- Email: [email protected]
On the problem of coexistence of superconductivity and ferromagnetism
ON THE PROBLEM SUPERCONDUCTIVITY 8. FISCHER lnstitut
de Physique
OF COEXISTENCE OF AND FERROMAGNETISM and M. PETER
de lu Matit?re Condense’e,
Universite’ de GenPve, Bd. d’Yvoy 32.
121 I GenPve 4, Switzerland
Synopsis
The suggestion of Jaccarino and Peter to compensate the exchange field in a ferromagnet by an external field to make possible the coexistence of ferromagnetism and superconductivity is analyzed. The coexistence is. however, only possible if the orbital critical field (without Pauli paramagnetism effects) is at least of the order of the mean exchange field. To reach high orbital critical fields one has to introduce disorders and impurities in the system. However in a ferromagnetic system this leads to strong exchange scattering& tending to reduce H,,. Only in superconductors with an extremely high K” does it seem possible to reach the critical fields necessary to realize the compensation.
1. Introduction. Several years ago it was found in EPR-measurements’) that the electron gas around a magnetic impurity often becomes polarized antiparallel to the impurity spin. On the other hand it is generally supposed that a ferromagnet does not become a superconductor because its conduction electrons are strongly polarized. It was suggested by Jaccarino and Peter2) that in cases where the polarization is antiparallel to J (the total magnetic moment) it may be compensated by an external magnetic field, thus allowing the ferromagnet to become superconducting in a range of very high magnetic fields. Coexistence of ferromagnetism and superconductivity was reported many years ago by Matthias et a1.3) in the system (Ce,_,Gd,)Ru,. Recent measurements on this system by Hillenbrand and Wilhelm4) on samples with good homogeneity did not show superconductivity in samples that had already become ferromagnetic. This indicates that in this concentration range the mean exchange field is too strong to allow coexistence of superconductivity and ferromagnetism, although one expects the spin-orbit scattering to be strong in such a system. The compensation method of Jaccarino and Peter could, in principle, be used here to realize the coexistence. However, to do this compensation, the magnetic field must penetrate the superconductor. In non-magnetic superconductors this is prevented at low fields by the interaction of the external field with the orbits of the electrons. In type-II superconductors the external field may partially penetrate the superconductor, but the interaction with the 597
598
8. FISCHER
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conduction electron orbits sets an upper limit on the held that the superconductor can stand. Unfortunately, these critical fields are, in the best cases, still somewhat lower than the fields needed to compensate the exchange polarization. In the following we always consider a dense ferromagnet, consisting of localized magnetic ions and a gas of conduction electrons. We suppose further that our system would have been superconducting if there were no exchangeinteraction between the conduction electrons and the magnetic ions. In section 2 we discuss the case of a well-ordered ferromagnet. We review the results of Avenhaus et ~1.~) who examined the possibility to compensate the Meissner-current and thus increase the critical field in a ferromagnetic superconductor with respect to the corresponding non-magnetic superconductor. In section 3 we proceed to the disordered ferromagnet. The critical field H,, of such a system is determined by two competing mechanisms: (1) nonmagnetic scatterings, raising N,,, (2), exchange scatterings decreasing T, and thus If,,. With increasing disorder the latter one becomes dominant and determines thus the maximum critical field that can be obtained for a certain substance. 2. Meissner efect in a regular dense ferromagnetic superconductor. The first question that arises is whether there are any interactions in a ferromagnetic system that may counteract in some way the interaction of the external field with the orbits of the conduction electrons. Between the magnetic ions and the electron gas there will be exchange interactions and the interaction of the conduction electrons with the dipole field produced by the spin and the angular momentum of the ions. As KitteP) has shown, the vector potential Aid of the dipole field contains a periodic part which does not give any contribution to the mean magnetization, and an aperiodic part, coming from the surface and which produces the whole mean magnetization. In the exchange interaction one normally only considers the spin dependence. However, as one knows from atomic physics7), the exchange interaction depends as well on the momentum of the particles involved. This is because the oneelectron wave function depends on momentum. The exchange interaction may therefore be written in the following form: 3V’ex =
z *
J(r-&)a&+
z D(r-l&)&-t 1
*. *,
(1)
where Si is the spin, Liis the angular momentum and Ri is the position of the ith ion. u, I and r are the corresponding values of the electrons. Since we are dealing with a model of localized magnetic ions, the functions J(r-Ri)and D (r - RJ (and also corresponding higher-order functions not written in eq. ( 1)) will be of short range. One may try to calculate the function D(r-Ri), using a Hartree-Fock procedure. However it seems wiser to see first if such an
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
interaction can have an effect on the Meissner-current. term in eq. (1) as
599
Writing the second
one may write the following Hamiltonian for our system:
+
z
V(r--Rd
+z
J(r-Ri)aSi
