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Superconductivity-induced softening of phonons in multilayered CuO2 superconductors
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Solid State Comnunications, Vol. 76, No. 2, pp. 145-148, 1990. Printed in Great Britain.
0038-i098/9053.00+.00 Pergamon Press plc
SUPERCONDUCTIVITY-INDUCED SOF1"~ING OF PHONONS IN HULTILAYERED CuO2 SUPERCONDUCTORS Fusayoshi J. OHIqAWA Department o f P h y s i c s , Hokkaido U n i v e r s i t y , Sapporo 060, Japan ( R e c e i v e d 28 June 1990 by ¥. S a s a k i ) The s o f t e n i n g of phonons can be d r i v e n by the a p p e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e in a s u p e r c o n d u c t i n g s t a t e . Such a t r a n s i t i o n of the second o r d e r i s p o s s i b l e in m u l t i l a y e r e d CuO~ s u p e r c o n d u c t o r s . The s o f t mode must be an o p t i c a l mode which v i b r a t e B CuO~ p l a n e s out o f p h a s e . The s o f t e n i n g can e x p l a i n a t i n y but s h a r p anomaly of the t r a n s v e r s e n u c l e a r q u a d r u p o l e r e l a x a t i o n r a t e around T=35K In d o u b l e l a y e r e d YBa_Cu307 ° w i t h c r i t i c a l t e m p e r a t u r e T •92K. However, the p r e s e n t t h e o r y p r e d i c t s the a b s e n c e of anomalie~ in s i n g l e - l a y e r e d compounds such as La2_xSrxCuO 4.
S i n c e t h e d i s c o v e r y of h i g h - T s u p e r c o n d u c t o r s w i t h q u a s i - t w o - d i m e n s i o n a l C u ~ layers, a lot of efforts have been made in oraer to elucidate their mechanism. It is a point of i s s u e w h e t h e r h i g h - T c s u p e r c o n d u c t i v i t y (SC) i s c o n v e n t i o n a l BCS SC, a n i s o t r o p i c SC s i m i l a r to s u p e r f l u i d i t y of l i q u i d 3He, 1 or e x o t i c SC such as SC o r i g i n a t i n g from f r a c t i o n a l s t a t i s t i c s and 2 so on. Measurements of the p e n e t r a t i o n d e p t h 3 and t h e t u n n e l i n g c u r r e n t s 4 seem to s u p p o r t c o n v e n t i o n a l BCS SC. They s u g g e s t t h a t the gap i s open e v e r y w h e r e on t h e Fermi s u r f a c e a t low t e m p e r a t u r e s . On the o t h e r hand, t h e r e a r e a l s o measurements which seem to s u p p o r t a n i s o t r o p i c SC. Only small enhancement j u s t below t h e c r i t i c a l t e m p e r a t u r e and the T3-dependence in a wide t e m p e r a t u r e r a n g e of the n u c l e a r m a g n e t i c r e l a x a t i o n r a t e , or the n u c l e a r quadrupole r e l a x a t i o n (NQR) r a t e 5 s u g g e s t t h a t a n i s o t r o p i c SC i s r e a l i z e d , and t h a t the gap v a n i s h e s in c e r t a i n p o r t i o n s on the Fermi s u r f a c e . E l e c t r o n i c Ran,an s c a t t e r i n g s 6 and n e u t r o n inelastic scatterings 7 also suggest that finite d e n s i t y of s t a t e s r e m a i n s w i t h i n the r e g i o n of the BCS gap. R e c e n t l y , two i n t e r e s t i n g o b s e r v a t i o n s in RBa~Cu~07_ A compounds have been r e p o r t e d , b e i n g On~ ~f r a r e - e a r t h i o n s . They become s u p e r c o n d u c t i n g around T=92K. However t h e i r lower c r i t i c a l f i e l d shows a l a r g e kink around 8 T=35K, which s u g g e s t s t h a t a n o t h e r s u p e r c o n d u c t i n g s t a t e shows up below 35K. The t r a n s v e r s e NQR I/T~ a l s o shows a t i n y but s h a r p ~,lO peak around T=35K; a peak r a t e a t T=35K i s ( l / T 2 ) ~ 2 . 0 x l O - 4 / s e c on t h e back ground r a t e 0 . 7 X l O - 4 / s e c . 9 No h y s t e r e s i s i s o b s e r v e d . Although small sample d e p e n d e n c e i s o b s e r v e d , the anomaly can be s e e n in any RBa2Cu~O7 compounds, f o r example, b o t h in F e - d o ~ e d - t n d undoped s a m p l e s . However no a n o m a l i e s can be found around 35K in t h e l o n g i t u d i n a l NQR I/T 1. Although t h e r e may be many p o s s i b l e
e x p l a n a t i o n s of the above e x p e r i m e n t a l r e s u l t s , one of the s i m p l e s t e x p l a n a t i o n s i s t h a t the gap v a n i s h e s a t high t e m p e r a t u r e s , w h i l e the gap i s open everywhere a t low t e m p e r a t u r e s as in s u p e r f l u i d i t y of l i q u i d 3He; s u p e r c o n d u c t i n g s t a t e s a r e d i f f e r e n t between above and below 35K. Two mechanisms of g a p - o p e n i n g have been p r o p o s e d a l o n g such a s i m p l e i d e a . A mechanism can o n l y be a p p l i e d to t e t r a g o n a l l a t t i c e s . The h i g h temperature phase is a superconducting s t a t e w i t h the d-symmetry, and t h a t the l o w - t e m p e r a t u r e phase i s the s o - c a l l e d s+id s t a t e , which i s the c o e x i s t e n c e between the s-symmetry and the d-symmetry whose