Semiconductor Wafermap Mathematics
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BACKGROUND Semiconductor wafer manufacturing produces an orthogonal pattern of
circuits on a semiconductor wafer. The
circuits are typically imaged onto the wafer using “Step and Repeat”
photo-lithography processing. “Step
and Repeat” photo-lithography involves exposing a group of one or more
circuits at a position on the wafer, moving to a new position on the wafer,
and repeating this process until the wafer is covered with an orthogonal
pattern. A single image of circuits is
termed a “Reticle”. There can be one
or more integrated circuits within a reticle.
Each integrated circuit is termed a “Die”. The orthogonal patterns of reticles and
dies can be described using Cartesian coordinate systems. Using a Cartesian coordinate system to
describe the reticles or die on a semiconductor wafer is termed a “wafermap”. A semiconductor wafer map can
be described with overlapping Cartesian coordinate systems for the reticles
and dies. In this example, the dark
squares containing nine smaller squares represent the reticles and the
smaller squares represent the die. The
circle represents the edge of the wafer.
Also, wafers will have a “notch” or “flat” marking. |
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The concept of wafermaps is key to analyzing data collected on semiconductor wafers. Unfortunately, the data is collected at many different points during the manufacturing process and the Cartesian coordinate system used at each data collection point can be different. The data collected in different Cartesian coordinate systems must be converted into a common Cartesian coordinate system for comparison. A good understanding of the Cartesian coordinate system and the mathematics related to it is essential to correctly performing data analysis of semiconductor wafers using wafer maps. AN OVERVIEW OF THE CARTESIAN COORDINATE SYSTEM The Cartesian coordinate system describes the set of points in a two dimensional plane. The most commonly used Cartesian coordinate system is diagrammed with the horizontal or “X” axis increasing to the right and the vertical or “Y” axis increasing to the top of the graph. The center, or “origin”, of the graph is the point X = 0, Y = 0. A point on the graph is described by an X (horizontal) and an Y (vertical) coordinate. |
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The graph is often subdivided into four quadrants labeled I, II, III, and IV. The “Top” of the graph refers to the region of the graph at the largest positive Y-axis value and the “Bottom” of the graph refers to the region of the graph at the smallest negative Y-axis value. Similarly, the “Right” of the graph refers to the region of the graph at the largest positive X axis value and the “Left” of the graph refers to the region of the graph at the smallest negative Y axis value. There are four possible Cartesian coordinate systems. The difference is the positive directions of the horizontal (X) or vertical (Y) axis. |
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Points in one coordinate system can be “mapped” to another coordinate system using simple equations. |
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For example, a point B(xB, yB) in the B coordinate system can be “mapped” to system point A(xA, yA) in the A coordinate system using the equations: xA = -xB, yA = yB. |
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However, the common equations used for translation and rotation in a Cartesian coordinate system are only applicable to the standard Cartesian coordinate system "A". Therefore, it is usually necessary to convert to the standard Cartesian coordinate system and perform all of the mathematics in that system. WAFERMAP MATHEMATICS The methodology of converting one or more semiconductor wafermaps into a common Cartesian coordinate system typically consists of four steps: 1. Mapping the original coordinate system to the standard coordinate system 2. Translating the wafermap so that the center of the wafermap is the origin point of the standard coordinate system 3. Rotating the wafermap around the origin point so that all wafer notches and flats are oriented the same 4. Translating the wafermap to it's final coordinates Two detailed examples will be presented that illustrate both the methodology and the mathematics of working with wafermaps. EXAMPLES OF WAFERMAP MATHEMATICS In the first example, a wafer has been probed with the notch up (0 degrees), the horizontal axis increases positively to the right, and the vertical axis increases positively downward. The final result should have the notch down (180 degrees), the horizontal axis increasing positively to the right, and the vertical axis increasing positively upward. The following diagram depicts the initial wafermap coordinate system. |
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The first step is to map the initial coordinate system to the standard coordinate system using the equations: x' = x and y' = -y. (These equations can be deduced by noting that the X-axis is already the standard X-axis and the Y-axis is the inverse of the standard Y-axis.) Applying these equations to each point in the wafermap results in the following wafermap. |
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The next step is to calculate the center of the wafermap. The center is calculated using the equations: x.center = INT((x.left + x.right)/2) and : y.center = INT((y.top + y.bottom)/2). The rule for the integer function, INT(), is to round up for values >= 0.5. |
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The next step is to translate the wafermap so that the center of the wafermap is equal to the origin of the coordinate system. The translation equations are: x' = x - x.center and y' = y - y.center. |
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The next step is to symmetrically rotate the translated wafermap about the origin. The rotation equations are: x' = x cos q - y sin q and y' = x sin q + y cos q. The angle of rotation is defined in a counter-clockwise direction. |
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The final step is to translate the wafermap to the final coordinates. In this case, we want the minimum final X and Y axis values to be equal to 1. The translation equations for this case are: x' = x + (x'.min - x.min) and y' = y + (y'.min - y.min). |
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In the second example, a wafer has been probed with the notch left (270 degrees), the horizontal axis increasing positively to the right, and the vertical axis increasing positively upward. We want the final result to have the notch up (0 degrees), the horizontal axis increasing positively to the right, and the vertical axis increasing positively upward. The following diagram depicts the initial wafermap coordinate system. |
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Since the wafer is already in the standard and final coordinate system and is centered on the origin, the first step is to rotate the wafermap -90 degrees counter-clockwise. The following diagram depicts the rotation equations and the rotated wafermap. |
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The final step is to translate the wafermap to its final coordinates. In this case we want the left most X-axis value equal 1 and the topmost Y axis value equal to -1. The translation equations are: x' = x + (x'.left - x.left) and y' = y + (y'.top - y.top). |
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After translation, the wafermap is positioned at the desired final coordinates. |
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A SUMMARY OF WAFERMAP MATHEMATICS Unless the wafermap is already in the standard coordinate system, the first step to be performed is to map the original wafermap coordinate system into the standard coordinate system. The equations to map to the standard coordinate system can be deduced from the original X and Y axis values using the following two rules: · If (x.right < x.left) then x' = -x Else x' = x · If (y.top < y.bottom) then y' = -y Else y' = y Unless the wafermap is already centered in the standard coordinate system, the second step to be performed is to center the wafermap in the standard coordinate system. First, the center of the wafermap is calculated using the equations: · x.center = INT((x.left + x.right)/2 · y.center = INT((y.top + y.bottom)/2) Where the rule for the integer function, INT(), is to round up for values >= 0.5. Next, the wafermap is translated so that the center of the wafermap is equal to the origin of the standard coordinate system using the translation equations: · x' = x - x.center · y' = y - y.center Unless the wafermap is already centered and rotated in the standard coordinate system, the third step to be performed is to rotate the wafermap using the equations: · x' = x cos q - y sin q · y' = x sin q + y cos q Where the angle of rotation, q, is defined in a counter-clockwise direction. Unless the wafermap is already centered, rotated, and within the bounds of the final coordinate range and system, the third step to be performed is to translate the wafermap to its final coordinate range and system. The equations are of the form: · x' = x + (x'.edge - x.edge) · y' = y + (y'.edge - y.edge) Where "edge" is the appropriate edge of the final coordinate range to align to. If the wafermap is in the standard coordinate system then the X and Y axis edges can be deduced from the final X and Y axis "minimum" values using the following two rules: · If (x'(y'=0) < x(y=0)) then edge = right Else edge = left · If (y'(x'=0) < y(x=0)) then edge = bottom Else edge = top |