A, is the vector potential of the external field. V is the periodic potential and G$?~,_~, is the electron-electron interaction, including Coulomb repulsion and electron-phonon interaction. The higher terms in eq. ( 1) were neglected. This system was discussed by Avenhaus et ~1.~). It was found that when the superconductivity is not destroyed by Pauli paramagnetism effects, the superconductor reacts on the external field as a non-magnetic superconductor in a field He+ JTidr where Rid = 47rM and M is the mean magnetization. (Demagnetization effects were neglected). This is because the Meissner-current only corresponds to the slowly varying part of the p-dependent interactions. Therefore the p-dependent part of the exchange interaction will not give any contribution to the Meissner-current. The same thing is true for the periodic part of the dipole field and the spin-orbit interaction (not written in eq. (2)). This result means however that there is in fact no possibility to compensate the Meissner-current. The critical field in a ferromagnetic superconductor will not be larger than in the corresponding non-magnetic superconductor, except for the fact that in a ferromagnetic superconductor one may compensate the effect of the spin paramagnetism on H,,. In this discussion of the Meissner-current the spin-dependent terms in Z’ were neglected, i.e., we supposed that we were in a compensated region. However, there remains a periodic variation of the exchange term &OS& J(r- R,) J). .i is the mean value of&r-&) evaluated at the Fermi surface. The situation is very similar to the one found in a simple antiferromagnetic system. For the latter case, Baltensperger and Strlsslel-8) have shown that a superconducting state with singlet pairing is possible. In the present case the construction used by them is not possible. However, since the period of the exchange interaction is normally much smaller than the coherence length, one does not expect such a periodic oscillation to have a strong influence on the superconducting proper-
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ties. The dynamics of the spin system will, on the other hand, have an influence on the effective interaction between the electrons. These effects have been investigated for the ferromagnetic case by Avenhaus et ~1.~). Their results confirm the results of Baltensperger and Strassler for the antiferromagnetic case, i.e., the effective interaction via spin waves is in the region of low temperature (T 4 0,) not strong enough to destroy superconductivity. 3. Effects of impurities and disorder in a dense ferromagnetic superconducfor. In the previous discussion we were concerned with a well-ordered system. However, the critical field for such a system is for all practical cases lower than the Clogston limitlo). Our assumption that the mean field on the spins is zero, therefore means that essentially .i = 0. To proceed to more realistic systems we have to introduce disorder and impurities into our system in order to raise H,, sufficiently so that we are able to compensate a real J # 0. In the last years many authors have investigated the effect of disorder and impurities on the critical field”-13), and recently Hake14) has published critical field limits for several high-field superconductors. These calculations were based on the work of Werthamer et u1.15), and showed that for the P-W structures these limits lie in the region of a few megagauss when one neglects the effect of Pauli-spin paramagnetism. Taking into account the latter one, and supposing strong spin-orbit scatterings the critical-field limits are reduced approximately by a factor 2 or 3. As an example, Hake finds for V,Ga a limit of 2540 kG without spinparamagnetism, but including the spinparamagnetism he finds 8 10 kG. Thus, since there exists where the mean exchange field is of the order of one megagauss, it seems at first glance that spin compensation should be possible and that one may gain a critical field compared with the non-magnetic case. However, these extremal critical fields can only be reached in very disordered systems. In a dense ferromagnet, such a disorder will necessarily involve strong exchange scattering, tending to decrease T, and thus H,,. We start with the ordered ferromagnet considered in the previous section. Since we are interested in the case where the net field acting on the spins of the conduction electrons is zero, we consider a hypothetical ferromagnetic system where the mean exchange field acting on the conduction electrons is zero, and where the external field only acts on the orbits of the conduction electrons. We then introduce disorder and impurities in the system and study the orbital critical field Hz2 as a function of the electron lifetime T. For cases where Hz2 turns out to be larger than the exchange field, H,, = .fS/gp and the latter one is negative, our hypothetical system will describe a real system in a narrow region of magnetic field H E=-H,,. Thus we are interested in finding the maximum critical field for a certain substance. For a non-magnetic system the corresponding calculation has been done by Helfand and Werthamer [HW]13). The result of their calculation is that Hz2
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
601
can be written as
-$F T,%(A),
Hz=
where h(A) is given by the equation
In% =5[&c
--m
(TITch”2)J(%) 1-
(A,h’/“).J(a,)
J((Y,) = 2 jd w exp (-w”) tan-l (Y,w;
I
;
(4)
(5)
0
T is the transport scattering time. In the non-magnetic case one may take T, as independent of h, but for the magnetic case this is no longer correct. Consider the perfect-ordered ferromagnet. If one replaces one magnetic ion by a non-magnetic ion, the system can be described as the periodic lattice plus a non-magnetic impurity minus a magnetic impurity. Since the periodic lattice is the unperturbed superconductor, the situation is nearly the same as for a magnetic impurity in a non-magnetic superconductor. The difference is that in the present case the impurity spins are all aligned. A similar argument holds for dislocations and for disorder in a compound. Now, as one can easily check, the calculation of Abrikosov and Gorkov14) also holds for aligned but randomly distributed spins, if one neglects the mean value of the exchange interaction. Therefore in absence of a magnetic field, the critical temperature will be given by: ln+=
+(4++)-+(i),
c
(7)
where
r
h,=L=_.
