Magnetic impurities in oxide superconductors: Evidence for intrinsic magnetic order in the superconducting compounds

~ ,~_~

Solid State Communications, Vol.65,No.9, pp.1013-1017, 1988. Printed in Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

MAGNETIC IMPURITIES IN OXIDE SUPERCONDUCTORS: EVIDENCE FOR INTRINSIC MAGNETIC ORDER IN THE SUPERCONDUCTING COMPOUNDS E.W. Fenton

Physics Division, National Research Council of Canada, Ottawa, Canada KIA OR6 (Received 19 October 1987 by R. Baffle)

With no ferromagnetism component present,

the only effect on T c and A

of rigid magnetism varying rapidly in space is replacement in the gap equation of the density of electron states for the nonmagnetic crystal by that of the magnetic crystal. In usual superconductors, very large reduction of T c caused by magnetic impurities in comparison to that from nonmagnetic impurities occurs because the spin direction of each impurity is free to rotate. In this case effects of magnetic impurities on T c in oxide superconductors, only as large as effects of nonmagnetic impurities, suggest that each impurity spin is not free to rotate. This would almost certainly be caused by some form of magnetic order intrinsic to the host crystal rather than by impurity-impurity interactions.

Recent experiments on high-T c oxide superconductors have shown that in contrast to the s i t u a t i o n for usual superconductors, effects of magnetic i m p u r i t i e s on the superconductivity t r a n s i t i o n temperature Tc are only as large as effects of nonmagnetic i m p u r i t i e s . I-4 In usual superconductors, which are s i m i l a r in that the p a i r o r b i t a l is nearly s-wave, 5 very much stronger reduction of Tc by magnetic impurities occurs due to s p i n - f l i p s c a t t e r i n g of electrons from Cooper pairs which are free to r o t a t e . In t h i s case the experiments suggest that impurity spins in the oxide superconductors are not free to r o t a t e . This can be caused only by s p i n - s i n g l e t ordering of the host c r y s t a l , or an i n t e r n a l magnetization f i e l d which is e i t h e r i n t r i n s i c to the host c r y s t a l or'due to spin-spin interactions among the magnetic impurities. Because of the large characteristic distance between dilute impurities, magnetic order of the host crystal appears much more likely to be occurring with the large ordering energy that would be required. This evidence for magnetic ordering is important because one of the major questions about oxide superconductors at this time is whether or not short-range magnetic order occurs in the compounds where superconductivity occurs. One statement in the above paragraph is, to our knowledge, not known or proven in the literature and must be proven here: that in usual superconductors the observed large depression of T o by magnetic impurities is due to spin-rotational freedom and would not occur if spin directions were fixed. In fact in the classic paper in this field by Abrikosov and Gor'kov, 6 the same rapid depression of T c by magnetic impurities would occur with impurity spins which are either fixed or free to rotate. As we will discuss, this result for rigid spins occurs because in ref. 6 the one-electron state was constrained to be a Pauli spinor which is

c o n s t a n t in space, and t h e p a i r s p i n was taken as z e r o and c o n s t a n t i n space. Both c o n s t r a i n t s are i n a p p r o p r i a t e , as d i s c u s s e d l o n g ago f o r a s i n g l e e l e c t r o n by A r r o t t 7 and f o r a Cooper p a i r by G o r ' k o v and Rusinov. 8 Taking s t r o n g d e p r e s s i o n of T c by magnetic i m p u r i t i e s i n u s u a l s u p e r c o n d u c t o r s as an e x p e r i m e n t a l f a c t , 6 to p r o v e t h a t t h i s i s due t o r o t a t i o n a l freedom of the impurity spins it is sufficient to prove that rigid-spin magnetic impurities do not depress T c more strongly than nonmagnetic impurities. The BCS H a m i l t o n i a n f o r an s-wave p a i r i n g p o t e n t i a l i n a pure c r y s t a l i s H= k,a

k~k',q

--

--

if,C( I

H0 + H v

(i)

where c~,Ck~ create or destroy electrons in k-wave Psuli-spinor Bloch eigenstates with ~pin-independent e n e r g i e s Ek.

