fem simulation of 2d mosfet
DESCRIPTION
Carried out with COMSOL 5.0TRANSCRIPT
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March 17TH, 2014 Andrea CuccatoDecember 3RD, 2012 Andrea CuccatoFriday, March 13, 2015
2D FEM simulation of a MOS
Field Effect Transistor
Ph.D. Course:
Numerical Methods for Electromagnetics
Authors:
Davide RESNATI
Francesco CECCARELLI
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
-
Friday, March 13, 2015
Metal Oxide Semiconductor Transistor
PLANAR NMOS TRANSISTOR:
Metal Oxide Silicon structure
Low bulk p-doping (~1017cm-3)
High source and drain n-doping (~1020cm-3)
WORKING PRINCIPLE:
In equilibrium drain and source are electrically
disconnected (n+-p-n+
junction)
VG =0V < VT
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Friday, March 13, 2015
Metal Oxide Semiconductor Transistor
PLANAR NMOS TRANSISTOR:
Metal Oxide Silicon structure
Low bulk p-doping (~1017cm-3)
High source and drain n-doping (~1020cm-3)
WORKING PRINCIPLE:
In equilibrium drain and source are electrically
disconnected (n+-p-n+
junction)
When the gate potential is raised above a threshold value VT a high electron concentration is induced near the oxide, connecting
source and drain (n+-n-n+ junction)
VG =1.4V > VT
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Friday, March 13, 2015
COMSOL mathematical model
=
Si=
q
Si p + ND n NA)
1
q = U
1
q = U
oxide
silicon
SEMICONDUCTOR MODEL:
Stationary condition assumed
Non-linear Poisson equation for electrostatics computation
Hole and electron continuity equations for currents computation
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Friday, March 13, 2015
COMSOL mathematical model
SEMICONDUCTOR MODEL:
Stationary condition assumed
Non-linear Poisson equation for electrostatics computation
Hole and electron continuity equations for currents computation
2 =
ox
oxide
silicon
OXIDE MODEL:
Stationary condition assumed
Homogeneous Poisson equation for electrostatics computation
No continuity equations (no carriers)
=
Si=
q
Si p + ND n NA)
1
q = U
1
q = U
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Friday, March 13, 2015
Boundary conditions
A
B C
DEF
G H
On AB CD:
= 0
= 0
= 0
oxide
silicon
SEMICONDUCTOR BOUNDARY:
Insulation on lateral surface
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Friday, March 13, 2015
Boundary conditions
pn = nieff2
p n + Nd+ Na
= 0
= V
A
B C
DEF
G H
= 0
= 0
= 0
On AB CD: On AF DE BC :
oxide
silicon
SEMICONDUCTOR BOUNDARY:
Insulation on lateral surface
Ohmic metal contact for source, drain and bulk terminals
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Friday, March 13, 2015
Boundary conditions
= 0
A
B C
DEF
G H
On GF HE:
oxide
silicon
OXIDE BOUNDARY:
Zero charge on lateral surface
SEMICONDUCTOR BOUNDARY:
Insulation on lateral surface
Ohmic metal contact for source, drain and bulk terminals
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Friday, March 13, 2015
= VG
A
B C
DEF
G H
= 0On GF HE:
On GH :
oxide
silicon
OXIDE BOUNDARY:
Zero charge on lateral surface
Fixed gate potential
SEMICONDUCTOR BOUNDARY:
Insulation on lateral surface
Ohmic metal contact for source, drain and bulk terminals
Boundary conditions
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Friday, March 13, 2015
Boundary conditions
OXIDE BOUNDARY:
Zero charge on lateral surface
Fixed gate potential
SEMICONDUCTOR BOUNDARY:
Insulation on lateral surface
Ohmic metal contact for source, drain and bulk terminals
INTERFACE:
Insulator interface, with continuous electric potential
and displacement
= = 0
0+ = 0)
=0+
=
=0
On FE :
A
B C
DEF
G Hoxide
silicon
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
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Friday, March 13, 2015
ID-VG static characteristics
Source and substrate grounded (V=0), drain biased at VDD=3.3V
Voltage sweep on gate terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
ID-VG static characteristics
VT
Source and substrate grounded (V=0), drain biased at VDD=3.3V
Voltage sweep on gate terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
ID-VG static characteristics
VT
Source and substrate grounded (V=0), drain biased at VDD=3.3V
Voltage sweep on gate terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
ID-VD static characteristics
Source and substrate grounded (V=0), gate biased at VG =1.4V > VT
Voltage sweep on drain terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
ID-VD static characteristics
Source and substrate grounded (V=0), gate biased at VG =1.4V > VT
Voltage sweep on drain terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
ID-VD static characteristics
Source and substrate grounded (V=0), gate biased at VG =1.4V > VT
Voltage sweep on drain terminal
Current evaluation on drain terminal
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
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Friday, March 13, 2015
Mesh design
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Friday, March 13, 2015
Mesh design
oxide
silicon
Finer mesh
corresponding with
abrupt n+-p
junctions (source
and drain)
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Friday, March 13, 2015
Mesh design
silicon
oxideFiner mesh
corresponding with
abrupt n+-p
junctions (source
and drain)
Finer mesh in the
channel region
(superficial
current)
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Friday, March 13, 2015
Mesh design
Finer mesh
corresponding with
abrupt n+-p
junctions (source
and drain)
Finer mesh in the
channel region
(superficial
current)
silicon
oxide
Refinement
at the drain-
channel
junction
(high fields)
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Friday, March 13, 2015
Numerical approaches
Finite Volume Method
Inital guess: Carriers and potential@ equilibrium:
p-doped silicon:
p = Na
n =ni
2
NaV = Ef
n-doped silicon:
n = Nd
p =ni
2
NdV = Ef
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Friday, March 13, 2015
Numerical approaches
Finite Volume Method
Inital guess: Carriers and potential@ equilibrium:
p-doped silicon:
p = Na
n =ni
2
NaV = Ef
n-doped silicon:
n = Nd
p =ni
2
NdV = Ef
Direct discretization of the conservation law:
1
=
, c={p,n}
Solution from FVM used as initialguess for successive voltage
