A Course in Luminescence Measurements and Analyses for Radiation Dosimetry. Stephen W. S. McKeever

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      Figure 2.1 (a) Idealized energy-band diagram for a perfect crystal at equilibrium, illustrating an empty conduction band and a filled valence band. For wide-band-gap insulators, the Fermi Level (EF) is located mid gap. (b) A more-realistic energy-band model in which energy levels exist in the forbidden gap. Depending upon the location of the energy level (specifically, their position with respect to the band edges and the Fermi Level) the levels may be considered as “traps” or as “recombination centers,” for either electrons or holes. The virtual demarcations between them are represented by Demarcation Levels, one for electrons (De) and one for holes (Dh).

      Point defects may be due to:

The archetypical materials for discussing defects in solids are the alkali halides, of the type A⁺B^–. In such a material, vacancies can exist on the A or the B sub-lattice. Both A and B ions can occupy interstitial positions. Impurities can reside on either the A or the B lattice, depending on their valency, and they can also occupy interstitial sites. Depending on the valency of the impurity compared to that of the host atom, vacancies of type A or B may be required to co-exist with the impurities to maintain charge neutrality. Charge neutrality may also be maintained by the creation of Frenkel defects where vacancies of type A are charge compensated by interstitials, also of type A. Similarly, Frenkel defects involving the B sub-lattice may also occur. Alternatively, vacancies in the body of the crystal, on the A (or B) sub-lattice, may be charge compensated through the creation of equal numbers of vacancies on both the B (or A) sublattices and these are known as Schottky defects.

      The local charge imbalance caused by vacancies, interstitials, and impurities may also be compensated by the localization of free charge carriers (electrons or holes) in cases where such delocalized free carriers exist. At equilibrium, there are negligible numbers of such carriers at normal temperatures (and none at zero Kelvin), but the defects can act as traps for whatever free charge carriers become available due to coulombic interactions between the free carriers and the traps. For example, trivalent rare-earth impurities (RE³⁺) in an alkaline-earth halide (e.g. CaF₂) may substitute for host positive ions (anions, in this case Ca²⁺). The charge imbalance results in a very strong coulombic attraction for free electrons forming divalent sites (i.e. RE³⁺ + e^– = RE²⁺). In cases where the electrons are localized at defect sites they can attain energies which are higher than the valence band energies, but smaller than the conduction band energies. Thus, the energy band diagram of a real crystal, containing these simple defect types, would be characterized by allowed energy levels in the forbidden gap, as illustrated in Figure 2.1b. The Fermi-Dirac function (Equation 1.1 in Chapter 1) demonstrates that the occupancy of energy level E, be it localized or delocalized, depends upon the temperature T and the value of E relative to the Fermi Level EF. If the band gap is such that Ec – Ev >> kT (Ev = top of the valence band; Ec = bottom of the conduction band), then all energy levels above EF are essentially empty of electrons at equilibrium, and all those below the Fermi level are essentially full.

Point defects allow a conceptual picture of what a defect might look like in a crystal. However, these simple descriptions are far from complete. They do not include ionic polarization effects and electronic interactions with electrons and nuclei in neighboring ions. Such effects mean that “point” defects in fact exert influences on the lattice out to several lattice spacings in all directions. Consider, for example, impurity X²⁺substituting for A⁺ in ionic compound A⁺B^–. In Figure 2.2a, a schematic ideal AB lattice is shown, typical of alkali halides. The alkali and halide sublattices are each face-centered-cubic and the picture shown in the figure is considered to stretch to infinity in all dimensions. Introduction of impurity X²⁺ substituting for one of the A⁺ host ions causes the surrounding A⁺ ions to be repelled and the B^– ions to be attracted to the X²⁺ ion (Figure 2.2b). Such polarization effects cause a distortion to the lattice, spreading out over several atomic spacings. The polarization effects are dependent upon the dielectric constant of the medium and decrease rapidly (1/r²) with inter-atomic spacing СКАЧАТЬ