The development of integration-based methods to extract parameters of two-terminal device models

Abstract

We present a historic overview of the initial motivating ideas, original foundations, and subsequent development, of integration-based methods which are currently used to extract semiconductor device model parameters, as well as to assess devices¿ and circuits¿ non- linearity. To illustrate these methods¿ capabilities, in this paper we review sample applications specifically focusing on two-terminal devices, such as non-ideal junctions, illuminated solar cells, and post-breakdown conduction through thin oxides. Additional applications of these integration-based extraction methods, pertaining to MOSFET models and harmonic distortion evaluation, are presented elsewhere in this conference.

Full text

Page 1

The Development of Integration-based Methods to Extract
Parameters of Two-Terminal Device Models

Francisco J. García-Sánchez, Adelmo Ortiz-Conde, Giovanni De Mercato
Solid State Electronics Laboratory, Simón Bolívar University, Apartado Postal 89000, Caracas 1080-A, Venezuela
Email: {fgarcia, ortizc}@ieee.org

Abstract

We present a historic overview of the initial motivating
ideas, original foundations, and subsequent development, of
integration-based methods which are currently used to extract
semiconductor device model parameters, as well as to assess
devices’ and circuits’ non- linearity. To illustrate these methods’
capabilities, in this paper we review sample applications
specifically focusing on two-terminal devices, such as
non-ideal junctions, illuminated
post-breakdown conduction through thin oxides. Additional
applications of these integration-based extraction methods,
pertaining to MOSFET models and harmonic distortion
evaluation, are presented elsewhere in this conference.

1. Origins
To the best of our knowledge, it was Araujo and
Sánchez who proposed for the first time, in 1982, an
integration-based method to extract device models’
parameters, specifically for the determination of parasitic
series resistance in solar cells [1]. Our own earliest
motivation for using an integration-based method dates
from 1993, and originated from the need to extract the
parameters of the basic but crucial model of an intrinsic
p-n junction with parasitic series resistance, given by the
modified single exponential Shockley’s equation:
() (
[
{
exp
−=
ss
RIVII
where IS is the reverse saturation current, V is the applied
voltage, Rs is the series resistance, Vth is the thermal
voltage kT/q, and n is the so called quality, or ideality,
factor of the junction.
The method we originally proposed [2-4] reduces the
problem of fitting the implicit exponential-linear
equation, to a much simpler problem of fitting an explicit
algebraic quadratic equation,
R
dVI
≈
∫
0
2
valid for I >> Is . The values of n and RS are directly
extracted from the coefficients of the quadratic explicit
equation of I in the rhs of (2), which is fitted to the
integrated experimental current. Then, the value of Is
may be readily calculated by substitution. An important
motivation for using integration-based methods, is that
integration of the experimental data has the inherent
advantage of acting as a low pass filter and, thus, they
have the potential to lessen the effect of measurement
errors on the extraction procedure.

2. The Integral Difference Function Concept
In 1995 and 1996 we proposed a different approach
[5,6] that does not start with the extraction of the
parasitic series resistance value. Instead, it does just the
opposite. The proposed method is based on calculating
solar cells, and
)} 1
−
th
Vn
]
, (1)
IVnI
th
S
V
+
2
, (2)
an auxiliary function, or rather an operator, whose
purpose is to eliminate the effect of the parasitic series
resistance, retaining only the intrinsic model parameters.
This new function was originally called “Integral
Difference Function,” it is denoted “Function D,” and is
defined as:
vi
∫∫
00
where D has units of “power.” The integrals with respect
to i and v are the device’s “Content” and “Co-content”,
respectively, as shown in Fig. 1. For simplicity’s sake
and without loss of generality, the lower limit of
integration in (3) was taken to be at the origin, but it may
equally be placed at any arbitrary point of interest along
the device’s characteristics. It was assumed that
Kirchoff’s Laws are satisfied along the path of
integration [7]. Notice that adding the Content and
Co-content, instead of subtracting them as in (3), yields
the device’s total power. From (3) we may further define
a “Difference Function Theorem” [5], that states that, for
any arbitrarily connected network of b linear and
nonlinear branch elements, where each port is taken as
one of the elements, the summation of the difference
function D, defined in (3), over all elements, is equal to
zero [5], that is,
⎛
=∑
∫
∑
==
k
k
The terms of (4) that correspond to linear branch
elements are in themselves equal to zero, since the
difference between Content and Co-content of a linear
element is zero. Conversely, only nonlinear branches
produce non-zero terms, and thus they are the only
elements that contribute to the total D.
iv divdvi ivdvidivivD
iv
∫∫
−=−=−≡
00
22),(
, (3)
0),(
11
=
⎟
⎠
⎞
⎜
⎝
−
∫
b
v
v
kk
i
i
kk
b
kk
kf
ki
kf
ki
dvi divivD
. (4)
v
0.00.20.40.60.8 1.0
i
0.0
0.2
0.4
0.6
0.8
1.0
Content
Co-content

