Calculation of the penetration depth and electromagnetic absorption edge for strong-coupling superconductors

CALCULATION OF THE ELECTROMAGNETIC STRONG-COUPLING

PENETRATION DEPTH AND ABSORPTION EDGE FOR SUPERCONDUCTORS?

J. C. SWIHART Department

of Physics,

Indiana

and W. SHAW

University,

Bloomington,

Indiana

47401 USA

Synopsis

We have calculated the electromagnetic response function for various cases for a model of lead, both for the pure metal and a dilute nonmagnetic alloy, and both for the weak-coupling and strong-coupling theories using Nam’s theory. With these results, we have calculated the lowfrequency penetration depth and the high-frequency surface resistance in the neighbourhood of the gap edge for these various cases. We find a penetration depth for strong-coupling pure lead of 406 A in better agreement with the experimental result of 390 A than previous calculations. The calculated absorption edge is steeper for the strong-coupling theory than for the weak-coupling theory and also is in good agreement with the experimental results of Leslie and Ginsberg and of Norman. The alloy calculation has a slightly steeper absorption edge than the calculation for the pure metal.

The response to an electromagnetic field is one of the most fundamental encompassing the zero resistance and properties of a superconductor, Meissner-Ochsenfeld effect. The first direct evidence for an energy gap in a superconductor was through the investigation of the interaction of high frequency electromagnetic waves with the superconductor’). The general behavior of a weak-coupling superconductor, such as Al, in an electromagnetic field is well understood in terms of the BCS theory2) as applied by Mattis and Bardeen3). However there are some discrepancies between the quantitative experimental results on certain superconductors and the MattisBardeen theory, as pointed out by Richards and Tinkham4), Ginsberg5), Palmer and Tinkham’j), and Norman7) among others. These discrepancies occur with the strong-coupling superconductors, particularly lead. And they involve an absorption edge that is much steeper than that of the theory, or for thin films a transmission coefficient that is larger than for the theory in the neighborhood of the gap edge. It has previously been shown6,8) that the application of the strong-coupling theory to the electromagnetic response of a superconductor as worked out by Nams) does account for the anomalies in the electromagnetic properties of Pb alloy films in the Pippard limit (limit of very large wave vector 4 % l/&,). t Research supported in part by a grant of the National Science Foundation. 678

PENETRATION

DEPTH AND ABSORPTION

EDGE FOR Pb

679

More recently Harris and GinsberglO) have shown that the strong-coupling theory also agrees with their transmission data on bismuth and gallium films in the Pippard limit. However there still remains the problem of strong-coupling superconductors not at one of the extreme limits. At the extreme limits, life is simpler due to the fact that the frequency and wavenumber dependent complex conductivity in the superconductor, a&q, o), when divided by the corresponding quantity in the normal state, u,(q, w), is independent of $.g). In fact, this frequency dependent ratio c~/w,(o), is the same in the London and extreme Pippard limits. However for bulk Pb one does not have the extreme Pippard limit no matter how long the mean free path. For although one can have 1% A if the purity is high, the purest lead has a coherence length to = (&v&rAo) = 780 A, while the penetration depth A = 390 All). Thus it is hardly true that A e to. Tinkham? and Ginsberg5) have taken note of this fact, and Ginsberg5) has carried out calculations of the absorption edge of dilute alloys of Pb using the weakcoupling theory. For this it was necessary to consider the q dependence as well as the frequency dependence of the conductivity. He also put in a mean free path dependence in order to compare with previous experiments of Leslie and Ginsberg13). Although Ginsberg’s calculations gave a steeper absorption edge than did the weak-coupling theory in either the London or extreme Pippard limits as Tinkham12) had predicted, his results were still not as steep as the experiments. If the theory gave a steeper edge than the experiments one would not be too concerned since this could always be understood in terms of experimental samples with inhomogeneities. Any inhomogeneity would tend to broaden the absorption edge, and it is impossible to have perfectly homogeneous samples. However the reverse situation in which the experiments give a steeper edge than the theory clearly points to an inadequacy in the theory. And this is the situation with regard to the weak-coupling theory when applied to lead. Norman7) has carried out experimental determinations of the absorption edge of pure thick films of superconducting Pb, Sn, In, Ta, Nb, and V. The mean free path seems to be long enough so that they are not in the London limit, and the films are thick enough so that they are not in the Pippard limit for Pb. He found that the initial slope of the absorption edge was dependent on T,/& of the superconductor. The smaller T,/&, materials had an edge that essentially agreed with the Mattis-Bardeen3) theory in the extreme Pippard limit, while with the larger ratio of T,/O, the experiments gave steeper edges than the theory. For Pb, the material with the largest T,/8, ratio, he found that the initial slope of the absorption edge was an order of magnitude greater than the weak-coupling theory. These results are suggestive of a strongcoupling effect. In an attempt to explain the results discussed above in terms of the strongcoupling theory, we have applied Nam’s theorys) to the calculation of the

