Constructing a Wafer Defect Diagnostic System by Integrating Yield

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________________________________________



:
Corresponding Author

The 11th Asia Pacific Industrial Engineering and Management Systems Conference

The 1
4
th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

7


10 Dec敭ber 201
0


Constructing a Wafer Defect Diagnostic System by Integrating Yield
Prediction and Defect Pattern Recognition


Li
-
Chang Chao


Graduate Institute

of Industrial Management,
Taiwan Shoufu
University

No.
168
, Nanshi Li, Madou Town, Tainan County 72153, Taiwan(R.O.C.)

Tel.: +886
-
6
-
5718888 ext.854; fax: +886
-
6
-
571
247
3

Email: fredchao@dwu.edu.tw


Li
-
I Chao

Computer Center,
Taiwan Shoufu

University

No.
168
, Nanshi Li, Madou Town, Tainan County
72153, Taiwan(R.O.C.)

Email: liichao@yahoo.com.tw



Abstrac
t

-

Wafer yield is a highly effective means of evaluating the process capability of integrated circuit
manufacturers. The defect number and cluster intensity of defects on a wafer are two critical factors
influencing wafer yield. As wafer sizes increase, the c
lustering phenomenon of defects increases. Clustered
defects cause the conventional Poisson yield model to underestimate actual waf
er yield. The cluster
parameter

α

of the negative binomial model can be very scattered and negative when the model is applied

to
predict yield. Compound Poisson yield models are complicated. The degree of fitness must be considered
when the regression methods are utilized to model the yield. Obtaining good prediction network requires
substantial effort to identify the parameters

of back
-
propagation neural network. Although some yield models
consider the effects of defect clustering on yield prediction, these models have some drawbacks. Furthermore,
the possible causes of process variation can be found by operators through analyzi
ng the defect pattern on a
wafer. Judging the process variation by operators is time
-
consuming and the accuracy of variation detecting
can be influenced resulting from the erroneous judgment. Although some recognizing methods considering
defect pattern on
a wafer were proposed, these recognizing methods still have some flaws. This study presents
a novel wafer defect diagnostic system that utilizes a general regression neural network integrating a multi
-
class support vector machines to predict the wafer yiel
d and recognize the defect pattern on a wafer. A
simulation study is utilized to demonstrate the effectiveness of the proposed method.


Keywords:

Defect, clustering phenomenon, yield model, general regression neural network, pattern
recognition, support ve
ctor machines
.




1. INTRODUCTION


As wafer sizes increase, the clustering phenomenon of
defects increases. Clustered defects cause

the conventional
Poisson yield model underestimate actual wafer yield, as
defects are no longer uniformly distributed over a wafer.

Although some yield models

consider the effects of defect
clustering on yield prediction, these models have some
drawbacks.

Furthermore,

Wafers are inspected during
manufacturing by retrieving information about defect
number and defect pattern by manually inspecting or
automatically classifying defects.

However, human
recognition of defect patterns can be time
-
consuming and
in
accurate.

Although some recognizing methods for defect
pattern on a wafer were proposed, these recognizing
methods still have some flaws.

Numerous mathematical models have been developed
for predicting wafer yield in the last 40 years
(
Cunningham,
J.

A. 19
90
)(
Stapper, C.

H.
1991
)(
Stapper, C. H. & Rosner,
R. J. 1995
)(
Tyagi, A.

& Bayoumi, A.

M. 1992
)
. The
Poisson model is the simplest model to use
.

However,
Stapper
(1985)
reported that defects are typically clustered
rather than dispersed randomly over a
wafer. Clustered
defects usually violate the independence assumption of the
Poisson model.

Under this scenario, numerous yield models
obtain more accurate yield predictions than the Poisson


The 11th Asia Pacific Industrial Engineering and Management Systems Conference

The 1
4
th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

7


10 Dec敭ber 201
0


model.
The
Compound Poisson yield models are
complicated

(
Cunningh
am, J.

A. 1990
)
. The cluster
parameter


of the negative binomial model can be very
scattered and negative when the model is applied to predict
yield
(
Cunningham, J.

