Electrodynamical properties of superconducting contacts

Electrodynamical properties of superconducting contacts

I. Phys. Chem. Solids Pergamon Press 1969. Vol. 30, pp. 509-520. ELECTRODYNAMICAL SUPERCONDUCTING Printed in Great Britain. PROPERTIES OF CONTACT...

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.I. Phys. Chem. Solids

Pergamon Press 1969. Vol. 30, pp. 509-520.

ELECTRODYNAMICAL SUPERCONDUCTING

Printed in Great Britain.

PROPERTIES OF CONTACTS*

G. DEUTSCHER, J. P. HURAULT* and P. A. van DALENt Laboratoire de Physique des Solides*, Fact&C des Sciences, 91-Orsay, France (Received

26 February

1968: in revisedform

19 July 1968)

Abstract-We have studied, expe~mentalfy and theoretically, the linear response to a small (d.c. or a.c.) magnetic field in the normal part of a normal-superconducting N/S binary layer. It has been shown that the tunneling characteristic of an N[S sandwich, measured on the N side, is sensitive to the values of the pair potential at points far away from the free surface of the N material. Similar nonlocai effects also appear in the electromagnetic response. The greater the temperature compared to the critical temperature of N, the bigger these effects. Some experimental confirmations of these theoretical predictions have been obtained. The theory also explains: (I) why a Meissner effect has not been observed in Cu for the Pb/Cu systems, down to 1°K; (2) why, on the other hand, the r.f. absorption is much smaller in Cu coated with Pb than in normal Cu. INTRODUCTION

WE CONSIDER a metal N backed by a superconductor S. Both materials are supposed to be ‘dirty’ and we assume the electrical contact between N and S to be good. At the temperature of the experiment, N is normal (T > TcN, TcN being the critical temperature of N) and S is superconducting (T < T,). In the normal metal, the pair potential decreases exponentially as e+*, where K is given by [ 1I$.

current j and the applied vector potential will have the local form: j(r,w)

= -Q(r,o)

A(r,o).

A

(1’)

Since A is small in N, we can perform expansions in powers of A for the evaluation of Q (r,w). A being solution, in N, of the equation

(2) we are led to a situation reminiscent of the gapless regime of ordinary dirty type II superconductors, and we might naively think that, as in this last case, Q(r,w) will depend only on the value of A at the point r of interest [2], But this is not true: A is not everywhere a solution of equation (2), due to the presence of the superconductor. A more careful treatment is needed then. As a result, we shall see that Q(r,w) depends on the values of A not only around r, but also in the superconducting metal S. We shall first treat, in Section 1, the zero frequency case (o = 0) and then in Section 2, the slightly more complicated case w # 0. In Section 3 we present some experi-

D being the diffusion coefficient in N: D = QvFl t+?being the Fermi velocity and I the mean free path. We apply a small d.c. or a.c. magnetic field and we want to evaluate the response of the system to this perturbation. For a ‘dirty’ system, the relation between the- resulting *Partially supported by the Centre National d’Etudes Spatiales. ton leave of absence from Philips Research Labora_ tories, Eindhoven, Netherlands. SLaboratdire associt au C.N.R.S. Qln the following, we adopt a system where h = k, = 1. 509

510

G. DEUTSCHER,

J. P. HURAULT

mental results obtained with very low frequencies, and corresponding practically to w = 0. SECTION

1: THE STATIC

RESPONSE

FUNCTION

We make the following approximations (to be discussed at the end of this section): (1) The pair potential A is small everywhere, even in S (T 5 T,,) . (2) Both materials are supposed to have the same density of states at the Fermi level, and also the same diffusion coefficient D. (3) Lastly, we assume a simplified form for A(r) shown on Fig. 1: if Ox is an axis perpendicular to the interface, we take

and P. A. VAN

DALEN

normal metal and the line drawn over the integrand is associated with an average over the impurity configurations. In order to do the calculation, it is convenient to go to the Fourier representation first without taking account of the presence of the field. So equation (4) becomes, without a magnetic field [5]: j(r)

