Effective mass of a free electron

Effective mass $m^*$ is the apparent mass of an electron in a crystal when it is accelerated by an external force; it includes the effect of the lattice's internal forces. It is defined as

$$m^* = \frac{\hbar^2}{d^2E/dk^2}$$

Free electron

For an electron moving freely in vacuum (or a "free" electron with $V = 0$), all its energy is kinetic:

$$E = \frac{1}{2}m_ev^2 = \frac{p^2}{2m_e}$$

By de Broglie, $p = \hbar k$, so

$$E = \frac{\hbar^2k^2}{2m_e}$$

This is a parabola in the E–k diagram. Differentiating twice:

$$\frac{dE}{dk} = \frac{\hbar^2k}{m_e}, \qquad \frac{d^2E}{dk^2} = \frac{\hbar^2}{m_e}$$

Substituting in the definition:

$$m^* = \frac{\hbar^2}{\hbar^2/m_e} = m_e$$

Hence, for a free electron the effective mass equals the ordinary mass of the electron in vacuum, $m_e = 9.1\times10^{-31}$ kg. This is expected: with no lattice there are no internal forces, so the only force is the external one and Newton's law $F = m_ea$ holds directly. In a real crystal the E–k curve is not exactly parabolic, so $m^*$ differs from $m_e$ (e.g. $1.08\,m_e$ for electrons in Si).