A. 1990
)
. Dupret and Kielbasa
(2004)

use the partial least square (PL
S) regression methods to
model the yield from measurements obtained during the
production. However, an advanced statistics is needed to
use the PLS regression methods.

Consequently, these
mathematical yield models have particular problems in
predicting waf
er yield.

S
tatistical approach, heuristic approach and
simulation approach

are
fundamental approaches to solv
e

pattern recognition problems

(
Nieddu, L. & Patrizi, G.
2000
)
.
A
n underlying statistical model for generating these
patterns

is

utilized to classify patterns by the
statistical
approach
.

The heuristic approach utilizes soft computing
schemes to perform pattern recognition.
However,

the

expensive evaluation processes
to achieve optimal solutions
need to overcome
(
Bhanu

et al
.

1995
)
.
The simulation
approach
subsequently lead
s

to a class of artificial neural
sys
tems termed neural networks
(
Jain

et al
.
2000
)

(
Nieddu,
L. & Patrizi, G.
2000
)
. However, neural networks
need to
adequately

determine the
parameters of the networks

(
Fiesler
, E.
1994
)
.


Constructing
wafer
yield

model

and c
onstructing

wafer defect
pattern
recognition
are

important issue
s

in
integrated circu
its (IC) manufacturing
.

This study presents
a novel wafer defect diagnostic system that utilizes a
general regression
neural network

(GRNN)

integrating a
multi
-
class support vector machines

(SVM)

to predict the
wafer yield and recognize the defect pattern on a wafer. A
simulation study is utilized to demonstrate the effectiveness
of the proposed method.


2.
RELATED
LITERATURE


The defect cluster indices
,

which
consider the effects
of defect clustering on yield prediction,

are introduced.
Approaches to
predicting wafer yield and approaches to
solving pattern recognition problems are then surveyed.


2.1
Defect Cluster
Index


Many cluster indices have been developed to
describe

the intensity of d
efects scattered on a wafer
(
Stapper, C.

H.
1973
) (
Tyagi, A. & Bayoumi, A. M.
1992
, 1994)
. The
ne
gative binomial yield model

utilizes a cluster parameter


to measure the intensity of defects clustered

(
Stapper,
C.H. 1973
)
. Tyagi and Bayoumi
(1992, 1994)

proposed a
variance/mean ratio
M
V
/

to evaluate the intensity of
defects clustered. Jun
et al
.
(1999)

proposed a cluster index
CI

to evaluate the intensity of defects clustered on a wafer.

Chao
(2009)

proposed a cluster index
CI
E

for depicting the
varying intensity of wafer cluster defects
.


2.
2

Y
ield
M
odels


The Poisson yield model
,

which is based on the
Poisson distribution

(
Ferris
-
Prabhu, A.

V. 1992
)
. The
Poisson yield model was sufficiently effective for small
chip sizes and tended to underestimate yields for larger chip
sizes
(
Cunningham, J. A. 1990
)
. To identify the clustering
properties of defects in the yield model, some

spatial
distributions, including compound Poisson distributions,
have been considered
(
Raghavachari, M.

et al
.

1997
)
. The
compound Poisson yield model replaces defect density,
which is assumed to be a constant in the Poisson yield
model, with a probabilit
y density function. The negative
binomial yield model, which is a widely applied yield
model, employs a gamma function for the distribution of
defect density
(
Okabe, T.

et al
.

1972
)(
Stapper, C.

H. 1973
)
.
The negative binomial model has been shown to be a
powerful prediction model in IC manufacturing. However,
reports also show that the cluster

parameter
in the negative
binomial model can be very scattered and negative when
the model is used to predict yield

(
Cunningham, J. A. 1990
)
.
Other yield models used

in various companies are
summari
zed in

the literature (
Stapper, C. H. & Rosner, R. J.
1995
)
.


2.
2.1

G
eneral
R
egression
N
eural
N
etwork


GRNN

is a

three
-
layer network model

(
Specht, D.

F.
1991
)
.

Input units are merely distribution units which
forward measurement variables to the pattern units in the
second (hidden) layer. This
hidden layer consists of one
neuron for each pattern in the training pattern.

