Journal of Low Temperature Physics, Vol. 82, Nos. 3/4, 1991
Properties of Proximity Systems Including Magnetic Impurities W. Stephan and J. P. Carbotte Physics Department, McMaster University, Hamilton, Ontario, Canada
(Received September 24, 1990)
We have calculated the thermodynamic critical field and deviation function for a proximity system with paramagnetic impurities in the strong scattering limit. For the proximity effect, we use the model of McMillan and for the impurity scattering the model of Shiba and Rusinov. Important deviations from BCS behaviour are predicted as is also the case when the specific heat difference is computed. Special attention is given to the jump in the specific heat at Tc and a previous calculation is corrected. 1. I N T R O D U C T I O N The proximity effect between different metals might be described as the "leakage" of the properties of one metal into another. In the particular case of an interface between a superconductor S and a normal metal N, the superconductivity may be described as "leaking" into the normal metal. This can be understood qualitatively to result from the long range coherence of the superconducting state; the superconducting coherence length is a measure of the spacial range of the special correlations that distinguish the superconducting state from a normal metal. In a rough way this coherence length may be said to reflect the spatial extent of the Cooper pair wavefunction. The superconducting correlations may vary only on length scales greater than, or of the order of, the coherence length, so that on the atomic length scale which is appropriate for a clean interface the superconducting order parameter cannot drop discontinuously to zero on the normal side of a junction. If there is a thin insulating film I between the normal and superconducting metals (S-I-N junction) with a finite probability for electron tunneling through the insulating barrier, then the amplitude of the superconducting correlations near the barrier on the N side will be reduced compared to that on the S side, but there will still be a finite Cooper pair amplitude, 145
oo22-2291/91/o2oo-o1455o6.5o/o © 1991 Plenum Publishing Corporation
146
W. Stephan and J. P. Carbotte
and this will decay in the normal metal on a length scale determined by the coherence length on the N side. In general the spatial variation of the superconducting order parameter in a junction greatly increases the complexity of the theoretical treatment, 1 with the exception of one special limit. It was first realized by McMillan 2 that if both S and N films are thinner than their respective coherence lengths, then to a very good approximation the superconducting order parameter will be constant in amplitude within each film. Within this approximation the spatial dependence of the order parameter is then replaced by two order parameters As and AN, which are determined selfconsistently in terms of the tunneling probability through the barrier. This approximation greatly reduces the computational complexity of the proximity effect problem, and is the model which will be used here. In Sec. 2, we review the necessary formalism which is not new. Section 3 deals with the free-energy difference based on a generalization of the original formula of Bardeen and Stephan. Numerical results for the free energy critical magnetic field and deviation function are to be found in Sec. 4. In Sec. 5, We give new analytic results for the jump in specific heat at To, and in Sec. 6, we present numerical results based on the derived formulas. A short conclusion is to be found in Sec. 7, and the appendix deals with some mathematical details associated with the derivation of the free-energy formula. 2. F O R M A L I S M Following McMillan 2 and Kaiser and Zuckermann, 3 who extended McMillan's model to include paramagnetic impurities on the N side, the thickness of the films will be denoted by ds and dN for the superconducting and normal sides, respectively. One main assumption of the McMillan model is that both N and S films are thinner than their respective coherence lengths, so that the superconducting properties may be treated as being independent of position within a given film. This typically restricts the validity of the McMillan model to films with thickness of the order of hundreds of angstroms or less, despite that it is often applied when this condition is not satisfied. In fact agreement may be essentially identical with experiment in both cases. 4 A second crucial assumption is that the effect of the interface between the N and S films may be treated in terms of tunneling through a thin insulating layer. This is described mathematically in terms of the tunneling Hamiltonian
+ bm.~a,,.+ H r = ~ T.m[a,,.~b.,t+b+_mja_,,+ +
+ a+.~b-m~J
(1)
tim
Here the operator a,~ + creates an electron in the one-electron state ~p. with spin o. on the normal metal side of the junction, and the operator b+~
Properties of Proximity Systems Including Magnetic Impurities
147
creates an electron in the one-electron state 4~,, with spin o- on the superconducting side. The Hamiltonian a therefore describes processes involving the transfer of electrons from one side of the insulating barrier to the other, with the matrix element T,,,. One further simplifying assumption made by McMillan is that the transfer matrix element Tnm is independent of the indices n and m, which means that all states on opposite sides of the barrier are coupled equally. This is obviously not true if the states qJ, are Bloch states, in which case states with wavevectors parallel or perpendicular to the plane of the boundary would not be expected to have equal tunneling probabilities. However, as pointed out by McMillan, 2 if the films are "dirty" in the sense first discussed by Anderson, 5 then the appropriate states ~, are no longer Bloch states, but rather linear combinations of Bloch states with contributions from all regions of the Fermi surface. In this case the transfer matrix element is, to a very good approximation, independent of the states involved. The approach as taken by McMillan is to include the contributions from the transfer Hamiltonian HT to the self-energies on either side of the barrier within second-order self-consistent perturbation theory. To include magnetic impurities on the normal side of the junction, the appropriate self-energy may simply be added to the proximity-effect contribution. This has been carried out for the AG case by Kaiser and Zuckermann. 5 The strong scattering limit has been investigated by Machida 6 within a model that treats the paramagnetic impurity scattering within an approximation which is equivalent to the ShibaT-Rusinov 8 (SR) model. Kaiser 9 has also considered the strong scattering limit and has attempted to include the Kondo effect by using an approximate form of the theory of MiillerHartmann and Zittartz (MZ) 1°'12. The approximation to the MZ theory results in a Green's function identical in structure to that of the SR model, with the Kondo effect resulting in the scattering parameter eo acquiring a temperature dependence. Mori j3 has calculated some transport properties for these models, both with and without including the approximate treatment of the K o n d o effect. If the pairing interaction on the superconducting side of the junction is treated within the BCS approximation, and the normal side is assumed to have no pairing interaction, then the gap equations which result in the SR case are those given by Machida. 6 There are two equations from the two matrix components of the Green's function on both sides of the junction, resulting in four coupled equations for the four functions AscN) and ZS~N~. On the real frequency axis they are 7~s((O) = Aeh + FsTXN(oJ)/DN(o~)
(2a)
Zs((O) = 1 + FsZN(w)/ DN((o)
(2b)
148
W. Stephan and J. P. Carbotte
7XN(to)=rNTXs(to)/Ds(w)+r2(to)7~N(to)/DN(to) ZN(to) = 1 +rNZs(to)/Ds(oO)+r,(to)ZN(w)/DN(to)
(2c) (2d)
where Dl(W) = [A~(w) - ¢o2Z~(w)] ~,
I = N or S
(3)
The square root in (3) must always be chosen to have a positive real part. The scattering rates Fs and FN are defined in terms of the tunneling matrix element T, the film thickness d and Area A, by F s = ~rA TZ dNNNo -
1
(4)
27s
and FN = IrAT2dsNso =
1
(5)
2~'N
which clearly satisfy (6)
Fs _ dNNNo F N dsNso
The relaxation time is estimated by McMillan 2 to be
(7)
~'N = L N / ( v ~ N ~ )
where LN is the average path length traveled between collisions within the barrier and o- is the barrier penetration probability. This gives
v~No"
(8)
FN - 4BdN
where 2BdN represents the average distance traveled between collisions with the barrier, with B taken to be a constant of order two for relatively clean films. These results imply that FN(s) is inversely proportional to dN~s). These parameters are usually fixed by matching the experimentally observed depression of Tc with varying thicknesses while keeping the ratio F s / F N constant as indicated by (6). In the above equations, AN(s), ZN(s~, N~(s~o, veN(s~ are, respectively, the pairing energy, renormalization, single spin density of electronic states and the Fermi velocity on the N ( S ) side. The frequency-dependent impurity scattering contributions F1 and F2 are given by n
r,(w)
oo
,~
~
(9a)
wZZ%(w) - ~ ( w )
2rr~VNo,~=o(21+l)(1-n'e') to2 2Z,v(W)-AN(to)s,'2
2
~o~z~(o,)-,~(o.,)
V:(~o) - 2 ~'?qNo t~o=(21+ 1)(e,- nt)e, w : Z ~ ( w ) - &~(~o)e~
(9b)
149
Properties of Proximity Systems Including Magnetic Impurities
where n is the impurity concentration and where the scattering parameters are el = c o s ( 6 [ - 37)
(10a)
~?t= cos(a[ + ~7)
(10b)
and
