Towards the possibility of increasing Tc of oxide superconductors through coherent interaction of planar defects

PHYSICA

Physica C 2 i 3 ( 1993 ) 421-426 North-Holland

Towards the possibility of increasing Tc of oxide superconductors through coherent interaction of planar defects Yu.A. Krotov and I.M. Suslov P.IV. Lebedev Physics Institute, Russian Academy of Sciences, Leninskij prospect 53, Moscow, GSP- I, 113308. Russian Federation

Received 27 March 1993

It is shown that the change in distance d between two linear defects in a two-dimensional superconductor can lead to the existence of periodical singularities Tc~ ( d - d , ) - l in the dependence of transition temperature on d; such singularities are the strongest among the ones known at present. The smoothing of these singularities is investigated with the following cutoff factors taken into account: inequality to zero of the coefficient of transmission through defects, quasiparticle attenuation, the finiteness of the temperature. The opportunity of raising T¢of oxide superconductors by inserting planar defects normal to the Cu-O planes is discussed.

1. Introduction

In the authors' recent p a p e r [ 1 ] the existence o f oscillations o f the s u p e r c o n d u c t o r transition temperature, T¢, with a change in distance between two p l a n a r defects inserted in a s u p e r c o n d u c t o r was shown: this effect is similar to the effect o f q u a n t u m oscillations which was observed in covered films [ 14 ] and was discussed by Kagan a n d D u b o v s k y [ 5 ]. According to ref. [ 1 ], to raise Tc one can use the coherent interaction o f the p l a n a r defects; in the present p a p e r this o p p o r t u n i t y is discussed in relation to high- Tc oxide superconductors [ 6 ]. We assume that the BCS-type theory with coupling constant 2o a n d frequency cutoff too is valid for the oxide superconductors; it does not m a t t e r what the coupling m e c h a n i s m between electrons ( p h o n o n , exciton and so o n ) is. Let us insert into the superconductors periodically repeated couples o f p l a n a r defects, n o r m a l to the C u - O planes: the distance d between defects in the couple is much smaller than a p e r i o d L #1, a n d L~<~, where ~o is the coherence length. Due to the p l a n a r defects the change in Tc

#1 On a qualitative level the obtained results are also valid in the case d~L. This is the simplest case for realization in experiment.

occurs. This change is defined by the following expression [ 7 ]: T~o - 2 3 L V~

dz [NoSN(z)+6N(z)

6N(z) =N(z) -No,

2] ,

( 1)

which is the exact consequence o f the G o r ' k o v equations [7] for the case o f localized space inhomogeneity; z-axis is n o r m a l to the plane defects, N ( z ) is the local density o f states at the F e r m i level [4 ], integration is carried out over the interval including one couple o f p l a n a r defects; T¢o, No and Vo are correspondingly transition temperature, the density o f states and BCS interaction constant for the pure sup e r c o n d u c t o r (we neglect the change o f V near the p l a n a r defects). I f we neglect the interaction between the C u - O layers, we m a y consider a two-dimensional superconductor with linear defects. In a three-dimensional superconductor containing p l a n a r defects with small transition coefficient the oscillations o f the transition t e m p e r a t u r e have a sawlike form [ 1 ]: the system is d i v i d e d by the defects into weakly coupled subsystems - films o f thickness do which have a quasicontinuous spectrum a n d the films o f thickness d with a spectrum consisting o f a set o f 2D bands (fig. 1 ). If thickness d is increased, the distance between bands decreases; when the bot-

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Yu.A. Krotov, LM. Suslov / Increasing Tc throughplanar defects

422

ErR)

tO

\

\ \

Anderson theorem [ 9 ] ). According to eq. ( 1 ), this is true in the case when in the integrand the first term is dominant (for example in the case fiN(z) << No); the doubling of the singularity power is the consequence of the second term (in eq. ( 1 ) ).

