4.4 Thermal conductivity

Thermal conductivity is a crucial property that determines how well materials transfer heat. It's essential for designing everything from electronics to insulation. Understanding thermal conductivity helps engineers create more efficient devices and structures.

Fourier's law describes heat flow in materials, while factors like temperature and impurities affect conductivity. In metals, electrons are the main heat carriers, while in insulators, it's phonons. Semiconductors have a mix of both, making their behavior more complex.

Thermal conductivity basics

• Thermal conductivity is a fundamental property that describes a material's ability to conduct heat
• Understanding thermal conductivity is crucial for designing materials with desired heat transfer properties in various applications, such as electronics, energy conversion, and thermal insulation

Fourier's law of heat conduction

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• Describes the relationship between heat flux ($q$) and temperature gradient ($\frac{dT}{dx}$) in a material
• Mathematically expressed as $q = -k \frac{dT}{dx}$, where $k$ is the thermal conductivity
• The negative sign indicates that heat flows from high to low temperature regions
• Assumes steady-state conditions and one-dimensional heat flow

Temperature gradient

• Represents the change in temperature per unit distance in a material
• Drives the flow of heat from high to low temperature regions
• Can be linear or non-linear depending on the material properties and boundary conditions
• Plays a crucial role in determining the magnitude and direction of heat flow

Thermal conductivity units

• Thermal conductivity is typically expressed in watts per meter-kelvin (W/m·K) or watts per meter-degree Celsius (W/m·°C)
• Other units include calories per second-centimeter-degree Celsius (cal/s·cm·°C) and British thermal units per hour-foot-degree Fahrenheit (BTU/h·ft·°F)
• The choice of units depends on the system of measurement and the application context
• Conversion factors are used to switch between different units of thermal conductivity

Thermal conductivity in solids

• In solids, thermal conductivity arises from two main mechanisms: phonons and electrons
• The relative contributions of phonons and electrons to thermal conductivity vary depending on the material type (metals, insulators, or semiconductors)

Phonons in thermal conduction

• Phonons are quantized lattice vibrations that carry heat through the material
• In insulators and semiconductors, phonons are the primary heat carriers
• Phonon thermal conductivity depends on factors such as phonon dispersion, scattering rates, and mean free path
• Phonon scattering mechanisms include phonon-phonon interactions, impurity scattering, and boundary scattering

Electrons in thermal conduction

• In metals, electrons are the dominant heat carriers due to their high mobility and large concentration
• Free electrons in metals can transport heat efficiently alongside electrical current
• The contribution of electrons to thermal conductivity is proportional to the electrical conductivity (Wiedemann-Franz law)
• Electron scattering mechanisms, such as electron-phonon interactions and impurity scattering, affect the electron thermal conductivity

Wiedemann-Franz law

• States that the ratio of thermal conductivity ($k$) to electrical conductivity ($\sigma$) is proportional to temperature ($T$) for metals
• Mathematically expressed as $\frac{k}{\sigma} = LT$, where $L$ is the Lorenz number
• The Lorenz number is a constant with a theoretical value of $2.44 \times 10^{-8}$ W·Ω/K²
• Deviations from the Wiedemann-Franz law can occur due to various factors, such as electron-electron interactions and inelastic scattering processes

Factors affecting thermal conductivity

• Several factors influence the thermal conductivity of materials, including temperature, impurities, defects, and microstructure
• Understanding these factors is essential for tailoring the thermal properties of materials for specific applications

Temperature dependence

• Thermal conductivity often varies with temperature, depending on the material type and the dominant heat transfer mechanism
• In metals, thermal conductivity generally decreases with increasing temperature due to increased electron-phonon scattering
• In insulators and semiconductors, thermal conductivity may increase or decrease with temperature, depending on the phonon scattering processes
• At low temperatures, thermal conductivity can exhibit different behavior, such as a peak in thermal conductivity for some materials

