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Bogoliubov-de Gennes method and its applications. (English) Zbl 1361.82007

Lecture Notes in Physics 924. Cham: Springer (ISBN 978-3-319-31312-2/pbk; 978-3-319-31314-6/ebook). xi, 188 p. (2016).
The lecture notes discuss the Bogoliubov-de-Gennes (BdG) method and its applications in superconductivity. The book is divided into seven chapters presented into framework of two parts devoted to the formalism of the BdG method and its applications, respectively. Chapter 1 introduces the correlated materials and briefly discusses the superconductivity discovery. The BdG equations are obtained in the continuum with consideration of their symmetry. The Green’s function method is discussed in connection with the BdG equations, in the framework of a tight-binding model. Physically measurable quantities are derived in terms of the BdG eigenfunctions. The connection of the BdG equations with the Abrikosov-Gorkov Green function method and the solution in the uniform case are treated. Chapter 3 considers the local electronic structure around a single impurity (either non-magnetic or magnetic) in conventional s-wave and unconventional d-wave superconductors. The Yu-Shiba-Rusinov state around a magnetic impurity and the Majorana zero-energy modes from a coupling to a spin chain in s-wave superconductors and the impurity resonance states in d-wave superconductors are discussed. The results are compared with STM measurements and the T-matrix method. Chapter 4 discusses the influence on macroscopic properties of disorder effects in both s-wave and d-wave superconductors. With this aim, the suppression of superconductivity is considered based on the response of superconducting order parameter transition temperature and superfluid density. Moreover, the localization/delocalization of gapless quasiparticles is treated in a disordered d-wave superconductor within a single-parameter scaling approach. The calculation of the local electronic structure in the presence of a magnetic field is discussed in Chapter 5. The continuum theory for a single vortex is developed and proceeds to the lattice version in s- and d-wave superconductors. The effect from a competing order in the quasiparticle states in a d-wave vortex state is explored. The Fulde-Ferrell-Larkin-Ovchinnikov state due to the spin Zeeman interactions of a magnetic field is treated. Chapter 6 considers the transport properties of hybrid superconductive structures based on the Blonder-Tinkham-Klapwijk theory. The transport factors are estimated on the base of scattering theory. The enhancement and the suppression of conductance due to, respectively, the Andreev reflection and the spin polarization are examined. The possible revelation of chiral Majorana modes through the tunneling differential conductance in a normal metal “ferromagnetic insulator” s-wave superconductor junction is treated. Chapter 7 is devoted to topological (Aharonov-Bohm) and quantum size effects, considering the periodicity of the supercurrent in multiply connected geometries at mesoscale and examing the superconducting properties in nanoscale superconductors, respectively. In total, these lecture notes written in a high mathematical level and based on strong physical grounds are sufficiently available for self-education of the BdG method being one of the important approaches in superconductivity. The book will be useful for gradient students and all those interested in moderate problems of superconductivity.

MSC:

82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
82D55 Statistical mechanics of superconductors
00A79 Physics
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