Fig. 1. Content and Co-content of a nonlinear function.
978-1-4244-2186-2/08/$25. 00 ©2008 IEEE

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This property embodies the essence of the function
D’s ability to eliminate parasitic resistances (linear
elements) from device models. For the benefit of
formality, observe that the Difference Function Theorem
might be thought of as a counterpart to Tellegen’s
Theorem of Power Conservation [8], since changing the
“minus” sign in (4) to a “plus” sign, directly yields
Tellegen’s Theorem.
It is important to notice and stress that function D
may be understood as a representation or measure of the
device’s amount of nonlinearity, which for a linear
element is obviously equal to zero. This description of
function D, in terms of linearity, led us to refer to this
function as the “Integral Non Linearity Function” (INLF),
and to use it to quantify the non-linear behavior of
devices and circuits in terms of distortion.

3. A Family of Potential Functions
It was soon recognized that the Integral Difference
Function, or function D, is only one member of a more
general family of “Potential Functions” [9] defined by:
(
−=
p
pi v ivF),(
)∫
+
v
dvi
0
1
. (5)
Let i(v) be a continuous function that can be adequately
represented by a polynomial of an order m:
∑
=
n
0
Substituting (6) into (5) yields:
∑
=
n
n
0
The resulting Potential Function FP is an m+1 order
polynomial in which the term of order n=p is missing.
This means that in general the Potential Function
(
pp
vaF
Therefore, it is possible to eliminate any pth order term
from a polynomially expressed function by calculating
its pth Potential Function. Accordinly, the previously
defined Integral Difference Function D corresponds to
the Potential Function of first order, that is F (v,i).
Notice that, again for simplicity’s sake and without
loss of generality, we have taken the lower limit of
integration in (5) to be at the origin, although it may
equally be placed at any arbitrary point along the
device’s characteristics.

4. Eliminating the series resistance
Real p-n junctions are frequently modeled by
equation (1) which includes a constant parasitic series
resistance. The extraction of the model’s parameters IS,
Rs, and n has been traditionally performed by graphical
analysis of ln(I) versus V plots. However, the presence of
a relatively large Rs can significantly reduce the linear
portion of these plots to such a degree that the
calculation of the values of IS, and n from them turns out
to be unreliable. Methods such as Norde’s [10] and
others based on similar ideas [11] have been proposed to
deal with the presence of parasitic series resistance. They
all depend on first trying to find the value of Rs and then
extracting the other two parameters IS, and n.
=
m
n
nvavi)(
. (6)
+
+
−
=
m
n
np
va
p
1
n
F
1
. (7)
) 0
=
p
. (8)
1
Eliminating the effect of Rs is therefore very
desirable for the extraction procedure, and it is the
underlying concept of the integration-based method that
we introduced in 1995 [5,6].
Restricting the analysis, as it is usually done, to the
region of the measured forward characteristics where
I>>IS, the substitution of (1) into (3) yields [5,6]:
[
ln
≈
th
VnID
which does not contain Rs. Dividing this equation by the
current yields an auxiliary function G of the form:
{
ln
≈≡
th
VnIDG
This function G is calculated from the experimental data,
using (3) divided by the current. When plotted against
ln(I), the resulting curve is, according to (10), a straight
line, whose intercept and slope allow the immediate
extraction of the values of IS, and n, respectively.

