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ECE 595, Section 10Numerical Simulations

Lecture 19: FEM for Electronic

Transport

Prof. Peter Bermel

February 22, 2013

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Outline

Recap from Wednesday

Physics-based device modeling

Electronic transport theory FEM electronic transport model

Numerical results

Error Analysis

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Recap from Wednesday

Thermal transfer overview

Convection

Conduction

Radiative transfer

FEM Modeling Approach

Numerical Results

Error Evaluation

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Physics-Based Device Modeling

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D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan &

Claypool , 2006.

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Electronic Transport Theory

Will assume electronic bandstructures known,and take a semiclassical approach

Electrostatics modeled via Poisssons equation:

Charge conservation is required:

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( )D AV p n N N+

= +

1

1

J

J

n n

p p

nU

t q

pU

t q

= +

= +

S. Selberherr: "Analysis and Simulation of

Semiconductor Devices, Springer, 1984.

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Electronic Transport Theory

Both p-type and n-type currents given by asum of two terms:

Drift term, derived from Ohms law

Diffusion term, derived from Second Law ofThermodynamics

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( ) ( )

( ) ( )

n n n

p p p

dnJ qn x E x qD

dx

dnJ qp x E x qD

dx

= +

=

S. Selberherr: "Analysis and Simulation of

Semiconductor Devices, Springer, 1984.

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FEM Electronic Transport Model

Much like in earlier work, will employ thefollowing 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:

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i (x,y,z,t)

p(x,y,z,t)

n (x,y,z,t)

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FEM Electronic Transport Model

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Regarding the grid set-up, there are several points that need to bemade:

In critical device regions, where the charge density variesvery rapidly, the mesh spacing has to be smaller than theextrinsic 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 pointsto achieve faster convergence

A regular grid (with small mesh aspect ratios) is needed forfaster convergence

2

maxeN

TkL B

D

=

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

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Poisson Solver

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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

dxE E

n n n Vk T

E Ep n n V

k T

= +

= =

= =

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

7/29/2019 2013.02.22-ECE595E-L19

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Poisson Solver

Perturbing potential by yields:

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( )

( )

( ) ( )

( )

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

en e e

= + +

+ +

+ = +

+

=

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

7/29/2019 2013.02.22-ECE595E-L19

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Poisson Solver

Renormalized form

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( ) ( )

( ) ( ) ( )

2

2

2

2

new old

new old

d

p n C p ndx

dp n p n C p n

dx

= + + +

+ = + +

=

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

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Poisson Solver

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Initialize parameters:

-Mesh size

-Discretizationcoefficients-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 maximumabsolute error update

Equilibrium solver

> tolerance

< tolerance

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

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Current Discretization

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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 neighboringgrid nodes. Since solutions are available only on the gridnodes, interpolation schemes are needed to determine thesolutions.

There are two schemes that one can use:

(a)Linearized scheme: V, n, p, and D vary linearlybetween neighboring mesh points

(b) Scharfetter-Gummel scheme: electron and holedensities follow exponential variation between meshpoints

peDExepxJneDExenxJ

ppp

nnn=

+=)()(

)()(

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

7/29/2019 2013.02.22-ECE595E-L19

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Nave Linearization Scheme

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Within the linearized scheme, one has that

This scheme can lead to substantial errors in regions ofhigh electric fields and highly doped devices.

2/12/11

2/12/12/1 +++

+++ +

= iii

iiiii neD

a

VVenJ

2

1 ii nn ++i

ii

a

nn +1

+

+

=

+++

+++++

ii

iiiii

i

i

i

iiiii

a

eD

a

VVen

a

eD

a

VVenJ

2/112/1

2/112/112/1

2

2

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

7/29/2019 2013.02.22-ECE595E-L19

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Scharfetter-Gummel Scheme

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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

11

)(and)(++

==iiii

nVnnVn

[ ]

=

=+=

+++

++

++

Vt

VVBn

Vt

VVBn

a

eDJ

e

eVgVgnVgnVn

iii

iii

i

ii

VtVV

VtVV

iiii

i

111

2/12/1

/)(

/)(

1

1

1)(),()(1)(

1

1)(

=

xe

xxB is the Bernouli function

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes, Arizona State

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ADEPT 2.0

Available on nanoHUB from Prof. Grays team:https://nanohub.org/tools/adeptnpt

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https://nanohub.org/tools/adeptnpthttps://nanohub.org/tools/adeptnpt

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ADEPT 2.0

Can customize all the calculation details:

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ADEPT 2.0

Outputs include electrostatic (Poisson) solution:

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ADEPT 2.0

Energy band diagram

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ADEPT 2.0

Carrier concentrations:

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ADEPT 2.0

And finally, realistic I-V curves:

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Next Class

Is on Monday, Feb. 25

Next time, we will cover electronicbandstructures

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