of 10
PHYSICAL REVIEW B
92
, 045210 (2015)
Measuring anisotropic resistivity of single crystals using the van der Pauw technique
Kasper A. Borup,
1
,
*
Karl F. F. Fischer,
1
David R. Brown,
2
G. Jeffrey Snyder,
2
,
3
and Bo B. Iversen
1
1
Center for Materials Crystallography, Department of Chemistry and iNANO, Aarhus University, DK-8000 Aarhus, Denmark
2
Materials Science, California Institute of Technology, Pasadena, California 91125, USA
3
ITMO University, Saint Petersburg, Russia
(Received 11 February 2015; revised manuscript received 16 June 2015; published 24 July 2015)
Anisotropy in properties of materials is important in materials science and solid-state physics. Measurement of
the full resistivity tensor of crystals using the standard four-point method with bar shaped samples requires many
measurements and may be inaccurate due to misalignment of the bars along crystallographic directions. Here
an approach to extracting the resistivity tensor using van der Pauw measurements is presented. This reduces the
number of required measurements. The theory of the van der Pauw method is extended to extract the tensor from
parallelogram shaped samples with known geometry. Methods to extract the tensor for both known and unknown
principal axis orientation are presented for broad applicability to single crystals. Numerical simulations of errors
are presented to quantify error sources. Several benchmark experiments are performed on isotropic graphite
samples to verify the internal consistency of the developed theory, test experimental precision, and characterize
error sources. The presented methods are applied to a RuSb
2
single crystal at room temperature and the results
are discussed based on the error source analysis. Temperature resolved resistivities along the
a
and
b
directions
are finally reported and briefly discussed.
DOI:
10.1103/PhysRevB.92.045210
PACS number(s): 72
.
20
.
i
,
81
.
70
.
q
,
84
.
60
.
Rb
I. INTRODUCTION
The directional dependence of transport and thermody-
namic properties is of interest to many fields of materials
science and solid-state physics [
1
]. Examples are magnetism,
the thermoelectric effects, thermal and electrical transport, and
mechanical properties [
2
]. The electrical conductivity or resis-
tivity is of high importance to any field dealing with electronic
materials, including photovoltaics, topological insulators, and
thermoelectrics. The precision of resistivity measurements is
high, i.e., low variation with repeated measurements. The
variation can often be less than 1% [
3
]. This makes resistivity
a good property for initially testing materials for anisotropy,
even when this is not the property of interest.
In thermoelectrics the resistivity is of high importance as
it accounts for efficiency reduction due to parasitic power
loss from Joule heating. Many thermoelectric materials of
interest possess complex structures which are often anisotropic
[
4
]. RuSb
2
is a narrow gap semiconductor of interest as an
isoelectronic and isostructural reference to FeSb
2
. FeSb
2
,in
turn, has a record high thermopower at cryogenic temperatures
and a highly correlated electron structure [
5
,
6
]. Both materials
have orthorhombic unit cells and are highly anisotropic at
low temperatures. Another important example is Bi
2
Te
3
based
materials, which are used in most commercial Peltier modules
and are being investigated as topological insulators [
7
9
].
These materials are anisotropic and directionally resolved
measurements are important for accurate characterization
[
10
12
]. A recent example is the novel high efficiency
thermoelectric material SnSe [
13
,
14
]. Accurate measurement
of thermoelectric transport properties is important to evaluate
the performance of materials [
3
,
15
,
16
]. Easy measurement of
anisotropic resistivity can increase the reliability of studies by
detecting anisotropic properties.
*
Corresponding author: kasperab@chem.au.dk
The most widely used method for measuring resistivity is
the four-point bar method. In this method, the resistance
R
of
a bar shaped sample with cross sectional area
A
and voltage
lead separation
l
is measured. From this the resistivity
ρ
is
calculated as
ρ
=
RA/l
. If the bar is cut along the
ˆ
x
unit
vector (usually from a single crystal) the measured resistivity
relates to the total resistivity tensor as
ρ
ˆ
x
=
ˆ
x
T
ρ
ˆ
x
. To extract
the full tensor at least as many uniquely oriented bars as unique
tensor elements are needed. This number is defined by the
crystal point group through Neumann’s principle [
2
,
17
,
18
].
The four-point bar method is known to potentially be imprecise
due to the difficulty of accurately measuring the geometric
A/l
factor [
15
]. In anisotropic materials an additional source
of error arises from aligning the bar along the direction of
interest.
Alternatively, methods using planar samples may be
adapted to extract the tensor elements. One such method is
the van der Pauw (vdP) method [
19
,
20
], which is extensively
used in thermoelectrics [
21
]. This method accepts planar
samples of arbitrary shape and uses four contacts on the sample
periphery. By posing requirements on the sample geometry
enough information can be obtained to extract the full in-plane
resistivity tensor. Little literature exists on this, and all studies
either assume the principal axis system of the resistivity tensor
to be parallel to the edges of rectangular samples [
22
,
23
]or
develop nonstandard measurement geometries [
24
28
]. The
most successful of these methods adds a fifth contact and an
additional resistance measurement to the vdP method [
28
].
This study focuses on expanding the theory of vdP mea-
surements to extract the full in-plane tensor. The main focus
is on single crystals where the principal axis system is fixed
by crystal symmetry in most crystal classes. The knowledge
of the principal axis system provides enough information to
extract the tensor. A complimentary method is presented where
the sample geometry is changed between two measurements.
Together these methods allow extraction of the resistivity
tensor in any anisotropic material where the tensor is constant
1098-0121/2015/92(4)/045210(10)
045210-1
©2015 American Physical Society
BORUP, FISCHER, BROWN, SNYDER, AND IVERSEN
PHYSICAL REVIEW B
92
, 045210 (2015)
throughout the sample. By using only the standard vdP geom-
etry, the techniques are more easily applied to measurements
using commercial setups. The requirements for the sample
geometry are relaxed from rectangles to parallelograms. The
only additional measurement is the angle of the parallelogram.
This is particularly useful in single crystals where the sample
size is often limited. The theory developed for parallelogram
shaped samples is directly applicable to the five-point method
[
28
]. The theory and methods developed herein are thoroughly
tested experimentally and numerical error simulations are
discussed. Both methods are applied to the crystallographic
(
a,b
) plane of a RuSb
2
single crystal. To the knowledge of the
authors, only the resistivity of polycrystalline materials and
along the crystallographic
c
axis has been reported.
II. THEORY
In the van der Pauw method, the four contacts A through
D are placed in order along the circumference of an arbitrarily
shaped disk without any holes. The resistance
R
1
is measured
by passing current between contact A and B (or C and D) and
measuring the voltage between the remaining contacts, see
Fig.
1
. The resistance
R
2
is equivalently measured by passing
current between A and D (or B and C). These resistances fulfill
the van der Pauw equation [
19
,
20
]
exp

