Thermal Conductivity in Solids
Field note — how heat moves through matter when there is no convection, only the patient push of vibrations and the restless flight of electrons.
The two channels
In solids, thermal energy is transported by two carriers:
- Phonons — quantized lattice vibrations. They dominate in insulators and semiconductors.
- Electrons — free charge carriers. They dominate in metals.
The total thermal conductivity is κ = κ_ph + κ_el. In a good electrical conductor (silver, copper, aluminum), κ_el ≫ κ_ph. In a good electrical insulator (glass, ceramic, diamond), κ_ph is everything. Some materials — silicon, graphite — are interesting because both channels contribute at comparable levels.
The kinetic formula
Both channels obey a similar transport equation:
κ = (1/3) · C · v · ℓ
where C is the volumetric heat capacity (J·m⁻³·K⁻¹), v is the characteristic carrier velocity (sound speed v_s for phonons, Fermi velocity v_F for electrons), and ℓ is the mean free path between scattering events. This formula captures the intuition: heat capacity tells you how much energy each carrier can store, velocity tells you how fast it moves, and mean free path tells you how far it gets before forgetting its direction.
Phonon thermal conductivity
In insulators, phonons carry the heat. Their mean free path is limited by:
- Phonon-phonon scattering (Umklapp processes). At high T, phonon population is large (n̄ ∝ T for the dominant modes), so scattering is frequent. ℓ ∝ 1/T. Since C_ph saturates at the Dulong-Petit limit, κ_ph ∝ 1/T. This explains why most insulators become worse thermal conductors as they heat up.
- Point defects and isotopes. Mass fluctuations from different isotopes scatter phonons. Natural silicon (mostly ²⁸Si, with ⁵% ²⁹Si and ⁰·³% ³⁰Si) has κ ≈ 150 W/m·K at room temperature. Isotopically pure ²⁸Si reaches κ ≈ 1850 W/m·K. That's 12× improvement from removing isotopes.
- Grain boundaries and dislocations. In polycrystalline materials, phonons scatter at every grain boundary. Nanostructuring reduces ℓ intentionally — this is the basis of thermoelectric materials, where low κ is desired.
- Boundaries. At very low T (T < 1 K), phonon-phonon scattering freezes out (n̄ → 0). ℓ is then set by the sample size itself. κ peaks and then falls, because C_ph ∝ T³.
The peak in κ_ph(T) is a signature feature. At intermediate temperatures, C_ph rises (more phonons to carry heat) but ℓ falls (more scattering). The maximum occurs where these two competing effects balance. For most crystals, the peak is between 10 and 30 K.
Electron thermal conductivity
In metals, electrons carry heat just as they carry charge. The Wiedemann-Franz law quantifies this:
κ_el / σ = L · T
where σ is the electrical conductivity and L ≈ 2.44 × 10⁻⁸ W·Ω·K⁻² is the Lorenz number. The ratio is universal (to leading order) because the same electrons transport both charge and heat, and both processes are limited by the same scattering time τ.
Derive it from the kinetic formula: C_el ∝ T (Sommerfeld result), v = v_F (temperature-independent), ℓ = v_F·τ. Then κ_el ∝ T·τ and σ ∝ τ, so κ_el/σ ∝ T. The Lorenz number is L = (π²/3)·(k_B/e)².
At room temperature, copper has κ_el ≈ 400 W/m·K and σ ≈ 6 × 10⁷ S/m. L = κ/(σT) ≈ 2.2 × 10⁻⁸, close to the theoretical value. The deviations (5–10%) come from electron-electron and electron-phonon corrections to the relaxation time approximation.
Why diamond is special
Diamond has two competing properties:
- High Debye temperature (Θ_D ≈ 2200 K), meaning stiff bonds and high sound velocity v_s ≈ 18 km/s.
- Low phonon scattering in ultrapure samples, giving ℓ on the order of micrometers even at room temperature.
Combined with moderate C_ph, this gives κ ≈ 2000 W/m·K — the highest of any bulk material at room temperature. (Graphene is higher but it is a 2D material.) The electron contribution is zero (diamond is an insulator), so κ = κ_ph.
Copper, by contrast, has κ ≈ 400 W/m·K, but that is entirely from electrons. The comparison is fair: both are excellent conductors, just by different mechanisms.
Thermal insulators
Glass and aerogels have very low κ (≈ 0.02–0.04 W/m·K) because:
- The disordered structure scatters phonons extremely efficiently (ℓ is on the order of atomic spacing).
- Aerogels add trapped gas, whose conduction is suppressed by the Knudsen effect at small pore sizes.
These materials work as insulators because they destroy the mean free path of their heat carriers.
Summary
Thermal conductivity in solids operates through two channels: phonons in insulators, electrons in metals. The kinetic formula κ = (1/3)Cvℓ unifies both. Phonon κ peaks at intermediate T (typically 10–30 K) and falls as 1/T at high T due to Umklapp scattering. Electron κ obeys the Wiedemann-Franz law, linking it directly to electrical conductivity. Material structure — purity, crystal quality, isotope composition — controls ℓ and therefore everything.