# 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:

[itex] a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE [itex]

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:

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

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)

Material Electron effective mass Hole effective mass
Silicon 0.36 me 0.81 me
Gallium arsenide 0.067 me 0.45 me
Germanium 0.55 me 0.37 me

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