Critical temperature and interface resistance of an NS bilayer

In experiments where the strongest proximity effect is desired, one tries to make as good a contact as possible between the normal and superconducting electrodes. The value of the resistance of the interface, which is then usually much smaller than the overall resistance of the structure, is not directly measurable. It is however a crucial piece of information because in this strong proximity regime, the degree of induced pairing depends highly on the transparency of the interface. This information can be derived from the value of the critical temperature of an NS bilayer with the same interface. We compute in the following the dependence of the critical temperature on the interface resistance, in a similar fashion as done in ref. [14] : nearTe ,the superconducting correlations are very small,

lei «

1, but the gap equation (5.12) has a solution for .6. ^=I=- O. The critical temperature is obtained by simplifying both sides of Eq. (5.12) by .6., and imposing

e

^{- t O.} We have used these results to determine the quality of an NS contact in the experiment presented in chapter 6.

Parabolic approximation

In bilayers of films thin enough, the derivatives _g:~ of order higher than two can be neglected.

We suppose that enough non-specular scattering occurs against the interfaces of the films for the electronic motion to be diffusive. Since

\el «

1, Eq. (5.14) can be linearized:

liDEPe

2

^ox2

+

iEe

+

^{.6.(x) = O.}

Here we have neglected spin-flip scattering. The pairing angle in the superconductor at the interface

e

s is then straightforwardly determined:

e

_s ^{= i.6.}_E

[1 _

_{NN +N}

^N

^N

1] .

s ^{1-iE/7 (5.29)}

The quantity

N

_x = Adxnx is the number of states per unit energy around the Fermi energy in the layer X of thickness dx . The coefficient 7-1 = 4n

/i,1!Js

~r;: is the product of the geometrical average of these integrated densities of states by the interface resistance in units of the quantum of resistance RK = ~ ~ 25.8 kO. Injecting the imaginary part of this pairing

115

-Chapter 5 Theoretical description of the proximity effect

e

vacuum N ^vacuum

o ^{d S}

x

Fig. 5.8. Spatial variations of the pairing angle in a NormaljSuperconducting bilayer.

(5.30)

5.4 Application to simple cases

angle into the gap equation Eq. (5.12) yields

1 ^l!iWD

^{dE [ NN}

1]

^E

- - - = - 1-

tanh---nsVeff 0 E NN +Ns ¹

+

E2/^T2 2k BTc'

The first term in the brackets leads to the integral appearing in the BCS gap equation, which determines the critical temperature T co of the bulk superconductor:

1

t:

^{dE E 1.13nwD}

- - - = - tanh ^rv In

-nsVeff ⁰ E 2k BTco - kBTco'

so that Eq. (5.30) can be rewritten as an integral equation forTc :

Tc

l!iWD

^{dE NN 1 E}

I n - = -

tanh---Tco ⁰ E NN +Ns ¹

+

E2/^T2 2kBTc·

(5.31)

(5.32) This result can be compared to the calculation of McMillan [13] who considered Nand S layers separated by a tunnel barrier. In McMillan's approach, the possibility for an electron to tunnel into the other slab translates into an additional contribution to the energies if proportional to the rate at which the electron escapes from one slab to enter into the other:

r = ~

^{ex: VxT}

TX dx '

where T is the probability to cross the barrier and Vx is the Fermi velocity in slab X. Ifthe Fermi velocities in the Nand S metals are comparable, we find that Eq. (5.32) coincides numer-ically with McMillan's result for interface resistances such that _(Rind2~2) > 1/ (100NkBTc) . However, in the case where the Fermi surface parameters of the Nand S layers differ, McMil-lan's simplified approach is incorrect.

We have plotted in Fig. 5.9 the value of the ratio Tc/Tco computed with Eq. (5.32) for the case where NN

=

N s (corresponding to similar Sand N materials of identical thicknesses d), as a function of the parameter kB8 DNRind RK

=

k

Be

Ddn)..~/ (8T) .The interface transmission T is related to the interface conductance through Gint = 2TNchGK , where Nch=A/ P.'F/2)2 is the number of effective channels in the barrier. In the particular case of our experiment on the proximity effect (see article reprinted in the next chapter), the critical temperature of an NS sandwich of 25 nm of Cu deposited over 20 nm of Al was inferior to 20 mK. This corresponds to a Tc/Tco ratio inferior to 0.02, yielding an interface transmission T greater than 0.01, or an interface conductance larger than 8 S for a surface of 100 nm x 100 nm.

In the following, we present analytical solutions of Eq. (5.32) in limiting cases.

117

-Chapter 5 Theoretical description of the proximity effect

1

0.1

TCiT Co

0.01

1E-3

0.01 0.1

1

¹⁰ 100 1000

Fig. 5.9. Ratio of the critical temperature of an NS sandwich over the critical temperature of the bulk superconductor, plotted versus a function of the transmission T of the NS contact. We have used the Debye temperature of Al

e

^D =^394K.

5.4 Application to simple cases

Critical temperature of a NS bilayer with a transparent interface For large values of T, Eq. (5.30) can be approximated by

~

^{[ 1 /}

2]~

- =

exp (-

v.: ) V

1

+ itu»

D) /T2

Teo ns eff

yielding for a perfect interface, i.^e. for ^{T rv} 00

if^T

»

kBTe , (5.33)

(5.34) 1

k^BTe=1.13fuuDex p - nSVeffJVS^IT H/(NN+JVS~r)'

as found by de Gennes [5]. In the limit of a perfect interface, the critical temperature of the bilayer is related in the usual way to the Debye frequency and the effective interaction, but this interaction is renormalized by a factor

N

_{s /}

N.

The main prediction is that the critical temperature is exponentially suppressed as the normal metal thickness increases.

Critical temperature of a NS bilayer with an opaque interface In the case of high tunnel resistances, the critical temperature is given by

Te

=

[exp ( _ ^{Jr )]} if^T

~

^kBTe,

Teo 8GKNsRintkBTe

thus the critical temperature is in this case only weakly modified, by a factor which is inde-pendent of the normal metal layer.

Dans le document QUASIPARTICLES IN A DIFFUSIVE CONDUCTOR: INTERACTION AND PAIRING (Page 120-124)