p h a s e s a r e d i f f e r e n t by ~n/2.11 However the s + i d s t a t e seems to be d i f f i c u l t to be r e a l i z e d , b e c a u s e a c t u a l CuO2 s u p e r c o n d u c t o r s are orthorhombic. The o t h e r mechanism can o n l y be a p p l i e d to h i g h - T c s u p e r c o n d u c t o r s w i t h m u l t i l a y e r s of CuO2 12 in u n i t c e l l s . Because one of the most c h a r a c t e r i s t i c f e a t u r e s of CuO~ s u p e r c o n d u c t o r s i s t h e i r weak i n t e r l a y e r c o u p l i n g , the c r i t i c a l t e m p e r a t u r e s must be n e a r l y d e g e n e r a t e between d i f f e r e n t t y p e s of s u p e r c o n d u c t i n g s t a t e s a s s o c i a t e d with l a y e r s . Therefore t h e i r c o e x i s t e n c e i s r a t h e r e a s y to be r e a l i z e d . One of the p u r p o s e s of t h i s p a p e r i s to show t h a t the s o f t e n i n g of phonons can be d r i v e n by the app e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e in a s u p e r c o n d u c t i n g s t a t e , or by an SC-SC transition. The s o f t mode i s an o p t i c a l mode which v i b r a t e s e l e c t r o n s in the CuO9 l a y e r s out of p h a s e ; i t makes d i f f e r e n t s u p e r c o n d u c t i n g o r d e r p a r a m e t e r s c o u p l e d . Then i t i s well known t h a t the s o f t e n i n g c a u s e s the d i v e r g e n c e of the t r a n s v e r s e NQR 1/T~, but no d i v e r g e n c e of the l o n g i t u d i n a l NQR 17T/. D o u b l e - l a y e r e d CuO~ s t r u c t u r e s a r e examined h e r e as the s i m p l e s t c a n e , b e c a u s e they i n c l u d e e s s e n t i a l f e a t u r e s of m u l t i - l a y e r e d s t r u c t u r e s . i t i s s t r a i g h t f o r w a r d to e x t e n d the p r e s e n t t h e o r y to m u l t i l a y e r e d s t r u c t u r e s with more than two l a y e r s . Let us examine a two-band Hubbard model, which i s one of the models of t w o - l a y e r e d CuO2 s u p e r c o n d u c t o r s :
145
146
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
~ = - t ~ ~ at a - V ~ (aTi~ t .._ nip nip oa2io+a2ioaliO ) n mjo iO T " 21-'U ~ ~ o ~ i o a n i o a n i - o a n i -
o
t a " ( b t + b ) + 8wbtb , ( l ) + W ~ ( a l i O a l i o - a 2 i o " 2iO) io w i t h a T . _ a c r e a t i o n o p e r a t o r of e l e c t r o n w i t h l a y e r ~m~l o r 2 ) , s i t e i , and s p i n O; t i n t r a l a y e r t r a n s f e r i n t e g r a l s , which a r e assumed o n l y between n e a r e s t n e i g h b o r s ; and V i n t e r l a y e r transfer integrals. The p r e s e n t s y s t e m can be c h a r a c t e r i z e d by i t s weak i n t e r l a y e r c o u p l i n g s such as Iv/tl<
~ (aTi ~ + a2i ~) ~ tl i O = ~/~
x~ t h e d i s t o r t i o n of l a t t i c e ; p>O, q>O, r>O, d>O, g>O, T ~>O, and AT>O w i t h o u t l o s s of generality. BecaUse g / d ~ ( V / t ) ( W / t ) and AT/T O ~ ( V / t ) 2 f o r s m a l l V / t , i t can be r e a s o n a b l y assumed t h a t g / d < < l and AT/T ~<<1 in the p r e s e n t s y s t e m , which i s c h a r a c t e r i z e d by i t s weak i n t e r l a y e r c o u p l i n g s ( ~ V / t l < < l ) . The phonons m o d u l a t i n g e l e c t r o n s in the two l a y e r s out of p h a s e make the two o r d e r p a r a m e t e r s P! and P~ c o u p l e d . Here i t s h o u l d be n o t e d t h a t Pl and P~ must b e l o n g to the same r e p r e s e n t a t i o n with r e s p e c t to the symmetry o p e r a t i o n s w i t h i n layers. The most p r o b a b l e one i s the d-symmetry in t e t r a g o n a l l a t t i c e s , because antiferromagn e t i c s u p e r e x c h a n g e i n t e r a c t i o n s between n e a r e s t - n e i g h b o r s f a v o r s t h e d-symmetry. On the o t h e r hand, t h e most p r o b a b l e one i s a mixed s t a t e of t h e s-symmetry and t h e d-symmetry in o r t h o r h o m b i c l a t t i c e s , b e c a u s e they s h o u l d be inevitably coupled. Because x e x p r e s s e s a d i s t o r t i o n of odd p a r i t y , i t makes Pl and P2 c o u p l e d ; dx2/2 is an e l a s t i c e n e r g y . I n f i n i t e s i m a l l y small e x t e r n a l s t r e s s , h, i s i n t r o d u c e d to c a l c u l a t e t h e s u s c e p t i b i l i t y d e f i n e d by x = [ d x / d h ] ~ ; i t s d i v e r g e n c e means the s o f t e n i n ~ of the ~ n o n . The f i r s t s u p e r c o n d u c t i n g t r a n s i t i o n o c c u r s at
,
Tcl=
Tc0 + AT ,
~ - (aTi a _ a2i T ~) . ~T 2i0 = ~
(3)
I t has a l r e a d y been p o i n t e d out t h a t the ant i f e r r o m a g n e t i c s u p e r e x c h a n g e i n t e r a c t i o n of t h e o r d e r of I t l 2 / U can work as a t t r a c t i v e i n t e r a c t i o n s between s i n g l e t Cooper p a i r s in s t r o n g l y correlated
systems ( [ t l / U < < l ) .
