Microelectronics Reliability 42 (2002) 583–596

www.elsevier.com/locate/microrel

A review of recent MOSFET threshold voltage

extraction methods
a,*
A. Ortiz-Conde anchez a, J.J. Liou
, F.J. Garcıa S b,1
, A. Cerdeira c,
M. Estrada c, Y. Yue d
a
Laboratorio de Electr
onica del Estado S olido (LEES), Universidad Simon Bolıvar, Apartado Postal 89000, Caracas 1080A, Venezuela
b
Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816-2450, USA
c
Secci
on de Electr
onica del Estado S olido (SEES), Departamento de Ingenierıa Electrica, CINVESTAV-IPN, Avenida IPN No. 2508,
Apartado Postal 14-740, 07300 DF, Mexico
d
Intersil Corporation, 2401 Palm Bay Road NE, Palm Bay, FL 32905, USA
Received 22 December 2001

Abstract
The threshold voltage value, which is the most important electrical parameter in modeling MOSFETs, can be ex-
tracted from either measured drain current or capacitance characteristics, using a single or more transistors. Practical
circuits based on some of the most common methods are available to automatically and quickly measure the threshold
voltage. This article reviews and assesses several of the extraction methods currently used to determine the value of
threshold voltage from the measured drain current versus gate voltage transfer characteristics. The assessment focuses
specially on single-crystal bulk MOSFETs. It includes 11 different methods that use the transfer characteristics mea-
sured under linear regime operation conditions. Additionally two methods for threshold voltage extraction under
saturation conditions and one specifically suitable for non-crystalline thin film MOSFETs are also included. Practical
implementation of the several methods presented is illustrated and their performances are compared under the same
challenging conditions: the measured characteristics of an enhancement-mode n-channel single-crystal silicon bulk
MOSFET with state-of-the-art short-channel length, and an experimental n-channel a-Si:H thin film MOS-
FET. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction There exist numerous methods to extract the value of

threshold voltage [10–41] and various extractor circuits
The threshold voltage (VT ) is a fundamental param- have also been proposed [42–44] to automatically mea-
eter for MOSFET modeling and characterization [1–6]. sure this parameter. Recently three books [1–3] and
This parameter, which represents the onset of significant three articles [4–6] have reviewed and scrutinized dif-
drain current flow, has been given several definitions [7– ferent available methods.
9], but it may be essentially understood as the gate The greater part of the procedures available to de-
voltage value at which the transition between weak and termine VT are based on the measurement of the static
strong inversion takes place in the MOSFET channel. transfer drain current versus gate voltage (ID –Vg ) char-
acteristics [10–35] of a single transistor. Most of these
ID –Vg methods use the strong inversion region [10–27],
* while only a few consider the weak inversion region [28–
Corresponding author. Fax: +582-9063631.
31]. Extraction is mostly done using low drain voltages
E-mail addresses: ortizc@ieee.org (A. Ortiz-Conde), jli@
ece.engr.ucf.edu (J.J. Liou), cerdeira@mail.cinvestav.mx (A. so that the device operates in the linear region [10–33].
Cerdeira), yyue@intersil.com (Y. Yue). However, VT extraction with the device operating in
1
Also at: Department of Electronics Science and Technol- saturation is also frequently carried out [34,35].
ogy, Huazhong University of Science and Technology, Wuhan A common feature present of most VT extraction
430074, P.R. China. methods based on the ID –Vg transfer characteristics is
0026-2714/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 6 - 2 7 1 4 ( 0 2 ) 0 0 0 2 7 - 6
584 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596

