Proximity effect: Numerical evaluation of the superconducting transition temperature

ANNALS

OF PHYSICS:

46,546-558

(1968)

Proximity Effect: Superconducting

Numerical Transition

Evaluation of the Temperature*

ROBERT H. T. YEH+ Department of Physics and Materials Research Laboratory, University of Illinois, Urbana, Illinois

De Genres’equation for the gap parameterin superposed filmsof normal and superconducting materialshave beensolvedvia a numericaltechnique.This is based upon a variation approachusedby Siivert and Cooper.The solutionindicatesthat the approximateanalyticalprocedureusedby Werthameris accuratefor thick-dirty films.However,for thinner cleanerfilms,our solutionsleadsto improvements asbig as 25% over Werthamer’smethod. This improvementis mainly a result of our better treatmentof the behaviorof the gapparameternearthe interface.

I. INTRODUCTION

Recently, there has been considerable theoretical and experimental (I), (2) interest in the “proximity effects”-that is, the mutual influence of superconductor and normal metal when they are in intimate contact. It is of interest not only as a clear manifestation of the large coherence length (6) in superconductor (3), but also can be used as a probe in determining effective electron-electron interaction, V, in normal metal (2), or to study the nature of magnetic impurity (4). We shall restrict our attention to the prediction of the transition temperature, T, , of this double-layer system. The basic model for this problem is Gorkov’s (5) self-consistent integral equation for the gap parameter d(r). T, is the highest eigenvalue of this equation. De Gennes and Guyon (6) applied this approach to a position-dependent electron-electron interaction V(r), however, exact solution is difficult to obtain (7), (8). Werthamer (9) has developed a very appealing trick to get a simple solution to this problem. He converted the integral equation for d(r) into an approximately equivalent differential equation. He has a very simple answer for T, . His results * Research supportedin part by ARPA SD-l 31. Thispaperisbasedon part of a thesissubmitted by the author in partial fulhllment of the requirements for the Ph. D. degreeat the University of Illinois, 1967. + Presentaddress:Departmentof Physics,University of Virginia, Charlottesville,Virginia, 22903. 546

PROXIMITY

547

EFFECT

agree very well with experiments in thick and dirty films (IO), but fails outside this limit (2), (II), (12). [Let d be the thickness and 4’ the electron mean free path of the film. A film is dirty (clean) if 4 > 4(.$ < e), also a film is thick (thin) if d > 5 (d < 01. In this work, we are interested in finding out the range of validity of Werthamer’s theory, and also for possible generalizations.

II.

THE

GAP

EQUATION

We shall follow closely de Gennes’ (2) treatment of the gap equation. Consider a double layer with the superconductor layer lies between x = 0 and x = ds , while the normal metal layer lies between x = -dN and x = 0. Then de Gennes’ Eq. (3.1) may be written as dx’ H&x,

x’, co,) d,(x’)

+ jr,

dx’ HsN(x, x’, q,) b(x?j, N

-=

dx’ HNN(X, x’, qJ d pk-4 + J; dx’ H~~(x, x’, w,) d s(xf) j ,

-dN

< x < 0,

(11.1)

where we use a notation in which a subscript N (normal metal), or S (superconductor) distinguishes the two layers. LN and Ls are the BCS-type frequency cutoffs which limit W* to be less than or of the order of the Debye frequency of each metal. In the dirty limit, H (x, x’) obeys the differential equation

\21 @JI + W) $1

H,(x, x’) = 27riv(x) 6(x - x’),

(11.2)

where w,, = 25-T,@ + 4) with p integer, and D(x) and N(X) are, respectively, the position-dependent diffusivity and density of states at the Fermi surface. The range in space in H is thus

and l/a,,+

is the coherence length 6.

548

YEH

Equation

(11.2) has the following solutions:

fMx, x’, 0) = 3

psl-‘l

+ ~s(r+m’-zds)]

+ As cash LX&& - X) cash ar,(d, - x’), + e--u~W~+2+5’)] x’, UJ)= $$N N [e--aNl5--2’1

H&X,

+ AN cash (Y&& + X) cash c+(& H&x,

x’, OJ)= H&x’,

+ x’),

x, w) = AsN cash CL&, - X) cash OI&& + x’).