πdR
1
ρ

+
exp

πdR
2
ρ

=
1
.
(1)
ρ
is the resistivity and
d
is the sample thickness.
Equation (
1
) can easily be proven for an infinite half-plane
with contacts along the edge and the general case follows
from the existence of a conformal mapping from the infinite
half-plane to arbitrarily shaped surfaces without holes.
The vdP method distinguishes itself from the four-point
bar method by being a two-dimensional (2D) technique
independent of geometry. The sample thickness
d
enters to
relate the three-dimensional (3D) resistivity to the 2D sheet
resistance. In the bar method a resistance measurement is
x
y
φρ
22
ρ
11
a
b
Linear
transformation
ρ
iso
x’
y’
a’
AB
C
D
AB
C
D
πα
b’
χ
FIG. 1. Effect of the change of coordinates on the sample
geometry and tensor. The sample (left) is anisotropic while the
transformation results in an equivalent isotropic representation
(right).
φ
is the angle between the tensor principal axis system and
the laboratory axes. Before the transformation the laboratory axes are
defined with
x
along the AB side and
y
perpendicular to this in the
AD direction.
χ
is the angle in the anisotropic parallelogram while
πα
is the angle in the isotropic parallelogram.
combined with a known geometry to obtain the resistivity.
In the vdP method, the geometry is substituted for a second,
independent resistance measurement.
In a general anisotropic crystal, the resistivity turns into
a tensor property with up to six independent elements. To
measure the full tensor using bar samples, six bars with
different orientations are needed. These bars cannot form a
single plane and any two of them cannot be collinear. van
der Pauw showed that the same approach is possible using
arbitrarily shaped disks; here the normal vectors can also not
form a single plane and any two cannot be collinear [
29
].
If the vdP method is performed on a sample with known
geometry, Eq. (
1
) is over determined and only one resistance
measurement is necessary, as in the bar method. If the sample
is anisotropic this extra information can be used to reduce the
number of required samples as two tensor elements can be
determined per measurement. If the orientation of the in-plane
principal axis system is known the full in-plane tensor can
be determined (three elements). This is the case when two of
the full 3D principal axes are in the sample plane and one is
exactly perpendicular to this. If this is not the case, e.g., when
one principal axis is not perpendicular to the plane or when
the principal axis system of the full tensor is not known (i.e.,
when the axes are not fixed by crystal symmetry), the in-plane
tensor can be extracted by performing a measurement on two
sample geometries in the same plane. In practice this can be
done by reshaping the sample (not a scaling) between the two
measurements. How to relate the full tensor to the in-plane
tensor is discussed in the Supplemental Material, Sec. S2 [
30
].
In the text only the in-plane tensor will be discussed from now.
To utilize this information, the sample geometry needs
to be explicitly related to the complex half-plane where the
resistances can be calculated, i.e., an explicit expression for
the conformal mapping must be known. In this paper a
parallelogram shaped sample is used, as this is often sufficient
to maximize the sample surface area of planes of single
crystals. The sample is mapped to the complex half-plane
in two steps: First, a linear mapping from the anisotropic
parallelogram to an equivalent isotropic parallelogram. Sec-
ond, an explicit expression is used for the conformal mapping
from the complex half-plane to the isotropic parallelogram.
The explicit expressions for the two transformations allow the
sample geometry to be related to the measured resistances and
resistivity tensor elements. From these relations, the tensor
elements can be calculated from measured quantities.
While a parallelogram is used here, other geometries can
be used provided an expression for the conformal mapping
to the isotropic equivalent geometry is known. The same
steps used here to derive explicit expressions can be used
for any geometry, both explicitly and numerically. The more
quantities necessary to describe a sample geometry (angles
and side lengths), the more complicated the expressions will
generally be. Parallelograms result in complicated but rea-
sonably tractable expressions; more complicated geometries
would likely be easier to treat fully or partially numerically.
A last note is on the behavior in magnetic fields. In this case
the total tensor becomes a sum of the magnetoresistivity (here
taken to include the zero-field resistivity) and Hall resistivity
tensor. The total tensor depends on the magnetic field direction
and strength and hence becomes rank-3 (can no longer strictly
045210-2
MEASURING ANISOTROPIC RESISTIVITY OF SINGLE . . .
PHYSICAL REVIEW B
92
, 045210 (2015)
be represented as a matrix). As the magnetic field is often
applied perpendicular to a flat sample, this can still be treated
as a rank-2 tensor. However, it is important to realize that
the tensor implicitly depends on the field direction. The Hall
resistivity tensor for a given field direction is antisymmetric
with zeroes on the diagonal and hence only has one element in
a planar sample. The magnetoresistivity tensor is symmetric
with nonzero diagonal elements. Experimentally, the two can
be separated by the field dependence: The magnetoresistivity
tensor is an even function of the magnetic field while the
Hall resistivity tensor is an odd function of the field strength
[
18
]. The contributions to a measured resistance from the
two tensors can be experimentally separated by reversing the
field direction or using the different reciprocity relations [
3
].
Without a magnetic field, interchanging the voltage and current
contacts in a resistance measurement gives the same resistance
(known as reciprocity). In a magnetic field, this causes the
Hall effect contribution to change sign and hence the magnetic
field will also need to be reversed (known as reverse field
reciprocity) [
31
,
32
]. It should be noted that only the symmetric
part of the tensor can be treated in the way presented here and
will need to be separated from the antisymmetric part. The
Hall resistivity part can be measured in the usual manner since
there is only one in-plane element. In a magnetic field the full
magnetoresistivity and Hall resistivity tensors can have up to
18 or 9 elements, respectively. For most inorganic compounds
this is significantly reduced by the crystal symmetry.
As is the case for the Hall resistivity tensor, the Hall
coefficient tensor will also be rank-3 and contain up to nine
elements. The tensors are related through
R
H
=
ρ
H
d/B
, where
R
H
is the Hall coefficient tensor,
ρ
H
is the Hall resistivity
tensor,
d
is the sample thickness, and
B
is the magnetic
field strength used in the measurement. This assumes all
measurements to have been carried out at the same field
strength. The Hall coefficient is often used to calculate the Hall
carrier concentration to estimate the true carrier concentration.
Since the Hall coefficient tensor in anisotropic materials can
have up to nine independent and distinct values, so can the Hall
carrier concentration. The true carrier concentration, however,
is the number of electrons or holes in the conduction or
valence bands, respectively. As such, it is a scalar number. The
two carrier concentrations are related by
n
=
r
H
n
H
,
where
r
H
is the Hall factor, and
n
and
n
H
are the true and Hall
carrier concentrations, respectively [
33
]. The Hall factor is
a band structure property accounting for carriers in bands
not behaving like a free electron gas. This property has a
directional dependence and is described by a tensor. Thus,
to avoid discrepancies between reports, it is important to
carefully describe the methodology employed when reporting
Hall carrier concentrations of anisotropic samples.
A. Isotropic equivalent sample
To relate the resistivity tensor to the sample geometry and
measured resistances, the Laplace equation for the sample
must be solved. To simplify this, the sample is mapped to the
upper complex half-plane where the resistances can be easily
calculated. While this can be done directly using a conformal
mapping, this would result in the tensor not being constant on
the complex half-plane, thus complicating the calculations. By
first mapping the anisotropic sample linearly to an equivalent
isotropic sample, this problem is removed. For reference,
this transformation is applied to the Laplace equation in the
Supplemental Material, Sec. S1 [
30
]. Here only the result is
used.
A sample shaped as a parallelogram with angle
χ
and side
lengths
a
and
b
will be used. Its resistivity is given by the
symmetric tensor
ρ
=
(
ρ
xx
ρ
xy
ρ
xy
ρ
yy
), which is the most general
case in two dimensions without a magnetic field. The principal
axis system of the tensor is rotated by
φ
(in the negative
direction) relative to the laboratory frame. This is illustrated in
the left of Fig.
1
. If the orientation is known the tensor in the
laboratory frame can be expressed in terms of its components
ρ
11
and
ρ
22
along the principal axes
e
1
and
e
2
. This will be
further explored later.
The transformation is done by the matrix
M
=
1
ρ
yy
ρ
iso

