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THREE DIMENSIONAL SIMULATION STUDY OF FULLY DEPLETED SILICON ON
INSULATOR MOSFET (SOI MOSFET) BY SEPARATION OF VARIABLE METHOD
Neha Goel1, Manoj Kumar Pandey2,
Bhawna agarwal3
1
(Research Scholar,SRM University NCR Campus Ghaziabad, India,[email protected])
2
(Department of ECE , SRM University NCR Campus Ghaziabad,India,
[email protected])
3
(Assistant prof. Birla Instt of Tech. Mesra. Ext(jaipur)
[email protected])
ABSTRACT
A three dimensional fully depleted Silicon-On-Insulator (SOI MOSFET) device is developed,
based on the numerical solution of three dimensional Poisson’s equation, is presented in this
paper. Separation of variable method is used to solve the three dimensional Poisson’s equation
analytically with necessary boundary conditions. In this Paper, Variation of Front Surface
potential wrt channel Length, channel Width are presented. In Addition to that threshold voltage
, Electric Field and Mobility Profiles wrt channel length and channel Width are also presented,
Keywords:
Silicon on insulator (SOI), Poisson’s Equation with boundary conditions, Front surface potential,
Threshold voltage. Mobility, Gate Width, Gate Length, Gate Oxide Thickness.
INTRODUCTION
Gain in integrability and speed are the main reason for the continuous improvements through
scaling in metal oxide semiconductor field effect transistors (MOSFETs), However Bulk CMOS
will remain as the main technology for submicron gate ULSI systems. Thin film fully depleted
silicon on insulators (SOI) MOSFETs have superior electrical performances due to better control
of short channel effects, excellent Latchup immunity, improved isolation & reduced parasitic
capacitances compared to bulk silicon technology[1].
In this paper, an analytical three dimensional model for small geometry SOI MOSFET is
presented by solving 3D poissons equation. The model is used to predict the subthreshold
behavior of small geometry MOSFET as the degradation of threshold voltage and the increase of
sub-threshold swing, are the dominant small geometry effects limiting the scaling of channel
dimensions and to give insights into device design and their scaling limits. In this paper, 3D
poissons equation is solved by separation of variables method.
SILICON ON INSULATOR (SOI) MOSFET
There are various characteristics of SOI MOSFET due to which it would be beneficial to switch
to SOI MOSFET technology. The main advantages of SOI technology are the following.
 Due to insulation layer, there is no parasitic bipolar devices, As a result, latch up can be
totally eliminated.
 SOI MOSFETs are having higher radiation tolerance.
 Due to the insulation layer above the substrate, these devices have smaller leakage
current.
 These devices have high speed of operation due to the lower capacitance between device
and substrate.
 Power dissipation of SOI MOSFET is small, because operated at lower voltages and
current levels [2].
There is a need to develop numerical device models for SOI-based MOSFETs as these are
suitable for circuit simulation. The analytical modeling of the threshold voltage of FD SOI
MOSFETs has already been reported by numerous authors [3]-[5].
Small device structures are difficult to described by one-dimensional or even twodimensional (2D) models, that’s why 3D numerical device models are developed for study of
the accurate electrical characterization of small geometry devices [7] – [8].
Fig; 1 Cross sectional view of SOI MOSFET along channel Length
Now, In order to analyze the structure shown in Fig. 1, we need to solve both Poisson’s equation
and current continuity equation. In this paper, we consider a fully depleted (FD) SOI film. The 3D Poisson’s equation in the FD SOI film region is given by,
d
2
2
dx
 ( x y  z) 
d
2
dy
2
 ( x y  z) 
d
2
2
dz
 ( x y  z) 
q  Na
si
-------------------------------------------------(1)
where NA is the doping concentration and ψ(x,y,z) is the potential at a particular point (x,y,z) in
the SOI film.
The 3-D Poisson’s Equation is numerically solved by using Separation of Variable method. The
boundary conditions used for the solution of Poisson’s equation are applicable only for bulk
MOSFETs. David Esseni [9] described a mobility model also for SOI MOSFET using solution
of 1D Poisson’s equation.
POISSON’S EQUATION WITH BOUNDARY CONDITIONS
Fig. 1 illustrates a three-dimensional view of a typical MOSFET structure with corresponding
device dimensions. The source–SOI film and drain–SOI film junctions are located at y=0 and
y=Leff, respectively,where, Leff is the effective channel length. The front and back Si–SiO
interfaces are located at x=0 and x=ts, where ts is the SOI film thickness. toxf and toxb are the front
and the back gate oxide thicknesses, respectively, where the applied potential to the front and
back gates are Vgf and Vgb.The vertical and the lateral directions are defined as x and y,
respectively, while the direction along the width of the transistor is defined as z. The sidewall Si–
SiO interfaces are located at z=0 and z=W.
In general, in order to analyze this structure, we need to solve the Poisson’s equation. In this
paper, we consider a fully depleted (FD) SOI film. The Separation of variables technique is used
to solve the 3D poisson’s equation analytically with appropriate boundary conditions. The 3-D
Poisson’s equation in the FD SOI film region is given by:
d
2
2
d
 ( x y  z) 
dx
2
dy
2
 ( x y  z) 
d
2
2
 ( x y  z) 
dz
q  Na
si
------------------------------------------------------------------(1)
In order to solve equation (1), it is separated into 1D Poisson’s equation, 2-&3-D Laplace equation as :
d
2
2
l( x) 
q  Na( x)
dx
d
2
2
 ( x y ) 
dx
d
2
2
si
d
2
dy
v ( x y  z) 
dx
----------------------------------------------------------------------------------(2)
2
s ( x y )  0
d
-----------------------------------------------------------------------------------(3)
2
dy
2
d
v ( x y  z) 
2
2
v ( x y  z)  0
dz
----------------------------------------------------------------(4)
Where, Ψi=Ψl(x)+Ψs(x,y)+Ψv(x,y,z) ---------------------------------------------------------------------------------------(A)
A: Solution of Ψl(x)
li  sb  Esb ts  x 