The GRNN
is essentially trained after one p
ass of the training patterns
and its activation function normally uses an exponential
function. The unique parameter of GRNN is the smoothing
factor


which influences the output value; that is, high
smoothing factors produce

increased

relaxed surface fits

throughout the data.

Unlike the conventional regression
model, GRNN can be defined through its joint continuous
probability density function, rather than utilizing a
specified function that must be determined in advance.

The
GRNN model utilizes a Parzen

window

(
Parzen, E.
1962
)
,
which is a nonparameter approach to estimating the joint
continuous probability density function
.

GRNN measures how far a given sample pattern is
from patterns in the training set. When a new pattern is
presented to the network,
the input pattern is compared to
all of the patterns in the training set to determine how far it
is from those patterns. The output that is predicted by the
network is a proportional amount of all of the outputs in the


The 11th Asia Pacific Industrial Engineering and Management Systems Conference

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4
th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

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training set. The proportion is based

upon how far the new
pattern is from the given patterns in the training set. GRNN
uses an algorithm to find appropriate individual smoothing
factors for each input as well as an overall smoothing factor.
The algorithm proceeds in two parts. The first part

trains
the network with the data in the training set. The second
part tests a whole range of smoothing factors. The method
will produce networks which work much better on the test
set.

The major difference between GRNN and other
supervised neural networks

is that GRNN can treat
continuous valued outputs and categorize data, and t
here
are fewer training parameters

are required. Moreover,
GRNN

can be used for any regression problem in which a
linearity assumption is violated, and it converges fast on the
opt
imal regression surface as the number of samples
becomes substantially large. The GRNN model, then, is
used in this study to
construct

wafer yield

model
.


2.3 Recognizing Defect Patterns


Many techniques used for wafer defect pattern
recognition are
statistical approach, heuristic approach and
simulation approach (Nieddu, L. & Patrizi, G. 2000). The
statistical approach can be viewed as determining a strategy
for classifying samples based on the measurement of
feature vector, such that classification
error is minimized.
The heuristic approach attempts to clarify the essential
problem and use available personal knowledge to solve it
with the assistance of soft computing schemes. But there
exists lots limitations (Bhanu et al. 1995) (Chen, C. L. &
Chang,

M. H. 1998). The simulation approach emulates the
computational paradigm of a biological system. Current
knowledge of cerebral processes is transferred from a
neuro
-
physiological medium to an electronic one. This
leads to neural networks. But there exists

lots drawbacks
(Jain et al. 2000) (Nieddu, L. & Patrizi, G. 2000).
SVM

have been widely used for pattern recognition in recent
years. Several studies report that the
SVM

classification is
more accurate than existing classification algorithms (Hsu,
C. W. &

Lin, C. J. 2002) (Joachims, J. 1998).

The
multi
-
class

SVM
, then, is used in this study to
r
ecogniz
e

wafer
d
efect
p
atterns
.


2.
3
.1

Support Vector Machines


SVM

techni
que was introduced by Vapnik
(
Cortes, C.
& Vapnik, V.
1995
)
. The original intent of the SVM
algorithm was to use a linear separating hyperplane to build
a classifier.
F
or all hyperplanes separating data, there exists
a unique optimal hyperplane distinguished by the
maximum margin of separation between any tr
aining

point
and the hyperplane
.

If the training set of instance
-
label
pairs are non
-
linearly separable, the linear
SVM

may not
work well again. The non
-
linear kernel can then solve the
classification problem. The most commonly applied non
-
linear kernels are the polynomial kernel, the Gaussian
kernel and the sigmoid kernel. The classification problem
can obtain reasonable
results when the Gaussian kernel is
applied to map samples into a higher dimensional space
(
Keerthi, S. S. & Lin, C. J.
2003
)
.

