In the past ten years we have witnessed a revival of, and subsequent rapid expansion in, the research on zinc oxide (ZnO) as a semiconductor. Being initially considered as a substrate for GaN and related alloys, the availability of high-quality large bulk single crystals, the strong luminescence demonstrated in optically pumped lasers and the prospects of gaining control over its electrical conductivity have led a large number of groups to turn their research for electronic and photonic devices to ZnO in its own right. The high electron mobility, high thermal conductivity, wide and direct band gap and large exciton binding energy make ZnO suitable for a wide range of devices, including transparent thin-film transistors, photodetectors, light-emitting diodes and laser diodes that operate in the blue and ultraviolet region of the spectrum. In spite of the recent rapid developments, controlling the electrical conductivity of ZnO has remained a major challenge. While a number of research groups have reported achieving p-type ZnO, there are still problems concerning the reproducibility of the results and the stability of the p-type conductivity. Even the cause of the commonly observed unintentional n-type conductivity in as-grown ZnO is still under debate. One approach to address these issues consists of growing high-quality single crystalline bulk and thin films in which the concentrations of impurities and intrinsic defects are controlled. In this review we discuss the status of ZnO as a semiconductor. We first discuss the growth of bulk and epitaxial films, growth conditions and their influence on the incorporation of native defects and impurities. We then present the theory of doping and native defects in ZnO based on density-functional calculations, discussing the stability and electronic structure of native point defects and impurities and their influence on the electrical conductivity and optical properties of ZnO. We pay special attention to the possible causes of the unintentional n-type conductivity, emphasize the role of impurities, critically review the current status of p-type doping and address possible routes to controlling the electrical conductivity in ZnO. Finally, we discuss band-gap engineering using MgZnO and CdZnO alloys.

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Fundamentals of zinc oxide as a semiconductor

We have performed a comprehensive first-principles investigation of native point defects in ZnO based on density functional theory within the local density approximation (LDA) as well as the $\mathrm{LDA}+U$ approach for overcoming the band-gap problem. Oxygen deficiency, manifested in the form of oxygen vacancies and zinc interstitials, has long been invoked as the source of the commonly observed unintentional $n$-type conductivity in ZnO. However, contrary to the conventional wisdom, we find that native point defects are very unlikely to be the cause of unintentional $n$-type conductivity. Oxygen vacancies, which have most often been cited as the cause of unintentional doping, are deep rather than shallow donors and have high formation energies in $n$-type ZnO (and are therefore unlikely to form). Zinc interstitials are shallow donors, but they also have high formation energies in $n$-type ZnO and are fast diffusers with migration barriers as low as $0.57\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$; they are therefore unlikely to be stable. Zinc antisites are also shallow donors but their high formation energies (even in Zn-rich conditions) render them unlikely to be stable under equilibrium conditions. We have, however, identified a different low-energy atomic configuration for zinc antisites that may play a role under nonequilibrium conditions such as irradiation. Zinc vacancies are deep acceptors and probably related to the frequently observed green luminescence; they act as compensating centers in $n$-type ZnO. Oxygen interstitials have high formation energies; they can occur as electrically neutral split interstitials in semi-insulating and $p$-type materials or as deep acceptors at octahedral interstitial sites in $n$-type ZnO. Oxygen antisites have very high formation energies and are unlikely to exist in measurable concentrations under equilibrium conditions. Based on our results for migration energy barriers, we calculate activation energies for self-diffusion and estimate defect-annealing temperatures. Our results provide a guide to more refined experimental studies of point defects in ZnO and their influence on the control of $p$-type doping.

Native point defects in ZnO

A single configuration model containing nonorthogonal magnetic orbitals is developed to represent the important features of the antiferromagnetic state of a transition metal dimer. A state of mixed spin symmetry and lowered space symmetry is constructed which has both conceptual and practical computational value. Either unrestricted Hartree–Fock theory or spin polarized density functional theory, e.g., Xα theory, can be used to generate the mixed spin state wave function. The most important consequence of the theory is that the Heisenberg exchange coupling constant J can be calculated simply from the energies of the mixed spin state and the highest pure spin multiplet.