.rrkB Tcr,
1’=-
Ts.crit
7s
2yt(l')
T,(r)
*
9t(“) =-* T,,(O) ’ 2Yfi
7s~crit = ~ TkJco
.
(8) (9) (10)
7s is the magnetic scattering time and In y is the Euler constant. For our system rs is proportional to the non-magnetic scattering time T. We write: rs =
a7.
(11)
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8. FISCHER
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M. PETER
The same relation also holds for a ferromagnetic compound where the magnetic ions only form a sublattice, if one supposes the impurities and the disorder to be randomly distributed. We now use the result from the previous section, that the ferromagnetic superconductor reacts on the external field as a non-magnetic superconductor if one includes the mean dipolefield Hid in the external field. As a first approximation we therefore calculate the critical field from the theory of [HW] but taking into account that T, decreases as T (and therefore 7,) decreases. Using eq. (3) we can write the enhancement h of H&T = 0) as
(12) From (6), (10) and ( I 1) we can write A as a function off: +a_.
r
(13)
4Y t(r)
The equations (7), (12) and (13) determine h”as a function of r. Note that the only parameter is a. Since we work with a hypothetical system we do not know the exact value of a. However, the order of magnitude can be estimated from experiments on magnetic impurities in pure and ordered superconductors. Using (lo), (11) and p = m/ne? we get a=
2yfLneZpcrtt nkBTcom
’
(14)
where perit is the normal state resistance at the critical concentration minus the resistance of the pure metal (for T = 0). With this formula one finds that a is typically of the order 100 for Gd3+ as an impurity atom. For the other rare earths a will be somewhat larger and for the transition metals somewhat smaller. For the system Th,_,Gd, we find, using the resistivity measurements of Peterson et ~1.‘~) and the T, measurements of Decker and Finnemore?, a = 170. A case where the exchange interaction is extremely small is the system Th,_,Er,. Andresls) has measured the dependence of T, on x and finds that the critical concentration is 6%. Together with the resistivity measurements15) this gives a = 1900. Note that for a system N,M, where the magnetic ions M only form a sublattice one expects a to be larger by a factor (x+ y)/x compared with the case discussed abovezO). To calculate l(F) we have to know h as a function of A. This calculation has essentially been given by [HW]. Using their values of h* defined by (15)
COEXISTENCE
and the formula,
OF SUPERCONDUCTIVITY
603
due to GorkoP)
we find that h is essentially
35 -
AND FERROMAGNETISM
linear in A. The result is shown in fig. 1.
h
30-
25.
20.
IS-
A 01 0
5 Fig.
I.
IO
15
20
25
. 30
35
4
The function h(A) as calculated from the works of refs. 13 and 2 1.
We finally calculated i(r) for the typical value a = 100 and for the more extreme value a = 1000. The results are shown in fig. 2. We find a maximum of h for T = 0.6, which corresponds to the values &,,(a haa&
= 100) = 4.2; = 1000) = 37.
The corresponding critical field Hz2 in a non-magnetic superconductor is approximately 2 times larger. For large a one may use the Maki-Fulde2*) theory (valid for large A) to calculate 5. The result is essentially the same as found abovez3). Knowing from experiment that for most structures the critical field for the pure and ordered case is not more than a few kgauss, it is clear that none of
604
8. FISCHER
0
a2
AND M. PETER
a4
0.6
0.8
1
r
Fig. 2. The reduced orbital critical field 6 of a ferromagnetic superconductor. are for the corresponding non-magnetic case.
The dashed curves
these structures can give a compensated ferromagnetic superconductor even if one was able to produce the right amount of disorder. The only possibility that remains for these cases is when the magnetic ions form a rather dilute but well-ordered sublattice. However, for certain structures very favourable for superconductivity, the critical field of the pure and well-ordered sample is very high. Let us take the example of the p-W structures, and suppose we were able to produce a ferromagnet with the electronic properties of V&a. Using the calculations of Hake we may calculate the critical field H,: for a given amount of disorder (i.e. for a given 7) as a function of a. We did that for the two cases where the orbital critical field H,* for a = +a is (a) 2500 kgauss (limit given by Hake)24) and (b) 1000 kgauss. The results are shown in fig. 3., and as one sees it is possible to reach relatively high Hz2 values also for low a’s (thus the magnetic scatterings are probably not the most serious difficulty in this case). However, one has to remember that our system is only realistic if the mean exchange field is nega-
COEXISTENCE
OF SUPERCONDUCTIVITY
500
0
50
100
a
1.50
AND FERROMAGNETISM
1
1000
200
605
250
a Fig. 3. Orbital critical field of a hypothetical ferromagnetic superconductor with the electronic properties of V,Ga. In the upper figure the transport-scattering time 7 corresponds to an orbital critical field in the non-magnetic case of 2,5 mgauss (limit of Hake). In the lower one this critical fieldis I MG.
