arXiv:cond-mat/0411318v1 [cond-mat.supr-con] 11 Nov 2004 Impurity-induced states in conventional and unconventional superconductors A. V. Balatsky ∗ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 I. Vekhter † Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 Jian-Xin Zhu ‡ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Dated: February 2, 2008) We review recent developments in our understanding of how impurities influence the electronic states in the bulk of superconductors. Our focus is on the quasi-localized states in the vicinity of impurity sites in conventional and unconventional superconductors and our goal is to provide a unified framework for their description. The non-magnetic impurity resonances in unconventional superconductors are directly related to the Yu-Shiba-Rusinov states around magnetic impurities in conventional s-wave systems. We review the physics behind these states, including quantum phase transition between screened and unscreened impurity, and emphasize recent work on d-wave superconductors. The bound states are most spectacularly seen in scanning tunneling spectroscopy measurements on high-Tc cuprates, which we describe in detail. We also discuss very recent progress on the states coupled to impurity sites which have their own dynamics, and impurity resonances in the presence of an order competing with superconductivity. Last part of the review is devoted to influence of local deviations of the impurity concentration from its average value on the density of states in s-wave superconductors. We review how these fluctuations affect the density of states and show that s-wave superconductors are, strictly speaking, gapless in the presence of an arbitrarily small concentration of magnetic impurities. Contents I. Introduction 2 A. Aim and scope of this article 2 B. Unconventional superconductivity 3 C. Outline 4 D. Other related work 5 II. A BCS theory primer 6 A. Bogoliubov transformation 7 B. BCS variational wave function 7 C. Green’s functions 8 III. Impurities in superconductors 8 A. Single impurity potential 8 B. Many impurities 9 C. The self-energy and the T -matrix approximation 10 D. Static and dynamic impurities 10 IV. Non-magnetic impurities and Anderson’s theorem11 V. Single impurity bound state in two-dimensional metals12 VI. Low-energy states in s-wave superconductors 13 A. Potential scattering 13 B. Classical spins 13 ∗ Electronic address: avb@viking.lanl.gov, http://theory.lanl.gov † Electronic address: vekhter@phys.lsu.edu ‡ Electronic address: jxzhu@viking.lanl.gov VII. Impurity-induced virtual bound states in d-wave superconducto A. Single potential impurity problem 15 B. Single magnetic impurity problem 17 C. Self-consistent gap solution near impurity 18 D. Spin-orbit scattering impurities 18 E. Effect of doppler shift and magnetic field 18 F. Sensitivity of impurity state to details of band structure19 VIII. Single impurity bound state in a pseudogap state of two-dimen IX. Quantum phase transition in S-wave superconductor with mag A. Introduction 23 B. Quantum phase transition as a level crossing 24 C. Particle and hole component of impurity bound state25 D. Intrinsic π phase shift for J 0 >J crit coupling 26 X. Kondo impurity 26 A. Kondo effect in fully gapped superconductors 26 1. Ferromagnetic exchange 27 2. Antiferromagnetic coupling 27 3. Anisotropic exchange and orbital effects 28 B. Kondo effect in gapless superconductors 28 XI. Dynamical impurities 30 A. Inelastic scattering from a single spin in d-wave superconductors30 B. Localized vibrational modes in d-wave superconductors32 XII. Interplay between collective modes and impurities in d-wave su