Effective mass (solid-state physics)

The effective mass ( m * mostly formula symbol ) is a term used in solid state physics (not to be confused with the reduced mass of the Newtonian mechanics or electromagnetic mass, which is referred to in some cases also as effective mass ).

It is the apparent mass of a particle in a crystal under a semiclassical description. It can be shown that the electrons and holes in a crystal to electrical and magnetic fields with similar characteristics, in many situations as if they were free particles in the vacuum, but with a changed composition. This effective mass is usually given in units of the electron mass ( me = 9.11 × 10-31 kg). Experimental methods for determining the effective mass use, among others, the cyclotron resonance.

The basic idea is that the energy-momentum relation (that is, the dispersion relation ) of a particle or quasi-particle in the vicinity of a local minimum as

With p can be developed for the pulse, and O for the higher terms. The quadratic term sees this as the kinetic energy of a particle of mass m * from.

Definition and properties

The effective mass is in analogy with Newton's second law defines (, acceleration equals force per unit mass ). A quantum mechanical description of the crystal electron in an external electric field E gives the equation of motion

Where a is the acceleration, Planck's constant, k is the wave number ( often referred to as a pulse somewhat lax, DA), the energy as a function of k ( dispersion relation), and q the electron charge. A free electron in vacuum, however, would the acceleration

Learn. Thus, the effective mass m * is the electron in the crystal

For a free particle, the dispersion relation is quadratic, and thus the effective mass would be constant ( and equal to the actual electron mass ). In a crystal, the situation is more complex: the dispersion relation is in general not square, which results in a speed-dependent effective mass, SA to the band structure. The concept of the effective mass is, therefore, most useful in the area of the minima or maxima of the dispersion relation, where it can be approximated by quadratic functions. The effective mass is proportional to the inverse of curvature of the tape edge. The interesting physics of the semiconductor plays in a minimum of the conduction band ( positive curvature = effective mass of electrons, positive) and in a maximum of the valence band ( negative curvature = effective mass of the electron negative) from. A Hole assigns one to the negative effective mass of electrons in the valence band, which is thus positive again.

For electron energies far away from such extremes, the effective mass can be negative or even infinite in the conduction band. This peculiar at first glance property one can be explained by the Bragg reflection in the one-dimensional lattice in the wave picture: With the Bragg condition

For reflection on the ion " flat ", and follows

For small amounts of k, the condition is hardly satisfied, the electrons move according to their body mass me. For larger amounts of k is increasingly reflected until effectively no acceleration by an electric field is possible. Is now. For even larger values ​​of k an acceleration leads by an external field by the action of internal forces ( interaction with phonons in the particle picture ) may opposed to an acceleration of the anticipated direction, the effective mass is therefore negative.

Effective mass than tensor

The effective mass is generally dependent on the direction (relative to the crystal axes ), and thus tensor quantities. For the effective mass tensor:

In particular this means that the acceleration of the electrons do not have to be parallel to the field vector in an electric field. In particular, it will be ( in analogy to the inertia tensor ) due to the symmetry of m * a main -axis system, in which takes (1 / m * ) ij diagonal form, with the associated eigenvalues ​​along the diagonal. If by then the electric field along one of these principal axes (which can be achieved by rotating the crystal in a constant field), only the corresponding eigenvalue is one. Since not all eigenvalues ​​must be equal, there is A., principal axes with large and small eigenvalue of the effective mass. Small eigenvalues ​​lead at constant electric field to a higher acceleration of the charge carriers. As temperature increases to the effective masses.

When calculating the density of states effective mass flows with. In order to maintain the shape of the isotropic case, we define a density of states mass

Wherein the degeneration factor N the number of equivalent minima indicates (N usually 6 or 8) and the eigenvalues ​​of the effective mass tensor.

The conductivity and mobility is proportional to the reciprocal effective mass. In anisotropic systems, a mean mobility can specify in which one uses the conductivity mass:

Effective mass for some semiconductors

Detailed values ​​for silicon

Conduction band

For electrons in the conduction band applies:, in. The two equal masses are called transverse mass and longitudinal mass. The density of mass () is at, at it is. The conductivity mass is at.

Valence band

In the valence band there is due to spin -orbit interaction (l = 1, s = 1/2) at the band edge of two sub-bands. One is the heavy holes ( "heavy holes" with and ), the other the light holes ("light holes" with and ). Both have different effective masses, is at and. Moreover, there is yet another subband ( "split off band" with ), which is energetically lowered relative to the valence band edge. When is. The density of states mass of the valence band is at and when it is.