Anisotropic polarons and high temperature superconductivity of copper oxides

Anisotropic polarons and high temperature superconductivity of copper oxides

Physica C 160 (1989) 202-216 North-Holland, Amsterdam ANISOTROPIC POLARONS AND HIGH TEMPERATURE SUPERCONDUCTIVITY OF COPPER OXIDES A.A. R E M O V A ...

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Physica C 160 (1989) 202-216 North-Holland, Amsterdam

ANISOTROPIC POLARONS AND HIGH TEMPERATURE SUPERCONDUCTIVITY OF COPPER OXIDES

A.A. R E M O V A a n d B.Ya. S H A P I R O Institute of Inorganic Chemistry, Academy of Sciences, Siberian Division, Novosibirsk 630090, USSR

Received 9 January 1989

A theory of high temperature superconductivity is constructed for crystals built of layers of copper oxides separated by those of calcium. The main point of the theory is prediction of the existence of two electron groups in these structures: free electrons in the copper-oxygen layers and a strongly correlated electron system between the copper-oxygen layers. It is shown that in the strongly correlated electron subsystem a new type of polaron can exist. The characteristic sizes of the polarons are different in different directions. The polaron in the layer has a button-like form. In the filament structure the polaron is a spindle-like solution. The button-like polarons may form a Wigner crystal. The frequency of the polaron oscillations is very high and may lead to the high temperature superconductivity (HTS). The frequencies of the spindle-like polaron oscillations are also high. The critical temperature of the superconducting transition due to the heavy plasmon oscillations has been obtained. The critical temperature TRis a rising function of the number of layers. The influence of magnetic field on the polaron is discussed.

1. Introduction High t e m p e r a t u r e superconductors with critical t e m p e r a t u r e Tk > 100 K which are known at the present time possess a layered structure with the c o p p e r - o x y g e n layer separating the layers o f other elements (fig. 1 ). The thallium a n d b i s m u t h layers play the d o p i n g role. In o u r o p i n i o n they suppress the Peierls instability in the c o p p e r - o x y g e n layers. T h e critical t e m p e r a t u r e is a function o f the n u m b e r o f c o p p e r - o x y g e n layers. This suggests that s u p e r c o n d u c t i v i t y in these systems is the consequence o f the hole transitions from one layer to another after interaction with a b o s o n m o d e between the layers [ 1 ]. T h e origin o f this m o d e m a y be d e t e r m i n e d b y the properties o f the layer separating the c o p p e r - o x y g e n layers. In particular, strong e l e c t r o n - p h o n o n interaction a n d easy polarization o f the oxygen sublattice in this layer m a y lead to the creation o f polarons in the layers between the c o p p e r - o x y g e n layers. The o x y g e n - c o p p e r filaments are the structure elements o f the y t t r i u m ceramics; the polarons m a y a p p e a r b o t h along these filaments a n d along the C axis ( d u e to the narrow b a n d in this d i r e c t i o n ) [ 2 ]. These l o w - d i m e n s i o n a l p o l a r o n s strongly differ from the usual Pekar polarons. Both for plane a n d for fila m e n t structures the p o l a r i z a t i o n is possible only along the layers a n d the filaments, respectively. The polarons in these structures m a y f o r m superlattices. These superlattices m a y be due b o t h to the superstructure along the layers (like in b i s m u t h a n d t h a l l i u m c e r a m i c s ) a n d to the W i g n e r crystal state (because o f the p o l a r o n repulsion). T h e characteristic frequencies o f the p o l a r o n structure are in the m i d d l e region between the electronic p l a s m o n s a n d ion frequencies. These oscillations m a y lead to the high critical t e m p e r a t u r e o f the ceramics. The external magnetic field p e r p e n d i c u l a r to the layers influences the structure a n d characteristics o f the polarons as it presents an a d d i t i o n a l cause o f the electron localization. In particular, for the plane polarons (in the layers) the critical magnetic field Hk exists destroying the polarons.

0 9 2 1 - 4 5 3 4 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

A.A. Remora, B. Ya. Shapiro/Polarons and high Tk superconductivity of copper oxides

t-



203

6

! (do K)

-i

t8

.q--

(lOS K)

0

I £q

02s K)

J

8

i

|

2324 ?

