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Received: 30 August 2016  收到日期：2016 年 8 月 30 日
Accepted: 23 November 2016
接受日期：2016 年 11 月 23 日
Published online: 02 February 2017
在线发表：2017 年 2 月 2 日

A generalized model for timeresolved luminescence of localized carriers and applications: Dispersive thermodynamics of localized carriers
局部载流子的时间分辨发光广义模型及其应用：局部载流子的色散热力学

Zhicheng Su & Shijie Xu
苏志成 & 徐士杰

For excited carriers or electron-hole coupling pairs (excitons) in disordered crystals, they may localize and broadly distribute within energy space first, and then experience radiative recombination and thermal transfer (i.e., non-radiative recombination via multi-phonon process) processes till they eventually return to their ground states. It has been known for a very long time that the time dynamics of these elementary excitations is energy dependent or dispersive. However, theoretical treatments to the problem are notoriously difficult. Here, we develop an analytical generalized model for temperature dependent time-resolved luminescence, which is capable of giving a quantitative description of dispersive carrier dynamics in a wide temperature range. The two effective luminescence and nonradiative recombination lifetimes of localized elementary excitations were mathematically derived as τ L = τ r 1 + τ r τ t r ( 1 − γ c ) e ( E − E a / k B T τ L = τ r 1 + τ r τ t r 1 − γ c e E − E a / k B T tau_(L)=(tau_(r))/(1+(tau_(r))/(tau_(tr))(1-gamma_(c))e^((E-E_(a)//k_(B)T:}))\tau_{L}=\frac{\tau_{r}}{1+\frac{\tau_{r}}{\tau_{t r}}\left(1-\gamma_{c}\right) e^{\left(E-E_{a} / k_{B} T\right.}} and τ n r = τ t r ( 1 − γ c ) e − ( E − E a ) / k B T τ n r = τ t r 1 − γ c e − E − E a / k B T tau_(nr)=(tau_(tr))/((1-gamma_(c)))e^(-(E-E_(a))//k_(B)T)\tau_{n r}=\frac{\tau_{t r}}{\left(1-\gamma_{c}\right)} e^{-\left(E-E_{a}\right) / k_{B} T}, respectively. The model is successfully applied to quantitatively interpret the time-resolved luminescence data of several material systems, showing its universality and accuracy.
对于无序晶体中的激发载流子或电子-空穴耦合对（激子），它们可能首先在能量空间中局部化并广泛分布，然后经历辐射复合和热转移（即通过多声子过程的非辐射复合）过程，直到最终返回到基态。早已知道，这些基本激发的时间动态是能量依赖或色散的。然而，理论处理这个问题 notoriously difficult。这里，我们开发了一种温度依赖的时间分辨发光的解析广义模型，该模型能够在广泛的温度范围内对色散载流子动态进行定量描述。局部化基本激发的两个有效发光和非辐射复合寿命分别被数学推导为 τ L = τ r 1 + τ r τ t r ( 1 − γ c ) e ( E − E a / k B T τ L = τ r 1 + τ r τ t r 1 − γ c e E − E a / k B T tau_(L)=(tau_(r))/(1+(tau_(r))/(tau_(tr))(1-gamma_(c))e^((E-E_(a)//k_(B)T:}))\tau_{L}=\frac{\tau_{r}}{1+\frac{\tau_{r}}{\tau_{t r}}\left(1-\gamma_{c}\right) e^{\left(E-E_{a} / k_{B} T\right.}} 和 τ n r = τ t r ( 1 − γ c ) e − ( E − E a ) / k B T τ n r = τ t r 1 − γ c e − E − E a / k B T tau_(nr)=(tau_(tr))/((1-gamma_(c)))e^(-(E-E_(a))//k_(B)T)\tau_{n r}=\frac{\tau_{t r}}{\left(1-\gamma_{c}\right)} e^{-\left(E-E_{a}\right) / k_{B} T} 。该模型成功应用于定量解释多个材料系统的时间分辨发光数据，显示了其普遍性和准确性。
Carrier localization (CL) in real crystalline solids due to various disorders, e.g., defects, impurities, composition fluctuation, lattice distortion etc. is a ubiquitous phenomenon which was theoretically treated by Anderson for the first time 1 1 ^(1){ }^{1}. To date, CL and related phenomena still remain as a subject of extensive interest primarily because of their scientific significance and profound impact on electrical, magnetic and optical properties of material systems 2 − 18 2 − 18 ^(2-18){ }^{2-18}. With the rapid development of the InGaN alloy based blue-green light emitting diodes, recently, the CL effect induced by structural imperfections has been increasingly addressed 19 − 21 19 − 21 ^(19-21){ }^{19-21}. For example, it has been well shown that localized carriers due to alloy disorder, especially indium content fluctuation, can produce efficient luminescence and unusual thermodynamic behaviors 19 − 28 19 − 28 ^(19-28){ }^{19-28}.
载流子局域化（CL）在真实晶体固体中由于各种缺陷、杂质、成分波动、晶格畸变等引起的现象是一种普遍存在的现象，首次由安德森理论上处理 1 1 ^(1){ }^{1} 。迄今为止，CL 及相关现象仍然是广泛关注的主题，主要是因为它们在科学上的重要性以及对材料系统的电学、磁学和光学性质的深远影响 2 − 18 2 − 18 ^(2-18){ }^{2-18} 。随着基于 InGaN 合金的蓝绿光发射二极管的快速发展，最近，结构缺陷引起的 CL 效应越来越受到关注 19 − 21 19 − 21 ^(19-21){ }^{19-21} 。例如，已经很好地表明，由于合金无序，特别是铟含量波动引起的局域载流子，可以产生高效的发光和异常的热力学行为 19 − 28 19 − 28 ^(19-28){ }^{19-28} 。
In order to interpret these unusual luminescence behaviors associated with the carrier localization, many attempts have been devoted. For example, Eliseev et al. proposed an empirical formula to interpret temperature-induced “blue” shift in peak position of luminescence 25 25 ^(25){ }^{25}. This model agrees well with experimental data at high temperatures, but does not work at low temperatures. Wang applied the pseudopotential approach to study the CL mechanism in different InGaN InGaN InGaN\operatorname{InGaN} systems 23 23 ^(23){ }^{23}, which mainly focuses on the contribution of component fluctuation and quantum-dot formation to the carrier localization. No temperature effect was taken into consideration in Wang’s theoretical work. Dal Don et al. employed Monte Carlo simulation method to simulate temperature dependent behaviors of peak width and peak position of PL in ZnCdSe quantum islands 29 29 ^(29){ }^{29}. Li et al. developed an analytical model for steady-state PL of localized state ensemble 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31}. By solving a rate equation taking into account several fundamental processes of localized carriers, they obtained an analytical distribution function for localized carriers and then built up a steady-state luminescence model. They named their formula set LSE (localized-state ensemble) luminescence model 30 30 ^(30){ }^{30}. The model not only quantitatively reproduces S-shape
为了解释与载流子局域化相关的这些异常发光行为，进行了许多尝试。例如，Eliseev 等人提出了一个经验公式来解释温度引起的发光峰位的“蓝移” 25 25 ^(25){ }^{25} 。该模型在高温下与实验数据非常吻合，但在低温下则不适用。王应用伪势方法研究不同 InGaN InGaN InGaN\operatorname{InGaN} 系统 23 23 ^(23){ }^{23} 中的 CL 机制，主要关注组分波动和量子点形成对载流子局域化的贡献。王的理论工作中没有考虑温度效应。Dal Don 等人采用蒙特卡罗模拟方法模拟 ZnCdSe 量子岛中 PL 的峰宽和峰位的温度依赖行为 29 29 ^(29){ }^{29} 。李等人开发了一个用于局域态集合的稳态 PL 的解析模型 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31} 。通过求解考虑局域载流子几个基本过程的速率方程，他们获得了局域载流子的解析分布函数，并建立了一个稳态发光模型。 他们将其公式命名为 LSE（局部态集合）发光模型 30 30 ^(30){ }^{30} 。该模型不仅定量再现了 S 形状。
Figure 1. Temperature induced luminescence peak shift of localized carriers in an InGaN / GaN InGaN / GaN InGaN//GaN\mathrm{InGaN} / \mathrm{GaN} quantum well. The experimental data (solid circles) were measured by Schömig et al. 24 24 ^(24){ }^{24}, whereas the solid line was a best fitting curve with Eqs (3) and (4).
