- Email: [email protected]
Lochon-induced interlayer pairing and gap anisotropy in cuprate superconductors
Solid State Communications, Vol. 99, No. 1I, p. 845-848, 1996 Copyright Q 1996 Published by E Psevier Science Ltd Printed in Great Britain. AU rights reserved 0038-1098/96 S12.00+ .OO
LOCHON-INDUCED
INTERLAYER PAIRING AND GAP ANISOTROPY SUPERCONDUCTORS
IN CUPRATE
K.P. Sinha and AS. Vyth~swaran Department
of Physics, Indian Institute of Science, Bangalore 560 012, India (Received 22 February 1996 by C.N.R. Rao)
The mecha~sm involving phonons and lochons (local charged bosons, local pairs) developed earlier for cuprate superconductors is applied for the study of the gap anisotropy in these systems. It is found that besides intraplanar pairing the lochon-induced interlayer pairing gives rise to an anisotropic s-wave gap which is capable of changing sign in some regions on the Fermi surface. Copyright 0 1996 Published by Elsevier Science Ltd
Indeed opinion is converging towards the view that oxide superconductors contain both the Fermi and The mechanism of pairing as well as the symmetry of Bose fluid features: the predo~nan~ of one or the the order parameter in cuprate superconductors are other depends on the degree of doping [20]. subjects of intense activity and discussion currently The purpose of the present paper is to examine the [l-3]. The existence of the antiferromagnetic order in question of gap anisotropy and interlayer pairing the insulating (undoped) state of cuprates prompted induced by lochons in these layered cuprate supersome authors to invoke spin-fluctuation exchange as conductors. The chemical species that can harbour the dominant pairing mechanism and ~onco~tantly (local pairs) lochons in the layers and in-between two to d-wave symmetry of the order parameter [4, 51. layers are Cu’+, 02-, O$-, O--Cu”-O-, Bi3+, Tl’+, However, this mechanism faces difficulty in explaining Pb2+, Hg, etc. [9-14, 181. These species can undergo the change of the isotope effect with oxygen content in double charge fluctuation and can sustain the bosonic YBCO [6]. Also the mechanism leads to very low character of the appropriate centres. The existence of values of the critical temperature 7’, N 1 K [7]. The lochons can give rise to charge density wave (static or mechanism involving electron-phonon (intralayer) dynamic). Indeed experimental evidence for these has interaction enhanced by interlayer Josephson pair been found in CuO chains of YBa2Cus07_~ [21]. tunneling gives rise to anisotropic s-wave symmetry Thus species exist in layered cuprates which lead to of the gap but does not change sign [8]. However, it is fermion-lochon interactions both intralayer and doubtful that Josephson pair tunneling can give high interlayer. enough T, in these systems. A combined m~hanism In what follows, we present the lochon-induced mediated by phonons and lochons (local charged intralayer and interlayer pairing in cuprates and disbosons, local pairs or bipolarons) for the pairing of cuss the gap anisotropy arising from the mechanism. fermions (holes or electrons) belonging to a wide band developed in the last few years provides expla2. FERMION-LOCHON MODEL nation of many properties of cuprate su~rconductors We consider cuprate systems having two conduct19-141. This model explains not only the high T, in these systems but deviation from ideal Fermi liquid ing CuO layers per unit cell besides the intervening dielectric (semiconducting) regions (chair or layer). behaviour, the variation of oxygen isotope exponent This model is adequate to describe two-layer comas a function of doping, the ratio of the gap parameter to critical temperature, etc. The model mediated by pounds, for example, Y( 12 3), Bi(2 2 12), Tl (2 2 12), etc. and can be easily extended to other multiple layer lochons was also developed as felon-boson interaction by several groups elsewhere [ 15-191. systems. 1. INTRODUCTION
845
846
GAP ANISOTROPY
IN CUPRATE SUPERCONDUCTORS
The Hamiltonian for the model is given by
1121.The transfo~ed
+
C
Vol. 99, No. 11
Hamiltonian is
Vii(kk’)c~TCit-#iCi,-klCikT
kk’ i=1,2