with 87 phase shifts. The parameter r/indicates the strength of the normal (nonexchange) scattering from the SR impurities and is most often taken to be unity (pure exchange scattering) as most superconducting properties are independent of normal scattering. There are, however, cases where the normal state electronic mean free path is important, as is the case for the electromagnetic coherence length and the experimentally more important upper critical magnetic field/'/~2. In this paper ~ will be assumed to be unity. The quantity AVh introduced in (2a) is the superconducting order parameter, which satisfies the usual BCS self-consistency equation
Aph=NsoVf/~'do~Re[ 7~-~-s(to)
[w2Z2(to)- ~2(to)]l/2j] tanh(flw/2)
(11)
where V is the pairing potential on the superconducting side and too the Debye energy. Two of the equations (2) may be eliminated by using the definitions
A, (to) = 7,,(to)/z, (to),
l=N,s
(12)
which result in FNAS(,O)
A~(to)- [A~(~)_
~],/~
[
1-t
FN r(to) ]-1 [A2(to)_(..o2]l/2 ~ [A~(f.o)_c.o2]I/2j
(13a)
and
As(to)= Aph + [A~q(to) _ to231/2j [. 1 4
(13b)
The frequency dependent pair-breaking parameters in (13a) is given by r(~)
= r,(~) n
- r~(~) ~o
to2_A~(to)
2~[NO,~O(21+l)(1-e~) to~2-aN(to)e,
(14)
In the AG limit eo + 1, in which case F is frequency independent. The pair of equations (13), together with the order parameter Eq. (11) written in terms of As(to), may be solved numerically for Aph and the functions as(tO) and AN(tO) by first assuming a value for Aph, solving for
150
W. Stephan and J. P. Carbotte
AN (to) and As (to) at various frequencies, and then integrating (11) to arrive at a new value for Aph. The entire procedure is then repeated until Aph has converged to a stationary value. However, in this case the original set of four coupled equations (2) were chosen for numerical solution, rather than the pair (13). Although this approach requires slightly more computational time, one appealing feature is that in (2) all square roots may always be chosen in one consistent manner, namely, to have a positive imaginary part; in the case of (13) one must choose that square root that is consistent with the convention in (2). In either case a further complicating factor is that although well behaved at most frequencies, in the neighborhood of any gap edge where the gap functions A1(to) change from being real to complex valued functions, the sets of equations (2) or (13) do not iterate to convergence. Following McMillan, 2 this problem was circumvented by deriving a set of first-order differential equations (by differentiating the equations with respect to frequency), which could then be integrated through the troublesome region using the Runge-Kutta 14 method. The accuracy of this procedure is confirmed following each use by checking for the convergence of the functions at the endpoint of the integration region; if the functions are not consistent with the true converged values to within some reasonably small error, then the entire procedure is begun again with a smaller step size in the Runge-Kutta integration. As an aside here one might mention that this problem is most apparent in the SR model with a small impurity concentration, where the number of "gap edges" may be as large as 2n + 1, where n is the number of scattering parameters included. This results in the computer code, as it currently stands, being most efficient for larger concentrations. Note that the result for the superconducting critical temperature derived by Kaiser and Zuckermann 3 also applies in this case, with only trivial modifications. The imaginary axis version of Eqs. (13) are expanded in powers of As and AN, keeping only the linear terms. The result is
A+-A_ _ [ 1 _~__]t)[2@T + ~ ] ] _ ~p[1;
(15a)
where
A ± = ½ [ a + r N + r s ] ± ~ [ [ ~ + r ~ + r s ] 2 - 4 a r s ] 1/~
(15b)
and OL -
-
-
"
-
~ (21+I)(1-~)
2~NNo t~o
(15c)
Properties of Proximity Systems Including Magnetic Impurities
151
The only difference between this result and the AG limit of Kaiser and Zuckermann3 is in the pair-breaking parameter a defined in (15c). Critical parameters, which reduce Tc to zero, may also be found exactly as done by KZ by taking the limit as Tc goes to zero in (15a). The critical parameters are then given by
ln(Ao/2)=[1-~++llnA+-[1-~_]lnA_ (16) where the critical value of the pair-breaking parameter ac must in general [ A+~ssA-]
be determined numerically, as A± also depend on ac, so that (16) cannot be solved explicitly for ac. As pointed out by KZ, (16) has no solution unless is less than or equal to some critical thickness given by Fs >- Ao/2. Here Ao is the energy gap. For very thick S films (Fs <
ds
3. THE FREE-ENERGY DIFFERENCE The reduction of the critical temperature of a thin superconducting film by a normal metal film through the proximity effect is only one part of a general modification of the thermodynamic properties. For example, the jump in the specific heat at the critical temperature is also reduced, as pointed out by Fulde and Moorman ~5within Ginzburg-Landau theory and Mohabir and Nagi 16 within the McMillan model, and as observed experimentally in eutectic alloys 17 (bulk material composed of alternating planes of two different metals). Although the composite materials examined up to now probably do not satisfy the condition of layers thinner than a coherence length as assumed in the McMillan model, it may still be of some interest to examine the predictions of this model for thermodynamic properties in general. The approach taken here to calculate the free-energy difference between normal and superconducting states of a proximity effect bilayer will be to generalize the strong-coupling calculations of Bardeen and Stephen 18 (BS) to include the proximity effect within the McMillan model. This involves using a modified form of the free energy functional used by BS, although most of the succeeding steps are then conventional. It is demonstrated in the appendix that the total free energy difference for the bilayer may be found by applying the conventional Bardeen-Stephan is formula for the free energy difference of a superconductor to each side, with the input being the gap and renormalization functions calculated within the McMillan model, and then adding the two contributions with the proper weighting, It should be stressed that only the total free-energy difference is given
152
W. Stephan and J. P. Carbotte
correctly by this procedure; the contributions from the two sides have no significance taken independently. In fact, if one insists on examining these contributions separately (although the division is not unique, there is a "natural" division that may be intuitively appealing), then a correction term to the BS formula must be added for each side. Although this correction is not small, it gives an equal correction to the two sides, with the final result that in calculating the total free energy difference no correction is required to the BS formula. The total free-energy difference per unit volume will then be given by
AF = vsAFs + vNAFN
(17)
where v~, I = N or S, is the volume fraction of a given material, and AFt is the free-energy difference per unit volume in material L The fractional volumes must obviously satisfy Vs _ ds VN dN
(18)
since the two films are assumed to share a common surface area A. Recalling the relationship between Fs and FN given by (6), (17) may be expressed as 1
Fs
Nso where
aF,
f' = N,0'
X = N, S
(20)
The values of fs and fN required in (19) may be calculated using the Bardeen-Stephen is formula for the free energy difference, which may be written in the form
f~s=-2~rT .>-o • [ [[~(n)+ 7~(n)]I/2-t~'(n)]
×[1
O(n)
ll
[o~(n)--g~(n)],/=3 3
(21)
Here f ] ~s is the free energy difference per unit volume between superconducting and normal states divided by the appropriate one-electron density of states N~o, I = N or S. The function io5~(n) = ioo,Zx(i~o,) is the renormalized Matsubara frequency, and itS°(n) is the normal state (A-.0) limit of this. The functions Zl(i~o,) and A~(n)=At(i~o,) are the imaginary axis renormalization and pairing functions for the material I = S or N. In our model these would be given by the imaginary axis equivalent of Eqs. (2).
Properties of Proximity Systems Including Magnetic Impurities
153
In the case of a pure superconductor in proximity with a pure normal metal (no magnetic impurities) the expression (21) or the real axis equivalent may simply be evaluated twice, once with the functions Zs and As, and then once again with the functions ZN and AN. However, if the normal side of the proximity junction contains magnetic impurities described by the SR model, then for the N side of the junction a correction term must be added to (21). This SR correction has been calculated by Yamamoto and Nagi ~9, and has the form co
f sR=-2rrT
E E (2l+l)a, n>--O I = 0
x (1-et)-2
2
-2
'/=
aZ(n) ~-( 1 ÷ ~2(n)e~+ 032(n) ~
)ln[,~
l
]]
(22)
where n
a,
27rNNo( 1 - e 2)
(23)
with et the various well depths in Shiba-Rusinov theory. The free-energy difference therefore becomes fs =fffs
(24a)
¢SR f N -_- J¢BS N -a--JN
(24b)
for the superconducting side and
for the normal side of the junction. One further step remains before the presentation of some results, that being the representation of the Matsubara frequency sums in Eqs. (21) and (22) in terms of real frequency axis integrals, as the gap equations (2) are being solvled for this case. Using the standard techniques, 2° that is, by first representing the sums as contour integrals and then deforming the contours, one obtains
×[lq
/°3° (~°)
II
[a2(o)) _ o32(~o)],/2.].] tanh [ ~ ]
(25)
where the functions A(o)) and o31(w) = ~oZ~(w) are given by the solution of (2) and the square roots are taken with a positive real part. The functions
154
W.
