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 �� phase shift for J0 Jcrit 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

2 XIII. Scanning tunneling microscopy results 35 A. STM results around a single impurity 35 B. Spatial distribution of particle and hole components 37 C. Fourier-transformed STM Measurement 37 D. Filter 38 XIV. Average density of states in superconductors with impurities39changes A. s-wave 40 1. Born approximation and the AG Theory 40 2. Shiba impurity bands 40 3. Quantum spins and density of states 41 B. d-wave 42 XV. Optimal fluctuation 42 A. Introduction 42 B. Tail states in semiconductors and optimal fluctuation43 C. S-wave superconductors 44 1. Magnetic and non-magnetic disorder 44 2. Diffusive limit, weak magnetic scattering 45 3. Diffusive limit, strong scattering 46 4. Ballistic limit, weak scattering 46 5. Ballistic regime, strong scattering 48 6. Summary 48 XVI. Summary and outlook 48 Acknowledgments 49 List of Symbols 50 References 50 Figures 55 Tables 58 I. INTRODUCTION A. Aim and scope of this article Real materials are not pure. Sometimes excessive impurities hinder observations of beautiful physics that would be there in cleaner systems. Magnetic disorder destroys the coherence of the superconducting state. At the very least, in conventional metals, impurities lead to higher resistivity. It is therefore very tempting to treat impurities as unfortunate obstacles to our understanding of the true underlying physics of the systems we study, strive to make cleaner and better materials, and ignore imperfections whenever possible. Yet sometimes impurities directly lead to the desired physical properties. They are crucial in achieving func- tionality of doped semiconductors: undoped semicon- ductors are just band insulators and not useful for ap- plications in electronics. The entire multi-billion dollar semiconducting electronics industry is based on the pre- cise control and manipulation of electronic states due to dopant (impurity) states. Consequently, sensitivity of a physical system to dis- order can be a blessing in disguise. It can lead not only to achieving new applications but also to uncovering the nature of exotic ground states, elucidating properties of electronic correlations, and producing electronic states that are impossible in the bulk of a clean system. Un- til recently this idea has not been emphasized enough in the study of correlated electron systems, but in recent years more and more efforts are focused on understand- ing the produced by disorder in a wide variety of strongly interacting electronic matter. One of the most promising directions is the study of disorder near quan- tum critical points, where several types of order compete and exist in delicate balance that impurities have the power to tip in favor of one of the orders (Millis, 2003). This is a review of the impurity effects on the elec- tronic states in superconductors. The main purpose of our article is to give a reader an appreciation of recent developments, review the current understanding and out- line further questions on impurity effects in conventional, and especially unconventional superconductors. Super- conductors present probably the first example of a non- trivial many-electron system where effects of disorder on the electronic states were studied experimentally and the- oretically, and this review focuses on these effects. The subject of impurity effects in superconductors is well established and well covered, see, for example, ex- cellent textbooks and reviews (Abrikosov et al., 1963 Annett, 1990 Fetter and Walecka, 1971 de Gennes, 1989 Schrieffer, 1964 Sigrist and Ueda, 1991 Tinkham, 1996). The main classical results, such as Abrikosov- Gor���kov theory of pairbreaking by magnetic impurities (Abrikosov and Gorkov, 1960), and Anderson theorem, that explains why non-magnetic impurities do not de- stroy conventional superconductivity, (Anderson, 1959) are well known from the 60s, and are now taught in graduate school. The need to review the subject arose since a) there are many new results b) the analyses of the classical papers have been substantially modified in applications to novel materials c) the emphasis of the study of the impurity effects shifted from macroscopic to microscopic length scales. From the early days of superconductivity, impurity doping was one of the most important tools to identify the nature of the pairing state and microscopic proper- ties. A classical study of the role of magnetic impuri- ties in conventional superconductors was carried out by Woolf and Reif, 1965, and followed by many detailed in- vestigations. Both magnetic and nonmagnetic impuri- ties are pairbreakers in unconventional superconductors, and often impurity suppression of superconductivity is a strong early hint of the unconventional pairing state. For example, the rapid suppression of the transition temper- ature, Tc, in Al doped SrRuO4 superconductor was the first and strong indication that it is a p-wave supercon- ductor (Mackenzie et al., 1998 Mackenzie and Maeno, 2003). In the past two decades we have witnessed a tremen- dous growth of the number of novel superconductors. Many of them belong to the general class of strongly correlated electron systems, and, as a result of Coulomb interaction, the superconductivity is unconventional, see

3 below. Study of the effect of impurities on unconven- tional superconductors is a still developing field, yet it is mature enough to warrant an overview. Sometimes superconducting state emerges from com- petition between different phases, such as magnetically ordered and paramagnetic in high-temperature cuprates, organic materials and heavy fermion systems. Experi- mentally, superconductivity often is the strongest when the two competing states are nearly degenerate, near quantum critical points. This is the case for example for Ce based heavy fermion materials (Sidorov et al., 2002) and UGe2 (Saxena et al., 2000). Study of impurity ef- fects in these materials allows (at least, in principle) to determine the nature of the superconducting state and reveal competing electronic correlations. This has driven in part the study of impurity effects in high-Tc superconductors. At present, despite much progress, there is no complete microscopic description and certainly no consensus in the community on the mechanism of superconductivity. Study of competing orders in the neighborhood of impurity atoms has the potential to reveal the nature and origin of the supercon- ducting state. The new states and structures that appear due to dis- order often are confined to micro- or mesoscopic length scales. They would remain in the realm of academic dis- cussion were it not for the development of new techniques and probes of disorder. At the time of classical work, ex- perimental interest lied solely in macroscopic properties of materials: transition temperature, Tc, specific heat, and the average density of states (obtained from planar junction tunneling measurements) were the experimen- tally measured quantities. With perfection of more local probes such as nuclear magnetic resonance (NMR), and especially with development of scanning tunneling mi- croscopy and spectroscopy (STM/STS), it became pos- sible to experimentally determine the structures on the atomic scales around the impurity sites. Therefore the emphasis of theoretical work also shifted to the study of these local properties. It is therefore timely and useful to review new results and ideas about impurity-generated states in supercon- ductors. We had to be selective about the topics that are in- cluded in this article. In the spirit of new approaches, our review primarily discusses the physics of the single impurity bound or quasi-bound states and the local elec- tronic effects in the vicinity of defects. We also discuss the physics on the mesoscopic scales, and the behavior of impurities in the presence of competing orders. In the specific case of high-Tc materials we discuss possible com- petition between superconducting state and some com- peting orders in the so called pseudogap state of these superconductors. In all our discussions we restrict ourselves to the study of the behavior of the density of states. A more compre- hensive review of all the effects that were studied experi- mentally and discussed theoretically is a much more diffi- cult task and would take substantially more space. We do not discuss the behavior of transport coefficients: while this is a subject of intense current interest and many im- portant results have been obtained there, it is beyond the scope of this article. To keep this review useful and relevant for people en- tering the field, we start with a simple Bardeen-Copper- Schrieffer (BCS) model for superconductivity, and use a modified version of this model throughout the arti- cle. We do not consider the corrections due to strong coupling that appear in the Eliashberg analysis in the known cases of electron-phonon interaction these correc- tions are quantitative rather than qualitative (Carbotte, 1990 Schachinger, 1982 Schachinger and Carbotte, 1984 Schachinger et al., 1980). In many unconventional materials dynamical glue in the self-consistent theory is not known. For example, there is an ongoing debate on the very nature of the normal state in the high-Tc cuprate superconductors. Yet most people agree that the super- conducting state of cuprates is less anomalous then the normal state, and has the superconducting gap of d-wave symmetry. We take a view that at low energies it could be described for the purposes of our article by BCS with d-wave pairing state. At the same time, while