where niko = &,cjka with c&,(cikO)denote the fermion creation (annihilation) operator in the state like), k = wave vector, g spin index in the i th conduction plane, ek being the corresponding single particle energy (relative to the chemical potential CL)which is taken independent of the layer index. Similarly, bf(bi) represent lochon creation (anni~lation) operators which, being composite have the forms b+1 = c+c+ It 11’
h
=
CliClTt
(2)
i.e., they create (destroy) a pair of fermions at the same local orbital state I&), the pair being in the spin singlet state. Here Et represents the lochon energy. The last two terms in equation (1) denote the fermionlochon interaction terms in which a lochon splits and delocalises into a pair of fermions and the reverse process of localisation of a fermion pair to give a lochon. The corresponding coupling constants can be written as effective two particle interaction (~l(l)#1(2)1V(12)l~,k(l)~i,-k(2))
=d(k)
(4 in a
g;(k) = gf (0) + 28; (nn)[cos(k,a) + cos(k,a)] + 4gf (nnn)[cos(k#)
cos(k,a)]
where
(7) (8) and El is the lochon energy and we have used the condition that (ek + e-k) < Et. 3. THE GAP PARAMETER Taking that Y, 1 = yZz = Y; VI2 = Y2, = V’ and hz&&kl)
(5)
a being the lattice constant. The summation over relative coordinates (R, - R2) of the pair has been taken from onsite (o), nearest nei~bours (nn) up to next to nearest neighbours (nnn). In the Hamiltonian in equation (1), the fermion-phonon and Coulomb interactions in the layers have not been written explicitly. Their effects will be incorporated later on. The intralayer and interlayer pairing interactions are obtained by the standard canonical transformation
=
+;k&,-ki)
(9)
the gap equation can be written as A(k) = c
~‘(~‘)A(~‘)
tanh~~/2)
w
(3)
where $iko = ~&CT) are the Bloch functions in the i th layer and &, is the localised orbital at site 1. Noting that the Bloch function in the tight binding representation can be expressed as
where RI2 = (Rt - Rt). For a square-lattice plane, this would give
(6) kk’ i#j
+ x
V(kk’)A(k’) tanh;F/2),
(10)
k’
where Ekp = (E: + A(k) ’ ] If2 and V(k, k’) = Vph(k,k’) + VI (k, k’)
(11)
and includes the phonon-indu~d intra-layer pairing interaction. The above gap equation may appear similar to the Josephson interlayer tunneling mechanism [S, 221. There are very important differences. Here the interlayer pairing is mediated by lochons located in the intervening regions. Further, the first term of equation (1) is not local in k as in [S, 221 but involves summation over k’. Also, the structure of I” is [as can be seen from equation (5)] very different from the q(k) chosen in [Xl, namely q(k)
= (Tj/l6)[cos(k,a)
- cos(kya)14.
(12)
The form (5) will be consistent with the band structure symmet~ where in the dispersion in the twodimensional plane has the form 1231 ek = e0 - 2tl [cos(k,a) + cos(k,,a)] + 4t2 cos(k,a) cos~k~~) - p p = chemical potential,
(13)
Vol. 99, No. 11
GAP ANISOTROPY
847
IN CUPRATE SUPERCONDUCTORS
where tt and t2 are respectively nearest neighbour and next nearest neighbour single particle hopping integrals in the layer. In order to see the effect of lochon-induced interlayer pairing on the gap anisotropy, we consider the solution of equation (1) at 7’ = 0 and the Fermi surface for which ek = 0. Then we have (14) where A0 is the intralayer contribution to the gap as taken in [8]. This is to compare our results with [8]. Note that our I’(kk’) = v,,(k, k’) I- V1(k, k’) and is more general; Vi (k, k’) may also show some dependence on the symmetry of the lattice. For computation, the two-dimensional Fermi surface given in [24] has been taken for hole concentrationp = 0.3. Then the summation over 95 points of kk in the first term of equation (14) gives (15) where g!(k) and g:(k’) are given in equation (5). We obtain A(k,) = A0 + l/2{Co + 2CJcos(k,a) + cos(k,a)] +
‘tC&OS(k,fZ)
cos(k,,a)J)
x (94Co - 53.1(2C1) - 7.67(4C&
(16)