图 1. 温度引起的局域载流子在 InGaN / GaN InGaN / GaN InGaN//GaN\mathrm{InGaN} / \mathrm{GaN} 量子阱中的发光峰位移。实验数据（实心圆点）由 Schömig 等人测量 24 24 ^(24){ }^{24} ，而实线是使用方程（3）和（4）得到的最佳拟合曲线。
temperature dependence of PL peak for different material systems 32 32 ^(32){ }^{32}, but also interprets V-shape temperature dependence of PL width and even temperature dependence of integrated luminescence intensity 30 , 31 30 , 31 ^(30,31){ }^{30,31}. Moreover, It was also proved that LSE model can be reduced to Eliseev et al.'s band-tail model at high temperatures 22 22 ^(22){ }^{22}. Li and Xu also proved that the integrated luminescence intensity formula of LSE model can be reduced to the well-known thermal quenching formula when the distribution parameter of localized states approaches zero 31 , 33 31 , 33 ^(31,33){ }^{31,33}. Nonetheless, the model is only applicable for steady-state luminescence of localized carriers, and it does not work for transient or time-resolved luminescence of localized carriers at all. Göbel et al. have done a good attempt to quantitative analysis of the time-resolved luminescence process in quantum well structures by numerically solving the rate-equation 34 34 ^(34){ }^{34}. But in their work temperature was not taken into consideration. To the best of our knowledge, an analytical model with well-defined physical quantities has not yet been established for temperature dependent time-resolved photoluminescence of localized carriers.
不同材料系统的光致发光（PL）峰值的温度依赖性 32 32 ^(32){ }^{32} ，同时还解释了 PL 宽度的 V 形温度依赖性，甚至集成发光强度的温度依赖性 30 , 31 30 , 31 ^(30,31){ }^{30,31} 。此外，还证明了 LSE 模型在高温下可以简化为 Eliseev 等人的带尾模型 22 22 ^(22){ }^{22} 。Li 和 Xu 还证明，当局部态的分布参数接近零时，LSE 模型的集成发光强度公式可以简化为著名的热淬灭公式 31 , 33 31 , 33 ^(31,33){ }^{31,33} 。然而，该模型仅适用于局部载流子的稳态发光，对于局部载流子的瞬态或时间分辨发光则完全不适用。Göbel 等人通过数值求解速率方程，对量子阱结构中的时间分辨发光过程进行了定量分析的良好尝试 34 34 ^(34){ }^{34} 。但在他们的工作中并未考虑温度。根据我们所知，目前尚未建立一个具有明确物理量的解析模型，用于局部载流子的温度依赖性时间分辨光致发光。
In this article, we attempt to fill the void by developing an analytical model for time-resolved photoluminescence of localized carriers with a substantial energy distribution. As derived and argued below, the model was formulated with two effective luminescence lifetime and nonradiative recombination lifetime. It was then applied to quantitatively interpret the experimental time-resolved luminescence data obtained by several groups, which enables us get deep insight into the recombination dynamics of localized carriers in real material systems.
在本文中，我们试图通过开发一个用于局部载流子具有显著能量分布的时间分辨光致发光的分析模型来填补这一空白。如下面所推导和论证的，该模型是通过两个有效的发光寿命和非辐射复合寿命来构建的。随后，该模型被应用于定量解释由多个研究小组获得的实验时间分辨光致发光数据，使我们能够深入了解真实材料系统中局部载流子的复合动力学。
For a luminescent system with a total density of states (DOS) of localized electronic states, ρ ( E ) ρ ( E ) rho(E)\rho(E), time evolution of the excited carrier concentration N ( E , T , t ) N ( E , T , t ) N(E,T,t)N(E, T, t) may be described by a partial differential equation 22 , 30 , 35 22 , 30 , 35 ^(22,30,35){ }^{22,30,35}
对于具有局域电子态总态密度（DOS） ρ ( E ) ρ ( E ) rho(E)\rho(E) 的发光系统，激发载流子浓度 N ( E , T , t ) N ( E , T , t ) N(E,T,t)N(E, T, t) 的时间演化可以用偏微分方程 22 , 30 , 35 22 , 30 , 35 ^(22,30,35){ }^{22,30,35} 来描述
∂ N ( E , T , t ) ∂ t = G + γ c N ′ Λ ρ ( E ) − N ( E , T , t ) τ t r e ( E − E a ) / k B T − N ( E , T , t ) τ r ∂ N ( E , T , t ) ∂ t = G + γ c N ′ Λ ρ ( E ) − N ( E , T , t ) τ t r e E − E a / k B T − N ( E , T , t ) τ r (del N(E,T,t))/(del t)=G+gamma_(c)(N^('))/(Lambda)rho(E)-(N(E,T,t))/(tau_(tr))e^((E-E_(a))//k_(B)T)-(N(E,T,t))/(tau_(r))\frac{\partial N(E, T, t)}{\partial t}=G+\gamma_{c} \frac{N^{\prime}}{\Lambda} \rho(E)-\frac{N(E, T, t)}{\tau_{t r}} e^{\left(E-E_{a}\right) / k_{B} T}-\frac{N(E, T, t)}{\tau_{r}}
where G , γ c , N ′ G , γ c , N ′ G,gamma_(c),N^(')G, \gamma_{c}, N^{\prime}, and Λ Λ Lambda\Lambda represent the generation (excitation) rate of carriers due to optical excitation, electrical injection etc., the re-capture coefficient of the thermally activated carriers, the total number of thermally activated carriers, and the total number of localized electronic states, respectively. In Eq. (1), E a E a E_(a)E_{a} stands for a distinct energetic position of materials, e.g., the location of a delocalized level to which the localized carriers can be thermally activated 30 , 31 30 , 31 ^(30,31){ }^{30,31}. Depending on material, τ t r τ t r tau_(tr)\tau_{t r} and τ r τ r tau_(r)\tau_{r} are the two time constants characterizing the thermal activation and radiative recombination processes of carriers, respectively. The latter process produces luminescence. Under the steady-state conditions, i.e., ∂ N / ∂ t = 0 ∂ N / ∂ t = 0 del N//del t=0\partial N / \partial t=0, one can get one solution described by 30 30 ^(30){ }^{30}
其中 G , γ c , N ′ G , γ c , N ′ G,gamma_(c),N^(')G, \gamma_{c}, N^{\prime} 和 Λ Λ Lambda\Lambda 分别表示由于光激发、电注入等引起的载流子生成（激发）率、热激活载流子的重新捕获系数、热激活载流子的总数以及局域电子态的总数。在公式 (1) 中， E a E a E_(a)E_{a} 代表材料的一个特定能量位置，例如局域载流子可以被热激活到的去局域化能级的位置 30 , 31 30 , 31 ^(30,31){ }^{30,31} 。根据材料的不同， τ t r τ t r tau_(tr)\tau_{t r} 和 τ r τ r tau_(r)\tau_{r} 是分别表征载流子的热激活和辐射复合过程的两个时间常数。后者过程产生发光。在稳态条件下，即 ∂ N / ∂ t = 0 ∂ N / ∂ t = 0 del N//del t=0\partial N / \partial t=0 ，可以得到由 30 30 ^(30){ }^{30} 描述的一个解。
