^{1}, S. W. Glunz

^{1}and W. Warta

^{1}

### Abstract

Differential light-biased dynamic measurements of charge carrier recombination properties in semiconductors have long been known to yield only differential rather than actual recombination properties. Therefore, the determination of injection-dependent recombination properties from such measurements was previously found to require integration over the entire injection range. Recent investigations of the phase shift between a time-modulated irradiation of silicon samples and excess carrier density reveal a striking analogy to the above findings: the phase shift is greater than the actual effective carrier lifetime in the case of a positive derivative of lifetime with respect to excess carrier density, and vice versa. This work attempts to rearrange the mentioned previous findings in a quantitative theoretical description of light-biased dynamic measurements of effective carrier lifetime. Both light-biased differential lifetime measurements as well as harmonically time-modulated methods without additional bias light are shown to represent a limiting case in a general treatment of light-biased dynamic lifetime measurements derived here. Finally, we sketch a way to obtain actual recombination properties from differential measurements—referred to as a *differential-to-actual* (*d*2*a*) lifetime analysis, which does not require integration over the entire injection range.

This work was funded by the Fraunhofer Gesellschaft within the project Silicon BEACON.

I. INTRODUCTION

II. THEORY

A. Structural change of the continuity equation

B. Theory of light-biased decay time

III. IMPLICATIONS OF LIGHT-BIASED DECAY THEORY

A. Discussion of limiting cases

B. Relation to time-modulated measurements

C. Interpretation of differential lifetime

IV. VALIDATION: NUMERICAL SIMULATIONS

A. Abrupt change of generation rate

B. Harmonic time modulation of generation rate

V. CONCLUSIONS

### Key Topics

- Carrier density
- 42.0
- Carrier generation
- 29.0
- Time measurement
- 27.0
- Carrier lifetimes
- 20.0
- Differential equations
- 6.0

## Figures

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 .

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 ).

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 ).

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 .

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* (*d*2*a*) 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).

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* (*d*2*a*) 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* (*d*2*a*) lifetime analyses to a nonlinear lifetime curve : Self-consistent and *d*2*a* 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.

Demonstration of the applicability of both the self-consistent ^{ 6 } and the *differential-to-actual* (*d*2*a*) lifetime analyses to a nonlinear lifetime curve : Self-consistent and *d*2*a* 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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