Semi-Classical Transport Theory.ppt

Semi-Classical Transport
Theory

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

Direct Solution of the Boltzmann
Transport Equation
 Particle-Based Approaches
 Spherical Harmonics
 Numerical Solution of the Boltzmann-Poisson
Problem
In here we will focus on Particle-Based
(Monte Carlo) approaches to solving the
Boltzmann Transport Equation

Ways of Solving the BTE Using MCT
Single particle Monte Carlo Technique
Follow single particle for long enough time to
collect sufficient statistics
Practical for characterization of bulk materials
or inversion layers
Ensemble Monte Carlo Technique
MUST BE USED when modeling
SEMICONDUCTOR DEVICES to have the
complete self-consistency built in
Carlo Jacoboni and Lino Reggiani, The Monte Carlo method for the solution of charge
transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645
- 705 (1983).

Path-Integral Solution to the BTE
 The path integral solution of the Boltzmann
Transport Equation (BTE), where L=Nt and
tn=nt, is of the form:
1
( )
0
( ) ( ') ( ', ( ) )
N
N m t
N m eff
m
f t t f p S p p eE N m t e

  

    

( ( ) )
m
g p eE N m t
  
K. K. Thornber and Richard P. Feynman,
Phys. Rev. B 1, 4099 (1970).

The two-step procedure is then found by
using N=1, which means that t=t, i.e.:
1 0
'
( ) ( ') ( ', ) t
eff
p
f t t f p S p p eE t e
   

0 ( )
g p eE t
 
Intermediate function that describes
the occupancy of the state (p+eEt)
at time t=0, which can be changed
due to scattering events (SCATTER)
Integration over a trajectory,
i.e.probability that no scattering
occurred within time integral t
(FREE FLIGHT)
+

Monte Carlo Approach to Solving the
Boltzmann Transport Equation
 Using path integral formulation to the
BTE one can show that one can
decompose the solution procedure into
two components:
1. Carrier free-flights that are interrupted by
scattering events
2. Memory-less scattering events that change
the momentum and the energy of the particle
instantaneously

Particle Trajectories in Phase Space
Particle trajectories in k-space and real space
y
k
x
k
-e
x
x
x x
x x
x x
x
x
x
E
y
x
y
x

Carrier Free-Flights
 The probability of an electron scattering in a small time interval dt is
(k)dt, where (k) is the total transition rate per unit time. Time
dependence originates from the change in k(t) during acceleration by
external forces
where v is the velocity of the particle.
 The probability that an electron has not scattered after scattering at t = 0
is:
 It is this (unnormalized) probability that we utilize as a non-uniform
distribution of free flight times over a semi-infinite interval 0 to . We
want to sample random flight times from this non-uniform distribution
using uniformly distributed random numbers over the interval 0 to 1.
 
 






t
t
t
d
n e
t
P 0
)
(
k
      
/
0 t
e
t B
v
E
k
k 




Generation of Random Flight Times
Hence, we choose a random number
 
 






t
t
t
d
i e
r 0
1
,
k
Ith particle first random number
We have a problem with this integral!
We solve this by introducing a new, fictitious scattering
process which does not change energy or momentum:
)
(
)
(
)
(
)
( k
k
x
x 





 

E
k S
ss

Generation of Random Flight Times
 
 






t
t
t
d
i e
r 0
1
,
k



i
i k
k )
(
)
( The sum runs over all the real
scattering processes. To this we
add the fictitious self-scattering
which is chosen to have a nice
property: 0


new





scatterers
real
i
ss k
k )
(
)
( 0

• The use of the full integral form of the free-flight probability
density function is tedious (unless k is invariant during the
free flight).
• The introduction of self-scattering (Rees, J. Phys. Chem.
Solids 30, 643, 1969) simplifies the procedure considerably.
• The properties of the self-scattering mechanism are that it
does not change either the energy or the momentum of the
particle.
• The self-scattering rate adjusts itself in time so that the total
scattering rate is constant. Under these circumstances, one
has that:
 
   
    dt
e
dt
e
dt
t
P
t
t t
t
d
self
t


 










 0
k
k
Self-Scattering

Self-Scattering
• Random flight times tr may be generated from P(t) above using
the direct method to get:
where r is a uniform random between 0 and 1 (and therefore r
and 1-r are the same).
•  must be chosen (a priori) such that > (k(t)) during the entire
flight.
• Choosing a new tr after every collision generates a random walk
in k-space over which statistics concerning the occupancy of the
various states k are collected.
   