Stephan and
J. P.
Carbotte
i~°(to) = iwZ°(w) are the normal state (A->0) limits of the renormalized frequencies. From (2)these are Z°N( to ) = 1 + i F N + F °
(26a)
to
and Z~(to) = 1 + i F s
(26b)
60
for to_> 0. The scattering rate F ° in (26a) is the normal state limit of the function Fl(w) defined in (9) and is given by
r °-
n
c~
E ( 2 i + 1 ) ( 1 - n,~,) 2 ~'Nso t=o
(27)
4. NUMERICAL RESULTS First consider the case of pure proximity junctions, that is, the McMillan model without any paramagnetic impurities. In this case there are two parameters that may be varied, Fs and FN. From (19) it is obvious that the total free-energy difference normalized to that of the pure superconductor may be expressed as a function of Fs and FN, with a ratio dN/ds coming in explicitly only in an overall normalization factor 1/(1 + dN/ds). It will therefore suffice to present results for the total free-energy difference with this normalization factor 1 / ( l + d N / d s ) calculated with the assumption Nso--NNo, which implies that 1 / ( l + d N / d s ) = l / ( l + F s / F N ) . If a particular junction composition is desired, then the ratio of thickness and Fs must of course be consistent with (6), which relates the F's to the densities of states and thicknesses. The total free energy or specific heat could then easily be calculated by multiplying the result calculated assuming Nso = NNo by the ratio [1 +Fs/FN]/[1 + dN/ds]. First consider the dependence of the total free-energy difference on the parameters Fs and FN. Figure l(a) shows, not surprisingly, that the magnitude of the free-energy difference per unit volume at low temperatures decreases with increasing normal film thickness, or with decreasing superconducting film thickness. Of more interest, however, may be the actual shape of the temperature dependence, which is most graphically displayed by the deviation function D(t) shown in Fig. l(b). Recall that D ( t ) for a pure BCS superconductor reaches a minimum value of approximately -0.037 at t 2--- 0.5, and that strong coupling superconductors such as lead or mercury have positive D ( t ) curves, which may be approximately attributed to the increase of the ratio 2Ao/T,. for these materials compared
Properties of Proximity Systems Including Magnetic Impurities
........
•r - - - - - - - - -
-i
. . . . . .
/" 7- .
Y " ~__ - - - - .
155
~
-0.2 -0.4
ZXF/AFo -0.6 -0.8
(a)
I
-I'.0
-0.02
I
0.2
:-.,. \
D(t-'O.04" -0.06
-.
0.4
I
~__
I
0.6
?
0.8
.... ,
/
.."/;,"
•. ,
o.-"
\
-0.08
"'..
..,...'"
~, "%
-0.10-"
...."
""
/"
/ ./ ",..,.
-O.H
.~.,.-" I
).0
/"
f
7
,,.-" \.
-0.1~
11
J /
J"
\
-0.14
ii
"~"
•
-0.12
1.0
0.2
t 0.4
(b) tz
I 0.6
I
0.8
1.0
Fig. 1. (a) The free-energy difference per unit volume of a proximity junction normalized to the free-energy difference per unit volume of the bulk s u p e r c o n d u c t o r versus reduced temperature, t = T~ To, for the proximity effect parameters: F s / T O= 0.10, F N / T O= 0.72 ( ); F s / T o = 0.72, F N / T o = 1.50 ( . - -); rsl T o = 0.72, F N / T o = 0.72 ( - - - ) ; V s / T o = 0.72, F N / T ~ = 0.10 ( - - . - - ) . (b) The deviation function, D(t)= Hc(t)tHc(O)-(1- t2), as a function of t z for the same parameters as in (a).
156
W. Stephan and J. P. Carbotte
to the BCS prediction of 3.53. Within this proximity effect model, for a given FN, as Fs varies from small values (large ds) to larger ones (small ds), the minimum of D(t) becomes more negative and also moves to lower reduced temperature t = T~ To. This behavior may be understood qualitatively in terms of the proximity effect system being an extra-weak coupling superconductor, with an "effective gap" that satisfies 2Ae~/Tc < 3.53.
0,81
't
I
l
l
0.6
AC/ACo
0.4 ,.o, .° °°" .°"
°.
,°.°
0.2
f
i
f
0.0 ......
-0.2'I
oo
t
I
02
0.4
I
o6
I
o8
I
io
t Fig. 2. The specific heat difference AC/ACo versus reduced temperature t = TI 7",. for a proximity effect junction. The specific-heat difference has been normalized to the specific heat difference at Tc for the pure superconductor ACo. The proximity-effect parameters are the same as in Fig. 1.
Properties of Proximity Systems Including Magnetic Impurities
157
Another property which might be considered here is the difference in specific heats between the normal and superconducting states of the junction. This is usually given by d2AF A C = - T. dT----T
(28)
Figure 2 shows some examples of the predicted temperature dependence
0.06
!
I
I
!
0.6
0.8
o,.*o% ;
•
0.04[ .."
0.03
0.02 \
\
\
\ x,
0.01
\ \ ,%
0.0
0.2
0.4
I.(3
t Fig. 3. T h e t e m p e r a t u r e d e p e n d e n c e o f the c o n t r i b u t i o n s o f the n o r m a l side o f a
proximity-effectjunction to the total free-energy difference. The proximity effect parameters are the same as in Figs. 1 and 2,
158
W. S t e p h a n and J. P. C a r b o t t e
-0.05 -0.10 AF/AFo -0.15 ,~os~
o,
-0.20 -0.25
(a)
0.0
I
I
I
I
0.2
0.4
0.6
0.8
I
!
1.0
t
-0.01 -0.02 D( t ) -0.03 -0.0"
1
'% "\ ' " -
4¢
,;N
\
L"
"\ %
\
I
•
"~.
•
%
"
t
t
"
°.
/..'1
i "f l ./J :"
" ":7 ~
I
,.°
-(3.0,'
-0.06 -O.OT
-0.08(~.0
I
I
0.2
0.4
t2
I
I
0.6
0.8
1.0
Fig. 4. (a) The temperature dependence of the total free-energy difference of a proximity effect junction containing paramagnetic impurities on the normal side. The proximity effect parameters are F s = F N = 0 . 3 6 T °, with eo=0, and: a = 0 . 0 1 T ° ( ); a = 0 . 1 0 T ° ( . . . ) ; a = 0 . 3 6 T ° ( - - - ) ; a = 1.00T ° ( - - . - - ) . (b) The deviation function D ( t ) as a function of the square of the reduced temperature, t = T~ T,,, for the same parameters as in (a).
Properties of Proximity Systems Including Magnetic Impurities
159
of the specific heat difference for the same parameters as in Fig. 1. Note first of all that the jump at Tc is strongly suppressed by decreasing superconducting film thickness. Secondly, the specific heat difference at low temperatures, 0.05 ~< T~ Tc <~0.5, is enhanced (becoming smaller in absolute value) for decreasing Fs (increasing ds). Figure 3 shows that fN, the contribution from (25) for the N side, has a small peak at low temperature T~ Tc = 0.1, while the contribution from the superconducting side fs (not O.30f
i
i
I
I
0.25 AC/L~Co 0.20
0.15
0.I0
0.05
0.0
I ~ ~ . / /
-
0
-
I
0
~ T/To
0
Fig. 5. The temperature dependence of the normalized specific-heat difference of a proximity effectjunction containing SR model paramagnetic impurities. The parameters are as given in Fig.4.
160
W. Stephan and J. P. Carbotte
shown) is always a monotonic function of temperature and is qualitatively the same as the total shown in Fig. 1. When the two contributions are properly combined the total free energy difference is also monotonic, but the structure in fN may show up in the temperature derivatives of the total. Figure 4 shows the dependence of the free-energy difference and D ( t ) on temperature for one set of proximity effect parameters and several 0.020
I
I
I
[
0.6
0.8
0.018
0.016 AFN/AFo 0.014
0.012
0.010
aO08
~006
0.00"
0.00;
0.(
0
0.2
0.4
1.0
t Fig. 6. The temperature dependence of the contribution of the normal side of a proximity effect junction containing paramagnetic impurities to the total free energy difference. The curves correspond to varying impurity concentrations, with parameters as in Figs. 4 and 5.
