Q2Electrical Materials

(a) Energy band diagrams of metal, semiconductor and insulator: in solids, the discrete atomic levels broaden into energy bands; electrical behavior is set by the arrangement of the highest filled (valence) band and the lowest empty (conduction) band. In a metal, the conduction band is either partially filled or overlaps the valence band, so there is no forbidden gap at the Fermi level; even an infinitesimal field can raise electrons into adjacent empty states, giving the enormous conductivity of metals at all temperatures. In a semiconductor, a narrow forbidden gap Eg of roughly 0.5–2 eV (Si: 1.1 eV, Ge: 0.67 eV) separates a nearly full valence band from a nearly empty conduction band; at absolute zero the material would be insulating, but at room temperature thermal energy excites a small but significant carrier population across the gap, giving intermediate conductivity that rises steeply with temperature and is highly controllable by doping. In an insulator, the gap is large (>3 eV, e.g. diamond ≈5.5 eV, SiO₂ ≈9 eV); thermal excitation across so wide a gap is negligible at ordinary temperatures, leaving essentially no free carriers and extremely high resistivity.

(b) Resistivity of the N-type semiconductor: both carrier types contribute to conduction, weighted by their densities and mobilities:

The resistivity of the sample is approximately 3.4 Ω·cm. Note how completely the electron term dominates: the hole contribution (5×10¹⁴) is nearly four orders of magnitude below the electron contribution (1.84×10¹⁸), both because electrons outnumber holes 16:1 in this N-type sample and because the stated electron mobility exceeds the hole mobility by a factor of 230 — a concrete illustration of why, in doped semiconductors, the minority-carrier contribution to conductivity is almost always negligible and σ ≈ enμe is an excellent approximation for N-type material.

This band-diagram framework and the resistivity calculation are directly connected: the band picture explains qualitatively why a semiconductor's conductivity is tunable and intermediate between metal and insulator, while this worked example demonstrates quantitatively how doping density and mobility — not the band gap directly — set the actual numerical conductivity of an extrinsic (doped) sample at room temperature, where the gap's role is already reflected in which carrier type dominates rather than in the magnitude of σ itself.

The Fermi level position in each of the three band pictures also differs characteristically and governs the temperature response. In a metal the Fermi level lies inside the partially filled band itself, so conduction electrons already exist at T = 0 K and conductivity actually falls slightly with rising temperature, because increased lattice (phonon) vibration scatters the already-abundant carriers more frequently — resistivity of metals therefore has a positive temperature coefficient. In the intrinsic semiconductor the Fermi level sits near mid-gap, and since carrier generation requires thermal excitation across Eg, the carrier population — and hence conductivity — increases exponentially with temperature (n, p ∝ exp(−Eg/2kT)), giving metals and semiconductors opposite signs of temperature coefficient of resistivity, a distinction often tested directly. In an insulator the Fermi level is likewise near mid-gap, but because Eg is so large compared with kT at ordinary temperatures (kT ≈ 0.026 eV at 300 K, versus Eg of several eV), the exponential term is vanishingly small and conductivity remains immeasurably low right up to temperatures approaching material breakdown or decomposition. This progression — metal (no gap, carriers always present), semiconductor (small gap, thermally activated carriers), insulator (large gap, negligible thermal activation) — is thus a single unified picture in which only the magnitude of Eg relative to kT changes, and it is this same unified picture that justifies treating the numerical resistivity problem in part (b) as an extrinsic-conduction case governed purely by doping rather than by thermal band-to-band generation.