r
r
t
e
r r
tr
ln
ln







 
 1
1
1

Free-Flight Scatter Sequence for
Ensemble Monte Carlo
 = collisions
1

n
2
3
4
5
6


N

     
    
      
      
     
     
     
      
1
,
i
t
1
,
i
t
However, we need a
second time scale,
which provides the
times at which the
ensemble is
“stopped” and
averages are
computed.
Particle time
scale

Free-Flight
Scatter
Sequence
dte=dtau
dte ≥ t?
no yes
dt2 = dte dt2 = t
Call drift(dt2)
dte ≥ t?
yes
dte2 = dte
Call scatter_carrier()
Generate free-flight dt3
dtp=t-dte2
dt3 ≤ dtp?
no yes
dt2 = dtp dt2 = dt3
Call drift(dt2)
dte2=dte2+dt3
dte=dte2
no
yes
dte < t ?
dte=dte-t
dtau=dte
R. W. Hockney and J. W.
Eastwood, Computer Simulation
Using Particles, 1983.

Choice of Scattering Event Terminating
Free Flight
o At the end of the free flight ti, the type of scattering which ends the
flight (either real or self-scattering) must be chosen according to the
relative probabilities for each mechanism.
o Assume that the total scattering rate for each scattering mechanism
is a function only of the energy, E, of the particle at the end of the
free flight
where the rates due to the real scattering mechanisms are typically
stored in a lookup table.
o A histogram is formed of the scattering rates, and a random number
r is used as a pointer to select the right mechanism. This is
schematically shown on the next slide.
        









 E
E
E
E pop
ac
i
self

Choice of Scattering Event Terminating
Free-Flight
We can make a table of the scattering processes at
the energy of the particle at the scattering time:
 
 
i
t
E
1

2
1 


3
2
1 




4
3
2
1 






Self
5
4
3
2
1 








0

1
3
2
4
5
Selection process
for scattering
0

r

2

1
 3

0
E

E

2
E

3


E

4
Look-up table of scattering rates:
Store the total
scattering rates
in a table for a
grid in energy
………

Choice of the Final State After Scattering
 Using a random number and probability
distribution function
 Using analytical expressions (slides that follow)
0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
3
4
5
6
x 10
4
Polar angle
Arbitrary
Units

1. Isotropic scattering processes
cos 1 2 , 2
r r
  
  
2. Anisotropic scattering processes (Coulomb, POP)
kx
ky
kz
k
0
0
Step 1:
Determine 0 and 0
kx’
ky’
kz’
k
Step 2:
Assume
rotated
coordinate
system
Step 3:
kx’
ky’
kz’
k
k’


k’≠k for
inelastic

kx’
ky’
Step 3:
perform scattering
     
 
0
2
0
2
1 1 2
cos ,
r
k k
k k
E E
E E

 
 
 

  
 
 
POP
2 2
2
cos 1
1 4 (1 )
D
r
k L r
  
 
Coulomb
=2r for both
kx’
ky’

Step 4:
kxp = k’sin cos, kyp = k’sin*sin, kzp = k’cos
Return back to the original coordinate system:
kx = kxpcos0cos0-kypsin0+kzpcos0sin0
ky = kxpsin0cos0+kypcos0+kzpsin0sin0
kz = -kxpsin0+kzpcos0

Representative Simulation Results From
Bulk Simulations - GaAs
k-vector
-valley
X-valley [100]
L-valley [111]
Conduction bands
Valence bands
-valley table
-Mechanism1
-Mechanism2
- …
-MechanismN
L-valley table
-Mechanism1
-Mechanism2
- …
-MechanismNL
X-valley table
-Mechanism1
-Mechanism2
- …
-MechanismNx
Define scattering mechanisms for each valley
Call specified
scattering mechanisms subroutines
Renormalize scattering tables
Simulation Results Obtained by
D. Vasileska’s Monte Carlo Code.

parameters initialization
readin()
scattering table construction
sc_table()
histograms calculation
histograms()
Free-Flight-Scatter
free_flight_scatter()
histograms()
write data
write()
?
carriers initialization
init()
t t t
   Time t exceeds maximum
simulation time tmax
yes
no
Optional
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Energy [eV]
Cumulative
rate
[1/s]
-6 -4 -2 0 2 4 6
x 10
8
0
5
10
15
20
25
30
35
40
Wavevector ky [1/m]
Arbitrary
Units
Initial Distribution of the
wavevector along the y-axis
that is created with the
subroutine init()
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
10
20
30
40
50
60
70
80
90
Energy [eV]
Arbitrary
Units
Initial Energy Distribution created
with the subroutine init()
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
10
10
11
10
12
10
13
10
14
10
15
Energy [eV]
Scattering
Rate
[1/s]
acoustic
polar optical phonons
intervalley gamma to L
intervalley gamma to X

