The crystal stabilities of ␣-and ␤-boron are studied theoretically by the density-functional calculations. The ground-state properties and thermodynamic properties are calculated by the pseudopotential method. These calculated thermodynamic properties include the effect of atomic disorder, observed experimentally, as well as the effect of phonons. The pressure dependence of the free energy is also studied. At zero temperature, it is found that ␣-boron is more stable than ␤-boron. This does not change even if the zero-point energy and atomic disorders are considered. The contribution of these effects to the energy is small at T = 0 K. However, these effects eventually cause a phase transition to ␤-boron at high temperatures. By considering the phonon contribution as the chief source of the temperature dependence of the free energy, 970 K is obtained as the transition temperature, which is in qualitative agreement with the experimental value of 1400 K. The difference between these values could be attributed to anharmonic effects. The effect of thermal expansion on the transition temperature is insignificant. At finite pressures, the stability of various polymorphs can be determined mainly using the atom density. The basic feature underlying all the above properties is that ␣-boron is dense, while ␤-boron is dilute. For ␤-boron, an energetic consideration shows that the disorder in the atom arrangement is inherent. The present calculations reveal a small change in bond length for specific intericosahedral bonds, which is caused by an atomic disorder.

The Raman spectra of numerous dodecaborides have been measured on high-quality single crystals at ambient conditions with high spectral resolution and signal-to-noise ratio. Besides the strong Raman-active modes, numerous Raman-inactive modes occur in the spectra, indicating distortions of the structures. Ab initio calculation of the phonon spectra on ZrB(12) excellently agrees with the experimental results. Force constants are theoretically calculated and force parameters are estimated from the Raman frequencies. The influence of the surface range on the Raman spectra is evident. The different isotopic effects (virtual crystal approximation, the polarization effect and the effect of isotopic disorder) on the phonon frequencies are determined, separated and discussed.

The thermoelectric power factor of SrTiO3 is unusually high with respect to its mobility and band gap. Good thermoelectrics usually have high mobility and a narrow band gap, but such properties are not found in SrTiO3. We have determined the mechanism behind the high power factor by calculating the transport coefficients. The key to understanding the power factor is that different effective masses contribute to different transport phenomena. The discrepancy between the effective mass for the conductivity and the thermoelectric power showed that the conductivity and thermoelectric power are conveyed by electrons with different effective masses in the Brillouin zone. Light electrons were responsible for the high conductivity, whereas heavy electrons were responsible for the high thermoelectric power. The high carrier concentrations of more than 1020 cm−3 did not reduce the thermoelectric power of SrTiO3 above the classical limit. This indicates that the electrons carrying the thermoelectric power were not degenerate. This is achieved by a decrease in the Fermi energy and the contribution of the heavy electrons to the Seebeck coefficient. The strong dielectric screening also contributed to the high power factor. The Coulomb scattering by ionized impurities, which would usually reduce the carrier mobility, was effectively screened. These results clarify the mechanism behind the contribution of different types of electrons, and show that high thermoelectric power does not necessarily reduce conductivity. Our findings provide a new direction for the band engineering of thermoelectric materials.

We have resolved long-standing discrepancies between the theoretical and experimental crystal structures of boron carbide B 13 C 2 . Theoretical studies predict that B 13 C 2 should be stoichiometric and have the highest symmetry of the boron carbides. Experimentally, B 13 C 2 is a semiconductor and many defect states have been reported, particularly in the CBC chain. Reconciling the disordered states of the chain, the chemical composition, and the lowest-energy state is problematic. We have solved this problem by constructing a structural model where approximately three-quarters of the unit cells contain (B 11 C)(CBC) and one-quarter of them contain (B 12 ) (B 4 ). This structural model explains many experimental results, such as the large thermal factors in x-ray diffraction and the broadening of the Raman spectra, without introducing unstable CBB chains. The model also solves the energy-gap problem. We show that there are many arrangements of these two types of unit cells, which are energetically almost degenerate. This demonstrates that boron carbides are well described by a geometrically frustrated system, similar to that proposed for β-rhombohedral boron.

