van der Pauw and Hall Voltage
Measurements with the 4200A-SCS
Parameter Analyzer––
APPLICATION NOTE

van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

Introduction

APPLICATION NOTE

Semiconductor material research and device testing often

van der Pauw Resistivity Measurement
Methodinvolve determining the resistivity and Hall mobility of a

The van der Pauw method involves applying a current

sample. The resistivity of semiconductor material is primarily

and meas­uring voltage using four small contacts on thedependent on the bulk doping. In a device, the resistivity

circumference of a flat, arbitrarily shaped sample of uniform

can affect the capacitance, the series resistance, and thethickness. This method is particularly useful for measuring­threshold voltage.

very small samples because geometric spacing of thecontacts is unimportant. Effects due to a sample’s size, which

The resistivity of the semiconductor material is often
determined using a four-point probe technique. With a four-is the approximate probe spacing, are irrelevant.

probe, or Kelvin, technique, two of the probes are used toUsing this method, the resistivity can be derived from a totalsource current and the other two probes are used to measureof eight measurements that are made around the periphery ofvoltage. Using four probes eliminates measurement errors

the sample with the configurations shown in Figure 1.

due to the probe resistance, the spreading resistance undereach probe, and the contact resist­ance between each metal

Once all the voltage measurements are taken, two values ofprobe and the semiconductor material. Because a highresistivity, ρA and ρB, are derived as follows:

impedance voltmeter draws little current, the voltage dropsπ		(V1 – V2 + V3 – V4)
ρA = ____ fAts _____________________
		ln 2		
4I

across the probe resistance, spreading resist­ance, andcontact resistance are very small.
One common Kelvin technique for determining the resistivityπ		(V5 – V6 + V7 – V8)
ρB = ____ fBts _____________________
		ln 2		
4Iof a semiconductor material is the van der Pauw (VDP)

where: ρA and ρB are volume resistivities in ohm-cm;

method. The 4200A-SCS Parameter Analyzer includes a

ts is the sample thickness in cm;

project for making van der Pauw measurements. Because ofV1–V8 represents the voltages measured by theits high input impedance (>1016Ω) and accurate low current

voltmeter;

sourcing, the 4200A-SCS with preamps is ideal for highI is the current through the sample in amperes;

resistance samples. This application note explains how tomake resist­ivity measurements of semiconductor materials
using the 4200A-SCS and the van der Pauw method.

fA and fB are geometrical factors based on sample symmetry.
They are related to the two resistance ratios QA and QB
as shown in the following equations (fA = fB = 1 for perfectsymmetry).

V5
1

2

4

3

V3

1

2

1

2

1

2

4

3

4

3

4

3

V7

V1
V6
1

2

4

3
V2

Figure 1. van der Pauw Resistivity Conventions

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V4

1

2

1

2

1

2

4

3

4

3

4

3

V8

van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

APPLICATION NOTEQA and QB are calculated using the measured voltages

The 4200A-SCS with four SMU instruments and four preamps

as follows:(for high resistance measurements) is an ideal solution for		V
1 – V2
QA = _________
		V3 – V4

meas­uring van der Pauw resistivity, and should enablemeasurements of resistances greater than 1012Ω. Since each
SMU instrument can be configured as a current source or as

		V
5 – V6
QB = _________
		V7 – V8

a voltmeter, no external switching is required, thus eliminatingleakage and offsets errors caused by mechanical switches.
This removes the need for additional instruments andAlso, Q and f are related as follows:
Q – 1		 _______f		_______=
arc coshQ + 1		 0.693		

(

e0.693/f_______
2)

programming.
For high resistance materials, a current source that can

A plot of this function is shown in Figure 2. The value of f can

output very small current with a high output impedance

be found from this plot once Q has been calculated.

is necessary. A differential electrometer with high input

Once ρA and ρB are known, the average resistivity (ρAVG) canbe determined as follows:

impedance is required to minimize loading effects on thesample. On the lowest current source ranges (1pA and 10pA)of the 4200A-SCS, the input resistance of the voltmeterρA + ρB
ρAVG = _________
		 2

is >1016Ω.