= -$&

A(x < 0) = const = ABC-(T)

and, for x > 0, using relation (1) A(x) = tA.,,(T)

exp(-Kx)

(2’)

where ABcs( T) is the equilibrium value of A in a bulk S material at temperature T. a = V,/V,, where V, and V, are the respective values, in S and N, of the BCS coupling constant [3]. The supercurrent resulting from the application of a small vector potential A is most easily evaluated using Green function techniques [41. j(r)

= ;T

x

(V,-V.)G,(r,r’)

-$A(r).

0

(3)

The notations are standard. w = (2n + l)nT, n being an integer, N the electronic density. Since we have assumed small values of A everywhere, we may expand G,(r,r’) in powers of A and evaluate j to the lowest order of A. So j(r) =kT

x CV+-V,)

p-+pr--

2eA c

the + sign depending upon whether it operates on A or A+. We are interested only in the lowest order term in A. Thus we neglect terms in A* and, to this order, IA(r)] is not changed by the field. Furthermore, we may choose a gauge where A is real. Finally we get j(r)

Ne2r A = -mcz a

g$$uP) ?lr-0

and since A depends on x only

0

XI G,“(r,l)A(l)G,o(m.r’) x A+(m)G,O(m,l) d3md31.

where r is the collision time, and A(p) the Fourier transform of A(r) . It has been shown by Maki[5] that the inclusion of field effects could be realized by the replacement

j(x)

=

-E$.$

(4) x2

Here G,” is the Green function of the pure

PI=0

ELECTRODYNAMICAL

PROPERTIES

We know that +m

+m

I

-m

f(q)g(q)

eeiqxg

= j

F(x’)G(x-x’)dx’

--m

where F and G are the respective Fourier transforms off and g. Thus we can write

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511

tration depth will be sensitive to the values of the pair potential not only in r, but also far from r. This is particularly true in our case where A(x.), for x’ < x, is an exponentially increasing function of Ix’ --xl. In fact, if we assume the following variation for A

e-ipx

A(x’)

=

A(x)

e-x(s’-x)

(11)

DP"

n+s+m = 1

(which is reasonable for x sufficiently large), we obtain the usual formula for h 161:

A(x’)~~(~-x’)

dx’

(7)

J--m

-= 1 h2(X)

with

+$AQ)

x

4 (4n2 _

K2)2

4PO

or - 1 q(n)

=

d(

(n+t)(4rrT/D));

(n 2 0).

Then, the final expression for j (x) becomes 47rTu

j(x) = -F;A

d2A dx2=

1 h”(x)A

$?$TsA2(x)

CC29 -._._A-+

eM-c7~)x

nrO 4n2-K2

E+ - 1 qn K-q,

(

2-!- (13) 11 qnZ

where a = V,/V,. Provided that K, as yielded by (I), be always smaller that the smallest of the qR, i.e. go, the second term in the bracket of equation (I 3) will become negligible as one gets very far from the interface. In fact, we can consider two regions: 1 -n P -: 90--K -X<-’

90 These

formula ( 12) is valid

1

fIo)

We see on formula (IO) that the local pene-

(12)

But, when we get closer to the interface, we must take account of the detailed shape of A, as depicted on Fig. 1, and some contributions to h will come from the superconducting region. In fact, if we take account, in (10). of relation (2’), we obtain --_1 X2(x) . ,

where CTis the normal state conductivity of the material. If we are now interested in the penetration law of the magnetic field, it is convenient for us to introduce a local penetration depth h(x). Since curl H = (4711c)j (assuming A to have nonzero components along the y axis, which allows us to choose a real value for A), we get

$!xi+(f_~),

=

A2(x)

_ K- formula (13) must be used.

