What is the usage of the squareness in the thin film magnetic property studies?
I am not sure what you are refering to...But I'll guess
Are you asking why certain properties are stated as "per square"? If so it just because it is the way the dimensions work out. "sq" is not a "proper" unit, it is just there as a reminder .
The easiest example would be sheet resistance which has the unit of Ohms/square (if you work out the dimensions you'll see that is "should" actually only be ohms).
Now, if you want to calculate the resistance (in ohms) between two points you have to multiply by the length and divide by the width; this ratio is dimensionless but it still needs to be there to get the right answer.
Hence, the answer to your question is that it is just there to remind you that it is a material parameter, and that it needs to be multipled by a geometric factor to give you something you can actually measure.
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