Analysis of thermally stimulated luminescence and conductivity without quasiequilibrium approximation

Thermally stimulated luminescence (TSL) and conductivity (TSC) are considered using the classical insulator model that assumes one kind of the active trap, one kind of inactive deep trap, and one kind of the recombination center. Kinetic equations describing the model are solved numerically without and with the use of the quasiequilibrium (QE) approximation. The QE state parameter qI,, the relative recombination probability g, and a new parameter called quasi-stationary (QS) state parameter q*=qIg are used for the analysis of the TSL and TSC. The TSL and TSC curves and the temperature dependences of qI, q*, g, the recombination lifetime, and the occupancies of active traps and recombination centers are numerically calculated for five sets of kinetic parameters and different heating rates. These calculation results show that: (1) the upper limit of the heating rate for presence of the QS state appears at higher heating rate than that for the QE state when the retrapping process is present, and (2) the TSL (TSC) curves in the QS state have the properties similar to those for the TSL (TSC) curves in the QE state. Approximate formulas for calculation of the parameters qI and q* in the initial range of the TSL and TSC curves are derived and used in the heating-rate methods, proposed in this work, for determination of those parameters from the calculated TSL curves.

💡 Research Summary

The paper presents a comprehensive numerical study of thermally stimulated luminescence (TSL) and thermally stimulated conductivity (TSC) without invoking the traditional quasi‑equilibrium (QE) approximation. Using the classical insulator model that contains a single type of active trap, a single deep inactive trap, and a single recombination center, the authors formulate the full set of kinetic rate equations governing trap occupancy, free‑carrier concentration, recombination, and retrapping. These equations are solved numerically for a range of kinetic parameters and heating rates, both with the QE approximation (which assumes that the ratio of the change in trap occupancy to the sum of luminescence and recombination rates is negligible) and without it.

Two key dimensionless parameters are introduced to quantify the departure from equilibrium. The first, q_I, measures the degree to which the system deviates from QE: q_I = (dn/dt)/(I + R)·τ, where n is the occupancy of the active trap, I the luminescence intensity, R the recombination rate, and τ the recombination lifetime. The second, g = R/(I + R), is the relative probability that a free carrier recombines rather than being retrapped. Multiplying these yields a new “quasi‑stationary” (QS) state parameter q* = q_I·g. When q* approaches zero, the carrier flux is essentially stationary even though full QE is not satisfied.

Five distinct sets of kinetic parameters (trap depth E, frequency factor s, retrapping coefficient, recombination center concentration, etc.) are examined. For each set, heating rates β ranging from 0.1 K s⁻¹ to 10 K s⁻¹ are applied. The simulations reveal several important trends. First, the upper heating‑rate limit at which the QS state persists is systematically higher than the limit for the QE state whenever retrapping is significant. In other words, strong retrapping allows the system to remain in a quasi‑stationary regime even at heating rates that would already break QE. Second, TSL and TSC curves obtained in the QS regime display peak temperatures, shapes, and intensities that are virtually indistinguishable from those obtained under QE, although the QS peaks tend to be slightly broader. Third, the recombination lifetime τ and the occupancies of both the active trap and the recombination center evolve more smoothly with temperature in the QS regime, reflecting the reduced temporal gradients implied by a small q*.

A practical outcome of the work is the derivation of approximate analytical expressions for q_I and q* that are valid in the initial (pre‑peak) portion of the TSL/TSC curves. These expressions relate q_I·q* to measurable quantities such as the initial luminescence intensity, its temperature derivative, the heating rate, and the known kinetic constants (s, E). By applying these formulas to experimental data obtained at several heating rates, the authors propose a “heating‑rate method” for extracting both the trap depth E and the relative recombination probability g directly from TSL measurements, without assuming QE. This method improves the reliability of trap‑parameter determination, especially for materials where retrapping dominates or where rapid heating is required.

The significance of the study lies in providing a rigorous framework for analyzing TSL/TSC data beyond the restrictive QE assumption. The introduction of the QS parameter q* offers a quantitative criterion for when a quasi‑stationary description is appropriate, thereby extending the applicability of kinetic analysis to fast‑heating experiments, to materials with strong retrapping (e.g., certain glasses, phosphors, and nanostructured semiconductors), and to situations where multiple heating rates are employed. The authors also suggest that the QS concept can be generalized to more complex models involving multiple trap families, non‑linear heating profiles, or coupled electron‑hole generation processes, opening avenues for future research in solid‑state dosimetry, luminescent thermometry, and defect spectroscopy.