ρ
yy
ρ
xy
0
ρ
iso

.
(2)
The definition
ρ
iso
=
det
ρ
=
ρ
xx
ρ
yy
ρ
2
xy
has been
used. When applying this to the Laplace equation, the
transformation rule
ρ

=
MρM
T
is found to apply to the
resistivity tensor (see Supplemental Material, Sec. S1 [
30
])
and results in
ρ

=
(
ρ
iso
0
0
ρ
iso
). The transformation results in
an isotropic representation of the sample with resistivity
ρ
iso
.
The sample geometry is transformed by transforming the
vectors representing the sample
a
and
b
sides to
a
=
M
a
and
b
=
M
b
. These are both nonstandard transformation rules,
which result from the change of variables in the Laplace
equation. After the transformation, the new side lengths are
a

=
a

ρ
yy
ρ
iso
,
(3)
b

=
b

cos
2
(
χ
)
ρ
yy
ρ
iso
+
sin
2
(
χ
)
ρ
xx
ρ
iso
2 cos
(
χ
)
sin
(
χ
)
ρ
xy
ρ
iso
.
(4)
The angle between
a

and
b

is defined by
cos(
πα
)
=
1
ρ
yy
cos
(
χ
)
ρ
yy
sin
(
χ
)
ρ
xy

cos
2
(
χ
)
ρ
yy
+
sin
2
(
χ
)
ρ
xx
cos
(
χ
)
sin
(
χ
)
ρ
xy
.
(5)
Equations (
3
)to(
5
) are derived in the Supplemental
Material, Sec. S3.a [
30
]. Since the change of variables was
applied to the Laplace equation, either representation of the
sample can be used to calculate or measure the electrostatic
potential. Any measurement performed on one sample would
give the same result if performed on the other, provided
the contact placement is also transformed correctly. As a
consequence, a normal vdP resistivity measurement performed
on an anisotropic sample will result in
ρ
iso
.
In a magnetic field,
ρ
becomes the magnetoresistivity tensor
and these elements are used to define
M
from Eq. (
2
). Applying
045210-3
BORUP, FISCHER, BROWN, SNYDER, AND IVERSEN
PHYSICAL REVIEW B
92
, 045210 (2015)
the transformation to the antisymmetric Hall resistivity tensor
does not change the tensor, i.e.
,Mρ
H
M
T
=
ρ
H
. This further
underlines the equivalence of performing measurements on the
two samples.
B. Conformal mapping
In the original derivation of the vdP equation [
19
,
20
], the
existence of a conformal mapping between some 2D areas
and an infinite half-plane was used. The criterion of physical
interest is that the outline of the area has to be singly connected,
which is to say there can be no isolated holes in the area.
A conformal mapping is any mapping that locally preserves
angles while distances are not necessarily conserved. van der
Pauw did not need an explicit expression for the mapping since
the contact placement on the infinite half-plane vanishes from
the vdP equation. Here an explicit expression is necessary since
the information contained in the sample geometry is needed to
extract more information from the measured resistances.
Figure
2
illustrates the conformal mapping defined by the
equation [
24
,
26
]
ξ

