Abstract
Within recent developments of density functional theory, its numerical implementation and of the superconducting density functional theory is nowadays possible to predict the superconducting critical temperature, , with sufficient accuracy to anticipate the experimental verification. In this paper we present an analytical derivation of the isotope coefficient within the superconducting density functional theory. We calculate the partial derivative of with respect to atomic masses. We verified the final expression by means of numerical calculations of isotope coefficient in monatomic superconductors (Pb) as well as polyatomic superconductors (CaC6). The results confirm the validity of the analytical derivation with respect to the finite difference methods, with considerable improvement in terms of computational time and calculation accuracy. Once the critical temperature is calculated (at the reference mass(es)), various isotope exponents can be simply obtained in the same run. In addition, we provide the expression of interesting quantities like partial derivatives of the deformation potential, phonon frequencies and eigenvectors with respect to atomic masses, which can be useful for other derivations and applications.
Highlights
The discovery of isotope effect (IE) in superconductors, that is the variation of the superconducting critical temperature (Tc) upon isotope substitution of the constituent atoms of the material, has had a fundamental historical role
In this paper we present an analytical derivation of the isotope coefficient within the superconducting density functional theory
We provide the expression of interesting quantities like partial derivatives of the deformation potential, phonon frequencies and eigenvectors with respect to atomic masses, which can be useful for other derivations and applications
Summary
The discovery of isotope effect (IE) in superconductors, that is the variation of the superconducting critical temperature (Tc) upon isotope substitution of the constituent atoms of the material, has had a fundamental historical role. The predicted isotope coefficients for simple metals [15, 43], allowed to highlight the origin of reduced isotope effect, underlining the need of a consistent treatment of electron–phonon and renormalized Coulomb electron–electron interaction which depends on the phonon frequency range. Due to its fundamental importance, the straightforward comparison with the experiments and to complete the form ulation of the SCDFT, in this paper we present an analytical derivation of the isotope coefficient within the SCDFT. This new approach will allow to obtain isotope coefficient as a postprocessing with a considerable gain in terms of computational time and precision. Present a numerical implementation and discuss results on Pb and CaC6 (section 4)
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