Q2Electrical Materials

(a) Resistance of the semiconductor sample at 500°C: the resistance of a fixed-geometry sample is inversely proportional to its conductivity, R = L/(σA) ∝ 1/σ. Evaluating the given conductivity function at both temperatures:

(the exponential term e⁻²⁵ ≈ 1.4×10⁻¹¹ of 10³, i.e. ~10⁻⁸, is utterly negligible). Using the ratio relationship with the given R(0°C) = 1000 Ω:

The sample's resistance at 500°C is therefore 990 Ω — slightly lower than at 0°C, consistent with semiconductor behavior in which conductivity rises (resistance falls) as temperature increases, here saturating toward the asymptotic value 10⁵ as the exponential term dies away.

Behaviour of the model across the full temperature range: the given function σ(t) = 10⁵ − 10³e^(−0.05t) is instructive precisely because of its asymptotic shape. At t = 0, the exponential term contributes its full weight of 10³, pulling σ down to 99,000 (1% below the asymptote); as t increases, the decaying exponential shrinks rapidly (its 1/e time constant is 1/0.05 = 20°C, so by t = 100°C the correction term has fallen to 10³e⁻⁵ ≈ 6.7, and by t = 500°C it is utterly negligible as shown above). Physically this means the sample's conductivity rises steeply over roughly the first 100°C above the reference temperature and then flattens out, approaching the constant value 10⁵ asymptotically — behaviour broadly consistent with an extrinsic (doped) semiconductor in which the initially low conductivity is limited by incomplete ionization of dopants at low temperature (freeze-out), rising quickly as thermal energy ionizes essentially all the dopant atoms, after which conductivity would, in a more complete physical model, eventually turn over and start falling again at very high temperature as intrinsic carrier generation and increased phonon scattering dominate — a regime beyond the given empirical formula's range of validity but worth noting as the qualitative next stage of real semiconductor R(T) behaviour.

(b) Why metallic resistance increases with temperature: in a metal the free-electron density n is essentially fixed by the atomic structure and does not change with temperature; what changes is the scattering rate. As temperature rises, the lattice ions vibrate with growing amplitude (increasing phonon population), presenting larger effective scattering cross-sections to the drifting electrons; the mean free path and relaxation time τ shorten, mobility μ = eτ/m falls, and since σ = neμ with constant n, conductivity falls — resistance rises. Over ordinary temperature ranges the phonon population grows roughly linearly with T, giving the familiar near-linear relation ρ(T) = ρ₀[1 + α(T−T₀)], where α is the temperature coefficient of resistance (≈0.004 /°C for copper). This mechanism is exactly opposite to the semiconductor of part (a): there, carrier generation dominates and conductivity rises with T; in the metal, carrier density is fixed and scattering alone dictates the trend.

Matthiessen's rule and the full resistivity picture in metals: more completely, a metal's total resistivity at any temperature is the sum of two independent contributions, ρ(T) = ρ_phonon(T) + ρ_residual, where ρ_phonon(T) is the temperature-dependent lattice-scattering term (linear at ordinary temperatures, falling toward zero as T → 0) and ρ_residual is a temperature-independent term set entirely by scattering from impurities, vacancies, dislocations and grain boundaries — the same crystalline-defect population discussed in the point/volume-defect context elsewhere in this paper. This additive rule (Matthiessen's rule) is why measured resistivity-vs-temperature curves for a family of samples of the same metal but different purity are parallel straight lines shifted vertically by their differing residual-resistivity intercepts, exactly as sketched below, and why measuring ρ_residual (via extrapolation to T → 0 or by low-temperature measurement) is a standard, highly sensitive technique for assessing the purity of a metal sample non-destructively.

Why different metals have different slopes: the temperature coefficient α is set by details of each metal's phonon spectrum (Debye temperature) and electronic structure (density of states at the Fermi level, number of free electrons per atom); metals with a higher density of conduction electrons per unit resistivity change, or a lower Debye temperature (softer, more easily excited lattice), tend to show a steeper ρ(T) slope. This is why practical resistance-thermometry elements (platinum RTDs) are chosen for a large, extremely linear and reproducible α, while precision resistor alloys (manganin, constantan) are specifically engineered — via alloying two metals whose opposing scattering contributions largely cancel — to have α close to zero over a wide temperature range, so their resistance value stays nearly constant despite ambient temperature fluctuations.

The sketch shows resistance/resistivity versus temperature for two different metals plotted on the same axes: both are essentially straight lines rising with temperature, but with different intercepts (residual resistivity, set by purity and defects) and different slopes (temperature coefficients α, a property of each metal — e.g., iron's α exceeds copper's, and alloys such as manganin are engineered for nearly zero slope precisely so their resistance stays constant with temperature for precision resistors). The contrast between these rising metallic lines and the falling resistance of the semiconductor in part (a) summarizes, in one picture, the opposite signs of the temperature coefficient of resistance in the two great classes of conducting materials.

Extending the comparison to very low and very high temperature: the near-linear ρ(T) relation for a metal is itself only an intermediate-temperature approximation. At very low temperature (below roughly the Debye temperature), phonon population and hence ρ_phonon fall off much faster than linearly (theoretically as T⁵ in the simplest Bloch-Grüneisen picture), so the curve flattens toward the residual resistivity ρ_residual rather than continuing linearly to zero — this is the origin of the practically important 'residual resistivity ratio' (RRR = ρ(300K)/ρ(4K)) used as a purity figure of merit for cryogenic conductors. At very high temperature approaching the melting point, ρ(T) can depart upward from linearity as thermal expansion and increased anharmonic lattice vibration add additional scattering beyond the simple linear phonon term. The semiconductor of part (a), by contrast, would in a fuller physical model eventually depart from its rising/saturating trend at sufficiently high temperature as well, once intrinsic (band-to-band) carrier generation begins to dominate over the fixed extrinsic dopant population — at which point its conductivity would resume rising exponentially with temperature rather than merely saturating, a regime not captured by the given empirical σ(t) expression but consistent with the general principle that semiconductor conductivity is ultimately carrier-generation-limited while metallic conductivity is scattering-limited across their respective full temperature ranges.