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A G R E A T E R M E A S U R E O F C O N F I D E N C E

Techniques for Measuring the Electrical

Resistivity of Bulk Materials

Mary Anne Tupta

November 18, 2010


A G R E A T E R M E A S U R E O F C O N F I D E N C E | w w w . k e I t h l e y . c o m

2

Resistivity Measurement Overview

• Definition of electrical resistivity of a material.

• Methods for making resistivity measurements of conductors,

insulators, and semiconductors.

• Key considerations for selecting equipment for measuring

resistivity.

• Sources of measurement errors and ways to optimize resistivity

measurements



3

What is the Electrical Resistivity of a Material?

The electrical resistance of a material……

• is a basic material property

• defines how well the material will conduct an electric current

• is a common electrical measurement.

Ohm’s law relates the current (I) and the applied voltage (V) to the

material resistance (R) as follows:

V=IR



4

What is the Electrical Resistivity of a Material?

Electrical Resistivity = longitudinal electrical resistance of a uniform

rod of unit length and unit cross-sectional area:

L

AR

V

L

A

Current Source

Voltmeter

ρ = resistivity (Ω-cm)

R = resistance: V/I (Ω)

A = cross-sectional area of

sample (cm2)

L = distance between two

leads of voltmeter (cm)



5

Typical Resistivity Values

The electrical resistivities of solid materials span over many magnitudes.

Three classifications of materials based on their resistivities:

Classification Type of Electrical

Conductor

Typical Resistivities

Metals Good electrical

conductors

10-6 Ω-cm

Insulators Low electrical

conductivity

109 to 1020 Ω-cm

Semiconductors Intermediate levels of

conductivity

10-3 to 107 Ω-cm



6

Comparison of Simplified Energy Bands

Conduction Band

Conduction Band

Conduction Band

Valence Band

Valence BandValence Band

Metal Insulator Semiconductor

Energy GapEnergy Gap

Energy



7

Measuring the Resistivity of

Conductors:Metals and Superconductors (very low resistance)

Graphene and other Nanomaterials

(low voltage and low power - use same techniques)



8

Measuring the Resistivity of Conductors

V

L

t

w

Voltmeter

Current Source

Sample

Basic Test Procedure:

1. Source current (I) through the

sample using one pair of leads.

2. Measure the voltage drop (V)

across a second pair of leads a

known distance (L) apart.

3. Calculate the resistivity (ρ) of

the sample using the cross-

sectional area (A=wt) and the

distance between the voltmeter

leads.

cmL

wt

I

V



9

Sources of Error When Measuring Low Resistance

• Test Lead Resistance

• Thermoelectric Voltages

• Low Frequency Noise

• External Noise Sources

• Johnson Noise

• Voltmeter Not Sensitive Enough



10

Eliminate Lead Resistance by Using the 4-Wire Method

V

RLeadRLead

RSample

I

V

RLeadRLead

RSample

I

Measured Resistance:

VM/I = RSample + 2RLead

Measured Resistance:

VM/I = RSample

2-Wire Method 4-Wire Method



11

Thermoelectric Voltages

Current Source

Sample – Metal B

V

Voltmeter

Copper Test Leads

Metal A

Thermoelectric voltages are generated when dissimilar metals (Metal A

and Metal B) in the circuit are at different temperatures (T1 and T2).

V

Voltmeter

Copper Test Leads

Metal A

T1 T2Sample

Metal B

Temperature Gradient



12

Ways to Reduce Thermoelectric Voltages

• Construct test circuits using the same materials for interconnects.

• Minimize temperature gradients within the test circuit

• Allow the test equipment to warm up

• Use an offset compensation method



13

Use the Current Reversal Method

to Eliminate Voltage Offsets (VEMF)

IRIRVIRVVV

V EMFEMFMMM

2

)(

2

Measurement with Positive Polarity Measurement with Negative Polarity

VM+ = VEMF + IR VM- = VEMF - IR

Voltage Measurement:

VEMF

R

VM+

VEMF

R

VM-I+ I-



14

Use the Delta Mode Method to Eliminate Voltage Offsets

and Noise

The Delta method consists of

alternating the current source

polarity and using a moving

average of voltage readings to

calculate the resistance.

Averaging reduces the noise

bandwidth and therefore the noise.



15

External Noise Sources

• External Noise Sources = interferences created by motors, computer

screens, or other electrical equipment

• Control these External Noise Sources by:

– Shielding and filtering

– Remove or turn-off the noise source

– When using DC instruments, integrate each measurement for an integer number of

power line cycles. The line cycle noise will “average out” when the integration time

is equal to an integration number of power line cycles.



16

Johnson Noise

• Johnson Noise – places a fundamental limit on resistance measurements.