N ( E , T ) = A ( T ) ⋅ f ( E , T ) ⋅ ρ ( E ) N ( E , T ) = A ( T ) ⋅ f ( E , T ) ⋅ ρ ( E ) N(E,T)=A(T)*f(E,T)*rho(E)N(E, T)=A(T) \cdot f(E, T) \cdot \rho(E)
where f ( E , T ) = 1 e ( E − E 0 ) / k B T + τ t r / τ T f ( E , T ) = 1 e E − E 0 / k B T + τ t r / τ T f(E,T)=(1)/(e^((E-E_(0))//k_(B)T)+tau_(tr)//tau_(T))f(E, T)=\frac{1}{e^{\left(E-E_{0}\right) / k_{B} T}+\tau_{t r} / \tau_{T}} represents a distribution function for localized carriers. The explicit expression of A ( T ) A ( T ) A(T)A(T) can be found in our previous publication 30 30 ^(30){ }^{30}. As argued previously by us, the lineshape of the steady-state luminescence spectrum of localized states, given by N ( E , T ) / τ r N ( E , T ) / τ r N(E,T)//tau_(r)N(E, T) / \tau_{r}, is essentially described by f ( E , T ) ⋅ ρ ( E ) f ( E , T ) ⋅ ρ ( E ) f(E,T)*rho(E)f(E, T) \cdot \rho(E). Under such circumstances, the peak position of the steady-state luminescence of localized states can be found by the following equation set 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31}
其中 f ( E , T ) = 1 e ( E − E 0 ) / k B T + τ t r / τ T f ( E , T ) = 1 e E − E 0 / k B T + τ t r / τ T f(E,T)=(1)/(e^((E-E_(0))//k_(B)T)+tau_(tr)//tau_(T))f(E, T)=\frac{1}{e^{\left(E-E_{0}\right) / k_{B} T}+\tau_{t r} / \tau_{T}} 表示局部载流子的分布函数。 A ( T ) A ( T ) A(T)A(T) 的显式表达式可以在我们之前的出版物 30 30 ^(30){ }^{30} 中找到。正如我们之前所论述的，局部态的稳态发光光谱的线形，由 N ( E , T ) / τ r N ( E , T ) / τ r N(E,T)//tau_(r)N(E, T) / \tau_{r} 给出，基本上由 f ( E , T ) ⋅ ρ ( E ) f ( E , T ) ⋅ ρ ( E ) f(E,T)*rho(E)f(E, T) \cdot \rho(E) 描述。在这种情况下，局部态的稳态发光的峰值位置可以通过以下方程组 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31} 找到。
E ( T ) = E 0 − ξ ( T ) ⋅ k B T , [ 1 ξ ( T ) ( σ k B T ) 2 − 1 ] e − ξ ( T ) = τ tr τ r × e − ( E 0 − E a ) / k B T , E ( T ) = E 0 − ξ ( T ) ⋅ k B T , 1 ξ ( T ) σ k B T 2 − 1 e − ξ ( T ) = τ tr τ r × e − E 0 − E a / k B T , {:[E(T)=E_(0)-xi(T)*k_(B)T","],[[(1)/(xi(T))((sigma)/(k_(B)T))^(2)-1]e^(-xi(T))=(tau_(tr))/(tau_(r))xxe^(-(E_(0)-E_(a))//k_(B)T)","]:}\begin{gathered} E(T)=E_{0}-\xi(T) \cdot k_{B} T, \\ {\left[\frac{1}{\xi(T)}\left(\frac{\sigma}{k_{B} T}\right)^{2}-1\right] e^{-\xi(T)}=\frac{\tau_{\mathrm{tr}}}{\tau_{\mathrm{r}}} \times e^{-\left(E_{0}-E_{a}\right) / k_{B} T},} \end{gathered}
where, E 0 E 0 E_(0)E_{0} and σ σ sigma\sigma are the parameters derived from ρ ( E ) = ρ 0 e − ( E − E 0 ) 2 / 2 σ 2 ρ ( E ) = ρ 0 e − E − E 0 2 / 2 σ 2 rho(E)=rho_(0)e^(-(E-E_(0))^(2)//2sigma^(2))\rho(E)=\rho_{0} e^{-\left(E-E_{0}\right)^{2} / 2 \sigma^{2}}, e.g., a standard Gaussian DOS for localized states. By using above equation set and taking into account the temperature induced bandgap shrinking usually described by Varshni’s empirical formula for ideal semiconductors, we can well reproduce the S-shape
其中， E 0 E 0 E_(0)E_{0} 和 σ σ sigma\sigma 是从 ρ ( E ) = ρ 0 e − ( E − E 0 ) 2 / 2 σ 2 ρ ( E ) = ρ 0 e − E − E 0 2 / 2 σ 2 rho(E)=rho_(0)e^(-(E-E_(0))^(2)//2sigma^(2))\rho(E)=\rho_{0} e^{-\left(E-E_{0}\right)^{2} / 2 \sigma^{2}} 中得出的参数，例如，局部态的标准高斯态密度。通过使用上述方程组，并考虑温度引起的带隙收缩，通常由 Varshni 的经验公式描述理想半导体，我们可以很好地重现 S 形。
temperature dependence of the steady-state luminescence of localized carriers in different materials 32 32 ^(32){ }^{32}. Here, let us show an example of application of the model to the experimental data obtained by Schömig et al. in an InGaN/ GaN quantum well sample 24 24 ^(24){ }^{24}. From Fig. 1, it can be seen that the experimental data was nearly perfectly reproduced by the steady-state LSE luminescence model.
不同材料中局部载流子的稳态发光的温度依赖性 32 32 ^(32){ }^{32} 。在这里，我们展示一个将该模型应用于 Schömig 等人在 InGaN/GaN 量子阱样品中获得的实验数据的例子 24 24 ^(24){ }^{24} 。从图 1 可以看出，实验数据几乎完美地被稳态 LSE 发光模型再现。
As addressed earlier, we are primarily interested in the solution of rate equation under transient conditions, e.g., pulsed optical or electrical excitation, for building up an analytical model for time-resolved luminescence of localized carriers in the present study. To develop such an analytical model, we need to do some analysis to some physical processes, i.e. the re-capture of already-thermally-activated carriers by localized states, and undertake necessary approximation. In the steady-state LSE model, the number of re-captured carriers per unit time is described by γ c N ′ ρ ( E ) Λ γ c N ′ ρ ( E ) Λ gamma_(c)N^(')(rho(E))/(Lambda)\gamma_{c} N^{\prime} \frac{\rho(E)}{\Lambda}, where N ′ = ∫ − ∞ + ∞ N ( E ′ , T , t ) τ e ( E ′ − E a ) / k B T d E ′ N ′ = ∫ − ∞ + ∞   N E ′ , T , t τ e E ′ − E a / k B T d E ′ N^(')=int_(-oo)^(+oo)(N(E^('),T,t))/(tau)e^((E^(')-E_(a))//k_(B)T)dE^(')N^{\prime}=\int_{-\infty}^{+\infty} \frac{N\left(E^{\prime}, T, t\right)}{\tau} e^{\left(E^{\prime}-E_{a}\right) / k_{B} T} d E^{\prime} represents the total number of thermally activated carriers. Please be noted that the re-capture rate or efficiency is assumed to be a constant for all localized states in the steady-state LSE model 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31}. However, such assumption may be no longer well justified for the localized state system under the transient excitation conditions (i.e., pulsed optical or electrical excitation) because of high carrier density at instant. It is obviously more reasonable to assume that the re-capture rate is a function of localized state energy 36 , 37 36 , 37 ^(36,37){ }^{36,37}.
