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Superconducting alloys with paramagnetic impurities I: Theory of the upper critical field
Physica C 159 (1989) 33-42 North-Holland, Amsterdam
SUPERCONDUCTING ALLOYS WITH PARAMAGNETIC IMPURITIES h THEORY OF THE UPPER CRITICAL FIELD W. P I N T Institut J~r Theoretische Physik, Technische Universit?it Graz, A-8010 Graz, Austria
E. S C H A C H I N G E R Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA
Received 27 March 1989 Revised manuscript received 20 April 1989
This paper presents a strong coupling theory of the upper critical field Be2(T) of isotropic superconducting alloys with dilute concentrations of paramagnetic impurities. We assume a random distribution of the impurity spins, treat them as classical spins and assume also that there are no correlations between them. The impurity spin orientation at low temperatures in the external magnetic field is described by "spontaneous" spin orientation. Numerical results show distinctive strong coupling effects for samples exceeding about half of the critical impurity concentration which go beyond the ( 1+ 2) renormalization of BCS-results, and the B2c(T) characteristic of alloys based on clean or dirty limit host materials shows very pronounced differences. Both effects are big enough to be accessible to experiment. Finally, for sufficiently large paramagnetic spin exchange potentials the theory predicts the possibility of a "reentrant"-type behaviour even for this simple kind of superconducting alloys.
1. Introduction The coexistence o f superconductivity with magnetic phenomena has always been a topic o f experimental and theoretical scrutiny. When the superconducting ternary c o m p o u n d s were found the question o f coexistence became a hot topic concentrating specifically on the two systems ErRhB4 and H o M o 6 S 6 [ 1-6 ]. On the other hand, we believe that superconducting alloys with paramagnetic impurities in the sense o f Abrikosov and G o r ' k o v [ 7 ] deserve further attention. These materials are characterized by very dilute concentrations o f magnetic ions which allows us to assume that the distribution o f the magnetic ions is r a n d o m throughout the sample and that there are no correlations between the impurity spins. The spins S o f the magnetic ions are treated as classical spin. The spin-flip interaction o f the electron spin with the spin of the magnetic ion is known to be most effective in breaking up the Cooper pairs responsible for superconductivity and there is a finite critical
concentration o f magnetic ions at which superconductivity is destroyed completely. It was shown later on by Schachinger et al. [ 9 ] that the original BCSbased theory overestimated somehow the effect o f the spin-flip interaction on the superconducting state and that strong coupling effects could increase the critical concentration o f paramagnetic impurities substantially. Further work by Schossmann and Schachinger [ 10 ], Marsiglio et al. [ 11 ] revealed that the upper critical field Be2 oftype-II superconductors was quite sensitive to strong coupling effects related to the actual shape o f the electron-phonon interaction spectral function ot(09)2F(oJ). These theoretical results were later on supported by the theoretical analysis o f experimental B¢2 (T) data o f niobium [ 12 ] and of Nb3Sn [13]. The experimental and theoretical investigation o f the full temperature dependence o f Be2 in superconducting alloys with different concentrations o f paramagnetic impurities could give an even better insight into the microscopic processes governing superconductivity because o f the effectiveness of such
0921-4534/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
34
W. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
impurities in destroying superconductivity. This paper develops the necessary theoretical background, starting with the derivation of the appropriate electron-impurity scattering self-energy term in section 2. Section 3 is devoted to the derivation of equations which will determine the temperature dependence of the upper critical field B~2(T) for isotropic systems and section 4 will discuss some numerical results. Finally, a short conclusion will finish this paper.
(Sx)=(Sv)=(S, Sj)=O; i~j={x,y,z},
(3)
(S~) = IS, ( 1 - c t a n h x ) ,
(4)
($2)=(S~)=
(5)
ISl---~2 ' ( c xt a n h x - 1 )
(S.Z-) = IS,2 I 1 - 2 (ctanh x - 1 ) ] ,
(6)
and
x = eISIIBI ~
2. The electron-impurity interaction self-energy This self-energy term has already been studied by Machida and Younger [ 14 ] using a different model. They used a short range interaction I to describe the spin-spin interaction between local moments which is responsible for magnetic ordering at low temperatures. Their model also assumed a translational symmetry of the spins built into the system in order to describe the situation in some ternary compounds. This model results in a magnetic relaxation time Zmwhich is temperature dependent. Nevertheless, the assumption of a translational symmetry certainly deviates from the original approach by Abrikosov and Gor'kov. As we are interested in the dilute limit of the concentration of magnetic impurities (nl < 1 at%) we assume the impurities to be randomly distributed with no correlation between the impurity spins. We therefore describe the "spontaneous" orientation of the spins at low temperatures and due to the externally applied magnetic field B by a Boltzmann distribution for the average spin ( S ) for a given spin distribution S(t2):
(7)
For T= 0 we find the limits: (Sz)=-ISI, ( $ 2 ) = ( S ~ ) =0;
(8)
(S~)=S(S+I).
(9)
If we adopt the standard 4 × 4 matrix notation introduced by Ambegaokar and Griffin [ 8 ] as an expansion of Nambu's 2 × 2 notation [ 15 ] we can define the matrix impurity potential as
VE(k,k')=Vo(k-k')z3ao+Vl(k-k')S.a~.
(10)
Vo(k) is the nonparamagnetic scattering potential of the impurity, V1(k) is the paramagnetic spin exchange potential and o~= ½( 1 + z3)~r+ ½( 1 - r3) 0-20"0-2
(11)
with the Pauli matrices zi, as, (i, j = 1, 2, 3). The assumption of randomly distributed impurity sites allows to recover the translational invariance of the electron system by calculating the average electron Green's function from the series: (G(x, x'; iogn) ) =Go(x, x'; io9,) - nl j d3y Go(x, y; i~n) Vl(0) Go (y, x'; io9~)
(S)=~
ds~S(~)exp-
(1)
+~
/'/l
J d3yd3zd3kGo(x,y; ion)
and
_f exp{
× Vl(k)Go(y, Z; i09,) J
(2)
where 12 is the polar angle, e is the charge on the electron and m is its mass. If we assume the magnetic field B to be parallel to the z-axis we find:
X V~(-k) Go(Z,x'; i09n) exp{ik- ( Y - Z ) } + ' " • (12) Only these first three terms are necessary to establish a first order Born's approximation [ 16]. Go(x, y; iog~) is the electron matrix Green's function of an
35
IV..Pint, E. Schachinger/ Superconducting alloys with paramagnetic impurities I
unperturbed electron system and ¢on=nT(2n+ I), n = 0, + 1, _+2, ...) are the Matsubara frequencies of the electron system. Eq. (12) can immediately be transformed to kspace and we find for the electron-impurity interaction self-energy in first order Born's approximation:
and kF is the momentum at the Fermi surface. The diagonal elements of the 4 × 4 electron matrix Green's function which describe the normal state are then given by: GI.2 (x, x'; kon ) = -
£', (k; ion.) = --rt I VI(O )
+ (--~)3 d3qVi(q)G(k-q;i°)n)V~(-q) •
kv
27tlx-x'lvv
ex@+l
-- V7 ( ] . / B B + F / I r I ( 0 ) ( S z > )
(13)
Usually, n~VI(0) is a constant contribution which is used to rescale the chemical potential/~. Nevertheless, in our particular problem n~V~(0) will change with decreasing temperature or increasing external magnetic field as the spin of the impurity atoms becomes more and more aligned.