Valence bond description of antiferromagnetic coupling in transition metal dimers

The surface and materials science of tin oxide

An approximate Xα functional is proposed from which the charge density fitting equations follow variationally. LCAO Xα calculations on atomic nickel and diatomic hydrogen show the method independent of the fitting (auxiliary) bases to within 0.02 eV. Variational properties associated with both orbital and auxiliary basis set incompleteness are used to approach within 0.2 eV the Xα total energy limit for the nitrogen molecule.

On some approximations in applications of Xα theory

Publisher Summary The limitations of applying an ab initio linear combination of atomic orbitals (LCAO) methods to complex molecules and solids are the size of the basis sets and the number of multicenter integrals or equivalent Hartree-Fock matrix elements In the self-consistent field (SCF)-Xα scattered-wave model that is also a first-principle technique, there is no basis-set problem because Schrodinger's equation for an Xα potential is numerically integrated There are no multicenter integrals and the model is practicable in both spin-restricted and spin-unrestricted forms for polyatomic systems of considerable stereochemical complexity The SCF-Xα scattered-wave technique uses only a small fraction of the computer time required by an ab initio Hartree–Fock LCAO method The applications of the scattered-wave method to polyatomic molecules and crystals are concerned with the generation of one-electron energy and wave functions While an SCF-Xα one-electron analysis leads to an accurate quantitative description of many chemical and physical properties, it is also very important to determine the total many-electron energy As the present method leads to a rapidly convergent numerical representation of the orbital wave functions, the accuracy of the theoretical model can easily be improved via perturbation theory, when necessary Finally, the original theoretical formalism can be extended to more general forms of superposed-atom Xα potentials by means of the generalized scattered-wave theory

Scattered-Wave Theory of the Chemical Bond

This paper describes a practical self-consistent-field (SCF) method of calculating electronic energy levels and eigenfunctions, adapted for polyatomic molecules and solids. The one-electron Schr\"odinger equation is set up for a so-called "muffin-tin" approximation to the true potential, spherically symmetrical within spheres surrounding the various nuclei, constant in the region between the spheres, spherically symmetrical outside a sphere surrounding the molecule. The method of solving this equation is a multiple-scattering method, equivalent to the Korringa-Kohn-Rostoker (KKR) method often used for crystals. Once the eigenfunctions and eigenvalues of this problem are determined, one assumes that the orbitals of lowest eigenvalue are occupied, up to a Fermi level. From the resulting charge densities, one can compute a total energy, using a statistical approximation for the exchange correlation. This approximation has an undetermined factor $\ensuremath{\alpha}$ (whence the name $X\ensuremath{\alpha}$ method). The spin orbitals and occupation numbers are varied to minimize this total energy, resulting in one-electron equations. The value of $\ensuremath{\alpha}$ for an isolated atom is determined by requiring that the total energy, using the statistical approximation, should equal the precise Hartree-Fock energy. This leads to very accurate spin orbitals. In a molecule or crystal, one uses the $\ensuremath{\alpha}'\mathrm{s}$ characteristic of the various atoms within the atomic spheres, and a suitable average in the region between. The computer programs for making these self-consistent calculations, for such radicals and polyatomic molecules as S${\mathrm{O}}_{4}^{\ensuremath{-}2}$, Cl${\mathrm{O}}_{4}^{\ensuremath{-}}$, Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$, and S${\mathrm{F}}_{6}$ have been worked out and calculations made. They are more than 100 times as fast as comparable programs using the LCAO (linear-combination-of-atomic-orbitals) method, and the results appear to be in better agreement with experiment than such LCAO results. For calculating the frequencies of optical transitions, one must make a self-consistent calculation, not for the initial or final state, but for what we call the transition state, in which occupation numbers are halfway between the initial and final states. Then it can be proved that the differences of eigenvalues of the $X\ensuremath{\alpha}$ method are more accurate than Hartree-Fock energy values, in that they take account of the modification or relaxation of the orbitals in going from the initial to the final states. These transition states, for a crystal, involve a localized perturbation at the site of the excited atom. The multiple-scattering method is adapted to the use of such perturbed crystals, as well as to isolated molecules, and to perfect crystals. It results, in such problems as x-ray absorption, in the use of localized orbitals rather than bandlike functions. The method is adapted to the calculation of magnetic problems, by use of a spin-polarized version of the method. The method can also be used for calculations of cohesive energy of crystals, and has been used successfully for several types of metals. Unlike most other SCF methods, long-range correlation is automatically included, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.