Mean-field

factorization of H v with zero pair spin as

required bI an+s-wave+ orbiial,V = A(i~y)~a,,--results in the BCS gap equation

A = V ~

& ~ tanh (c~+A2) ~

In t h i s

equation it

-2kBZ

is essential

C_k_~ f o r t h e two s t a t e s

(2)

that

eke =

i n the Cooper p a i r i n g ,

and t h i s i s ensured by t i m e - r e v e r s a l i n v a r i a n c e of the H a m i l t o n i a n f o r a nonmagnetic c r y s t a l .

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1014

MAGNETIC IMPURITIES IN OXIDE SUPERCONDUCTORS

We now i n t r o d u c e nonmagnetic i m p u r i t i e s which are r i g i d and a l l o w o n l y e l a s t i c s c a t t e r i n g of e l e c t r o n s , H0 + HO + HI . This Hemiltonian still has o n e - e l e c t r o n e i g e n s t a t e s . As f i r s t discussed by Anderson, 9 a p h y s i c a l l y c l e a r s o l u t i o n of the t o t a l H a m i l t o n i a n H0 + HI + HV f o r the impure c r y s t a l i s o b t a i n e d by f i r s t s o l v i n g f o r e l e c t r o n e i g e n s t a t e s of t h e one-body p a r t H0 + HI and then r e - e x p r e s s i n g the many-body p a r t Hv i n terms of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s b~,b n f o r the impure-crystal eigenstates: H = [Enbn+bn - ½ ~ V ( n l , n ~ ) b + b + .b .b nI nI n2 n2 n nl,n 2

H =

E+(k)a~,+,~ak,±, ~ - ½ _

-

-

-

-

- -

-

-

~ k,k,,~

(3)

d~nsity of states N(E k) for eq. 2 is replaced by N(~n). In this case ~n]y small changes of T c and & are caused by introducing a small concentration of nonmagnetic impurities. In usual superconductors the pairing interaction in the pure crystal is nearly but not exactly s-wave, which results in further changes to T C and A as it becomes more s-wave when nonmagnetic impurities are introduced. These changes are the same order of magnitude as the effects of N(~ k) + N(E n) and will in general occur as well for--oxide superconductors when pairing is nearly but not exactly s-wave. We have shown recently I0-12 that a similar situation to the Anderson theory occurs when the nonmagnetic impurities are replaced by a rigid antiferromagnetism field, so that H = H 0 + HAF + HV, where

Mk is the magnitude of the e f f e c t

period 2a of the antiferromagnetism field. At the new magnetic Brillouin zone boundary which halves the first Brillouin zone (since the lattice period has been doubled), the constant-in-space Pauli spinors (~) and (~) of the nonmagnetic Bloch functions have zero amplitude in the magnetic eigenfunctions, whereas far from this magnetic zone boundary they dominate. As before for the impurity problem, after first solving for the one-electron part of the Hamiltonian H 0 + HAF , structure factors for four eigenstates in H V are used to express H V in terms of creation and annihilation operators for electrons in magnetic eigenstates, 7,8 resulting in

(5)

v(M)(k,k ' a+ = ---- ) k,±,Jtk,±,_=ak,,±,_=a_k,,±,:

The s-wave pairing interaction in eq. i above represents a delta function in ~-space. Using structure factors for matrix elements of the interaction in terms of the four impure-crystal eigenfunctions represented in H V results again in a constant V. Because H 0 + H I is still invariant under time reversal, a time-reversed state n* exists for every n state with the same energy ~n" In this case exactly the gap function of eq. 2 is obtained again 12 but with ~k ÷ En' which means that the Bloch state

HAF = ~M~(~.~)~6C~+Q Ck~ + h . c .

