Hole pairing in antiferromagnetically correlated layered oxides

Hole pairing in antiferromagnetically correlated layered oxides

Physica C 153-155 (1988) 1227-1228 North-Holland, Amsterdam HOLE P A I R I N G IN A N T I F E R R O M A G N E T I C A L L Y C O R R E L A T E D L A Y...

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Physica C 153-155 (1988) 1227-1228 North-Holland, Amsterdam

HOLE P A I R I N G IN A N T I F E R R O M A G N E T I C A L L Y C O R R E L A T E D L A Y E R E D OXIDES

N arendra K U M A R D e p a r t m e n t of Physics, Indian Institute of Science, Bangalore -560 012, India

We have considered a two-band Hubbard model having i n t e r l a c e d Cu-Sd(x2 _y2) and O-2p(x,y) orbitals representing the C u e 2 square planes. Simple C u Q ~ - c l u s t e r c a l c u l a t i o n suggests t h a t the additional holes created by doping stay mainly on oxygen. Motibn of an oxygen hole i n t e r l a c i n g with the a n t i f e r r o m a g n e t i c a l l y c o r r e l a t e d background of copper spins, creates a string of high energy spin c o n f i g u r a t i o n of f i n i t e length giving mass r e n o r m a l i z a t i o n . Another hole of opposite spin can now anneal this string tension providing a traingular pairing p o t e n t i a l for large pair momentum. The l a t t e r implies unusual Bose condensation of the wake-bound c o m p a c t Bose-like pairs on a non-zero m o m e n t u m shell. E f f e c t of disorder favouring condensation at the ,-nobility edge is pointed out.

1.

INTRODUCTION The recent discovery by Bednorz and M'Llller(1) of high t e m p e r a t u r e s u p e r c o n d u c t i v i t y in L a - B a - C u - O with the t r a n s i t i o n t e m p e r a t u r e T c - 30K, closely f o l l o w e d by reports of still higher t r a n s i t i o n t e m p e r a t u r e -901< in Y - B a - C u - O and the r e l a t e d family of oxides having layered perovskite structure has caused serious r e - t h i n k i n g on the conventional pairing mechanism and at ' a still deeper level, on the universality of the BC5 state itself. The near absence of oxygen isotope e f f e c t a n the high abselute value of transition t e m p e r a t u r e T c = T n , t h e Neel t e m p e r a t u r e , strongly suggest and e l e c t r o n i c / m a g n e t i c pairing mechanism. The normal- state resistive and magnetic properties as a function of doping and the known oxide chemistry of this class of compounds suggest that these are odd-electron Mott-Hubbard insulators, rendered correlated metals by doping or oxygen deficiency that creates holes in the lower Hubbard band. There are compelling reasons to believe that the essential physics is contained in the weakly coupled CuO~ square p ~ n e s of f o u r f o l d o~+ygen •

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coordinated copper Cu .Absence of Cu in photoemisiion studies indicates that the additional holes due to doping reside mainly on oxygen (O-or. d i m e r i z e d O~.-)z . (2).. In. this . work we reexamine an e l e c t r o n i c paLrlng mechanism for these i t i n e r a n t holes inherent in the Hubbard model with t w o interlaced bands where the oxygen holes move in the background of a n t i f e r r o magnetically correlated Cu-spins. The basic mechanism has been suggested by several workers almost simultaneously(5,4). It is consistent with the general observation that p r o x i m i t y to m e t a l - i n s u l a t o r transition, low dimensionality, low spin(half) and i n t e r a t o m i c i n t e r m e d i a t e ( m i x e d )

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valency of c o p p e r - o x y g e n favour s u p e r c o n d u c t i v i t y in these oxides. 2.