2. Form of Tc oscillations without cutoff of the singularities Fig. 1. When the coefficient of transmission through defects is small the system is divided by defects into weakly coupled subsystems of two types - films of thickness do which have a quasicontinuous spectrum and films of thickness d with a spectrum consisting o f a set of 2D-bands (on the fight ).

Let the C u - O layer lie in the plane (y; z); separating the variables we get one-particle wave functions and eigenvalues En in the form

~.(y,z) =~os(z)exp(ikly) , tom of some band intersects the Fermi level, Tc j u m p s up due to the j u m p of the density of states. In the two-dimensional case, I D bands appear instead of the 2D ones in fig. 2. In the vicinity of the b o t t o m of a I D band the density of states depends on the energy e as e-~/2, so one can expect the appearance of periodical singularities ~ ( d - dn) - ~/2 in To. The periodical singularities do appear (fig. 2), but they are stronger than the ones discussed above, Tc~ ( d - d n ) -1. The problem is that, in the intuitive arguments given above, we consider Tc of the spaceinhomogeneous system as a function of the average density of states only (situation corresponding to the

E.=~s+k~/Zm,

(2)

where kll is the longitudinal q u a s i m o m e n t u m and s is the transversal quantum number. It is supposed that the two-dimensional spectrum has the form e(k)=kZ/2m, and that linear defects have a negligible thickness in the transversal direction and are situated at the points z = _+d/2 (according to eq. ( 1 ), it is enough to consider only one couple of defects). The boundary conditions for the wave function of traversal motion ~0(z) at point z = d/2 have the form ~0(d/2+0) = ~ 0 ( d / 2 - 0 ) ,

4

Fig. 2. Form o f To oscillations with c h ~ n g d ( f o r d = const) without cutoff o f the singularities; l and 2 are the envelopes o f maxima when x is finite ( ] ) and l or T is finite (2); 3 - asymptotic form o f the envelope o f minima for the case n >> l.

Yu.A. Krotov, I.M. Suslov / Increasing T~ through planar defects

~ o ' ( d / 2 + 0 ) - ~ o ' ( d / 2 - 0 ) =x~o(d/2),

(3)

and at point z = - d / 2 the conditions are analogous. Using the definition o f the local density o f states:

423

kv

m f dq 2q 2 N(z)= -~ a (k 2_q2),/2 x2 o

1 - cos 2qz cos qd' X sinZqd,+4q4/x 4 .

N(E, r) = ~ I % ( r ) I Z 6 ( e - E . )

(7)

n

-

x/~ 2n ~l~°s(z)

0(c-Es) 1 2 ~ '

(4)

calculating ~0s(z) and t~, transferring from summation by s to integration and neglecting terms ~ a/L (a is the interatomic space ), we get for N ( z ) -= N( ~V; y, z): kv

m ~

N(z) = -~

)--~- - - ~ ( ~

× ]r l + ~

d'=d+ 2 / x ,

(8)

and one can approximate it by the set of 6-functions. The same situation occurs in the case Izl > d/2. One can get

N(z)

dq

(2q 2 1 + 2qZ/x 2+ UI (q)cos 2qz

kk

q,=nn/d',

(k2F __q2)I/2 o

.

It is clear that the integrand is localized near the points

. . . .

cos 2qz, + ~

,

I

..~

IZl

m

~

1-(-1)~cos2q,,z

nd~,~l---~kT_--q~,)~/T--,


~No[1-Jo(2kFZ')],

sin 2qz,],

Izl
z'=lzl-d/2>O,

(9a) (9b)

V(q) = (sin qd+ ( 2 q / x ) c o s qd) 2+ 4 (q/1¢) 4 ,

where No=m/2n, M= [kFd'/n], Jo(x) is the Bessel function. According to eq. (9b), the integral over the region Izl > d/2 in eq. ( 1 ) does not depend on d a n d it may be calculated for d = 0 , i.e. for the case when two defects merge into one; the integral on the region I zl < d/2 gives the oscillational part o f To:

Ui ( q ) = ( 2q/ x )sin q d - cos qd ,

8T¢(d) = STy(O) + (ST¢(d) ) ....