The elastic constants of the icosahedral boron crystals have been studied by the formulation of Born and Huang. First of all, a technique of symmetry decomposition has been developed for general crystals possessing molecular units in order to see the relaxation mechanism by the internal shift. It is proven that if a librational mode is Raman active, which is often the case, the mode is able to relax the external strain considerably. For ␣ boron, when only central forces are assumed, the c 44 component completely vanishes. A shear strain 4 induces rotations of icosahedra, which cancel the shear strain completely. This gives a qualitative account for why this crystal is metastable. The rotations of icosahedra frequently happen in order to relax other types of strain too. This rotation-induced relaxation mechanism is looked upon as a special example of the above general property. The cancellation for 4 would remain in boron carbide, if only central forces are assumed, even though additional elements are introduced in the unit cell. In this case, the stability of the crystal has been ascribed to large noncentral forces, which emerge from the covalent bonds of the linear chain in the unit cell. Another way of stabilizing the crystal structure of ␣ boron is suggested: the surface contact of icosahedra, which is realized in the crystal of ␤ boron. In this family of crystals, the only direction in which a rotational motion is not induced is the z direction. The deformity of the icosahedron, instead, leads to an unexpected effect on the elasticity of boron carbide. The crystal is shown to be less stiff in the c axis than in the ab plane, despite the strongest interatomic forces being oriented parallel to the c axis. The rhombohedral site slightly deviates from the lattice vector, and this geometry gives rise to a great relaxation in the compression along the c axis.

Recent discoveries of supposedly pure α-tetragonal boron require to revisit its structure. The system is also interesting with respect to a new type of geometrical frustration in elemental crystals, which was found in β-rhombohedral boron. Based on density functional theory calculations, the present study has resolved the structural and thermodynamic characteristics of pure α-tetragonal boron. Different from β-rhombohedral boron, the conditions for stable covalent bonding (a band gap and completely filled valence bands) are almost fulfilled at a composition B 52 with two 4c interstitial sites occupied. This indicates that the ground state of pure α-tetragonal boron is stoichiometric. However, the covalent condition is not perfectly fulfilled because non-bonding ingap states exist that cannot be eliminated. The half occupation of the 4c sites yields a macroscopic amount of residual entropy, which is as large as that of β-rhombohedral boron. Therefore, α-tetragonal boron can be classified as an elemental crystal with geometrical frustration. Deviations from stoichiometry can occur only at finite temperatures. Thermodynamic considerations show that deviations δ from the stoichiometric composition (B 52+δ ) are small and positive. For reported high-pressure syntheses conditions δ is predicted to be about 0.1 to 0.2. An important difference between pure and C-or N-containing α-tetragonal boron is found in the occupation of interstitial sites: the pure form prefers to occupy the 4c sites, whereas in C-or N-containing forms a mixture of 2a, 8h, and 8i sites are occupied. The present article provides relations of site occupation, δ values, and lattice parameters, which enable us to identify pure α-tetragonal and distinguish the pure form from other ones.

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The vibrations of boron carbide have been studied by the valence force model. There have been so far several ambiguities in the interpretation of experimental spectra, which made theoretical study difficult. This paper solves the main difficulties, that is, characterization of the highest band and most of the Raman bands, providing reasonable interpretations for low-frequency modes. The band is ascribed to the anti-symmetric stretching of the linear chain. The necessity of large angle-bending forces of carbon atom arose for better fitting, which fact can be regarded as evidence for formation of strong covalent bonds around this atom. The present analysis also throws a new light on the Longuet-Higgins interpretation of the role of carbon incorporation. The coexistence of the strong covalent bonds of carbon atoms and the relatively weak icosahedral bonds results in such a balance of bonding that the intra-icosahedral and rhombohedral bonds are further weakened, and the covalent bonds of the carbon chain are instead reinforced.