Effective Mass (solid-state Physics) - Definition

Definition

When an electron is moving inside a solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of an electron in Newton's law. The effective mass can be negative or different depending on the circumstances. Generally, in the absence of an electric or magnetic field, the concept of effective mass does not apply.

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, the acceleration a_ℓ along coordinate direction ℓ is:

where ћ = h/2π is reduced Planck's constant, k is the wave vector (often loosely called momentum since k = p / ћ for free electrons), ε (k) is the energy as a function of k, or the dispersion relation as it is often called.

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, for example:

where the minimum is assumed to occur at k=0. In many semiconductors the minimum does not occur at k=0. For example, in silicon the conduction band has six symmetrically located minima along the Δ = symmetry lines in k-space. The constant energy surfaces at these minima are ellipsoids oriented along the k-space axes (see figure).

In contrast, the holes at the top of the silicon valence band are classified as light and heavy and the band energies for the two types are given by a complicated relation:

leading to what is termed warped energy surfaces. Parameters A, B and C are wavevector independent constants. This behavior is introduced here to alert the reader that it is common for carriers to have rather non-parabolic energy-wavevector relations.

A simplification can be made, however, for electrons which have energy close to a minimum, and where the effective mass is the same in all directions, the mass can be approximated as a scalar :

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 many 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 be understood only with quantum mechanics.