(q)

(q)

J

z'= I z l - d / 2 > 0,

(5)

where

Uz (q) = - sin 2qd- (2q/K) sin 2qd

Substituting eq. (9a) in eq. ( 1 ), we get ( [... ] - integer part o f the number)

+ (2q2/x z) ( 1 - 3cosZqd) + (2q3/x3)sin 2qd, U3(q) = ( 2q/x)sin2qd+ ( 3qZ/x2)sin 2qd + (4q3/x 3)coszqd.

(10)

(6)

The expected increase o f Tc is large in the case o f the strong defect, i.e. under condition I xl >> kF. Below we shall discuss the case x>> kF only, because mean-field theory is not valid in the case - x >> kv #2. The expression for local density o f states in the region Izl >a it is destroyed by fluctuations. Thus, in the two-dimensional case (in opposition to 3D), the existence of T a m m states has no qualitative effect and the results for the case - x > > kv should be similar to the ones for K>> kF.

--

T~o [a]

r~2o L

-- 2S1 +

1

S,=,=1 ~ (ot2-n2) l/z' kvd'

or-- - -

S 2 +

[a]

$2

1

S z = , =~I a : - n : ' (11)

In the case when o~ is not very large, $1 and $2 consist o f several terms and they are calculated straightforwardly; in particular, (8To)osc = 0 for 0 < o~< 1. In the case o~>> 1, one can express 8T~ in terms o f periodical functions ~v({ol}) and f ( o 0 . (ST~)os~

1

Too

n2okFL

Yu.A. Krotov, I.M. Suslov/ Increasing T~throughplanardefects

424

6Tc(0) with logarithmic accuracy is obtained from eq. ( 1 8 ) by the substitution l ~ L .

f(a)=

~ cos(2nn{a}-~/4) n=l

V/~

a=--kFd' ~

~

'

3. The cutoff of the singularities (12)

where ~u(z) is the logarithmic derivative o f the g a m m a function [10], {...} is the fractional part o f the number. In the vicinity o f the nth singularity we obtain in the leading a p p r o x i m a t i o n in n:

Consider the main factors leading to the cutoff o f the singularities. ( 1 ) The finiteness o f x. If x is finite, then in the region Iz[
((STc)osc , ~ , _ 1 T~o =2okFL

N(z)= ~ A.[1-(-1)"cos2q.z], (a_rt)l/2 +3

a--n

)' (13)

where f m ~ = - 1.03 is the m i n i m u m o f the function f ( a ) ; the first term in brackets gives the envelope o f the m i n i m a o f the oscillations (fig. 2). Substituting eq. ( 9 b ) in eq. ( 1 ), one can calculate the value o f T¢(0); the integral o f (8N) 2 diverges logarithmically for large I z l. F o r dirty superconductors the cutoff distance is the electron m e a n free path l. The finite value o f I m a y be taken into account by the substitution

where

m A"=~ 5

-- qn

~

dq 2q~/x 2 (k~-(q,+q)2)'/2(qd')Z+4q~/K4"

-- qn

(18) The singularities a p p e a r due to the M t h term o f sum ( 1 7 ) (qM ~ kF). Separating this term and assuming ~ c = ~ in the others, we obtain for Tc the expression (STc)osc ~ , Tco

_

1 ~2okFL

× 2nfmi"x/n+ 2n3nAM+m -6rc4nA2 ~ Mlj . ~(e-E,)o

Calculating AM for finite x we get

8N(z)--*SN(z)e -21~11l, I=VF/y.

