International Journal of Electrical & Computer Sciences IJECS-IJENS Vol: 12 No: 02
23
SEMICONDUCTOR MATERIAL AND PROCESS CHARACTERIZATION USING
THREE PROBES
Ateeq Ahmad Khan
College of Engineering,Salman Bin Abdul Aziz University,Al Kharj, Saudi Arabia(ateeq@ksu.edu.sa)
ABSTRACT
One of the most powerful techniques of
semiconductor material and process characterization
is the use of three probes in spreading resistance
configuration. The raw data thus obtained needs
analysis for extracting profiling information. The
problem of poor spatial resolution of Spreading
Resistance measurements on shallow doped layers
has been taken up. An analysis of the raw spreading
resistance data is made and a technique is developed
for the correction of these data and their utilization in
profiling the doped samples. The technique includes
both the areas of application, namely the lateral
profiling across the surface of the doped samples and
the depth profiling of the doped layers using beveled
samples. The mathematical model developed for the
analyses is in conformity with the physical set up of
the experimental measurements. The results of the
analysis in both the above stated cases are presented.
The resulting corrected spreading resistance data are
in good agreement with the values obtained by the
other well established methods.
KEYWORDS
Spreading resistance, spatial resolution,
semiconductor material, doping, and diffusion.
pressed against the specimen surface. The measuring
current is passed through the two outer probes and
the voltage across the two inner probes is measured
using basic voltmeter-ammeter circuitry. The method
directly gives the sheet resistance, a measure of the
carrier concentration per unit surface area of the
sample. For depth profiling this technique is applied
along with the successive layer removal.
The technique however, has two drawbacks. Firstly it
has a poor resolution and secondly it requires care
and too much time for the accurately controlled oxide
growth during successive layer removal. The
Spreading Resistance technique which is the area of
investigation in the present paper is particularly
attractive compared to other well established methods
due to its better resolution, accuracy, lesser time
consumption and ease in its implementation. The
technique has gained popularity in both the areas of
application, namely the lateral profiling across the
surface of the doped samples and the depth profiling
of the doped layers using beveled samples. However,
the technique is still under the process of
development. Earlier workers in this field have
already studied and attempted many of the relevant
problems. The problem related to the resolution of
the technique is dealt with in the present paper.
2.
1.
THE MEASUREMENT TECHNIQUES
INTRODUCTION
Semiconductor material and process characterization
are an integral part of the IC technology. The basic
measurement made on a semiconductor is its sheet
resistivity which is a measure of its impurity content
or carrier concentration. The measurement is usually
made on the check slices prepared simultaneously
during each diffusion step of IC fabrication. The
profile showing the lateral variation of resistivity
along the surface or the variation of resistivity along
the depth characterizes the specimen. With the
advancement of technology, the importance of
reliable, accurate and efficient measurement
technique has considerably increased. The most
widely used and probably the best known technique
so far is the four point probe resistance technique in
which four probes, equally spaced in straight line, are
For a given semiconductor sample the spreading
resistance at a point on its surface due to a given
probe may be defined as the resistance appearing at
the near tip of the probe when it is pressed against the
surface at that point. Thus, if a small diameter probe,
radius r, making a point contact with a sample
surface, is kept at a potential V and a current I passed
into the sample through the probe, then the quotient
V/I gives the spreading resistance (Rs) at the contact
point. It can very well be used to characterize a
semiconductor sample as it has a definite relation
with its conductivity. The measurements may be
made with the usual voltmeter-ammeter circuitry.
Several probe configurations have been suggested. In
the 2-probe configuration, the voltage across the two
current probes is measured and quotient V/I is the
sum of the spreading resistances due to the two
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probes. The spreading resistance under each probe is
to be separated which is often difficult in a practical
set up.
The difficulty is eliminated in the 3-probe
configuration in which a third probe, the potential
probe, is introduced to measure the potential
difference between the sample and one of the current
probes, designated as the critical probe. A remarkable
advantage of the spreading resistance technique is the
ease with which depth profiles are obtained. Here the
much time consuming procedure of the successive
removal of layers is not required. Instead, the sample
surface is beveled at a shallow angle, usually less
than one degree, and the spreading resistance
measurements are made on the greatly magnified
surface thus exposed. As the relation between the
spreading resistance and the resistivity depends upon
a number of physical conditions, it is necessary to
calibrate the apparatus by measuring the spreading
resistance of known resistivity samples. The
calibration curve so obtained is unique for a given
probe. The problem of reproducibility of
measurements has already been taken up and
necessary measures suggested. The error caused by
variation of resistivity within the sampling depth has
also been studied and the correction models
suggested.
