High-Tc superconductors and interaction between pairs

Physica C 211 (1993) 380-392 Norlh-Holland

High-Tc superconductors and interaction between pairs

Huei Peng a and L. Luo b • Institute of Applied Mathematics, Beijing, China b Department of Physics, University of Alabama, Birmingham, AL 35205. USA

Received 24 September 1992 Revised manuscript received 12 January 1993

We propose that there is dynamic interaction between pairs (called the pair interaction ). The pair interaction is independent of the formation mechanism of pairs, and, thus, may be introduced into different models of superconductivity as long as there are pairs in those models. To show the effects of the pair interaction, we introduce it into the BCS theory, and find that the energy gap in terms of 2d/keTc varies from zero to 8 and higher. To test for the existence of the pair interaction, we derive a general expression for the x-dependence of the pair interaction, where x denotes all factors which may affect the pair interaction and the energy gap, such as the mass of the ion (an isotope-like effect), external pressure, composition, etc.

1. Introduction

High-To superconductors (HTSC) [ 1 ] have [2], compared with the predictions of the BCS model [3 ], some unusual properties, such as larger energy gaps, a smaller or zero isotope effect, coexistence of superconductivity and antiferromagnetism, etc. The fundamental question has been asked whether the BCS theory is totally inappropriate for HTSC's [4]. Although the absence of the isotope effect does not exclude the phonon mechanism, it appears that there is more than one interaction which leads to HTSC's. A first concern is the energy gap which is twice as large as that in the BCS model in terms of 2A(0)/ kBTc [ 5 ]. New mechanisms for strongly bound pairs are needed.

Most efforts have been focused on seeking new pairing mechanisms which would explain among others how the strongly bound pairs of HTSC's are formed. A variety of pairing mechanisms of HTSCs has been proposed but the list is still incomplete [ 2].

In addition to seek new pairing mechanisms, we suggest a different approach: there may be supplementary mechanism (s) not responsible for forming pairs (thus independent of pairing mechanisms), but strengthening (under certain conditions) the binding energy of pairs. The avantages of introducing

supplementary mechanism(s) are the following. Once a supplementary mechanism proves correct, it may bc combined with different pairing mechanisms. With the help of supplementary mechanism(s), a pairing mechanism is not necessarily strong coupling in order to explain a large energy gap. As an example, we will show that the combination of the BCS phonon mechanism and a supplementary mechanism yields a large energy gap. Some of the unusual phenomena of HTSC's, which do not relate to how pairs are formed, may be explained separately.

To answer the above-mentioned question, in section 2, we study once more the BCS theory, because of two reasons: (1) properties of conventional metallic superconductors (SC) are well accounted for by the BCS model, and some basic phenomena of superconductivity, such as the energy gap, still exist in HTSC's. Thus, in our opinion, the BCS theory contains the ingredient of a complete theory; (2) so far, the experimental data of HTSCs has been mostly compared with the BCS model. We empha- size an unnoticed crucial approximation in the BCS model and other models (e.g., the Eliashberg model [16]) and show that this approximation actually forbids one to apply these models to the HTSC's. Thus we must distinguish the BCS model from the

0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

Huei Peng, L. Luo / Interaction between pairs 381

BCS theory. The BCS theory is referred to by the following concepts; pairing, quasi-particles, the phonon mechanism, and the BCS reduced hamiltonian with its general solutions. On the other hand, by the BCS model, we mean particular solutions of the BCS the- pry with several assumptions introduced in ref. [ 3 ]. Then, without the use of the crucial approximation of the BCS model, a general expression for the energy gap of the BCS theory is derived. The BCS the- pry is richer than the BCS model and still not sufficient to explain HTSC's, but it provides an example for a new model to compare with.

In section 3 we propose a possible supplementary mechanism, the interaction between pairs, hereafter called the pair interaction. The pair interaction may be responsible for some of the non-BCS phenomena. Even when the pair interaction is not dominant, it may affect properties of superconductors (SC's) considerably. Since the pair interaction is independent of the pairing mechanisms, it may be introduced into other models of HTSC's as long as there are pairs. To show how the pair interaction works, in section 4 we introduce it into the BCS theory. We find that the energy gap in terms of 2J(0)/kBT~ varies from zero to 8 and higher. To test the existence of the pair interaction and its effects on HTSC's, a general expression which shows how the pair interaction is affected by the surrounding is derived. Par- ticularly, the pressure effect and isotope-like effect are presented.

2. The energy gaps of the BCS model and BCS theory

To show the difference between the BCS model and the BCS theory, let us first write down the energy gap of the BCS model:

23(0) -~3.528, (1) kBT¢

which implies that the energy gap is the same for different SC's. The fact is, however, that, even for low- T¢ SC's, the energy gap is material-dependent. Equa- tion ( 1 ) fails to explain this fact. In the derivation of eq. ( 1 ), several approximations have been used, such as the weak coupling, N(0) V<< 1, and the constant electron-phonon potential V.