680

J. C. SWIHART

AND

W. SHAW

absorption edge of pure Pb and a dilute nonmagnetic alloy of lead at T = 0. We have also carried out a calculation of the zero temperature penetration depth of lead, since a widely quoted theoretical value14) for this quantity of 480 A seemed somewhat high compared to the experimental value of 390 _.&I]). In any case, it is of interest to determine the strong-coupling effect on the magnitude of this quantity. For our model of lead we have used the square of the electron-phonon interaction times the phonon density of states, a2F(w), which was determined from tunneling by McMillan and RowelP5). We solved the non-linear energygap equation16) for the complex energy-gap function A(w) and renormalization function 2, (0) at zero temperature. A(w) = AI(w) +iA,(o); Z,(w) = Z,,(w) +iZ,,(w).

(1)

The Coulomb interaction tI, was taken to be 0.127 158 and the cutoff in energy in the integral equation was 104.0 meV or more than ten times the position of the phonon peak of largest energy. Corrections were still made for the contributions of A1(w) and Z1 (w) for energies larger than the cutoff. We then adjusted the over-all strength of o12F(w) to give an energy gap of 1.3405 meV. We found that multiplying the McMillan-Rowe11 02F(w) by 1.105 would give this energy gap. We also calculated the complex renormalization function for the normal state, Z,(w), with this ~-y~F(w). Our calculated gap function A(w), which is given in fig. 1, is almost identical to McMillan and Rowell’s15).

I

I

I

1

p\

I ‘_.“,

3-

\ \

IL

I. The complex

frequency

dependent

linear gap equation

I

Pb

\

I

Fig.

I

I

I

T=O

20 meV

energy-gap

function

using cu’F(o) of McMillan

for lead calculated and RowelI’“).

from the non.

PENETRATION

DEPTH

AND ABSORPTION

EDGE

FOR Pb

681

With these functions, we have calculated the wave-number and frequency dependent complex response function K(q, w) of Namg) for various cases in the superconducting and normal state. K relates the current density] to the vector potential A. j(q, w) = - (+r)K(q,

o)A (41 w) -

The complex electrical conductivity

(2)

u is closely related to K by

&Gic0)

u(q,w)=-i

The most general equation relating j to A gives K as a tensor, but for a cubic material such as Pb this reduces to the scalar quantity of eq. (2). We have previouslyE,’ determined C&,,(W) for Pb at zero temperature in the Pippard and London limits, and from this had calculated oSZ/rr,(o) in the region &J s 2A, by the Kramers-Kronig relation*). Nam’s original curve for oSse/a, for Pbg) is too large and the corrected value was given only in the region of the gap edge’*). We have calculated this quantity over the entire frequency range both directly and from o&,,((oj by the Kramers-Kronig relation. Figure 2 gives the results of this calculation. We see that aSp changes 1.4r

I

1.2-

1

-

I

,

,

Pb LONDON LIMIT

D

f

6

,

,

,

__--~~ PI ‘54

*=.a b”

,

T=O OR PIPPARD LIMIT

1 ---

MATTIS-BARDEEN

4

Q’ b-.2

Fig. 2. The real and imaginary frequency dependent conductivity (normalized by the normal conductivity) for lead in the London and Pipparci limits. The solid curves are for the strong-coupling theory while the dashed curves are the Mattis-Bardeen3) weak-coupling theory.

sign at about 60 = 2_5(2A,,) indicating that the superconducting system goes from inductive to capacitive. We found that Namg) had dropped a term in his expression for aSJ~,,(o) for no > 2A,, which is in his expression for the response function. This term, which we denote by SUM3, vanishes for weakcoupling superconductors.

