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Consequences of Fermi surface geometry
Physiea C 162-164 (1989) 769-770 North-Holland
~ENCES
OF FERMI SURFACE GEOMETRY
Walter A. HARRISON* Department of Applied Physics, Stanford University, Stanford, CA 94305-4090, USA It is assumed that a Fermi surface exists in the high-Tc superconductors and that the interaction V(r) between electrons may be regarded as static. In the tight-binding context, the matrix element of V(r) between band states contains a term equal to the onsite repulsion U , which seriously suppresses superconductivity, especially for low dimensions and short-range interaction. It is seen how this difficulty can be removed in the context of a BCS-like superconducting state, iTk(Uk+vkbk+)/O>, for various Fermi-surface geometries: that for sheets of chains, that for the square geometry appropriate to CuO2 planes, for Fermi surface pockets at the corners of a square antiferromagnetic Brillouin Zone, and for the planar surfaces appropriate to the cubic bismuth-oxide superconductors. The Hamiltonian relevant to superconductivity is that for the non-interacting electrons plus the electron-electron interaction, taken as V(r). The variational calculation (which does not assume weak coupling) using the BCS form of the wavefunction leads to the condition Z~k =
--~. , ~
-2~--~,/
V l~- k A k,
2 • VE~+Zt~k.
i
(1)
in terms of the composite variational parameter Ztk, which turns out to be the energy-gap parameter. The ek represent the energy bands which we describe in a tight-binding framework1, 2 and the Vq is the corresponding Fourier coefficient of the electron-electron interaction,
Vq = ,~j e -iq'rjV(rj).
(2)
This sum contains the j = 0 term, equal to the large repulsive intra-atomic repulsion U . *
This may well even make Vq positive definite 2 , ruling out superconductivity, according to Eq. 1, without variations in the sign of Ak over the region of integration. We seek first a solution for the simplest structure illustrative of our central conclusions. It is a sheet of atomic chains, each lying in a y-direction, as in the 123compound. Even if there is no electronic overlap between chains we may construct two-dimensional bands, and then the energy will not depend upon kx ; the Fermi surface consists of two lines of constant ky . T h e sum over q along one line becomes an integral Jdqx in Eq. 1 . Now we allow the A k to vary along the Fermi line as exp(ikxal)where a is the spacing between chains, and I must be an integer (possibly zero) so that A k is singlevalued at the Brillouin Zone edges. The sum for Vq is substituted and the integral performed, eliminating all terms except for those with the x-component of rj equal to
27r1. The Cooper pair consists of electrons
This work was supported by the Office of Naval Research under Contract N00014-85-0167.
0921-4534/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
770
W.A. Harrison /Consequences of Fermi surface geometry
on Ith-neighbor chains. We see that for / ~ 0, the onsite repulsion is eliminated so any attractive interaction for neighboring sites can produce superconductivity2. A real order parameter, varying as sin kxal does not yield as low an energy. The energy bands for the CuO2 planes in the cuprate superconductors also yield Fermi surfaces consisting of approximately straight lines.2, 3 However, in this case the lines extend from the center of each edge of the square Brillouin Zone to the center of a neighboring edge. A variation of 27c/ is required over a combination of two segments2 (e. g., lower to right and left to top centers). Exactly the same argument as for the chains suggests nonzero / . Required relations between Ak and Z~-k for this case lead to a modification of the usual k, -k pairing and association of triplet pairing with odd 1.2 The cuprates are close to an antiferromagnetic insulating state which can be described in terms of a band gap at the Fermi surface (which becomes the new Brillouin Zone) of the half-filled band given above for the CuO2 planes 4,5. Upon doping with holes, small pockets of Fermi surface might arise, for example at the center of the new Brillouin-Zone edges, as suggested by Schrieffer, et a1.5 Again the contribution of the intrasite repulsion is eliminated if the average of the order parameter over the Fermi surface vanishes. Schrieffer, et al.,5 suggested alternate signs for Ak at alternate pockets, which does eliminate the onsite coupling and is thus a favorable choice for this Fermi-surface geometry. If the Fermi surface is pulled uniformly by doping from the antiferromagnetic Zone, the squareFermi-surface analysis given above obtains. Finally we consider the cubic bismuth-
oxide superconductors which also have an approximately half-filled band 6, as in the other cases. The Fermi surface is the threedimensional counterpart of the square CuO2 Fermi surface. It consists of (111) planes bisecting the vectors to the corners of the cubic Brillouin Zone; it has the same shape as an fcc Brillouin Zone inscribed in the true cubic Zone. The three-dimensional generalization of our CuO2 Ztk is a variation e ik'a/, with a any cube edge. Indeed, with planar Fermi surfaces the electrons are onedimensional in just the sense they were for the chains described above, but consisting of four sets moving in different [111] directions, just as the planes had two sets moving in different [11] directions. The electronic structure and superconductivity seem remarkably similar in these disparate systems. 1. See, for example, W. A. Harrison, Electronic Structure and the Properties of Solids, (Freeman, New York, 1980); reprinted by Dover (New York, 1989). . For superconductivity, W. A. Harrison, Phys. Rev. B38, 270 (1988). . More accurate bands have been given by L. F. Mattheiss, Phys. Rev. Lett. 58, 1028 (1987). . W. A. Harrison, in Novel Superconductivity, edited by Stuart A. Wolf and Vladimir Z. Kresin, Plenum Press, (New York, 1987), p. 507. . J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Phys. Rev. Lett 60, 944 (1988), and private communication. 6. Brent A. Richert and Roland E. Allen, Proceedings of the 19th International Conference on the Physics of Semiconductors (Warsaw, 1988).