=
w
(
ξ
)
=
a

A
α,k

ξ
0
f
α,k
(
z
)
d
z
,
(6)
where
f
α,k
(
z
)
=
z
α
1
(1
z
)
α
(1
k
z
)
α
1
(7)
and
A
α,k
=

1
0
f
α,k
(
z
)
d
z
=

(
α
)

(
1
α
)
2
F
1
(
1
α,α
;1;
k
)
.
(8)
w
(
ξ
) takes a point
ξ
=
x
+
iy
on the half-plane and maps it
to a point (
x

,y

) in the isotropic parallelogram. The point is
represented by the complex number
ξ

=
x

+
iy

.
α
is the
angle in radians of the isotropic parallelogram divided by
π
.
k
1
is the position of contact C along the real axis on the
half-plane.

and
2
F
1
are the Gamma function and Gaussian
hypergeometric function, respectively.
k
and
α
can be related to the geometry of the isotropic
equivalent sample by considering the length of the mapped
BC side. This can be calculated from the conformal mapping
ρ
iso
x’
y’
a’
b’
AB
C
D
Conformal
mapping
Im(
ξ
)
Re
(
ξ
)
ABC
D
011/
k
πα
FIG. 2. Illustration of the conformal mapping with contact place-
ments. Contact D corresponds to infinity in any direction on the upper
complex half-plane.
k
has to be in the range 0
<k<
1 for contact C
to be placed between contacts B and D.
as [
27
,
34
]
b

=
a

A
α,k

k
1
1
f
α,k
(
z
)
d
z
=
a

A
α,
1
k
A
α,k
.
(9)
A
α,k
is a normalization constant and in the following only
ratios such as in Eq. (
9
) will appear. Hence it is convenient to
define
B
α,k
=
A
α,
1
k
A
α,k
=
2
F
1
(
α,
1
α
;1;1
k
)
2
F
1
(
α,
1
α
;1;
k
)
.
(10)
This can easily be calculated using standard mathematical
software such as MATLAB
TM
or from an infinite series (see
Supplemental Material, Sec. S4.a [
30
]).
The infinite (complex) half-plane provides a system that
allows easy calculation of the electrostatic potential differ-
ences between the contacts when a current is passed through
the sample. As no scaling is performed, the resistivity of the
infinite half-plane sample is still
ρ
iso
.
R
AB
,
DC
denotes the ratio
of the voltage between contact D and C when a current is
passed from A to B. In a vdP measurement including Hall
effect, three such resistances are measured:
R
AB
,
DC
=
U
D
U
C
I
AB
=
ρ
iso
πd
ln
(
1
k
)
,
(11)
R
BC
,
AD
=
ρ
iso
πd
ln
(
k
)
,
(12)
R
AC
,
BD
=
ρ
iso
πd
ln(
k
1
1)
.
(13)
d
is the sample thickness. The derivation can be found in
the Supplemental Material, Sec. S3.b [
30
], and is done in the
absence of magnetic fields. These are mutually dependent and
one measurement is enough to determine
k
while
α
cannot be
obtained from these resistance measurements. The resistance
in Eq. (
13
) [which is the difference between the resistance in
Eqs. (
11
) and (
12
)] corresponds to the Hall effect measurement
and is a resistive offset to the Hall signal. Both resistances in
Eqs. (
11
) and (
12
) are necessary for the vdP measurement and
are used to calculate
ρ
iso
from the vdP equation. Equation (
13
),
however, provides no new information.
Since the Hall resistivity tensor only contributes a voltage
perpendicular to the current direction, and no current can
flow perpendicular to the sample edge at the edge, a voltage
measurement between two consecutive contacts will not be
affected by the Hall effect. Hence, since the method presented
here only relies on the resistances in Eqs. (
11
) and (
12
),
this is directly applicable to the magnetoresistivity tensor.
The Hall effect will add a contribution of
ρ
H
xy
/d
to Eq. (
13
),
where
ρ
H
xy
is the off-diagonal element of the Hall resistivity
tensor for the sample. This assumes the magnetic field points
out of the sample plane as defined in Figs.
1
and
2
.The
contribution to
R
BD
,
AC
will be equal but with opposite sign,
and both contributions will change sign if the magnetic field is
reversed.
C. Calculating the tensor elements
After the measurement
ρ
iso
and
k
are known but no
information on
α
is obtained. Determining
α
will be the focus
of the following sections. Using the definition for
ρ
iso
together
045210-4
MEASURING ANISOTROPIC RESISTIVITY OF SINGLE . . .
PHYSICAL REVIEW B
92
, 045210 (2015)
with Eqs. (
3
)–(
5
) and (
9
) the tensor elements can be expressed
in terms of
α
and known parameters
ρ
xx
=
1
sin(
χ
)
rB
α,k
λ
α
+
cos(
χ
)
sin(
χ
)
λ
α
cos(
χ
)
r
1
B
1
α,k
2 cos(
πα
)
,
(14)
ρ
yy
=
sin(
χ
)
r
1
B
1
α,k
λ
α
,
(15)
ρ
xy
=
cos
(
χ
)
r
1
B
1
α,k
λ
α
λ
α
cos
(
πα
)
.
(16)
r
=
a/b
is the ratio of the anisotropic sample
a
and
b
sides
and
λ
α
=
ρ
iso
/
sin(
πα
). The derivation of Eqs. (
14
)–(
16
)is
quite lengthy and can be found in the Supplemental Material,
Sec. S3.c [
30
].
Initially, simply assuming
πα
=
χ
provides useful infor-
mation. The assumption is equivalent to assuming the principal
axis system to be identical to the laboratory frame
φ
=
0. The
estimated anisotropy (ratio of
ρ
11
to
ρ
22
)fromthiswillnever
be further from 1 than the true anisotropy and can thus be used
as a limit. If the sample is anisotropic it is still possible, but
unlikely, that the sample will be found to be isotropic. This
will only happen for
φ
=
45
.
1. Known principal axis system
In single crystals, the orientation of the crystallographic
axes can be determined using standard diffraction techniques.
In all crystal classes except triclinic and monoclinic this
also implies knowledge of the principal axes orientation (see
Supplemental Material, Sec. S2, for a discussion [
30
]) and
hence
φ
is known, if two principal axes are in the sample plane.
The tensor can then be expressed in terms of the elements
along the principal axes and the orientation as
ρ
=
(
R
φ
)
T
ρ
0
R
φ
.
ρ
0
is the tensor in the principal axis system which has only
diagonal elements and
R
φ
is the matrix rotating vectors
φ
in the
positive direction (see Supplemental Material, Sec. S1 [
30
]).
Now only two tensor elements need to be determined. This can
be combined with Eqs. (
14
)–(
16
) to give an equation relating
α
to measured parameters:
0
=

