1. Field theoretical methods in transport theory  F. Flores  A. Levy Yeyati  J.C. Cuevas

2. Plan of the lectures  Compact review of the non-equilibrium perturbation formalism  Special emphasis on the applications: transport phenomena  Background on standard equilibrium theory  Some knowledge on Green functions General aim Stumbling block

3. Plan of the lectures (more detailed)  L1 : Summary of equilibrium perturbation formalism (minimal)  L2 : Non-equilibrium formalism  L3 : Application to transport phenomena  L4 : Transport in superconducting mesoscopic systems

4. L1: Summary of equilibrium theory Background material in compact form  Model Hamiltonians in second quantization form  Representations in Quantum Mechanics: the Interaction Representation  Green functions (what is actually the minimum needed?)  Compact review of equilibrium perturbation formalism

5. L2:Non-equilibrium theory: Keldish formalism  Non equilibrium perturbation theory  Keldish space & Keldish propagators  Application to transport phenomena:  Simple model of a metallic atomic contact

6. L3:Application to transport phenomena  Current through an atomic metallic contact  Shot noise in an atomic contact  Current through a resonant level  Current through a finite 1D region  Multi-channel generalization:  Concept of conduction eigenchannel

7. L4:Superconducting transport  Superconducting model Hamiltonians: Nambu formalism  Current through a N/S junction  Supercurrent in an atomic contact  Finite bias current: the MAR mechanism

8. A sample of transport problems that will be addressed Current through an atomic metallic contact STM fabricatedMCBJ technique AI  V  d.c. current through the contact  contact conduction channels  conductance quantization

9. Resonant tunneling through a discrete level resonant level L R Quantum Dot MM

10. N/S superconducting contact d Tunnel regime Contact regime Conductance saturation

11. Normal metal Superconductor Andreev Reflection ProbabilityTransmitted charge

12. SQUID configuration t ransmission S/S superconducting contact Josephson current

13. Conduction in a superconducting junction 22 22 I eV 22 E F,L E F,L - E F,R = eV > 2  22 E F,R I Standard tunnel theory

14. Superconductor Andreev reflection in a superconducting junction eV>  I eV 22  ProbabilityTransmitted charge

15. Superconductor Multiple Andreev reflection eV > 2  /3 I eV 22  2  /3 ProbabilityTransmitted charge

16. Background material I. Model Hamiltonians in second quantized form In terms of creation & destruction field operators : first quantization Single electron Hamiltonian Interaction potential

17.  A more useful expression in terms of a one-electron basis  Expanding the field operators.  The operators create or destroy electrons in 1-electron states |i   H exhibits explicitly all non vanishing 1e and 2e scattering processes

18.  Example 1 Non-interacting free electron gas first quantization second quantization  Example 2 Non-interacting tight-binding system first quantizationsecond quantization local basis second quantization

19.  Tight-binding basis: especially suited for systems in the nanometer scale: atomic contacts  Example 3 Tight-binding linear chain Electronic and transport properties t 00 00 00 00 00 ttt graphical representation Importance of atomic structure Use of a local orbital basis

20. II. Representations in Quantum Mechanics A) Schrödinger representation Usual one, based on the time-dependent Schrödinger equation: In equilibrium Background material  Example Free electron

21. B) Heisenberg representation Unitary transformation from Schrödinger representation: Equation of motion for the operators: In equilibrium  Example Free electron gas

22. C) Interaction representation  Necessary for perturbation field theory non interacting electrons perturbation  Unitary transformation from Schrödinger representation: transformationsequations of motion

23.  Dynamics of operators in the interaction representation It is the same that in the non-perturbed system  Example Free electron gas with interactions irrespective of V

24.  Dynamics of wave function in the interaction representation It is the perturbation V I that controls the evolution  Connection between and (unperturbed ground state) Adiabatic hypothesis If V is adiabatically switched on (off) at t =  It is generally possible to identify:

25. Adiabatic evolution of the ground state

26.  The temporal evolution operator Without solving explicitly A formal expression for S is obtained: Transforming back to interaction: From which the following properties are easily derived:  S -1 =S +  S(t,t)=1  S(t,t’)S(t’,t’’)=S(t,t’’)  S(t,t’)=S(t’,t) -1

27.  Perturbation expansion of S An explicit expression for S is obtained from: by iteration Integral equation Zero order First order

