Effective Mass
The effective mass m* of an electron (or hole) in a crystal is the mass it appears to have when it responds to external electric or magnetic forces, defined by the curvature of its energy band: 1/m* = (1/ℏ²) d²E/dk². It encodes the effect of the periodic lattice potential, so it can be much smaller or larger than the free-electron mass, depend on direction, or even be negative near the top of a band.
Curious
Inside a solid an electron is never alone: the regular rows of atoms constantly act on it. Rather than track all those forces, physicists absorb them into a single number, the effective mass. With it, an electron in a crystal can be treated almost like a free particle in Newton's second law, with the external force producing an acceleration a = F/m* (see Newton's Second Law).
In gallium arsenide, conduction electrons behave as if they had only about 7% of the free electron mass, which helps make GaAs devices very fast. Near the top of a filled band, the effective mass of an electron is negative; it is simpler to describe the missing electron as a positively charged “hole” with positive mass (see Semiconductors).
Exploring
| Material and carrier | Effective mass (units of me) |
|---|---|
| GaAs, electrons | 0.067 |
| Si, electrons (longitudinal / transverse) | 0.98 / 0.19 |
| Ge, electrons (longitudinal / transverse) | 1.64 / 0.082 |
| Heavy-fermion metals | up to several hundred |
Deep Dive
Derivation. In the semiclassical model, a wavepacket moves at the group velocity v = (1/ℏ) dE/dk, and an external force changes crystal momentum as ℏ dk/dt = F. Differentiating v gives a = (1/ℏ²)(d²E/dk²) F, so 1/m* = (1/ℏ²) d²E/dk². In three dimensions this is a tensor, (M−1)ij = (1/ℏ²) ∂²E/∂ki∂kj, which is why silicon has different longitudinal and transverse masses. For the 1D tight-binding band E = E0 − 2t cos(ka), the mass at the band bottom is m* = ℏ²/(2ta²); with t = 1 eV and a = 3 Å, m* ≈ 0.42 me (computed): wide bands (large t) give light carriers.
k·p theory. Perturbation theory near a band edge shows that m* is small when the band gap is small and the coupling to the nearby band is strong, explaining why narrow-gap semiconductors such as InSb (m* ≈ 0.014 me) have very light electrons.
Measurement and limits. Effective masses are measured by cyclotron resonance (the orbital frequency in a magnetic field is eB/m*), and by heat capacity and quantum oscillation experiments. In strongly correlated materials, interactions can raise the effective mass hundreds of times (heavy fermions), beyond the reach of single-electron band theory. The concept assumes slowly varying fields and fails near band crossings or under very strong fields.
Concept Relationship Map
Requires: energy bands, Newton's second law
Enables: semiconductor devices, p-n junction
Commonly confused with: a real change in the electron's mass
Sources
- Ashcroft, N.W., Mermin, N.D. (1976). Solid State Physics. Holt, Rinehart and Winston.
- Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley.
- Dresselhaus, G., Kip, A.F., Kittel, C. (1955). “Cyclotron Resonance of Electrons and Holes in Silicon and Germanium Crystals.” Physical Review, 98, 368–384.