this is a review of recent work on impurity effects in unconventional superconductors, it is emphatically not a comprehensive review of impurity effects in high-Tc cuprates. Nature of superconducting state, detailed microscopic description and competing or- ders in the cuprates are still a subject of intense debate at present. There is a number of excellent reviews of physics of cuprates, including scanning tunneling microscopy (STM) (Fischer and et al, 2004), angle-resolved photoe- mission spectroscopy (ARPES) (Campuzano et al., 2004 Damascelli et al., 2003) and on nature of pseudogap state (Timusk and Statt, 1999). Reader is referred to these re- views for the in depth discussion of the issues specific to high-Tc materials. B. Unconventional superconductivity Examples of exotic superconductors discovered in the last two decades include high-Tc, heavy fermion super- conductors, organic superconductors, SrRuO4. The com- mon feature of all of them is that they are unconven- tional, i.e. the pairing symmetry is not s-wave, in con- trast to conventional materials, such as lead. Here any superconductor with the gap function that transforms according to a trivial representation of the point group of the crystal will be called an s-wave super- conductor. We call a superconducting order parameter unconventional if it transforms as a nontrivial represen- tation of the symmetry group. To be more precise, su- perconductivity is characterized by an order parameter, that describes pairing of fermions with time-reversed mo- menta, k and ���k, ��(k)��,�� = angbracketleft��k,�������k,��angbracketright, (1.1)

4 where ��,�� are spin indices of the paired fermionic states. We distinguish between the spin singlet pairing (total spin of the pair S = 0), for which ��(k)��,�� = ��(k)(i������), (y where ��(y) is the Pauli matrix in spin space, and spin triplet state (S = 1), when ������ is a symmetric spinor in ��,��. Since the order parameter has to be antisym- metric with respect of permutation of fermion operators in Eq. (1.1), the spatial part of ��(k)��,�� is even for spin singlet superconductors and odd in the spin-triplet case. Expanding in eigenfunctions of orbital momentum, it fol- lows that spin singlet pairing corresponds to even orbital function of momentum k and hence we call it s- (for l = 0), d-wave (for l = 2), etc. superconductor in analogy with the notation for the atomic states. For spin triplet superconductor, the orbital part is an odd function of k, and hence spin triplet superconductor can be p-wave (l = 1), f-wave (l = 3) etc. More rigorously one would characterize pairing states by the irreducible representa- tion of the symmetry of the crystal lattice, including the spin-orbit interaction (Blount, 1985 Sigrist and Ueda, 1991 Volovik and Gor���kov, 1984). Characterization in terms of orbital moment is an oversimplification, and we will use this terminology with understanding that the correct symmetries are used for a given crystal structure. The above classification is given for BCS-like or even fre- quency superconductors. This classification is opposite for odd-frequency pairing, where, for example, spin sin- glet state has odd parity because pairing wave function is odd function of time (Balatsky and Abrahams, 1992 Berezinskii, 1974). We will focus on BCS like or even- frequency superconductors here. A reasonable definition of unconventional pairing state, that we adopt here, is that the order parameter average over the Fermi surface vanishes : summationdisplay k ��(k)���� = 0. (1.2) Hence superconductors with the constant or nearly con- stant order parameter on the Fermi surface are s-wave, while p-, d- or higher wave states, where Eq. (1.2) holds, are signatures of an unconventional superconduc- tor. There are several excellent recent reviews that ad- dress the unconventional nature of superconducting pair- ing states in specific compounds, such as p-wave super- conductivity in SrRuO4 (Mackenzie and Maeno, 2003) and d- wave state in high-Tc materials (Annett, 1990 Harlingen, 1995 Tsuei and Kirtley, 2000). C. Outline We start with the general overview of BCS-like super- conductivity. To review the effects of impurities we need to discuss the properties of superconductors in general. In cuprates, as well as in some heavy fermion systems and other novel superconductors, there is some evidence for the existence of an order competing with superconductiv- ity on all or parts of the Fermi surface. The exact nature of the competing order parameter is only conjectured. A general feature of all such models is the enhancement of the competing order once superconductivity is destroyed, for example in the vicinity of a scattering center. It has been suggested that the reaction of the system to the introduction of impurities can be an important test of the order, or even growing correlations towards such an order, in the superconducting state. The prerequisite for such a test is the detailed un- derstanding of the behavior of ���simple��� superconductors with impurities. Work aimed at developing this under- standing spans a period of more than 40 years, and some of the very recent results continue to be fresh and un- expected. Therefore we devote a large fraction of this review to the discussion of the properties of supercon- ductors with impurities in the absence of any competing order. In this case, from a theoretical standpoint, before discussing the impurity effects we need to agree upon methods to describe the very phenomenon that makes the impurity effects so interesting: superconductivity. Even in the most exotic compounds investigated so far the su- perconducting state itself is not anomalous, in that it results from pairing of fermionic quasiparticles, and in that these Cooper pairs may be broken by interaction with impurities or external fields. Impurity effects in conventional superconductors were subject of the very early studies by Anderson, so called ���Anderson theorem���, (Anderson, 1959) and by Abrikosov and Gor���kov (Abrikosov and Gorkov, 1960), hereafter AG. This pioneering work laid the foundation for our understanding of impurity effects in conventional and un- conventional superconductors, described in terms of elec- tron lifetime due to scattering on an ensemble of impuri- ties. AG predicted the existence of the gapless supercon- ductivity that was subsequently observed in experiments (Woolf and Reif, 1965). The brief summary of the AG theory and its extensions to non-s-wave superconductiv- ity is given Table I where effect of impurities on the su- perconducting state on average, or globally is listed. After intense interest in the early days of the BCS the- ory, the subject was considered ���closed��� in mid-60s, with most experimentally relevant problems solved. However, as often happens, recently there has been a revival of the interest in the studies of ���traditional��� low-temperature s- wave superconductors with magnetic and non-magnetic impurities, with many new theoretical and experimental results changing our perspective on this classical prob- lem. A special place in this review is devoted to the study of impurity induced local bound states or resonances. This is an old subject, going back to the 60���s when the bound states near magnetic impurities in s-wave superconduc- tors were predicted in a pioneering work of Yu, Shiba and Rusinov (Rusinov, 1969 Shiba, 1968 Yu, 1965). They considered pairbreaking by a single magnetic impurity in a superconductor, and found that there are quasipar- ticle states inside the energy gap that are localized in the vicinity of the impurity atom. The corresponding

5 gap suppression occurs locally and the concept of lifetime broadening is inapplicable. In general, in this situation it is more useful to focus on local quantities, such as lo- cal density of states (LDOS), local gap etc., rather than on average impurity effects (which vanish for the single impurity in the thermodynamic limit). Yet it is clear that this local physics at some finite concentration of im- purities suppresses superconductivity completely. This connection was discussed in (Rusinov, 1969 Shiba, 1968 Yu, 1965). In particular, formation of the intragap bound state and impurity bands due to magnetic impurity leads to filling of the superconducting gap, and therefore con- nects to the AG theory (Abrikosov and Gorkov, 1960). At the time there were no experimental techniques to directly observe single impurity states. As a result the entire subject was largely forgotten until the STM was applied to study the impurity states by Yazdani et al. (Yazdani et al., 1997). This reinvigorated the field and lead to a firm shift in the interest from global to lo- cal properties. Soon afterwards STM was used to observe local impurity states near vacancies and impurities in the high-Tc cuprates (Hudson et al., 2001, 1999 Pan et al., 2000b Yazdani et al., 1999). These discoveries opened a new field of research where impurities open a window into the study of electronic properties of exotic materials with atomic spatial resolution. As a first test of theories this allowed a direct comparison of the local electronic features in tunneling characteristics with the theoretical predictions for the density of states. We start by briefly reviewing the BCS theory in Sec. II. Our main goal there is to review three approaches that will be used to analyze the impurity effects: direct di- agonalization of the hamiltonian via Bogoliubov-Valatin transformation, variational wave function of the original BCS paper, and the Green���s function method which is well suited to the analysis of multiple impurity problems. Then we define different