Germanium has a band gap of ~0.67 eV; silicon has a gap of ~1.1 eV. As temperature rises from near 0 K, which material becomes electrically conductive first?
ASilicon, because a larger gap means more electrons are available to jump into the conduction band
BGermanium, because its smaller gap requires less thermal energy to promote electrons from the valence band to the conduction band
CNeither — band gaps permanently prevent both materials from ever conducting
DThey transition at the same temperature because both are Group 14 elements with similar crystal structures
Thermal excitation promotes electrons across the band gap. A smaller gap (Ge, 0.67 eV) is bridged at lower temperatures than a larger gap (Si, 1.1 eV), so germanium becomes conducting first. Option 0 reverses the logic. Option 2 confuses insulators (large gap, e.g., diamond ~5.5 eV) with semiconductors — semiconductor conductivity increases strongly with temperature precisely because thermal energy bridges the gap. Option 3 ignores the quantitative gap difference.
Question 2 Multiple Choice
What is the fundamental criterion that determines whether a solid is a metal, insulator, or semiconductor in band theory?
AThe total number of electrons in the material
BThe temperature of the material at room conditions
CHow the available electrons fill the allowed energy bands relative to the size of band gaps
DWhether the material has a crystalline or amorphous structure
The filling state of the highest occupied band is the key. Partially filled band → metal (electrons can absorb small energy increments). Completely filled band with large gap → insulator (no nearby empty states, gap too large for thermal bridging). Completely filled band with small gap → semiconductor (thermally bridgeable at room temperature). Temperature and crystal structure matter for quantitative behavior, but the fundamental classification comes from band filling — which is determined by electron count and band structure.
Question 3 True / False
Conduction electrons in metals are freed largely from atomic binding forces and behave like a classical gas of free particles.
TTrue
FFalse
Answer: False
This is the classic misconception. Conduction electrons are quantum states delocalized over the entire crystal — they are not bound to individual atoms, but they are still bound to the material as a whole. The correct picture is that they occupy extended Bloch states (standing waves in the periodic crystal potential) within a partially filled band. Their mobility comes from having nearby empty states to move into, not from being classically 'free.' This matters: a classical free electron gas cannot explain the band gap, conductivity changes with temperature, or semiconductor behavior.
Question 4 True / False
A completely filled energy band does not conduct electricity even though it contains many electrons, because those electrons have no nearby empty states to move into when an electric field is applied.
TTrue
FFalse
Answer: True
This is the heart of band theory's explanation for insulators. For an electron to accelerate in response to an electric field, it must transition to a slightly higher energy state. In a completely filled band, every state is occupied — there is nowhere to go within that band. The only option is to jump the gap to the next band, which requires energy equal to the band gap. If the gap is large (as in diamond), a small electric field cannot supply this energy. Electrons are present in abundance but immobile. The Pauli exclusion principle is what blocks the movement.
Question 5 Short Answer
Why does a completely filled energy band not allow electrical conduction, even though it contains many electrons?
Think about your answer, then reveal below.
Model answer: For an electron to respond to an electric field and accelerate, it must absorb a small amount of energy and move to a slightly higher energy state. In a completely filled band, all available quantum states are occupied — there are no nearby empty states to move into. The electrons are blocked by the Pauli exclusion principle: two electrons cannot occupy the same state. To conduct, an electron would need to jump across the band gap to the next empty band, which requires energy equal to the full gap. If the gap is large (as in insulators), neither thermal energy nor weak electric fields can supply this. A partly filled band always has empty states just above the filled ones, allowing easy energy absorption and conduction.
This distinguishes the band theory picture from the classical intuition that 'more electrons = better conductor.' What matters is not the number of electrons but whether they have accessible empty states nearby. A metal with a half-filled band has billions of filled states and billions of empty states at essentially the same energy level — electrons can move freely. A filled band is immobile despite containing just as many electrons.