1
× [ x - x ' [ sgn(~)}
{ x-x'(l~.l+ 1 (Sz)~ vv -"L'ml ISI /
x exp-
-ie;dsA(s)}
(19)
x
3. Theory of Bc2(T)
where the " + " sign is significant for the Green's function G, (x, x'; iw~),
If we replace eq. ( 12 ) of ref. [ 10 ] by our eq. ( 13 ) and use the impurity potential (10) we can follow strictly the procedure outlined in ref. [ 10 ]. As a first result we find an equation for the renormalized normal state Matsubara frequencies 69n: 63~=o9~+xT
~
m ~ -oo
°
d ~ ~2------~2 ~ ( 0 ) 2F(~) ,
(15)
0
1 1 t+ -- 2~,rtr, t_ -- 27C.Cm,
(16)
where
1
- 2~n~N(0)
and
1 --~'ml =2~nIN(0) f ~dQ
× Vo(kv, k'F)V,(kF, k'F)[SI . (14)
which is just eq. (6) ofref. [9] with
;
(20)
2(COn--Ogre)sgn(69m) + x ( t + + t _ ) sgn(6)n)
2(o)n) = 2
G3,4 (x, x'; ion) = -- [GI,2 (X, x'; ioJn)]*
-r-- I Vo(kv, k~) 12
(17)
Ttr
(21)
vv is the Fermi velocity calculated from the isotropic Fermi gas picture,/ta is Bohr's magneton and [... ]* denotes the complex conjugate. The result (19) differs from eq. (15a) of ref. [10] only by the additional contributions coming from the orientation of the z-component of the impurity spin (Sz). We can therefore expect the off diagonal elements of the electron Green's function, denoted by ] (io&) to be complex as was the case in ref. [ 10 ]. We find the relations: 3,,2 ( i o . ) = COc
defines the inverse transport relaxation time,
g T Z [,~(t.On-tOm)-tl*]Z,,2(~),n)~,,z(iogn) m
I=2xn,N(O)S(S+I) ~m
da -~ IV~(kv, k'v)l
(18)
defines the inverse magnetic relaxation time, N(0) is the electronic density of states at the Fermi surface
+nt,Z,.2 (~,)J~.2 (itn,) + lttzZ,.2 (03,)J2., (i09n) (22) with
36
W. Pint, E. Schachinger/ Superconducting alloys withparamagnetic impuritiesI
dqe -q2
,~1,2 ((.On) ~---
×tan-I
I~.l-Ti(#uB+niV~(O)(S~))sgn(&.)
'
(23) 2a=eBc2(T)v 2 , t, =
2(S~) t+ - t _ S ( S + 1------~ ; t2 = t _ -s ( s-+ 1 )
(24)
2 ( S 2) + (S~) = S ( S + I ) ,
=
[Z~(&.) 1"
eq. (28) turns into
(26)
there are two possible choices for the gap function: (i) A2(io9.) = [~t (io9.) ]* and (ii) A2(iog.) = - [A~ (io9.) ]*. In the case of nonmagnetic impurities both choices are valid and give the same result. In the presence of magnetic impurities only one solution, namely A2(io9n)=-[A~(io9.)]* will recover the basic symmetry properties of Green's functions [ 17 ]: G(k, o9) = [ G(k, -o9) ]*
(27)
We find finally: ~(iog.) = ~ T ~ [2 (o9~ - o9,~) - #* ]X(&,.)A(iog~) m
+TtZ(~.)
t+ - t_ S ( S + 1--------~
- t _ S2((S$+2 )1----'-~[j'(iog.) 1.1
(28)
with z(6o.)
2 idqe_q2
=~
(30)
(25)
/~* is the Coulomb interaction pseudo potential and ogc is some cut-off frequency usually chosen to he an integer multiple of the Debye frequency O9DBecause of Z2(69,)
paramagnetic limiting (PPL) term #BB in eq. (29). It may even happen that the contribution which comes from spin orientation effects will compensate the PPL term completely. In such a case, Z(&.) will become real and so will ](io9.). This in turn will eliminate any contribution to the over all behaviour which comes from spin orientation effects or from Pauli limiting effects; because of
o
V
Xtan-~ L lCb. I --i(UBB+ n~qx/-a V~(0) (S~)) sgn(rJ.)] (29) It is important to recognize that according to eq. (5) (Sz) has a negative sign and therefore the term nlV~ ( 0 ) ( S ~ ) compensates at least partly the Pauli
zt(io9n)= ~T ~ [~( o9n --o9m) -- lA* ])~( (~_)m)~(io9m) m
+re(t+ - t_)Z(~,)3(io9,).
(31)
This equation would have been found by neglecting PPL and spin orientation effects from the beginning. We also note in passing that eqs. (28) and (29) reduce to eq. (5) of ref. [9] for T=T~ and B2¢(T~) =0.
