4.4 Thermal conductivity

Thermal conductivity basics

Thermal conductivity describes how effectively a material transfers heat. It connects directly to how we design everything from CPU heat sinks to building insulation, and understanding the underlying physics lets you predict how different solids will behave under thermal loads.

In solids, two carriers do the heavy lifting: electrons (dominant in metals) and phonons (dominant in insulators). Semiconductors involve both, and the balance between them shifts with temperature and doping. This section covers the governing law, the microscopic mechanisms, and the factors that raise or lower conductivity in each class of solid.

Fourier's law of heat conduction

Fourier's law is the starting equation for heat transport. It relates the heat flux $q$ (energy per unit area per unit time) to the temperature gradient:

$q = -k \frac{dT}{dx}$

where $k$ is the thermal conductivity of the material. The negative sign tells you heat flows from hot to cold, opposite to the direction of increasing temperature. This form assumes steady-state, one-dimensional heat flow.

Temperature gradient

The temperature gradient $\frac{dT}{dx}$ is the change in temperature per unit distance through the material. It's what drives heat flow: steeper gradients mean larger heat flux for a given $k$.

• The gradient can be linear (uniform material, constant $k$) or nonlinear (temperature-dependent $k$, composite structures)
• Both the magnitude and direction of heat flow are set by this gradient

Thermal conductivity units

Thermal conductivity is most commonly expressed in watts per meter-kelvin (W/m·K). You'll also encounter:

• cal/s·cm·°C (CGS system)
• BTU/h·ft·°F (imperial, common in engineering applications)

Since 1 W/m·K ≈ 0.578 BTU/h·ft·°F, keep conversion factors handy when reading older or engineering-oriented references.

Thermal conductivity in solids

Heat in a solid is carried by two populations: lattice vibrations (phonons) and conduction electrons. Their relative importance depends on the material class.

Phonons in thermal conduction

Phonons are quantized lattice vibrations. Think of them as wave packets of vibrational energy propagating through the crystal. In insulators and most semiconductors, phonons are the primary heat carriers.

Phonon thermal conductivity depends on:

• Phonon dispersion (how phonon frequency relates to wave vector, which sets group velocities)
• Scattering rates from phonon-phonon interactions, impurities, and boundaries
• Mean free path (average distance between scattering events)

Higher group velocities and longer mean free paths both increase $k$.

Electrons in thermal conduction

In metals, the large concentration of mobile conduction electrons makes them the dominant heat carriers. These same electrons also carry electrical current, which is why good electrical conductors tend to be good thermal conductors.

Electron thermal conductivity is limited by scattering from phonons, impurities, and other electrons. The tight link between electrical and thermal transport in metals is formalized by the Wiedemann-Franz law.

Wiedemann-Franz law

This law states that for metals, the ratio of thermal to electrical conductivity is proportional to temperature:

$\frac{k}{\sigma} = LT$

where $L$ is the Lorenz number, with a theoretical value of $L = 2.44 \times 10^{-8} \; \text{W·Ω/K}^2$.

The physical reason: both heat and charge are carried by the same electrons, so scattering that impedes one impedes the other proportionally. Deviations from this law show up when inelastic scattering (where electrons exchange different amounts of energy and momentum) becomes important, such as at intermediate temperatures where electron-phonon scattering is neither purely elastic nor fully inelastic.

Factors affecting thermal conductivity

Temperature dependence

Temperature affects $k$ differently depending on the material:

• Metals: $k$ generally decreases with rising temperature because electron-phonon scattering intensifies. At very low temperatures, $k$ is limited instead by impurity scattering and rises with $T$ (since electronic heat capacity grows linearly with $T$), producing a peak before the high-temperature decline.
• Insulators: At low $T$, $k$ rises as more phonon modes become populated. It reaches a peak (often near $T \sim \Theta_D / 10$, where $\Theta_D$ is the Debye temperature), then falls at higher $T$ as Umklapp scattering dominates.
• Semiconductors: Behavior can resemble insulators at low $T$ but becomes more complex as thermally excited carriers add an electronic contribution at higher $T$.

Impurities and defects

Any disruption to the perfect crystal scatters heat carriers and lowers $k$:

• Point defects (vacancies, substitutional atoms) scatter phonons through mass and force-constant differences. Even small concentrations can matter: alloying Cu with just a few percent Zn noticeably reduces $k$.
• Dislocations and extended defects introduce strain fields that scatter both phonons and electrons.
• The impact scales with defect concentration, type, and spatial distribution.

Grain boundaries and interfaces

Grain boundaries force phonons and electrons to scatter due to changes in crystal orientation and bonding environment. The thermal resistance at an interface between two different materials is called Kapitza resistance (or interfacial thermal resistance), and it arises from the mismatch in phonon spectra and densities of states across the boundary.

