β = gβ,β + Xβ,α Gα,β , ˆ (6) ˆ ˆ ˆ ˆ ˆ ˆ A = I − gα,α tα,a ga,a ta,α ˆ ˆ ˆ ˆ ˆ ˆ B = I − gβ,β tβ,b gb,b tb,β ˆ Xα,β = gα,β tβ,b gb,b tb,β ˆ ˆ ˆ ˆ (8) (10) ˆ Xβ,α = gβ,α tα,a ga,a ta,α . ˆ ˆ ˆ ˆ (11) (7) with (9) Then we have ˆ ˆ ˆ ˆ ˆ ˆ ˆ Gα,β = Y −1 gα,β + Y −1 Xα,β B −1 gβ,β ˆ ˆ ˆ ˆ ˆ ˆ Gβ,β = B −1 gβ,β + B −1 Xβ,α Gα,β , (12) ˆ ˆ ˆ ˆ ˆ Y = A − Xα,β B −1 Xβ,α . (14) (13) where we used the notation ˆ +,− ˆ +,− The Keldysh Green’s functions Ga,α (ω) and Gα,a (ω) are next obtained from the Dyson-Keldysh equations31,33 ˆ ˆ a,α ˆ ˆ+,− ˆ ˆ α,α I + GR (ω)tα,a ga,a (ω)ta,α GA (ω) ˆ +,− Ga,α (ω) = ˆ a,β ˆ ˆ+,− ˆ ˆ β,α + GR (ω)tβ,b gb,b (ω)tb,β GA (ω) (15) ˆ +,− ˆ α,α ˆ ˆ ˆ α,α ˆ ˆ+,− Gα,a (ω) = GR (ω)tα,a ga,a (ω) I + ta,α GA (ω) ˆ α,β ˆ ˆ+,− ˆ ˆ β,α + GR (ω)tβ,b gb,b (ω)tb,β GA (ω), (16) where we Fourier transformed from time t to frequency ω. Note that ω is conjugate to the time difference in Eq. (1), ˆ and thus we consider zero frequency noise. The notations GA,R (ω) are used for the advanced and retarded fully i,j dressed Green’s functions connecting i and j at frequency ω. The method discussed here allows to treat an arbitrary distance between the normal electrodes and contains all information about quantum interference effects34 , unlike quasiclassics35. We did not find qualitative changes of the crossed conductance and of the sign of noise cross-correlations when crossing over from R > ξ to R < ξ within the ∼ ∼ Green’s function approach and therefore we present the results only for R > ξ, in which case we recovered for the ∼ crossed conductance the behavior of Refs. 20,23,24. B. Blonder Tinkham Klapwijk (BTK) approach to noise cross-correlations The Blonder, Tinkham, Klapwijk (BTK36 ) approach was previously applied to non local transport24 , i.e. computing the current voltage characteristics in the different leads (see also Ref. 16). Here it is extended to address noise crosscorrelations. Considering a one dimensional (1D) Na SNb structure (see Fig. 2b), three regions are connected to each other: (i) normal electrode Na at coordinate x < 0; (ii) superconducting region S for 0 < x < R, and (iii) normal electrode Nb for x > R. The two-component wave-function in electrode Na takes the form ψa (x) = 1 0 + see aa exp (ikF x) + seh aa 1 0 exp (−ikF x). 0 1 exp (ikF x) (17) 5 The two-component wave-function in electrode S takes the form ψS (x) = c u0 v0 exp (ikF x) exp (−x/ξ) + d v0 u0 exp (−ikF x) exp (−x/ξ) + c′ u0 v0 exp (−ikF (x − R)) exp ((x − R)/ξ) + d′ v0 u0 (18) exp (ikF (x − R)) exp ((x − R)/ξ), and in electrode Nb it is given by ψb (x) = seh ab 0 1 exp (−ikF (x − R)) + see ab 1 0 exp (ikF (x − R)). The amplitudes are specified by matching the electron and hole wave functions and their derivatives at both boundaries. They describe Andreev reflection (seh ), normal reflection (see ), crossed Andreev reflection (seh ) and aa aa ab elastic cotunneling (see ), as well as transmission in the superconductor with/without branch crossing of evanescent ab states localized at the left/right interfaces (coefficients c, d, c′ and d′ ). Under standard assumptions, the zero frequency (k,l) noise cross-correlations (the time integral of the real time crossed correlator) are given by18,19 Sa,b = k,l Sa,b , where k, l label electrodes Na and Nb , and where (k,l) Sa,b 2π = 0 d(kF R) 2π α,β,γ,δ∈{e,h} qα qβ h dωAkγ,lδ (aα)Alδ,kγ (bβ)fkγ (1 − flδ ) (19) where Greek indices denotes the nature (e for electrons, h for holes) of the incoming/outgoing particles with their associated charges qα , while Latin symbols l, k identify the leads. fkγ is a Fermi function for particles of type γ in reservoir k at energy hω. The matrix ¯ † Akγ,lδ (iα) ≡ δik δil δαγ δαδ − sαγ sαδ , il ik (20) enters the definition of the current operator, and contains all the information about the scattering process which were described above. Note that the noise cross-correlations in Eq. (19) are averaged over the Fermi phase factor kF R accumulated while traversing the superconductor, which is a standard procedure for evaluating non local transport through a superconductor20 . C. Comparison between the two methods We carried out systematically all calculations with the two approaches (microscopic Green’s functions and BTK) in order to determine which are the generic feature of the noise cross-correlation spectra. We find such modelindependent variations of the noise cross-correlation