conclusions

are similar to those of

G. DEUTSCHER,

512

J. P. HURAULT

Mauro in his study of tunneling effect [7]. Our corrections to formula (12) will be important when the region where it is significant is large. This will be the case if T is much larger than the critical temperature of the bulk normal sample Tcw On the contrary, when T is of order of TcN, it is correct to use formula (12) as we have already done in preceding papers [6,8]. The case T 9 TcN (or, more generally, IV,] smalI)* is of particular interest as it is associated with the study of Meissner effect in some systems like Pb/Cu, and with the determination of the transition temperature, if it exists, of Cu or other noble metals. So let us now study the case 1V,l small. IV,1 small

As K + q,,, the leading term in the summation of equation (13) is the term n = 0. If we consider now the region in the proximity of the interface (x < l/ ]K - q,,l), we can practically write 1679’ o1 = .,TC2A2(x) h*(X)

&_E+-

nr,, qn*[ qn

1 2 qn+K I

(14)

or

and P. A. VAN DALEN

To evaluate the summation over n, we shall make the following approximation: we replace F, by FL such that Fh=Fq

y and K’* being determined in such a way that F. = FA and F, = F;. In this way we find y = (Y+o.7 K’* -- 0.2

g--

C F, n

(Y

The sum over the FL* is then easily expressed in terms of digamma functions. Finally, for x < l/ IK - q,,l &=-&(~+05)zJI’(t)A2(x).

(15)

We can make the following comments about this last formula: (1). A crucial parameter is the value of the local penetration depth at the interface (A(x = 0)). It has been shown in [6] that the screening properties in N depended mainly on the value, at the interface, of the Landau-Ginzburg like parameter defined as K(0)

1 = sA*(r) X2(x)

*IK,* n

=

$$ .

(14’)

If K(O) s 1, screening currents are effective in N. If K(O) > 1, the magnetic field penetrates freely into N up to the interface. _At a given temperature, since aA(0) = A&T), K(O) is proportional to K( 1 - 1/2a). If we consider a series of materials N, the coupling *When N cannot become superconducting at any constant of which increases from a small temperature, i.e. if for instance the BCS coupling constant VN is small and negative, then,K is greater than q,, negative value to a small positive value, I/a and lower than q, and the solution of the linearized gap decreases and K increases. Thus the Landauequation yields Ginzburg parameter K(O) increases continuously when the material becomes more 1 _-_zLy>@ repulsive. Nevertheless, provided that we are K--q, qo at low enough temperatures, K(O) may be Then 1I (K - qo) s l/q, and we can perform the same lower than 1 and Meissner effect can be analysis as when V, is small and positive: the only difference comes from the fact that a changes its sign. observed in such a case. Furthermore, if the

