The document discusses various computational models for semiconductor device transport simulation. It begins by describing semiclassical transport theory and approaches like drift-diffusion, hydrodynamics, and particle-based Monte Carlo methods. It then covers topics like inclusion of tunneling effects, quantum corrections, and particle-based and quantum transport simulations. Specific models are discussed for generation-recombination mechanisms, low-field and field-dependent mobility, inversion layer mobility, and the hydrodynamic approach for including velocity overshoot effects.

1. QUALITATIVE MODEL OF
TRANSPORT
R.Rokhit
17D071

2. 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

3. 1. Increased costs for R&D and production facilities, which are becoming too large for
any one company or country to accept.
2. Shorter process technology life cycles.
3. Emphasis on faster characterization of manufacturing processes, assisted by
modeling and simulation.
The need for semiconductor device modeling
With permission from Intel Corp.

4. What Transport Models exist?
• Semiclassical FLUID models
(ATLAS, Sentaurus, Padre)
– Drift – Diffusion
– Hydrodynamics
1. PARTICLE DENSITY
2. velocity saturation
effect
3. mobility modeling
crucial
10
6
10
7
1 10 100
Current simulations
Yamada simulations
Canali exp. data
Driftvelocity[cm/s]
Electric field [kV/cm]
1. Particle density
2. DRIFT VELOCITY, ENERGY DENSITY
3. velocity overshoot effect
problems
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
8
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.2 ps
0.3 ps

5. What Transport Models Exist?
• Semiclassical PARTICLE-BASED
Models:
– Direct solution of the BTE Using Monte
Carlo method
• Eliminates the problem of Energy
Relaxation Time Choice
• Accurate up to semi-classical limits
• One can describe scattering very well
• Can treat ballistic transport in devices

6. Semi-Classical Transport Theory
• It is based on direct or approximate solution of the
Boltzmann Transport Equation for the semi-classical
distribution function f(r,k,t)
which gives one the probability of finding a particle in
region (r,r+dr) and (k,k+dk) at time t
• Moments of the distribution function give us information
about:
– Particle Density
– Current Density
– Energy Density
    3
1
1 1
8
k k kk
F
kk
k r k k k k k k k
f V
E f f d f f f f
t 
   

          
 h h
D. K. Ferry, Semiconductors, MacMillian, 1990.

7. Semi-Classical Transport Approaches
1.Drift-Diffusion Method
2.Hydrodynamic Method
3.Direct Solution of the Boltzmann Transport
Equation via:
– Particle-Based Approaches – Monte Carlo method
– Spherical Harmonics
– Numerical Solution of the Boltzmann-Poisson
Problem
C. Jacoboni, P. Lugli: "The Monte Carlo Method for Semiconductor Device Simulation“,
in series "Computational Microelectronics", series editor: S. Selberherr; Springer, 1989, ISBN: 3-211-
82110-4.

8. 1. Drift-Diffusion Approach
Constitutive Equations
• Poisson
• Continuity Equations
• Current Density Equations
1
1
J
J
n n
p p
n
U
t q
p
U
t q

   


    

 D AV p n N N  
      
( ) ( )
( ) ( )
n n n
p p p
dn
J qn x E x qD
dx
dn
J qp x E x qD
dx


 
 S. Selberherr: "Analysis and Simulation of
Semiconductor Devices“, Springer, 1984.

9. Numerical Solution Details
• Linearization of the Poisson equation
• Scharfetter-Gummel Discretization of the Continuity
equation
• De Mari scaling of variables
• Discretization of the equations
– Finite Difference – easier to implement but requires more
node points, difficult to deal with curved interfaces
– Finite Elements – standard, smaller number of node points,
resolves curved surfaces
– Finite Volume

10. Linearized Poisson Equation
φ→φ + δ where δ= φnew - φold
• Finite difference discretization:
– Potential varies linearly between mesh points
– Electric field is constant between mesh points
• Linearization → Diagonally-dominant
coefficient matrix A is obtained
   
   
 
2
/ / / /
2
2
/ / / /
2
/ /
/
/
old old old old
T T T T
old old old old
T T T T
old old
T T
new
V V V V V V V Vi i
i
new
V V V V V V V Vnewi i
i
T
V V V V oldi
T
new old
en end V
e e C n e e
dx
en end V
e e V e e C n
dx V
en
e e V
V
V V

 
 


 
 

     
      
 
 

11. Scharfetter-Gummel Discretization of the
Continuity Equation
• Electron and hole densities n and p vary exponentially
between mesh points → relaxes the requirement of using very
small mesh sizes
• The exponential dependence of n and p upon the potential is
buried in the Bernoulli functions
1/2 1 1/2 1 1/2 1 1/2 1
1 12 2 2 2
1/2 1 1/2 1 1/2 1
12 2 2
n n n n
i i i i i i i i i i i i
i i i i
T T T T
n n n
i i i i i i i i i
i
T T T
D V V D V V D V V D V V
B n B B n B n U
V V V V
D V V D V V D V V
B p B B
V V V
       
 
     

           
           
           
      
     
      
