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arXiv:cond-mat/9811312 · Physical review. B, Condensed matter

Bound states for a magnetic impurity in a superconductor

Zorawar S. Bassi, André LeClair

Published Jul 1, 1999• 3 citations
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Abstract

We discuss a solvable model describing an Anderson-like impurity in a BCS superconductor. The model can be mapped onto an Ising field theory in a boundary magnetic field, with the Ising fermions being the quasiparticles of the Bogoliubov transformation in BCS theory. The reflection S matrix exhibits Andreev scattering and the existence of bound states of the quasiparticles with the impurity lying inside the superconducting gap.

DOI
10.1103/physrevb.60.615

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We discuss a solvable model describing an Anderson-like impurity in a BCS superconductor. The model can be mapped onto an Ising field theory in a boundary magnetic field, with the Ising fermions being the quasiparticles of the Bogoliubov transformation in BCS theory. The reflection S matrix exhibits Andreev scattering and the existence of bound states of the quasiparticles with the impurity lying inside the superconducting gap.
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We consider a localized impurity atom that interacts with a cloud of fermions in the paired state. We develop an effective scattering length description of the interaction between an impurity and a fermionic atom using their vacuum scattering length. Treating the pairing of fermions at the mean-field level, we show that the impurity atom acts like a magnetic impurity in the condensed matter context, and leads to the formation of a pair of Shiba bound states inside the superconducting gap. In addition, the impurity atom can lead to the formation of deeply bound states below the Fermi sea.
We discuss a very simple and essentially exactly solvable model problem which illustrates some nice features of Andreev bound states, namely, the trapping of a single Bogoliubov quasiparticle in a neutral s -wave BCS superfluid by a wide and shallow Zeeman trap. In the quasiclassical limit, the ground state is a doublet with a splitting which is proportional to the exponentially small amplitude for “normal” reflection by the edges of the trap. We comment briefly on a prima facie paradox concerning the continuity equation and conjecture a resolution to it.
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The low-energy quasiparticle states of a disordered d-wave superconductor are investigated theoretically. A class of such states, formed via tunneling between the Andreev bound states that are localized around extended impurities (and result from scattering between pair-potential lobes that differ in sign), is identified. Its (divergent) contribution to the total density of states is determined by taking advantage of connections with certain one-dimensional random tight-binding models. The states under discussion should be distinguished from those associated with nodes in the pair potential.