There are five problems, for a total of 100 points, covering aspects of lithography and oxidation. You are required to submit your answers with detailed derivations, either handwritten or typed, and, where necessary, attach plots and code to your solutions.

While you are permitted to discuss your approach with classmates, you are not allowed to work together to obtain solutions, or to write code collaboratively. You are also welcome to contact me at office hours to discuss how to approach these problems.

If you are unsure about expectation regarding the honor code on this assignment, please speak to me directly. Do not underestimate the amount of time it takes to complete this homework. It will require constant effort for the next 3 weeks to finish this
assignment worth 8% of your homework grade.

Problem 1:

The resolution of lithography is limited by the diffraction optics of the lithography machine. It is also limited in the direction perpendicular to the wafer plane because of defocusing of the mask image projected away from the focal plane of the lens.

Before exposure, the wafer and resist coating must be aligned with the mask accurately, and often times this process is conducted using alignment marks located on both surfaces.

However, there will still be residual misalignments between the lens, mask, and wafer planes. In addition, a magnification ratio of 10:1 or 5:1 is often used to reduce the size of the mask onto the wafer. In such cases, a "stepper" motion must be undertaken in order to expose the entire wafer. Answer the following:

1) Explain why image reduction is useful, and discuss in general terms how many and where you would place alignment marks on the wafer (e.g. in the center, around the periphery, etc.).

2) Calculate the relationship between the optical resolution and the depth of field of the imaging system. What kind of scaling does this relationship follow and what does it tell us about trying to shrink the size of the resulting microstructures?

3) Assuming constants K1 and K2 from the equation in b) are K1=0.25, K2=0.6, and the exposure wavelength is λ=193nm, calculate the maximum resist thickness that can be exposed using a perfect mask-wafer alignment in order to achieve a structure resolution of 1µm and 500nm, respectively.

4) Assuming that a 6" wafer is misaligned to the focusing lens by small tilt angles α and β but that the center of the wafer is in focus, calculate how the maximum resist thickness in the previous question would change with values of α and β. Conversely, what are the maximum tilt angles that can be tolerated by a given resist thickness T? Plug in different values for resist thickness and discuss the results.

5) Does the stepper require high resolution, accuracy or repeatability of motion in order to ensure the entire wafer is exposed as accurately as possible?

Problem 2:

A <1 0 0> Silicon wafer has 400 nm oxide on its surface. This wafer will be subjected to additional time and temperature oxidation process using water vapor. To solve this problem, you need to consult are oxidation charts, as well as tables with values for oxidation coefficients from your textbook.

a) Write down the equations governing the oxide thickness growth as a function of time for this continued oxidation process. Explain the meaning of parameters in your equation as a function of time and temperature.

b) How long will it take to grow am additional 1 micron of oxide in Wet Oxygen at 1050 C? Compare graphical and mathematical results.

c) What is the color of the final oxide under vertical illumination by white light?

d) Explain what differences you would expect in the oxidation process if the wafer was heavily doped with Boron prior to oxidation .

Attachment:- Table 3.2.rar