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Solution of the

Poisson/Laplace Equation

Dragica Vasileska

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Electromagnetic Fields

Time Varying Fields

Jt

DH

t

BE

B

D

0

SSC

SC

S

VS

sdJsdDt

ldH

sdBt

ldE

sdB

dvsdD

0

Maxwells Eq. in Diff. Form Maxwells Eq. in Integral Form

EJHBED

,,

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Poissons Equation

We have the differential form of Gauss law

Using D = E and E = - in the above equation, we

get

This is the Poissons Equation

D

V

2D = V =-E V

2V

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Poisson/Laplace Equation Solution

Poisson/Laplace Equation

No knowledge of solving of PDEs

Method of images

With knowledge for solving of PDEs

Theoretical Approaches

Numerical Methods:

finite difference

finite elementsPoisson

Greens function method

Laplace

Method of separation of variables(Fourier analysis)

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Numerical Solutions Procedures

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

ADI

SOR

Gauss-Seidel

Krylov

Multigrid

DAE

Ui(x,y,z,t)

p(x,y,z,t)

T(x,y,z,t)

or

(,,, )

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Poisson Equation Application

Electromagnetic Problems

Electronics Devices

Biological Systems

.

.

.

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Electronics Devices

Understand the operation of different semiconductor devices:

- Diodes

- BJTs

- MOSFETs

- MESFETs

- HEMTs

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Physical Device Simulation

There are two main components inany physical device simulator:

- Characterization of chargemotion due to driving forces anddiffusion process (transport)

- Fields due to charge distribution

and motion

Recessed MOSFET represented on 3D mesh

over finite domain (courtesy of S. M. Goodnick)

Initialize Data

Field Solver

Transport Kernel

yes

no Criterionsatisfied?

START

STOP

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Biological Ion Channels

Proteins that form nanoscopic aqueous tunnels in cell membrane

Enormous range of Biological functions

- regulate ion flow and composition inside cell

- control electrical signaling in the nervous system

- muscle contraction, drug delivery

Disease malfunctioning channels

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Channels are naturally occurring device elements

Nanoscale devices with ~0.2-15nm in diameter and between 0.3-10nm

length

self-assembled

perfectly reproducible

have many specific built-in features and functions can be mutated

Channels can be designed with specific conductances, selectivities and

functions

So, why are we interested in ion channels ??

http:// www.rcsb.org/pdb

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nanoHUB.orgonline simulations and more

Network for Computational Nanotechnology

A pn-diode has NA=ND=1016 cm-3 doping and a length of 1 um of both p and n-

regions. Calculate analytically and verify via simulations the values of the:(a) Built-in voltage Vbi(b) Total depletion region width W.

(c) Maximum electrical field at the metallurgical junction.

Analytically, the built-in voltage

is calculated using:

2ln 0.69A Dbi T

i

N NV V V

n

One can also extract thebuilt-in voltage from the energy

band diagram using:

1

bi Cp CnV E Eq

Vbi

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nanoHUB.orgonline simulations and more

Network for Computational Nanotechnology

Example 1: Charge Density

W

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nanoHUB.orgonline simulations and more

Network for Computational Nanotechnology

Example 1: Electric Field Profile

Emax