
Thermal conductivity is the property of a material to conduct heat, determining how efficiently thermal energy transfers through substances like thin films or superconductors.Thermal conductivity (k) is the proportionality coefficient in Fourier's law: heat flux density q⃗ = –k ∇T, where ∇T is the temperature gradient. It characterizes the ability of a substance to transfer the kinetic energy of microscopic particles without macroscopic movement of the substance. In metals, heat transfer is mainly by conduction electrons; in dielectrics, it is by phonons (quanta of crystal lattice vibrations). Silver has the highest thermal conductivity among pure metals (k ≈ 430 W/(m·K)).
The law of thermal conduction was formulated by Joseph Fourier in 1822 in his 'Analytical Theory of Heat'. For over a century, thermal conductivity remained a phenomenological constant. The kinetic theory of gases (James Clerk Maxwell, Ludwig Boltzmann) made it possible to relate it to molecular motion. In the 20th century, the development of quantum solid-state theory (Peter Debye, Rudolf Peierls) led to the phonon picture and the prediction of lattice thermal resistance. At the end of the century, materials with extreme thermal conductivity (diamond, graphene, nanotubes) were discovered.
In solids, heat transfer occurs via two main carriers: conduction electrons and phonons. Phonons — collective lattice vibrations — behave like a gas of quasiparticles and obey Bose–Einstein statistics. The distribution of phonons over frequencies and wave vectors is described by dispersion relations depending on crystal symmetry. Thermal resistance arises because the phonon mean free path is finite, limited by anharmonicity of interatomic potentials (three-phonon processes), isotopic disorder, defects, and grain sizes. At high temperatures, scattering on other phonons (so-called Umklapp processes) dominates. In nanostructures, the mean free path can be limited by geometric dimensions, leading to reduced thermal conductivity (phonon ballistic regime).
Copper pipes in a radiator quickly release heat into the room, while a wool sweater retains body heat due to low thermal conductivity of air in fluffy fibers.
Diamond has a thermal conductivity five times higher than copper, although diamond is an electrical insulator. The reason is a rigid crystal lattice and light carbon atoms, along which phonons travel quickly.
| k | variable | thermal conductivity — thermal conductivity watt per metre per kelvin |
| C | variable | heat capacity per unit volume — volumetric heat capacity joule per cubic metre per kelvin |
| v | variable | average particle speed — speed metre per second |
| l | variable | mean free path — mean free path metre |
| S_a^{1/2} | variable | Spectral density of acceleration noise (square root) — acceleration noise spectral density metre per second squared per root hertz |
| T | variable | Temperature — temperature kelvin |
| k | variable | Spring constant of the resonator — spring constant newton per metre |
| m | variable | Mass of the resonator — mass kilogram |
| ω | variable | Angular resonance frequency — angular frequency radian per second |
| Q | variable | Quality factor of the resonator — quality factor |
| k_B | constant | Boltzmann constant, relates thermal energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin |
| \sigma | variable | electrical conductivity — electrical conductivity siemens per metre |
| \sigma_0 | variable | pre-exponential factor (conductivity at infinite temperature) — electrical conductivity siemens per metre |
| E_a | variable | activation energy for conduction — energy joule |
| T | variable | absolute temperature — temperature kelvin |
| k_B | constant | Relates temperature to energy in statistical mechanics. — boltzmann constant = 1.380649e-23 joule per kelvin |
| σ_total | variable | total conductivity — electrical conductivity siemens per metre |
| σ_cond | variable | conductivity from condensed (coherent) carriers — electrical conductivity siemens per metre |
| σ_fluct | variable | conductivity from fluctuating (incoherent) carriers — electrical conductivity siemens per metre |
| G | variable | Electrical conductance of a mesoscopic conductor — conductance siemens |
| T(E) | variable | Transmission probability for an electron at energy E — transmission probability |
| E | variable | Energy of the electron — energy joule |
| e | constant | Magnitude of the charge of an electron — elementary charge = 1.602176634e-19 coulomb |
| h | constant | Planck constant, fundamental quantum of action — planck constant = 6.62607015e-34 joule second |
| \sum | operator | Sum over all transmission channels or energy levels — summation |



