t h r e e t y p e s of s i n g l e t possible:
13
Therefore
Cooper p a i r s a r e
1, ~ t t t . -~ -" ~ n i t ~ n j . ~ - ~ n i ~ n j .t;>
PI2 = ( p / q ) ( T c l - T) ,
and P_=x=O below T . . The jump of the s p e c i f i c h e a t ~t Tcl i s g i v ~ by 1( 2 , "T ACt = 2 P /q~ c l
t
2tali?ff.2j~-~li~O~_jT)> , between a b o n d i n g e l e c t r o n
(5)
> 0 ,
are satisfied. I t can be seen t h a t i n t e r l a y e r coupling constant g enhances Tc,. Below the o r d e r p a r a m e t e r s a r e g~ven ~y To2'
and an a n t i b o n d i n g
2 _ P P2 q+r_g2/d
4
1 ^2~2
(ll)
q+r-g2/d
i n d e p e n d e n t . 12 Thus the f o l l o w i n g Landau f r e e energy is obtained:
4
(10)
q-r+g2/d =
2 Pl
1
(9)
i f b o t h Tc2>O and
e l e c t r o n , and they are odd w i t h r e s p e c t to the operation. A m i c r o s c o p i c c a l c u l a t i o n shows t h a t two types of i n t r a b a n d Cooper p a i r s a r e not
F(T) = ~p(T-Tco-AT)P 1 +
"
To2 = Tco - AT/~ ,
hand, the t h i r d ones a r e i n t e r b a n d Cooper p a i r s g i v e n by t
(8)
(4)
'
( n = t ) , or between a n t i b o n d i n g e l e c t r o n s (n=2). They are i n t r a b a n d Cooper p a i r s , and they a r e even w i t h r e s p e c t to the o p e r a t i o n . Here < - - - > stands f o r the thermal a v e r a g e . On the o t h e r
t
to be f i n i t e ;
There i s no anomaly of t h e s u s c e p t i b i l i t y x at c Tcl. On the o t h e r hand, the second t r a n s i t i o n occurs at
a r e Cooper p a i r s between b o n d i n g e l e c t r o n s
1, t
(7)
(2)
where o n l y Pl s t a r t s and
VO1. 76, NO. 2
P(T-Tc0+AT) 1
+ 4-q(Pl+P2 ) + ~ r P l P 2 - gPlP2 x + ~dx
2
- hx , (6)
in the gauge i n v a r i a n t form, w i t h Pl the o r d e r p a r a m e t e r o f even p a r i t y , P2 of odd p a r i t y , and
-
P q+r_g2/d
(T
u
(~2
-
T)
- T) ,
(12)
(13)
and x = ( g / d ) P . P _ w i t h T =T A*(AT/~). Not o n l y c two s u p e r c o n d u c t i n g o r d e r ~ a r a m e t e r s c o e x i s t below T c . , but t h e r e a l s o e x i s t s f i n i t e l a t t i c e distortion; two CuP 2 p l a n e s a r e d i s t o r t e d o u t of phase. Because the f r e e - e n e r g y g a i n i s l a r g e r , o t h e r o r d e r p a r a m e t e r s P2 and x can show up
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
VOI. 76, No. 2
below T_, in the p r e s e n c e of P , . T h e r e f o r e the thermodynamic c r i t i c a l f i e l d , ~ h i c h i s p r o p o r t i o n a l to [ - F ( T ) ] I / 2 , s h o u l d show a r a p i d e r i n c r e a s e w i t h d e c r e a s i n g t e m p e r a t u r e s below T_, than above T_,. The jump of t h e s p e c i f i c hea~ L a t Tc2 i s c a ~ u l a t e d as AC2 = ~(Tc2/Tcl)AC 1 . The s u s c e p t i b i l i t y
×c
(14)
shows d i v e r g e n c e a t Tc2 as
= 5(g_.g._)2(q/p)(q+r_g?/d) ~ 1 ~ d T-Tc2
+ ---,
(15)
above Tc2, and
Xc
l = -1( g 12 ( q / 2 p ) ( q + r _ g 2 / d ) ~ ~ d Tc2-T
+ ---,
(16)
below T~ .~ Because t h e d i v e r g e n c e of the s u s c e p t i b i Y ~ t y means t h a t t h e r e s t o r i n g f o r c e v a n i s h e s , i t means n o t h i n g but the s o f t e n i n g of the phonon a t T ~. I t s h o u l d be n o t e d t h a t the s o f t e n i n g i s d r ~ e n by the a p p e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e P2" I t i s o b v i o u s t h a t the s o f t e n i n g can c a u s e an anomaly of the t r a n s v e r s e NQR I / T ~ . Energyl e v e l s of n u c l e a r s p i n s w i t h I=3/2 oT Cu i o n s a r e s p l i t o f f i n t o l Z =~1/2 and [ =±3/2 in CuO~ s u p e r c o n d u c t o r s , and t h e i r q u a d r ~ p o l e s p l i t t i n g s h o u l d depend on t h e l a t t i c e d i s t o r t i o n , x. In the p r e s e n t model, the q v e c t o r of the s o f t mode i s assumed as Q_=(0,O,Q ) w i t h Q =0. By assuming Qn, t h e dynamical s ~ s c e p t i b i ~ i t y can be w r i t t e n p h e n o m e n o l o g i c a l l y as 1
g
×c(iW,q) =
(
L
~
,
d
K(q-Qo)2+~+iw
w i t h K and L c o n s t a n t s ,
and E~IT-Tc2 I:
(17)
I T2
w~O (g/d) 2 A -
I
-
(iS)
}T-Tc2 11/2
'
with A quadrupole coupling constant. Because the s o f t mode i s odd w i t h r e s p e c t to the symmotry o p e r a t i o n of c h a n g i n g l a y e r s , no anomaly can be e x p e c t e d in the NQR l / T 2 of Cu i o n s in the s o - c a l l e d c h a i n s i t e s of RBa_Cu307 . . It is a l s o o b v i o u s t h a t t h e s o f t e n i n g ~an n e v e r c a u s e any d i v e r g e n c e of t h e l o n g i t u d i n a l I~R I/T l . So f a r , i t has been assumed t h a t no d m s t o r t i o n o f l a t t i c e e x i s t s in t h e normal s t a t e , and t h a t the two l a y e r s a r e e q u i v a l e n t . However, when t h e y a r e not e q u i v a l e n t , t h a t i s , x#0, in t h e normal s t a t e , the second t r a n s i t i o n a t T . i s r e d u c e d to a c r o s s o v e r . The two o r d e r p ~ a m e t e r s Pl and P~ c o e x i s t even j u s t below the f i r s t c r n t i c a l ~ e m p e r a t u r e T , , and P, and x become r a p i d l y l a r g e around T , . ~ l n this~case, a s m a l l e r anomaly i s e x p e c t e d ~ r o u n d Tc2 even f o r Cu ions in the c h a i n s i t e s . I f the s o f t e n i n g of t h e phonon i s r e a l l y
147
r e a l i z e d in RBa.Cu307 ., many a n o m a l i e s can be e x p e c t e d such a~ a jum~ of the s p e c i f i c h e a t , the l a t t i c e d i s t o r t i o n and so on. However no o t h e r a n o m a l i e s have been r e p o r t e d so f a r b e s i d e s a t i n y but s h a r p peak of the NQR I/T~ and a k i n k of the lower c r i t i c a l f i e l d . ThereTore the p o s s i b i l i t y of o b s e r v i n g o t h e r a n o m a l i e s s h o u l d be c r i t i c a l l y examined. Because of weak i n t e r l a y e r c o u p l i n g s , ~T/Tco(
148
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
to s e t t l e e x p e r i m e n t a l l y whether the second t r a n s i t i o n e x i s t s or not in La2_xSrxCUO4.
