fem simulation of 2d mosfet

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

Friday, March 13, 2015

Summary

Problem description

Performed simulations

Numerical issues and solutions

Localized charge effects

Short channel effects


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


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


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


COMSOL mathematical model

SEMICONDUCTOR MODEL:


Non-linear Poisson equation for electrostatics computation

Hole and electron continuity equations for currents computation

2 =

ox

oxide

silicon

OXIDE MODEL:


Homogeneous Poisson equation for electrostatics computation

No continuity equations (no carriers)

=

Si=

q

Si p + ND n NA)

1

q = U

1

q = U


Boundary conditions

A

B C

DEF

G H

On AB CD:

= 0

= 0

= 0

oxide

silicon

SEMICONDUCTOR BOUNDARY:

Insulation on lateral surface


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



Ohmic metal contact for source, drain and bulk terminals


Boundary conditions

= 0

A

B C

DEF

G H

On GF HE:

oxide

silicon

OXIDE BOUNDARY:

Zero charge on lateral surface





= VG

A

B C

DEF

G H

= 0On GF HE:

On GH :

oxide

silicon

OXIDE BOUNDARY:


Fixed gate potential




Boundary conditions


Boundary conditions

OXIDE BOUNDARY:


Fixed gate potential




INTERFACE:

Insulator interface, with continuous electric potential

and displacement

= = 0

0+ = 0)

=0+

=

=0

On FE :

A

B C

DEF

G Hoxide

silicon


Summary

Problem description






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


ID-VG static characteristics

VT

Source and substrate grounded (V=0), drain biased at VDD=3.3V

Voltage sweep on gate terminal



ID-VD static characteristics

Source and substrate grounded (V=0), gate biased at VG =1.4V > VT

Voltage sweep on drain terminal



Summary

Problem description






Mesh design


Mesh design

oxide

silicon

Finer mesh

corresponding with

abrupt n+-p

junctions (source

and drain)


Mesh design

silicon

oxideFiner mesh

corresponding with

abrupt n+-p

junctions (source

and drain)

Finer mesh in the

channel region

(superficial

current)


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)


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



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



Finite Volumes,

0th order base

functions



Finite Volumes,

0th order base

functions

Finite Elements,

1st order base

functions



Finite Volumes,

0th order base

functions

Finite Elements,

1st order base

functions

Finite Elements,

2nd order base

functions



Finite Volumes,

0th order base

functions



Finite Volumes,

0th order base

functions

Finite Elements,

1st order base

functions



Finite Volumes,

0th order base

functions

Finite Elements,

1st order base

functions

Finite Elements,

2nd order base

functions



Different localbehaviour, but

almost identical

global behaviour

in standard

operation

conditions


Summary

Problem description







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






the MOS structure

Simplified modeling of charge traps:

bulk trap: closed shape in the oxidewith a surface charge condition:

=

silicon

oxide






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



The trapped charge lowersthe potential at the silicon

surface

exponential variation of electron concentration and

of current density

surface charge



The same amount of charge induces a higher threshold

shift if it is closer to the

channel Higher VT


surface


of current density

surface charge


Higher VT


The same amount of charge induces a higher threshold

shift if it is closer to the

channel


surface


of current density

Problems in deep-subthreshold regimes (low

carrier concentrations)

surface charge


Summary

Problem description







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 =

, >






=,,

=

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


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



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



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


Conclusions

Thank you for

your attention

fem simulation of 2d mosfet

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