The concept of electron and hole effective mass in semiconductors has been developed starting from the mobility of carriers (be them electrons or holes). When an external electricc field `E` (or equivalent a potential difference) is applied to the semiconductor it will start to conduct electric current, by the motion of electrons and holes. The carries will accelerate into the electric field, and will be decelerated by different collisions (scientific term is different scattering mechanisms) that happen in their motion through the semiconductor. The average drift velocity of the carriers is of course proportional to the field applied.

`v_d = mu*E`

Here above `mu` is the mobility of the carriers (different for electrons or holes). This mobility in turn is conceivable to be inverse proportional to the mass (and charge) of the carriers (as Newton second law states `F = m*a` , the accelerating force here is the electric force) and inverse proportional to the time `tau` between two collisions (that decelerate the carriers).

`mu = q/m*tau`

Here above `m` is the effective mass of the carriers. Again. this concept of effective mass relates to the fact that carriers are accelerated by the electric force and they need to obey the Newton second law.

In solid state physics there have been derived methods to experimentally measure the value of carrier mobility `mu`, by measuring the Hall coefficient and the resistivity of the material. Thus, from these experimental determinations, usually for semiconductors the mass of holes has been determined to be greater than the mass of electrons. This means that in electric field the holes accelerate less than electrons.

Now for the theoretical explanations.

Everybody learns that in semiconductors there are bands of energy. Thus the last two bands are called valence band and conduction band. Usually when we draw on paper these bands we represent them as two horizontal lines separated by a gap (forbidden band). What is little known is that in fact from quantum calculus the shape of these bands is not linear. The bands are like two rotation ellipsoids having a minimum distance at one point between them (the forbidden band). It has been mathematically shown, that the radius of curvature of the ellipsoids representing the bands are proportional to the effective mass of carriers. (The conduction band curvature accounts for the electron mass and the valence band curvature for the holes mass). Hence the mass of holes is greater than the mass of electrons by quantum computations of allowed bands energy (which in turn comes from the structure of the crystal and the structure of the composing atoms themselves).

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