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# Effective mass

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10-31 kg).

Effective mass is defined by analogy with Newton's second law F=m a. Using quantum mechanics it can be shown that for an electron in an external electric field E:

$a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE$

where a is acceleration, h is Planck's constant, k is the wave number (often loosely called momentum since k = p / h), ε(k) is the energy as a function of k, or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of qE, where q is the charge. Hence under the model that only the external electric field acts, effective mass m* becomes:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 \varepsilon} \over {d k^2}} \right]^{-1}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes ), is a tensor. However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics.

## Effective mass for some common semiconductors (for density of states calculations)

<td alme
Material Electron effective mass Hole effective mass
Silicon 0.36 me 0.81 me
Gallium arsenide 0.45 me
Germanium 0.55 me 0.37 me