Weak proximity effect

7.1 Weak proximity effect

In this section we reformulate the theory of the proximity effect in the case of a small pairing angle in the normal metalleNI

«

1, and a superconducting angle near its BCS value,

e

s ^rv e_{BC S '} In addition, we concentrate on energies much smaller than the superconducting gap 6. The Usadel equations are greatly simplified in this case, even in non strictly one-dimensional geometries relevant for the experimental situations.

7.1.1 Green functions in the perturbative limit

The pairing angle in the superconductor can be developed around the BCS value as es (E)

=

eB CS (E)

+ a

(E) , with

lal «

1

and the retarded Green function

R

_s ^is

A A ( - sin e_BCS(E) e-i'Pscos e_BCS(E) )

Rs(x,E) ~ ^RBCS(E)

+

^{a (E) ei'Pscose}_BCS^{(E) sine}_{Bc s} ^{(E) .}

(7.1a)

(7.2) In the normal metal the pairing angle is small, so that the retarded Green function

R

N

(formula (5.8)) is parametrized by a single complex function r (x,E), that we call the pairing parameter:

r(x,E) = e(x,E)e-i'P,

Irl«

¹

A ( 1 . ;E) ) .

R^N (x,E) c:::: r (x,-E)

The advanced Green function ANis in this limit

A ( 1 -_r

1

(x,E ) ).

AN(x,E)c:::: -r(x,-E)

7.1.2 Pairing parameters in the weak proximity effect in the one-dimensional case

(7.3) (7.4)

The weak proximity effect regime in an infinite one-dimensional geometry was considered in chapter 5. The value of the pairing parameter on both sides of an interface of tunnel conductance G_{T ,} at zero energy, was given (see Eq. (5.44) and Eq. (5.45a)). The pairing

169

-Chapter 7 Simplified theory of the proximity effect in the limit of small pair correlations

parameters are, to first order in the tunnel conductance G_{T :}

and

e

(E ⁼ 0)=

~

^{_ R (L6.) GT}

s 2

V2'

(7.5)

(7.6) whereR (L_sf)is the resistance of a lengthLsf of the normal wire, and is R (L6.) is the resistance of a lengthL6.⁼ J/iDs

/,6.

of the superconducting wire in its normal state. These expressions clearly indicate the range of validity of the weak proximity effect approximation:

R (Lsf ) GT

«

^{1, and} R (L6.) GT

«

^1.

Equation (7.5) shows how the pairing induced in the normal metal results from the balance between the input through the tunnel barrier of pair correlations, and the loss of coherence in the normal metal.

Since the length L6. is usually shorter than the coherence length Lsf , the modifications induced in the superconductor are often neglected in comparison with those induced in the normal metal.

7.1.3 Expression for the current at an NS tunnel junction

7.1.3.1 General expressions

Given the simplified forms of the Green functions, the quasiparticle, Andreev and Josephson contributions to the current through an NS tunnel junction deduced from Eq. (5.61, 5.63, 5.65) are:

7.1 Weak proximity effect

Because of the high interface resistance, the currents through the structures are small, so that it is justified to take the thermal distributions as occupation factors. In the normal metal one has:

and in the superconductor:

{ JON 'c

~

^(tanh

(:,.:~) +

^tanh

(:,~~»)

{

fos

=

^tanh_2k~T

[is = O.

(7.10)

(7.11)

(7.12)

(7.13) 7.1.3.2 Example of the zero voltage conductances at zero temperature in the one-dimensional infinite system

Given the expressions (7.5) and (7.6) of the pairing parameters, the Andreev conductance and quasiparticle conductance at zero voltage and zero temperature are, in second order in GT :

( ^~~)

_v=o ⁼ y2

^~R(Lsr)G~

( 8I^{qp )} 1 2

8V = j()R (Lt::,.) GT

v=o y2

The expression of the Andreev conductance is of no surprise by now. The expression of the quasiparticle conductance is non-zero, in contrast with the case of a strictly unperturbed superconductor. The quasiparticle conductance is related to the departure of the supercon-ductor from the BCS state at the interface: the density of states Re cos

e

s is no longer zero at low energies.

We now turn to the analysis of the weak proximity effect in more experimentally relevant configurations.

171

-Chapter 7 Simplified theory of the proximity effect in the limit of small pair correlations

7.1.4 Linearized Usadel equations in a normal wire with planar junctions to superconductors.

In the following, we solve the Usadel equation for the pairing parameter to first order in tunnel conductances.

To this order, the spectral current at the barrier, which provides the boundary conditions for the Usadel equation, can be taken as the spectral current between unperturbed electrodes.

Furthermore, we limit the resolution to energies much smaller than the superconducting gap, so that the spectral current is independent of energy, and given by (see Eq. (5.26))

(7.14) where ii is a unit vector normal to the interface and 9T is the barrier tunnel conductance per unit area. Given these approximations, the superconductor provides a constant source of pairing, proportional to the tunnel conductance of the barrier and with the phase factor of the superconductor. The spectral current conservation is readily incorporated as a constant source term in the Usadel equation of the normal metal. We hereafter give the form of the solution for a realistic geometry in which the tunnel junctions between normal metal and superconductor are formed at the overlap of the two metals (see Fig. 7.1).

In the region of the NS junction, the variation of the spectral current in the normal metal is due to the spectral current flowing into the normal metal through the NS tunnel barrier, that is

Here h is the thickness of the normal film. The linearized Usadel equation in a normal metal covered by a superconductor over a finite area is therefore

nu

^{fJ2 r (. It )} ltD 9Te-i<Ps(x)

- - +

1,E-- r = - - - II (x)

2 fJx2 Tsf 2 (J h '

where II(x) = 1 in the region of the junction and 0 elsewhere.

172

-(7.17)

7.1 Weak proximity effect

7' / I I J

/ / /

/

j(x)

N

f----I-H+H-+---"'---/

dx

Fig. 7.1. Sketch of the typical layout of an NS junction: the junction is formed at the overlap of the normal and superconducting electrodes. In the weak proximity effect limit, the superconductor constitutes a constant source of spectral current over the area of the junction.

173

-Chapter 7 Simplified theory of the proximity effect in the limit of small pair correlations

7.2 Solution of the linearized Usadel equation in terms

Dans le document QUASIPARTICLES IN A DIFFUSIVE CONDUCTOR: INTERACTION AND PAIRING (Page 174-179)