This document describes an experiment using the van der Pauw method to measure the sheet resistance of a thin film of graphite made from pencil lead on paper. Electrical contacts were made on a cloverleaf-shaped sample and resistances were measured in different configurations. The sheet resistance was calculated to be 25.7 ± 1.4 kΩ, indicating the pencil lead film has similar resistivity to solid pencil lead samples. The ratio of vertical and horizontal resistances was close to 1, showing the sample had relatively uniform conductivity. This demonstrates the van der Pauw method can accurately characterize the electrical properties of non-uniform thin films like graphite.

1. Sheet resistance measurement of thin graphene for use in
Hall Effect calculations
Edward Burt Driscoll∗
Department of Physics, North Carolina State University, Raleigh, North Carolina 27607, USA
(Dated November 25, 2014)
Abstract
Using the van der Pauw method of measuring sheet resistance, we are
able to find the resistances associated with a thin film of graphite. With a
2.5 cm diameter cloverleaf sample of 2B pencil lead on paper, we measured
vertical and horizontal resistances of 5.46 ± 0.28 kΩ and 5.87 ± 0.36 kΩ
respectively. The sheet resistance is than found to be 25.7 ± 1.4 kΩ, and
the ratio R
R
= 1.075 ± .086.
Introduction
The van der Pauw method is a tool derived in 1959 to
measure the resistivity and Hall coefficient of samples, re-
gardless of their shape[1]. Van der Pauw derived a method
for calculating sheet resistivity for arbitrarily shaped sam-
ples. Besides the resistivity of the material, by exposing
the sample to a magnetic field normal to the sample, and
then the same magnetic field flipped. This allows for cal-
culations of the doping of the conducter (p-type or n-type,
whether the majority carrier is a hole or an electron), the
sheet density of the majority carrier, and the mobility of the
majority carrier. Additionally, the van der Pauw method
can be used to find the Hall coefficient of a sample as well
using the described magnetic fields.
Despite these further possible calculations, we will focus
on the calculation of the sheet resistance, and the ratio of
the vertical and horizontal resistances.
The optimization of the measurement involves mini-
mizing the contact area for the leads, uniformity of the
lead materials and properties, and Ohmic properties of the
lead materials. Larger contact areas lead to lower resistance
measurements due to the shorting of the contact circuit. Al-
though this is not extremely significant with zero magnetic
field applications, such as simple van der Pauw resistivity,
but is exacerbated during Hall effect calculations under
magnetic fields[2].
The use of pencil lead on paper to simulate thin film
graphite is notable as it is extremely variable (relative to
silicon) in its surface and consistency. The resistance of this
pencil lead, 2B, for a solid sample on paper runs at 25 ± 5
Ω for a length of 2 cm, and 35 ± 5 Ω for a length of 2.5 cm,
each having a width of 0.5 cm. This will be a good estimate
for the scale of sheet resistances in the samples used.
Experimental Methods
The sample was created on graph paper using a mechan-
ical pencil with 2B graphite lead. The shape, a cloverleaf, is
used to minimize the error attributed with the finite size of
the contacts[1]. This cloverleaf was a 2.5 cm diameter circle
with 0.5 cm × 0.75 cm rectangles missing. The graphite
was applied by hand, layering it on by scribbling until the
entire area of the cloverleaf was covered and dark, along
with being reflective.
The paper was folded on itself, to insure that the alliga-
tor clips would not rip through the single layer of paper and
destroy the connection. This setup is shown in Figure 1.
These four leads then ran to a breadboard, where they were
inserted into rows 1-4 accordingly. These four rows would
be used to provide interchangeable input/outputs for use in
the rest of the circuit.
Figure 11: The setup of the measurements. The paper on which
the sample was created was folded back upon itself, with alligator
clips creating contacts at the tips of each leaf. To prevent sam-
ple degredation, these alligator clips were held constant through
measurement, so the wires were switched about further down
when needed.