parameters initialization
readin()
scattering table construction
sc_table()
histograms()
Free-Flight-Scatter
free_flight_scatter()
histograms()
write data
write()
?
carriers initialization
init()
t t t
   Time t exceeds maximum
simulation time tmax
yes
no
Optional
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
10
10
11
10
12
10
13
10
14
10
15
Energy [eV]
Scattering
Rate
[1/s]
acoustic
polar optical phonons
intervalley gamma to L
intervalley gamma to X
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Energy [eV]
Cumulative
rate
[1/s]
-6 -4 -2 0 2 4 6
x 10
8
0
5
10
15
20
25
30
35
40
Wavevector ky [1/m]
Arbitrary
Units
Initial Distribution of the
wavevector along the y-axis
that is created with the
subroutine init()
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
10
20
30
40
50
60
70
80
90
Energy [eV]
Arbitrary
Units
Initial Energy Distribution created
with the subroutine init()

Transient Data
0 1 2
x 10
-11
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
5
time [s]
velocity
[m/s]

Steady-State
Results
0 1 2 3 4 5 6 7
x 10
5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
5
Electric Field [V/m]
Drift
Velocity
[m/s]
0 1 2 3 4 5 6 7
x 10
5
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Electric Field [V/m]
Conduction
Band
Valley
Occupancy
gamma valley occupancy
L valley occupancy
X valley occupancy
k-vector
-valley
X-valley [100]
L-valley [111]
Conduction bands
Valence bands
Gunn Effect

Particle-Based Device Simulations
In a particle-based device simulation
approach the Poisson equation is
decoupled from the BTE over a short time
period dt smaller than the dielectric
relaxation time
Poisson and BTE are solved in a time-marching
manner
During each time step dt the electric field is
assumed to be constant (kept frozen)

Particle-Mesh Coupling
The particle-mesh coupling scheme consists of the
following steps:
- Assign charge to the Poisson solver mesh
- Solve Poisson’s equation for V(r)
- Calculate the force and interpolate it to the
particle locations
- Solve the equations of motion:
 
   


r
1 E
k
k
r
k
q
dt
d
t
E
dt
d


 ;
Laux, S.E., On particle-mesh coupling in Monte Carlo semiconductor device simulation,
Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on,
Volume 15, Issue 10, Oct 1996 Page(s):1266 - 1277

Assign Charge to the Poisson Mesh
1. Nearest grid point scheme
2. Nearest element cell scheme
3. Cloud in cell scheme

Force interpolation
The SAME METHOD that is used for the
charge assignment has to be used for the
FORCE INTERPOLATION:
     ,
r r r r
 

F E
i i p p
p
q W
xp-1 xp xp+1
e
p
x

w
p
x













 

w
p
p
p
e
p
p
p
x
V
V
x
V
V 1
1
2
1
p
E

Treatment of the Contacts
From the aspect of device physics, one can distinguish
between the following types of contacts:
(1) Contacts, which allow a current flow in and out of the
device
- Ohmic contacts: purely voltage or purely current
controlled
- Schottky contacts
(2) Contacts where only voltages can be applied

Calculation of the Current
The current in steady-state conditions is
calculated in two ways:
 By counting the total number of particles that
enter/exit particular contact
 By using the Ramo-Shockley theorem
according to which, in the channel, the current
is calculated using
1
( ),
N
x
i
e
I v i
dL 
 

Current Calculated by Counting the Net Number of
Particles Exiting/Entering a Contact
Electrons that naturally came out in time interval dt (N1)
Electrons that were deleted (N2)
Electrons that were injected (N3)
dq = q(N1+N2-N3), q(t+dt)=q(t) + dq, current equals the slope of q(t) vs. t
Source Drain
Gate
Mesh node
Electron
Dopant

Device Simulation Results for MOSFETs:
Current Conservation
0
1000
2000
3000
4000
5000
6000
7000
1 1.5 2 2.5 3
source contact
drain contact
net
#
of
electrons
exiting/entering
contact
time [ps]
W
G
= 0.5 mm
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
Current
I
D
[mA/mm] Distance [nm]
(b)
VG=1.4 V, VD=1 V
Cumulative net number of particles
Entering/exiting a contact for a 50 nm
Channel length device
Drain contact
Source contact
Current calculated using Ramo-Schockley
formula
1
( ),
N
x
i
e
I v i
dL 
 
X. He, MS, ASU, 2000.

Simulation Results for MOSFETs: Velocity
and Enery Along the Channel
0
5x10
6
1x10
7
1.5x10
7
2x10
7
2.5x10
7
0 50 100 150
Drift
velocity
[cm/s]
Distance [nm]
(a)
Mean Drift Velocity
Along the Channel
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
Average
energy
[eV]
Distance [nm]
(b)
Average Kinetic Energy
Along the Channel
VD = 1 V, VG = 1.2 V
Velocity overshoot effect observed
throughout the whole channel length of
the device – non-stationary transport.
For the bias conditions used average
electron energy is smaller that 0.6 eV
which justifies the use of non-parabolic
band model.

