Infinite effective mass and its E-k explanation

Effective mass $m^* = \hbar^2/(d^2E/dk^2)$ is the mass an electron in a crystal appears to have under an external force. Infinite effective mass means $d^2E/dk^2 = 0$ (the inflection point of the E–k curve): there the external force produces no acceleration at all, because the force from the lattice exactly balances it.

Derivation of $m^*$

The electron is a wave packet moving with group velocity ($E = \hbar\omega$):

$$v_g = \frac{d\omega}{dk} = \frac{1}{\hbar}\frac{dE}{dk}$$

An external force $F$ acting for time $dt$ does work $dE = F v_g\,dt$:

$$\frac{dE}{dk}dk = F\cdot\frac{1}{\hbar}\frac{dE}{dk}dt \quad\Rightarrow\quad F = \hbar\frac{dk}{dt}$$

Acceleration:

$$\begin{aligned} a &= \frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2}\frac{dk}{dt} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}F \end{aligned}$$

Comparing with $F = m^*a$:

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

Positive, negative and infinite $m^*$ from the E–k diagram

In a crystal the band has a cosine-like shape. Taking $E = E_0 - 2\gamma\cos ka$,

$$\frac{dE}{dk} = 2\gamma a\sin ka, \qquad \frac{d^2E}{dk^2} = 2\gamma a^2\cos ka$$

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

   v_g ~ dE/dk        (b) velocity
   ^           max at +pi/2a
   |             .--.
   |           /      \
   +----------*--------*---------> k
   |  \      /  0      +pi/a
   |    '--'

• Lower half of band ($|k| < \pi/2a$): curve concave up, $d^2E/dk^2 > 0$, $m^* > 0$. Velocity rises with $k$; the electron accelerates in the direction of the force, like a normal particle.
• Inflection point ($|k| = \pi/2a$): $d^2E/dk^2 = 0$, $m^* = \infty$. Velocity is maximum; increasing $k$ does not increase $v_g$. The applied force gives zero acceleration.
• Upper half of band ($\pi/2a < |k| < \pi/a$): curve concave down, $d^2E/dk^2 < 0$, $m^* < 0$. Velocity decreases as $k$ increases, so the electron accelerates opposite to the applied force. Physically, the electron is being Bragg-reflected by the lattice and the lattice pushes back harder than the external force.
• At $k = \pm\pi/a$ (band edge), $v_g = 0$: a standing wave.

Significance

• Near the bottom of the conduction band electrons have positive $m^*$ and behave like free electrons.
• Near the top of the valence band electrons have negative $m^*$. An electron missing from there behaves as a hole with positive charge and positive mass, which is the basis of hole conduction in semiconductors.