the strong influence of the source and drain parasitic respect to Vg [12]; (5) ratio method (RM), which finds the
series resistances and the channel mobility degradation gate voltage axis intercept of the ratio of the drain
on the resulting value of the extracted VT . This situation current to the square root of the transconductance [13–
is highly undesirable because the correct value of the 18]; (6) transition method [33]; (7) integral method [32];
extracted VT should not depend on parasitic components (8) Corsi function method [21]; and (9) second derivative
nor mobility degradation. In order to eliminate the in- logarithmic (SDL) method, which determines VT at the
fluence of these unwanted effects some methods have minimum of the SD of logðID Þ–Vg [31]; (10) linear co-
been proposed which are based on measuring capaci- factor difference operator [22] (LCDO) method, and (11)
tance as a function of voltage [36,37]. However these C– non-linear optimization [23,24].
V methods have the disadvantage of requiring elaborate This article will also review the following two meth-
high-resolution equipment to measure the small capac- ods to extract the VT of single-crystalline MOSFETs,
itances present in MOSFETs, particularly in very small operating in the saturation region: (1) extrapolation in
geometry state-of-art devices. Other approaches to elimi- the saturation region (ESR) method, which finds the
nate the influence of parasitic series resistances are based gate voltage axis intercept of the linear extrapolation of
on measuring the ID –Vg transfer characteristics of vari- the ID0:5 –Vg characteristics at its maximum first deriva-
ous devices having different mask channel lengths tive (slope) point [1,2]; and (2) G1 function extraction
[38,39], or on measuring several devices connected to- method [34,35].
gether [40,41]. Although such multi-device approaches Finally, we will review and discuss some amorphous
offer interesting solutions to this problem, they require TFT specific procedures which have been recently pro-
additional work and the availability of several supple- posed to extract the threshold voltage of these non-
mentary special devices. Another recently proposed crystalline devices [45,46].
method that requires repeated measurements is based on
a proportional difference operator [26,27].
The extraction of VT in non-crystalline MOSFETs is 2. Extraction from the ID –Vg curve of MOSFETs biased
more conveniently performed using the drain current in in the linear region
saturation, considering that these devices present much
smaller currents than single-crystalline devices. Amor- In order to critically assess and compare the different
phous and polycrystalline thin film transistors (TFTs) linear region extraction methods reviewed here, we will
introduce the additional difficulty that the saturation apply them all to extract the value of the threshold
drain current in strong inversion is usually modeled by a voltage from the measured transfer characteristics of a
power law with an exponent which can differ from 2 state-of-the-art bulk single-crystal silicon enhancement-
[45,46]. Because of this behavior, using conventional VT mode n-channel MOSFET with a 5 lm mask channel
extraction methods developed for single-crystal devices width, a 0.18 lm mask channel length, and a 32A gate
will generally produce values of VT that are unacceptable oxide thickness. For this group of methods the device is
or at least not very accurate. Therefore the extraction biased to operate in the linear regime by applying a
method must be capable of extracting the value of the drain voltage of 10 mV. Fig. 1 presents the output
unknown power-law exponent parameter and take it characteristics of this device for general reference pur-
into consideration in the extraction process. To that end, poses.
methods have been proposed that are specific for non-
crystalline thin MOSFET TFTs [45,46] and thus allow 2.1. Constant-current method
to extract their threshold voltage correctly.
This article will review and scrutinize the following The CC method [1–6] evaluates the threshold voltage
existing ID –Vg methods for extracting VT in single-crystal as the value of the gate voltage, Vg , corresponding to a
MOSFETs, biased in the linear region: (1) constant- given arbitrary constant drain current, ID and Vd < 100
current (CC) method, which defines VT as the gate mV. A typical value [20] for this arbitrary constant drain
voltage corresponding to a certain predefined practical current is ðWm =Lm Þ  107 , where Wm and Lm are the
constant drain current [1–6,10,11]; (2) extrapolation in mask channel width and length, respectively. This
the linear region (ELR) method, which finds the gate method is widely used in industry because of its sim-
voltage axis intercept of the linear extrapolation of the plicity. The threshold voltage can be determined quickly
ID –Vg characteristics at its maximum first derivative with only one voltage measurement, as shown in Fig. 2.
(slope) point [1–6]; (3) transconductance linear extrap- In spite of its simplicity, this method has the severe
olation (GMLE) method, which finds the gate voltage disadvantage of being totally dependent of the arbi-
axis intercept of the linear extrapolation of the gm –Vg trarily chosen value of the drain current level. This is
characteristics at its maximum first derivative (slope) evident by the results in Fig. 2, where different gate
point [19,20]; (4) second derivative (SD) method, which voltages can be taken at different drain current values to
determines VT at the maximum of the SD of ID with represent the threshold voltages.
 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596 585

fining the previously arbitrary drain current level used

to define the threshold voltage at the drain current
where d2 ID =dVg2 presents a maximum. This amounts to a
combination of the CC method and the second-deriva-
tive method, which will be presented latter.

2.2. Extrapolation in the linear region method

The ELR method [1–6] is perhaps the most popu-

lar threshold-voltage extraction method. It consists of
finding the gate-voltage axis intercept (i.e., ID ¼ 0) of the
linear extrapolation of the ID –Vg curve at its maximum
first derivative (slope) point (i.e. the point of maximum
transconductance, gm ), as illustrated in Fig. 3. The value
of VT is calculated by adding Vd =2 to the resulting gate-
voltage axis intercept, which for the device at hand
happens to be 0.51 V. The main drawback of this other-
wise useful method is that the maximum slope point
might be uncertain, because the ID –Vg characteristics can
deviate from ideal straight line behavior at gate voltages
even slightly above VT , due to mobility degradation ef-
fects and to the presence of significant source and drain
Fig. 1. Measured ID –Vd output characteristics at five values of series parasitic resistances [2]. Therefore, the threshold
gate bias for the test bulk single-crystal n-channel MOSFET voltage value extracted using this method, often referred
with 5 lm mask channel width and 0.18 lm mask channel
to as the extrapolated VT , can be strongly influenced by
length.

