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BOUNDARY CONDITIONS FOR THE USADEL EQUATIONS AND PROPERTIES OF DIRTY S-N-S SANDWICHES

1978; EDP Sciences; Volume: 39; Issue: C6 Linguagem: Inglês

10.1051/jphyscol

2777-3418

Z. G. Ivanov, M. Yu. Kupriyanov, K. K. Likharev, О.В. Снигирев,

Rare-earth and actinide compounds

Boundary conditions f o r the Usade equatsons a t a plane wall conducting interface are it found to cons is t i n continuity of F and ND ( iZeA/$)F. These conditions are used t o f ind the effec t ive boundary conditions fo r the Ginzburg-Landau equations a t the S-N interface, as well as the current-phase relat ionship fo r S-N-S sandwiches with a short electron mean f r ee path. INTRODUCTION.The Usadel equations/ l / give one a convenient tool f o r calculat ing the dc properties of superconducting s t ruc tures i n the d i r t y l i m i t . However, it was s t i l l unclear which boundary cond i t ions a r e to be used fo r these equations i n the analysis of the propert ies of S-N and S-N-S structures. In t h i s paper the proper boundary conditions as well as the r e su l t s of t he i r applicat ion t o two basic problems are reported. BOUNDARY CONDITIONS.S tar t ing from the Eilenberger equations/2/ one can show tha t within the d i r t y l i m i t these are the Usadel functions F(r,w) which a r e t o be continuous a t the plane(*) in ter face between two metals i n the case of small e lec t ron ref l ec t i on from the boundary. Taking i n t o account the conservation of current , we f ind tha t the product -t ND(9 iZeG/Hc)F must a l so be continuous. Here N is the Fermi surface density of s t a t e s , i s the vector potent ia l , D = v R/3 i s the diffusion coefF f i c i e n t , and R i s the electron mean f r ee path which i s assumed to be much l e s s than the charac ter i s t ic coherence length 5 = (HD12rkT) *'. The Usadel equations, together with the boundary conditions given above, are va l id a t a rb i t ra ry temperatures. However, i n t h i s work we r e s t r i c t ourselves to t h e i r applicat ion t o the s i tua t ions where the normal metal with a zero c r i t i c a l tempera ture contacts the superconductor with T % T. The Ginzburg-Landau (GL) equations can be used i n t h i s case only a t distances much la rger than 5 from the interface, and a more general theory i s t o be used to obtain the ef fec t ive boundary conditions fo r the -t GL order parameter A(r) 131. Using the Usadel equat ions we can get these conditions i n a much eas ier way than with the help of more complex in tegra l equations discussed by Zaitsev/4/. S-N INTERFACE.I f the thickness d of the normal layer i s large (d>>S ), the supercurrent through N the boundary is absent, A can be taken to be rea l , and the ef fec t ive boundary condition f o r the GL equations has the form/3/ %A/A = b-' (1 The parameter b appears to be dependent on the following r e l a t i on Y = N ~ D ~ / N ~ D ~ * ~ = (uN/cN) / (oS/cS) (2) A t y /SS> = ( ~ / ~ ) ~ ~ / C T ~ T ~ ** (4) can be of the order of unity. However, i f y i s about or more than unity, the r e su l t (3) i s wrong. A t y>>l we get the other simple expression S(T)/bll = n ~ o ( 2 n + l ~ 2 / n=O P (2n+l)-& 2 1.117 (5) (*)The products NF and D(# i 2 e b i c ) ~ are t o be continuous a t the in ter face with the d i f fuse electron sca t te r ing . but f o r y % 1 the Usadel equations are t o be solved numerically. Figure I shows the r e su l t of the calculation. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786250 Fig. 1 : Dependence of the e f f e c t i v e length b ( see i n s e r t ) , determining the e f f e c t i v e boundary condit i o n ( 1 ) f o r the Ginzburg-Landau equat ion, on the parameter y (2) f o r t h e plane S-N in te r face . S-N-S SANDWICH.Analysing the Josephson e f f e c t i n d i r t y (R >maxp , c N / q ) t h e order parameter a t the boundaries of the i n t e r l a y e r is reduced because of the proximity e f f e c t . I f the thickness d is much l a r g e r than EN, t h i s suppression of I A ~ is independent of $ and the Is($) re la t ionsh ip i s s inusoidal . On the contrary, a t d/EN < 1 the superconductors feel the phase of each o ther through the i n t e r l a y e r , so the suppression is s t ronges t a t $T and weakest a t $ 0. Since Is is proport ional t o I A ~ ' , t he dependence I (4) deviates S from the s inuso ida l form and has a maximum a t $<.rr/2 d Is/IO = 9-P s in$ 25J2 p2cos2 ($;2)+q2sin2 ($12) Ic 'L (TC-T) ' (8) I f the i n t e r l a y e r is t h i n (d/<N<<I',I'-l), the supercurrent through the sandwich can be s t rong enough t o cause the depair ing e f f e c t i n the electrodes. The corresponding c r i t i c a l cur ren t is a l s o achieved when $<n/2. Figure 2 shows the t r a n s i t i o n from the VTB l i m i t (7) t o the depair ing l i m i t (9) with decreasing sandwich thickness a t the f ixed value of F. These