Determination of Resistivity using Four Probe method
Vatsal Sinha , Roll no. 22134007
Int.Ph.D ,National Institute of Science Education and Research
January 18, 2023
1
Abstract
In our world filled with electronic devices , electrical
property like resistivity plays a large part in selection
of material for manufacturing of these devices. Similarly band gap of semiconductor is magical property
which drives our modern silicon era. So in our experiment we study and determine the value of resistivity of different materials and also determine the
bandgap of the Germanium using the Four probe
method. During this experiment consequentially we
will look to temperature dependence of resistivity of
semiconductor
2
through the outer probes without bothering about
the change in resistance(due to contact resistance).
Since the current flowing through is constant the
voltage across the inner probe remains almost
constant and used for accurate measurement of
resistivity of sample. The Constant Current Source
has the range 20-200 mA
Introduction
Figure 1: Constant Current Source (CCS-01)
Four Probe is four terminal device which is used to
determine the resistivity of the material very precisely. This precision is achieved by the four probe by
eliminating the contribution of contact resistances
and thus the measurement resistivity for wide of materials is possible using the four probe method. Here
in this experiment we will measure the resistivity of
the Aluminium & silicon with conducting base using
the four probe methodology, Also we will determine
the band gap of the germanium using four probe.
3
3.1
Low Current Source:- Low Current Source (LCS)
is constant current source but it is capable of generating small current (in range of microampere). This
current source is used particularly for the sample
material which is having high resistivity value.
The Low current Source has the range from 2 microampere to 2 milliampere with the resolution of
1 nanoampere as the device has 3 decimal precision
displayed on 31/2 digit LCD display screen.
Theory
Current Source and Voltmeter
In this experiment we use two different types of
current source and voltmeter’s depending on the
Figure 2: Low Current Source (LCS-02)
material used for the experiment.
Constant Current Source:- Constant current
source (CCS) is used to get the constant current MicroVoltmeter:- The microvoltmeter is used to
measure the voltage between the inner two probes in
order to determine the resistivity of the sample material. Microvoltmeter is required as the current flowing through the sample is ranging from few nanoampere’s to hundred’s of milliampere depending on the
sample. So this versitile voltmeter is used as range
knob of voltmeter reads from 1mV to 10V with the
resolution of microvolts.
Figure 3: Digital Microvoltmeter
PID Controller :- The term PID stands for
proportional integral derivative and it is one kind of
device used to control different process variables like
pressure, flow, temperature, and speed in industrial
applications. In this controller, a control loop
feedback device is used to regulate all the process
variables.
With the use of a low cost simple ON-OFF controller, only two control states are possible, like
fully ON or fully OFF. It is used for a limited
control application where these two control states
are enough for the control objective. However
oscillating nature of this control limits its usage and
hence it is being replaced by PID controllers.
• P-Controller:-Proportional or P- controller
gives an output that is proportional to current
error e (t). It compares the desired or set point
with the actual value or feedback process value.
The resulting error is multiplied with a proportional constant to get the output. If the error value is zero, then this controller output
is zero.This controller requires biasing or manual reset when used alone. This is because it
never reaches the steady-state condition. It provides stable operation but always maintains the
steady-state error.
• I-Controller:-Due to the limitation of pcontroller where there always exists an offset
between the process variable and setpoint, Icontroller is needed, which provides necessary
action to eliminate the steady-state error. It integrates the error over a period of time until the
error value reaches zero. It holds the value to
the final control device at which error becomes
zero.ntegral control decreases its output when a
negative error takes place. It limits the speed of
response and affects the stability of the system.
• D-Controller:-I-controller doesn’t have the capability to predict the future behavior of error. So it reacts normally once the setpoint is
changed. D-controller overcomes this problem
by anticipating the future behavior of the error.
Its output depends on the rate of change of error
with respect to time, multiplied by derivative
