Quantum Transport
Outline:
 What is Computational Electronics?
 Semi-Classical Transport Theory
 Drift-Diffusion Simulations
 Hydrodynamic Simulations
 Particle-Based Device Simulations
 Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators
 Tunneling Effect: WKB Approximation and Transfer Matrix Approach
 Quantum-Mechanical Size Quantization Effect
 Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum
Moment Methods
 Particle-Based Device Simulations: Effective Potential Approach
 Quantum Transport
 Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical
Basis of the Green’s Functions Approach (NEGF)
 NEGF: Recursive Green’s Function Technique and CBR Approach
 Atomistic Simulations – The Future
 Prologue
Quantum Transport
Direct Solution of the Schrodinger Equation:
Usuki Method (equivalent to Recursive Green’s
Functions Approach in the ballistic limit)
NEGF (Scattering):
Recursive Green’s Function Technique, and
CBR approach
Atomistic Simulations – The Future of Nano
Devices
Description of the Usuki Method
Wavefunction and potential defined on
discrete grid points i,j
j=M+1
transmitted
waves
incident
waves
y
reflected
waves
x
j=0
i=0
Usuki Method slides
provided by Richard
Akis.
i=N
i th slice in x direction - discrete problem
involves translating from one slice to the next.
Grid spacing: a<< lF
Obtaining transfer matrices from the discrete SE
apply Dirichlet boundary conditions on upper and lower boundary:
j=M+1
 i , j 0   i , j  M 1  0
Wave function on ith slice
can be expressed as a vector
Discrete SE now becomes a matrix equation
relating the wavefunction on adjacent slices:
(1b)




H 0i i  t i 1  t i 1  E i
  i,M 


i
,
M

1


  . 
i  

.


 . 



 i ,1 
j=M
j=1
j=0
i
t
0
(Vi , M  4t )

t
(Vi , M  4t )  t

where: H0i  












 t (Vi ,2  4t )
 t 
0
t
(Vi ,1  4t )
(1b) can be rewritten as:

Combining this with the trivial equation
(2)
where


 i 
 i 1 
   Ti   
 i 1 
 i 
I
0

Ti   I  H0i  E 



t



Modification for a perpendicular
magnetic field (0,0,B) :
B enters into phase factors
important quantity:
flux per unit cell
H0i  E   
 i  i 1
t




 i   i one obtains:
 i 1  
 0
Ti    P 2


Is the transfer
matrix relating
adjacent
slices
I

 P( H 0i  E ) 


t


Pi , j  ei 2   j i , j ,
Ba 2

  /0
h/e


 1 
 1  yields the modes on the
Solving the eigenvalue problem: T1     l   
 0 
 0  left side of the system

um (  ) 

Mode eigenvectors have the generic form:
l (  )u (  )
redundant
m
 m

There will be M modes that propagates to the right (+) with eigenvalues:
ikm a
propagating
lm ( )  e
, m  1,, q
lm ( )  e  ma , m  q  1,, M
evanescent
There will be M modes that propagates to the left (+) with eigenvalues:
lm ( )  e ikma , m  1,, q
propagating
lm ( )  e ma , m  q  1,, M
evanescent
defining


U   u1 (  )  um (  ) and
Complete matrix of eigenvectors:
U tot
l  diag l1 (  )  lm (  )
 U

l U 
U 
l U  
Transfer matrix equation for translation across entire system
Unit matrix
waves incident
from left have unit
amplitude
Transmission matrix
t
I
1
0  Utot TN 1TN 2 T1 Utot r 
 
 
Zero matrix
no waves incident
from right
Recall:
Converts back to
mode basis
2e 2
G
h
2
vn
tn , m
v
m ,n m

reflection
matrix
Converts from mode basis
to site basis
In general, the velocities must
be determined numerically
Variation on the cascading scattering matrix technique method
Usuki et al. Phys. Rev. B 52, 8244 (1995)
C1(0,0)  I, C(0,0)
0
2
Boundary condition- waves of
unit amplitude incident from right
Iteration scheme
for interior slices
C1(i 1,0) C(i2 1,0) 
C1(i,0)

  Ti 
I 
 0
 0
0
I
Pi  
,

Pi1 Pi2 
Pi1  Pi2Ti21C1(i,0) ,

C(i,0)
2
 Pi
I 
plays an
analogous
role
to Dyson’s
equation in
Recursive
Greens
Function
approach
1
Pi2  [Ti21C(i,0)