In most of the cases, the potential distribution is
determined by considering the non-uniform profiles
of interest to be a multilayered structure of different
resistivity values. The resulting computer routines are
fairly complex and involve large CPU time. For very
shallow layers, the spreading resistance has been
shown to be related to the sheet conductivity as,
ρ = (2-1)
An attractive feature of the spreading resistance
technique is its excellent spatial resolution as
expected from the fact that practically all the voltage
drop V from probe to specimen occurs within a
distance of a few probe radii. But it was observed
that, even with a uniform resistivity sample, the
measured spreading resistance is much higher near an
insulating edge. Typically, with a 2 micron diameter
probe; an increase of 10% at a distance of 400 micron
and of 60% at a distance of 10 micron from the
insulating edge has been reported. It follows that the
spatial resolution is poor, contrary to what was
expected earlier. A theoretical analysis has therefore
been made and a correction procedure is developed
here.
3.
24
THE DATA ANALYSIS
For both the uniform and non-uniform resistivity
samples, the mathematical analysis of the spreading
resistance technique is presented as a mathematical
model which is in conformity with the physical set up
of the experimental measurements.
Figure 1 shows the arrangement of the probe on the
sample surface and the co-ordinate system used in the
analysis. The probes have a spacing ‘d’ and are at a
distance ‘x’ from the insulating edge. Probes 1, 2 and
3 are located at (0,-d/2), (0, d/2) and (0, 3d/2)
respectively the current I enters at probe 1 and leaves
at probe 2, and the potential of probe 1, the critical
probe, is sensed relative to probe 3. The spreading
resistance measurement thus reduces to measuring
the potential of the critical probe relative to probe 3
when the measuring current I flows through the
former, the probe 2 being ignored.
y
Insulating Edge
3
d
-I
-I
Image
2
X
+I
d
+I
1
Figure 1: The Three Probe Arrangement
Assuming a large semi-infinite sample, a lateral
variation of sheet conductivity in the xdirection only is considered. Starting from first
principle, the relevant equations are as follows.
= . = −. ∇
∇. = 0
Therefore,∇. −. ∇ = 0
(3-1)
(3-2)
(3-3)
may be replaced by ,the two being directly
related . So the equation 3-3 changes to
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x
International Journal of Electrical & Computer Sciences IJECS-IJENS Vol: 12 No: 02
∇. − . ∇ = 0
(3-4)
This is the extended Laplace’s equation which can be
expanded as
! !
+ + ! " !
=0
+ ! " !
=0
(3-6)
= 0%& = −'
(3-7)
( = −2*% . ∇;
(3-8)
V=0,y=0
(3-9)
→ ∞%0 → ∞%-, → ∞ )
(3-10)
At, + , − - = % ,
Where V is the potential V(x,y) at any point P(x,y)
and is the current density in x-direction.
For a uniform resistivity sample with a constant sheet
conductance , equation 3-6 is solved analytically
as well as numerically. The analytical solution yields
=
1
23
41 +
!
!
relationship.
(3-12)
The numerical methods can, now, readily be applied.
(3-5)
Equation 3-6 is to be solved under the following
boundary conditions
#$ − the
= 68 7-7
25
(3-11)
The equation is solved numerically using the wellknown ‘Finite-Element’ method. The entire domain
of integration is divided into a number of regions of
finite dimensions, called finite elements. Each
element has several nodes. The local solutions of the
equation over each element to describe the local
behavior of that region are written with the help of
Galerkin’s method. Thus a matrix relationship
between various problem variables at each node is
established over each element. The system equations
are then assembled from the contributions of the
respective elements. The boundary conditions are
then inserted into these equations which are
subsequently solved numerically.
