Interface transparency

To study the transport through NM/S [45] or F/S [46] bilayers, the Usadel’s equation is used in the dirty limit. By considering the coherence length ξF in F metal, which is determined by exchange energy Eex, the NM/S bilayer can be easily adapted to the F/S case. In the general situation, the exchange energy is much large than the superconducting gap, and this situation makes ξF virtually independent of temperature.

From the current continuity requirement, the boundary conditions for the anomalous Green’s functions at interface are derived by Kuprianov and Lukichev. [47] The interface transparency parameter

) /

(

B

AR ρ ξ

γ =

is proportional to the interface resistance when superconductor is in the normal state.

The boundary conditions are justified only when the exchange field in the F is much smaller than the Fermi energy. For strong F like Co in our case, appropriate boundary conditions for the Usadel’s equations need to be worked out. [48] Recently, the quasiclassical formalism, or the Eilenberger’s equations, has been employed for the Andreev conductance of NM/S [49] and F/S [50] interfaces. Vodopyanov and Tagirov have derived boundary conditions for strong F case. [50] The quantum mechanical transmission and reflection coefficients for the two spin channels were discussed in the normal and superconducting states. However, the interface transparency was not taken into account.

Perfect transmission coefficient T=1 of the boundary conditions to the Usadel’s equations was assumed in the work of Radovic et al. [4] Lots of experimental works on the F/S junctions in the CIP geometry has applied the theory of Radoic et al. to the explanation of the data. However, more and more reports have pointed out that the

inconsistency between data and calculation could be traced back to the assumption of continuity of the wave functions at the F/S interface. Analyses and procedures for fitting experimental results have to take the finite transparency into account. For example, Aarts et al. were the first to observe the importance and presented experimental evidence of the intrinsically reduced interface transparency in the V/VxFe1-x multilayers. [51] They have explained the non-monotonic behavior in Tc as the competing effects of increasing attenuation depth ξF of the order parameter in the F material and the decreasing transparency of the F/S interface for the penetration of Cooper pairs. Lazar et al. have fitted their results by introducing interface transparency and pointed out its relation to the angular average of the transmission coefficient. [52] Kim et al. [53] have reported the F layer thickness dependence of the Tc behaviors in bilayer F/S structures, determined with CIP resistance measurements.

Quantitative analyses were made form these literatures. For example, the interface resistance at the Ni/Nb and Cu0.4Ni0.6/Nb boundary estimated from the best fit γB

values were 2AR ~ 2.4 fΩ m² for both Ni and Cu0.4Ni0.6. The estimated values are comparable to our CPP measurements with S in the normal state. Experimentally, γB is usually treated as an adjustable parameter to describe and modify the behavior of critical temperature dependence on the thickness for S or F.

We can estimate the interface transparency parameter γB without spin-flip scatterings directly from our results. The characteristic spatial scale is given by

* which corresponds to the actual penetration depth of the Cooper-pairs in the F. While

*

ξF is the Cooper-pairs penetration depth in normal metal without considering the exchange field. Both diffusion constant and ξ^ex of Co were derived to be

5 2/

DF = cm s and ξ_F^ex =1.2nm . [3] These quantities allow us to obtain the following parameters in our F/S CPP multilayers: characteristic spatial length

* 8.1

F nm

ξ ≈ and transparency parameter γ_B≈1.6 for Co/Nb, ξ_F^* ≈8.3nm 1.2γ_B ≈ for Co/Nb0.4Ti0.6, and ξ_F^* ≈9.3nm 1.6γ_B ≈ for Co/Nb0.6Ti0.4 when S is in the normal state. These finite transparency parameters justify the boundary conditions we used to describe the TC dependence on S thicknesses with current parallel to plane by Radovic’s model. [3, 4] Numerical studies also showed insignificant discrepancy of the Tc(dS) behavior when using the boundary condition of high-quantum-mechanical transparency and of finite transparency introduced by Lazar et al. [39] and Tagirov [54] as γ_B is small. For comparison, γ_B =0.7 for Ni/Nb bilayers [53], and

B 0.5

γ = and 1.15 in CoFe/Au and Au/Nb interface, respectively [55], for CoFe/Au/Nb trilayers, were derived by fitting TC(dF) curves. The given values for γ_B depend on the way by which ξ_F^* is extracted from the TC that may be somewhat different in multilayers or in single films. We know that the transmission coefficient for the Cooper pairs in the F/S proximity effect theory is close to the smaller one between the transparency coefficients T_↓ for spin-down and T_↑ for spin-up for quasiparticles to form Cooper pairs. [52, 56] But this is not the only mechanism, since from our study the transparency can be varied by adjusting x as a result of changing compositional disorder or the changing lattice parameter between the Co and NbxTi1-x

interface. The spin-flip scattering is another mechanism which can lead to a large interface resistance. The interface spin-flips physically come from the following mechanisms: (1) inelastic electron scattering in the intermixed level between the magnetic and non-magnetic layers; (2) the direction of magnetization changed locally near the interface; and (3) spin-orbit scattering at the interface induced by the polarization in magnetic layer. The spin triplet symmetry can also be induced in a

superconductor surrounded by ferromagnets with non-collinear magnetizations and spin flip processes. [57, 58] Thus, the value of interface resistance between the ferromagnet and the superconductor both in normal and superconducting states can provide a lot of physical information in F/S heterostructures.