By: Math Nerd (math.nerd.delete@this.nerds.com), June 27, 2022 5:05 am

Room: Moderated Discussions

> It is a tad more expensive but not a significant cost in relation to a finished wafer on a modern

> process. The biggest reason you don't want to do it is the loss of edge area for I/O.

The perimeter of a rectangle with a 2:1 aspect ratio is only 14% more than the perimeter of a regular hexagon with the same area. The perimeter of a square is only 7% more than the perimeter of a regular hexagon with the same area. If anyone would make a hexagonal die, it would be for the largest die, because they have the most wasted area on a circular wafer. The largest die today all use flip chip area bonding, not edge bonding. There seems to be a trend to use chiplets today instead of large dies and that reduces the wasted area near the circumference of the wafer.

In any case, the ability to make non-rectangular die is very interesting and surprising, which I’m sure was the main reason you mentioned it. One semi-practical use for this technology could be making microscopic chips where it would be better to not have sharp rectangular corners, such as RFID tags for pets or sensors implanted under human skin.

For the record, here is how I calculated that the perimeter of a rectangle with a 2:1 aspect ratio is 14% more than the perimeter of a regular hexagon with the same area:

Perimeter of 2:1 aspect ratio rectangle with smaller side a is 6a (rectangle is a x 2a).

Perimeter of regular hexagon with side b is 6b.

Ratio is perimeters is a/b.

Area of a 2:1 aspect ratio rectangle with smaller side a is 2 a^2.

Area of a regular hexagon with side b is b^2 (3/2) sqrt[3].

Set areas equal: 2 a^2 = b^2 (3/2) sqrt[3]

a/b = sqrt[ (3/4) sqrt[3] ] = 1.14

> process. The biggest reason you don't want to do it is the loss of edge area for I/O.

The perimeter of a rectangle with a 2:1 aspect ratio is only 14% more than the perimeter of a regular hexagon with the same area. The perimeter of a square is only 7% more than the perimeter of a regular hexagon with the same area. If anyone would make a hexagonal die, it would be for the largest die, because they have the most wasted area on a circular wafer. The largest die today all use flip chip area bonding, not edge bonding. There seems to be a trend to use chiplets today instead of large dies and that reduces the wasted area near the circumference of the wafer.

In any case, the ability to make non-rectangular die is very interesting and surprising, which I’m sure was the main reason you mentioned it. One semi-practical use for this technology could be making microscopic chips where it would be better to not have sharp rectangular corners, such as RFID tags for pets or sensors implanted under human skin.

For the record, here is how I calculated that the perimeter of a rectangle with a 2:1 aspect ratio is 14% more than the perimeter of a regular hexagon with the same area:

Perimeter of 2:1 aspect ratio rectangle with smaller side a is 6a (rectangle is a x 2a).

Perimeter of regular hexagon with side b is 6b.

Ratio is perimeters is a/b.

Area of a 2:1 aspect ratio rectangle with smaller side a is 2 a^2.

Area of a regular hexagon with side b is b^2 (3/2) sqrt[3].

Set areas equal: 2 a^2 = b^2 (3/2) sqrt[3]

a/b = sqrt[ (3/4) sqrt[3] ] = 1.14