(15)

1

- n2okvL [ l n ( 4 k F l ) -

n]

"

(6Tc)os~ T¢o , ~ , , -

l n20kFL

X{2~fminNiInJt.=3/2~FNlffHF(u)JpB"K2 2) --f~F F(u) ~'

Calculating the integral in eq. ( 1 ) we get ~3

T¢o

(19)

(14)

1 y n (e_En)2..k~, 2

in eq. ( 4 ) , from which

aTe(O)

(17)

n=l

(16)

In the case o f a clean superconductor, the result for #3 The derivation of eq. ( 1) for clean superconductors also remains valid in the dirty limit, but we must use the average (K(z, z') ) instead of the kernel K(z, z') (average is taken over impurity distribution ). Instead of N ( z ), the average quantity (N(z)) appears in eq. ( 1), which we can express in terms of the imaginary part of the average Green's function ( G(z; z') ). The quasipartical attenuation is taken into account by the substitution e~e +i7, which is equivalent to eq. (14).

g

K 2

u= -~ k~v v (a-n)

,

(20)

where the function F ( u ) describes the form o f a s m o o t h e d singularity and is defined by

±~ 0x

F(u)=

1

J x/~ (u-x)e+ 1 0

l

( U " ~ (U2"[ -

_ x/~k ,.

1)1/2~ l/2

u~-++1-

.}

(21)

Yu.A.Krotov,LM. Suslov/ IncreasingT¢throughplanardefects Substituting F ( u ) --'Fm~x= ( 3 / 4 ) 3/4 ~~0.81, one can obtain the envelope of the maxima. (2) The quasiparticle attenuation. The substitution (14) in eq. (4) takes into account the finiteness of the electron mean free path. In consequence one can obtain N(z) in the form (17) with

(~Tc)os~

G

425

1

c~n- ~'okF L

X{2/~fmin f~+ ~J'O + 3202 , F 1 GE(u)~

4 Tn

2m 1/2 kZ_q 2 A. = 2--~; ( ~ - ) F ( ~ ) , 1

(22'

and the function F(u) is the same as that in eq. (21). Using eq. (19) we get (8T~)o~:

(

1

f 'F "kl/2

X~2~fminX/%+2~7 )

~1 F(U)+6

} F(u) 2 ,

G(u)=

~ dx t h ( x - u )

J,/% 0

(27)

x-u

The c u t o f f o f the singularities occurs at {c¢}/c¢~ T~ 'F; in the interval T/,F<< { a } / ~ << CO0/,F the singularities have no pure power behavior due to log-

arithmic corrections. The function G(u) describes the form of the maxima; we may obtain the envelope of maxima from eq. (27) by the substitution

G(u)~Gmax~.3.8.

2e F o / - t/

u= -- -y n

(23)

The envelope of the maxima is obtained by the substitution F ( u ) ~ F~x. It is interesting that its dependence on n differs from eq. (20). (3) The finiteness of the temperature. When deriving eq. ( 1 ) in ref. [ 7 ], it was supposed that N(,; z) changes slowly on the scale COo,and can be taken in the p o i n t , = 'F. Taking into account that N(z) arises from the sum rule for the kernel,

f K(r, r') d r ' = V(r)N(r)ln

1.140)o T '

(24)

it is easy to understand that in the general case we must use Neff(z) = 2 o T z~ ~ d, N(''z------~),2+O)2

(25)

instead of N(z), where summation is taken on Matsubara frequencies 0)~=rcT(2s+ 1 ), (I0)~1 <0)0); it is taken into account that T,~/'co. Transforming Neff in the form (17), we get x/~

'v a - n

J

--C°d,

An-- 27¢d ~_gZST,2or ~, ,o ! ~

1

[~+(qZ--k2F)/2m]2+0)z" (26)

If {Ot}/a>>0)O/'F we return to eq. (13). In the opposite case, in accordance with eq. (19), we get

4. Discussion of the results

From eqs. (20), (23 ) and (27) we obtain that the highest increase in Tc is reached under conditions

d~a,

x>kF(~o/a) ~/2, I>~o,

(28)

and it has a value ~T c T~o

EF a T~L"

(29)

