thermodynamics of superconducting magnetic alloys in the weak coupling theory

Journal c~[Low Temperature Physics, Vol. 19, Nos. 5/6, 1975

Thermodynamics of Superconducting Magnetic Alloys in the Weak Coupling Theory

John Lam

Tile Dikewood Corporation, Unicersit), Research Park , Albuquerque, N e w Mexico 87106

(Received November 15, 1974)

The thermodynamics of superconducting magnetic alloys is studied in the weak coupling theory presented in a previous paper. This theory describes the effect of magnetic impurity scattering by means of a constant conduction electron lifetime. The variation of the superconducting order parameter with the pair- breaking parameter and the temperature is obtained numerically. The free energy, the critical magnetic field, the entropy, and the electronic specific heat are calculated. The critical magnetic field at zero temperature is found to decrease linearly with increasing impurity concentration. The specific heat uaries linearly with temperature at near-zero temperatures, due to a finite density of states at the Fermi lerel.

i. I N T R O D U C T I O N

The properties of superconductors can be systematically altered by the addition of small quantities of magnetic impurities. This effect is believed to arise from an exchange interaction between the conduction electrons and the spins of the impurity atoms. The first theory to explain this phenomenon was advanced by Abrikosov and Gor ' kov 1 (AG), who concluded that such superconductors can become gapless. Although the predictions of this theory are generally in fair qualitative agreement with experiment, serious deviations do frequently occur. In particular, its expression for the density of states about the Fermi level does not fit the available data.

In a recent publication, 2 hereafter referred to as paper 1, we presented a theory of superconducting magnetic alloys in the weak coupling limit. It is valid when the electron-impurity interaction is much weaker than the superconducting pairing interaction, as is the case when the characteristic temperature TK of the spin exchange interaction for the magnetic alloy in the normal phase is many times smaller than the transition temperature of

467 �9 1975 P l e n u m Pub l i sh ing C o r p o r a t i o n , 227 West 17th Street , N e w York , N .Y. 10011. N o par t o f this pub l i ca t ion may be reproduced~ s tored in a retr ieval sys tem, or t r ansmi t t ed , in any fo rm or by any means~ electronic, m e c h a n i c a l p h o t o c o p y i n g , mic ro f i lming , record ing , o r o therwise , w i thou t wri t ten pe rmis s ion of the publ i sher .

468 John Lam

the pure superconducting host. The electron-impurity interaction can then be treated in the lowest order, and gives rise to a pure lifetime effect. The content of this theory is completely summarized by the boundary values on the real energy axis of the pair of normal and anomalous two-time retarded thermodynamic Green's functions at temperature T:

+ ice + e k

- A ( ( a _ h - * = 2 __ m 2 ak'f))o~+ie ( ~ __+ i ~ ) 2 _ ~3 k

(1)

Here e k is the conduction electron energy relative to the Fermi energy, and A is the superconducting order parameter. The width ~ due to electron- impurity scattering, or the pair-breaking parameter, is calculated from a one- impurity spin exchange model in the lowest order and reads

= � 8 8 + 1) (2)

where c is the impurity concentration, S is the spin of the impurity, J is the exchange coupling constant, and p is the density of electronic states per electron per spin near the Fermi level.

The results of our theory are often qualitatively similar to those in the AG theory, and we actually obtain identically the same law of transition temperature depression :

(3)

where T~ is the transition temperature of the superconducting magnetic alloy at pair-breaking parameter ~, T~o is the corresponding quantity for the pure superconductor at ~ = 0, and q; is the digamma function. However, we obtain an entirely different expression for the density of states. It is strictly nonzero, and hence the superconductivity of magnetic alloys is always gapless. This result is completely at odds with that in the AG theory, where a well-defined energy gap persists over a wide range of impurity concentra- tions right up to shortly before the destruction of superconductivity. But it compares very well with the conductance data on the alloy LaCe. 3

In this paper we study the thermodynamic properties of superconducting magnetic alloys in our theory. The free energy is calculated in detail and used to derive the critical magnetic field, the entropy, and the electronic specific heat. The results are presented in graphical form for easy comparison with

Thermodynamics of Superconducting Magnetic Alloys 469

experiment. Recently Smith 4 has published an extensive set of data on the alloy ZnMn. The characteristic temperature T K of ZnMn in the normal phase is 0.24 K as determined in magnetic susceptibility measurements, while the superconducting transition temperature of pure Zn is about 0.86 K, so that a weak coupling situation prevails. The data show large deviations from the AG theory. On comparison we find that our results in general lie roughly midway between the data and the predictions of the AG theory.

