Читать книгу A Course in Luminescence Measurements and Analyses for Radiation Dosimetry - Stephen W. S. McKeever - Страница 19

Insulators and semiconductors are characterized by an energy gap between the uppermost filled energy band (the valence band) and the next empty band (the conduction band). At absolute zero, the valence band is completely full and the conduction band is completely empty. For a “perfect” crystal, no energy states are allowed in the energy gap between the top of the valence band (at energy Ev) and the bottom of the conduction band (at energy Ec). That is, if Z(E) is the density of available states at any energy E, then Z(E) = 0 for Ev < E < Ec, and the energy gap (Ec – Ev) is known as the “forbidden” gap or zone. However, real crystals contain defects such that Z(E) ≠ 0 in this forbidden zone. Energy states E can exist for which Ev < E < Ec and Z(E) > 0, and electrons can occupy energy states that are above the valence band but below the conduction band. Since such energy levels arise because of defects (e.g., impurities, vacancies, interstitials, and larger defect complexes), these states are localized at specific lattice sites within the crystal whereas the conduction and valence bands are delocalized. As a result, excitation of valence band electrons to one of these higher energy states, through the absorption of energy from a radiation field, requires not just a transition to a higher, excited energy level, but it also requires transport of the electron from one atomic or molecular site to another within the host crystal. That is, movement through the crystal is needed. This can only occur via a “transfer state” – in other words, via the conduction band (Figure 1.2a). Once the excited electrons have been transported to their new positions in the lattice, they relax into lower energy levels E, where Ev < E < Ec.

Figure 1.2 (a) Excitation from the equilibrium state (valence band) to the metastable state, via the conduction (“transport”) band. Stimulation from the metastable state results in recombination and relaxation to the equilibrium state, again via the conduction band. (b) The metastable state can be thought of as two energy levels within the energy band gap, one above the Fermi Level and one below. At equilibrium, all energy levels above the Fermi Level are empty and all levels below the Fermi Level are full. Excitation of electrons to the conduction band results in “trapping” at localized states, above the Fermi Levels. Similarly, holes are localized (“trapped”) at states below the Fermi Level. This is a non-equilibrium condition and represents the system in a metastable condition. Stimulation of the electron (say) from the localized state results in its recombination with the localized hole and the return of the system to equilibrium. Transitions: (1) Excitation (radiation); (2) Localization (trapping); (3) Stimulation (heat or light); (4) Relaxation (recombination). Animated versions of Figures 1.2a and 1.2b are available on the web site, under Exercises and Notes, Chapter 1.

This description is only partially complete, however. Since electrons have been excited out of the valence band, delocalized electronic holes are created. These positive charge species can move via the valence band states until they too become localized at defects within the lattice. In effect, this can be considered as an electron from the defect transitioning to the valence band, or as a hole from the valence band transitioning to the defect. The net result is that a hole, that is, a lack of an electron, now exists at that localized state.

If the two localized states just described – i.e., the localized electron state and the localized hole state – are the same, that is to say at the same defect, then the electron and hole will recombine and the whole system will return directly to its equilibrium state. However, if the two localized states are at different defects (different defect types) then they will remain localized and the system will no longer be in equilibrium. This is the metastable state.

The situation is illustrated in Figure 1.2b. (Animated versions of Figures 1.2a and 1.2b are available on the web site under Exercises and Notes, Chapter 1.) The two localized energy states, one above the Fermi Level and one below, localize excited electrons from the conduction band and free holes from the valence band, respectively. When localized in this way, the system is in a metastable condition. Absorption of energy from an external stimulus can free electrons (say) from the trap causing a transition to the conduction band, and these may subsequently recombine with the trapped holes, returning the system to equilibrium.

There may be multiple localized states available for electrons and holes. Consider an arbitrary distribution of available states Z(E). According to Fermi-Dirac statistics, the occupancy of any energy level E, at temperature T, is given by the distribution function f(E), where:

(1.1)

where EF is the Fermi Level and k is Boltzmann’s constant. At equilibrium (and at T = 0 K), f(E < EF) = 1 (all states full), and f(E > EF) = 0 (all states empty). The situation is illustrated in Figure 1.3a, for an arbitrary distribution Z(E).

Figure 1.3 (a) Arbitrary distributions of available states Z(E), at equilibrium and T = 0 K, with the Fermi-Dirac occupancy function f(E). f(E) = 1 for E < EF, and f(E) = 0 for E > EF. (b) After irradiation some electrons occupy states above the Fermi level, and some states below this level are empty. Two quasi-fermi Levels can be defined, one for electrons EFe_, where EF < E < EFe and one for holes EFh, where EF > E > EFh, as shown. (c) During the return to equilibrium (i.e., during stimulation), the quasi-fermi Levels move toward EF as the occupancy of the localized states changes. (d) Eventually, the system returns to equilibrium.

After irradiation (also at T = 0 K) the occupancy function f(E) changes, as illustrated by the red line in Figure 1.3b. In this view, two new energy levels can be defined, known as quasi-Fermi levels, one for electrons EFe and one for holes EFh. EFe is defined such that all localized states at energy level E are full when EF < E < EFe, and are empty when EF > E > EFh.

During stimulation after irradiation, the states above EF empty while those below EF fill, and the two quasi-Fermi Levels move closer to the original Fermi Level, EF (Figure 1.3c). Eventually, when all localized states above EF are empty of electrons and all those below are full, EFe = EF = EFh and the system has returned to equilibrium (Figure 1.3d).

The above picture describes the broad, conceptual notions describing the perturbation of a system from equilibrium due to irradiation, and the return of the system to equilibrium during either thermal stimulation or optical stimulation. If the final relaxation processes are radiative, TL and OSL result. In the chapters to follow, the equations describing the changes in occupancy of the various energy levels during excitation and stimulation will be examined. First, however, related processes including radiophotoluminescence, RPL, are introduced.