Impurities and defects

• Impurities and defects in the crystal structure can scatter phonons and electrons, reducing thermal conductivity
• Point defects, such as vacancies and substitutional impurities, introduce additional scattering centers
• Dislocations and other extended defects can also scatter heat carriers and lower thermal conductivity
• The impact of impurities and defects on thermal conductivity depends on their concentration, type, and distribution within the material

Grain boundaries and interfaces

• Grain boundaries and interfaces between different materials can act as barriers to heat flow
• Phonons and electrons can scatter at grain boundaries due to changes in crystal orientation and atomic structure
• Interfacial thermal resistance (Kapitza resistance) arises from the mismatch in phonon and electron properties across interfaces
• Nanostructured materials, with a high density of grain boundaries and interfaces, often exhibit reduced thermal conductivity compared to their bulk counterparts

Thermal conductivity of metals

• Metals are excellent thermal conductors due to their high concentration of free electrons
• The free electron model provides a framework for understanding the thermal conductivity of metals

Free electron model

• Assumes that valence electrons in metals behave as a gas of free particles
• Electrons are treated as non-interacting particles that move freely through the metal lattice
• The heat capacity of the electron gas is described by the Sommerfeld model, which accounts for the Fermi-Dirac distribution of electron energies
• The free electron model provides a good approximation for the thermal conductivity of simple metals (sodium, copper)

Electron mean free path

• Represents the average distance an electron travels between scattering events
• Determined by various scattering mechanisms, such as electron-phonon interactions, impurity scattering, and electron-electron interactions
• A longer electron mean free path generally leads to higher thermal conductivity
• The electron mean free path can be estimated from the electrical conductivity using the Drude model

Matthiessen's rule

• States that the total resistivity of a metal is the sum of the resistivities arising from different scattering mechanisms
• Mathematically expressed as $\rho_{total} = \rho_{phonon} + \rho_{impurity} + \rho_{defect} + ...$
• The thermal conductivity can be related to the electrical resistivity through the Wiedemann-Franz law
• Matthiessen's rule allows for the separation and analysis of different contributions to the thermal conductivity of metals

Thermal conductivity of insulators

• In insulators, heat is primarily carried by phonons, as there are few free electrons available for thermal conduction
• The thermal conductivity of insulators is generally lower than that of metals and is strongly influenced by phonon scattering processes

Phonon scattering mechanisms

• Phonon-phonon scattering: Interactions between phonons can lead to the redistribution of energy and momentum, limiting the phonon mean free path
  • Normal processes conserve the total phonon momentum and do not directly contribute to thermal resistance
  • Umklapp processes do not conserve phonon momentum and are the primary source of thermal resistance at high temperatures
• Impurity scattering: Phonons can scatter off impurities and defects in the crystal structure, reducing the phonon mean free path
• Boundary scattering: Phonons can scatter at the boundaries of the material, such as grain boundaries and interfaces, limiting the phonon mean free path

Umklapp processes

• A type of phonon-phonon scattering that does not conserve the total phonon momentum
• Occurs when the sum of the wave vectors of the interacting phonons falls outside the first Brillouin zone
• The resulting phonon has a wave vector that is "folded back" into the first Brillouin zone, effectively reversing its direction
• Umklapp processes are the dominant scattering mechanism at high temperatures and are responsible for the decrease in thermal conductivity with increasing temperature in many insulators

Phonon mean free path

• Represents the average distance a phonon travels between scattering events
• Determined by the various phonon scattering mechanisms, such as phonon-phonon interactions, impurity scattering, and boundary scattering
• A longer phonon mean free path generally leads to higher thermal conductivity
• The phonon mean free path can be estimated from the phonon dispersion relations and scattering rates using models such as the Debye model or the Callaway model

Thermal conductivity of semiconductors

• Semiconductors exhibit intermediate thermal conductivity values between those of metals and insulators
• The thermal conductivity of semiconductors is influenced by both phonons and electrons, depending on factors such as temperature and doping