2. O R D E R P A R A M E T E R

From the second Green's function in (1) we can derive the parametric equation

1 ('~'~ dx 2 Im ~ + + i (x 2 (4) 1 J_wo ix 2 + A2)

where fi = 1/kBT, co D is the Debye energy, and V (> 0) is the coupling constant of the superconducting pairing interaction. This describes implicitly the dependence of the order parameter A on ~ and T. For ~ = 0 it reduces to the Bardeen-Cooper-Schrieffer (BCS) integral equation in A and T for a pure superconductor. In paper I we have already calculated analytically the variation of A with ~ at T = 0. Here we consider the case of general T, by numerical methods. The thermodynamic quantities can be obtained once A is known as a function of ~ and T.

Equation (4) can be converted into a form suitable for numerical calcula- tion by first rewriting it as follows:

P - - I m ~ + i - 0 + ~ + i

= P A2) l /a lm~ + ~ + i (x 2 ov X2 -'}-

dX Im o + + i

-~ A2) 1/2)

15)

where ricO = 1/kBT~o. We have extended coo to infinity whenever the integrals converge. The left-hand side can be evaluated exactly by contour integration as described in Section 4 of paper I, and the right-hand side by using the following series representation of the digamma function :

' 1) - - + - 0 . 5 7 7 2 . . . (6) ~ , (z ) : z ,=1 n + z

4 7 0 J o h n L a m

Then (5) becomes

In (-~To)+ q/(~ + 2~ ) - 6 ( ~ )

tI( 1 = n + ~ + 2 ~ t + n=O 2~-~/ J - n + ~ + 2 ~ (7)

which is equivalent to (4) and reduces to (3) at A = 0. For T close to zero or To, (7) can be solved in analytical form. At

T --- T~, A is small and the right-hand side of (7) becomes

~0 / ~ + ~ ~ +o(a 4) (8/

Thus we obtain the solution

At T ~ 0, ]~ is large and we have asymptotically

) I r a0 + ~ + i (x 2 +A2) 1/2

. , . tan_l( '(x2 ~ 2 ) 1 / 2 ) - 1 (27r] 2 a (x2+A2) 1/'2 12/ fl I (X2 -}" A2 q- ~2)2

so that (4) can be rewritten as

~[A(O)] - ~[A(T)I -

T -~ T~ (9)

+ 0(3- 4) (10)

6 [A2(0) -}- 0~2] 3/~(7~kBT)2 O (T 4) + (11)

where

~(A) = (x 2 ~- A2)1/2 tan -1 (12)

In Section 5 of paper I this integral is worked out analytically and the result is

2o3 D qS(A) = In c~ + (A 2 + ~2)1/2 (13)

Combining (1 l) and (13), we obtain the solution

A(T) 1 ~ (rckBT) 2 - 1 T -~ 0 (14) A(0) 6 A2(0) _it_ 0~2 [A2(0) .~ ~2~1/2 _ 0~'


We recall f rom (36) of paper 1 that

%(0) (] 5)

where Ao(0 ) is the order pa rame te r of the pure superconductor at T = 0 and % = Ao(0)/2 is the critical pa i r -breaking parameter .

The expression (14) is valid as long as c~ r 0. For the pure superconduc tor with c~ = 0, the expansion (10) does not hold since the point T = 0 is an essential singularity. In fact we have in this case

- ~ t a n h [ ~ ( x : 2 _ +A2)1"21

This yields the well-known exponential dependence on 1/T for a pure superconductor , entirely different from (14). The difference is of course due to the fact that in our theory superconduct ing magnet ic alloys are always gapless.

j ~ - I k [ I ] - - I

/

0 0

T/TOO

Fig. I. The order pa ramete r A as a function of the t empera tu re T and the pair-breaking parameter ~. T,. o and AdO) are the transition temperature and the order parameter at T = 0 of the pure superconductor, and ac = 40(0)/2.