spectra in the limit of small interface transparencies30 and for highly transparent interfaces. The two approaches do not coincide at the cross-over between small and large interface transparencies because different asumptions are made about the geometry. We complement these numerical results by an explanation from the Anatram-Datta18 noise cross-correlation formula of why noise cross-correlations are positive for highly transparent interfaces and at small bias. For microscopic Green’s functions we calculate the non local conductance and the noise cross-correlations of a single channel contacts, in the spirit of the approach by Cuevas et al.33 for a S-S break junction, and averaged over the microscopic Fermi phase factors. The electrons (or pairs of electrons) tunneling through such weak links have to reduce in size down to atomic dimensions in order to traverse the junction, resulting as in optics in diffraction. In this approach, the tunneling Hamiltonian transfers electrons from one point in the normal metal lead to one point in the superconductor. The contribution of wave functions from all momenta in the normal metal are extracted to 6 104 Sa,b(Vb) Va/∆=0.2, Antiparallel (AP) alignement (a) 2 1 0 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 Vb/∆ 1 Va/∆=0.2, parallel (P) alignement 104 Sa,b(Vb) 0 -1 (b) -2 -0.75 -0.5 -0.25 T=0.1 T=0.2 T=0.3 T=0.4 T=0.5 T=0.6 T=0.7 T=0.8 0 0.25 0.5 0.75 Vb/∆ 1 T=0.9 T=1.0 FIG. 3: Noise Sa,b (eVb /∆) of a Fa SFb three-terminal structure with ferromagnetic electrodes Fa and Fb with antiparallel spin orientations (a), and with parallel spin orientations (b), for interface transparencies ranging from T = 0.1 to T = 1. The voltage on electrode Fa is fixed to Va /∆ = 0.2. The distance R between the contacts is such that exp (−R/ξ0 ) = 10−2 , with ξ0 the BCS coherence length at zero energy. The dependence of the coherence length on energy is supposed to have the BCS form for a ballistic superconductor. The noise spectra on this figure are obtained from microscopic Green’s functions. 7 tunnel into a point in the superconductor therefore transferring to all momenta states in the latter (and vice versa). On the opposite, in the 1D BTK model, a single (transverse) mode on the normal side is converted into a single mode in the superconductor. Diffraction effects are absent for the one dimensional BTK model. As an example, for perfect interfaces, a hole cannot be forward-scattered in the 1D BTK model. A natural interpretation of this effect for the 1D BTK model23 is momentum conservation for perfect transmission. III. A. RESULTS Half-metal/superconductor/half-metal structures The dominant transmission channel can be controlled by the relative spin orientation in the case of half-metals (containing only majority spin electrons). More precisely, as mentioned in the Introduction, two processes contribute to the non local conductance in this case (see Fig. 1): elastic cotunneling (transmission of a spin-up electron from electrode Fb as a spin-up electron in electrode Fa ) and crossed Andreev reflection (transmission of a spin-up electron from electrode Fb as a hole in the spin-down band in electrode Fa with a pair left in the superconductor). The dominant non local transport channel can be selected by the relative spin orientation of half-metals: elastic cotunneling in the parallel (P) alignment and crossed Andreev reflection in the antiparallel (AP) alignment. The resulting noise cross-correlations are shown on Fig. 3 for half-metals in the parallel (Fig. 3a) and antiparallel (Fig. 3b) alignments and for different values of the transmission coefficients ranging from tunnel to highly transparent interfaces. We obtain a characteristic sign of noise cross-correlations for the two spin orientations (negative crosscorrelations for elastic cotunneling, and positive cross-correlations for crossed Andreev reflection), and a characteristic dependence on the voltage Vb at fixed Va . For each setup (antiparallel/parallel alignment), there is a special value of voltage where the noise cross-correlations vanish. For instance, the vanishing of the noise cross-correlations occurs at Vb = −Va for the antiparallel configuration. For half-metals, a (singlet) Cooper pair cannot be transmitted as a whole in the right or left lead because of the opposite spin orientation of the two leads. The two electrons of a Cooper pairs