constant. It gives the kick start for the output
thereby increasing system response. Response
of D, the controller is more, compared to the
PI controller, and also settling time of output is
decreased. It improves the stability of the system by compensating for phase lag caused by
I-controller.So finally we observed that by combining these three controllers, we can get the
desired response for the system.
PID controller maintains the output such that there
is zero error between the process variable and setpoint/ desired output by closed-loop operations.
3.2
Four Probe Arrangement
The arrangement consists of four equally spaced
collinear probes. These probes are attached to
spring load so that excess load do not damage the
sample. The probes are mounted in a teflon bush,
which ensure a good electrical insulation between
the probes. A teflon spacer near the tips is also
provided to keep the probes at equal distance.
The probe arrangement is mounted in a suitable
stand, which also holds the sample plate and RTD
sensor. This stand also serves as the lid of PID
Controlled Oven. Proper leads are provided for
current, Voltage & Temp. measurement with their
universal connectors. For current measurement
there is three pin connector which can be connected
to the CCS-01/ LCS-02 as per requirement of
sample. For voltage measurement BNC connector is
used connected to DMV-001 unit. For temperature
measurement, a two pin connector is provided for
connection with PID- Controlled oven unit PID-200
at connector marked as Temperature Sensor.
1. The resistivity of materials in area of measurement is uniform
2. There is no injection of minority charge carriers in the semiconductor due to current flowing
throught four probe
3. The surface on which the probes rest is flat with
no surface leakage.
4. The distance between probes of four should be
equal and they should collinear.
5. Diameter of the four probe contact should be
small compared to the distance between the
probes.
Figure 4: Four Probe Arrangement
Three levelling screws are provided in Four Probe arrangement by which we can adjust the level of plateform to make it horizontal. A probe holding screw is
provided at the collar of the arrangement. Initially it
should be in loose position, to allow free movement
of Probe Pipe. After placing the sample the Probe
Pipe should be lowered so that all four pins touches
the sample. Further Press the pipe very lightly so
that the assured firm contact is made of all Four Pins
with the sample. Tighten the Probe Holding Screw
at this position. The Probe Arrangement is ready
with the sample for the experiment.
6. Surface of the material can be conducting or
non-conducting
3.4
Resistivity Measurement of the
large sample
From the design of the four probe the diameter of
probes is very small compare to that of distance between the probes so we can consider that the probes
are far away from any other surfaces of the sample
and we assume semi-infinite volume of uniform resistivity material[fig]. Four probes are spaced S1 ,S2
and S3 apart. Current I passes through the outer
probes(1 and 4) and the floating potential is measured across the inner pair of probes 2 and 3.
The floating potential Vf a distance r from an elec3.3 Working of Four Probe
trode carring a current I in a material of resistivity
Four probe contains of four terminals which are ρ0 is given by
placed on the surface of the sample. The outer two
ρ0 I
Vf =
probes are used to pass the current by using the DC
2πr
constant current source and remaining two probes In the fig- there are two current carrying electrodes,
are used measure the potential difference with the 1 and 4, and the floating potential V ,at any point
f
help of millivoltmeter. So here we can see that the Y in the semiconductor is the difference between the
use different probe for measuring the voltage and potential induced by each of the electrode, since they
current eliminates the contribution from the contact carry currents of equal magnitude but in opposite
resistance as the contact resistance will be experi- direction thus:
enced by the probes used for passing current and
since current is not used in calculating the resistivρ0 I 1
1
Vf =
−
(1)
ity, thus the value of resistivity becomes independent
2π r1 r4
of contact resistance.
Now while using the Four probe for measurement we where r1 = distance from probe number 1 and r4 =
have to make some assumptions. These assumptions distance from probe number 4.
The floating potentials at probe 2, Vf 2 and at probe
are :
3, Vf 3 can be calculated from (1) by substituting the The resistivity then becomes
proper distance as follows:
ρ0
ρ=
(6)
G6 ( W
)
1
ρ0 I 1
S
−
Vf 2 =
2π S1 S2 + S3
where resistivity ρ0 is computable from (2 and 3)
can be used if the point spacing are different, but
) is comapproximately equal. the function G6 ( W
S
ρ0 I
1
1
puted from
Vf 3 =
−
2π S1 + S2 S3