• In any resistance, thermal energy produces the motion of charged particles. This charge movement results in Johnson noise.

• The formula for the voltage noise generated:

where k=Boltzmann’s constant, T= temp in K,

B=noise bandwidth in Hz, R=resistance of sample in ohms

• Reduce by:

1. reduce the measurement bandwidth –digital filtering (averaging readings) or analog filtering

2. reduce temperature of the device

3. reduce the sample resistance (usually not practical)

kTRBVrms 4



17

Use a Nanovoltmeter to Measure Voltage Drops

When measuring the resistances of conductors or other low power

materials, very small voltages are measured, typically in the microvolt and

nanovolt range.

To measure these very small voltage drops, use a sensitive voltmeter such

as a nanovoltmeter.Verify the product specifications to

make sure the measurement

resolution and accuracy will be

able to perform the sensitive

measurement of your application.

Model 2182A Nanovoltmeter



18

Review of Measurement Considerations for Measuring

the Resistivity of Conductors

• Make a 4-wire measurement to eliminate the lead resistance from affecting the measurement accuracy

• Use an offset compensation technique to eliminate voltage offsets and reduce noise: Current Reversal Method or Delta Method

• Eliminate external noise sources and use line cycle integration.

Continued……

V

L

Voltmeter

Current Source

Sample



19

Continued…..Review of Measurement Considerations

• Reduce Johnson noise by reducing the

measurement bandwidth, averaging

readings, and reducing the temperature

of the sample

• Use a sensitive voltmeter, such as a

nanovoltmeter, to measure the low

voltages

• Use a low noise, bipolar current source

that can perform current reversals

Model 2182A

Nanovoltmeter

kTRBVrms 4

Model 6221 AC

+ DC Current

Source

Model 6220 DC

Current Source



20

Measuring the Resistivity of Insulators

(high resistance)



21

Resistivity Measurements of Insulators

The resistance of an insulator

is measured by:

1. Applying a voltage to the

sample for a specified

time period

2. Measuring the resulting

current

3. Calculating the resistivity

using Ohm’s Law and

geometrical considerations

A

Sample

Resistance

AmmeterVoltage

Source

I

R

V

I

VR



22

Volume Resistivity of Insulator

t

A

I

V

Test Procedure for Volume Resistivity:

1. Please sample between 2 electrodes of area (A).

2. Apply potential difference (V) between the 2

electrodes.

3. Wait specified time (60 seconds) and measure

current (I) using sensitive ammeter.

4. Calculate resistivity based on the area of the

electrodes and thickness of the sample (t). Units

are Ω-cm.

Volume Resistivity is a measure of the leakage

current directly through a material.

HI

LO

HI

LO

Voltage

Source

Electrode

Sample

t

A



23

Surface Resistivity of Insulator

L

w

I

V

Surface Resistivity is defined as the electrical resistance of the surface of an

insulator.

Test Procedure for Surface Resistivity:

1. Place two electrodes a known distance (L) apart on sample.

2. Apply potential difference (V) between electrodes.

3. Wait specified time and measure current (I) with ammeter.

4. Calculate resistivity using width (w) of material and distance (L) between the

electrodes. The units are ohms or ohms per square.

ElectrodeHI

LO

HI

LO

wSample

L

A

I



24

Sources of Error When Measuring High Resistance

• Improper Measurement Instrumentation

• Electrification Time

• Test Voltage

• Background Currents

• Electrodes and Geometrical Considerations

• Electrostatic Interference

• Humidity



25

Use Sensitive Ammeter and Proper Cabling

Measuring the resistance of insulators usually involves measuring current in the

1E-9 and 1E-12 current range.

When measuring very small current, <100nA, it is important to use a sensitive

ammeter, such as an electrometer or picoammeter.

Electrometers and picoammeters have sub-picoamp (1E-12) sensitivity.

Model 6517B Electrometer/

Voltage Source:

The 6517B has 1E-15 A

Sensitivity!



26

Time Dependency of High Resistance Measurement

250V Step Response of Antistatic Bag

•Apply 250V

•Wait 60s

•Measure Current

•Calculate Resistivity

At 5 s, current=

1.5x10-10

At 60 s, current=

2.5x10-12



27

Test Voltage

• Resistivity measurements of insulators are dependent on the applied voltage.

• In general, as the voltage increases

the resistivity decreases

V

• Sometimes the voltage may be varied intentionally to determine

the voltage dependence of an insulator.



28

Background Currents

When measuring very high resistances, background currents can cause

erroneous readings. Background may be due to:

•charge stored in the material

•static or triboelectric charge

•piezoelectric effects

Background currents can be equal or greater than the current stimulated by the

applied voltage.