types of impurity scattering in Sec. III. We pay special attention to distinguishing be- tween magnetic and non-magnetic impurities, and dif- ferentiating between static and dynamic scatterers. The basic features of non-magnetic scattering in s-wave su- perconductors are outlined in Sec. IV. To keep in tune with our intention to make the review readable by graduate students and researchers entering the field, we begin the discussion of the localized states by considering an example of an impurity bound state in a two-dimensional (2D) metal in Sec. V. Then we discuss the low-energy bound state in s- and d-wave supercon- ductors in Sec. VI and Sec. VII respectively. Changes in the ground state of a superconductor containing a classical spin as a function of the coupling strength be- tween the spin and conduction electrons are discussed in Sec. IX. We proceed to consider the situations when the impu- rities have their own dynamics, so that their effect on the electrons is complicated, see Sec. XI, and the com- bined influence of the collective modes and impurities, Sec. XII. We briefly touch upon possible existence of impurity resonances in different models of the pseudo- gap state of the cuprates in Sec. VIII, and discuss recent STM measurements on both conventional and unconven- tional superconductors in Sec. XIII. The final two parts of our review are devoted to the discussion of the effects on impurities on meso- and macroscopic scale. For com- pleteness, we briefly review the basics ideas of computing the average density of states for a macroscopic sample in Sec. XIV. For lack of space we cannot do justice to this very rich subject and use it largely to discuss new results on the impurity effect on the scales small compared to the sample size, but large relative to the superconduct- ing coherence length. In that situation there are dramatic consequences of local impurity realizations that may be different from the average, and we overview the results for the density of states in Sec. XV. We conclude with the summary in Sec. XVI. D. Other related work In focusing largely on the properties of impurities on atomic or mesoscopic scales, we cannot give due atten- tion within the confines of this review to several other questions that have been important in the studies of im- purities. One of these is how exactly does the impu- rity band grow out of bound states on individual impu- rity sites, i.e. what is the effect of interference between such sites is in real space. We briefly review some of re- cent work in Sec. XIV, but do not discuss the subject in depth. The answer to this question is still somewhat un- settled even in general: while the usual finite lifetime approach (Gorkov and Kalugin, 1985 Hirschfeld et al., 1986 Schmitt-Rink et al., 1986) gives a constant density of states at the Fermi level in a d-wave superconduc- tor, and even though the same result has been obtained in field theoretical models of Dirac fermions in two di- mensions, mimicking the d-wave superconductor (Ziegler, 1996 Ziegler et al., 1996), it has also been argued that this DOS diverges (Pepin and Lee, 1998, 2001), or van- ishes. Vanishing DOS can occur with different power laws in energy depending on the approach one uses to treat disorder (Nersesyan and Tsvelik, 1997 Nersesyan et al., 1995 Senthil and Fisher, 1999) (see also (Bhaseen et al., 2001)). The vanishing itself can be traced to level re- pulsion when the system is treated within random ma- trix theory (Altland and Zirnbauer, 1997). Detailed self- consistent numerical studies indicate, however, the the behavior of the DOS depends on the details of the impu- rity scattering and electronic structure (Atkinson et al., 2000 Zhu et al., 2000b). In particular, the divergence only occurs in perfectly particle-hole symmetric systems, and generically Atkinson et al. find that there is a non- universal suppression of the density of states over a small energy scale close to the Fermi level. The interference between many impurities have been investigated recently (Atkinson et al., 2003 Zhu et al., 2003, 2004b) with the eye on the importance of these

6 effects for the interpretation of the features in the STM data on the high-Tc cuprates collected over a large area of the sample. The interference is also responsible for the formation of the impurity bands and therefore is crucial for determining the trans- port properties, which we do not address in this re- view. Within the framework of the t-matrix approxi- mation transport properties of unconventional supercon- ductors in general (Arfi et al., 1988 Graf et al., 1996 Hirschfeld et al., 1989, 1986, 1988 Pethick and Pines, 1986a Schmitt-Rink et al., 1986), and high-Tc cuprates in particular (Duffy et al., 2001 Graf et al., 1995 Hirschfeld and Goldenfeld, 1993 Hirschfeld et al., 1994, 1997 Quinlan et al., 1996, 1994) have been extensively discussed, and the experiments on both microwave, opti- cal, and thermal conductivity are used to extract prop- erties of impurity scattering, see (Timusk and Statt, 1999) for a review as well as very recent results in both experiment (Carr et al., 2000 Chiao et al., 2000 Corson et al., 2000 Hill et al., 2004 Hosseini et al., 1999 Lee et al., 2004 Segre et al., 2002 Tu et al., 2002 Turner et al., 2003) and theory (Berlinsky et al., 2000 Chubukov et al., 2003 Hettler and Hirschfeld, 1999 Howell et al., 2004 Nicol and Carbotte, 2003). The question of localization in both s-wave (Ma and Lee, 1985) and d-wave (Atkinson and Hirschfeld, 2002 Lee, 1993 Senthil and Fisher, 2000 Senthil et al., 1998 Vishveshwara et al., 2000 Yashenkin et al., 2001) con- tinues to be investigated. Some of these results have been summarized in recent reviews on high-Tc sys- tems (Timusk and Statt, 1999). We also do not touch upon the rich phenomena related to the surfaces play- ing the role of extended impurities that can also lead to the formation of the bound states (Aprili et al., 1998 Blonder et al., 1982 Buchholtz and Zwicknagl, 1981 Covington et al., 1997 Fogelstr��om et al., 1997 Hu, 1994 Kashiwaya and Tanaka, 2000). By now there are also few reviews available on the subject of impurity states. Joynt (Joynt, 1997) reviewed early work on the impurity states within the t-matrix theory focusing on anomalous transport due to finite life- time of the quasibound states around impurities. By- ers, Flatte and Scalapino , contributed substantially to studies of the detailed electronic structure of the reso- nance state and interference patterns (Byers et al., 1993 Flatt��e and Byers, 1997a,b Flatte and Byers, 1998), and reviewed their and related work (Flatt��e and Byers, 1999). An excellent review of thermal and transport properties of low-energy quasiparticles in nodal super- conductors was recently given by Hussey (Hussey, 2002). The subject is so rich and well developed that it does not seem possible to do justice to both local quasiparticle properties around a single impurity site and the questions of interference and transport within the confines of a sin- gle paper. With this in mind we now are ready for a main discussion. II. A BCS THEORY PRIMER We begin by reviewing the Bardeen-Cooper-Schrieffer (BCS) theory. This section only briefly summarizes the results pertinent to our discussion many excel- lent textbooks provide an in-depth view on the theory (de Gennes, 1989 Ketterson and Song, 1999 Schrieffer, 1964 Tinkham, 1996). Consider a general hamiltonian HBCS = hatwide H 0(r) + Hint, where hatwide H 0 (r) = summationdisplayintegraldisplay �� ddr����(r)[��(���i���r) ��� ��� ��]����(r) (2.1) is the band hamiltonian of quasiparticles with dispersion ��(k), �� is the chemical potential, and the interaction part Hint = ��� 1 2 summationdisplayintegraldisplay ��,�� ��,�� ddrddr�������(r)����(r���)V��������(r, ��� ��� r���)����(r���)����(r). (2.2) Here r is the real space coordinate, �� and �� are the spin indices, and ����� and �� are the fermionic creation and an- nihilation operators respectively. The mean field approx- imation consists of decoupling the four-fermion interac- tion into a sum of all possible bilinear terms, so that Hint = summationdisplayintegraldisplay ��,�� ddrddr��� braceleftbigg tildewide V ���� (r, r���)����(r)����(r���) ��� (2.3) +������(r, r���)����(r)����(r���) ��� ��� + ������(r, ��� r���)����(r)����(r���) bracerightbigg . The effective potential, tildewide V ���� (r, r���) is the sum of the Hartree and Fock (exchange) terms, and the last two terms account for superconducting pairing. The pairing field, ��, is determined self-consistently from ������(r, r���) = 1 2 V��������(r, r���)angbracketleft����(r���)����(r)angbracketright. (2.4) The pairing occurs only below the transition tempera- ture, Tc above Tc the average of the two annihilation op- erators in Eq. (2.4) vanishes, and therefore ������ = 0. In contrast, Hartree and Fock terms are finite at all temper- atures, and can be incorporated in the quasiparticle dis- persion, ��(k). These terms do change below Tc, upon en- tering the superconducting state. Their relative change, however, is of the order of the fraction of electrons par- ticipating in superconductivity, and therefore is small for weak coupling superconductors (��� ��/W ��� 1, where W is the electron bandwidth). Therefore the effective po- tential, tildewide, V is not explicitly included in the following dis- cussion except where specified. Therefore we start with a reduced mean field BCS hamiltonian, HBCS = summationdisplayintegraldisplay �� ddr����(r) ��� hatwide H 0(r)����(r) (2.5) + summationdisplayintegraldisplay ��,�� ddrddr��� braceleftbigg ������(r, r���)����(r)����(r���) ��� ��� + h.c. bracerightbigg .