Nanostructured materials exploit this: by packing in a high density of grain boundaries, you can dramatically suppress $k$ while preserving electrical conductivity. This is a key strategy in thermoelectric material design.

Thermal conductivity of metals

Metals are the best thermal conductors among solids, with $k$ values ranging from roughly 20 W/m·K (stainless steel) to over 400 W/m·K (copper) at room temperature. Free electrons are responsible for most of this.

Free electron model

The free electron (Drude-Sommerfeld) model treats valence electrons as a gas of non-interacting fermions moving through the lattice:

• Electrons obey Fermi-Dirac statistics, so only those near the Fermi energy participate in thermal transport.
• The electronic heat capacity is linear in $T$ (from the Sommerfeld expansion), much smaller than the classical prediction but crucial for thermal conductivity.
• This model works well for simple metals like Na, Cu, and Ag, where the Fermi surface is nearly spherical.

Electron mean free path

The electron mean free path $\ell_e$ is the average distance an electron travels between collisions. Longer $\ell_e$ means higher $k$. It's set by whichever scattering mechanism is strongest:

• Electron-phonon scattering dominates at high $T$
• Impurity scattering dominates at low $T$
• Electron-electron scattering is usually a minor correction

You can estimate $\ell_e$ from the Drude relation: $\sigma = \frac{ne^2 \ell_e}{m_e v_F}$, where $v_F$ is the Fermi velocity.

Matthiessen's rule

Matthiessen's rule says the total electrical resistivity is the sum of independent contributions from each scattering mechanism:

$\rho_{\text{total}} = \rho_{\text{phonon}}(T) + \rho_{\text{impurity}} + \rho_{\text{defect}} + \cdots$

Since the Wiedemann-Franz law connects $k$ and $\sigma$, you can use Matthiessen's rule to separate and analyze different contributions to thermal conductivity. The temperature-independent piece ($\rho_{\text{impurity}} + \rho_{\text{defect}}$) is called the residual resistivity, and it's what limits $k$ at the lowest temperatures.

Thermal conductivity of insulators

With few or no free electrons, insulators rely entirely on phonons for heat transport. Their $k$ values are generally lower than metals (diamond is a famous exception, at ~2000 W/m·K, thanks to its stiff bonds and light atoms giving very high phonon group velocities).

Phonon scattering mechanisms

Three main processes limit the phonon mean free path:

• Phonon-phonon scattering:
  • Normal (N) processes conserve total crystal momentum. They redistribute energy among phonon modes but don't directly create thermal resistance.
  • Umklapp (U) processes do not conserve crystal momentum and are the primary source of intrinsic thermal resistance at elevated temperatures.
• Impurity scattering: Mass disorder from impurities or isotopic variations scatters phonons, especially high-frequency ones (Rayleigh-type scattering, rate $\propto \omega^4$).
• Boundary scattering: At low temperatures where phonon mean free paths become very long, scattering from the physical surfaces or grain boundaries of the sample limits $k$. This is why $k$ depends on sample size at low $T$.

Umklapp processes

Umklapp scattering is the key mechanism that gives insulators finite thermal resistance at high temperatures. It occurs when two phonons interact and their combined wave vector falls outside the first Brillouin zone. The resulting wave vector gets mapped back into the zone by a reciprocal lattice vector $\mathbf{G}$:

$\mathbf{k}_1 + \mathbf{k}_2 = \mathbf{k}_3 + \mathbf{G}$

This effectively reverses the direction of phonon momentum flow, creating thermal resistance. Umklapp processes require phonons with large enough wave vectors, so they freeze out exponentially at low temperatures (rate $\propto e^{-\Theta_D / bT}$, where $b$ is a constant of order unity). That's why $k$ in insulators rises steeply as you cool below the Debye temperature.

Phonon mean free path

The phonon mean free path $\ell_{ph}$ is determined by whichever scattering mechanism dominates at a given temperature:

• High $T$: Umklapp scattering dominates, $\ell_{ph}$ is short, and $k \propto 1/T$
• Intermediate $T$: $\ell_{ph}$ grows as Umklapp processes freeze out, producing the conductivity peak
• Low $T$: $\ell_{ph}$ saturates at the sample or grain size (boundary scattering limit)

Models like the Debye model (single phonon branch, isotropic dispersion) and the Callaway model (which distinguishes N and U processes) are used to estimate $\ell_{ph}$ and predict $k(T)$.

Thermal conductivity of semiconductors

Semiconductors sit between metals and insulators, with room-temperature $k$ values typically in the range of 1–150 W/m·K. Silicon, for instance, has $k \approx 150$ W/m·K at 300 K, while bismuth telluride ($\text{Bi}_2\text{Te}_3$) is around 1.5 W/m·K.

Phonon vs. electron contributions

• At low temperatures and in undoped (intrinsic) semiconductors, phonons dominate thermal transport almost entirely.
• As temperature rises, thermally excited electrons and holes appear across the band gap. Their concentration grows exponentially with $T$, and they begin contributing meaningfully to $k$.
• The crossover temperature depends on the band gap: narrow-gap semiconductors see electronic contributions at lower $T$.

