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Acta Physica Polonica A

Zero Temperature Phase Diagram of a d-Wave Superconductor with Anderson Impurities

Lech S. Borkowski

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We study the model of a d-wave superconductor interacting with finite concentration of the Anderson impurities at zero temperature.The interaction between impurity and conduction electrons is taken into account within the large-N approximation.We discuss the obtained phase diagram and its dependence on the main energy scales.
DOI
10.12693/aphyspola.115.120

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The effect of conduction-electron interactions for an Anderson impurity is investigated in one dimension using a scaling approach. The flow diagrams are obtained by solving the renormalization-group equations numerically. It is found that the Anderson impurity case is different from its counterpart, the Kondo impurity case, even in the local moment region. The Kondo temperature for an Anderson impurity shows nonmonotonic behavior, increasing for weak interactions but decreasing for strong interactions. The implication of the study to other related impurity models is also discussed.
What are zero energy excitations in high-TcT_{c}Tc​ superconductors?
The long disputed issue of the limiting value of quasiparticle density of states $\r(0) = \r (\e \to 0)$ in a \emph{d-}wave superconductor with impurities (\emph{vs} its linear vanishing, $\r_0(\e) \propto |\e|/\D$, near the nodal point $\e = 0$ in a pure system with the gap parameter $\D$) is discussed. Using the technique of group expansions of Green functions in complexes of interacting impurities, it is shown that finite (˚0)\r(0)(˚​0) value is possible if the (finite) impurity perturbation VVV is spin-dependent (magnetic). The found value has a power law dependence on the impurity concentration ccc: $\r(0) \propto \r_N c^{n}$, where $\r_N$ is the normal metal density of states and n≥2n \geq 2n≥2 is the least number of impurities in a complex that can localize nodal quasiparticle. This result essentially differs from the known predictions of self-consistent approximation: $\r(0) \propto \r_N \sqrt {c/\r_N\D}$ (for the unitary limit V→∞V \to \inftyV→∞) or $\r(0) \propto (\D/cV^2)\exp(-\D/cV^2\r_N)$ (for the Born limit $|V|\r_N \ll 1$). We predict also existence of a narrow interval (mobility gap) around the Fermi energy, where all the states are localized on proper impurity clusters, leading to exponential suppression of low-temperature kinetics.
We derive a solvable resonant-level-type model, to describe an impurity spin coupled to zero-energy bound states localized at the edge of a one-dimensional d-wave superconductor.This results in a two-channel Kondo effect with a quite unusual low-temperature thermodynamics.For instance, the local impurity susceptibility yields a finite maximum at zero temperature (but no ln-divergence) due to the splitting of the impurity in two Majorana fermions.Moreover, we make comparisons with the Kondo effect occurring in a two-dimensional d-wave superconductor.
The long disputed issue of the limiting value of quasiparticle density of states $\r(0) = \r (\e \to 0)$ in a \emph{d-}wave superconductor with impurities (\emph{vs} its linear vanishing, $\r_0(\e) \propto |\e|/\D$, near the nodal point $\e = 0$ in a pure system with the gap parameter $\D$) is discussed. Using the technique of group expansions of Green functions in complexes of interacting impurities, it is shown that finite (˚0)\r(0)(˚​0) value is possible if the (finite) impurity perturbation VVV is spin-dependent (magnetic). The found value has a power law dependence on the impurity concentration ccc: $\r(0) \propto \r_N c^{n}$, where $\r_N$ is the normal metal density of states and n≥2n \geq 2n≥2 is the least number of impurities in a complex that can localize nodal quasiparticle. This result essentially differs from the known predictions of self-consistent approximation: $\r(0) \propto \r_N \sqrt {c/\r_N\D}$ (for the unitary limit V→∞V \to \inftyV→∞) or $\r(0) \propto (\D/cV^2)\exp(-\D/cV^2\r_N)$ (for the Born limit $|V|\r_N \ll 1$). We predict also existence of a narrow interval (mobility gap) around the Fermi energy, where all the states are localized on proper impurity clusters, leading to exponential suppression of low-temperature kinetics.
The twofold degenerate Anderson impurity model [1-4] is investigated in normal and superconducting states, and the strong electronic correlations of d-electrons of impurity ion are taken into account by elaborating a suitable diagram technique. We discuss the properties of the Slater-Kanamori model [2-4] of d-impurity electrons. After finding the eigenfunctions and eigenvalues of all 16 local states, we determine the local one-particle propagator. Then we construct the perturbation theory around the atomic limit of the impurity ion and obtain the Dyson type equations between impurity electron propagators and normal and anomalous correlation functions. By summing infinite series of ladder diagrams, the approximation for correlation functions is established. The criterion for appearance of a superconducting state of the model is discussed.