如前所述，我们主要关注瞬态条件下速率方程的解，例如脉冲光学或电激发，以建立本研究中局部载流子的时间分辨发光的分析模型。为了开发这样的分析模型，我们需要对一些物理过程进行分析，即已经热激活的载流子被局部态重新捕获，并进行必要的近似。在稳态 LSE 模型中，每单位时间重新捕获的载流子数量由 γ c N ′ ρ ( E ) Λ γ c N ′ ρ ( E ) Λ gamma_(c)N^(')(rho(E))/(Lambda)\gamma_{c} N^{\prime} \frac{\rho(E)}{\Lambda} 描述，其中 N ′ = ∫ − ∞ + ∞ N ( E ′ , T , t ) τ e ( E ′ − E a ) / k B T d E ′ N ′ = ∫ − ∞ + ∞   N E ′ , T , t τ e E ′ − E a / k B T d E ′ N^(')=int_(-oo)^(+oo)(N(E^('),T,t))/(tau)e^((E^(')-E_(a))//k_(B)T)dE^(')N^{\prime}=\int_{-\infty}^{+\infty} \frac{N\left(E^{\prime}, T, t\right)}{\tau} e^{\left(E^{\prime}-E_{a}\right) / k_{B} T} d E^{\prime} 代表热激活载流子的总数。请注意，在稳态 LSE 模型中，重新捕获率或效率被假定为所有局部态的常数 22 , 30 , 31 22 , 30 , 31 ^(22,30,31){ }^{22,30,31} 。然而，在瞬态激发条件下（即脉冲光学或电激发），由于瞬时的高载流子密度，这种假设可能不再合理。显然，假设重新捕获率是局部态能量的函数 36 , 37 36 , 37 ^(36,37){ }^{36,37} 更为合理。
In the present study we thus assume that the re-capture efficiency is proportional to the number of unoccupied localized states left by carriers which are thermally activated away, as expressed by
在本研究中，我们因此假设重新捕获效率与被热激活离开的载流子留下的未占用局部态的数量成正比，如下所示
η = N ( E , T , t ) e ( E − E a ) / k B T ∫ − ∞ + ∞ N ( E ′ , T , t ) e ( E ′ − E a ) / k B T d E ′ η = N ( E , T , t ) e E − E a / k B T ∫ − ∞ + ∞   N E ′ , T , t e E ′ − E a / k B T d E ′ eta=(N(E,T,t)e^((E-E_(a))//k_(B)T))/(int_(-oo)^(+oo)N(E^('),T,t)e^((E^(')-E_(a))//k_(B)T)dE^('))\eta=\frac{N(E, T, t) e^{\left(E-E_{a}\right) / k_{B} T}}{\int_{-\infty}^{+\infty} N\left(E^{\prime}, T, t\right) e^{\left(E^{\prime}-E_{a}\right) / k_{B} T} d E^{\prime}}
Then the number of re-captured carriers per unit time can be described by
然后每单位时间内重新捕获的载体数量可以用以下公式描述：
∂ N r c ∂ t = γ c N ′ η = γ c N τ t r e ( E − E a ) / k B T . ∂ N r c ∂ t = γ c N ′ η = γ c N τ t r e E − E a / k B T . (delN_(rc))/(del t)=gamma_(c)N^(')eta=gamma_(c)(N)/(tau_(tr))e^((E-E_(a))//k_(B)T).\frac{\partial N_{r c}}{\partial t}=\gamma_{c} N^{\prime} \eta=\gamma_{c} \frac{N}{\tau_{t r}} e^{\left(E-E_{a}\right) / k_{B} T} .
Under such an assumption and the transient excitation conditions, the rate equation may be re-written as
在这样的假设和瞬态激励条件下，速率方程可以重新写为
∂ N ( E , T , t ) ∂ t = G ( E , t ) − ( 1 − γ c ) N ( E , T , t ) τ t r e ( E − E a ) / k B T − N ( E , T , t ) τ r . ∂ N ( E , T , t ) ∂ t = G ( E , t ) − 1 − γ c N ( E , T , t ) τ t r e E − E a / k B T − N ( E , T , t ) τ r . (del N(E,T,t))/(del t)=G(E,t)-(1-gamma_(c))(N(E,T,t))/(tau_(tr))e^((E-E_(a))//k_(B)T)-(N(E,T,t))/(tau_(r)).\frac{\partial N(E, T, t)}{\partial t}=G(E, t)-\left(1-\gamma_{c}\right) \frac{N(E, T, t)}{\tau_{t r}} e^{\left(E-E_{a}\right) / k_{B} T}-\frac{N(E, T, t)}{\tau_{r}} .
Eq. (7) can be simplified as
公式（7）可以简化为
∂ N ( E , T , t ) ∂ t + P ( E , T ) N ( E , T , t ) = G ( E , t ) , ∂ N ( E , T , t ) ∂ t + P ( E , T ) N ( E , T , t ) = G ( E , t ) , (del N(E,T,t))/(del t)+P(E,T)N(E,T,t)=G(E,t),\frac{\partial N(E, T, t)}{\partial t}+P(E, T) N(E, T, t)=G(E, t),
where  哪里
P ( E , T ) = ( 1 − γ c ) e β ( E − E a ) / τ t r + 1 / τ r , P ( E , T ) = 1 − γ c e β E − E a / τ t r + 1 / τ r , P(E,T)=(1-gamma_(c))e^(beta(E-E_(a)))//tau_(tr)+1//tau_(r),P(E, T)=\left(1-\gamma_{c}\right) e^{\beta\left(E-E_{a}\right)} / \tau_{t r}+1 / \tau_{r},
and  和
G ( E , t ) = g ( t ) ρ ( E ) . G ( E , t ) = g ( t ) ρ ( E ) .  G(E,t)=g(t)rho(E)". "G(E, t)=g(t) \rho(E) \text {. }
Here β = 1 / k B T β = 1 / k B T beta=1//k_(B)T\beta=1 / k_{B} T. Under such circumstance, a solution of Eq. (8) can be found as
在这种情况下，可以找到方程（8）的解。
N ( E , T , t ) = ρ ( E ) e − P ( E , T ) ⋅ t ∫ g ( t ) e P ( E , T ) ⋅ t d t . N ( E , T , t ) = ρ ( E ) e − P ( E , T ) ⋅ t ∫ g ( t ) e P ( E , T ) ⋅ t d t . N(E,T,t)=rho(E)e^(-P(E,T)*t)int g(t)e^(P(E,T)*t)dt.N(E, T, t)=\rho(E) e^{-P(E, T) \cdot t} \int g(t) e^{P(E, T) \cdot t} d t .
