Usadel equations

Solutions of the Usadel equations in hybrid systems with F and S layers show that triplet components of / are generated near the F/S boundaries and decay into both the ferromagnetic and superconducting materials. When the boundaries are sufficiently close to each other, the triplet component propagates over the whole system and can lead to new interesting effects. 

The proximity effect in diffusive metals can be described by the quasiclassi-cal theory based on the Usadel equations [5, 6, 7, 8]. In the usual 9 para-metrization, the complex angle 6(e,x) is related to the pair amplitude as F(e,x) = —ism0(e,x). The local density of states is expressed as ... 

In order to find the Green s function g, we consider the diffusive case when the Usadel equation is applicable. This equation can be used provided the condition Jr momentum relaxation time). Of course, this condition can hardly be satisfied for strong ferromagnets like Fe and in this case one should use a more general Eilenberger equation for a quantitative... 

The Usadel equation is a nonlinear equation for the 4x4 matrix Green s function [15]. The Usadel equation is complemented by the boundary conditions at the S/F interface [12]... 

The Usadel equation can be solved in some limiting cases [15]. In Fig. 2 we present the spatial dependence of the condensate function (singlet and triplet). One can see that the SC penetrates the F layer over a short distance of the order whereas the TC penetrates over a long distance = y/Df/2itT. The amplitude of the long-range part of the TC has a maximum at a = 7r/4. 

Using the solution of the Usadel equation [15], one obtains after simple transformations Is = Ic sin p, where... 

We studied odd, s-wave, triplet superconductivity that may arise in S/F multilayered structures with a non-collinear orientation of magnetizations. It was assumed that the orientation of the magnetization is not affected by the superconductivity (e.g. the energy of the magnetic anisotropy is much larger than the superconducting energy). The analysis was carried out in the dirty limit (Jr -C 1) when the Usadel equation is applicable. 

It would be interesting to carry out experiments on S/F structures with non-collinear magnetization in order to observe this new type of superconductivity. As follows from a semiquantitative analysis, the best conditions to observe the Josephson critical current caused by the TC are high interface transparency (small y ) and low temperatures. These conditions are a bit beyond our quantitative study. Nevertheless, all qualitative features predicted here (angle dependence, etc) should remain in a general case when one has to deal with the non-linear Usadel equation. 

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