Modeling Photoluminescence Dynamics with Rate Equations

Hello! Welcome to the final lesson in our module on exciton dynamics.

In our previous session, we established the link between microscopic scattering processes and the homogeneous linewidth of an excitonic transition. We saw that the total scattering rate, $Γ_{scat}$ , determines the coherence lifetime ( $τ_{c} = 1/ Γ_{scat}$ ), which is observable as a spectral broadening of $ℏ Γ_{scat}$ .

Today, we shift our focus from coherence to population. Instead of asking how long an exciton state remains phase-coherent, we will ask how long the population of excitons survives before recombining. This is governed by the population lifetime, and the primary experimental technique to measure it is Time-Resolved Photoluminescence (TRPL). Our learning outcome is to model time-resolved photoluminescence data using rate equations to extract population dynamics and relaxation rates. This is a crucial skill for any experimentalist in the field, as it bridges the gap between raw data and physical parameters.

Recombination Pathways

After an ultrashort laser pulse creates a population of excitons, their number, $N (t)$ , begins to decrease. This decay is driven by several competing processes, broadly categorized as radiative and non-radiative.

Exciton Recombination Mechanisms in Semiconductors — This diagram illustrates the primary recombination pathways for charge carriers and excitons in a semiconductor. **(a)** Radiative recombination, where an electron-hole pair annihilates to produce a photon. This is the source of the photoluminescence signal. **(b)** Trap-assisted (Shockley-Read-Hall) non-radiative recombination, where a carrier is trapped by a defect state within the bandgap. **(c)** Auger non-radiative recombination, a three-particle process where the energy from an e-h pair recombination is transferred to a third carrier, ejecting it to a higher energy state. These processes are the physical basis for the rate equations we will develop.

The Simplest Model: Mono-exponential Decay

Let's start with the most fundamental model. We consider a single population of excitons, $N (t)$ , that can decay either radiatively with a rate $Γ_{R} = 1/ τ_{R}$ or non-radiatively with a rate $Γ_{N R} = 1/ τ_{N R}$ . The total decay rate is $Γ_{t o t} = Γ_{R} + Γ_{N R}$ . The change in the exciton population over time is described by a simple first-order differential equation:

\frac{d N ( t )}{d t} = - (Γ_{R} + Γ_{N R}) N (t) = - Γ_{t o t} N (t)

The solution is a straightforward exponential decay:

N (t) = N_{0} e^{- Γ_{t o t} t} = N_{0} e^{- t / τ_{t o t}}

where $τ_{t o t} = 1/ Γ_{t o t}$ is the total population lifetime.

A crucial assumption in TRPL is that the measured photoluminescence intensity, $I (t)$ , is directly proportional to the rate of radiative recombination. The radiative recombination rate is, in turn, proportional to the number of excitons available to recombine, so $I (t) \propto Γ_{R} N (t)$ . This gives us the classic mono-exponential decay for the PL signal:

I (t) = I_{0} e^{- t / τ_{t o t}}

This simple model already allows for powerful analysis. If we can measure both the total lifetime $τ_{t o t}$ from a TRPL experiment and the photoluminescence quantum yield (PLQY), $η = \frac{photons emitted}{photons absorbed}$ , we can solve for the individual rates.

Since $η = \frac{Γ _{R}}{Γ _{R} + Γ _{N R}}$ and $τ_{t o t} = \frac{1}{Γ _{R} + Γ _{N R}}$ , we find:

τ_{R} = \frac{τ _{t o t}}{η} and τ_{N R} = \frac{τ _{t o t}}{1 - η}

The following video provides an excellent introduction to these concepts and the basics of TRPL measurements.

Intro to TCSPC - Time Correlated Single Photon Counting - by Jeff DuBose

This video from the Kamat Lab at the University of Notre Dame offers a clear and practical introduction to photoluminescence decay and Time-Correlated Single Photon Counting (TCSPC), the most common technique for measuring it.

Please watch the first 5 minutes of the video (00:00 - 05:00). Pay close attention to: The relationship between the lifetime au and the radiative (k_R) and non-radiative (k_{NR}) rates. The derivation of the mono-exponential decay from the basic rate equation. The assumption that links the excited state population N(t) to the measured intensity I(t).