where Co = gf (O)/ E1)“2, Ci = gf (nn)/(E1)“2 and C2 = g! (nnn)/(Ei)’ i 2, using equation (5). At this stage, we should like to remark that the carriers in cuprates are subject to strong correlation. This means that the probability of two holes starting from the same site or ending at the same site in a layer is almost negligible. Thus Co can be treated to be almost equal to zero or small negative (repulsive). The value of Ei is taken to be of the order of 400 meV [ 121 and A0 N 3meV [S]. The calculation shows that the value of A(k,) varies from point to point on the Fermi surface both in magnitude and sign. For a few points the values are: for Co = 0, Ci > 0 and C2 > 0 and C2 > Ci, we have A(k,) > 0 for all points on the Fermi surface for which cos(k,a) cos(k,a) < 0. On the other hand A(kF) < 0 for some points for which cos(k,a) cos(k,,a) > 0, for example near points between approximately (105”, 112”) and (I 12”, 105’). Further, for the condition Co = 0, C, < 0 and C, > 0, we get negative A(k,) at four different regions, namely (90”, 136”) to (135”, 89”), (244”, 260”) to (259’, 264”) and (127”, 270”) to (900,226”). In other regions of the (k+z, k,a) plane the gap is positive, the
lJ
2T
KXa
Fig. 1. The thin nearly circular line represents the Fermi Surface (FS) as in [24]. The thick line shows A(k,) at the FS for Co = 0, C, = -0.2 (meV)“’ and C2 = 1.571 (meV) ‘I2 . The region outside the FS has positive sign for A(kF) and inside the FS negative sign. The dq$ed line shows the A4kF) for Co = 0, Ci = 0.09 ” The dashed line shows $::I for+~o~o~?).(lm($!V)i/2 C - 0 and C = 0.9 (meV) l/2 . In all cases the gap &k,) has nigative values for some regions of FS. The maximum positive values for the three cases are 3 1.84 meV, 30.5 meV and 30.17 meV respectively. maximum value being 31.84meV. In Fig. 1, we give a plot of the gap parameter against (k,a, k,,a) along with Fermi surface contour. The precise values of the parameters Co, C, , C2 are shown in the caption of the figure. The above results show that the correlation effect plays an important role in the anisotropic behaviour of the gap. The probability of a pair existing on the same site is inhibited (i.e. Co is either zero or negative implying repulsion). Even Ci has a smaller value than C2. It means that the strongest pair attraction is due to next nearest neighbour interaction in the plane. The fact that we can get a negative sign for the gap parameter for some points at the Fermi surface supports the role of inter-layer pairing as well as correlation in the plane. The attempt by some authors to explain the change in sign of the gap function arising from plane-chain coupling with interaction in the latter being repulsive is not required [7, 251. That the next nearest neighbour interaction interaction is more important is in agreement with the finding of the phenomenolo~cal model of some workers 1261. A final point will be in order. We have taken the same values of C,, Ci and C2 for planes 1 and 2. It is
848
GAP ANISOTROPY
IN CUPRATE SUPERCONDUCTORS
probable that they may have different values in different planes if the lochons are not placed symmetrically between the planes. This situation will render the mechanism more effective in offering various possibilities. In conclusion, we would like to stress that lochoninduced interlayer pairing along with intraplanar interaction involving phonons and lochons can give the gap function which is in accord with the symmetry of the lattice. The possible coexistence of s- and dwave condensates having the same superconducting transition temperature is allowed for in deviations from tetragonal symmetry [27]. However, in light of the above analysis it is difficult to say whether or not it is really necessary. Ac~n5~ledge~e~~~-The authors would like to thank T.V. Ramak~shnan for discussion. They are grateful to the Council of Scientitic and Industrial Research, New Delhi for financial support for this work. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Dynes, R.C., Solid State ~5~~un. 92, 1994, 53. Schrieffer, J.R., Satid State Commun. 92, 1994, 129. Van Harlingen, D.J., Revs. Mad. Phys. 67, 1995, 515. Monthoux, P.’ and Pines, D., Phys. Rev. B49, 1994,426l. Monthoux, P. and Scalapino, D.J., Phys. Rev. Lett. 72, 1994, 1874. Liechtenstein, A.I., Mazin, 1.1.and Andersen, O.K., Phys. Rev. Lett. 74, 1995, 2303. Pashitskii, E.A., JETP. Lett. 61, 1995, 275. Chakravarty, S., Sudbo, A., Anderson, P.W. and Strong, S., Science 261, 1993, 337.