If background current is same polarity as measured current, then resultant

current reading will be much higher than the true value.

If background current is the opposite polarity, these unwanted currents may

cause a reverse polarity current reading. This can cause the calculated

resistance to be negative!



29

Alternating Polarity Method

Positive Voltage

Applied and

Exponential

Current Measured.

Negative Voltage

Applied and

Exponential

Current Measured.



30

Electrodes

Electrodes for use on an insulating material:

• Has good contact to the material: paint on electrodes or use flat metal plates with conductive rubber

• Should be much lower resistance than the sample and should not contaminate the sample

• Enables easy calculation of resistivity from the geometrical considerations

Rt

AelectrodeWhere:

ρ=Volume Resistivity

A=area of electrode on sample

t=thickness of sample

R=measured resistance V/I



31

Volume Resistivity with the Model 8009 Resistivity Test

Fixtures

Model 8009

Resistivity Test Chamber

Test

Sample

A

Voltage

SourceAmmeter

Guard0V

Electrodes

HI

LO

HI

LO

Volume resistivity is a measure of the leakage

current through the material, between the top

electrode and center bottom electrode. The

outside ring electrode is guard.

Top Electrode

Center Electrode

Ring



32

Surface Resistivity with the Model 8009

Test

Sample

AAmmeter

Guard0VElectrodes

HI

LO

HI

LO

R

The Surface Resistivity is

measured by placing two

electrodes on the surface of the

test sample, applying a potential

difference between them, and

measuring the resulting current.

Notice the surface resistivity (R)

is measured between the bottom

Center Electrode and the Ring

Electrode.



33

Model 65 High Resistivity Measurement Package

8009

Resistivity

Test Fixture

6517B Electrometer/

Voltage Source

6524

Software



34

Electrostatic Interference and Shielding

Shield

HI

LO

Voltage

SourceAmmeter

HI

LO

Sample

Connect Shield to LO terminal of Picoammeter or Electrometer



35

Review of Measurement Considerations for Measuring

the Resistivity of Insulators

1. Use a picoammeter or electrometer for low current measurements

2. Use the same electrification time for each test to compare results

3. Use the same applied voltage for test comparisons

4. Use the Alternating Polarity Technique to reduce the effects of background currents

5. Use proper electrodes and take geometrical considerations into account

6. Use electrostatic shielding to avoid errors due to electrostatic interference

7. Use an environmentally controlled room

Model 8009

Model 6517B



36

Measuring the Resistivity of

Semiconductors



37

Measuring Resistivity of Semiconductors

The two most common methods for measuring the resistivity of

semiconductor materials are the

• Four-Point Collinear Probe Method

• van der Pauw Resistivity Method

Both of these methods use a 4-wire method to eliminate both the

lead resistance and the contact resistance from affecting

measurement accuracy.

Photo courtesy of Lucas Signatone



38

4-Point Collinear Probe Method

tkI

V

2ln

Test Procedure:

1. Place probes in center of wafer.

2. Source current from probes 1 to 4.

3. Measure voltage between probes 2 and 3.

4. Calculate resistivity:

Where:

ρ= volume resistivity (ohm-cm)

V=voltage measured between 2 and 3

I=source current (A)

t = sample thickness (cm)

k=correction factor

V

4-Point

Collinear

Probe1 2 3 4

Source Current

From 1 To 4

Measure

Voltage

Between 2

And 3

HI

HI LO

LO

Wafer



39

Sources of Measurement Error

1. Lead and contact resistance

2. Voltage offsets (use current reversal method to reduce)

3. Instrumentation

4. Issues with high resistance materials



40

Test Set-Up Showing Circuit Resistances

V

1 2 3 4

Source Current

Measure Voltage

Between 2 And 3

HI

HI LO

LO

RL1 RL2 RL3RL4

RC1

RS2RS1 RS3

V

RC2 RC3 RC4

RL=Lead Resistance

RC=Contact Resistance

RS=Semiconductor Resistance

Only the voltage drop due to RS2 is

measured by the voltmeter.



41

Instrumentation for Mid-Range Resistances

(100mohm to <1Mohm)

Use a standard DC current source (bipolar to perform current reversals) to

force the current and a DMM to measure the voltage drop.

OR

Use a SourceMeter which can source the current and measure the voltage

drop.

ALSO

Use a commercially available 4-point probe head.




