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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