9.
10. 11. 12.
15. 16.
19. 20. 21. 22. 23. 24. 25. 26. 27.
Vol. 99, No. 11
Sinha, K.P. and Singh, M., J. Phys. C.21, 1988, L231. Singh, M. and Sinha, K.P., SoiidState Commun. 70, 1989, 49. Sinha, K.P., Physica B163, 1990, 664. Sinha, K.P. and Singh, M., Phys. Status. Solidi (b) 159, 1990, 787. Sinha, K.P., Solid State Cammun. 79,1991,735. Sinha, K.P. and Rastogi, A., National Acad. Sci. Lett. 18, 1995, 221. Friedberg, J.R., Lee, T.D. and Ren, H.C., Phys. Lett. 152A, 1991, 417 (and references therein). Ranninger, J. and Robin, J.M., Physica C253, 1995, 279 (and references therein). Bar-Yam, Y., Phys. Rev. B43, 1991,359,2601. Sinha, K.P. and Kakani, S.L., High Temperature Superconductivity: Current Results and Novel mechanisms. Nova Science Publishers, New York, 1994 for other references. Dzhumanov, S. and Khabibullaev, P.K., Pramana J. Phys. 45, 1995, 385. Alexandrov, A.S. and Mott, N.F., Supercond. Sci. Technol. 6, 1993, 215. Edwards, H.L., Barr, A.L., Markert, J.T. and delozanne, A.L., Phys. Rev. Lett. 73, 1994, 1154. Muthukumar, V.N. and Sardar, M., Solid State Commun. 97, 1996,289. Yu, J. and Freeman, A.J., J. Phys. Chem. Soli& 52, 1991, 1351. Krasheninnikov, A.V., Opyonov, L.A. and Elesin, V.F., JETP Lett. 62, 1995, 59. Combescot, R. and Leyronas, X., Phys. Rev. Lett. 75, 1995, 3732. Fehrenbacher, R. and Norman, M.R., Phys. Rev. Lett. 74, 1995, 3884. Muller, K.A., Nature 377, 1995, 133.
LOCHON-INDUCED
INTERLAYER PAIRING AND GAP ANISOTROPY SUPERCONDUCTORS
IN CUPRATE
K.P. Sinha and AS. Vyth~swaran Department
of Physics, Indian Institute of Science, Bangalore 560 012, India (Received 22 February 1996 by C.N.R. Rao)
The mecha~sm involving phonons and lochons (local charged bosons, local pairs) developed earlier for cuprate superconductors is applied for the study of the gap anisotropy in these systems. It is found that besides intraplanar pairing the lochon-induced interlayer pairing gives rise to an anisotropic s-wave gap which is capable of changing sign in some regions on the Fermi surface. Copyright 0 1996 Published by Elsevier Science Ltd
Indeed opinion is converging towards the view that oxide superconductors contain both the Fermi and The mechanism of pairing as well as the symmetry of Bose fluid features: the predo~nan~ of one or the the order parameter in cuprate superconductors are other depends on the degree of doping [20]. subjects of intense activity and discussion currently The purpose of the present paper is to examine the [l-3]. The existence of the antiferromagnetic order in question of gap anisotropy and interlayer pairing the insulating (undoped) state of cuprates prompted induced by lochons in these layered cuprate supersome authors to invoke spin-fluctuation exchange as conductors. The chemical species that can harbour the dominant pairing mechanism and ~onco~tantly (local pairs) lochons in the layers and in-between two to d-wave symmetry of the order parameter [4, 51. layers are Cu’+, 02-, O$-, O--Cu”-O-, Bi3+, Tl’+, However, this mechanism faces difficulty in explaining Pb2+, Hg, etc. [9-14, 181. These species can undergo the change of the isotope effect with oxygen content in double charge fluctuation and can sustain the bosonic YBCO [6]. Also the mechanism leads to very low character of the appropriate centres. The existence of values of the critical temperature 7’, N 1 K [7]. The lochons can give rise to charge density wave (static or mechanism involving electron-phonon (intralayer) dynamic). Indeed experimental evidence for these has interaction enhanced by interlayer Josephson pair been found in CuO chains of YBa2Cus07_~ [21]. tunneling gives rise to anisotropic s-wave symmetry Thus species exist in layered cuprates which lead to of the gap but does not change sign [8]. However, it is fermion-lochon interactions both intralayer and doubtful that Josephson pair tunneling can give high interlayer. enough T, in these systems. A combined m~hanism In what follows, we present