A material has an even number of electrons per unit cell. Is it necessarily an insulator?
AYes — an even electron count always means completely filled bands
BNo — bands from different zones can overlap in energy (band overlap), leaving both partially filled even with an even electron count, as in divalent metals like magnesium and calcium
CNo — an even number of electrons always makes a metal
DIt depends only on the crystal structure, not the electron count
An even electron count is necessary but not sufficient for an insulator. If the valence and conduction bands overlap in energy (which depends on the band structure and Brillouin zone geometry), both are partially filled even though the total electron count would fill bands completely if there were a gap. This is precisely what happens in divalent metals like Mg, Ca, and Zn — band overlap makes them metallic despite having two electrons per atom. Conversely, an odd electron count in a simple band picture must produce a metal (odd filling cannot completely fill a band).
Question 2 Multiple Choice
At room temperature, silicon has a resistivity ~10^3 Ω·m while copper has ~10^-8 Ω·m — a difference of 11 orders of magnitude. What is the fundamental band-theory explanation?
ASilicon atoms are heavier and scatter electrons more
BCopper has partially filled bands with a high density of carriers at E_F, while silicon's 1.1 eV gap means only ~10^10 cm^-3 thermally excited carriers versus copper's ~10^23 cm^-3
CSilicon has a stronger crystal potential
DCopper has more electrons per atom
The carrier density difference is the dominant factor. Copper's Fermi level sits in the middle of a band, providing ~10^23 conduction electrons per cm^3. Silicon's 1.1 eV gap means the carrier density at 300K follows n ∝ exp(-E_g/2k_BT) ≈ 10^10 cm^-3 — thirteen orders of magnitude fewer carriers. Even though silicon's carriers may have comparable mobility to copper's, the enormous density difference determines the conductivity ratio.
Question 3 True / False
Diamond (5.5 eV gap) is an insulator, silicon (1.1 eV) and germanium (0.67 eV) are semiconductors, and tin (alpha-Sn, 0 eV gap) is a semimetal. All four are Group IV elements with the same crystal structure. The gap decreases monotonically with atomic number.
TTrue
FFalse
Answer: True
All four crystallize in the diamond structure. Going down Group IV, atoms get larger and the bond lengths increase. Larger inter-atomic spacing means more orbital overlap between bonding and antibonding states (the bands get wider), and the gap between them shrinks. By tin, the gap has closed entirely, producing a semimetal with overlapping bands. This trend — gap shrinks with increasing atomic size in isostructural materials — is a general principle that reflects the connection between orbital overlap and band width.
Question 4 Short Answer
Explain the physical distinction between a semimetal and a semiconductor with zero gap.
Think about your answer, then reveal below.
Model answer: A semiconductor with exactly zero gap (like a gapless semiconductor or zero-gap semiconductor) has the valence band maximum and conduction band minimum touching at the same energy but not overlapping — the density of states vanishes at the Fermi level. A semimetal has band overlap: the bottom of one band dips below the top of another, so both bands are partially occupied and there are carriers of both electron and hole character even at T = 0, with a small but nonzero density of states at E_F. Examples of semimetals include bismuth and graphite. The distinction matters because semimetals always have carriers (metallic-like, though with low density), while a zero-gap semiconductor has zero carriers at T = 0.
Graphene is the most famous zero-gap semiconductor: its conduction and valence bands touch at the Dirac points, but do not overlap, and the density of states vanishes linearly at the Fermi energy.