F i n a l l y , a p r e d i c t i o n made e l s e w h e r e 12 s h o u l d be mentioned t h a t the gap can s t a r t to open e v e r y w h e r e on the Fermi s u r f a c e below T , o n l y in m u l t i l a y e r e d CuO~ s t r u c t u r e s l a r g e l y c~ d i s t o r t e d from the t e t r a R o n a l symmetry such as RBapCu207 6" T h e r e f o r e i t has been p r e d i c t e d t h a t b ~ h b ~ l o r s of a n i s o t r o p i c SC s h o u l d c o n t i n u e down to t h e zero K e l v i n in m u l t i l a y e r e d CuO~ s t r u c t u r e s with the t e t r a g o n a l symmetry, an~ s i n g l e - l a y e r e d CuO~ s t r u c t u r e s . To summarize Yhe p r e s e n t p a p e r , i t i s shown t h a t the second t r a n s i t i o n i s p o s s i b l e in m u l t i l a y e r e d CuO~ s u p e r c o n d u c t o r s between s u p e r c o n d u c t i n g s t a l e s , which b e l o n g to d i f f e r e n t r e p r e s e n t a t i o n s w i t h r e s p e c t to a symmetry o p e r a t i o n of e x c h a n g i n g l a y e r s ; weak i n t e r l a y e r c o u p l i n g s make the second t r a n s i t i o n p o s s i b l e . The t r a n s i t i o n i n d u c e s the s o f t e n i n g o f phonons, and the
Vol. 76, NO. 2
lattice distortion in the low t e m p e r a t u r e phase. The s o f t mode is an o p t i c a l mode which v i b r a t e s CuO~ l a y e r s o u t of phase. The s o f t e n i n g can e x p T a i n an observed t i n y but sharp anomaly of the t r a n s v e r s e n u c l e a r q u a d r u p o l e r e l a x a t i o n r a t e around T=35K in d o u b l e - l a y e r e d YBa~CuqO7 with critical t e m p e r a t u r e T =92K. Alth~ugE [h~ c t r a n s i t i o n around T=35K can cause many o t h e r anomalies such as a jump of the s p e c i f i c h e a t , a k i n k of the lower c r i t i c a l f i e l d and so on, t h e i r anomalies should be weak because of weak interlayer couplings.
Acknowledgement: The a u t h o r i s t h a n k f u l to K. Kumagai f o r showing h i s r e s u l t s p r i o r to publication. He o b t a i n e d the key i d e a of t h i s work, when he s t a y e d in CNRS, Grenoble. He is v e r y g r a t e f u l to J . F l o u q u e t and C. L a c r o i x f o r their hospitality. CNRS, G r e n o b l e i s l a b o r a t o i r e a s s o c i ~ ~ l ' U n i v e r s i t e Joseph F o u r i e r de G r e n o b l e .
References 1. K. A. B r u e c k n e r et a l . Phys. Rev. 118, 1142 (1960). 2. For example, R. B. L a u g h l i n , S c i e n c e 242, 525 (1988). 3. A. T. F i o r y e t a l . Phys. Rev. L e t t . 61, 1149 (1988); L. Krusin-Elbaum et al. Phys. Bey. Lett. 62, 217 (1988); D. R. Harshman et al. Phys. Rev. 1339, 851 (1989). 4. P. J. M. van Bentum et al. Phys. Rev. B36, 843 (1987); G. A. Thomas et al. Phys. Rev. Lett. 61, 1313 (1988). 5. T. Imai et al. J. Phys. Soc. Jpn. 57 (1988), 2280; K. Ishida et al., J. Phys. Soc. Jpn. 57, 2897 (1988); W. W. Warren et al., Phys. Rev. Left. 62, 1193 (1989). 6. S. L. Cooper e t a l . Phys. Bey. B37, 5920 (1988); A. Yamanaka e t a l . Jpn. J . Appl. Phys. 27, L1902 (1988). 7. J. R o s s a t - M i g n o t e t a l , to appear in Dynamics of Magnetic F l u c t u a t i o n s in High-T c M a t e r i a l s , e d s . G. R e i t e r , P. Horsh and G. P s a l t a k i s , Plenum P r e s s , 1990.