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(4) of the

a ~ t i f e r r o m a g n e t i s m on the e l e c t r o n s end n i s the p o l a r i z a t i o n d i r e c t i o n of the magnetizatTon. The w o v e - v e c t o r Q of the a n t i f e r r o m a g n e t i s m i s taken to double The p e r i o d of the h o s t c r y s t a l in one direction, Q = ~/a. Because the antiferromagnetism is rigid in this model, H 0 + HAF is a one-electron Hami]tonian which has one-electron eigenstates. These have been solved for explicitly and exactly lO-13 and do not have the constant-in-space Pauli spinor form o£ the B]och functions for the nonmagnetic host crystal. Instead the electron two-component function varies strongly in ~-space with the

a+,a are creation and annihilation operators for magnetic eigenstates with energy E+(k). ± refers to two magnetic bands which-r~su]t from one bond of the nonmagnetic host crystal when the antiferromagnetism halves the Bri]louin zone. Because of complicated spin structure of the one-electron eigenstates, v~M)(k,k ' ) " has complicated dependence on ¢, k, and--kT. I0 This leads to spin and wave-vector--depend~nce of the two-by-two matrix gap function ~(k) which has both icy and iOy~.~ components. 7

The two-space

is not Pau]i spinor space but a magnetic eigenfunction space. However ~(k) satisfies A+(k)&(k) = l&121 N This combined with the fact that the energies of (~,±,¢) and (-~,±,-~) mognetic states are the some, due to invariance of the crystal and the Hamiltonian when r ÷ r + a followed by time reversal (and usually invariance under a spatial inversion followed by time reversal), means that again the gap equation of eq. 2 is obtained. This has been derived explicit]y. I0 In this case ~k + E±(~) in eq. 2 and N(Ek) + N(E±(~)). Again Where is ]ittle change in T c and A from Tc and A for the nonmagnetic and pure host crystal, and for an s-wave interaction all of that change again comes about because of a change in the electron density of states. Although ~+(k)~(k) has no k dependence, which means the magnitude of the gap is constant over the Fermi surface, each of the icy and iOy~-~ components in ~(k) has strong dependence, with the former vanishing at the magnetic Bri]louin zone boundary and the ]otter vanishing with ~ sufficiently far from that boundary. This is reminiscent of the Balian-Werthamer state for helium-three, where three p-wave components in the pairing each have strong k dependence but the magnitude of the total gap function is constant over the Fermi surface. The present problem is somewhat mere complicated in that sing]et and triplet correlations in the two-by-two space, here not a spin space but magnetic eigenfunction space, must have even-in-k and odd-in-k orbital

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MAGNETIC IMPURITIES IN OXIDE SUPERCONDUCTORS

functions respectively, due to quantum symmetry which requires change of sign when two pairing electrons are interchanged. Nevertheless this structure and all magnetic-Umklapp-vector processes have been included exactly in an explicit solution. I°'II To see what is happening in spin space it is necessary to transform the entire theory from magnetic eigenfunction space back to k-wave Pauli-spinor Bloch function space. II Whe~ this is done one sees that there are both constant-in-space and cosQ.~ pairing components, with spin-singlet and spin-triplet components in both cases, and with even-parity orbitals for spin-singlet pairing and odd-parity orbitals for spin-triplet pairing. One can see that the overall charge-density symmetry of the Cooper pairing remains s-wave, because introducing isotropic scattering by nonmagnetic impurities to this case results in no suppression of T c s n d A~ 12 which can occur only for s-wave palrlng. To summarize what happens when Cooper pairing occurs due to an s-wave pairing interaction in the presence of a rigid background of antiferromagnetism, the antiferromagnetism changes the spin-density symmetry of both the host crystal and the Cooper pairing, but does not change the charge-density symmetry of either. No suppression of T c and A in the pure antiferromagnet occurs due to orbital currents or Zeeman-energy splitting of paired-state energies. The former are absent because the wavelength of the sntiferromsgnetism is much smeller than the size of a Cooper pair, and the latter does not occur because of the symmetry in the Hamiltonian discussed earlier which maintains degeneracy of paired states. Just as in Anderson's theory for the case with a rigid field of nonmagnetic impurities present, 9 the only changes that occur in T o and A are due to replacing the density of electron states N(E k) for the pure and nonmagnetic host crystal wit~ the density of states which occurs in the presence of the added background. In the antiferromagnetism case, N(E k) ÷ N(E±(~)). I0-12 In both cases, changes of Tc--and A are small. We cannot in fact properly solve the problem of Cooper pairing due to an s-wave interaction in the presence of a random distribution of rigid-spin magnetic impurities. That this problem is nontrivial can be seen from the fact that an explicit solution for a single magnetic impurity results in an oscillatory triplet-spin component of Cooper pairing which radiates outward from the impurity. 13 Any form of averaging over the positions of impurities as in the Abrikosov and Gor'kov theory 6 will immediately eliminate this pairing component in the theory. In this case the complete treatment of all spin and parity structure and magnetic scattering that was required to recover the form of the gap equation in eq. 2 for the antiferromagnetism case I0-I2 will be lost: no particular solid with a random distribution of magnetic impurities resembles the impurity-averaged state. Nor can we simply solve for a single rigid-spin magnetic impurity and then multiply effects on T c by the number of widely-spaced and independent impurities that are present: it is impossible to have a single rigid-spin magnetic impurity with no

