# Silicon crystal orientation

I'm confused about how  direction is determined for (100), (110) or (111) wafers. I found a book chapter which just confused me even more. From the image below, I understand how  is determined on the (110) wafer but not the other two. I'm also having a hard time understanding what different planes would look like on the (111) and (110) wafer. I would appreciate a resource for silicon wafers specifically (not necessarily crystallography). Here are things I'm not understanding.

How is the  direction determined and why is it different for each of the three wafers below? For the (110) wafer  is between x and y axis which makes sense. But for (100) it seems to go from z to y and for (111) it's between x and z. From my understanding  intercepts x and y axis at 1 and doesn't intercept z axis but that doesn't seem to be the case for (100) and (111) wafers.

Does the definition of x,y and z axis change when we're talking about (100), (111) or (110) wafers? I know that whichever wafer it is, that direction will be pointing out from the wafer surface but when I draw planes with respect to  and  (for (111) wafer), I don't know where the (100) plane should be. (110) plane would be perpendicular to the  direction and would be normal to the surface and (111) would be 35° from it. I can't find a drawing for it which makes me think we don't need to show the (100) plane? But if not then what if I need to align my mask to (100) plane on (111) wafer?

How would I define higher miller index planes on the 3 wafers? For example I need to align my mask to (122) or (411) plane, how would I start with it? I understand that this might become clear once I learn about the primary planes on the wafers. I would appreciate a resource for silicon wafers specifically (not necessarily crystallography).

This video is fun to watch (the difference between a  and a  wafer is striking) and it points at further resources.

This interactive Jmol site lets you select a plane while also showing the unit cell orientation. For the image below (which is an interactive 3D model on the live site), first select to show 5x5x5 unit cells. Then, enter (1,1,1) for the plane, increase the thickness with the slider and click "draw atoms in the plane". Once you have a model of your wafer, change to another plane and click "show HKL plane". If you look closely, you can see the unit cell in the center in thin black lines (you have to turn it on manually, by right clicking and selecting Style->Unicell or Axes), allowing you to verify the orientation of the two planes. How is the  direction determined and why is it different for each of the three wafers below?

Silicon has cubic symmetry, so the three directions , , and  are equivalent. What is confusing is that the images in the question show three different planes intersecting with the wafers: (0,1,1), (1,1,0), (1,0,1) from left to right, but they are labeling all of them <1,1,0> because they are indistinguishable once you take away the axis notation.

Does the definition of x,y and z axis change when we're talking about (100), (111) or (110) wafers?

No, in all three pictures, the x axis goes along , the y-axis along , and the z-axis along .

For example I need to align my mask to (122) or (411) plane, how would I start with it?

Let $$\vec{a}$$ be a vector perpendicular to the surface of the waver, and $$\vec{b}$$ be a vector perpendicular to the surface used in the cutoff, and $$\vec{c}$$ be the vector perpendicular to the desired alignment plane. The cross product of $$\vec{a}$$ and $$\vec{b}$$ is parallel to the cutoff edge. The cross product of $$\vec{a}$$ and $$\vec{c}$$ is parallel to intersection of the wafer surface and the desired alignment plane. To get the angle between the two, you use the usual dot product method.

For a (111) wafer (rightmost diagram), for example, the cutoff line is parallel to (1,1,1) x (1,0,1) = (1,0,-1). The intersection between the wafer surface and the (122) plane is a line parallel to (1,1,1) x (1,2,2) = (0,-1,1). The angle between these two is 120 degrees. Because of the symmetry of the wafer, other solutions are 0 or -120 degrees. (You would get those by dialing in the (212) or the (221) plane.)