Doping effects on thermal conductivity

Doping has competing effects on $k$:

• Electronic contribution increases: More free carriers (electrons for n-type, holes for p-type) means a larger electronic thermal conductivity.
• Phonon contribution decreases: Dopant atoms act as point-defect scattering centers for phonons, reducing the lattice component of $k$.

At typical doping levels in thermoelectric materials ($\sim 10^{19}$ to $10^{20} \; \text{cm}^{-3}$), the phonon reduction often outweighs the electronic gain, giving a net decrease in total $k$.

Bipolar thermal conductivity

In narrow-bandgap semiconductors or at high temperatures, significant populations of both electrons and holes coexist. They can diffuse together (ambipolar diffusion), carrying energy from the hot side to the cold side even without a net electrical current. This bipolar thermal conductivity adds on top of the lattice and single-carrier electronic contributions.

Bipolar conductivity is a major concern in thermoelectric design because it increases $k$ without improving the power factor, degrading the figure of merit $ZT$. Strategies to suppress it include widening the band gap or using band engineering to separate electron and hole contributions.

Measuring thermal conductivity

Experimental methods fall into two broad categories based on whether the measurement is done under equilibrium or time-varying conditions.

Steady-state methods

These establish a constant temperature gradient across the sample and measure the resulting heat flux:

• Guarded hot plate: A flat sample is sandwiched between a heater and a heat sink, with guard heaters to minimize lateral losses. Best for low-to-moderate $k$ materials (insulation, polymers).
• Heat flow meter: Similar geometry but uses a calibrated reference to determine heat flux. Faster but less accurate than the guarded hot plate.

Steady-state methods need large, flat samples and careful control of heat losses, which makes them less practical for small or high-$k$ specimens.

Transient methods

These apply a brief heat pulse and track how temperature evolves over time:

• Laser flash: A short laser pulse heats one face of a thin disc; an IR detector records the temperature rise on the opposite face. Measures thermal diffusivity $\alpha$, from which $k = \alpha \rho c_p$. Very common for solids across a wide $k$ range.
• 3ω method: An AC current at frequency $\omega$ heats a thin metal line on the sample surface. The third-harmonic voltage (at $3\omega$) is sensitive to the thermal conductivity of the substrate. Excellent for thin films.
• Transient plane source (TPS): A flat sensor acts as both heater and thermometer. Good for bulk materials and can handle anisotropic samples.

Transient methods work well with small samples and span a wide range of $k$ values.

Thermal conductivity spectroscopy

This technique applies a periodic heat input and measures the frequency-dependent thermal response. By sweeping the modulation frequency, you probe phonons with different mean free paths: high frequencies are sensitive to short-mean-free-path phonons, low frequencies to long-mean-free-path ones.

This gives you a "spectrum" of how different phonon populations contribute to $k$, which is valuable for understanding nanostructured materials where certain phonon groups are selectively scattered.

Applications of thermal conductivity

Thermoelectric materials

Thermoelectric devices convert heat to electricity (Seebeck effect) or pump heat using electricity (Peltier effect). Their efficiency is governed by the dimensionless figure of merit:

$ZT = \frac{S^2 \sigma T}{k}$

where $S$ is the Seebeck coefficient. You want high $S$, high $\sigma$, and low $k$. Since $\sigma$ and the electronic part of $k$ are linked by Wiedemann-Franz, the main lever is reducing the lattice thermal conductivity. Common strategies:

• Nanostructuring to introduce boundary scattering
• Alloying to increase mass-disorder scattering
• Complex crystal structures (e.g., skutterudites, clathrates) with "rattler" atoms that scatter phonons

Heat sinks and thermal management

Electronics generate concentrated heat that must be spread and dissipated quickly. Heat sink materials need high $k$:

• Copper ($k \approx 400$ W/m·K) and aluminum ($k \approx 237$ W/m·K) are standard choices.
• Advanced solutions like heat pipes and vapor chambers use phase-change transport to move heat over longer distances with very low thermal resistance.

As devices shrink and power densities rise, thermal management increasingly limits performance, making high-$k$ materials and clever geometries critical.

Thermal insulation materials

Here the goal is the opposite: minimize $k$. Effective insulators suppress all three heat transfer modes (conduction, convection, radiation):

• Fiberglass and cellulose trap air in small pockets, exploiting the low $k$ of still air (~0.025 W/m·K).
• Aerogels push this further with extremely porous structures ($k$ as low as ~0.015 W/m·K).
• Vacuum insulation panels eliminate gas-phase conduction entirely, achieving $k$ values below 0.005 W/m·K.

The choice of insulation depends on the operating temperature range, mechanical requirements, moisture resistance, and cost.