Simulation Results for MOSFETs:
IV Characteristics
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
Drain
current
[mA/mm]
Drain voltage V
D
[V]
1.2 V
0.8 V
1.0 V
Silvaco simulations
VG
= 1.0 V
VG
= 1.4 V
2D MCPS
VG
= 1.4 V
The differences between
the Monte Carlo and the
Silvaco simulations are
due to the following
reasons:
• Different transport
models used (non-
stationary transport is
taking place in this device
structure).
• Surface-roughness and
Coulomb scattering are
not included in the
theoretical model used in
the 2D-MCPS.
X. He, MS, ASU, 2000.

Simulation Results For SOI MESFET
Devices – Where are the Carriers?
Nd = 1019
Na =3-10x 1015
Nd = 1019
Si Substrate
Lg =60, 100nm
Lg =60, 100nm
Oxide Layer
Nd = 1019
Nd = 3-10x1015
Nd =1019
Si Substrate
Oxide Layer
Fig. 3.1 Schematic cross-sections of a) the SOI MOSFET and b) the SOI
MESFET devices that have been simulated.
Fig. 3.2. The electron distributions in c) the 60 nm SOI MOSFET and d) the 60 nm
SOI
MOSFET
SOI
MESFET
Applications:
Low-power RF electronics.
T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985).
Micropower circuits based
on weakly inverted
Implantable
cochlea and retina
Digital Watch
Pacemaker
MOSFETs

Proper Modeling of SOI MESFET Device
Gate current calculation:
 WKB Approximation
 Transfer Matrix Approach
for piece-wise linear
potentials
Interface-Roughness:
 K-space treatment
 Real-space treatment
Goodnick et al., Phys. Rev. B 32, 8171
(1985)

10
7
10
8
10
9
10
10
10
11
10
12
0.1 1 10 100 1000
Lg =25nm simulated results
Lg = 90nm simulated results
Lg = 0.6um Experimental results
Lg = 50nm Projected Experimental results
Drain Current I
d
[ µA/µm]
Cutoff
Frequency
f
T
[Hz]
Output Characteristics and Cut-off
Frequency of a Si MESFET Device
-1000
-800
-600
-400
-200
0
200
400
-0.2 0 0.2 0.4 0.6 0.8 1
Vgs
= 0.3V
Vgs
= 0.4V
V
gs
=0.5V
V
gs
= 0.6V
I
d
[
µ
A/
µ
m]
V
ds
[Volts]
Tarik Khan, PhD, ASU, 2004.

Output Characteristics and Cut-off
Frequency of a Si MESFET Device
-1000
-800
-600
-400
-200
0
200
400
-0.2 0 0.2 0.4 0.6 0.8 1
Vgs
= 0.3V
Vgs
= 0.4V
V
gs
=0.5V
V
gs
= 0.6V
I
d
[
µ
A/
µ
m]
V
ds
[Volts]
10
7
10
8
10
9
10
10
10
11
10
12
0.1 1 10 100 1000
Lg = 90nm simulated results
Lg = 0.6um Experimental results
Lg = 50nm Projected Experimental results
Drain Current I
d
[ µA/µm]
Cutoff
Frequency
f
T
[Hz]
Tarik Khan, PhD, ASU, 2004.

Modeling of SOI Devices
When modeling SOI devices lattice
heating effects has to be accounted for
In what follows we discuss the following:
 Comparison of the Monte Carlo, Hydrodynamic
and Drift-Diffusion results of Fully-Depleted SOI
Device Structures*
 Impact of self-heating effects on the operation
of the same generations of Fully-Depleted SOI
Devices
*D. Vasileska. K. Raleva and S. M. Goodnick, IEEE Trans. Electron Dev., in press.