ferromagnetism component present. What we will do is argue that all the essential physics for the random distribution of rigid-spin magnetic impurities is present in the problem of Cooper pairing due to an s-wave interaction in the presence of regularly spaced "impurity spins", i.e. antiferromagnetism. If the reader then agrees that the essential physics is the same in t h e two cases, i t i s proven at once that nonmagnetic or rigid-spin-magnetic impurities affect T c and A in the same way, changing only the electron density of states in the gap equation. In this case the experiments on oxide superconductors suggest that magnetic impurity spins are not free to rotate, in contrast to magnetic impurities which rapidly depress T c in usual superconductors. We start with two rigid-spin magnetic impurities in a crystal which has inversion symmetry, with spins oppositely directed, positioned so that spatial inversion about the midpoint between them followed by time inversion iesves the crystal invarisnt, and spaced from each other by a distance small compared to the size of a Cooper pair. In this case no orbital currents are caused by the magnetism. Secondly, the Bloch-wave pairing of degenerate states in the host crystal is replaced by pairing of n (M) and n (M)' eigenstates which are degenerate because of the symmetry in the crystal discussed a b o v e . 14 I t i s c l e a r i n t h i s c a s e t h a t a gap e q u a t i o n of t h e f o r m i n e q . 2 w i l l a g a i n be obta ine d, as in the a ntife rrom a gne tis m I0-12 or nonmagnetic impurity 9 cases,

N(~ k) + N(E~M)).

w i t h Ek + E~M) and

This impurity-pai~ situation

can--be repeated with a large number of similar pairs spaced sufficiently far from each other so as to be in every sense independent. This is still not exactly the same as a random distribution of magnetic impurities. However for a random distribution, Zeeman-energy splitting of paired electron states will be negligihle to roughly the same extent that any ferromagnetism component is negligible on the scale of length of the Cooper pairing. In this case with no depression of T c due either to orbital currents or Zeeman-energy splitting of p a i r e d s t a t e s , we expect e x a c t l y the same u l t i m a t e r e s u l t as when the r i g i d spins a r e r e g u l a r l y spaced i n the a n t i f e r r o m a g n e t i s m case: a gap e q u a t i o n of the form i n eq. 2 but w i t h Ck + E~H) and N(ek) + N ( ~ M ) ) .