Before answering the fundamental question mentioned in the introduction, it is necessary to elimi- nate as many as possible of these approximations, and derive a general solution for the BCS theory. Without a general solution of the BCS theory, it is not easy to tell whether the introduced approximations put a constraint on the application of the BCS model, or the BCS theory is partially valid, or the BCS theory does not work at all. Actually, some of these approximations have been modified even before the discovery of the HTSC's, such as in the Eliashberg strong coupling model [6 ], and through non-constant attractive potentials [7]. Because of the large energy gap of HTSC's, the Eliashberg strong coupling model has been tried and failed as an ex- planation of HTSC's.

Now, let us study a crucial approximation of the BCS model. It is a familiar manipulation in the BCS theory to replace the sum by the integral

htOD tic hO)D/2

~ - , f d, or f d ( ,6c ' /2 ) , (2) - - htOD - - ]tc htOD / 2

where flc=-(kBTc) -t. The crucial approximation, then, was introduced into the BCS model and other models, that

htOD - - > > 1 , 2kB Tc

the limit of the integral

flchtOo/ 2 ---,09, ( 3 )

which implies that, since T¢ is much smaller than the Debye temperature 0D, the integral limit fl~ho~/2 can be replaced by infinity. We refer to this approximation as the low-To approximation which makes the calculation considerably simpler, and most of the predictions of the BCS model, such as the energy gap (eq. ( 1 ) ), are based on it. Therefore, the BCS model is really the low-To approximate solution of the BCS theory. The low-To approximation actually forbids one to apply the BCS model to those SC's which do not satisfy eq. (3).

For the low-T¢ SC's, even though the low-T¢ approximation is true and the BCS model works, the low-T¢ approximation brings a disadvantage: im- portant information about the htODfl¢-dependence of properties of SC's is lost; instead, one obtains just

382 Huei Peng, L. Luo /Interaction between pairs

constant. This can be seen by considering an arbi- trary integral

F(h~D~)~ htODi~:/2

f (x) dx, 0

where F is a function of htODfl c. By replacing the upper limit by ~ , if ho~flc/2 is very large, we have

oo

F--- f f (x) dx=cons tan t . 0

On the other hand, for HTSC's with a typical De- bye temperature 0D= 300-450 K and T~-80-125 K, the low-T~ approximation no longer holds; thus, the BCS model and any model employing the iow-T¢ approximation cannot be applied to the HTSC's. It is more appropriate to derive general solutions for the BCS theory and compare properties of HTSC's with the general solutions, and then the fundamental question mentioned in the introduction may be an- swered more precisely.

To derive the general expression of the energy gap in the BCS theory without using the low-Tc approximation, we start from the ground state

Ak.(O) (4) Ak(0)=-- ~ Vk,k 2Ek--'-- ~ ,

where Ek= [~2+A2(0)],/2. We still adopt the following two assumptions of the BCS model: Ak-A and V~,k= - V; the latter actually implies that the nature of the attractive potential (s) has not been taken into aecount, i.e., the BCS model contains all kinds of attractions (including the electron-phonon interaction) as long as the attractions may be represented as a constant. The discussion of a non-constant potential is beyond the scope of the present paper (see papers in, for example, refs. [2] and [8] for HTSC's, and ref. [7] for low-T~ SC's). The one has, as in the BCS model,

~¢.OD

i ! d~ N(O)V- [E2+A2(0) 11/2 " (5)

To find the relation between A(0) and To, considering the excitation state, one has, with Ak(T) --A(T) and fl-= ( kBT) -l,

A(T) ( 1 - 2J~) (6) A ( T ) = V ~ 2Ek(T---- ~

where fk ~ {exp [flEk (T) ] + 1 } - ~ and Ek (T) = [ ~ +A~(T)] 1/2.

Replacing the sum by the integral, and letting T~ To, eq. (6) becomes

,&:h~D/2

1 f tanh Xdx, (7) N(O)V- x

0

which shows that the ratio of the Debye temperature to the T¢, hwD/kaT¢, is determined by the coupling N(O)V.

As is well known, ifeq. (7) is integrated by parts and the upper limit is replaced by ~ , one obtains the expression of the BCS model, kaT¢=l.14hogD exp[-l/N(O)V], which, combining with eq. (5), yields the energy gap of the BCS model, eq. ( 1 ).

If we do the integral without replacing the upper limit by co, the combination of eqs. (5) and (7) yields

2A(0) 2hC0Dflc - - - - - ( 8 )

sinh r ' kBT~

where

~.. 4 f flchtOD ) z-- arctan , (9)

,=o (2n+ l )n ~, (2--n-~)n

and we did not make the weak coupling approximation, N(0)V<< 1.

The numerical solutions of eq. (8) are listed be- low, see table 1.