(11.3)

Since it is required that (d/&c) H(x, x’, 0) vanishes when x falls on either boundary between metal and vacuum. We also assume that the normal metal and superconductor have a sharp interface. Without an oxide layer in between (i.e., perfect transmission at the interface). This implies that [l/N(x’)] H( x, x’, w) and D(x’)(d/dx’) H(x, x’, w) must be continuous when x’ crosses the N-S interface (2). From this the coefficients A’s can be easily determined (7) as A = 2.rrNse-+s s

A

=%rNNelXNdN . N

ASN

WDS ~NDN

asDsNs cash aNdN - aNDNNNsinh aNdN asDsNs sinh a,d, cash aNdN+ aNDNNNcash asdssinh aNdN’ aNDNNNcash a,ds - asDsNs sinh a,d, cusDsNssinh ol,ds cash aNdN+ CLNDNNN cash a,ds sinh LyNdN’

LhrN,N, = aNDNNN cash asds sinh aNd, + QDNN~ sinh asdscash aNdN ’

III. THE WERTHAMER’S

(11.4)

SOLUTION

Terms of H(x, x’, UJ) in Eq. (11.3) can be divided into two kinds. One is the particular solution of Eq. (11.2), e-~l”-“‘I, which is not small so long as ] x - x’ ] 5 cy-l. This term we refer to as volume term. The rest of H(x, x’, co) are complementary solution of Eq. (11.3), and all decay exponentially from some boundaries with characteristic length 01-l. These we refer to as the surface terms. Note that the volume term is translational invariant in space, while the surface term is not. The surface terms arise as a result of the existance of the surface barriers. Werthamer (9) suggested a simpler way to handle the surface barriers. He chose to drop all the surface terms in Eq. (ILI), so the gap parameter in superconductor d,(x), is no longer coupled with the gap parameter in normal metal d,(x).

PROXIMITY

549

EFFECT

Because of the translational invariance of the volume term, he can then approximately convert Eq. (11.1) into a differential equation of d(x), namely, x (-fz ,$ = [~~J/6&~7’~]~/~ and

-$)

A(x) = In [J$Q-]

(III. 1)

d(x).

is the effective coherence length, suitable for the dirty limit, Ofx
GO’

Tcs > TeN .

By this, Werthamer has implicitly assumed that both metals of the double layer are superconductors. We refer to “superconductor” as the one with higher transition temperature T,, , and “normal metal” as the one with lower transition temperature. The actual observed transition temperature T, lies always in between T,, and TcN . The gap equation (11.1) is certainly not suitable for metal with repulsive electron-electron interaction (I). Also, x is defined as x(4 = w

(111.2)

+ Bz) - WI,

Where Y is the digamma function W>

= $ln

J%) = -y

- i + n$ c - -J--) z+n

(Z

# -1, -2,...):

y = 0.57721 is the Euler’s constant. De Gennes (2) showed that, in the dirty limit, d(x)/N(x)

V(x) and N(x) D(x)/&)

* dA(x)/dx

are continuous across the superconductor-normal metal interface, provided that there is no oxide layer in between. Also dd(x)/dx must vanish at metal-insulator or metal-vacuum surfaces as there is no current across these boundaries. With these conditions, Werthamer obtained the following solution of ,4(x):

where ks , kN are defined, respectively, by x(ks”&2) = In y,

c

x(-kNatN2)

= h %. r:

(111.4)

550

YEH

T, is given implicitly

by Nsfs2ks tan ks ds = NJN2kN

tanh kN dN .

(111.5)

Werthamer’s method has been generalized to arbitrary mean free path by Moormann. (13) Also, in the thin-film limit, the earlier theory of Cooper (3) is very successful.

IV.

VARIATIONAL

SOLUTION

OF

THE

GAP

EQUATION

To improve Werthamer’s solution, we use a variational principle suggested by Silver? and Cooper. (14) They computed Q, the difference of the thermodynamic potential between superconducting and normal states, as a functional of the gap parameter d(T). They showed that, the requirement that the variation of 1;2with respect to A be stationary, i.e., Sln[A, TJ/SA = 0

(IV. 1)

is equivalent to the integral gap equation. In our notation,

Q[A, z-j = ,z y(x)rA(x) - W) ~(41 dx,

(IV.2)

s

Y(X)= 23j

Jz, H,(x,

x’) A(x’) dx’ = C yy(x). Y

This principle is useful because with some appropriate Werthamer’s solution of A(x), the equation