1/2 1
12
n
i i i
i i i
T
D V V
p B p U
V
 

   
    
   
( )
1x
x
B x
e


Bernoulli function:

12. Scaling due to de Mari
Variable Scaling Variable Formula
Space Intrinsic Debye length (N=ni)
Extrinsic Debye length (N=Nmax)
2
Bk T
L
q N


Potential Thermal voltage
* Bk T
V
q

Carrier concentration Intrinsic concentration
Maximum doping concentration
N=ni
N=Nmax
Diffusion coefficient Practical unit
Maximum diffusion coefficient
2
1
cm
D
s

D = Dmax
Mobility
*
D
M
V

Generation-Recombination
2
DN
R
L

Time 2
L
T
D


13. Numerical Solution Details
Governing
Equations
ICS/BCS
Discretization
System of
Algebraic
Equations
Equation
(Matrix)
Solver
Approximate
Solution
Continuous
Solutions
Finite-Difference
Finite-Volume
Finite-Element
Spectral
Boundary Element
Hybrid
Discrete
Nodal
Values
Tridiagonal
SOR
Gauss-Seidel
Krylov
Multigrid
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes,
Arizona State University, Tempe, AZ.

14. (A) Low-Field Models for Bulk Materials
Phonon scattering:
- Simple power-law dependence of the temperature
- Sah et al. model:
acoustic + optical and intervalley phonons
combined via Mathiessen’s rule
Ionized impurity scattering:
- Conwell-Weiskopf model
- Brooks-Herring model

15. (A) Low-Field Models for Bulk
Materials (cont’d)
Combined phonon and ionized impurity scattering:
- Dorkel and Leturg model:
temperature-dependent phonon scattering +
ionized impurity scattering + carrier-carrier
interactions
- Caughey and Thomas model:
temperature independent phonon scattering +
ionized impurity scattering

16. (A) Low-Field Models for Bulk
Materials (cont’d)
- Sharfetter-Gummel model:
phonon scattering + ionized impurity scattering
(parameterized expression – does not use the
Mathiessen’s rule)
- Arora model:
similar to Caughey and Thomas, but with
temperature dependent phonon scattering

17. (A) Low-Field Models for Bulk
Materials (cont’d)
Carrier-carrier scattering
- modified Dorkel and Leturg model
Neutral impurity scattering:
- Li and Thurber model:
mobility component due to neutral impurity
scattering is combined with the mobility due to
lattice, ionized impurity and carrier-carrier scattering
via the Mathiessen’s rule

18. (B) Field-Dependent Mobility
The field-dependent mobility model provides smooth transition
between low-field and high-field behavior
vsat is modeled as a temperature-dependent quantity:



 /1
0
0
1
)(
















satv
E
E  = 1 for electrons
 = 2 for holes
cm/s
600
exp8.01
104.2
)(
7









L
sat
T
Tv

19. (C) Inversion Layer Mobility Models
• CVT model:
– combines acoustic phonon, non-polar optical
phonon and surface-roughness scattering (as an
inverse square dependence of the perpendicular
electric field) via Mathiessen’s rule
• Yamaguchi model:
– low-field part combines lattice, ionized impurity
and surface-roughness scattering
– there is also a parametric dependence on the in-
plane field (high-field component)

20. (C) Inversion Layer Mobility Models
(cont’d)
• Shirahata model:
– uses Klaassen’s low-field mobility model
– takes into account screening effects into the
inversion layer
– has improved perpendicular field dependence for
thin gate oxides
• Tasch model:
– the best model for modeling the mobility in MOS
inversion layers; uses universal mobility behavior

21. Generation-Recombination Mechanisms
Classification
Two
particle
One step
(Direct)
Two-step
(indirect)
Energy-level
consideration
• Photogeneration
• Radiative recombination
• Direct thermal generation
• Direct thermal recomb.
• Shockley-Read-Hall
(SRH) generation-
recombination
• Surface generation-
recombination
• Shockley-Read-Hall
(SRH) generation-
recombination
• Surface generation-
recombination
Three
particle
Impact
ionization
Auger
• Electron emission
• Hole emission
• Electron capture
• Hole capture
Pure generation process

22. Hydrodynamic Modeling
• In small devices there exists non-stationary
transport and carriers are moving through the
device with velocity larger than the saturation
velocity
– In Si devices non-stationary transport occurs
because of the different order of magnitude of the
carrier momentum and energy relaxation times
– In GaAs devices velocity overshoot occurs due to
intervalley transfer
T. Grasser (ed.): "Advanced Device Modeling and Simulation“, World Scientific Publishing Co.,
2003, ISBN: 9-812-38607-6 M.
M. Lundstrom, Fundamentals of Carrier Transport, 1990.

23. Velocity Overshoot in Silicon
-5x10
6
0
5x10
6
1x107
1.5x10
7
2x10
7
2.5x10
7
0 0.5 1 1.5 2 2.5 3 3.5 4
1 kV/cm
5 kV/cm
10 kV/cm
50 kV/cm
time [ps]
Driftvelocity[cm/s]
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2 2.5 3 3.5 4
1 kV/cm
5 kV/cm
10 kV/cm
50 kV/cm
Energy[eV]
time [ps]
Scattering mechanisms:
• Acoustic deformation potential scattering
• Zero-order intervalley scattering (f and g-
phonons)
• First-order intervalley scattering (f and g-
phonons)
g
f
kz
kx
ky
g
f
g
ff
kz
kx
ky
X. He, MS Thesis, ASU, 2000.