tive and smaller than Hz2. If we suppose S = &this gives for the first case IJ 1 < 0.046 eV and for the second case JJI < 0.018 eV25). On the other hand there remains, of course, the problem of producing the right amount of disorder. Therefore, although there is a possibility to realize the compensation in a bulk material, the preparation of the sample seems to involve some difficult problems. 4. Conclusions. The orbital critical field in a ferromagnetic superconductor is influenced by two effects. (1) Non-magnetic scatterings, tending to increase H,* and (2) magnetic, scatterings decreasing T, and thus H$ As the degree of
606
8. FISCHER
AND M. PETER
disorder and impurity content increases the latter one tends to dominate thus defining a maximal orbital critical field, that can be obtained for a certain substance. One may distinguish two cases: 1) The ferromagnet would in absence of exchange interaction have been a low-~,, superconductor. In this case there seems no hope to produce high critical fields in a dense ferromagnet. To realize the coexistence one has to produce a compound where the magnetic ions form a rather dilute but nevertheless a well-ordered sublattice. 2) The ferromagnet would in absence of exchange interactions have been an extreme high-K0 superconductor. In these cases one need not introduce too much disorder to reach orbital critical fields necessary to compensate the mean exchange field, thus these cases seem to be the only ones where the compensation suggested by Jaccarino and Peter may be possible. Note that our discussion may also be applied to .the antiferromagnetic case. Although the possibility of coexistence of superconductivity and antiferromagnetism has been demonstrated*), a few percent of disorder in the antiferromagnetic lattice will be sufficiently to destroy superconductivity. This may be one reason why so far no antiferromagnetic superconductor has been found. Acknowledgments. Muller for interesting on this paper.
We want to thank Professors B. Giovannini and J. discussions and H. Jones for many valuable comments
REFERENCES 1) Jaccarino, V., Matthias, 9. T., Peter, M., Suhl, H. and Wernick, J. H., Phys. Rev. Letters 5(196O)Y524. 2) Jaccarino, V., Peter, M., Phys. Rev. Letters 9 (1962)290. 3) Matthias, 9. T., Suhl, H. and Corenzwit, E., Phys. Rev. Letters l(1958) 449. 4) Hillenbrand, 9. and Wilhelm: to be published in J. Phys. Chem. Solids. 5) Avenhaus, R., Fischer, Id., Giovannini, 9. and Peter, M., to be published in Helv. Phys. Acta. 6) Kittel, C., Phys. Rev.,Letters 10 (1963) 339. 7) See e.g. Slater, J.,: Quantum Theory of Atomic Structure, McGraw-Hill, N.Y. (1960). 8) Bahensperger, W. and Striissler, S., Phys. kondens. Materie l( 1963) 20. 9) Avenhaus, R., Enz, C. P., Heim, Klose, W. and Peter, M., unpublished. 10) Clogston, A. M., Phys. Rev. Letters 9 (1962) 266. Chandrasekhar, 9. S., Appl. Phys. Letters 1(1962)7. 1 I) Maki, K., Physics 1 (I 964) 2 I. 12) De Gennes, P. G., Phys. kondens, Materie 3 (1964) 79. 13) Helfand, E. and Werthamer, N. R., Phys. Rev. 147 (1966) 288. 14) Hake, R. R., Appl. Phys. Letters 10 (1967) 189. 15) Werthamer,N. R., kelfand, E. and Hohenberg, P. C.; Phys. Rev. 147 (1966) 295. 16) Abtikosov, A. A. and Gorkov, L. PI, Soviet Physics-JETP 12 (1961) 1243. 17) Peterson, D. T., Page, D. F., Rump, R. 9. and Finnemore, D. K., Phys. Rev. 153 (1967) 701.
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
607
18) Decker, W. R. and Finnemore, D. K., Phys. Rev. 172 (1968)430. 19) Andres, K., Tagung der schweiz. Phys. Gesellschaft, Bern (May, 1969). 20) Note however that our example of rare earth in Th is a favourable case where the exchange interaction is weak. 21) Gorkov, L. P., Soviet Physics-JETP 10 (1960) 998. Gorkov does not give explicitly this formula, but it can be obtained from his eqs. (15), (I 9) and (2 I). 22) Fulde, P. and Maki, K., Phys. Rev. 141(1966) 275. 23) It follows from the work of Fulde and Maki for the case of magnetic impurities in nonmagnetic superconductors that H,,(T = 0) decreases linearly with concentration and somewhat faster than T,. However since we are concerned with orders of magnitude rather than exact values, this does not change the essential results. 24) Note, however. that for this case the coherence length will be of the order interatomic distances, thus the periodic variations of the exchange interaction may in this case have a strong influence on the superconductivity. 25) We supposed here that in AsB only the B atoms are magnetic.
de Physique
OF COEXISTENCE OF AND FERROMAGNETISM and M. PETER
de lu Matit?re Condense’e,
Universite’ de GenPve, Bd. d’Yvoy 32.