The classification problem mentioned above refers to
binary classification. Many real
-
world problems, however,
have more than two

classes.
A multi
-
class
SVM

can be
employed to solve the classification problem that have
more than two classes. Many methods have been developed
to solve multi
-
class
SVM

such as the
one
-
against
-
all
method
(
Bottou

et al
. 1994),
one
-
against
-
one method
(
Kre
B
el
, U. 1999
),
Directed Acyclic Graph method

(
Platt

et
al
. 2000) and c
onsidering all
d
ata at a
o
nce

(
Vapnik, V.
1998
).
This study utilizes a multi
-
class
SVM

for wafer
defect pattern recognition.
Because the training time of the
one
-
against
-
one method is th
e shortest of these methods
(
Hsu, C. W. & Lin, C. J. 2002
)
, this method is used for
wafer defect pattern recognition in this study.


3
.
PROPOSED APPROACH


Wafers must be further analyzed to determine whether
a specific defect pattern causes the medium or l
ow yield.
Therefore, the factors affecting yield are selected as the
features for
the yield model and

the
pattern

recognition in
the
wafer defect diagnostic system
.

This study
construct
s a

wafer defect diagnostic system that utilizes a
GRNN
network

integrating a multi
-
class
SVM

to predict the wafer
yield and recognize the defect pattern on a wafer.


3
.
1

Feature

Selection


Yield models
can be described as

)
,
,
(
K
A
D
f
Y




(
1
)

w
here

D

represents
the average number of
defects per unit
area
,

K

represents an empirical correction factor for chip
area
A

(
Cunningham, J. A. 1990
)
.

The average number of
defects per unit area

D

can be used to describe the intensity
of the defect
-
dense areas on a wafer. The average number
of defects per unit area
D

can be used as a feature factor
for
the
wafer defect diagnostic system
.

The defect number and cluster intensity of defects on a
wafer

are two critical factors that may influence wafer yield.

T
he angle variation
A
CV

and the distance variation
D
CV

obtained by measuring the angle variation and the distance
variation of the individual defect on a wafer
are also
utilized as feature factors. The
A
CV

and
D
CV

can be
derived as follows:

Step 1: Determine the positive angle
i

,
which is the


The 11th Asia Pacific Industrial Engineering and Management Systems Conference

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th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

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angle
between the coordinates of individual defect and the
x
-
axis. The
i


can be described as

n
i
x
y
i
i
i
,...,
2
,
1
,
tan
1















(
2
)

where
i
x

and
i
y

denote the x and the y coordinates,
respectively, of the
i
-
th defect in the x
-
y plant. Sorting
i


in ascending order obtains
)
(
i

.
A sequence of angle
differences is defined as

n
i
A
i
i
i
,...,
2
,
1
,
)
1
(
)
(









(
3
)

where
0
)
0
(


.

Step 2: Determine
i
L

as the distance
between the
individual defect and the origin in the coordinate axes. The
i
L
can be described as

n
i
y
x
L
i
i
i
,...,
2
,
1
,
2
2






(
4
)

Sorting
i
L

in ascending order obtains
)
(
i
L
.
The
sequence of distance differences is defined by

n
i
L
L
D
i
i
i
,...,
2
,
1
,
)
1
(
)
(






(
5
)

where
0
)
0
(

L
.

Step 3: The
A
CV

and
D
CV

are defined
as

A
S
CV
A
A



(
6
)

D
S
CV
D
D



(
7
)

where
A

and
2
A
S

denote the sample mean and the
sample variance of
i
A
, respectively, and
D

and
2
D
S

denote the sample mean and the sample variance of
i
D
,
respectively. The variations of the ang
le differences and the
distance differences are smaller when defects are randomly
distributed than when defects are clustered. One of these
two variations is increased regardless of the defect pattern.
Therefore, the wafer map presents certain patterns of
defect
clusters as long as one of these differences posses a large
variation.
Therefore,

A
CV

and
D
CV

can provide feature
factors for
the wafer defect diagnostic system.


Moreover, t
he cluster index
E
CI

is

utilized to
be a
feature factor

and can be described as:






s
i
i
i
s
E
p
p
p
p
p
CI
1
2
2
1
))
1
(
log
(
)
,
,
,
(



(
8
)

where
s

represents the number of defect clusters;
i
p

represents the proportion of defects in the
i
th

cluster to total
number of wafer defects.

The more profound the cluster
phenomenon, the larger the

E
CI
.

