effectivemass

The effective mass, sometimes written as effectivemass, is a quantity in solid-state physics that describes how charge carriers respond to external forces within a crystal, taking into account the interaction with the periodic lattice. It replaces the free-electron mass in equations of motion and enters equations for transport and optical properties. In practice, m* is determined by the curvature of the electronic energy bands and can vary with direction, band, and carrier type.

Mathematically, near a band extremum at wave vector k0, the energy can be expanded as E(k) ≈ E0 + (ħ^2/2) (k - k0)^T (M)^{-1} (k - k0), where M is the effective mass tensor. In one dimension, 1/m* = (1/ħ^2) d^2E/dk^2. In three dimensions, m* is generally a tensor that may be anisotropic.

The effective mass is often approximated as energy-independent only close to the band edge; bands that are strongly nonparabolic require energy-dependent mass or models like Kane’s. In many materials, the electron and hole masses differ and may be labeled light, heavy, or transverse and longitudinal masses.

Experimentally, m* is inferred from measurements such as cyclotron resonance, Shubnikov–de Haas oscillations, and optical absorption, or calculated from electronic band structure using k·p perturbation theory or density functional theory.

The mass influences mobility μ = qτ/m* and conductivity σ = nqμ, and it also affects the density-of-states via a DOS effective mass. Anisotropic crystals yield a tensor m*, leading to direction-dependent transport.

While the concept works well for conventional semiconductors, in materials with linear dispersion such as graphene, the usual mass notion is modified; carriers can behave as massless Dirac fermions and the cyclotron mass becomes energy-dependent.