The low-T¢ SC's and HTSC's correspond approx-

Table 1

p~tOD 2 3 4 5 6

2d(O) 3.85 3.75 3.68 3.63 3.60 kBTc

7 8 9 10 20

3.58 3.57 3.56 3.55 3.53


imately to the ratios fl~htoo> 12 and flchOgD"2-10, respectively. It is readily seen from eq. (8) that the maximum value of 2zJ/kBT~ is 4 when flchtoo~O, and that there is a minimal value of 2A/kaTc, 3.528, when fl~htoD satisfies the following condition:

coth z tanh(fl~hmo/2) = I, (10)

i.e., p~tOD--*OO. The BCS theory is applicable to those SC's which have an energy gap 23/kBT~ between 3.528 and 4.

SC's with an energy gap 2zl(O)/kBTc either >4 or < 3.53 are out of the range of the BCS theory, unless the BCS theory is modified. The BCS model is the limit of the BCS theory on the low side:

BCS theory

o 3.s 4 = i ' t

BCS model k~T¢

The BCS model works for one point on the axis of 2d/kBTc, while the BCS theory covers a certain length of the 2z~/kBT~ axis. A better theory or model should cover the 2zt/kBT~>4 part of the axis, at least up to 2~/kBT~ ~- 8 for the present HTSC's.

Comparing the energy gap, eq. (8), of the BCS theory with that of the BCS model, eq. ( 1 ), the difference is the following. In the BCS theory, the energy gap 2~(O)/(kaT¢) varies with the ratio flchOJD (eqs. (8) and (9) ) which is determined by the coupling N ( 0 ) I / ( e q . (7) ) . This provides an explana- tion of why the energy gap 2J(0) / (kBTc) varies with different materials.

There is no significant difference in the numerical predictions of the energy gap between the BCS model and the BCS theory. Now, there is no theoretical constraint on the application of eq. (8) to HTSC's, and the BCS theory (like the BCS model) cannot deal with HTSC's. New mechanism(s) , either new pairing mechanism (s) or supplementary mechanism (s), are needed.

However, the general solutions of the BCS theory provide an example for a new model to compare with and make us alert to avoiding the use of the Iow-T~ approximation in dealing with HTSC's.

3. Pair interaction

There are several reasons for introducing the pair

interaction into micro-theories of SC's. First, pairing is still a basic phenomenon of HTSC's and the mechanism of superconductivity. The pair interaction which is proportional to the density of pairs has been introduced into the Ginzburg-Landau theory (GL) and works very well [9]: not only reproducing the time-independent GL theory, but also in automati- cally providing the time-dependent GL theory. The- oretically, it is better if a micro-theory contains the concepts of the macro-theory. Experimentally, it is recognized that the carriers concentration is an im- portant parameter in determining Tc of HTSC's, and that an external pressure may affect Tc [ 1 ].

To our knowledge, the pair interaction has not been taken into account in micro-theories of SC's and was only mentioned in the BCS theory [ 10]. The BCS theory treats SC's as if dynamical interactions ex- isted only between members of a pair. The interaction between pairs in real metals is considered to be entirely due to the Pauli principle rather than due to true dynamical pair interaction [ 10].

However, it is natural to ask: (1) Is there a dynamic interaction between pairs (especially between those pairs which are near each other)? (2) Is there scattering between pairs? If so, what are the effects of the scattering between pairs?

The answer to the first question is that, since pairs carry the same charges, there must be a Coulomb repulsive interaction between pairs. A repulsive scattering of a pair by another pair is due to their Cou- lomb fields. Therefore, we argue that the pair interaction (which is a Coulomb interaction) tran- scends the Pauli principle and should be introduced into superconductivity.

The Coulomb interaction is the real mechanism of the pair interaction. In principle, the pair interaction can be described by a summation (over all pairs) of Coulomb potentials between any two pairs. But it is difficult to solve for the Coulomb interaction between all of the pairs mathematically. In such cases it is usually advantageous to consider pairs as a con- tinuous fluid with only macro-properties. Therefore, to deal with the pair interaction, in this paper we adopt a fluid model, i.e., treat the cloud of pairs as a fluid. Then we use fluid quantities, such as the density of pairs, pressure due to the scattering between pairs, etc, to describe the properties of the pair fluid.

384 Huei Peng. L. Luo /Interaction between pairs

As one does in fluid mechanics, we take a pressure model of the pair fluid saying that the pressure is proportional to the density of pairs, n.

The pair interaction is not a pairing mechanism, but rather a supplementary mechanism to either en- hance or reduce the energy gap of existing pairs, no matter what the mechanisms of pairing are; it is re- lated to the density and size of the pairs as shown later.

