Analytical high frequency channel thermal noise modeling in deep sub-micron MOSFETs

Abstract

A simple high frequency channel thermal noise model was developed for MOSFETs in strong inversion region. Short channel effects such as channel length modulation effect and mobility degradation due to vertical field were taken into account in the current-voltage model and channel thermal noise model. It was found that the long channel Tsividis' noise model is still valid for short channel devices by including the proposed effective mobility model and the channel length modulation effect. Good agreement has been obtained between the simulated and measured results across different frequencies, gate biases and drain biases.

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This document is downloaded from DR-NTU, Nanyang TechnologicalThis document is downloaded from DR-NTU, Nanyang Technological
University Library, Singapore.University Library, Singapore.
Title
Analytical high frequency channel thermal noise modeling
in deep sub-micron MOSFETs.
Author(s)
Ong, Shih Ni.; Yeo, Kiat Seng.; Chew, Kok Wai.; Chan,
Lye Hock.; Loo, Xi Sung.; Do, Manh Anh.; Boon, Chirn
Chye.
Citation
Ong, S. N., Yeo, K. S., Chew K. W. J., Chan, L. H. K.,
Loo, X. S., Do, M. A., et al. (2009). Analytical high
frequency channel thermal noise modeling in deep sub-
micron MOSFETs. Proceedings of the 12th International
Symposium on Integrated Circuits, (pp.306-309)
Singapore.
Date2009
URLhttp://hdl.handle.net/10220/6354
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ISIC 2009
Analytical High Frequency Channel Thermal Noise
Modeling in Deep Sub-micron MOSFETs

S.N. Ong1,2, K.S. Yeo1, K.W.J. Chew1,2, L.H.K. Chan1, X.S. Loo1,2, M.A. Do1 and C.C. Boon1
1Centre for Integrated Circuits & Systems
Nanyang Technological University
Nanyang Avenue, Singapore 639798
2Chartered Semiconductor Manufacturing Ltd
Woodlands, Singapore 738406.


model was developed for MOSFETs in strong inversion region.
Short channel effects such as channel length modulation effect
and mobility degradation due to vertical field were taken into
account in the current-voltage model and channel thermal noise
model. It was found that the long channel Tsividis’ noise model is
still valid for short channel devices by including the proposed
effective mobility model and the channel length modulation
effect. Good agreement has been obtained between the simulated
and measured results across different frequencies, gate biases
and drain biases.
Abstract— A simple high frequency channel thermal noise

Index Terms— Channel length modulation, current-voltage
model, effective mobility model, high frequency channel thermal
noise, mobility degradation, Tsividis’ model.
I.
INTRODUCTION
Complementary metal oxide semiconductor (CMOS) has
emerged as leading candidates for wireless applications due to
the advantages of being low cost while offering a high level of
integration and low power consumption. The continued
scaling to deep sub-micrometer dimension has further
improved the RF performance of CMOS devices resulting in
extremely high unity-gain frequencies of tens to hundreds of
gigahertz [1]. However, a physics-based accurate model of
high frequency noise of deep sub-micron MOSFETs is crucial
for developing a low noise RF circuit. Simple analytical
equation of the noise model is needed to shorten the design
cycle.
At RF frequencies, the channel thermal noise is the
dominant source of noise. The channel thermal noise in short-
channel MOSFET has been studied in many publications [2-
11]. The short channel MOSFETs noise model deviate from
long channel noise model because of short channel effects
suggested such as velocity saturation effect, carrier heating
effect, and channel length modulation effect. In this work, our
approach to derive a channel thermal noise model, presented in
section V in this paper, is similar to Tsividis’ model while
considering the mobility reduction due to the vertical field and
the channel length modulation effect. The current-voltage
model is discussed in section III. The channel length
modulation effect and effective mobility model are elaborated
in section IV. After that, comparisons were made in section VI
between the proposed model and the existing model together
with the measurement data. The measurement setup is
discussed in section II.
II.
MEASUREMENT
In this paper, the device under test (DUT) and OPEN
structures were fabricated using Chartered Semiconductor
Manufacturing Ltd’s 0.13μm RFCMOS technology process.
The devices measured are NMOS transistors with channel
width per finger ??? 5?? and number of finger ??? 4,
while the channel length
? ? 0.13??,0.25?? and 0.50?? . The current-voltage
characteristic was measured using Keitley 4200 semiconductor
characterization analyzer. Besides, on-wafer scattering
parameters and noise parameters measurement was performed
using HP8510 network analyzer and ATN NP5B Microwave
Noise Parameter System. The frequency used in this paper is
ranging from 2GHz to 25GHz, with gate biases from 0.4V to
1.2V and drain biases from 0.6V to 1.2V.
?
was varies with
Upon obtaining measurement data, those data were de-
embedded in order to minimize errors due to pad parasitic and
interconnect parasitic [12]. Thereafter the power spectral
density of channel thermal noise ??? was extracted from the de-
embedded noise measurement data.
III.
CURRENT-VOLTAGE MODEL
The drain current of MOSFET is dominated by drift
current in strong inversion, and generally can be expressed as