Properties of Proximity Systems IncludingMagnetic Impurities
161
parametric impurity concentrations with eo = 0. As in the non-proximity case, the minimum of D(t) increases with increasing impurity concentrations in the gapless region. The temperature dependence of the specific heat difference for the same parameters is shown in Fig. 5, where a small structure at t = 0.1 is seen to disappear with increasing impurity concentration. This is consistent with the behavior of the N side contribution to the free energy difference, Fig. 6, where the peak at t - 0.1 disappears with increasing impurity concentrations. Please note once more that the function fN plotted in Figs. 3 and 6 was calculated using only the BS formula, (25), and does not include the correction given by (A.21). As previously mentioned, within the present model the two contributions to the total free-energy difference have no physical significance when taken separately. The function fN is shown merely to indicate the source of the large curvature of the specific heat difference at t = 0.1. 5. RESULTS FOR THE JUMP 5.1. Analytic Results
At this point it is useful to reconsider the jump in the specific heat at the critical temperature, as this may be calculated analytically. This serves two purposes: first, and primarily, it is a very good check of the accuracy of the previously numerical real-axis calculation, and second, this allows the dependence of the jump on the relevant parameters to be examined with only a trivial amount of numerical computation, which is more efficient than the numerical procedure that must be used at arbitrary temperatures. This also will allow the present approach to be compared with the previous calculation of the specific heat jump of pure proximity effect junctions by Mohabir and Nagi. ~6Although the McMillan model of the proximity effect was used, their calculation of the specific heat jump was very different from the present approach; Mohabir and Nagi ~6performed and integration over the coupling constant directly, which is accomplished in the present approach through the use of the result of Luttinger and Ward) 1 The results will be compared after the discussion of the present calculation. Recall first the Bardeen-Stephen ~8 free-energy formula on the imaginary frequency axis, (21) or (25)
f~s=-2crT .>_oE[{[~(n)+ 7~2(n)]'/~-~,(n)) /
-1]
(29)
162
W. Stephan and J. P. Carbotte
For temperatures approaching Tc the pairing function A goes to zero; (29) as well as the gat~ equations may therefore be expanded in powers of this function. To order A4 the summand in (29) (call it hi(n)) becomes
h,(n)=[[~(n)+y~(n)]a/2_~,(n)][ 1 1
-2uZ(n)[~,(n)-~°(n)]
&___~(n____))
]
[o3~(n) + z~(n)]'/2J
1
8u4(n)[~,(n)-3~°(n)]
(30)
In (30) the function uI(n)= ~(n)/Ar(n) is inversely proportional to AI, so that in order to find h~(n) to order A4 one must insert ~(n)-~°(n) to order Az and o3~(n) -3o3°(n) to zeroth order in A. This may easily be done by expanding the gap equations in powers of the A's. The imaginary frequency axis version of the set of Eqs. (13) is given by the limit oJ ~ -iw,., which gives
7~s(n) = Aph + FsY~u(n)/ DN(n) ~s(n) = w. + Fs~N(n)/ DN(n) AN(n) =FNAs(n)/Ds(n)+F2(n)AN(n)/DN(n) ~N(n)=o~.+FN~s(n)/Ds(n)+Fl(n)~N(n)/DN(n)
(31a) (31b) (31c) (31d)
where
D,(n)=[A~(n)+~(n)] 1/2,
I=N
or S
(31e)
The frequency-dependent impurity scattering contributions F1 and F2 are given by
~(n)+A~(n) Fl(n) = 3" /~ (21+ 1)(1 - 7/,e,) -2 ~z z t=o oN(n)q-AN(n)et Fz(n) = 3" ,~o (21+ 1)(e,-
~%(n)+~Z(n) ~T,)et~2(n ) + ~xZu(n)~
(32a)
(32b)
with y ' = n/2"rrNNo. To order A2 the renormalization frequencies from (31) are
[ l]
G(n)=o~°+rs 1 2u~(n)
(33a)
and ~N(n)=w.+FN[1
2u~n)l+y'(1--eo)[l+~[[l--e~]--~ll
(33b)
where the limit of only a single scattering parameter eo, and no normal scattering 0% = 1, this cancels out of the final result in any event) has been taken for simplicity.
Properties of Proximity Systems Including Magnetic Impurities
163
Upon substituting the results (33) into (30) and retaining terms to fourth order in A, the summands for the free-energy difference become w. + Fs
Fs
hs(n)-4u4(n) 4uZ(n)u~(n )
(34a)
and t0.+FN
hN(n)-4u~(n)
FN
y ' ( 1 - e o ) ( 1 - e o2) 4U2s(n)u~(n)+ 2u~(n)
(34b)
At this point the other contribution to the free-energy difference on the normal side of the junction should also be considered, that is, the SR correction of Yamamoto and Nagi, 19 which has the form (22)
TSR=--2~T • ~ (21+l)at n ~ O 1=O
F
A~(n) [ o3N(n) 1] × [_(1- el) A~(n)e~+~(n) [.[¢52(n)+7~(n)] ~/2 •
'
+ ~/(.~(.)(l'i'-f-~t2)
In L ~
~
jj
(35)
where a = y'(1 - ~). Proceeding exactly as in the previous case by expanding to fourth order in A, (35) becomes
fSR=_2~rT ~ y,[--(l--eo)(l--ez)+(1--e~) 2] .->o
2u%(n)
(36)
When the above result is combined with the normal side Bardeen-Stephan term f~s given by (29), (30), and (34b), one finds
fN =f~s +fsR
=-2~rT Z_>oL4--~(~
4u2s(n)u2u(n)4 2u4(n) j
(37a)
where a = y(1 - e~) is the pair-breaking parameter when only S-wave scattering is included, and
fs =fffs = -27rT .~->0
+rs
4u~(n-~u2( n)Fs
]
(37b)
Note that in the limit of no paramagnetic impurities (~-~0), the two expressions (37) differ only by the interchange of N and S, as would be expected. Also, in the AG limit of eo~ 1 the SR correction term in fN explicitly goes to zero.
164
W. Stephan and J. P. Carbotte
Now that the free-energy difference near Tc has been expressed in terms of these sums, the remaining major steps are to determine the functional forms of us(n) and uN(n) as explicit functions of the Matsubara frequencies ico,, and then the evaluation of the sums. The first of these is very simple; the gap equations (31) may be linearized in the A's near T~, allowing the self-consistent pair of equations AN(n) = rNAs(n)[w. +FN
(38a)
+ 0¢] - 1
and As(W) = [A pho~. + rsZXN(n)][o~. + r s ] - '
(38b)
to be solved analytically. Substituting (38b) into (38a) gives
uN(n)_~_aN(n) COn
=FNAph[[w.+FN+a][w.+Fs]-FsFN] -1
(39)
which may then be inserted into (38b) to obtain
us(n)_ 1_as(n) (.O n
= a~h[co. + r N + =][[co. + r N + ~][o,. + r s ] - r s r N ]
-1
(40)
The next step is not quite as easy as the previous one; frequency sums of the terms of the form [co. + r s ] U(T)A4h = 21rT .~o L ~ J
(41a)
w"+Fu] V( T)A4h =27rT ~o5" [Lu4(n) J
(418)
1 ] W( T)A4h= 2rrT,>_o y" .~o y~ [ U2s(nfu~(n)
(41c)
Y(T)A4h=2~rT ~
(41d)
must be performed, where the functions uN(n) and us(n) are given by (29)-(40). These sums may be performed analytically in terms of polygamma functionsff which involves first expanding the summands in terms of partial fractions. The result may be expressed in terms of the variables A:~ =½[a +FN +Fs] +½[[5 +FN +Fs]2-4aFs] 1/2
(42)
Properties of Proximity Systems Including Magnetic Impurities
165
previously introduced in the Tc Eq. (15) and in terms of the new auxiliary functions E . = [[a + FN + Fs] 2 - 4aFs ] -I/ 2[FN
+a -A±]
(43)
The results may then be written in the form
E~(rs-A_) U(T)- -6-~-~T~ ~k(3)(/i-)-~ E3 [E_
EL(Fs - A + ) 6(27rT) 3 ~p(3)(fi~+)
] g,{2)(A_)
2(2rrT) 2
E3 [E+ 2(2rrT):
2E2-E+ [_2E__(2E_+3E+)Fs -A- ]
+ 2~rT(A+ - A_)
2EZ+E- [
4 21rT-~_-A+)
4
A+ - A_J ~b{l)('4-)
-2E+-(2E++3E_)r~-a+] A _ _ A+j qt°)('4+)
(A+-A_) 3
1 (A+-A_) 2 k A+-A_
_] 3
[¢,(A_)- ¢,(A+)] ,[ r N /',[FN-A_ F~-A+ v(r)=tAfSA_ J [ 6-~f~ q,(3'(a-)~ 6(2~'T) 3 ~o("(A+) x
1[ 1[
2(2,/rT) 2 1 2(2,n.T) 2 1
(44a)
4(FN -- A - ) ] ~(2)(,~_) A+-A_ J 4(AF~_NS~+++)] ~b(e)(,4+ )
2~'T(A+-A_) 21rT(A_-A+) +
2 - 5 (--~_-
~o)(,~+)