7 The spatial and spin structure of ������(r, r���) determines the type of superconducting pairing. In most of this re- view we consider singlet pairing, when �� has only the off-diagonal matrix elements in spin space, and it is com- mon to write ������(r, r���) = (i��y)������(r, r���), where �� is now a scalar function, see previous section. In a uniform superconductor the interaction depends only on the relative position of the electrons, so that V (r, r���) = V (�� ��� r ��� r���). Therefore in the absence of impurities, the structure of the order parameter in real space depends on the symmetry properties of V (��). These are easier to consider in momentum, rather than coordinate, space. In models with local attraction, when V (��) = V0��(��), the Fourier transform of the interaction is featureless, and ��(k) = ��0 an example of an isotropic, or s-wave superconductor. In the remainder of this section we overview the main methods solving the BCS hamiltonian since the same methods are commonly applied to the studies of impu- rity effects in superconductors. The approaches that we consider are: a) direct diagonalization via Bogoliubov- Valatin transformation b) variational determination of the ground state energy from the trial wave function and c) Green���s function formalism. A. Bogoliubov transformation Since the effective hamiltonian of Eq. (2.5) is bilinear in fermion operators, �� and �����, it can be diagonalized by a canonical transformation of the form ����(r) = summationdisplaybracketleftbigg n un��(r)��n + vn��(r)��n ��� bracketrightbigg , (2.6) subject to condition |un��(r)|2 + |vn��(r)|2 = 1. The re- sulting equations on the coefficients u and v are Eu��(r) = H0(r)u��(r) + integraldisplay ddr���������(r, r���)v��(r���(2.7) ), ���Ev��(r) = H0 ��� (r)v��(r) + integraldisplay ddr���������(r, ��� r���)u��(r���(2.8) ). Here we suppressed the label n for brevity. Clearly, when �� = 0, coefficients u and v do not couple, and there is no particle-hole mixing. For each n there are four functions, u���(r),u���(r),v���(r),u���(r) that need to be determined. However, for a singlet superconductor, for example, the matrix ������ is off-diagonal in the spin indices, so that u��� (u���) couples only to v��� (v���), so that in practice only two of the equations are coupled. In the presence of the impurity potential, however, in general all four components become interdependent. Equations (2.7)-(2.8), are coupled integro-differential equations for the functions un��(r) and vn��(r). They have to be complemented by the self-consistency equations on ������, which can be obtained directly from Eq. (2.4) to be ������(r, r���) = ��� 1 2 V��������(r, r���) summationdisplay n bracketleftbigg un��(r���)vn��(r)f(En)��� +vn��(r���)un��(r)(1 ��� ��� f(En)) bracketrightbigg . (2.9) Here the Fermi function f(E) = [exp(E/T) + 1]���1. In a uniform superconductor the Fourier transform of the Bogoliubov equations, Eqs. (2.7)-(2.8), into the mo- mentum space gives (��k ��� Ek)uk�� + ������(k)vk�� = 0, (2.10) (��k + Ek)vk�� + ������(���k)uk�� ��� = 0, (2.11) where ��k is the bare quasiparticle energy, measured with respect to the chemical potential, ��k = ��(k) ��� ��. In a singlet superconductor (��k ��� Ek)uk��� + ��(k)vk��� = 0, (2.12) (��k + Ek)vk��� ��� �����(k)uk��� = 0, (2.13) and recover the familiar energy spectrum Ek = radicalbig ��k 2 + |��(k)|2, with the coefficients u and v given by parenleftBigg uk 2 vk 2 parenrightBigg = 1 2 bracketleftBigg 1 �� ��k Ek bracketrightBigg . (2.14) B. BCS variational wave function Superconductivity originates from the instability of the Fermi sea towards pairing of time-reversed quasiparticle states. Therefore a variational wave function approach, originating with the classic BCS paper, is to restrict the trial wave function to the subspace of either empty or doubly occupied states, |��(r)angbracketright = productdisplay n (an + bncn���c��� ��� ����� n )|0angbracketright, (2.15) and to minimize the energy, EBCS = angbracketleft��|H|��angbracketright. This is a legitimate approximation at T = 0, and is a very good approach at low temperatures. In Eq. (2.15) the vacuum state |0angbracketright denotes the filled Fermi sea, and cn��� ��� (c��� ����� n ) cre- ates a quasiparticle with spin up (down) and with the wave function ��n(r) (��n(r)) ��� that is the eigenfunction of the single particle Hamiltonian. Normalization requires that |an|2 + |bn|2 = 1. In the absence of impurities these eigenfunctions can be labeled by the same indices, k and ��, as in the pre- vious section. Consequently, the variational approach is completely equivalent to the Bogoliubov analysis with the choice un(r) = an��n(r), and vn(r) = bn��n(r). In general, however, interaction with impurities may lead to the appearance of the single particle states in the ground state wave function, see Sec. IX. Moreover, it is worth

8 remembering that energy of the state described by the BCS wave function is greater or equal to that of the exact ground state obtained by solving the Bogoliubov equations. C. Green���s functions The third approach that we will use in this work is the Green���s function method, which originates with the work of Gor���kov. Following Nambu we introduce a 4- vector that is a spinor representation of the particle and hole states, �����(r) = (�����,�����,�����,�����). ��� ��� (2.16) The matrix Green���s function is defined as the imaginary- time ordered average hatwide(x,x���) G = ���angbracketleftT����(x)�����(x���)angbracketright, (2.17) where the four-vector x = (r,��) combines the real space coordinate, r, and the imaginary time, ��. The time evolution of the creation and annihilation operators in the Heisenberg approach is given by �����/����� = [HBCS ��� ��N,��]. For a singlet homogeneous superconductor the Hamil- tonian of Eq. (2.5) in the Nambu notation takes the form, HBCS = integraldisplay dr�����(r)(��(���i���)��3 + ����1��2)��(r), (2.18) and we find (Maki, 1969) hatwide���1(k,��) G 0 = i��n ��� ��(k)��3 ��� ��(k)��2��1. (2.19) Here ��n = ��T(2n + 1) is the Matsubara frequency, ��i are the Pauli matrices acting in spin space, ��i are the Pauli matrices in the particle-hole space, and ��i��j de- notes a direct product of the matrices operating in the 4-dimensional Nambu space. The self-consistency equa- tion for a single superconductor takes the form ��(k) = ���T summationdisplayintegraldisplay ��n dk���V (k, k���)Tr[��1��2G0]. (2.20) In BCS the interaction is restricted to a thin shell of electrons near the Fermi surface, and therefore ��(hatwide ��) = ���TN0 summationdisplayintegraldisplay ��n dhatwide���V �� (hatwide, �� hatwide���)Tr �� bracketleftbigg ��1��2 integraldisplay d��kG0 bracketrightbigg , (2.21) where hatwide �� denotes a direction on the Fermi surface, and N0 is the normal state density of states. The off-diagonal component of hatwide G 0 , is often called the Gor���kov���s anomalous F, (Gor���kov) Green���s functions since it describes the pairing average F����(x,x���) = ���angbracketleftT������(x)����(x���)angbracketright. (2.22) In general F����(x,x���) = g����F(x,x���), where g is the ma- trix describing the spin structure of the superconducting order. For the singlet pairing g = i��(y), where ��(y) is the Pauli matrix. Therefore in a singlet spatially uniform su- perconductor normal and anomalous components of hatwide G 0 are G(��n, k) = i��n + ��k (i��n)2 ��� ��k 2 ��� |��(k)|2 , (2.23) F(��n, k) = ��(k) (i��n)2 ��� ��k 2 ��� |��(k)|2 . (2.24) The connection with the Bogoliubov���s transformation is provided by rewriting the Green���s functions as G(��n, k) = uk 2 i��n ��� Ek + vk 2 i��n + Ek , (2.25) F(��n, k) = ukvk ��� parenleftbigg 1 i��n ��� Ek ��� 1 i��n + Ek parenrightbigg ,(2.26) where uk and vk are given by Eq. (2.14). The three approaches discussed above are complemen- tary and equivalent in the case of homogeneous super- conductors. However, some of them are better suited for addressing specific questions in the presence of impuri- ties. In particular, we will see that the Green���s function method is sometimes advantageous for determining the thermodynamic properties of a material and averaging over many impurity configurations. For inhomogeneous problems, where we are interested in the spatial varia- tions of the superconducting order and electron density, both Bogoliubov equations and Green���s functions are of- ten used. In general, the choice of a specific methods depends on the type of question asked in the presence of impurities, and we briefly describe the basic models and issues related to impurity scattering in superconductors below. III. IMPURITIES IN SUPERCONDUCTORS A. Single impurity potential If we are to address theoretically the question of what defects do to superconductivity, we must describe the defects and superconductivity in the same framework. Grain and surface boundaries, twinning planes, and other structural inhomogeneities scatter conduction electrons and therefore affect the resulting order parameters. How- ever, here we focus on only one type of imperfection: im- purity atoms. a. Potential scattering. First and foremost an impurity atom has a different electronic configuration than the host solid, and therefore interacts with the density of conduction electrons via a Coulomb potential. Himp = summationdisplayintegraldisplay �� dr����(r)Upot(r)����(r). ��� (3.1)