472 John Lam

TABLE I

Variation ofthe Order Parameter A(T) in Units ofAo(0)withT/~ and ~/%

x ~ , / ~ c 0.0 0.2 0.4 0.6 0.8 0.9 T/~ \ \

0.0 1.0000 0.8944 0.7746 0.6325 0.4472 0.3162 0.1 1.0000 0.8939 0.7735 0.6309 0.4455 0.3149 0.2 0.9999 0.8918 0.7697 0.6259 0.4403 0.3104 0.3 0.9971 0.8860 0.7614 0.6161 0.4310 0.3030 0.4 0.9850 0.8713 0.7451 0.5997 0.4170 0.2920 0.5 0.9569 0.8429 0.7174 0.5743 0.3969 0.2770 0.6 0.9070 0.7960 0.6749 0.5377 0.3694 0.2570 0.7 0.8288 0.7252 0.6127 0.4861 0.3323 0.2300 0.8 0.7110 0.6207 0.5229 0.4132 0.2813 0.1940 0.9 0.5263 0.4568 0.3851 0.3039 0.2061 0.1425 1.0 0.0 0.0 0.0 0.0 0.0 0.0

For general T we solve (7) by iteration. It must be noted that we have to take several hundred terms of the sum in order to obtain accuracy in the fourth decimal place. The results are plotted in Fig. 1 as a function of T/T~o and c~/c~ c. We also tabulate them as a function of T/T~ and ~/~c for future calculations in Table I. The dependence of TJT~o on ~ was previously calculated and shown in Table 1 and Fig. 1 of paper I.

(Hi . t ) -- _ NAZ/V

Therefore (17) can be written as

OA21 V] dA2

Using (4) and (6), we obtain after some algebra

3. F R E E E N E R G Y

The difference between the free energies in the superconducting and the normal states is given by

cv dV AF = F s - F,, = J o T ( H I ' ) (17)

where Him is the BCS four-fermion superconducting pairing interaction Hamiltonian. In the Hartree-Fock approximation we have

(18)

O ( )1 -~2-~P(fl)2 ~ I (n+~+~l/3~) 2 + [flA]Z] -3/2 aX 2 P = .=o / 2 ~ / _1

(19)

(2o)


by which (19) becomes

AF(T) = -Np{ fi7~) 2

• 2 n + ~ + 2 ~ + ~ - n + ~ + y ~ n = 0

• ,,+5+~ +12~lj (21)

This expression is we]] suited for computing AF when A is given by Table I. For T - T~ and A _~ 0, (21) becomes

1 N [~c 12 .11 AF(T)~- g pl}-~l ~ 1~ + fl2~)A 4 (22)

or by (9)

( ~7) 2 [1 -- (flc~189 + (flc~/2g))~ 2 AF{T) = 2No ~b"(�89 + (fl~/2~))

x 1 - , T -~ T~ (23)

At T "" 0, (4) becomes, according to (10),

1 1 :~ = ~b(A) 6 (A 2 + 0~2) 3/2(TrkBT)2 + o ( r 4 ) (24) v,

Substituting (24) into (19) and applying (13) and (14), we obtain

AF(T) = AF(0) + ~gp 1 - [A2(0) + 7211,'2 (rCkB r)2' r ~- 0 (25)

where

AF(0) = -�89 Npa{[A2(0) + cd]':2 _ ~}

= -�89 - (a/ar (26)

by (15). In Fig. 2 we plot AF in (21) in units of �89 against T/T~o. It is also shown in Table II as a function of T/T~ and :r for future calculations.

474 John Lam

%o

<~

a l ~ c = 0

- I L- 0

T / T CO

Fig. 2. The free energy difference between the superconducting and the normal states as a function of T and c~.

TABLE II

Variation of the Free Energy Difference AF(T) in Units of �89 with T/T~ and ~/'a c

~ / a ~ 0.0 0.2 0.4 0.6 0.8 0.9

0.0 -1.0000 -0.6400 -0.3600 -0.1600 -0.04000 -0.010000 0.1 -0.9788 -0.6262 -0.3521 - 0.1564 -0.03911 - 0.009797 0.2 -0.9155 - 0.5850 -0.3287 -0.1460 -0.03652 -0.009140 0.3 -0.8117 -0.5184 -0.2912 - 0.1294 -0.03245 -0.008153 0.4 -0.6761 -0.4321 -0.2429 -0.1082 -0.02725 -0.006867 0.5 -0.5228 -0.3345 -0.1884 -0.08410 -0.02132 -0.005406 0.6 -0.3671 -0.2353 -0.1329 -0.05955 -0.01519 -0.003881 0.7 -0.2239 -0.1438 -0.08140 -0.03661 -0.009415 -0.002409 0.8 -0.1069 -0.06881 -0.03908 - 0.01763 -0.004576 -0.001177 0.9 - 0.02847 -0.01839 -0.01044 -0.004761 -0.001247 -0.0003299 1.0 0.0 0.0 0.0 0.0 0.0 0.0