4. Numerical results
In this chapter we will discuss results of the theory on the basis of the rather weak coupling system LaA12 having a Tco of about 3.3 K [18]. An a(og)2F(o9) spectrum can be derived from the generalized phonon density of states reported by Yeh et al. [ 19 ] by simple rescaling to give a mass enhancement factor 2=2(0) =0.55 which results in a/1" of about 0.1 for the given Tco. It is possible to make rather clean single and ploycrystals from this material [ 20 ] and we therefore call it a clean limit host system. Our numerical results for this system will be contrasted by calculations involving a dirty limit host system having a very small transport relaxation time and the same values for Tco, 2 and ~*. Typical dirty limit host systems would be amorphous metals [ 21 ] but they have higher critical temperatures (Tc~ 8 K) and higher mass enhancement factors (2--- 1.0). In the first step we compare the decrease in critical temperature Tc with increasing concentration of magnetic impurities (increasing value of t_ ) as calculated from the full Eliashberg equations with the results of the weak coupling limit equation [22 ]:
W.. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
Tc0
t_
a.-
,32,
where ~,(x) is the digamma function. Results are shown in fig. 1 with the dotted curves corresponding to data calculated from eq. (32). The full strong coupling calculation (solid line) predicts a higher critical concentration and the critical concentration and at the critical concentration predicted by eq. (32), the full theory shows that the alloy is still superconducting with a reduced critical temperature of tc = Tc/ Too~0.15. The weak coupling limit of eqs. (14), (28) and (29) is obtained by using the square well model described by Allen and Mitrovi~ [23], following closely the steps outlined by Schachinger et al. [24] and neglecting PPL and spin orientation effects: In
"~ )
(36)
t+
t*+- 1+2
(37)
J ( a * ) = 2 f dqe-q2 tan -1 (qa*) . 0
Next, we compare the results for the initial slope IdB¢2(T)/dT[ r=r~ calculated from eq. (33) with the results of a full strong coupling calculation, eq. (31), in fig. 2 for two limiting cases using the same value o f 0 . 3 2 × 106 m / s for the Fermi velocity VF: (a) the clean limit of the host material (t+ = 0 ) and (b) the dirty limit of the host material (simulated by t+ = 100 meV). In both cases we find differences between weak coupling limit results and the full strong coupling theory; they are very pronounced in the dirty limit. The general feature is that for higher concentrations of magnetic impurities (tc<0.6) the weak coupling limit equations begin to predict steeper initial slopes than the full theory. This becomes more pronounced with further increasing impurity concentrations and affects the reduced upper critical field
(33)
h*= eBc2(T)(v~)2 ,
vv
oo
with 2~2T2o
(35)
and
ra~O
(T/Tcox/~) J(a*,.)
x/~ ( l + 2 ) l o g , l+n(t*+ +t*_) '
v~= 1+2'
= 2 --
-1 - ( (t-~+---~-_) ~ ) ( a ~ , )
37
(34)
4.0.
3.0.
%
2.0-
1.0
0.0 0.00
0.02
0.04
t
,
0.06
0.08
meV
Fig. 1. Critical temperature vs. impurity parameter t_ ocnl, the concentration of magnetic impurities. The solid line presents results of a full Eliashberg type calculation while the dotted line corresponds to results found by solving eq. (32).
38
W. Pint, E. Schachinger/ Superconducting alloys with paramagnetic impurities I 0.03
5,00
3.75
Y
0.02
I.-II E-
II I-
2.50'
ra
0.01
1.25'
0'000.0
0:5
a
1.0
to = T J T o
0"000. 0
0:5
b
1.0
to = T J T o o
Fig. 2. (a) Initial slope of the upper critical field vs. the reduced critical temperature for the clean limit case of the host material ( t + = 0 ). Solid and dotted curves present the results of a full strong coupling calculation and of a weak coupling limit calculation respectively. (b) Same as fig. 2a but for the dirty limit of the host material (t+ = 100 meV). Note the pronounced differences between the two theories.
BCz(T) bc2(T) = ]dBc2( T) /dTIT=T~Tc "
(38)
This dimensionless quantity offers the advantage not to d e p e n d on the F e r m i velocity. We present results for bc2(t), t = T / T ~ in fig. 3 for the clean a n d the dirty limit o f the host m a t e r i a l a n d for three reduced critical temperatures, n a m e l y te= 1.0, 0.6 a n d 0.3. At first we note that the r e d u c e d u p p e r critical field curve changes substantially with increasing concentrations of magnetic impurities. The fact that be2 (t) is decreasing with increasing i m p u r ity concentration implies that the u p p e r critical field Bcz(T) is m o r e strongly suppressed at lower temperatures than it is suppressed close to To. I f this were not the case the bcz(t) characteristic should be the same for all concentrations o f magnetic impurities. We conclude that magnetic i m p u r i t i e s b e c o m e m o r e effective in breaking up C o o p e r pairs at low tern-
peratures even in cases where we assume that spin orientations effects are negligible. Figure 3 also compares the predictions o f the full strong coupling theory, eq. (31 ), (solid lines) to those o f the weak coupling limit theory ( d o t t e d lines). In the clean and dirty limit o f the host material without magnetic impurities we see no or little difference between the two theories. Since there are differences in the initial slope we have to conclude that the weak coupling limit theory underestimates Be2 ( T ) in the region o f low t e m p e r a t u r e s but gives a qualitatively correct description o f the t e m p e r a t u r e d e p e n d e n c e o f Be2. The sample with a reduced critical t e m p e r a t u r e & = 0 . 3 on the other h a n d shows a very p r o n o u n c e d difference between the predictions o f the full strong coupling theory a n d its weak coupling limit. This beh a v i o u r can be related to the overestimation o f the initial slope by the weak coupling limit theory. Nevertheless, the difference seems to be p r o n o u n c e d
IV.. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I .~
0.8
39
0.8 "r
t t
0.6
0.6 _
= 1.0
= 1.0
0.6
0.6-
0.3
0.3
0.4-
0.4.
",.°°o
",**
0.2. °'%°
o o Ia 0.0
0;5
FI~
1.o
T/T
0.2
o.o
b
o.o
~°°'°
o15
1.0
T/To
Fig. 3. (a) Reduced upper critical field bat(t) vs. the reduced temperature T/Tc for samples in the clean limit of the host material and with reduced critical temperatures to= To~Too=1.0, 0.6 and 0.3. The solid lines represent results calculated from a full strong coupling theory while the dotted lines correspond to predictions of the weak coupling limit approach. Note the pronounced differences in the two approaches for the sample with a to= 0.3. (b) Same as fig. 3a but for the dirty limit of the host material.