9 In good metals the Coulomb interaction is screened at the length scales comparable to the lattice spacing, and therefore the resulting scattering potential is often as- sumed to be completely local, Upot(r) = U0��(r ��� r0), with the impurity at r0. The resulting scattering occurs only in the isotropic, s-wave, angular momentum chan- nel. If finite range of the interaction is relevant, scatter- ing in l = 0 channels needs to be considered. In that case the treatment is similar to that of magnetic scattering in conventional superconductors, see Sec. VI, and was ap- plied to unconventional superconductors in, for example (Balatsky et al., 1994 Kampf and Devereaux, 1997). In the 4-vector notation of the previous section the po- tential scattering has to have the same matrix structure as the chemical potential, or ��(k) in Eq. (2.19), so that Himp = integraldisplay dr�����(r)Upot(r)��3��(r), (3.2) or, in Nambu notation, hatwide U pot = U0��3��(r ��� r0) (3.3) b. Magnetic scattering. In addition to the electrostatic interactions, if the impurity atom has a magnetic mo- ment, there is an exchange interaction between the local spin on the impurity site and the conduction electrons, Himp = summationdisplayintegraldisplay ���� drJ(r)����(r)S ��� �� ����������(r). (3.4) The range of interaction here is determined by the quan- tum mechanical structure of the electron cloud asso- ciated with the localized spin. Again, in reality we often consider a simplified exchange hamiltonian with J(r) = J0��(r ��� r0), which captures the essential physics of the problem. In the 4-vector notations of the previous section the electron spin operator becomes �� = 1 2 bracketleftbigg (1 + ��3)�� + (1 ��� ��3)��3����3 bracketrightbigg . (3.5) Therefore Himp = integraldisplay dr�����(r)J(r)S �� ����(r), (3.6) or, in Nambu notation, hatwide U mag = J(r)S �� ��. (3.7) c. Anderson impurity. However, even if the ground state of an isolated impurity has an electron spin, the result of putting such an impurity into a host matrix may modify the spin configuration as the impurity electrons couple to the conduction band. Therefore a realistic model for an impurity site is based on the Anderson model, with the Hamiltonian HA = summationdisplay E0d��d�� ��� + Und���nd��� + Hsd, (3.8) Hsd = summationdisplay�� k,�� Vsdck,��d�� ��� + h.c. (3.9) Here E0 is the position of the impurity level relative to the Fermi energy, d��� and d operate on the impurity site, U is the Coulomb repulsion for the electrons localized on the impurity site, and ck,ck ��� create and annihilate the conduction electrons. This Hamiltonian allows the elec- trons to hop on and off the impurity site, resulting in a finite width of the impurity level, �� = ��|Vsd|2N0. The model describes the potential scattering, when U ��� ��. On the other hand, when E0 ��� EF , E0 + U ��� EF , and U ��� ��, we expect the local levels to remain split, so that the impurity state is singly occupied and has a local spin. Therefore it allows a natural interpolation between potential and magnetic scattering, as well as the study of the mixed valence regime. The price to pay for such a rich behavior of the Anderson impurities is the difficulty of studying them analytically, and therefore in practice many results have been obtained in the simplified mod- els above, although a number of very thorough numerical renormalization group studies of Anderson impurities in superconductors exist. We will review some of them for completeness, but will not focus on those extensively. B. Many impurities In all of our discussions we assume noninteracting im- purities, so that the net impurity potential is simply hatwide U imp(r) = summationdisplay hatwide U imp(r ��� ri) (3.10) = integraldisplayi dr�����imp(r���)hatwideimp(r U ��� r���). (3.11) Here hatwide U denotes the matrix structure of the potential in both spin and particle-hole space, and we introduced the impurity density, ��(r) = summationdisplay i ��(r ��� ri). (3.12) We also assume the dilute impurity limit of the average impurity concentration ni ��� 1, where ni = integraldisplay dr V ��(r). (3.13) For magnetic scatterers it was explicitly shown that the effect of the RKKY interaction between scatter- ing centers on the superconducting properties is small (Galitskii and Larkin, 2002 Larkin et al., 1971). If we now compute a local physical quantity, such as the density of states measured at the position r by the

10 STM, it will depend on the distance from the nearby im- purities, and therefore will be different for different real- ization of impurity distributions. In contrast, thermody- namic quantities, such as the density of states measured in planar junctions, or the specific heat, average the den- sity of states over many random local configurations of impurities. Therefore in computing their values we av- erage over all impurity configurations (Abrikosov et al., 1963), so that, for example, ��(��n, G k) = Ni productdisplay i=1 bracketleftbigg 1 V integraldisplay driG(��n, k, r1,..., rNi ) bracketrightbigg . (3.14) Here a bar denotes such an impurity average. By definition ��imp �� = ni. We also assume an uncorre- lated, or random, impurity distribution, which means ��(r)��(r���) ��� Ni productdisplay i=1 bracketleftbigg 1 V integraldisplay dri��(r, r1,..., rNi )��(r���, r1,..., rNi ) bracketrightbigg = ni��(r ��� r���) + ni 2 . (3.15) Since the impurities are dilute, ni 2 ��� ni, and we neglect the second term compared to the first. In Sec. XIV we implement this impurity averaging procedure to deter- mine the average density of states. C. The self-energy and the T-matrix approximation In practice to compute the Green���s function in the pres- ence of impurities we will often employ the T-matrix ap- proximation. This method is described in detail in many original articles and reviews (Hirschfeld and Goldenfeld, 1993 Hirschfeld et al., 1986, 1988 Hotta, 1993 Hussey, 2002 Mahan, 2000), and we only briefly summarize it. For a single impurity with the scattering potential hatwide U k,k��� in the momentum space (given by one of the models discussed at the beginning of this chapter), the T-matrix accounts exactly for multiple scattering off of one impu- rity. In the language of Feynman diagrams, the corre- sponding process is shown in Fig. 1. Here, and through- out the review, the hat over a letter means that it denotes a matrix in Nambu space. Therefore the full Green���s function is hatwide(k, G k���) = hatwide G 0(k) + hatwide G 0(k)hatwidek,k��� U hatwide G 0(k���) (3.16) + summationdisplay k������ hatwide G 0 (k)hatwidek,k������ U hatwide G 0 (k������)hatwidek������,k��� U hatwide G 0 (k���) + .... Here we suppressed the frequency index in the Green���s function as the scattering is elastic. The series can be summed to write (see Fig. 1) hatwide(k, G k���) = hatwide G 0(k) + hatwide G 0(k) hatwide T k,k��� hatwide G 0(k���), (3.17) where the T-matrix is given by hatwide T k,k��� = hatwide U k,k��� + summationdisplay ������ hatwide U k,k������ hatwide G 0(k������)hatwidek������,k��� U + ...(3.18) = hatwidek,k��� U + summationdisplayk k������ hatwidek,k������ U hatwide0(k������)hatwidek������,k���. G T (3.19) This equation needs to be solved for hatwide. T If the impu- rity scattering is purely local, hatwide(r U ��� r���), the scattering is isotropic, hatwide U k,k��� = hatwide, U greatly simplifying the process of solving the equation for the T-matrix, as hatwide T depends only on frequency. Notice that we could draw the set of diagrams in Fig. 1 in real space, and write the corresponding set of equa- tions for the T-matrix and Green���s function hatwide(r, G r���) in complete analogy with Eq. (3.19). The main observation here is that, in the vicinity of the impurity, the trans- lational invariance is broken, and the Green���s function depends on two momenta, k and k���. hatwide(r, G r��� ��) = hatwide G 0 (r, r��� ��) + hatwide G 0 (r, r0 ��) hatwide(��)hatwide0(r0, T G r��� ��) (3.20) The T-matrix lends itself easily to describe the effect of an ensemble of impurities. The so called self-consistent T-matrix approach (SCTM) considers multiple scattering on a single site of an electron that has already been scat- tered on all other impurity sites (Hirschfeld et al., 1986, 1988). This results in replacing the bare Green���s func- tion in Eq. (3.19) by its impurity-averaged counterpart, hatwide(k,��). G Notice that after averaging over the random impurity distribution the translational invariance is re- stored, so that the Green���s function depends on a single momentum k. The combined effect of impurities is given by the self energy, hatwide ��( k,��) = nimp hatwide T k,k, so that hatwide���1(k,��) G = hatwide���1(k,��) G 0 ��� hatwide ��( k,��). (3.21) In contrast to the single impurity case where Eq. (3.17) with the T-matrix given by Eq. (3.19) is the exact so- lution of the problem, the Green���s function given above is an approximation, and much recent research is moti- vated by questions about how accurately it describes the properties of nodal superconductors with impurities. D. Static and dynamic impurities So far we only discussed the static impurities, and most of our review addressed such a situation. However, even for purely potential scattering a situation is possi- ble when a vibrational mode leads to a time-dependent modulation of the charge on an impurity site, and, as a result, Upot acquires a characteristic frequency. Such a mode can be extended, as a phonon, or local. Influence of the dynamical impurity on the local properties of a su- perconductor is a relatively new subject of research and we summarize recent results in Sec. XI For magnetic scattering the situation is more complex even in a normal metal. The degeneracy between the spin-up and spin-down states on the impurity site and the non-trivial commutation relations between different spin components ensure that quantum dynamics of the impurity is generated even if the exchange constant is purely static. In simple words, if the scattering process,