i
q

 Na ts  x
2si
2
i
---------------------------------------------------------------(5)
B: Solution of Ψs(x,y)
1
   si 

 
 
 sinh   Leff   Vs  sinh   y  Vr sinh  Leff j  y  sin   x  ox toxf    cos   x 
j 

--------------(6)
s ( j)  
Vs  Vr 


si
1

Vds    1  cos  ts   
 toxf sin  ts 
ox


iDnum
------------------------------------------------------(7)
inum1 
Vr 
si
ox
 toxf    inum2
iDnum
1
iDnum 
4 
  2ts  sin 2 ts   
q Na
inum1 
si
-------------------------------------------------------------(8)
3
2
 si  toxf     1   2ts  sin 2ts    si toxf   1  cos  2 ts  


2 ox
 ox
 4 
  1  cos  ts    Esb
sin ts 

  Vbi  sb  Esb ts 
q

2
 Na ts
2si
2 1
   Vbi  sb 

-------(9)
cos  ts 

-----(10)
q Na
inum2 
si
3
 sin ts   Esb cos
 ts    Esb  q Na ts   1   Vbi  sb  sin ts 

 2  
si  2


 
------------------(11)
C: Solution of Ψv(x,y,z)
v  Psr sinh sr ( W  Z)  sinh  sr Z 
sin s  ( y  Leff )
cos  s  Leff 


  sin  r x 
si
ox

------------(12)
 toxf  r cos  r x 
Nsr
Psr 
1  coshsr W
-----------------------------------------------------------------------------(13)

iDnum 
inum
Nsr 
 1   cosh  sr W   
sinh  sr W   2
inum 
si
ox
cos  s  Leff   1
si
ox
 toxw sr cosh  sr W  

s  cos  s  Leff  
  i1 
si
ox


 toxf  r i2 


 toxw sr sinh  sr W 
2
 si  toxw sr   sinh sr W 


 ox

----------------------------(14)
1

i3
cos  s  Leff  sinh r Leff 
 ( Vs  i4  Vr i5)
-------------------(15)
2
 1

1
sin 2 s  Leff  
 si  toxf  r   1   2 r ts  sin 2 r ts    si toxf   1  cos  2 r ts    
  2 r ts  sin 2 r ts    
 Leff 



2 ox
2 s
 4 r
 ox
 4 r
  2 cos 2  s  Leff 

iDnum  
--------------------------(16)
i1 
q Na
3
si r
  1  cos  r ts    Esb
sin r ts 
2
r
  Vgf  Vfb  sb  Esb ts 

q
2 si
 Na ts
2 1
   Vgf  Vfb  sb 
 r
cos  r ts 
r
i2 
q Na
3
 sin r ts   Esb
si r
i3 
1
4 r
  2 r ts  sin 2 r ts   
r

