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 electron hole in a crystal respond to electric field 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 kilogram). Effective mass is defined by analogy with Newton's laws of motion 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 axis), 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 ''m_e''	0.81 ''m_e''
Gallium arsenide	0.067 ''m_e''	0.45 ''m_e''
Germanium	0.55 ''m_e''	0.37 ''m_e''

==External link== *[http://www.ioffe.rssi.ru/SVA/NSM/Semicond/ NSM archive] Solid state physics Mass

Effective mass

This article says that an electron hole in silicon has effective mass around 1.00me. But Electron hole says it's typically about 0.36me. Which is closer to the true value ? :0.36 -- User:Tim Starling 08:57, Jan 17, 2004 (UTC) This article uses the density of states effective mass, while the other article you mention just uses the heavy hole effective mass. It might be good to add a few paragraphs to distinguish these. Also, to my eye, the effective mass quoted for electrons in GaAs in this article is just plain wrong... Am I missing something? This article uses the density of states effective mass, while the other article you mention just uses the heavy hole effective mass. It might be good to add a few paragraphs to distinguish these. Also, to my eye, the effective mass quoted for electrons in GaAs in this article is just plain wrong... Am I missing something?

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