A Modified EKV Model for Circuit Simulation
Abstract- A revised model for the current-voltage (I-V) characteristics based on the current EKV
model is presented. A EKV model uses a linear combination of two logarithmic functions to
interpolate between two regions of operation to generate a single-piece model which allows
continuous derivatives with respect to the external bias voltages. This model is rather convenient
for circuit simulation for its simplicity and continuous nature. However, the original EKV model
does not have the required precision compared to the simplified, symmetrical two-piece model,
which is commonly used in SPICE. Thus, a modified EKV model has been developed to obtain
the precision required as well as preserving the original advantages of such a model.
I.
INTRODUCTION
To simulate MOS transistors in a circuit, we need a model which can generate results in a
reasonable amount of time while the accuracy is maximized. Although the complete chargesheet model can produce the best accuracy, it is rather computational intensive attribute to the
fact that it contains a high-order of polynomial as well as implicit expressions for the surface
potentials. And, to be sure, the latter requires iterative techniques to be evaluated, which is
completely unsuitable for circuit simulation.
Till now, one of the most popular SPICE MOSFET Level-3 model is the Simplified
Symmetric Model [1] – [4]. This model as named suggested is simple and reasonably accurate
within a certain range of operation, namely, the weak and strong inversion. Yet, this model
completely ignores the moderate inversion have we not stretched the boundary between strong
and moderate inversions. Because of the nature of this model, a discontinuity occurs when we
go from the weak inversion into the strong inversion. In addition, even inside the strong
inversion, there are two equations required to describe saturation and non-saturation regions.
Hence, in this regard, this multi-piece model does not really simplify the matter for it
complicates the derivatives of the current with respect to the external biases.
A EKV model solves the problem. The initiative of an EKV model is not to improve the
accuracy but to be able to collapse the simple model discussed above into a single-piece which
has a continuous derivative with each of the externally applied bias voltage, and most important
of all, it describes the moderate inversion. To do so, it interpolates between the strong and weak
inversion to approximate what is happening in the complete charge-sheet model. Hence, it is
rather a mathematical effort as opposed to physics to arrive such a model. Nevertheless, the
present EKV model, though simple, is biased either toward the strong inversion, depending on
some parameters. But if we can fix this problem by mathematical means, an EKV model is
reasonably suitable for circuit simulation if the precision expectation does not exceed that from
the symmetrical model.
In the following section, the revised EKV model is presented, and compared to the
complete charge-sheet model as well as the simplified symmetrical model. In section III, the
results and precision issues from the modified EKV model will be discussed.
II.
The Model
The analytical model presented in this paper is based on the assumptions used to derive the
simplified symmetric model. This includes the assumptions: first of all, a gradual channel
approximation is considered. Second, once velocity saturation occurs near the drain end of the
channel, further increase in the drain current is only due to channel length modulation. Next, an
uniform substrate doping concentration. Last, the negligence of the gate current in order to make
the model simple.
Ids EKV
2
VgbV X  n `Vsb
VgbV X  n `Vdb 2 



 
W

 Coxt 2  2( n`  )  ln( 1  e 2 n`t )  ln( 1  e 2 n`t  
L
 
 



Equation 1
where
Cox
W
L
t
n`
n` 1 
where
Oxide Capacitance per unit area
Width of the channel
Length of the channel
thermal voltage
a slope function given by

2  0  V `P
 0  2f  (3t )(tanh( Vgb  VHB )  1)

2

 Vgb  Vfb ) 2   0
2
4
V X  Vfb   0    0
1
  (tanh( Vgb  V MB )  1)
2
V `P  (
where
0
f
VHB
VMB
V`P
VX


Vfb
[1.1]
[1.2]
[1.3]
[1.4]
[1.5]
an interpolating surface potential function between 2f and 2f  6t
intrinsic fermi potential of the substrate
the boundary between the strong and moderate inversion in Vgb
the boundary between the moderate and weak inversion in Vgb
the pinch-off voltage function at surface potential  0
an analogous function to VT0, but with a dynamic surface potential
a dummy function interpolating between 2 to -2
channel length modulation coefficient
flat-band voltage
When the value of Vgb is small, i.e. in the weak inversion region, equation 1 can be easily
reduced to the simplified symmetric model in the weak inversion given by:
Ids sw
VgbVT 0  nVsb

