The discovery of superconductivity in the thallium barium calcium copper oxide (TBCCO) system generated intense interest and led to numerous structural studies and measurements of physical properties on these compounds. This chapter discusses brief description of the information obtained from specific-heat measurements on conventional superconductors, and a summary of the measurements on YBCO and their interpretation. Furthermore, analysis of the shape of the anomaly is always complicated by uncertainty in the subtraction of the background or lattice specific heat. The early specific-heat measurements on YBCO provided evidence of a bulk superconducting transition near 90 K, but the observed anomalies were relatively broad, with heights considerably lower than those observed in later experiments. The chapter discusses better specific-heat measurements on better samples, especially single crystals, should help to remove some of the ambiguities in the data.

Twenty two years after the opening of the field of high temperature superconductivity by Georg Bednorz and Alex Müller a new addition to this class of materials, namely the iron-pnictides and chalcogenides, perplexes the scientific community. Making use of the accumulated wisdom, a great deal of their physics has been “mapped” in record time using all available measurement techniques. Among these, specific-heat has been proven a powerful and unique technique in the study of physical properties of bulk materials, especially of superconductors. We report here specific-heat measurements on the nearly optimally doped iconic iron-arsenide superconductor Ba0.59K0.41Fe2As2. We use a new method, based on direct comparisons of α-model expressions for the electron contribution with the measured total specific heat, to extract the electron contribution. It circumvents the need in the conventional analyses for an independent, necessarily approximate, determination of the lattice contribution, which is subtracted from the total to obtain the electron contribution, and it eliminates the consequent uncertainties in the electron contribution. For Ba0.59K0.41Fe2As2 the electron density of states is comprised of contributions from two electron bands with superconducting-state energy gaps differing by a factor 3.8, with 77% coming from the band with the larger gap. The vortex-state specific heat suggests nodeless gaps. Comparison of the normal-state density of states with band-structure calculations shows an extraordinarily large effective mass enhancement, for which there is no precedent in simple metals and no theoretical explanation.

We report specific-heat measurements on Ba0.59K0.41Fe2As2, one of a family of high-T-c Fe-pnictide superconductors for which the pairing interaction is of special interest. We use a new method, based on direct comparisons of alpha-model expressions for the electron contribution with the measured total specific heat, to extract the electron contribution. It circumvents the need in the conventional analyses for an independent, necessarily approximate, determination of the lattice contribution, which is subtracted from the total to obtain the electron contribution, and it eliminates the consequent uncertainties in the electron contribution. For Ba0.59K0.41Fe2As2 the electron density of states is comprised of contributions from two electron bands with superconducting-state energy gaps differing by a factor 3.8, with 77% coming from the band with the larger gap. The vortex-state specific heat suggests nodeless gaps. Comparison of the normal-state density of states with band-structure calculations shows an extraordinarily large effective mass enhancement, for which there is no precedent in simple metals and no theoretical explanation.

We report measurements of the specific heat of Ba$_{0.59}$K$_{0.41}$Fe$_{2}$As$_{2}$, an Fe-pnictide superconductor with $T_c$ = 36.9 K, for which there are suggestions of an unusual electron pairing mechanism. We use a new method of analysis of the data to derive the parameters characteristic of the electron contribution. It is based on comparisons of ${\alpha}$-model expressions for the electron contribution with the total measured specific heat, which give the electron contribution directly. It obviates the need in the conventional analyses for an independent, necessarily approximate, determination of the lattice contribution, which is subtracted from the total specific heat to obtain the electron contribution. It eliminates the uncertainties and errors in the electron contribution that follow from the approximations in the determination of the lattice contribution. Our values of the parameters characteristic of the electron contribution differ significantly from those obtained in conventional analyses of specific-heat data for five similar hole-doped BaFe$_{2}$As$_{2}$ superconductors, which also differ significantly among themselves. They show that the electron density of states is comprised of contributions from two electron bands with superconducting-state energy gaps that differ by a factor 3.8, with 77$\%$ coming from the band with the larger gap. The variation of the specific heat with magnetic field is consistent with extended $s$-wave pairing, one of the theoretical predictions. The relation between the densities of states and the energy gaps in the two bands is not consistent with a theoretical model based on interband interactions alone. Comparison of the normal-state density of states with band-structure calculations shows an extraordinarily large effective mass enhancement, for which there is no precedent in similar materials and no theoretical explanation.

We report measurements of the specific heat of two samples of carbon-doped MgB2, Mg(B1−xCx)2, x = 0.05 and 0.1, in magnetic fields to μ0H = 9 T and at temperatures from ∼1 K to somewhat above the critical temperature for superconductivity for each sample. The carbon doping reduced the critical temperature from 39 K for MgB2 to 31.4 K and 19.7 K for the x = 0.05 and 0.1 samples, respectively. The results give the electron–phonon coupling and the electron density of states, including the individual contributions of the π and σ bands. These quantities are compared with theoretical calculations. The results also give the energy gaps on the π and σ bands, which are compared with other experimental determinations, and also with theoretical calculations that include predictions of the "merging" of the two gaps as a consequence of the band filling and increased interband scattering associated with doping.

Heat-capacity measurements on exceptionally high-quality samples of Ba1-xKxFe2As2 give the values of a number of parameters relevant to the electronic structure, the structural/magnetic transition in the undoped parent compound, and the superconducting transition in the near optimally doped, x = 0.41 sample. In BaFe2As2 the changes in the lattice and magnetic structure appear as a single, sharp first-order transition at 140 K; the Sommerfeld coefficient is gamma = 5.1 mJ K-2 mol(-1). For the superconducting, x = 0.41, sample, the residual density of electron states in the superconducting states is essentially zero. The specific-heat anomaly at T-c suggests extreme strong coupling, with Delta(0)/k(B)T(c) approximate to 2.6, and a normal-state Sommerfeld coefficient gamma(n) approximate to 45 mJ K-2 mol(-1).