5. Determination of double exponential diode model
parameters at very low forward voltage.
Different conduction mechanisms might be dominant
at various forward voltage ranges of a junction’s forward
I-V characteristics. Therefore, the experimentally
measured characteristics frequently exhibit a more
complex behavior than that which can be represented by
a single exponential expression with constant reverse
saturation current and ideality factor, as in (1). Thus, it is
usual to express the junction current by a summation of
two or more exponential expressions, each one of them
with values of reverse saturation current and ideality
factors that attempt to model the different conduction
mechanisms prevalent, such as thermionic emission,
diffusion of carriers,
tunneling, leakage, high injection, etc. For example, the
following expression represents
mechanisms:
()
[]
{}
1exp
11
+−=
ths
VnVII
which might correspond to carrier diffusion and to
generation-recombination in the space-charge region,
modeled by the values of n1=1 and n2=2, and Is1< Is2,
respectively.
Traditional diode model parameter extraction
methods, including those mentioned in previous sections,
assume that the “-1'' term in (1) or (11) may be neglected.
That is, they extract the diode’s model parameters from
values of the experimentally measured current
considerably larger than the reverse saturation current.
Consequently, those methods are restricted to values of
the junction forward voltage greater than a few thermal
voltages. Therefore, when a junction exhibits a
conduction mechanism dominant only at very low
voltage, its parameters n and IS, pertaining to that
mechanism, may not be extracted using conventional
methods. To circumvent this obstacle, we proposed in
1999 a technique that is able to extract the reverse
saturation current and ideality factor parameters of
semiconductor junctions from the low forward voltage
region of the device's characteristics [12].
Consider the integration of the forward current from
zero up to a maximum voltage value of interest.
Assuming a single exponential model with Rs=0 and
without neglecting the “-1'' term, the following two
auxiliary functions may be defined:
()
] 2
−
s
II
, (9)
( )
I
( )
I
[]
}
2 ln
+−
s
. (10)
generation-recombination,
two conduction
()
[
V
]
{} 1
−
exp
22ths
VnI
, (11)

Page 3

(
IVInVIdVIA
s th
V
−=≡∫0
) , (12)
and
()
sth
V
IVI nVVdVIB
−=≡∫0
, (13)
According to (12) a plot of A as a function of V/I
produces a straight line with slope IS and ordinates axis
intercept of nVth. Likewise, (13) indicates that a plot of B
as a function of I/V also produces a straight line with
slope nVth and ordinates axis intercept of IS. Therefore,
either one of these two equations may be used to esxtract
the parameters at low forward voltage.

6. Extraction under the presence of parasitic series and
shunt resistance.
Sometimes real junctions exhibit, not only parasitic
series resistance, but considerable parasitic shunt
conductance as well. This added phenomenon is most
commonly accounted for by including a parallel
resistance in the modified single exponential Shockley’s
equation model:
⎛
⎪
⎨
⎢⎢
⎣
⎠
⎝
Since auxiliary function G, defined in (10), may not be
applied directly to (14) because of the significant Rsh, we
proposed [13] to first calculate an estimated current ID (V)
flowing through the branch containing the ideal diode
junction:
VIVI
−=
)()(
where Rshe is an estimated guess value of Rsh. The
procedure now consists of applying the auxiliary
function G to the previously calculated ID (V) data.
Plotting G for different values of Rshe allows to select as
the correct value of Rsh that Rshe which produces the best
straight line. This might seem at first a crude procedure,
but it is not. The procedure is highly sensitive.
Figure 2 presents plots of auxiliary function G versus
ID, for several values of the estimated shunt resistance
Rshe, as calculated using synthetic I-V data. Notice that the
plot corresponding to Rshe =1MΩ is indeed a straight line,
at current values greater than the reverse saturation
current.
()
sh
th s
sh
s
s
R
V
nVIR
R
R
VII
+
⎟
⎠
⎟
⎞
⎜
⎝
⎜
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎧
⎥⎥
⎦
⎤
⎡
−
⎟⎟
⎞
⎜⎜
⎛
+=
11exp
. (14)
sheD
RV
, (15)
ID (A)
10-1210-1110-10
10-9
10-8
10-7
10-6
10-5
G (V)
0.0
0.2
0.4
0.6
0.8
Rshe = 1 MΩ
0.99 MΩ
0.9 MΩ
1.1 MΩ
1.01 MΩ

Fig. 2. Auxiliary function G versus the logarithm of the
estimated current ID, using
characteristics data, calculated with parameters IS =10pA,
n=1.5, Rs=1kΩ, and Rsh =1MΩ.
synthetic junction I-V
The figure also clearly demonstrates the high
sensitivity of the procedure to the estimated value of Rshe,
since it reveals that a difference of a mere 1% produces
plots of considerable deviation from straight line
behavior. This procedure may be automated to yield the
value of Rshe that produces the least error of a linear fit to
G.. Another procedure to extract the parameters IS, and n,
in the presence of significant shunt resistance Rsh, was
presented in [14]. It is based on using the difference
function D, as a function of voltage, to eliminate the
effect of the shunt resistance. The reverse saturation
current is first extracted from plots of estimated reverse
saturation current versus forward voltage, calculated
from the experimental data, for different assumed values
of the ideality factor n.