If the excitation is continuous and constant, i.e., the optical excitation or injection current was kept a constant for time, g ( t ) = g 0 g ( t ) = g 0 g(t)=g_(0)g(t)=g_{0}, then
如果激励是连续且恒定的，即光学激励或注入电流在时间 g ( t ) = g 0 g ( t ) = g 0 g(t)=g_(0)g(t)=g_{0} 内保持恒定，则
N ( E , T , t ) = τ t r g 0 ρ ( E ) ( 1 − γ c ) e β ( E − E a ) + τ t r / τ r . N ( E , T , t ) = τ t r g 0 ρ ( E ) 1 − γ c e β E − E a + τ t r / τ r . N(E,T,t)=(tau_(tr)g_(0)rho(E))/((1-gamma_(c))e^(beta(E-E_(a)))+tau_(tr)//tau_(r)).N(E, T, t)=\frac{\tau_{t r} g_{0} \rho(E)}{\left(1-\gamma_{c}\right) e^{\beta\left(E-E_{a}\right)}+\tau_{t r} / \tau_{r}} .
By setting γ c = 0 γ c = 0 gamma_(c)=0\gamma_{c}=0, Eq. (12) is reduced to the steady-state solution described by Eq. (2).
通过设置 γ c = 0 γ c = 0 gamma_(c)=0\gamma_{c}=0 ，方程(12)简化为由方程(2)描述的稳态解。
It is well known that in time-resolved luminescence measurements, pulsed excitation was used. For a pulsed excitation, time-dependent creation of carriers may be mathematically stated as 36 36 ^(36){ }^{36}
众所周知，在时间分辨的发光测量中，使用了脉冲激发。对于脉冲激发，载流子的时间依赖性生成可以数学上表述为 36 36 ^(36){ }^{36} 。
g ( t ) = g 0 e − ( t − t 0 ) 2 2 σ t 2 . g ( t ) = g 0 e − t − t 0 2 2 σ t 2 . g(t)=g_(0)e^((-(t-t_(0))^(2))/(2sigma_(t)^(2))).g(t)=g_{0} e^{\frac{-\left(t-t_{0}\right)^{2}}{2 \sigma_{t}^{2}}} .
Here σ t σ t sigma_(t)\sigma_{t} is an important parameter governing the generation process of carriers. Under such pulsed excitation, a solution of Eq. (8) may be written as
这里 σ t σ t sigma_(t)\sigma_{t} 是一个重要参数，控制载流子的生成过程。在这种脉冲激发下，方程（8）的解可以写成
N ( E , T , t ) = π 2 σ t g 0 ρ ( E ) { erf [ t − ( t 0 + σ t 2 P ( E , T ) ) 2 σ t ] + 1 } e σ t 2 P ( E , T ) 2 / 2 + ( t 0 − t ) P ( E , T ) N ( E , T , t ) = π 2 σ t g 0 ρ ( E ) erf t − t 0 + σ t 2 P ( E , T ) 2 σ t + 1 e σ t 2 P ( E , T ) 2 / 2 + t 0 − t P ( E , T ) N(E,T,t)=sqrt((pi)/(2))sigma_(t)g_(0)rho(E){erf[(t-(t_(0)+sigma_(t)^(2)P(E,T)))/(sqrt2sigma_(t))]+1}e^(sigma_(t)^(2)P(E,T)^(2)//2+(t_(0)-t)P(E,T))N(E, T, t)=\sqrt{\frac{\pi}{2}} \sigma_{t} g_{0} \rho(E)\left\{\operatorname{erf}\left[\frac{t-\left(t_{0}+\sigma_{t}^{2} P(E, T)\right)}{\sqrt{2} \sigma_{t}}\right]+1\right\} e^{\sigma_{t}^{2} P(E, T)^{2} / 2+\left(t_{0}-t\right) P(E, T)}
Then, time evolution of luminescence intensity of localized carriers may be formulated as
然后，局部载流子的发光强度的时间演化可以表述为
Figure 2. Calculated TRPL spectra (image) of a localized state system with Eq. (15). Obviously, the theoretical luminescence lifetime shows a distinct dependence on energy (right figure), while luminescence spectrum exhibits an interesting dependence on the delay time (top figure).
图 2. 使用公式(15)计算的局部态系统的 TRPL 光谱（图像）。显然，理论发光寿命对能量（右图）表现出明显的依赖性，而发光光谱则对延迟时间（上图）表现出有趣的依赖性。
Figure 3. Temperature dependence (solid circles) of the nonradiative lifetime for localized excitons of InGaN InGaN InGaN\operatorname{InGaN} MQWs, adopted from ref. 40. The solid line represents a best fitting with Eq. (18).
图 3. 从参考文献 40 中采用的 InGaN InGaN InGaN\operatorname{InGaN} MQWs 局域激子的非辐射寿命的温度依赖性（实心圆点）。实线表示与方程（18）的最佳拟合。
I L ( E , T , t ) = N ( E , T , t ) τ r ∝ ρ ( E ) ⋅ { erf [ ( t − t 0 ) − σ t 2 / τ L 2 σ t ] + 1 } e − ( t − t 0 ) / τ L , I L ( E , T , t ) = N ( E , T , t ) τ r ∝ ρ ( E ) ⋅ erf t − t 0 − σ t 2 / τ L 2 σ t + 1 e − t − t 0 / τ L , I_(L)(E,T,t)=(N(E,T,t))/(tau_(r))prop rho(E)*{erf[((t-t_(0))-sigma_(t)^(2)//tau_(L))/(sqrt2sigma_(t))]+1}e^(-(t-t_(0))//tau_(L)),I_{L}(E, T, t)=\frac{N(E, T, t)}{\tau_{r}} \propto \rho(E) \cdot\left\{\operatorname{erf}\left[\frac{\left(t-t_{0}\right)-\sigma_{t}^{2} / \tau_{L}}{\sqrt{2} \sigma_{t}}\right]+1\right\} e^{-\left(t-t_{0}\right) / \tau_{L}},
where  哪里
τ L ( E , T ) = 1 P ( E , T ) = τ r 1 + τ r τ t r ( 1 − γ c ) e ( E − E a ) / k B T . τ L ( E , T ) = 1 P ( E , T ) = τ r 1 + τ r τ t r 1 − γ c e E − E a / k B T . tau_(L)(E,T)=(1)/(P(E,T))=(tau_(r))/(1+(tau_(r))/(tau_(tr))(1-gamma_(c))e^((E-E_(a))//k_(B)T)).\tau_{L}(E, T)=\frac{1}{P(E, T)}=\frac{\tau_{r}}{1+\frac{\tau_{r}}{\tau_{t r}}\left(1-\gamma_{c}\right) e^{\left(E-E_{a}\right) / k_{B} T}} .
In fact, Eq. (16) can be simplified as
实际上，公式（16）可以简化为
τ L = τ r 1 + e α ( E − E m ) , τ L = τ r 1 + e α E − E m , tau_(L)=(tau_(r))/(1+e^(alpha(E-E_(m)))),\tau_{L}=\frac{\tau_{r}}{1+e^{\alpha\left(E-E_{m}\right)}},
which is widely adopted in literature 37 − 39 37 − 39 ^(37-39){ }^{37-39}. Here, E m E m E_(m)E_{m} was defined by Oueslati et al. as a specific energy at which the recombination rate equals the transfer rate, and α α alpha\alpha was a model dependent parameter with the unit of reverse energy 38 38 ^(38){ }^{38}. Note that our model takes into account the temperature effect. In other words, Eq. (16) gives a quantitative description of dispersive thermodynamics of localized carriers for luminescence.