Fig. 1. The HTS structure with different number of copper-oxygenlayers.

2. Basic equation The Lagrange function of the electron interacting with the polar substance (in the classical approximation) under strong electron-phonon interaction, has the form: •

2

2\ /l=affEt~l,

a t = a o / 4 ~ t o 2,

Ot]

p2

~

ao=(E-l(~)--e-l(0))/47z.

-

-

-

V(r) i ~12],

-

(1)

Here ~ is the electron wave function, P is the polarization vector of the substance, A is the vector potential of the magnetic field, D is the electrical induction, to is the frequency, e (to) is the dielectric susceptibility, too is the highest optical mode, r are the spatial coordinates, V(r) is the potential energy of electron, h is the Planck constant, e is the electron charge, c is the speed of light, t is the time, m is the electron mass and o are the vector components. The Poisson equation is

0D..____~=4~lel 17,12. 0r~

(2)

The condition for the wave function is

f l ~ul 2 d r = 1 .

(3)

In the adiabatic approximation the polarization and modulus of the wave function of the polaron are connected with its motion as a whole. Therefore, introducing the radius-vector of the polaron R (t) we obtain after transformation [ 3 ]:

204

A.A. Remova, B. Ya. Shapiro / Polarons and high Tk superconductivity o f copper oxides

a new Lagrangian l ( _ i h O ~ x ~ - e A.(x) ) ~, 2- - V ( x + R ) I~ul2--e Le= f dx { -2--ram

# { OPt. \2

p2 }_eak~R~Rk +

Rk)

°-

m

.2

TRy,

Here ek~j is the Levi-Chivita symbol, H is the magnetic field. Using (5) one can obtain the set of equations for the electron wave function in the polaron well. Obviously, the solution depends on the specific form of the potential V(r).

3. Single polaron in the thin layer

We shall define the thin layer with the superstructure by the potential energy of the electron in the form (fig. 2):

V(r) ~- O(rll ) U(r± ),

U" (0) 2

O(rll ) ~- Uo + ~

rH ,

(6)

where the potential energy changes much faster along the z axis than along the layers. Potential U(r H) can be due to the superstructure along the layers or due to the polaron repulsion in the layers (Wigner crystal). The wave function of the electron in this potential may be looked for in the form:

V(x)=z(p)epo(Z),

x = (p, z ) .

(7)

Here q~o(z) is the wave function of the ground state of the Schr6dinger equation with potential U(z+ R ± )Uo: hZ dZf°°(z) 2m dz 2

UoU(z+R±)epo(z)=Eo~oo(Z).

(8)

The Lagrange equations have the form:

0

Fig.2. The potential energy of the electron in the layered structure. Here U, Oare the potentials in the directions perpendicular and parallel to the layers, respectively.

A.A. Remora, B. Ya. Shapiro / Polarons and high 7", superconductivity of copper oxides

0t0R.-OR.

' 0z0~.-0~.'

-0z

' q~-=(x, P, D),

205

(9)

z-p.

Here .W* is determined by the relation: ~*=

f

dx

{ 2# ( 0 P . )2 p2 -~XkRk + P . D . - - -2Oto -UdU(rx)

(p+R,,)2 } Ix(P) 12~,g(z) 2

2

+ f dp [ - 2---ram(1-ih0--~, - e A . ( p ) ) X ] - E o f I z l 2 d p + -zm ~ R E - - - ae~ k R ~ R k c

+ f dxA(x) \Ox, -4~lel IX(P)I

Ixl clp

(10)

Substituting (10) in (9) we obtain the set of equations describing the polaron (magnetic field H is directed along the z axis): Mu = - g R , , + ceeozRit~H=,

(11)

R(t)=(RvR±) •

(It can be easily shown that for the polaron motion in the direction perpendicular to the layers we have: k . = 0, R± =0.) (12)

[,z+H(P)Z=-¢X,

h2 1 O(p~p) £ = 2m p Op

e2 H2p 2 - ~mc 2

e=Eo-E '

H (p ) = 4XtXoe2 J- K( IP-P' I )IX(P' ) 12do', Mu=4=(41~e)zal

f OlzI2 OlzI 2 Op,

Op----SK ( I p - p ' I ) d p d p '

'