(IV.3)

trial function such as

Q(a[Awerthsmer(T)l = 0

(IV.4)

yields a very good approximate solution to T, . Besides, the solution of Eq. (IV.4) is always a lower bound of T, , so definite comparisons with other theories can be made. To see this, note that sZ[A, T] is not only stationary for small variation of A, but is also a minimum when A is a solution of the gap equation, or

WtrdT)l

B QGdT)I,

Wmct(Tc)l = 0,

(IV.5)

where A,,t is an exact solution of the gap equation. Now suppose we solve T from Q[A(T)] = constant < 0, from Eq. (IV.5), it is clear that at this T, we also have Q[A,,t(T)] < 0, or T < T, . Numerical solution shows that @A(T)] is

PROXIMITY

551

EFFECT

monotonic in T for T near T, , so @d(T)] = 0 yields the best lower bound of T, . A different argument and a new variation principle is proposed elsewhere (12). To see why Eq. (IV.4) works, (as Eq. (IV.1) is not a variational principle for T,), we note that for a fixed trial function d(T), we have 6Q[d(T)]

= W(T)]

W(T,)l = g (

-

0” - TJ,

(IV.6)

Artxea

where T, is the true transition temperature. Let T be the solution of Q[A(T)] = 0. Numerical solution shows that aQ/aTfA is very large (of order several hundred). Also, by the nature of variational principle, In[A(T,)] M Q[d,,t(T,)] = 0. So it is clear that T - T, is very small, or solution of T from L&l(T)] = 0 is a very good approximation to T, . To compute Q[&‘)], it is natural to use Werthamer’s solution of gap parameter as our trial function. From Eq. (IV.3), substitute d(x’) by Werthamer’s solution [Equation (111.3)], we get

(IV.7)

where C, and Ck are determined by the following conditions (2):

I

ds Y,(X)

dx

=

&

I

-dN u(x)

Ns

=- Y(X) NN z+o+

xN

4-d

NO

dx,

(IV@ z-‘O-

’

which gives C,(NN[N

sinh aNdN cash cusd, -j- N,&, sinh a,d, cash aNdN)

= 13’ + 1 1NNtN sinh aNdN - [ , 2v + :y +

(t

2~ + 1 1>1/2cash W&

+ , 2V +?ivNkN8tNa

k af s 2

- [ , 2v + ~~ks2ts

NNk&?

tmh kvd,] ,

1 2V + $‘k~‘f~”

Nsks&2 tan k&s

I

552

YEH

C:(NNgN sinh aNdN cash ar,d, + Ns& sinh a,d, cash (yNdJ

= I 2~ + 1 I N&s sinh w& I , 2v + :F

+(I2~ + 1 lP2 cash4s

ks2gs2-

[ , 2v + 57

12v + fyV”kN2&2

ks2gs2N&A2 tan MS

1

- , 2v + ??kN2gN2 NNkNSN2tanh kNdN. Substituting

2

v=l

2V -

NNVN 1 - kN2gN2

GXtanh

g

2v _

yy;

(kNi%tanh

g

2 gl

&

-TV;cs2gs2

2~

-

NsVs 1 + ks2gs2

gs

c” v=l (2v -

+2; "El

- SN,V,%

-SNNVNd”z

t2v

W&d

'b&r)

Ij1,2

f

G:

2v - 1

N2 N2

+

(IV.9)

back into Eq. (IV.2), we finally get

-2;

+

1

A

('&,

.!

h k&

1)1/Z

G,tanhkolsds

tan

k&

(2v -

1)

G

CO

GUI2

l)2[e”sds + e-@s12

G:a gN v=l (2v - 1)2[eW%J + e-+N]2

LS - 2NJsA

vz

LN (2v

~~jJ,2

tad

4s

~NNVN

-

c2v

vz

?21)6i2

tanh

aids

- (2~’ - 1)lj2 tanh aids] -

4N,v,

LN

v-l

,c,

&

(2v

-$$,

_

1) tv

1

vt> [(2v

-

1)1'2

tanh

aNdN

- (2~’ - l)lj2 tanh ct&dN],

(IV.10)

PROXIMITY

553

EFFECT

where

I,[ ] denotes the integer part of the number in the bracket. u.@ and or) are Debye frequencies in superconductor and normal metal region respectively. G, and G: are given by G, = (2~ - l)lla [tanh a&N + x tanh q&l-1 x

I(2u -

d l)‘/” tanh OL N N

_ ( - NSVS 1 $ ksa&2 2V

Ws5s) tan Ws

+ k 2v - y,,&~

NN

vN

- 1 - k,“&? )