在文献中被广泛采用 37 − 39 37 − 39 ^(37-39){ }^{37-39} 。在这里， E m E m E_(m)E_{m} 被 Oueslati 等人定义为重组速率等于转移速率的特定能量，而 α α alpha\alpha 是一个依赖于模型的参数，其单位为反向能量 38 38 ^(38){ }^{38} 。请注意，我们的模型考虑了温度效应。换句话说，公式(16)对局部载流子的色散热力学进行了定量描述，以用于发光。
By adopting parameters of σ t = 1.5 ps , γ c = 0.148 , τ t r = 0.625 ns , τ r = 16.557 ns , E a = 2.718 eV , E 0 = 2.684 eV σ t = 1.5 ps , γ c = 0.148 , τ t r = 0.625 ns , τ r = 16.557 ns , E a = 2.718 eV , E 0 = 2.684 eV sigma_(t)=1.5ps,gamma_(c)=0.148,tau_(tr)=0.625ns,tau_(r)=16.557ns,E_(a)=2.718eV,E_(0)=2.684eV\sigma_{t}=1.5 \mathrm{ps}, \gamma_{c}=0.148, \tau_{t r}=0.625 \mathrm{~ns}, \tau_{r}=16.557 \mathrm{~ns}, E_{a}=2.718 \mathrm{eV}, E_{0}=2.684 \mathrm{eV}, σ = 50 meV σ = 50 meV sigma=50meV\sigma=50 \mathrm{meV} and T = 300 K T = 300 K T=300KT=300 \mathrm{~K}, we calculate time-resolved PL (TRPL) spectra of a localized state system with Eq. (15). The calculated TRPL spectra are illustrated in a two-dimensional image in Fig. 2. The horizontal axis of the image stands for energy while the vertical axis represents delay time. The image contrast displays luminescence intensity, i.e., red color means strong emission. The top figure shows several theoretical PL spectra at different delay times, whereas the right figure depicts three luminescence intensity decaying traces for three different photon energies. Obviously, the theoretical TRPL spectra based on Eq. (15) exhibit interesting evolution tendency upon delay time. For example, luminescence lifetime shows a distinct dependence on energy and luminescence spectrum exhibits an interesting dependence on the delay time, i.e., fast redshift of the peak position at an early stage of the delay time and rapid narrowing of the lineshape at higher energy.
通过采用参数 σ t = 1.5 ps , γ c = 0.148 , τ t r = 0.625 ns , τ r = 16.557 ns , E a = 2.718 eV , E 0 = 2.684 eV σ t = 1.5 ps , γ c = 0.148 , τ t r = 0.625 ns , τ r = 16.557 ns , E a = 2.718 eV , E 0 = 2.684 eV sigma_(t)=1.5ps,gamma_(c)=0.148,tau_(tr)=0.625ns,tau_(r)=16.557ns,E_(a)=2.718eV,E_(0)=2.684eV\sigma_{t}=1.5 \mathrm{ps}, \gamma_{c}=0.148, \tau_{t r}=0.625 \mathrm{~ns}, \tau_{r}=16.557 \mathrm{~ns}, E_{a}=2.718 \mathrm{eV}, E_{0}=2.684 \mathrm{eV} 、 σ = 50 meV σ = 50 meV sigma=50meV\sigma=50 \mathrm{meV} 和 T = 300 K T = 300 K T=300KT=300 \mathrm{~K} ，我们利用公式(15)计算了局部态系统的时间分辨光致发光（TRPL）光谱。计算得到的 TRPL 光谱在图 2 中以二维图像的形式展示。图像的横轴代表能量，而纵轴表示延迟时间。图像的对比度显示了发光强度，即红色表示强发射。顶部图形显示了在不同延迟时间下的几个理论 PL 光谱，而右侧图形描绘了三种不同光子能量的发光强度衰减轨迹。显然，基于公式(15)的理论 TRPL 光谱在延迟时间上表现出有趣的演变趋势。例如，发光寿命对能量表现出明显的依赖性，而发光光谱则对延迟时间表现出有趣的依赖性，即在延迟时间的早期阶段峰位快速红移，以及在较高能量下线形迅速变窄。
Figure 4. Time-resolved photoluminescence traces (various solid symbols) of a single-layer InGaN alloy measured by Satake et al. for different energy 41 41 ^(41){ }^{41}. The solid lines are the fitting curves with Eq. (15).
图 4. Satake 等人测量的单层 InGaN 合金在不同能量 41 41 ^(41){ }^{41} 下的时间分辨光致发光轨迹（各种实心符号）。实线为使用公式(15)拟合的曲线。
Figure 5. Various lifetimes (solid symbols) of localized carriers in Ag-coated InGaN/GaN quantum wells measured by Okamoto et al. 42 42 ^(42){ }^{42} (a) for different energy at room temperature and (b) for different temperatures. The solid lines are the calculated results with Eqs (16) and (18), independently or jointly, while the dashed lines represent the theoretical curves for overall luminescence lifetime with different parameters (a) and the nonradiative lifetime (b) respectively. The short dashed line in (b) indicates constant radiative lifetime.
图 5. Okamoto 等人测量的 Ag 涂层 InGaN/GaN 量子阱中局部载流子的不同寿命（实心符号） 42 42 ^(42){ }^{42} ，（a）在室温下不同能量的情况，以及（b）在不同温度下的情况。实线是根据方程（16）和（18）独立或联合计算的结果，而虚线分别表示不同参数下的整体发光寿命的理论曲线（a）和非辐射寿命（b）。（b）中的短虚线表示恒定的辐射寿命。
E ( eV ) E ( eV ) E(eV)E(\mathrm{eV}) 3.08 3.12 3.16 3.20
τ L ( p s ) τ L ( p s ) tau_(L)(ps)\tau_{L}(p s) 932.91 712.54 510.19 366.48
σ t ( p s ) σ t ( p s ) sigma_(t)(ps)\sigma_{t}(p s) 19.49 19.42 19.43 18.99
E(eV) 3.08 3.12 3.16 3.20 tau_(L)(ps) 932.91 712.54 510.19 366.48 sigma_(t)(ps) 19.49 19.42 19.43 18.99| $E(\mathrm{eV})$ | 3.08 | 3.12 | 3.16 | 3.20 | | :--- | :--- | :--- | :--- | :--- | | $\tau_{L}(p s)$ | 932.91 | 712.54 | 510.19 | 366.48 | | $\sigma_{t}(p s)$ | 19.49 | 19.42 | 19.43 | 18.99 |
Table 1. Parameters for the solid curves in Fig. 4.
表 1. 图 4 中实线的参数。
Figure 6. Luminescence lifetimes (solid symbols) of InGaAsN epilayer measured by Mair et al. for (a) different energy and (b) different temperatures 37 37 ^(37){ }^{37}. The solid lines are the theoretical curves with Eq. (16).
