Comprehensive Overview on Yu-Shiba-Rusinov Bound State in Superconductor-Magnetic Impurity Hybrids

As quasiparticle excitations arise at the superconductor-magnetic impurity interfaces, Yu-Shiba-Rusinov (YSR) bound states serve as an ideal platform for exploring fascinating quantum behaviors, which are crucial for both fundamental physics and designing high-performance low-power quantum components. Generally, YSR states are induced by the interaction between local magnetic moments and superconducting Cooper pairs. In recent years, this field has attracted increasing attention from researchers across various disciplines, including physics, chemistry, materials science, electronic science, and so on. This review provides a comprehensive and structured overview of the origin, behavior, and applications of YSR states, which emerge from the competition between Kondo screening and spin-singlet Cooper pairing. Drawing on the Anderson impurity model solved with the numerical renormalization group method, as well as experimental techniques such as scanning tunneling microscopy and spectroscopy, we examine the evolution of YSR states across a range of quantum impurity systems. Covering single and double-impurity setups, hybrid quantum architectures, and recent experimental advances, we show how key parameters offer direct control over YSR spectral properties, including Coulomb repulsion, hybridization strength, impurity level, external magnetic field, and exchange coupling. These tunable features enable the design of ground-state configurations and provide insights into quantum phase transitions. By integrating theoretical, numerical, and experimental perspectives, this review serves as a foundational reference for researchers seeking to harness YSR states in topological quantum computation, magnetic sensing, and hybrid superconducting devices.