(14)

g=U~ J- U(r±)~2(z) dz, q~2(z){Oo(Z') dzdz'

K(IP-P'I) =

(13)

x/(P-P')Z+

(z-z')2

(15)

'

where i, j are the components in the layer plane. Determining the wave function of the electron in the plane Z(P) from (12) one can obtain both the polaron structure and its effective mass 3. I. The plane polaron under zero magnetic fieM

Under the condition H=0, eq. (12) may be represented in the form: (16)

£oZ+ (e+ Ho(p) )X+ W(p)z=O ,

/.oXo+(eo+Ho(p))Xo=0, Ho(p) ~

47~[~0 e2

E=Eo+~e*,

) dr' [P-P'I '

/o=/,(H=0),

X2(p '

H(p) ~-Ho(p) +O( dz/dp) .

W(p)=H(p)-Ho(p),

(17) (18) (19)

Here dz and dp are the characteristic sizes of the polaron in the different directions (d=/dp << 1 ). In the main,

A.A. Remova, B. Ya. Shapiro/Polarons and high Tk superconductivity of copper oxides

206

on dz/dp order (limiting anisotropic case) one can obtain the wave function of the ground state by expanding Ho(p) in powers ofp. Substituting/-/o(P) by the model function 2

(20)

/lo (p) = / l ~ ( p ) = / l o ( O ) + ~6' 8~ we obtain from ( 17 )

£oXo(P) + goXo(P) - m--~2p2Xo(P)= 0 ,

(21)

oo

rns~2=-16~2e2~ o

[email protected]' , go=eo+8gae2c~o

X2(p ') dp' .

(22)

0

It is easy to obtain the nonlinear equation in order to determine go and for the oscillator we have: rag'22 _

(4~e) 2 4X/~

Xo(P). Using the well known result

OZo(mg'2) 3/2 ,

(23)

Xo(P) = (dEn) -1/2 e x p ( - p E / E d 2 ) ,

(24)

dp=(ml2) -'/2 .

(25)

From ( 2 2 ) - ( 2 5 ) we obtain t2 and dp go=4~3/2e2Oto/ dp .

(26)

In the main, in the dz/dp approximation we obtain Eo=0 for the model potential (20). Taking into account the difference between the exact potential Ho (p) and H~ (p) we obtain for the lowest energy level of the electron in the polaronic well (appendix A) using the perturbation theory (fig. 3):

8~=- f X~(p)(Ho(p)-H"d(p))dp,

4/~3/20~0 e2

~i~=-0.2Ho(0),

/70(0)=-

dp

'

I1[~(O)=IIo(O)/d 2 (27)

Mp,1 I

2

!

~

H=HK ~o

H.o Fig. 3. The position of the energy level of the electron in the polaronic well.

z - Mp Fig. 4. The mass and the characteristic size of the plane polaron vs. polarization coefficient (curves 2 and 1, respectively).

A.A. Remova, B. Ya. Shapiro/Polarons and high Tk superconductivityof copperoxides

207

(8¢ is the electron energy in the polaronic well). Using the next term of dz/dp in (16) we obtain corrections for the energy level in the polaron well associated with the three-dimensionality of the system (anisotropy is decreasing). The energy level is increased (fig. 3). 8¢*= - J Z~(P) W(p)dp= -4ne2Oto J Z2(p)z2(p ' )~o2(z)~OoZ(Z' )

1

× Ix/ (P-p')~+(z-z') 2

1]

IP--P'I dz dz' dp dp' .

(28)

Calculating integral (28 ) using (24) (see appendix B) we have for 8¢*

8¢*= 4ne2a°d± A ,

(29)

A = J I ( - ( ' I~02(()~02(( ' ) d ( d ( ' .

(30)

From (23) and (25) we obtain the expression for da (fig. 2):

dp=X/~ao

4n2ot-----~ >> ao.

(31 )

To obtain the polaron mass one has to substitute the wave function Zo(P) (24), (25) into (13). As a result we have for M

Mij=x/-~ o4 ( 4 n e ) 2 8 j d 3 >>m.

(32)

Thus, in the thin layers the plane heavy polarons may exist. The polaron mass as a function of polarization coefficient is shown in fig. 4.