E f f e c t s per

s p i n oF T c and A s h o u l d i n f a c t be somewhat s m a l l e r then e f f e c t s of a n t i f e r r o m a g n e t i s m . In the l a t t e r case, coherent s c a t t e r i n g from r e g u l a r l y - s p a c e d r i g i d spins r e s u l t s i n an energy gap at the magnetic B r i l l o u i n zone boundary end d e n s i t y - o f - s t a t e s peaks at i t s edges, which may i n t e r s e c t t h e Fermi s u r f a c e . Apart from t h i s coherence e f f e c t , t h e r e i s no d i f f e r e n c e i n t h e p h y s i c s of r i g i d spins spaced r e g u l a r l y or i r r e g u l a r l y , p r o v i d e d t h a t no f e r r o m a g n e t i s m component i s p r e s e n t i n both cases and the l e n g t h s c a l e f o r v a r i a t i o n of the m a g n e t i z a t i o n f i e l d i s s m a l l compared t o the Cooper p a i r s i z e . The e x p l i c i t r e s u l t s f o r an a n t i f e r r o m a g n e t i0 - 12 show t h a t T c f o r superconductivity

due t o a p u r e l y - c h a r g e

a-wave

1016

MAGNETIC IMPURITIES IN OXIDE SUPERCONDUCTORS

pairing interaction is not depressed by rigid short-wavelength magnetism. T h i s is due to the fact that the magnetism i t s e l f does not change the charge-density symmetry of this pairing, 12 combined with the fact that there are one-electron eigenstates due to the r i g i d i t y . These are not of the usual constant-in-space Pauli spinor form.13 I f Pauli spinor s t a t e s are imposed in the theory, then these electron spins are flipped by scattering from rigid-spin magnetic impurities. In tbe Abrikosov and Gor'kov theory, 6 lifetime effects in these incorrect single particle states do not cancel in the gap equation with effects of the impurity-mediated interaction between up-spin and down-spin Pauli-spinor states of the zero-spin Cooper pair. In this case scattering from rigid-spin magnetic impurities depresses Tc for zero-spin pairin 9 of Pauli-spinor states. The same effect is entirely absent when the correct two-space one-electron ei~enstates are combined in degenerate pairs, lO-Iz However if the magnetic impurity spins are free to rotate, then there are no one-electron eigenstates of the crystal with the magnetic impurities present. In this case exactly the results obtained by Abrikosov and Gor'kov must occur, with T c strongly reduced by free-spin magnetic

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interaction), experiments on the high-T c oxide superconductors suggest that magnetic impurity spins in the superconducting compounds are not free to rotate. This would be due to some magnetization field which has an ordering energy large compared to T O for the superconductivity. This may be present in the pure host crystal or occur due to spin-spin interactions among magnetic impurities. Consider the latter in comparison to the situation which exists in usual superconductors which have much lower T c. At the large distances between impurities which occur for sufficiently dilute concentration and in three dimensions, the RKKY interaction VRKKY between spins decreases with increasing 3 spin-spin distance according to (~-R4)- . At the same time the number of free-spi~Vmagnetic impurities required to reduce T c by ten percent or by one hundred percent is in each case proportional to Tc.6 In each case the characteristic distanceTR_i---_R-~_R j between f r e e spins

is proportional to To-II3. The r a t i o of the average-VRKKy-between-spins to TO i s t h e r e f o r e independent of Tc. (The same r e s u l t occurs i n two dimensions.) In t h i s case the c o n d i t i o n t h a t K-~Ky V ~ be sufficiently small compared to

impurities (or free-spin magnetic moments of the host) as is observed in usual superconductors. More generally, the spin-rotation degree of freedom of the magnetic impurities in usual superconductors would be best treated in Eliashberg equations, with the magnetic fluctuation spectrum of the impurities acting as an electron-electron interaction which causes both one-electron lifetime effects and a pair potential. This magnetic-fluctuation interaction is repulsive in s-wave pairing, opposing the attractive interaction due to phonons (or other quanta) and strongly reducing TO • From the arguments above, which are in some general sense within the context of but not precisely restricted to the usual Cooper pairing theory (not restricted to the usual pairing