There are two effects due to the pressure. (1) The smooth-pair-distribution-pressure effect. This effect is independent of the size of pairs and can be explained in both languages, the Coulomb repulsive interaction and fluid mechanics. When there is a perturbation in the density of pairs, there is a higher Coulomb repulsive field in a higher-pair-density region. The higher Coulomb repulsive field will push pairs to move to a lower-pair-density region in which there is a lower Coulomb field. However, as mentioned above, we will not describe the effect in terms of the Coulomb interaction. On the other hand, in terms of fluid mechanics, there appears an interaction which is due to the gradient of the pressure. Since the pressure is proportional to the density of pairs, the pair interaction is proportional to the gradient of the density, Vn, which acts like a force on a pair and pushes pairs to spread out as uniformly as posiblc in space. The work done by this interaction for one pair is proportional to the density of pairs, flc;n [9]. The coupling parameter tic is material dependent. We call it the smooth-pair-distribution-pressure interaction. For a set of pairs, the hamiltonian is

Hp= ~ ~. ( f lcn)kb~bk, (11 ) k

where

b~ c * * = k C _ k ,

bk m C - k C k ;

b~ and bk are the creation and annihilation operators for pairs, respectively. The factor ½ is due to the fact that the hamiltonian for a set of pairs counts each pair twicc. The smooth-pair-pressure interaction has been employed to explain the non-linear term of the GL theory [9], fllgtl 2. We did not absorb the factor

into the coupling parameter fiG, because fig and the parameter B of the GL theory have the same interpretation.

(2) The pair-squeezing-pressure effect which occurs only when the size of pairs is small as shown later. This effect can be understood by considering a pair as a mass-spring-mass system, where two masses are the two members of the pair, and the spring stands for attractive forces between the two members. When the spring is so short that the amount of pairs in between the pair under consideration is much smaller than that of pairs surrounding and interacting with the pair under consideration, the net Coulomb interaction on the pair points inward, i.e., the Cou- lomb interaction, pushes two members of the pair toward each other. In the terms of fluid mechanics, the pressure on the pair due to its scattering with other pairs will squeeze the pair under consideration. This pair-squeezing-pressure interaction acts like an effectively attractive (EA) potential and is proportional to the pressure and, thus, to the density of pairs, Qsn, where Qs is the coupling parameter yet to be determined. Since the pair-squeezing-pressure potential is an attractive potential, the hamiltonian of the EA potential of a set of pairs is

Heap=- ~ (Qsn)k.~.b~,bk, (12) k ' k

where Q~n > 0, and

( Qsn )k,k = ( k, - k l Qsnl k', - k ' ) , (13)

which represents the scattering by Q~n of the pair oc- cupying the state I k', - k' ) into the other pair state Ik, - k ) .

The pair interaction does not depend on a specific pairing mechanism, so it is reasonable to expect that the pair interaction may be introduced into all of the pairing models of superconductivity. As mentioned before, we expect that the BCS theory contains the integredients of a complete theory of SC's. To show how the pair interaction works, in this paper we introduce the pair interaction into the BCS theory.

4. The pair interaction in the BCS theory

In the BCS theory, only the kinetic energy of the pairs and the electron-phonon attractive energy between two members of pairs are taken into account. The well-known BCS reduced hamillonian for the ground state is


Hr,d=2 ~ ¢kb~bk+ ~. Vk.kb~.bk. (14) k k 'k

Introducing two effects of the pair interaction, i.e., combining H~¢a, Hp, and H¢, v, we have the total hamiltonian

H = 2 ~ ~kb~b, + ~ Vk.kb~,.bk k k 'k

+~ ~ (flon)kb'~bk- ~ (Qsn)k.,b~.bk. (15) k k 'k

Following the BCS theory, the ground state should be described by the product of the occupation operators for all pair states,

I~'>= I-I (uk+v~b~)lO>, k

where 10> is the vacuum state, v 2 and Uk 2 are the probabilities of pair occupancy and pair vacancy, respectively, and v~ + u~, = 1. The energy of the super- conducting ground state relative to the normal ground state is the expectation value of the hamiltonian,

k k k '

+ Z Vk'kUk'Vk'UkVk--~. (Q~ Z V~,,,)k'kUk'Vk'UkVk, k'k k 'k k"

(16)

where n=Yv~, has been used. On minimizing w, Ow/Ov~ =0, we find

' ( ' ~ ) v ~ = ~ l - - ~ k k , (17)

U~=~ 1 + , (18)

A(0) (19) UkVk= 2Ek(0) '

where

E~(0) =-x/[C~ +A~(0) ] , (20)

(k -= ~k + Lk(0) - Fk(0) , (21 )

Ak(0) =ABcs.k(0) + Gk(O) , (22)

Gk(O)- ~, (Q~n)k'kUk'Vk', (23) k '

ABCS,k(O) -- -- ~. V~.~u~.v~., (24) k '

L~(O )= ½ (flGn )k , (25)

Fk(O)=---~ ~ (Q~k)k.k-Uk'Vk'Uk"Vk-. (26) k'k"