FD-SOI Devices:
Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
feature 14 nm 25 nm 90 nm
Tox 1 nm 1.2 nm 1.5 nm
VDD 1V 1.2 V 1.4 V
Overshoot
EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD
with series resistance
153%/96% 108%/67% 39%/26%
Source/drain doping = 1020
cm-3
and 1019
cm-3
(series resistance (SR) case)
Channel doping = 1E18 cm-3
Overshoot= (IDHD-IDDD)/IDDD (%) at on-state
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm

FD-SOI Devices:
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
VDD 1V 1.2 V 1.4 V
Overshoot
EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD
153%/96% 108%/67% 39%/26%
cm-3
and 1019
cm-3
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
25 nm

FD-SOI Devices:
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
VDD 1V 1.2 V 1.4 V
Overshoot
EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD
153%/96% 108%/67% 39%/26%
cm-3
and 1019
cm-3
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Drain
Current
[mA/um]
DD SR
HD SR
Monte Carlo
14 nm

FD-SOI Devices:
Why Self-Heating Effect is Important?
1. Alternative materials (SiGe)
2. Alternative device designs (FD SOI, DG,
TG, MG, Fin-FET transistors

FD-SOI Devices:
Why Self-Heating Effect is Important?
dS
L ~
300nm
A. Majumdar, “Microscale Heat Conduction
in Dielectric Thin Films,” Journal of Heat
Transfer, Vol. 115, pp. 7-16, 1993.

Conductivity of Thin Silicon Films –
Vasileska Empirical Formula
300 400 500 600
20
40
60
80
Temperature (K)
Thermal
conductivity
(W/m/K)
experimental data
full lines: BTE predictions
dashed lines: empirical model
thin lines: Sondheimer
20nm
30nm
50nm
100nm
/2
3
0
0
2
( ) ( ) sin 1 exp cosh
2 ( )cos 2 ( )cos
a a z
z T d
T T

   
   
 
   

  
 
   
   
 

0
( ) (300/ )
T T
 

0 2
135
( ) W/m/K
T
a bT cT
 
 
0 2 4 6 8 10
10
5
7.5
10
12.5
15
17.5
20
20
Distance from Si/gate oxide interface (nm)
Thermal
conductivity
(W/m-K)
300K
400K
600K
Si
BOX
10nm
0 20 40 60 80 100
0
10
20
30
40
50
60
70
80
80
Distance from Si/gate oxide interface (nm)
Thermal
conductivity
(W/m-K)
20nm
30nm
50nm
100nm

Particle-Based Device Simulator That
Includes Heating
Average and smooth: electron
density, drift velocity and electron
energy at each mesh point
end of MCPS
phase?
Acoustic and Optical Phonon
Energy Balance Equations Solver
end of
simulation?
end
no
yes
Define device structure
Generate phonon temperature
dependent scattering tables
Initial potential, fields, positions and
velocities of carriers
t = 0
t = t + t
Transport Kernel (MC phase)
Field Kernel (Poisson Solver)
t = n t?
yes
Average and smooth: electron
density, drift velocity and electron
energy at each mesh point
end of MCPS
phase?
end of MCPS
phase?
Acoustic and Optical Phonon
Energy Balance Equations Solver
end of
simulation?
end of
simulation?
end
no
yes
t = 0
t = t + t
t = n t?
t = 0
t = t + t
t = n t?
t = n t?
yes

Heating vs. Different Technology
Generation
25 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
10 20 30 40 50 60 70
3
8
13
13
20 40 60 80 100 120
3
12
21
21
20 40 60 80 100 120 140 160 180
3
15
27
27
300
400
500
300
400
500
300
400
500
source contact drain contact
source region drain region
50 100 150 200
3
20
35
35
50 100 150 200 250
3
21
39
39
50 100 150 200 250 300
3
23
43
43
300
400
500
600
300
400
500
300
400
500
0 120 240 360
360
4
28
52
52
50 100 150 200 250 300 350 400
4
33
60
60
x (nm)
100 200 300 400 500
4
41
76
76
400
600
400
600
300
400
500
600
10 20 30 40 50 60 70
3
8
13
13
20 40 60 80 100 120
3
12
21
21
20 40 60 80 100 120 140 160 180
3
15
27
27 300
400
500
300
400
500
300
400
500
600
source contact
drain contact
50 100 150 200
3
20
35
35
50 100 150 200 250
3
21
39
39
50 100 150 200 250 300
3
23
43
43 300
400
500
600
400
600
300
400
500
600
50 100 150 200 250 300 350
4
28
52
52
50 100 150 200 250 300 350 400
4
33
60
60
x (nm)
100 200 300 400 500
4
41
76
76
400
600
400
600
300
400
500
600
700
T=300K on gate T=400K on gate
Acoustic Phonon Temperature Profiles

Higher Order Effects Inclusion in Particle-
Based Simulators
 Degeneracy – Pauli Exclusion Principle
 Short-Range Coulomb Interactions
 Fast Multipole Method (FMM)
V. Rokhlin and L. Greengard, J. Comp. Phys., 73, pp. 325-348
(1987).
 Corrected Coulomb Approach
W. J. Gross, D. Vasileska and D. K. Ferry, IEEE Electron Device
Lett. 20, No. 9, pp. 463-465 (1999).
 P3M Method
Hockney and Eastwood, Computer Simulation Using Particles.

