10.2 Specific Heat and Thermal Conductivity

Specific heat and thermal conductivity are crucial tools for understanding superconductors. These measurements reveal key information about electronic and phonon behavior, helping scientists identify phase transitions and characterize superconducting properties.

By analyzing specific heat jumps and temperature dependence, researchers can determine critical temperatures, gap symmetries, and competing orders. Thermal conductivity measurements provide insights into electron and phonon contributions, scattering mechanisms, and unconventional superconductor behavior.

Specific Heat for Superconducting Materials

Specific Heat Measurements and Superconductivity

• Specific heat measures the amount of heat required to raise the temperature of a material by one degree Celsius per unit mass
• In superconductors, specific heat measurements provide information about the electronic and phonon contributions to the total heat capacity
• The electronic specific heat in superconductors is strongly suppressed below the critical temperature (Tc) due to the formation of Cooper pairs, resulting in a characteristic jump in the specific heat at Tc
  • The magnitude of the specific heat jump at Tc relates to the superconducting gap and the density of states at the Fermi level

Advanced Specific Heat Analysis in Superconductors

• The temperature dependence of the specific heat below Tc reveals information about the symmetry of the superconducting order parameter and the presence of nodal structures
  • For example, an exponential decay in specific heat suggests an s-wave symmetry, while a power-law behavior indicates a d-wave symmetry
• Specific heat measurements can detect the presence of multiple superconducting gaps or other exotic phenomena, such as the coexistence of superconductivity and magnetism
  • Additional peaks or anomalies in the specific heat data may indicate the existence of competing orders, such as charge density waves or magnetic ordering

Interpreting Specific Heat Data

Identifying Phase Transitions and Superconducting Properties

• The specific heat of a superconductor exhibits a sharp jump at the critical temperature (Tc), indicating the onset of superconductivity and the formation of Cooper pairs
• The magnitude of the specific heat jump at Tc is proportional to the superconducting gap and the density of states at the Fermi level, providing insights into the strength of the superconducting pairing
• The temperature dependence of the specific heat below Tc distinguishes between different types of superconductors, such as s-wave (exponential decay) or d-wave (power-law behavior) symmetries
• The presence of additional peaks or anomalies in the specific heat data may indicate the existence of multiple superconducting gaps or competing orders

Analyzing Normal-State Specific Heat and Theoretical Modeling

• The analysis of the normal-state specific heat above Tc provides information about the electronic and phonon contributions, as well as the presence of strong electron-phonon coupling or other many-body effects
  • The electronic contribution to the specific heat is typically linear with temperature, while the phonon contribution follows a T^3 dependence at low temperatures
• Comparing the measured specific heat data with theoretical models, such as the BCS theory or its extensions, helps determine the microscopic parameters of the superconducting state
  • These parameters include the coupling strength, the density of states, and the superconducting gap symmetry
• Fitting the specific heat data to appropriate models allows for the extraction of important quantities, such as the Sommerfeld coefficient (γ) and the Debye temperature (ΘD)

Thermal Conductivity in Superconductors

Principles and Techniques for Thermal Conductivity Measurements

• Thermal conductivity measures the ability of a material to conduct heat and is determined by the contributions from electrons and phonons
• In superconductors, the electronic contribution to thermal conductivity is strongly suppressed below Tc due to the formation of Cooper pairs, while the phonon contribution remains relatively unchanged
• The thermal conductivity of superconductors is typically measured using steady-state methods, such as the four-probe technique, where a temperature gradient is applied across the sample and the resulting heat flow is measured
  • Transient methods, such as the 3ω method or the time-domain thermoreflectance (TDTR) technique, can also be used, particularly for thin films or nanostructured materials
• The choice of measurement technique depends on factors such as the sample geometry, the temperature range of interest, and the desired accuracy and resolution

Factors Influencing Thermal Conductivity Measurements

• Proper sample preparation, including the control of sample dimensions, surface quality, and electrical contacts, is crucial for accurate thermal conductivity measurements
  • Factors such as sample purity, the presence of defects or impurities, and the influence of magnetic fields or other external perturbations need to be carefully considered
• The interpretation of thermal conductivity data requires accounting for various scattering mechanisms, such as electron-phonon scattering, impurity scattering, and boundary scattering
• The application of magnetic fields can significantly alter the thermal conductivity of superconductors by breaking Cooper pairs and introducing additional scattering channels for quasiparticles

Thermal Conductivity vs Temperature in Superconductors

Temperature Dependence of Thermal Conductivity

• In conventional superconductors, the electronic contribution to thermal conductivity is strongly suppressed below Tc due to the formation of Cooper pairs, leading to a rapid decrease in thermal conductivity with decreasing temperature
• The residual thermal conductivity at low temperatures is primarily determined by the phonon contribution, which can be influenced by factors such as sample quality, impurity scattering, and boundary scattering
  • The temperature dependence of the phonon contribution to thermal conductivity typically follows a T^3 behavior at low temperatures
• The temperature dependence of the thermal conductivity above Tc provides insights into the scattering mechanisms and the strength of electron-phonon coupling in the normal state
  • The electronic contribution to thermal conductivity in the normal state is often described by the Wiedemann-Franz law, which relates the thermal conductivity to the electrical conductivity through the Lorenz number

Unconventional Superconductors and Theoretical Modeling

• In unconventional superconductors, such as high-Tc cuprates or iron-based superconductors, the temperature dependence of thermal conductivity may exhibit additional features, such as a peak or a plateau
  • These features can be related to the presence of nodal structures in the superconducting gap or competing orders, such as charge density waves or magnetic ordering
• The analysis of the thermal conductivity data in conjunction with other measurements, such as electrical resistivity or specific heat, helps disentangle the contributions from different quasiparticle excitations and provides a more comprehensive understanding of the superconducting state
• Theoretical models, such as the BCS theory or its extensions, can be used to interpret the temperature dependence of thermal conductivity and extract relevant parameters
  • These parameters include the superconducting gap, the quasiparticle mean free path, and the electron-phonon coupling strength
• Comparing the experimental data with theoretical predictions allows for the validation of existing models and the identification of novel phenomena in superconducting materials