How to derive the effective mass approximation from band structure

[aaaa202]

I have noticed that in a lot of theoretical modelling of semiconductors you assume that the electrons living in the bottom of the conduction band obey a free particle Hamiltonian:

H = p^2/2m*

, where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I am not sure how this is derived rigourously. I suppose you solve the band structure and show that as a function of k the band structure is parabolic in k about the minimum of the conduction band:

E ≈ E0 + ħ^2k^2/2m*

But how do you rigorously go from this expression, which contains the wave numbers k = (kx,ky,kz) back to differential operators? I hope you understand my question.

 

[Louis Fry]

I am not sure how it is rigorously defined, no doubt it has to do with the potential in the semiconductor. Perhaps you must find the actual wave numbers and approximate the dispersion relation to a parabolic one within the band of interest, with mass adjusted as fitting parameter? For the experimentalists the effective mass approximation is useful for describing results, so in some cases the aim for a theorist would be to obtain a matching effective mass value in a model that explains their experimental results.