ja:有効質量

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 m_e (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:

<math> a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE <math>

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:

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

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 alm_e

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

External link

• NSM archive (http://www.ioffe.rssi.ru/SVA/NSM/Semicond/)