have then to end up into two electrodes with different chemical potentials, and one has then determine whether the incoming and outgoing states are available (see Fig. 4). For the value of voltages Vb = −Va , electrons with opposite energies with respect to the superconducting chemical potential are not available, and the same is true for pairs of holes with opposite energies. Therefore the cross-correlations vanish for this value of voltage, but they increase gradually for voltages Vb > −Va or Vb < −Va as Pauli blocking effects disappear. B. Normal metal/superconductor/normal metal structures We consider now a one dimensional Na SNb structure within the Green’s function approach as well as within the one dimensional BTK approach. The dependence of the noise cross-correlations Sa,b (Va , Vb ) on eVb /∆ for a fixed eVa /∆ at zero temperature is shown in Figs. 5 and 6, for tunnel interfaces to highly transparent interfaces. For the BTK approach, the scattering coefficients see , seh , seh , and see are parameterized by a single interface parameter z (for aa aa ab ab this symmetric system), z ≡ 2mH/¯ 2 kF , where m is the band mass, and H is the strength of the delta potential at h the interface. The parameter z is related to the transmission probability in the normal state z2 = 1 − T0 T0 (21) The expected Bignon et al.30 tunnel limit for noise cross-correlations is recovered in both cases. In this case, the noise cross-correlations vary almost linearly if the voltage Vb is smaller than Va (in absolute value) are recovered for small interface transparencies, in both plots. In the other limiting case of high transparencies, the overall shape of the curves is of an inverted pyramid with its apex at Vb = −Va . As the transparency is increased the base of the pyramid enlarges, and the height decreases. Similar predictions are obtained in the two approaches both in the limits of small and large interface transparencies. C. Interpretation For high interface transparencies, it is well established20,23,24 that the crossed conductance is negative, with the same sign as normal electron transmission from one electrode to the other across the superconductor. 8 Energy Vb ∆ F F Va Filled state (a) Elastic cotunneling Vb Filled state µS S F −∆ ∆ F S µS (b) Crossed Andreev reflection Va −∆ FIG. 4: Pauli blocking as exemplified in (a) electron-cotunneling setup and (b) for a crossed Andreev reflection setup. For (a), an electron incoming from Fb cannot be transmitted in Fa due to Pauli blocking for Vb = Va . For (b) a pair from the superconductor cannot split in electrodes Fa and Fb if Vb = −Va . Let us now consider noise crossed correlations. We show that positive noise correlations are expected from the Anatran-Datta18 noise formula. We focus on the case of a small applied voltage eVb ≪ ∆. First if k and l in Eq. (19) both belong to the same electrode Na we find Aaγ,aδ (aα)Aaδ,aγ (bβ) = (δα,γ δα,δ − saγ;aα saα;aδ ) (−saδ;bβ sbβ;aγ ) . (22) (23) At zero temperature, the terms containing δα,γ δα,δ do not contribute to noise cross-correlations because of the Fermi occupation factors in Eq.(19). For high transparencies, the term saγ;aα saα;aδ encodes local Andreev reflection changing electrons into holes, with therefore α = −γ = −δ. Taking into account the four s-matrix coefficients and the prefactor containing the sign of the transmitted carriers, we deduce that this contribution to noise cross-correlation is vanishingly 9 Va/∆=0.2, N-S-N structure T=1 T=0.8 T=0.6 T=0.4 T=0.2 1 0 5 10 Sa,b(Vb) 2 -1 -2 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 Vb/∆ 1 FIG. 5: Noise Sa,b (eVb /∆) of a Na SNb three-terminal structure with normal electrodes Na and Nb for moderate and high interface transparencies (normal transmissions T = 0.2, 0.4, 0.6, 0.8, 1), within microscopic Green’s functions. The tunnel limit30 with Sa,b (eVb /∆) linear in Vb in the window −Va < Vb < Va is recovered for small T . The distance R between the contacts is such that exp (−R/ξ0 ) = 10−2 , with ξ0 the BCS coherence length at zero energy. (k,l) small. It can be shown by similar arguments that the term in Sa,b with k and l labeling the different electrodes Na and Nb respectively leads to positive noise cross-correlations for high interface transparencies, in agreement with Fig. 6. More precisely, in terms of scattering matrices, advanced and retarded propagations correspond to the