More Semiconductor Wafermap Mathematics

By Michael Hackerott; Published 3/23/2004 9:53 AM

A wafermap can be thought of as either a linear orthogonal “grid” or a “point” coordinate system. Grid wafermaps are most often used for reticle and die maps. Point wafermaps are most often used for defect and bit maps. Therefore, it is important to understand both coordinate systems and how to relate them in order to spatially correlate different types of semiconductor manufacturing data.

The following diagram specifies a simple Wafermap using a linear orthogonal “grid” coordinate system. The positive x axis is to the right, the positive y axis is down. The Wafermap is a grid consisting of a “cell” for each intersection of the x and y axis coordinates. The cells are identified by their respective x and y axis coordinates using the notation “X,Y”.

The same simple Wafermap could be specified using a linear orthogonal “point” coordinate system. The grid cells are now identified by the respective left, top, right, and

Bottom of the cell verticies using the notation “L,T,R,B”.

Many other notations can be used. The key idea is that in the “grid” case we have a very simple coordinate system that identifies a cell but does not provide any additional spatial information about the cell; whereas in the more complex “point” case the identifiers themselves are spatial information about the cell. In the point coordinate system, the cells width and height can

Be calculated directly from the cell coordinates: width = right – left, height = bottom – top.

MINIMAL GRID WAFERMAPS

In order to produce a valid wafermap from a set of coordinates and data requires some minimal “metadata” about the coordinate system in addition to the coordinate data. For example, the following hard bin data set:

X Y HBIN

1 1 1

2 1 1

3 1 c

1 2 1

2 3 a

3 2 2

1 3 1

2 3 3

3 3 f

Does not contain the additional metadata to produce the wafermap representation of the data.

The minimum set of metadata required to produce the wafermap is the x and y axis positive directions, and the minimum and maximum cell coordinates ASSUMING that the X is always the horizontal axis and the Y is always the vertical axis (otherwise, the orientation of the x and y axis also must be specified.).

AxisPositiveX=LEFT|RIGHT

The positive direction of the x axis.

AxisPositiveY=UP|DOWN

The positive direction of the y axis.

CellMinimumX=COORDINATE

The minimum x axis coordinate value.

CellMaximumX=COORDINATE

The maximum x axis coordinate value.

CellMinimumY=COORDINATE

The minimum y axis coordinate value.

CellMaximumY=COORDINATE

The maximum y axis coordinate value.

However, although this minimum set of metadata allows a valid grid wafermap to be produced, it is not sufficient to represent the alignment of the wafer to the wafermap. In order to align the wafer to the wafermap additional metadata is required. This additional set of metadata is:

WaferToGridOrientation=UP|RIGHT|DOWN|LEFT

The orientation of the wafer flat or notch to the wafermap grid.

WaferToGridCenterCellX=COORDINATE

The x axis grid coordinate that contains the point at the physical center of the wafer.

WaferToGridCenterCellY=COORDINATE

The y axis grid coordinate that contains the point at the physical center of the wafer.

WaferToGridCenterCellAnchor=TOP_LEFT|TOP_RIGHT|BOTTOM_RIGHT|BOTTOM_LEFT|CENTER

The vertex or center point of the cell that is the point closest to the physical center of the wafer.

MINIMAL POINT WAFERMAPS

Similarly given the following hard bin data set:

L T R B HBIN

1 1 2 2 1

2 1 3 2 1

3 1 4 2 c

1 2 2 3 1

2 3 3 3 a

3 2 4 3 2

1 3 2 4 1

2 3 3 4 3

3 3 4 4 f

The following additional metadata is required:

AxisPositiveX=LEFT|RIGHT

The positive direction of the x axis.

AxisPositiveY=UP|DOWN

The positive direction of the y axis.

CellMinimumX=COORDINATE

The minimum x axis coordinate value.

CellMaximumX=COORDINATE

The maximum x axis coordinate value.

CellMinimumY=COORDINATE

The minimum y axis coordinate value.

CellMaximumY=COORDINATE

The maximum y axis coordinate value.

WaferOrientation=UP|RIGHT|DOWN|LEFT

The orientation of the wafer flat or notch to the wafermap grid.

The previous metadata is the same for both grid and point data. However, the metadata to relate the wafer to the data is different.

WaferToPointCenterCellL=COORDINATE

The x axis left coordinate of the cell that contains the point at the physical center of the wafer.

WaferToPointCenterCellT=COORDINATE

The y axis top coordinate of the cell that contains the point at the physical center of the wafer.

WaferToPointCenterCellR=COORDINATE

The x axis right coordinate of the cell that contains the point at the physical center of the wafer.

WaferToPointCenterCellB=COORDINATE

The y axis bottom coordinate of the cell that contains the point at the physical center of the wafer.

WaferToPointCenterCellAnchor=TOP_LEFT|TOP_RIGHT|BOTTOM_RIGHT|BOTTOM_LEFT|CENTER

The vertex or center point of the cell that is the point closest to the physical center of the wafer.

ALIGNMENT OF GRID TO POINT WAFERMAPS

These minimum sets of metadata are sufficient to produce accurate graphical representations of a grid or point wafermap with the wafer aligned to the grid. However, it is still insufficient to align the grid or point data to the true physical center of the wafer. The ability to exactly overlay the grid and point coordinate systems to the wafer is critical to the correlation of data from these coordinate systems.

In order to align or “register” the grid or point coordinate systems to the wafer requires additional metadata to exactly align the center cell with the physical center of the wafer. This requires knowing the physical dimensions of a cell and a vector from the cell anchor to the physical center of the wafer:

CellSizeX=DISTANCE

The size of the cell along the x axis.

CellSizeY=DISTANCE

The size of the cell along the x axis.

CellUnits=UNIT

The units of the cell size and distance.

CellAnchorToWaferCenterX=DISTANCE

The distance from the cell x anchor point to the physical center of the wafer.

CellAnchorToWaferCenterY= DISTANCE

The distance from the cell y anchor point to the physical center of the wafer.

GRID WAFERMAP EXAMPLE 1

A wafermap with an odd number of rows and columns is simple since the physical wafer center is coincident with the center die.

The metadata is:

AxisPositiveX=RIGHT

AxisPositiveY=DOWN

CellMinimumX=1

CellMaximumX=5

CellMinimumY=1

CellMaximumY=5

WaferToGridOrientation=RIGHT

WaferToGridCenterCellX=3

WaferToGridCenterCellY=3

WaferToGridCenterCellAnchor=CENTER

GRID WAFERMAP EXAMPLE 2

A wafermap with an even number of rows and columns demonstrates the need for the WaferToGridCenterCellAnchor metdata.