Multi-exponential Decays and the Realities of Data Fitting

In practice, a single exponential decay is the exception rather than the rule, especially in complex systems like the nanostructures you've worked with. TRPL data often exhibit multi-exponential behavior:

I (t) = i \sum α_{i} e^{- t / τ_{i}}

This can arise from several physical situations:

Multiple, distinct populations of excitons (e.g., in different quantum wells of a multi-quantum-well structure).
The presence of various trap states, each with its own characteristic trapping and de-trapping time.
Cascading relaxation processes through a series of intermediate states.

Modeling this data involves fitting the experimental curve to a sum of exponentials. However, this is not just a mathematical exercise; it requires careful physical interpretation.

Intro to TCSPC - Time Correlated Single Photon Counting - by Jeff DuBose

Let's continue with the same video, which now delves into the critical aspects of data analysis and interpretation.

Please watch from 26:06 to the end (26:06 - 50:05). This section is dense with practical wisdom. Focus on: The importance of goodness-of-fit parameters (\chi^2, residuals) and the visual inspection of the fit. The critical warning: you can almost always improve a fit by adding more exponential terms, but you must have a physical justification for doing so. The mathematical correlation between the amplitudes (\alpha_i) and lifetimes ( au_i), which can lead to non-unique fitting results. The examples from the literature, which show how tracking the evolution of the fitted parameters (\alpha_i, au_i) as a function of an external variable (e.g., temperature, composition, quencher concentration) can provide robust physical insights despite fitting ambiguities.

The key message here is that while fitting provides parameters, the real physics is uncovered by understanding how those parameters change with systematic variations of your experiment. This is directly applicable to your research in magneto-PL, where you will analyze how lifetimes and amplitudes change with magnetic field.

Physically-Motivated Rate Equations

While multi-exponential fitting is a versatile tool, a more rigorous approach is to build a model from the underlying physics, which may not result in a simple sum of exponentials. Let's explore two powerful examples.

1. The `An + Bn²` Model

In many semiconductors, especially at higher excitation densities, excitons can dissociate into a plasma of free electrons and holes. The recombination of these free carriers is a bimolecular process. The rate equation now includes a term proportional to the product of electron and hole concentrations, n*p. If n=p, this is an n² term. Including the standard monomolecular non-radiative decay (An), the rate equation for the carrier density n(t) becomes:

\frac{d n}{d t} = - A n - B n^{2}

Here, $A$ is the Shockley-Read-Hall (SRH) coefficient for non-radiative monomolecular decay, and $B$ is the coefficient for radiative bimolecular recombination. The PL intensity is now proportional to the bimolecular term: $I (t) \propto B n (t)^{2}$ .

This differential equation can be solved, leading to a more complex, non-exponential decay function for $I (t)$ . The following paper provides a clear derivation.

A novel model on time-resolved photoluminescence ...

This paper by Zhou et al. develops a model specifically for the non-exponential TRPL curves observed in polar InGaN/GaN quantum wells. Their derivation of a decay curve from the An + Bn² rate equation is a perfect example of a physically-motivated model.

Please read the 'Theoretical Model' section. Follow the derivation starting from Equation (3), rac{dn}{dt} = -An - Bn^2, to the final expression for the TRPL decay curve, Equation (7). Note how fitting the data to this specific functional form allows the direct extraction of the physical coefficients A and Bn₀.

By fitting the experimental data to the derived function (Eq. 7 in the paper), one can extract the values for the monomolecular non-radiative coefficient $A$ and the initial radiative rate $B n_{0}$ , providing direct insight into the competition between these two processes. This is a significant step beyond simply fitting abstract exponential components.

2. The Full ABC Model and the Logarithmic Derivative Method

We can extend this model to include three-body Auger recombination, represented by a term $C n^{3}$ . This gives the full "ABC" rate equation, which is a cornerstone of semiconductor physics:

\frac{d n}{d t} = - A n - B n^{2} - C n^{3}

While solving this equation analytically is cumbersome, a remarkably elegant method exists to analyze the data and extract the A, B, and C coefficients graphically. This method involves plotting the instantaneous recombination rate, $r_{P L} (t)$ , against the PL intensity.