Fig. 2. CC method implemented on the ID –Vg transfer charac-

teristics of the test bulk device measured at Vd ¼ 10 mV. This
method evaluates the threshold voltage as the value of the gate
voltage corresponding to a given arbitrary constant drain cur-
rent. Fig. 3. ELR method implemented on the ID –Vg characteristics
of the test bulk device measured at Vd ¼ 10 mV. This method
consists of finding the gate-voltage axis intercept (i.e., ID ¼ 0) of
Recently Zhou and his group have proposed [10,11] the linear extrapolation of the ID –Vg curve at its maximum slope
an improvement to the CC method. It consists on de- point.
586 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596

parasitic series resistances and mobility degradation maximum slope of the gm –Vg characteristics offers a
effects. better description of VT .

2.3. Transconductance extrapolation method in the linear

2.4. Second-derivative method
region
The SD method [12], developed to avoid the depen-
A seldom used method is the transconductance ex-
dence on the series resistances, determines VT as the gate
trapolation method in the linear region (GMLE) which
voltage at which the derivative of the transconductance
was proposed in 1998 [19,20]. This method suggests that
(i.e., dgm =dVg ¼ d2 ID =dVg2 ) is maximum. The origin of
the threshold voltage corresponds to the gate voltage
this method can be understood by analyzing the fol-
axis intercept of the linear extrapolation of the gm –Vg
lowing ideal case of a MOSFET modeled with a simple
characteristics at its maximum first derivative (slope)
level ¼ 1 SPICE model, where ID ¼ 0 for Vg < VT and ID
point. This method is based on the following arguments
is proportional to Vg for Vg > VT . Using the previous
when the device is biased in the linear region. (1) In weak
simplifying assumption, dID =dVg becomes a step func-
inversion, the transconductance depends exponentially
tion, which is zero for Vg < VT and has a positive con-
on gate bias; (2) For strong inversion, if the series re-
stant value for Vg > VT . Therefore, d2 ID =dVg2 will tend to
sistance and mobility degradation are negligible, the
infinity at Vg ¼ VT . Since for a real device such simpli-
transconductance tends to a constant value; (3) The
fying assumptions are obviously not exactly true,
transconductance decreases slightly with gate bias due to
d2 ID =dVg2 will of course not become infinite, but will
the series resistance and mobility degradation; (4) In the
instead exhibit a maximum at Vg ¼ VT .
transition region between weak and strong inversion, the
As Fig. 5 indicates, the implementation of this
transconductance depends linearly on gate bias. Fig. 4
method is highly sensitive to measurement error and
presents the application of this method to the gm –Vg
noise, because the use of the SD amounts to applying a
characteristics producing an apparent value for VT of
high-pass filter in the measurement. Notice in this figure
only 0.44 V. The following method also based on the
that the maximum value of d2 ID =dVg2 occurs at about
Vg ¼ 0:54 V due to the measurement noise present;

Fig. 4. Transconductance extrapolation method (GMLE) im-

plemented on the gm ¼ dID =dVg versus Vg characteristics of the
test bulk device measured at Vd ¼ 10 mV. This method suggests Fig. 5. SD method implemented on the plot of d2 ID =dVg2 versus
that the threshold voltage corresponds to the gate voltage axis Vg of the test bulk device measured at Vd ¼ 10 mV. This method
intercept of the linear extrapolation of the gm –Vg characteristics consists of finding the gate-voltage at which d2 ID =dVg2 exhibits a
at its maximum slope point. maximum value.
 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596 587

whereas if the noise is suppressed the maximum appears

to be around Vg ¼ 0:50 V.