10(FN -- Fs - a) } (A+ - A_) 3 [ ~b(5._) - ~b(/T.+)]
(44b)
166
W. Stephan and J. P. Carbotte
W(T) = [A~--~-A_J [6(2~-T) 3 @(3)(~_)4 6(2rrT) 3 #/3)(~+) 1
+~[ + ~1 [
E
E+E
-
,~+-,a_ j
#~(a+)
_-
3E2_ + E2++6E_E+ + 2 7 r T ( A + - A - ) 2 g,(1)(#_) 3E2++ E2_ +6E_E+ + 2~T(A+_A_)= 2
+
2+
4(E_+E+
~(~)("~+)
3E_E+)"A
-(~-a_-L7
'-~(.,~+)]}
(44c)
f~ -)
and Y(T)
=f[ A +_rN ,_ 3 [@(3)(,~_)+ @(3)(,~+)] - A _ JFI[6(2~rT) +
2 10
20
-I 21rT(A+_ A_12 [#j(')(A_)+ O(')(/~+)] 4 (A+_ A_)3
x [~(#_) - O(Ai+)]}
(44d)
The functions O(m)(A±) are polygamma 22 functions and the arguments are defined by A± = A ~ / 2 ~ T + 1/2. Recalling the definitions of these functions (41), the result (37) for the free-energy difference becomes
[~.+r~ 4,4(,,~(,,) rs ]
Ss=f~s=-2r;T ~oL~-~) -
(mph)4
4
(45a)
[ U(T) - FsW(T)]
and - - ¢'BS.4_ ~ S R
f N - - J N --dN
f~o+r~
= -2~-T -
fo L
~
r~
~(I-4)I
4,4(,,),,L(,,) + 2,,%(,,) J
(~h)~4 [V(T)-rNW(T)+2a(1 -So2) Y(T)] 4
(45b)
Properties of Proximity Systems Including Magnetic Impurities
167
The next step leading toward the calculation of the jump in the specific heat is the determination of the temperature dependence of the order parameter Aph near To. This is most easily accomplished starting from the imaginary axis representation of the gap equation (11), which is
Nc
Aph =
As(n )
NsoV2zrT,=o ~ [0)2"[- m2(/'l)] 1/2
(46)
where the cutoff on the sum is given by (2n + 1)TrT<_ WD- NOW (46) may be expanded in powers of As near To, resulting in 1
c
NsoV =27rTy~.=o• S . -
z
(Aph)2S3.J "~ l.~(A4h)
(47)
where the function S. is given by 1
S.-
Aehus(n) tOn + F N + Ot
(.,m + r u + ~)(,o° + r s ) - r s r N
-
(48)
Now note that with the definitions N¢
1
R(T)=27rT E S . - - .=o Nso V
(49a)
P ( T ) = 7rT E $3. n=0
(49b)
and
(47) may be solved for the order parameter to get
A2h( T) = R ( T)/ P( T) (50) The functions R(T) and P(T) may be evaluated in terms of polygamma functions just as in the previous cases, resulting in
R(T)=ln[T--~-~]+O(1/2)-E_O(.4_)+E+q~(,4+)
(51a)
and
P(T)
1
4(2~T)2 [-E-~0(2)("~-) + E+~O~2)(fi,+)]
3E_E+ -~4~T(A_ - A+) [E-0~1)(A-) + E+~0(I)(A+)] 3E_E+(E_ + E+) 2(A__ A+) 2
[~O(A_)- ~O(A+)]
(51b)
168
W. Stephan and J. P. Carbotte
Also note that the function R ( T ) is actually the equation which determines Tc through R(Tc)--0. This is important to recognize as it greatly simplifies the determination of the jump in the specific heat at T~. That is, recalling that the difference in specific heat is related to the free energy difference by
[ d2AF]
(52)
and that the free-energy difference is proportional to A4h(T), one realizes that the only term which survives after twice differentiating is that which is proportional to [R'(T~)] 2 (using R(T~) = 0). Using (45) and (50)-(52) the specific-heat difference becomes
hCs
~
= (vc/2)[
[ R'(Tc)'] 2
[
(53a)
and
ACN NNo
FR'(T~) "12
= (T~/2) Lp - - - ~ ) ] [ V(T~) - FNW(T~) + 2a (1 - eo2) Y( T~)]
(53b)
where the derivative of R is given by 1
R'(T) = - - ~ + ~
E_A_
(,
-
E+A+
qJ (A_)--~-~--y O(1)(A+)
(54)
This essentially completes the calculation of the specific heat jump. Taking the limit F -->0, FN -> oo, and a = 0 of (53) the result for the jump of a pure BCS superconductor 167r 2 Tc0 ACo = --NNo ~0(2)(1/2)- 1.43yT °
(55)
is recovered, where 'Y~-
- -
0
is the coefficient of the linear term in the electronic heat of the normal state. It will be convenient to normalize the calculated values in the proximity effect case by dividing by this amount. The normalized specific heat jump is therefore AC
ACo
- [1 +
dN/ds]-' FACs+ (Fs/FN) L Nso
ACN]
(56)
NNOJ
with the contributions from the two sides of the junction given by (53), and of course the results (47) needed in the evaluation of these expressions.
Properties of Proximity Systems Including Magnetic Impurities
169
5.2. Numerical results
Figure 7 shows the dependence of the specific heat jump at Tc on Fs for various values of FN. This may be compared directly with Fig. 3a of Mohabir and Nagi, ~6 as the identical parameters have been employed. Although the results are qualitatively in agreement, they differ significantly in a quantitative sense. For small FN the present results show a much greater l,O
I
0.8
i
AClACo
I
I
I
o. i,i
-
I1"
i i
0.4? '\ "..
li )
\
i
\
"~ 0.(
L
q).0
""
\ \
"'....
\ ~
,,
" ~ . . - - %. " , . I . - - .
1.0
""%'°°
""..... ~' "1' ~ --'-" ......... I..............
2.0
3.0
4.0
rs/T~ Fig. 7. The specific heat j u m p at T,. as a function of F s for a pure proximity-effect junction with: F N / T ° = 5.0 ( - - ) ; F N / T ° = 1.0 ( . . . ) ;
r~/~=o.5 (---); r~/~=o.l
(--.--); r~/~=o.ol
(. . . .
).
170
W. Stephan and J. P. Carbotte
suppression of the specific heat jump for a given Fs than the results of Mohabir and Nagi. 16 A small peak in the jump very near the critical value of Fs in Fig. 3a of Ref. 16 is also absent in the results presented here. The source of these discrepancies may be found at the starting point of the calculation of Mohabir and Nagi, 16 their equation (28)
(3F)s = ffs 3(1/gs)A~s
(57)
together with the same expression with S replaced by N for the normal side. This ansatz immediately implies that if the pairing interaction on the normal side of the junction is zero (gN ~ 0), the normal side will make no explicit contribution to the free energy difference or the specific heat jump. The error in this procedure is that the two sides of the junction are being treated as if they were entirely independent systems and not strongly coupled by the fact that they share a common pool of conduction electrons. The difference between the present results and those of Mohabir and Nagi 16 may to a good approximation be attributed to their method's incorrect handling of the normal side in the limit of no pairing interaction, as in shown in Fig. 8, where the specific heat jumps for the two sides are plotted for the same parameters as in Fig. 7. The contributions from the N side are always negative, resulting in a decrease in the size of the jump. Note that the contribution from the superconducting side shown in Fig. 8 is very close to the total jump predicted by Mohabir and Nagi, 16 including the small peak near the critical value of Fs. This peak cancels with a peak from the normal side when the total jump, the only value with any physical significance, is calculated. Figure 9(a) shows the predicted behavior for the specific heat jump as a function of FN for various values of Fs, with no paramagnetic impurities. Also shown for reference purposes in Fig. 9(b) is the reduced critical temperature Tc/T °, as well as in Fig. 9(c) the ratio [AC/ACo][T°/Tc], which would be predicted to be equal to unity by the law of corresponding states for non-proximity superconductors. The fact that the jump is reduced by a larger amount than is Tc (as indicated by the curves in Fig. 9(c) always being less than or equal to one) is in disagreement with the prediction of Zaitlin23 for the Cooper limit (film thickness much less than the coherence length). Zaitlin23 has solved the Bogoliubov equation for the excitation spectrum of a bilayer within certain approximations and has found the jump in this limit to be given by the BCS form, with the replacement of the energy gap A by a parameter Aea. Zaitlin assumes, however, that the ratio 2Aefr/Tc = 3.53, the BCS value, without any real justification. Zaitlin also does not allow for the existence of superconductivity in the normal
Properties o f P r o x i m i t y Systems Including M a g n e t i c Impurities
1 .~0 ~
i
I
\ ..~
0.2
171
i
"'"'"'"" \
"'... %.