J. C. SWIHART

682

SUM3(o)

= d

j-

AND W. SHAW

do’

A,/&-W

+Im

(4)

W+O’ [(o+~‘)~-A(,+,‘)~-j”~

[CLP-

II*

A(w’) 12_ A(Wf)2]1/2 Cm

In fig. 3 we show C&~(W) calculated with and without SUM3 in the region where it makes a contribution. Only the expression with SUM3 included agrees with u&r,, from o,Jon by the Kramers-Kronig relation. Note’ that the scale on the ordinate is expanded in fig. 3 compared to fig. 2. Thus this is a small effect probably beyond the range of accuracy of present experiments.

.I6

I

,

I

,

LONDON

,

,

,

,

,

Pb T=O LIMIT OR PIPPARD

,

I

LIMIT

$04L bN

0 WITHOUT

SUM 3

-.04WITH

FREQUENCY

SUM 3

hw42A,)

Fig. 3. An expanded scale drawing of the imaginary part of the conductivity for lead according to the strong-coupling theory in the frequency range where u,, becomes negative. For the term SUM3 see the text.

The response of the forms)

function

hF(q,

K(q, o) is calculated

E, I’){&,

w*x)

A 11,

from a sum of integrals, each

(5)

where (6)

PENETRATION

DEPTH

AND ABSORPTION

EDGE

FOR Pb

683

is the London penetration depth. N(0) is the electronic band density of states for one spin at the Fermi surface, vF is the Fermi velocity [both without the phonon enhancement] t

F(q,E,r) S=

=&F 2S+

(1-P)

In=

=

(7)

(8)

(E-iiT)/(h+q);

&yx’)

1;

XX’+ A (x)A(x’) [xZ_~(x)2]1/2[(x~)2_~(x’)2]1/2’

(9)

E and r are sums of the real and imaginary parts respectively of Z(o) [o* Am] 1/Zfor the argument o replaced by x and by o +x. We see that a natural function to consider is hL2K(q, w) which is dimensionless. For our calculations which are in neither limit we have lifetime effects. For the strong-coupling case the lifetime r or energy width T of the quasiparticle occurs as fi/~ = r(w)

= Im {Z(W) [w~-A(w.)~]~‘~},

(10)

which is frequency dependent and is due to the decay of the quasiparticle to a lower energy state combined with the creation of a phonon. This does not occur in the weak-coupling theory. In both theories there can be a lifetime due to scattering of the quasiparticle by ncbnmagnetic impurities. This is expressed in terms of a mean free path 1 and gives a contribution to the energy width of

rl = ti2+/1,

(11)

which is independent of w. For the strong-coupling theory, the total width is the sum of eqs. (10) and (11). Our pure lead calculations are for 1= 03or Tl = 0, while our alloy calculations are for I= 1.7 10e5 cm, the value taken by Ginsberg5) in his calculation for the one per cent thallium in lead alloy of

t The density of states N(0) in eq. (6) arises in Nam’s theory from changing a sum over oneelectron k states to an integration over band energies. Thus one should use the band density of states which is 0.7 times the free electron density of states [see ref. 191 or is I/Z&J = 0) = l/2.32 times the specific heat density of states [see ref. 201. The former gives a density of states of 4.61 IV3 (erg cm3)-’ while the latter gives 5.8 1 1P3 (erg cm3)-l. The Fermi velocity in Nam’s theory is the band velocity averaged over the actual Fermi surface (i.e. one should not divide by the area of the free electron sphere to determine the average). Since the Pb Fermi surface is very freeelectron like over the directions for which it exists [see ref. 211 we have used the free-electron velocity of vF = 1.83 108cm/s. This gives London penetration depths of AL= 173 A with the first density of states and AL= 154 A for the second density of states.

684

J. C. SWIHART

AND W. SHAW

Leslie and Ginsberg13). With this value of 1 we can compare with the previous resultst. At frequencies below the gap (w < 2A0), the imaginary part of the response function, K2, vanishes, and the real part K, is relatively independent of w for all 9. In fact K1 becomes completely independent of o for o small enough. Figure 4 gives plots of K1 as a function of 4 at low w for various cases.

hvrq meV Fig. 4. The wave-number

dependence of the real part of the low frequency response function for lead for various calculations.

