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 fabricated MCBJ technique

AI

V

d.c. current through the contactcontact conduction channels conductance quantization

Resonant tunneling through a discrete level

resonant level

L R

Quantum Dot

M M

N/S superconducting contact

-3 -2 -1 0 1 2 30

1

2

G(V

)/G

0

eV/

= 1 = 0.9 = 0.5

)exp( dt

d Tunnel regime

Contact regime

0

1

h

eGG

2

0

42

eV

Conductance saturation

1

Normal metal Superconductor

Andreev Reflection

Probability 2Transmitted charge e2

12

SQUID configuration

transmission

S/S superconducting contact

Josephson current21 2 /cos

sen

h

e)(I s

Conduction in a superconducting junction

2 2

I

eV2

EF,L

EF,L - EF,R = eV > 2

2EF,R

I

Standard tunnel theory

Superconductor

Superconductor

Andreev reflection in a superconducting junction

eV>

I

eV2

Probability 2Transmitted charge e2

Superconductor

Superconductor

Multiple Andreev reflection

eV > 2/3

I

eV22 /3

Probability 3Transmitted charge e3

Background material

)r,r(V)r(hH j

N

jii

N

ii

11 2

1

','' )r()'r()'r,r(V)'r()r('drdr

)r()r(h)r(drH

21

I . Model Hamiltonians in second quantized form

In terms of creation & destruction field operators: )r(),r(

first quantization

Single electron Hamiltonian Interaction potential

ij 'ijkl

l'k'jiijkljiij ccccVcchH2

1

A more useful expression in terms of a one-electron basis i

i

ii c)r()r(

Expanding the field operators.

The operators create or destroy electrons in 1-electron states |i ii c,c

H exhibits explicitly all non vanishing 1e and 2e scattering processes

jhihij l,'kV'j,iVijkl

Example 1 Non-interacting free electron gas

N

i m

)r(H

1

22

2

k

kkk ccH

ikrk e

V)r(

1

first quantization

second quantization

Example 2 Non-interacting tight-binding system

)r(Vm

)r(H lattice

2

22

first quantization

ij

jiij cctH

second quantization

)Rr( ilocal 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

)cccc(tnH iii i

iii

110

t0 0 0 0 0

t t t

graphical representation

Importance of atomic structure

Use of a local orbital basis

II . Representations in Quantum Mechanics

A) Schrödinger representation

)t()t(t

i SS

H

Usual one, based on the time-dependent Schrödinger equation:

)t(HH

)t(SS OO

)(e)t( Sti

S 0 H

In equilibrium

In equilibrium

Background material

Example Free electron m

k,

m

)(k 22

2222

r

H

)(e)t( ti k 0kk

V/)iexp(k kr

B) Heisenberg representation

Unitary transformation from Schrödinger representation:

)()t(e SSti

H 0 H

tiS

tiH ee)t( HH OO

HOO ,)t(t

i HH

k

kkk ccH

Equation of motion for the operators:

In equilibrium

Example Free electron gas

tie)()t( kkk cc

0

C) Interaction representation

Necessary for perturbation field theory

VHH 0

non interacting electrons

perturbation

Unitary transformation from Schrödinger representation:

)t(e)t( Sti

I 0H

tiS

tiI ee)t( 00 HH OO 0HOO ,)t(

ti II

)t()t(t

i III

V

transformations equations of motion

Dynamics of operators in the interaction representation

0HOO ,)t(t

i II

It is the same that in the non-perturbed system

Example Free electron gas with interactions

VccHk

kkk

tie)()t( kkk cc

0

irrespective of V

Dynamics of wave function in the interaction representation

)t()t()t(t

i III

V

It is the perturbation VI that controls the evolution

Connection between and )t(I 0 (unperturbed ground state)

Adiabatic hypothesis

telim

VV

0

0 )(I

If V is adiabatically switched on (off) at t =

It is generally possible to identify:

0 0

)t(I

t t0t

t

Adiabatic evolution of the ground state

0 0

)t(I

t t0t

t

The temporal evolution operator

)t()t,t()t( II 00 S

Without solving explicitly )t()t()t(t

i III

V

A formal expression for S is obtained:

)t(ee)t(e)t( S)tt(iti

Sti

I 0000 HHH

Transforming back to interaction: )t(S 00000

0ti)tt(iti eee)t,t( HHHS

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:

)t()t()t(t

i III

V by iteration

t

t

IIII )t()t(Vdti)t()t(0

1110 Integral equation

Zero order

)t()t( II 0

First order

)t()t(dti)t( I

t

t II 0110

1

V

)t(Vdt)....t(Vdt)t(Vdt)i()t,t( nI

t

t nI

t

tn

I

t

t

n n

1

0

1

0022110 1S

Noticing that time arguments verify: 021 tt.......ttt n

n

nII

t

t

t

t n

n

)t()t(dt...dt!n

)i()t,t( VVTS 110

0 0

1

t

t I )t(dtiexp)t,t(0

110 VTS

T is the time ordering or chronological operator: 01 tt...tt n

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

)(i)(g LL Close to the Fermi energy (linear regime)

Background material

What is actually the minimum needed for most practical applications?