2V

+ 2v -y-2N2tN2

('h!%

mh

k,dN))J

G: = (2~ - 1)1/z [tanh aNdN+ X tanh c+d,]-* x (2v - l)l/2 X tanh asds (2y _ 7:: I h(k&)

tan

2e 2 s s

ksds -

2~

NNVN

2V - 1 -

NNvN - 1 - k,“f,’

)

kNafNa (kNeN) tid

k,d,j\ (IV.1 1)

We remark that, for the case of clean films, it is desirable to make the Silvert’s (15) generalization in our expression of Q’[d]. All we need is to replace & and & by the frequency-dependent [is), 6:“) quantities in all relevant formulas where p

= &(l

+ 2 I w, I 78)- l/2 9

,gN)

=

+

eN(l

V. NUMERICAL

2 1 w, 1 TNpz,

RESULTS

(IV.12) 7-l

=

UP/e.

AND DISCUSSIONS

Werthamer’s solution [Eq. (III.S)] and our solution [Eq. (IV.4)] are programmed on IBM 7094 computer for all ranges of values of film thickness and electron mean free path. Lead is taken as the superconductor; for the normal metal, TcN is either known (aluminum) or computed with BCS weak-coupling formula from the assumed value of (NvN. (Copper, platinum). Our theory does not take into account the spin-dependent interaction (e.g., as in the presence of magnetic impurities) or oxide-layer effects. But when the experiments meet the criteria

554

YEH 7

T

I

r

I

I

I

I

I

I

6-

5PB-AL 0 EXF! POINTS

Tc(“K)

I-

--1

0

I I

0

.A-.--

.I-L--1-__1---.-L2

3

__L dPb1102

4

ii)6

7

B

I 9

I IO

II

FIG. 1. Our theory compared with experiment on Pb-Al (IO).

Ppb a I.8 x10-50hn-cm 5Pcu *I I 16’

ohm-cm

4-

T, (“K) 3-

2--

n

I-

OL

I I

0

I

I

I

I

I

I

1

I

!

2

3

4

5

6

7

8

9

IO

d,$02& FIG.

2. Our theory compared with experiment on Pb-Cu (IO).

II

PROXIMITY

555

EFFECT

T, (“K) 3-

G-PI-PC 0 EXP

d,b=

2-

FVINTS (HAUSER 0 (N’&= 0.09

P,,, = 3 x 16’ I-

et. al.)

300A,

ohm-cm

P~.l.8xldJohn-cm I I

0 0

1 2

I 3

1

I 4 c&y

FIG.

1 IlO2

I 7

1 6

1 9

t IO

II

iI

3. Our theory compared with experiment on Pb-Pt (10).

mentioned in Section II, the agreement between our theory and experiments is very good. (Figs. l-3). We find agreement with Werthamer’s solution in the thick dirty limit. We also did an iteration expansion (12) of the integral gap equation, using Werthamer’s solution of d(T) as zeroth-order approximation. This expansion is stable if 7’ is close to Werthamer’s solution of To ; otherwise it diverges. We find large discrepancy with Werthamer’s solution in thin clean limit. (Figs. 4 and 5). The relative difference can be as much as 25 % (absolute difference of order 1°K). Since our prediction of T, always lies above Werthamer’s prediction, from what we proved in Section IV, ours is an improvement. However, the combination of very clean with very thin film limit may be unrealistic, as boundary scattering will certainfy limit the electron mean-free-path length. More realistic value yields some what smaller difference. (--O.YK). (Figs. 6 and 7) We find that both copper and platinum have attractive electron-electron interaction. For our method, the d,-To plot is not sensitive to the value of V,, while the drTc plot is very useful. Best fit of experimental data is obtained if we set (NV), = +0.08 f 0.02, (Nv),t = +0.09 f 0.02. Both correspond to a mill°ree transition temperature. However, these values should not be taken too seriously, as at the present time, we still cannot measure all the parameters (e.g. Glm thickness) to within 10 % error. Also, the fit is still fair within the error range of f0.02, which is quite large. Our method, although tedious, is capable of generalizations to cover various

556

YEH

Tc (OK)

PRESENT

WORK

1

!

b

Ib

/2

/4

/6

it3

$0

:2

;4

A

&

d,, (to2 i, 4. Our theory compared with Werthamer’s theory for P&AI,

FIG.

PRESENT /

6

T,d,

plot.

