Resistivity and Seebeck measurements

Daniel Harada

August 18 2010

Resistivity

lAR

l

A

I

ResistivityTwo-point probe Four-point probe

Measures sample only

I

I

V

I

I

V

Rcontact Rcontact

Rsample

VRcontact Rcontact

Rsample

I

V

I

I

Measures sample + contact resistance + probe resistance

In four-point probe, negligible current flows through the voltmeter, the only voltage drop measured is across Rsample.

ResistivityTwo-point probe Four-point probe

Measures sample only

Rcontact Rcontact

Rsample

VRcontact Rcontact

Rsample

VMeasures sample + contact resistance + probe resistance

Collinear Contacts

Typical pellet and contact sizes for collinear contacts

d s

t

D

d = 1 mm

s = 2 mm

t = 1-2 mm

D = 12.7 mm

Collinear contacts

I

I

V

sIVs 2

Collinear contacts

I

I

V

s FIVs 2

F corrects for sample thickness, sample diameter, edge effects, and temperature.

Collinear contacts

For samples thinner than the probe spacing s, F can be written as a product of three independent correction factors.

321 FFFF

•F1 corrects for sample thickness

•F2 corrects for lateral sample dimensions

•F3 corrects for placement of probes near edges

Collinear contacts

For non-conducting substrates:

For conducting substrates replace sinh with cosh.

Collinear contacts

F11 is for non-conducting substrates

F22 is for conducting substrates

t/s for pellets ~0.75 - 1 F11 ~0.4 – 0.6

Collinear contacts

For circular wafers of diameter D:

3)/(3)/(ln)2ln(

)2ln(

2

22

sDsD

F

D/s ~ 6

F2 ~ 0.8

Collinear contacts

F3 accounts for contacts placed near sample edges

d/s ~ 1.7

F31 ~ 0.9

Parallel:

Perpendicular:

d/s ~ 3

F32 ~ 1

conducting substrates

non-conducting substrates

Van der Pauw method

Van der Pauw found a method to determine the resistivity of an arbitrarily shaped sample subject to the following conditions:

a) the contacts are at the circumference of the sampleb) the contacts are infinitely small (point contacts)c) the sample has uniform thicknessd) the surface of the sample is singly connected, i.e., the sample does not have isolated holes

Van der Pauw method

#1

#2 #3

#4

ab

cdcdab I

VR ,

fRR

t22ln

41,2334,12

41,23

34,12

RR

ff

Van der Pauw method

f must satisfy the relation:

2]/)2exp[ln(arccosh

2ln41,2334.12

41,2334.12 ffRRRR

For symmetric contacts f = 1

34,122lntR

34,12532.4 tR

Van der Pauw method

The previous equations were formulated assuming that contacts are point contacts. For contacts of finite size d on a circular disc of diameter D with d/D << 1, the percent increase in resistivity per contact can be found:

2

2

2ln41

Dd

van der Pauw, Philips

Research Reports 13 pg. 1-9

for d/D ~ 0.08 Δρ/ρ ~ -0.000559

total change for 4 contacts ~-0.224%

Porosity Correction

Pressed pellets are generally not going to achieve full theoretical density, and thus will contain non-conducting pores which will increase the measured conductivity. Our cold pressed pellets are typically ~80% of theoretical density.

Two models that are used to correct for porosity are the Bruggeman Effective Media model, and Minimal Solid Area.

Porosity Correction

The effective media model is discussed by McLachlan, Blaszkiewicz, and Newnham. For a pellet 90% dense or higher, it gives a correction factor of:

ff2311 2/3

where f is the volume fraction of spherical pores. At 90% dense this gives a correction of 0.85.

D. McLachlan, M. Blaszkiewicz, R. Newnham. Electrical Resistivity of Composites. Journal of the American Ceramic Society, 73 (8) 2187-2203 (1990)

Porosity Correction

The Minimal Solid Area model was presented by Rice. It assumes that fluxes through a medium will be limited by the smallest cross sectional area they pass through. This model gives a correction of:

fbewhere b is a factor that depends on the type of pores contained in the medium.

R. Rice. Evaluation and extension of physical property-porosity models based on minimum solid area. Journal of Materials Science, 31 102-118 (1996)

Porosity Correction

For spherical pores, b = 3, giving a correction factor of ~0.55 for 80% dense pellets. This model will give reasonable results for pellets ~70% dense and higher.

This is the model that will typically be used as our pellets are not dense enough to use the Bruggeman model.

Resistivity Summary

Collinear contacts: Van der Pauw contacts:

FIVs 2

F1 ~0.4 – 0.6

F2 ~ 0.8

F3 ~ 0.9

34,12532.4 tR

Contact correction is negligible

All equations and plots taken from Semiconductor Material and Device Characterization Third ed. by D.K. Schroder, unless otherwise noted.

Porosity correction = e-3f

Seebeck measurements

Thermoelectric effect: When a temperature gradient is maintained across a material, a voltage arises.

Seebeck effect: When two dissimilar conductors are joined together, and their junctions are held at different temperatures, a current flows.

TH TC

V≠0

TH TC

I

Seebeck measurements

n-type material in thermal equilibrium:

- -- -

--

--

- --

-

nleft = nright

Tleft= Tright

Seebeck measurements

apply a temperature gradient:

- -- -

--

--

- --

-

nleft > nright

Tleft> Tright

TH TC------ -

Seebeck measurements

free carriers diffuse from high concentration to low, leaving a net charge:

+ ++ -

--

--

- --

-

nleft = nright

Tleft> Tright

TH TC+

+

E

-- -

-

-

Seebeck measurements

VS

TH TCdTTSTSV

H

C

T

TCusampleS )()(

)()( CHCusampleS TTSSV

If S(T) does not vary much with temperature, then:

sampleCu SS TVS S

sample

+-

S should be negative for n-type materials