MODEL We consider a strongly h y b r i d i z e d t w o band Hubbard model appropriate to the C u e _ square planar n e t w o r k ^ with tight-bindin~ Wannier orbitals Cu:3d(xZ-y z) and O:2p(x), 2p(y) c o n s t i t u t i n g our H i l b e r t space. ~/e w~t'k . . . i+ lu lng_the~ hole r e p r e s e n t a t i o n w i t h Cu (3d ), O k (2p U) as hole vacuum. The essential point to note is that the hole on copper(oxygen) can move only via oxygen (copper) orbitats (i.e. bands are interlaced). The H a m i l t o n i a n in obvious n o t a t i o n is (5) H=ZE(d)d+(ia)d(io)+Z E(p)p+(Io)p(Io) + U(d)Z n(i+) n(i4,) +Z (t(il) d ' ( i o ) p(la)+ h.c.), where the m i x i n g m a t r i x element t(il) is w r i t t e n with due regard to o r b i t a l phases, and / t ( i l ) / = t for neighbours. Also, E(p)>E(d)=0, E(d)+ U(d)>E(p) so that the f i r s t hole is nominally on copper while the second hole added by doping is nominally on oxygen. To ascertain the valence of copper, consider a CuO 2 cluster as basic unit. Thus we have the s y m m e t r y adapted linear combination of the oxygen orbitals~(g)= (pa(x) + pg(y))/ / 2 and ~b (u) = (-p1(x)+ pg(y))/ ¢*2. g N o w g } ( u ) h y b r i d i z e s strongly with the ~ Cu:Sd(x% y~) giving bonding and antibonding levels while ~ (g) is non-bonding The single hole per unit lies in the antibonding state. The added hole on doping forms a singlet with the p r e - e x i s t i n g hole in the ground state: -k + + ~P(2 holes)= cose d~'o d + +cr +sin0 cosq0~(u~)~ (u+)+ sine sin fl0 ((d + (+) ¢ + (u+) -d + (+)~ + (u4'))/ V2)/vac:~ This singly charged object moves with a transfer m a t r i x e l e m e n t t(eoff): t(eff~)= -t(sjn 26 cos~0 +side sin2q~ / / 2 + s i n 2 e s i ~ / / 2 + sin~e sin'q), w i t h t a n O = -cos£0/ cos 2£0 and

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tan 20 = /4t sin £0/(U(d)-2t sin2q~) for U(d):~,(p)-E(d). We can now find the r e l a t i v e w e i g h t f o ~ the Cu ~+ c o n f i g u r a t i o n . This is given by C o s ' 0 . For a reasonable choice of parameters this is about 10 per cent. This is too high to be acceptable (2). We , t h e r e f o r e , assume the other possibility of the added hole being on the non-bonding oxygen orbital ¢ (g). This would correspond to the O- species. The l a t t e r may be s t a b i l i z e d by a d i r e c t transfer m a t r i x element_ connecting the t w o oxygen orbitals in the unit. We will now consider the m o t i o n of this f e r m i o n i c oxygen hole of calculable bare mass m , via the v i r t u a l CuZ+/Cu/+, levels (th~ interlacing of bands).Here the copper Cu z+ spins form the background with antifer~'omagnetic superexchange coupling .] ~ /4t /U(d) E(p), assumed u n a f f e c t e d by a dilute c o n c e n t r a t i o n of added holes. 3. HOLE P A I R I N G M E C H A N I S M As the hole moves r e - o r i e n t i n g its spin in the process, it creates a string of upturned copper spins of wrong o r i e n t a t i o n with respect to the spins adjacent to the string. The energy per unit string length (tension) is 2J/a, where 'a'is l a t t i c e constant. The background spins, however, r e - o r i e n t on a t i m e scale ~'h/43. Thus the m a x i m u m string length is (~t~v(k)/4.], where v(k) is the hole speed for wavenumber k ( ~ the F e r m i speed). The associated energy is fav(k)/2a. This gives for the hole energy E(k): E(k) =fiv(k)/2a + lq2k2/2m*. Recalling that the group speed v(k) = 0E(k)/3k)/f% the above d i f f e r e n t i a l equation may be solved and the r e n o r m a l i z e d mass calculated. Now, another hole of opposite spin can follow in the wake and anneal the string tension providing thereby a linear confining p o t e n t i a l V(r): V(r) =4Or/a, for r < ~ , 4 J ~ a for r > ( . As the hole is on oxygen, the hard-core repulsion may not be pronounced. 4. PAIRED STATE A N D BOSE C O N D E N S A T I O N if we crudely t r e a t V(x) as a t w o - b o d y central p o t e n t i a l , we could e x p e c t an s-wave or a d-wave spin-singlet Cooper pair f o r m a t i o n with m a x i m u m binding energy for optimal doping or k~-(3). The nature of the w a k e - b i n d i n g , however, suggests the following form for a pair wavefunction: (~ k(Xl,X 2) = exp(i2k'(Xl+X2)/2) ' X k(Xl-Xg) witt~ large centre of--mass momentuf~ (2kO. Thus~ if we regard the c o m p a c t pair as a Bose- like object, it can not condense in the zero m o m e n t u m (k=0) state because the pair unbinds for small values of pair m o m e n t u m .