with

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513

Finally, we would like to discuss the various N layer has a thickness of order K-l and if approximations which have been used in these s 0.4, one can expect the existence of a thermodynamical critical field relative to N, calculations. (1). We have supposed A to be the breakdown field Hb[6, 91, even if N is a small everywhere, which limits the validity of repulsive material. (2). For a given N material, our estimations to the region T > T,, and our with a critical temperature TcN, and for a set of results cannot be used for very low temperavalues of T such that T > 2T,, and T > T:, tures. This is particularly true for formula ( 15). h(0) goes d/T, which is quite different from When )V,( is small, (15) predicts that h(0) + 0 the temperature dependence predicted by [ 121. when. T + 0. In fact, in this case, Ah,must go (3). The ratio of h(O), as given by (15>, to its to /3X, when T + 0, As being the penetration value given by (12), is depth in the bulk superconductor. p is a coefficient which takes account of (a) the $‘(+ DK2/4rT) eventual lowering of A and S near the S/N 2 interface; (b) the fact that the contribution to ff+o*s J( j(0) coming from N is very small. Thus, Taking account of the fact that V, is small, and A(0) 3 2As when T -+ 0. (2). We have remembering (l), this ratio is roughly equal to assumed the same N(0) and the same D in lIN(0)Vs. Thus K(O), as given by (15), is both materials. In fact, we could have used a larger than its former determination (12). For correlation function method, as developped in a Pb/Cu system, for instance, (12) predicted [3] and [7], to evaluate j(r). It can then be l”K, in the most favourable K-1 for Tshown that j”(u), as defined by (7), obeys case. This meant that a very weak but detectdiffusion type and sum rule equations. .&(u) able Meissner effect could be expected. On can easily be calculated when DS # DN and the other hand, the improved calculation ( 15) when NN(0) # N,(O). The same result for predicts that K(O) will be rather about 2.5, A(x) is found for x = 0 and practically the which explains why no screening properties same for x > 0. So, even if the diffusion cohave been observed in such systems!. (4). efficients and the densities at the Fermi level are different in the two materials, (13) remains valid. (3). The last approximation was to assume a spatial variation for A as drawn on Fig. I. Such an approximation will remain (16) valid as long as the extrapolation length as The only dependence of K(O) on (Y will be defined in [3] (Fig. 2) is greater than 1.5 t(T), t(T) being the coherence length in S*. The through the term 1 -O.~/(Y. Thus a determination of N (0) V, for weakly attractive or re- only modification for our formulae will be to replace ABCS(T) by the true value A(O-) of the pulsive systems will be possible if a precision better than 0*5/a in the measurements is reached. Practically this represents a precision better than 10 per cent. K(O)

pLe-Kx *T, is the temperature below which the expansion in powers of A is no more valid in S. From reference [lo]. T, is such that (ABcs(T,))/T, - 3. tin the case of a Pb/Cu system, when T = i”K, T is lower than T, and we are not in the theoretical range of validity of (15). Nevertheless, as we shall see in Section 4. the Y’T law for h(0) seems to be valid down to 1°K for the PblCd system.

;

Fig. 1. Schematic representation of the pair potential inside the binary layer used in the calculation of j (x, 0). *This last condition is fulfilled in N is sensibly dirtier than S(DN + 0~).

514

G. DEUTSCHER,

J. P. HURAULT

pair potential in S, for x = 0. (Fig. 2(a)). If this is not the case, the variation of A in S can be approximated by a straight line (Fig. 2(b). This implies a slight modification of the F, in (14) and of the coefficients y and K’* which appear in the determination of the Fk (14’). exact.shape

forA

and P. A. VAN

DALEN

This expression being symmetrical in V, we can choose v > 0. We have to consider the same integrals as in (6). Thus, as before, we sum expressions as the ones concerned in formula (13). Provided that the distance x from the interface is less than l/jq,--KI, it is sufficient to evaluate I (x, oO) for x = 0. The value of Z(x, oO) for x > 0 is obtained by replacing A (0) + A (0) eeKz.

(17’)

Then Z (0, w,,) has the form 0

X

Fig. 2. Possible schematic shapes of the pair potential to be used in the calculation of j(x.0). (2~): D, < Ds; (26): D, > D.F.

c F:+,+c

Z(O,o,) =%[2

n-

0 v-1 +

j(r) = - Q(r, o) A(r, w) = :

FnFv-n+i WL]

0

SECTION 2: THE ELECTROMAGNETIC RESPONSE AT FINITE FREQUENCIES

We make the same approximation as previously and are only interested in the case T % TcN. We write

C

F,F,,, 0

(18)

n

the F, being defined as in (14’). As before, we replace F, by F,. The summation in (18) can be expressed using polygamma functions and we get

[- io f 4?TT [email protected],

011A(r,WI.

