Intrinsic electronic disordering defects

In order to exclude the influence of gaseous phase at this stage, it is essential to take into consideration a simple example (e.g., silicon) as a semiconducting material with vacancies. Assuming that possible defects of the crystal structure are vacancies and electron defects, the processes of intrinsic electronic disordering, vacancy formation, and vacancy ionization can be written, respectively, as ... 

Let us assume that AH2 > AHi > AH3, as the energy of atomic disorder is normally larger than that needed for intrinsic electronic disordering in turn, the latter is higher with respect to the energetic effects related to defect charging (see Figures 3.3 and 3.4). Under this assumption, it is possible to consider two alternative approximations. 

Perfect electronic order is achieved only at a temperature of OK, where all electrons are in the lowest possible energy levels under the constraint of the Pauli exclusion principle. Any excitation of electrons from their ground state to higher energy levels results in electronic disorder. In ceramics, however, intrinsic electronic disorder refers to the formation of free electrons in the conduction band and holes in the valence band. An intrinsic electronic defect thus consists of a free electron in the conduction band and a free electron hole in the valence band. The concentrations of free electrons (e ) and electron holes (h ) are determined by the band gap and temperature. According to Fermi statistics, the probability of an electron occupying an energy level E, P(E), is expressed as... 

In view of the many types of point defects that may be formed in inorganic compounds and that each type of defect may have varying effective charge, numerous defect reactions may in principle be formulated. In the following, a few simple cases will be treated as examples. First, we will consider defect stmcture situations in stoichiometric compounds (Schottky, Frenkel and intrinsic electronic disorders) and then defect structure situations in nonstoichiometric oxides will be illustrated. Finally, examples of defect reactions involving foreign elements will be considered. 

In this situation the concentration of defect electrons decreases and that of holes increases with increasing O The two terms cross at n = p = K], and assuming that Kj is independent of the doping, this is necessarily the same level as that of the intrinsic electronic disorder in the pure, undoped material. At this point also the doped material may be said to be stoichiometric it contains oxygen vacancies exactly matching the presence of the lower-valent cation (acceptor) If we consider the doped material to consist of M2O3 and MIO, both these constituents are stoichiometric, and the additional presence of reduced and oxidised states (n and p) is effectively zero at this point. The doped oxide can be said to be stoichiometric with respect to the valence of its constituents. 

The electrical properties of the titanate-based pyrochlores can be described by point defect models in which the acceptor (A) and donor (D) impurities are compensated by oxide ion vacancies, or oxide ion interstitials, respectively. The principal defect reactions inclnde the redox reaction, Equation (5.67), the Frenkel disorder, Equation (5.57), dopant ionization, intrinsic electronic disorder, Equation (5.60), and the electroneutrality relation. For these compounds the total electroneutrality condition given by Equation (5.55) is, on the one hand, reduced, taking Equation (5.57) into account, and is, on the other hand, extended to include acceptor and donor impurities. In addition, defect association is included explicitly. 

The defect structure of these materials is far from simple, with a number of coincident defect equilibria contributing to the observed behavior. First, we must consider the intrinsic defect processes for a hypothetical mixed conductor Lai xSrxBOsi, which undergoes Schottky disorder together with intrinsic electronic disorder. Following an analysis given in [16] ... 

The compound will be stoichiometric, with an exact composition of MX10ooo when the number of metal vacancies is equal to the number of nonmetal vacancies. At the same time, the number of electrons and holes will be equal. In an inorganic compound, which is an insulator or poor semiconductor with a fairly large band-gap, the number of point defects is greater than the number of intrinsic electrons or holes. To illustrate the procedure, suppose that the values for the equilibrium constants describing Schottky disorder, Ks, and intrinsic electron and hole numbers, Kc, are... 

Fig. 2. (a) Sketch of the relations between defect concentrations and partial pressure (Brouwer diagram) of a pure oxide MO In regime II the intrinsic Schottky disorder determines the concentration, whereas in I and III non-stoichiometry prevails, (b) Dependence of the hole and electron concentration on the frozen-in oxygen vacancy concentration in a negatively (acceptor) doped oxide. 

We have formulated defect reactions which describe intrinsic ionic and electronic disorder, nonstoichiometry, variable ionisation of point defects, and substitutional dissolution of aliovalent cations and ions and anions. Ahovalent elements may be compensated by electronic defects or by point defects, of which the former involve red-ox-reactions. 

While intrinsic disorder of the Schottky, Frenkel, or anti-Frenkel type frequently occurs in binaiy metal oxides and metal halides, i.e., Equations (5.1), (5.3), and (5.5), Schottky disorder is seldomly encountered in temaiy compounds. However, in several studies Schottky disorder has been proposed to occur in perovskite oxides. Cation and anion vacancies or interstitials can occur in ternary compounds, but such defect stractures are usually to be related with deviations from molecularity (viz. Sections II.B.2 and II.B.3), which in fact represent extrinsic disorder and not intrinsic Schottky disorder. From Figures 5.3 and 5.4 it is apparent that deviations from molecularity always influence ionic point defect concentrations, while deviations from stoichiometry always lead to combinations of ionic and electronic point defects, as can be seen from Figures 5.2 and 5.5. 

The intrinsic disorder of the continuous network is less easily classified in terms of defects. The network has many different configurations, but provided the atomic coordination is the same, all these structures are equivalent and represent the natural variability of the material. Since there is no correct position of an atom, one cannot say whether a specific structure is a defect or not. Instead the long range disorder is intrinsic to the amorphous material and is described by a randomly varying disorder potential, whose effect on the electronic structure is summarized in Section 1.2.5. 

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