121 I GenPve 4, Switzerland
Synopsis
The suggestion of Jaccarino and Peter to compensate the exchange field in a ferromagnet by an external field to make possible the coexistence of ferromagnetism and superconductivity is analyzed. The coexistence is. however, only possible if the orbital critical field (without Pauli paramagnetism effects) is at least of the order of the mean exchange field. To reach high orbital critical fields one has to introduce disorders and impurities in the system. However in a ferromagnetic system this leads to strong exchange scattering& tending to reduce H,,. Only in superconductors with an extremely high K” does it seem possible to reach the critical fields necessary to realize the compensation.
1. Introduction. Several years ago it was found in EPR-measurements’) that the electron gas around a magnetic impurity often becomes polarized antiparallel to the impurity spin. On the other hand it is generally supposed that a ferromagnet does not become a superconductor because its conduction electrons are strongly polarized. It was suggested by Jaccarino and Peter2) that in cases where the polarization is antiparallel to J (the total magnetic moment) it may be compensated by an external magnetic field, thus allowing the ferromagnet to become superconducting in a range of very high magnetic fields. Coexistence of ferromagnetism and superconductivity was reported many years ago by Matthias et a1.3) in the system (Ce,_,Gd,)Ru,. Recent measurements on this system by Hillenbrand and Wilhelm4) on samples with good homogeneity did not show superconductivity in samples that had already become ferromagnetic. This indicates that in this concentration range the mean exchange field is too strong to allow coexistence of superconductivity and ferromagnetism, although one expects the spin-orbit scattering to be strong in such a system. The compensation method of Jaccarino and Peter could, in principle, be used here to realize the coexistence. However, to do this compensation, the magnetic field must penetrate the superconductor. In non-magnetic superconductors this is prevented at low fields by the interaction of the external field with the orbits of the electrons. In type-II superconductors the external field may partially penetrate the superconductor, but the interaction with the 597
598
8. FISCHER
AND
M. PETER
conduction electron orbits sets an upper limit on the held that the superconductor can stand. Unfortunately, these critical fields are, in the best cases, still somewhat lower than the fields needed to compensate the exchange polarization. In the following we always consider a dense ferromagnet, consisting of localized magnetic ions and a gas of conduction electrons. We suppose further that our system would have been superconducting if there were no exchangeinteraction between the conduction electrons and the magnetic ions. In section 2 we discuss the case of a well-ordered ferromagnet. We review the results of Avenhaus et ~1.~) who examined the possibility to compensate the Meissner-current and thus increase the critical field in a ferromagnetic superconductor with respect to the corresponding non-magnetic superconductor. In section 3 we proceed to the disordered ferromagnet. The critical field H,, of such a system is determined by two competing mechanisms: (1) nonmagnetic scatterings, raising N,,, (2), exchange scatterings decreasing T, and thus If,,. With increasing disorder the latter one becomes dominant and determines thus the maximum critical field that can be obtained for a certain substance. 2. Meissner efect in a regular dense ferromagnetic superconductor. The first question that arises is whether there are any interactions in a ferromagnetic system that may counteract in some way the interaction of the external field with the orbits of the conduction electrons. Between the magnetic ions and the electron gas there will be exchange interactions and the interaction of the conduction electrons with the dipole field produced by the spin and the angular momentum of the ions. As KitteP) has shown, the vector potential Aid of the dipole field contains a periodic part which does not give any contribution to the mean magnetization, and an aperiodic part, coming from the surface and which produces the whole mean magnetization. In the exchange interaction one normally only considers the spin dependence. However, as one knows from atomic physics7), the exchange interaction depends as well on the momentum of the particles involved. This is because the oneelectron wave function depends on momentum. The exchange interaction may therefore be written in the following form: 3V’ex =
z *
J(r-&)a&+
z D(r-l&)&-t 1
*. *,
(1)
where Si is the spin, Liis the angular momentum and Ri is the position of the ith ion. u, I and r are the corresponding values of the electrons. Since we are dealing with a model of localized magnetic ions, the functions J(r-Ri)and D (r - RJ (and also corresponding higher-order functions not written in eq. ( 1)) will be of short range. One may try to calculate the function D(r-Ri), using a Hartree-Fock procedure. However it seems wiser to see first if such an
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
interaction can have an effect on the Meissner-current. term in eq. (1) as
599
Writing the second
one may write the following Hamiltonian for our system:
+
z
V(r--Rd
+z
J(r-Ri)aSi