I (r, o) can be obtained through an analytic continuation of the quantity Z(r,oo), where w0 = 2~ rT, v being an integer[4,2]. I(r,o,,) can be expressed in terms of the Green functions G, and F,. Assuming small values of A we can expand in powers of A. We go over the Fourier representation and perform, as previously, the averaging procedure on the impurity sites. This has already been done by Maki [ 111. The result is

Z(O,wo) = *(~)z(CY+o.7).[,++~+~)

+

(19)

The analytic

continuation

of .Z(0, wo) can

ELECTRODYNAMICAL

PROPERTIES

be made by replacing o. -+ -iw. So finally:

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pro~~ional to AZ, we will reprace by an average value SQ such that

(24)

Formula (20) will be valid if Aecsf2~T e 1, which restricts the domain of temperature to the vicinity of the critical temperatue of S. (20) can be expanded in the three following cases: (a)o/2rT < l/So. Then, provided that ff*,l

Q(o,o,=-iwj[l+5a(A~)2(l+~)2 x,~(~)]+*~~(I+~~~~(f)

(21)

A,(x) being the vector potential in the system for T P TON,.+ In order of magnitude, SQ(x,o) is proportional to A&- (c~~K(x-~~)~c~~K~~) in N and to A&, in S. We now suppose that 2Kg s 1, S being the skin depth in N (practically, this condition means that we are not in the Pippard, but in the London limit). Formula (24) shows that the contribution to 62 coming from N, 6eN, is essentially due to the points in the vicinity of the interface: thus, in view of the calculation of the surface impendance relative to N, we can adopt the formula 6Q(x, w) = SQ(0, o)eW=

(b)&

&e

(25)

1.

(c)

x

(1+!E2> a



(23)

We are interested in the surface impedance 2 relative to the normal side, the thickness of which being & We have to solve the Maxwelt equations in N with a spatially varying Q(x, w), as defined by (17’) and (20). If SQ(x,o) is the part of Q&o) which is

everywhere in N, even for large values of x(-r > l/JK-%I). Then, two cases are possible. (1). D, D, * SQN is proportional to A&-s/2&S emdN’*,as soon as & > K-r, and SQS, the contribution to sQ coming from S, is proportional to A&s eVxt8. Thus, according to our hypothesis (2Kfi 9 l), it is essentially the S region which contributes to 83 and this remains true for arbitrarily large values of r&. We must note that the form we have adopted for the pair potential is not correct for DN &. Instead of being constant in S, A goes as aA(O’)( I- Kx) in S, and, in N, A = A(O+) emKk.A(O+) is determined using the results of reference [3]. This leads us to change the values of the F, which appear in (18). Thus

1

(26)

G. DEUTSCHER,

516

F;=D

2?rT

J. P. HURAULT

a-l-2-2 0.57 qi?+&--4

-

G-59

But, concerning SQ, our last result remains valid and

(28) (2). DN + 0. When DN 4 0, the radiofrequency field penetrates very weakly into S, SQs goes to zero and, if dN s 6

0) [email protected] + SQ(0, (3KdJ2



(29)

and P. A. VAN DALEN

surface impedance, Meissner effect, the dependence on A will mainly concern A in S, and not A in N. Thus the determination of N(O)V,, using such experiments, will be very difficult. Further, we expect formally identical conclusions for the ultrasonic attenuation, heat conductivity or the nuclear spin relaxation time 7,. A local dependence on A (in fact the value of A at the free surface of IV) could be reached using surface impedance measurements in the anomalous regime 6 < i, 1 being the mean free path and 2K6 < 1. Nevertheless, for this last condition to be fulfilled, one would have to use very high frequencies ZJ(V - lOI3 Hz). Finally, let us quote the results obtained for Z in the limit D.y -+ 0 and dx 5 8.