enough to be clearly detectable by experiment and we conclude that samples with impurity concentrations exceeding about half of the critical concentration should allow us to study strong coupling effects which go beyond the (1 + 2 ) renormalization of the standard BCS-type results. As a final point, we investigate the effect of spin orientations on the reduced upper critical field be2(t) by solving the full set of eqs. (14), (28) and (29). For this purpose we concentrate on a L a A l 2 alloy with 0.1 at% Gd having a Tc of about 2.9 K [20] as a model system. The spin of the Gd-atom is 7/2 and we are interested in the effect different values of V~(0) have on the reduced upper critical field. The results of our calculations are shown in fig. 4a for a clean limit host system. As we have to assume that the Gd impurities will also result in some decrease in the transport relaxation time we set t+ =0.09 in order to describe the effect of the mo-
mentum scattering part Vo of the impurity potential. The appropriate value of t is calculated from the decrease in critical temperature. The figure presents bc2(t) curves for a set of values I/1(0)=0.25, 0.5, 1.0, 5.0 and 10.0 meV and hi=0.1 (full lines). The results are compared with calculations neglecting spin orientation (dotted line). We see that bc2(t) reaches a maximum at some temperature which depends on the actual value of the potential V~(0). Thus, in trying to fit experimental results we can retrieve some information on the actual microscopic paramagnetic spin exchange potential. This results reported in this figure are quite similar to results discussed earlier by Machida and Younger [ 14 ]. For sufficiently large, but probably unrealistic, values of V~(0) ( V~(0) > 5 MeV) even our system shows a "reentrant"-type behaviour, i.e.: a temperature region close to absolute zero where B c e ( T ) =0.
W. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
40
o.8 la
0.8 V,(O) =
b
0.25 meV 0.5 1.0
0.6
~'...
5.0
V,(O) =
1.0
meV
0.6 _ , , _ , , . , ~ ~ 0.5
10.0 0.25 0.4
f,~
__ "%%
0 . 0 II 0.0
""""%%
0.2 •
0.2-
0:5
1.o
T/To
0.0o.o
0:5
~.o
T/To
Fig. 4. (a) Reduced upper critical field bc2(t) vs. the reduced temperature for a LaA12 sample with 0.1 at% Gd. The host material is in the clean limit. The full lines describe the effect of spin orientation for different values of V, (0), namely 0.25, 0.5, 1.0, 5.0 and 10.0 meV and with an impurity concentration nt = 0.1. The dotted line corresponds to the result one would get for a system without spin orientation effects. (h) Same as fig. 4a but for a dirty limit host material. The dashed line describe the influence of Pauli paramagnetic limiting alone.
The situation changes quite remarkably if alloys based on a dirty limit host material are studied. The results of our calculations can be found in fig. 4b for the same values of V, (0) and n, as in fig. 4a. The dotted line presents the solution of eq. (31 ) for this material and the dashed line shows the additional influence of PPL on the reduced upper critical field. In dirty limit materials we have to assume rather high upper critical fields which will then result in rather pronounced PPL effects suppressing the effective upper critical field quite substantially. Finally, the solid lines in this figure present the result which include impurity spin orientations effects. For large spin exchange potentials 1.'1(0) >/5 meV b¢2 (t) is almost completely suppressed and we find a "reentrant"-type behaviour. The case V, (0) = 1 meV is special because in this case the PPL term and
the spin orientation term of eq. (29) cancel each other at about t ~ 0.5 which brings the bc2(t) characteristic close to the one we found from solving eq. (31 ). The oscillations of the solid line around the dotted line show very nicely the delicate balance between PPL and spin orientation effects. Making V, (0) even smaller results in an only partial cancellation of paramagnetic limiting and the sample will show a higher upper critical field as one would expect from neglecting spin orientation effects at all. Thus alloys based on dirty limit host materials should display a behaviour which is distinctly different from the behaviour predicted for alloys based on clean limit materials. Using different paramagnetic elements and studying alloys with different impurity concentrations should provide enough vari-
V~ Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
ation o f the factor nl Vl (0) tO observe the described effects experimentally.
5. Conclusion We studied the effect o f magnetic impurities on the upper critical field o f superconducting alloys, We assumed that the impurity spin was classic and the impurity concentration very dilute which resulted in two further assumptions: (i) a r a n d o m distribution of the impurity sites throughout the sample and (ii) the absence of correlations between impurity spins. The only effect which temperature and external magnetic field were allowed to have on the impurity spin was described by a "spontaneous" orientation of these spins against the direction o f the external magnetic field represented by a Boltzmann distribution. These assumptions resulted in a temperature independent magnetic relaxation time and a spin orientation effect which is similar to the Pauli paramagnetic limiting term. But in contrast to this term which depends only on the magnetic field Be2 (T), the spin orientation term has an additional, explicit temperature dependence. Thus the effect o f spin orientation on the upper critically field will become more pronounced with decreasing temperatures and increasing magnetic fields. A numerical investigation o f the theory resulted in two interesting results: (i) for concentrations exceeding about half of the critical concentration, strong coupling effects going beyond the ( 1 + 2) renormalization become important and pronounced enough to be detected by experiment and (ii) the behaviour o f alloys based on clean limit or dirty limit host systems will show completely different features as far as the influence o f spin orientation is concerned. Depending on the actual strength of the paramagnetic spin exchange potential V~(0) the alloy based on a clean limit host material will show some m a x i m u m in Be: (T) with decreasing temperature and in extreme cases on may even find a "reentrant"-type behaviour having B~2 (T) = 0 close to absolute zero. This is in contrast to the behaviour o f alloys based on dirty limit host materials where we can observe a delicate balance between Pauli paramagnetic limiting effects and the influence o f the impurity spin orientation. Again, in the case o f very large magnetic exchange
41
interaction potentials a "reentrant"-type behaviour could be observed. These results are in strong contrast to models with temperature dependent magnetic relaxation times because they cannot describe such differences between alloys based on clean or dirty limit host materials. They will predict about the same kind o f behaviour for both types o f alloys.
Acknowledgements This research was supported in part by the Office of Naval Research under grant No. N00014-89-J1088. The authors want to thank Dr. H.W. Weber for many fruitful discussions and his continuous interest in this work. One o f us (E.S.) also wants to thank the members of the Center of Theoretical Physics at the Physics Department of the Texas A&M University for the warm welcome and their hospitality.
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W. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
[ 14 ] K. Machida and D. Younger, J. Low Temp. Phys. 35 (1979) 449. [ 15 ] Y. Nambu, Phys. Rev. 117 (1960) 648. [16]A.A. Abrikosov and L.P. Gor'kov, Soy. Phys. - J E T P 9 (1959) 220. [17] A.A. Abrikosov, L.P. Gor'kov and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Dover Publications, New York, 1975 ). [ 18 ] B. Vlcek, E. Seidl and H.W. Weber, Jpn. J. Appl. Phys. 263 (1987) 967. [19] C.T. Yeh, W. Reichart, B. Renker, N. Niicker and M. Loewenhaupt, J. de Physique 42 ( 1981 ) C6-371. [20] B. Vlcek, H.W. Weber, W. Pint and E. Schachinger (to be published).