1
r

s
2
q Na ts  1
si
  Vgf  Vfb  sb 

 r2
sin r ts 
r
2
 si  toxf  r   1   2 r ts  sin 2 r ts    si toxf   1  cos  2 r ts  


2 ox
 ox
 4 r
 sinh  r Leff 
r
i4 
i5 
2
  Esb 
r
sin  s  Leff 
s
cos  r ts 
 s 
 
 r 
2
 cosh  s  Leff   sinh   r Leff  
1
sin  s  Leff   cosh  r Leff 
 s 
 
 r 
r
2
Main Equation of Surface Potential (Ψi) can be calculated by putting values of Ψl,Ψs and Ψv in Equation
A.
RESULTS
Variation of Surface Potential, Threshold Voltage, Electric Field and Mobility with respect to Channel
Length and width can be seen as below.
SURFACE POTENTIAL
The basic 3D Poisson’s equation (1) is solved using Separation of Variable Method to determine the
surface potential for fixed value of gate voltage and assumed value of the drain voltage.
The variation of front surface potential at the front Si-SiO2 interface (i.e., x=0) of a uniformly doped SOI
MOSFET for different values of channel length is shown in fig.2a.
In this figure, we determine the Variation of front surface potential for n-channel SOI MOSFETs along
the different values of channel length at the front Si-SiO2 interface and z=w/2. The values we have taken
here are: toxf=3nm ,ts=70nm, toxb=400nm, NA=1×1017/cm3 at Vgf=Vgb=0 & Vds=1.5V
front surface potential
1.4
1.2
i
1
0.8
0.6
0
2 10
8
4 10
6 10
8
8
8 10
8
yi
position along channel length (y/Leff)
Fig. 2(a)
Fig.2a. Variation of the front Surface Potential along different values of the channel length at the front
Si-SiO2 interface
From this graph we found that the front surface potential decreases linearly near the source end
and increases linearly near drain end. This is due the fact that the high electric field near the
drain causes the conductivity rapidly. Due to this it is expected that the electric field near the
drain end reaches the critical field for high drain voltage and hence causes the velocity
saturation.
Similarly, the variation of front surface potential at the front Si-SiO2 interface of a uniformly doped SOI
MOSFET for different values of channel width is shown in fig.2b.
front surface potential
In this figure, we determine the Variation of front surface potential for n-channel SOI MOSFETs along
the different values of channel width at the front Si-SiO2 interface and z=w/2. The values we have taken
here are: for toxf=3nm,ts=70nm,toxb=400nm,NA=1×1017/cm3 at Vgf=Vgb=0 & Vds=50mV
0.2
i
0.1
0
0
2 10
7
4 10
7
6 10
7
8 10
7
1 10
6
Zi
position along the channel width (z/W)
Fig.2(b)
Fig.2b. Variation of the front Surface Potential along different values of the channel width at the front
Si-SiO2 interface
THRESHOLD VOLTAGE OF A SMALL GEOMETRY FDSOI MOSFET
The threshold voltage of the short channel MOSFET[5,11] is defined as the gate voltage at which
the minimum surface potential in the channel is the same as the channel potential at threshold for
a long channel device, i.e., at threshold.
The variation of threshold voltage with respect to Channel length and Channel Width is shown in
Fig 3(a),3(b) resp,
Fig 3(a). shows the variation of threshold voltage for n-channel SOI MOSFETs with different
values of channel length having these values: Leff=1µm,toxf=10nm,toxb=400nm,toxw=15nm,
for Vds=50mV and Vgb=0V.
Fig 3(b). shows the variation of threshold voltage for n-channel SOI MOSFETs with different
values of channel width having these values: W=1.2µm,toxf=10nm,toxb=400nm,toxw=15nm,
for Vds= 50 mV and Vgb=0V .
VTF of a small geometry SOI MOSFET is defined as:
VTF=Vgf when Ψ(0,y,W/2) =2φb --------------------(B)
Now, putting (B) in main Equation of surface potential, and using (5), (6)–(15), we have
VTF  VTFO  VTF1  VTFW
i
i ---------------------------------------------------------------------------------------------(17)
Where,
VTFO  Vfb   1 

Cs
q Na ts
  2b  Coxf sb  2Coxf
Coxf 


 Vs  sinh    y  Vr sinh   ( Leff  y)
Coxf sinh    Leff 
 sr W i  sins  ( y  Leff )
2si

