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Quasilowdimensionality in the weak coupling limit
PhysicaC153-155 (1988) 1610-1616 North-HoHand, Amsterdam
QUASILOWDIMENSIONALITY
Jacques
IN THE WEAK
COUPLING
LIMIT
FRIEDEL
P h y s i q u e des Solides, Orsay, F r a n c e
Universit~
Paris-Sud,
Formation
Associ~e
The three q u e s t i o n s of q u a s i l o w d i m e n s i o n a l i t y , weak c o u p l i n g c o u p l i n g are d i s c u s s e d for the new oxyde s u p e r c o n d u c t o r s . INTRODUCTION In a field where t h e o r e t i c a l d i s c u s sion seems still open, e v e r y b o d y comes with p r e c o n c i e v e d ideas. My own stem from an e a r l i e r i n t e r e s t in c h a i n - l i k e s u p e r c o n d u c t o r s , first the A15 c o m p o u n d s with Labb4 and B a r i s i c (I),(2), then the o r g a n i c family (TMTSF)2X d i s c o v e r e d in O r s a y nine years ago by Bechgaard, J 4 r o m e and R i b a u l t (3) and much s t u d i e d since then (4). In the (TMTSF) 2X's, the flat u n s a t u r a ted o r g a n i c m o l e c u l e s are p i l e d in columns a l o n g w h i c h the v a l e n c e e l e c t r o n s jump much more r a p i d l y than from c o l u m n to column. The c o r r e s p o n d i n g t r a n s f e r integrals are such that t0 >> t& . (I) This s i t u a t i o n is d e s c r i b e d as q u a s i l o w d i m e n s i o n a l (5). It implies a h i e r a r c h y such that it is useful to t h i n k first of the columns as s e p a r a t e units, and then to treat their i n t e r a c t i o n s as p e r t u r b a tions. In these o r g a n i c s u p e r c o n d u c t o r s , it is also clear that the e f f e c t i v e c o u p l i n g V leading to s u p e r c o n d u c t i v i t y is smaller than the band width, a c o n d i t i o n w h i c h reads there as
Ivl < 2 t . .
t2~
It is t h e r e f o r e m e a n i n g f u l to start from e l e c t r o n s s t r o n g l y d e l o c a l i s e d along the columns. I will call this a w e a k c o u p l i n g limit, a l t h o u g h the e f f e c t of V m i g h t be somewhat too large to be q u a n t i t a t i v e l y g iven by the first p e r t u r b a t i v e term. Thus this c o u p l i n g m i g h t include e f f e c t s similar to the 'strong c o u p l i n g limit' of lead. It does not include s i t u a t i o n s where the e l e c t r o n s are s t r o n g l y loealised by their e l e c t r o s t a t i c i n t e r a c t i o n s or by very strong e l e c t r o n - p h o n o n couplings. F i n a l l y the p h a s e d i a g r a m of these o r g a n i c s u p e r c o n d u c t o r s show that the most stable c r i t i c a l t e m p e r a t u r e s are o b s e r v e d next to an a n t i f e r r o m a g n e t i c
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
au CNRS,
91405
and e l e c t r o n
phonon
transition, w h e n p r e s s u r e or c o m p o s i tion are v a r i e d (4). This and a c a r e f u l study (6), (7) of the f l u c t u a t i o n s above T c s t r o n g l y suggests that V is due to a d l r e c t e l e c t r o n - e l e c t r o n interaction, and is not p h o n o n m e d i a t e d (8). It is then but n a t u r a l to ask w h e t h e r or not any part of such a scheeme a p p l i e s to the n e w o x y d e s u p e r c o n d u c t o r s . It is, I feel, with d e c r e a s i n g c e r t a i n l y that one can answer the three q u e s t i o n s of q u a s i l o w d i m e n s i o n a l i t y , weak c o u p l i n g and e l e c t r o n or p h o n o n m e d i a t e d interactions. This p a p e r s u m m a r i s e s and c o m p l e t e s three p r e v i o u s ones (9),(10),(11). It is m o s t l y c o h e r e n t w i t h p a p e r s by Labb4, Bok and C o m b e s c o t (12), (13), (14) by B a r i s i c and B a t i s t i c (15),(16), by J 4 r o me et al (17),(18) and by P o u g e t and N oguera(19).Hirschand ScalaDino(20).Schulz(59), D z a l o s h i n s k y (21) and B a r d e e n (22) have e x p r e s s e d views w h i c h in part start from similar p o i n t s of view. But a number of p o i n t s made here o r i g i n a t e from many d i s c u s s i o n s , e s p e c i a l l y also with Schulz, H 4 r i t i e r and Lederer. I. Q U A S I L O W D I M E N S I O N A L I T Y . This is i m p l i c i t in most t h e o r e t i c a l a n a l y s i s nowadays, and indeed seems c o n f i r m e d by a v a i l a b l e e x p e r i m e n t s on the a n i s o t r o p y of r e s i s t i v i t y or of pen e t r a t i o n depth. Both lead, for Y B a 2 C u 3 0 7 , to (23),(24), (35)
tlt/ t~ ~ 102 . (3) A n i s o t r o p y of p l a s m a f r e q u e n c y in L a 2 N i O 4 g i v e s a lower limit of the same o r a e r [25). One can then define mean field intra and i n t e r b l o c k couplings, m e a s u r e d by their gaps 4,, A& or their c r i t i c a l temp e r a t u r e s Tit , T& . The three d i m e n s i o nal c r i t i c a l t e m p e r a t u r e T is an average of T;. and TI w h i c h d e p e n d s on d i m e n sionality. By ~ n a l o g y w i t h c o r r e s p o n d i n g
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
magnetic roughly,
problems,
on can I/2
T c ~ (~; T~) for stacked c ~ a i n s Tc ~ ~ Td
say that,
very
(4) (26),
and (5)
for stacked planes (27), where I<~<£n (~//T~).~ is thus at most a few units. One should then expect a zone of 2d s u p e r c o n d u c t i v e f l u c t u a t i o n s (28) in the range T << T < T~land, possibly, from b elow TC up to near T~ , a p s e u d o g a p ~;l larger ~han the BCS v a l u e I/2 3.5 k_T even if the BCS r e l a t i o n applies b e t w e e n Atl and TII (29), (30), (31). B e c a u s e of s i z e a b l e gaps A, one expects small c o h e r e n c e lengths ~ VM/A ~ a t/A, thus s u p e r c o n d u c t i v i t y of th~ second kind. The c o h e r e n c e length should be a n i s o t r o p i c with, typically, ~l / ~
~ a tlt/c ~ >>
I
(6)
Ratios of that order are d e d u c e d from m e a s u r e m e n t s (32) (33) of Hc. near T . ' Z . C A c o n s e q u e n c e is that v o r t e x l~nes should have c y l i n d r i c a l cores of radius ~l >> a when lying normal to the Cu 02 p~anes. In the absence of defects, such v o r t e x lines should g l i d e easily and take their e q u i l i b r i u m c o n f i g u r a t i o n s u n d e r a p p l i e d field. When lying parallel to the planes, the v o r t e x lines should have e l o n g a t e d c o r e s of size ~j~ along the planes, but t h i c k n e s s ~ of atomic d i m e n s i o n s normal to the Cu 09 planes. Their line tensions ~ and ~,/s~ould t h e r e f o r e be s t rongl y anisotropic, with (28)
T,,/q ~ q /~t/ (7) There should be a strong t e n d a n c y for the v o r t e x lines, w h e n p a r a l l e l to the Cu 02 planes to lie along the w e a k e r s u p e r c o n ducting links, thus b e t w e e n the Cu O 2 planes in the La~ Sr CuO. compounas z-x x . 4-y and along the Y planes in Y B a 2 C u 3 0 7 x" This should lower L~ from (7). A I s o suc~ v o r t e x lines should g l i d e easily along the Cu 02 planes, in p r e f e r e n c e to the normal d l r e c t i o n w h e r e ~ should vary periodically, with sizeable m a x i m a between its p o s i t i o n s of m i n i m u m v a l u e (34). As a c o n s e q u e n c e , for single crystals, H is s t r o n g l y a n i s o t r o p i c (28) and w e a k f~ fields a p p l i e d p a r a l l e l to the Cu 02 planes (35), the ratio /H c being given by ~ / T ~ . HcI~ I~ In p o l y c r y s t a l s , v o r t e x lines introduced by weak a p p l i e d fields should present, at equilibrium, a z i g z a g g i n g form, so as to pass t h r o u g h a larger n u m b e r of g r a i n s where the v o r t e x lines can lie p a r a l l e l to the Cu 09 planes. For a r a n d o m texture, a direct e x [ e n s i o n of the strong p i n n i n g
1611
of d i s l o c a t i o n s by i m p u r i t i e s (36) shows the zigzags to be short but m a r k e d (App e n d i x A). This can e x p l a i n the strong h y s t e r e s i s o b s e r v e d (37). In a powder, the same e f f e c t can e x p l a i n a r o t a t i o n of the g r a i n s so as to a l i g n p r e f e r e n t i a l l y w i t h Cu O^ p l a n e s p a r a l l e l to the a p p l i e d fiel~ (38). Finally, in the limit of w e a k interunit interactions, i.e. if t~ is not too large c o m p a r e d with 2A~ , the s u p e r c o n d u c t i v e c o u p l i n g of the units will be by e x c h a n g e of C o o p e r p a i r s as in a J o s e p h s o n junction. A p o w e r law of the type 2~
~
t~ / 2 All
will then hold, (39) .