Tofor i m p u r i t y spins to be r o t a t i o n a l l y f r e e i s as w e l l s a t i s f i e d in high-T o superconductors as in usual superconductors. In both cases, at d i l u t e c o n c e n t r a t i o n the distance between i m p u r i t i e s i s too l a r g e f o r spins to be f i x e d (on the energy scale of the s u p e r c o n d u c t i v i t y ) by i m p u r i t y - i m p u r i t y i n t e r a c t i o n s . In t h i s case the e x p e r i m e n t a l evidence I - 4 suggests that spins of magnetic i m p u r i t i e s i n high-T c o x i d e superconductors must be f i x e d by some form of magnetic o r d e r i n g i n t r i n s i c t o the host c r y s t a l s i n which s u p e r c o n d u c t i v i t y occurs. I f so, the magnetic o r d e r i n g must e x i s t at temperatures much l a r g e r than Tc as w e l l as i n the superconducting s t a t e , and i n p r i n c i p l e could occur at temperatures r i g h t up to c r y s t a l l o g r a p h i c t r a n s f o r m a t i o n or m e l t i n g of the c r y s t a l .

References i. 2.

3.

4.

5.

S.B. Oseroff et el, to be published. G. Xieo, F.H. Streitz, A. Gavrin, Y.W. Du, and C.L. Chien, Phys. Rev. B3__55,8782 (1987). J.M. Terasoon, L.H. Greene, P. Barboux, W.R. McKinnon, G.W. H u l l , T.P. Orlando, K.A. D e l i n , S. Foner, and E.J. MoNiff, Proo. Berkeley S u p e r c o n d u c t i v i t y Conference, June, 1987 (Plenum). C r y s t a l f i e l d e f f e c t s seem t o f u l l y account f o r r e d u c t i o n of To by t r a n s i t i o n metal i m p u r i t i e s , M.W.C. Dharma-wardana, to be published. S u p e r c o n d u c t i v i t y i n oxide superconductors p e r s i s t s i n extremely d i r t y compounds, which i s not compatible w i t h p-wave, d-wave, o r s i m i l a r a n i s o t r o p i o o r b i t a l s ; and the London p e n e t r a t i o n depth i s c o m p a t i b l e w i t h s-wave but not w i t h a n i s o t r o p i c p a i r i n g , D.R. Marshman e t a l , Phys. Rev. 366, 2386 (1987).

6. 7. 8. 9. 10. 11. 12. 13.

14.

A.A. Abrikosov and L.P. Gor'kov, Soviet Physics JETP 12, 1243 (1961). A. A r r o t t , Magnetism l i B (Ed. Redo and S u h l ) , 296-416 (1966)? See eq. 3.4 on p. 519. L.P. Gor'kov and A . I . Rusinov, Soviet Physics 3ETP 19, 922 (1964). P.W. Anderson, Phys. Chem. S o l i d s 11, 26 (1959). E.W. Fenton, S o l i d S t a t e Comm. 54, 6}3 (1985). E.W. Fenton, S o l i d S t a t e Comm. 56, 1033 (1985). E.W. Fenton, Solid State Comm. 5__77,241 (1986). E.W. Fenton, Prog. Theor. Phys. Suppl. 80, 94 (1984). For s t a n d i n g waves a p p r o p r i a t e when i m p u r i t i e s are present, t - ~ n i n g nf~ the impurity field changes (n ~ , n(~)) p a i r i n g of degenerate m~gnetio i m p u r i t y

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MAGNETIC IMPU~ITIES IN OXIDE SUPERCONDUCTORS

states into (k~,-k-~) pairing of standing Bloch waves. This occurs because for standing waves spatial inversion reverses k and time reversal only reverses spin. For-standing Bloch waves (k~,k-~) pairing of time-reversed states corresponding to Anderson's time-reversed (n,n*) impurity states is the same as (k~,-k-~) pairing.

1017

Spatial inversion about a particular point followed by time reversal leaves the infinite crystal invariant for antiferromagnetism with period incommensurate with the lattice constant of the host crystal. In this case again pairs of degenerate eigenstates occur end a gap equation of the form of eq. 2 is obtained.