Читать книгу A Course in Luminescence Measurements and Analyses for Radiation Dosimetry - Stephen W. S. McKeever - Страница 30

Discussion of the electronic transitions that occur during TL, OSL, and RPL rely upon the simplified energy band model already discussed in Chapter 1 and illustrated again, in Figure 2.1. The depiction of a valence band completely full of electrons and a conduction band completely empty of electrons, with a constant band gap width throughout the crystal (Figure 2.1a) is a theoretical and ideal construct. It implies that all host atoms are exactly located in their equilibrium lattice positions dictated by their size and the nature of the bonds between them. No atoms other than the host atoms are assumed; that is, there are no impurity atoms. The conceptual picture also assumes an infinite crystal, with no surfaces. Of course, no such material exists in nature. Real materials contain defects, of which there are many possible types.

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:

 vacancies, where a host atom is missing;

 interstitials, where a host atom occupies an off-lattice position between other host atoms;

 anti-site defects, where in a host of type AB, A atoms occupy B sites, and vice-versa;

 substitutional impurities, where a host atom is replaced by an impurity atom;

 interstitial impurities, where an impurity atom is located in an interstitial, off-lattice position;

 complex clusters of the above.

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 (r). The polarization energy affects the optical and electronic properties of the material and moving electronic charges also cause subsequent changes to the polarization and displacements (Hayes and Stoneham 1985).

Figure 2.2 (a) An idealized lattice for an ionic crystal of the type A⁺B^–. (b) Stylized polarization effects caused by the substitution of an A⁺ ion with a divalent impurity ion X²⁺.

The schematic illustration in Figure 2.2b is for illustrative purposes only. Modern density functional theory (DFT) calculations are able to calculate the positions of the ions when a divalent impurity is introduced into an A⁺B^– lattice. An example is shown in Figure 2.3 for LiF doped with Mg²⁺, with the Mg ion in an interstitial position. For another example, Masillon et al. (2019) reveal that when Mg²⁺ substitutes for Li⁺ there is a resultant displacement of six nearest F atoms symmetrically in pairs, by 0.14 Å, 0.15 Å and 0.18 Å, while one F atom close to the Li-vacancy moves by 0.15 Å. At the same time, three neighboring Li atoms move away from their original position; two by 0.11 Å and one by 0.14 Å.

Figure 2.3 Two views, (a) and (b), of density functional theory calculations for the distortions in the lattice caused by the addition of an interstitial Mg²⁺ ion impurity into LiF. Li⁺ ions are in grey; F^– ions are in green, and the Mg²⁺ impurity ion is in orange. (Original data kindly provided by Guerda Masillon, © Guerda Masillon, UNAM, Mexico.)

The conclusion from these considerations is that a “point” defect in a lattice can exert influence over several lattice spacings and, in the certain cases, over several thousand surrounding host ions. Indeed, a “point defect” is not a “point” at all (Townsend 1992).

In a dilute system each defect can be considered “unseen” by other defects. In this definition, no matter over how many lattice spacings the defect can exert influence, there are no other defects within this sphere of influence. However, this is not always the case. For example, a divalent impurity substituting for an alkali ion can be charge compensated locally through coulombic attraction with an alkali vacancy. Using the popular TL dosimetry material LiF:Mg as an example, Mg²⁺ ions that substitute for host Li⁺ ions are charge compensated by Li vacancies, which are effectively negatively charged and occupy nearest-neighbor positions along the <110> direction in the lattice. The clustering process forms dipolar complexes (Figure 2.4a), which are revealed by DFT calculations in LiF:Mg (Masillon et al. 2019). However, at room temperature this process of dipole formation does not lead to the final thermodynamic equilibrium state of the system, and further reduction in the system’s free energy can be attained by further clustering of the dipoles to form trimers, consisting of three dipoles in one of several possible configurations, an example of which is shown in Figure 2.4b (Taylor and Lilley 1982a; McKeever 1985; Gavartin et al. 1991).

Figure 2.4 (a) Schematic view of a LiF lattice with Mg²⁺ impurity substituting for a Li⁺ host ion and charge compensated by a Li-vacancy in a <110> direction, forming a dipolar complex; (b) example trimer cluster of three Mg²⁺-Li_vac dipoles.

An additional consideration, not indicated in the conceptual Figures 2.2 and 2.4, is that the radius of the impurity ions generally do not match those of the host ions for which they substitute. For example, the radius of a Mg²⁺ ion is 86×10^–12 m, whereas the radius of Li⁺ is 90×10^–12 m. This small change in radius (<5%) causes a much larger change in ionic volume in that part of the lattice and substituting a Li⁺ ion with a Mg²⁺ ion results in a decrease of ionic volume by ~15%. Similarly, Ti⁴⁺ results in a ~76% volume decrease, while Y³⁺ in CaF₂ causes a ~69% decrease when substituting for Ca²⁺. These effects immediately cause lattice distortions over and above distortions due to coulombic forces.¹

Radiation-dosimetry-quality LiF does not contain only Mg impurities; the TL sensitivity of the material is strongly enhanced by the inclusion of Ti⁴⁺ impurities. Charge compensation in this case appears through the incorporation of OH^– impurities, forming Ti-OH complexes. The role of the Ti-OH complexes in the production of TL is still not precisely determined but they are believed to be active in the recombination processes occurring in this material. In particular, much evidence has accumulated in recent years to suggest that the main TL peak from this material (the so-called peak 5) is the result of a Mg-trimer/Ti-OH complex of an as-yet-undetermined crystalline structure. That is, the Ti-OH complexes and the Mg-Li_vac trimers appear to be spatially associated in the LiF lattice and many of the characteristic TL properties of peak 5, and the TL glow curve in general, are uniquely affected by this association.

It is the breakdown in the lattice periodic potential caused by defects that gives rise to energy levels within the forbidden gap. The wavefunctions of electrons in a crystal with perfect periodicity are delocalized and extend throughout the material. States where the electron wavefunction is localized are not allowed. It is when the periodicity breaks down due to the presence of a defect that localized wavefunctions occur. These decay with distance away from the center of the defect over several lattice sites and the corresponding energies reside within the band gap.

Whether they are relatively simple defects or complex defects, interactions between the localized electrons and holes can take place either, or both, non-locally (i.e. via the delocalized bands) or locally (e.g. tunneling of charge between localized states) depending upon both the energy and the spatial association between the defects. The energy levels may be discrete (i.e. characterized by a single energy value) or, because of their complexity, can be distributed in energy with the exact energy value depending upon the nature of the surrounding environment and the presence of other defects. This is especially true of non-crystalline materials such as glasses, in which the surrounding lattice may display short- or long-range disorder, resulting in a range of energies for a particular defect type.