A theoretical formalism to calculate the single crystal elastic constants for orthorhombic crystals from first principle calculations is described. This is applied for ${\mathrm{TiSi}}_2$ and we calculate the elastic constants using a full potential linear muffin-tin orbital method using the local density approximation (LDA) and generalized gradient approximation (GGA). The calculated values compare favorably with recent experimental results. An expression to calculate the bulk modulus along crystallographic axes of single crystals, using elastic constants, has been derived. From this the calculated linear bulk moduli are found to be in good agreement with the experiments. The shear modulus, Young’s modulus, and Poisson’s ratio for ideal polycrystalline ${\mathrm{TiSi}}_2$ are also calculated and compared with corresponding experimental values. The directional bulk modulus and the Young’s modulus for single crystal ${\mathrm{TiSi}}_2$ are estimated from the elastic constants obtained from LDA as well as GGA calculations and are compared with the experimental results. The shear anisotropic factors and anisotropy in the linear bulk modulus are obtained from the single crystal elastic constants. From the site and angular momentum decomposed density of states combined with a charge density analysis and the elastic anisotropies, the chemical bonding nature between the constituents in ${\mathrm{TiSi}}_2$ is analyzed. The Debye temperature is calculated from the average elastic wave velocity obtained from shear and bulk modulus as well as the integration of elastic wave velocities in different directions of the single crystal. The calculated elastic properties are found to be in good agreement with experimental values when the generalized gradient approximation is used for the exchange and correlation potential.

This review summarizes recent first-principles investigations of the electronic structure and magnetism of dilute magnetic semiconductors (DMSs), which are interesting for applications in spintronics. Details of the electronic structure of transition-metal-doped III-V and II-VI semiconductors are described, especially how the electronic structure couples to the magnetic properties of an impurity. In addition, the underlying mechanism of the ferromagnetism in DMSs is investigated from the electronic structure point of view in order to establish a unified picture that explains the chemical trend of the magnetism in DMSs. Recent efforts to fabricate high-$T_C$ DMSs require accurate materials design and reliable $T_C$ predictions for the DMSs. In this connection, a hybrid method (ab initio calculations of effective exchange interactions coupled to Monte Carlo simulations for the thermal properties) is discussed as a practical method for calculating the Curie temperature of DMSs. The calculated ordering temperatures for various DMS systems are discussed, and the usefulness of the method is demonstrated. Moreover, in order to include all the complexity in the fabrication process of DMSs into advanced materials design, spinodal decomposition in DMSs is simulated and we try to assess the effect of inhomogeneity in them. Finally, recent works on first-principles theory of transport properties of DMSs are reviewed. The discussion is mainly based on electronic structure theory within the local-density approximation to density-functional theory.

We report on ab initio calculations of the two-dimensional systems ${\text{MoS}}_2$ and ${\text{NbSe}}_2$, which recently were synthesized. We find that two-dimensional ${\text{MoS}}_2$ is a semiconductor with a gap which is rather close to that of the three-dimensional analog, and that ${\text{NbSe}}_2$ is a metal, which is similar to the three-dimensional analog of this compound. We further computed the electronic structure of the two-dimensional hexagonal (graphene-like) lattices of Si and Ge and compared them with the electronic structure of graphene. It is found that the properties related to the Dirac cone do not appear in the case of two-dimensional hexagonal germanium, which is metallic, contrary to two-dimensional hexagonal silicon, also known as silicene, which has an electronic structure very similar to the one of graphene, making them possibly equivalent.

The magnetoelectric behavior of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ has been explored on the basis of accurate density functional calculations. We are able to predict structural, electronic, magnetic, and ferroelectric properties of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ correctly without including any strong correlation effect in the calculation. Unlike earlier calculations, the equilibrium structural parameters are found to be in very good agreement with the experimental findings. In particular, the present calculation correctly reproduced experimentally observed elongation of cubic perovskitelike lattice along the [111] direction. At high pressure we predicted a pressure-induced structural transition from rhombohedral $\left(R3c\right)$ to an orthorhombic $\left(Pnma\right)$ structure. The total-energy calculations at expanded lattice show two lower energy ferroelectric phases (with monoclinic $Cm$ and tetragonal $P4mm$ structures), closer in energy to the ground-state phase. Spin-polarized band-structure calculations show that $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ will be an insulator in $A$- and $G$-type antiferromagnetic phases and a metal in $C$-type antiferromagnetic, ferromagnetic configurations, and in the nonmagnetic state. Chemical bonding in $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ has been analyzed using partial density of states, charge density, charge transfer, electron localization function, Born-effective-charge tensor, and crystal orbital Hamiltonian population analyses. Our electron localization function analysis shows that stereochemically active lone-pair electrons are present at the Bi sites which are responsible for displacements of the Bi atoms from the centrosymmetric to the noncentrosymmetric structure and hence the ferroelectricity. A large ferroelectric polarization of $88.7\mu \mathrm{C}∕{\mathrm{cm}}^2$ is predicted in accordance with recent experimental findings, but differing by an order of magnitude from earlier experimental values. The strong spontaneous polarization is related to the large values of the Born-effective charges at the Bi sites along with their large displacement along the [111] direction of the cubic perovskite-type reference structure. Our polarization analysis shows that partial contributions to polarization from the Fe and O atoms almost cancel each other and the net polarization present in $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ mainly $(>98\%)$ originates from Bi atoms. We found that the large scatter in experimentally reported polarization values in $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ is associated with the large anisotropy in the spontaneous polarization.

The five different elastic constants of all the hexagonal 4d transition metals (Y, Zr, Tc, and Ru) and the 5d transition metals Re and Os have been calculated by means of first-principles electronic-structure calculations using the full-potential linear muffin-tin orbital method. The calculated data agree with the experimental values within ∼30%. We demonstrate, using experimental data, that the hexagonal transition metals obey the Cauchy relations much better than the cubic ones. This is due to the fact that the shape of the density of states for the hexagonal materials retains its form to a larger extent, for all types of shears, than it does for the cubic metals. We introduce normalized elastic constants ${\mathrm{C}}_{\mathrm{ij}}^{\prime }$=${\mathrm{C}}_{\mathrm{ij}}$/B, where B is the bulk modulus, which show a regular behavior for the hexagonal transition metals, in contrast to the cubic transition metals, where large irregularities are observed. These regular as well as irregular behaviors are well reproduced by the full-potential calculations.

Conventional methods to calculate the thermodynamics of crystals evaluate the harmonic phonon spectra and therefore do not work in frequent and important situations where the crystal structure is unstable in the harmonic approximation, such as the body-centered cubic (bcc) crystal structure when it appears as a high-temperature phase of many metals. A method for calculating temperature dependent phonon spectra self-consistently from first principles has been developed to address this issue. The method combines concepts from Born’s interatomic self-consistent phonon approach with first principles calculations of accurate interatomic forces in a supercell. The method has been tested on the high-temperature bcc phase of Ti, Zr, and Hf, as representative examples, and is found to reproduce the observed high-temperature phonon frequencies with good accuracy.