1. Ellipsometer B93901007 許恭銓

2. Problem Overview We have a thin layer of oxide on a Si substrate and we want to find its thickness by using an ellipsometer. How?

3. Overview on Procedure 1/4

4. Overview on Procedure 2/4 Light beam given by a light source passes through a combination of tunable polarizer and compensator. The light then enters the thin layer, the exiting beam on the other side should be a combination of reflected beam by the air- oxide surface and the oxide-substrate surface

5. Overview on Procedure 3/4 The light beam then passes an analyzer made to block any linearly-polarized beam. The Detector detects the beam We tune the polarizer to make the detected beam having minimum light intensity (i.e. The exiting light linearly-polarized and blocked by the analyzer)

6. Overview on Procedure 4/4 The Ellipsometer uses a fitting technique to find the complex refractive index and the thickness of the oxide

7. Discussion 1/10

8. Discussion 2/10 The total reflection coefficient

9. Discussion 3/10 The measurement of the Ellipsometer First of all, let’s see how the Ellipsometer detect this: We can get a relationship in matrix form

10. Discussion 4/10 Tunable Polarizer matrix be at state S Tunable Compensator matrix be at state S Tunable Analyzer Matrix be which acts to block a certain direction of linear-polarized light

11. Discussion 5/10 Then if we tune the Polarizer and compensator to State such that the detected beam is with minimal energy (i.e. 0, when the reflected electric fields are linearly polarized) So we are able to solve this and get and, and so is

12. Discussion 6/10 We know that So, basically, we want to solve the equation for a set of parameters being the solution but apparently, not unique.

13. Discussion 7/10 So we must change certain parameters to get more equations so that we’ll come up with a unique solution Since the materials are not-changeable, the only choices are the incident angle, and the type of light beam (changing ) Changing the incident angle seems to be better

14. Discussion 8/10 And since each equation runs down to two equations actually (real and imaginary part), so we need 4 different measurements?? Let’s say, changing the incident angle 4 times, which come up with 8 equations and 7 unknowns, which seems an unique solution solvable

15. Discussion 9/10 Although we have the equations needed in hand, but with so many unknowns and the equations are terribly non- linear. So it’s hard to solve directly. What Ellipsometer do next is a “Fitting technique”: Given an initial set of parameters, the algorithm takes this set into the equation. Then by observing the errors, it tries another “more possible” set of parameters By doing this “fitting” over and over again, the Ellipsometer would stop once it thinks the error is small enough. The solution set is estimated!

16. Discussion 10/10 Note that Ellipsometer itself do the fitting on and, where and Such that