Between these two extreme cases DN & and DN --, 0, we must use in (24) the exact Z== R-i-ix. form of&(x). (30) When the ratio DslDN increases, 6QN18Qs If R,, and X,, are the values of R and X also increases but high values of Ds/DN for T 2 TcNs and if we only consider the must be reached for SQN to be comparable cases of practical interest w + 2?rT, we get tosQ* But the main result is the following: 83 is always proportional to Az(O-), and not to A2(O+). This can explain qualitatively the results obtained by Fischer and Klein in a Pb/Cu system[l2]. These authors .observe, in the case DN 4 I&, an important reduction of the real part of 2, compared with the (31) situation when the induced superconductivity in Cu has been destroyed by a magnetic field. (The thickness of the Cu layer is comparable to the skin depth in the experiment.) We also note that in their case 6QN - SQ$_ (32) But we stop the discussion here since the lead, in [ 121, is very clean and, in S, the electromagnetic wave obeys the relations of the anomalous skin depth regime [ 131. Thus, the correction to the surface imped~zee due to the induced superconductivity by proximity effect in a normal metal will be R=Ro lproportional to AYO-). This conclusion, [ the same as for the study of the local penetration depth, is somewhat negative: for (33) experiments such as tunneling characteristics,

ELECTRODYNAMICAL

PROPERTIES

x=x,[ I++$-(*+$-p(o -&(

1 +~);w].

(34)

In these last formulae, A(O-Y’ ” = (2Kd#’

(35)

SECTION 3: EXPERIMENTAL RESULTS IN THE ZERO FREQUENCY CASE

In a recent letter[8], experiments were reported which showed that a normal layer could display efficient screening currents well above its own critical temperature. We now present a more detailed discussion of these experiments, together with some new data. A weak magnetic field, applied parallel to the axis of a hollow glass cylinder on which the S and N layers have been deposited, penetrates into the N layer up to a distance 1 from the interface. 1 has the following properties: (or) At a given temperature, 1 increases with dA,, but goes to a finite limiting value p when dN+ ~0. Cp) p is a decreasing function of H. p + p. when H + 0. (-y) p0 decreases with increasing T. In the case of the InBi/Zn system, p. + 0 for T = TM = 3.3”K (TM - 4.5 X TcJ. Properties (a!) and @) were in agreement with theory. Moreover, the field and distance dependence of p agreed with the generalized Landau-Ginzburg theory [4,5], and it was possible, with the simple set of the two parameters X(O) and K-l, to describe correctly the variation of p with dN and H. So, from this kind of experiments, we could deduce h(0) and K-l. The values of K-l were in agreement within 20 percent with theory but the experimental values of A(0) seemed to by systematically higher (by a factor of order 2) than the theoretical values. Later[7], it was

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claimed that this disagreement could be partly explained performing a more careful estimation on X(0) as done in the first part of this paper. Our present aims are as follows: (1) to study a situation in which the value of TM is not limited by Tcs, as seems to be the case for the InBi/Zn system; (2) to check formula (15), especially the remarkable temperature dependence of A(0); (3) to see if X(0) is a very sensitive function of TcN, and of the boundary conditions at the interface. Choice of the materials In view of points (1) and (2), Pb is a convenient S material because of its rather high critical temperature. Further, Aacs in Pb is practically constant below 4*2”K: thus the intrinsic temperature dependence of A(0) appears in a straightforward way in this temperature range. In order to study point (3), we choose Zn and Cd as normal metals. Thus we can compare two situations where (a) the normal layer is the same (InBi/Zn and Pb/Zn); (b) the superconducting layer is the same (Pb/Zn and Pb/Cd). Furthermore, we know that no reciprocal solubility is expected in the case of Pb/Zn and Pb/Cd. In our experiments, the Pb layer is clean (I 9 to) and the Zn and Cd layers have mean free paths 1, of the order of magnitude of qo-l. Thus the normal layer cannot be considered as being really dirty. Undoubtedly, this represents a considerable disadvantage, with respect to an exact comparison with theory. Nevertheless, (i) theoretical results for the dirty case should apply at least in a semi-quantitative way to the case I - qo-‘; (ii) if the normal layers were really dirty (I - qo-l/l 0), the screening currents would be much weaker at a given temperature (A(0) - l/V1 and K(O) - l/1). Thus measurements should be carried on at much lower temperatures (T < 0.5”K). This unfortunately was not possible at the moment in our laboratory.