at least
(8) approximately
2. W E A K C O U P L I N G ? I w a n t to e x p l a i n w h y this is a possible and even likely model, d e s p i t e its p r e s e n t lack of p o p u l a r i t y . a - Why weak c o u p l i n g ? The main a r g u m e n t in favour of weak c o u p l i n g stems from band c o m p u t a t i o n s a s s u m i n g i n d e p e n d e n t d e l o c a l i s e d electrons. These lead to c o n d u c t i o n b a n d s of s i z e a b l e w i d t h w, of o r d e r of (40) at least 5 eV. This v a l u e is l a r g e r than in t r a n s i t i o n a l metal o x y d e s : the lower e n e r g y E3~ in C o p p e r i n c r e a s e s t,.~ t ~ 7 ( E ~ - - E ^ ) w h e r e E^ refers , Jez . z zp the oxygen. ~ l S ~ e s u l t is coherent w i t h e x p e r i m e n t a l e v i d e n c e s from p h o t o e m i s s i o n (41) and X rays data : Cu X rays a b s o r p t i o n and e m i s s i o n s p e c t r a lead to o c c u p i e d and e m p t y p a r t s of the 3d c o n d u c t i o n band l a r g e r (42) than 2 eV. M o r e o v e r , 0 soft X rays a b s o r p t i o n from the 2s level shows a d e f i n i t e line due to the 2p state, s h o w i n g that the O 2p state are not full in the g r o u n d state (43). The fact that, in the g r o u n d state, e l e c t r o n holes e x p l o r e the O 2 p states as well as the Cu 3d ones is coherent w i t h the band c o m p u t a t i o n s , w h i c h describe a partly occupied covalent Cu 3d O 2 p band. The study of c o r r e l a tions then r e q u i r e s the use of an ext e n d e d H u b b a r d model ; and l o c a l i s a t i o n of one e l e c t r o n (or hole) per Cu O^ cell is o b t a i n e d only if both the ~n site U_~ucu _ and U__ .. .uu but also. the inter.-r---slEe u~ ~ r e p u l s i o n s are larger tnan the b a ~ w i d t h w. W i t h w ~ 5 eV, this seems v e r y u n l i k e l y in a c o n d u c t i v e compound. In c o h e r e n c e w i t h this view, the c o r r e l a t i o n s a t e l l i t e s o b s e r v e d in A u g e r (44) and p h o t o e m i s s i o n (41) spectra, due to double i o n i s a t i o n of Cu atoms, are s h i f t e d in e n e r g y by an amount
1612
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
w h i c h can be e x p l a i n e d u s i n g an on site r e p u l s i o n U of o r d e r 2 eV if the b a n d w i d t h is a s s u m e d to be a b o u t 7 eV. V a l u e s of U a n d w of the same o r d e r c a n be u s e d to e x p l a i n the s t a b i l i t y of the a n t i f e r r o m a ~ n e t i c p h a s e (I I) w i t h Sr d o p i n g in La^ Sr CuO. a n d the v a l u e of the obse~vXd Xtom~cYmoment (45). All t h e s e e x p e r i e n c e s are t h e n c o h e r e n t , on a q u a n t i t a t i v e basis, w i t h the idea t h a t e l e c t r o n s are d e l o c a l i s e d and i n t e r a c t r e l a t i v e l y w e a k l y (U < w). T h e y c a n n o t h o w e v e r be t a k e n as s u c h as c o m p l e t e p r o o f s , as o t h e r a n a l y s e s in the s t r o n g c o u p l i n g l i m i t U > w are a l s o p r o b a b l y p o s s i b l e (46) . F i n a l l y t h e r e are a n u m b e r of o b j e c t i o n s to w e a k c o u p l i n g w h i c h m u s t be c o m m e n t e d on. Hall m e a s u r e m e n t s (47) on p o l y c r y s t a l l i n e La^ Sr CuO. have g i v e n a -x .x n u m b e r of ~ a r r l e r s ~h~les) p r o p o r t l o n a l to d o p i n g x. H o w e v e r no e x p e r i m e n t seems to h a v e b e e n c a r r i e d so far on s i n g l e c r y s t a l s , as t h e y h a v e b e e n in Y B a g C u g 0 7 , w h e r e the e f f e c t i v e n u m b e r of c a r r ~ e r ~ (electrons) is l a r g e p a r a l l e l to the Cu 02 p l a n e s (24). Photoemission s p e c t r a i n d i c a t e an apparently vanishingly small d e n s i t y of s t a t e s at the F e r m i level (41), (48). However photoemission explores layers n e a r the s u r f a c e , w h i c h are o f t e n a l t e r e d by s u r f a c e r e a c t i o n s (49). Specific heat measurements lead to y v a l u e s of the o r d e r of m a g n i t u d e e x p e c t e d for m e t a l l i c p h a s e s (50) (51). H o w e v e r a y t e r m has b e e n o b s e r v e d in the a n t i f e r r o m a g n e t i c phase of Y B a ^ C u ~ O _ (but n o t La~ Sr C u O _ ) ; z J I -x it has a l s o b e e n o b s e r v e ~ in ~ h e ~ u ~ e r c o n d u c t i v e p h a s e s of the c o r r e s p o n d i n g c o m p o u n d s . T h i s has l e a d to s u g g e s t i o n s of a c o n t r i b u t i o n of n o n c o n d u c t i v e fertalons in the RBV b e l o w T c (51) (64), or of l o c a l i s e d t u n n e l l i n g m o d e s due to i m p e r fections below t or T _ ( 5 2 ) . M o r e mea• c s u r e m e n t s on s l n g l e c r y s t a l s w o u l d be appropriate. In c o n c l u s i o n , the p r e s e n t s i t u a t i o n is r e m i n i s c e n t of the long b a t t l e for the d c h a r a c t e r of t r a n s i t i o n a l m e t a l s . P a r a m e t e r s U and w a r e i n d e e d of the same o r d e r of m a g n i t u d e in the d e l o c a l i s e d p i c t u r e (53). As for t h e s e m e t a l s , f e r m i o l o g y (54) and a n g u l a r d e p e n d e n t p h o t o emission might provide crucial proofs. b - Superconductivity p l i n g limit.
in the w e a k
cou-
B e c a u s e of q u a s i l o w d i m e n s i o n a l i t y , one m u s t f i r s t d i s c u s s the m e a n f i e l d superconductive c o u p l i n g w i t h i n e a c h unit, t h e n l o o k at p e r t u r b a t i o n s due to
their interactions. In the w e a k c o u p l i n g limit, it w a s p o i n t e d o u t by L a b b 4 for c h a i n s (I) and by H i r s c h and S c a l a p i n o for p l a n e s (21) that, if the F e r m i l e v e l is n e a r to the corresponding infinite Van Hove singular i t y (55), the m e a n f i e l d c r i t i c a l t e m p e r a t u r e Tj! w i l l be a n o m a l o u s l y large. The m a x i m u m e f f e c t d e p e n d s on d i m e n s i o nality, with 2kB~ ~ V2/8t, (9) for c h a i n s
and
4~25" I 2kB~ ! $
4 tll e x p - ( - - - - ~
/2
(10)
for p l a n e s (I) , (10) , (12) . lon~,g is i n d e e d n o t m u c h d e c r e a s e d as as the F e r m i level is s h i f t e d f r o m the V a n H o v e s i n g u l a r i t y by at m o s t a few t i m e s the m e a n f i e l d s u p e r c o n d u c t i ve gap An, or a l s o if the V a n H o v e sing u l a r i t y is b r o a d e n e d by t e r m s less than a f e w ~ s (I), (I0), (11). The r e a son is t h a t the BCS i n t e g r a l s e x p l o r e the d e n s i t y of s t a t e s w i t h a l o t e n t z i a n of w i d t h All and long wings. T h e r e is of c o u r s e b r o a d e n i n g f r o m the c o u p l i n g of the u n i t s t h r o u g h t~ ; but it is small b e c a u s e of (3). The b r o a d e n i n g due to i n t r a b l o c k s c a t t e r i n g s is a l s o small b e c a u s e e l e c t r o n s n e a r a V a n H o v e s i n g u l a r i t y h a v e v e r y small v e l o c i t i e s , t h u s small r e l a x a t i o n freq u e n c i e s e v e n if the m e a n free p a t h s are short. It is t h e r e f o r e r e a s o n a b l e (11) to n e g l e c t t h e s e b r o a d e n i n g in c o m p u t i n g T~. Finally, comparing (9) a n d (10) to the c l a s s i c a l 3d BCS formula s h o w s t h a t ~I i n c r e a s e s w i t h d e c r e a s i n g d i m e n s i o n a lity. The d e c r e a s e of T w i t h r e s p e c t to T.~ a l s o i n c r e a s e s w i t h ~ e c r e a s i n g d i m e n slonality (cf (4), (5)) . It is t h e n v e r y e a s y to show, u s i n g (8), t h a t in the lim i t t/l/t~ >> I, ~ u a s i t w o d i m e n s i o n a l i t y g i v e s b Y far the b e s t c o m p r o m i s e (10) for T . W e a k c o u p l i n g s are t h e n c o m p a c . t i b l e w l t h h i g h T . F o r i n s t a n c e , for La^ Sr CuO. , t~e m a x i m u m v a l u e T c Z ~ X 4 0 ~ i s 4 e ~ s i l y e x p l a i n e d w i t h (11) depending in (5).
