Illustration of the discrepancy (ratio) between a measurable decay time and actual lifetime as a function of a bias light parameter with and . Three situations are plotted: a positive derivative of lifetime with respect to excess carrier density with , a negative derivative with , and an injection-independent lifetime. In any case, the discrepancy is most pronounced in the limit .
Depiction of excess carrier density as a function of time after an abrupt change of generation rate at . Here, three scenarios of are discussed, the actual lifetime curves (as used for the simulation of ) are depicted in Fig. 3 as solid lines, respectively. The relative change of excess carrier density is greater than the relative change of generation rate if , and vice versa. This gives rise to a discrepancy between decay time and lifetime for light-biased measurements ( ), as can be seen in Fig. 3 . The resulting curves also quantitatively confirm the validity of Eq. (14) (cf. Fig. 1 ).
Comparison of measurable decay time according to Eq. (22) (symbols) and actual lifetime (solid lines) for the scenarios of an abrupt change of generation rate at as shown in Fig. 2 . All results quantitatively agree with Eq. (14) . For , decay time and actual lifetime coincide. For , decay time is greater than actual lifetime if and smaller than actual lifetime if . The greater , the greater is the discrepancy between decay time and actual lifetime (cf. Fig. 1 ).
Depiction of relative generation rate and excess carrier density in the case of a harmonically time-modulated irradiation. The excess carrier density curves shown here represent numeric solutions of the time-dependent continuity equation for different linear curve shapes of . Three linear lifetime scenarios are plotted: a positive derivative of lifetime with respect to excess carrier density , a negative derivative , and a constant lifetime of . All lifetime curves intersect at , which also corresponds to the maximally achieved excess carrier density in each scenario (due to ). The upper plot (a) depicts a total modulation while the lower plot (b) focuses on the most relevant peak area. Plot (b) shows that phase shifts can exactly be predicted by Eq. (14) in the limit .
Injection-dependent phase shifts between harmonically time-modulated generation rate and excess carrier density were compared to true injection-dependent lifetime for the three linear scenarios of discussed here. All results of clearly confirm Eq. (14) in the limit . Additionally, two injection-dependent lifetime analyses were performed—a self-consistent lifetime analysis 6 and an advanced self-sufficient analysis—referred to as differential-to-actual (d2a) lifetime analysis here. This analysis applies the theory of light-biased decay time (Eq. (20) ) to injection-dependent phase shifts. 7 Both analyses accurately yield true injection-dependent lifetime in any of the discussed scenarios. Therefore, integration over the entire injection range is not necessary in order to correctly determine injection-dependent lifetimes from measurements of decay time (or phase shift).
Demonstration of the applicability of both the self-consistent 6 and the differential-to-actual (d2a) lifetime analyses to a nonlinear lifetime curve : Self-consistent and d2a lifetime analyses were performed on simulated dynamic photoluminescence measurements on the basis of the true carrier lifetime curve shown in the plot. Two measurements were simulated at injection conditions such that they covered the Shockley-Read-Hall ascent and the Auger descent of carrier lifetime, respectively. It can be seen that phase shifts substantially differ from actual carrier lifetime. Despite the nonlinearity of , both lifetime analyses accurately yield true injection-dependent effective carrier lifetime.
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