the lochon-induced mediated by phonons and lochons (local charged intralayer and interlayer pairing in cuprates and disbosons, local pairs or bipolarons) for the pairing of cuss the gap anisotropy arising from the mechanism. fermions (holes or electrons) belonging to a wide band developed in the last few years provides expla2. FERMION-LOCHON MODEL nation of many properties of cuprate su~rconductors We consider cuprate systems having two conduct19-141. This model explains not only the high T, in these systems but deviation from ideal Fermi liquid ing CuO layers per unit cell besides the intervening dielectric (semiconducting) regions (chair or layer). behaviour, the variation of oxygen isotope exponent This model is adequate to describe two-layer comas a function of doping, the ratio of the gap parameter to critical temperature, etc. The model mediated by pounds, for example, Y( 12 3), Bi(2 2 12), Tl (2 2 12), etc. and can be easily extended to other multiple layer lochons was also developed as felon-boson interaction by several groups elsewhere [ 15-191. systems. 1. INTRODUCTION
845
846
GAP ANISOTROPY
IN CUPRATE SUPERCONDUCTORS
The Hamiltonian for the model is given by
1121.The transfo~ed
+
C
Vol. 99, No. 11
Hamiltonian is
Vii(kk’)c~TCit-#iCi,-klCikT
kk’ i=1,2
where niko = &,cjka with c&,(cikO)denote the fermion creation (annihilation) operator in the state like), k = wave vector, g spin index in the i th conduction plane, ek being the corresponding single particle energy (relative to the chemical potential CL)which is taken independent of the layer index. Similarly, bf(bi) represent lochon creation (anni~lation) operators which, being composite have the forms b+1 = c+c+ It 11’
h
=
CliClTt
(2)
i.e., they create (destroy) a pair of fermions at the same local orbital state I&), the pair being in the spin singlet state. Here Et represents the lochon energy. The last two terms in equation (1) denote the fermionlochon interaction terms in which a lochon splits and delocalises into a pair of fermions and the reverse process of localisation of a fermion pair to give a lochon. The corresponding coupling constants can be written as effective two particle interaction (~l(l)#1(2)1V(12)l~,k(l)~i,-k(2))
=d(k)
(4 in a
g;(k) = gf (0) + 28; (nn)[cos(k,a) + cos(k,a)] + 4gf (nnn)[cos(k#)
cos(k,a)]
where
(7) (8) and El is the lochon energy and we have used the condition that (ek + e-k) < Et. 3. THE GAP PARAMETER Taking that Y, 1 = yZz = Y; VI2 = Y2, = V’ and hz&&kl)
(5)
a being the lattice constant. The summation over relative coordinates (R, - R2) of the pair has been taken from onsite (o), nearest nei~bours (nn) up to next to nearest neighbours (nnn). In the Hamiltonian in equation (1), the fermion-phonon and Coulomb interactions in the layers have not been written explicitly. Their effects will be incorporated later on. The intralayer and interlayer pairing interactions are obtained by the standard canonical transformation
=
+;k&,-ki)
(9)
the gap equation can be written as A(k) = c
~‘(~‘)A(~‘)
tanh~~/2)
w
(3)
where $iko = ~&CT) are the Bloch functions in the i th layer and &, is the localised orbital at site 1. Noting that the Bloch function in the tight binding representation can be expressed as
where RI2 = (Rt - Rt). For a square-lattice plane, this would give
(6) kk’ i#j
+ x
V(kk’)A(k’) tanh;F/2),
(10)
k’
where Ekp = (E: + A(k) ’ ] If2 and V(k, k’) = Vph(k,k’) + VI (k, k’)
(11)
and includes the phonon-indu~d intra-layer pairing interaction. The above gap equation may appear similar to the Josephson interlayer tunneling mechanism [S, 221. There are very important differences. Here the interlayer pairing is mediated by lochons located in the intervening regions. Further, the first term of equation (1) is not local in k as in [S, 221 but involves summation over k’. Also, the structure of I” is [as can be seen from equation (5)] very different from the q(k) chosen in [Xl, namely q(k)
= (Tj/l6)[cos(k,a)
- cos(kya)14.