2.5. Ratio method

The RM [13–18], developed to avoid the dependence

of the extracted VT value on mobility degradation and
parasitic series resistance, proposes that the ratio of the
drain current to the square root of the transconduc-
0:5
tance (ID =gm ) behaves as a linear function of gate bias,
whose intercept with the gate-voltage axis will equal the
threshold voltage. This method was originally published
independently in 1988 by Jain [13] and by Ghibaudo
[14]. Jain demonstrated that if the mobility degradation
0:5
were negligible, the function ID =gm would be indepen-
dent of parasitic series resistance [13]. On the other
hand, Ghibaudo showed that if the parasitic series re-
0:5
sistance were negligible, the function ID =gm would not
depend on mobility degradation [14]. In 1995, Fikry and
0:5
his coworkers proved [15] that the function ID =gm is
independent of mobility degradation, parasitic series
resistance and velocity saturation effects. The RM was
further improved in 2000 [18] to account for a more
general mobility degradation model. Fig. 6. RM implemented on the plot of the ratio of the drain
0:5
Summarizing the RM developments, the drain cur- current to the square root of the transconductance (ID =gm )
rent ID in the linear region can be expressed as [1–3] versus Vg of the test bulk device measured at Vd ¼ 10 mV. This
method evaluates VT from the intercept to the lateral axis of its
W straight line fit.
ID ¼ lCo ðVGS  VT ÞVDS ; ð1Þ
Leff

where W is the channel width, Co is the oxide capaci-

tance per unit area, l is the effective free-carrier mobil- where gm is the transconductance and
ity, and VGS and VDS are the intrinsic gate–source and  
drain–source voltages, respectively. The intrinsic volt- Lm  DLeff  lv0satVd
s¼ : ð6Þ
ages can be related to the external gate–source and W l0 Co Vd
drain–source voltages (Vg and Vd ) by
1=2
Then, by plotting the ID =gm versus Vg curve the values
VGS ¼ Vg  ID RD ð2Þ
of VT and s can be extracted from the intercept and the
slope of its straight line fit. Fig. 6 shows the results of
and
applying this method to the present test device. As can
VDS ¼ Vd  ID ðRS þ RD Þ: ð3Þ be observed, in the present case it is not clear where to
do the linear approximation to be extrapolated to the Vg
Here RD and RS represent the drain and source parasitic axis to extract the value of VT . The ID =gm 1=2
versus Vg
series resistances, respectively. According to Fikry et al. curve for the present test device shown in Fig. 6 does not
[15], the velocity saturation effect is imbedded in the appear to totally fulfill this method’s assumptions, since
following free-carrier mobility model: it does not clearly behave in the linear manner expected.
l0 Therefore, the linear fit shown is just a guess, amidst the
l¼

 ; ð4Þ evident non-linearity and the noise present, significantly
1 þ h Vg  VT 1 þ Lleff0 Vvsat
d

enhanced by dividing the current by the square root of

its first derivative (gm ).
where l0 is the low-field mobility, h is the mobility de-
gradation factor due to the vertical field, and vsat is the
saturation velocity of the carriers. Using (1)–(4) and the 2.6. Transition method
approximation Vg ¼ VGS , it can be proved that
This method uses the sub-threshold-to-strong inver-
ID

1=2
¼ s1=2 Vg  VT ; ð5Þ sion transition region of the MOSFET’s transfer char-
gm acteristics to extract the threshold voltage. It is based on
588 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596

an auxiliary operator that involves integration of the necessary values of voltage and current in an integral
drain current as a function of gate voltage. function D defined as
In order to extract VT , the drain current is measured Z y0 Z x0
versus Vg below and above threshold with zero body bias Dðx; yÞ ¼ x dy  y dx; ð8Þ
0 0
and a small constant value of drain voltage. Next the
following function G1 is numerically calculated from and after substituting and performing algebraic manip-
measured data [33]: ulations the following function can be obtained:
R Vga
ID ðVg ÞdVg 2Vgb Vgb
V D1 ðVgb ; Rm Vgb Þ ¼ þ
G1 ðVg ; ID Þ ¼ Vg  2 gb ; ð7Þ K KðVmax  Vgb  VT Þ
ID  
2ðVmax  VT Þ Vgb
where Vgb and Vga are the lower and upper limits of þ ln 1  ;
K Vmax  VT
integration corresponding to gate voltages below and ð9Þ
above threshold, respectively.
A plot of G1 versus ln ID should result in a straight where Vgb ¼ Vmax  Vg and Vmax is a constant parameter
line below threshold, where the current is dominated by equal to the maximum gate voltage under consideration.
diffusion and consequently it is predominantly expo- When D1 is plotted versus Vgb , the value of VT is
nential. As soon as Vg ¼ VT function G1 should drop obtained using a procedure similar to extracting the
abruptly. This is what is observed with the present test ideality factor and saturation current of a junction di-
device, as revealed in Fig. 7. It can be shown that the ode, as explained in [47,48]. Fig. 8 illustrates the appli-
maximum value of G1 corresponds to the threshold cation of this method to the test device producing a VT
voltage of the device [33], which for this case happens to value of 0.51. Notice that D1 also permits the extraction
be 0.49 V. It should be noted that parasitic resistance of parameter K. Although this method gives accurate
and mobility degradation effects influence the shape of results, is it quite cumbersome to implement.
the above-threshold G1 , but not significantly its maxi-
mum value, unless those effects are highly pronounced. 2.8. Corsi function method