If L~~o, then we obtain ~Tc~Tco. If ~Tc>T¢o, the initial eq. ( 1 ) is not true, but from the physical sense one can expect further increase in Tc when L becomes smaller than ~o. The upper estimate (29) may not be reached because ( 1 ) the planar defects are not perfect; (2) in the vicinity of a singularity of the states the Fermi level is unstable [ 11 ], etc. In the limit x-,oo, a two-dimensional superconductor is divided into 1D strips, in which superconductivity is destroyed by fluctuations. The fluctuations are not important if J ± > T [12] (J± is the transversal overlap integral). Substituting Jx~ EF(Xa)-~ we get in the case d~a)

X
(30)

426

Yu.A. Krotov, LM. Suslov / Increasing Tc through planar defects

One can create the considered structures (fig. 1 ) by the use of." ( a ) the control o f the twinning processes; ( b ) the insertion o f layers o f alien a t o m s (a-oriented superlattices o f the oxide superconductors with the p e r i o d 100 A already created [ 13 ] ); (c) deposition o f thin films ( a b o u t several C u - O layers) on substrata with artificial periodicity. It is interesting that the effect o f increasing Tc occurs even in the case o f r a n d o m position o f the planar defects (with the average distance L ) . Assuming that the distance d between defects fluctuates on the scale 8 d ( k y I << 8d<< d, L ) a n d taking the average o f the eqs. (20), ( 2 3 ) and ( 2 7 ) by d in the interval from 0 to L, we get (taking into account eqs. ( 1 0 ) and (16)): 8To _ Too 7t2

3In m a x FL

,

, y

+ln max(k/, kL) +In kd+O( 1)], i.e. the estimate 8 T J T ~ a / L is valid with a logarithmically large factor. This result is qualitatively valid i f S d ~ d ~ L, when the occurrence o f the defects is absolutely r a n d o m . It is possible that the heightened T values which were i r r e p r o d u c i b l y observed in the beginning o f the high-To s u p e r c o n d u c t o r investigation were stipulated by a high concentration o f the p l a n a r defects or their m u t u a l disposition.

Acknowledgement This work was s u p p o r t e d by a Soros F o u n d a t i o n G r a n t a w a r d e d by the A m e r i c a n Physical Society.

References [ 1 ] Yu.A. Krotov and I.M. Suslov, Zh. Eksp. Teor. Fiz. ( 1992 ). [2] V.M. Golianov, M.N. Miheeva and M.B. Tsetlin, Zh. Eksp. Teor. Fiz. 68 (1975) 736; V.M. Golianov and M.N. Miheeva, Zh. Eksp. Teor. Fiz. 70 (1976) 2236. [3] K.A. Osipov, A.F. Orlov, V.P. Dmitriev and A.K. Milay, Sov. Solid State 19 (1977) 1226. [4] H. Sixl, Phys. Lett. A 53 (1975) 333. [5] Yu.M. Kagan and L.B. Dubovsky, Zh. Eksp. Teor. Fiz. 72 (1977) 647. [6] J.G. Bednorz and K.A. Miitter, Z. Phys. B 64 (1986) 189; M.K. Wu et al., Phys. Rev. Lett. 58 (1987) 908. [ 7 ] I.M. Suslov, Zh. Eksp. Teor. Fiz. 95 ( 1989 ) 949; idem, Supercond. Phys. Chem. Technol. 4 ( 1991 ) 2093. [8]A.A. Abricosov, L.P. Gor'kov and I.E. Dzyaloshinskii, "Methods of Quantum Field Theory in Statistical Physics" (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963). [9] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. [10] G.A. Korn and T.M. Korn, "Mathematical Handbook" (McGraw-Hill, 1968). [ 11 ] I.M. Suslov, Pisma V Zh. Eksp. Teor. Fiz. 46 ( 1987 ) 402. [ 12 ] "The Problem of High-Temperature Superconductivity", eds. V.L. Ginsburg and D.A. Kirzhnits (Nauka, Moscow, 1977) pp. 261,315. [ 13] C.B. Eom, A.F. Marshall, J.M. Triscone et al., Science 251 (1991) 780.