∞
The potential difference V between probes 2 and 3


1
W
S X
1

G6
=1+4
(−1)n 
− r
 r 2

2
S
W n=1
is then
S
S
2
2
+ (2n)
2W
+ (2n)
W
ρ0 I 1
1
1
1
(7)
V = Vf 2 −Vf 3 =
+
−
−
2π S1 S3 S2 + S3 S1 + S2
3.6
and the resistivity is given as
V
I
ρ0 =
2π
1
S1
+
1
S3
−
1
S2 +S3
1
S1 +S2
−
(2)
Resistivity Measurement on a
Thin Slice-Non-Conducting Bottom Surface
In this case we will the process as case , except the
When the point spacings are equal, that is, bottom surface of the slice is nonconducting. This
S1 = S2 = S3 = S the equation written above means that all the images of figure 3 have the same
simplifies to :
charge as the current source. Thus all the images
on a row have equal charges and eq.5 describes the
potential difference between inner probes, if (−1)n
V
(3) is removed from the equation. Then,
ρ0 = × 2πS
I
3.5
Resistivity Measurement of the
bottom conducting surface
In this case thin film of conducting material is
deposited on to the silicon wafer thus forming the
boundary in between the conducting thin film and
non-conducting silicon wafer.Therefore, we have
two boundary condition and these boundaries are
parallel and thus the solution is taken by using the
method of images. In this solution each current
source has infinite series of images along a line normal to the plane and passing through the current
source.
The model for this case is shown in Fig. 9. The
side surface of the slice is assumed to be far from
the area of measurement and, therefore, only the
effect of the bottom surface needs to be considered.
In this analysis equal probe spacing S shall be
assumed. The width of the slice is W. The array
of images needed is indicated in Fig. 9. where the
polarity and spacing of the first few images are as
shown.
The floating potential Vf2 at electrodes 2 is
ρ0
G7 ( W
)
S
ρ=
(8)
where


∞
S X
r
G7 (W/S) = 1 + 4
W n=1  S
W
1
2

1

− r

2
S
+ (2n)2
2W
+ (2n)2
(9)
For smaller values of W/S the function becomesG7 =
2S
loge 2
W
(10)
for a typical sample W/S ¡0.25 the correction factor
can be obtained from equation(10) directly.
3.7
Determination of Band Gap of
Germanium
In figure four probes are spaced S1, S2, S3 and S4
apart. Current I is passed through the outer probes
2π n=−∞
S 2 + (2nW )2
(2S)2 + (2nW )2
n=−∞
(4)
(1 & 4) and the floating potential V is measured
Likewise, the floating potential at electrode (3) can approx the inner pairs of probes 2 & 3. The potetial
be given by
difference between 2& 3 can given by
"
#
Vf 2 =
ρI
V = Vf 2 =

