Energy Levels in a Crystalline Solid — Question 7

Question 7

One and two electrons per cell lead to different band occupations

A crystal has N primitive cells and one relevant spatial orbital per cell, producing a single band of capacity $2N$. Ignore magnetic order, interactions, and band overlap.

1. Find the band filling fractions with one or two valence electrons per cell.

2. If a separated next band lies above, explain which filling can support a band-insulating ground state and count empty slots in the other case.

Question 7 - Solution

One and two electrons per cell lead to different band occupations

(a) Method and calculation

$$\nu=1:\quad f=N/(2N)=\boxed{1/2}.$$$$\nu=2:\quad f=2N/(2N)=\boxed{1}.$$

(b) Interpretation and further calculation

$$\boxed{\nu=2:\ \text{full band plus gap permits a band insulator}}.$$$$\nu=1:\quad N_{\mathrm{empty}}=2N-N=\boxed{N}.$$

Physical check

Half filling permits metallic behavior in this noninteracting model but does not guarantee it in real materials. Correlations, symmetry breaking, and structural changes can open gaps.