8. J. P. S t r ~ b e l et a l . , P h y s i c a C 153-155, 1537 (1988); N. Kobayashi et a l . P h y s i c a C 153-155, 1525 (1988); T. I s h i i and T. Yamada, P h y s i c a C 159, 483 (1989). 9. M. T e l , H. T a k a i , K. Mizoguchi and K. Kume, Proc. of ISNQRS (Takayama, Aug. 1989), to be p u b l i s h e d in Z. N a t u r f o r s c h . ( a ) ; and K. Kume, P a r i t y , v o l . 5 N o . l (1990) p.49 ( i n Japanese). 10. K. Kumagai, Y. Nakamichi, Y. Nakamura, T. T a k a t s u k a and H. Nakajima, to be p u b l i s h e d in J. Phys. Soe. Jpn. 11. G. K o t l i a r , Phys. Rev. B37, 3663 (1988). 12. F. J. Ohkawa, to be p u b l i s h e d in Phys. Rev. B. 13. J. E. H i r s c h , Phys. Rev. L e t t . 54, 1317 ( ] 9 8 5 ) ; P. W. Anderson, Science 235, 1196 (1987). 14. M. Sato, et a l . , S o l i d S t a t e Comm. 76, 689 (1889).
Solid State Comnunications, Vol. 76, No. 2, pp. 145-148, 1990. Printed in Great Britain.
0038-i098/9053.00+.00 Pergamon Press plc
SUPERCONDUCTIVITY-INDUCED SOF1"~ING OF PHONONS IN HULTILAYERED CuO2 SUPERCONDUCTORS Fusayoshi J. OHIqAWA Department o f P h y s i c s , Hokkaido U n i v e r s i t y , Sapporo 060, Japan ( R e c e i v e d 28 June 1990 by ¥. S a s a k i ) The s o f t e n i n g of phonons can be d r i v e n by the a p p e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e in a s u p e r c o n d u c t i n g s t a t e . Such a t r a n s i t i o n of the second o r d e r i s p o s s i b l e in m u l t i l a y e r e d CuO~ s u p e r c o n d u c t o r s . The s o f t mode must be an o p t i c a l mode which v i b r a t e B CuO~ p l a n e s out o f p h a s e . The s o f t e n i n g can e x p l a i n a t i n y but s h a r p anomaly of the t r a n s v e r s e n u c l e a r q u a d r u p o l e r e l a x a t i o n r a t e around T=35K In d o u b l e l a y e r e d YBa_Cu307 ° w i t h c r i t i c a l t e m p e r a t u r e T •92K. However, the p r e s e n t t h e o r y p r e d i c t s the a b s e n c e of anomalie~ in s i n g l e - l a y e r e d compounds such as La2_xSrxCuO 4.
S i n c e t h e d i s c o v e r y of h i g h - T s u p e r c o n d u c t o r s w i t h q u a s i - t w o - d i m e n s i o n a l C u ~ layers, a lot of efforts have been made in oraer to elucidate their mechanism. It is a point of i s s u e w h e t h e r h i g h - T c s u p e r c o n d u c t i v i t y (SC) i s c o n v e n t i o n a l BCS SC, a n i s o t r o p i c SC s i m i l a r to s u p e r f l u i d i t y of l i q u i d 3He, 1 or e x o t i c SC such as SC o r i g i n a t i n g from f r a c t i o n a l s t a t i s t i c s and 2 so on. Measurements of the p e n e t r a t i o n d e p t h 3 and t h e t u n n e l i n g c u r r e n t s 4 seem to s u p p o r t c o n v e n t i o n a l BCS SC. They s u g g e s t t h a t the gap i s open e v e r y w h e r e on t h e Fermi s u r f a c e a t low t e m p e r a t u r e s . On the o t h e r hand, t h e r e a r e a l s o measurements which seem to s u p p o r t a n i s o t r o p i c SC. Only small enhancement j u s t below t h e c r i t i c a l t e m p e r a t u r e and the T3-dependence in a wide t e m p e r a t u r e r a n g e of the n u c l e a r m a g n e t i c r e l a x a t i o n r a t e , or the n u c l e a r quadrupole r e l a x a t i o n (NQR) r a t e 5 s u g g e s t t h a t a n i s o t r o p i c SC i s r e a l i z e d , and t h a t the gap v a n i s h e s in c e r t a i n p o r t i o n s on the Fermi s u r f a c e . E l e c t r o n i c Ran,an s c a t t e r i n g s 6 and n e u t r o n inelastic scatterings 7 also suggest that finite d e n s i t y of s t a t e s r e m a i n s w i t h i n the r e g i o n of the BCS gap. R e c e n t l y , two i n t e r e s t i n g o b s e r v a t i o n s in RBa~Cu~07_ A compounds have been r e p o r t e d , b e i n g On~ ~f r a r e - e a r t h i o n s . They become s u p e r c o n d u c t i n g around T=92K. However t h e i r lower c r i t i c a l f i e l d shows a l a r g e kink around 8 T=35K, which s u g g e s t s t h a t a n o t h e r s u p e r c o n d u c t i n g s t a t e shows up below 35K. The t r a n s v e r s e NQR I/T~ a l s o shows a t i n y but s h a r p ~,lO peak around T=35K; a peak r a t e a t T=35K i s ( l / T 2 ) ~ 2 . 0 x l O - 4 / s e c on t h e back ground r a t e 0 . 7 X l O - 4 / s e c . 9 No h y s t e r e s i s i s o b s e r v e d . Although small sample d e p e n d e n c e i s o b s e r v e d , the anomaly can be s e e n in any RBa2Cu~O7 compounds, f o r example, b o t h in F e - d o ~ e d - t n d undoped s a m p l e s . However no a n o m a l i e s can be found around 35K in t h e l o n g i t u d i n a l NQR I/T 1. Although t h e r e may be many p o s s i b l e