Using the 4200A-SCS to Measure
Resistivity with the van der Pauw
Method1.0
0.9

Each terminal of the sample is connected to one SMU
0.8

f

instrument, so a 4200A-SCS with four SMU instruments isrequired. The van der Pauw Project, which has four tests,

0.7

is in the Projects Library and can be found by selecting theMaterials box from the Select pane. A screen capture of the0.6

van der Pauw project is shown in Figure 3.
Each test has a different measurement setup. A diagram

0.5

of how the four SMU instruments are configured in the
0.4

1

10

100

Q
Figure 2. Plot of f vs. Q

Test Equipment

four tests is shown in Figure 4. For each test, three of the
SMU instruments are configured as a current bias and a
voltmeter (VMU). One of these SMU instruments applies thetest current and the other two SMU instruments are usedas high impedance voltmeters with a test current of zeroamps on a low current range (typically 1nA range). The fourth
SMU instrument is set to common. The voltage difference is

The electrical measurements for determining van dercalculated between the two SMU instruments set up as highPauw resistivity require a current source and a voltmeter.

impedance voltmeters. This measurement setup is duplicatedTo automate measurements, one might typically use aaround the sample, with each of the four SMU instruments

programmable switch to switch the current source and thechanging functions in each of the four tests. The test current

voltmeter to all sides of the sample. However, the 4200A-SCS

and voltage differences between the terminals from theis more efficient than this.

four tests are used to calculate the resistivity in the subsite
calc sheet.

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van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

APPLICATION NOTE

Figure 3. van der Pauw Project

Test Name:
i4-v12

Test Name:
i2-v34
CommonCurrent Bias (+)

SMU1

SMU2

CommonCurrent Bias (–)

SMU1

SMU2

Voltmeter

Voltmeter

Voltmeter

SMU1

SMU2

SMU1

V12

2

1

2

1

2

1

2

4

3

4

3

4

3

4

3

SMU4

SMU3

V34

Voltmeter

SMU4

SMU3

V34

Voltmeter

Voltmeter

SMU4

SMU3

Current Bias (+)

CommonVoltmeter

CommonVoltmeter

Common

SMU1

SMU2

SMU1

SMU2

1

Voltmeter

2

4

V41

3
SMU3

Current Bias (+)

SMU4

Voltmeter

Current Bias (+)

SMU2

2

1

4

3

4
SMU3

Current Bias (–)

Voltmeter

SMU1

1

Figure 4. SMU Instrument Configurations for van der Pauw Measurements

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SMU4

SMU3

Current Bias (–)

CommonTest Name:
i1-v23

Test Name:
i3-v41

SMU4

SMU2

V12

1

Voltmeter

V41

Voltmeter

SMU4

CommonCurrent Bias (–)

2
3

V23
SMU3

Voltmeter

Voltmeter

SMU1

SMU4

Common

SMU2
1

2

4

3

V23
SMU3

Voltmetervan der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

APPLICATION NOTE

Test Configurations for Measuring van der Pauw
Resistivity

are averaged using the AVG function with an equation, such

In each of the four tests, one SMU instrument is configured

located on the Test Settings pane, the average voltage (V23)

as the current source using the current list sweep function,is selected so that it is sent to the subsite data sheet, where it

two SMU instruments are configured as voltmeters, and one

will be used in the resistivity calculation. The magnitude of theis configured as a common. Here is the configuration for thecurrent source must also be sent to the subsite data sheet, so

i1-v23 test:

click that check box (I1) as well.

Terminal 1 – SMU1: Set to a two-point Current List Sweep.

This same procedure is repeated in all four tests so that

Enter both the positive and negative values of the appropriate

as V23=AVG(ABSV23). In the Output Values d­ ialog box

all sides of the sample are measured, as shown previously

source current. Enter the Compliance level and use the Bestin Figure 1. The average voltages from each test are thenFixed source range. Averaging voltage measurements (fromused to calculate the resist­ivity on the subsite Calc sheet.

SMU2 and SMU3) taken at both a positive and negative test

A diagram showing how the SMUs are set up in each test is

current will correct for voltage offsets in the circuit.

shown in Figure 4.

Terminal 2 – SMU2: Set to Current Bias (VMU) with a test

Adjusting the Source Current

current of 0A. Set the appropriate compliance voltage, andThe source current value will need to be modified according

check the Measure Voltage box (VB). Even though 0A willto the expected sample resistance. Adjust the current so thatbe output, select an appropriate current source range. The

the voltage difference will not exceed 25mV (approximately).

input impedance of the voltmeter is directly related to theIn each of the four tests (i2-v34, i3-v41, i4-v12, i1-v23), enter

current source range. The lower the current source range is,both polarities of the test current. The same magnitude mustthe higher the input impedance will be. However, the lower

be used for each test.

the current source range, the slower the measurement time
will be. For most applications, the 1nA range may be used.
However, for very high resistance measurements, use a lower
current range.

