Direct evaluation of the isotope effect within the framework of density functional theory for superconductors

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.

Special issue on novel superconducting and magnetic materials

“…The contribution by Lüders and coworkers reports methodological advances within the superconducting density functional theory 7 , proposing an analytical procedure to efficiently compute the isotope coefficient, a key quantity to decipher the onset of a phonon-mediated superconducting state. 11 In fact, the observation of a finite isotope effect, i.e. a shift of T C observed upon isotopic substitution, is considered to be the definite "smoking gun" for conventional superconductivity.…”

“…Sandro has been one of the driving forces in the development of more and more accurate methods to treat conventional superconductors, such as high-pressure hydrides and MgB 2 , where superconductivity is mediated by phonons (lattice vibrations). The contribution by Lüders and coworkers reports methodological advances within the superconducting density functional theory [7], proposing an analytical procedure to efficiently compute the isotope coefficient, a key quantity to decipher the onset of a phonon-mediated superconducting state [11]. In fact, the observation of a finite isotope effect, i.e.…”