2.7. Integral method Corsi and coworkers have proposed [21] a method
based on the following function:
The integral method was developed in [32] to be in-
sensitive to the effect of drain and source parasitic series ID
Beta ¼ ; ð10Þ
resistances. It was demonstrated that substituting the Vg  VP

Fig. 7. Transition method implemented on the plot of function Fig. 8. Integral method implemented on the plot of function D1
G1 versus ID of the test bulk device. This method evaluates the versus Vgb of the test bulk device. This method evaluates the
threshold voltage from the maximum value of G1 . threshold voltage by doing a curve fitting of function D1 .
 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596 589

Fig. 9. Corsi function method implemented on the plot of the Fig. 10. SDL method implemented on the plot of d2 lnðID Þ=dVg2
Corsi function versus Vg of the test bulk device for several ar- versus Vg of the test bulk device measured at Vd ¼ 10 mV. This
bitrary values of VP . This method evaluates the threshold method consists of finding the gate-voltage at which d2 ID =dVg2
voltage by finding the plot for which the minimum just disap- exhibits a minimum value.
pears and for this particular case VP ¼ VT .

where Vg > VP and VP is a parameter chosen in the region about 0.5 V, if measurement noise and error are sup-
of expected values of VT . Fig. 9 shows plots of this pressed.
function versus Vg , for several values of VP , as derived
from the experimental transconductance characteristics 2.10. Linear cofactor difference operator method
of the test device. The minimum is related to a value
of Vg ¼ VT þ ða=2ÞVd , where a is a parameter dependent This method (LCDO), recently developed by He and
on small channel effects and the body effect. It can co-workers to avoid the dependence of the extracted VT
be demonstrated that the minimum disappears when value on mobility degradation, proposes to use the fol-
VP ¼ VT . In practice this method appears not to be very lowing auxiliary function [22]:
precise for determining the value of VT and in our
opinion it offers no particular advantages. DID  Gx Vg  ID ; ð11Þ

2.9. Second derivative logarithmic method where Gx is an arbitrary constant. The drain current,
neglecting parasitic series resistance, is modeled by


The SDL method was proposed by Aoyama in 1995 G d Vg  VT
[31]. The threshold voltage is determined as the gate ID ¼
; ð12Þ
1 þ h Vg  VT
voltage at which the second difference of the logarithm
of the drain current takes on a minimum value. It cor- where Gd  ðW =Leff ÞlCo Vd is a constant of the device
responds to the gate voltage at which drift and diffusion with units of conductance, h is the mobility reduction
drain currents are equal to each other. The authors factor due to the vertical electric field in the channel,
claim that this definition of VT overcomes the disad- and other parameters have their usual meaning. Sub-
vantages of the CC method, which requires measuring stituting (12) into (11) and taking the first derivative,
the effective channel length, and that it is more accurate it can be proved that DID will present a minimum
than ELR, which can be applied only to the low drain value located at Vg ¼ Vgp and DID ¼ DIDP . The evalua-
voltage region, or than the already described transcon- tion of this minimum value allows to extract VT and h by
ductace method. However, similarly to other methods using:
based on taking SDs, this method is highly sensitive
to experimental measurement noise and error. Fig. 10 "
1=2 #
DIDP Gx
shows the implementation of this method for the present VT ¼ 1=2
þ 1 Vgp ð13Þ
ðG x G d Þ Gd
test device. It produces a reasonable value for VT of
590 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596


b VGS  VT  VDS
2
VDS
ID ¼ ; ð15Þ
1 þ hðVGS  VT Þ
where b ¼ ðW =Leff ÞlCo is the transconductance param-
eter, h is the mobility reduction factor due to the vertical
electric field in the channel, and other parameters have
their usual meaning. For the MOSFET biased in the
strong inversion region with a small drain voltage, and
assuming the voltage drop in the source and drain series
resistances is small compared to the gate bias, the drain
current can be rewritten as
Vg  b
ID ¼ a Vd ; ð16Þ
Vg  c

where
b
a¼ ; ð17Þ
h þ bRDS

Vd
b ¼ VT þ  VT ; ð18Þ
Fig. 11. LCDO method implemented on the plot of function
2
DID versus Vg of the test bulk device measured at Vd ¼ 10 mV. and
1
c ¼ VT  : ð19Þ
and h þ bRDS