Notice that the alloy has a smaller value of K, at each q value than for the pure metal whether in the weak-coupling or strong-coupling theory. The penetration depth is determined from a q integration of a function of K, as we shall discuss below. A smaller K1 over all 4 leads to a larger penetration depth in agreement with experiments on alloys. We also see that alloying tends to broaden the peak at 4 = 0 as well as depressing it. In fact if the mean free path is decreased sufficiently, K, flattens out to become independent of q on the scale of fig. 4. This is in the London limit. The term London limit is also used in a slightly different way to mean the function K,(q) = constant and equal to the actual K,(q) at q = 0. We have indicated the London limit for the weak coupling pure case by the horizontal line in fig. 4. t Actually, it is hvF/l that is chosen in calculating XLZKas a function of fL+q for a given frequency as we see from eqs. (5) (7) (8). and (11). The value we chose is iiv,/l = 3.10 1O-15erg corresponding to I= 1.7 10e5 cm for vr = 0.5 lO’cm/sec, the value of vr we thought Ginsberg used and the one we initially used. It was the results with this vF that were reported at the Superconductivity Conference. We have since decided that it is the unrenormalized uF = 1.83 108cm/set rather than the renormalized vF that should be used. We have not changed Vivpil;so we now have I= 6.22 low5 cm for the alloy. Ginsberg has pointed out to us that he also used the free electron value of vr. Note added in proof: We now believe, after discussiok with Bardeen and Ginsberg, that Tr in eq. (11) should have an additional factor E/E. This would change our alloy calculations slightly.

PENETRATION

DEPTH

AND ABSORPTION

EDGE

FOR Pb

685

At large q, K,(q) goes as l/q. The function a/q with a! chosen so this function coincides with K,(q) at large q is called the Pippard limit. In fig. 4 we have drawn the Pippard limit for the weak-coupling pure case. For this case the constant (Y is simply2) hL2hvFa = ~~~~~ = 9.923 for Pb. We consider q very large in eq. (5), and the only place it enters is in F. In the limit of large q, F of eq. (7) becomes simplyg)t lim F(q, E, r) = Y--+x &q’

(l-2)

which is independent of r. The mean free path enters only in r; so we have the result that the Pippard limit is independent of mean free path. This is seen in fig. 4 where the pure and alloy K,‘s come together both in the weak-coupling and strong-coupling cases faster than the weak coupling approaches the strong coupling. The definition of Pippard Limit above is the same as the Pippard limit meaning 1 + A and &, P A. The scale of q in determining the penetration depth by the integration over q is determined by l/AL (this point is marked in fig. 4). On the other hand, the region where K, changes from a constant function of q at small q to the function a/q is in the neighborhood of 115, this last value for the pure weak-coupling model of Pb also being marked in fig. 4. If 6 % AL, keeping A,> fixed, then in fig. 4 the function K1 would be a/q over the entire range except for an infinitesimal region about q = 0. This latter region being infinitesimal would not contribute to the integral for A and thus A would be determined by only the 1/q or Pippard limit of K1. But if AI, 4 ,$, we would find that also A G 5. But since l/t - l/&,+p/f where p is a constant of order 1, we can have 4 % A only if both &, Z+A and I S= A. Hence these two usages of the term Pippard limit are equivalent. We see that of the four cases calculated, the weak coupling pure Pb case is the closest to being in the Pippard limit over the entire range, but even here, t,, > AL,and K1 is not close to the Pippard limit at small values of q. Going from weak coupling to strong coupling has a similar effect on K1 for small q as does decreasing the mean free path. This is due to the fact that the quasiparticles in the strong-coupling theory have a nonzero width and this is added to fil(v,l) in the F function. The penetration depth and surface impedance for bulk samples are determined from the response function by an appropriate integral over q. In this, the type of scattering of quasiparticles at the surface must be considered. Two extreme cases that have been handled in the past22. 12)are diffuse scattering and specular scattering, with the former probably being closer to the truth t This limit is valid only for r > 0. This latter is always true even for r + 0 since n/K is the reciprocal relaxation time.