1
tan
(
2
χ
)
+
1
tan
(
2
φ
)

2 cos
(
χ
)
r
1
B
1
α,k
+
1
sin
(
χ
)
rB
α,k

1
tan
(
χ
)
+
1
tan
(
2
φ
)

2 cos
(
πα
)
.
(17)
This equation can be solved to get
α
. The derivation can be
found in the Supplemental Material, Sec. S3.d [
30
]. In some
cases, the equation may have two solutions which both result
in valid but different tensors. This occurs when the parameters
(
φ
,
χ
,
k
, and
r
) lie close to a region in the parameter space
that does not correspond to any physically valid situation.
Experimental noise in the parameters may shift them into the
region without any solutions. This is further discussed in the
Supplemental Material, Sec. S4.a [
30
].
The equation needs to be solved numerically. A
MATLAB
TM
script for this can be supplied by contacting the
corresponding author. Once
α
has been obtained the tensor in
the laboratory frame can be calculated from Eqs. (
14
)–(
16
)
and the principal elements can be obtained by a rotation of the
laboratory frame tensor. An algorithm for solving Eq. (
17
)is
found in the Supplemental Material, Sec. S4.a [
30
].
2. Unknown principal axis system
If the orientation of the tensor elements is not known more
information needs to be obtained by other means. This could
be when the principal axis system is not fixed by crystal
symmetry, the sample is not perpendicular to a principal
axis, or the crystallographic orientation is unknown. It is
possible to perform a five-point analog to the vdP measurement
[
27
], however, this requires modification of the measurement
system. This is not easily done with commercial systems but
custom setups may be more easily adapted to accommodate
a fifth contact [
21
]. The five-point method can be directly
combined with the theory in this work.
If such modification is not feasible extra information can be
obtained by modifying the sample, as indicated in Fig.
3
.An
initial measurement is performed (middle sample, called A),
the sample geometry is changed, and a second measurement
is performed (such as on the left or right samples, called B). If
the sample is homogenous the tensors in A and B are identical
except for possibly a rotation. If
ρ
iso
changes more than a
few percent the sample is most likely inhomogeneous or other
errors are affecting the measurement. After the measurement
on A, the parameters
r
A
,
k
A
,
χ
A
, and
ρ
iso
are known and the
tensor can be calculated as a function of only
α
,
ρ
(
α
). In B
these parameters are different but still known and the tensor
principal axis system has been rotated by
θ
(which may be
negative, zero, or positive). The new tensor
ρ

(
β,θ
) expressed
in the A reference frame can hence be calculated as a function
of the new
α
, now called
β
, and
θ
.
Rotating the tensor in B by
θ
will result in the same tensor
as in A. This gives the equation
ρ
(
α
)
=
ρ