[21 ] see for instance: D.B. Kimhi and T.H. Geballe, Phys. Rev. Lett. 45 (1980) 1039. [22] P.B. Allen, in Lecture Notes for the XVI Karpacz Winter School of Theoretical Physics, Poland, 1979, ed. J. Lopuszaflski (University of Wroctaw, Wroctaw, Poland, 1978). [23] P.B. Alien and B. Mitrovi~, in Solid State Physics, vol. 37, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1982) p. 1. [24] E. Schachinger, H.G. Zarate, M. Schossmann and J.P. Carbotte, J. Low Temp. Phys. 63 (1986) 1.
SUPERCONDUCTING ALLOYS WITH PARAMAGNETIC IMPURITIES h THEORY OF THE UPPER CRITICAL FIELD W. P I N T Institut J~r Theoretische Physik, Technische Universit?it Graz, A-8010 Graz, Austria
E. S C H A C H I N G E R Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA
Received 27 March 1989 Revised manuscript received 20 April 1989
This paper presents a strong coupling theory of the upper critical field Be2(T) of isotropic superconducting alloys with dilute concentrations of paramagnetic impurities. We assume a random distribution of the impurity spins, treat them as classical spins and assume also that there are no correlations between them. The impurity spin orientation at low temperatures in the external magnetic field is described by "spontaneous" spin orientation. Numerical results show distinctive strong coupling effects for samples exceeding about half of the critical impurity concentration which go beyond the ( 1+ 2) renormalization of BCS-results, and the B2c(T) characteristic of alloys based on clean or dirty limit host materials shows very pronounced differences. Both effects are big enough to be accessible to experiment. Finally, for sufficiently large paramagnetic spin exchange potentials the theory predicts the possibility of a "reentrant"-type behaviour even for this simple kind of superconducting alloys.
1. Introduction The coexistence o f superconductivity with magnetic phenomena has always been a topic o f experimental and theoretical scrutiny. When the superconducting ternary c o m p o u n d s were found the question o f coexistence became a hot topic concentrating specifically on the two systems ErRhB4 and H o M o 6 S 6 [ 1-6 ]. On the other hand, we believe that superconducting alloys with paramagnetic impurities in the sense o f Abrikosov and G o r ' k o v [ 7 ] deserve further attention. These materials are characterized by very dilute concentrations o f magnetic ions which allows us to assume that the distribution o f the magnetic ions is r a n d o m throughout the sample and that there are no correlations between the impurity spins. The spins S o f the magnetic ions are treated as classical spin. The spin-flip interaction o f the electron spin with the spin of the magnetic ion is known to be most effective in breaking up the Cooper pairs responsible for superconductivity and there is a finite critical
concentration o f magnetic ions at which superconductivity is destroyed completely. It was shown later on by Schachinger et al. [ 9 ] that the original BCSbased theory overestimated somehow the effect o f the spin-flip interaction on the superconducting state and that strong coupling effects could increase the critical concentration o f paramagnetic impurities substantially. Further work by Schossmann and Schachinger [ 10 ], Marsiglio et al. [ 11 ] revealed that the upper critical field Be2 oftype-II superconductors was quite sensitive to strong coupling effects related to the actual shape o f the electron-phonon interaction spectral function ot(09)2F(oJ). These theoretical results were later on supported by the theoretical analysis o f experimental B¢2 (T) data o f niobium [ 12 ] and of Nb3Sn [13]. The experimental and theoretical investigation o f the full temperature dependence o f Be2 in superconducting alloys with different concentrations o f paramagnetic impurities could give an even better insight into the microscopic processes governing superconductivity because o f the effectiveness of such
0921-4534/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
34
W. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
impurities in destroying superconductivity. This paper develops the necessary theoretical background, starting with the derivation of the appropriate electron-impurity scattering self-energy term in section 2. Section 3 is devoted to the derivation of equations which will determine the temperature dependence of the upper critical field B~2(T) for isotropic systems and section 4 will discuss some numerical results. Finally, a short conclusion will finish this paper.
(Sx)=(Sv)=(S, Sj)=O; i~j={x,y,z},
(3)
(S~) = IS, ( 1 - c t a n h x ) ,
(4)
($2)=(S~)=
(5)
ISl---~2 ' ( c xt a n h x - 1 )
(S.Z-) = IS,2 I 1 - 2 (ctanh x - 1 ) ] ,
(6)
and
x = eISIIBI ~
2. The electron-impurity interaction self-energy This self-energy term has already been studied by Machida and Younger [ 14 ] using a different model. They used a short range interaction I to describe the spin-spin interaction between local moments which is responsible for magnetic ordering at low temperatures. Their model also assumed a translational symmetry of the spins built into the system in order to describe the situation in some ternary compounds. This model results in a magnetic relaxation time Zmwhich is temperature dependent. Nevertheless, the assumption of a translational symmetry certainly deviates from the original approach by Abrikosov and Gor'kov. As we are interested in the dilute limit of the concentration of magnetic impurities (nl < 1 at%) we assume the impurities to be randomly distributed with no correlation between the impurity spins. We therefore describe the "spontaneous" orientation of the spins at low temperatures and due to the externally applied magnetic field B by a Boltzmann distribution for the average spin ( S ) for a given spin distribution S(t2):
(7)
For T= 0 we find the limits: (Sz)=-ISI, ( $ 2 ) = ( S ~ ) =0;
(8)
(S~)=S(S+I).
(9)
If we adopt the standard 4 × 4 matrix notation introduced by Ambegaokar and Griffin [ 8 ] as an expansion of Nambu's 2 × 2 notation [ 15 ] we can define the matrix impurity potential as
VE(k,k')=Vo(k-k')z3ao+Vl(k-k')S.a~.
