# Effective Mass Approximation

I am looking at this energy dispersion relationship that has the form

$$\varepsilon = -\alpha k_y + \sqrt{\beta k_y^2 + \gamma k_x^2 }$$

Here is my conundrum - I am able to find the minima of the energy dispersion (happens to be at (0,0)) and using the definition

$$[M_{ij}]^{-1} = \left(\frac{1}{\hbar^2}{\frac{\partial^2 \varepsilon}{\partial k_{i}\partial k_{j}}}\right)\bigg|_{k_x=0, k_y =0}$$

I encounter singularities in finding the effective mass.

So I took a different approach and tried to find the Taylor expansion of the second term, but I believe this is also a futile effort. Any advice on how to proceed?

The effective mass can be nulled out in specific case (like in graphene, some k values will lead to an M$$_{ij}$$=0). It can also reach infinite values in specific case (see this free article: https://aip.scitation.org/doi/abs/10.1063/1.3488833#:~:text=An%20electron%20rotating%20under%20a,to%20an%20azimuthal%20electric%20field.) It can also be negative (one can see this looking at the dispersion/energy curves if I recall correctly).
In other words, I think your answer that M$$_{i}^{-1}$$=0 (since you said you obtain infinite M$$_{ij}$$, I assumed that you get this result as well) is plausible if the minima is at k$$_{x}$$=k$$_{y}$$=0.
However, after a quick plot of your energy function (I might have done the plot a bit fast), I'm not sure the minimum is at k$$_{x}$$ and k$$_{y}$$=0, but at something closer to k$$_{x}$$=0 and k$$_{y}$$<0 (depending on the size of $$\alpha$$,$$\beta$$ and $$\gamma$$ the minima could move).