1. Matt Brown Alicia Allbaugh Electrodynamics II Project 10 April 2001
Ellipsometry
Matt Brown
Alicia Allbaugh
Electrodynamics II Project
10 April 2001

2. Ellipsometry
A method of probing
surfaces with light.

3. Introduction History Methodology Theory Types of Ellipsometry
Applications
Summary

4. History
Fresnel derived his equations which determine the Reflection/Transmission coefficients in early 19th century. Ellipsometry used soon thereafter.
Last homework assignment Electrodynamics I.
Ellipsometry became important in 1960’s with the advent of smaller computers.

5. Methodology
Polarized light is reflected at an oblique angle to a surface
The change to or from a generally elliptical polarization is measured.
From these measurements, the complex index of refraction and/or the thickness of the material can be obtained.

6. Theory Determine r = Rp/Rs (complex)
Find r indirectly by measuring the shape of the ellipse
Determine how e varies as a function of depth, and thickness L of transition layer.
Note: We will focus on the case of very thin films. In this case, only the imaginary part of r matters.

7. Maxwell’s equations for a wave incident 1 2 x z y
Maxwell’s equations for a wave incident
On a discontinuous surface. (Gaussian Units)
Boundary Conditions

8. Derivation of Drude Equation
Fundamentals of Derivation
Concept: Integrate a Maxwell Equation along z over transition region of depth L. Result will be a new Boundary Condition.
Fundamental Approximations: a.
b. We assume certain field components , which vary slowly along z, are constant. Y
Example: Since Hx+= Hx-, and l/L<<1, Hx1~Hx2.

9. Derivation of Drude Equation
Assumption that is uniform
With respect to y
Integrate along z over L

10. Derivation of Drude Equation
Assumption that varies little:
Since , = constant.
and
Substituting
Rearrangement yields

11. ; Y
Integrate
and vary
little over L
where

12. Similarly, we now find new B.C. for and
New complete Boundary Conditions
Where Y

13. Y We now solve Maxwell’s equations with these new Boundary Conditions
Relate H and E
Form of E field (to satisfy Maxwell eq.) Y
Continuity

14. Again solve Maxwell’s equations with these new Boundary Conditions
Note on notation:
Subscript p refers to component parallel to incident plane (x-z plane), and subscript s refers to perpendicular (same as y) component.
Boundary Condition
Relate H and E
Form of E field (to satisfy Maxwell eq.)
Continuity y

15. This results in 4 relations between , , and .

16. Algebraically eliminate transmission terms.
Example: Parallel components
where
Notice that if we assume p and q terms to be
Proportional to L, the imaginary parts of top and
Bottom are proportional to

17. Approximation for when L<<l such that terms in second order of L/l can be neglected.

18. Set polarization at 45 degrees. Then
Using Snell’s Law,
We get
Again, keeping only terms to first order in L/l, and using binomial expansion,
where

19. This is the Drude Equation.
Recall that at Brewster’s angle Ep is minimized
So near Brewster’s Angle, we get
This is the Drude Equation.
For thin films, we often take to be the dielectric constant
Of air, to be that of our substrate, and to be constant
in the film. Then

20. Types of Ellipsometry Null Ellipsometry Photometric Ellipsometry
Phase Modulated Ellipsometer
Spectroscopic Ellipsometry

21. Null Ellipsometry
We choose our polarizer orientation such that the relative phase shift from Reflection is just cancelled by the phase shift from the retarder.
We know that the relative phase shifts have cancelled if we can null the signal with the analyzer

22. Example Setup
Phase modulated ellipsometer

23. How to get r,an example. Phase Modulated Ellipsometry

24. How to get r,an example. Phase Modulated Ellipsometry
The polarizer polarizes light to
45 degrees from the incident plane.

25. How to get r,an example. Phase Modulated Ellipsometry
The birefringment modulator introduces a time varying phase shift.

26. How to get r,an example. Phase Modulated Ellipsometry
Upon reflection both the parallel and perpendicular components are changed in phase and amplitude.
For a discontinuous interface,
For a continuous interface,

27. How to get r,an example. Phase Modulated Ellipsometry

28. How to get r,an example. Phase Modulated Ellipsometry
Photomultiplier Tube measures intensity.
Note: The J’’s are the Bessel Functions

29. At the Brewster Angle,

30. How to get r,an example. Phase Modulated Ellipsometry
We find the Brewster angle by adjusting until
Which is where
Now we can use a calibration procedure to
Find the proportionality of

31. Applications Determining the thickness of a thin film
Focus of this presentation

32. Applications - Continued
Research
Thin films, surface structures
Emphasis on accuracy and precision
Spectroscopic
Analyze multiple layers
Determine optical constant dispersion relationship
Degree of crystallinity of annealed amorphous silicon
Semiconductor applications
Solid surfaces
Industrial applications in fabrication
Emphasis on reliability, speed and maintenance
Usually employs multiple methods

33. Ellipsometry Ellipsometry can measure the oxide depth.
Intensity doesn’t vary much with film depth but D does.

34. Other Methods Reflectometry Microscopic Interferometry
Mirau Interferometry

35. Reflectometry Reflectometry
Intensity of reflected to incident (square of reflectance coefficients).
Usually find relative reflectance.
Taken at normal incidence.
Relatively unaffected by a thin dielectric film.
Therefore not used for these types of thin films.

36. Ellipsometry Ellipsometry can measure the oxide depth.
Intensity doesn’t vary much with film depth but D does.

37. Reflectometry

38. Reflectometry
Can be more accurate for thin metal films.

39. Microscopic Interferometry
Uses only interference fringes.
Only useful for thick films and/or droplets
Thickness h>l/4

40. Mirau Interferometry Accuracies to 0.1nm
Dx is less than present ellipsometry
At normal incidence.
Kai Zhang is constructing one for use at KSU.

41. Ellipsometry Allows us to probe the surface structure of materials.
Makes use of Maxwell’s equations to interpret data.
Drude Approximation
Is often relatively insensitive to calibration uncertainties.

42. Ellipsometry Accuracies to the Angstrom
Can be used in-situ (as a film grows)
Typically used in thin film applications
For more information and also this presentation see our website:
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43. Bibliography
Bhushan, B., Wyant, J. C., Koliopoulos, C. L. (1985). “Measurement of surace topography of magnetic tapes by Mirau interferometry.” Applied Optics 24(10):
Drude, P. (1902). The Theory of Optics. New York, Dover Publications, Inc., p
Riedling, K. (1988). Ellipsometry for Industrial Applications. New York, Springer-Verlag Wein, p.1-21.
Smith, D. S. (1996). An Ellipsometric Study of Critical Adsorption in Binary Liquid Mixtures. Department of Physics. Manhattan, Kansas State University: 276, p
Tompkins, H. G. (1993). A User's Guide to Ellipsometry. New York, Academic Press, Inc.
Tompkins, H. G., McGahan, W. A. (1999). Spectroscopic Ellipsometry and Reflectometry: A User's Guide. New Your, John Wiles & Sons, Inc.
Wang, J. Y., Betalu, S., Law, B. M. (2001). “Line tension approaching a first-order wetting transition: Experimental results from contact angle measurements.” Physical Review E 63(3).