Basics of Green function technique and its applications

Dr Igor Poboiko (KIT, Karlsruhe) and Prof Mikhail Feigel'man (CENN Nanocenter)

Course description....


1. Introduction

1.1. Second quantization formalism: free bosons and fermions

1.2. Density matrix, Gibbs ensemble, Wick’s theorem

1.3. Evolution operator, Liouville-von-Neumann equation

2. Basics of Keldysh technique

2.1. Keldysh Green functions for free particles (operator formalism)

2.2. Keldysh rotation, retarded, advanced and Keldysh Green functions. Equilibrium relation

2.3. Grassmanian algebra, path integral representation

2.4. Generating functional. Linear response theory and Kubo formula

2.5. Wigner transformation, semiclassical distribution function

3. Diffusive Fermi systems (diagrams)

3.1. White noise random potential as a model for weak impurities

3.2. Probability propagation, diffusion

3.3. Drude formula

4. Disordered Fermi systems (NLSM)

4.1. Hubbard-Stratanovich transformation, Q-matrix

4.2. Saddle point manifold, NLSM action

4.3. Inhomogeneous case: Usadel equation

5. Superconductivity

5.1. BCS model, Hubbard-Stratanovich transformation. Nambu space

5.2.Vicinity of critical temperature: Cooper instability, Ginzburg-Landau functional

5.3. Self-consistency equation below Tc

6. Diffusive superconductors

6.1. NLSM for superconducting systems. Saddle point approximation, angular parametrization

6.2. Usadel equation. Boundary conditions (superconductor-insulator, superconductor-metal, tunneling contact)

6.3. Usadel equation out of equilibrium: the use of Keldysh technique

7. Applications to superconducting heterostructures

7.1. One-dimensional S-N-S junction: Density of States and tunneling conductivity

7.2. Two-dimensional S-N-S array: superconducting islands on graphene

7.3. Interplay of Coulomb anomaly and superconductivity

7.4. Superconductor-ferromagnet-superconductor junctions

7.5. Non-equilibrium effects: critical current of nanowire in presence of an injection