~\
\
• .
.
\ ~ . f \. "
0.0
--
...
~
_ __5_ .
°'.°.
\\~ .
"........
"~" ~ .
.
.
.
.
"°°,,.
"..'r- .......... ~"
-0.2 I
0.0
1.0
I 2.0
I 3.0
4.0
rs/T Fig. 8. The contributions to the specific heat jump from the two sides of a proximity effect junction as a function of F s for a pure proximity effect junction with the same parameters as in Fig. 7: F N / T O= 5.0 ( ); F N / T O= 1.0 (- • .); FN/T~ = 0.5 ( - - - ) ; F u / T O= 0.1 ( - - . - - ) ; F N / T ~ = 0.01 (. . . . ). The contributions from the S side are positive, while the contributions from the N side for the same parameters are given by the same type of line for negative AC. Note that the final three curves for the N side are indistinguishable from the axis AC = 0 on this scale. Also note that when the two contributions are combined to find the total jump they must be added with the correct weighting for their relative volumes.
W. Stephan and J. P. Carbotte
172
1.0 0
i
O.8
(a)-
0.6 <~
I
I
...... ................ •............
0.4
. .,.°....°,..,°.°°°'"°'°"°°°"
0.2
.... .
........
~
~
"
.........
o.(
.... r .............
i .............
(bl
0.( ~
.....................................................................
o0. ( =~
~
.°°,....'"'""
. . . . . . •. . . . . . . .
0 . 4 .."°o
~
;°
0.2 (3.0
~~
--
~
.-~" .
<;2
.,~,
" k- -'\
~ 0"8" 'I
.-. _ .......
=
o
.
.
.
.
r----)......... ~
.~°
"" ...... . . . . . . . . .
.......
~-.
u
............... 2""'2"2;'"~ .....~::"'2-~"'"-
",,
• "~..
~ ...,..-.-"; "_'-..L~
~" ~'I'~"~":;':;'"";"~"
~) 0.60.4i
.o
~ o.2,. ~
~
~
..*°
(c)
°
-o.o
.
1.0
2.0 rN/Te*
3.0
4.0
Fig. 9. (a) The specific heat jump at Tc as function of F N for a pure proximity effectjunctionwith:Fs/TO=O.l( );Fs/TO=0.8(. • .);Fs/ Tco_- 1.4 ( - - - ) ; F s / T ~ = 2 . 0 ( - - . - - ) ; F s / T ° = 4 . 0 (. . . . ). (b) The reduced critical temperature for the same parameters as in a). (c) The function (AC/Tc)/(ac°/T °) versus FN for the same parameters as in (a) and (b).
m e t a l r e g i o n , w h i c h is n o t a g o o d a p p r o x i m a t i o n i n t h i s e x t r e m e t h i n - f i l m limit. The present results raise some doubts about the conclusions arrived a t b y Z a i t l i n 23 f o r t h i s e x t r e m e limit. E x a m i n i n g Fig. 9 c l o s e l y , o n e m i g h t n o t i c e t h a t t h e r e s u l t s m a y b e d i v i d e d i n t o t w o c a t e g o r i e s : t h e t w o c u r v e s f o r t h i c k s u p e r c o n d u c t i n g films ( F s / T O= 0.1, a n d 0.8) a r e q u a l i t a t i v e l y d i f f e r e n t f r o m t h o s e f o r F s / T ° > 1.0
Properties of Proximity Systems Including Magnetic Impurities
1"/3
(the actual division is at Fs = dto/2 = T~/1.13). The source of this qualitative difference is that within the McMiUan model for Fs < Ao/2, Tc is finite for all values of FN, while there is a critical value of FN for larger values of Fs. As shown in Fig. 9(c), the corresponding states prediction of [AC/ACo][T°/Tc] = 1 holds only in the limits T ~ 0 or Tc-~ T O(FN ~ ) .
1.0 o
0.8
I
I
I
.................
la)
0.6 0.4 O.2 0.0 0,8
I
'-~"
i---==_-C=_ --:---.;::.;-;;;-;_-:-.;:;--.-=::
~_o O.6 "'"";';";'i.............. '""'T .......... ;......:............... ".":':....... T:'""--"",'":'"'""':
~_o 0.4 0.2 0.0
lb} :._:.._.._.
I
I
.....
-
J
~" 0.8 -~ 0.6 ~_o
0.4 (c)
o.2
0.(
,
0
I. . . . . .
1.0
I,
2.0
,
_
l
3.0
4.0
a/T2 Fig. 10. (a) the specilicheatjump at T,. as a function of paramagnetic impurity concentration for a proximity effect junction with Fs/T°=0,5, SR impurity scattering parameter % = 0
and with the following: F N / T ~ = 0 . 1
(
);
rN/~=o.8 (..-); r ~ / ~ = 1 . 4 (---); r~/~=2.0 (~.--); rN/~=4.0 (. . . . ). (b) The reduced critical temperature for the same parameters as in (a). (c) The function (AC/T,)/(AC°/T~) versus impurity concentrations for the same parameters as in (a) and (b).
174
W. Stephan and J. P. Carbotte
Turning now to the influence of paramagnetic impurities on the specificheat jump, there will again be two regimes with qualitatively different behavior: Fs < A0/2, where T~ is always finite, and Fs > Ao/2, where Tc goes to zero for some combination of FN and a. The behavior predicted in the first case is shown in Fig. 10, where the jump, reduced critical temperatures, and the ratio of the two are plotted as a function of a (for eo = 0) for 1.0
0.8 o O
<]
0.6
~
o.4
I
I
I
.
(o)
-
U
<1 0.2 0.0
..................I
....
l" . . . . .
i ......... (b)
0.8
Re o.e-
i"
0.0
~°0. 6 0.41~ 0.2
-
°°'oo
I
"'~
"-. ...... "-..
"..
-.
"--..... 2.0 '
3.0
4.0
alto* Fig. 11. (a) The specific heat jump at Tc as a function of paramagnetic impurity concentration for a proximity effect junction with Fs/TO= 1.5, SR impurity scattering parameter S o = 0 and with the following: F N / T ~ = 0 . 8 ( );
FNI~= 1.5 (-" '); FNIT°=3.0 ( - - - ) ; £N/T~=5.0 ( - - . - - ) ; r ~ / r o = ~0.0 (. . . . ). (b) The reduced critical temperature for the same parameters as in (a). (c) The function (AC/Tc)/(AC°/T~) versus impurity concentrations for the same parameters as in (a) and (b).
Properties of Proximity Systems Including Magnetic Impurities
175
Fs = T°/2.0, and various FN. Note that in this case the influence of the paramagnetic impurities is not very great, with a large concentration being required to suppress either Tc or the specific-heat jump appreciably. The case of a thin superconducting film with Fs/T°~= 1.5 is shown in Fig. 11, In this case the specific heat jump is again more strongly suppressed
I
|
I
(a)
0.8 0
0.6
0.4~ <1
0.2
O.C
I
~o
0.6
~.o
o.4~
-'--" . . . . .
0.2
~
~
*~ "~'-~
(b)
O.~"
0.0
" ~"
~
~
.___
~
.
I
.
.
I
0.8 " ~ ' ~ ' ' ~ ' ' " - ' - ~ - - ~
(c )
[-° o.6 o.4
f f 0.2
°'°'.o-o
1.0
' 2.0
&O
4.
(~/Tg Fig. 12. (a) The specific-heatjumpofaproximity effectjunctionas afunction of SR impurity concentrations for the parameters: F s / T ~ = 1.5, FN/T~ = 3.0, Co=0.0 ( - ) ; r s / T ~ = l . 5 , F N / T ° = 3 . 0 , e o = l . 0 ( . . . ) ; Fs/T°=l.5, F N / T ~ = 10.0, e o = 0.0 ( - - - ) ; U s / T o = 1.5, F N / T o = 10.0, e o = 1.0 ( - - - - - ) . (b) The reduced critical temperature for the same parameters as in (a). (c)
The function
(AC/Tc)/(AC°/T~) versus
same parameters as in (a) and (b).
impurity concentrations for the
176
W. Stephan and J. P. Carbotte
than the critical temperature, although both go to zero at a critical concentration ac. Figure 12 shows that all of these results depend only very weakly on the SR scattering parameter e0, as the cases eo = 0 and eo = 1 are nearly identical.