elements of the scattering matric s or to their complex conjugate respectively. To the advanced and retarded labels are added here the labels encoding propagation forward or backward in time for the two branches of the Keldysh contour (see (k,l) Sec. II A). Let us consider the term in Sa,b with k and l labeling the different electrodes Na and Nb respectively. We deduce that, for this term, propagation forward in time is the product of an advanced non local propagation from Nb to Na , and a retarded local Andreev process at the interface SNa . Combining the advanced and retarded terms, propagation forward in time has the net result of changing a spin-up electron from electrode Na as a hole in the spin-down band in electrode Nb , with a pair left in the superconductor (as shown on Fig. 7), which corresponds to crossed Andreev reflection with positive noise crossed correlations (see Fig. 6). Pauli blocking is the same as for crossed Andreev reflection in a Fa SFb structure in the antiparallel alignment. This picture holds for highly transparent interfaces at low energy. IV. CONSEQUENCES FOR INTERACTIONS IN THE LEADS (PHENOMENOLOGICAL APPROACH) Before concluding, we discuss briefly the inclusion of Coulomb interactions within the (one dimensional) leads connected to the superconductor. In normal metal junctions, it is known37 that electron-electron interactions change qualitatively the current voltage characteristics. Such Luttinger liquid behavior can be approached from the strong interaction limit, or alternatively from the weak interaction limit38,39 . In the latter, a perturbative renormalization 10 Va/∆=0.2, N-S-N structure z=0.0 z=0.4 z=0.6 z=0.8 z=1.0 z=3.0 0.0 5 10 Sa,b(Vb) 5.0 -5.0 -1.0 0.0 -0.5 0.5 1.0 Vb/∆ FIG. 6: Noise Sa,b (eVb /∆) of a Na SNb three-terminal structure with normal electrodes Na and Nb from transparent (z = 0.0) to tunneling (z = 3.0) interfaces, from the BTK approach. The tunnel behavior30 with Sa,b (eVb /∆) linear in Vb in the window −Va < Vb < Va is recovered for large z in the tunnel limit. The distance R between the contacts is such that exp (−R/ξ0 ) = 10−2 , with ξ0 the BCS coherence length at zero energy. group (RG) scheme was used to derive the renormalized scattering matrix coefficients. The interaction-induced dependence of the transmission probability on energy ω reads39 : T (ω) = ω T0 | D0 |α ω 1 − T0 (1 − | D0 |α ) (24) where T0 is the transparency in the absence of interaction, 0 < α < 1 quantifies the electron-electron interaction strength (α = 0 corresponds to no interactions), and D0 is a high-energy cut-off determined by the energy bandwidth of the electronic states. For normal metal/superconductor interfaces, this approach was adapted40,41 to predict that electron-electron interactions induce a suppression of the Andreev conductance with a non-ideal transparency. However, the Andreev conductance remains unaffected by interactions for a purely transparent NS interface. For the microscopic Green’s function approach of a three terminal Na SNb device, the bare transmission amplitude is thus replaced by its energy dependent value in order to obtain the noise cross correlations in the presence of interactions in the leads. For the BTK approach, we follow the phenomenological treatment of Ref. 41, which amounts to replacing T0 by T (ω) to account for the presence of interactions. The interaction parameter is changed to z → zee : 2 zee = | ω −α 2 ω −α 1 − T0 | =| | z D0 T0 D0 (25) When z = 0 (i.e., T0 = 1), the result ze−e = 0 implies that for a transparent interface electron-electron interactions have no effect on electronic transport40,41 . In the BTK approach zee is simply inserted in the expression for the scattering matrix coefficients, and the noise cross correlations are computed subsequently. We discuss briefly in the conclusion the limitations of these approaches (for the Green’s function and the BTK situation). 