3.2. Plane polaron in magnetic field If the external magnetic field is perpendicular to the layers, then the electron energy in the polaron well decreases (the well bottom has risen (fig. 3) ). The condition an>> dp (an is the magnetic length) remains valid up to the complete vanishing of the polaron well. However, condition an >> dp permits use of the perturbation theory in order to calculate the electron level in the polaron well under nonzero magnetic field. Using eqs. ( 11 ), (12) and the ground state wave function in the form (24) we have for AE

e2H 2 ("

1 e2H 2

A~(H)-2n~mc~ J z°(P)P @= 8 mc 2 d 2 '

(33)

Ae(H) ~ - BE,

(34)

_

H~~~ood2p,

2

3

(oo=hc/e.

(35)

The condition au >> dp is still correct.

3.3. The spindle-like polaron (solitons) If the electron in the layer is in the two-dimensional potential, then the polaron may be created only along the z axis perpendicular to the layers. (In the special case of the yttrium ceramics the electron is localized by the two-dimensional potential too; however the z axis is directed along the layers.) The potential in Lagrangian (5) has the form:

V(r)

""

Cr(r± ) U(rpl ) .

(36)

208

A.A. Remora, B. Ya. Shapiro / Polarons and high Tk superconductivity of copper oxides

(Here superstructure is along the z axis. ) Looking for the wave function in the form

~U=Xo(p)q~(z), x=(p,z)

(37)

we obtain the SchriSdinger equation in the two-dimensional potential well h2 - ~---~m/tpXo- U, (P)Zo =Eo)~o,

(38)

e2n 2 UI (p) = U(p+RH) Uo+ ~mczP 2 •

(39)

The Langrangian function ~ * is

£P*=

dx

mR~

+ --2-

+P~D. - 20t----~ P~ -U(rll)U" 2Xg(P)l~(z)l 2 + ~

-~k

ea,kRkR,+ E C

-~z dz-Eo

[~ol2dz

\-b-~-x~-4rclelx2(p)l~°(z)12 ) f 1~012dz+f dx2(x) (oo

(40)

Substituting (40) in the equation of the polaron motion (9) we have, after replacing ~(X, P, D) by ~(~o, P, D) with z = z, the set of equations instead of (10), ( 11 ): h 2 02

£ 1 - 2 m O z 2'

Ill(Z)=4rceZa°

K(lz-z'l)l~(z')12dz'

M=41t(4ne)2°q f l O[~(z)[20l~(z,)12K( Iz-z' I) d z d z ' 0z

£1q~(z)+H~(z)qT(z)=-f¢~(z), ,

8z'

(41) (42)

K ( I z - z ' l ) = (, x Z~(p)z~(p')dpdp' / ( p _ p , ) 2 + ( z _ z , ) 2 , g=U" f U(rll))C~)(p) dp,

(43)

MR'± = - g R ± .

(44)

In this case the classical motion in the direction perpendicular to the z axis is absent: Rll = 0, Rtl = 0. Taking into account the condition dz >> dp we transform (43) to the form:

K( Iz-z'l ) ~ 1 / I z - z ' l •

(45)

Substituting (45) into (42) we obtain the equation for the wave function of the electron in the polaron well: £1 ~(z) +4xeEozo(

i

I~(z')lZdz' Iz-z'l

)

~0(z) = - 8E ~0(z).

(46)

Taking into account that the integral in (46) is determined, mainly, by the small value of I z - z ' l , we obtain instead of (46) with logarithmic exactness £1 ~0(z) + 4ne2Oto I~0(z) Iz~o(z)2 I n / z = - 8E ~o(z),

(47)

ao

where lz is the distance between the layers. (This is the equation for a "condenson" in a strong magnetic field [4] or in a medium with linear inhomogeneity [ 5].) The solution of eq. (47) is the solution:

A.A. Remova,B. Ya. Shapiro/ Polaronsand high Tksuperconductivityof copperoxides

N/1

209

1

~0(z) =

2dz chz/dz

dz=(-h2/2mS¢) 1/2,

(48) 8¢=-72rn/4h2,

(49)

y = 8ne2ao In lz/ao.

(50)

Substituting ( 4 8 ) - ( 5 0 ) into (41) we obtain for the polaronic mass:

M=

87t (4he) 2 15 d-----~al lnlz/ao.