ABCS.k is the BCS energy gap due to the phonon mechanism. Gk is the additional energy gap attributed to the pair-squeezing-pressure potential. Be- cause a pair cannot tell the origins of the binding energies, and because, practically, one can only mea- sure the total energy gap, Ak is the effective energy gap to be actually determined. We do not need to distinguish two attractive potentials, Vk'k and (Qsn)k.k. The/:k can be interpreted as follows: while the pair-squeezing-pressure interaction enhances the effective binding energy of a pair, it also reduces the kinetic energy of two members of the pair by an amount of 2Fk. As mentioned above, the smooth- pair-pressure interaction forces pairs to distribute as smoothly as possible. This interaction increases the kinetic energy of a pair, represented by 2Lk= (flGn)k. Therefore, the total kinetic energy of a pair is 2 ( ~ k - - F k + L k ) .

To solve eq. (22), we make some assumptions: for I (k I < hwr~, Vkk = -- V, Aacs.~ = Aacs = constant, (Qsn)k'k. = Qsn, Gk = G = constant, Fk = F = constant, and Lk=L=constant. Thus dk=d=constant. Sub- stituting eqs. (19), (20), (23), and (24) into eq. (22), we obtain

fitOD

2 ~ dc N(0)-~, = ( 1 + D ) (62+A2)t/2, (27)

-- hood

where N(0) is the density of stales, and

D=- Q,n (28) V '

which is the ratio of the pair-squeezing-pressure potential to the phonon attractive potential.

To obtain the energy gap in terms of 2A (0)/kB To, one has to find the hamiltonian of the excitation state. When we consider the situation where T~T¢, we have, with/Yc- = l/kBTo

flehtODI2

1 _ ] t a n h x d x - z , (7) N(O)V x

0

which is exactly the same as in the BCS theory, because of the fact that at T ~ T¢, the energy gap and the effects of the pair interaction, G, F, and L. vanish.


Now we finally obtain the energy gap by combining eqs. (27) and (7):

2/I (0) 2hWDflc

- - T

2

( ' ))) × 1 - ~ tanh I+(Q~n/V) ' (29)

o r

2A( O ) 2hWDfl¢ kBT¢ . , [ I / N ( O ) V "~

s'nn , l ;

l / L , [ l / N ( O ) V "1"~2"~ × ( 1 - ~kh-~D tannkl ~ ) J , ] ) , (30)

which is a generalization of eq. (8) of the BCS theory; when the pair interaction is neglected, eq. (29) reduces to eq. (8). The products, Q,n and L = ~ n / 2, play a role, not only n. In the derivation ofeq. (29) we did not use the low-To approximation, and small terms are ignored.

Equations (7) and (29) show the following. (1) There are effects of the electron-phonon coupling N(0) V. The N(0) V has two roles: first, to de- termine the ratio htOD/(kaT¢) (fig. I ) for both low-

Tc SC's and HTSC's in the same way: Iow-T¢ SC's have large htOD/kBTc which corresponds to the coupling N(0)V---0.2 to 0.35, while HTSC's with 3 < hwDk, Tc < 12 correspond to the coupling 0.8>N(0) V>0.4, approximately. The electron- phonon potential of HTSC's is nearly twice as large as that of low-T¢ SC's; second, it affects the energy gap. In the BCS theory of low-T~ SC's, the effect of the coupling N(0) V on the energy gap, via the ratio hwD/k.T~, is described by eq. (8). For HTSC's, the effects of N(0) V are more complicated (eq. (29) ). (2) Then we have the effects of the pair interaction. The pair interaction has nothing to do with the ratio htOD/kaT¢, but affects the energy gap via L( =~cn/ 2) and Qsn.

Note: an imporiant feature ofeq. (29) is the dependence ofthe energy gap 2A(0)/k.T¢ on the ratios of the pair-squeezing-pressure attractive potential to the phonon mechanism (Qsn/V), and of the smooth- pair-pressure potential to the Debye energy (fl~n/ htoo), and hwt>/k,T¢. Q~n/V shows to what per- centage V is dominant over Q~n.

5. Spec ia l s i tuations of energy gaps

Here we only show two extreme situations of eq. (29).

N(O)V

°'6r i ' I

0.55 t !

f' 0.5 ~

O.45 i

0.4 i

I 0.35 -

t

0.25 ~

- G r a p h i c - , -

\ \

20 ~ 60 80 100

Fig. 1.