Potential, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.
MOTIVATION

Length [nm]
100
110
120
130
140
150
Width
[nm]
60 80 100 120 140
Current Stream Lines, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.
MOTIVATION

Discrete Impurities
(Short-Range Interaction)
Efficient 3D Poisson
Equation Solvers
3D Monte Carlo
Transport Kernel
Effective Potential
(Space Quantization)
3DMCDS
3DMCDS
Discrete Impurities
(Short-Range Interaction)
Efficient 3D Poisson
Equation Solvers
3D Monte Carlo
Transport Kernel
Effective Potential
(Space Quantization)
3DMCDS
3DMCDS
The ASU Particle-Based Device
Simulator
(1) Corrected
Coulomb Approach
(2) P3M Algorithm
(3) Fast Multipole
Method (FMM)
(1) Ferry’s Effective
Potential Method
(2) Quantum Field
Approach
Statistical Enhancement:
Event Biasing Scheme
Short-Range Interactions
and Discrete/Unintentional
Dopants
Quantum Mechanical
Size-quantization Effects
Boltzmann Transport Equations
(Particle-Based Monte Carlo Transport Kernel)
Long-range Interactions
(3D Poisson
Equation Solver)

Significant Data Obtained
Between 1998 and 2002

MOSFETs - Standard Characteristics
0
50
100
150
200
250
300
350
40 60 80 100 120 140
V
D
=1.0 [V], V
G
=1.0 [V]
V
D
=0.5 [V], V
G
=1.0 [V]
Electron
energy
[meV]
Distance [nm]
L
G
= 80 nm
0
5x10
6
1x10
7
1.5x10
7
2x10
7
40 60 80 100 120 140
V
D
=0.5 [V], V
G
=1.0 [V]
V
D
=1.0 [V], V
G
=1.0 [V]
Drift
velocity
[cm/s]
Distance [nm]
L
G
=80 nm
(a)
 The average energy of the carriers increases when going from the
source to the drain end of the channel. Most of the phonon scattering
events occur at the first half of the channel.
 Velocity overshoot occurs near the drain end of the channel. The
sharp velocity drop is due to e-e and e-i interactions coming from the
drain.
W. J. Gross, D. Vasileska and D. K. Ferry, "3D Simulations of Ultra-Small MOSFETs with Real-Space
Treatment of the Electron-Electron and Electron-Ion Interactions," VLSI Design, Vol. 10, pp. 437-452 (2000).

MOSFETs - Role of the E-E and E-I
0
100
200
300
400
100 110 120 130 140 150 160 170 180
with e-e and e-i
mesh force only
Electron
energy
[meV]
Length [nm]
V
D
=1 V, V
G
=1 V
channel drain
0
100
200
300
400
500
600
700
800
0.12 0.13 0.14 0.15 0.16 0.17 0.18
Energy
[meV]
Length [nm]
0
100
200
300
400
500
600
700
800
0.12 0.13 0.14 0.15 0.16 0.17 0.18
Length [nm]
Energy
[meV]
mesh force
only
with e-e and e-i
Individual electron
trajectories over
time
-1x10
7
-5x10
6
0
5x10
6
1x10
7
1.5x10
7
2x10
7
2.5x10
7
0 40 80 120 160
with e-e and e-i
mesh force only
Drift
velocity
[cm/s]
Length [nm]
V
D
=V
G
=1.0 V
source drain
channel

MOSFETs - Role of the E-E and E-I
Mesh force only With e-e and e-i
Short-range e-e and e-i interactions push some
of the electrons towards higher energies
10
-3
10
-2
0 50 100 150 200 250 300 350 400
source
channel
drain
Electron
distribution
(arb.
units)
Energy [meV]
V
G
=0.5 V, V
D
=0.8 V
10
-3
10
-2
0 50 100 150 200 250 300 350 400
Source
Channel
Drain
Electron
distribution
(arb.
units)
Energy [meV]
V
G
=0.5 V, V
D
=0.8 V
D. Vasileska, W. J. Gross, and D. K. Ferry, "Monte-Carlo particle-based simulations of deep-submicron n-
MOSFETs with real-space treatment of electron-electron and electron-impurity interactions," Superlattices
and Microstructures, Vol. 27, No. 2/3, pp. 147-157 (2000).