11 which flips both the spin of the conduction electron and the impurity spin, is relevant, the dynamics of the lo- cal spin flips becomes essential. This dynamics leads to Kondo screening of the impurity spin in a metal, and in Sec. XI we briefly discuss the current status of the yet not fully understood problem of Kondo effect in a super- conductor. In the limit of large impurity spin, however, the change of the impurity spin by 1 during the spin flip scattering is not relevant, and its dynamics does not play a major role. In this limit of classical spin the static local and global density of states is discussed in Sec. VI and Sec. XIV respectively. Such a spin acquires dynamics only when placed in an external magnetic field, which also affects the superconducting state. IV. NON-MAGNETIC IMPURITIES AND ANDERSON���S THEOREM One of the most important early results was the ro- bustness of the conventional superconductivity to small concentrations of non-magnetic impurities. Theoretical underpinning of this result is known as Anderson���s theo- rem (Anderson, 1959). Anderson���s observation was that, since superconductivity is due to the instability of the Fermi surface to pairing of time-reversed quasiparticle states, any perturbation that does not lift the Kramers degeneracy of these states does not affect the mean field superconducting transition temperature. This is most clearly seen from the BCS analysis, which we carry out following Ma and Lee, 1985. We consider an isotropic pairing potential, V��������(r, r���) = V ��(r ��� r���). In the absence of a magnetic field the coefficients an = sin ��n and bn = cos ��n can be taken real without loss of general- ity, so that the self-consistency condition, Eq. (2.9) reads ��n = V summationdisplay m=n ��m (��m 2 + ��m) 2 integraldisplay ddr��n(r)��m(r). 2 2 (4.1) Here ��n = integraldisplay ddr��(r)��n(r). 2 (4.2) As noted above, in the BCS approach �����s are the eigen- functions of the single particle hamiltonian. In the ab- sence of impurities the system is translationally invariant, so that ��(r) = ��n = ��0. The most important assump- tion underlying Anderson���s theorem is that the super- conducting order parameter can be taken to be uniform, ��(r) = ��1, even in the presence of impurities. In that case the individual eigenfunctions of the single particle hamiltonian including impurities are rather complicated. However, the gap equation, Eq. (4.1), takes the form 1 V = integraldisplay d��radicalbig N(��, r) ��2 + ��1 2 , (4.3) equivalent to that of a pure superconductor provided the density of states N(��, r) = summationdisplay n ��m(r)��(�� 2 ��� ��m), (4.4) is unchanged compared to the pure metal, N(��, r) ��� ��0. If this condition is satisfied, the solution ��1 of the gap equation Eq. (4.3) must be identical to that of the BCS equation in the absence of impurities, and therefore the transition temperature and the gap are insensitive to the impurity scattering at the mean field level. Anderson���s theorem helped explain why superconduc- tivity was robust to impurities in many early experi- ments. It is important to realize however that it is an approximate statement about the thermodynamic aver- ages of the system. Beginning with the next section we will analyze in more detail the changes that impurities create in superconductors in their immediate surrounding as well as on average. We will see that even purely poten- tial scattering does induce changes in the local properties of superconductors, albeit the corresponding change in the transition temperature remains minimal. Anderson���s theorem brings to the fore the need to separate the study of impurity effects on different lengths scales, from lattice spacing to the coherence length, to sample size. Before we proceed to study the local properties we dis- cuss the extensions of the Anderson���s treatment of im- purities. In weakly disordered systems the density of states remains nearly constant as a function of disor- der. Ma and Lee (Ma and Lee, 1985) argued that An- derson���s theorem remains valid in the form above even in a strongly disordered system provided the localization length, L ��� (��0��0)1/d. In that case there is a large number of states localized within energy ��0 of the Fermi surface, and these state form a local superconducting patch. The Josephson interaction between the patches then leads to global phase coherence at T = 0. More- over, they argued that the theorem holds all the way to the limit of site localization. It is important to note that the superfluid stiffness, i.e. the ability of the superconductor to screen out the mag- netic field, is affected by disorder. In particular, when the quasiparticle lifetime, ��, becomes sufficiently short, ��0�� ��� 1, the superfluid density ��s ��� ��0��. Conse- quently the superconductor is sensitive to the local phase fluctuations of the order parameter, and the experimen- tally observed transition temperature may be severely suppressed compared to the mean field Tc, as it is, for example, in granular superconductors. Approaches ex- tending beyond the mean field picture are largely outside the scope of this review. Therefore for dilute impurities Anderson���s theorem is valid provided the superconducting order parameter can be taken to be nearly uniform. Since the ���heal- ing length��� of ��(r) over which it can change apprecia- bly is the coherence length, ��0 ��� planckover2pi1vF /��0, where vF is the Fermi velocity, while the Coulomb screening length for the charged impurities in metals is of the order of

12 the lattice spacing, a, for ��0 ��� a the order parame- ter remains essentially uniform, and Anderson���s theo- rem holds. Much work has been done recently on the effect of the ultrashort coherence length on the impu- rity scattering in superconductors. In particular, it has been shown that when the superconducting pairing is of the order of the electron bandwidth, Anderson���s theorem is violated (Ghosal et al., 1998 Moradian et al., 2001 Tanaka and Marsiglio, 2000). Ghosal et al. (Ghosal et al., 1998), and Xiang and Wheatley (Xiang and Wheatley, 1995) have explored in detail the discrepancy between the single particle exci- tation gap and the superconducting order parameter as a function of disorder in these circumstances. Beyond Anderson���s regime of the constant density of states, both quantities decrease at first, since the disorder depletes the density of states. Then, however, the spectral gap persists while the superconducting order vanishes. As pointed out by Ma and Lee (Ma and Lee, 1985) in the limit of strong disorder the models with on-site pair- ing, such as those studied by Tanaka and Marsiglio, and Ghosal et al., show the on-site spectral gap (so-called Anderson negative-U glass) without the off-diagonal long range order and without symmetry breaking. In most experimentally relevant situations, however, the corrections to the main statement of Anderson���s the- orem are quantitative rather than qualitative. This is generally true of most results pertaining to the impurity scattering in superconductors, and therefore it is very instructive to consider this problem in BCS-like systems. V. SINGLE IMPURITY BOUND STATE IN TWO-DIMENSIONAL METALS Before we proceed to calculate the effect of impurity in a d-wave superconductor it is instructive to review a simpler problem of a single impurity in a metal. We show here a T-matrix calculation for finding the bound states due to a single impurity in d dimensions with an attractive potential U0 ��� 0. The Hamiltonian for the problem is H = summationdisplay k [��(k) ��� ��]ck,��ck,�� ��� + summationdisplay k,k��� U0ck,��ck���,�� ��� (5.1) the U0 term describes the on-site energy change of elec- tron density n(r) in external potential U(r) = U0��(r). We consider a single particle (�� = 0), although the cal- culation for the normal metal with a finite density of states follows simply by replacing ��(k) ��� ��(|bfk) in the following. The bare Green���s function for a free particle is G0(��, k) = [�� ��� ��(k)]���1. (5.2) Since the vertex of the impurity interaction, U0 is mo- mentum independent, the equation for the T-matrix is particularly simple and follows from Eq. (3.19), T(��) = U0 + U0 summationdisplay k G0(��, k)T(��) T(��) = U0 1 ��� U0 ��� k G0(��, k) (5.3) Summation over momentum in ��� k is easily performed using the density of states N(��) = summationdisplay k ��(�� ��� ��(k)) = ��d�� d 2 ���1, (5.4) where ��d is a constant dependent on dimension. There- fore g0(��) = summationdisplay k G0(��, k) = integraldisplay W 0 d��N(��) �� ��� �� ��� �����d�� d���2 2 ,(5.5) for d = 2, where W is the electron half bandwidth. In two dimensions g0 ��� �����2 ln(W/|��|). Consequently, the T-matrix for d = 2 is given by T = U0 1 ��� gd�� d���2 2 (5.6) where gd = ���U0��d is the effective coupling constant, and by the same expression with the obvious substitution of ln(W/��) for d = 2. Since the Green���s functions in the presence of impurity scattering is G = G0 + G0TG0, see Eq. (3.17), poles of the T-matrix are the new poles of G that are not poles of G0, signifying the appearance of new states. We can find this pole, ��0, from Eq. (5.6) for different dimension d. The bound state (��0 0, see Fig. 2) is formed for an arbitrarily small potential |U0| in d = 1, 2, but requires a critical coupling for d = 3. The energy of this state is given by ��0 ��� (g1)2, if d = 1 (5.7) ��0 = W exp(��� 1 g2 ), if d = 2 (5.8) ��0 ��� [g3 1 2 ��� g3c], 1 2 g3 ��� g3c, if d = 3, (5.9) where the d = 3 critical coupling g3c ��� W ���1/2. We focus in more detail on the two-dimensional case, when g2 = ��2|U0| and ��2 = m 2�� is the electron density of states. The bandwidth, W ��� planckover2pi12 2ma2 is the ultravio- let cutoff corresponding to the lattice parameter a for free particle. This result can be compared to the solu- tion of the Schr��odinger���s equation for the particle in the 2D attractive potential U0 (Landau and Lifshitz, 2000), Ch. 45. For an arbitrary potential U(r) the solution obtained using the T-matrix is asymptotically correct if the scattering length is greater than a. For shallow potential the bound state energy �����0 is small, and the characteristic extent of the bound state wave function is l0 = ( planckover2pi12 2m��0 )1/2 ��� a Therefore in this limit we can safely approximate U(r) = U0��(r), where U0 = integraltext U(r)dr.