~w~

k i t e


( I ) Q,n/V~_O and flGn/hooD~O, i.e., the pair interaction only contributes to the motion of pairs, not to the squeezing of pairs. Equation (29) reduces to

kBT¢ - s inhr - ~ ~ t a n h r . (31)

Therefore the energy gap is reduced, i.e., pairs are more easily broken. For htor,/kaT¢> 12, tanh r= 1, so eq. (31 ) becomes

2410) 2 ooofl ( l - l ( L kBT¢ - s inhr ~ \ ~ ] ] . (32)

As mentioned above, there are two barriers in the energy gap of the BCS theory; one is 3.528 at the low value side, the other is 4 at the high value side. The comparison between eq. (8) of the BCS theory and eq. ( 31 ) shows that the pair interaction L breaks the low barrier of the BCS theory and allows one to deal with small energy gaps:

Eq. 1311

BCS theory 1

3.5 4 2~

BCS model kBT ¢

For example, consider a metal SC with L/htOD". 0.5 and hooD/knT¢~-290; then eq. (32) yields 2A/ kate ~- 3.07. Some of metal SC's have similar param- eters. For a higher value of L/hooD, one obtains a smaller energy gap. An extreme example is a SC with L/ho.~ ~- ~ and htoD/knT¢ > 12; then eq. (32) yields

2A(0) ~ 0 , k,,T~

which implies a gapless SC. What are the conditions under which the pair in-

teraction L/hooD will play a role? From the defini- tion of the coherence length ~o,

( ,o- hA ' (33)

we obtain, from eq. (31), 2

( & ) ( s i n h r ~ ) =2 1 n(o (c°th r)2 " (34)

The bigger Go is, the larger is L/kBT¢. When [vrsinh r/(n(,oooD)] > 1, there will be no smooth- pair-distribution-pressure effect.

We have shown that eq. (31) works for SC's with large-size pairs and energy gaps vary from zero to 4. (2) Q~n/V#O and flGn/htOD"O, i.e., the pair interaction only contributes to the pair-squeezing-pressure potential, not to the motion of pairs (or only negligibly). In this situation, the energy gap must be enlarged. Equation (31) becomes

2A(0) 2hooDfl¢ (35)

sinh l + ( Q s n / V )

How does the effective attractive potential, Qsn, depend on the size of the pairs? We have, from eq. (35),

Qs n r - - 1 , ( 3 6 )

V arcsinh(nooDG~ k VF /

which shows the Go-dependence of the ratio Q,n/V: when z> arcsinh (hooD(o/v~), there is the pressure- squeeze-pair potential due to the pair interaction; the ratio Q~n/V< 0 implies the non-existence of pair interaction. For example, consider a low-To SC with Debye temperature 0D-~ 400 K, T¢-~ 1.28 K, v~--- 106 m/s, and a relatively larger ~o ~ _ 1.8× 10 - 6 m, then

r=5 .8 7 ,

arcsin (nO.Op) = 6.32 ;

therefore, there is no pair interaction to squeeze pairs. On the other hand, for a HTSC with 0D" 400 K, T¢-~ 100 K, vr= 105 m/s, and a relatively smaller ~o --~ 10 -9 m, the ratio is

Q,n "-0.2. V

The small Go corresponds to the existence of the pair- squeezing-pressure effect, and, thus, enhances the energy gap.

Equation (35) shows that the energy gap depends on both the electron-phonon coupling N(0)V and the pair-squeezing-pressure interaction Q,n, and is very sensitive to the ratio Q,n/V (fig. 2). Figure 3 is a magnified picture of the low Q~n/V part of fig. 2. We consider three special cases of eq. (35). ( 1 ) For SC's with a given Q,n/v, say 0.4, i.e., the electron-phonon attraction V is 2.5 times the pair-


2~(o)

kBT¢ I 15

~D

kBT e

15

12.

-Graphics3D-

Fig. 2.

squeezing-pressure potential (actually, for a given SC, the ratio Q,n/Vis to be determined by eq. (35) ), we observe the energy gap 2A(O)/kBT~ and hCOD/kBT¢ to satisfy table 2 (from fig. 2).

This table shows that the pair-squeezing-pressure potential breaks the high barrier, 4, of the BCS theory. Figures 2 and 3 and the comparison of tables 1 and 2 show the role of the pair-squeezing-pressure interaction. In the BCS theory, when flchO~D increases, the energy gap 2zt/kBT¢ decreases. The change in the gap, 8(2A/kBT¢), depends on the change of #~hogD, 8 (flchogD):

~(\k--~]2A ~= s i ~ h r ( l _ c o t h r t a n h ( f f ~ ) )

× 8(#¢htOD) . (37)

~sl ]

V

Since for flchO)D<GO, coth rtanh(flchtOD/2)> 1, the energy gap 2A/kBT¢ decreases when fl~htoD increases. When flc~/O)D--~OO, coth rtanh(flcfi~Oo)-*l, and 2d/ kaT¢ reaches the constant 3.528 of the BCS model.

The pair-squeezing-pressure potential, Qsn, re- verses this situation: when fl~fi~OD and D satisfy the following relation:

t a n h ( ~ ) = (1 +D)tan (l--+-D) ' (38)

the energy gap 2A/kBT¢ has a minimal value. Passing the minimal value, then, lhe energy gap begins to increase with flJTtOo. On the other hand, for a largcr value of D, eq. (38) no longer holds, and thus the energy gap 2A/kBT~ always increases with increase of


~D

k e T c

2A(0)

kBTc

2.

