Taskin Kocak

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Mathematical and Physical Modeling of Image
Formation in Scanning Electron Microscopes

• Taskin Kocak
• School of Electrical Engineering and
• Computer Science
• Applications of Calculus
• Derivatives of Trigonometric Functions

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Example problem

• Calculus Topic Derivatives of Trigonometric
  Functions
• Section 3.5 15 Differentiate

   
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Example problem (cont.)
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Learning Objectives

• Determine the characteristics of SEM output
  images for semiconductor fabrication
• Examine the physical problem of locating defects
  in semiconductor metrology
• Model image intensity as a function of the
  feature profile

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SEM

• SEM Scanning Electron Microscope

A microscope is a tool that lets the user see
objects at a magnification greater than the
actual specimen. The most common type of
microscope is a magnifying glass, which uses a
ground lens to focus the light reflecting off an
object into a larger image. More complex light
microscopes use a series of lenses to further
magnify the object.
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Example SEM Image
Source http//remf.dartmouth.edu/images/insectPar
t1SEM/source/20.html
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How SEM works?

• http//www.mos.org/sln/SEM/works/slideshow/semmov.
  html

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Principles of SEM
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How chips are manufactured?

• http//www.necel.com/v_factory/en/index.html

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Semiconductor Fabrication
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SEM Images
Top-down
Cross-section
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Assessment of Learning Objective 1

• 1.  To grasp the size of the test samples in this
  work, can you give other examples of submicron
  (i.e., less than 10-6 m) size?
• 2.  Which type of image (top-down or
  cross-section) would give more information about
  the topography of the feature?

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Problem Statement

• Abnormal shapes due to over or less exposure
• Footing or T-topping
• Cross-section images can be used but the process
  is destructive!

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Image Pre-processing (top-down)
An image may be defined as a two-dimensional
function, f(x,y), where x and y are spatial
(plane) coordinates, and the amplitude of f at
any pair of coordinates (x,y) is called the
intensity or gray level of the image at that
point.
Take average of the intensity for each horizontal
location of the region of interest
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Image Pre-processing (cross-section)
Edge detection
Height vector (Profile)
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Data set 1 - normal
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Data set 2 - abnormal
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Assessment of Learning Objective 2

• 1. In groups of two students, discuss the
  properties of the intensity waveform and the
  profile given in Figs. 4 and 5. What are the
  differences?
• 2. (5-minute paper) explain which orientation
  (top-down or cross-section) is the preferred
  method of metrology, and why.

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A Simple Mathematical Model
We assume that SEM intensity waveform, I(x), is
affected by the interaction thickness The
thickness is approximately proportional to the
absolute value of tan(ß), where ß is the slope
angle. tan(ß) equals to the derivative of the
profile, P(x), which is P(x).
Assume the following equation
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Mathematical model including diffusion and
shadowing

• We notice that SEM intensity curves are convex
  upward at the bottom of the feature sidewall.
• The secondary electrons were emitted in all
  directions.
• Also, the diffusion at the top corner of the
  feature is very strong which means more secondary
  electrons are detected around the top corner

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Assessment of Learning Objective 3

• 1.  Which equation describes the relationship
  between the image intensity and the profile more
  accurately?
• 2.  (5-minute paper) Assume that c20 and tan(ß)
  equals to the derivative of the profile. Find the
  image intensity in terms of the slope angle, ß
  using Eq. 2. (assume dß/dxm)

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Data Assignment
Calculate the image intensity as a function of
incident angle
The total emission ? is given as the sum of
secondary electrons yield ? and back-scattered
electrons coefficient yield ?. Dependence of ? on
the incident angle, ?, is given by  
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Data Assignment
 ? is expressed with an empirical formula below
where Z is the atomic number.  
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Data Assignment
 
where B and C are constant coefficients. Since
image intensity is proportional to the total
emission yield, ?. We can express  
Further, we can substitute ? and ?
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Assignment

• Assume B 1, ?0 0.05, C 1.
• Find the image intensity in terms of the incident
  angle and the atomic number.
• Differentiate I with respect to the incident
  angle (dI/d?). For two different elements with
  Z19 and Z227, contrast the changes in image
  intensity
• Assign ycos? . First, write I as a function of
  y. Then differentiate I with respect to ? using
  the chain rule. Verify your result with your
  answer to the second question. (Hint
• )

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Rubric for Assessment