NUMERICAL SIMULATION OF THE ISOTOPE EFFECT IN THE HIGH-T_c SUPERCONDUCTORS

Abstract

The results of a numerical simulation are presented for the oxygen isotope effect in the high-T c oxides by solving the Schrödinger equation for a local vibrational mode of oxygen ions in a double well anharmonic potential of the form −Ax2/2+Bx4/4. The features of the energy spectrum, eigenfunctions and the matrix elements of the dipole and quadrupole moments are discussed. The applicability region of a two-level approximation to describe the electron-phonon interaction is given. The constant of the isotope effect α is determined to be comparable to that of La 2−x Sr x CuO 4 but in disagreement with the up-to-date experimental observations for the YBaCuO system.

Anomalous Isotope Effects of Fulleride Superconductors

“…Large positive Oxygen isotope effects α > 1/2 beyond the BCS value 1/2 had been found in HTSC material La 2−x Sr x CuO 4 (α=0.75) at doping level x=0.12 [7] and in fullerides Rb 3 C 60 (α > 1.0) with different ratios of substitutions of 12 C with 13 C [8,9,10]. For La 2−x Sr x CuO 4 , near x=0.12 the large coefficient of isotope effect is probably due to the strong an-harmonic vibrations companying with structural transitions.…”

“…(8) derived in this paper. 13 substitution by 13 C is obtained from the old one before isotope substitution by simply shifting energy of phonon and scaling it to satisfy the constraint The an-harmonic parameters A * are calculated in terms of the shifts of phonon energies. For the case (2) in table (2), we get α = 1.207 which is very close to the already known maximum value 1.275 in experiments with uncompleted substitution (α=2-2.25, p=0.60) [8].…”

“…The parameter of electron-phonon interaction λ=0.67 is defined by the experimental transition temperature T c about 29.5(K) using parameters µ * =0.1 and Ω P =65 meV. We calculate the η = η 12 for 12 C. The phonon energies after 13 C substitutions are unknown because the phonon energies are dependent on the ratios of 13 C substitutions. We choose the nine possible energies for phonons in table (2), which are all smaller than Ω P =65 meV because the energies (or frequency) decrease with increasing mass of atoms.…”

“…For La 2−x Sr x CuO 4 , near x=0.12 the large coefficient of isotope effect is probably due to the strong an-harmonic vibrations companying with structural transitions. Similarly, the transition from partial substitution to completed substitution of 12 C by 13 C for Rb 3 C 60 leads to the decrease of α from 1.275 or 1.189 [8,9] to 0.30 [11] because the anharmonic effect is relatively weak at completed substitution. The Eliashberg theory combined with different models of an-harmonic lattice vibrations had been used to qualitatively explain these anomalous isotope effects [12,13,14].…”

“…Similarly, the transition from partial substitution to completed substitution of 12 C by 13 C for Rb 3 C 60 leads to the decrease of α from 1.275 or 1.189 [8,9] to 0.30 [11] because the anharmonic effect is relatively weak at completed substitution. The Eliashberg theory combined with different models of an-harmonic lattice vibrations had been used to qualitatively explain these anomalous isotope effects [12,13,14]. Beyond Migdal's theorem, the coefficient of isotope effect α > 1/2 had also been obtained by including non-adiabatic effect [15].…”