(
β,θ
)
=
(
R
θ
)
T
ρ
(
β
)
R
θ
.
(18)
φ
ρ
22
ρ
11
a
b
φ-θ
ρ
22
ρ
11
a’
b’
φ
ρ
22
ρ
11
θ
χ’
χ’
a’
b’
FIG. 3. Reshaping the sample between two measurements can
provide enough information to determine
α
. After the first measure-
ment the middle sample is cut along either the dotted or dashed lines.
In the right sample with the dotted line the tensor principal axes has
been rotated relative to the lab frame, indicated by the angle
θ
.Inthe
left sample, the sample geometry has changed but the tensor principal
axes have the same orientation. The sample can be reshaped in many
more ways which will give the right result provided attention is paid
to the rotation of the tensor principal axes.
045210-5
BORUP, FISCHER, BROWN, SNYDER, AND IVERSEN
PHYSICAL REVIEW B
92
, 045210 (2015)
This corresponds to three equations, one for each of the
tensor elements. To solve this, the vector
d
(
α,β
)
=
ρ
xx
(
α
)
ρ
yy
(
α
)
ρ
xy
(
α
)
ρ
xx
(
β,θ
)
ρ
yy
(
β,θ
)
ρ
xy
(
β,θ
)
=
0
(19)
is used to reduce the system of equations to one equation.
Equation (
19
) can then be solved by finding the minimum in
d
2
(
α,β
). This can be solved using Newton’s method or the
method of steepest descent. A discussion of this is found in
the Supplemental Material, Sec. S4.b [
30
], and a MATLAB
TM
script can be supplied by contacting the corresponding author.
This method is equivalent to van der Pauw’s approach but
with the extra information on the sample geometry taken into
account.
It is not necessary to change
χ
for the method to work
and
θ
may also be 0. A change in
r
alone is sufficient. Any
modification of the sample except a scaling is acceptable. The
fundamental criterion is that at least one of the parameters
determining
ρ
are changed, that is
χ
,
k
,or
r
. If the sample is
simply scaled, any value of
α
=
β
will solve Eq. (
19
).
D. Numerical error analysis
An important source of error is the contact placement. If
the contacts are not placed exactly on the corners, errors in
the tensor elements are expected while
ρ
iso
is not expected
to change. Finding a general analytic expression for the error
from misplaced contacts is difficult. Instead, it can easily be
simulated numerically by moving a contact on the complex
half-plane away from the correct position and mapping this
to the sample using the two transformations in Eqs. (
6
) and
(
2
). As with the normal vdP method, small errors from several
misplaced contacts are expected to be additive [
19
,
20
].
The A and C as well as B and D corners of the sample are
mathematically identical and give the same error for contact
displacements. Considering only the distinct A and B corners
is hence sufficient. An example of an error simulation of the
two distinct corners is shown in Fig.
4
. The angles of the
corresponding corners of the isotropic equivalent sample are
noted in the figure. The error in each corner is calculated for
displacements towards both adjacent corners, as noted. For the
calculations a tensor with principal axis elements
ρ
11
=
1 and
ρ
22
=
2
.
73 (arbitrary units) and orientation
φ
=
54
.
08
was
used. The sample had aspect ratio
r
=
1 and angle
χ
=
90
.
This was chosen as it corresponds to a sample used later in
this paper. The calculated change in tensor elements when
moving a contact was divided by
ρ
iso
since this sets the scale
for the tensor elements. If the change is calculated relative to
the individual elements errors in elements with low values are
overemphasized.
From the figure it is seen that the error in corners with
high angle (in the isotropic equivalent sample) is significantly
larger than in low angle corners and hence these should be
avoided, i.e., it is recommended to keep
α
close to 0.5. This is a
more general requirement than can be made for the anisotropic
sample itself since the current paths will depend strongly on
the tensor. Keeping the error in contact placement less than a
few percent is possible even for small samples and the total
Relative change in tensor elements (%)
(a)
(b)
63.55°
116.45°
Relative displacement
ρ
xx
ρ
yy
ρ
xy
←A
B
C→
←D
A
B→
0.1
0.05
0
0.05
0.1
0.10
0.05
0
-0.05
-0.10
3
2
1
0
-1
-2
-3
-4
-5
FIG. 4. (Color online) Tensor element error relative to
ρ
iso
for
displacement of the A (a) and B (b) contacts toward the adjacent
corners. The displacement is relative to the
a
and
b
side lengths.
error from this source are expected to be less than 3%–5% of
ρ
iso
.
Another source of error is from the angles in the parallel-
ogram, either errors in
χ
or deviations from a parallelogram
towards a trapezoid or general quadrilateral. Errors in
χ
can
easily be simulated numerically while the later has to be tested
experimentally since the conformal mapping no longer holds
(another expression is needed).
A simulation of the error as a function of the error in
χ
is shown in Fig.
5
. The error from
χ
is less sensitive to the
angle of the corner in the isotropic equivalent than for contact
displacements. The angle of the sample can be measured
with high accuracy with a digital microscope or equivalent
and the error will usually not exceed 0
.
5
. Hence, the error
from measuring the angle is expected not to exceed 1%–2%
of
ρ
iso
.
The errors from
χ
in the two corners are related but not
identical. The sample has been rotated 90
, thus interchanging
ρ
xx
and
ρ
yy
and changing sign of
ρ
xy
. Additionally, a positive
error in (a) corresponds to a negative error in (b). If the
x
axis
in (b) is inverted and the sign of the error in
ρ
xy
is changed the
two plots are very similar. They are only completely identical
for isotropic, square samples due to the 90
rotation.
Apart from these errors, the requirements for the vdP
method still apply: The sample has to be flat, of uniform
thickness, and there can be no holes. For round samples it has
been found that samples can be considered flat if the thickness
is less than the radius [
35
]. Extending this to other geometries
is not trivial; however, keeping the thickness well below the
shortest distance across a sample and between any two contacts
045210-6
MEASURING ANISOTROPIC RESISTIVITY OF SINGLE . . .
PHYSICAL REVIEW B
92
, 045210 (2015)
Relative change in tensor elements (%)
(a)
(b)
63.55°
116.45°
-2
-1
0
1
2
-2
-1
0
1
2
3
4
Error in
χ
(°)
-1.5
-0.5
0.5
1.5
-3
-4
-2
-1
0
1
2
3
4
-3
-4
ρ
xx
ρ
yy
ρ
xy
FIG. 5. (Color online) Relative change in tensor elements as a
function of error in
χ
. The simulation was carried out using the same
sample as in Fig.
4
, and normalized with
ρ
iso
.
should provide a reasonable rule of thumb. When this limit is
approached a more accurate estimate is required, which can be