~ENCES
OF FERMI SURFACE GEOMETRY
Walter A. HARRISON* Department of Applied Physics, Stanford University, Stanford, CA 94305-4090, USA It is assumed that a Fermi surface exists in the high-Tc superconductors and that the interaction V(r) between electrons may be regarded as static. In the tight-binding context, the matrix element of V(r) between band states contains a term equal to the onsite repulsion U , which seriously suppresses superconductivity, especially for low dimensions and short-range interaction. It is seen how this difficulty can be removed in the context of a BCS-like superconducting state, iTk(Uk+vkbk+)/O>, for various Fermi-surface geometries: that for sheets of chains, that for the square geometry appropriate to CuO2 planes, for Fermi surface pockets at the corners of a square antiferromagnetic Brillouin Zone, and for the planar surfaces appropriate to the cubic bismuth-oxide superconductors. The Hamiltonian relevant to superconductivity is that for the non-interacting electrons plus the electron-electron interaction, taken as V(r). The variational calculation (which does not assume weak coupling) using the BCS form of the wavefunction leads to the condition Z~k =
--~. , ~
-2~--~,/
V l~- k A k,
2 • VE~+Zt~k.
i
(1)
in terms of the composite variational parameter Ztk, which turns out to be the energy-gap parameter. The ek represent the energy bands which we describe in a tight-binding framework1, 2 and the Vq is the corresponding Fourier coefficient of the electron-electron interaction,
Vq = ,~j e -iq'rjV(rj).
(2)
This sum contains the j = 0 term, equal to the large repulsive intra-atomic repulsion U . *
This may well even make Vq positive definite 2 , ruling out superconductivity, according to Eq. 1, without variations in the sign of Ak over the region of integration. We seek first a solution for the simplest structure illustrative of our central conclusions. It is a sheet of atomic chains, each lying in a y-direction, as in the 123compound. Even if there is no electronic overlap between chains we may construct two-dimensional bands, and then the energy will not depend upon kx ; the Fermi surface consists of two lines of constant ky . T h e sum over q along one line becomes an integral Jdqx in Eq. 1 . Now we allow the A k to vary along the Fermi line as exp(ikxal)where a is the spacing between chains, and I must be an integer (possibly zero) so that A k is singlevalued at the Brillouin Zone edges. The sum for Vq is substituted and the integral performed, eliminating all terms except for those with the x-component of rj equal to
27r1. The Cooper pair consists of electrons
This work was supported by the Office of Naval Research under Contract N00014-85-0167.
0921-4534/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
770
W.A. Harrison /Consequences of Fermi surface geometry
on Ith-neighbor chains. We see that for / ~ 0, the onsite repulsion is eliminated so any attractive interaction for neighboring sites can produce superconductivity2. A real order parameter, varying as sin kxal does not yield as low an energy. The energy bands for the CuO2 planes in the cuprate superconductors also yield Fermi surfaces consisting of approximately straight lines.2, 3 However, in this case the lines extend from the center of each edge of the square Brillouin Zone to the center of a neighboring edge. A variation of 27c/ is required over a combination of two segments2 (e. g., lower to right and left to top centers). Exactly the same argument as for the chains suggests nonzero / . Required relations between Ak and Z~-k for this case lead to a modification of the usual k, -k pairing and association of triplet pairing with odd 1.2 The cuprates are close to an antiferromagnetic insulating state which can be described in terms of a band gap at the Fermi surface (which becomes the new Brillouin Zone) of the half-filled band given above for the CuO2 planes 4,5. Upon doping with holes, small pockets of Fermi surface might arise, for example at the center of the new Brillouin-Zone edges, as suggested by Schrieffer, et a1.5 Again the contribution of the intrasite repulsion is eliminated if the average of the order parameter over the Fermi surface vanishes. Schrieffer, et al.,5 suggested alternate signs for Ak at alternate pockets, which does eliminate the onsite coupling and is thus a favorable choice for this Fermi-surface geometry. If the Fermi surface is pulled uniformly by doping from the antiferromagnetic Zone, the squareFermi-surface analysis given above obtains. Finally we consider the cubic bismuth-
oxide superconductors which also have an approximately half-filled band 6, as in the other cases. The Fermi surface is the threedimensional counterpart of the square CuO2 Fermi surface. It consists of (111) planes bisecting the vectors to the corners of the cubic Brillouin Zone; it has the same shape as an fcc Brillouin Zone inscribed in the true cubic Zone. The three-dimensional generalization of our CuO2 Ztk is a variation e ik'a/, with a any cube edge. Indeed, with planar Fermi surfaces the electrons are onedimensional in just the sense they were for the chains described above, but consisting of four sets moving in different [111] directions, just as the planes had two sets moving in different [11] directions. The electronic structure and superconductivity seem remarkably similar in these disparate systems. 1. See, for example, W. A. Harrison, Electronic Structure and the Properties of Solids, (Freeman, New York, 1980); reprinted by Dover (New York, 1989). . For superconductivity, W. A. Harrison, Phys. Rev. B38, 270 (1988). . More accurate bands have been given by L. F. Mattheiss, Phys. Rev. Lett. 58, 1028 (1987). . W. A. Harrison, in Novel Superconductivity, edited by Stuart A. Wolf and Vladimir Z. Kresin, Plenum Press, (New York, 1987), p. 507. . J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Phys. Rev. Lett 60, 944 (1988), and private communication. 6. Brent A. Richert and Roland E. Allen, Proceedings of the 19th International Conference on the Physics of Semiconductors (Warsaw, 1988).