Negative and infinite effective mass

The effective mass $m^*$ of an electron in a crystal is the mass that makes Newton's law $F_{ext} = m^*a$ hold for the external force alone. It includes the effect of the internal forces from the lattice, which are too complex to write directly.

Expression

With group velocity $v_g = \frac{1}{\hbar}\frac{dE}{dk}$ and $F_{ext} = \hbar\frac{dk}{dt}$:

$$a = \frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2}\frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}F_{ext}$$ $$m^* = \frac{\hbar^2}{d^2E/dk^2}$$

So $m^*$ is set by the curvature of the E–k curve. For a free electron ($E = \hbar^2k^2/2m_e$), $m^* = m_e$.

E–k diagram of a band

   E
   ^   __                           __
   |     \   m* < 0       m* < 0   /
   |      \                       /
   |       x  <- m* = infinite -> x
   |        \   (inflection)     /
   |         \                  /
   |          `-.__ m* > 0 __.-'
   +----|--------------|--------------|---> k
      -pi/a            0            +pi/a

Negative effective mass

Near the top of a band (close to $k = \pm\pi/a$) the curve bends downward, $d^2E/dk^2 < 0$, so $m^* < 0$. Here the electron wave is close to the Bragg condition and is strongly reflected by the lattice. When a force pushes it, the lattice pushes back even harder, so the electron accelerates opposite to the applied force, as if its mass were negative. An electron with negative charge and negative mass behaves like a positive charge with positive mass: this is the basis of the hole concept for the top of the valence band.

Infinite effective mass

At the inflection point of the curve ($k = \pm\pi/2a$ for a cosine band) $d^2E/dk^2 = 0$, so $m^* \to \infty$. The velocity $v_g$ is maximum there; an applied force cannot increase it, so the acceleration is zero. The external force is exactly balanced by the lattice force; the electron behaves as an infinitely heavy particle.

Negative and infinite mass are not real changes in the electron's mass; they are ways of describing how the lattice modifies its response to an external force.