SIiVIULATION OF SEiVIICONDUCTOR DEVICES AND PROCESSES Vol.

6
Edited by H. Ryssel, P. Pichler - September 1995

A Method for Extracting the Threshold Voltage

of MOSFETs Based on Current Components
K. AOYAMA
NTT LSI Laboratories,
3-1, Morinosato Wakamiya, Atsugi-shi, Kanagawa Pref., 243-01 JAPAN

Abstract
A new method for extracting the threshold voltage of MOSFETs is presented.
The threshold voltage is the gate voltage at which the second difference of the
logarithm of the drain current takes a minimum value. The method is applied to
a 0.6-pm NMOSFET. The threshold voltage characteristics are compared with
ones measured with previous methods and it is shown that the proposed method
overcomes previous problems. The threshold voltage is extracted based on a
physical background verified with 2D device simulation and shows a transition
voltage at which drift and diffusion components in the drain current are equal.

1. Introduction
A serious problem is that threshold voltage definitions in measurements differ from
the one in compact MOSFET models. The threshold voltage in compact models is
defined as the gate voltage at which the surface potential 4,of the channel reaches
the double Fermi potential in the bulk 2df. This definition is very popular but has the
disadvantages that the position where 4, = 2dj in the channel is not clear and it is
difficult to measure 4, in MOSFETs. On the other hand, threshold voltages in measurements are extracted by the constant current (CC), the linear extrapolation (LE)
and the transconductance change (TC) methods [I], [2]. With CC, the difficulties are
measuring the effective channel length and width and defining a. constant value of the
drain current. LE should be applied only in the low drain voltage region. In the high
drain voltage region, however, the square root of current extrapolation (SRE) should
be applied. With extrapolation methods such as LE and SRE, the continuity in all
operation voltages is lost. The threshold voltage characteristics with T C are strange
for drain voltage changes, as shown in Fig. 1. A threshold voltage definition that
overcomes the above disadvantages is needed. This paper presents a new method for
extracting the threshold voltage: the second difference of the logarithm of the drain
current minimum (SDLM).

2. Method
The diffusion component and the drift component can be respectively approximated
by an exponential function and a polynomial expression, as shown in Fig. 2. The
first difference of the logarithm of the drain current almost stays constant when

K. Aoyama: A Method for Extracting the Threshold Voltage of MOSFETs

119

the diffusion component is dominant and gradually approacl~eszero when the drift
component is dominant, as shown in Fig. 3 (a). Fig. 3 (b) s.hows the second difference
takes a minimum value at the unique transition voltage in an NMOSFET. In this work,
this voltage is defined as the threshold voltage. This threshold voltage is shown in
Fig. 1 for various drain voltages compared to ones extracted with previous methods.
The characteristics are reasonable even for drain voltage changes.

3. Verification
With a view to certify that the dominant component in the drain current changes
from diffusion to drift at the threshold voltage extracted with the above method, the
rate of each component was calculated by 2D device simulation. In the simulation, an
NMOSFET with L, = 1.Opm was used with Vd, = 3.OV and Vsb= OV. As a result, the
minimum value is calculated with this method as shown in Fig. 4 like in measurements
and the potential distributions at the interfaceof Si and SiOz obtained are those shown
in Fig. 5. The drift component
and diffusion conlponent [ids(dij/)]are

where p is the electron mobility, n is the electron density, $ is the conduction band
potential, (4-[,)
is the relative potential and [, is the electron quasi-Fermi potential.
At each position at the interface, the rates of drift and diffusion compete with the
rates of (d$/dx) and [-d($ - (,)/dx]. Fig. 6 shows the distribution of the rate of
current components through the channel. The average rate of the drift component in
the channel region
is

where x, and xd are the source edge and the drain edge and L is the channel length.
Fig. 7 shows the rate of the drift component in the channel increases as the gate
voltage increases. At V,, = 0.215V, drift equals diffusion. Therefore, the threshold
voltage defined with this method shows the unique gate voltage that is the transition
voltage in the dominant component in the drain current.

4. Summary
A method for extracting the threshold voltage with the second difference of the logarithm of the drain current was proposed and verified from the physical point of view
using 2D device simulation. The extracted threshold voltage has the following features: It is extracted from only the drain current without both the effective channel
length and width and has a physical meaning. It shows the gate voltage at which
drift and diffusion in the drain current are equal each other. These features may be
useful for unifying the threshold voltage in compact models and in measurement.

References
[I] M. J. Deen and Z. X. Yan, "A new method for measuring the threshold voltage of
small-geometry MOSFETs from subthreshold conduction", Solid-State Electronics, vol. 33, no. 5, pp. 503-511, 1990.
[2] H. -S. Wong, M. H. White, T. J. Krutsick and R. V. Booth, "Modeling of transconductance degradation and extraction of threshold voltage in thin oxide MOSFETs", Solid-State Electronics, vol. 30, no. 9, pp. 953-968, 1987.

K. Aoyama: A Method for Extracting the Threshold Voltage of MOSFETs

120

NMOSFET Lp=Odpm Wpell.OW V s W V

.
1

.
1

.
I

transition
voltage

polynomial
expression

diffusion
exponential
function

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

gate voltage: Vgs
drain voltage: Vds(V)
Fig. 2. The idea of extracting the
Fig. 1. Comparison of the threshold voltages threshold voltage in this work.
extracted with previous methods and SDLM
for various drain voltages using the measured
drain current in the NMOSFET.

12

$g-10

a
>

8-20

zi
3
A
V1

V1

-a

-30

M
0

0
4 -40
0.2 0.4 0.6 0.8 1.0 1.2
0.2 0.4 0.6 0.8 1.0 1.2
gate voltage: Vgs(V)
gate voltage: Vgs(V)
(a)
(b)
Fig. 3. Results for this method when applied to a submicron NMOSFET for the
various substrate voltages. (a) The first difference-ofthe drain current. (b) The
second difference of the drain current.
16
,20--

2 il
M

a 10

-20

8
$ 6
. 4

-35
\

-40

V1

V1

8;

-60

N
H
0

4 -80
0.2 0.4 0.6 0.8 1.0
0 0.2 0.4 0.6 0.8 1.0
gate voltage: Vgs(V)
gate voltage: Vgs(V)
(a)
(b)
Fig. 4. The derivatives of the drain current calculated from 2D device simulation.
(a) The first difference of the drain current. (b) The second difference of the drain current.

K. Aoyarna: A Method for Extracting the Threshold Voltage of MOSFETs

4.0

..
..

.
..

.
..

.
..

..

12 1

..

&ativd potedtial....'.....hk

_
4.5
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
distance: x(pm)
Fig. 5. The conduction band potential and the relative
potential distribution at the interface at Vds=3.0V and
Vgs=02V using 2D device simulation results. The device
structure used in the simulation is also shown.
&conduct~on
band potential
I

Vth extracted with SDLM method

1.3 1.4 1.5 1.6 1.7 1.8 1.9

distance: x(pm)
Fig. 6. The rate of the two drain current
components in the channel calculated
using the conduction band potential and
the relative potential at the interface of Si
and Si02.

0.15

0.20 0.25 0.30 0.35

gate voltage: Vgs(V)
Fig. 7. The RATE of the drift current
component at the interface for various
gate voltages. At Vgs=0.215V, the drift
component is equal to the diffusion
component.