2.2 Band structure approximation and electronic properties of materials
2.2.6 Charge neutralization and amphoteric defect model
In the surfaces or interfaces of practical material or heterostructured material systems, the perfect periodicity of lattice translational symmetry is often broken. The breaking symmetry of the crystals creates distinct states localized at the sur- face or interface, giving rise to the distribution of space charge. For example, the truncated surface of a material has atoms which are usually missed the nearest neighbors. This leads to surface reconstruction via atomic rearrangements (e.g.
atomic dangling bonds) and/or chemisorption to minimize the total energy. Thus, the associated lattice defects (e.g. atomic interstitials and vacancies) or other ex- trinsic impurities cause charge depletion or accumulation layers at the surface and interface of the materials.
Attempts to approach such a space charge states in electronic aspects, the complex band structure of a one-electron lattice within the nearly-free electron ap- proximation can be utilized solving the Schr¨odinger equation. In the Hamiltonian of such a model, a small periodic potential is used in terms of Bloch functions. This is added as a perturbation term satisfying the Bragg diffraction (Brillouin zone) boundary condition, 2ki·G = |G|2, where G is the reciprocal lattice vec- tor and ki = (π/a) + iq is wavevectors with real values. Solving the Schr¨odinger
equation within the nearly-free electron model together with the dispersion and pe- riodic potential opens an energy gap at the Brillouin zone boundary. Meanwhile, to describe the localized states, at which the periodicity of a crystal is broken, the imaginary part,iq, express the evanescent states that decay exponentially both into the vacuum and the bulk of the solid [113]. The solutions of the associated complex dispersion with real energies are present within the band gap of the materials, but are not able to be normalized in the bulk. Therefore, these states can be termed virtual gap states (ViGS) [114, 115]. The ViGS derive from the conduction and va- lence band states of the bulk, and thus their sign/nature depends on the Fermi level position relative to the branch point at which their character changes from predom- inantly donor-like close to the valence band to predominantly acceptor-like close to the conduction band [113]. In other words, the branch point energy of the ViGS is a state which has equal donor- and acceptor-character, and hence it is also described as the charge neutrality level (CNL). The CNL or branch-point energy is situated below the mid-gap energy atqmaxin the one-dimensional model. In order to predict the position of the CNL, many theoretical approaches have been developed. For example, empirical tight-binding calculations for many semiconductors have been performed and determined a linear relationship between the branch-point energy of the ViGS and dielectric band gap energy with a slop parameter 0.449±0.007 below the averaged mid-gap energy of the semiconductors. This was formed in good agree- ment with the one-dimensional model [113]. Later, Tersoff identified the averaged mid-gap energy, ( ¯Emid), using a semi-empirical method as [116]
¯
Emid = 1
2( ¯EC+ ¯EV). (2.58)
Based on this approach, the CNL is universal in most semiconductors. Thus, the role of the CNL is justification for competence to determine the preferential type of the semiconductors (either n-type or p-type). Unlike conventional semiconductors (e.g.Si and GaAs), where the CNL lies within the band gap [113, 117], most metal oxides have a large size and electronegativity mismatch between the metal cations (much larger, less electronegative atom) and the oxygen anion (smaller, high elec-
tronegative atom) [14, 17, 118]. These induce a single low lying conduction bands at the Γ-point, leading ton-type characteristics without intentional doping and the following difficulties for the realization of p-type conduction in many metal oxides. The origin of the n-type properties of the undoped metal oxides is primarily due to the formation of donor-like unintentional defects, e.g., cation interstitials, anion vacancies, and impurities and associated localized states near the CBM [50]. How- ever,n-type conductivity in the metal oxides is limited by increasing the prevalence of compensating defects as the Fermi level reaches the CNL in the conduction band. Such charge neutralization processes are associated with a self-compensation effect between the donor-like defects (dopants) and the acceptor-like defects (acceptors) in the surface or interface or the bulk of materials, based on an amphoteric model. This defect model, suggested by Walukiewicz, explains that variation in the forma- tion energy of preferential defects relative to oppositely charged defects is dependent on the Fermi level with respect to the CNL [14]. Namely, the formation energy for donor (acceptor) native defects increases (decreases) with increasing the Fermi level, leading to the formation of donor (acceptor) defects is most favourable as the Fermi level is below (above) the Fermi stabilization energy as shown in Fig. 2.12. This model can be applied to interpret the energetic of native defect formation and dop- ing limits in dilute semiconductors. Consequently, the concept and implication of the CNL and associated amphoteric defect formation are of primarily importance and widely applicable for understanding the electronic properties, bulk doing, and surface/interface space-charges in different semiconductor materials, and for design- ing/modifying their characteristics.