Interband Pairing Theory (IBP)
(A Brief Summary)
Jamil Tahir-Kheli and Jason K. Perry, First Principles Research, Inc.
Electronic Structure
LDA band structure calculations dating back fourteen years have served as a starting point for virtually all theories of superconductivity in the cuprates. The consensus from these calculations is that the very 2D Cu dx2-y2/O ps (x2-y2) band is the only band to cross the Fermi level in materials such as La2-xSrxCuO4. While this picture appears to be consistent with a number of key experiments (e.g. the ARPES Fermi surface), it does not immediately explain many anomalous details of these same experiments (e.g. the ARPES pseudogap or background), nor does it offer any explanation of multiple other experiments (e.g. the NMR, neutron scattering, resistivity, etc.). Theorists have turned to very exotic models to explain the many unusual normal state properties of these materials, yet all have used the basic LDA band structure as a starting point. In particular, the vast majority of Hubbard models only consider the effect of correlation on the Cu dx2-y2/O ps (x2-y2) band in isolation. In a series of papers, we have argued that this starting point may be invalid and the lack of progress in our understanding of these materials is probably linked to faulty assumptions about their electronic structure made more than a decade ago.
One of the most glaring failures of the LDA is that it could not produce the 2.0 eV gap in the undoped antiferromagnetic state of La2CuO4. Recently we showed that the more sophisticated U-B3LYP functional provided an excellent description of the undoped state, quantitatively reproducing the insulating gap (Paper 8). The conclusion from this work is that the paramagnetic state (LDA or restricted spin B3LYP) is improperly described, overestimating the extent of on-site repulsion in the x2-y2 band. In contrast, the antiferromagnetic state (unrestricted spin U-B3LYP) is properly described, producing a state which is 0.5 eV more stable.
While it is well recognized that correlation missing in the LDA always lowers the total energy, what is often overlooked is that the x2-y2 band benefits most from correlation and is lowered relative to other bands on the order of 1.0 eV (half the band gap). By throwing away the other bands, standard Hubbard models have totally missed the dominant effect of correlation (Paper 4). The primary conclusion from this work is that a second band having z2 character is also important at the Fermi level, as can be seen from the density of states for the U-B3LYP antiferromagnetic state shown below.
Furthermore, we have shown with the U-B3LYP functional that explicitly doped La2-xSrxCuO4 (x = 0.125, 0.25, 0.50) provides an even more compelling case for z2 hole formation (Paper 9). In contrast to LSDA, we find the z2 holes are inhomogeneously distributed in the cuprate, localized in the vicinity of the impurity. This finding is consistent with NQR data which supports formation of localized holes. It is also consistent with XAFS data which supports significant structural distortions around the Sr impurity, quantitatively reproduced with z2 hole formation.
While the U-B3LYP calculations lead to a good description of the undoped antiferromagnetic state, the doped state is unrealistically highly ordered. A description of the doped state where the impurities are positioned in random locations is needed. We are in the process of developing such a model. In earlier models, we treated the z2 impurities as a delocalized band. We developed a new tight binding band structure to describe the paramagnetic state which explicitly includes the Cu dz2 (z2) band in addition to the standard x2-y2 band (Paper 2). By correcting the energy of an orbital in the limit of a strongly correlated system as shown below, the z2 band is brought to the Fermi level. This is a chemically intuitive result that completely invalidates the LDA single band picture as a starting point for a superconductivity theory.
The band dispersion for La1.85Sr0.15CuO4 (LASCO) is shown below with and without correlation:
Symmetry allows a crossing of the two bands along the (0,0) - (p,p) direction. Elsewhere the bands repel. With the incorporation of 3D coupling into the Hubbard model (Paper 5), the narrow z2 band gains measurable anisotropic z-axis dispersion and thus the energy at (0,0,2p /c) is raised above the Fermi level opening up a new hole-like Fermi surface. The new Fermi surface for LASCO is shown below. (The standard Fermi surface looks like the first panel over all kz.)
Note in particular, the crossing that occurs in panel three. Symmetry allowed band crossings occur all the time. However, the probability of a band crossing occuring at the Fermi level is virtually zero. The cuprates are unique in that a crossing can be tuned to the Fermi level by doping.
Despite the fact that many have concluded that band theory cannot explain high Tc phenomena, it may simply be the case that the wrong band structure was used to make this conclusion.
Interband Cooper Pairing
In the vicinity of the Fermi level band crossing, a new kind of Cooper pair can form comprised of a k a electron from one band and a -k b electron from the other band (Interband Cooper Pairs). These will couple with standard BCS-like Cooper pairs away from the band crossing (Paper 3).
The many anomalous properties associated with the cuprate superconductors can be derived using standard physics, given the above band structure and the postulate of interband pairing near the crossing.
Gap Symmetry
Interband Cooper pairing leads to the observed D-Wave tunneling since the pair is NOT time-reversal invariant with itself as is the case with BCS-like pairs (Paper 1). Consider tunneling of interband pair (k1a,-k2 b) to a standard BCS-like Cooper pair (p a,-p b) across a junction (1 and 2 are band indices).
The supercurrent is of the form, (T is the single electron tunneling matrix element)
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IBP: Orbital band character also contributes to the phase of the supercurrent.
For tunneling of BCS-like pairs to BCS-like pairs, the time reversal symmetry of the Cooper pairs leads to,
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BCS: The phase of the supercurrent is due completely to gap symmetries.
D-Wave!!
Generalizing the argument to include BCS-like pairs away from the band crossing leads to a net D-Wave order parameter.
ARPES
Angle-Resolved Photoemission Spectroscopy (ARPES) "sees" only a single band. However, the z2 states have dispersion normal to the planes comparable to their planar dispersion. This leads to a z2 k state linewidth that is dominated by the linewidth of the photoelectron rather than the interesting photohole. Thus, these states do not lead to resolvable quasiparticle peaks.
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The observed pseudogap is due to the rapidly changing character of the two bands near the crossing rather than the hypothetical existence of Cooper pairs above Tc. The approximate D-wave symmetry of the gap arises naturally from the symmetry of the crossing.
Schematic of ARPES observed BiSCO(2212) Fermi surface
In addition, the anomalous "step-function" background is well reproduced by calculating the inelastic scattering associated with this band structure (Paper 7).
NMR
The NMR anomalies are explained automatically by this band structure without anti-ferromagnetic spin fluctuations. The primary reason for the difference between Cu and O NMR is that there are two relevant orbitals on Cu (x2-y2 and z2) and only one (ps) on the O. It is important to note that in the vicinity of a band crossing it is no longer correct to assume the bare density of states is constant over kT (must keep the DOS in the integral) (Paper 6).
Calculated Cu and O NMR spin relaxation rates
Other Properties
Temperature dependence of Hall effect arises from the strong change in the curvature of the two bands in the vicinity of the crossing in the range plus/minus a few kT due to band repulsions. We obtain the correct factor of two decrease of Hall coefficient from 100-300K (Paper 3).
Manuscripts describing the mid-IR absorption, neutron scattering, X-ray absorption, and resistivity are in preparation.
Bottom Line
Consensus: The prevailing assumption is there is only one relevant band or band theory is invalid. This forces one to conclude that the physics must be very exotic in order to explain the experimental data.
We believe: The CHEMISTRY of these materials is more complex than previously thought, leading to two bands that cross at the Fermi level. The experimental data can then be explained with very conventional PHYSICS. The Interband Pairing Theory is simply a twist on the conventional BCS theory that brings the band orbital characters to the forefront.