The circuit used is shown in Figure 2. The NI USB-6009 pro-
vides a constant 5 V input in to the system, which runs directly
into the resistor which then runs to the according breadboard
number. On the negative end of the input voltage, a lead runs
from the breadboard row corresponding to the neighboring lead.
The first NI USB-6009 input is a voltage input across the resistor.
The second input is the voltage difference across the two remain-
∗e-mail: ebdrisco@ncsu.edu
1Alligator clip image fetched from http://cdn3.stanleysupplyservices.com/images/p/126-211.01_s500_p1._V937f1f58_.jpg
1

2. ing leads, which are opposite of the first two. The beginning run
is diagramed in Figure 2, and then each lead is rotated for the
next measurement, and then flipped to include all eight possible
combinations.
Figure 2: The circuit diagram of the sample. Vi is set to a con-
stant 5 V using an NI USB-6009 running from LabView. Ri is
known to be 1017350 ± 50 Ω, measured with a Keithley Digital
Multimeter. The four ”leaves” of the cloverleaf-shaped sample
are rotated around and flipped as needed. V and the voltage
across Ri are measured with the NI USB-6009.
For each run, a seperate resistance will be calculated. The
two inputs are voltages averaged over 10 seconds, recorded at
100 Hz. The standard deviation of each measurement is also
recorded. To begin, the voltage across the resistor is converted
into a current using the simple resistance formula:
Iij =
VRi
Ri
, (1)
where Iij is the current from i to j, VRi
is the voltage across
the resistor, and Ri is the resistance of the known resistor, which
was found to be 1017350±50 kΩ with a Keithley Digital Multime-
ter. The error of this calculation is found applying the uncertainty
propogation formula, which leads to:
σ2
Iij
= σ2
VRi
1
R2
i
+ σ2
Ri
V 2
Ri
R4
i
, (2)
where σIij
is the uncertainty of the current calculation, σVRi
is the standard deviation of the voltage measurement across the
resistor, and σRi
is the uncertainty of the resistance.
The second input, Vkl, is the voltage difference from leads
k to k. This is used in the resistance calcuation through the
resistance formula as used in Formula 1,[1]
Rij,kl = Vkl
Iij
. (3)
The uncertainty of this value is found using the error pro-
pogation formula in a similar way as in Formula 2:
σ2
Rij,kl
= σ2
Vkl
1
I2
ij
+ σ2
Iij
V 2
kl
I4
ij
. (4)
The vertical and horizontal resistances are calculated using
weights as the inverse of variance as followed:
Rvert = 1
Σi
1
σ2
i
R12,34
σ2
R12,34
+
R21,43
σ2
R21,43
+
R43,21
σ2
R43,21
+
R34,12
σ2
R34,12
(5)
and
Rhor = 1
Σi
1
σ2
i
R24,13
σ2
R24,13
+
R42,31
σ2
R42,31
+
R31,42
σ2
R31,42
+
R13,24
σ2
R13,24
, (6)
for which
Σi
1
σ2
i
(7)
is the normalization constant, the sum of the weights. The
error formula for these measurements are easily calculated to be
σ2
Rvert/hor
= 1
Σi
1
σ2
i
2
Σi,j,k,l
1
σ2
Rij,kl
, (8)
which is the sum of the product of the square of the weights
and the variance, and for which
σ2
Rij,kl
(9)
is calculated for each resistance used for Rvert and Rhor re-
spectively. Using these two resistances, the sheet resistance and
ratio R
R
can be calculated as so, where R’ is the maximum of
Rvert and Rhor, while R” is the remaining resistance. The error
of this is found by:
σ2
R
R
= σ2
R
1
R 2 + σ2
R
R 2
R 4 . (10)
The sheet resistance is found using a formula derived by van
der Pauw[3]:
e
−π
R12,34
RS + e
−π
R23,41
RS = 1. (12)
which is then formatted to allow for the inclusion of Rvert
and Rhor as better estimations:
e
−π
Rvert
RS + e
−π
Rhor
RS = 1. (12)
This is the most simple way to formulate the sheet resistance,
as there is no direct calculation available. RS is found using Ex-
cel, editing the value of RS until the value of 1 is significantly
found. The error of this calculation is
σ2
RS
= π2σ2
Rvert
+ π2σ2
Rhor
. (13)
Along with these measurements resistance calculations, a
base measurement for each signal input is recorded without wires
in the NI USB-6009 device.