Self-Consistent-Field X α Cluster Method for Polyatomic Molecules and Solids

A theoretical description of the chemical bonding of transition metals and other complex metal atoms in molecules and solids has been formulated in terms of a spin-unrestricted self-consistent-field cluster model. This model permits the accurate calculation, from first principles, of electronic energy levels and wave functions for the cluster, but requires relatively little computer time. The cluster, which may be a free polyatomic molecule, part of a macromolecule, or a polyatomic complex in an ordered or disordered solid, is geometrically partitioned into contiguous atomic, interatomic, and extramolecular regions. The one-electron Schr\"odinger equation is numerically integrated within each region in the partial-wave representation for spherically averaged and volume-averaged potentials which include Slater's $X\ensuremath{\alpha}$ statistical approximation to exchange correlation. The wave functions and their first derivatives are joined continuously throughout the cluster via multiple-scattered-wave theory similar to that developed originally by Korringa. The effects of a particular environment on the cluster are described by boundary conditions, e.g., the matching of the solutions of Schr\"odinger's equation in the extramolecular region to those within the atomic regions at an artificial spherical boundary surrounding the entire cluster. This numerical procedure is repeated, using the wave functions obtained at each iteration to generate a charge density and new potential, until self-consistency is attained. The model is illustrated for the tetrahedrally coordinated permanganate ion (Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$) in the stabilizing field of a typical crystalline environment KMn${\mathrm{O}}_{4}$. The spin-unrestricted cluster calculation leads to a ground-state electronic structure of Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$ which is consistent with the observed Van Vleck paramagnetism of permanganate crystals. Computer-generated contour maps of the cluster wave functions and charge densities are presented, showing the formation of directed chemical bonds. Both $\ensuremath{\pi}"$ and "$\ensuremath{\sigma}$ bonding" between oxygen $2p$ electrons and manganese $3d$ electrons are shown to be important. In conjunction with Slater's transition-state theory, which accurately describes the effects of orbital relaxation in optical transitions, the cluster model is used to interpret the measured optical properties of permanganate crystals. The characteristic purple color of KMn${\mathrm{O}}_{4}$ is shown to originate from the induced transfer of electrons between the oxygen ligands and central manganese atom in the Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$ cluster. Finally, with the use of larger clusters, the theoretical model is extended to the chemical bonding of transition-metal impurities in semiconductors and transition metals which are the biologically active centers of certain enzymes and proteins.

Chemical Bonding of a Molecular Transition-Metal Ion in a Crystalline Environment

The theoretical foundation of the self‐consistent statistical exchange (X α) multiple scattering method based on overlapping atomic spheres is presented. After discussing the principal assumptions and approximations, the theory is illustrated by a study of the electronic structure of the TCNQ molecule. Satisfactory results are obtained for the energy level structure and electronic charge distribution if the atomic sphere radii are chosen in accordance with the following guidelines: (a) the calculated first ionization energy should agree with experiment; and (b) the calculated virial coefficient should be equal to −2. The present work suggests that the overlapping sphere model can be used to obtain realistic electronic structures for large planar organic molecules as well as for arrays of such molecules.

Multiple scattering method based on overlapping atomic spheres with application to the TCNQ molecule

The electronic structure and optical properties of a red phosphor, ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$:Eu, have been studied using the first-principles molecular-orbital and band-structure methods. Using the calculated one-electron energy levels, several properties of the host material and properties of the phosphor based on host-impurity interactions have been explained. However, it has been found that the luminescence properties of ${\mathrm{Eu}}^{3+}$ require the spin-orbit effects to be included in the computational methods.

Keith H. Johnson

Papers

Scattered-Wave Theory of the Chemical Bond

Self-Consistent-Field X α Cluster Method for Polyatomic Molecules and Solids

Chemical Bonding of a Molecular Transition-Metal Ion in a Crystalline Environment

Multiple scattering method based on overlapping atomic spheres with application to the TCNQ molecule

Electronic structure and optical properties of europium-activated yttrium oxide phosphor.