518

G. DEUTSCHER,

J. P. HURAULT

and P. A. VAN

DALEN

(iii) It is not very convenient to have an S material much dirtier than the N one since the extrapolation length b is small then, which reduces proximity effects in N. The experiment We measure essentially the dynamic susceptibility X(V) of our cylinder, using the Shawlow-Devlin device [ 131. We recall that the cylinder is placed in a coil whose impedance is L. If X(V) changes by an amount dx due for instance to an entry of flux, a change dL, hence a change du of the resonance frequency of an l.c. circuit will be detected: 0

dv = dL a dx. Such a variation of L is obtained for instance when we apply a static magnetic field parallel to the axis of the cylinder. In such conditions, it is shown elsewhere that Av(H) = V(H)--(H==O)

a p(H)--p(O) +*af(H) aH

Fig. 3. Example

100

H(c

of a v(H) plot given by the ShalowDelvin device.

These thicknesses were determined by resistivity measurements. The change of Av was calibrated using the determination of Av between zero applied field and the thermodynamical critical field of the S layer, the thickness ds of the S layer being known. Most of the experiments were carried out between 1.2” and 4*2”, well above the transition temperature of Zn (0.92”K) and that of Cd (0.56“K). Some experiments were made above 4.2”K.

because the flux which enters the sandwich goes as - H p(H). Thus we obtained v(H) vs. H curves, an example of which is given on Fig. 3. The field was swept at a rate of about O-5 Oe/sec and Methods of determining A(0) and K-’ Using the above information, the lengths the v(H) vs. H curve was reached by means of a digital-analog converter and an X-Y K-r and h(0) can in principle be obtained recorder. Lastly, the resonance frequency was by various procedures: low enough to neglect the imaginary part of (a) At a giVen teDIperatUR& Hb iS an exponentially decreasing function of dN and can Q(o) compared to the real part. be observed if K(0) < 044[7,5]. Thus we obtained for each binary layer the following information: the screening distance p as a function of H and T; the temperature Hb = Ho exp(- K dN). T*(d,)[9], below which a first-order transition can be observed at the breakdown field The slope of the straight line Log Hb = dependence of Hb f(dN) gives K-l. The intersection of this Hb; the temperature for T < T*. line with the H axis yields Ho (Fig. 4): About 10 samples of each S/N system were 40 prepared, ds being kept constant and dN Ho = 1.9 (36.a) varying typically from 2000 to 20,000 A;. 2?r X(0) K-1’

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\ 0

05

1

W

Fig. 4. Determination of K-l, using the experimental values of Hh and equation (36a) in the case of a Pb/Cd binary layer.

In this way, we obtain X(0). (b) When dN + m, the p(H) curve may be used[8] to find the values of X(0) and K-l. Nevertheless, this requires tedious calculations, so that this procedure was used only to get a confirmation of some results. (c) As shown in reference [6], p. is a simple function of X(0) and K-l (Fig. 5):

We used (36.b) to get h(O), the value of K-l being deduced by procedure (a)?. (d) The T*(dJ experimental curve, together with K-‘(T), has shown to lead to another determination of X(0). This procedure was not used here as we found it rather indirect. Moreover, the value of T* could be very sensitive to trapped flux. In particular, the presence of trapped flux was found to result in an apparent lowering of T*, and we are not sure that the measured values of T&, are correct (T&+, = 2.5% for the Pb/Cd system, and T,&, = 3aO”K for the Pb/Zn one). We must recall here that the determination of tit may be emphasized that the value of X(0) obtained in this way is relatively independent of determination (a), because p. is measured on semi-infinite layers (& > &), whereas HB is measured on films of finite thickness (c& - K-l).