A, is the vector potential of the external field. V is the periodic potential and G$?~,_~, is the electron-electron interaction, including Coulomb repulsion and electron-phonon interaction. The higher terms in eq. ( 1) were neglected. This system was discussed by Avenhaus et ~1.~). It was found that when the superconductivity is not destroyed by Pauli paramagnetism effects, the superconductor reacts on the external field as a non-magnetic superconductor in a field He+ JTidr where Rid = 47rM and M is the mean magnetization. (Demagnetization effects were neglected). This is because the Meissner-current only corresponds to the slowly varying part of the p-dependent interactions. Therefore the p-dependent part of the exchange interaction will not give any contribution to the Meissner-current. The same thing is true for the periodic part of the dipole field and the spin-orbit interaction (not written in eq. (2)). This result means however that there is in fact no possibility to compensate the Meissner-current. The critical field in a ferromagnetic superconductor will not be larger than in the corresponding non-magnetic superconductor, except for the fact that in a ferromagnetic superconductor one may compensate the effect of the spin paramagnetism on H,,. In this discussion of the Meissner-current the spin-dependent terms in Z’ were neglected, i.e., we supposed that we were in a compensated region. However, there remains a periodic variation of the exchange term &OS& J(r- R,) J). .i is the mean value of&r-&) evaluated at the Fermi surface. The situation is very similar to the one found in a simple antiferromagnetic system. For the latter case, Baltensperger and Strlsslel-8) have shown that a superconducting state with singlet pairing is possible. In the present case the construction used by them is not possible. However, since the period of the exchange interaction is normally much smaller than the coherence length, one does not expect such a periodic oscillation to have a strong influence on the superconducting proper-
600
0. FISCHER
AND
M. PETER
ties. The dynamics of the spin system will, on the other hand, have an influence on the effective interaction between the electrons. These effects have been investigated for the ferromagnetic case by Avenhaus et ~1.~). Their results confirm the results of Baltensperger and Strassler for the antiferromagnetic case, i.e., the effective interaction via spin waves is in the region of low temperature (T 4 0,) not strong enough to destroy superconductivity. 3. Effects of impurities and disorder in a dense ferromagnetic superconducfor. In the previous discussion we were concerned with a well-ordered system. However, the critical field for such a system is for all practical cases lower than the Clogston limitlo). Our assumption that the mean field on the spins is zero, therefore means that essentially .i = 0. To proceed to more realistic systems we have to introduce disorder and impurities into our system in order to raise H,, sufficiently so that we are able to compensate a real J # 0. In the last years many authors have investigated the effect of disorder and impurities on the critical field”-13), and recently Hake14) has published critical field limits for several high-field superconductors. These calculations were based on the work of Werthamer et u1.15), and showed that for the P-W structures these limits lie in the region of a few megagauss when one neglects the effect of Pauli-spin paramagnetism. Taking into account the latter one, and supposing strong spin-orbit scatterings the critical-field limits are reduced approximately by a factor 2 or 3. As an example, Hake finds for V,Ga a limit of 2540 kG without spinparamagnetism, but including the spinparamagnetism he finds 8 10 kG. Thus, since there exists where the mean exchange field is of the order of one megagauss, it seems at first glance that spin compensation should be possible and that one may gain a critical field compared with the non-magnetic case. However, these extremal critical fields can only be reached in very disordered systems. In a dense ferromagnet, such a disorder will necessarily involve strong exchange scattering, tending to decrease T, and thus H,,. We start with the ordered ferromagnet considered in the previous section. Since we are interested in the case where the net field acting on the spins of the conduction electrons is zero, we consider a hypothetical ferromagnetic system where the mean exchange field acting on the conduction electrons is zero, and where the external field only acts on the orbits of the conduction electrons. We then introduce disorder and impurities in the system and study the orbital critical field Hz2 as a function of the electron lifetime T. For cases where Hz2 turns out to be larger than the exchange field, H,, = .fS/gp and the latter one is negative, our hypothetical system will describe a real system in a narrow region of magnetic field H E=-H,,. Thus we are interested in finding the maximum critical field for a certain substance. For a non-magnetic system the corresponding calculation has been done by Helfand and Werthamer [HW]13). The result of their calculation is that Hz2
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
601
can be written as
-$F T,%(A),
Hz=
where h(A) is given by the equation
In% =5[&c
--m
(TITch”2)J(%) 1-
(A,h’/“).J(a,)
J((Y,) = 2 jd w exp (-w”) tan-l (Y,w;
I
;
(4)
(5)
0
T is the transport scattering time. In the non-magnetic case one may take T, as independent of h, but for the magnetic case this is no longer correct. Consider the perfect-ordered ferromagnet. If one replaces one magnetic ion by a non-magnetic ion, the system can be described as the periodic lattice plus a non-magnetic impurity minus a magnetic impurity. Since the periodic lattice is the unperturbed superconductor, the situation is nearly the same as for a magnetic impurity in a non-magnetic superconductor. The difference is that in the present case the impurity spins are all aligned. A similar argument holds for dislocations and for disorder in a compound. Now, as one can easily check, the calculation of Abrikosov and Gorkov14) also holds for aligned but randomly distributed spins, if one neglects the mean value of the exchange interaction. Therefore in absence of a magnetic field, the critical temperature will be given by: ln+=
+(4++)-+(i),
c
(7)
where
r
h,=L=_.
