Thermal conductivity [109] is the heat flow rate across a unit cross section
perpendicular to the heat flow rate and absolute temperature. Thermal energy can flow through a material by means of electrical carriers (electrons or holes), lattice waves (phonons), electromagnetic waves, spin waves, or other excitations. Most of the thermal transport in metals is done by electrons or holes while insulators use phonons as the dominant heat carrier. The total κ is a sum of all these components. κ can vary greatly from material to material and is dependent on single crystal versus polycrystalline, lattice defects or imperfections, dislocations, anharmonicity of the lattice forces, carrier concentrations, interaction between the carriers and the lattice waves, interactions between the magnetic ions and lattice waves, and so forth. Thus, being able to isolate
42
the thermal transport in a material and then trying to tailor it to specific properties is very complicated and difficult. The total κ can usually be represented simply by the sum
of the electronic and thermal contributions.
κT = κL+ κE
Useful κ relationships can be found using simple kinetic theory and focusing solely on the contributions from the electrons and phonons. Thermal conductivity, κ, is the diffusion of thermal energy from one end of a material to the other end due to a
temperature difference. For conduction in gases by kinetic theory, c is the heat capacity of each particle, and n is the concentration of particles. If there exists a temperature gradient of ΔT, for a particle to travel with velocity v, its E must change at the rate of
𝑑𝐸
𝑑𝑡 = 𝑐𝜈Δ𝑇
By calculating the average distance this particle would travel, vτ (τ is relaxation time).
Summing over all the particles, the average total heat flow rate per unit area is Q = 1
3 𝑛𝑐𝜏𝜈2Δ𝑇
Combining these equations we have
43
κ = 1
3 𝑛𝑐𝜏𝜈2 = 1 3 𝐶𝜐 𝑙
where C = nc the total heat capacity and l = vτ is the particle mean free path. If heat conduction were not limited by scattering mechanisms and were allowed to flow unobstructed, the l could be infinite and therefore κ would be infinite. This solution can be used in good approximation for various excitations and can be generalized by summing over all the different excitations.
All the portions of the lattice and electrical contributions of the thermal conductivity can follow Matthiessen’s rule of electrical resistance. This works only when there is one dominant scattering mechanism in the equation, otherwise the cross terms will
complicate the data analysis. Therefore it works as a first order approximation. The
thermal resistances are inversely related to the thermal conductivities.
𝑊 = 𝜅−1
For the different thermal resistances, it can be written
𝑊𝐿 = 𝑊𝑙𝑎𝑡−𝑙𝑎𝑡+ 𝑊𝐵+ 𝑊𝑖𝑚𝑝+ 𝑊𝑑𝑖𝑠+ 𝑊𝑙𝑎𝑡−𝑒𝑙
where the contributions come from lattice-lattice interactions, boundaries, impurities,
44
dislocations, and lattice-electron interaction respectively. This can also be found for the
thermal resistances for the electrical portion as
𝑊𝐸 = 𝑊𝑒𝑙−𝑒𝑙+ 𝑊𝑖𝑚𝑝+ 𝑊𝑒𝑙−𝑙𝑎𝑡
where the contributions come from electron-electron, electron-impurities, and electron-phonon interactions respectively.
In looking at the different scattering mechanisms, it is assumed that the scattering processes are not influenced by each other. This is not strictly true, but this treatment can give a good first order approximation of the behavior. The predominant scattering mechanisms for phonons reviewed will be phonon-phonon interactions, boundary
scattering, point defect scattering, and scattering by dislocations.
κL is most dominant in nonmetals and can dominate a significant portion of the
thermal conductance for semiconductors and alloys as well. In a solid the atoms of the material vibrate around their equilibrium position in the crystal lattice. They are coupled to their neighbors and thereby the crystal lattice vibrations can be characterized by the normal modes produced. The quanta of the crystal vibrations are called phonons. When a temperature gradient is present, the thermal energy propagates by wave packets consisting of various normal modes, or phonons. The phonons can be found to have
45
either acoustic or optical modes. The acoustic phonons are low frequency and
correspond to the atoms in the unit cell moving in the same phase. The optical phonons are high frequency and occur when the atoms in the unit cell move in opposite phases.