??? ??????????? (1)
where ? is the total channel width of device, ?????? is the
inversion channel charge per unit area, and ???? is the drift
velocity of carrier in channel. The inversion channel charge
per unit area ?????? can be expressed as

?????? ? ???????? ?????? (2)

???? ???? ??? (3)
where ??? is the gate oxide capacitance per unit area, ??? is
the gate-to-source voltage, ??? is the threshold voltage, ????
is the surface potential along the channel at distance ? from
source, which varies from 0 at source side to ??? at drain side,
and ? is the bulk charge effect coefficient.
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As the channel length of MOSFET shrinks, short channel
effects such as velocity saturation effect and channel length
modulation effect should be considered in the drain current
equation. The velocity saturation effect due to high lateral
electric field is one of the reasons which caused the electrical
characteristic of short channel devices deviated from long
channel device [11], and can be included in the carrier drift
velocity equation [11].

???? ?
????????
??????
??
(4)
where ???? is the effective surface mobility, ???? is the lateral
electric field along channel, ?? is the critical electric field,
which is equal to 2???? ????
⁄
.
Finally the drain current equation can be shown as [13]
???
?
?
?
?
?
?
?????
???
???????
???????
?
?????????????
?
??, ???? ?????
?
???????
??????
?
????????????????????
?
?
?, ???? ?????
(5)
where ? is the total channel length, ????? is the drain saturation
voltage, which can be expressed as [13]

??????
?
?????
???
????
?? (6)
and ???? is the effective channel length for the gradual channel
region.
IV. CHANNEL LENGTH MODULATION AND EFFECTIVE
MOBILITY MODEL
A. Channel Length Modulation
In short channel devices, the effect of channel length
modulation become more prominent and cannot be neglected.
The length of gradual channel region no longer closes to the
drawn length or the total channel length. Thus, the effective
channel length for the gradual channel region need to be
considered, and can be expressed as [2,15]

????? ? ? ΔL (7)
where ?? is the length of velocity saturation region, which is

Δ? ?
?
???????????????????
??
? (8)

??? ???1 ? ?????????????
??
?
? (9)

? ? ??
?
?
???
??????? (10)
with ?? is the junction depth, and ? is the channel length
modulation coefficient obtained from drain current versus
drain voltage plot.
B. Effective Mobility Model
In MOSFET, the carriers flow in the silicon layer very
close to the silicon-silicon dioxide interface which is rougher
than the silicon layer in the bulk. Hence, the carriers’ mobility
is degraded to be lower than those in bulk silicon. The
mobility of carriers in the inversion layer is affected by the
vertical electric field, ??. The mobility reduction due to
vertical field is defined as surface mobility or effective
mobility, ????, and is approximated by [14]

?????
??
?????????
? (11)
where ?? is the low-field surface mobility, ?? is fitting
parameter, and ???? is the average electric field in ? -
direction. ???? can be approximated by

