Anharmonic phonons and the isotope effect in superconductivity

Anharmonic interionic potentials are examined in an Einstein model to study the unusual isotope-effect exponents for the high-${\mathit{T}}_{\mathit{c}}$ oxides. The mass dependences of the electron-phonon coupling constant λ and the average phonon frequency √〈${\mathrm{\omega }}^2$〉 are computed from weighted sums over the oscillator levels. The isotope-effect exponent is depressed below 1/2 by either a double-well potential or a potential with positive quadratic and quartic parts. Numerical solutions of Schrödinger’s equation for double-well potentials produce λ’s in the range 1.5–4 for a material with a vanishing isotope-effect parameter α. However, low phonon frequencies limit ${\mathit{T}}_{\mathit{c}}$ to roughly 15 K. A negative quartic perturbation to a harmonic well can increase α above 1/2. In the extreme-strong-coupling limit, α is 1/2, regardless of anharmonicity.