Degradation of Output Characteristics
0
10
20
30
40
50
60
70
80
0.0 0.2 0.4 0.6 0.8 1.0 1.2
with corrected Coulomb
mesh force only
Drain
current
I
D
[mA]
Drain voltage V
D
[V]
increasing V
G
LG = 35 nm, WG = 35 nm, NA = 5x1018 cm-3,
Tox = 2 nm, VG = 11.6 V (0.2 V)
 The short range e -e and e -i interactions have significant influence on
the device output characteristics.
 There is almost a factor of two decrease in current when these two inte-
raction terms are considered.
LG = 50 nm, WG = 35 nm, NA = 5x1018 cm-3
Tox = 2 nm, VG = 11.6 V (0.2 V)
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2
with corrected Coulomb
mesh force only
Drain
current
I
D
[mA]
Drain voltage V
D
[V]
increasing V
G
W. J. Gross, D. Vasileska and D. K. Ferry, "Ultra-small MOSFETs: The importance of the full Coulomb
interaction on device characteristics," IEEE Trans. Electron Devices, Vol. 47, No. 10, pp. 1831-1837 (2000).

Mizuno result:
(60% of the fluctuations)
Stolk et al. result:
(100% of the fluctuations)
Fluctuations in the
surface potential
Fluctuations in the
electric field
Depth-
Distribution
of the charges

MOSFETs - Discrete Impurity Effects
0
20
40
60
80
100
0 1 2 3 4 5
s
Vth
=> approach 1
s
Vth
=> approach 2
s
Vth
=> our simulation results
s
Vth
[mV]
Oxide thickness T
ox
[nm]
0
5
10
15
20
25
30
35
40
1x10
18
3x10
18
5x10
18
7x10
18
s
Vth
=> approach 1
s
Vth
=> approach 2
s
Vth
s
Vth
[mV]
Doping density N
A
[cm
-3
]
10
20
30
40
50
60
20 40 60 80 100 120 140
s
Vth
=> approach 1
s
Vth
=> approach 2
s
Vth
s
Vth
[mV] Device width [nm]
eff
eff
A
ox
ox
A
B
Si
B
B
Si
Vth
W
L
N
T
N
q
q
/
T
k
q 4
4 3
4
3
4













s
Approach 2 [2]:














s
i
A
B
B
eff
eff
A
ox
ox
B
Si
Vth
n
N
ln
q
T
k
;
W
L
N
T
q 4
4 3
2
Approach 1 [1]:
[1] T. Mizuno, J. Okamura, and A. Toriumi, IEEE Trans. Electron Dev. 41, 2216 (1994).
[2] P. A. Stolk, F. P. Widdershoven, and D. B. Klaassen, IEEE Trans. Electron Dev. 45, 1960
(1998).

0
20
40
60
80
100
120
140
160
180
200
160
170
180
190
200
210
220
230
240
250
260
270
Number of Atoms in Channel
Number
of
Devices
5 samples at
maximum
5 samples at
minimum
5 samples of average
Depth Correlation of sV
T
 To understand the role that the position
of the impurity atoms plays on the
threshold voltage fluctuations, statistical
ensembles of 5 devices from the low-end,
center and the high-end of the
distribution were considered.
 Significant correlation was observed
between the threshold voltage and the
number of atoms that fall within the first 15
nm depth of the channel.
0.9
1
1.1
1.2
1.3
1.4
160 180 200 220 240 260 280 300
Threshold
voltage
[V]
Number of channel dopant atoms
low-end
center
high-end
(a)
L
G
=50 nm, W
G
=35 nm
N
A
=5x10
18
cm
-3
, T
ox
=3 nm
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
V
T
correlation
Depth [nm]
depth
Moving slab
range
ND ND
(b) L
G
=50 nm, W
G
=35 nm, T
ox
=3 nm
N
A
=5x10
18
cm
-3
Number of atoms in the channel
Number
of
devices
Depth [nm]
V
T
correlation
Threshold
voltage
[V]