Clearly, the
E
CI

possesses the advantage of accurately detecting the
intensity of clustering defects.

E
CI

is employed

as a
feature factor
for
the wafer defect diagnostic system.



3
.
2

Constructing Diagnostic System


A major cause affecting yield is the degree to which
defects are clustered (Friedman
et al
. 1997) (Stapper
et al
.
1983). In addition to the random pattern, common wafer
defect clustering patterns include bull’s eye pattern,
crescent moon pattern, bottom pattern and edge pattern
(Friedman
et al
. 1997).

Four feature factors (
D
,
A
CV
,
D
CV

and
E
CI
) are
suggested for
the wafer defect diagnostic system
.
The
GRNN

network predicts

wafer yield

by

employing these
four feature factors as inputs

and t
he actual yield of the
wafer
as

the only
output of
the GRNN yield model. The
percentage of the chip without defects on a wafer is used as
the actual yield value of the wafer
. T
hen
,

a

multi
-
class
SVM classifies wafer defect patterns by
employing these
four feature factors as inputs and one of five defect pa
tterns
as output
.

The relationships
among

these feature factors
,
yield
s

and defect patterns can be constructed by presenting
the adequate training and testing samples in the
wafer
defect diagnostic system.

T
he proposed approach for the
w
afer
d
efect
d
iagnostic
s
ystem

can be described as follows:

Step 1: Obtain the simulated defect wafer map. Utilize
Borland Delphi programming language to simulate all
possible defect clustering patterns for 8
-
inch wafers.

Step 2: Calculate
the values of
all
feature f
actors

for
each
wafer
. For each defect clustering pattern on a wafer,
calculate the
se

f
our feature factors (
D
,
A
CV
,
D
CV

and
E
CI
)
.

Step 3: Build a GRNN yield model. Input the
feature
factors

in Step 2 into the GRNN yield model. The actual
yield of the wafer is the only output of the GRNN yield
model. The percentage of the chip without defects on a
wafer is used as the actual yield value of the wafer. In this
study, the neural n
etworks package NeuroShell 2 is
employed to train and test the GRNN network.

Step 4
:
Build a
multi
-
class SVM classifi
er
.

These f
our

identical

feature factors are suggested for recognizing
defect patterns. A multi
-
class SVM classifies wafer defect
patterns
by employing these four feature factors as inputs
and one of five defect patterns as output.

In this study, the
LIBSVM
is employed to train and test the

multi
-
class SVM



The 11th Asia Pacific Industrial Engineering and Management Systems Conference

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th Asia Pacific Regional Meeting of International Foundation for Production Research

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(
Chang, C. C. & Lin, C. J. (2004)
(
Hsu, C. W. & Lin, C. J.
2002
)
.

Step 5:
D
iagnose

the wafers with

medium or low yield
.
Input the wafers
with

medium or low yield
predicted in
Step 3 to
the trained
multi
-
class SVM

in Step 4.

Wafers
can

be further analyzed to determine whether a specific defect
pattern causes the medium or low yield.



4
.
IMPLEMENTATION

4
.1
Simulation Study


This study employs three design factors to simulate
defect cluster patterns

in 8
-
inch wafers
: defect number,
percentage of defects located in grey regions and size of
grey regions.

Defect number

is the

number of defects
distributed over the entire wafer. Five factor levels for 25,
50, 100, 200 and 300 defects are simulated.

Percentage of
defects located in grey region represents the defect
-
dense
areas on a wafer. In the four clustering patterns,
four

fa
ctor
levels for 80%, 85%, 90% and 95% are simulated, and the
remaining d
efects are distributed randomly
.

Three sizes of
grey regions considered
are

25, 49 and 81 cm
2
.


According to the above three design factors. Each
trial of factor
-
level combination is r
eplicated five times, to
obtain 12
2
5 simulation trials. Specifically, there are 12
2
5
simulated wafer maps.