11 Normal metal −e R− A− h Superconductor +A +A R −A e e −A Normal metal +A +R +R h FIG. 7: Schematic representation of the process contributing to the noise cross-correlations of a Na SNb structure. This process in noise cross-correlations corresponds to crossed Andreev reflection: propagation forward in time of an electron from the left normal metal converted as a hole in the right normal metal with a pair transmitted in the superconductor, and the reverse process backward in time. Propagation forward (backward) in time are denoted by “+” (“-”) on the figure and by solid blue (dashed red) lines. We have shown on the figure the pair transmitted inside the superconductor for propagation forward in time, but, for clarify, we have not represented the corresponding process for propagation backward in time. Similarly we have shown the “advanced” (A) and “retarded” (R) labels for forward propagation in time. In Fig. 8a (Fig. 8b) the parameter α is gradually switched on to the full interaction value (α = 1) for the Green’s function calculation (for the BTK approach). For almost transparent contacts (Fig. 9, left panel) we obtain a change from completely positive crossed correlations throughout the range to alternating between negative and positive as α is increased. The weak interactions results are analogous to their non-interacting counterparts: single valley at Vb = −Va . However for strong interactions two valleys are located at Vb = ±Va . Further increase of interactions leads to a gradual smoothing of these features, eventually leading to the result for tunnel interfaces30 . This is in contrast with the case of semi-transparent contacts wherein noise cross-correlations which are uniformly negative (as in Fig. 9, right panel) turn positive as interactions are increased in the positive voltage range. Indeed, for this situation, interactions change a transparent contact into a tunneling contact. V. CONCLUSIONS Experiments on non local Andreev reflection5–7 have already been achieved, and they have generated a lot of excitement in the mesoscopic physics community. Noise cross-correlation experiments in normal metal/superconducting hybrid structures are the next on the list and experiments are under way. Such experiments could allow to observe the splitting of Cooper pairs from a superconductor into two distinct normal metal leads. This would constitute a milestone toward probing entanglement in the context of nanophysics. A renewed theoretical interest into such questions, taking into account experimental constraints (the separation between contacts is large compared to the superconducting coherence length) is simply warranted. In the beginning of this work, we computed the noise cross-correlation spectra as a function of voltage with varying interface transparencies for half-metal/superconductor/half-metal systems. For oppositely polarized halfmetals noise cross-correlations are positive because crossed Andreev reflection is then the only possible channel of non local transport. Crossed Andreev reflection leads to positive cross-correlations as electron and holes detected at separate electrodes arise from the same Cooper pair in the superconductor. However, for half-metals polarized in the parallel alignment, cross-correlations are negative because elastic cotunneling is the only means of transport. Elastic cotunneling leads to negative cross-correlation since electrons detected at the separate electrodes do not share any information. These results, which display the physics of Pauli Blocking, have an obvious interpretation in the tunneling limit. When the interface transmission is increased, we find that the the overall sign of the noise correlations is unchanged. However, both for the parallel and anti-parallel configuration, the amplitude of noise correlations are much more sensitive to the applied voltage, and the slope (in absolute value) around the vanishing of such correlations is increased. But the most important part of this work concerns the results for normal metal leads, which are more accessible experimentally, but for which the splitting of Cooper pair cannot be achieved by projecting the spin. As a first check, we recover the tunneling limit results30 , where the sign of cross correlations goes from positive to negative by varying the voltages attributed to each lead, and thus favoring cross Andreev processes or electron cotunneling processes. As the transparency is gradually increased, we observe a crossover to positive correlations. This result is especially 12 Va/∆=0.2, T0=0.9, N-S-N structure 1 0 5 10 Sa,b(Vb) 2 -1 (a) Microscopic Green’s functions -2 1 0 5 10 Sa,b(Vb) 2 -1 -1 (b) BTK -2 -1 -0.75 -0.5 α=0 α=0.2 α=0.4 -0.25 0 Vb/∆ 0.25 0.5 0.75 1 α=0.6 α=0.8 α=1 FIG. 8: Noise Sa,b (eVb /∆) of a Na SNb three-terminal structure with normal electrodes Na and Nb , for an interface transparency T = 