4. C R I T I C A L M A G N E T I C F I E L D

The critical magne t i c field He(T) is given by

(~/8rr)H2(T) = - AF(T) (27)

where D is the vo lume of the supe rconduc to r , or al ternatively,

Hc(T),/Hco(O ) = [ - 2AF(T)/NpA~(0)] I/2 (28)

where Hco(0) is the critical magne t ic field at T = 0 for the pure superconduc to r . Fo r T -~ T~ we have by (23)

H~(T) 4TO 1 -- (fl~/2Zt)~b'( 1 + (flcc~/2rc))

H~o(0) flr ) [ _ ~j,,(�89 + (fl~a/29))]~!2

x 1 - , r -~ T~ (29)

An d for r ~ 0 we have by (25), (26), and (15)

H~(T) _ 1 ~ 2b 2 % (T~o) 2 Hco(0) ~ 3 2~c - ~ , T - 0 (30)

where b = 1.781 . . . is the logar i thm of Euler ' s cons tan t and bAo(0) = rck n T~o. By using Table II we easily evaluate (28), and the results are p lo t ted

in Fig. 3. We note that at T = 0, (30) yields the simple relat ion

H~(O)/H~o(O ) = 1 - (~/~c) (31)

I I I I I

H (T) N N ~ a C /ac Hco(O)

O I T / T

c o

Fig. 3. The critical magnetic field as a function of Tand c~.

476 J o h n L a m

[ I I I

Hc{O)

Hco(O)

0 0 TC/TCO

Fig, 4. The critical magnetic field at T = 0 versus TJTco,

F i g u r e 4 s h o w s a p lo t of th i s r a t i o as a f u n c t i o n of TjT~ 0 s ince e x p e r i m e n t a l l y

t he t r a n s i t i o n t e m p e r a t u r e c a n be d e t e r m i n e d w i t h g r e a t e r a c c u r a c y t h a n

t he i m p u r i t y c o n c e n t r a t i o n . It is to be c o m p a r e d w i t h t he d a t a in Fig. 5

of Ref. 4.

T h e c r i t i ca l m a g n e t i c field is o f t en a n a l y z e d w i t h the f u n c t i o n

T A B L E I!I Variation of the Critical Magnetic Field Deviation Function D(T/Tc) with T/T~ and ~/%

x',~/% 0.0 0.2 0.4 0.6 0.8 0.9 T/T~

\

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 -0.0006 -0.0009 -0.0011 -0.0012 -0.0013 -0,0002 0.2 -0.0032 -0,0039 -0.0045 -0.0048 -0.0045 -0,0040 0.3 -0.0091 -0.0100 -0.0107 -0.0108 -0.0094 -0.0071 0.4 -0.0177 -0.0184 -0,0186 -0.0179 -0.0147 -0.0113 0.5 -0.0269 -0.0271 -0,0266 -0.0250 -0.0200 -0.0147 0.6 -0.0341 -0.0337 - 0.0325 - 0.0299 -0.0239 -0.0170 0.7 -0.0368 - 0.0360 -0.0345 -0.0317 -0.0248 - 0.0192 0.8 -0.0331 - 0.0321 -0.0305 -0.0281 -0.0218 -0.0170 0.9 -0.0213 -0.0205 -0.0197 -0.0175 - 0.0135 - 0.0084 1.0 0.0 0.0 0.0 0.0 0.0 0.0


which shows up the deviation of the temperature variation of He(T) from that in the Gorter-Casimir two-fluid model. We tabulate D(T/T~) in our theory in Table III. Except at small T/T~, the deviation diminishes numerically with increasing ~. At ~ = 0.9:~ c, D(T/T~) has only about one-half its BCS value. This is in reasonable accord with Fig. 4 of Ref. 4 in view of the considerable amount of scatter in the data. Our result is in marked contrast with that in the AG theory as calculated and reported by Decker and Finne- more. 5 The latter shows an increase of the deviation with :t. These authors find excellent agreement between their calculations and their data on ThGd alloys.

5. ENTROPY

The difference between the entropies in the superconducting and the normal states is

AS = S ~ - S, = - ( O / g T ) AF (33)

The normal state entropy S, can be given its free Fermi gas value

S,(T) = 7T, ? = ~-Np(rckn) 2 (34)

since the width a in (1) produces a correction only of order of :r over the Fermi energy.

From (33) we obtain the normalized entropy

S~(T) T [T~ol 2 e [ 2AF(T) I

where

q = �88 2 = 0.2364... (36)

We evaluate the derivative by numerical differentiation of Table II, and plot the entropy against T/T~ in Fig, 5, It is to be compared with the data in Figs. 6 and 7 of Ref. 4.