(12)
The form (5) will be consistent with the band structure symmet~ where in the dispersion in the twodimensional plane has the form 1231 ek = e0 - 2tl [cos(k,a) + cos(k,,a)] + 4t2 cos(k,a) cos~k~~) - p p = chemical potential,
(13)
Vol. 99, No. 11
GAP ANISOTROPY
847
IN CUPRATE SUPERCONDUCTORS
where tt and t2 are respectively nearest neighbour and next nearest neighbour single particle hopping integrals in the layer. In order to see the effect of lochon-induced interlayer pairing on the gap anisotropy, we consider the solution of equation (1) at 7’ = 0 and the Fermi surface for which ek = 0. Then we have (14) where A0 is the intralayer contribution to the gap as taken in [8]. This is to compare our results with [8]. Note that our I’(kk’) = v,,(k, k’) I- V1(k, k’) and is more general; Vi (k, k’) may also show some dependence on the symmetry of the lattice. For computation, the two-dimensional Fermi surface given in [24] has been taken for hole concentrationp = 0.3. Then the summation over 95 points of kk in the first term of equation (14) gives (15) where g!(k) and g:(k’) are given in equation (5). We obtain A(k,) = A0 + l/2{Co + 2CJcos(k,a) + cos(k,a)] +
‘tC&OS(k,fZ)
cos(k,,a)J)
x (94Co - 53.1(2C1) - 7.67(4C&
(16)
where Co = gf (O)/ E1)“2, Ci = gf (nn)/(E1)“2 and C2 = g! (nnn)/(Ei)’ i 2, using equation (5). At this stage, we should like to remark that the carriers in cuprates are subject to strong correlation. This means that the probability of two holes starting from the same site or ending at the same site in a layer is almost negligible. Thus Co can be treated to be almost equal to zero or small negative (repulsive). The value of Ei is taken to be of the order of 400 meV [ 121 and A0 N 3meV [S]. The calculation shows that the value of A(k,) varies from point to point on the Fermi surface both in magnitude and sign. For a few points the values are: for Co = 0, Ci > 0 and C2 > 0 and C2 > Ci, we have A(k,) > 0 for all points on the Fermi surface for which cos(k,a) cos(k,a) < 0. On the other hand A(kF) < 0 for some points for which cos(k,a) cos(k,,a) > 0, for example near points between approximately (105”, 112”) and (I 12”, 105’). Further, for the condition Co = 0, C, < 0 and C, > 0, we get negative A(k,) at four different regions, namely (90”, 136”) to (135”, 89”), (244”, 260”) to (259’, 264”) and (127”, 270”) to (900,226”). In other regions of the (k+z, k,a) plane the gap is positive, the
lJ
2T
KXa
Fig. 1. The thin nearly circular line represents the Fermi Surface (FS) as in [24]. The thick line shows A(k,) at the FS for Co = 0, C, = -0.2 (meV)“’ and C2 = 1.571 (meV) ‘I2 . The region outside the FS has positive sign for A(kF) and inside the FS negative sign. The dq$ed line shows the A4kF) for Co = 0, Ci = 0.09 ” The dashed line shows $::I for+~o~o~?).(lm($!V)i/2 C - 0 and C = 0.9 (meV) l/2 . In all cases the gap &k,) has nigative values for some regions of FS. The maximum positive values for the three cases are 3 1.84 meV, 30.5 meV and 30.17 meV respectively. maximum value being 31.84meV. In Fig. 1, we give a plot of the gap parameter against (k,a, k,,a) along with Fermi surface contour. The precise values of the parameters Co, C, , C2 are shown in the caption of the figure. The above results show that the correlation effect plays an important role in the anisotropic behaviour of the gap. The probability of a pair existing on the same site is inhibited (i.e. Co is either zero or negative implying repulsion). Even Ci has a smaller value than C2. It means that the strongest pair attraction is due to next nearest