= V / t ~ ~ 0.4 on the e x a c t v a l u e
taken
for
3. P H O N O N S O R E L E C T R O N S ? I w o u l d f i n a l l y like to e x p l a i n why, at l e a s t in the w e a k c o u p l i n g limit, V seems m o r e l i k e l y to be p h o n o n m e d i a t e d t h a n due to d i r e c t e l e c t r o n - e l e c t r o n interactions. If we c o m p a r e the p h a s e d i a g r a m s of the n e w o x y d e s w i t h t h o s e of the o r g a n i c superconductors, we f i n d in b o t h c a s e s
J, Friedel / Quasi-low-dimensionality in the weak coupling limit
a n t i f e r r o m a g n e t i c s p h a s e s (56),(57). This was a p o w e r f u l i m p e t u s to try and e x t e n d to the o x y d e s the e l e c t r o n med i a t e d s c h e e m e s that seem to a p p l y to the o r g a n i c c o m p o u n d s . The p h a s e diag r a m s are h o w e v e r v e r y d i f f e r e n t : for the o x y d e s , t h e r e is no t e m p e r a t u r e range of c o e x i s t e n c e of the two phases, but on the c o n t r a r y T and T. v a n i s h at or b e f o r e any c o n t a c ~ b e t w e eN n the two phases. A n t i f e r r o m a g n e t i c f l u c t u a t i o n s seem also to d e c r e a s e s h a r p l y in range, i n t e n s i t y and t e m p e r a t u r e , o u t s i d e the d o m a i n of s t a b i l i t y of the t h r e e d i m e n sional a n t i f e r r o m a g n e t i c phase (57). Interference between antiferromagnetism and s u p e r c o n d u c t i v i t y , if any, seems t h e r e f o r e d e s t r u c t i v e , and not c o n s t r u c tive as in the o r g a n i c s u p e r c o n d u c t o r s . I n d e e d p h a s e d i a g r a m s such as those of the o x y d e s are k n o w n in t r a n s i t i o n a l alloys, w h e r e s u p e r c o n d u c t i v i t y is p h o non m e d i a t e d , and d i r e c t e l e c t r o n - e l e c tron i n t e r a c t i o n s w e a k e n s o m e w h a t T . c But, as in t h a t case, one c o u l d e x p e c t this w e a k e n i n g of T to be reduced, c owing to the d i f f e r e n c e in f r e q u e n c i e s of p h o n o n s and p l a s m o n s (58). In fact, if there was no i n t e r f e r e n ce from a n t i f e r r o m a g n e t i s m , one w o u l d expect, in La~ Sr CuO. , T to d e c r e a z-x x o v e r~-v.. se as x increases, all ~he r a n g e of s t a b i l i t y of s u p e r c o n d u c t i v i t y . This is b e c a u s e the Fermi level shifts away f r o m the V a n Hove s i n g u l a r i t y , w h i c h stays at half filled b a n d owing to the e q u a l i t y of Cu Cu b o n d s in the o r t h o r h o m bic p h a s e (19) (the w e a k Jahn T e l l e r inst a b i l i t y that w o u l d split the V a n Hove s i n g u l a r i t y by m a k i n g the Cu Cu b o n d s u n e q u a l is e a s i l y s u p p r e s s e d by the small b r o a d e n i n g of the Van Hove s i n g u l a r i t y (11)). P o s i t i v e i n t e r f e r e n c e s b e t w e e n a n t i f e r r o m a g n e t i s m and s u p e r c o n d u c t i v i t y w o u l d s t r e n g t h e n this d e c r e a s e of T w i t h i n c r e a s i n g x (21),(59). The o n l y way to e x p l a i n that T d e c r e a s e s for x d e c r e a s i n g f r o m O.15 ~s then to a s s u m e a n e g a t i v e i n t e r f e r e n c e of s u p e r c o n d u c t i v i t y w i t h the weak and short range a n t i f e r r o m a g n e t i c f l u c t u a t i o n s still e x i s t i n g in this range. This is p o s s i b l e if the s u p e r c o n d u c t i v e c o u p l i n g is m e d i a ted by the full s p e c t r u m of phonons. If now we a s s u m e a p h o n o n m e d i a t e d ~ n t e r a c t i o n V, low but non v a n i s h i n g v a l u e s of the i s o t o p e e f f e c t find a natural e x p l a n a t i o n . It was p o i n t e d out by L a b b 4 (I) that, in the w e a k c o u p l i n g limit, the p r e s e n c e of a d i v e r g i n g V a n Hove a n o m a l y n e a r the F e r m i level s h o u l d s t r o n g l y r e d u c e the i s o t o p e effect. The r e a s o n is that the c u t o f f o f f e r e d by the
1613
steep d e c r e a s e of the d e n s i t y of s t a t e s w i t h e n e r g y b e c o m e s m o r e e f f e c t i v e than the c u t o f f by the D e b y e f r e q u e n c y w D. I n d e e d e s t i m a t e s (9) and (IO) do not contain ~_ at all ( 1 0 ) (11). M o r e r e f i n e d e s t i m a t e s (I) , (IO),~12), (14) i n t r o d u c e some c o r r e c t i o n s w h i c h for p h o n o n m e d i a ted V ' s , w o u l d g i v e a small i s o t o p e effect. M o r e p r e c i s e l y (60), it is l a r g e r for p l a n e s than for chains, and i n c r e a ses c o n t i n u o u s l y t o w a r d s the c l a s s i c a l v a l u e w h e n the F e r m i level s h i f t s away f r o m the V a n H o v e s i n g u l a r i t y by m o r e than s e v e r a l Aj;s. Then in La^ Sr CuO_ , for x ~ 0.15, z~x x 4-y w h e r e the F e r m l level Is at a d l s t a n c e f r o m the V a n Hove s i n g u l a r i t y , one exp e c t s (11), as o b s e r v e d (61),a small but s i z e a b l e i s o t o p e effect, w h i c h s h o u l d d e c r e a s e w i t h d e c r e a s i n g x. In Y B a 2 C u ~ O _ w h e r e the C u O c h a i n s p r o b a b l y take soNe/' of the e l e c t r o n s of the o t h e r w i s e half f i l l e d d b a n d of the C u O ~ planes, one e x p e c t s the Fermi level £o be b o t h n e a r the s i n g u l a r i t y of the c h a i n s and v e r y n e a r the lower half of the s i n g u l a r i t y of the CuO^ planes, split in the o r t h o r h o m b i c p h a s e (9), (11),(14). Thus c o n d i t i o n s for lower i s o t o p e e f f e c t (62) and for h i g h e r T (63) w o u l d be fullc filled. CONCLUSIONS. The c l a s s i c a l a p p r o a c h s k e t c h e d h e r e seems to o f f e r a f a i r l y c o h e r e n t p i c t u re of the n e w o x y d e s u p e r c o n d u c t o r s . It s u g g e s t s at least two w a y s of i m p r o v i n g T : - C b e t t e r 'super s a n d w i c h e s ' , r e s p e c t i n g the f u n d a m e n t a l q u a s i t w o d i m e n s i o n a l char a c t e r and a F e r m i level near to V a n H o v e s i n g u l a r i t i e s , but c o n t a i n i n g a h i g h e r p r o p o r t i o n of c h a i n s perhaps. - b e t t e r c o m p o u n d s w i t h l a r g e r t,~s and thus l a r g e r A,. A l o n g this s e c o n ~ line, the c o u p l e C u O is a l r e a d y f a v o u r a b l e . But o t h e r couples, u s i n g S, Se or Te or Ag, thus s m a l l e r r e l a t i v e s t a b i l i t i e s of the d and p states, w i t h s i m i l a r t o p o l o g i e s , m i g h t be of interest. There are a number of experimental difficulties in this scheme, which should be checked on single crystals. F i n a l l y , e v e n if c o n f i r m e d by f u r t h e r e x p e r i m e n t s , e x p e c i a l l y on fermi or a n g u l a r d e p e n d e n t p h o t o e m i s s i o n , this a p p r o a c h c o u l d be improved. Thus the a s s u m e d r a t i o s of U / w and V / w are s i z e a ble. M o r e r e f i n e d t r e a t m e n t s of antif e r r o m a g n e t i s m w o u l d take into a c c o u n t h i g h e r p o w e r s (53) of U or use v a r i a t i o nal m e t h o d s (65) ; s c a l i n g m e t h o d s w o u l d be in o r d e r for s u p e r c o n d u c t i v i t y (cf
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J. Friedel / Quasi-low-dimensionality in the weak coupling limit
(4) , (8) , (21) . .) . C r u d e e s t i m a t e s of meanfield T s h o u l d b e c o r r e c t e d to inc . c l u d e c o r r e c t l o n s d u e to t h e D e b y e c u t o f f (59). D e v i a t i o n s f r o m r i g i d b a n d a p p r o x i m a t i o n s h o u l d de t a k e n i n t o a c c o u n t i n d o p i n g , e s p e c i a l l y for s m a l l c o n c e n t r a t i o n s in a n t i f e r r o m a g n e t i c La. Sr Z--X ,X CuO• o r for t h e e f f e c t o f 0 v a c a n c i e s
(11~? y F i n a l l y it is n o t c l e a r w h e t h e r s t r o n g interplane phase fluctuations (solitons) h a v e b e e n i n c l u d e d in t r e a t i n g t h e XY m a g n e t i c c o u p l i n g e x p e c t e d in t h e a n t i f e r r o m a g n e t i c p h a s e (cf A p p e n d i x B) a n d the nearly twodimensional superconduct i v i t y . T h e y m i g h t l o w e r T. a n d T from s t a n d a r d t h e o r i e s (66), t h u s i n c r e a s e in e q u a t i o n (5) f r o m v a l u e s n e a r to u n i t y , a n d e x p l a i n the s t r o n g f l u c t u a t i o n s o b s e r v e d w e l l a b o v e T N in t h e a n t i f e r r o m a g n e t i c c a s e (57). A P P E N D I X A. Z I G Z A G G I N G V O R T E X L I N E S A POLYCRYSTAL WITH RANDOM TEXTURE•
IN
A v o r t e x l i n e c a n l o w e r its e n e r g y b y z i g z a g g i n g if t h e l i n e t e n s i o n 6t gained by passing through grains with p l a n e s p a r a l l e l to the v o r t e x is l a r g e r than the_energy lost against the line tension ~ by zigzagging. L e t L b e t h e a v e r a g e l e n g t h of a zigzag b e t w e e n t w o ' f a v o u r a b l e ' g r a i n s , m e a s u r e d a l o n g the g e n e r a l d i r e c t i o n of t h e v o r t e x , H the a v e r a g e h e i g h t of the zigzag and D the average grain diameter. T h e c h a n g e AT in a v e r a g e l i n e t e n s i o n d u e to z i g z a g g i n g is s u c h t h a t (L+D)AT = ( / ~ z ~ 2 - L ) ~ D6T. (A.I) The favourable grains must have their p l a n e s l y i n g w i t h i n t h e a n g l e 0 of t h e zigzag. Thus D 3 . L H 2 ~ 2n s i n e (A.2) with tg 8 m H/L. If e w a s small, o n e c o u l d w r i t e I
H 2
D
with D 3
2~0 ~ 2~ H LH 2 L Hence H ~ (2~) - 1 1~3 D.
