By Jian-Xin Zhu
The function of this publication is to supply an uncomplicated but systematic description of the Bogoliubov-de Gennes (BdG) equations, their distinct symmetry houses and their relation to Green’s functionality concept. in particular, it introduces readers to the supercell approach for the ideas of the BdG equations, in addition to different similar suggestions for extra swiftly fixing the equations in useful applications.
The BdG equations are derived from a microscopic version Hamiltonian with a good pairing interplay and completely trap the neighborhood digital constitution via self-consistent suggestions through unique diagonalization. This procedure has been effectively generalized to review many points of traditional and unconventional superconductors with inhomogeneities – together with defects, sickness or the presence of a magnetic box – and turns into an excellent extra beautiful selection whilst the first-principles details of a standard superconductor is included through the development of a low-energy tight-binding version. extra, the lattice BdG strategy is vital whilst theoretical effects for neighborhood digital states round such defects are in comparison with the scanning tunneling microscopy measurements.
Altogether, those lectures supply a well timed primer for graduate scholars and non-specialist researchers, whereas additionally supplying an invaluable reference advisor for specialists within the field.
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Additional info for Bogoliubov-de Gennes Method and Its Applications
61) with eigenvalue EnQ1 . We take the complex conjugation to Eq. r0 ; r/. r/. r// D EnQ1 . 66) is just identical to Eq. r/ ! r/ Á ! 67) and EnQ2 Á EnQ1 . This property has a significant consequence. , Eq. 60), instead of both Eqs. 61), and obtain all information essential for the measurable. It saves the computational time significantly, which is desirable for the real-space inhomogeneous or disordered problems, even in a collinear spin-polarization. Note again here that this simplification exists in the absence of the spin-orbit coupling and other spin-flip scattering terms in the Hamiltonian.
For the dx2 y2 -wave superconductors, the quasiparticle gap is closed at some special momentum directions ' D 4 ; 34 ; 54 ; 74 . These gapless quasiparticles are called nodal quasiparticles. 0/ vk D ! 0/ uk ! 110) Ek . Here the BdG wave function amplitude is 0s Â Ã1 ! 111) 28 1 Bogliubov-de Gennes Equations for Superconductors in the Continuum Model and 'k is the phase angle of for the pair potential s D k. 113) for a two-dimensional dx2 y2 -wave superconductor with the quasiparticle excitation q 2 2 energy Ek D k C .
102), both positive and negative energy eigenvalues are still taken into account. The profile of the energy gap around the Fermi surface for these two types of pairing symmetry is depicted in Fig. 4. 7 Solution to the BdG Equations in Homogeneous Systems (a) (b) ky 27 ky kx kx Fig. 4 A schematic drawing of quasiparticle excitation gap for an s-wave (a) and a dx2 pairing superconductor (b) y2 -wave surface. For the dx2 y2 -wave superconductors, the quasiparticle gap is closed at some special momentum directions ' D 4 ; 34 ; 54 ; 74 .