T
]
2
i22
Final transmission matrix for
entire structure is given by


tU λ

 1
C
N 1
1



U U λ


1
 1
A similar iteration gives the reflection matrix
After the transmission problem has been solved,
the wave function can be reconstructed
It can be shown that:




PN 2  ψ N   N ,1   N ,k   N ,M

wave function on column N resulting from the kth mode
One can then iterate
backwards through the structure:
ψ i  Pi1  Pi 2 ψ i 1
The electron density at each point is then given by:
q
n( x, y)  n(i, j )    ijk
k 1
2
First propagating mode for an irregular potential
u1(+) for B=0.7 T
u1(+) for B=0 T

un   n ( y )
u1j
confining
potential
0
vm 
j
e
h

j
40
2
2t sin( 2km  2 j ) umj
Mode functions no longer
simple sine functions
80
general formula for velocity of mode m
obtained by taking the expectation
value of the velocity operator with
respect to the basis vector.
Conduction band [eV]
Example – Quantum Dot Conductance as a Function of Gate voltage
0.8
Simulation gives comparable
2D electron density to that
measured experimentally
Conduction band profile Ec
0.6
Energy of the
ground subband
0.4
2
3D
11
2
N
(
E

E
)
~
4

10
cm
F
0
2m*
0.2
0.0
Fermi level EF
-0.2
0.00
0.02
0.04
0.06
0.08
0.10
z-axis [mm]
Potential felt by 2DEG- maximum of electron distribution ~7nm below interface
Vg= -1.0 V
Vg= -0.9 V
Vg= -0.7 V
Potential evolves smoothly- calculate a few as a function of Vg, and
create the rest by interpolation
1
EXPERIMENT (+0.6)
0.01 K
0.8
-0.897 V
conductance fluctuation (e2/h)
0.6
0.4
0.2
-0.923 V
-0.951 V
0
-0.2
Same simulations also reveal that certain scars may
-0.4
RECUR as gate voltage is varied. The resulting
0.4 mm
THEORY
periodicity agrees WELL with that of the conductance
-0.6
-1
-0.9
-0.8
-0.7
-0.6
oscillations
* Persistence of the scarring at zero magnetic field
gate voltage (volts)
Subtracting out a background that removes
the underlying steps you get periodic
fluctuations as a function of gate voltage.
Theory and experiment agree very well
indicates its INTRINSIC nature
 The scarring is NOT induced by the application of
the magnetic field
Magnetoconductance
B field is perpendicular to plane of dot
classically, the electron trajectories
are bent by the Lorentz force
Conductance as a function of magnetic field also
shows fluctuations that are virtually periodic- why?
Green’s Function Approach:
Fundamentals
 The Non-Equilibrium Green’s function approach for
device modeling is due to Keldysh, Kadanoff and Baym
 It is a formalism that uses second quantization and a
concept of Field Operators
 It is best described in the so-called interaction
representation
 In the calculation of the self-energies (where the
scattering comes into the picture) it uses the concept of
the partial summation method according to which
dominant self-energy terms are accounted for up to
infinite order
 For the generation of the perturbation series of the time
evolution operator it utilizes Wick’s theorem and the
concepts of time ordered operators, normal ordered
operators and contractions
Relevant Literature
 A Guide to Feynman Diagrams in the Many-Body
Problem, 2nd Ed.
R. D. Mattuck, Dover (1992).
 Quantum Theory of Many-Particle Systems,
A. L. Fetter and J. D. Walecka, Dover (2003).
 Many-Body Theory of Solids: An Introduction,
J. C. Inkson, Plenum Press (1984).
 Green’s Functions and Condensed Matter,
G. Rickaysen, Academic Press (1991).
 Many-Body Theory
G. D. Mahan (2007, third edition).
 L. V. Keldysh, Sov. Phys. JETP (1962).
Schrödinger, Heisenberg and Interaction
Representation
 Schrödinger picture

