Anomalous normal state and high temperature superconductivity in the cuprates

It is generally believed that a high electronic density-of-states n(E_F), and therefore a low Fermi velocity v_F, are required to obtain a high transition temperature, since T_c ≅ ω_ph exp(-1/λ) and λ = n(E_F)V. However, V = < I² >/Mω_ph², and Bardeen showed that I ≅ E_Fk_F = (1/2)k_F²v_F. Thus one may expect that λ should increase with v_F. While it may not be feasable to increase the one-electron velocity v_F signifcantly, the velocity may increase greatly as a result of renormalization by electron-electron interactions. Such a renormalization exists in Hartree-Fock theory for an unscreened electron-gas. We found that for a medium-density electron-gas (r_s ≅ 10-20) imbedded in a background with a dielectric constant e{open}(ω) such that e{open}/e{open}∞ > 10, there is a significant increase of v_F by renormalization, even when screening is taken into consideration. The peak of v(k) at k = k_F is very narrow, the half width being somewhat less than the frequency ω₀ at which e{open}(ω) falls by a factor of 2. When e{open}(ω) is due to ionic polarization, ω_o is a typical phonon frequency. The height of the peak v_F/v_F⁰ is of order E_F/ω₀, and the width in units of momentum is: δk/k_F ≅ (ω₀/E_F)². This velocity peak is associated with a minimum in the screening constant at E_p. Its sharpness causes the normal state properties to be highly anomalous; namely the conductivity is exceptionally high, with an anomalous temperature dependence; the conductivity anisotropy, thermoelectric power, and Hall constant are anomalous. Direct determination of v(k) in YBCO by several methods indeed suggests a large, sharp peak, the width being of order 10 me V. This peak manifests itself in the tunneling and point-contact spectroscopy I-V curves. We suggest that the reduced screening at the Fermi level, associated with this velocity peak, is responsible for the high T_c of the cuprates and several other "exotic" superconductors. Thus, the high T_c is a reflection of the anomalous normal-state properties.