Determining the Settling TimeFor high resistance samples, it will be necessary to
determine the settling time of the measurement. This can

Terminal 3 – SMU3: Set up the same as Terminal B.

be accomplished by sourcing current into two terminals ofTerminal 4 – SMU4: Set the Operation Mode to Common.

the other two terminals. The settling time can be determined

In this test, the current is sourced between Terminals 1 and
4 (SMU1 to SMU4). SMU2 will measure the voltage fromthe sample and measuring the voltage difference between
by graphing the voltage difference versus the time of themeasurement.

Terminal 2 to Terminal 4. SMU3 will measure the voltage from

Using the 4200A-SCS, the voltage versus time graph can

Terminal 3 to Terminal 4.

easily be created by copying and pasting one of the tests

The Formulator calculates the voltage difference between
SMU2 and SMU3 for both the positive and negative test
current. This can be done using an equation such as
V23DIFF=V2-V3. The absolute values of these numbers(from both the positive and negative test current) aretaken using the ABS function with an equation such as
V23ABS=ABS(V23DIFF). The two voltage difference values

described previously. In the Test Settings pane, take a few
hundred or so readings with a sweep time of one second.
Make sure that the Timestamp box is selected. After thereadings are done, plot the voltage difference versus timeon the graph. (You can choose the parameters to graph byusing Graph Settings.) The settling time is determined fromthe graph. A timing graph of a very high resistance material isshown in Figure 5.

WWW.TEK.COM | 5van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

Figure 5. Voltage vs. Time Graph of Very High Resistance Sample

Figure 6. Subsite Data “Calc” Sheet with Resistivity Displayed

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APPLICATION NOTE

van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

Determine the settling time by visually inspecting the voltage
difference vs. time graph. Once the settling time has been
determined, use this time as the Sweep Delay (in the Test
Settings pane) for the four resistivity measurement tests
listed previously. This settling time procedure will need to berepeated for different materials; however, it is not necessary
for low resistance materials since they have a short

APPLICATION NOTEHall Voltage Measurements
Hall effect measurements are important to semiconductormaterial characterization because from the Hall voltage,
the conductivity type, carrier density, and mobility can bederived. With an applied magnetic field, the Hall voltage canbe measured using the configurations shown in Figure 7.

settling time.

Calculating the Resistivity
To view the calculated resistivity in the subsite level, selectthe subsite plan in the Project Tree. The Output Values

V(voltage differences and test current) will appear on the data

1
4

2
3

V

1

2

4

3

sheet at the subsite level. The resistivity is calculated on the
Calc sheet from the cell references on the Data sheet. The
thickness, coefficients, and correction factors are also input
on the Calc sheet for the resistivity equation.

Figure 7. Hall Voltage Measurement Configurations

With a positive magnetic field, B, apply a current between

These values can be updated in the Calc sheet in theterminals 1 and 3, and measure the voltage drop (V2–4+)

subsite level. Select the subsite “vdp-test.” Go to thebetween terminals 2 and 4. Reverse the current and measure

Subsite Data tab. It contains the output values of the voltage

the voltage drop (V4–2+). Next, apply current between terminals

differences and the test current. From the Calc Sheet tab,2 and 4, and measure the voltage drop (V1–3+) between

the thickness can be adjusted. The default thickness is 1cm.

terminals 1 and 3. Reverse the current and measure theIf necessary, a correction factor can also be applied to thevoltage (V3–1+) again.

resistivity equation.

Running the Project
The van der Pauw Project (vdp-resistivity) must be run atReverse the magnetic field, B, and repeat the procedureagain, measuring the four voltages: (V2–4–), (V4–2–), (V1–3–),
and (V3–1–).

the subsite level. Make sure that all boxes in the Project are

From the eight Hall voltage measurements, the average Hall

checked and select the vdp-test subsite. Execute the project

coefficient can be calculated as follows:

by using the Run button. Each time the test is run, the subsite

		t____________________________________s (V4–2+ – V2–4+ + V2–4– – V4–2–)
RHC =
		
4BI

data is updated. The voltage differences from each of the four
tests (i2-v34, i3-v41, i4-v12, i1-v23) will appear in the Subsite
Data “vdp-device” sheet. The resistivity will appear in theSubsite Data “Calc” sheet as shown in Figure 6.