(A.4)
Optimizing 9~with H / L ~ (2~) " J 6 T / ~ The pinning energy is t h e n 2 L A~ ~ D 6T
respect per
In f a c t ~-- ~ 2 - -
• ~ ?'~"
to L g i v e s (A.5) favourable grain (A.6) is n o t
a small
q u a n t i t y . T h e e q u l ± i D r i u m 0 is l a r g e a n d s h o u l d be d e d u c e d d i r e c t l y f r o m (A.I) to (A.3) . B u t r e s u l t s w o u l d b e s i m i l a r , l e a d i n g to s t r o n g a n d s h o r t z i g z a g s (H ~ L m D) a n d p i n n i n g e n e r g i e s of o n e
zigzag
of o r d e r
D
6 t.
A P P E N D I X B. X Y M A G N E T I S M TIVE OXYDES.
IN S U P E R C O N D U C -
T h e m a g n e t i c a n i s o t r o p y of C u 0 9 p l a nes i n v o l v e s v i r t u a l e x c i t a t i o n s ~f Cu 3d e l e c t r o n s f r o m r o u g h l y h a l f f i l l e d x 2 - y 2 s t a t e s to o t h e r f u l l 3d s t a t e s . T h i s is i n d e p e n d e n t o n w h e t h e r the m a g n e t i c e l e c t r o n s of C u a r e l o c a l i s e d or delocalised. A c l a s s i c a l a n a l y s i s (67) s h o w s t h a t t h e a n i s o t r o p y is m a i n l y a x i a l w i t h r e s p e c t to t h e n o r m a l z to t h e C u O 9 p l a n e s • Thus, to s e c o n d o r d e r in s p i n o ~ b i t c o u p l i n g I Z s, t h e d i f f e r e n c e in e n e r g y b e t w e e n an e l e c t r o n s p i n o r i e n t e d a l o n g z o r n o r m a l to z is E z - Exy=
~-~2
L
Eo_E2
]
Eo_E3
- E^ a n d E - E. a r e r e s p e c t i v e l y ~e l a t t i c e s ~ l i t t ~ n g s of t h e x 2 - y 2 s t a te w i t h the x y a n d x z o r y z ones. W i t h E -E^ < E -E~ < 20 eV, a n d I ~ 0 . 0 7 eV, .o. z " o ~ " .. t n l s e n e r g y is n e c e s s a r 1 ± y p o s i t i v e ; w i t h i n a f l u c t u a t i o n a r e a ~2, it is l a r g e c o m p a r e d w i t h t h e r m a l e n e r g y u p to at least room temperature (57). A n y a n i s o t { o p y in t h e X Y p l a n e is d u e to t e r m s in I~ w i c h a r e o b v i o u s l y n e g l i g i b l e at room temperature• REFERENCES (I) J. L a b b 4 , S. B a r i s i c a n d J. F r i e d e l , Phys. Rev. L e t t . V o i . 1 9 , 1039 (1967)• (2) S. B a r i s i c a n d J. L a b b 4 , J. Phys. Chem. Sol. V o i . 2 8 , 2477 (1967) ; J. L a b b 4 , P h y s . Rev. Vol. 172, 45 (1968). (3) D. J 4 r o m e , A. M a z a u d , M. R i b a u l t a n d K. B e c h g a a r d , C o m p t e R e n d u s v o i . 2 9 0 B, 27 (1980) • (4) D. J 4 r o m e a n d H.J. S c h u l z , Adv. P h y s . V o i . 3 1 , 229 (1982). (5) J. F r i e d e l , Phil. T r a n s . Roy. Soc. vol. A 3 1 4 , 189 (1985). (6) F. C r e u z e t , D. J 4 r o m e , K. B e c h g a a r d a n d A. M o r a d p o u r , J. P h y s i q u e vol. 45, L 7 5 5 (1984). (7) F. C r e u z e t , T h ~ s e O r s a y (1987). (8) C. B o u r b o n n a i s a n d L.G. C a r o n , E u r o p h y s . L e t t . vol. 5, 209 (1988) a n d references therein• (9) J. F r i e d e l , C o m p t e s R e n d u s A c a d . Sci. vol. 305 II, 543 (1987). (10)J. F r i e d e l , J. P h y s i q u e , v o i . 4 8 , 1787 (1987), a n d E r r a t u m , ibid, u n d e r p r e s s (1988). (11)J. F r i e d e l , J. P h y s i q u e u n d e r p r e s s (1988) . (12) J. L a b b 4 a n d J. Bok, E u r o p h y s . L e t t . v o l . 3 , 1225 (1987).
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
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(35) A. Umezawa, G.W. Crabtree, J.Z. Liu, T.J. Moran, S.K. ~alik, L.H. Numez, W.L. Kwok and C.H. Sowers, Phys. Rev. Lett. under press (1988). (36) J. Friedel, D i s l o c a t i o n s , P e r g a m o n L o n d o n (1964). (37) cf for i n s t a n c e M. Oussena, S. S e n o u s s i and G. Collin, Europhys. Lett. vol. 4, 625 (1987). (38) B. Farnoux, R. Kahn, A. Brule, G. C o l l i n and J.P. Pouget, J. P h y s i q u e vol. 48, 1623 (1987). (39) H. Schmidt, Z P h y s i k voi.216, 336 (1968). (40) L.F. Mattheiss, Phys. Rev. Lett. vol. 58, 1028 (1987). (41) A. Samsavai, T. Miller, T.C.Chiang, B.G. Pazol, T.A. F r i e d m a n n and D. M. Ginzberg, to be published. (42) T. Gourieux, G. Krill, M. Maurer, M.F. Ravet, A. Menney, H. T a l e n t i n o and A. Fontaine, Phys. Rev. B under press (1988). (43) A.G. Schrott, R. Park and C . C . T s u e i to be p u b l i s h e d ; P. Thiry, G.Rossi, Y. Petroff, A. R e v c h o l e v s k i and J. Jegondez, Europhys. Lett. vol.5, (1988). (44) A. Balzarotti, M. de Crescenzi, C. Gionella, P. Messi, N. Motta, F. P a t e l l a and A. S p a r l a t a under press. (45) H.J. Schulz, this conference. (46) G.A. Sawatsky, priv. comm. (47) S. Uchida, H. Takagi, H. Ishii, H. Eisaki, T. Yab4, S. Tojima and S. Tanaka, Jap. J. Appl. Phys. voi.26, L 440 (1987). (48) Y. Petroff, P. Thiry, G. Rossi, A. R e v c h o l e v s k i and J. Jegondez, A d r i a t i c o Conf. on High T supercond. T r i e s t e (1987). c (49) H.W. Zandbergen, R . G r o n s k y and G. Thomas, under press (1988). (50) C. Ayache, B. Barbara, B. Bonjour, P. Burlet, R. Calemczuk, M. Coupch, M.J.G. Jugens, J.Y. Henry and J. Rossat Mignot, u n d e r press. (51) K. Kumagai, Y. Nakamichi, I.Watanabe, Y. Nakamura, H. Nakajima, N. Wada, and P. Lederer, Techn. Rep. ISSP, ISSN ISSN 0 0 8 2 - 4 7 9 8 A, 1890 (1988). (52) F.J. W a l k e r and A.C. Anderson, Phys. Rev. Vol. 29, 588 (1984). (53) J. Friedel and C.M. Sayers, J. Phys. Lett.vol. 39, L59 (1978). (54) L. Hoffmann, A.A. Manuel, M. Peter, E. W a l k e r and M.A. Damento, Europhys. Lett. under press (1988) ; L. S m e d s k j a e r to be p u b l i s h e d (1988). (55) L. Van Hove, Phys. Rev. voi.89, 1189 (1953). -
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(57
(58) (59) (60) (61)
(62)
(63)
(64) (65)
(66)
(67)
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
I.D. Vaksien, S.K. Sinha, D.E. Monston, D.C. Sohnston, J.M. Newsam, C.R. Sofinya and H.E. King, Phys. Rev. Lett. v o i . 5 8 , 2 8 0 2 (1987) ; G. Shirane, Y. Endoh, R. J. Birgenan, M.A. Kastner, Y. Hidata, M. Oda, M. Suzuki and T. Murakami, Phys. Rev. Lett. vol. 59 (1987). G. Shirane, Y. Endoh, R.J.Birgenau, M.A. Kastner, Y. Hidaka, M. Oda, M. Suzuki and T. Murakami, Phys. Rev. Lett. voi.59, 1613 (1987). D.J. Scalapino, in S u p e r c o n d u c t i v i ty, ed. R.D. Parks, New Y o r k (1969). H.J. Schulz, Europhys. Lett. vol.4 609 (1987). J. Labb4 and Combescot, this conference. T.A. Faltens, W.K. Hans, S.W. Keller, K.J. Leary, J.N. Michaels, A.M. Stacy, H.C. Zur Loye, D.E. Morris, T.W. Barber III, L.C. Bourne, M.L. Cohen, S. Hoen and A. Zettl, Phys. Rev. Lett. voi.59, 915 (1987). H.K. Zur Loye, S.W. Keller, W.K. Ham, T.A. Faltens, J.M. M i c h a e l s and A.M. Stacy, S c i e n c e v o l 228, 1358 (1987). C.W. Chu, R.H. Hor, R.L, Meng, Z.J. Huang and Y.Q. Wang, Phys. Rev. Lett. voi.58, 405 (1987). P.W. Anderson, Science vol 235, 1196 (1987). P. Fulde, P h y s i c a Scr~pta under press (1988) ; A.M. Ol~s, J. Zaanen and P. Fulde, Yamada Conf. XVIII Sendai (1987). P. R e g n a u l t and J. Rossat M i g n o t in M a g n e t i c p r o p e r t i e s of l a y e r e d t r a n s i t i o n metal compounds, ed. L.S. Jough and R.D. Willet, Reidel Publi. Cy in press. J. Kanamori, in M a g n e t i s m I, ed. G.T. Rado and H. Suhl, A c a d e m i c Press N e w Y o r k (1963).