∞
X

ρI
1
2π
S
1
n
(−1)
+
∞
X
−
p
1
n
(−1)
n=1
p
S 2 + (2nW )2
∞
X
−
∞
X

p
4
n
(−1)
n=1

4
n
(−1)
p
(2S)2 + (2nW )2
(5)
V =
Iρ0
2πs
Where ρ0 is the resistivity of the material, I is 5
Graph
the amount of current passing through the material.
Therefore,
5.1 Resistivity for Silicon
V
ρ0 = 2πs
S=0.2± 2% cm
I
G7 (W/S)=5.545[from eq(10) ]
Since the thickness of the crystal is very small compared to the probe distance a correction factor for it
has to be applied.
ρ=
ρ0
ρ0
=
G7
f ws
Now substituting the values,
V
V
= 1.256
I
I
and the correction factor G7 i.e. f ws is 5.89
ρ0 = 2 × 3.14 × 0.2 ×
Figure 5: Graph 1: Voltage vs Current for silicon
1.256 V
ρ0
=
5.89
5.89 I
V
ρ = 0.213
I
ρ=
∴
5.2
Resistivity of Germanium
Thus ρ may be calculated for various temperatures.
1
Now, if we plot log10 ρ vs. , we get a curve which is
T
linear at higher
temperatures.
We know resistivity,
Eg
ρ = C exp
, where C is a constant. From
2KT
Eg 1
+ ln C
this expression we can have: ln ρ =
2K T
Therefore, width of the energy gap may be determined from the slope of the linear portion of the
experimental curve:
Figure 6: Graph 2 : Voltage vs Current for germanium
∆ log10 ρ
∆ ln ρ
1
Eg
=
=
×
1
1
2.303 2K
∆
2.303 × ∆
T
T
Thus we have
Eg = 2.303 × 2K
4
∆ log10 ρ
1
∆
T
(11)
Apparatus
PID Controller, Constant Current Source, Low current source, Four Probe, Silicon , Germanium ,
Oven, Microvoltmeter, Four Probe stand
5.3
Resistivity of Alumunium
Figure 7: Graph 2 : Voltage vs Current for alumunium
5.4
Temperature Dependence of Resistivity of Germanium
Figure 8: Graph 3 :Temperature dependence of resistivity of Germanium
6
6.1
Calculations
let slope = m
∴
For Silicon
ρ=
mπW
loge 2
From the graph 1 we get the slope = 136.2 and thus
using the value to calculate the resistivity ρ0 & ρ,
Now Error in resistivityThus we get,
ρ0 = Slope of graph 1 ×2πS=171.15×2π ×
s
2 2
0.2=130.898 Ω.cm
∂ρ
∂ρ
∆ρ =
∆m +
∆W
∴ Corrected resistivity of Si is=171.15/5.545=
∂m
∂W
30.866 Ω.cm
(12)
(13)
∴
6.2
For Germanium
From the graph 2 we get the slope = 86 and thus
using the value to calculate the resistivity ρ0 & ρ,
Thus we get,
ρ0 = Slope of graph 1 ×2πS=86×2π × 0.2=108.07
Ω.cm
∴ Corrected resistivity of Si is=108.07/5.545= 19.48
Ω.cm
6.3
For Alumunium
s
∆ρ =
πW
∆m
loge 2
2
+
mπ
∆W
loge 2
2
(14)
For si ∆m = 0.135Ω.cm,
∆W = 2% × 0.05cm=0.001cm
∴ ∆ρ for si =0.473 Ω.cm
For Ge ∆m = 0.061Ω.cm ,
∆W = 2% × 0.05cm=0.001cm
For Al ∆m = 0.0002Ω.cm ,
∆W = 2% × 0.05cm=0.0001cm
From the graph 2 we get the slope = 0.0003 and
thus using the value to calculate the resistivity ρ0 & From Eq. 14 ∆ρ for ge =0.355 Ω.cm
ρ, Thus we get,
ρ0 = Slope of graph 1 ×2πS=0.0003×2π × Now we know band gap 0.2=0.00188 Ω.cm
log ρ
∴ Corrected resistivity of Si is=0.00188/5.545=
Eg = 2k 1e
T
0.681×10− 6Ω.cm
6.4
Band gap of Germanium
From the graph 3 we found that the slope of graph
is 3822.5, Thus using equation 11 to find the band
gap energy of the germanium , thus we get, Eg =
∆ log10 ρ
2.303 × 2K
1
∆
T
so Eg = 2k × slope of graph = 2 ∗ 8.6 ∗ 10−5 ∗ 3822.5
⇒ 0.657eV
7
Error Analysis
We know that from Eq (8)
ρ0
2πS
ρ=
=(slope of V vs I graph)× 2S
W
G7 ( S )
loge 2
W
Slope × π × W
ρ=
loge 2
⇒ Eg = 2KB × (slope of graph 3)
⇒ ∆Eg = 2KB × ∆slope (graph-3)=0.0078 eV
8
Result and Conclusion
From this four probe experiment we successfully determined the value of resistivity of different materials
and band gap energy of germanium with high degree
of precision.
1. The experimentally measure value of resistivity
of silicon is, ρSi = 23.606 ± 0.473) Ω.cm.
2. The experimentally measure value of resistivity
of Germanium 17.73 ±0.355) Ω.cm.
3. The experimentally measure value of Bandgap
of Germanium is,Eg = (0.7022 ± 0.0078)eV.
9
Source of Error
1. Material (Al) used in the foil is commercial
grade, while standard resistance is for pure Al.
2. The thickness of Al foil is very small and thus
prone to non-uniformity in thickness
3. Variation of doping can affect the result of the
experiment
4. The resistivity of used cross section can show
non-uniformity
10
Precautions
• Probes are sharp so minimal pressure should be
applied so that it does not damage sample.
• The oven need to be given some amount of time
to achieve constant temperature.
• The electrical connection should not be loose.
• The probes should be at equal heights
• The probe pipe need to be stable and probes
need to be as sharp as possible
11
References
• https://www.iiserkol.ac.in/~ph324/
ExptManuals/ResistivityFourProbe.pdf
• https://www.pveducation.org/
pvcdrom/characterisation/
four-point-probe-resistivity-measurements
• https://www.ossila.com/pages/
sheet-resistance-measurements-thin-films