Heat capacity and thermal transport diagnostics

For a conventional three-dimensional metal at temperatures well below relevant electronic and Debye scales, a useful molar model combines a linear electronic heat capacity with a cubic acoustic-phonon term. Plotting heat capacity divided by temperature against temperature squared separates intercept and slope. The electronic coefficient reflects a thermodynamic density of states, including appropriate quasiparticle renormalization in a Fermi-liquid interpretation. It is not the same weighted quantity as a Hall number or one photoemission intensity.

Magnetic excitations, nuclear or impurity Schottky anomalies, low-dimensional modes, glassy defects, and additional transitions can invalidate that two-term fit. Addenda from the platform and grease must be subtracted, the sample amount must be known, and thermal equilibration must be demonstrated. Experiments often measure heat capacity at constant pressure; comparison with a constant-volume model requires that the thermal-expansion correction be negligible or included. Changing the fit window is a physical stability test, not merely a way to improve a regression coefficient.

In a steady one-dimensional diffusive bar, heat current density is minus thermal conductivity times temperature gradient. A heater power and a measured temperature difference give conductance, while conductivity additionally requires length and cross section. The power entering the sample can differ from electrical heater power because of heat leaks through wires, radiation, and the mounting structure. A contact temperature drop can also make the apparent sample gradient too large and the inferred conductivity too small.

Use thermometers calibrated in the same field and temperature regime as the measurement, check linearity with heater power, and allow a stationary profile to develop. At very low temperatures the electronic and lattice subsystems or the thermometers can decouple, so the reported temperature may not represent every heat-carrying subsystem. Boundary-limited or ballistic transport can require a description beyond a bulk local conductivity. In a magnetic field, transverse heat flow and thermoelectric corrections can require a tensor treatment rather than a scalar bar formula.

For degenerate conventional charged quasiparticles dominated by elastic scattering, a low-temperature electronic Wiedemann–Franz benchmark relates thermal and electrical conductivity through the Sommerfeld Lorenz number. The thermal conductivity is the open-circuit electronic value; phonons and neutral excitations add heat without carrying the same electrical current. Inelastic scattering, multiband structure, finite-temperature thermoelectric effects, and nonlocal transport can alter a naive comparison. The benchmark is not an unrestricted identity for all quantum matter.

A residual intercept of thermal conductivity divided by temperature depends on how phonons and other contributions are extrapolated. A fitted intercept can change with boundary scattering or a mistaken temperature calibration, while a genuine electronic intercept need not determine a unique gap structure. Combine temperature and field trends with calorimetric entropy and independent gap probes. Shared geometry errors can partly cancel in a Lorenz ratio when exactly the same segment carries both currents, but mismatched contacts and different heat paths can destroy that cancellation.