		t____________________________________s (V3–1+ – V1–3+ + V1–3– – V3–1–)
RHD =
		
4BI
where: RHC and RHD are Hall coefficients in cm3/C;
ts is the sample thickness in cm;
V represents the voltages measured by the voltmeter;
I is the current through the sample in amperes;
B is the magnetic flux in Vs/cm2 (1 Vs/cm2 = 108 gauss)
Once RHC and RHD have been calculated, the average Hall
coefficient (RHAVG) can be determined as follows:

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van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

APPLICATION NOTE

		R
HC + RHDRHAVG = ____________
		
2

Leakage Current

From the resistivity (ρAVG) and the Hall coefficient (RHAVG), themeasurements. The leakage current is due to the insulation

mobility (μH) can be calculated:		| RH |
	µH = _____		
ρAVG

For high resistance samples, leakage current may degrade
resistance of the cables, probes, and test fixturing. Leakage
current may be minimized by using good quality insulators, byreducing humidity, and by using guarding.

Using the 4200A-SCS to Measure theHall Voltage

A guard is a conductor connected to a low impedance point

The setup to measure the Hall voltage is very similar to theimpedance lead being guarded. The inner shield of the triax

setup for measuring resistivity. The difference is the location

connector of the 4200-SMU is the guard terminal. This guardof the current source and voltmeter terminals. Figure 7

should be run from the 4200-SMU to as close as possible toillustrates the setup for Hall voltage measurements. (Figure 4

the sample. Using triax cabling and fixturing will ensure that

illustrates the setup for resistivity measurements.) If the user-

the high impedance terminal of the sample is guarded. The

supplied electromagnet has an IEEE-488 interface, a program

guard connection will also reduce measurement time since

can be written in KULT (Keithley User Library Tool) to control

the cable capacitance will no longer affect the time constantthe electromagnet using the 4200A-SCS. This would requireof the ­measurement.

that the 4200A-SCS have the 4200-COMPILER option.

Sources of Error and Measurement
Considerations
For successful resistivity measurements, the potential

in the circuit that is nearly at the same potential as the highLight
Currents generated by photoconductive effects can degrade
measurements, especially on high resistance samples. To
prevent this, the sample should be placed in a dark chamber.

sources of errors need to be considered.

Temperature

Electrostatic Interference

Thermoelectric voltages may also affect measurement

Electrostatic interference occurs when an electricallycharged object is brought near an uncharged object. Usually,
the effects of the interference are not noticeable because
the charge dissipates rapidly at low resistance levels.
However, high resistance materials do not allow the charge
to decay quickly and unstable measurements may result.
The erroneous readings may be due to either DC or AC
electrostatic fields.
To minimize the effects of these fields, an electrostatic shield
can be built to enclose the sensitive circuitry. The shield ismade from a conductive material and is always connectedto the low impedance (FORCE LO) terminal of the SMU
instrument.

accuracy. Temperature gradients may result if the sample
temperature is not uniform. Thermoelectric voltages may
also be generated from sample heating caused by thesource current. Heating from the source current will more
likely affect low resistance samples because a higher test
current is needed to make the voltage measurements easier.
Temperature fluctuations in the laboratory environment
may also affect measurements. Since semiconductors
have a relatively large temperature coefficient, temperature
variations in the laboratory may need to be compensated forby using correction factors.

Carrier Injection
To prevent minority/majority carrier injection from influencing
resistivity measurements, the voltage difference between the

The cabling in the circuit must also be shielded. Low noise

two voltage sensing terminals should be kept at less thanshielded triax cables are supplied with the 4200A-SCS.

100mV, ideally 25mV, given that the thermal voltage, kt/q, isapproximately 26mV. The test current should be kept to as
low as possible without affecting the measurement precision.

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van der Pauw and Hall Voltage Measurements withthe 4200A-SCS Parameter Analyzer

APPLICATION NOTE

Bibliography
ASTM, F76-08. Standard Method for Measuring Hall
Mobility and Hall Coefficient in Extrinsic Semiconductor
Single Crystals.
van der Pauw, L. J. A Method of Measuring SpecificResistivity and Hall Effects of Discs of Arbitrary Shape. Philips
Rec. Repts., 1958: 13 1.
Schroder, Dieter K. Semiconductor Material and Device
Characterization. John Wiley & Sons, Inc., 3rd Edition, 2015.
Low Level Measurements, Keithley Instruments, Inc.,
Cleveland, Ohio, 2014.

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APPLICATION NOTE

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