sweeps computed with FEM
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Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
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Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
Finite Elements,
1st order base
functions
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Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
Finite Elements,
1st order base
functions
Finite Elements,
2nd order base
functions
-
Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
-
Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
Finite Elements,
1st order base
functions
-
Friday, March 13, 2015
Numerical approaches
Finite Volumes,
0th order base
functions
Finite Elements,
1st order base
functions
Finite Elements,
2nd order base
functions
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Friday, March 13, 2015
Numerical approaches
Different localbehaviour, but
almost identical
global behaviour
in standard
operation
conditions
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
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Friday, March 13, 2015
Localized charge effects
Localized charges are caused by defects in the oxide
Electrostatic interaction of trapped electrons with the channel strongly
impacts the threshold voltage of
the MOS structure
silicon
oxide
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Friday, March 13, 2015
Localized charge effects
Localized charges are caused by defects in the oxide
Electrostatic interaction of trapped electrons with the channel strongly
impacts the threshold voltage of
the MOS structure
Simplified modeling of charge traps:
bulk trap: closed shape in the oxidewith a surface charge condition:
=
silicon
oxide
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Friday, March 13, 2015
Localized charge effects
Localized charges are caused by defects in the oxide
Electrostatic interaction of trapped electrons with the channel strongly
impacts the threshold voltage of
the MOS structure
Simplified modeling of charge traps:
bulk trap: closed shape in the oxidewith a surface charge condition:
=
surface trap: segment of the silicon-oxide interface with a surface charge
condition
silicon
oxide
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Friday, March 13, 2015
Localized charge effects
The trapped charge lowersthe potential at the silicon
surface
exponential variation of electron concentration and
of current density
surface charge
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Friday, March 13, 2015
Localized charge effects
The same amount of charge induces a higher threshold
shift if it is closer to the
channel Higher VT
The trapped charge lowersthe potential at the silicon
surface
exponential variation of electron concentration and
of current density
surface charge
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Friday, March 13, 2015
Higher VT
Localized charge effects
The same amount of charge induces a higher threshold
shift if it is closer to the
channel
The trapped charge lowersthe potential at the silicon
surface
exponential variation of electron concentration and
of current density
Problems in deep-subthreshold regimes (low
carrier concentrations)
surface charge
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Friday, March 13, 2015
Summary
Problem description
Performed simulations
Numerical issues and solutions
Localized charge effects
Short channel effects
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Friday, March 13, 2015
Short channel effects
Very large scale integrated MOSFETs are of paramount importance today
Constant field scaling (Dennard et al. 1974) implies reducing voltages
(reduction of VT, compatibility)
=,,
=
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Friday, March 13, 2015
Short channel effects
Very large scale integrated MOSFETs are of paramount importance today
Constant field scaling (Dennard et al. 1974) implies reducing voltages
(reduction of VT, compatibility)
=,,
=
Voltages are less scaled than the dimensions
short channel effects =
, >
-
Friday, March 13, 2015
Short channel effects
Very large scale integrated MOSFETs are of paramount importance today
Constant field scaling (Dennard et al. 1974) implies reducing voltages
(reduction of VT, compatibility)
=,,
=
Voltages are less scaled than the dimensions
short channel effects =
, >
NUMERICAL REMARKS:
The mesh has been defined in a scalable way, so that it is not necessary to change it
Sweeping on a geometry parameter is very challenging because every step can not use the previous solution as initial guess
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Friday, March 13, 2015
LOCAL POINT OF VIEW:
Lateral contributions increase their influence on the channel
electrostatics when the channel
length is shrinking (channel is
controlled also by the drain)
The potential barrier seen by electrons between source and
drain is lowered
Short channel effects
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Friday, March 13, 2015
LOCAL POINT OF VIEW:
Lateral contributions increase their influence on the channel
electrostatics when the channel
length is shrinking (channel is
controlled also by the drain)
The potential barrier seen by electrons between source and
drain is lowered
GLOBAL POINT OF VIEW:
The threshold is lower than the one in a long channel device Current with zero biased gate
increase exponentially;
Subthreshold slope of the curve degrades Worse Ion/Ioff ratio
Short channel effects
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Friday, March 13, 2015
An MOS Field Effect Transistor (MOSFET) model has been developed using COMSOL Multiphysics 5.0
Several numerical issues have been faced and solved (electrical current conservation, minority carriers concentration, etc...).
Unfortunately numerical problems are often trade-offs between
computational speed and solution accuracy
The model has been verified and used in two realistic cases, in order to study the effects produced by localized charges in the
oxide or by scaling the channel length
Conclusions
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Friday, March 13, 2015
Conclusions
Thank you for
your attention