J. Mod. Phys., 2010, 1, 340-347
doi:10.4236/jmp.2010.15048 Published Online November 2010 (http://www.SciRP.org/journal/jmp)
A Theoretical Study on Van Der Pauw Measurement Values
of Inhomogeneous Compound Semiconductor Thin Films
Toru Matsumura, Yuichi Sato
Graduate School of Engineering and Resource Science, Akita University,
Tegata-gakuen, Akita, Japan
Received August 16, 2010; revised October 19, 2010; accepted October 25, 2010
Abstract
The influence of intermixing heterogeneous regions that have different electrical properties from the base
materials on van der Pauw measurement values was theoretically studied by computer simulation using the
finite-element method. The measurement samples selected were thin films of inhomogeneous semiconduc-
tors. Calculated electrical properties, such as resistivity, carrier density, and mobility of the thin films, varied
in predictable ways when heterogeneous regions were dispersed in wide ranges over the samples. On the
other hand, the mobility of the thin films showed a different change when heterogeneous regions were lo-
cally concentrated in the measurement samples.
Keywords: Semiconductor, Hall Effect, Van Der Pauw Method, Inhomogeneous Semiconductor
1. Introduction
The van der Pauw method [1] is one of the most util-
ized measurement methods for the evaluation of electri-
cal properties in semiconductor materials, such as resis-
tivity, carrier density, and mobility. The van der Pauw
method can be used to measure samples of arbitrary
shape, although several basic sample conditions must be
satisfied to obtain accurate measurements, such as the
thickness of the sample must be constant, point contacts
placed at the edges of the samples must be used for the
measurements, and the sample quality has to be homo-
geneous. Most semiconductor samples satisfy these con-
ditions, so that this convenient measurement method has
been widely utilized.
Compound semiconductors have various useful prop-
erties for the realization of a large number of high per-
formance devices in fields such as electronics and opto-
electronics. Many novel compound semiconductors, es-
pecially in the form of thin films, are continually being
developed. However, a tendency in the fabrication of
compound semiconductor thin films is that inhomoge-
neous samples are often obtained. Unintentional inho-
mogeneity occurred in thin films may be easy or difficult
to recognize instantly. Examples of the unintentional
appearance of inhomogeneity in thin films appear as me-
tallic droplets in III-V nitride semiconductors such as
InN [2,3], phase separation in ternary alloys such as In-
GaN thin films [2,4], and inhomogeneous doping in p-
type impurity-doped ZnO thin films [5]. Most of these
compound semiconductor thin films are under develop-
ment to realize high performance novel devices.
The van der Pauw method is a convenient measure-
ment method for the evaluation of semiconductor thin
films. Therefore, newly prepared thin films will be some-
times measured using the van der Pauw method to de-
termine film quality as a reference, regardless of their
homogeneity, although the method is said not to be able
to measure inhomogeneous samples. If tendencies and
degrees of deviations in measurements obtained by the
van der Pauw method for inhomogenous samples from
those of homogeneous samples are made clear, then the
utility of the measurement method and reliability of the
measurement values will be increased. There have been
several reports on the influence of inhomogeneity on van
der Pauw measurements of such unintentional inhomo-
geneous samples [6-11]. However, concrete values on
such deviations have not been sufficiently reported.
Therefore, in this work, the influence of intermixing het-
erogeneous regions in semiconductor thin films on van
der Pauw measurements was theoretically studied by
computer simulation using the finite-element method
(FEM).

T. MATSUMURA ET AL. 341
2. Simulation Method
Thin film compound semiconductors were selected as
measurement samples. The shape of the samples is
square with point ohmic electrodes are contacted at the
four corners. The arrangement of the electrodes with
schematics of the van der Pauw method are shown in
Figures 1(a) and (b). For resistivity measurements, a
voltage is applied to flow current IAB along one side of
the square sample and the voltage along the opposite side,
VCD, is measured. For Hall effect measurements, a volt-
age is applied between the contacts placed at diagonally
opposite corners and the current I flowing between them
is measured. In addition, a magnetic field B is applied in
the direction of the sample thickness, and the change in
voltage VH, between the point contacts placed at diago-
nally opposite corners is measured. Resistivity ρ, carrier
density n, and mobility μ, are calculated from the meas-
ured values, applied magnetic fields and the film thick-
ness d, of the measured sample. Equations (1) to (5) are
used to calculate the electrical properties:
,,
()
ln 22
AB CDBCDA
RR
d
f