T
he 12
2
5 simulated wafer maps
were utilized as samples for constructing the
wafer defect
diagnostic system
.
The 12
2
5 wafers were divided into two
par
ts: one part containing
98
0 wafers used to train the
diagnostic system
; the second part containing 2
4
5 wafers
used to test the accuracy of the
diagnostic system
.
F
our
feature factors (
D
,
A
CV
,
D
CV

and
E
CI
)

are obtained
for each simulation wafer by simple calculation. These f
our

feature factors
are utilized as inputs
,
the resp
ective
yield
and
defect pattern of the 12
2
5 wafer maps are utilized as
outputs for the proposed
diagnostic

system
. The trained
diagnostic system

can then be
further analyzed to determine
whether a specific defect pattern causes the medium or low
yield.


Software utilized in this study for
GRNN network
was
the neural networks package NeuroShell 2

and for
multi
-
class SVM
was

LIBSVM
(
Hsu, C. W. & Lin, C. J.
2002
).

To obtain the generalization results, five
-
fold cross
-
validation was used to determine optimal parameter
combinations.
In this study, the unique parameter of GRNN
network, that is, the smoothing factor,

is set at 0.07
22, and
t
he penalty parameter and the kernel parameter for multi
-
SVM

were
8192

and 0.125, respectively.

Extra
1
0

wafer maps

are simulated to show

the
reproductive performance of the proposed d
iagnostic
s
ystem
. Table
1

summarizes the
attributed values

for th
ese
10

wafer
s.
The attributed value of GRNN predicted yield
column show that
w
afer

1, 2, 4, 6, 8, 10

present
the
medium or low yield
.

T
hese wafers must

be further
analyzed to determine whether a specific defect pattern
causes the
ph
enomenon

of yield down
.

Table
2

shows
the
actual defect pattern and the respective pattern recognized
by the multi
-
class SVM

for th
ese 6

wafer
s.
Table
2

reveals
that the proposed approach produces
a good

diagnostic

for

the wafers
with

medium or low yield
.


5
.
CONCLUSION


This study presents a novel
diagnostic system

that
utilizes

a GRNN network for predicting wafer yield

and

utilizes
a
multi
-
class
SVM

for recognizing wafer defect
patterns. A simulated case is applied to demonstrate the
effectiveness of the proposed model
.

The merits of the proposed approach are summarized
as follows:

1
.

The proposed method utilizes
f
our

relevant
feature
factors
:

D
,
A
CV
,
D
CV

and
E
CI

as input for
constructing the wafer defect
diagnostic system
. The
diagnostic

result
s

show that the proposed
method

achieves
accurate
diagnostic
.

2.
The proposed system can be integrated with KLA
inspection machines to
d
iagnose

wafers presenting medium
or low yield.


Table 1
:
Attributed values

for th
ese extra 10

wafer
s


















Table 2
:
Actual defect pattern and
the
multi
-
class SVM
recognized
pattern

for th
ese 6

wafer
s










The 11th Asia Pacific Industrial Engineering and Management Systems Conference

The 1
4
th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

7


10 Dec敭ber 201
0





ACKNOWLEDGMENT


The authors would like to thank the National Science
Council of the Republic of China, Taiwan, for financially
supporting this research under Contract No. NSC
99
-
2218
-
E
-
434
-
001
-
.


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The 11th Asia Pacific Industrial Engineering and Management Systems Conference

The 1
4
th Asia Pacific Regional Meeting of International Foundation for Production Research

Melaka
,

7


10 Dec敭ber 201
0


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AUTHOR BIOGRAPHIES

Li
-
Chang Chao

is a
Assistant Professor

in
Department of
Industrial
Management
,
Diwan University
.

He received a
Doctoral Degree

from the
Department of Industrial
Engineering and Management

at
National Chiao Tung
University
,
Taiwan (R.O.C)
in
200
9
. His teaching and
research interests include operations research
, quality
management and data mining
.

His email address is
<
fredchao@dwu.edu.tw
>


Li
-
I

Chao

is a
Lecturer
in
Computer Center
,
Diwan
University
.

He received a
Master

Degree

from the
Department of
Information

Management

at
National
Kaohsiung

University

of Applied Sciences
,
Taiwan (R.O.C)
in
200
8
. His teaching and research interests include
data
mining and knowledge management
.

His email address is
<
liichao@yahoo.com.tw
>