图 6. Mair 等人测量的 InGaAsN 外延层的发光寿命（实心符号），分别对应于（a）不同能量和（b）不同温度 37 37 ^(37){ }^{37} 。实线为公式（16）的理论曲线。
It is well known that the measured luminescence time constant is usually written as τ L − 1 = τ n r − 1 + τ r − 1 τ L − 1 = τ n r − 1 + τ r − 1 tau_(L)^(-1)=tau_(nr)^(-1)+tau_(r)^(-1)\tau_{L}^{-1}=\tau_{n r}^{-1}+\tau_{r}^{-1}. By using this relationship and Eq. (16), we can derive an explicit expression of τ n r τ n r tau_(nr)\tau_{n r} :
众所周知，测得的发光时间常数通常写作 τ L − 1 = τ n r − 1 + τ r − 1 τ L − 1 = τ n r − 1 + τ r − 1 tau_(L)^(-1)=tau_(nr)^(-1)+tau_(r)^(-1)\tau_{L}^{-1}=\tau_{n r}^{-1}+\tau_{r}^{-1} 。通过使用这个关系和公式 (16)，我们可以推导出 τ n r τ n r tau_(nr)\tau_{n r} 的显式表达式：
τ n r ( E , T ) = τ t r ( 1 − γ c ) e − ( E − E a ) / k B T . τ n r ( E , T ) = τ t r 1 − γ c e − E − E a / k B T . tau_(nr)(E,T)=(tau_(tr))/((1-gamma_(c)))e^(-(E-E_(a))//k_(B)T).\tau_{n r}(E, T)=\frac{\tau_{t r}}{\left(1-\gamma_{c}\right)} e^{-\left(E-E_{a}\right) / k_{B} T} .
Above physical quantity may be regarded as an effective nonradiative recombination time of localized carriers. Eq. (18) tells us that the effective nonradiative recombination time of localized carriers exhibits a distinct exponential dependence on temperature and energy, predominantly governing the luminescence lifetime. The temperature dependence of nonradiative lifetime for localized excitons of InGaN MQWs, measured by Narukawa et al. 40 40 ^(40){ }^{40}, can be quantitatively interpreted with Eq. (18), as shown in Fig. 3.
上述物理量可以视为局部载流子的有效非辐射复合时间。公式（18）告诉我们，局部载流子的有效非辐射复合时间对温度和能量表现出明显的指数依赖性，主要主导发光寿命。Narukawa 等人测量的 InGaN 多量子阱局部激子的非辐射寿命的温度依赖性可以用公式（18）进行定量解释，如图 3 所示。
Now we apply the model developed in the present work to the time-resolved photoluminescence traces of a single-layer InGaN alloy measured by Satake et al. 41 41 ^(41){ }^{41}. In such InGaN ternary alloy, localized excitons were proposed to give a major spontaneous emission 20 , 41 20 , 41 ^(20,41){ }^{20,41}. In Fig. 4, various symbols represent the experimental data from ref. 41, while the solid lines are the theoretical curves with Eq. (15). Parameters used in the calculation are listed in Table 1. The rise time keeps almost constant of τ rise ≈ 2.5631 σ t = 49.80 p s τ rise  ≈ 2.5631 σ t = 49.80 p s tau_("rise ")~~2.5631sigma_(t)=49.80 ps\tau_{\text {rise }} \approx 2.5631 \sigma_{t}=49.80 p s, while the decay time decreases as the luminescence energy increases. From Fig. 4, its can be seen that very good agreement between theory and experiment is achieved.
现在我们将本研究中开发的模型应用于 Satake 等人测量的单层 InGaN 合金的时间分辨光致发光曲线 41 41 ^(41){ }^{41} 。在这种 InGaN 三元合金中，局域激子被提出作为主要的自发发射来源 20 , 41 20 , 41 ^(20,41){ }^{20,41} 。在图 4 中，各种符号代表参考文献 41 中的实验数据，而实线是使用公式(15)得到的理论曲线。计算中使用的参数列在表 1 中。上升时间几乎保持不变 τ rise ≈ 2.5631 σ t = 49.80 p s τ rise  ≈ 2.5631 σ t = 49.80 p s tau_("rise ")~~2.5631sigma_(t)=49.80 ps\tau_{\text {rise }} \approx 2.5631 \sigma_{t}=49.80 p s ，而衰减时间随着发光能量的增加而减少。从图 4 中可以看出，理论与实验之间达成了非常好的一致性。
In addition to InGaN alloy, InGaN/GaN quantum wells, which usually act as the active layers of blue/green LED’s 19 19 ^(19){ }^{19}, also exhibit strong localization effect. Okamoto et al. experimentally measured the PL lifetimes of localized carriers in Ag-coated InGaN/GaN quantum wells for different energy and different temperatures 42 42 ^(42){ }^{42}. The solid symbols in Fig. 5(a) and (b) are their experimental data. For the luminescence lifetimes, one can see that they exhibit an interesting dependence on energy. In Fig. 5(a), solid and dashed lines are two theoretical curves with Eq. (16). For the solid line, parameters of γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns gamma_(c)=0.148,E_(a)=2.718eV,tau_(tr)=0.625ns\gamma_{c}=0.148, E_{a}=2.718 \mathrm{eV}, \tau_{t r}=0.625 \mathrm{~ns} and τ r = 5.273 ns τ r = 5.273 ns tau_(r)=5.273ns\tau_{r}=5.273 \mathrm{~ns} were adopted, while parameters of γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns gamma_(c)=0.148,E_(a)=2.718eV,tau_(tr)=0.625ns\gamma_{c}=0.148, E_{a}=2.718 \mathrm{eV}, \tau_{t r}=0.625 \mathrm{~ns} and τ r = 16.557 ns τ r = 16.557 ns tau_(r)=16.557ns\tau_{r}=16.557 \mathrm{~ns} for the dashed curve. Obviously, Okamoto et al.'s experimental data can be well reproduced with Eq. (16) when the first set of parameters was adopted. For the second set of parameters, agreement between theory and experiment becomes unideal, especially in the lower energy region. The key reason causing this deviation between theory and experiment in the low energy region is that τ r τ r tau_(r)\tau_{\mathrm{r}} in the second set of parameter is much larger. However, when this value is adopted, temperature dependence of the average luminescence lifetime of localized carriers can be clearly elucidated, as shown by the solid line in Fig. 5(b). That is, the luminescence lifetime is predominantly determined by constant radiative lifetime ( τ r = 16.557 ns τ r = 16.557 ns tau_(r)=16.557ns\tau_{r}=16.557 \mathrm{~ns} ) for low temperatures < 100 K < 100 K < 100K<100 \mathrm{~K}, while it is gradually controlled by temperature-dependent nonradiative lifetime for higher temperatures. It shall be noted that in Fig. 5(a), the dispersive luminescence lifetimes τ L τ L tau_(L)\tau_{L} (i.e., energy dependent) was measured at room temperature ( ∼ 300 K ∼ 300 K ∼300K\sim 300 \mathrm{~K} ), whereas the average luminescence
除了 InGaN 合金，InGaN/GaN 量子阱通常作为蓝色/绿色 LED 的活性层 19 19 ^(19){ }^{19} ，也表现出强烈的局域化效应。冈本等人实验测量了不同能量和不同温度下银涂层 InGaN/GaN 量子阱中局域载流子的光致发光寿命 42 42 ^(42){ }^{42} 。图 5(a)和(b)中的实心符号是他们的实验数据。对于发光寿命，可以看到它们对能量表现出有趣的依赖性。在图 5(a)中，实线和虚线是基于方程(16)的两条理论曲线。对于实线，采用了参数 γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns gamma_(c)=0.148,E_(a)=2.718eV,tau_(tr)=0.625ns\gamma_{c}=0.148, E_{a}=2.718 \mathrm{eV}, \tau_{t r}=0.625 \mathrm{~ns} 和 τ r = 5.273 ns τ r = 5.273 ns tau_(r)=5.273ns\tau_{r}=5.273 \mathrm{~ns} ，而虚线则采用了参数 γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns γ c = 0.148 , E a = 2.718 eV , τ t r = 0.625 ns gamma_(c)=0.148,E_(a)=2.718eV,tau_(tr)=0.625ns\gamma_{c}=0.148, E_{a}=2.718 \mathrm{eV}, \tau_{t r}=0.625 \mathrm{~ns} 和 τ r = 16.557 ns τ r = 16.557 ns tau_(r)=16.557ns\tau_{r}=16.557 \mathrm{~ns} 。显然，当采用第一组参数时，冈本等人的实验数据可以很好地用方程(16)重现。对于第二组参数，理论与实验之间的吻合变得不理想，特别是在低能量区域。造成低能量区域理论与实验之间偏差的关键原因是第二组参数中的 τ r τ r tau_(r)\tau_{\mathrm{r}} 要大得多。 然而，当采用该值时，局域载流子的平均发光寿命的温度依赖性可以清晰地阐明，如图 5(b)中的实线所示。也就是说，发光寿命主要由低温下的恒定辐射寿命（ τ r = 16.557 ns τ r = 16.557 ns tau_(r)=16.557ns\tau_{r}=16.557 \mathrm{~ns} ）决定，而在高温下则逐渐受到温度依赖的非辐射寿命的控制。需要注意的是，在图 5(a)中，分散的发光寿命（ τ L τ L tau_(L)\tau_{L} ，即能量依赖）是在室温下测量的（ ∼ 300 K ∼ 300 K ∼300K\sim 300 \mathrm{~K} ），而平均发光寿命
lifetimes for all interested energy were measured at different temperatures in Fig. 5(b). For localized carriers, their radiative recombination lifetime usually does not change with temperature in low temperature region 40 40 ^(40){ }^{40}. At high temperatures, τ n r τ n r tau_(nr)\tau_{n r} decreases significantly and predominantly determines the luminescence lifetime τ L τ L tau_(L)\tau_{L}. In Okamoto et al.'s experiment, plasmonic effect of Ag coating thin layer may be a strong function of temperature, which causes very different radiative recombination lifetimes of localied carriers. That is why different values of radiative recombination time τ r τ r tau_(r)\tau_{r} should be used to reproduce the energy dependence of luminescence lifetime τ L τ L tau_(L)\tau_{L}.
