Questions: Semiclassical Electron Dynamics: Effective Mass and Holes

Question 1

For the band ε(k) = −2t cos(ka), what is the electron's velocity at the zone boundary k = π/a?

• A. 2ta/ħ
• B. Zero
• C. ħπ/(ma)
• D. Infinite

Answer: (B)

v = (1/ħ)dε/dk = (2ta/ħ) sin(ka). At k = π/a, sin π = 0, so v = 0. The band is flat at the zone boundary.

Question 2

A completely filled energy band carries no net electric current, even when an electric field is applied.

• T. True
• F. False

Answer: True

In a filled band, for every state k there is a state −k with opposite velocity, and the field only relabels states within the band. The velocities sum to zero, so there is no current.

Question 3

What is the effective mass at the bottom of the band ε(k) = −2t cos(ka), at k = 0?

• A. ħ²/(2ta²)
• B. 2ta²/ħ²
• C. −ħ²/(2ta²)
• D. The free-electron mass

Answer: (A)

1/m* = (1/ħ²)d²ε/dk² = 2ta² cos(ka)/ħ². At k = 0 this gives m* = ħ²/(2ta²). At the band top, k = π/a, m* has the same size but negative sign.

Question 4

In a perfect crystal under a constant electric field, the semiclassical model predicts that an electron oscillates in position rather than accelerating forever. Name the effect and explain why it is not seen in ordinary metals.

Model answer: Bloch oscillations. In ordinary metals, electrons scatter off impurities and phonons long before completing one oscillation period, h/(eEa), which is very long for typical fields and lattice constants.

Bloch oscillations were observed in semiconductor superlattices, whose large period a shortens the oscillation period, and in ultracold atoms in optical lattices.