It is shown that a single, strongly scattering impurity produces a bound or a virtual bound quasiparticle state inside the gap in a d-wave superconductor.The explicit form of the bound state wave function is found to decay exponentially with angle-dependent range. These states provide a natural explanation of the second Cu NMR rate arising from the sites close to Zn impurities in the cuprates. Finally, for finite concentration of impurities in a d-wave superconductor, we reexamine the growth of these states into an impurity band, and discuss the Mott criterion for this band.
The consequences of localized, classical magnetic moments in superconductors are explored and their e ect on the spectral properties of the intragap bound states is studied. Above a critical moment, a localized quasiparticle excitation in an s-wave superconductor is spontaneously created near a magnetic impurity, inducing a zero-temperature quantum transition. In this transition, the spin quantum number of the ground state changes from zero to 1 2 , while the total charge remains the same. In contrast, the spin-unpolarized ground state of a d-wave superconductor is found to be stable for any value of the magnetic moment when the normal-state energy spectrum possesses particle-hole symmetry. The e ect of impurity scattering on the quasiparticle states is interpreted in the spirit of relevant symmetries of the clean superconductor. The results obtained by the non-self-consistent (T matrix) and the self-consistent mean-eld approximations are compared and qualitative agreement between the two schemes is found in the regime where the coherence length is longer than the Fermi length.
We report the experimental observation of intrinsic dynamically localized vibrational states in crystals of the highly nonlinear halide-bridged mixed-valence transition metal complex ͕͓Pt͑en͒ 2 ͔ ͓Pt͑en͒ 2 Cl 2 ͔ ͑ClO 4 ͒ 4 ͖, where en ethylenediamine. These states are identified by the distinctive structure and strong redshifts they impose upon the overtone resonance Raman spectra. Quantitative modeling of the observed redshifts is presented based on a nonadiabatic coupled electron-lattice model that self-consistently predicts strong nonlinearity and highly localized multiquanta bound states.[S0031-9007(99)08915-2]
Scanning tunneling microscopy can provide a probe for the detailed study of quasiparticle states in high-T c superconductors. We propose that it can also be used to acquire specific information about impurity-induced quasiparticle states and the superconducting order-parameter structure. In particular, the local density of states is found to be sensitive to impurity-induced resonances and to the symmetry of the order parameter. [S0031-9007(96)
Balatsky and Salkola Reply: In their Comment, Aristov and Yashenkin [1] claim that we [2] have calculated the impurity-induced wave function incorrectly and hence missed a power-law term that decays as 1͞r for all directions. Contrary to their claim, we have explicitly noted the presence of this term in the impurity wave function as is evident from the discussion following Eq. (1) in [2]. Moreover, we have explored the spatial dependence of the impurity state in [3], where all of the terms were kept (see also Ref. [4]). That there is a 1͞r term for all directions is a simple observation which follows, for instance, from the calculation with the linearized quasiparticle spectrum near the nodes. Nonetheless, we decided to neglect it in our subsequent discussion of localization for the following reasons.First, if we take into account the 1͞r decay of the wave function for all directions, the impurity state is not normalizable. The fact that the single-particle states are already extended makes the whole issue of localization more subtle. However, we did not address this question, because the hybridization of impurity states with the quasiparticle continuum regularizes the problem naturally by introducing a cutoff at large distances.Second, for D 0 ø E F , the numerical solution of the impurity state clearly shows the strong cross-shaped anisotropy. Indeed, with good approximation, the impurity wave function is concentrated in the cross-shaped tails as shown in Fig. 2 Third, at nonzero impurity concentrations, the lifetime effects cut off the power-law behavior at finite lengths. As an illustration, consider a d-wave superconductor close to half-filling so that the Fermi surface is square. Let n be the normal to the two sheets of the Fermi surface which are most nearly normal to r, and define r Ќ jn ? rj and r k j͑e 3 3 n͒ ? rj; by definition, r Ќ $ r k . Thenwhere K 1 is the modified Bessel function of the first order, j Ќ a͑W ͞D 0 ͒ is the coherence length, and l Ќ j Ќ ͑D 0 ͞g͒ and l k a͑D 0 ͞g͒ are the cutoff length scales.Here, g is the inverse lifetime, W is the half-bandwidth, and a is the lattice spacing. Note that, for D 0 ø W , the anisotropy of the impurity state is fully developed because j Ќ ¿ a. Furthermore, the decay rate along the nodal directions is much slower than in the other directions: l Ќ ¿ l k . For r Ќ , l Ќ and r k , l k , we obtain the usual power-law behavior, and, for r Ќ . l Ќ or r k . l k , the impurity state is exponentially small to all directions due to the factors e 2r Ќ ͞l Ќ and e 2r k ͞l k . Thus, we may conclude that the power-law decay for all but the nodal directions is less important. As a consequence, the original problem of long-range hoppings with the exponential cutoff is mapped in the limit of a dilute concentration of unitary impurities onto an effective tight-binding model with the hopping range l Ќ ¿ j Ќ . In this model, weak localization leads to an extremely long localization length compared to the bare one in the absence of the impurity-induced quasiparticle states. ...
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