(51)

This type of polaron is shown in fig. 5. The dependence of the longitude size of the polaron dz and its mass on t~o is shown in fig. 6.

4. Wigner crystal If the polaron concentration in the layers is increasing then one has to take into account the collective phenomena in these structures. Let us show that there is a repulsion between the heavy polarons. The Hamiltonian of the interacting polarons is (see (1)):

p2 ) dr-

H= f ( PD- ~

~

f dp 2mh2 ~7Xot~Tx~"l-~~,

f e2lx~(p) E ( ~ ) lElx~'(P')12dpdp', lp-p'l

r

(52)

(Here the last two terms have been integrated to the z coordinate ( 8 ) ) . The last term takes into account the Coulomb interaction between the electrons. After the minimization of (52) on P we have:

D=P/ceo.

(53)

After replacing P and D from the equation

D='el ~ f lX,~12~Oo(Z'(171r~r,l)dr',

(54,

into Hamiltonian (52) we have the expression:

rn I

"

Fig. 5. The spindle-likepolaron.

Fig. 6. The longitude size of the spindle-like polaron.

210

A.A. Remova, B. Ya. Shapiro / Polarons and high Tk superconductivity of copper oxides

2dpdp'- ~ f dp~-~m h2 VXxVZ*+o~,~, 121x~(p,) 12dp dp,. H= ~,~'~" a°4~e2Iz"(p)I2z~'(P')IIp_p,I ~ f dlx~(p) e(oo)lp-p'l

(55)

The Schrbdinger equation for this Hamiltonian has the form:

_~mApZ~+C~o4~e2z~ f lz~(P' e2 12 IP-P'I)12 dp, - ~--~Z~, ~, f ]Z~'(P')lp_p,[ dp'=-Ez~.

(56)

The third term in (56) is small due to the weak overlapping of the wave functions of the polarons (the small parameter is ratio dp/rs, rs is the distance between the charges). Using the perturbation theory and chosing the wave function of the single polaron as a ground state wave function, we have for 5E: e2 f Iz°~'(p)121X°~(P')12dpdp'. E(O---)~ IP-P'I

8E=E-¢=

(57)

After integrating in (57) we obtain for 8E in the main order on

BE=

- -e2

3"

dp/rs:

1

(58)

e(O)~'Za, Ip%, I

Here p O _ p _ p ~ , is the distance between the charges. From (58) we have 8 E > 0 . Hence the polarons mutually repel. The gas of the plane polarons behaves like the two-dimensional electronic gas and the Wigner crystal state is possible in this system. Indeed, under the condition kBT
(59)

r s - _ N s 1/2

(N~ is the polaron concentration, kB is the Boltzmann constant, T is the temperature) the polarons are crystallized into a hexagonal lattice [6] (fig. 7). For this structure the static potential energy Eo is Eo -~ - (2.21

[E(O)r~)e2eaao.

(60)

The characteristic frequencies of the polaron oscillations at the equilibrium point are [ 7 ]:

o)~+ = ½{~xx(k) + ~yy(k) +_[ ( ~xx(k) - Cl)yy(k))2 + 4 ~ y ( k )

1,/2},

Fig. 7. The polaron Wigner crystal.

(61)

A.A. Remova, B. Ya. Shapiro / Polarons and high Tk superconductivity of copper oxides

q~.,(k) = ~ q~,~p(q) e - ' * ' p °

,

pO_pO ~/-- - - pO j,

211

(62)

J

1 0

0 v(p) p=po

i#j,

(63)

--Mk¢il ~. OPo, ~ aPp ~ v(p)p=p~k i=j,

(64)

v(p) =eZ /pe(O) . Here a, fl are the components of the two-dimensional vector p, i, j are the sites of the lattice. The dispersion law to(k) was determined in refs. [8,9], where it was shown that under condition kp ° << 1 we have for longitudinal "phonons": to2(k) -~

2~Nse2 k

(65)

M

'

and for the transverse mode to(k) ~ k .

(66)

In the reverse limit kp ° >> 1, we have from (62) the nonzero terms under condition mation we have for the lattices of the crystal group Cn

q'xx(k) =

q~xAk)= 0 .

q~yy(k),

i=j (p°-O).