QUASILOWDIMENSIONALITY
Jacques
IN THE WEAK
COUPLING
LIMIT
FRIEDEL
P h y s i q u e des Solides, Orsay, F r a n c e
Universit~
Paris-Sud,
Formation
Associ~e
The three q u e s t i o n s of q u a s i l o w d i m e n s i o n a l i t y , weak c o u p l i n g c o u p l i n g are d i s c u s s e d for the new oxyde s u p e r c o n d u c t o r s . INTRODUCTION In a field where t h e o r e t i c a l d i s c u s sion seems still open, e v e r y b o d y comes with p r e c o n c i e v e d ideas. My own stem from an e a r l i e r i n t e r e s t in c h a i n - l i k e s u p e r c o n d u c t o r s , first the A15 c o m p o u n d s with Labb4 and B a r i s i c (I),(2), then the o r g a n i c family (TMTSF)2X d i s c o v e r e d in O r s a y nine years ago by Bechgaard, J 4 r o m e and R i b a u l t (3) and much s t u d i e d since then (4). In the (TMTSF) 2X's, the flat u n s a t u r a ted o r g a n i c m o l e c u l e s are p i l e d in columns a l o n g w h i c h the v a l e n c e e l e c t r o n s jump much more r a p i d l y than from c o l u m n to column. The c o r r e s p o n d i n g t r a n s f e r integrals are such that t0 >> t& . (I) This s i t u a t i o n is d e s c r i b e d as q u a s i l o w d i m e n s i o n a l (5). It implies a h i e r a r c h y such that it is useful to t h i n k first of the columns as s e p a r a t e units, and then to treat their i n t e r a c t i o n s as p e r t u r b a tions. In these o r g a n i c s u p e r c o n d u c t o r s , it is also clear that the e f f e c t i v e c o u p l i n g V leading to s u p e r c o n d u c t i v i t y is smaller than the band width, a c o n d i t i o n w h i c h reads there as
Ivl < 2 t . .
t2~
It is t h e r e f o r e m e a n i n g f u l to start from e l e c t r o n s s t r o n g l y d e l o c a l i s e d along the columns. I will call this a w e a k c o u p l i n g limit, a l t h o u g h the e f f e c t of V m i g h t be somewhat too large to be q u a n t i t a t i v e l y g iven by the first p e r t u r b a t i v e term. Thus this c o u p l i n g m i g h t include e f f e c t s similar to the 'strong c o u p l i n g limit' of lead. It does not include s i t u a t i o n s where the e l e c t r o n s are s t r o n g l y loealised by their e l e c t r o s t a t i c i n t e r a c t i o n s or by very strong e l e c t r o n - p h o n o n couplings. F i n a l l y the p h a s e d i a g r a m of these o r g a n i c s u p e r c o n d u c t o r s show that the most stable c r i t i c a l t e m p e r a t u r e s are o b s e r v e d next to an a n t i f e r r o m a g n e t i c
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
au CNRS,
91405
and e l e c t r o n
phonon
transition, w h e n p r e s s u r e or c o m p o s i tion are v a r i e d (4). This and a c a r e f u l study (6), (7) of the f l u c t u a t i o n s above T c s t r o n g l y suggests that V is due to a d l r e c t e l e c t r o n - e l e c t r o n interaction, and is not p h o n o n m e d i a t e d (8). It is then but n a t u r a l to ask w h e t h e r or not any part of such a scheeme a p p l i e s to the n e w o x y d e s u p e r c o n d u c t o r s . It is, I feel, with d e c r e a s i n g c e r t a i n l y that one can answer the three q u e s t i o n s of q u a s i l o w d i m e n s i o n a l i t y , weak c o u p l i n g and e l e c t r o n or p h o n o n m e d i a t e d interactions. This p a p e r s u m m a r i s e s and c o m p l e t e s three p r e v i o u s ones (9),(10),(11). It is m o s t l y c o h e r e n t w i t h p a p e r s by Labb4, Bok and C o m b e s c o t (12), (13), (14) by B a r i s i c and B a t i s t i c (15),(16), by J 4 r o me et al (17),(18) and by P o u g e t and N oguera(19).Hirschand ScalaDino(20).Schulz(59), D z a l o s h i n s k y (21) and B a r d e e n (22) have e x p r e s s e d views w h i c h in part start from similar p o i n t s of view. But a number of p o i n t s made here o r i g i n a t e from many d i s c u s s i o n s , e s p e c i a l l y also with Schulz, H 4 r i t i e r and Lederer. I. Q U A S I L O W D I M E N S I O N A L I T Y . This is i m p l i c i t in most t h e o r e t i c a l a n a l y s i s nowadays, and indeed seems c o n f i r m e d by a v a i l a b l e e x p e r i m e n t s on the a n i s o t r o p y of r e s i s t i v i t y or of pen e t r a t i o n depth. Both lead, for Y B a 2 C u 3 0 7 , to (23),(24), (35)
tlt/ t~ ~ 102 . (3) A n i s o t r o p y of p l a s m a f r e q u e n c y in L a 2 N i O 4 g i v e s a lower limit of the same o r a e r [25). One can then define mean field intra and i n t e r b l o c k couplings, m e a s u r e d by their gaps 4,, A& or their c r i t i c a l temp e r a t u r e s Tit , T& . The three d i m e n s i o nal c r i t i c a l t e m p e r a t u r e T is an average of T;. and TI w h i c h d e p e n d s on d i m e n sionality. By ~ n a l o g y w i t h c o r r e s p o n d i n g
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
magnetic roughly,
problems,
on can I/2
T c ~ (~; T~) for stacked c ~ a i n s Tc ~ ~ Td
say that,
very
(4) (26),
and (5)
for stacked planes (27), where I<~<£n (~//T~).~ is thus at most a few units. One should then expect a zone of 2d s u p e r c o n d u c t i v e f l u c t u a t i o n s (28) in the range T << T < T~land, possibly, from b elow TC up to near T~ , a p s e u d o g a p ~;l larger ~han the BCS v a l u e I/2 3.5 k_T even if the BCS r e l a t i o n applies b e t w e e n Atl and TII (29), (30), (31). B e c a u s e of s i z e a b l e gaps A, one expects small c o h e r e n c e lengths ~ VM/A ~ a t/A, thus s u p e r c o n d u c t i v i t y of th~ second kind. The c o h e r e n c e length should be a n i s o t r o p i c with, typically, ~l / ~
~ a tlt/c ~ >>
I
(6)
Ratios of that order are d e d u c e d from m e a s u r e m e n t s (32) (33) of Hc. near T . ' Z . C A c o n s e q u e n c e is that v o r t e x l~nes should have c y l i n d r i c a l cores of radius ~l >> a when lying normal to the Cu 02 p~anes. In the absence of defects, such v o r t e x lines should g l i d e easily and take their e q u i l i b r i u m c o n f i g u r a t i o n s u n d e r a p p l i e d field. When lying parallel to the planes, the v o r t e x lines should have e l o n g a t e d c o r e s of size ~j~ along the planes, but t h i c k n e s s ~ of atomic d i m e n s i o n s normal to the Cu 09 planes. Their line tensions ~ and ~,/s~ould t h e r e f o r e be s t rongl y anisotropic, with (28)
T,,/q ~ q /~t/ (7) There should be a strong t e n d a n c y for the v o r t e x lines, w h e n p a r a l l e l to the Cu 02 planes to lie along the w e a k e r s u p e r c o n ducting links, thus b e t w e e n the Cu O 2 planes in the La~ Sr CuO. compounas z-x x . 4-y and along the Y planes in Y B a 2 C u 3 0 7 x" This should lower L~ from (7). A I s o suc~ v o r t e x lines should g l i d e easily along the Cu 02 planes, in p r e f e r e n c e to the normal d l r e c t i o n w h e r e ~ should vary periodically, with sizeable m a x i m a between its p o s i t i o n s of m i n i m u m v a l u e (34). As a c o n s e q u e n c e , for single crystals, H is s t r o n g l y a n i s o t r o p i c (28) and w e a k f~ fields a p p l i e d p a r a l l e l to the Cu 02 planes (35), the ratio /H c being given by ~ / T ~ . HcI~ I~ In p o l y c r y s t a l s , v o r t e x lines introduced by weak a p p l i e d fields should present, at equilibrium, a z i g z a g g i n g form, so as to pass t h r o u g h a larger n u m b e r of g r a i n s where the v o r t e x lines can lie p a r a l l e l to the Cu 09 planes. For a r a n d o m texture, a direct e x [ e n s i o n of the strong p i n n i n g
1611
of d i s l o c a t i o n s by i m p u r i t i e s (36) shows the zigzags to be short but m a r k e d (App e n d i x A). This can e x p l a i n the strong h y s t e r e s i s o b s e r v e d (37). In a powder, the same e f f e c t can e x p l a i n a r o t a t i o n of the g r a i n s so as to a l i g n p r e f e r e n t i a l l y w i t h Cu O^ p l a n e s p a r a l l e l to the a p p l i e d fiel~ (38). Finally, in the limit of w e a k interunit interactions, i.e. if t~ is not too large c o m p a r e d with 2A~ , the s u p e r c o n d u c t i v e c o u p l i n g of the units will be by e x c h a n g e of C o o p e r p a i r s as in a J o s e p h s o n junction. A p o w e r law of the type 2~
~
t~ / 2 All
will then hold, (39) .