  VgbV2Tn0t nVdb  
W


2
2 nt
 Coxt  ( n  1)  e
  e

L

 
 


Equation 2
where
 0  2f
[2.1]
On the other hand, when the value of Vgb is large, i.e. in the strong inversion region, equation 1
corresponds to the simplified symmetric model in the strong inversion given by:
Idsn s 
W
n


Cox Vgb  VT 0 Vdb  Vsb  (Vdb 2  Vsb 2 )
L
2


Equation 3
where
 0  2f  6t
[3.1]
Hence, between the strong and weak inversion, we can see that the EKV model must provide an
interpolation solution to the moderate inversion for equation 1 is continuous both in itself as well
as its derivative correspondent.
In the original EKV model, equations [1.1] – [1.5] are simplified reduced to their correspondent
expressions when the surface potential is either in the strong [3.1] or weak inversion [2.1]. In
addition, beta function is reduced to 0 at the strong inversion and to –1 for the weak inversion.
The technique used here is the hyperbolic function of tangent which gives the following
characteristics:
Y(x) = tanh(x)
1
Y( x)
1
0
1
1
10
10
5
0
x
5
10
10
By choosing different shifting for x and y, we can interpolate between any two points with a
reasonable sharp edge, i.e. approximate a step function. Thus, the interpolating functions from
[1.1] to [1.5] are used in order to produce the correct simplification when IdsEKV is in either the
strong or weak inversion.
III.
Result and Discussion
In the following, several aspects of modified EKV model is presented against either the complete
charge-sheet model or the simplified symmetric model or both. More specifically, the discussion
is divided into three parts: 1) I-V curve 2) derivative of I vs. V 3) accuracy evaluation. Each of
which contains characteristic plots against all external bias voltages, Vgb, Vdb, and Vsb. The
purpose of these plots is to demonstrate the accuracy of the modified EKV model.
1) I-V Characteristics
a) Ids vs. Vgb
i) Strong Inversion
6.260302  10
4
1 10
3
Ids ( Vgb vdb  vsb  1.5  1.5 )
1 10
Ids EKV( Vgb vdb  vsb )
3.085611  10
5
1 10
4
5
1.2
1.382219
1.4
ii) Moderate Inversion
1.6
1.8
2
2.2
Vgb
2.4
2.6
2.8
3
2.999219
3.267234  10
5
1 10
1 10
4
5
Ids ( Vgb vdb  vsb  1.2  1.2 )
1 10
Ids EKV ( Vgb vdb  vsb )
1 10
5.351348  10
8
1 10
6
7
8
0.7
0.782219
0.8
0.9
1
1.1
1.2
1.3
Vgb
1.4
1.382219
iii) Weak Inversion
0.0000001
1 10
1 10
1 10
1 10
7
8
9
10
Ids ( Vgb vdb  vsb  0.5  0.5 )
1 10
Ids EKV ( Vgb vdb  vsb )
1 10
1 10
1 10
2.910421  10
15
1 10
11
12
13
14
15
0.2
0.215821
b) Ids vs. Vdb
c) Ids vs. Vsb
0.3
0.4
0.5
Vgb
0.6
0.7
0.8
0.782211
6.265204  10
4
7 10
6 10
5 10
4 10
Ids ( vgb Vdb  vsb  3  3 )
4
4
4
4
Ids EKV( vgb Vdb  vsb )
3 10
2 10
1 10
4
4
4
0
0
0
0.5
1
1.5
0.3
4
6.265204  10
7 10
6 10
5 10
4 10
2
2.5
Vdb
3
3
4
4
4
4
Ids ( vgb vdb  Vsb  3  3 )
Ids EKV( vgb vdb  Vsb )
3 10
2 10
1 10
4
4
4
0
4.295202  10
20
1 10
4
0
0.3
2) Derivative of I vs. V
a) I`ds vs. Vgb
i) Strong Inversion
ii) Moderate Inversion
iii) Weak Inversion
b) I`ds vs. Vdb
c) I`ds vs. Vsb
0.5
1
1.5
Vsb
2
2.5
3
3
3) Accuracy Evaluation
4) Error of Modified EKV Model, compared to the complete charge-sheet model
a) Error of Symmetric Model, compared to the complete charge-sheet model