6. C O N C L U S I O N S A variety of properties of proximity effect junctions have been considered within the McMillan 2 model. The free-energy difference is found to be given by applying the standard Bardeen-Stephan 18 result to both sides of the junction and adding the contributions weighted with the appropriate volume fraction. The temperature dependence of the free-energy difference is found to differ significantly from that of a BCS superconductor, with the deviation function becoming much more negative than the BCS prediction. In disagreement with a previous calculation, 16 the jump in the specific heat at Tc is predicted to have a significant (negative) contribution from the normal side of the junction, resulting in a more rapid depression of the jump with increasing normal film thickness than previously predictedfl 6 Both the free-energy difference and the specific-heat jump are suppressed by the addition of paramagnetic impurities, although these properties are relatively insensitive to the details of scattering as described by the SR scattering eo.
APPENDIX:
FREE-ENERGY
DIFFERENCE
O F A PROXIMITY J U N C T I O N The free-energy difference between the superconducting and normal states of a strong-coupling superconductor was cast into a very useful form by Bardeen and Stephens 18 (BS), who generalized the work of Eliashberg 24 and Luttinger and Ward 2~ to this particular case. The work of BS has likewise been generalized to include further complications. For example, Mitrovi6 and Carbotte 25 have derived an analogous result for cases where the normal-state electronic density of states has structure on a frequency scale of the order of the Debye frequency. Yamamoto and Nagi 19 have found that a correction term is necessary when parametric impurities in the model of Shiba 7 and Rusinov 8 are included. Following the spirit of the calculation of Yamamoto and Nagi, 19 it will be shown in this appendix that the convention BS a8 equations may be used to calculate the free-energy difference of a proximity bilayer within the McMillan 2 model, with "correction" terms canceling exactly to zero. This calculation will exclude SR
Properties of Proximity Systems Including Magnetic Impurities
177
model impurities for the sake of brevity; the generalization required to include this effect follows exactly the calculation of Yamamoto and Nagi. 19 The starting point of the calculation will be the free-energy functional -1 =2-# _k,. ~] Tr[ln[~sl(-k' n)]+Xs(_k, n)~Js(_k, n)] 1
23 ~
Tr[ln[~J;)(_k, n)]+EN(_k, n)~JN(_k, n)]
1
+~-~ g,,~ V_kq_Tr[p3~s(_k, n)p3~s(q, m)] q,m
1
+T-~2 ~ r~q Tr[p3C~s(_k, n)p3q3N(q, n)] 23 _k,n -
(A.1)
q
The functions ~s(_k, n), q3N(k, n), Es(_k, n), and E(k, n) are the 4 x 4 matrix Green's functions and self-energies where the subscripts S and N refer to the superconducting and normal sides of the junction respectively, and by Tr the trace of the matrix is meant. Note that the pairing interaction has been used in the BCS form, so that phonon contributions need not be considered. By construction this functional f~ is stationary with respect to variations of the self-energies about their true values, and reduces to the functional of BS in the nonproximity limit. These are the necessary and sufficient conditions of Luttinger and Ward 2~ for this functional to yield the correct free energy, within of course the approximation that only these most important self-energy contributions are being retained. Explicitly using Dyson's equation ~ ; ' ( k n) = (~°)-'(_k, n) - E , ( k , n),
I = n or S,
(A.2)
it is easily verified that requiring the first-order variation of the functional (A.1) be zero ( S f / = 0) results in exactly the self-energies 1
~s(_k, n ) - - ~ ~] Vk_qP3~Js(_q,m)p3+ T2~ p3qJN(q, n)p 3 _q.m q = EgCS(_k, n)+Esr(_k, n)
(A.3a)
and EN(_k, n)= T2y. pards(q, n)p3 = ~(_k, n) _q
(A.3b)
(p, or are Pauli matrices), which are the self-energies of the pure McMillan model.
178
W. Stephan and J. P. Carbotte
To begin with, use the identity Tr[ln A] = ln[Det(A)]
(A.4)
for A any Hermitian matrix, with Det signifying the determinant of the matrix (this is trivially true for A diagonal, and in the general case A may be diagonalized by a unitary transformation), in order to rewrite one contribution to (A.1) in the form Tr[ln ~)-~(k, n)] = ln[Det[~-l(k, n)]]
(A.5)
Recalling the general form for the inverse of the matrix Green's function ~7l(_k, n) = ko,Zl(_k, n)--(eg+X_k)P3--q~l(k , n)p2o-2
(A.6)
(with small notational changes: e k + g k = ~ , and ¢i(_k, n)=A(_k, n)) and introducing the functions Gi(k,n)=[(~x(_k,n)]11 =
ito,Zl(n)+(e_g+X_k), N~(k,n)
(A.Ta)
and
F~(_k, n)= [~,(k, n)]l,=
¢,(_k, n) N,(_k, n )
(A.7b)
where
N,(k, n) = to~.Z~(_k, n) + ( ~ + X_~)2+ ¢~(_k, n)
(A.8)
together with the definitions Y.1,(k, n ) = []~l(k, n)]11
(A.9a)
~2, (k, n ) = []~/(_k, n)]14
(A.9b)
Det[ ~-l(k, n)] = ¢~(k, n)
(A.10)
and
one may write
with Ct(k, n ) = [[/too- e_k--~ll(k, n)][ito, + ek + Xl*l(_k, n ) ] - Y-~t(_k,n)] (A.11) Equation (A.10) immediately gives ln[Det[ ~)-~(_k, n)]] = 2 In ¢,(k, n)
(A.12)
179
Properties of Proximity Systems Including Magnetic Impurities
Using the above definitions once more, and the fact that the Pauli matrices are traceless, the traces of the other terms which are needed in the evaluation of the functional fl are readily evaluated. For example, Tr[~,(_k, n) ~,(k, n)] -1 - Nt(_k, n~Tr [[iron[1- Zt (_k, r/)]-I-)(kp3-t-~b, (_k, n)p20r2] x [io~°Z,(k_, n) + [~_k+ X_~]p3+ 4,i (_k, n)p2~2]] -4
- N,(k, n~ [ito,Zr(_k, n ) i t o , [ l - ZI (_k, n ) ] + (e_k+X_k)X_k+~b~(_k,n)] = 4[Re[G1(_k, n)?Bl,(_k, n ) ] - Ft (k, n)E2,(k, n)]
(A.13)
Note that the band-structure renormalization Xk is to a good approximation equal to zero, and in the conventional approximations (which will also be made here) is set to zero. In this case it is not really necessary to take the real part (Re) of GIEI as is done in (A.13), as this quantity is real for Xk = 0. In a similar fashion one finds
Tr[p3G1(_k, n)p3Gj( q, m)] = 4[Re[G,(_k, n)Gs(q_, m ) ] - F1(_k, n)Fj(q, m)]
(A.14)
where ! and J may be N or S as required. Using the results (A.12)-(A.14), the free-energy functional (A.1) may be written in the form 1 II = - f l _k.,E[ln[~bs(_k, n)] + 2[Re[Nls(k, n)Gs(k, n ) ] - E 2 s ( k , n)Fs(k, n)]] 1
- - - E [ln[~bN(_k, n)] fl _k,n +2[Re[?B1N(k, n)GN(k_, n)]--~2N(_k, n)FN(_k, n)]] 1 +--f12 _k,, • Vkq[Re[Gs(_k, n)Gs(q_, m)]-Fs(k, n)Fs(q_, m)] q_,m
2 +-- ~, T2kq[Re[Gs(k_,n)GN(q, m)]-Fs(k, n)FN(o, m)] _k, n
--
-
(A.15)
-
q, r n
The momentum sums in (A.15) could now in principle be evaluated, but as there is a significant amount of cancellation among the terms which may be easily seen at this point, it is useful to first manipulate (A.15) slightly.
