QB_PH_SEC7_015: Band Theory of Solids & Semiconductors

In the Kronig-Penney model with periodic delta-function potential barriers $V(x) = \sum_n V_0 b \delta(x - n a)$, the condition for energy band edges corresponds to

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Kronig-Penney Transcendental Equation**: $P \frac{\sin(\alpha a)}{\alpha a} + \cos(\alpha a) = \cos(ka)$ where $\alpha = \sqrt{2mE}/\hbar$ and $P = \frac{m V_0 b a}{\hbar^2}$. 2. **Propagating State Condition**: For an electron to propagate through the lattice, the crystal momentum $k$ must be purely real, requiring: $-1 \le \cos(ka) \le +1$ 3. **Band Extremities**: The boundaries of the allowed energy intervals (band edges) occur at the extreme mathematical limits: $\cos(ka) = +1 \implies ka = 0, 2\pi, \dots$ $\cos(ka) = -1 \implies ka = \pm \pi, \pm 3\pi, \dots$ 4. **Conclusion**: Band edges occur at $\cos(ka) = \pm 1$. Option (A) is correct.