7. Extraction from explicit analytic solutions of the I-V
characteristics.
We have always known that the best way to extract
the model parameters of real junctions, with series and
shunt parasitic resistances, would be by using exact
explicit analytical solutions of their otherwise implicit
I–V characteristics. Unfortunately such solutions were
not available until recently. Nowadays we know that
such explicit solutions may be expressed in terms of
Lambert’s W function [15]. We applied the difference
function D, defined in (3), to these explicit analytical
solutions and, after several
manipulations, were able to conveniently express
function D as a purely algebraic equation of the form
[16,17]:
),(DIDVDVID
VIV
++=
where the five coefficients are given in terms of the
junction’s model parameters, as defined in Table 2 of
[17]. The procedure consists of fitting (16) to function D
calculated from the experimental I-V characteristics of
the junction. This quadratic bivariate fitting process turns
out to be a fast and accurate extraction procedure that is
theoretically exact, its exactness depending only on the
numerical computational precision used.

8. Post-breakdown current in MOS structures
The dielectric breakdown of thin gate dielectrics in
MOS devices brings about the formation of a current
path between gate and substrate. Attempts to understand
the conduction mechanism and to model the I-V
characteristics under this condition have led to simple
models based on an ideal diode analog with series
resistance, in the case of n-type silicon substrate, or to
more complicated models consisting on a combination of
two diodes and resistors, in the case of p-type silicon
substrates. The simplest model is given by:
(
[
{
exp
0
−=
VII
α
where I is the terminal post-breakdown current, V is the
terminal voltage, RBD is the series breakdown resistance,
I0 is a saturation current, and α is a constant that
characterizes the mechanism.
The parameters of these post-breakdown current
models may be extracted, as in the case of junction
models, using function D-based methods [18]. The use
of these methods is of special importance in this case
since the experimentally measured post-breakdown
current usually tends to be particularly noisy.
substitutions and
2
2
2
21111
IDVD VI
IVI
++
, (16)
)
]
} 1
−
DB
RI
, (17)