在图 5(b)中，测量了所有感兴趣能量在不同温度下的寿命。对于局域载流子，它们的辐射复合寿命通常在低温区域 40 40 ^(40){ }^{40} 下不会随温度变化。在高温下， τ n r τ n r tau_(nr)\tau_{n r} 显著降低，并主要决定了发光寿命 τ L τ L tau_(L)\tau_{L} 。在冈本等人的实验中，银涂层薄层的等离子体效应可能是温度的强函数，这导致局域载流子的辐射复合寿命非常不同。这就是为什么应该使用不同的辐射复合时间 τ r τ r tau_(r)\tau_{r} 来重现发光寿命 τ L τ L tau_(L)\tau_{L} 的能量依赖性。
In addition to InGaN/GaN materails system, this model is also capable of giving quantitative interpretation to the experiemntal data reported by different groups for very different materials with localization effect, such as InGaAsN 37 InGaAsN 37 InGaAsN^(37)\operatorname{InGaAsN}{ }^{37}, InAs QDs 43 43 ^(43){ }^{43} and GaInP 44 44 ^(44){ }^{44}. As shown in Fig. 6, for example, the temperature and energy dependent recombination dynamics of InGaAsN InGaAsN InGaAsN\operatorname{InGaAsN} epilayer can be well fitted with Eq. (16) using the parameters of γ c = 0.220 γ c = 0.220 gamma_(c)=0.220\gamma_{c}=0.220, E a = 1.244 eV , τ t r = 0.129 ns E a = 1.244 eV , τ t r = 0.129 ns E_(a)=1.244eV,tau_(tr)=0.129nsE_{a}=1.244 \mathrm{eV}, \tau_{t r}=0.129 \mathrm{~ns}, and τ r = 0.342 ns τ r = 0.342 ns tau_(r)=0.342ns\tau_{\mathrm{r}}=0.342 \mathrm{~ns}. Again, excellent agreement between experiment and theory is achieved.
除了 InGaN/GaN 材料系统外，该模型还能够对不同组报告的具有局部化效应的非常不同材料的实验数据进行定量解释，例如 InGaAsN 37 InGaAsN 37 InGaAsN^(37)\operatorname{InGaAsN}{ }^{37} 、InAs 量子点 43 43 ^(43){ }^{43} 和 GaInP 44 44 ^(44){ }^{44} 。如图 6 所示，例如， InGaAsN InGaAsN InGaAsN\operatorname{InGaAsN} 外延层的温度和能量依赖的复合动力学可以很好地用方程(16)拟合，使用的参数为 γ c = 0.220 γ c = 0.220 gamma_(c)=0.220\gamma_{c}=0.220 、 E a = 1.244 eV , τ t r = 0.129 ns E a = 1.244 eV , τ t r = 0.129 ns E_(a)=1.244eV,tau_(tr)=0.129nsE_{a}=1.244 \mathrm{eV}, \tau_{t r}=0.129 \mathrm{~ns} 和 τ r = 0.342 ns τ r = 0.342 ns tau_(r)=0.342ns\tau_{\mathrm{r}}=0.342 \mathrm{~ns} 。再次实现了实验与理论之间的良好一致性。
In conclusion, an analytical generalized model for time-resolved luminescence of localized carriers is developed, which is capable of giving quantitative description of dispersive thermodynamics of localized carriers. The explicit expressions of effective luminescence lifetime and nonradiative recombination lifetime of localized carriers were derived for the analytical model. The formulas were used to quantitatively interpret temperature- and energy-dependent time-resolved luminescence data measured by several groups. The model and its applications enable us obtain state-of-the-art understanding of time-resolved luminescence processes of localized carriers, especially temperature and energy dependence of these dynamic processes in semiconductors.
总之，开发了一种用于局部载流子时间分辨发光的分析广义模型，该模型能够对局部载流子的色散热力学进行定量描述。为该分析模型推导了局部载流子的有效发光寿命和非辐射复合寿命的显式表达式。这些公式被用于定量解释多个研究小组测得的温度和能量依赖的时间分辨发光数据。该模型及其应用使我们能够获得对局部载流子的时间分辨发光过程的最先进理解，特别是这些动态过程在半导体中的温度和能量依赖性。

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Acknowledgements  致谢

The study was financially supported by HK-RGC-GRF Grants (Grant No. HKU 705812P), Natural Science Foundation of China (Grant No. 11374247), and in part by a grant from the University Grants Committee Areas of Excellence Scheme of the Hong Kong Special Administrative Region, China (Project No. [AoE/P-03/08]).
本研究得到了香港研究资助局一般研究基金（资助编号：HKU 705812P）、中国自然科学基金（资助编号：11374247）的财政支持，并部分得到了中国香港特别行政区大学资助委员会卓越领域计划的资助（项目编号：[AoE/P-03/08]）。

Author Contributions  作者贡献

S.J.X. conceived and supervised the project. Z.C.S. and S.J.X. wrote the manuscript.
S.J.X. 设计并监督了该项目。Z.C.S. 和 S.J.X. 撰写了手稿。

Additional Information  附加信息

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  1. Department of Physics, Shenzhen Institute of Research and Innovation (SIRI), and HKU-CAS Joint Laboratory on New Materials, The University of Hong Kong, Pokfulam Road, Hong Kong, China. Correspondence and requests for materials should be addressed to S.X. (email: sjxu@hku.hk)
    深圳研究与创新院物理系，香港大学新材料 HKU-CAS 联合实验室，香港薄扶林道，香港，中国。有关通讯和材料请求应发送至 S.X.（电子邮件：sjxu@hku.hk）