?????
?
????,???????? ??,????? ?
?
????????
?
? ??? (12)
where ???? is the total inversion layer charge and ?????
?????,???? , ?? is the total depletion layer charge and
??? ?????,????. Thus, the average electric field is function
of both gate bias and drain bias.
Conventionally, the effective mobility is approximated to
be [14,16]

????_?????????????
??
??????? (13)
with the drain bias dependency effect is assumed to be
negligible.
However, as the device channel length continues to scale
down, the effect of drain bias in the inversion charge density
and depletion charge density are no longer negligible. As drain
bias increases in short channel devices, the increment in
inversion charge and depletion charge will become significant,
and thus need to be taken into account. Furthermore, the
effective mobility will influence the critical electric field,
which will further affect the effective channel length, ????
calculation. Hence, the effect of this drain bias dependency of
the effective mobility model can significantly influence the
total channel thermal noise.
As a result, the effective mobility is proposed to be
approximated by

????_?????????
??
???????????????????? (14)
where ?? and ?? are fitting parameters. This proposed
effective mobility model includes the drain bias dependency
term, which caused by the increment of the inversion charge
and depletion charge.
The comparison of the conventional mobility model and
proposed mobility model usage in the channel thermal noise
model will be presented in section VI.
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V.
CHANNEL THERMAL NOISE MODEL
To calculate the channel thermal noise, the linear part of
the channel is divided into small sections of length ??. Each
section is assumed to act as a resistor of value ?? which can
be obtained as ?? ????
⁄
, where ???? is the local channel
conductance. In thermal equilibrium, the voltage noise spectral
density of each segment ?? is given by the Nyquist expression
4????. With an assumption that the noise sources of different
channel segments are uncorrelated, the power spectral density
of the channel thermal noise ??? is the sum of the drain current
noise generated from each section. Therefore,

????
???
??? ???
?
?
(15)
Similarly, it has been shown by Tsividis that [14]

???? 4??
?
????????? (16)
where

????? ?????????????
????????
????? (17)

? ? ?1 ?
???
?????, ???? ?????
0, ???? ?????
(18)
In short channel devices, ? can be replaced by ???? in (14)
and ? can be replaced by ????. Thus, the channel thermal
noise is proposed to be expressed as

???? 4??
????
????
?
??????? (19)
VI.
RESULTS AND DISCUSSIONS
The effect of conventional effective mobility equation in
(13) and proposed effective mobility equation in (14) on
channel thermal noise model in (19) was compared in Figures
1 and 2. The “proposed model” denotes the model using (19)
and including effect of (14), while the “conventional model”
indicates the model using (19) and including effect of (13).
Figure 1 illustrates that the channel thermal noise model
using either the conventional effective mobility equation of
(13) or the proposed effective mobility equation of (14) can
predict the gate bias dependency of channel thermal noise.
However, from Figure 2, it can be seen that the channel
thermal noise model with the proposed effective mobility
equation can predict the drain bias dependency of channel
thermal noise more accurately than the channel thermal noise
model with conventional effective mobility equation. As
channel length shrinks, the improvement becomes more
prominent by including the proposed effective mobility
equation in channel thermal noise model.
Figures 3, 4 and 5 show the comparison between the
proposed model with modified Tsividis’ model and the
original Tsividis’ model for channel length ? ? 0.13??
across frequencies, gate biases and drain biases. In this paper,
Tsividis’ model is modified by including the conventional
effective mobility model and the channel length modulation
effect discussed in section IV in order to model the channel
thermal noise in short channel MOSFETs. Another well-
known model, the model of Chen and Deen [2], includes the
channel length modulation effect and the hot carrier effect.
However, the hot carrier effect observed to be negligible
experimentally. In this case, model of Chen and Deen is
similar with the modified Tsividis model and thus will not be
shown in Figures 3 to 5.
Figures 3 to 5 indicate that by substituting the proposed
effective mobility equation and effective channel length
equation in Tsividis model, this simple channel thermal noise
model is able to predict the channel thermal noise behaviors for
deep sub-micron MOSFETs across frequencies, gate biases and
drain biases.