Page 4

[3] A. Ortiz-Conde, F.J. García-Sánchez, J.J. Liou, J.
Andrian, R.J. Laurence, P.E. Schmidt, A method to
extract parameters in a generalized two-terminal device,
Proc. of South Conf., 262-265, March (1994).
A. Ortiz-Conde, F.J. García-Sánchez, J.J. Liou, J.
Andrian, R.J. Laurence, P.E. Schmidt, A generalized
model for a two-terminal device and its application to
parameters extraction, Solid-State Electron., 38, 1,
265-266 (1995).
F.J. García-Sánchez, A.Ortiz-Conde, G. De Mercato, J.J.
Liou, L. Recht, Eliminating parasitic resistances in
parameter extraction of semiconductor device models,
Proc. First IEEE ICCDCS, 298-302, December (1995).
[6] F.J. García-Sánchez, A. Ortiz-Conde, J.J. Liou, A
parasitic series resistance-independent method for
device-model parameter extraction,, IEE Proc. -Cir. Dev.
and Sys., 143, 1, 68-70 (1996).
[7] L. Chua, Stationary principles and potential functions for
nonlinear networks, J. Franklin Inst., 296, 91-114 (1973).
[8] B.D. Tellegen, A general network theorem, with
applications, Philips Res. Reports., 7, 259-269 (1952).
[9] F.J. García Sánchez,, A. Ortiz-Conde, G. De Mercato, J.
J. Liou, “Parameter Extraction and Signal Processing
using Potential Functions”, Univ. Ciencia Tecnol., 4,
123-136 (2000).
[10] H. Norde, A modified forward I-V plot for Schottky
diodes with high series resistance, J. Appl. Phys., 50,
5052-5053 (1979).
[11] T.C. Lee, et al, A systematic approach to the
measurement of ideality factor, series resistance, and
barrier height of Schottky diodes, J. Appl. Phys., 72,
4739-4742 (1992).
[12] J.C. Ranuarez, F.J. García-Sánchez, A. Ortiz-Conde,
Procedure for the determination of diode model
parameters at very low forward voltage, Solid-St.
Electron., 43, 2129-2133 (1999).
[13] J.C. Ranuárez, A. Ortiz-Conde, F. J. García-Sánchez, A
new method to extract diode parameters under the
presence of parasitic series and shunt resistance,
Microelectron. Reliability, 40, 55-358 (2000).
[14] A. Ortiz-Conde, M. Estrada, A. Cerdeira, F.J. García
-Sánchez, G. De Mercato, Modeling real junctions by a
series combination of two ideal diodes with parallel
resistance and its parameter extraction, Solid-State
Electron., 45, 223-228, (2001).
[15] A. Ortiz-Conde, F.J. García-Sánchez, J. Muci, Exact
analytical solutions of the forward non-ideal diode
equation with series and shunt parasitic resistances,
Solid-State Electron., 44, 1861-1864 (2000).
[16] A. Ortiz-Conde, F.J.García-Sánchez, J.J. Liou, Extracting
Model Parameters of Non-Ideal Junctions Based on
Explicit Analytical Solutions of I-V Characteristics, Proc.
of the7th ICSICT, Beijing, China, October 18-21 (2004).
[17] A. Ortiz-Conde, F.J. García-Sánchez, Extraction of
non-ideal junction model parameters from the explicit
analytic solutions of its I-V characteristics, Solid-State
Electron., 49, 465-472, (2005).
[18] E. Miranda, A. Ortiz-Conde, F.J. García-Sánchez, and E.
Farkas-Sosa, Postbreakdown Current in MOS Structures:
Extraction of Parameters Using the Integral Difference
Function Method, IEEE Trans. Device Materials
Reliability, 6, 2, 190-196 (2006).
[19] A. Ortiz-Conde, F.J. García-Sánchez, J Muci, New
method to extract the model parameters of solar cells
from the explicit analytic solutions of their illuminated
I-V characteristics, Solar Energy Mat & Solar Cells, 90,
352-361, (2006).
[20] A. Ortiz-Conde, F.J. García-Sánchez, R. Salazar, On
Integration-based Methods
Parameter Extraction, Proc. of the 9th ICSICT, Beijing,
China, October 20-23 (2008).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
V (V)
Fig. 3 Extracted single-exponential I-V characteristics (line)
over the original experimental data points (symbols) from an
illuminated organic solar cell.

9. Solar cell model parameters.
The I-V characteristics of illuminated solar cells are
customarily described by the use of a lumped parameter
equivalent circuit which requires considering the
presence of parasitic series resistance and shunt
conductance, as well as reverse saturation current and
photogenerated current. In 2006 we proposed a method
based on calculating the Co-content from the Lambert W
function-based explicit analytical solutions of the
illuminated I-V characteristics [19]. The resulting
Co-content is expressed as a purely algebraic function of
current and voltage from whose coefficients the intrinsic
and extrinsic model parameters can be readily
determined by bivariate fitting. Figure 3 presents the
measured characteristics of an experimental organic
solar cell together with the model calculated using the
extracted parameters indicated.

10. Summary
Since we first started in 1993 using integration-based
methods to extract electron device model parameters, we
have developed the original
methodological concepts. The most significant perhaps
was the introduction of the Family of Potential Functions
and its very useful member, the Integral Difference
Function, function D, or INLF, as well as several other
derived auxiliary functions. The range of applications
where these mathematical concepts can be used has also
been growing steadily. At present it spans from simple
extraction of junction’s parameters in the presence of
parasitic resistances, solar
post-breakdown conduction, to more complex MOSFET
compact model parameter extraction methods [20],
assessing the nonlinearity and measuring harmonic
distortion of circuits and devices, etc.

References
[1] G. Araujo, E. Sánchez, A New Method for Experimental
Determination of the Series Resistance of a Solar Cell,
IEEE Trans. Electr. Dev., 29, 1511-1513, (1982).
[2] A. Ortiz-Conde, F.J. García Sánchez, P.E. Schmidt, R.J.
Laurence, Extraction of diode parameters from the
integration of the forward current, Proc. of the Intl
Semiconductor Research Symposium, 2, 531-534,
Charlottesville, Virginia, USA, December (1993).
I (mA.cm-2 )
−8
−6
−4
−2
0
Gp = 5.07 mS cm-2
Rs = 8.59 Ω
Iph = 7.94 mA/cm2
n = 2.31
Io= 13.6 nA/cm2
Ω cm2

[4]
[5]
ideas into new
cells, thin oxide
for MOSFET Model