24. How is the Velocity Overshoot Accounted
For?
• In Hydrodynamic/Energy balance modeling
the velocity overshoot effect is accounted for
through the addition of an energy
conservation equation in addition to:
– Particle Conservation (Continuity Equation)
– Momentum (mass) Conservation Equation

25. Hydrodynamic Model due to Blotakjer
Constitutive Equations: Poisson +• More convenient set of balance equations is in terms of n, vd
and w:
 
 
 
coll
d
d
B
dd
coll
d
dd
dd
coll
d
t
w
e
vm
w
k
n
n
w
t
w
tm
e
vnmnw
nm
m
mt
t
n
n
t
n

































































)(
2
*
3
2
)(
*
*
2
1
*3
2
)*(
*
)(
2
2
vE
vv
vE
v
vv
v

26. Closure
• To have a closed set of equations, one either:
(a) ignores the heat flux altogether
(b) uses a simple recipe for the calculation of the heat flux:
)(*2
5
,
2
wvm
nTk
Tn Bq
• Substituting T with the density of the carrier energy, the
momentum and energy balance equations become:
 
 
ddd
co
d
ddd
d
vmw
k
nwn
t
nw
t
n
envnmnwn
t
n


































*
2
1
3
2
)(
*
2
1
3
2
)(
2
2
vv
p
Epv
p

27. Momentum Relaxation Rate
• The momentum rate is determined by a steady-state MC
calculation in a bulk semiconductor under a uniform bias
electric field, for which:
d
p
dp
coll
dd
vm
eE
w
w
m
e
tm
e
t
*
)(
0)(
**












v
EvEv
K. Tomizawa, Numerical Simulation Of
Submicron Semiconductor Devices.

28. Energy Relaxation Rate
• The emsemble energy relaxation rate is also determined by a
steady-state MC calculation in a bulk semiconductor under a
uniform bias electric field, for which:
0
0
)(
0)(
ww
e
w
wwe
t
w
e
t
w
d
w
wd
coll
d














vE
vEvE
K. Tomizawa, Numerical Simulation Of
Submicron Semiconductor Devices.

29. Validity of the Hydrodynamic Model
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%
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
DrainCurrent[mA/um]
DD
EB
HD
DD SR
EB SR
HD SR
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm device
SR = series resistance

30. Validity of the Hydrodynamic Model
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%
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
DrainCurrent[mA/um]
DD
EB
HD
DD SR
EB SR
HD SR
Silvaco ATLAS simulations performed by Prof. Vasileska.
25 nm device
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
Drain Voltage [V]
DrainCurrent[mA/um]
DD
HD
EB
DD SR
EB SR
HD SR
SR = series resistance

31. 0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
2.5
Drain Voltage [V]
DrainCurrent[mA/um]
DD
EB
HD
DD SR
EB SR
HD SR
Validity of the Hydrodynamic Model
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%
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
Drain Voltage [V]
DrainCurrent[mA/um]
DD
HD
EB
DD SR
EB SR
HD SR
Silvaco ATLAS simulations performed by Prof. Vasileska.
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Drain Voltage [V]
DrainCurrent[mA/um]
DD
EB
HD
DD SR
EB SR
HD SR
14 nm device
SR = series resistance

32. Failure of the Hydrodynamic Model
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.3 ps
0.2 ps
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm
25 nm
14 nm
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
8
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.2 ps
0.3 ps
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
Drain Voltage [V]
DrainCurrent[mA/um] 10
19
cm
-3
10
20
cm
-3
0.1 ps
0.2 ps
0.3 ps

33. Failure of the Hydrodynamic Model
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
Drain Voltage [V]
DrainCurrent[mA/um] 10
19
cm
-3
10
20
cm
-3
0.1 ps
0.2 ps
0.3 ps
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm
25 nm
14 nm
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.3 ps
0.2 ps
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
8
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.2 ps
0.3 ps

34. Failure of the Hydrodynamic Model
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
8
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.2 ps
0.3 ps
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
Drain Voltage [V]
DrainCurrent[mA/um] 10
19
cm
-3
10
20
cm
-3
0.1 ps
0.2 ps
0.3 ps
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm
25 nm
14 nm
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
Drain Voltage [V]
DrainCurrent[mA/um]
1020
cm-3
1019
cm-3
0.1 ps
0.3 ps
0.2 ps

35. Summary
• Drift-Diffusion model is good for large MOSFET devices, BJTs,
Solar Cells and/or high frequency/high power devices that
operate in the velocity saturation regime
• Hydrodynamic model must be used with caution when
modeling devices in which velocity overshoot, which is a
signature of non-stationary transport, exists in the device
• Proper choice of the energy relaxation times is a problem in
hydrodynamic modeling