Tuning of quantum entanglement in a superconductor with transition-metal and rare-earth impurities: Effect of potential scattering on quantum phase transitions

Bipartite quantum entanglement of electron spins of a Cooper pair in a dirty $s$-wave superconductor is investigated at absolute zero temperature in terms of the exchange interaction and potential scattering as well as the relative distance between the two electrons. In the case of transition-metal impurities, we utilize the T-matrix approach to obtain the relevant Green's functions. We employ the two-electron spin-space density matrix, which is associated with the perturbed Green's functions. We show that a two-spin state can be described by the Werner state for dirty superconductors as well as for clean ones. We find that both the first and second quantum phase transitions (QPTs) are generated by the competition between potential scattering and other interactions. Meanwhile, it is not possible to determine the allowable value for each interaction individually. In the case of rare-earth impurities, concurrence, as a measure of bipartite entanglement, is discussed in terms of the relative distance and collision times for all finite and infinite Debye frequencies. The results confirm that quantum correlation and QPTs can be tuned by impurities.