Systems we are interested in:

M Melectronic 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 )x(Vdx

d

mH

2

22

2

)x(E)x()x(H

)'xx()'x,x(G)x(HE Definition for a general non-interacting electron system

1 HIG )(

matrix Green function in frequency (energy) space:

In a particular one-electron basis the different Green functions will be:

j)(i)(Gij G

k

kkk ccH Example Free electron gas

k

k

1

),(G IGHI )(

V/)exp( rkk

Example Two site tight-binding model

t0 0

1 2

10

01

2221

1211

0

0

GG

GG

t

t

220

2112 t)(

t)(G)(G

220

02211 t)(

)(G)(G

Precise definition as a complex function

1 HIG )i()(a,r

Retarded (Advanced) Green functions:

Relation of imaginary part to the electronic density of states

)(GIm)(

),(GIm),(

a,riiii

a,r

1

1

rr

i local basis

Proof

1 HIG )i()(a,r

IGHI )()i( a,r

k One-electron basis set that diagonalize H

Inserting k

kk 1

k k

a,r

i

kk)(

G

Poles: one-electron energy eigenvalues

k k

a,r

i

kk)(

G

k

k )(kk)( ρ

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

)()(Im a,r ρG

)(GIm)( a,riiii

1

i local basis

Relation of real and imaginary parts of )(G a,r

'

)'(GIm'd)(GRe

a,ra,r 1

Hilbert transform

This a direct consequence of its pole structure:

k k

a,r

i

kk)(

G

The “wide band” approximation

'

)'('d

'

)'(GIm'd)(GRe ii

a,riia,r

ii

1

In the limit of a broad and flat band:

)()( iiii

)(i)(GIm)(G iia,r

iia,r

ii

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:

VHH 0

Where the green functions of H0 are known. then

1110

VGVHIG (0) )()(

)()()()( )()( VGGGG 00 Particular instance of the Dyson equation (more general)

self-energy

)()()()()( )()( GΣGGG 00

Example Surface Green function and density of states

(important for transport calculations)

Simple model: semi-infinite tight-binding chain

t0 0 0 0

t t

1234

surface site

Assume a perturbation consisting in coupling an identical level at the end:

0

t

01

1001 ccccV t

As the final system is identical to the initial one: )(G)(G )( 01100

Then using the Dyson equation:

)()()()( )()( VGGGG 00

)(GV)(G)(G

)(GV)(G)(G)(G)(

)()(

00100

1110

10010

000

0000

Taking into account that:

tVV

)/()(G

)(G)(G)(

)(

0110

00

00

01100

1

The following closed equation is obtained:

01000200

2 )(G)()(Gt

Solving the equation we have for the retarded (advanced green function):

2

00 21

2

1

ti

tt)(G a,r 00

2

00 21

1

tt)(

semi-elliptic density of states

-4 -2 0 2 4

00GRe

00GIm

t/kacostk 20

local Green functions

local density of states

Example Quantum level coupled to an electron reservoir

0

Rt

metallic electrode

RRRt ccccnHH R

0000

0

000

1

)(G )( )(

)(G

000

00

1

uncoupled dot green function

perturbation

coupled dot green function

dot selfenergy

0

Rt

metallic electrode

R

0

)()()()( )()( VGGGG 00 RRR tVV 00

)(GV)(G)(G)(G RR)()( 00

000

00000

)(GV)(G)(G R)(

RRR 0000

0

)(G)(Gt)(G)(G )(RRR

)( 00020

0000

)(Gt)(G

)(RRR

02

000

1

)(Gt)( )(

RRR 0200

In the wide band limit:

i)(ti)( RR 200

i)(G

000

1

220

00

1

)(

)(

Lorentzian density of states

0

Lt Rt

L R

0 RL

RL ii)(G

000

1

)(t RLL 2

)(t RRR 2

In addition to )(G),(G ra

The causal Green function

)(GRe)(GRe)(GRe cra

),(GIm)(GIm

),(GIm)(GImrc

ac

needed in equilibrium perturbation theory

1 HIG ))sgn(i()(c

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

HH )'t'()t( rΨrΨ HjiH )'t()t( cc

Hole propagation:

HH )'t'r()rt( ΨΨ

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

The retarded Green function

HjiH

HijjiHrij

)'t(),t()'tt(i

)t()'t()'t()t()'tt(i)'t,t(G

cc

cccc

)'t'()t()'tt(i)'t',t(G r rΨrΨrr

The advanced Green function

HjiHaij )'t(),t()t't(i)'t,t(G

cc

The causal Green function

HjiHcij )'t()t(i)'t,t(G

ccT

HijH

HjiHcij

)t()'t()t't(i

)'t()t()'tt(i)'t,t(G

cc

cc

Different ways of combining the same electron and hole parts

Example The free electron gas

k

kkk ccH

ti

ti

e)t(

e)t(k

k

kk

kk

cc

cc

H ground state: Fermi sphere

Then, for instance, the retarded Green function

HHr )t()'t()'t()t()'tt(i)'tt,(G

kkkk cccck

)'tt(ir e)'tt(i)'tt,(G kk

i),(G r

k

k1

Transforming Fourier to frequency space:

)sgn(i),(Gc

k

k1

Perturbation theory in equilibrium

Compact summary in six steps

1) Interaction representation

VHH 0

non interacting electrons

perturbation

Assume an electronic system of the form:

0H full quantum mechanical knowledge

We want to calculate averages in the ground state:

HH

HHH )t(A

A

change to interaction representation

)t()t(

)t()t()t(A

II

III

A

The time evolution of is known (non-interacting system))t(IA

2) Adiabatic hypothesis (theorem)

Switching on an off the perturbation at ttelim

VV

0

If is the unperturbed stationary ground state0

0),t()t(I S

Does it all make sense when ?0

?

0 0

)t(I

t t0t

t

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 0 as

Example: two site tight-binding system

/ie

t00 0

1 2

exact solutiontett 00

212

100

0000 /ite),()( S

Solution:)t(

)t(lim

),(

),(lim

0

000

0

0 0

0

S

S

Problem: symmetry breaking!! 000

Time symmetry (equilibrium)

00

00

),t()t,(

),t()t()t,( I

SS

SASA

00

00

),t()t,(

),t()t()t,( I

SS

SASA

0 0

)t(I

t t0t

t

00

00

),(

),()t(I

S

SATA

This is not true out of equilibrium!!

3) Expansion of S

0101000

1

)t()t()t(dt...dt!n

)i(

S nIIInn

n

VVATA

From a formal point of view this would be all

From a practical this is not the case at all !!

n

nIIn

n

)t()t(dt...dt!n

)i(),( VVTS 111

4) Wick’s theorem

General statement of statistical independence in a non-interacting electron system

In the perturbation expansion the averages have the form

010 )t()t()t( nIII VVAT

0210 )'t()t()t()t( j'k'ni ccccT

Wick’s theorem: the average decouples in all possible factorizations of elemental one-electron averages (only two fermion operators)

00 )'t()t( jiccT

Example (Wick’s theorem)

0210 )'t()t()t()t(I lkji ccccT

020010

020010

)t()t()'t()t(

)'t()t()t()t(I

iklj

lkij

ccTccT

ccTccT

Factorizations containing averages

000 )'t()t( li ccT

00210 )t()t( kj ccT

are not included

)t,t(iG)t()t( c)(jiij 10

010 ccT

The “elementary unit” in the decoupling procedure is the causal Green function of the unperturbed system

0210 )'t()t()t()t(I lkji ccccT

)t,t(G)'t,t(G)'t,t(G)t,t(GI )(ki

)(jl

)(kl

)(ji 2

01

02

01

0

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 )'t,t(G

Dyson equation

)'t()t(i)'t,t(G jiij ccT

0101000

1 )'t()t()t()t(dt...dt

!n

)i()'t,t(G jnIIin

n

n

ij

cVVcTS

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 )(V r

)()(V)(d rΨrrrΨV

Graphical conventions:

)'t',t(G rr )'t',t(G )( rr0 )(V r

Terms in the expansion of )'t',t(G rr as given by Wick’s theorem

Zero order )'t',t(G )( rr0

First order )'t',t(G)(V)t,t(G )()( rrrrr 110

1110

Second order

)'t',t(G)(V)t,t(G)(V)t,t(G )()()( rrrrrrrr 220

222110

1110

intermediate variables integrated

tr

't'r

11tr

Perturbative expansion in diagrammatic form

)'t'r,t(G)(V)t,t(G)'t',t(G)'t',t(G )()(11111

00 rrrrrrrr

)()()()( )()()( GVGGG 000

Dyson equation

)()()()()( )()()( GΣGGG 000 general validity

one-electron potential

Coulomb interaction

)(V r

)(ee