G1=2
d  Gx
1=2 Fig. 12 shows measured ID versus Vg characteristics
h¼
: ð14Þ (solid lines) for Vd ¼ 10 mV of the same test device
G1=2
x Vgp  VT
previously described. The fit (closed circles) to the sim-
ulated results were obtained by using the optimized
Fig. 11 shows the results of applying this method to the
values of a ¼ 12:4 mA/V, b ¼ VT ¼ 0:57 V and c ¼
present test device. As can be observed, the location of
0:24 V such that the following parameter e has the
the minimum value changes for different Gx . According
minimum value:
to this method, the values of VT and h should be inde-
pendent on the selected value of Gx . In contrast to this
assumption, our results indicates that for variations of
Gx from 40 to 60 lS, VT changes from 0.35 to 0.45 V and
h goes from 0.53 to 0.38 V1 . Therefore this method
does not seem to be very appropriate for short-channel
devices.

2.11. Non-linear optimization method

The non-linear optimization method [23,24] extracts

VT based on optimization techniques applied to the
MOSFET current–voltage characteristics. It has two
main advantages: (1) the consistent determination of all
the model parameters because of the simultaneous ex-
traction; and (2) the reduction of the effects of the noise
on the experimental data due to the optimization tech-
niques. There are two main disadvantages, however: (1)
non-physical parameter values can be obtained because
of the pure fitting scheme, and (2) the requirement of a
long computational process.
The development of this method, proposed by Kar- Fig. 12. Non-linear optimization method implemented on the
lsson and Jeppson [24], is as follows; The drain current measured ID –Vg characteristics of the test bulk device measured
for the MOSFET is expressed as at Vd ¼ 10 mV.
 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596 591

XN
2
Vg  b
e ID  a Vd : ð20Þ
i¼1
Vg  c

Then, the following three parameters can be calculated

from the values of a, b and c:
a
b¼ ; ð21Þ
b  c V2d

1
h þ bRDS ¼ ; ð22Þ
b  c  V2d

ðLm  DLeff Þ
b1 ¼ ðlCo Þ1 : ð23Þ
W

3. Extraction from the ID –Vg curve of MOSFETs biased

in the saturation region

To extract the saturation threshold voltage VTsat the

drain current must be measured as a function of gate
voltage with the drain connected to the gate, to guarantee Fig. 13. Extrapolation method in the saturation region (ESR)
that the device is operating in the saturation regime. implemented on the measured ID0:5 –Vg characteristics of the
test bulk device measured at Vd ¼ Vg . This method consists
3.1. Extrapolation method in the saturation region of finding the gate-voltage axis intercept (i.e., ID0:5 ¼ 0) of the
linear extrapolation of the ID0:5 –Vg curve at its maximum slope
point.
The ESR method, determines the threshold voltage
0:5
from the gate voltage axis intercept of the IDsat –Vg
characteristics linearly extrapolated at its maximum first
derivative (slope) point [1–3] as illustrated in Fig. 13. Substituting (25) and (26) into (24) and solving for Vg ,
The value of VTsat calculated for the present device re- we obtain:
sults to be 0.46 V.
1=2
2IDsat 2 2
Vg ¼ VT þ Rt IDsat þ þ Rh IDsat ; ð27Þ
Ko
3.2. G1 function method
where
This method [34,35] considers that the device is op-
h
erating in the saturation region and under strong in- Rh  ð28Þ
version. The gate and drain terminals are connected Ko
together to ensure saturation operation. The saturation is an effective resistance due to the free-carrier mobility
drain current may be expressed as [1–3] degradation in the channel, and
K
IDsat ¼ ðVGS  VT Þ2 ; ð24Þ Rt  Rs þ Rh ð29Þ
2
where VT is the threshold voltage, is the total effective resistance.
Using the approximation, ð2IDsat =Ko Þ R2h IDsat
2
, Eq.
VGS ¼ Vg  IDsat Rs ð25Þ
(27) is simplified to
is the intrinsic gate–source voltage, Vg is the extrinsic
1=2
2 1=2
gate–source voltage, Rs is the source series resistance, Vg  VT þ Rt IDsat þ IDsat : ð30Þ
Ko
and
b Based on an approach developed previously [47,48], we
K¼ : ð26Þ have proposed to use the following function to suppress
1 þ hðVGS  VT Þ
the linear term of IDsat in (30):
In the previous equation, h is the mobility degradation Z Vg
parameter and b ¼ ðW =Leff ÞlCo is the transconductance
 2
G1 Vg ; IDsat ¼ Vg  IDsat ðVg ÞdVg : ð31Þ
parameter. IDsat 0
592 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596