VTFW 
 Psr sinh 
 r
i
Coxf
 2  cos  s  Leff 
VTF1 
si
Cs  Cit 
front threshold voltage,VTF
0.11176
0.11174
0.11172
VTFi
0.1117
0.11168
0.11166
0
2 10
8
4 10
6 10
8
8
8 10
8
yi
channel Length
Fig. 3(a)
Fig.3(a) Variation of threshold voltage for n-channel SOI MOSFETs with respect to channel
length.
Here Fig 3(a) shows that Threshold Voltage Increases linearly near the source end and Decreases
linearly near drain end.
front channel threshold Voltage(VTF)
0.5
0
VTFi 0.5
1
1.5
0
5 10
7
1 10
6
1.5 10
6
2 10
6
Wi
Channel Width
Fig.3(b)
Fig.3(b). Variation of threshold voltage for n-channel SOI MOSFETs with respect to channel
length.
ELECTRIC FIELD:
The electric field distribution along the channel length and width is also obtained and it is shown
in figures 4(a),4(b). The electric field along the length of the channel (Ey) is dominant over the
electric field along channel width (Ez)
The electric field increases rapidly near the drain end. This is due to the fact that the carrier
density near the drain end experiences a rapid decrease in surface concentration which calls for a
rapid increase in the electric field to maintain the constant drain current.
.
7 10
Electric Field,E
6
6 10
6
Ey i 5 106
4 10
6
3 10
6
0
2 10
8
4 10
6 10
8
8
8 10
8
yi
channel Length
Fig. 4(a)
Fig.4(a).Variation of Electric Field along the length of channel
1 10
Electric Field,E
5
Ez i 5 10
4
0
0
2 10
7
4 10
7
6 10
7
8 10
7
1 10
6
Zi
Channel Width
Fig. 4(b)
Fig.4(b).Variation of Electric Field along the width of channel
MOBILITY:
The value of mobility (velocity per unit electric field) is influenced by several factors. The
mechanisms of conduction through the valence and conduction bands are different, and so the
mobility associated with electrons and holes are different. As the density of dopants increases,
more scattering occurs during conduction. Mobility therefore decreases as doping increases. At
low temperatures, electrons and holes gain more energy than the lattice with increasing T,
therefore mobility increases. At high temperatures, lattice scattering dominates, and thus
mobility falls.
We can find out the mobility of charge carriers from the electric field values. The mobility along
channel length & channel width is given by,
y  Ey si ox
--------------------------------------------------------------------------(18)
&
z  Ez si ox
--------------------------------------------------------------------------(19)
Where, Ey=Electric field along channel length
Ez=Electric field along channel width
εsi=silicon permittivity
εox=oxide permitivity
The mobility variation along the channel length & channel width is shown in Fig 5(a) & 5(b)
resp.
Mobility
Mobility is directly proportional to the electric field. So the shape of the mobility curve along the
channel length and width is same as that of the electric field curve along channel length and
width resp.
6 10
17
5 10
17
4 10
17
3 10
17
y i
0
2 10
8
4 10
6 10
8
8
yi
channel Length
Fig. 5(a)
8 10
8
Fig. 5(a) Variation of mobility with respect to length of channel
Mobility
510
19
zi
0
0
210
7
410
7
610
7
810
7
Zi
Channel Width
Fig. 5(b)
110
6
Fig. 5(b) Variation of mobility with respect to width of channel
Conclusion:
A threshold voltage model for fully depleted (FD) SOI MOSFET based on numerical solution of
3-D Poisson’s equation is presented by using Separation of variables method. The Study of
Surface potential distribution, Threshold Voltage, electric Field and Mobility is done with
respect to channel length and width in this paper.
References:
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the 0.1-0.025pm MOSFET,” in I993 Symp. VLSI Technol. Dig. Tech. Papers, 1993, pp. 33--34.
[2] Guruprasad Katti , Nandita DasGupta , Amitava DasGupta , “Threshold Voltage Model for Mesa –
Isolated Small Geometry Fully Depleted SOI MOSFETs Based onAnalytical Solution of three
Dimensional Poisson’s Equation”, IEEE Transactions On Electron Devices, Vol. 51, No. 7, July 2004.
[3] Jason C. S. Woo, Kyle W. Terrill, Prahalad K. Vasudev, “ Two – Dimensional Analytic Modeling of
Very Thin SO1 MOSFET’s ”, IEEE Transactions on Electron Devices , Vol. 37. No.9. September 1990.
[4] Hans van Meer, Kristin De Meyer , “A 2-D Analytical Threshold Voltage Model for Fully-Depleted
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