ECE 595, Section 10
Numerical Simulations
Lecture 19: FEM for Electronic
Transport
Prof. Peter Bermel
February 22, 2013
Outline
• Recap from Wednesday
• Physics-based device modeling
• Electronic transport theory
• FEM electronic transport model
• Numerical results
• Error Analysis
2/22/2013 ECE 595, Prof. Bermel
Recap from Wednesday
• Thermal transfer overview
– Convection
– Conduction
– Radiative transfer
• FEM Modeling Approach
• Numerical Results
• Error Evaluation
2/22/2013 ECE 595, Prof. Bermel
Physics-Based Device Modeling
2/22/2013
D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan &
Claypool , 2006.
ECE 595, Prof. Bermel
Electronic Transport Theory• Will assume electronic bandstructures known,
and take a semiclassical approach
• Electrostatics modeled via Poissson’s equation:
• Charge conservation is required:
2/22/2013 ECE 595, Prof. Bermel
( )D AV p n N Nε + −∇ ⋅ ∇ = − − + −
1
1
J
J
n n
p p
nU
t q
pU
t q
∂= ∇ ⋅ +
∂
∂= − ∇ ⋅ +
∂
S. Selberherr: "Analysis and Simulation of
Semiconductor Devices“, Springer, 1984.
Electronic Transport Theory
• Both p-type and n-type currents given by a
sum of two terms:
– Drift term, derived from Ohm’s law
– Diffusion term, derived from Second Law of
Thermodynamics
2/22/2013 ECE 595, Prof. Bermel
( ) ( )
( ) ( )
n n n
p p p
dnJ qn x E x qD
dx
dnJ qp x E x qD
dx
µ
µ
= +
= −
S. Selberherr: "Analysis and Simulation of
Semiconductor Devices“, Springer, 1984.
FEM Electronic Transport Model
• Much like in earlier work, will employ the following strategy:
– Specify problem parameters, including bulk and boundary conditions
– Construct finite-element mesh over spatial domain
– Generate a linear algebra problem
– Solve for key field variables:
2/22/2013 ECE 595, Prof. Bermel
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
FEM Electronic Transport Model
2/22/2013 ECE 595, Prof. Bermel
• Regarding the grid set-up, there are several points that need to be
made:
� In critical device regions, where the charge density varies
very rapidly, the mesh spacing has to be smaller than the
extrinsic Debye length determined from the maximum doping concentration in that location of the device
� Cartesian grid is preferred for particle-based simulations
� It is always necessary to minimize the number of node points
to achieve faster convergence
� A regular grid (with small mesh aspect ratios) is needed for
faster convergence
2
maxeN
TkL B
D
ε=
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Poisson Solver
2/22/2013 ECE 595, Prof. Bermel
• The 1D Poisson equation is of the form:
( )2
2
exp exp( / )
exp exp( / )
D A
F ii i T
B
i Fi i T
B
d ep n N N
dx
E En n n V
k T
E Ep n n V
k T
ϕ
ε
ϕ
ϕ
= − − + −
−= =
−= = −
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Poisson Solver
• Perturbing potential by δ yields:
2/22/2013 ECE 595, Prof. Bermel
( )
( )
( ) ( )
( )
2/ /
2
/ /
2/ / / /
2
/ /
/
/
T T
T T
T T T T
T T
V Vii
V Vi
V V V Vnewi ii
V V oldi
new old
ende e C n
dx
ene e
en ende e e e C n
dx
ene e
ϕ ϕ
ϕ ϕ
ϕ ϕ ϕ ϕ
ϕ ϕ
ϕ
ε
δε
ϕϕ
ε ε
ϕε
δ ϕ ϕ
−
−
− −
−
= − − + +
+ +
− + = − − + −
− +
= −
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Poisson Solver
• Renormalized form
2/22/2013 ECE 595, Prof. Bermel
( ) ( )
( ) ( ) ( )
2
2
2
2
new old
new old
dp n C p n
dx
dp n p n C p n
dx
ϕδ
ϕϕ ϕ
δ ϕ ϕ
= − − + + +
− + = − − + − +
= −
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Poisson Solver
2/22/2013 ECE 595, Prof. Bermel
Initialize parameters:
-Mesh size-Discretization coefficients-Doping density-Potential based on charge neutrality
Solve for the updated potential given the forcing function using LU decomposition
Update:- Central coefficient of the linearized Poisson Equation- Forcing function
Test maximum absolute error update
Equilibrium solver
> tolerance
< tolerance
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Current Discretization
2/22/2013 ECE 595, Prof. Bermel
• The discretization of the continuity equation in conservative form requires the knowledge of the current densities
on the mid-points of the mesh lines connecting neighboring grid nodes. Since solutions are available only on the grid nodes, interpolation schemes are needed to determine the solutions.
• There are two schemes that one can use:
(a)Linearized scheme: V, n, p, µ and D vary linearly between neighboring mesh points
(b) Scharfetter-Gummel scheme: electron and hole densities follow exponential variation between mesh points
peDExepxJ
neDExenxJ
ppp
nnn∇−=
∇+=µµ)()(
)()(
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Naïve Linearization Scheme
2/22/2013 ECE 595, Prof. Bermel
• Within the linearized scheme, one has that
• This scheme can lead to substantial errors in regions of high electric fields and highly doped devices.
2/12/11
2/12/12/1 +++
+++ ∇+−
−= iii
iiiii neD
a
VVenJ µ
2
1 ii nn ++i
ii
a
nn −+1
+
−−
+
−−=
+++
+++++
i
i
i
iiii
i
i
i
iiiii
a
eD
a
VVen
a
eD
a
VVenJ
2/112/1
2/112/112/1
2
2
µ
µ
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
Scharfetter-Gummel Scheme
2/22/2013 ECE 595, Prof. Bermel
• One solves the electron current density equation
for n(V), subject to the boundary conditions
• The solution of this first-order differential equation leads to
x
V
V
neD
a
VVen
x
neD
a
VVenJ
ii
iii
ii
iiii
∂
∂
∂
∂+
−−=
∂
∂+
−−=
++
+
++
++
2/11
2/1
2/11
2/12/1
µ
µ
11
)(and)(++
==iiii
nVnnVn
[ ]
−−
−=
−
−=+−=
+++
++
−
−
++
Vt
VVBn
Vt
VVBn
a
eDJ
e
eVgVgnVgnVn
iii
iii
i
ii
VtVV
VtVV
iiii
i
111
2/12/1
/)(
/)(
11
1)(),()(1)(
1
1)(
−=
xe
xxB is the Bernouli function
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State
ADEPT 2.0
• Available on nanoHUB from Prof. Gray’s team:
https://nanohub.org/tools/adeptnpt
2/22/2013 ECE 595, Prof. Bermel
ADEPT 2.0
• Can customize all the calculation details:
2/22/2013 ECE 595, Prof. Bermel
ADEPT 2.0
• Outputs include electrostatic (Poisson) solution:
2/22/2013 ECE 595, Prof. Bermel
ADEPT 2.0
• Energy band diagram
2/22/2013 ECE 595, Prof. Bermel
ADEPT 2.0
• Carrier concentrations:
2/22/2013 ECE 595, Prof. Bermel
ADEPT 2.0
• And finally, realistic I-V curves:
2/22/2013 ECE 595, Prof. Bermel
Next Class
• Is on Monday, Feb. 25
• Next time, we will cover electronic
bandstructures
2/22/2013 ECE 595, Prof. Bermel