180
W. Stephan and J. P. Carbotte
Recalling the self-energies (A3) and (A15) become 1 fI=--fl_k,n ~ [ln[q~s(-k' n)] +2[Re[~:ls(_k, 1
n)Gs(_k, n)]-~2s(_k , n)Fs(_k, n)]]
_k,~[ln[~bu(_k, n)]
+2[Re[Xlu(_k,
n)aN(_k, n)]-X2N(_k, n)FN(_k, n)]]
1
+ - - X [Re[as(k,_ /3 _k,,
n)y.~cs(k_,_n)]-Fs(_k, n~EBCStk, 2s ,_, n)]
1
+ - - Y, Y, [Re[Gs(_k, n)~rls(_k, n)]-Fs(_k , n)E~'s(_k, n) /~ _k,n_k,n + Re[aN(_k, n)~,r~N(k, n)]-FN(k, n)EfN(k, n)]
(A.16)
In (A.16) the terms on the last three lines combine exactly to cancel out one half of the terms of the form 2IRe GtEI~ -FIE21] in the first two lines, resulting in the functional /3 k.n [ln[~bs(_k, n)] + Re[~:ls(_k, n)as(_k, n)]-Ees(_k, n)Fs(k, n)] a=-LE 1
- - - Y~ [ln[ON(_k, n)] /3 _k,n + Re[E~N(_k, n)aN(_k, n)]-~12~(_k,
n)FN(_k, n)]
(A.17)
At this point the conventional approximation is for evaluating the normal state limit of (A.17) may be made by taking the limit of FI and E2~ going to zero, without, however, replacing the diagonal Green's function G~ by its true normal state value. That is, if one denotes the normal state limits by a superscript 0, G O is approximated by simply taking the limit ~b~-->0. This is reasonable because of the stationary property of the free-energy functional; a small error in G O will induce only higher order errors in the free-energy difference. The difference in free energies between the superconducting and normal states of the junction then takes the form /3 ~, -E°s(_k,
n)as(k, n)
[~b](_k,-n-) +Re[Zls(k,
n ) a°s(_k, n)]-E2s(_k, n) Fs(_k, n ) ]
-ly~[ln[~-~)]+Re[EiN(_k,n)aN(_k,n) 3 _~,
L 4,N(_k,
-
-
-E°u(k, n)a°(_k, n)]-EzN(_k, n)FN(k, n) /
3
(A.18)
Properties of Proximity Systems Including Magnetic Impurities
181
At this point (A.18) appears to be simply the BS result, repeated for the two sides of the junction N and S. There is, however, a significant difference. The terms involving E1~GI and the normal-state limits of this may be written in the form El1Gi - E lol G to -_ [GI + G to] [ Z ~ , - E ~o, ] + C, (A.19) where C, = G,Y. ° - G ° E ,
(A.20)
and the arguments (_k, n) of all functions in (A.79) and (A.20) have been suppressed. In the nonproximity case the correction Ct may easily be shown to be zero after performance of the momentum sum by using the equations for the self-energy in terms of the Green's function. In the proximity case however, these become
Cs =
T 2 ~ [Re[Gs(k, _q
n)G°N(q,n) -- G°(_k,n)GN(q, n)]] # 0
(A.21a)
n)G°s(q, n) - G°(k, n)Gs(q, n)]]
(A.21b)
and CN = T 2 Y~[Re[GN(k, _q
Taken separately, neither Cs nor CN are zero after summation over k, but they are equal and opposite and therefore cancel exactly. In the numerical evaluation of the free-energy difference it is obviously useful to take advantage of this cancellation from the outset, which amounts to using the BS functional with no corrections. The only caveat is that without the inclusion of these corrections the separate contributions to the free energy difference from the two sides have no physical significance at all, only the total has meaning. With the inclusion of the corrections, one might consider examining the separate contributions from N and S sides, although on very general grounds it could also be argued that this division is not at all unique. In general, the energies of the components of an interacting system can only be defined in an arbitrary manner. Using (A.18) and (A.19) the free-energy difference may be written in the form In k ~ _ k , n - ) x {l~,s(k,
+ R e [{Gs(_k,_ n)+G°s(k,_n)}
n)-N~s(_k, n)}]-1~2s(_k, n)Fs(_k,n) I
- 1 ~ [lnr4N(-k'~)]+Re[{GN(k_,n)+G°N(k,n)} #
L 4,°(_k,
-
_
x{X,N(_k,n) - ~ uo( _ k , n)}]-~2N(_k, n)FN(k, n) /
J
(A.22)
182
W. Stephan and J. P. Carbotte
Now the functions (A.7)-(A.9) may be substituted into (A.22) and the integrals 7/"
1
f ~ de 8 2 + A
2 --
A
ERe(A) > 0]
(A.23a)
and
de In
A
e2-T~
= ¢r(A-B)
ERe(A, B ) > 0]
(A.23b)
used, in order to rewrite the free-energy difference (A.22) in the form ~_f~o=
fl
NsodsA ~ [2{[o3~(n) + ~b~(n)] '/2 -IO3s(n)J}
_ f,.
~(n)
[[oS~(n) + 6~(n)] '/2
sgn(to°)}(~s-~g}
[o;~(n) + 4~(n)]'/=3
-~ NNodNA ~ [2{[o32(n)+ q~(n)]'/'-[~N(n)I} _[
,~,,,(")
[ [ o ~ 2 ( n ) -4- q ~ Z ( n ) ] l / 2
¢~(n)
sgn(w.) }{~SN-- ~5°}
,-~
(A.24)
The notation o3f(n)= to.Z1(n) and o3°(n)= to.Z°l(n) has been introduced in (A.24), as well as the function sgn(to.), which is 1 if to. > 0, and -1 if to. < 0. This result may be further rearranged to the form
_ l~o= _27r NsodsA /~
E [{[a3~(n) + ¢~(n)] '/2- [O3s(n)l}
n~0 /
x 1-[t~2s(nl+cb2s(n)],/2jj -2"n'NNodNA Z [{[t~(n)+ fb~(n)]t/2--]t~N(n)l} n>O
x{1
-°3°(n)
II
[oS~,(n)+ ~,(n)]'/=J3
(A.25)
Properties of Proximity Systems Including Magnetic Impurities
183
which is the form required in the text. The extension of the above work to include paramagnetic impurities within the SR model on the normal side of the junction may, upon examination of the calculation of Yamamoto and Nagi, 19 be performed by adding the correction, their Eqs. (27) and (22) here, to the above result. REFERENCES 1. G. Deutscher and P. G. de Gennes, in Superconductivity, R. D. Parks, ed. Marcel Dekker, New York, 1964), vol. 2, p. 1005. 2. W. L. McMillan, Phys. Rev. 175, 537 (1968). 3. A. B. Kaiser and M. J. Zuckermann, Phys. Rev. B 1, 229 (1970). 4. P. W. Wyatt, R. C. Barker and A. Yelon, Phys. Rev. B 6, 4169 (1966). 5. P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959). 6. K. Machida, J. Low Temp. Phys. 27, 737 (1977); K. Machida and L. Dumoulin, J. Low Temp. Phys. 31, 143 (1978). 7. H. Shiba, Prog. Theor. Phys. 40, 435 (1968). 8. A. I. Rusinov, Zh. Eksp. Teor. Fiz. 56, 2043 (1969) [Soy. Phys. JETP 29, 1101 (1969)]. 9. A. B. Kaiser, Phys. Rev. B 22, 2323 (1980). 10. J. Zittarz and E. MiJller-Hartmann, Z. Phys. 232, 11 (1970). 11. E. Miiller-Hartmann and J. Zittarz, Z. Phys. 234, 58 (1970). 12. J. Zittarz, Z. Phys. 237, 419 (1970). 13. N. Mori, J. Low Temp. Phys. 40, 275 (1980); 43, 107 (1981). 14. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956). 15. P. Fulde and W. Moormann, Phys. Kondens. Materie 6, 403 (1967). 16. S. Mohabir and A. D. S. Nagi, J. Low Temp. Phys. 36, 307 (1979). 17. J. Lechevet, J. E. Neighbour, and C. A. Shiffman, Z Low Temp. Phys. 27, 407 (1977). 18. J. Bardeen and M. Stephan, Phys. Rev. B 6, 1485 (1964). 19. H. Yamamoto and A. D. S. Nagi, Phys. Rev. B 30, 1573 (1984). 20. J. Schrieffer, Theory of Superconductivity (W. A. Benjamin, New York, 1964). 21. J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1418 (1960). 22. P. J. Davis, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1965), p. 253. 23. M. P. Zaitlin, Phys. Rev. B 25, 5729 (1982). 24. G. M. Eliashberg, Zh. Eksp. Teor. Fiz., 43, 1005 (1962) Soy. Phys, JETP 16, 780 (1963)]. 25. B. Mitrovi6 and J. P. Carbotte, Can. J. Phys. 61,872 (1983).