  , (1)
,,
,
CD
D
A
AB CDBCDA
AB BC
VV
RR
I
I
 , (2)
H
H
Vd
R
I
B
, (3)
1
H
nRe , (4)
H
R


, (5)
where RH is the Hall coefficient, e is the elementary
charge and f is the correction function.
Electric potential distributions in the measurement
samples were obtained by calculation using the FEM to
obtain values for these electrical properties. The square
shaped model of the sample was divided into about
13000 triangular elements, and electric potentials at each
node of the divided elements were calculated using the
continuity of current equation:
0J (6)
where J is the current density. This equation is trans-
formed by the relationship ==V

 
J
E in the fol-
lowing equation:
[] 0
TV

 , (7)
where

is the conductivity, E is the electric field and
V is the electrical potential. The conductivity tensor
[]
T

is defined below and changes according to the
applied magnetic field.
I
AB
V
CD
A
C
B
D
B
I V
H
A
C
B
D
(a) (b)
Figure 1. Schematic illustrations of (a) resistivity and (b)
Hall effect measurements by the van der Pauw method.
2
1
[] ()1
1( )
T
B
en
B
B








 
(8)
In this calculation, we assumed that the sizes of the
samples were 1 cm square and 1μm thick. The voltage of
1 V was applied between the electrode terminals in both
resistivity and Hall effect measurements, and subse-
quently a magnetic field of 0.3 T was applied in Hall
effect measurements.
3. Simulation Results and Discussion
Firstly, the electrical properties of inhomogeneous semi-
conductor samples, in which the inhomogeneity was
dispersed over the entire sample area, were simulated.
The distribution patterns of the heterogeneous regions in
the samples are shown in Figure 2. The area ratio of one
heterogeneous region was fixed as 1.5 × 10−4 to the entire
sample area, and the ratios of the total area to the entire
sample area were varied. In this case, the carrier density
and mobility of the homogeneous base region was fixed
as 1 × 1018 cm−3 and 1000 cm2/V·s, respectively. On the
other hand, the electrical properties of the intermixed
heterogeneous regions were varied, as shown in Table 1.
We assumed that the conduction of the semiconductor
sample was n-type and only electrons contributed to the
electrical conduction and; there was no rectifying con-
duction in the inhomogeneous regions. In addition, we
assumed that only the transverse directions of the sam-
ples were inhomogeneous, because the samples were thin
films and the sizes of individual heterogeneous regions
were comparable to or larger than the thickness of the
samples.
Figures 3(a)-(c) show the simulation results for the
electrical properties. The resistivities of the heterogene-
ous regions are the same values as the base regions (Ta-
ble 1); therefore, no change in resistivity was observed,
despite intermixing of heterogeneous regions, as shown
in Figure 3(a). On the other hand, the carrier densities

T. MATSUMURA ET AL. 342
(a) (b) (c)
Figure 2. Distribution patterns of the heterogeneous regions (black areas) in the measurement samples. The
total ratio of heterogeneous regions are (a) 0.04, (b) 0.13, and (c) 0.26. The area ratio of one heterogeneous re-
gion is 1.5 × 10−4 to the entire sample area. The electrical properties of each region in these models are shown
in Tables 1 and 2.
Figure 3. Electrical properties obtained from simulations using the samples shown in Figure 2 and Table 1. (a)
Resistivity, (b) carrier density and (c) mobility changes as a function of the total heterogeneous region ratio.