After sum-

(67)

As a result we obtain e2

1

¢0~+ = ~ (pO)3¢(O) M' t

o

~

(68)

~z

m M¢(O)r~"

(69)

Here z is the number of the "nearest neighbours". These oscillations are the heavy plasmons in the polaron crystal.

5. The polaron oscillations Along with the Wigner crystal oscillations described above, in the polaron gas may exist the boson modes due to the superstructure in these systems. In particular, such superstructure was observed experimentally in the superconductors B i - S r - C a - C u - O and T1-Ca-Ba-Cu-O. In order to obtain the characteristic frequencies of the polarons in the superstructure one has to use the equation for polaron motion in the potential ( 11 ), (44). For small oscillations we have for different polarons:

a) Plane polarons "QO "~

(De (DO

4X 102ca / a ~ , ea = ~ 2 / 2 m a ° 2 '

toe = x//g-/m •

(70) (71)

212

A.A. Remova, B. Ya. Shapiro / Polarons and high Tk superconductivity of copper oxides

b) Spindle-like polaron g2o-

co, coo /a21n2 l~ ~a X 102 /

(72)

ao "

It can be easily seen from ( 7 0 ) - ( 7 2 ) that the frequency of the "heavy" plasmons satisfies the condition: 090 << g2o << co¢

(73)

(here co~~ cop is the electronic plasmon frequency). These oscillations are the "heavy" optical olasmons and they may play a very important role in the high temperature superconductivity. 6. The critical temperature of the superconductivity

As was pointed out before, the superconducting electrons belong to the copper-oxygen layers. On the other hand, the critical temperature increases with increasing number of layers [ 10 ]. This fact has a natural explanation in the framework of the model [ 1 ] in which the superconductivity is due to the hole transition from one copper-oxygen layer to another. This transition takes place as a result of the hole interaction with the polaron oscillations. We think that the origin of the high temperature superconductivity lies in the interaction between the free electrons inside the copper-oxygen layers and the strongly correlated electrons between them. The Hamiltonian for this process has the form [ 11 ]: / t = H o +Hi,t ; Ho=He+Hosc,

(74)

Hose= ~ 12qb~ bq,

(75)

q

He= ~ ~s(k)a~ask, s,k

R(q)=(2K2qM)-~/2(bq+b+),

s = 1, 2, 3...

Hint= a 2 R(q)(a=~a~'.k-q+a+~'ka~,k-q)

(76) (77)

"

Here a is the layer number ( a = 1, 2, ..., N - 1 ) ( a ' = ot + 1 ); ak, bq are the operators of the Fermi and Bose fields, Q is the value of the hole-oscillation interaction, t2q are the polaron frequencies and es(k) is the dispersion law for the holes in the copper-oxygen layers. After reduction of the Hamiltonian ( 7 4 ) - ( 7 7 ) we obtain a new Hamiltonian in which the polaron oscillations are averaged [ 12 ]. In the limit of the weak coupling between the holes of the copper-oxygen layers and the polaron oscillations we obtain as usual the Gor'kov set of equations for the multiband model:

G,~(x,x,)=GO(x_x,)+A,~p f Go,(x,x o . .)Fp(x . . . . , x )F,~(x + ,,, x , ) d 4 x " '

(78)

F~(x, x') =A~ B f G°(x, x" )F~(x", x" )G=(x", x') d4x " ,

(79)

F,~=- T~(q/,~(x)q/~(x' ) ) ,

(80)

£ = - ~0 +Es(p)-Uo, £ G ° ( x - x ' ) = ~ ( x - x ' ) ,

(81)

A~=-Q2/122M,

(82)

a, fl= 1, 2 ..... N ,

fl=a+l,

where z is the Matzubara time. Here £2o is either the limiting frequency of the Wigner crystal or the optical oscillation mode of the polarons in the superstructure. From ( 7 8 ) - ( 8 2 ) we have the equation for the calculation of the critical temperature Tk (fig. 8):

A.A. Remova, B. Ya. Shapiro/ Polaronsand high Tksuperconductivityof copperoxides

213

. = * C ~

-: ,+ • =x:f°). •

.



<2> +



,

.

?+

x: . . . . o





.

(83)

[email protected]

.=F3(o)

= G,°(p,~) ......

G~(p.o,,)

~ = .