686

J. C. SWIHART

AND

W. SHAW

for our cases. The pertinent integrals over 4 for these two cases then are’“) Z(o) =?r{Tdqln

[l+K(q,w)/$]}-’

0

(13)

= rrAL[

7d(A,q) ln [I+

(~L2~)/(~L~)21)-1~

0

and

m I(w) =z

dq 7r I q2+K(4,

w)

0

(14) d(A,q) (A,q)‘+ (AL~K)

'

0

for diffuse and specular cases respectively. We have written the second form in each equation to show explicitly what we contended above, that the proper scale for q in the integral when considering the functional form of AL2K is 1/A,. The penetration depth A and surface impedance Z(o) are then A = Z(w = 0)

(15)

and Z(w) = R(o) +iX(o)

= 47riol(o)/c2.

(16)

For o < 2Ao, K = K, is real and is nearly independent of o. The same is true for f(o), and A will be real and can be closely approximated by I(w). Table 1 gives A for the various cases considered. We used o = 0.5 meV and o = 0.2 meV (2& = 2.681 meV) for these calculations. The difference in A for the two frequencies was less than 1%. The value for AL used was 173 A. The best calculated value to compare with experiment is the exact calculation for pure lead in the strong-coupling theory with diffuse scattering. This value of 406 A is within the range of error of Lock’s experimental value?. The effect of the strong coupling is to increase A, and it is seen that the strongcoupling A fits the experiment better than the weak-coupling A. The reason we obtain a smaller value of A than the previous theoretical calculation14) is due to the fact that we have used different values of AL and +. Our values of AL = 173 A and uF = 1.83 lo* cm/s are the unrenormalized values (without phonon enhancement) corresponding to the band density of states and the free electron Fermi velocity. They are what is called for in Nam’s theory and are quite different from the experimental values of AlAand I+. We have also calculated A for the weak-coupling pure Pb case with Al.= t

Lock

(1 -t4)-“‘.

determined This

A, by extrapolating

his measured A(T) to T = 0 by assuming too high.

would tend to give a A, that is slightly

h(T) = A”

345 294

Diffuse Specular

173 173

London 287 255

Pippard

Experimental value”) A = 390& 30 A

Exact

Scattering

Pure Lead

361 308

Exact

Weak-coupling theory

216 216

London

Alloy

287 255

Pippard 406 348

Exact 269 269

London

Pure Lead

313 278

Pippard

425 366

Exact

Strong-coupling theory

Calculated penetration depth for various models of lead

TABLEI

302 302

London

Alloy

313 278

Pippard

688

J. C. SWIHART

AND W. SHAW

370 A and ur = 0.5 108cm/s, the values used by Bardeen and Schrieffer14) and the values which are approximately the experimental values, and we obtain A = 480 A in agreement with their calculation. The surface resistance at the gap edge has been calculated for the same models of lead for both the superconducting and normal states. The ratio RJR, for diffuse scattering at various frequencies is plotted in fig. 5 together with the experimental data of Norman7) for two lead samples. The solid line is drawn through the calculated points for the alloy. The curves for specular scattering which are omitted from fig. 5 are more steep. We also plot this

FREQUENCY hd(2AJ Fig. 5. The ratio of the surface resistance in the superconducting state to that in the normal state near the gap edge, fro = 2&. The experimental curves are for two different samples of Pb determined by Norman’). The calculated points are for diffuse scattering and the solid line is drawn through the calculated alloy points.

quantity for the weak-coupling Pippard limit and the strong-coupling Pippard limit. From this it is seen that about half of the increase in steepness is brought about by the strong coupling and about half is from the 4 dependent response function not in the Pippard limit. Also the alloy is slightly steeper than the pure Pb model. Both the pure lead and the lead alloy models fit Norman’s data quite well. These calculations can also be compared with the data of Leslie and Ginsberg13) and we do this in fig. 6. The frequency dependent function P(o) is defined by53 13) P(0)

=

R,(o) --R,(o) +R,(o)/K’

R,W,l~)

(17)

where K is an experimentally determined quantity and is K = 4.673. 1 -P(w) is roughly equal to RJR, in the neighborhood of the gap. The calculated points in fig. 6 are for the alloy caset both with diffuse and specular scattering at the boundary. The solid curve is drawn through the diffuse points. Ginsberg’s calculation5) for the q dependent weak-coupling t See footnote on page 684.

