Effective Mass Theory in Solids from a Many-Particle Standpoint

Effective-mass equations for electrons and holes in solids are usually derived by means of the independent-particle model. It is shown in the present paper that for the case of an extra electron or a hole in an insulator, or semiconductor, moving under the action of a weak Coulomb field, an effective-mass equation, $\left(-\frac{{\hslash }^2}{2m^{*}}{\nabla }^2-\frac{\mathrm{qe}}{\kappa r}\right)F=EF,$ can be derived from some very general properties of the entire many-particle system ($m^{*}=\mathrm{effective}\mathrm{mass}$, $q=\mathrm{external}\mathrm{charge}$, $\kappa =\mathrm{static}\mathrm{dielectric}\mathrm{constant}$). The effects of exchange and correlation are automatically included without approximation. It is suggested that the same type of result can be derived for the motion of electrons and holes in arbitrary slowly varying and weak electromagnetic fields.