Results
Using LabView to collect data and calculate measure-
ments and their standard deviation, we ended up with a base
(no circuit connected) measurement of background noise
and error to be 0.8 ± 2.4 nA for the current input, and
0.2±2.5 mV for the voltage input. This measurement is dis-
played in Figure 3a,c. The eight resistances Rij,kl are then
derived from the input currents and voltages using Equa-
tions 3 and 2, as displayed in Figure 4. The four values
for the vertical resistances are 6.75 ± 0.57 kΩ, 4.07 ± 0.58
kΩ, 7.16 ± 0.52 kΩ, 3.55 ± 0.56 kΩ; and the four values for
the horizontal resistances are 5.9 ± 1.1 kΩ, 6.4 ± 1.0 kΩ,
3.68 ± 0.61 kΩ, 7.55 ± 0.56 kΩ. The raw data for these mea-
surements are displayed in Figure 3b,d, and the calculated
values are shown in Figure 4.
2

3. Figure 3: Plots of measured current Iij through the sample and voltage Vkl of the sample over time. Plots 3a and 3c are measured
with no input to the NI USB-6009, showing the background and mechanical error. 3b and 3c are two of the sixteen datasets collected
during measurement, showing the current through the sample and the potential difference of the sample over time.
Using Formulas 5 and 6, Rvert is found to be 5.46 ± 0.28 kΩ,
and Rhor is found to be 5.87 ± 0.36 kΩ. These can be furthered
derived to find R
R
, which is 1.075 ± 0.086. The final calcula-
tion we derive is for the sheet resistance, which is found to be
25.7 ± 1.4 kΩ.
Figure 4: The individual resistances Rij,kl calculated during each
run, along with the derived vertical and horizontal resistances.
Discussion
The sheet resistance of 25.7±1.4 kΩ is very much on the
same scale of the pencil lead resistance measurements for
the 2 cm × 0.5 cm and 2.5 cm × 0.5 cm 2B lead on paper
setups, which were 25±5 kΩ and 35±5 kΩ respectively. The
ratio of the Rvert and Rhor also shows a well-distributed
sample, as it is very close to 1.0000, within one standard de-
viation. The error of the voltage measurements ranged from
2.3 to 2.7 mV, which is extremely similar to the noise de-
rived from the base measurements, 2.5 mV. For the current
calculations, the average error of 2.7 nA is also very close to
the measured base current error of 2.4 nA. These two facts
show the accuracy of the measurements and calculations,
and that there was no significant uncertainty besides me-
chanical uncertainties attributed to the NI USB-6009 device.
Summary
Although the sample used in this experiment was very
low-quality and cheap, it still yielded precise results, show-
ing its performance ability in other van der Pauw calcula-
tions, such as measuring Hall voltages, carrier doping, car-
rier mobility, and sheet carrier density. This experiment
optimized certain aspects of creating a simple, cheap van
der Pauw measurement of sheet resistance. The ratio of
the two resistances can also be used to estimate the unifor-
mity of the sample, which in this case was found to be not
significantly off from a 1:1 ratio. This means that materi-
als such as graphite could be optimized for use in van der
Pauw calculations that are usually reserved for uniform and
well-tested semiconducting substances like silicon[1].
References
[1] L.J. van der Pauw, Philips Tech. Rev., 20 (1959) 220.
[2] R. Chwang, B.J. Smith and C.R. Crowell, Solid-State
Elect., 17 (1974) 1217.
[3] L.J. van der Pauw, Philips Res. Rep., 13 (1958) 1.
3