.rrkB Tcr,
1’=-
Ts.crit
7s
2yt(l')
T,(r)
*
9t(“) =-* T,,(O) ’ 2Yfi
7s~crit = ~ TkJco
.
(8) (9) (10)
7s is the magnetic scattering time and In y is the Euler constant. For our system rs is proportional to the non-magnetic scattering time T. We write: rs =
a7.
(11)
602
8. FISCHER
AND
M. PETER
The same relation also holds for a ferromagnetic compound where the magnetic ions only form a sublattice, if one supposes the impurities and the disorder to be randomly distributed. We now use the result from the previous section, that the ferromagnetic superconductor reacts on the external field as a non-magnetic superconductor if one includes the mean dipolefield Hid in the external field. As a first approximation we therefore calculate the critical field from the theory of [HW] but taking into account that T, decreases as T (and therefore 7,) decreases. Using eq. (3) we can write the enhancement h of H&T = 0) as
(12) From (6), (10) and ( I 1) we can write A as a function off: +a_.
r
(13)
4Y t(r)
The equations (7), (12) and (13) determine h”as a function of r. Note that the only parameter is a. Since we work with a hypothetical system we do not know the exact value of a. However, the order of magnitude can be estimated from experiments on magnetic impurities in pure and ordered superconductors. Using (lo), (11) and p = m/ne? we get a=
2yfLneZpcrtt nkBTcom
’
(14)
where perit is the normal state resistance at the critical concentration minus the resistance of the pure metal (for T = 0). With this formula one finds that a is typically of the order 100 for Gd3+ as an impurity atom. For the other rare earths a will be somewhat larger and for the transition metals somewhat smaller. For the system Th,_,Gd, we find, using the resistivity measurements of Peterson et ~1.‘~) and the T, measurements of Decker and Finnemore?, a = 170. A case where the exchange interaction is extremely small is the system Th,_,Er,. Andresls) has measured the dependence of T, on x and finds that the critical concentration is 6%. Together with the resistivity measurements15) this gives a = 1900. Note that for a system N,M, where the magnetic ions M only form a sublattice one expects a to be larger by a factor (x+ y)/x compared with the case discussed abovezO). To calculate l(F) we have to know h as a function of A. This calculation has essentially been given by [HW]. Using their values of h* defined by (15)
COEXISTENCE
and the formula,
OF SUPERCONDUCTIVITY
603
due to GorkoP)
we find that h is essentially
35 -
AND FERROMAGNETISM
linear in A. The result is shown in fig. 1.
h
30-
25.
20.
IS-
A 01 0
5 Fig.
I.
IO
15
20
25
. 30
35
4
The function h(A) as calculated from the works of refs. 13 and 2 1.
We finally calculated i(r) for the typical value a = 100 and for the more extreme value a = 1000. The results are shown in fig. 2. We find a maximum of h for T = 0.6, which corresponds to the values &,,(a haa&
= 100) = 4.2; = 1000) = 37.
The corresponding critical field Hz2 in a non-magnetic superconductor is approximately 2 times larger. For large a one may use the Maki-Fulde2*) theory (valid for large A) to calculate 5. The result is essentially the same as found abovez3). Knowing from experiment that for most structures the critical field for the pure and ordered case is not more than a few kgauss, it is clear that none of
604
8. FISCHER
0
a2
AND M. PETER
a4
0.6
0.8
1
r
Fig. 2. The reduced orbital critical field 6 of a ferromagnetic superconductor. are for the corresponding non-magnetic case.
The dashed curves
these structures can give a compensated ferromagnetic superconductor even if one was able to produce the right amount of disorder. The only possibility that remains for these cases is when the magnetic ions form a rather dilute but well-ordered sublattice. However, for certain structures very favourable for superconductivity, the critical field of the pure and well-ordered sample is very high. Let us take the example of the p-W structures, and suppose we were able to produce a ferromagnet with the electronic properties of V&a. Using the calculations of Hake we may calculate the critical field H,: for a given amount of disorder (i.e. for a given 7) as a function of a. We did that for the two cases where the orbital critical field H,* for a = +a is (a) 2500 kgauss (limit given by Hake)24) and (b) 1000 kgauss. The results are shown in fig. 3., and as one sees it is possible to reach relatively high Hz2 values also for low a’s (thus the magnetic scatterings are probably not the most serious difficulty in this case). However, one has to remember that our system is only realistic if the mean exchange field is nega-
COEXISTENCE
OF SUPERCONDUCTIVITY
500
0
50
100
a
1.50
AND FERROMAGNETISM
1
1000
200
605
250
a Fig. 3. Orbital critical field of a hypothetical ferromagnetic superconductor with the electronic properties of V,Ga. In the upper figure the transport-scattering time 7 corresponds to an orbital critical field in the non-magnetic case of 2,5 mgauss (limit of Hake). In the lower one this critical fieldis I MG.