6. SPECIFIC HEAT

The difference between the electronic specific heats in the superconducting and the normal states is

AC = Cs - C, = - T ( ~ 2 / ~ T 2 ) A F (37)

From (23) we obtain readily the discontinuity in the electronic specific

478 John Lam

SIT.___2) Y T c

I [

/ / /

/ / .9

/ / / / i z i2

l TIT C

Z / /

Fig. 5. The normalized entropy as a function of T/T~ and ~. The broken line represents the entropy of normal metal.

heat at the transition temperature

AC(T~) T~ q/'(�89 FI /?c~ " /1 3~1-]2 (38) ACo(T~o) - T~0 ~"(�89 -~ ( ~ / 2 ~ ) ) [ - ~ - ~ u / i + 2n ] J

where ACo is its value for the pure superconductor. We plot the numerical results in Fig. 6, which is to be compared with Fig. 9 of Ref. 4.

For T --- 0 we derive from (25) and (37)

AC = - C , 1 - [A2(0) + ~2]1/2 (39)

since C, = 7T. In paper I we have shown that the density of states of the superconducting magnetic alloy is given by

co+ ice p(co) = p Re [(co + i~)2 _ A211/2 (40)

Therefore (39) becomes simply

C~ c~ p(0) - - = - , T - 0 ( 4 1 ) C, [A2(0) + ~2]~/2 p

This means that at T ~ 0 the difference between C~ and C, is solely due to the difference in the density of states at the Fermi level at T = 0.


0

Fig. 6. The discontinuity in the specific heat at the transition t empera tu re versus TJTr

By (15) we can also write

Cs/C . = ~/(2~ c - ~), T -~ 0 (42)

This ratio has been plotted against ~/~c in Fig. 3 of paper I. Here we plot it against TJT~o in Fig. 7 for ready comparison with the experimental data in Fig. 8 of Ref. 4.

I

C 5

Cn

o I J I f 0

T C ITco

Fig. 7. The ratio of the specific heats in the superconduc t - ing and the normal states at T ~ 0 versus T~/T~o.

480 John Lam

7. DISCUSSION

In paper I we found that our theory of superconducting magnetic alloys is similar to the Abrikosov-Gor 'kov theory in yielding the same law of transition temperature depression, but dissimilar in predicting an entirely different density of states. The thermodynamic quantities calculated in this paper provide further area for comparison. In many respects the two theories are qualitatively alike. But often there exist considerable quantitative deviations. At least for the alloy ZnMn, on which systematic measurements have been performed by Smith, 4 the deviations of our results from those of the AG theory are in the right direction indicated by experiment. In fact, our results often lie roughly midway between the predictions of the AG theory and the experimental data.

The significance of the quantitative difference between two qualitatively similar sets of results should perhaps not be overemphasized at this stage, since we have at hand only a very limited amount of data and the situation can change from one alloy to another. There is, however, an irreconcilable difference between the two theories which is brought out regularly at T ~- 0, and this can be ultimately attributed to the basic difference in the density of states. Recall that in the AG theory there is a well-defined gap in the density of states which persists up to ~ -~ 0.91~. Consequently the temperature dependence of the thermodynamic quantities at T ~- 0 is exponential, just as for the pure superconductor. In our theory no such gap exists, and the density of states everywhere varies continuously with ~. Such a difference shows up dramatically in the ratio of the specific heats Cs/C, at T ~- 0. The prediction of our theory is shown in Fig. 7. In the AG theory this ratio is strictly zero except in the neighbourhood of Tr = 0. The data of Smith 4 on ZnMn weigh overwhelmingly in our favor. He found that the ratio tends to a nonzero limit as T tends to zero. Since Smith's results were obtained by critical magnetic field measurements, it will be of interest to check them by measuring independently the density of states of ZnMn in electron tunneling experiments.

REFERENCES

1. A. A. Abrikosov and L. P. Gor'kov, Soviet Phys.- JETP 12, 1243 (1961). 2. J. Lain, J. Low Temp. Phys. 19, 113 (1975), 3. A. S. Edelstein, Phys. Rev. Lett. 19, 1184 (1967), 4. F. W. Smith, J. Low Temp. Phys. 5, 683 (1971). 5. W. R. Decker and D. K. Finnemore, Phys. Rev. 172, 430 (1968).

thermodynamics of superconducting magnetic alloys in the weak coupling theory

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