(10)
Vo(k) is the nonparamagnetic scattering potential of the impurity, V1(k) is the paramagnetic spin exchange potential and o~= ½( 1 + z3)~r+ ½( 1 - r3) 0-20"0-2
(11)
with the Pauli matrices zi, as, (i, j = 1, 2, 3). The assumption of randomly distributed impurity sites allows to recover the translational invariance of the electron system by calculating the average electron Green's function from the series: (G(x, x'; iogn) ) =Go(x, x'; io9,) - nl j d3y Go(x, y; i~n) Vl(0) Go (y, x'; io9~)
(S)=~
ds~S(~)exp-
(1)
+~
/'/l
J d3yd3zd3kGo(x,y; ion)
and
_f exp{
× Vl(k)Go(y, Z; i09,) J
(2)
where 12 is the polar angle, e is the charge on the electron and m is its mass. If we assume the magnetic field B to be parallel to the z-axis we find:
X V~(-k) Go(Z,x'; i09n) exp{ik- ( Y - Z ) } + ' " • (12) Only these first three terms are necessary to establish a first order Born's approximation [ 16]. Go(x, y; iog~) is the electron matrix Green's function of an
35
IV..Pint, E. Schachinger/ Superconducting alloys with paramagnetic impurities I
unperturbed electron system and ¢on=nT(2n+ I), n = 0, + 1, _+2, ...) are the Matsubara frequencies of the electron system. Eq. (12) can immediately be transformed to kspace and we find for the electron-impurity interaction self-energy in first order Born's approximation:
and kF is the momentum at the Fermi surface. The diagonal elements of the 4 × 4 electron matrix Green's function which describe the normal state are then given by: GI.2 (x, x'; kon ) = -
£', (k; ion.) = --rt I VI(O )
+ (--~)3 d3qVi(q)G(k-q;i°)n)V~(-q) •
kv
27tlx-x'lvv
ex@+l
-- V7 ( ] . / B B + F / I r I ( 0 ) ( S z > )
(13)
Usually, n~VI(0) is a constant contribution which is used to rescale the chemical potential/~. Nevertheless, in our particular problem n~V~(0) will change with decreasing temperature or increasing external magnetic field as the spin of the impurity atoms becomes more and more aligned.
1
× [ x - x ' [ sgn(~)}
{ x-x'(l~.l+ 1 (Sz)~ vv -"L'ml ISI /
x exp-
-ie;dsA(s)}
(19)
x
3. Theory of Bc2(T)
where the " + " sign is significant for the Green's function G, (x, x'; iw~),
If we replace eq. ( 12 ) of ref. [ 10 ] by our eq. ( 13 ) and use the impurity potential (10) we can follow strictly the procedure outlined in ref. [ 10 ]. As a first result we find an equation for the renormalized normal state Matsubara frequencies 69n: 63~=o9~+xT
~
m ~ -oo
°
d ~ ~2------~2 ~ ( 0 ) 2F(~) ,
(15)
0
1 1 t+ -- 2~,rtr, t_ -- 27C.Cm,
(16)
where
1
- 2~n~N(0)
and
1 --~'ml =2~nIN(0) f ~dQ
× Vo(kv, k'F)V,(kF, k'F)[SI . (14)
which is just eq. (6) ofref. [9] with
;
(20)
2(COn--Ogre)sgn(69m) + x ( t + + t _ ) sgn(6)n)
2(o)n) = 2
G3,4 (x, x'; ion) = -- [GI,2 (X, x'; ioJn)]*
-r-- I Vo(kv, k~) 12
(17)
Ttr
(21)
vv is the Fermi velocity calculated from the isotropic Fermi gas picture,/ta is Bohr's magneton and [... ]* denotes the complex conjugate. The result (19) differs from eq. (15a) of ref. [10] only by the additional contributions coming from the orientation of the z-component of the impurity spin (Sz). We can therefore expect the off diagonal elements of the electron Green's function, denoted by ] (io&) to be complex as was the case in ref. [ 10 ]. We find the relations: 3,,2 ( i o . ) = COc
defines the inverse transport relaxation time,
g T Z [,~(t.On-tOm)-tl*]Z,,2(~),n)~,,z(iogn) m
I=2xn,N(O)S(S+I) ~m
da -~ IV~(kv, k'v)l
(18)
defines the inverse magnetic relaxation time, N(0) is the electronic density of states at the Fermi surface
+nt,Z,.2 (~,)J~.2 (itn,) + lttzZ,.2 (03,)J2., (i09n) (22) with
36
W. Pint, E. Schachinger/ Superconducting alloys withparamagnetic impuritiesI
dqe -q2
,~1,2 ((.On) ~---
×tan-I
I~.l-Ti(#uB+niV~(O)(S~))sgn(&.)
'
(23) 2a=eBc2(T)v 2 , t, =
2(S~) t+ - t _ S ( S + 1------~ ; t2 = t _ -s ( s-+ 1 )
(24)
2 ( S 2) + (S~) = S ( S + I ) ,
=
[Z~(&.) 1"
eq. (28) turns into
(26)
there are two possible choices for the gap function: (i) A2(io9.) = [~t (io9.) ]* and (ii) A2(iog.) = - [A~ (io9.) ]*. In the case of nonmagnetic impurities both choices are valid and give the same result. In the presence of magnetic impurities only one solution, namely A2(io9n)=-[A~(io9.)]* will recover the basic symmetry properties of Green's functions [ 17 ]: G(k, o9) = [ G(k, -o9) ]*
(27)
We find finally: ~(iog.) = ~ T ~ [2 (o9~ - o9,~) - #* ]X(&,.)A(iog~) m
+TtZ(~.)
t+ - t_ S ( S + 1--------~
- t _ S2((S$+2 )1----'-~[j'(iog.) 1.1
(28)
with z(6o.)
2 idqe_q2
=~
(30)
(25)
/~* is the Coulomb interaction pseudo potential and ogc is some cut-off frequency usually chosen to he an integer multiple of the Debye frequency O9DBecause of Z2(69,)
paramagnetic limiting (PPL) term #BB in eq. (29). It may even happen that the contribution which comes from spin orientation effects will compensate the PPL term completely. In such a case, Z(&.) will become real and so will ](io9.). This in turn will eliminate any contribution to the over all behaviour which comes from spin orientation effects or from Pauli limiting effects; because of
o
V
Xtan-~ L lCb. I --i(UBB+ n~qx/-a V~(0) (S~)) sgn(rJ.)] (29) It is important to recognize that according to eq. (5) (Sz) has a negative sign and therefore the term nlV~ ( 0 ) ( S ~ ) compensates at least partly the Pauli
zt(io9n)= ~T ~ [~( o9n --o9m) -- lA* ])~( (~_)m)~(io9m) m
+re(t+ - t_)Z(~,)3(io9,).
(31)
This equation would have been found by neglecting PPL and spin orientation effects from the beginning. We also note in passing that eqs. (28) and (29) reduce to eq. (5) of ref. [9] for T=T~ and B2¢(T~) =0.