...

G;(p,~)

..

n =0,+-1 . . . .

co"n =¢cg(en+O

Fig. 8. The diagram form of the Gor'kov equations near the critical temperature.

Here p is the m o m e n t u m , T is the temperature. For three layers we have from (83) the set of linear equations. d2p FI = ~n J Kll (P, O)n)F2

(T~52 '

F2= ~ ~ KEE(P, og.)Fl ( -dEp ~)z+~

f KEE(P, o9.)F3 (2~) d2p 2 ,

F 3 = ~ j" K33(P, oo.)Fz dEp (2n) 2'

K.=AoG°(p, oJ.)G°(p, - c o . ) , Ao

~---~AI

1 --~'~'22

= A 3 3

(84)



After integrating in (84) we obtain the condition of consistency for linear equations:

T~

+

n

Fig. 9. The critical temperature as a function of the copper-oxygen number layers. The dependence has a saturation due to the dielectrization of the layers far from the doping layers.

214

A.A. Remova. B. Ya. Shapiro / Polaronsand high Tk superconductivityof copperoxides 1.14/2o -AoN1 (0) In---T---

1 _ _ -AoN2(0) In 1.14£2o T 0

0 1 14K2o -AoNz(0) l n ~

1 --AoN3 (0) In

1.14/20 T

(85)

1

We have for Tk in this case

l-g2g3 ln2

1.14£2o Tk

g2gl ln2

1.1412o = 0 Tk "

(86)

As a result Tk = 1.1400 exp ( ~--g

t g21+ g2 ~33),

gi=Ni(O)Ao,

i=1,2,3.

(87)

Here N~(0) is the density of states at the Fermi level for the nth layer. For n layers we have Tk = 1.14Oo exp( -- 1/x/En_s~ ' g,g,+l ) •

(88)

Tk VS. n is shown in fig. 9. It is clear the copper-oxygen layers which are far from the doping layers have a density of states (DOS) lower than the DOS in the nearest layers. Thus, DOS is an inhomogeneous function of the coordinate perpendicular to the ab plane. In this case we have saturation for Tk in (87) (fig. 9). 7. Conclusions The main results of this paper are: 1. The heavy anisotropic polarons may appear in the layered structures due to the strong electron-photon interaction between the copper-oxygen layers. These polarons may take different spacial forms. 2. The characteristic frequencies of the polaron oscillations have been determined. These frequencies are much higher than the characteristic optical mode for these systems. 3. It is shown that the plane polarons may form a Wigner crystal. 4. The influence of the magnetic field on the conditions of formation of the polaron has been studied. The critical magnetic field destroying polarons Hk has been calculated. 5. The critical temperature of the superconductivity is obtained. We supposed that in these systems there are two electron groups: the free electrons of the copper-oxygen layers and strongly correlated electrons between these layers. The latter are the source of the boson mode for superconductivity. The free electrons may become coupled due to the interaction with these modes. If under this interaction the electrons travel from one copperoxygen layer to another, then the critical temperature increases with increasing number of layers n (fig. 9). (The thallium layers in T I - B a - C a - C u - O and the copper-oxygen chains in Y - B a - C u - O systems suppress the Peierls instability in the copper-oxygen layers. Far from the thallium layers the copper-oxygen layers are in the dielectric state. Therefore, the dependence of Tk on n tends to saturate. ) 6. The spindle-like polarons may appear and play a role in the superconductivity of the yttrium ceramics. These polarons may be caused by the electrons from copper-oxygen chains moving along the C axis (the polaron is created due to polarization of the copper-oxygen layers.) (fig. 5). The superconductivity in this case occurs in the copper-oxygen layers only. This is the usual BCS pairing in which the role of the phonon frequency is played by the frequency of the oscillating polarons.

A.A. Remora, B. Ya. Shapiro/ Polaronsand high Tksuperconductivityof copperoxides

215

7. The numerical estimates for the parameters of the different types of polaron are: a) For spindle-like polarons with parameters of the substance Oto--~0.01,

O9o-103K,

ao---0.5~,,

lz~_5 ~

(89)

we have - ~E---4000K,

C2o---2500K ,

dz---2.5A.