Phonon vs electron contributions

• At low temperatures, the thermal conductivity of intrinsic semiconductors is dominated by phonons
• As the temperature increases, the contribution of electrons to thermal conductivity becomes more significant due to the increased concentration of charge carriers
• The relative contributions of phonons and electrons to thermal conductivity can be estimated using models such as the Callaway model or the two-band model

Doping effects on thermal conductivity

• Doping semiconductors with impurities can significantly impact their thermal conductivity
• N-type doping (with electron donors) increases the concentration of electrons, enhancing the electronic contribution to thermal conductivity
• P-type doping (with electron acceptors) increases the concentration of holes, which can also contribute to thermal conductivity
• However, dopants can also act as scattering centers for phonons, reducing the phonon contribution to thermal conductivity

Bipolar thermal conductivity

• In narrow-bandgap semiconductors or at high temperatures, both electrons and holes can contribute to thermal conductivity
• The presence of both types of charge carriers leads to bipolar thermal conductivity
• Bipolar thermal conductivity arises from the ambipolar diffusion of electrons and holes, which can transport heat
• The bipolar contribution to thermal conductivity is often significant in thermoelectric materials and can limit their thermoelectric efficiency

Measuring thermal conductivity

• Various experimental techniques are used to measure the thermal conductivity of materials
• These methods can be broadly classified into steady-state and transient methods

Steady-state methods

• Measure thermal conductivity under conditions where the temperature distribution in the sample is time-independent
• Examples include the guarded hot plate method and the heat flow meter method
• Steady-state methods are suitable for materials with relatively high thermal conductivity and large sample sizes
• Require careful control of boundary conditions and minimization of heat losses

Transient methods

• Measure thermal conductivity by analyzing the temperature response of a sample to a time-dependent heat input
• Examples include the laser flash method, the 3ω method, and the transient plane source method
• Transient methods are suitable for materials with a wide range of thermal conductivities and can accommodate small sample sizes
• Require precise measurement of temperature changes and accurate modeling of heat transfer processes

Thermal conductivity spectroscopy

• A technique that measures the frequency-dependent thermal conductivity of materials
• Based on the analysis of the temperature response to a periodic heat input
• Provides information on the contributions of different phonon modes to thermal conductivity
• Useful for studying the phonon dynamics and scattering processes in materials
• Requires specialized experimental setups and data analysis techniques

Applications of thermal conductivity

• Understanding and controlling thermal conductivity is crucial for various technological applications
• Some key areas where thermal conductivity plays a critical role include thermoelectrics, thermal management, and thermal insulation

Thermoelectric materials

• Materials that can convert heat into electricity (Seebeck effect) or electricity into cooling (Peltier effect)
• Ideal thermoelectric materials have high electrical conductivity, low thermal conductivity, and a high Seebeck coefficient
• Low thermal conductivity is essential to maintain a large temperature gradient across the thermoelectric device
• Strategies to reduce thermal conductivity in thermoelectric materials include nanostructuring, alloying, and introducing phonon scattering centers

Heat sinks and thermal management

• Efficient removal of heat is crucial for the performance and reliability of electronic devices and systems
• Heat sinks are designed to dissipate heat from electronic components to the surrounding environment
• Materials with high thermal conductivity, such as copper and aluminum, are commonly used for heat sinks
• Advanced thermal management solutions, such as heat pipes and vapor chambers, rely on the efficient transport of heat through materials with high thermal conductivity

Thermal insulation materials

• Materials with low thermal conductivity are used for thermal insulation in various applications, such as buildings, refrigeration, and aerospace
• Examples of thermal insulation materials include fiberglass, cellulose, aerogels, and vacuum insulation panels
• Low thermal conductivity is achieved by minimizing the conduction, convection, and radiation of heat through the material
• Thermal insulation materials often have a porous or multi-layered structure to suppress heat transfer mechanisms
• The choice of thermal insulation material depends on factors such as the operating temperature range, mechanical properties, and cost