The MN+1AXN phases: A new class of solids

The Lorenz number, which was assumed to be a universal factor, was later found to be deviating from its theoretical Sommerfeld value. Our understanding of the physics of the Weidemann-Franz-Lorenz law is based on a large number of investigations. In view of its importance in the understanding of thermal and electrical conductivity (particularly the electronic contribution), an attempt has been made to review the experimental determination of Lorenz number in metals, semi-metals, alloys and degenerate semiconductors.

Experimental determinations of the Lorenz number

The heat capacities, thermal-expansion coefficients, thermal and electrical conductivities of Nb2AlC (actual Nb:Al:C mole fractions: 0.525±0.005, 0.240±0.002, and 0.235±0.005, respectively), Ti2AlC and (Ti, Nb)2AlC (actual Ti:Nb:Al:C mole fractions: 0.244±0.005, 0.273±0.005, 0.240±0.003, and 0.244±0.005, respectively) were measured as a function of temperature. These ternaries are good electrical conductors, with a resistivity that increases linearly with increasing temperatures. The resistivity of (Ti, Nb)2AlC is higher than the other members, indicating a solid-solution scattering effect. The thermal-expansion coefficients, in the 25 °C to 1000 °C temperature range, are comparable and fall in the narrow range of 8.7 to 8.9 × 10−6 K−1, with that of the solid solution being the highest. They are all good conductors of heat, with thermal conductivities in the range between 15 to 45 W/m K at room temperature. The electronic component of the thermal conductivity is the dominant mechanism at all temperatures for Nb2AlC and (Ti, Nb)2AlC. The conductivity of Ti2AlC, on the other hand, is high because the phonon contribution to the conductivity is nonnegligible.

Thermal and electrical properties of Nb2AlC, (Ti, Nb)(2)AlC and Ti2AlC

Heat energy can be transmitted through solids via electrical carriers (electrons or holes), lattice waves (phonons), electromagnetic waves, spin waves, or other excitations. In metals electrical carriers carry the majority of the heat, while in insulators lattice waves are the dominant heat transporter. Normally, the total thermal conductivity can be written as a sum of all the components representing various excitations: 1⁄4 X ; ð1Þ

Theory of Thermal Conductivity

Transition-metal carbides☆

Phonon Scattering by Conduction Electrons and by Lattice Vacancies in Carbides of the Transition Metals

Transition metal carbides, such as TiC, exhibit very low thermal conductivities (K) at low temperatures. The lattice conductivity at such temperatures is severely limited by phonon scattering by conduction electrons. At high temperatures, point-defect scattering of phonons by carbon vacancies in the non-stoichiometric compounds dominate K. Scattering of electrons by the vacancies limits the electronic contribution. Above ~1000oC the thermal conductivity of the carbides is increased by ambipolar diffusion.

Thermal Conductivity of Transition Metal Carbides

The unusually low lattice thermal conductivity of the transition-metal carbides at low temperatures has been attributed to the scattering of phonons by conduction electrons. An experiment which confirms this interpretation is reported. Thermal conductivity measurements on single-crystal Nb${\mathrm{C}}_{0.96}$ through its superconducting critical temperature (9.8\ifmmode^\circ\else\textdegree\fi{}K) show that the lattice component of the thermal conductivity, ${K}_{l3}$, increases greatly below ${T}_{c}$ because of decreased phonon-electron scattering. The maximum increase in lattice thermal conductivity from this effect occurs at 3\ifmmode^\circ\else\textdegree\fi{}K with $\frac{{K}_{\mathrm{ls}}}{{K}_{\mathrm{ln}}}$ equal to 160. The theories of Bardeen, Rickayzen, and Tewordt and of Klemens and Tewordt for the influence of electrons and point defects on lattice conductivity provide a quantitative interpretation of the effect. The best fit to the Nb${\mathrm{C}}_{0.96}$ data is for $\frac{2\ensuremath{\epsilon}(o)}{k{T}_{c}}=4.0$. The behavior of Nb${\mathrm{C}}_{0.96}$ is in contrast to that of Nb${\mathrm{C}}_{0.76}$, which remains in the normal state throughout the temperature interval studied and shows no increase in thermal conductivity.

Lattice Thermal Conductivity of Superconducting Niobium Carbide

The first estimate of the superconducting energy gap in a transition-metal carbide was recently made from thermal conductivity data on Nb${\mathrm{C}}_{0.96}$ The quantity $\frac{2\ensuremath{\epsilon}(0)}{k{T}_{c}}$ was treated as an adjustable parameter in fitting the data to Bardeen-Rickayzen-Tewordt (BRT) theory, and a good fit was found for $2\ensuremath{\epsilon}(0)k{T}_{c}=4.0$. The resulting value of the gap energy has now been verified by tunneling measurements using both probe and thin-film techniques. The best value is $2\ensuremath{\epsilon}(0)=3.2\ifmmode\pm\else\textpm\fi{}0.1$ meV. The result supports the applicability of BRT theory to the transitionmetal carbides having low vacancy concentrations and demonstrates that under favorable circumstances, which may be unique to nonstoichiometric compounds, thermal conductivity data can be used to evaluate the gap energy of a superconductor.

L. G. Radosevich

Papers

Phonon Scattering by Conduction Electrons and by Lattice Vacancies in Carbides of the Transition Metals

Thermal Conductivity of Transition Metal Carbides

Lattice Thermal Conductivity of Superconducting Niobium Carbide

Gap energy of superconducting niobium carbide.

Low‐Temperature Thermal Conductivity of Uranium Nitride