1 and an interaction parameter α ranging from 0 to 1. We use D0 /∆ = 100 for the parameter D0 in Eq. (25). (a) and (b) correspond to microscopic Green’s functions and to the BTK approach respectively. The distance R between the contacts is such that exp (−R/ξ0 ) = 10−2 , with ξ0 the BCS coherence length at zero energy. robust close to ideal transparency, which constitutes the main result of this paper. This is especially surprising in the light of previous work on the non-local conductance. We conclude that a negative crossed conductance is compatible with positive noise cross-correlations for highly transparent interfaces in Na SNb structures and this is a consequence of the different underlying process which contribute to the noise and in the non local conductance. For the noise cross-correlations, an electron from Nb is converted into a hole in Na with a pair transmitted in the superconductor, as in non local (cross) Andreev reflection. For the non local conductance, a pair of electrons from Nb tunnels through the superconductor into electrode Na effectively in the form of two-electron cotunneling. This means that for transparent interfaces, the negative crossed conductance is not expected to be a signature of entanglement whereas the positive noise crossed correlation signal provides an evidence for the presence of entangled electrons between the two leads. We found it informative to address the issue of interactions, if the normal metal leads are one dimensional. Rather than providing a detailed derivation of the renormalization procedure for this system with two interfaces, we chose a phenomenological approach41 which “only” renormalizes the hopping amplitudes/interaction parameters at the two interfaces, rather than treating the two leads system as a whole. We find that for near ideal interfaces, interactions reduce the amplitude of the positive noise cross correlation signal and can even reverse its sign for strong interactions. For intermediate to low interfaces, upon renormalization the result coincides with the tunneling limit as it should be. 13 0.06 0.03 6.0 z=3.0 0 0.00 -0.03 4.0 -0.1 -0.06 -1.0 -0.5 0.0 0.5 1.0 4 -0.2 10 Sa,b z=1.0 z=0.1 -0.3 2.0 α=0.0 α=0.2 α=0.4 α=0.0 α=0.2 α=0.6 -0.4 0.0 -1.0 -0.5 0.0 Vb/∆ 0.5 -0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Vb/∆ FIG. 9: The same as Fig. 8 but with z = 0.1 (almost transparent) while the right panel is for z = 1.0 (semi-transparent). In the inset of the left panel the plot for the tunneling limit coincides with that of Ref.[30] (linear behavior in between −Va < Vb < Va ). The distance R between the contacts is such that exp (−R/ξ0 ) = 10−2 , with ξ0 the BCS coherence length at zero energy. Granted, these constitute preliminary results, because in particular the renormalization of the transmission phase is not described rigorously by renormalization group equations, as in Refs.[40,42]. However, as far as the current for a single interface is concerned, results from the exact works are reproduced with this phenomenological approach, as seen from Refs. [41,42]. One could expect that the separation between the two normal metal/superconductor interfaces could complicate things (compared to a single interface), but many round trips between the two interfaces are prohibited by our assumption of a large contact separation (compared to the coherence length). Moreover, for single interfaces, the renormalization of the Andreev reflection phase is predicted40 to affect only transport through geometries containing 2 superconductors (Josephson effect). Nevertheless, the renormalization treatment of the full two interface problem would constitute an extension of the present work. Effects associated with one dimensional leads could be experimentally probed when connecting a superconductor to carbon nanotube wires, which could exhibit Luttinger liquid behavior. Experiments with a single nanotube split into two leads by an overlapping superconducting contact have been recently achieved in Ref. 43. A connection between the work of Ref. 44 and the present situation with interacting one dimensional leads can be envisioned, due to the fact that the physics of dynamical Coulomb blockade37 is intimately tied to the physics of tunneling between Luttinger liquids (excitation of “many” bosonic modes during an electron tunneling event). 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