Let me begin by explaining what I mean in the term “effective mass.” It’s a term used to make math easier and allows you to think of electrons moving through solids as a particle that has a specific motion and position (“semiclassical” Physics) in an assumption that moving charge carriers are located close to the edge of the band.
Taking an approximation, rather than directly taking the E-k relationship we want to define a term “m*” or “effective mass” that gets us to something where a semiclassical approach is close enough.
The approximation follows the standard Taylor’s series extension.
E(k) ~ E(0) + k dE(k0)/dk + k^2/2 d^2 E(k0)/dk^2 + …
At the edge of the band dE/dk = 0.
This is what we get: E(k) (k) = E(0) + k2/2 d2 E(k0)/dk^2.
It’s interesting… it appears quite a lot like the formula used in classical studies to determine the relation between energy and momentum
E = p2 2m
We already know by definition that p = Hbar * and we already know that p = hbar *.
By directly substituting the words and rearranging them leaves us with
m* = hbar^2 / (d^2 E(k0)/dk^2).
The mass that is effective comes completely out of the structure.
Examining some tables of effective electron and hole mass, I see that it’s not always the instance that electrons are significantly lighter than hole. Germanium particularly has extremely light holesthe m* value is 0.04 to be exact for hole, as opposed to the m* value of 1.64 in the case of electrons. Other semiconductors also contain light holes and heavy electrons.