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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:
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:
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 tensorFor more technical Wiki articles on tensors, see the section later in this article. In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independen. However, for most calculations the various directions can be averaged out.
Effective mass should not be confused with reduced massReduced mass is a concept that allows one to solve the two-body problem of mechanics as if it were a one body problem. Given two bodies, one with mass and the other with mass , they will orbit the barycenter of the two bodies. The equivalent one-body prob, which is a concept from Newtonian mechanicsMechanics ( Latin mechanicus from the Greek mechanikos "one skilled in machines") is a variety of specialised sciences pertaining to the functions and routine operations of machines, machine-like devices or objects. When preceded by a qualifier, mechanics. Effective mass can only be understood with quantum mechanics.
| Material | Electron effective mass | Hole effective mass |
|---|---|---|
| SiliconSilicon is the chemical element in the periodic table that has the symbol Si and atomic number 14. A tetravalent metalloid, silicon is less reactive than its chemical analog carbon. It is the second most abundant element in the Earth's crust, making up 25 | 0.36 me | 0.81 me |
| Gallium arsenideGallium arsenide (GaAs) is a chemical compound composed of gallium and arsenic. It is an important semiconductor, and is used to make devices such as microwave frequency integrated circuits, infrared light-emitting diodes and laser diodes. Properties Gene | me | 0.45 me |
| GermaniumGermanium is a chemical element in the periodic table that has the symbol Ge and atomic number 32. This is a lustrous, hard, silver-white, metalloid that is chemically similar to tin. Germanium forms a large number of organometallic compounds and is an im | 0.55 me | 0.37 me |