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

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      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:

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).