at least
(8) approximately
2. W E A K C O U P L I N G ? I w a n t to e x p l a i n w h y this is a possible and even likely model, d e s p i t e its p r e s e n t lack of p o p u l a r i t y . a - Why weak c o u p l i n g ? The main a r g u m e n t in favour of weak c o u p l i n g stems from band c o m p u t a t i o n s a s s u m i n g i n d e p e n d e n t d e l o c a l i s e d electrons. These lead to c o n d u c t i o n b a n d s of s i z e a b l e w i d t h w, of o r d e r of (40) at least 5 eV. This v a l u e is l a r g e r than in t r a n s i t i o n a l metal o x y d e s : the lower e n e r g y E3~ in C o p p e r i n c r e a s e s t,.~ t ~ 7 ( E ~ - - E ^ ) w h e r e E^ refers , Jez . z zp the oxygen. ~ l S ~ e s u l t is coherent w i t h e x p e r i m e n t a l e v i d e n c e s from p h o t o e m i s s i o n (41) and X rays data : Cu X rays a b s o r p t i o n and e m i s s i o n s p e c t r a lead to o c c u p i e d and e m p t y p a r t s of the 3d c o n d u c t i o n band l a r g e r (42) than 2 eV. M o r e o v e r , 0 soft X rays a b s o r p t i o n from the 2s level shows a d e f i n i t e line due to the 2p state, s h o w i n g that the O 2p state are not full in the g r o u n d state (43). The fact that, in the g r o u n d state, e l e c t r o n holes e x p l o r e the O 2 p states as well as the Cu 3d ones is coherent w i t h the band c o m p u t a t i o n s , w h i c h describe a partly occupied covalent Cu 3d O 2 p band. The study of c o r r e l a tions then r e q u i r e s the use of an ext e n d e d H u b b a r d model ; and l o c a l i s a t i o n of one e l e c t r o n (or hole) per Cu O^ cell is o b t a i n e d only if both the ~n site U_~ucu _ and U__ .. .uu but also. the inter.-r---slEe u~ ~ r e p u l s i o n s are larger tnan the b a ~ w i d t h w. W i t h w ~ 5 eV, this seems v e r y u n l i k e l y in a c o n d u c t i v e compound. In c o h e r e n c e w i t h this view, the c o r r e l a t i o n s a t e l l i t e s o b s e r v e d in A u g e r (44) and p h o t o e m i s s i o n (41) spectra, due to double i o n i s a t i o n of Cu atoms, are s h i f t e d in e n e r g y by an amount
1612
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
w h i c h can be e x p l a i n e d u s i n g an on site r e p u l s i o n U of o r d e r 2 eV if the b a n d w i d t h is a s s u m e d to be a b o u t 7 eV. V a l u e s of U a n d w of the same o r d e r c a n be u s e d to e x p l a i n the s t a b i l i t y of the a n t i f e r r o m a ~ n e t i c p h a s e (I I) w i t h Sr d o p i n g in La^ Sr CuO. a n d the v a l u e of the obse~vXd Xtom~cYmoment (45). All t h e s e e x p e r i e n c e s are t h e n c o h e r e n t , on a q u a n t i t a t i v e basis, w i t h the idea t h a t e l e c t r o n s are d e l o c a l i s e d and i n t e r a c t r e l a t i v e l y w e a k l y (U < w). T h e y c a n n o t h o w e v e r be t a k e n as s u c h as c o m p l e t e p r o o f s , as o t h e r a n a l y s e s in the s t r o n g c o u p l i n g l i m i t U > w are a l s o p r o b a b l y p o s s i b l e (46) . F i n a l l y t h e r e are a n u m b e r of o b j e c t i o n s to w e a k c o u p l i n g w h i c h m u s t be c o m m e n t e d on. Hall m e a s u r e m e n t s (47) on p o l y c r y s t a l l i n e La^ Sr CuO. have g i v e n a -x .x n u m b e r of ~ a r r l e r s ~h~les) p r o p o r t l o n a l to d o p i n g x. H o w e v e r no e x p e r i m e n t seems to h a v e b e e n c a r r i e d so far on s i n g l e c r y s t a l s , as t h e y h a v e b e e n in Y B a g C u g 0 7 , w h e r e the e f f e c t i v e n u m b e r of c a r r ~ e r ~ (electrons) is l a r g e p a r a l l e l to the Cu 02 p l a n e s (24). Photoemission s p e c t r a i n d i c a t e an apparently vanishingly small d e n s i t y of s t a t e s at the F e r m i level (41), (48). However photoemission explores layers n e a r the s u r f a c e , w h i c h are o f t e n a l t e r e d by s u r f a c e r e a c t i o n s (49). Specific heat measurements lead to y v a l u e s of the o r d e r of m a g n i t u d e e x p e c t e d for m e t a l l i c p h a s e s (50) (51). H o w e v e r a y t e r m has b e e n o b s e r v e d in the a n t i f e r r o m a g n e t i c phase of Y B a ^ C u ~ O _ (but n o t La~ Sr C u O _ ) ; z J I -x it has a l s o b e e n o b s e r v e ~ in ~ h e ~ u ~ e r c o n d u c t i v e p h a s e s of the c o r r e s p o n d i n g c o m p o u n d s . T h i s has l e a d to s u g g e s t i o n s of a c o n t r i b u t i o n of n o n c o n d u c t i v e fertalons in the RBV b e l o w T c (51) (64), or of l o c a l i s e d t u n n e l l i n g m o d e s due to i m p e r fections below t or T _ ( 5 2 ) . M o r e mea• c s u r e m e n t s on s l n g l e c r y s t a l s w o u l d be appropriate. In c o n c l u s i o n , the p r e s e n t s i t u a t i o n is r e m i n i s c e n t of the long b a t t l e for the d c h a r a c t e r of t r a n s i t i o n a l m e t a l s . P a r a m e t e r s U and w a r e i n d e e d of the same o r d e r of m a g n i t u d e in the d e l o c a l i s e d p i c t u r e (53). As for t h e s e m e t a l s , f e r m i o l o g y (54) and a n g u l a r d e p e n d e n t p h o t o emission might provide crucial proofs. b - Superconductivity p l i n g limit.
in the w e a k
cou-
B e c a u s e of q u a s i l o w d i m e n s i o n a l i t y , one m u s t f i r s t d i s c u s s the m e a n f i e l d superconductive c o u p l i n g w i t h i n e a c h unit, t h e n l o o k at p e r t u r b a t i o n s due to
their interactions. In the w e a k c o u p l i n g limit, it w a s p o i n t e d o u t by L a b b 4 for c h a i n s (I) and by H i r s c h and S c a l a p i n o for p l a n e s (21) that, if the F e r m i l e v e l is n e a r to the corresponding infinite Van Hove singular i t y (55), the m e a n f i e l d c r i t i c a l t e m p e r a t u r e Tj! w i l l be a n o m a l o u s l y large. The m a x i m u m e f f e c t d e p e n d s on d i m e n s i o nality, with 2kB~ ~ V2/8t, (9) for c h a i n s
and
4~25" I 2kB~ ! $
4 tll e x p - ( - - - - ~
/2
(10)
for p l a n e s (I) , (10) , (12) . lon~,g is i n d e e d n o t m u c h d e c r e a s e d as as the F e r m i level is s h i f t e d f r o m the V a n H o v e s i n g u l a r i t y by at m o s t a few t i m e s the m e a n f i e l d s u p e r c o n d u c t i ve gap An, or a l s o if the V a n H o v e sing u l a r i t y is b r o a d e n e d by t e r m s less than a f e w ~ s (I), (I0), (11). The r e a son is t h a t the BCS i n t e g r a l s e x p l o r e the d e n s i t y of s t a t e s w i t h a l o t e n t z i a n of w i d t h All and long wings. T h e r e is of c o u r s e b r o a d e n i n g f r o m the c o u p l i n g of the u n i t s t h r o u g h t~ ; but it is small b e c a u s e of (3). The b r o a d e n i n g due to i n t r a b l o c k s c a t t e r i n g s is a l s o small b e c a u s e e l e c t r o n s n e a r a V a n H o v e s i n g u l a r i t y h a v e v e r y small v e l o c i t i e s , t h u s small r e l a x a t i o n freq u e n c i e s e v e n if the m e a n free p a t h s are short. It is t h e r e f o r e r e a s o n a b l e (11) to n e g l e c t t h e s e b r o a d e n i n g in c o m p u t i n g T~. Finally, comparing (9) a n d (10) to the c l a s s i c a l 3d BCS formula s h o w s t h a t ~I i n c r e a s e s w i t h d e c r e a s i n g d i m e n s i o n a lity. The d e c r e a s e of T w i t h r e s p e c t to T.~ a l s o i n c r e a s e s w i t h ~ e c r e a s i n g d i m e n slonality (cf (4), (5)) . It is t h e n v e r y e a s y to show, u s i n g (8), t h a t in the lim i t t/l/t~ >> I, ~ u a s i t w o d i m e n s i o n a l i t y g i v e s b Y far the b e s t c o m p r o m i s e (10) for T . W e a k c o u p l i n g s are t h e n c o m p a c . t i b l e w l t h h i g h T . F o r i n s t a n c e , for La^ Sr CuO. , t~e m a x i m u m v a l u e T c Z ~ X 4 0 ~ i s 4 e ~ s i l y e x p l a i n e d w i t h (11) depending in (5).