4. Numerical results
In this chapter we will discuss results of the theory on the basis of the rather weak coupling system LaA12 having a Tco of about 3.3 K [18]. An a(og)2F(o9) spectrum can be derived from the generalized phonon density of states reported by Yeh et al. [ 19 ] by simple rescaling to give a mass enhancement factor 2=2(0) =0.55 which results in a/1" of about 0.1 for the given Tco. It is possible to make rather clean single and ploycrystals from this material [ 20 ] and we therefore call it a clean limit host system. Our numerical results for this system will be contrasted by calculations involving a dirty limit host system having a very small transport relaxation time and the same values for Tco, 2 and ~*. Typical dirty limit host systems would be amorphous metals [ 21 ] but they have higher critical temperatures (Tc~ 8 K) and higher mass enhancement factors (2--- 1.0). In the first step we compare the decrease in critical temperature Tc with increasing concentration of magnetic impurities (increasing value of t_ ) as calculated from the full Eliashberg equations with the results of the weak coupling limit equation [22 ]:
W.. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
Tc0
t_
a.-
,32,
where ~,(x) is the digamma function. Results are shown in fig. 1 with the dotted curves corresponding to data calculated from eq. (32). The full strong coupling calculation (solid line) predicts a higher critical concentration and the critical concentration and at the critical concentration predicted by eq. (32), the full theory shows that the alloy is still superconducting with a reduced critical temperature of tc = Tc/ Too~0.15. The weak coupling limit of eqs. (14), (28) and (29) is obtained by using the square well model described by Allen and Mitrovi~ [23], following closely the steps outlined by Schachinger et al. [24] and neglecting PPL and spin orientation effects: In
"~ )
(36)
t+
t*+- 1+2
(37)
J ( a * ) = 2 f dqe-q2 tan -1 (qa*) . 0
Next, we compare the results for the initial slope IdB¢2(T)/dT[ r=r~ calculated from eq. (33) with the results of a full strong coupling calculation, eq. (31), in fig. 2 for two limiting cases using the same value o f 0 . 3 2 × 106 m / s for the Fermi velocity VF: (a) the clean limit of the host material (t+ = 0 ) and (b) the dirty limit of the host material (simulated by t+ = 100 meV). In both cases we find differences between weak coupling limit results and the full strong coupling theory; they are very pronounced in the dirty limit. The general feature is that for higher concentrations of magnetic impurities (tc<0.6) the weak coupling limit equations begin to predict steeper initial slopes than the full theory. This becomes more pronounced with further increasing impurity concentrations and affects the reduced upper critical field
(33)
h*= eBc2(T)(v~)2 ,
vv
oo
with 2~2T2o
(35)
and
ra~O
(T/Tcox/~) J(a*,.)
x/~ ( l + 2 ) l o g , l+n(t*+ +t*_) '
v~= 1+2'
= 2 --
-1 - ( (t-~+---~-_) ~ ) ( a ~ , )
37
(34)
4.0.
3.0.
%
2.0-
1.0
0.0 0.00
0.02
0.04
t
,
0.06
0.08
meV
Fig. 1. Critical temperature vs. impurity parameter t_ ocnl, the concentration of magnetic impurities. The solid line presents results of a full Eliashberg type calculation while the dotted line corresponds to results found by solving eq. (32).
38
W. Pint, E. Schachinger/ Superconducting alloys with paramagnetic impurities I 0.03
5,00
3.75
Y
0.02
I.-II E-
II I-
2.50'
ra
0.01
1.25'
0'000.0
0:5
a
1.0
to = T J T o
0"000. 0
0:5
b
1.0
to = T J T o o
Fig. 2. (a) Initial slope of the upper critical field vs. the reduced critical temperature for the clean limit case of the host material ( t + = 0 ). Solid and dotted curves present the results of a full strong coupling calculation and of a weak coupling limit calculation respectively. (b) Same as fig. 2a but for the dirty limit of the host material (t+ = 100 meV). Note the pronounced differences between the two theories.
BCz(T) bc2(T) = ]dBc2( T) /dTIT=T~Tc "
(38)
This dimensionless quantity offers the advantage not to d e p e n d on the F e r m i velocity. We present results for bc2(t), t = T / T ~ in fig. 3 for the clean a n d the dirty limit o f the host m a t e r i a l a n d for three reduced critical temperatures, n a m e l y te= 1.0, 0.6 a n d 0.3. At first we note that the r e d u c e d u p p e r critical field curve changes substantially with increasing concentrations of magnetic impurities. The fact that be2 (t) is decreasing with increasing i m p u r ity concentration implies that the u p p e r critical field Bcz(T) is m o r e strongly suppressed at lower temperatures than it is suppressed close to To. I f this were not the case the bcz(t) characteristic should be the same for all concentrations o f magnetic impurities. We conclude that magnetic i m p u r i t i e s b e c o m e m o r e effective in breaking up C o o p e r pairs at low tern-
peratures even in cases where we assume that spin orientations effects are negligible. Figure 3 also compares the predictions o f the full strong coupling theory, eq. (31 ), (solid lines) to those o f the weak coupling limit theory ( d o t t e d lines). In the clean and dirty limit o f the host material without magnetic impurities we see no or little difference between the two theories. Since there are differences in the initial slope we have to conclude that the weak coupling limit theory underestimates Be2 ( T ) in the region o f low t e m p e r a t u r e s but gives a qualitatively correct description o f the t e m p e r a t u r e d e p e n d e n c e o f Be2. The sample with a reduced critical t e m p e r a t u r e & = 0 . 3 on the other h a n d shows a very p r o n o u n c e d difference between the predictions o f the full strong coupling theory a n d its weak coupling limit. This beh a v i o u r can be related to the overestimation o f the initial slope by the weak coupling limit theory. Nevertheless, the difference seems to be p r o n o u n c e d
IV.. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I .~
0.8
39
0.8 "r
t t
0.6
0.6 _
= 1.0
= 1.0
0.6
0.6-
0.3
0.3
0.4-
0.4.