acquired from an experimental test or numerical simulation.
MATLAB
TM
scripts for performing the numerical error
analysis presented here can be supplied by contacting the
corresponding author.
III. EXPERIMENTAL TEST ON ISOTROPIC GRAPHITE
Several test systems were employed to test the rigidity
of the theory and assess experimental noise. All geometric
parameters were measured ten times. Angles were measured
with a digital microscope and dimensions were measured
with a micrometer with 10
μ
m delimitation. The standard
deviation never exceeded 0
.
5
for angles and 40
μ
mfor
dimensions. Generally, deviations were below 0
.
15
and
10
μ
m, respectively. To ensure all samples could be considered
flat, the thickness never exceeded 1
/
3 of the shortest side
length.
To test the validity of the conformal mapping and mea-
surement precision, pseudoanisotropic samples were prepared.
Polycrystalline graphite with close to isotropic resistivity was
shaped as the isotropic equivalent of an imagined anisotropic
sample. The tensor was then independently calculated from
(1) the measured resistances and (2) using only the geometry
of the graphite. This allows testing of the conformal mapping
since (2) depends only on sample geometries and
ρ
iso
(average
of all measurements on each sample) while (1) also relies
on the conformal mapping. To simulate different degrees of
anisotropy the aspect ratio of the graphite was progressively
ρ
xx
ρ
yy
ρ
xy
0.50
1.00
0.75
1.25
1.50
-1
0
1
2
3
4
1.0
1.5
2.0
2.5
3.0
3.5
0.5
Tensor elements (mΩ-cm)
Aspect ratio
(a) Rectangle
(b) Parallelogram
FIG. 6. (Color online) Tensor elements as a function of aspect
ratio of the isotropic equivalent sample. The points are from the
measured resistances and the lines are calculated from the aspect
ratio and average resistivity. The graphite was shaped as a rectangle
in (a) and as a parallelogram with
πα
corresponding to 63
.
55
in (b).
increased in small steps from about 0.5 to about 1.5. This
changes
k
, while angles were kept fixed. Figure
6
shows the
tensor elements from (1), shown as points, compared to (2),
shown as lines. As seen in Fig.
6
, there is a good agreement
between the tensor elements from the two methods, supporting
the conformal mapping approach.
Two samples were used for this test: One was a rectangle
with
α
=
0
.
497 and the other was a parallelogram with
α
=
0
.
353. In both cases the imagined anisotropic sample was
assumed to have
χ
=
90
and
r
=
1.
A. Reproducibility and errors
The reproducibility was tested by mounting and measuring
two graphite samples 20 times each. These samples were the
graphite samples used for Fig.
6
with
r
=
0
.
81 (rectangle) and
r
=
0
.
86 (parallelogram). The samples were rotated between
measurements, resulting in continued cyclic permutation of
the contacts. This ensures low correlation of errors between
successive measurements. From this two types of errors can
be calculated: a parallel shift of the elements corresponding
to an error in
ρ
iso
and an antiparallel shift of
ρ
xx
and
ρ
yy
corresponding to an error in geometry. The latter error is most
likely from bad contact placement. The errors relative to
ρ
iso
(average of all measurements on each sample) are shown in
Fig.
7
.The
ρ
iso
error is seen to be much smaller than the
geometric error. This is expected since the vdP resistivity
measurement is precise while accurate contact placement is
045210-7
BORUP, FISCHER, BROWN, SNYDER, AND IVERSEN
PHYSICAL REVIEW B
92
, 045210 (2015)
0
5
10
15
20
Measurement number
Relative error (%)
-8
-6
-4
-2
0
2
4
6
8
ρ
iso
error
Geometric error
-8
-6
-4
-2
0
2
4
6
8
(a) Rectangle
(b) Parallelogram
FIG. 7. (Color online) Errors in tensor elements relative to
ρ
iso
for
20 repeated measurements. The graphite was shaped as a rectangle
in (a) and as a parallelogram with
πα
corresponding to 63
.
55
in (b)
with aspects ratios
r
=
0
.
81 and
r
=
0
.
86, respectively.
difficult. The vdP resistivity measurement is not affected by
contact positions as long as they are on the sample edge.
The errors are three times larger for the parallelogram than
the rectangle. This is due to a combination of
πα
being far
from 90
and that parallelograms are slightly harder to mount
correctly. The error is the deviation from the tensor elements
calculated from the geometry and average resistivity divided
by
ρ
iso
. This corresponds to the lines in Fig.
6
. The errors
shown in Figs.
4
and
5
were calculated for the sample used in
Fig.
7(b)
. Assuming errors up to 5% on the contact placement
and 0
.
5
in
χ
, the expected maximum error of roughly 6%
fit well with that observed in Fig.
7(b)
. The 6% estimate is
obtained by adding roughly 2% for the B and D contacts,
negligible contribution from the A and C contacts, and 2%
from
χ
. The same error analysis for the sample in Fig.
7(a)
gives a 2% error which is similar to the scatter observed in the
figure. This indicates that these error plots give good estimates
of the expected maximum error in the method, provided the
vdP method is correctly and carefully applied.
A type of error, which cannot easily be estimated by nu-
merical simulations, is deviation from parallelograms towards
a trapezoid, i.e., opposing sides not being completely parallel.
In this case the expression for the conformal mapping is no
longer valid. The effect of this error was tested on a graphite
sample with
χ
=
90
.
27
and
r
=
0
.
996. The angle between
two opposing sides was progressively increased by grinding
away wedges of one side. The change in the tensor elements
-4
-2
0
2
4
01234
Deviation from parallelogram (°)
Relative error (%)
ρ
xx
ρ
yy
FIG. 8. (Color online) Error in tensor elements relative to
ρ
iso
as
a function of deviation from a parallelogram in degrees. The lines are
guides to the eye.
relative to
ρ
iso
was calculated. This error is plotted against
the measured deviation in Fig.
8
. Ensuring parallel sides in a
parallelogram poses little practical challenge and this deviation
is usually expected to be much less than 1
. Hence, the error
from this effect is expected to be less than 1% if appropriate
care is taken.
B. Using two samples
Graphite was also used to test the extraction of
α
from
Eq. (
19
). A rectangular piece of graphite was shaped in
five steps into a progressively more skewed parallelogram.
Measurements were performed on all samples and all unique
combinations of the five measurements were used to extract
α
from Eq. (
19
) and calculate the tensor elements. The results
can be seen in Fig.
9
. The reference frame is the same in all
samples (i.e.,
θ
=
0) since the AB side was unchanged.
If the two samples are too similar, Eq. (
19
) will have a