WCRK __---

T, (OK)

Pb-Gu d,,=3OOA (NWcU = +0.08 Ppb = Pcu=l.6

"WI

6

8'

I IO

I 12

I 14

I I6

I 18

I 20

I 22

1 24

x dohm-cm

I 26

d&lo2 ii, FIG.

5. Our theory compared with Werthamer’s theory for Pb-Cu, T&S plot.

I 28

’

PROXIMITY

I

7-

I

I

,

I

557

EFFECT

!

I

1

I

I

I

I

I

I

6-

i

/ II

0

0

Pb-Al 4, =9ooi ppb’pI, =l.ElxIdohm-cm

I

I

I

I

I

2

4

6

8

IO

I

12

I

I

I

14

16

I8

I

I

L

I

/

20

22

24

26

28

dpb(io2 i, FIG. 6. Our theory compared with Werthamer’s _.theory for Pb-AI, T,-&

plot.

Pb-ti (m%”

* +o.oe v3ooi

d,

Pm= Pcu ~7.2 x 1G7 ohm-cm

2-

01

0

2I

41

6I

8I

I

,

I

I

I

IO

I2

I4

I6

I8

I

I

I

I

20

22

24

26

d,,uo2 i, FIG.

7. Our theory compared with Werthamer’s theory for Pb-Cu, To-&., plot.

I 28

558

YEH

limits. Preliminary comparisons (7) show good agreement with direct numerical solution of the integral gap equation. We attribute our improvement over Werthamer’s solution of T, as a result of better handling of surface barriers effects. Note that all the surface terms of H are included in Eq. (IV.3). It is clear that in the surface barrier phenomena (such as H,, , tunneling, barrier field), the inclusion of surface terms is even more important. Unlike classical electromagnetic field theory, the usual Ginzberg-Landau-type equation is not valid near the surface barrier. Hence the boundary conditions derived from them is not exact (8). However, surface term effects can be included in the differential equation scheme via an iteration method (12). Research is under progress to discuss various barrier effects from this point of view. We conclude that Werthamer’s simple differential equation is very useful but fails outside the thick dirty limit. While a variational principle approach leads to definite improvement and can be quite useful in explaining surface barrier phenomena. ACKNOWLEDOMBNT~ The author wishes to express his gratitude to Professor Leo P. Kadanoff for suggesting and guiding this work. He also wishes to thank Professor D. M. Ginsberg, Allan E. Jacobs, and Reiner C. Ktimmel for valuable discussions. RECEIVED: August 3, 1967 REFERENCES 1. For review and references, see G. DEUTSCHERAND P. G. DE GENNES. Proximity effects, to be published in “Superconductivity,” R. Parks (Ed.) Dekker, New York. 2. P. G. DE GENNJB, Rev. Mod. Phys. 36,225 (1964). 3. L. N. COOPER, Phys. Rev. Letters 6, 89 (1961). 4. J. J. HAUSER, H. C. T.HEUERER,AND N. R. WERTHAMER, Phys. Rev. 142, 118 (1966). 5. L. P. GORKOV, Sovief Phys.-JETP 7, 505 (1958). 6. P. G. DE GENNE~ AND E. GUYON, Phys. Letters 3, 168 (1963). 7. For numerical solution of the gap equation, see A. JACOBS, private communication (to be published). 8. See R. 0. ZAWSEV, Soviet Phys.-JETP 21,426 (1965) for asymptotic solutions and boundary conditions. See also R. 0. ZAIWW, Soviet Phys.-JETP21,1178 (1965); ibid. 23,702 (1966). 9. N. R. WERI-HAMIB, Phys. Rev. 132, 2240 (1963). A similar method has been shown to be valid for bulk dirty type II superconductor for applied field H near H,, by K. Maki, Physics 1, 21 (1964) and P. G. de Germs, Phys. Condensed Mutter 3, 79 (1964). 10. J. J. HAUSER AND H. C. THEIJE~~R, Phys. Letters 14, 270 (1965). Also J. J. HAUSER, H. C. Theuerer, and N. R. Werthamer, Phys. Rev. 136, A637 (1964). II. G. V. MINNXXRODE, 2. Physik 192,379 (1966). 12. R. H. T. Ym, Unpublished doctoral dissertation, University of Illinois, 1967. 13. W. MOORMANN, Z. Physik 197, 136 (1966). 14. W. SILVFRT AND L. N. COOPER, Phys. Rev. 141,336 (1966). 15. W. SILVERT, Physics 2, 151 (1966).