T. MATSUMURA ET AL. 343
Table 1. Electrical properties of the homogeneous base re-
gion and heterogeneous regions in the inhomogeneous sam-
ples shown in Figure 2.
Carrier density
(cm−3)
Mobility
(cm2/V·s)
Resistivity
(Ω·cm)
Homogeneous base
region 1 × 1018 1000 6.24 × 10−3
Heterogeneous
regions 1 × 1019～1021 1～100 6.24 × 10−3
and mobilities obtained by simulation showed large
variations that were dependent upon the total heteroge-
neous region ratios, as shown in Figures 3(b) and (c).
The calculated carrier density increased with the increase
in the heterogeneous region. The degree of change was
almost the same, regardless of the carrier density of the
heterogeneous region, although the increase in the calcu-
lated carrier density was slightly larger when the carrier
density of the heterogeneous region was high. The cal-
culated mobility decreased with the increase in the het-
erogeneous region. The degree of change was almost the
same, regardless of the mobility of the heterogeneous
region, although the degree of decrease was slightly large
when the mobility of the heterogeneous region was low.
From the results shown in Figures 3(b) and (c), devia-
tions of the electrical properties of the inhomogeneous
samples from those of the homogeneous base region
were evaluated to be within several percent when the
total ratio of the heterogeneous region was smaller than
approximately 10%; therefore, the influence of inter-
mixing of heterogeneous regions was not so large. How-
ever, the deviations of the electrical properties from
those of the homogeneous base region became signifi-
cantly larger when the total ratio of the heterogeneous
region was much larger than 10%.
Heterogeneous regions that have more varied electri-
cal properties and different resistivities from the homo-
geneous base region were then intermixed in a different
base sample. The electrical properties of the homogene-
ous base region and heterogeneous regions used in these
simulations are shown in Table 2. The distribution pat-
terns of the heterogeneous regions were the same as the
previous simulations. The calculated resistivities were
changed by the intermixing, as shown in Figure 4(a).
That is, the resistivities increased with increasing total
ratio of the heterogeneous region when the resistivity of
the heterogeneous region was higher than that of the
homogeneous base region, while the resistivities de-
creased when its resistivity was lower than that of the
base region. The calculated carrier density and mobility
also largely changed by intermixing of heterogeneous
regions that had various carrier densities and mobilities,
as shown in Figures 4(b) and (c). The degree of change
in the calculated carrier densities tended to be large when
the mobilities of the heterogeneous regions were high.
Similarly, the degree of change in the calculated mobili-
ties tended to be large when the carrier densities of the
heterogeneous regions were high.
In the above simulations, the heterogeneous regions
were distributed over the entire sample area. The changes
in the calculated electrical properties observed by the
intermixing of heterogeneous regions were the expected
ones. On the other hand, different changes were observed
when the heterogeneous regions were concentrated lo-
cally in the samples. The distribution patterns of the het-
erogeneous regions are shown in Figures 5(a) and (b).
The size of one heterogeneous region is the same as that
used in the previous simulation models. The electrical
properties examined in this simulation are shown in Ta-
ble 3. The calculated resistivity and carrier density in
these models changed similarly to those in the previous
simulations in which the heterogeneous regions were
dispersed over the entire sample area. That is, the
changes were simply dependent upon the properties of
the heterogeneous regions.
However, the mobilities calculated in this case re-
vealed a different change. The calculated mobilities for
the samples with locally concentrated heterogeneous
regions are shown in Figure 6. The calculated mobilities
became larger than that of the homogeneous base mate-
rial within a range of the total ratio of the heterogeneous
region, although the mobility of the heterogeneous re-
gion was definitely not higher than that of the homoge-
neous base region. It is reasonable to expect that mobility
will decrease with an increase of intermixing of hetero-
Table 2. Electrical properties of the homogeneous base re-
gion and heterogeneous regions in the inhomogeneous sam-
ples shown in Figure 2.
Carrier density
(cm−3)
Mobility
(cm2/V·s)
Resistivity
(Ω·cm)
Homogeneous base
region 1 × 1019 100 6.24 × 10−3
Heterogeneous
regions 1 × 1017～1021 10～1000 6.24 × 10−5～10−1
Table 3. Electrical properties of the homogeneous base re-
gion and heterogeneous regions in the inhomogeneous sam-
ples shown in Figure 5.
Carrier density
(cm−3)
Mobility
(cm2/V·s)
Resistivity
(Ω·cm)
Homogeneous base
region 1 × 1019 100 6.24 × 10−3
Heterogeneous
regions 1 × 1020, 1021 100 6.24 × 10−4, 10−5