42

Commercially Available Automatic Resistivity System




43

Issues for High Range Resistances (up to 1E12 ohms)

• Use a 4-point collinear probe that has excellent isolation between

the probes to avoid leakage current errors

• Use a current source with high output impedance (1E14) to avoid

loading errors

• Use a current source with a built-in guard to reduce the effects of

shunt capacitance

• Use voltmeters with high input impedance (1E14 ohms)

• Use shielding to avoid errors due to electrostatic interference

• Use differential electrometer method to avoid issues of common

mode current



44

Common Mode Current Errors When Measuring High R

RIN

V

Voltmeter

RV

AC

HI LO

2 3

RC1

1

RC3 RC4

RC

AC

LO

CurrentSource

HI

CommonMode

Current

i

RC2

RS1 RS2 RS3

4

When the resistance of the

sample, RS2, becomes on the

same order of magnitude as

the isolation spec (input LO to

chassis) of the current source

and voltmeter, then common

mode current will flow

affecting the measurement

accuracy.Sample Resistance

Contact

Resistance



45

Making Differential Four-Point Probe Measurements

(For Very High Resistances >1Mohm)

V

Voltmeter

21

LOHI

X1Buffer

X1Buffer

HI LO

HI LO HI LO

RC3 RC4RC2RC1

RS1 RS2 RS3

43



46

van der Pauw Resistivity Method

V

1

2

3

4

Force

Current

Measure

Voltage

van der Pauw resistivity is a 4-probe

technique that involves applying a current

and measuring a voltage using four small

contacts on a circumference of a flat,

arbitrarily shaped sample.

Test Procedure:

•Force Current (I) on adjacent terminals

•Measure Voltage (V) on an opposite pair of

adjacent terminals

•Repeat measurements around sample

•Calculate resistivity



47

van der Pauw Resistivity Method

• vdp configuration is

useful for measuring

very small samples

• Easy to measure Hall

voltage using an

electromagnet to apply

the B field

• Force I and Measure V

on opposite terminals

V

1

2

3

4

Force

Current

Measure

Voltage

Hall Configuration

B



48

van der Pauw Resistivity

V7

V8

V1

V2

V4

V5

V6

V3

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34

A series of 8 measurements are performed around the periphery of the sample

to compensate for offsets and are combined mathematically to compute the

resistivity.



49

Switching to Perform vdp Measurements

V

1

1

1

2

2

2

3

3

3

4

4

4

Sample

Current

Source

Voltmeter

Switch Matrix

Use a switch matrix to

automatically switch the current

source and voltmeter between all

four terminals of the sample.

Choose a switch system that will

not degrade the measurements.

Use offset compensation

technique – current reversals – to

eliminate voltage offsets due to

the switch.

Use same configuration to

measure the Hall voltage using

an electromagnet.



50

High Resistance vdp Measurements

Model 7065 Hall Effect

Card Configuration

Matrix Card that switches

current source and

voltmeter to the 4 terminals

of the sample.

Has unity gain buffers on

the card to avoid problems

with isolation or needing to

use an electrometer to

measure the voltage drops



51

Using Four SMUs to Measure High R Samples

SMU2

V Measure

SMU3

V Measure

SMU4

Common

SMU1

I Source

1 2

34

V Difference

i

Model 4200-SCS with 4 SMUs and 4 preamps

– Input impedance >1016

– Accurate low current sourcing, pA

– No leakage errors due to

mechanical switches

– Includes software to automate

measurements and calculate

resistivity

Model 4200-SCS Semiconductor

Characterization System



52

Making Good Measurements on High Resistance

Samples

• Use electrostatic shielding to minimize electrical interference– Shield the DUT and all sensitive circuitry

– Use shielded cabling

– Connect the shield to the low terminal of the system

• Use guarding to reduce the effects of leakage current in system– Guarded current source

– Guarded voltmeters

– Use triax cable instead of coax cable

• Allow sufficient settling time– Source I and measure V as a function of time to determine appropriate

settling time

– A diamond sample can take several minutes for settling



53

Settling Time of a 1012Ω Resistance Sample

Wait at least

20 seconds

for a settled

measurement



54

Summary

Use an appropriate method for measuring the

resistivity. The method will depend on if the material is

a conductor, insulator, or semiconductor.

Choose the appropriate instrumentation.

Apply the proper measurement techniques to avoid

measurement errors.



55

Reference Materials

www.keithley.com

Low Level Measurements Handbook, 6th

Edition

Application Notes: Hall effect, van der

Pauw method, four-point collinear probe

method, insulator resistivity, low voltage

measurements, low current

measurements, etc.

White Papers: Delta method, current

reversal techniques, etc.



56

Contact Keithley for Further Information

Worldwide Headquarters

Within the USA: 1-888-KEITHLEY

Outside the USA: +1-440-248-0400

Email: [email protected]

Additional offices: www.keithley.com

Europe:

Germany: (+49) 89 849 307 40

Great Britain: (+44) 118 929 7500

Asia:

China: (+86) 10-8447- 5556

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