",.°°o
",**
0.2. °'%°
o o Ia 0.0
0;5
FI~
1.o
T/T
0.2
o.o
b
o.o
~°°'°
o15
1.0
T/To
Fig. 3. (a) Reduced upper critical field bat(t) vs. the reduced temperature T/Tc for samples in the clean limit of the host material and with reduced critical temperatures to= To~Too=1.0, 0.6 and 0.3. The solid lines represent results calculated from a full strong coupling theory while the dotted lines correspond to predictions of the weak coupling limit approach. Note the pronounced differences in the two approaches for the sample with a to= 0.3. (b) Same as fig. 3a but for the dirty limit of the host material.
enough to be clearly detectable by experiment and we conclude that samples with impurity concentrations exceeding about half of the critical concentration should allow us to study strong coupling effects which go beyond the (1 + 2 ) renormalization of the standard BCS-type results. As a final point, we investigate the effect of spin orientations on the reduced upper critical field be2(t) by solving the full set of eqs. (14), (28) and (29). For this purpose we concentrate on a L a A l 2 alloy with 0.1 at% Gd having a Tc of about 2.9 K [20] as a model system. The spin of the Gd-atom is 7/2 and we are interested in the effect different values of V~(0) have on the reduced upper critical field. The results of our calculations are shown in fig. 4a for a clean limit host system. As we have to assume that the Gd impurities will also result in some decrease in the transport relaxation time we set t+ =0.09 in order to describe the effect of the mo-
mentum scattering part Vo of the impurity potential. The appropriate value of t is calculated from the decrease in critical temperature. The figure presents bc2(t) curves for a set of values I/1(0)=0.25, 0.5, 1.0, 5.0 and 10.0 meV and hi=0.1 (full lines). The results are compared with calculations neglecting spin orientation (dotted line). We see that bc2(t) reaches a maximum at some temperature which depends on the actual value of the potential V~(0). Thus, in trying to fit experimental results we can retrieve some information on the actual microscopic paramagnetic spin exchange potential. This results reported in this figure are quite similar to results discussed earlier by Machida and Younger [ 14 ]. For sufficiently large, but probably unrealistic, values of V~(0) ( V~(0) > 5 MeV) even our system shows a "reentrant"-type behaviour, i.e.: a temperature region close to absolute zero where B c e ( T ) =0.
W. Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
40
o.8 la
0.8 V,(O) =
b
0.25 meV 0.5 1.0
0.6
~'...
5.0
V,(O) =
1.0
meV
0.6 _ , , _ , , . , ~ ~ 0.5
10.0 0.25 0.4
f,~
__ "%%
0 . 0 II 0.0
""""%%
0.2 •
0.2-
0:5
1.o
T/To
0.0o.o
0:5
~.o
T/To
Fig. 4. (a) Reduced upper critical field bc2(t) vs. the reduced temperature for a LaA12 sample with 0.1 at% Gd. The host material is in the clean limit. The full lines describe the effect of spin orientation for different values of V, (0), namely 0.25, 0.5, 1.0, 5.0 and 10.0 meV and with an impurity concentration nt = 0.1. The dotted line corresponds to the result one would get for a system without spin orientation effects. (h) Same as fig. 4a but for a dirty limit host material. The dashed line describe the influence of Pauli paramagnetic limiting alone.
The situation changes quite remarkably if alloys based on a dirty limit host material are studied. The results of our calculations can be found in fig. 4b for the same values of V, (0) and n, as in fig. 4a. The dotted line presents the solution of eq. (31 ) for this material and the dashed line shows the additional influence of PPL on the reduced upper critical field. In dirty limit materials we have to assume rather high upper critical fields which will then result in rather pronounced PPL effects suppressing the effective upper critical field quite substantially. Finally, the solid lines in this figure present the result which include impurity spin orientations effects. For large spin exchange potentials 1.'1(0) >/5 meV b¢2 (t) is almost completely suppressed and we find a "reentrant"-type behaviour. The case V, (0) = 1 meV is special because in this case the PPL term and
the spin orientation term of eq. (29) cancel each other at about t ~ 0.5 which brings the bc2(t) characteristic close to the one we found from solving eq. (31 ). The oscillations of the solid line around the dotted line show very nicely the delicate balance between PPL and spin orientation effects. Making V, (0) even smaller results in an only partial cancellation of paramagnetic limiting and the sample will show a higher upper critical field as one would expect from neglecting spin orientation effects at all. Thus alloys based on dirty limit host materials should display a behaviour which is distinctly different from the behaviour predicted for alloys based on clean limit materials. Using different paramagnetic elements and studying alloys with different impurity concentrations should provide enough vari-
V~ Pint, E. Schachinger / Superconducting alloys with paramagnetic impurities I
ation o f the factor nl Vl (0) tO observe the described effects experimentally.
5. Conclusion We studied the effect o f magnetic impurities on the upper critical field o f superconducting alloys, We assumed that the impurity spin was classic and the impurity concentration very dilute which resulted in two further assumptions: (i) a r a n d o m distribution of the impurity sites throughout the sample and (ii) the absence of correlations between impurity spins. The only effect which temperature and external magnetic field were allowed to have on the impurity spin was described by a "spontaneous" orientation of these spins against the direction o f the external magnetic field represented by a Boltzmann distribution. These assumptions resulted in a temperature independent magnetic relaxation time and a spin orientation effect which is similar to the Pauli paramagnetic limiting term. But in contrast to this term which depends only on the magnetic field Be2 (T), the spin orientation term has an additional, explicit temperature dependence. Thus the effect o f spin orientation on the upper critically field will become more pronounced with decreasing temperatures and increasing magnetic fields. A numerical investigation o f the theory resulted in two interesting results: (i) for concentrations exceeding about half of the critical concentration, strong coupling effects going beyond the ( 1 + 2) renormalization become important and pronounced enough to be detected by experiment and (ii) the behaviour o f alloys based on clean limit or dirty limit host systems will show completely different features as far as the influence o f spin orientation is concerned. Depending on the actual strength of the paramagnetic spin exchange potential V~(0) the alloy based on a clean limit host material will show some m a x i m u m in Be: (T) with decreasing temperature and in extreme cases on may even find a "reentrant"-type behaviour having B~2 (T) = 0 close to absolute zero. This is in contrast to the behaviour o f alloys based on dirty limit host materials where we can observe a delicate balance between Pauli paramagnetic limiting effects and the influence o f the impurity spin orientation. Again, in the case o f very large magnetic exchange
41
interaction potentials a "reentrant"-type behaviour could be observed. These results are in strong contrast to models with temperature dependent magnetic relaxation times because they cannot describe such differences between alloys based on clean or dirty limit host materials. They will predict about the same kind o f behaviour for both types o f alloys.
Acknowledgements This research was supported in part by the Office of Naval Research under grant No. N00014-89-J1088. The authors want to thank Dr. H.W. Weber for many fruitful discussions and his continuous interest in this work. One o f us (E.S.) also wants to thank the members of the Center of Theoretical Physics at the Physics Department of the Texas A&M University for the warm welcome and their hospitality.
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