Transcript Quantum Transport Outline:  What is Computational Electronics?  Semi-Classical Transport Theory  Drift-Diffusion Simulations  Hydrodynamic Simulations  Particle-Based Device Simulations  Inclusion of Tunneling.

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)




H0i i  t i 1  t i 1  E i
t
0
(Vi , M  4t )

t
(Vi , M  4t )  t

where: H 0i  



  i,M 


i
,
M

1


  . 
i  

.


 . 



 i ,1 







  t (Vi ,2  4t )
 t 
 0
t
(Vi ,1  4t )
j=M
j=1
j=0
i
(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
H 0i  E  




 i
i 1
t




 i   i one obtains:
 i 1  
 0
Ti   P2


Is the transfer
matrix relating
adjacent
slices
I

 P(H0i  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:
propagating
l (  )  eikm a , m  1,, q
m
lm (  )  e  m a , m  q  1,, M
evanescent
There will be M modes that propagates to the left (+) with eigenvalues:
lm (  )  e ikm a , m  1,, q
propagating
lm (  )  e m a , m  q  1,, M
evanescent
defining


U  u1()  um () and
Complete matrix of eigenvectors:
l  diagl1()  lm ()
 U
U tot  
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  UtotTN 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) C(i,0)

2

  Ti 
 Pi
I 
I 
 0
 0
plays an
0
I
Pi  
,
analogous

P
P
role
i2 
 i1
to Dyson’s
(i,0)
equation in
Pi1   Pi2Ti21C1 ,
Pi2  [Ti21C
Final transmission matrix for
entire structure is given by


tU λ
(i,0)
2

 1
Recursive
Greens
Function
approach
1
 Ti22 ]
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
2 m*
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)  Hˆ o  Hˆ 1 S (t)
t


 Interaction picture

i I (t)  Hˆ 1 t I (t)
t
 ˆ
i OS  0
t
Oˆ S  Oˆ S

ˆ t
ˆ t 

i
H

i
H
o ˆ
o
Oˆ I (t)  e
OSe
i Oˆ I (t)  Oˆ I , Hˆ o
t
 Heisenberg picture
ˆ t ˆ  iHˆ t


i
H
ˆ
i H (t)  0
O H (t)  e
OSe
t
i
 ˆ
O H (t)  Oˆ H , Hˆ H
t
ˆ (t,0) (0) U
ˆ time evolution operator
S (t)  U
H
 ˆ
ˆ
ˆ
HU  i U
t



Time Evolution Operator
ˆ t,0 (0)
I (t)  U
I
I

 iHˆ o t ˆ  iHˆ o t
ˆ
ˆ
i I (t)  H I (t)I (t) H I (t)  e
H1e
t
 ˆ
 ˆ
ˆ
ˆ
ˆ t,0
i U I t,0I (0)  H I (t)U I t,0I (0)
i U I t,0  Hˆ I (t)U
I
t
t
Time evolution operator representation
as a time-ordered product
 ˆ
t
ˆ
ˆ
ˆ (t')
i  dt' U I nt',0
i  dt'H
t   t dt' H It(t')U I t',0
I


i

t'
0
ˆ0 (t,0)
ˆ (t )H
ˆ (t )...H
ˆ (t )  Te
0 dt ... dt T H
U
dt
1
2
n
I
1
I
2
I n
n! 0 0 t 0
ˆ t,0  U
ˆ 0,0  dt' Hˆ (t')U
ˆ t',0
iU
I
I
I
I

t

t


0

F
Contractions and Normal Ordered
Products
A  B  T AB - N AB
ˆa k (t 2 )aˆ l (t1 )  aˆ k (t 2 )aˆ l (t1 ) - - 11 aˆ l (t1 )aˆ k (t 2 )




ˆ
ˆ
ˆ
 a k a l  a l aˆ k e
 δ kl e
 i k t 2  i l t1
e
 i k  t 2  t1 
t2  t1
ˆa k (t 2 )aˆ l (t1 )  aˆ l (t1 )aˆ k (t 2 ) - - 11 aˆ l (t1 )aˆ k (t 2 )
0
t2  t1
t1  t 2
t1  t 2
Wick’s Theorem
 Contraction (contracted product) of operators



i

t

t



k
2
1


ˆ
ˆ
b k (t 2 )b l (t1 )  δ kl e
bˆ  (t )bˆ  (t )  0
k
2
l
1
t2  t1
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 [ UVW...XY.Z]
 Take matrix element over Fermi vacuum
0 T [UVW...XYZ] 0  0 N[UVW...XYZ] 0  ... 0 N[UVW...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