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Slope of the superconducting gap function in
$Bi_2Sr_2CaCu_2O_{8+\delta}$ measured by vacuum tunneling
spectroscopy

Phys. Rev. B59, 11962 (1999)

Reproducible scanning tunneling microscope (STM) spectra of $Bi_2Sr_2CaCu_2O_{8+\delta}$
consistently exhibit asymmetric tunneling characteristics, with the higher peak conductance
corresponding to a negatively biased sample. We consider various possible sources of this asymmetry
that are not intrinsic to the superconducting state, including energy dependence of the normal state
densities of states of sample and/or tip, existence of bandwidth cutoffs, unequal work functions of tip
and sample, and energy-dependent transmission probability. None of these effects can explain the sign
and temperature dependence of the observed asymmetry. This indicates that the observed asymmetry
reflects an intrinsic property of the superconducting state: an energy-dependent superconducting gap
function with non-zero slope at the Fermi energy. Such a sloped gap function will also give rise to a
thermoelectric effect in STM experiments, resulting in a $positive$ thermopower. We discuss the
feasibility of observing this thermoelectric effect with an STM and conclude that it is easily observable.
An analysis of thermoelectric currents and voltages together with the tunneling spectra as function of
temperature and tip- sample distance would allow for accurate determination of the slope of the gap
function. It is suggested that it would be very worthwhile to perform these experiments, because the
slope of the gap function reflects a fundamental property of the superconducting state. The theory of
hole superconductivity has predicted the existence of such a slope, of universal sign, in all
superconductors. It is furthermore argued that recent experimental results on vortex lattice imaging
provide further strong evidence for the existence of the gap slope discussed here.

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