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

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

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

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

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

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

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

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

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

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

Normal metal Superconductor Andreev Reflection ProbabilityTransmitted charge

SQUID configuration t ransmission S/S superconducting contact Josephson current

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

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

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

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

 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

 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

 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

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

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

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

 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

 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:

Adiabatic evolution of the ground state

 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

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

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

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

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:

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

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

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

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

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

 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

 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

 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:

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

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

local Green functions local density of states

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

00 metallic electrode

 In the wide band limit: Lorentzian density of states

00 LR

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

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:

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

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

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

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

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

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

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

Solution: Problem: symmetry breaking!!

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

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

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)

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

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

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

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:

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

Perturbative expansion in diagrammatic form Dyson equation

general validity one-electron potential Coulomb interaction