4.2

-Graphics3D-

0.02

Fig. 3.

Qs n

v

Table 2

~ O D

kB T¢

2d kB T,

4

6.1

5

6.31

6

6.52

7

6.72

8

6.92

9

7.11

10

7.29

Table 3

h o ~

kB~

2d(O)

kB~

3

9.01

4

9.58

5

10.18

6

10.78

7

11.37

8

11.94

9

12.49

10

13.03

390

Table 4

Huei Peng, L. Luo / Interaction between pairs

2A 4 4.5 5 5.5 6 6.5 7 7.5 8

kB~

~"" 0.05 0.134 0.216 0.299 0.383 0.467 0.552 0.638 0.729 v

Table 5

htoD 4 5 6 7 8 9 10

ks T~

N(0 ) Y 0.658 0.576 0.521 0.483 0.453 0.430 0.412 Q,n - - 0.552 0.505 0.466 0.435 0.409 0.388 0.370

v

It is interesting to see what is going to happen when the pair-squeezing-pressure potential and the electron-phonon interaction are in the same order, Q,n ~- v: it is possible, theoretically, that the energy gap may exceed that of the present HTSC's (table 3). (2) For a given value ofhooD/kBT~, say 4 (e.g., SC's with 0 D ~---400 K and Tc = 100 K, or 0D = 200 K and T~=50 K), eq. (7) shows that the chosen value of htoD/kBT¢ corresponds to N(0)V=0.658. One may have a variety of SC's with different energy gaps 2J / kBT'~ corresponding to different pair interactions Q~n/ V (table 4).

Recently discovered HTSC's have energy gaps

2J(O) - - - ~ 6 - 8 . kBT~

Table 4 shows that the combination of both an electron-phonon coupling N(0)V-,-0.66 and pair interactions from Q~n~-O.38v to Q,n~-O.73v or from Qsn~-O.25/N(O) to Qsn~-O.48/N(O) is sufficient to explain the large energy gaps of HTSC's. The coupling N(0)V=0.66 is stronger than that of low-T¢ SC's, N(0) V-~0.2 to 0.3, but weaker than the strong coupling N ( 0 ) V > I of the Eliashberg model [6]. Therefore, the pair interaction Qsn is a supplementary mechanism; with its help, a medium electron- phonon coupling may explain large energy gaps.

The energy gap 2A/kBT¢ is very sensitive to Qsn/ v. A change in Q,n/V causes a bigger change in 2A/ kBTc:

T

× [l+(Q~n/V)]2 . (39)

In the range of present HTSC's, for Q~n/v ~ - 0.35, we have

kB T¢ = \ k-~-~c ] =5.98~5 ,

hoaD kBT¢ = 5 - - , 8 ( ~ ) ~ - 6 . 6 4 ~ (~-~) ,

htob 6_ .8 (2J (0 ,~ (_Q~) kB T¢ = k, k-~-T-c-~ J "--7.28 8 ,

htoD _ 7~6{2A(0)x~ ~ ( _ ~ ) kB Tc -- \ kB Tc ] ~- 7.89 ,

which implies that in the HTSC range, a small change of the ratio Qsn/v, say, 0.1, will cause a 6-8 times larger change in the energy gap 2A/kBTc, 0.6-0.8. (3) For SC's with a given energy gap, say, 2A(0)/ kBT~ ~- 7, different values of the ratio hogo/kBTc (corresponding to different values of the electron-phonon coupling N(O)V) have different pair-squeezing- pressure effects (table 5).

6. Test of pair interaction: isotope-like and pressure effects

In this paper we are devoted to HTSC's, so we only


consider the test of the pair-squeezing-pressure interaction. To test the existence of this interaction, we derive a general expression of the x-dependence of the pair-squeezing-pressure interaction, where x stands for all factors which may affect the interaction, such as the mass of the ion, external pressure, density o f state, coherence length, composition, etc.

0In(D+ 1) (0 In mo Olnx = \ - O l n x

(0 In toD - ( 1 + D ) \ -Oi-nx

0 In T~'~ htoD 0 / n x ] t anh(2k- -~)

01nA'X . [ r o-- -Tn x) tannlt } •

(40)

Now consider the pressure and isotope-like effects. The isotope effect was the strong evidence prior to

the BCS theory that the electron-phonon interaction was responsible for the superconductivity. The isotope effect in the BCS model,

ToM'S=constant, a = ½ , (41)

was derived in the low-To approximation. Since the introduction of the pair interaction, the

isotope effect no longer holds. Following the same idea, we derive an isotope-like effect. In the new model, the energy gap is no longer proportional to either T¢ or too. Thus we obtain an expression of the isotope-like effect depending on A(0), D, and htoD/ kaTe. Let toDMt/2=constant and TdV/"=constant; eq. (40) yields

(1 +D)tanh( l - -~D )

a = i - tanh(fl~htoo/2) + O ~ ]

r 01n ( l+D) + tanh(fl~fitoD/2) 01nM ' (42)

where M is the ionic mass. The deviation from the isotope effect may reflect

the effect of the pair interaction. However, we should compare the isotope-like effect, eq. (42), with that of the BCS theory, not with eq. (41) of the BCS model. In the BCS theory "or= ~" is no longer the criterion of the isotope effect, because it was derived by the use of the low-To approximation. The isotope effect in the BSC theory (hereafter called the generalized isotope effect) has different expressions.