The acoustic phonons are the main heat carriers because the optical phonons have low group velocities, and therefore are not effective heat carriers. However, they may interact with the acoustic phonons and therefore retard the heat conduction. At low temperature, boundary scattering dominates, then point-defect, and at high temperature Umklap scattering is dominant.
Interactions between phonons are one of the most significant contributions to the lattice thermal conductivity. If thermal vibrations were harmonic, the mean free path of the phonons could be infinite in a pure crystal. As temperature rises, the lattice
vibrations grow more anharmonic which causes the mean free path to vary inversely with temperature at high temperatures. The scattering events caused are of two kinds, normal (N) processes and umklapp (U) processes. N-processes conserve momentum and energy and do not directly lead to any thermal resistance, but redistribute the
momentum within the phonon system. By changing the momentum of the phonons, N-processes can significantly change the scattering effects of other process and therefore make a notable impact on the thermal conductivity. U-processes in which
46
momentum is not conserved are responsible for the observed finite thermal conductivity.
For a U-process to occur the frequency of one of the phonons has to be greater than half
the maximum frequency of the lattice as predicted by Debye, ℎ 𝜐max= 𝑘𝐵𝜃𝐷
Therefore one incident phonon must have at least half of νmax as shown 1
2 𝜐max= 𝑘𝐵𝜃𝐷 2ℎ
At low temperature (T ≤ θD) U-processes are improbable and vary as κL ≈ exp �𝜃𝐷
𝑇 �
but as temperature increases (T ≥ θD) there is an increasing probability of U-processes because the wave numbers are large and κL varies as
𝜅𝐿 ≈ 1 𝑇
Phonons can also be scattered by point defect scattering which includes impurities, vacant lattice sites, interstitial atoms, and other crystal defects. These defects disrupt the periodicity of the lattice. At low temperature, the average distance between the
impurities is smaller than the mean free path of the long wavelength phonons. At high
47
temperature the scattering is less frequency dependent because the phonon wavelength is short and is therefore more nearly temperature independent. When defects are dominant at higher temperatures the lattice thermal conductivity is given by
𝜅𝐿 ≈ 𝑇−32
Boundary scattering can also play a significant role in the κ. As temperature decreases the phonon mean free path increases therefore reducing scattering by point mechanisms because the wavelengths are much larger than the defects. At this point the mean free path is constrained only by crystal size or sample dimensions. At a certain point, even as the temperature decreases further, there is no longer an increase in mean free path due to size constraints. The temperature dependence of κ is due solely to the
temperature dependence of C and varies as 𝜅𝐿 ≈ 𝑇3
The free electron theory of electron conduction in solids in the first instance considers each electron moving in a periodic potential produced by the ions and other electrons without disturbance, and then regards the deviation from the periodicity due to the vibrations of the lattice as a perturbation. If we take the general statement for the
48
thermal conductivity
𝜅 = 1 3 𝐶𝜐 𝑙
and impose the condition that the mean free path of the electrons and of the phonons are similar ,the equation can be modified via some classical equations. The mean free path
between collisions, 𝑙, can replaced by
𝑙 = 𝜐𝜏𝑝ℎ
where ν is the velocity and τ is the phonon relaxation time. The heat capacity can be replaced by
𝐶𝑒𝑙= 𝜋2𝑛𝑘𝐵2𝑇 𝑚𝜐2
Because the mean free path of the electrons and phonons are similar, τe= τph. If the heat
capacity and mean free path are substituted into the original equations, it becomes 𝜅𝐸 = 𝜋2𝑛𝑘𝐵2𝑇𝜏
Knowing that the electrical conductivity is σ = 𝑛𝑒𝑒𝜏
𝑚
49
the electrical contribution of the thermal conductivity becomes 𝜅𝐸 = 𝐿0𝜎𝑇
This gives us the Wiedemann-Franz relationship which illustrates that the κE is related to the electrical conductivity of the material. This relationship holds as long as the mean free path of the phonons and electrons are almost the same. This relation is well obeyed at higher temperatures (T ≥ θD) but not in the intermediate temperature regimes where it does not account for the inelastic scattering of the charge carriers.
In summary the thermal conductivity is made up of a lattice and electrical contribution. The scattering processes are summed and can be assumed not to be influenced by each other for a good approximation. At lower temperature boundary scattering dominates and at higher temperature scattering by U-process dominate.
50