The instantaneous rate is defined as the logarithmic derivative of the intensity:

r_{P L} (t) = - \frac{d}{d t} [ln I (t)] = - \frac{1}{I ( t )} \frac{d I ( t )}{d t}

For a simple mono-exponential decay, $I (t) = I_{0} e^{- t / τ}$ , this gives $r_{P L} (t) = 1/ τ$ , a constant, as expected. For more complex decays, $r_{P L}$ will be time-dependent.

Exact determination method of the recombination mode ...

The paper 'Exact determination method of the recombination mode...' by S.P. Lepkowski and collaborators presents this powerful logarithmic derivative method. Given your mathematical background, you will appreciate the elegance of this approach.

Please read Section 2, 'Theoretical background'. This is the core of the lesson. Focus on the derivation that leads to Equation (11) (or (14)). This equation shows that for a system described by the ABC model where radiative recombination is bimolecular (I \propto n^2), a plot of the instantaneous rate r_{PL} versus \sqrt{I_{PL}} yields a parabola whose coefficients are directly related to A, B, and C.

The key result you derived is:

r_{P L} (t) = - \frac{d}{d t} [ln I_{P L} (t)] = 2 A + 2 B n_{0} I_{P L} (t) + 2 C n_{0}^{2} I_{P L} (t)

(assuming $I_{P L}$ is normalized to 1 at $t = 0$ ).

This is a powerful result. By calculating the instantaneous rate from the TRPL data and plotting it against the square root of the intensity, you can fit a polynomial. The coefficients of this polynomial directly give you the relative strengths of the monomolecular ( $A$ ), bimolecular ( $B$ ), and trimolecular ( $C$ ) recombination processes.

Test your understanding!

Imagine you use the logarithmic derivative method on a TRPL decay curve and find that the plot of $r_{P L}$ vs. $I_{P L}$ is nearly a horizontal line. What does this imply about the dominant recombination mechanisms in your sample under these conditions?

Show answer

A horizontal line means that the instantaneous rate $r_{P L}$ is constant and does not depend on the intensity $I_{P L}$ . Looking at the equation $r_{P L} = 2 A + 2 B n_{0} I_{P L} + 2 C n_{0}^{2} I_{P L}$ , this implies that the coefficients of the $I_{P L}$ and $I_{P L}$ terms are negligible. Therefore, the recombination is dominated by the monomolecular process described by the coefficient $A$ . This could be SRH recombination or, in the low-density limit, excitonic recombination. Bimolecular and Auger recombination are insignificant in this case.

This method allows you to "watch" the dominant recombination mechanism change in real time as the carrier density drops during the decay. For example, a decay might start out dominated by the $C$ and $B$ terms at high intensity ( $t$ near 0) and then transition to being dominated by the $A$ term at low intensity (large $t$ ).

Conclusion

In this lesson, we have built a framework for understanding and modeling the decay of exciton populations as measured by time-resolved photoluminescence.

Key Takeaways:

TRPL measures the decay of the exciton population lifetime, which is limited by radiative and non-radiative recombination events.
The simplest model assumes a single population with a constant decay rate, leading to a mono-exponential decay of the PL intensity.
Real data is often multi-exponential. While fitting these curves can provide characteristic lifetimes, physical interpretation must be done with care, ideally by studying the trends of fitted parameters with external variables.
More advanced models based on physical rate equations (e.g., the ABC model) can describe non-exponential decays arising from multi-particle interactions like bimolecular and Auger recombination.
The logarithmic derivative method is a powerful analytical tool that transforms a TRPL curve into a plot that can graphically separate and quantify the contributions of monomolecular (A), bimolecular (B), and Auger (C) processes.

Preview of the Next Lesson

This concludes our module on exciton dynamics in the absence of external fields. We are now well-equipped to tackle one of the core topics of this course and your research: the behavior of excitons in magnetic fields. In our next lesson, we will begin Module 7 by deriving the exciton Hamiltonian in a uniform magnetic field and defining the conserved pseudomomentum. We will then analyze the system in the weak-field regime, exploring the origins of the diamagnetic shift and Zeeman splitting.

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