Figure 1 Comparison between proposed model and conventional model
with extracted data from noise measurement across gate biases at
frequency ? ? 5??? and ??? ? 1.2?. The comparison is performed for
device with total width ? ? 4 ? 5?? , and channel length ? ?
0.13??, 0.25?? and 0.50??. The symbols represent the extracted data
from noise measurements, while the solid line represents the proposed
model, and the dash line represent the conventional model.


Figure 2 Comparison between proposed model and conventional model
with extracted data from noise measurement across drain biases at
frequency ? ? 5??? and ??? ? 1.2?. The comparison is performed for
device with total width ? ? 4 ? 5?? , and channel length ? ?
0.13??, 0.25?? and 0.50??. The symbols represent the extracted data
from noise measurements, while the solid line represents the proposed
model, and the dash line represent the conventional model.
0
10
20
30
40
50
60
70
80
90
00.20.40.6
VGS(V)
0.811.21.4
Sid(x10-23) (A2/Hz)
Frequency = 10GHz
VDS = 1.2V
Measured
Proposed model
Conventional model
L = 0.13μm
L = 0.25μm
L = 0.50μm
0
20
40
60
80
100
120
140
0.60.70.80.911.11.21.3
Sid(x10-23) (A2/Hz)
VDS (V)
Measured
Proposed model
Conventional model
Frequency = 10GHz
VGS = 1.2V
L = 0.13μm
L = 0.25μm
L = 0.50μm
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VII.
CONCLUSION
In this paper, a simple analytical high frequency channel
thermal noise model is presented. The model is based on
Tsividis’ noise model, with the inclusion of proposed effective
mobility model and channel length modulation effect. The
proposed model has been verified with data extracted from
noise measurements and is found to agree well. We found that
the proposed effective mobility model has more prominent
effect for shorter channel length devices. By using this
effective mobility model in the channel thermal noise model,
the deep sub-micron MOSFET noise performance can be
predicted for different frequencies, gate biases and drain biases
without complex calculation. Thus this model can be easily
implemented by IC designer in a simulation environment.
ACKNOWLEDGMENT
The authors are grateful to Chartered Semiconductor
Manufacturing Ltd for fabricating the test structures.
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]

Figure 3 Comparison between the proposed model with the modified
Tsividis’ and Tsividis’ model across frequencies for channel length
? ? 0.13?? and channel width ? ? 4 ? 5?? device. The symbol
(◊) represents extracted noise measurement data. The line (--)
represents calculated data using the model.
Figure 4 Comparison between the proposed model with the modified
Tsividis’ and Tsividis’ model across gate biases for channel length
? ? 0.13?? and channel width ? ? 4 ? 5?? device. The symbol
(◊) represents measurement data. The line (--) represents calculated
data using the model.
160
Figure 5 Comparison between the proposed model with the modified
Tsividis’ and Tsividis’ model across drain biases for channel length
? ? 0.13?? and channel width ? ? 4 ? 5?? device. The symbol
(◊) represents measurement data. The line (--) represents calculated
data using the model.
0
20
40
60
80
100
120
140
160
180
0.005.0010.0015.0020.0025.0030.00
Sid(x10-23) (A2/Hz)
Frequency( GHz)
Measured
Proposed model
Modified Tsividis
Tsividis
VGS = 1.2V
VDS = 1.2V
0
10
20
30
40
50
60
70
80
90
00.20.40.60.811.21.4
Sid(x10-23) (A2/Hz)
VGS(V)
Frequency = 10GHz
VDS = 1.2V
Measured
Proposed model
Modified Tsividis
Tsividis
0
20
40
60
80
100
120
140
0.60.70.80.911.11.21.3
Sid(x10-23) (A2/Hz)
VDS (V)
Measured
Proposed model
Modified Tsividis
Tsividis
Frequency = 10GHz
VGS = 1.2V
309
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