T. MATSUMURA ET AL. 344
Figure 4. Electrical properties obtained from simulations using the samples shown in Figure 2 and Table 2. (a)
Resistivity, (b) carrier density and (c) mobility changes as a function of the total heterogeneous region ratio.
geneous regions; however, on the contrary, the calculated
mobility increased. Figure 7 shows the calculated mo-
bilities as a function of the mobility of the heterogeneous
regions in similar inhomogeneous samples shown in
Figure 5(b). In this case, the carrier density of the het-
erogeneous region was set to 1 × 1021 cm–3 and mobility
of the heterogeneous region was gradually reduced from
the value of the base region. As a result, the calculated
mobilities were higher than that of the base region until
the mobility of the heterogeneous region was reduced to
a value of approximately one order lower.
Therefore, there were cases in which the calculated
mobility became larger than the value of the base mate-
rial, despite the intermixing of heterogeneous regions
with lower mobilities than that of the base material. The
mobility μ is calculated from the ratio between the values
of the Hall coefficient RH and resistivity ρ in Hall-effect
measurements, as shown in Equation (5). If the degree of

T. MATSUMURA ET AL. 345
(a) (b)
Figure 5. Concentration of the heterogeneous regions (black areas, checkered pattern) in local ar-
eas of the measurement sample. The total ratio of the heterogeneous regions are (a) 0.06, and (b)
0.15.
Figure 6. Mobilities obtained from simulations using the samples shown in Figure 5 and Table 3, as
a function of the total heterogeneous region ratio; the heterogeneous regions were concentrated lo-
cally in the samples.
Figure 7. Mobilities obtained from simulations using the sample shown in Figure 5(b) as a function
of the mobility of the heterogeneous region. The ratio of the total heterogeneous region was fixed to
0.15.

T. MATSUMURA ET AL. 346
Figure 8. Changes in the resistivity ρ and Hall coefficient RH obtained from the simulations using
the samples shown in Figure 5(b) as a function of the mobility of the heterogeneous regions.
(a)
(b)
Figure 9. Typical examples of the calculated electrical potential distributions on the resistivity (left
sides) and Hall effect (right sides) measurements of the (a) homogeneous and (b) inhomogeneous
samples in which heterogeneous regions were locally concentrated. The carrier density and mobil-
ity of the homogeneous base region were 1 × 1019 cm−3 and 100 cm2/V·s, respectively, while those of
the heterogeneous regions were 1 × 1020 cm−3 and 100 cm2/V·s.

T. MATSUMURA ET AL. 347
variation of the Hall coefficient and resistivity by inter-
mixing of heterogeneous regions are not same, then the
value of mobility calculated from the ratios will fluctuate
depending upon the ratios. The degree of change (de-
crease) in resistivity by intermixing of heterogeneous
regions is larger than the degree of change (decrease) in
the Hall coefficient in this case, as shown in Figure 8.
Therefore, mobilities calculated from the ratio become
larger than that of the base material.
We have simulated about more various inhomogene-
ous samples and confirmed that samples in which het-
erogeneous regions locally concentrated showed the
similar tendencies described above. Typical electrical
potential distributions in such sample are shown in Fig-
ure 9 with those of a homogeneous sample. The poten-
tial distribution is markedly distorted in the locally het-
erogeneous region concentrated sample. The measured
voltage between the terminals C and D in the resistivity
measurement (left sides) become considerably small
when the resistivities of the heterogeneous regions are
smaller than that of the homogeneous base region. On
the other hand, the Hall voltages between the terminals A
and C in the Hall effect measurement (right sides) are
almost same about the both samples. Therefore, the value
of the mobility calculated from the values of Hall coeffi-
cient and resistivity become larger than that of the ho-
mogeneous sample.
4. Conclusions
The effect of intermixing of heterogeneous regions in
compound semiconductor thin films on the electrical
properties obtained by the van der Pauw method were
investigated by simulation using the FEM. The calcu-
lated electrical properties varied predictably according to
the electrical properties of the heterogeneous regions
when the regions were dispersed over the entire sample
area. The deviations of electrical properties of inhomo-
geneous samples from those of the homogeneous base
materials were not very large when the total ratio of the
heterogeneous region was lower than approximately 10%
in such cases. On the other hand, the calculated mobility
tended to increase unexpectedly by the intermixing of
heterogeneous regions when the regions were concen-
trated locally in the samples, even though the mobilities
of the intermixed heterogeneous regions were lower than
that of the base material. Thus, the electrical properties
of inhomogeneous compound semiconductor thin films
obtained by the van der Pauw method are dependent
upon the distribution patterns of the heterogeneous re-
gions, and unexpected values may be obtained in cases in
which the heterogeneous regions are concentrated locally
in the measurement samples.
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