1

0

2

519

y--b

,

3

4

VK

Fig. 5. The distance from the interface of the flux front, in the limit of zero applied d.c. field, nO, for Pb/Cd, PblZn and InBi/Zn systems. T,&,

could be of interest since for T = T,&,,

K(0) = 0.44. The results

For the PblZn and PblCd systems, the independent determinations (a) and (c) give results in close agreement with one another. These results are shown on Figs. 6 and 7. The observed temperature dependence of A(O) seems to us to be a good proof of the validity of our estimations. This is particularly clear in the case of Pb/Cd, where x is seen to vary as q/T within the experimental accuracy.

I’

A ,/ .=$ /’ i

m 0

_/ ,’

,/’ ,’

I’

I’

,/’

,’

I’

,’ 1

2

lk

Fig. 6. The value of the enetration depth at the interface, A(O), as a function of J T for a Pb/Zn system. As can be seen, the g/T law for A(O) is verified for high temperatures, but a systematic deviation from the Z/T law is observed for low temperatures (T - TcN:).

520

G. DEUTSCHER,

J. P. HURAULT

and P. A. VAN DALEN

on the N side of an N/S binary layer, and seems to be applicable also in the case of the ultrasonic attenuation, the heat conductivity or the nuclear spin relaxation time. Consequently, it will be necessary to perform very precise and sensitive experiments in view of determining the BCS interaction NF’ of materials whose critical temperature is unknown. would like to thank Professor P.-G. de Gennes, K. Maki, G. Fischer and R. Klein and the members of the Orsay Group for numerous stimulating discussions. One of the authors (PAVD) takes this opportunity to express his appreciation to the direction of Philips Research Laboratories for giving him the possibility to work at Service de Physique des Solides in Orsay, particularly to Professor Rathenau for his stimulating interest.

Acknowledgements-We

Fig. 7. h(O) as a function of v/T for a Pb/Cd system. The law is seen to be valid for a large domain of temperature (1°K < T < 4°K).

VT

In the Pb/Zn case, there seems to be a systematic deviation from formula (15) at low temperatures. In fact, in the vicinity of TcN (case of Zn) the screening distance p,, becomes very large and may be of the order of magnitude or greater than l/IK-q,l. So the local penetration depth h(x) is less sensitive to the values of A at the interface, which results in an apparent decrease of X(X),which is effectively observed. Conclusions The study of the response to a small d.c. or a.c. magnetic field on the N part of an N/S binary layer has shown that the physical observable quantities (Meissner elect in N, surface impedance on the N side) were mainly dependent on the values of the pair potential not in N, but in the superconductor S, for N materials with very low critical temperatures. Such a result was previously obtained in the study of the tuneling characteristic

REFERENCES 1. DE GENNES P. G. and GUYON E., Phys. Lett. 3, 168 (1963). 2. See for instance “Gapless Superconductivity” by MAKI K., To be published in Superconductrvity (Edited by R. D. Parks). Marcel Dekker, New York. 3. DE GENNES P. G., Rev. Mod. Phvs. 36.225 (1964). 4. See for instance ABRIKOSOV A. A., GOR’KOV L. P. and DZYALOSHINSKI 1. E., Methods of Quantum

Field

Theory

in

Statistical

Phvsics.

Prentice-Hall, London (1963). 5. MAKI K., Physics 1,21 (1964). 6. Orsay groupon superconductivity, Proc. 1965 Sussex Univ. Symp., Quantum Fluids (Edited by D. Brewer), North-Holland, Amsterdam (1966). 7. DE GENNES P. G. and MAURO S., Solid Stare Commun.

3,381

(1965).

8. DEUTSCHER G. and HURAULT J. P., Phys. Left. 24A, 102 (1967). 9. Orsay group on superconductivity, Phys. condensed Matter 6,307

(1967).

10. SILVERT W. and COOPER L. N., Phys. Rev. 141, 336 (1966). 11. MAKI K., Phys. Rev. 141,33 l(1966). 12. FISCHER G. and KLEIN R., Preprint. A. A., GOR’KOV L. P. and 13. ABRIKOSOV KHALATNIKOV I. M., Sov. Phys. J.E.T.P.