Heat-capacity, X-ray diffraction, and resistivity measurements on a high-quality BaFe$_{2}$As$_{2}$ sample show an evolution of the magneto-structural transition with successive annealing periods. After a 30-day anneal the resistivity in the (ab) plane decreases by more than an order of magnitude, to 12 $\mu\Omega$cm, with a residual resistance ratio $\sim$36; the heat-capacity anomaly at the transition sharpens, to an overall width of less than K, and shifts from 135.4 to 140.2 K. The heat-capacity anomaly in both the as-grown sample and after the 30-day anneal shows a hysteresis of $\sim$0.15 K, and is unchanged in a magnetic field $\mu_{0}$H = 14 T. The X-ray and heat-capacity data combined suggest that there is a first order jump in the structural order parameter. The entropy of the transition is reported.

Na0.3CoO2 ∣1.3H2O progressively changes its properties with ambient temperature ageing from two-gap, layered superconductivity to charge-density-wave order as the number of O vacancies in the layers increase. The charge-density-wave order was theoretically predicted on the basis of band-structure calculations, independently of knowledge of the specific-heat measurements.

We report ambient-pressure magnetization, heat capacity, and thermal-expansion measurements of the ferromagnetic superconductor ${\text{UGe}}_{2}$ in high magnetic fields. An analysis of the magnetic heat capacity derived from both magnetization and specific-heat data shows that ${\text{UGe}}_{2}$ is well described in the framework of the molecular-field theory. Our heat-capacity and thermal-expansion results reveal a clear crossover regime, a feature that illustrates the proximity to the quantum critical end point of a first-order boundary between two different ferromagnetic phases. Furthermore, we show that the ferromagnetic contribution to these thermodynamic quantities can be split into two terms with distinct Gr\"uneisen parameters.

Na0.3CoO2 center dot 1.3H(2)O progressively changes its properties with ambient temperature ageing from two-gap, layered superconductivity to charge-density-wave order as the number of O vacancies in the layers increase. The charge-density-wave order was theoretically predicted on the basis of band-structure calculations, independently of knowledge of the specific-heat measurements.

We report ambient-pressure magnetization, heat capacity, and thermal-expansion measurements of the ferromagnetic superconductor UGe2 in high magnetic fields. An analysis of the magnetic heat capacity derived from both magnetization and specific-heat data shows that UGe2 is well described in the framework of the molecular-field theory. Our heat-capacity and thermal-expansion results reveal a clear crossover regime, a feature that illustrates the proximity to the quantum critical end point of a first-order boundary between two different ferromagnetic phases. Furthermore, we show that the ferromagnetic contribution to these thermodynamic quantities can be split into two terms with distinct Gruumlneisen parameters.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi­ bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer­ ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom­ mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Three samples of Na0.3CoO2⋅1.3H2O that differed in sample age showed different ordering: after three and five days, superconductivity with T c∼4.5 K but different superconducting condensates; after 40 days, no superconductivity but charge-density wave order at T CDW∼7 K. A pair-breaking action that progresses with sample age and acts preferentially in one of two electron bands produces the changes in the superconducting condensate, and ultimately destroys the superconductivity. Theoretical calculations predicted the charge-density wave transition.

The specific heat of Na0.3CoO2·1.3H2O gives evidence of two electron bands with comparable densities of states and different superconducting-state energy gaps. A non-magnetic pair-breaking action, which acts preferentially in the band with the smaller gap, progresses with sample age. The two bands and non-magnetic pair breaking have implications for possible pairing mechanisms; the effects of the pair breaking constitute the “sample dependence” of the properties of this material. In concert with structural changes reported by others, the specific heat suggests dual roles for O vacancies —tuning the carrier concentration to favor superconductivity, and pair breaking to destroy it.

The specific heat of three samples of Na0.3CoO2 center dot 1.3H(2)O shows an evolution of the superconductivity and its eventual disappearance with increasing sample age. The specific heat of two superconducting samples is characteristic of a superconductor with two energy gaps, which implies contributions of two electron bands to the Fermi surface. The changes in the specific heat are associated with a nonmagnetic pair-breaking action that progresses with sample age and acts preferentially in the band with the smaller gap to produce an increasing "residual" electron density of states and a shift in the relative contributions of the bands to the superconducting condensate. For the nonsuperconducting sample the pair breaking has weakened the superconducting-state electron pairing to the point that it has given way to a competing order. The similarity of the time scale for these changes to that recently reported for the formation of 0 vacancies suggests a relation between the two effects and the identification of the 0 vacancies as the pair.-breaking scattering centers. Together, these effects provide an understanding, of the strong sample dependence of the properties of this material. They also suggest an unusual competition between two effects of the 0 vacancies: enhancement of the superconductivity at low concentrations by adjusting the carrier concentration and destruction of the superconductivity at high concentrations by pair breaking. Comparison of the coefficient of the normal-state conduction-electron specific heat, gamma(n) = 16.1 mJ K-2 mol(-1), with band-structure calculations supports the existence of the controversial e'(g) hole pockets in the Fermi surface, in addition to the well established a(1g) surface. The onset of the transition to the vortex state is independent of magnetic field, suggesting the presence of unusually strong fluctuation effects. The specific-heat results and their implications for band structure and symmetry of the superconducting-state order parameter are compared with other experimental and theoretical results.