PENETRATION

DEPTH AND ABSORPTION

EDGE FOR Pb

689

FREOUENCY tiw/(ZA,l

Fig. 6. Frequency dependence of 1 -P(o) in the neighborhood of the gap edge. 1 -P(w) is defined by Ginsbergs) and is roughly RJR,. The calculation of Ginsbergs) is a 4 and mean free path dependent weak-coupling calculation in an attempt to fit the experimental results (X points) of Leslie and Ginsberg13). The present calculations are for diffuse and specular scattering in the alloy. The solid curve is drawn through the calculated points with diffuse scattering.

case with finite mean free path is also given+ together with the Pippard limit for weak coupling and for strong coupling. Once again we see that the strongcoupling case is steeper than weak coupling and the q dependent case is steeper than the Pippard limit. The best calculation, strong-coupling q dependent with diffuse scattering is slightly steeper than the experimental data. But since the theory should be at least as steep as experiment, we feel that the experimental data are explainable in terms of this theory. Acknowledgments. The authors wish to thank Dr. D. M. Ginsberg, Dr. W. L. McLean, and Dr. M. Tinkham for valuable discussions. We are also indebted to Dr. D. H. Douglass for making available a copy of Dr. Norman’s thesis before publication and to Dr. D. M. Ginsberg for detailed information on his calculation and the experiments of Leslie and Ginsberg.

REFERENCES 1) Glover, III, R. E. andTinkham, M., Phys. Rev. 108(1957) 243. 2) Bardeen, J., Copper, L. N., and Schrieffer, J. R., Phys. Rev. 108 (1957) 1175. 3) Mattis, D. C. and Bardeen, J., Phys. Rev. lll(l958) 412. 4) Richards, P. L. and Tinkham, M., Phys. Rev. 119 (1960) 575. 5) Ginsberg, D. M., Phys. Rev. 151(1966) 241. 6) Palmer, L. H. and Tinkham, M., Phys. Rev. 165 (1968) 588. 7) Norman, S. L., Phys. Rev. 167 (1968) 393. 8) Shaw, W. and Swihart, J. C., Phys. Rev. Letters 20 (1968) 1000. $ We have reproduced Ginsberg’s calculation of 1 -P(W) with our computer program for two energy points to better than I percent for the diffuse case and to approximately 1 percent for the specular case.

690

9) 10) I 1) 12) 13) 14) 15) 16) 17) 18) 19) 20) 2 1) 22)

PENETRATION

DEPTH

AND ABSORPTION

EDGE

FOR Pb

Nam, S. B., Phys. Rev. 156 (1967) 470,487. Harris, R. E. and Ginsberg, D. M., Phys. Rev. 188 (1969) 737. Lock, J. M., Proc. Roy. Sot. A208 (195 1) 391. Tinkham, M., in Optical Properties and Electronic Structure of Metals and Alloys, ed. by F. Abeles, North-Holland Publishing Co., (Amsterdam, 1966), p43 I. Leslie, J. D. and Ginsberg, D. M., Phys. Rev. 133 (1964) A362. Bardeen, J. and Schrieffer, J. R., in Progress in Low Temperature Physics, ed. by C. J. Gorter North-Holland Publishing Co., (Amsterdam, 1961), Vol. 111,~243. McMillan, W. L. and Rowell, J. M., Phys. Rev. Letters 14 (1965) 108. Superconductivity, R. D. Parks, ed. Marcell Dekker, Inc. (New York, 1969) Vol. 1, p. 561. Scalapino, D. J., Schrieffer, J. R. and Wilkins, J. W., Phys. Rev. 148 (1966) 263. Swihart, J. C., Schrieffer, J. R. and Scalapino, D. J., unpublished; reported at the 1965 Annual A.P.S Meeting, Bull. Amer. Phys. Sot. Ser. II, 10 (I 965) 7. Nam, S. B., private communication. Nam’s corrected u&r,, in the region of the gap edge is given in ref. 6. Smith, F. W. and Cardona, M., Solid State Commun. 6 (1968) 37. Swihart, J. C., Scalapino, D. J. and Wade, Y., Phys. Rev. Letters 14 (1965) 106. Anderson, J. R. and Gold, A. V., Phys. Rev. 139 (1965)A 1459. Reuter, G. E. and Sondheimer, E. H., Proc. Roy. Sot. (London) Al95 (1948) 336.