flat minimum. In this case, the algorithm for solving the
equation converges slower. The low gradient also increases
the effect of experimental noise on
α
and the tensor elements.
ρ
xx
ρ
yy
02468101214161820
1.00
1.02
1.04
1.06
0.98
0.96
0.94
Relative tensor elements
Change in χ (°)
FIG. 9. (Color online) Tensor elements divided by
ρ
iso
as a
function of the change in
χ
in the parallelogram.
α
was extracted
from Eq. (
19
). While
ρ
xy
is not strictly 0 this is omitted since it is
small and has a low variation.
045210-8
MEASURING ANISOTROPIC RESISTIVITY OF SINGLE . . .
PHYSICAL REVIEW B
92
, 045210 (2015)
Determining the minimum change in
χ
required for the effect
of experimental noise to be tolerable is not straightforward. In
most cases, a change of 15
or more should suffice and not
pose major practical problems.
IV. RuSb
2
SINGLE CRYSTAL
An approximately 20 mm
3
RuSb
2
single crystal was grown
from an antimony self-flux. The starting material was a
mixture of 3.4161 g small RuSb
2
crystals from a previous
synthesis, 0.2168 g Ru (99
.
9% metals basis), and 22
.
9863 g Sb
(99
.
9999% metals basis) and was placed in an Al
2
O
3
crucible
in a quartz ampule. This gives a roughly 6:94 molar Ru:Sb ra-
tio. The ampule was sealed at a pressure of less than 10
4
mbar
and placed inside an Al
2
O
3
tube filled with thermally insulating
ceramic wool. The insulated ampule was heated to 1100
C
at 150 K
/
h and soaked for 5 h, cooled to 640
Cat1K
/
h,
after which the furnace was turned off. To remove the excess
flux, the crucible was placed upside down in another ampoule
with ceramic wool (collects the solid) on top of quartz shards
(provides a reservoir for the flux) in the bottom. This ampule
was heated to approximately 690
C for 30 min and centrifuged
while hot. While heating and centrifuging, the ampoule was
placed in a special high thermal mass and insulated stainless
steel receptacle to prevent the flux from solidifying before the
centrifugation had completed. The synthesis produced crystals
at a variety of sizes, some of which were agglomerated. The
largest isolated crystal with well-defined facets on all sides
was selected for the experiments. The facets were indexed
in a powder x-ray diffractometer and the angles between the
facets were consistent with a single crystal. RuSb
2
has the
marcasite structure (space group
Pnnm
) with lattice constants
of
a
=
5
.
95
̊
A,
b
=
6
.
67
̊
A, and
c
=
3
.
18
̊
A[
36
].
From the single crystal a parallelogram in the (
a,b
) plane
was cut. This sample had
r
A
=
0
.
705,
χ
A
=
96
.
2
, and
φ
A
=
45
.
7
. The measurement resulted in
ρ
iso
=
101 m
cm and
k
=
0
.
116. When Eq. (
17
) is solved with this input, two values
of
α
result: 0.128 and 0.659. This results in the two tensors
(
562
0
018
.
3
) and (
66
.
90
0
154
)m
cm. The first solution indicates
the
b
direction to be more resistive and the second solution
indicates the
a
direction to be more resistive. In this sample this
cannot be used to determine the correct solution; however, this
may be possible in some samples such as layered compounds.
To distinguish the two solutions, a new sample with a
geometry defined by
r
B
=
0
.
957,
χ
B
=
79
.
6
, and
φ
B
=
62
.
3
was cut from the first. The two samples are related by
θ
=−
16
.
6
. From the measurement
ρ
iso
=
116 m
cm and
k
=
0
.
287 was obtained. Only one
α
, 0.553, is obtained
from the single crystal equation. This results in the tensor
(
78
.
70
0
170
)m
cm. From this it seems likely that the second
solution in the first sample is the correct one even though
they differ somewhat. This is due to a 15% increase in
ρ
iso
between the two measurements. The anisotropy, defined as
ρ
a
b
, is 2.29 in the first sample second solution and 2.16 in
the second sample. This corresponds to less than 6% decrease
in anisotropy, much less than the change in
ρ
iso
. The reason
for the lower resistivity in the first sample is most likely
bad measurements due to, e.g., excessively large contacts,
contacts not being on the circumference, etc. These errors all
0
50
100
150
200
-200
-100
0
100
200
300
Temperature (°C)
Tensor elements (mΩ-cm)
ρ
a
ρ
b
1.0
1.5
2.0
2.5
ρ
a
/ ρ
b
3.5
-200
0
200
Temperature (°C)
FIG. 10. (Color online) Temperature resolved measurement of
ρ
a
and
ρ
b
of RuSb
2
. The single crystal method and Eq. (
17
) with
constant, temperature independent
φ
were used. The inset shows
ρ
a
b
.
result in a decrease in measured resistivity [
19
,
20
]. Repeated
measurements on the second sample gave comparable results.
The method of changing the sample geometry can strictly
not be applied due to the change in resistivity. Nonetheless, this
is included for comparison. Using
ρ
iso
=
109 m
cm (average
of the two samples) gives
α
=
0
.
639 and
β
=
0
.
540, which is
similar to the values obtained from the single crystal method.
This results in the tensor (
76
.
70
0
154
)m
cm and anisotropy
2.00. Despite the large change in
ρ
iso
the tensor is quite similar
to the ones calculated from the single crystal method. This is
expected since both
k
and
α
depend on the anisotropy rather
than the actual resistivity. Since the anisotropy only changes
slightly,
k
and the extracted
α
and
β
are still reasonably
accurate.
The second sample was used to measure the resistivity from
185 to 300
C. The data are shown in Fig.
10
. Since the unit
cell orientation is independent of temperature,
φ
is constant
during temperature resolved measurements. This allows for
easy application of the single crystal method to temperature
resolved measurements.
The resistivity has a metallic region at low temperatures
and shows activated behavior at higher temperatures. The
temperature dependence below 100
C is similar to that
reported by Sun
et al.
[
6
] along the
c
direction. Fitting an
Arrhenius type expression to the activated region (75–300
C)
gives a thermal band gap of 0
.
34 eV, consistent with previously
reported values. Below
185
C another activated region has
been reported [
6
], the onset of which is visible as a positive
curvature in resistivity. Hall coefficient measurements (not
shown) indicate an
n
-to
p
-type transition at 30
C. Below
this temperature the material is
n
type.
The anisotropy seems to stabilize at around 1.4 and is almost
independent of temperature above 150
C. This indicates that
the relative mobility of the
a
and
b
directions is constant. In
conventional semiconductors this ratio is mainly determined
by the Fermi surface. Hence, the conduction band Fermi
surface has a constant shape from 150 to at least 300
C.
The anisotropy increases at lower temperatures and reaches
a plateau in the low temperature metallic region. Another
045210-9