Chapter 8
Theory
of
High-Temperature
Superconductivity
Overlap of Wannier Functions and the Role of Dielectric Screening Daniel C. Mattis Physics Department, University of Utah, Salt Lake City, UT 84112 Virtual charge fluctuations in one conducting plane can interact with virtual fluctuations in adjacent, parallel, conducting planes and can, under certain conditions, overcompensate the intraplanar Coulomb repulsion. We prove this using the Lindhard dielectric function formalism. Surprisingly, the physical overlap of Wannier functions on neighboring atoms within each plane is highly favorable to High-T superconductivity, while analogous overlap between planes is detrimental to it. Thus, small intraplanar m* favors High-T superconductivity, and the hard-to-achieve combination of high interplanar resistivity and small interplanar distance, optimizes it. c
c
This Chapter outlines a new concept, and introduces a theory for the High-T superconductors (1-3) based on interactions between virtual charge fluctuations on adjacent, parallel metallic planes. The mechanism can be superficially understood by analogy with the well-known van der Waals interaction between neutral atoms and molecules, with one essential difference: the attractive potential in the present instance encourages condensation of the metallic electrons into a BCS (4) superconducting state within each plane. c
0097-6156/88/0377-0103$06.00/0 ° 1988 American Chemical Society
104
CHEMISTRY OF HIGH-TEMPERATURE SUPERCONDUCTORS H
The inclusion of a small amount of interplanar hopping does not significantly affect the results, but ensures that superconducting long-range order in the present model (in contradistinction with purely intraplanar schemes) will not violate the important Mermin-Wagner theorem (5). This theorem forbids long-range order in one and two dimensions at finite temperature, permitting only superconducting fluctuations ( para"superconductivity) to survive. With a small amount of interplanar hopping, the correlation length in the vertical direction increases from zero to a finite value. But in many other respects, the present mechanism is inherently low-dimensional. Insofar as the electronic overlap from plane to plane becomes comparable to that within the planes, the attractive mechanism is lost. 1 therefore do not expect this mechanism to work in isotropic, three-dimensional materials. But would it help to go to lower dimensions still? Although a similar mechanism may conceivably be optimized in certain arrays of polymers, or in other quasi-one-dimensional systems, there are technical reasons, related to a substantial decrease in the relevant phase space, which would prevent T from becoming as dramatically M
c
large in ID as in 2D. Thus the discovery of Berdnoz and lluller (1) seems to be singularly related to the two-dimensional, layered structure of the LaSrCuO material they chose to examine, and not to any other unusual properties that it (or the other high-T materials c
(2,3)) might incidentally exhibit, such as Jahn-Teller distortion, superstructures, antiferromagnetism, twinning, or the like. 1 present the theory in two stages. In the first, an intuitive picture is drawn and a simple explanation of the phenomenon is given. In the second, a formulation is sketched out which, aside from one or two lengths that can be experimentally obtained, is practically free of adjustable parameters. Theessentialrole played by the Wannier states' overlap and by dielectric screening is explained. Explicit formulas are exhibited for such properties of the superconductors as T . c
The Simple Physical Picture Even in the best of metals, a space- and time-dependent charge distribution 8p(r,t) cannot be efficiently screened out at wavelengths short compared with the screening length nor at frequencies high
8.
MATTIS
Theory of
105
High-ΊSuperconductivity
compared with the plasma frequency. The screening response of an electrical conductor to a test-charge is measured by the dielectric function £(q,o)), a complex function of wave-vector and frequency (6) which also depends on such electronic parameters as the Fermi wave-vector kp . In the present theory, we are only concerned with relatively small wave-vectors
q*2kp and low frequencies ΐ ι ω « Ε ρ =
fi kp /2m*, corresponding to a low-density gas of fermions. Given m* 2
2
and lattice parameters a~b and d=interplanar separation (the vertical cell dimension cmay be a multiple of d) it becomes possible to estimate quantitatively the optimal parameters for high-T c
superconductivity. At first, let us seek possible sources of electron-electron attractions (a principal requirement, if the BCS theory is to be applicable) within the Coulomb interactions themselves. The real part of the static dielectric function for a charged gas of fermions, interacting by ordinary Coulomb forces (e /rjj), on a single 2
planar conducting sheet, was formulated by F.Stern (7): e(q,0) = [ I • (2/qa *) ]
, q < 2k
0
(la)
F
= [ I • (2/qa *)(Hl-(2k / q ) ] F
0
2
1/2
)], q > 2k
F
(lb)
where q=(q ,qy), a *= fi /m*e being the Bohr radius, the background x
2
0
2
dielectric constant κ being taken here to be I. This formula shows that screening is highly effective only for small q. The Hamiltonian for any two parallel conducting sheets is, Η = H, • H • Η· 2
(2)
1>2
with Hj containing the kinetic (and potential) energy operators of the Fermi fluid over the Ν eel Is (each of area a ) in the i-th plane, and 2
H'i.2 = (1/N) Σ ς W (q;d) p,(q)p (-q) 0
2
(3)
is the "bare" interaction connecting charge fluctuations ρ in the two planes, with the operators pj(q) given as: pj(q) = Σκα V q , o , i k , a , i c
c
( 4 )
106
CHEMISTRY OF HIGH-TEMPERATURE SUPERCONDUCTORS II
in the language of fermion anihilation and creation operators, a (=îori) being the spin parameter. W (q;d) is obtained as follows: 0
W (q;d)= (e /a )J d r e ^ r [d * r ] " 2
2
2
2
2
1/2
0
This integration yields: (5a)
W (q;d)= e~ l[2Tre /qa ] d(
2
2
0
We note that this is an tf^/restimate of the interaction, as both p and p should be accompanied by their respective //7/ir
