Analytical solution of 2d poisson’s equation using separation of variable method for fdsoi mosfet

International Journal of Electronics and Communication Engineering & Technology (IJECET),
INTERNATIONAL JOURNAL OF ELECTRONICS AND
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Special Issue (November, 2013), © IAEME

COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)

ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Special Issue (November, 2013), pp. 150-154
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Analytical Solution of 2d Poisson’s Equation Using Separation of
Variable Method for FDSOI MOSFET
Prashant Mani1, Manoj Kumar Pandey2
Research Scholar1, Director2
SRM University, NCR Campus Ghaziabad, India
1prashantmani29@gmail.com, 2mkspandey@gmail.com

ABSTRACT: A two–dimensional (2-D) analytical model for the surface potential variation along
the channel in fully depleted silicon-on-insulator MOSFETs is developed .Our Approach to solve
poisson’s equation using suitable boundary conditions, results high accuracy to calculate the
potential of the channel as compare to various approaches. A Simple and accurate analytical
expression for surface potential in channel are derived. The proposed model will help to
reduce Short Channel Effect, Drain Induced Barrier Lowering in Fully Depleted SOI MOSFETs
etc

KEYWORDS: SOI MOSFET, Poisson Equation, 2D Solution
I.

INTRODUCTION

FULLY DEPLETED single-gate SOI MOSFETs are expected to become next-generation devices
due to their superior short-channel immunity and ideal subthreshold characteristics. There
have been many reports on their modeling of devices [1]–[11]. The assumption of constant
surface potential used in the charge sharing models is invalid for submicrometer channel
lengths. The solution of 2-D Poisson’s equation has been obtained using various approaches.
Pseudo-2-D solutions of Poisson’s equation [12], [13] and a quasi-2-D technique [14], [15] have
been reported in literature.
The solution of 2-D Poisson’s equation by power series approach has also been obtained
[16].However, the solution of 2-D Poisson’s equation by power series approach is obtained by
neglecting the higher order terms and, hence, it is not as accurate as the other approaches.
Analytical solution of 2-D Poisson’s equation by means of Green’s function technique [17] is
another method to solve 2-DPoisson’s equation. This model solves 2-D Poisson’s equation
using the separation of variables method. However, the boundary conditions used for the
solution of Poisson’s equation are applicable only for bulk MOSFETs. This model solve the 2 D
POISSON’S equation by using separation of variable method .First split the 2D Poisson’s
equation in to one dimensional poisson’s equation and 2D laplace equation .
International Conference on Communication Systems (ICCS-2013)
B K Birla Institute of Engineering & Technology (BKBIET), Pilani, India

October 18-20, 2013
Page 150


II.

SOLUTION OF 2D POISSON’S EQUATION

Fig. 1: Cross-sectional View

MOSFET

The cross-sectional view of an n-channel SOIMOSFET along the channel length is shown in Fig.
1. The source–SOI film and drain–SOI film junctions are located at y=0 and y= Leff, where Leff
is the effective channel length .The front and back interfaces are located at Si-SiO2 at x=0 and
x= ts, where ts is SOI film thickness. toxf and toxb are the thickness of frontgate an backgate
oxide thickness Vgf and Vgb are the applied potential.
In this paper, we consider a fully depleted (FD) SOI film. The Poisson’s equation inthe FD SOI
film region is given by
(

)

(

)

( )

where NA is the doping concentration and is the potential at a particular point (x,y) in the SOI
film. The boundary conditions required to solve the 2-D Poisson’s equation are
(

(

)

)

(2)
Where x=0

(

(

)

)

(3)
Where x=ts

(
(

)

(4)
)

Where y=0

(5)
Where y=Leff

The separation of variable method applied on equation (1), following step involve



Split the 2D Poisson’s Equation in two parts, first one is one dimensional poisson’s
equation and two dimensional laplace equation.
Now 2D Laplace equation has all variable in one side.


October 18-20, 2013
Page 151


( )

( )

(

)

Where

(

(

)

)

( )

( )

(

)

In equation (8) ( ) is the solution of equation (6) and
(7) with boundary conditions.

III.

(8)
(

) is the solution of equation

SOLUTION OF ΨL(X)

In equation
below

l is the solution of 1 D POISSON equation, using the boundary condition given

(9)

(10)
Now after solving the above boundary condition we find the solution in the form of

(11)

The following graph is plotted between l and xi. From the graph we find out the result as in
parabolic form. As the thickness of the silicon layer increases from 0 to 1.25*10-8 nm the
surface potential decreases, but after more increase in si thickness the surface potential also
increase.
1.62

1.61

1.6

1.59
li
1.58

1.57

1.56

1.55
0

4.17 10

9

8.33 10

9

1.25 10

8

1.67 10

8

2.08 10

8

2.5 10

8

xi

Fig. 2: Potential variation along the channel thickness vary from ts=0 to ts=2.5*10-8nm

October 18-20, 2013
Page 152


Where l is the potential of one dimensional poission equation and xi is the thickness of silicon
film.

IV.

SOLUTION OF ΨS (X,Y)

In equation (8) Ψs is the solution of 2 D Laplace equation, the boundary conditions required to
solve the Laplace equation are given below.
(

(

(

)

)

(

)

(

)
Where x=0

)

(

)

(

)

(

)

Where x=ts

(14)
(15)

Where y=0
Where y=Leff

(16)
The solution comes in the form of summation of sin and sin hyperbolic terms. Here the terms
can be explained as below .Where the Vr and V’r are defined as given below.
inum1 
Vr 

si
ox

 toxf   inum2

iDnum


Vds  1   1  cos   ts   si  toxf sin   ts 
 

ox
 

V’r Vr 
iDnum

179.8942

179.89415

s j

179.8941

179.89405

179.894
7
1 10

1.5 10

7

2 10

7

2.5 10

7

3 10

7

3.5 10

7

Leff j

Fig. 3: The Potential Variation along the Channel length. Leff =0 to Leff =3.5 10-7 nm
The graph shows as the effective channel length increase the potential decrease.

October 18-20, 2013
Page 153


V.

RESULT AND FUTURE WORK

SOI MOSFET based on an analytical solution of 2-D Poisson’s equation is presented. The effect
of change in thickness of silicon film can be analyzed by fig. 2 and fig. 3 shows the outcome of
potential as changes the effective channel length of proposed model. The 2D Poisson’s
equation is solved analytically using the separation of variables technique. The solution will
then extend to obtain the 3D Poisson’s equation of the small geometry SOI MOSFET.

REFERENCES
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[16] K.-W. Su and J. B. Kuo, “Analytical threshold voltage formula including narrow channel
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