# Calculating the Lattice Mismatch of GaN/Sapphire

1. May 27, 2014

### maxxlr8

Dear Physics Forum

It has been reported that the lattice mismatch of GaN/Sapphire is ~13.9%.

I have tried the following formula, but got the wrong answer:

[(GaN-Al2O3)/GaN] x 100%

where:
GaN = 3.189 Angstroms
Al2O3 = 4.765 Angstroms

Obviously I am missing something huge.

Thank you.

2. May 28, 2014

### Cthugha

In crystal growth things depend strongly on things like crystal orientation, which surface you choose, crystal growth direction, in-plane orientation and such stuff. So you need to know which planes you are talking about. 13.8% is what you get if you use the (10-10) and (11-20) planes of GaN and sapphire.

See: http://www.icdd.com/resources/axa/vol43/v43_047.pdf

3. May 28, 2014

### f95toli

Just to add to what was said above: Sappbhire is extremely anisotropic meaning there are many ways you can cut it to make a substrate.
The most common type of substrate used is R-cut; which is actually cut at an "angle" with respect to the "z-axis"(note that you actually need four indices to define a plane in sapphire). You can also buy for example c-cut and r-cut (note that R and r are different cuts)

Note that this "flexibility" is one reason why sapphire is so popular, you can make it "fit" to a large number of other materials by just using the right cut.

4. May 28, 2014

### maxxlr8

Thank you for pointing out that they have calculated this by using the GaN(10-10) and Al2O3(11-20)

I have calculated this using the same formula and have arrived at a value of 14%, which should be tolerable since my database and C. J. Rawn's would exhibit different degree of strain.

The interplanar spacing for GaN(10-10) and Al2O3(11-20) is 2.762 angstroms and 2.375 angstroms, respectively.

So, one can't just choose to use the length of a unit cell to calculate the lattice mismatch. The plane relationship must be determined first.

Thank you again.

Anyway, below is an image of how (I think) the hexagonal cells are arranged.

ps: most papers also report that the GaN has no problem aligning its c-axis along with the substrate's, but the a-axis (and therefore, b-axis) will usually twist 60 degree with respect to the substrate.

Last edited: May 28, 2014
5. May 28, 2014

### maxxlr8

Thank you for the information.

I did not realise that r-cut and R-cut are prepared with a different plane face.

Some reports said that they used sapphire (1-102) for their substrates. Now I am not sure if this is an r- or R-sapphire.