ANALYSIS OF PROXIMITY EFFECTS BETWEEN SPIN SINGLET AND SPIN TRIPLET SUPERCONDUCTORS

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ANALYSIS OF PROXIMITY EFFECTS BETWEEN

SPIN SINGLET AND SPIN TRIPLET

SUPERCONDUCTORS

K. Scharnberg, D. Fay, N. Schopohl

To cite this version:

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JOURNAL DE PHYSIQUE Cofloque C6, supplkment au

no

8, Tome 39, aolit 1978, page C6-481

ANALYSIS OF PROXIMITY EFFECTS BETWEEN S P I N SINGLET AND S P I N T R I P L E T SUPERCONDUCTORS K. Scharnberg, D. Fay and N. Schopohl

AbteiZung fiir Theoretische Festkbrperphysik, Universitlit Hamburg, Jungiusstrasse 11, 0-2000 HAMBURG 36, Germany

Rdsum6.- Sur la base du modSle de McMillan de l'effet de proximitd supraconducteur, il est montrb pour que l'existence possible d'une interaction btat-p attractif conduisant B une supraconductivitd d'dtat-p B basse tempdrature, ne peut pas 8tre testde expdrimentalement en utilisant l'effet de pro- ximit6.

Abstract.- Using McMillan's tunneling model of the superconducting proximity effect we show that the possible existence of an attractive p-wave interaction, leading to p-state superconductivity at some low temperature, cannot be tested experimentally using the proximity effect.

It has been suggested that certain metals /l/, which show no superconducting transition even at the

lowest temperatures presently attainable, may in fact be p-state superconductors.

Here we discuss some problems involved in in- vestigating the existence of p-state superconductivity by means of the proximity effect. Our discus- sion is based on McMillan's widely used /2/ tunneling model 131. It is very helpful1 to note that this model is formally identical to a special case (no interband coupling, no intraband scattering) of the two-band model of superconductivity /4/, when the effect of nonmagnetic impurities is included /5/.

Starting from the pairing Hamiltonian

(1)

and including as perturbation the tunneling Hamiltonian

where the subscripts

R

and r refer to the left and right hand side of the sandwich, it is straightfor- ward to derive the equation of motion

stand for

R

or r.

In order to obtain an explicit expression for 2(g,w ) we need to make an ansatz for the order pa-

1 n

rameter. Among the various possible expressions for

-+

.ii(k) in the presence of spin triplet pairing we choose that of Balian and Werthamer 171. In the absence of strong coupling effects this has been shown to be the favoured state /7/. However, since the proximity of an S-state superconductor induces S-wave pairing in any normal metal, we shall consi- der mixed S- and p-state pairing right from the

beginning.

This leads us to the following ansatz / 7 , 8 / :

..

-+

(g)

= A (k) T3T1 +

A

(k) ikr T2 and

R

sR PP,

-

-+

k = k/k is a unit vector in the direction of quasi- particle momentum, and Ask,

ApR

an:

Asr

are real scalar functions of k =

121 .

Since

At($)

is

not

pro-

-0 +

portional to a unitary matrix, obtaining GR(k,wn)is somewhat tedious, Omitting the subscript R we find the result (w?c2+A2+~i) ( ~ w ~ P ~ T ~ + E P , T ~ )

-

A~(w>E~+&:-A~)

p2T2

+

I U ~ E ~ + ( A ~ + A

)2}{~2~2+(~s-~

-+ -+ 2 + + P P

1

T . .(q,k)p3ro G.(k,q1,wn) = 6 ~ 1 3 ( 3 ) -+ lJ k A - + ( w ~ + E ~ + A ~ - A ~ ) ~ ~ T , n + 2A (iw p T + E P ~ T ~ ) + A k-a S n o 0 a + +

for the 4 X 4 matrix Green function G(q,ql,w ) in- P

n { w ~ E ~ + ( A ~ + A ~ ) P 21{w>~2+(~2-~2) S P

'1

troduced by Maki 161. p" and rv are Pauli matrices ₍₅₎ operating on electron-hole states and ordinary spin

-+

states respectively 161. The subscripts i and j where we have defined cr = (p3r1, pOr2, p 3 ~ 3 ) / 6 / .