13 Finding the energy of the bound state, Eq. (5.8), is only one part to the solution. We also want to determine the corrections to the local Density of States due to bound state. We write the equation for the Green���s function in real space, Eq. (3.20), G(r, r��� ��) = G0(r, r��� ��) + G0(r, 0 ��)T(��)G0(0, r��� ��) and read off the position dependent Density of states (DOS) N(r,��) = ��� 1 �� ImG(r, r ��) = N0(r,��) + ��N(r,��). (5.10) Here the first term is the DOS of a clean system, and the second is the correction due to the bound state. We focus on the energy range close to the bound state energy, �� ��� ��0. Since the bound state is below the bottom of the band, the unperturbed Green���s function G0 has no imaginary part in this range (N0 = ���Img0(�� = 0)/��). Therefore the only contribution to the imaginary part of the full Green���s function, Eq. (5.10) comes from the T-matrix : ImT(��) = Im 1 1/g2 ��� log[W/(�����)] = Imln���1[ �� + i�� ��0 ] = ����(�� ��� ��0), (5.11) and the correction to the DOS of a clean system is : ��N(r,��) = |G0(r,��0)|2��(�� ��� ��0) (5.12) with G0(r,��) = N0J0(kF r)ln[W �� ] is the real part of Green���s finction in 2D systems. Equations (5.9) and (5.12) are the main results of this section. They establish the logic we will adhere to in finding impurity induced bound states: a) find the poles of the T matrix in the and the poles of the dressed Green���s function Eq. (5.9), b) find the inhomogeneous DOS due to impurity induced state, Eq. (5.12). One should keep in mind that this approach is just one of many one can implement in a search for scattering induced bound states. Alternatively one can use for example the exact numerical solution of a finite system. As we will argue for superconducting case the self-consistency condition can not be implemented an- alytically and the numerical solution remains the only method available. VI. LOW-ENERGY STATES IN s-WAVE SUPERCONDUCTORS A. Potential scattering Even though the potential scattering does not change the bulk properties of isotropic superconductors, it does affect the local density of states (Machida and Shibata, 1972 Shiba, 1973). Let us consider the Anderson im- purity model, Eqs. (3.8)-(3.9) in the limit U = 0 (res- onance scattering). As discussed above the localized state acquires a finite width, �� = ��|Vsd|2N0, due to hy- bridization with the conduction band. The Green���s func- tion of the conduction electrons can be evaluated in the T-matrix approach, with the result at real frequencies (Machida and Shibata, 1972 Shiba, 1973) hatwide(��) T = |Vsd|2��3 bracketleftbigg �� ���E0��3 ���|Vsd|2��3 summationdisplay k hatwide G 0 (k,��)��3 bracketrightbigg���1 ��3. (6.1) The poles of the T-matrix determine the location of the bound states ��2 bracketleftbigg 1 + 2�� ��� ��2 ��� ��2 bracketrightbigg = E0 2 + ��2. (6.2) In most physical situations �� ��� ��, so that the bound states are located at ��0 = ����(1 ��� 2��2(��Nd(0))2), (6.3) where Nd(0) = �����1��/(��2 +E0 2) is the density of states of the resonant impurity level. For typical densities of states ��Nd(0) ��� 10���3, so that the bound states lies essentially at the gap edge. Shiba considered a finite but small value of the Coulomb repulsion and allowed for the induced pairing on the impurity site (Shiba, 1973). He concluded that, even though there may be a shift of the bound state to lower energies, it still lies within 10���3�� of the mean field gap edge, and therefore can be neglected in the discussions of physical properties. B. Classical spins If the substitution atoms have a magnetic moment, the time-reversal symmetry is violated, and therefore super- conductivity will be suppressed. We consider the mag- netic scattering, Eq. (3.6), which we rewrite in the mo- mentum space as Hex = 1 2N summationdisplay k,k��� ���� J(k ��� k���)ck,�������� ��� �� Sck�����. (6.4) We first review a simplified version of this problem, where we do not need to consider Kondo screening. We re- view scattering on classical spins first studied indepen- dently at about the same time by Shiba, Rusinov, and Yu (Rusinov, 1968, 1969 Shiba, 1968 Yu, 1965). Quan- tum mechanical properties of spin can be neglected when S ��� ���, and we simultaneously take J ��� 0 so that the product JS = const. In this limit the localized spin acts as a local magnetic field. Therefore we study the effect of the impurity with the potential U(r) = U0 + Uex, or Himp = Himp + Hex, on a

14 BCS s-wave superconductor with the unperturbed hamil- tonian of the form H0 = summationdisplay k�� ��kck,��ck�� ��� + ��0 summationdisplay k {ck���c���k��� ��� ��� + c���k���ck���}. (6.5) This problem serves as a starting point for all subsequent analysis of the resonance states in superconductors. To find a localized state with energy 0 E ��0 near a single paramagnetic impurity we perform a Bogoliubov transformation (Rusinov, 1968 Yu, 1965) to find Eu��(r) = ��(k)u��(r) + i��������v��(r) y + U����(r)u��(r), (6.6) Ev��(r) = �����(k)v��(r) ��� i��������u��(r) y ��� U����(r)v��(r)(6.7) . This system is solved by Fourier transforming the equa- tions and expanding the impurity potential in spherical harmonics, and has solutions with energies El ��0 = 1 + (��N0Vl)2 ��� (��N0JlS/2)2 radicalbig [1 + (��N0Vl)2 ��� (��N0JlS/2)2]2 + 4(��N0JlS/2)2 , (6.8) where N0 is, again, the density of states at the Fermi energy in the normal state. This result can be written in a more elegant form if we introduce the phase shifts, ��l, of scattering for up (+) and down (-) electrons, in each angular channel, tan ��l �� = (��N0)(Vl �� JlS/2). (6.9) Then the energies of the states in the gap become ��l = El ��0 = cos(��l + ��� ��l ��� ). (6.10) Clearly, for purely potential scattering (��l + = ��l ���) the spectrum begins at the gap edge, and there are no in- tragap states. However, as the magnetic scattering in- creases, a series of low-energy states below the gap edge appear. Purely magnetic scattering corresponds to ��l + = �����l ���, and strong scattering (large phase shift) yields a localized state deep in the gap, while weak scattering (small phase shift) results in the bound state very close to the gap edge. The same result can be obtained using the Green���s function formulation (Rusinov, 1969 Shiba, 1968) and solving the single impurity problem using the T-matrix method described above. With the impurity hamiltonian of Eq. (3.6) in the Nambu notations the matrix Green���s function for the system is hatwide(k, G k��� ��) = hatwide G 0(k,��)��(k���k���)+ hatwide G 0(k,��) hatwide(k, T k���) hatwide G 0(k���,��). (6.11) Here the T-matrix is computed as in Sec. III, and we sum over the indices of the matrix �� in each vertex. The l-th angular component of the T-matrix satisfies the matrix equation (for a spherical Fermi surface and isotropic gap) hatwide T l (��) = hatwide U l + hatwide U l integraldisplay d�� hatwide G 0 (k,��) hatwide T l (��). (6.12) The full expressions for the T-matrix for both poten- tial and magnetic scattering in all angular channels is straightforward to obtain (Rusinov, 1969) but is rather cumbersome, so that we don���t give it here. Even the case of only spherically symmetric scattering (l = 0) with both U0 = 0 and J = 0 the T-matrix is simple yet lengthy (Okabe and Nagi, 1983). The main results for the en- ergy of the Shiba states remains the same, of course as Eq. (6.10). In the particular case of purely magnetic spherically symmetric exchange, J(k ��� k���) = J, only