i S (t)  Ĥ o  Ĥ1 S (t)
t


 Interaction picture

i I (t)  Ĥ1 t I (t)
t

i ÔS  0
t
ÔS  ÔS
Ô I (t)  e
 iĤ o t
ÔSe
 iĤ o t
 Heisenberg picture

i H (t)  0
Ô H (t)  e  iĤt ÔSe  iĤt
t


i Ô I (t)  Ô I , Ĥ o
t
i

Ô H (t)  Ô H , Ĥ H
t
S (t)  Û(t,0) H (0) Û time evolution operator
ĤÛ  i

Û
t



Time Evolution Operator
I (t)  Û I t,0 I (0)

 iĤ o t  iĤ o t
i I (t)  Ĥ I (t) I (t) Ĥ I (t)  e
Ĥ1e
t


i Û I t,0 I (0)  Ĥ I (t) Û I t,0 I (0)
i Û I t,0   Ĥ I (t) Û I t,0 
t
t
Time evolution operator representation
as a time-ordered product

t
i  dt' Û I nt' ,0
i  dt'Ĥ I (t')
t   t dt' Ĥ It(t' )Û I t' ,0 


i
0 dt ... dt T Ĥ (t )Ĥ (t )...Ĥ (t )  Te 0
Û0 (t,0)t'
dt
1
2
n
I 1
I 2
I n
n! 0 0 t 0
i Û I t,0   Û I 0,0   dt' Ĥ I (t' )Û I t' ,0
t

t


0

F
Contractions and Normal Ordered
Products
A  B  T AB - N AB
â (t 2 )â (t 1 )  â k (t 2 )â (t 1 ) - - 1 â l (t 1 )â k (t 2 )

k

l

l
 â k â l  â l â k e
 δ kle
1
 i k t 2  i  l t 1
e
 i k  t 2  t1 
t 2  t1
â (t 2 )â (t 1 )  â (t 1 )â k (t 2 ) - - 1 â (t 1 )â k (t 2 )

k

l

l
0
t 2  t1
1

l
t1  t 2
t1  t 2
Wick’s Theorem
 Contraction (contracted product) of operators
 i k  t 2  t1 


b̂ k (t 2 )b̂ l (t 1 )  δ kle
t 2  t1
b̂ k (t 2 )b̂ l (t 1 )  0
t1  t 2
 For more operators (F 83) all possible pairwise contractions of
operators
 Uncontracted, all singly contracted, all doubly contracted, …
T [UVW...XYZ ]  N [UVW...XYZ ]  N [ UVW...XYZ]  N [UVW... XYZ] 
N [ UVW...XYZ]  ...  N [ UV W...XY.Z]
 Take matrix element over Fermi vacuum
0 T [UVW...XYZ ] 0  0 N[UVW...XYZ ] 0  ...  0 N[UV W...XY.Z] 0
 All terms zero except fully contracted products
Propagator
Partial Summation Method
Example: Ground State Calculation
GW Results for the Band Gap
Definitions of Green’s Functions
* 1 = x1,t1
Time ordered
Allows perturbation theory
(Wick’s theorem)
Retarded, Advanced
Simple analitycal structure
and spectral analysis
Correlation functions
Direct access to observable
expectation values
Equilibrium Properties of the System
Gr, Ga, G<, G> are enough to evaluate all the GF’s
and are connected by physical relations
General identities
Fluctuation-dissipation th.
Spectral function
Just one indipendent GF
See eg:
H. Haug, A.-P. Jauho
A.L. Fetter, J.D. Walecka
Non-Equilibrium Green’s Functions
See eg: D. Ferry, S.M. Goodnick
H.Haug, A.-P. Jauho
J. Hammer, H. Smith, RMP (1986)
G. Stefanucci, C.-O. Almbladh, PRB (2004)
• Time dep. phenomena
• Electric fields
• Coupling to contacts at
different chemical potentials
 Contour-ordered perturbation theory:
Gr, Ga, G<, G> are all involved in the PT
2 of them are indipendent
No fluctuation dissipation
theorem
Contour ordering
Constitutive Equations
 Two Equations of Motion
In the time-indipendent limit
Dyson Equation
Keldysh Equation
Gr, G< coupled via the self-energies
Computing the (coupled) Gr, G< functions
allows for the evaluation of transport properties
Summary
 This section first outlined the Usuki method as a
direct way of solving the Schrodinger equation in
real space
 In subsequent slides the Green’s function
approach was outlined with emphasis on the
partial summation method and the self-energy
calculation and what are the appropriate Green’s
functions to be solved for in equilibrium, near
equilibrium (linear response) and high-field
transport conditions