We are thus led to considering a reduced H a m i | t o n i a n for holes that favours pairs with large centre-of-mass momentum: Hre d = ~: E(k) C+(ko ) C(ko )-l/2~:V(k,k',q,q'). C+(W+q',a )C+(W-q',-o )C(k-q, -d ) C(k+q,o ). Here the i n t e r a c t i o n m a t r i x e l e m e n t V is non-zero on an energy shell. O n e c an now define the anomalous e x p e c t a t i o n k~q, < V(k,k',q,q') C ( k ' - q ' , -o ) C ( k ' + q ' , ~ ) ~ A (kq). We can proceed t o c a l c u l a t e the e l e m e n t a r y e x c i t a t i o n spectrum and A self-consistently. This is in progress and will be r e p o r t e d elsewhere. 5. DISCUSSION The above pairing mechanism requires only that the background spins be a n t i f e r r o m a g n e t i c a l l y c o r r e l a t e d over a distance larger than the pair size . Such an in-plane c o r r e l a t i o n , may survive the destruction of 3 - d i m e n s i o n a l i t y driven anti-ferromagnetic longrange order seen at low doping ~3 per cent(6). That pairing requires large centre-of-mass momentum rules out the conventional condensation in zero m o m e n t u m state, instead one may have Bose condensation on a m o m e n t u m shell in k-space enclosing the normal Fermi sea. The radius of the Fermi sphere is d e t e r m i n e d by quasi-chemical equilibrium of the Bose and Fermi systems. The Fermi system can contribute the usual linear specific heat. The O D L R O associated with such a condensation is n o n - t r i v i a l . F i n a l l y we note that disorder can have interesting e f f e c t on such a Bose condensation. Inasmuch as the size of the l o c a l i z e d tail states does not scale with the system size, we can not have a t r u l y macroscopic occupation of these states even for the weakest hard core repulsion. Indeed in a martree a p p r o x i m a t i o n one would e x p e c t the l o c a l i z e d energy levels to get renormalised upwards progressively towards the m o b i l i t y edge where e v e n t u a l l y the true Bose condensation takes place. Thus the chemical p o t e n t i a l is zero for a wide range of t e m p e r a t u r e even above T . This should have i m p o r t a n t e f f e c t on the Cnormal state properties. ACKNOWLEDGEMENTS i would like to thank Prof.T.V. Ramakrishnan, P r o f . H . R . Krishna Murthy and D r . M . M . Mohan for discussions~ and UGC and DST for support. REFERENCES 1. 3.G.Bednorz and K.A.Mu"l]er, Z.Phys. B64(1986)189. 2. D.D.Sarma,K.Sreedhar,P.Gangu!y and C.N.R. Rao,Phys.Rev.36B (1987) 2371 , A . B i a n c o n i e t al. Z.Phys.B Condensed M a t t e r 67 (1987) 307. 3. M.M.Mohan and N . K u m a r , 3.Phys.C.:Solid State Physics C20 (1987) L527. 4. 3.E.Hirsh, P h y s . R e v . L e t t . 59 (1987) 228. 5. F.C.Zhang and T.M.Rice,(1987) P r e p r i n t . 6. G.Shirane et al, 59,(1987) 1613.