1. Resistivity and Seebeck measurements
Daniel Harada
August

2. Resistivity I

3. Resistivity Two-point probe Four-point probe Measures sample only
Measures sample + contact resistance + probe resistance
Rcontact
Rsample
Rcontact
Rsample
In four-point probe, negligible current flows through the voltmeter, the only voltage drop measured is across Rsample.

4. Resistivity Two-point probe Four-point probe Measures sample only
Measures sample + contact resistance + probe resistance
Rcontact
Rsample
Rcontact
Rsample

5. Typical pellet and contact sizes for collinear contacts
d = 1 mm
s = 2 mm
t = 1-2 mm
D = 12.7 mm

6. Collinear contacts

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

8. Collinear contacts
For samples thinner than the probe spacing s, F can be written as a product of three independent correction factors.
F1 corrects for sample thickness
F2 corrects for lateral sample dimensions
F3 corrects for placement of probes near edges

9. Collinear contacts For non-conducting substrates:
For conducting substrates replace sinh with cosh.

10. Collinear contacts F11 is for non-conducting substrates
F22 is for conducting substrates
t/s for pellets ~
F11 ~0.4 – 0.6

11. Collinear contacts
For circular wafers of diameter D:
D/s ~ 6
F2 ~ 0.8

12. Collinear contacts F3 accounts for contacts placed near sample edges
conducting substrates
Parallel:
d/s ~ 1.7
F31 ~ 0.9
Perpendicular:
d/s ~ 3
F32 ~ 1
non-conducting substrates

13. 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 sample
b) the contacts are infinitely small (point contacts)
c) the sample has uniform thickness
d) the surface of the sample is singly connected, i.e., the sample does not have isolated holes

14. Van der Pauw method #4 #1 #2 #3

15. Van der Pauw method f must satisfy the relation:
For symmetric contacts f = 1

16. 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:
van der Pauw, Philips Research Reports 13 pg. 1-9
for d/D ~ Δρ/ρ ~
total change for 4 contacts ~-0.224%

17. 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.

18. 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:
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) (1990)

19. 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:
where 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, (1996)

20. 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.

21. Resistivity Summary Collinear contacts: Van der Pauw contacts:
Contact correction is negligible
F1 ~0.4 – 0.6
F2 ~ 0.8
F3 ~ 0.9
Porosity correction = e-3f
All equations and plots taken from Semiconductor Material and Device Characterization Third ed. by D.K. Schroder, unless otherwise noted.

22. Seebeck measurements
Thermoelectric effect: When a temperature gradient is maintained across a material, a voltage arises.
V≠0 TH TC I
Seebeck effect: When two dissimilar conductors are joined together, and their junctions are held at different temperatures, a current flows. TH TC

23. n-type material in thermal equilibrium:
Seebeck measurements
n-type material in thermal equilibrium: - - - - - - - - - - - -
nleft = nright
Tleft= Tright

24. apply a temperature gradient:
Seebeck measurements
apply a temperature gradient: - - - - - - TH - - - - TC - - - - - - - - -
nleft > nright
Tleft> Tright

25. Seebeck measurements
free carriers diffuse from high concentration to low, leaving a net charge: + - + - - - TH + - - - TC - - + + - - - - -
nleft = nright
Tleft> Tright

26. Seebeck measurements TH TC
If S(T) does not vary much with temperature, then: - + VS
S should be negative for n-type materials