neighbour interaction in the plane. The fact that we can get a negative sign for the gap parameter for some points at the Fermi surface supports the role of inter-layer pairing as well as correlation in the plane. The attempt by some authors to explain the change in sign of the gap function arising from plane-chain coupling with interaction in the latter being repulsive is not required [7, 251. That the next nearest neighbour interaction interaction is more important is in agreement with the finding of the phenomenolo~cal model of some workers 1261. A final point will be in order. We have taken the same values of C,, Ci and C2 for planes 1 and 2. It is
848
GAP ANISOTROPY
IN CUPRATE SUPERCONDUCTORS
probable that they may have different values in different planes if the lochons are not placed symmetrically between the planes. This situation will render the mechanism more effective in offering various possibilities. In conclusion, we would like to stress that lochoninduced interlayer pairing along with intraplanar interaction involving phonons and lochons can give the gap function which is in accord with the symmetry of the lattice. The possible coexistence of s- and dwave condensates having the same superconducting transition temperature is allowed for in deviations from tetragonal symmetry [27]. However, in light of the above analysis it is difficult to say whether or not it is really necessary. Ac~n5~ledge~e~~~-The authors would like to thank T.V. Ramak~shnan for discussion. They are grateful to the Council of Scientitic and Industrial Research, New Delhi for financial support for this work. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Dynes, R.C., Solid State ~5~~un. 92, 1994, 53. Schrieffer, J.R., Satid State Commun. 92, 1994, 129. Van Harlingen, D.J., Revs. Mad. Phys. 67, 1995, 515. Monthoux, P.’ and Pines, D., Phys. Rev. B49, 1994,426l. Monthoux, P. and Scalapino, D.J., Phys. Rev. Lett. 72, 1994, 1874. Liechtenstein, A.I., Mazin, 1.1.and Andersen, O.K., Phys. Rev. Lett. 74, 1995, 2303. Pashitskii, E.A., JETP. Lett. 61, 1995, 275. Chakravarty, S., Sudbo, A., Anderson, P.W. and Strong, S., Science 261, 1993, 337.
9.
10. 11. 12.
15. 16.
19. 20. 21. 22. 23. 24. 25. 26. 27.
Vol. 99, No. 11
Sinha, K.P. and Singh, M., J. Phys. C.21, 1988, L231. Singh, M. and Sinha, K.P., SoiidState Commun. 70, 1989, 49. Sinha, K.P., Physica B163, 1990, 664. Sinha, K.P. and Singh, M., Phys. Status. Solidi (b) 159, 1990, 787. Sinha, K.P., Solid State Cammun. 79,1991,735. Sinha, K.P. and Rastogi, A., National Acad. Sci. Lett. 18, 1995, 221. Friedberg, J.R., Lee, T.D. and Ren, H.C., Phys. Lett. 152A, 1991, 417 (and references therein). Ranninger, J. and Robin, J.M., Physica C253, 1995, 279 (and references therein). Bar-Yam, Y., Phys. Rev. B43, 1991,359,2601. Sinha, K.P. and Kakani, S.L., High Temperature Superconductivity: Current Results and Novel mechanisms. Nova Science Publishers, New York, 1994 for other references. Dzhumanov, S. and Khabibullaev, P.K., Pramana J. Phys. 45, 1995, 385. Alexandrov, A.S. and Mott, N.F., Supercond. Sci. Technol. 6, 1993, 215. Edwards, H.L., Barr, A.L., Markert, J.T. and delozanne, A.L., Phys. Rev. Lett. 73, 1994, 1154. Muthukumar, V.N. and Sardar, M., Solid State Commun. 97, 1996,289. Yu, J. and Freeman, A.J., J. Phys. Chem. Soli& 52, 1991, 1351. Krasheninnikov, A.V., Opyonov, L.A. and Elesin, V.F., JETP Lett. 62, 1995, 59. Combescot, R. and Leyronas, X., Phys. Rev. Lett. 75, 1995, 3732. Fehrenbacher, R. and Norman, M.R., Phys. Rev. Lett. 74, 1995, 3884. Muller, K.A., Nature 377, 1995, 133.