Generally, there are two approaches to derive the generalized isotope effect. (1) Start from eq. (7); this approach yields a coupling-model-dependence expression of the isotope effect; (2) start from eq. (8): we obtain an energy-gap-dependence expression of the isotope effect in the BCS theory:

1 tanhr (~ 01nA(0)'~ aBCS= ~ -- tanh(p~htoo/2) + ~n-M ]"

(43)

The advantage of this equation is that it does not depend on the coupling models one may choose for N(0) V. Once A(0), too, and T¢ are measured, aBcs is determined. A point to note is that the generalized isotope effect is valid provided that eq. (43) is satisfied.

By determining 0 In A/O In M, fl~htoD, and or, eq. (42) provides information on the M-dependence of D. The comparison between eqs. (42) and (43) shows the effect, attributed to the pair interaction, on the generalized isotope effect of the BCS theory.

The pressure effect on SC's plays a key role in the discovery and development of understanding HTSC's [ l ]. The effect of the density of the carriers on the Tc has been reported in recent works, see ref. [ 2 ]. The pressure effect will reveal the density-dependence of the pair interaction Q,n.

In the new model, the pressure effect is described by the following equation, obtained from eq. (40),

0 In T¢ 0in WD 0 In Vv 0 In Vv

[01nJ(0) 0In top) + t - 0 1 n ~ 0 In V-~-v tanh (pchtor~/2)

0 1 n ( l + D ) - (44)

tanh(l~chtoD/2) 0in Vv '

where Vv is the volume. For comparison, the pressure effect of the BCS theory is, from eq. (8),

0 In Tc 0 In too Oln Vv - Oln Vv

/a lnA(0) 0 In toO'~ tanhr . (45) ~-0 In V--~v 0 In Vv] tanh(pcho~)

392 Huei Peng, L. Luo / Interaction between pairs

7. S u m m a r y

We show that there is a theoretical constraint on the application of the BCS model and any model, in which the low-To approximation is employed, to HTSC's. The general solution of the BCS theory is not subject to the constraint, but still cannot explain HTSC's: it works for SC's with an energy gap of 3.53 to 4.

We propose that there is a pair interaction. As a supplementary mechanism to pairing mechanisms, the pair interaction changes the binding energy of a pair and, thus, changes the energy gap. To show how this works, we introduce it into the BCS theory and derive an equation o f the energy gap which breaks the barriers o f the BCS theory in both directions,

new model c

BCS theory

0 3.5 4 8 2~

BCS model kBT ¢

i.e., the e lect ron-phonon pairing mechanism combined with the supplementary mechanism (the pair interaction) explains small and large energy gaps. Thus, we suggest that the model proposed in this paper contains the ingredient o f a complete theory, and

that the pair interaction would play a role in other models of HTSC's in which there are pairs.

The pair interaction may be tested by the isotope- like effect (eq. (42) ), the pressure effect (eq. (44) ) , and the coherence length Go-dependence of the pair interaction (eqs. (34) and (36) ) . The isotope-like effect, eq. (42), will also reveal whether a new pairing mechanism other than the e lectron-phonon mechanism is needed for HTSC's.

References

[ 1 ] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189; C.W. Chu et al., Phys. Rev. Left. 58 (1987) 405; M.K. Wu et al., Phys. Rev. Lett. 58 (1987) 908; P.H. Hor el al., Phys. Rev. Letl. 58 (1986) 911.

[ 2 ] For recent works, see, for example, papers in Physica C 185- 189 (1991); V.L. Ginzburg, preprint, 1992.

[3] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.

[ 41 J.R. Schrieffer, Physica C 185-189 ( 1991 ) 17. [5] K.A. Muller, Physica C 185-189 (1991) 3. [6] G. Eliashberg, Sov. Phys. JETP I 1 (1960) 696. [7]Papers in Superconductivity, ed. R.D. Parks (Marcel

Dekker, New York, 1969). [ 8 ] J.E. Crow and N.P. Ong, in: High temperature superconduc-

tivity, ed. J.W. Lynn (Springer, New York, 1990). [9 ] H. Peng and K. Wang, Can. J. Phys. 69 ( 1991 ) 1399.

[10IJ.R. Schrieffer, Theory of superconductivity (W.A. Benjamin, New York, 1964).

Documents

High-Tc superconductors and interaction between pairs