Properties of a localized magnetic impurity in a superconducting host: The Anderson–Holstein–BCS model

•Effect of superconductivity on a magnetic impurity is studied with AH-BCS model.•Lang–Firsov transformation and Cluster-variation method are used to solve the problem.•Magnetic moment decreases with electron–phonon coupling constant (λ)(λ).•Binding energy (BE) between impurity and conduction electrons increases with λ.•BE, however, decreases with increasing gap parameter.

A symmetric Anderson–Holstein model with a BCS interaction term is considered to investigate the effect of local electron–phonon interaction on a magnetic impurity in a superconductor. The Kikuchi–Morita Cluster variation (CV) method is used to calculate the local impurity magnetic moment and the binding energy between the impurity and the conduction electrons in the superconductor. The effect of electron–phonon interaction and the order parameter on the bound state is discussed.