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We rewrite (3) in the form of a Dyson equation and introduce the same approximations for the selfenergy as McMillan /3/. We also take the tunne-

+ +

ling matrix elements T (k,kY) to be constant sothat Erz

with respect to momenta Gi is diagonal and the selfenergy is constant. We then find

These two matrix eauations reduce to four scalar e- quations for E: and

,

:

X

and the full Green function

11

-0

is then obtained from G (5) by the replacement

G i % F

a

+

:i

= an + "(W.,) and

Asi

+

Asi

=

Psi

+ Ii(wn).

A is not affected. Introducing the quantities

PR

we can reduce the four selfenergy equations tothree. Inserting the full anomalous Green function into the

selfconsistency equation we obtain :

Apt

= AdRNR(0)V

PE du tanh

g

Re

where V and V are the S-and p-wave components of the pairing interaction V(k,kY,cosOkk,) and where the analytically continued functions u(w) are to be determined from

UrAsr

l

The upper (lower) sign has to be used, when

'L

A -AsR(un) > 0 (< 0). A, di, Ni(0), and T~ are PR

defined in McMillanYs paper /3/.

PR

E

= -U:, SO that this For V = 0 we have U-

system of equations is reduced to the one obtained by McMillan / 3 / . It is identical to the system of selfconsistency equations found for impure two-band superconductors 151.

If we assume all three order parameters to tend to zero at the same temperature we find two coupled equations, involving A and A from which

S

R

sr'

the TcNS of the sandwich is determined, and a sepa- rate equation determining the critical temperature

T for the formation of p-wave pairs. This means CP

that, as long as T cp < TcNS, there is no effect at all of the attractive p-wave interaction on the observable superconducting transition temperature T c ~ ~ ' When T c ~ > TcNS, i.e. a p-state superconductor in contact with a normal metal, we find a reduction of T which is given by an expression identical

=P

with that describing the reduction of the transition temperature of an S-state superconductor due to ma- gnetic impurities / 6 / , or of a p-state superconductor due to nonmagnetic impurities /g/.

T in the presence of a S-gap in the sand- CP

wich can be obtained from

with

Although the solution of this equation requires numerical calculation, it seems to be clear that T is reduced in the presence of the S-state con-

CP

densate. But contrary to the case of mixed S- and p-state superconductivity in clean bulk materials

/ 7 / , it is not obvious to what extent the S-state

superconductivity induced in the left hand side of the sandwich suppresses the formation of p-wave pairs.

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References

/l/ Appel, J. and Heyszenau, H., Phys. Rev.

188

(1969) 755 Foulkes, I.F. and Gyorffy, B.L., Phys. Rev. B 15 (1977) 1395 Fay, D. and Appel, J., Phys. Rev. B

16

(1977) 2325

/2/ Deutscher, G. and de Gennes, P.G., in "Superconductivity" (Edited by Parks, R.D., Marcel Dekker, New York) 1969, p. 1005

Gilabert, A., Ann. Phys. (Paris)

2

(1977) 203

Toplicar, J.R. and Finnemore, D.K., Phys. Rev. B

16

(1977) 2072

/3/ McMillan, W.L., Phys. Rev.

175

(1968) 537

/ 4 / Suhl, H., Matthias, B.T. and Walker, L.R., Phys. Rev. Lett. 3 (1959) 552

-

151 Schopohl, N. and Scharnberg, K., Solid State Comun.

22

(1977) 371

161 Maki, K., in "Superconductivity'' (Edited by Parks R.D., Marcel Dekker, New York) 1969, p. 1035

/7/ Balian, R. and Werthamer, N.R., Phys. Rev.

131

(1963) 1553 /8/ Leggett, A.J., Rev. Mod. Phys.

5

(1975) 331