The discussion so far refers to 2-dimensional
problem. A conductivity variation (z) with respect to
depth, however, makes the problem a 3-dimensional
one which would be more difficult to solve. For a
shallow junction and a small bevel angle, this
difficulty is overcome by approximating the problem
to a 2-dimensional sheet conductivity problem using
4.
EXTRACTION
OF
CONDUCTIVITY
FROM SPREADING RESISTANCE DATA
Prior to applying any of the computational methods, a
careful selection of the mesh of finite elements is
necessary. The location of mesh points in the x and y
directions are on the basis of equal potential
difference rather than equal spacing. The potential
variation being very near to logarithmic, the node
spacing varied almost logarithmically. Secondly, a
finer mesh gives more accuracy at the cost of an
increased computation time and a compromise is
made between these two factors. The accuracy of the
numerical solution with a particular mesh is tested by
obtaining the results numerically as well as
analytically with a uniform conductivity of any
assumed value (say, unity) and then comparing the
two results.
The discussions so far are related to determining the
spreading resistance values 9 (x) if the conductivity
variation is known. What is required in
practice is the opposite of this, i.e. determination of
variation from knowledge of 9 (x) data
obtained experimentally rather than the other way
round. As such there is no well-defined mathematical
procedure for such a reverse calculation. Here a
method using simple iterative technique is presented.
A spreading resistance variation 9 (x) is first
assumed and the corresponding sheet conductance
variation is obtained from equation 2-1. Using
this assumed , the voltage variation V(x,y) due
to the critical probe is determined by solving
numerically the extended Laplace’s equation under
appropriate boundary conditions using finite element
method and taking in to consideration various
corrections required. The spreading resistance is then
calculated as
9 (X) = [V (0,-d/2)-V (0, 3d/2)]/I
(4-1)
The spreading resistance values so obtained are then
compared with the experimental values. Based on the
errors so obtained, corrections are applied to the
assumed Rs(x) values, and subsequently to the
corresponding
variation,
using
an
optimization routine. The same procedure is repeated
several times till the calculated spreading resistance
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IJECS-IJENS
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0
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values coincide with the experimental values within
the required limits of accuracy, few iterations are
usually sufficient.
160
DISCUSSION
In the case of lateral profiling,, the data for the
analysis is obtained from the spreading resistance
measurements made on a boron diffused silicon
wafer, up to a distance of 300 micron from the
insulating edge. For the analysis in the case of depth
profiling,
g, the data used are obtained from the
measurements on a phosphorus diffused pp-type
substrate beveled at 0.5 degree, along a lateral
distance of 210 micron corresponding to the depth
from the surface to junction. In each case, the probe
tip diameter and spacing
acing are 2 micron and 635
micron respectively.. The meshes used for numerical
solutions are optimized to give an error within 0.67%
in the spreading resistance values with respect to the
values obtained analytically, with a uniform
resistivity.A comparison of the raw spreading
resistance data (measured values) with the corrected
data obtained from thee analysis in both the cases is
demonstrated graphically in figures 5-11 and 5-2.
SPREADING RESISTANCE(OHMS)
5.
150
RAW
140
CORRECTED
130
120
110
100
50
300
550
800
1050
DISTANCE FROM THE EDGE (MICRO METER)
Figure 2:: Spreading Resistance Profiles (Lateral)
RAW
370
CORRECTED
330
SPREADING RESISTANCE(OHMS)
It is observed in the case of lateral profiling that the
measured spreading resistance
istance values do not give
correct conductivity variation due to poor spatial
resolution. They underestimate the surface
concentration by more than 50% near the insulating
edge. Thus the raw spreading resistance data are
practically of no use and the appli
application of the
correction technique presented is essential. In the
case of depth profiling on beveled sample, the
resolution problem appears to exist more near the
surface and near the junction than in the middle
portion where the corrected values coincide with the
measured values. This is because of the higher
resistivity on one side and lower resistivity on the
other side which bring the weighted average very
near to the local value. The corrected profile is in
good agreement with that obtained from the we
wellestablished incremental
al sheet resistance technique
except near the junction. An explanation for the
discrepancy near the junction has been given in terms
of carrier depletion effect.
290
250
210
170
130
90
50
10
0
0.5
1
1.5
DEPTH FROM THE SURFACE(MICRON)
Figure 3:: Spreading Resistance Profiles (Depth)
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