e x p l a n a t i o n s of the above e x p e r i m e n t a l r e s u l t s , one of the s i m p l e s t e x p l a n a t i o n s i s t h a t the gap v a n i s h e s a t high t e m p e r a t u r e s , w h i l e the gap i s open everywhere a t low t e m p e r a t u r e s as in s u p e r f l u i d i t y of l i q u i d 3He; s u p e r c o n d u c t i n g s t a t e s a r e d i f f e r e n t between above and below 35K. Two mechanisms of g a p - o p e n i n g have been p r o p o s e d a l o n g such a s i m p l e i d e a . A mechanism can o n l y be a p p l i e d to t e t r a g o n a l l a t t i c e s . The h i g h temperature phase is a superconducting s t a t e w i t h the d-symmetry, and t h a t the l o w - t e m p e r a t u r e phase i s the s o - c a l l e d s+id s t a t e , which i s the c o e x i s t e n c e between the s-symmetry and the d-symmetry whose p h a s e s a r e d i f f e r e n t by ~n/2.11 However the s + i d s t a t e seems to be d i f f i c u l t to be r e a l i z e d , b e c a u s e a c t u a l CuO2 s u p e r c o n d u c t o r s are orthorhombic. The o t h e r mechanism can o n l y be a p p l i e d to h i g h - T c s u p e r c o n d u c t o r s w i t h m u l t i l a y e r s of CuO2 12 in u n i t c e l l s . Because one of the most c h a r a c t e r i s t i c f e a t u r e s of CuO~ s u p e r c o n d u c t o r s i s t h e i r weak i n t e r l a y e r c o u p l i n g , the c r i t i c a l t e m p e r a t u r e s must be n e a r l y d e g e n e r a t e between d i f f e r e n t t y p e s of s u p e r c o n d u c t i n g s t a t e s a s s o c i a t e d with l a y e r s . Therefore t h e i r c o e x i s t e n c e i s r a t h e r e a s y to be r e a l i z e d . One of the p u r p o s e s of t h i s p a p e r i s to show t h a t the s o f t e n i n g of phonons can be d r i v e n by the app e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e in a s u p e r c o n d u c t i n g s t a t e , or by an SC-SC transition. The s o f t mode i s an o p t i c a l mode which v i b r a t e s e l e c t r o n s in the CuO9 l a y e r s out of p h a s e ; i t makes d i f f e r e n t s u p e r c o n d u c t i n g o r d e r p a r a m e t e r s c o u p l e d . Then i t i s well known t h a t the s o f t e n i n g c a u s e s the d i v e r g e n c e of the t r a n s v e r s e NQR 1/T~, but no d i v e r g e n c e of the l o n g i t u d i n a l NQR 17T/. D o u b l e - l a y e r e d CuO~ s t r u c t u r e s a r e examined h e r e as the s i m p l e s t c a n e , b e c a u s e they i n c l u d e e s s e n t i a l f e a t u r e s of m u l t i - l a y e r e d s t r u c t u r e s . i t i s s t r a i g h t f o r w a r d to e x t e n d the p r e s e n t t h e o r y to m u l t i l a y e r e d s t r u c t u r e s with more than two l a y e r s . Let us examine a two-band Hubbard model, which i s one of the models of t w o - l a y e r e d CuO2 s u p e r c o n d u c t o r s :
145
146
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
~ = - t ~ ~ at a - V ~ (aTi~ t .._ nip nip oa2io+a2ioaliO ) n mjo iO T " 21-'U ~ ~ o ~ i o a n i o a n i - o a n i -
o
t a " ( b t + b ) + 8wbtb , ( l ) + W ~ ( a l i O a l i o - a 2 i o " 2iO) io w i t h a T . _ a c r e a t i o n o p e r a t o r of e l e c t r o n w i t h l a y e r ~m~l o r 2 ) , s i t e i , and s p i n O; t i n t r a l a y e r t r a n s f e r i n t e g r a l s , which a r e assumed o n l y between n e a r e s t n e i g h b o r s ; and V i n t e r l a y e r transfer integrals. The p r e s e n t s y s t e m can be c h a r a c t e r i z e d by i t s weak i n t e r l a y e r c o u p l i n g s such as Iv/tl<
~ (aTi ~ + a2i ~) ~ tl i O = ~/~
x~
,
Tcl=
Tc0 + AT ,
~ - (aTi a _ a2i T ~) . ~T 2i0 = ~
(3)
I t has a l r e a d y been p o i n t e d out t h a t the ant i f e r r o m a g n e t i c s u p e r e x c h a n g e i n t e r a c t i o n of t h e o r d e r of I t l 2 / U can work as a t t r a c t i v e i n t e r a c t i o n s between s i n g l e t Cooper p a i r s in s t r o n g l y correlated
systems ( [ t l / U < < l ) .
