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

Ellipsometry A method of probing surfaces with light.

Introduction History Methodology Theory Types of Ellipsometry Applications Summary

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.

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.

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.

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

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.

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

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

; Y Integrate and vary little over L where

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

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

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

This results in 4 relations between , , and .

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

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

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

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

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

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

Example Setup Phase modulated ellipsometer

How to get r,an example. Phase Modulated Ellipsometry

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

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

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,

How to get r,an example. Phase Modulated Ellipsometry

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

At the Brewster Angle,

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

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

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

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

Other Methods Reflectometry Microscopic Interferometry Mirau Interferometry

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.

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

Reflectometry

Reflectometry Can be more accurate for thin metal films.

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

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

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.

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: html://www.phys.ksu.edu/~allbaugh/ellipsometry

Bibliography Bhushan, B., Wyant, J. C., Koliopoulos, C. L. (1985). “Measurement of surace topography of magnetic tapes by Mirau interferometry.” Applied Optics 24(10): 1489-1497. Drude, P. (1902). The Theory of Optics. New York, Dover Publications, Inc., p. 287-292. 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. 18-27. 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).