smaller drain currents than conventional single-crystal

bulk devices. Amorphous TFTs introduce the following
additional difficulties for VT extraction: First, the satu-
ration drain current in strong inversion is usually mod-
eled by an equation of the form [49]
IDsat ¼ K ðVGS  VT Þm ; ð33Þ

where K is a conductance parameter with units of A Vm

and m an empirical parameter which can be different
from 2, the value used in conventional MOSFET mod-
els. Second, the value of parameter m cannot be easily
extracted from a simple plot of logðIDsat Þ versus logðVg Þ
because practical operation values of Vg are usually not
large enough to validate the approximation: ðVGS 
VT Þ  Vg . Third, it is not clear at what point the IDsat
versus Vg plot could be linearly extrapolated, since the
curve does not present an inflexion point because the
mobility of these devices raises as Vg is increased.
A method to extract the threshold voltage of amor-
phous thin film MOSFETs, that circumvents some of
Fig. 14. G1 function method in the saturation region im- these difficulties, is based on the following function
plemented using the plot of the G1 function versus ID0:5 of the
which can be numerically computed from the measured
test bulk device measured at Vd ¼ Vg . This method consists of
finding the gate-voltage axis intercept (i.e., ID0:5 ¼ 0) of the linear
IDsat ðVg Þ characteristics:
extrapolation of the ID0:5 –Vg curve. R Vg

 IDsat ðVg ÞdVg
H Vg ¼ 0 ; ð34Þ
IDsat
The function defined in the previous equation, with units
where the upper limit of integration is any suitable value
of V, can be numerically computed from the measured
greater than the threshold voltage.
IDsat ðVg Þ characteristics. It can be proved, after using
The integral in (34) is negligible for values of Vg such
integration by parts and doing algebraic manipulations,
that the device is operating in the strong inversion re-
that (31) becomes
gion. Thus, H ðVg Þ may be approximated by

1=2 R Vg

 1 2 1=2
G1 Vg ; IDsat ¼ VT þ IDsat : ð32Þ
 V
IDsat ðVg ÞdVg
3 Ko H Vg  T : ð35Þ
IDsat
Therefore, the value of Ko can be obtained from the
1=2 After substitution of (33) into (35), and assuming that
slope of G1 versus IDsat plot, and VT can be determined
the variation of K with respect to Vg is insignificant, we
from the intercept of the linear extrapolation of the G1
obtain:
curve to the y-axis.

The value of VTsat is extracted from the G1 axis in-
 Vg  VT
tercept of the linear fit of the calculated G1 versus IDsat0:5 H Vg ¼ ; ð36Þ
mþ1
curve, extrapolated in the region of the curve where the
square root of the saturation current has a linear de- which means that H ðVg Þ behaves linearly in the strong
pendence on the gate voltage. That region is clearly inversion region. Therefore, a plot of function H versus
shown in Fig. 13 around the maximum slope point. The Vg has a slope that defines the value of m and a Vg axis
result of applying this method to the present test device intercept which gives the sought after value of VT . Be-
is presented in Fig. 14, indicating a value of VTsat close to cause of the low-pass filter nature of integration, this
0.45 V. method offers the additional advantage of inherently
reducing the effects of experimental errors.
After having found m and VT , the remaining pa-
4. Extraction from the ID –Vg curve of non-crystalline rameter in (33), K, may be easily evaluated from
MOSFETs biased in the saturation region IDsat
K¼
m : ð37Þ
Vg  VT
The extraction of VT in non-crystalline MOSFETs is
more conveniently performed from the drain current in This extraction procedure will be applied to an experi-
saturation, considering that these devices present much mental n-channel a-Si:H thin film MOSFET having: a
 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596 593

Fig. 16. Measured IDsat (symbols) versus gate bias for the ex-
perimental n-channel amorphous TFT. A 0.5 V gate-to-source
voltage step was used with the drain connected to the gate. Also
Fig. 15. Measured ID –Vd characteristics at three values of gate shown (continuous line) are the simulated results using the
bias for the experimental n-channel amorphous TFT. extracted set of parameter values: m ¼ 3:07, VT ¼ 3:25 V and
K ¼ 3:2 nA Vm .