tive and smaller than Hz2. If we suppose S = &this gives for the first case IJ 1 < 0.046 eV and for the second case JJI < 0.018 eV25). On the other hand there remains, of course, the problem of producing the right amount of disorder. Therefore, although there is a possibility to realize the compensation in a bulk material, the preparation of the sample seems to involve some difficult problems. 4. Conclusions. The orbital critical field in a ferromagnetic superconductor is influenced by two effects. (1) Non-magnetic scatterings, tending to increase H,* and (2) magnetic, scatterings decreasing T, and thus H$ As the degree of
606
8. FISCHER
AND M. PETER
disorder and impurity content increases the latter one tends to dominate thus defining a maximal orbital critical field, that can be obtained for a certain substance. One may distinguish two cases: 1) The ferromagnet would in absence of exchange interaction have been a low-~,, superconductor. In this case there seems no hope to produce high critical fields in a dense ferromagnet. To realize the coexistence one has to produce a compound where the magnetic ions form a rather dilute but nevertheless a well-ordered sublattice. 2) The ferromagnet would in absence of exchange interactions have been an extreme high-K0 superconductor. In these cases one need not introduce too much disorder to reach orbital critical fields necessary to compensate the mean exchange field, thus these cases seem to be the only ones where the compensation suggested by Jaccarino and Peter may be possible. Note that our discussion may also be applied to .the antiferromagnetic case. Although the possibility of coexistence of superconductivity and antiferromagnetism has been demonstrated*), a few percent of disorder in the antiferromagnetic lattice will be sufficiently to destroy superconductivity. This may be one reason why so far no antiferromagnetic superconductor has been found. Acknowledgments. Muller for interesting on this paper.
We want to thank Professors B. Giovannini and J. discussions and H. Jones for many valuable comments
REFERENCES 1) Jaccarino, V., Matthias, 9. T., Peter, M., Suhl, H. and Wernick, J. H., Phys. Rev. Letters 5(196O)Y524. 2) Jaccarino, V., Peter, M., Phys. Rev. Letters 9 (1962)290. 3) Matthias, 9. T., Suhl, H. and Corenzwit, E., Phys. Rev. Letters l(1958) 449. 4) Hillenbrand, 9. and Wilhelm: to be published in J. Phys. Chem. Solids. 5) Avenhaus, R., Fischer, Id., Giovannini, 9. and Peter, M., to be published in Helv. Phys. Acta. 6) Kittel, C., Phys. Rev.,Letters 10 (1963) 339. 7) See e.g. Slater, J.,: Quantum Theory of Atomic Structure, McGraw-Hill, N.Y. (1960). 8) Bahensperger, W. and Striissler, S., Phys. kondens. Materie l( 1963) 20. 9) Avenhaus, R., Enz, C. P., Heim, Klose, W. and Peter, M., unpublished. 10) Clogston, A. M., Phys. Rev. Letters 9 (1962) 266. Chandrasekhar, 9. S., Appl. Phys. Letters 1(1962)7. 1 I) Maki, K., Physics 1 (I 964) 2 I. 12) De Gennes, P. G., Phys. kondens, Materie 3 (1964) 79. 13) Helfand, E. and Werthamer, N. R., Phys. Rev. 147 (1966) 288. 14) Hake, R. R., Appl. Phys. Letters 10 (1967) 189. 15) Werthamer,N. R., kelfand, E. and Hohenberg, P. C.; Phys. Rev. 147 (1966) 295. 16) Abtikosov, A. A. and Gorkov, L. PI, Soviet Physics-JETP 12 (1961) 1243. 17) Peterson, D. T., Page, D. F., Rump, R. 9. and Finnemore, D. K., Phys. Rev. 153 (1967) 701.
COEXISTENCE
OF SUPERCONDUCTIVITY
AND FERROMAGNETISM
607
18) Decker, W. R. and Finnemore, D. K., Phys. Rev. 172 (1968)430. 19) Andres, K., Tagung der schweiz. Phys. Gesellschaft, Bern (May, 1969). 20) Note however that our example of rare earth in Th is a favourable case where the exchange interaction is weak. 21) Gorkov, L. P., Soviet Physics-JETP 10 (1960) 998. Gorkov does not give explicitly this formula, but it can be obtained from his eqs. (15), (I 9) and (2 I). 22) Fulde, P. and Maki, K., Phys. Rev. 141(1966) 275. 23) It follows from the work of Fulde and Maki for the case of magnetic impurities in nonmagnetic superconductors that H,,(T = 0) decreases linearly with concentration and somewhat faster than T,. However since we are concerned with orders of magnitude rather than exact values, this does not change the essential results. 24) Note, however. that for this case the coherence length will be of the order interatomic distances, thus the periodic variations of the exchange interaction may in this case have a strong influence on the superconductivity. 25) We supposed here that in AsB only the B atoms are magnetic.