Finding a constant in nonparabolic dispersion relation in conduction band

Question

The dispersion relation for a semiconductor is $$frac{hbar^2 k^2}{2m^*}=sqrt{alpha E+E_g^2}-E_g$$ where $E$ is the energy from the conduction band edge, $E_g$ is the bandgap energy, $m^*$ is the effective mass at $k=0$, $alpha$ is a constant in the unit of energy. The given questions are:
(a) Find the expression of $alpha$.
(b) Find the expression of effective mass m(E) as a function of E.
My questions are
(1) How do we do part (a)?
(2) For part (b), I am not sure why the effective mass m(E) depends on E. Shoudn't the effective mass be always associated with the point of minimum energy so that it is not E-dependent?

boyfarrell · Answer

Definition of effective mass is,
$$
frac{1}{m^*} = frac{1}{hbar^2}frac{partial^2 E}{partial k^2}
$$
So for part 1) you need to re-arrange your equation for $E$, differentiate it twice, and then solve for $alpha$. For part 2) use the result from part 1) and the equation above.

Finding a constant in nonparabolic dispersion relation in conduction band

One Answer

Add your own answers!

Ask a Question