Silicon crystal orientation

I'm confused about how [110] 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 [110] 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 [110] direction determined and why is it different for each of the three wafers below? For the (110) wafer [110] 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 [110] 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 [110] and [111] (for (111) wafer), I don't know where the (100) plane should be. (110) plane would be perpendicular to the [110] 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.


[![enter image description here](https://i.stack.imgur.com/lhWUp.png)](https://i.stack.imgur.com/lhWUp.png)





> 
> I would appreciate a resource for silicon wafers specifically (not necessarily crystallography).
> 
> 
> 


[This video](https://youtu.be/vXk6Uhq74nU) is fun to watch (the difference between a [111] and a [100] wafer is striking) and it points at further resources.


This [interactive Jmol site](http://lampx.tugraz.at/%7Ehadley/memm/materials/silicon/silicon.php) 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.


[![enter image description here](https://i.stack.imgur.com/KYITo.png)](https://i.stack.imgur.com/KYITo.png)



> 
> How is the [110] direction determined and why is it different for each of the three wafers below?
> 
> 
> 


Silicon has cubic symmetry, so the three directions [100], [010], and [001] 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.


[![enter image description here](https://i.stack.imgur.com/m1ZeP.png)](https://i.stack.imgur.com/m1ZeP.png)
Source: <https://journals.aps.org/authors/crystallographic-notation-h1>



> 
> 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 [100], the y-axis along [010], and the z-axis along [001].



> 
> 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.)


### Why does it matter?


From uhoh's comments, slightly edited:



> 
> In one of my previous lives I had to align contact photomasks to silicon wafers in order to open holes in photoresist, used to etch through a thin silicon nitride layer (10 to 100 nm) which was then used as a mask for directional wet etching of the silicon. KOH can have a 100:1 or larger wet etch anisotropy for certain crystal directions, and you can get pyramidal or even very vertical (wrt wafer surface) sidewalls from anisotropic wet etches if you rotate your mask to get specific orientations of the pattern edges with respect to crystal directions. [...] [Anisotropic wet etching](https://en.wikipedia.org/wiki/Etching_(microfabrication)#Anisotropic_wet_etching_(Orientation_dependent_etching)) (Orientation dependent etching)
> 
> 
> 


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