l = 0 compo- nents are non-vanishing and the T-matrix has a partic- ularly simple form (Shiba, 1968). The diagonal in spin indices component is, T (1)(��) = 1 N (JS/2)2hatwide0(��) g I ��� (JShatwide0(��)/2)2 g . (6.13) Here hatwide0 g is the local matrix Green���s function, hatwide0(��) g = 1 N summationdisplay k hatwide G 0(k,��) = �����N0 �� + ��0��2��2 radicalbig ��0 2 ��� ��2 . (6.14) The bound state energy ��0 = E0 ��0 = 1 ��� (JS��N0/2)2 1 + (JS��N0/2)2 . (6.15) The wave functions of the bound states at El can be computed using the Bogoliubov equations above. In the simplest case of isotropic scattering at distances r ��� pF ���1, both u(r) and v(r) vary as (Fetter, 1965 Rusinov, 1969) sin(pF r ��� ��0 ��) pF r exp(���r/��0| sin(��0 + ��� ��0 ���)|, (6.16) that is, the state is localized near the impurity site at distances r0 ��� ��0 | sin(��0 + ��� ��0 ���)| = ��0 radicalbig 1 ��� ��02 . (6.17) The square of these coefficients gives the spatial depen- dence of the amplitude of the particle and hole com- ponents of the density of states at a given position r (Yazdani et al., 1997). The analysis above was carried out under the assump- tion that the variation of the superconducting order pa- rameter, ��, around the impurity site does not change the position of the resonance low energy state. There are several characteristic length scales for this variation, ����(r). Far away from the impurity, r ��� ��0, at tempera- tures close to Tc, where this variation can be determined perturbatively, ����(r)/��0 ��� 1/(pF r) (Heinrichs, 1968 Rusinov, 1968). This power law is insensitive to the phase shifts of scattering on the impurity. At low temperatures a fully self-consistent treatment is required, which leads to ����(r) decaying as (pF r)���3 and oscillating on the scale of ��0��0/��D, where the Debye temperature ��D sets the

15 scale for the interaction between electrons (Schlottmann, 1976). In the immediate vicinity of impurity, vF /��D ��� r ��� ��0, the variation of the order parameter is ����(r)/��0 ��� 1/(pF r)2 in the linear response approximation (Rusinov, 1968). In the fully self-consistent treatment at distances r ��� ��0��D/EF , this dependence was found to acquire an oscillating factor sin2 pF r (Schlottmann, 1976). In all these cases, since the suppression of the order parameter is determined by the Fermi wavelength, the effect is negligible in determination of the position of the bound state. VII. IMPURITY-INDUCED VIRTUAL BOUND STATES IN d-WAVE SUPERCONDUCTORS We are now ready to extend our discussion to impu- rity induced states in d-wave superconductors. Scalar (non-magnetic) impurities are pair-breakers for ���higher- orbital-momentum��� states, such as a d-wave pairing state. This occurs because change of the momentum of particles in the Cooper pair upon scattering disrupts the phase assignment for particular momenta in a nontriv- ial pairing (Anderson, 1959 Markowitz and Kadanoff, 1963 Tsuneto, 1962). More rigorously this follows from the analysis of the normal and anomalous self-energies due to scattering within the Abrikosov-Gorkov theory (Abrikosov et al., 1963). We also note that one of the first arguments about pairbreaking effects of potential scattering was given by Larkin (Larkin, 1965). As we have emphasized, for pairbreaking impurities the local properties of the superconductor near an im- purity site, such as the local density of states and the gap amplitude, will be modified dramatically. To cap- ture these modifications, we use a variation of the Yu- Shiba-Rusinov approach (Rusinov, 1968 Shiba, 1968 Yu, 1965), which treats magnetic impurities in the strong scattering limit, see Sec. VI. We restrict our consider- ation to the s-wave scatterers with the phase shift close to the unitarity limit, ��0 ��� ��/2, when the bound state has energy away from the gap edge. In contrast to the s-wave superconductors, in d-wave systems the density of states below the gap maximum is non-zero, and varies linearly with energy in a pure system. Consequently, the overlap with the particle-hole continuum only allows the formation of virtual bound states with a finite lifetime. We focus in this section on point-like defects, and use the T-matrix approach. A closely related method uses quasiclassical approximation and picture of Andreev scattering ideas to reproduce the results of T-matrix calculation (Chen et al., 1999 Choi and Muzikar, 1990 Shnirman et al., 1999). Even more interesting results are obtained within the quasiclassical formalism for extended defects. For example, it has been shown that index theo- rem dictates the existence of the low energy quasi-bound state (Adagideli et al., 1999). Zn substitutions in cuprates are one example of non- magnetic atoms that are predominantly potential scat- terers in high-Tc superconductors. Although Zn ions are nominally non-magnetic, Tc is strongly suppressed by Zn substitution of Cu in the planes (Hotta, 1993 Ishida et al., 1991). Therefore, it is reasonable to as- sume that Zn ions are non-magnetic unitary scatterers, see below. We analyze virtual impurity-bound states in a d- wave superconductor and, within this framework, ex- plore possible implications of the assumption that the pairing in cuprates is in the dx2���y2 channel. We model cuprates as a 2D d-wave superconductor, based on strong anisotropy of electronic transport. Our results, can be easily extended for any nontrivial pairing state and may be relevant, e.g. for heavy-fermion superconductors with impurities. Here we closely follow the references (Balatsky et al., 1995 Buchholtz and Zwicknagl, 1981 Salkola et al., 1996, 1997 Stamp, 1987). Main results of this section are as follows: (i) A strongly-scattering scalar impurity is a requirement for a localized, virtual or virtually bound state ( or reso- nance) to exist in a d-wave superconductor. It is intu- itively obvious that any strong enough pair-breaking im- purity ��� magnetic or non-magnetic ��� will induce such a state. Indeed, the low-lying quasiparticle states close to the nodes in the energy gap will be strongly influ- enced even by a non-magnetic impurity potential, result- ing in a virtual bound state in the unitary limit. (ii)This should be compared to the fact that, in s-wave super- conductors, both magnetic and resonant non-magnetic impurities produce bound states inside the energy gap (Machida and Shibata, 1972). The energy ����� and the decay rate �������� of this state are given by �� ��� ����� + i�������� = �����0 ��c/2 log(8/��c) bracketleftbigg 1 + i�� 2 1 log(8/��c) bracketrightbigg (7.1) where c = cot ��0. These results are computed assuming logarithmic accuracy is sufficient, with log(8/��c) ��� 1. In the unitary limit, defined as ��0 ��� ��/2 (c ��� 0), the virtual bound state becomes a resonance at �� ��� 0 with ��������/����� ��� 0. In the opposite case of weak scattering with c lessorsimilar 1, the energy of the virtual bound state formally approaches ����� ��� ��0 and the state is ill-defined because �������� ��� ����� (see Fig. 10. The wave function of the bound state is found to decay as a power law: ��(r) ��� 1/r and is not normalizable. This is consistent with the virtually bound state being not really a bound state. Wave func- tion is localized along the directions of the vanishing gap, so called nodal directions. A. Single potential impurity problem Consider the single scalar impurity problem with Hint = summationdisplay kk����� U0ck��ck����� ��� (7.2)