28. Noticing that time arguments verify:  T is the time ordering or chronological operator:

29. III. Compact survey on Green functions  The current depends strongly on the local density of states in the junction region  Both local density of states and current are closely related to the local Green functions in the junction region  Close to the Fermi energy (linear regime) Background material  What is actually the minimum needed for most practical applications?  Systems we are interested in: MM electronic transport

30. Green functions of non-interacting electrons Green functions are first introduced in the solution of differential equations, like the Schrödinger equation:  Example Electron in 1D  Definition for a general non-interacting electron system matrix Green function in frequency (energy) space: In a particular one-electron basis the different Green functions will be:

31.  Example Free electron gas  Example Two site tight-binding model t 00 00

32.  Precise definition as a complex function Retarded (Advanced) Green functions:  Relation of imaginary part to the electronic density of states local basis

33. Proof One-electron basis set that diagonalize H Inserting Poles: one-electron energy eigenvalues

34.  The imaginary part is related to the density operator (matrix): local basis

35.  Relation of real and imaginary parts of Hilbert transform  This a direct consequence of its pole structure:

36.  The “wide band” approximation  In the limit of a broad and flat band:  Reasonable for a range of energies close to  Transport in the linear regime

37.  Master equation for Green function calculations: The Dyson equation On many occasions it is hard to obtain the GF from a direct inversion Let H be a 1-electron Hamiltonian that can be decomposed as: Where the green functions of H 0 are known. then Particular instance of the Dyson equation (more general) self-energy

38.  Example Surface Green function and density of states (important for transport calculations) Simple model: semi-infinite tight-binding chain t 00 00 00 00 tt surface site Assume a perturbation consisting in coupling an identical level at the end: 00 t As the final system is identical to the initial one:

39. Then using the Dyson equation: Taking into account that: The following closed equation is obtained:

40. Solving the equation we have for the retarded (advanced green function): semi-elliptic density of states

41. local Green functions local density of states

42.  Example Quantum level coupled to an electron reservoir 00 metallic electrode uncoupled dot green function perturbation coupled dot green function dot selfenergy

43. 00 metallic electrode

44.  In the wide band limit: Lorentzian density of states

45. 00 LR

46.  In addition to The causal Green function needed in equilibrium perturbation theory

47. Green functions in time space  Green functions in time space are related with the probability amplitude of an electron propagating: In space from one state to another  Hole propagation:

48.  Green functions in time space (propagators) are defined combining the electron and hole propagation amplitudes The retarded Green function

49.  The advanced Green function  The causal Green function Different ways of combining the same electron and hole parts

50.  Example The free electron gas ground state: Fermi sphere Then, for instance, the retarded Green function Transforming Fourier to frequency space:

51. Perturbation theory in equilibrium Compact summary in six steps 1) Interaction representation non interacting electrons perturbation Assume an electronic system of the form: full quantum mechanical knowledge

52. We want to calculate averages in the ground state: change to interaction representation The time evolution of is known (non-interacting system)

53. 2) Adiabatic hypothesis (theorem) Switching on an off the perturbation at If is the unperturbed stationary ground state Does it all make sense when ? ?

54. Commentaries on the adiabatic hypothesis  What really happens is that the have function acquires a phase while evolving In time This phase factor diverges when as Example : two site tight-binding system t0t0 00 00 exact solution

55. Solution: Problem: symmetry breaking!!

56. Time symmetry (equilibrium) This is not true out of equilibrium!!

57. 3) Expansion of S  From a formal point of view this would be all  From a practical this is not the case at all !!

58. 4) Wick’s theorem  General statement of statistical independence in a non-interacting electron system  In the perturbation expansion the averages have the form  Wick’s theorem: the average decouples in all possible factorizations of elemental one-electron averages (only two fermion operators)

59.  Example (Wick’s theorem) Factorizations containing averages are not included

60.  The “elementary unit” in the decoupling procedure is the causal Green function of the unperturbed system  Example (Wick’s theorem)

61. 5) Expansion of the Green function  Wick’s theorem allows to write the perturbation expansion in terms of the unperturbed causal Green function  It is interesting to analyze the expansion of Dyson equation Wick’s theorem Diagrams Dyson equation

62. 5) Feynman diagrams and Dyson equation  The different contributions produced by Wick’s theorem are usually represented by diagrams Example: External static potential Graphical conventions:

63. Terms in the expansion of as given by Wick’s theorem Zero order First order Second order intermediate variables integrated

64. Perturbative expansion in diagrammatic form Dyson equation

65. general validity one-electron potential Coulomb interaction