= V / t ~ ~ 0.4 on the e x a c t v a l u e
taken
for
3. P H O N O N S O R E L E C T R O N S ? I w o u l d f i n a l l y like to e x p l a i n why, at l e a s t in the w e a k c o u p l i n g limit, V seems m o r e l i k e l y to be p h o n o n m e d i a t e d t h a n due to d i r e c t e l e c t r o n - e l e c t r o n interactions. If we c o m p a r e the p h a s e d i a g r a m s of the n e w o x y d e s w i t h t h o s e of the o r g a n i c superconductors, we f i n d in b o t h c a s e s
J, Friedel / Quasi-low-dimensionality in the weak coupling limit
a n t i f e r r o m a g n e t i c s p h a s e s (56),(57). This was a p o w e r f u l i m p e t u s to try and e x t e n d to the o x y d e s the e l e c t r o n med i a t e d s c h e e m e s that seem to a p p l y to the o r g a n i c c o m p o u n d s . The p h a s e diag r a m s are h o w e v e r v e r y d i f f e r e n t : for the o x y d e s , t h e r e is no t e m p e r a t u r e range of c o e x i s t e n c e of the two phases, but on the c o n t r a r y T and T. v a n i s h at or b e f o r e any c o n t a c ~ b e t w e eN n the two phases. A n t i f e r r o m a g n e t i c f l u c t u a t i o n s seem also to d e c r e a s e s h a r p l y in range, i n t e n s i t y and t e m p e r a t u r e , o u t s i d e the d o m a i n of s t a b i l i t y of the t h r e e d i m e n sional a n t i f e r r o m a g n e t i c phase (57). Interference between antiferromagnetism and s u p e r c o n d u c t i v i t y , if any, seems t h e r e f o r e d e s t r u c t i v e , and not c o n s t r u c tive as in the o r g a n i c s u p e r c o n d u c t o r s . I n d e e d p h a s e d i a g r a m s such as those of the o x y d e s are k n o w n in t r a n s i t i o n a l alloys, w h e r e s u p e r c o n d u c t i v i t y is p h o non m e d i a t e d , and d i r e c t e l e c t r o n - e l e c tron i n t e r a c t i o n s w e a k e n s o m e w h a t T . c But, as in t h a t case, one c o u l d e x p e c t this w e a k e n i n g of T to be reduced, c owing to the d i f f e r e n c e in f r e q u e n c i e s of p h o n o n s and p l a s m o n s (58). In fact, if there was no i n t e r f e r e n ce from a n t i f e r r o m a g n e t i s m , one w o u l d expect, in La~ Sr CuO. , T to d e c r e a z-x x o v e r~-v.. se as x increases, all ~he r a n g e of s t a b i l i t y of s u p e r c o n d u c t i v i t y . This is b e c a u s e the Fermi level shifts away f r o m the V a n Hove s i n g u l a r i t y , w h i c h stays at half filled b a n d owing to the e q u a l i t y of Cu Cu b o n d s in the o r t h o r h o m bic p h a s e (19) (the w e a k Jahn T e l l e r inst a b i l i t y that w o u l d split the V a n Hove s i n g u l a r i t y by m a k i n g the Cu Cu b o n d s u n e q u a l is e a s i l y s u p p r e s s e d by the small b r o a d e n i n g of the Van Hove s i n g u l a r i t y (11)). P o s i t i v e i n t e r f e r e n c e s b e t w e e n a n t i f e r r o m a g n e t i s m and s u p e r c o n d u c t i v i t y w o u l d s t r e n g t h e n this d e c r e a s e of T w i t h i n c r e a s i n g x (21),(59). The o n l y way to e x p l a i n that T d e c r e a s e s for x d e c r e a s i n g f r o m O.15 ~s then to a s s u m e a n e g a t i v e i n t e r f e r e n c e of s u p e r c o n d u c t i v i t y w i t h the weak and short range a n t i f e r r o m a g n e t i c f l u c t u a t i o n s still e x i s t i n g in this range. This is p o s s i b l e if the s u p e r c o n d u c t i v e c o u p l i n g is m e d i a ted by the full s p e c t r u m of phonons. If now we a s s u m e a p h o n o n m e d i a t e d ~ n t e r a c t i o n V, low but non v a n i s h i n g v a l u e s of the i s o t o p e e f f e c t find a natural e x p l a n a t i o n . It was p o i n t e d out by L a b b 4 (I) that, in the w e a k c o u p l i n g limit, the p r e s e n c e of a d i v e r g i n g V a n Hove a n o m a l y n e a r the F e r m i level s h o u l d s t r o n g l y r e d u c e the i s o t o p e effect. The r e a s o n is that the c u t o f f o f f e r e d by the
1613
steep d e c r e a s e of the d e n s i t y of s t a t e s w i t h e n e r g y b e c o m e s m o r e e f f e c t i v e than the c u t o f f by the D e b y e f r e q u e n c y w D. I n d e e d e s t i m a t e s (9) and (IO) do not contain ~_ at all ( 1 0 ) (11). M o r e r e f i n e d e s t i m a t e s (I) , (IO),~12), (14) i n t r o d u c e some c o r r e c t i o n s w h i c h for p h o n o n m e d i a ted V ' s , w o u l d g i v e a small i s o t o p e effect. M o r e p r e c i s e l y (60), it is l a r g e r for p l a n e s than for chains, and i n c r e a ses c o n t i n u o u s l y t o w a r d s the c l a s s i c a l v a l u e w h e n the F e r m i level s h i f t s away f r o m the V a n H o v e s i n g u l a r i t y by m o r e than s e v e r a l Aj;s. Then in La^ Sr CuO_ , for x ~ 0.15, z~x x 4-y w h e r e the F e r m l level Is at a d l s t a n c e f r o m the V a n Hove s i n g u l a r i t y , one exp e c t s (11), as o b s e r v e d (61),a small but s i z e a b l e i s o t o p e effect, w h i c h s h o u l d d e c r e a s e w i t h d e c r e a s i n g x. In Y B a 2 C u ~ O _ w h e r e the C u O c h a i n s p r o b a b l y take soNe/' of the e l e c t r o n s of the o t h e r w i s e half f i l l e d d b a n d of the C u O ~ planes, one e x p e c t s the Fermi level £o be b o t h n e a r the s i n g u l a r i t y of the c h a i n s and v e r y n e a r the lower half of the s i n g u l a r i t y of the CuO^ planes, split in the o r t h o r h o m b i c p h a s e (9), (11),(14). Thus c o n d i t i o n s for lower i s o t o p e e f f e c t (62) and for h i g h e r T (63) w o u l d be fullc filled. CONCLUSIONS. The c l a s s i c a l a p p r o a c h s k e t c h e d h e r e seems to o f f e r a f a i r l y c o h e r e n t p i c t u re of the n e w o x y d e s u p e r c o n d u c t o r s . It s u g g e s t s at least two w a y s of i m p r o v i n g T : - C b e t t e r 'super s a n d w i c h e s ' , r e s p e c t i n g the f u n d a m e n t a l q u a s i t w o d i m e n s i o n a l char a c t e r and a F e r m i level near to V a n H o v e s i n g u l a r i t i e s , but c o n t a i n i n g a h i g h e r p r o p o r t i o n of c h a i n s perhaps. - b e t t e r c o m p o u n d s w i t h l a r g e r t,~s and thus l a r g e r A,. A l o n g this s e c o n ~ line, the c o u p l e C u O is a l r e a d y f a v o u r a b l e . But o t h e r couples, u s i n g S, Se or Te or Ag, thus s m a l l e r r e l a t i v e s t a b i l i t i e s of the d and p states, w i t h s i m i l a r t o p o l o g i e s , m i g h t be of interest. There are a number of experimental difficulties in this scheme, which should be checked on single crystals. F i n a l l y , e v e n if c o n f i r m e d by f u r t h e r e x p e r i m e n t s , e x p e c i a l l y on fermi or a n g u l a r d e p e n d e n t p h o t o e m i s s i o n , this a p p r o a c h c o u l d be improved. Thus the a s s u m e d r a t i o s of U / w and V / w are s i z e a ble. M o r e r e f i n e d t r e a t m e n t s of antif e r r o m a g n e t i s m w o u l d take into a c c o u n t h i g h e r p o w e r s (53) of U or use v a r i a t i o nal m e t h o d s (65) ; s c a l i n g m e t h o d s w o u l d be in o r d e r for s u p e r c o n d u c t i v i t y (cf
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J. Friedel / Quasi-low-dimensionality in the weak coupling limit
(4) , (8) , (21) . .) . C r u d e e s t i m a t e s of meanfield T s h o u l d b e c o r r e c t e d to inc . c l u d e c o r r e c t l o n s d u e to t h e D e b y e c u t o f f (59). D e v i a t i o n s f r o m r i g i d b a n d a p p r o x i m a t i o n s h o u l d de t a k e n i n t o a c c o u n t i n d o p i n g , e s p e c i a l l y for s m a l l c o n c e n t r a t i o n s in a n t i f e r r o m a g n e t i c La. Sr Z--X ,X CuO• o r for t h e e f f e c t o f 0 v a c a n c i e s
(11~? y F i n a l l y it is n o t c l e a r w h e t h e r s t r o n g interplane phase fluctuations (solitons) h a v e b e e n i n c l u d e d in t r e a t i n g t h e XY m a g n e t i c c o u p l i n g e x p e c t e d in t h e a n t i f e r r o m a g n e t i c p h a s e (cf A p p e n d i x B) a n d the nearly twodimensional superconduct i v i t y . T h e y m i g h t l o w e r T. a n d T from s t a n d a r d t h e o r i e s (66), t h u s i n c r e a s e in e q u a t i o n (5) f r o m v a l u e s n e a r to u n i t y , a n d e x p l a i n the s t r o n g f l u c t u a t i o n s o b s e r v e d w e l l a b o v e T N in t h e a n t i f e r r o m a g n e t i c c a s e (57). A P P E N D I X A. Z I G Z A G G I N G V O R T E X L I N E S A POLYCRYSTAL WITH RANDOM TEXTURE•
IN
A v o r t e x l i n e c a n l o w e r its e n e r g y b y z i g z a g g i n g if t h e l i n e t e n s i o n 6t gained by passing through grains with p l a n e s p a r a l l e l to the v o r t e x is l a r g e r than the_energy lost against the line tension ~ by zigzagging. L e t L b e t h e a v e r a g e l e n g t h of a zigzag b e t w e e n t w o ' f a v o u r a b l e ' g r a i n s , m e a s u r e d a l o n g the g e n e r a l d i r e c t i o n of t h e v o r t e x , H the a v e r a g e h e i g h t of the zigzag and D the average grain diameter. T h e c h a n g e AT in a v e r a g e l i n e t e n s i o n d u e to z i g z a g g i n g is s u c h t h a t (L+D)AT = ( / ~ z ~ 2 - L ) ~ D6T. (A.I) The favourable grains must have their p l a n e s l y i n g w i t h i n t h e a n g l e 0 of t h e zigzag. Thus D 3 . L H 2 ~ 2n s i n e (A.2) with tg 8 m H/L. If e w a s small, o n e c o u l d w r i t e I
H 2
D
with D 3
2~0 ~ 2~ H LH 2 L Hence H ~ (2~) - 1 1~3 D.