Fluctuations in High-Field
Characteristics
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
average velocity
correlation
drain current
correlation
Correlation
Depth [nm]
(c)
LG
=50 nm
WG
=35 nm
Tox
=3 nm
NA
=5x10
18
cm
-3
 Impurity distribution in the channel also
affects the carrier mobility and saturation
current of the device.
 Significant correlation was observed
between the drift velocity (saturation
current) and the number of atoms that fall
within the first 10 nm depth of the
channel.
0
5 10
6
1 10
7
1.5 10
7
160 180 200 220 240 260 280
Drift
velocity
[cm/s]
(a)
low-end
center
high-end
VG
=1.5 V, VD
=1 V
LG
=50 nm, WG
=35 nm
NA
=5x10
18
cm
-3
0
5
10
15
20
160 180 200 220 240 260 280
Drain
current
[mA]
low-end
center
high-end
(b)
Drift
velocity
[cm/s]
Depth [nm]
Correlation
Drain
current
[mA]
VG = 1.5 V, VD = 1 V

Current Issues in Novel
Devices – Unintentional Dopants

Results for SOI Device
Size Quantization Effect (Effective Potential):
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
2 4 6 8 10 12 14 16
Channel Width [nm]
Threshold
Voltage
[V]
Experimental
Simulation
S. S. Ahmed and D. Vasileska, “Threshold voltage shifts in narrow-width SOI devices due to
quantum mechanical size-quantization effect”, Physica E, Vol. 19, pp. 48-52 (2003).

Due to the unintentional dopant both the electrostatics
and the transport are affected.
-10000
10000
30000
50000
70000
90000
110000
130000
0 20 40 60 80
Distance Along the Channel [nm]
Average
Velocity
[m/s]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Average
Kinetic
Energy
[eV]
Velocity
Energy
Dip due to the presence of
the impurity. This affects the
transport of the carriers.

Unintentional Dopant:
D. Vasileska and S. S. Ahmed, “Narrow-Width SOI Devices: The Role of Quantum Mechanical
Size Quantization Effect and the Unintentional Doping on the Device Operation”, IEEE
Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.

Channel Width = 10 nm
VG = 1.0 V
VD = 0.1 V
32.54%
47.62%
33.01%
26.98%
42.85%
27.18%
11.90%
26.19%
11.11%
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50
Distance
Along
the
Width
[nm]
.
Source
Drain 34.13%
47.62%
42.06%
16.66%
42.85%
26.19%
9.93%
26.19%
19.46%
0
1
2
3
4
5
6
7
0 10 20 30 40 50
Distance
Along
the
Depth
[nm]
.
Source
Drain

86.30%
96.76%
86.52%
87.39%
96.09%
86.96%
69.57%
88.26%
67.39%
0
1
2
3
4
5
0 10 20 30 40 50
Distance
Along
the
Width
[nm]
Source
Drain
81.09%
96.76%
88.48%
79.78%
96.09%
88.26%
59.78%
88.26%
76.09%
0
1
2
3
4
5
6
7
0 10 20 30 40 50
Distance
Along
the
Depth
[nm]
Source
Drain
Channel Width = 5 nm
VG = 1.0 V
VD = 0.1 V

Impurity located at the very source-end, due to the availability of Increasing
number of electrons screening the impurity ion, has reduced impact on the
overall drain current.
0%
10%
20%
30%
40%
50%
60%
0 10 20 30 40 50
Current
Reduction
Impurity position varying along the
center of the channel
V G = 1.0 V
V D = 0.2 V
Source end Drain end

-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
2 4 6 8 10 12 14 16
Channel Width [nm]
Threshold
Voltage
[V]
Experimental
Simulation (QM)
Discrete single dopants
D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005
Page(s):227 – 236.
S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005
Page(s):465 – 471.
D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

Electron-Electron Interactions:
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
0 0.2 0.4 0.6 0.8 1
Electron Kinetic Energy [eV]
Distribution
Function
[a.u.]
PM
FMM
V G = 1.0 V
V D = 0.3 V
0.0E+00
5.0E+04
1.0E+05
1.5E+05
2.0E+05
2.5E+05
3.0E+05
0 20 40 60 80 100
Electron
Velocity
[m/s]
PM
FMM
V G = 1.0 V
V D = 0.3 V
Source Drain
D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005
Page(s):227 – 236.
S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005
Page(s):465 – 471.
D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

Summary
 Particle-based device simulations are the most desired
tool when modeling transport in devices in which velocity
overshoot (non-stationary transport) exists
 Particle-based device simulators are rather suitable for
modeling ballistic transport in nano-devices
 It is rather easy to include short-range electron-electron
and electron-ion interactions in particle-based device
simulators via a real-space molecular dynamics routine
 Quantum-mechanical effects (size quantization and
density of states modifications) can be incorporated in
the model quite easily with the assumption of adiabatic
approximation and solution of the 1D or 2D Schrodinger
equation in slices along the channel section of the device

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