t h r e e t y p e s of s i n g l e t possible:
13
Therefore
Cooper p a i r s a r e
1, ~ t t t . -~ -" ~ n i t ~ n j . ~ - ~ n i ~ n j .t;>
PI2 = ( p / q ) ( T c l - T) ,
and P_=x=O below T . . The jump of the s p e c i f i c h e a t ~t Tcl i s g i v ~ by 1( 2 , "T ACt = 2 P /q~ c l
t
2tali?ff.2j~-~li~O~_jT)> , between a b o n d i n g e l e c t r o n
(5)
> 0 ,
are satisfied. I t can be seen t h a t i n t e r l a y e r coupling constant g enhances Tc,. Below the o r d e r p a r a m e t e r s a r e g~ven ~y To2'
and an a n t i b o n d i n g
2 _ P P2 q+r_g2/d
4
1 ^2~2
(ll)
q+r-g2/d
i n d e p e n d e n t . 12 Thus the f o l l o w i n g Landau f r e e energy is obtained:
4
(10)
q-r+g2/d =
2 Pl
1
(9)
i f b o t h Tc2>O and
e l e c t r o n , and they are odd w i t h r e s p e c t to the operation. A m i c r o s c o p i c c a l c u l a t i o n shows t h a t two types of i n t r a b a n d Cooper p a i r s a r e not
F(T) = ~p(T-Tco-AT)P 1 +
"
To2 = Tco - AT/~ ,
hand, the t h i r d ones a r e i n t e r b a n d Cooper p a i r s g i v e n by t
(8)
(4)
'
( n = t ) , or between a n t i b o n d i n g e l e c t r o n s (n=2). They are i n t r a b a n d Cooper p a i r s , and they a r e even w i t h r e s p e c t to the o p e r a t i o n . Here < - - - > stands f o r the thermal a v e r a g e . On the o t h e r
t
to be f i n i t e ;
There i s no anomaly of t h e s u s c e p t i b i l i t y x at c Tcl. On the o t h e r hand, the second t r a n s i t i o n occurs at
a r e Cooper p a i r s between b o n d i n g e l e c t r o n s
1, t
(7)
(2)
where o n l y Pl s t a r t s and
VO1. 76, NO. 2
P(T-Tc0+AT) 1
+ 4-q(Pl+P2 ) + ~ r P l P 2 - gPlP2 x + ~dx
2
- hx , (6)
in the gauge i n v a r i a n t form, w i t h Pl the o r d e r p a r a m e t e r o f even p a r i t y , P2 of odd p a r i t y , and
-
P q+r_g2/d
(T
u
(~2
-
T)
- T) ,
(12)
(13)
and x = ( g / d ) P . P _ w i t h T =T A*(AT/~). Not o n l y c two s u p e r c o n d u c t i n g o r d e r ~ a r a m e t e r s c o e x i s t below T c . , but t h e r e a l s o e x i s t s f i n i t e l a t t i c e distortion; two CuP 2 p l a n e s a r e d i s t o r t e d o u t of phase. Because the f r e e - e n e r g y g a i n i s l a r g e r , o t h e r o r d e r p a r a m e t e r s P2 and x can show up
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
VOI. 76, No. 2
below T_, in the p r e s e n c e of P , . T h e r e f o r e the thermodynamic c r i t i c a l f i e l d , ~ h i c h i s p r o p o r t i o n a l to [ - F ( T ) ] I / 2 , s h o u l d show a r a p i d e r i n c r e a s e w i t h d e c r e a s i n g t e m p e r a t u r e s below T_, than above T_,. The jump of t h e s p e c i f i c hea~ L a t Tc2 i s c a ~ u l a t e d as AC2 = ~(Tc2/Tcl)AC 1 . The s u s c e p t i b i l i t y
×c
(14)
shows d i v e r g e n c e a t Tc2 as
= 5(g_.g._)2(q/p)(q+r_g?/d) ~ 1 ~ d T-Tc2
+ ---,
(15)
above Tc2, and
Xc
l = -1( g 12 ( q / 2 p ) ( q + r _ g 2 / d ) ~ ~ d Tc2-T
+ ---,
(16)
below T~ .~ Because t h e d i v e r g e n c e of the s u s c e p t i b i Y ~ t y means t h a t t h e r e s t o r i n g f o r c e v a n i s h e s , i t means n o t h i n g but the s o f t e n i n g of the phonon a t T ~. I t s h o u l d be n o t e d t h a t the s o f t e n i n g i s d r ~ e n by the a p p e a r a n c e of a n o t h e r s u p e r c o n d u c t i n g s t a t e P2" I t i s o b v i o u s t h a t the s o f t e n i n g can c a u s e an anomaly of the t r a n s v e r s e NQR I / T ~ . Energyl e v e l s of n u c l e a r s p i n s w i t h I=3/2 oT Cu i o n s a r e s p l i t o f f i n t o l Z =~1/2 and [ =±3/2 in CuO~ s u p e r c o n d u c t o r s , and t h e i r q u a d r ~ p o l e s p l i t t i n g s h o u l d depend on t h e l a t t i c e d i s t o r t i o n , x. In the p r e s e n t model, the q v e c t o r of the s o f t mode i s assumed as Q_=(0,O,Q ) w i t h Q =0. By assuming Qn, t h e dynamical s ~ s c e p t i b i ~ i t y can be w r i t t e n p h e n o m e n o l o g i c a l l y as 1
g
×c(iW,q) =
(
L
~
,
d
K(q-Qo)2+~+iw
w i t h K and L c o n s t a n t s ,
and E~IT-Tc2 I:
(17)
I T2
w~O (g/d) 2 A -
I
-
(iS)
}T-Tc2 11/2
'
with A quadrupole coupling constant. Because the s o f t mode i s odd w i t h r e s p e c t to the symmotry o p e r a t i o n of c h a n g i n g l a y e r s , no anomaly can be e x p e c t e d in the NQR l / T 2 of Cu i o n s in the s o - c a l l e d c h a i n s i t e s of RBa_Cu307 . . It is a l s o o b v i o u s t h a t t h e s o f t e n i n g ~an n e v e r c a u s e any d i v e r g e n c e of t h e l o n g i t u d i n a l I~R I/T l . So f a r , i t has been assumed t h a t no d m s t o r t i o n o f l a t t i c e e x i s t s in t h e normal s t a t e , and t h a t the two l a y e r s a r e e q u i v a l e n t . However, when t h e y a r e not e q u i v a l e n t , t h a t i s , x#0, in t h e normal s t a t e , the second t r a n s i t i o n a t T . i s r e d u c e d to a c r o s s o v e r . The two o r d e r p ~ a m e t e r s Pl and P~ c o e x i s t even j u s t below the f i r s t c r n t i c a l ~ e m p e r a t u r e T , , and P, and x become r a p i d l y l a r g e around T , . ~ l n this~case, a s m a l l e r anomaly i s e x p e c t e d ~ r o u n d Tc2 even f o r Cu ions in the c h a i n s i t e s . I f the s o f t e n i n g of t h e phonon i s r e a l l y
147
r e a l i z e d in RBa.Cu307 ., many a n o m a l i e s can be e x p e c t e d such a~ a jum~ of the s p e c i f i c h e a t , the l a t t i c e d i s t o r t i o n and so on. However no o t h e r a n o m a l i e s have been r e p o r t e d so f a r b e s i d e s a t i n y but s h a r p peak of the NQR I/T~ and a k i n k of the lower c r i t i c a l f i e l d . ThereTore the p o s s i b i l i t y of o b s e r v i n g o t h e r a n o m a l i e s s h o u l d be c r i t i c a l l y examined. Because of weak i n t e r l a y e r c o u p l i n g s , ~T/Tco(
148
SUPERCONDUCTIVITY-INDUCED SOFTENING OF PHONONS
to s e t t l e e x p e r i m e n t a l l y whether the second t r a n s i t i o n e x i s t s or not in La2_xSrxCUO4.