gate oxide thickness of 0.3 lm; an intrinsic a-Si:H layer

thickness of 0.3 lm; 0.1 lm thick nþ drain and source
regions with impurity concentrations of 1018 cm3 ;
channel width of 600 lm and channel length of 40 lm.
The measured ID versus Vd output characteristics for
several values of Vg are presented in Fig. 15. Examina-
tion of this figure indicates that the threshold voltage
must be smaller than 10 V, since it shows that there is a
reasonable drain current flowing already at Vg ¼ 10 V.
Fig. 16 presents the measured IDsat versus Vg transfer
characteristics with linear and logarithmic scales for the
vertical axis. The drain current was measured using 0.5
V gate-to-source voltage steps, with the drain connected
to the gate to insure operation in the saturation regime.
Fig. 16 also presents the results of simulating the device
using (33) with the values of parameters that will be
extracted by the present method. It is clear from this
figure that the plot of IDsat does not show evidence of any
inflexion point and, thus, a plot of gmsat ¼ dIDsat =dVg will
always rise as Vg is increased. If we were to apply the
commonly used so-called ‘‘constant current definition’’
for threshold voltage as being the gate bias correspond-
ing to an arbitrary value of drain current, for instance Fig. 17. Function H ðVg Þ of the experimental n-channel amor-
0.1 lA, we would obtain VT ¼ 6 V, a value which is far phous TFT calculated from the ID –Vg characteristics presented
from being correct, as we shall see. Likewise, using the in the previous figure. The slope of the straight line for strong
plot of logðIDsat Þ versus Vg would give the false impres- inversion is 0.246, which according to (5) implies m ¼ 3:07. The
sion that the transition from weak to strong inversion intercept of the straight line to the gate bias axis is 3.25 V, which
occurs at about 8 V, a value that is an even worse esti- implies VT ¼ 3:25 V.
mation of the threshold voltage.
Fig. 17 shows a plot of the numerical calculation of slope of 0.246 and a Vg axis intercept (threshold voltage)
H ðVg Þ according to (34). For strong inversion the curve of 3.25 V. Furthermore, according to (36), this slope
is seen to behave approximately as a straight line with a implies a power-law empirical exponent m ¼ 3:07. It is
594 A. Ortiz-Conde et al. / Microelectronics Reliability 42 (2002) 583–596

worth noting here that an alternate but laborious way Table 2

to extract m for the strong inversion region would be Threshold voltage values obtained from two extraction meth-
to find, through trial and error, the value of m which ods for a short-channel single-crystal bulk device (Lm ¼ 0:18
1=m
produces the maximum linearity of IDsat , evaluated lm) biased in the saturation region
through a linear regression coefficient. Such procedure Method Threshold voltage (V)
was applied and it yields m ¼ 3:06, with a linear re- ESR 0.46
gression coefficient of 0.999797, which matches very well G1 0.45
the value previously extracted through the present pro-
cedure and thus confirms its accuracy.
Synthetic IDsat curves were simulated using (33) with values for this device. As can be observed in this table,
the extracted parameter values: m ¼ 3:07, VT ¼ 3:25 V seven out of the eleven methods presented to extract
and K ¼ 3:2 nA Vm . They are presented in Fig. 16 to- threshold voltage under linear region bias produce very
gether with the original experimental data. The excellent similar results, of about 0.5 V.
match obtained between the resulting characteristics for Two additional methods were applied under satura-
strong inversion, simulated using the extracted param- tion regime operation to the same single-crystal bulk
eters, and the measured data clearly validates the pro- device. The saturation threshold voltage values extracted
cedure. by either method are very close, as shown in Table 2.
Finally, we can also conclude that the results of applying
the non-crystalline MOSFET specific method to an ex-
perimental n-channel a-Si:H TFT has revealed that this
5. Conclusions method is better suited for accurate threshold voltage
extraction of this type of device than other more con-
We have presented, reviewed and critically compared ventional methods.
several extraction methods currently used to determine
the threshold voltage value of bulk single-crystal and
non-crystalline thin film MOSFETs from their drain
current versus gate voltage transfer characteristics mea- Acknowledgements
sured either in linear or saturation operation regimes.
The relative performance of the presented methods was This work was supported by ‘‘Universidad Sim on
illustrated and compared under the same conditions by Bolıvar’’, by CONICIT (Venezuela) through grant S1-
applying them to the measured characteristics of two real 98000567, and by CONACYT (Mexico), project N1
test devices: (a) an enhancement-mode n-channel single- 34400-A.
crystal silicon bulk MOSFET with state-of-the-art 0.18
lm channel length, and (b) an experimental n-channel a-
Si:H thin film MOSFET. Eleven methods that use the References
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