(A.4)
Optimizing 9~with H / L ~ (2~) " J 6 T / ~ The pinning energy is t h e n 2 L A~ ~ D 6T
respect per
In f a c t ~-- ~ 2 - -
• ~ ?'~"
to L g i v e s (A.5) favourable grain (A.6) is n o t
a small
q u a n t i t y . T h e e q u l ± i D r i u m 0 is l a r g e a n d s h o u l d be d e d u c e d d i r e c t l y f r o m (A.I) to (A.3) . B u t r e s u l t s w o u l d b e s i m i l a r , l e a d i n g to s t r o n g a n d s h o r t z i g z a g s (H ~ L m D) a n d p i n n i n g e n e r g i e s of o n e
zigzag
of o r d e r
D
6 t.
A P P E N D I X B. X Y M A G N E T I S M TIVE OXYDES.
IN S U P E R C O N D U C -
T h e m a g n e t i c a n i s o t r o p y of C u 0 9 p l a nes i n v o l v e s v i r t u a l e x c i t a t i o n s ~f Cu 3d e l e c t r o n s f r o m r o u g h l y h a l f f i l l e d x 2 - y 2 s t a t e s to o t h e r f u l l 3d s t a t e s . T h i s is i n d e p e n d e n t o n w h e t h e r the m a g n e t i c e l e c t r o n s of C u a r e l o c a l i s e d or delocalised. A c l a s s i c a l a n a l y s i s (67) s h o w s t h a t t h e a n i s o t r o p y is m a i n l y a x i a l w i t h r e s p e c t to t h e n o r m a l z to t h e C u O 9 p l a n e s • Thus, to s e c o n d o r d e r in s p i n o ~ b i t c o u p l i n g I Z s, t h e d i f f e r e n c e in e n e r g y b e t w e e n an e l e c t r o n s p i n o r i e n t e d a l o n g z o r n o r m a l to z is E z - Exy=
~-~2
L
Eo_E2
]
Eo_E3
- E^ a n d E - E. a r e r e s p e c t i v e l y ~e l a t t i c e s ~ l i t t ~ n g s of t h e x 2 - y 2 s t a te w i t h the x y a n d x z o r y z ones. W i t h E -E^ < E -E~ < 20 eV, a n d I ~ 0 . 0 7 eV, .o. z " o ~ " .. t n l s e n e r g y is n e c e s s a r 1 ± y p o s i t i v e ; w i t h i n a f l u c t u a t i o n a r e a ~2, it is l a r g e c o m p a r e d w i t h t h e r m a l e n e r g y u p to at least room temperature (57). A n y a n i s o t { o p y in t h e X Y p l a n e is d u e to t e r m s in I~ w i c h a r e o b v i o u s l y n e g l i g i b l e at room temperature• REFERENCES (I) J. L a b b 4 , S. B a r i s i c a n d J. F r i e d e l , Phys. Rev. L e t t . V o i . 1 9 , 1039 (1967)• (2) S. B a r i s i c a n d J. L a b b 4 , J. Phys. Chem. Sol. V o i . 2 8 , 2477 (1967) ; J. L a b b 4 , P h y s . Rev. Vol. 172, 45 (1968). (3) D. J 4 r o m e , A. M a z a u d , M. R i b a u l t a n d K. B e c h g a a r d , C o m p t e R e n d u s v o i . 2 9 0 B, 27 (1980) • (4) D. J 4 r o m e a n d H.J. S c h u l z , Adv. P h y s . V o i . 3 1 , 229 (1982). (5) J. F r i e d e l , Phil. T r a n s . Roy. Soc. vol. A 3 1 4 , 189 (1985). (6) F. C r e u z e t , D. J 4 r o m e , K. B e c h g a a r d a n d A. M o r a d p o u r , J. P h y s i q u e vol. 45, L 7 5 5 (1984). (7) F. C r e u z e t , T h ~ s e O r s a y (1987). (8) C. B o u r b o n n a i s a n d L.G. C a r o n , E u r o p h y s . L e t t . vol. 5, 209 (1988) a n d references therein• (9) J. F r i e d e l , C o m p t e s R e n d u s A c a d . Sci. vol. 305 II, 543 (1987). (10)J. F r i e d e l , J. P h y s i q u e , v o i . 4 8 , 1787 (1987), a n d E r r a t u m , ibid, u n d e r p r e s s (1988). (11)J. F r i e d e l , J. P h y s i q u e u n d e r p r e s s (1988) . (12) J. L a b b 4 a n d J. Bok, E u r o p h y s . L e t t . v o l . 3 , 1225 (1987).
J. Friedel / Quasi-low-dimensionality in the weak coupling limit
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(35) A. Umezawa, G.W. Crabtree, J.Z. Liu, T.J. Moran, S.K. ~alik, L.H. Numez, W.L. Kwok and C.H. Sowers, Phys. Rev. Lett. under press (1988). (36) J. Friedel, D i s l o c a t i o n s , P e r g a m o n L o n d o n (1964). (37) cf for i n s t a n c e M. Oussena, S. S e n o u s s i and G. Collin, Europhys. Lett. vol. 4, 625 (1987). (38) B. Farnoux, R. Kahn, A. Brule, G. C o l l i n and J.P. Pouget, J. P h y s i q u e vol. 48, 1623 (1987). (39) H. Schmidt, Z P h y s i k voi.216, 336 (1968). (40) L.F. Mattheiss, Phys. Rev. Lett. vol. 58, 1028 (1987). (41) A. Samsavai, T. Miller, T.C.Chiang, B.G. Pazol, T.A. F r i e d m a n n and D. M. Ginzberg, to be published. (42) T. Gourieux, G. Krill, M. Maurer, M.F. Ravet, A. Menney, H. T a l e n t i n o and A. Fontaine, Phys. Rev. B under press (1988). (43) A.G. Schrott, R. Park and C . C . T s u e i to be p u b l i s h e d ; P. Thiry, G.Rossi, Y. Petroff, A. R e v c h o l e v s k i and J. Jegondez, Europhys. Lett. vol.5, (1988). (44) A. Balzarotti, M. de Crescenzi, C. Gionella, P. Messi, N. Motta, F. P a t e l l a and A. S p a r l a t a under press. (45) H.J. Schulz, this conference. (46) G.A. Sawatsky, priv. comm. (47) S. Uchida, H. Takagi, H. Ishii, H. Eisaki, T. Yab4, S. Tojima and S. Tanaka, Jap. J. Appl. Phys. voi.26, L 440 (1987). (48) Y. Petroff, P. Thiry, G. Rossi, A. R e v c h o l e v s k i and J. Jegondez, A d r i a t i c o Conf. on High T supercond. T r i e s t e (1987). c (49) H.W. Zandbergen, R . G r o n s k y and G. Thomas, under press (1988). (50) C. Ayache, B. Barbara, B. Bonjour, P. Burlet, R. Calemczuk, M. Coupch, M.J.G. Jugens, J.Y. Henry and J. Rossat Mignot, u n d e r press. (51) K. Kumagai, Y. Nakamichi, I.Watanabe, Y. Nakamura, H. Nakajima, N. Wada, and P. Lederer, Techn. Rep. ISSP, ISSN ISSN 0 0 8 2 - 4 7 9 8 A, 1890 (1988). (52) F.J. W a l k e r and A.C. Anderson, Phys. Rev. Vol. 29, 588 (1984). (53) J. Friedel and C.M. Sayers, J. Phys. Lett.vol. 39, L59 (1978). (54) L. Hoffmann, A.A. Manuel, M. Peter, E. W a l k e r and M.A. Damento, Europhys. Lett. under press (1988) ; L. S m e d s k j a e r to be p u b l i s h e d (1988). (55) L. Van Hove, Phys. Rev. voi.89, 1189 (1953). -
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(57
(58) (59) (60) (61)
(62)
(63)
(64) (65)
(66)
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J. Friedel / Quasi-low-dimensionality in the weak coupling limit
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