F i n a l l y , a p r e d i c t i o n made e l s e w h e r e 12 s h o u l d be mentioned t h a t the gap can s t a r t to open e v e r y w h e r e on the Fermi s u r f a c e below T , o n l y in m u l t i l a y e r e d CuO~ s t r u c t u r e s l a r g e l y c~ d i s t o r t e d from the t e t r a R o n a l symmetry such as RBapCu207 6" T h e r e f o r e i t has been p r e d i c t e d t h a t b ~ h b ~ l o r s of a n i s o t r o p i c SC s h o u l d c o n t i n u e down to t h e zero K e l v i n in m u l t i l a y e r e d CuO~ s t r u c t u r e s with the t e t r a g o n a l symmetry, an~ s i n g l e - l a y e r e d CuO~ s t r u c t u r e s . To summarize Yhe p r e s e n t p a p e r , i t i s shown t h a t the second t r a n s i t i o n i s p o s s i b l e in m u l t i l a y e r e d CuO~ s u p e r c o n d u c t o r s between s u p e r c o n d u c t i n g s t a l e s , which b e l o n g to d i f f e r e n t r e p r e s e n t a t i o n s w i t h r e s p e c t to a symmetry o p e r a t i o n of e x c h a n g i n g l a y e r s ; weak i n t e r l a y e r c o u p l i n g s make the second t r a n s i t i o n p o s s i b l e . The t r a n s i t i o n i n d u c e s the s o f t e n i n g o f phonons, and the
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lattice distortion in the low t e m p e r a t u r e phase. The s o f t mode is an o p t i c a l mode which v i b r a t e s CuO~ l a y e r s o u t of phase. The s o f t e n i n g can e x p T a i n an observed t i n y but sharp anomaly of the t r a n s v e r s e n u c l e a r q u a d r u p o l e r e l a x a t i o n r a t e around T=35K in d o u b l e - l a y e r e d YBa~CuqO7 with critical t e m p e r a t u r e T =92K. Alth~ugE [h~ c t r a n s i t i o n around T=35K can cause many o t h e r anomalies such as a jump of the s p e c i f i c h e a t , a k i n k of the lower c r i t i c a l f i e l d and so on, t h e i r anomalies should be weak because of weak interlayer couplings.
Acknowledgement: The a u t h o r i s t h a n k f u l to K. Kumagai f o r showing h i s r e s u l t s p r i o r to publication. He o b t a i n e d the key i d e a of t h i s work, when he s t a y e d in CNRS, Grenoble. He is v e r y g r a t e f u l to J . F l o u q u e t and C. L a c r o i x f o r their hospitality. CNRS, G r e n o b l e i s l a b o r a t o i r e a s s o c i ~ ~ l ' U n i v e r s i t e Joseph F o u r i e r de G r e n o b l e .
References 1. K. A. B r u e c k n e r et a l . Phys. Rev. 118, 1142 (1960). 2. For example, R. B. L a u g h l i n , S c i e n c e 242, 525 (1988). 3. A. T. F i o r y e t a l . Phys. Rev. L e t t . 61, 1149 (1988); L. Krusin-Elbaum et al. Phys. Bey. Lett. 62, 217 (1988); D. R. Harshman et al. Phys. Rev. 1339, 851 (1989). 4. P. J. M. van Bentum et al. Phys. Rev. B36, 843 (1987); G. A. Thomas et al. Phys. Rev. Lett. 61, 1313 (1988). 5. T. Imai et al. J. Phys. Soc. Jpn. 57 (1988), 2280; K. Ishida et al., J. Phys. Soc. Jpn. 57, 2897 (1988); W. W. Warren et al., Phys. Rev. Left. 62, 1193 (1989). 6. S. L. Cooper e t a l . Phys. Bey. B37, 5920 (1988); A. Yamanaka e t a l . Jpn. J . Appl. Phys. 27, L1902 (1988). 7. J. R o s s a t - M i g n o t e t a l , to appear in Dynamics of Magnetic F l u c t u a t i o n s in High-T c M a t e r i a l s , e d s . G. R e i t e r , P. Horsh and G. P s a l t a k i s , Plenum P r e s s , 1990.
8. J. P. S t r ~ b e l et a l . , P h y s i c a C 153-155, 1537 (1988); N. Kobayashi et a l . P h y s i c a C 153-155, 1525 (1988); T. I s h i i and T. Yamada, P h y s i c a C 159, 483 (1989). 9. M. T e l , H. T a k a i , K. Mizoguchi and K. Kume, Proc. of ISNQRS (Takayama, Aug. 1989), to be p u b l i s h e d in Z. N a t u r f o r s c h . ( a ) ; and K. Kume, P a r i t y , v o l . 5 N o . l (1990) p.49 ( i n Japanese). 10. K. Kumagai, Y. Nakamichi, Y. Nakamura, T. T a k a t s u k a and H. Nakajima, to be p u b l i s h e d in J. Phys. Soe. Jpn. 11. G. K o t l i a r , Phys. Rev. B37, 3663 (1988). 12. F. J. Ohkawa, to be p u b l i s h e d in Phys. Rev. B. 13. J. E. H i r s c h , Phys. Rev. L e t t . 54, 1317 ( ] 9 8 5 ) ; P. W. Anderson, Science 235, 1196 (1987). 14. M. Sato, et a l . , S o l i d S t a t e Comm. 76, 689 (1889).