(90)

b) For plane polarons with parameters of the substance Oto-~0.01 ,

¢Oo--- 103K,

ao--0.5 ~,,

lz--- 3.25 ~,

(91)

dz~_lz,

-~-5X103K.

(92)

one obtains (fig. 2)

dp~-ZA,

~2o= 2 X 103K,

c) For the polaron Wigner crystal using the condition dp < rs we have

rs>ao, N~
(93)

The highest frequency of the Wigner crystal is (69) ¢ooo"" %

(M)w2(ao~ 3/2 -kr~ /

-~2X103K.

(94)

It may be seen from the formulas for Tk that the frequencies (90), (92), (94) are sufficiently high to obtain Tk_~200K.

Acknowledgements We would like to thank Prof. A.Z. Patashinsky for his attention. We are grateful to Prof. V.I. Belenicher and Prof. E.B. Amitin for helpful discussion. This paper was supported by Grant No. 198 of the Soviet Academy of Sciences.

Appendix A Let us calculate the function Ho(p) in eq. (27) from (18): Z=Zo(p) = (d 2 n)-t/2 e x p ( - p 2 1 2 d 2 ) , llo(p ) -

4a°e2 2 f e-(a-a')2/d~ dp d~°=rcx/~ --alp e -v 2"'~/~aP I° ~ 2dp P2~)",~

(A.1) (A.2)

Here Io(x) is the Bessel function. Using the expansion

IIG(p) =/70(0) +zr~; (0)p2/2,

(A.3)

we obtain the following result: x/~4°t°e2 f (e-P2rb/2a~lo(P2 ~-l+2P-~d2p)e-p2/a~dp=(x/~l)n3/240to e2

nd 3

,

\ 2dp ]

(A.4)

216

A.A. Remora,B. Ya. Shapiro/ Polaronsand high Tksuperconductivityofcopperoxides

Appendix B Let us calculate the integral from (28)

i X~(x,y)z~(x',y') d.xdydx' dy' I= -~ x/ (x_x ')2+ (y_y, )2+ (z-z' )2 ' Zo =

1 (Kd2) 1/2

exp(-p2/2d2),

(B.1)

p = ~ 2 .

(B.2)

Substituting (B.2) into (B. 1 ) we have after transformation

i e-a(x-x')2-2axx' ; e-a¢2-2~'~-2ax" -~ x/(x_x ,)2+a dxdx' = ,/~

d~dx'=

x

f e-~2/2 d

~=x-x'

,

(B.3)

--oo

1!

e-a2/2d~ (B.4)

For new variables (B.5) we obtain the result:

I=-~e(Z-~')2/2d2 _=

; e-'~2 d q Iz--z'l/x/~da

n 1 -z- z,,"2'2d ]c~_~_e, / ,2 e r f c ( I Z - Z _ ' I ] N/Zdp \ dpx/2 ,/"

(B.6)

(B.7)

References [ 1 ] B.Ya. Shapiro, I st Soviet Conference on HTS, Sverdlovsk, 1987. [2] B.Ya. Shapiro, Phys. Lett. 127 (1988) 239. [3] B.D. Laikhtman, Zh. Exsp. Teor. Fiz. 68 (1975) 1806. [ 4 ] L.S. Kukushkin, Pisma Zh.E.T.F. 7 ( 1968 ) 251. [5] A.M. Kosevich, Fiz. Nizk. Temp. 4 (1978) 902. [ 6 ] L. Bonsall and A.A. Maradudin, Surf. Sci. 58 (1976) 312. 17 ] T. Ando, A. Fowler and F. Stern, Rev. Mod. Phys. 54 ( 1982 ) 1. [8] L. Bonsall and A.A. Maradudin, Surf. Sci. 58 (1976) 312. [9] G. Meissner, H. Namaizawa and M. Voss, Phys. Rev. B 13 (1976) 1370. [ 10] Y. Shimakawa, Y. Kubo, T. Manako, Y. Nakabayashi and H.G. Igarashi, Physica C 156 (1988) 97. [ 11 ] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinsky, Metodi Kvantovoi Teorii Polya v Statisticheskoi Fizike (Nauka, Moscow, 1962). [ 12 ] A.V. Svidzinskii, Prostranstvenno-neodnorodnie zadachi teorii sverkhprovodimosti (Nauka, Moscow, 1982).