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 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 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 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 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}. 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}. 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}. 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}. 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} 。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} 。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} 。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} 。
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}. 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\operatorname{InGaN} systems ^(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}. Li et al. developed an analytical model for steady-state PL of localized state ensemble ^(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}. 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} 。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\operatorname{InGaN} 系统 ^(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} 。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} 。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} 。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\mathrm{InGaN} / \mathrm{GaN} quantum well. The experimental data (solid circles) were measured by Schömig et al. ^(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\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} ,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}, but also interprets V-shape temperature dependence of PL width and even temperature dependence of integrated luminescence intensity ^(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}. 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}. 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}. 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} ,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} 。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} 。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} 。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} 。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, rho(E)\rho(E), time evolution of the excited carrier concentration N(E,T,t)N(E, T, t) may be described by a partial differential equation ^(22,30,35){ }^{22,30,35} 对于具有局域电子态总态密度(DOS) rho(E)\rho(E) 的发光系统,激发载流子浓度 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} 来描述
where G,gamma_(c),N^(')G, \gamma_{c}, N^{\prime}, and 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} 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}. Depending on material, tau_(tr)\tau_{t r} and 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., del N//del t=0\partial N / \partial t=0, one can get one solution described by ^(30){ }^{30} 其中 G,gamma_(c),N^(')G, \gamma_{c}, N^{\prime} 和 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} 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} 。根据材料的不同, tau_(tr)\tau_{t r} 和 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 del N//del t=0\partial N / \partial t=0 ,可以得到由 ^(30){ }^{30} 描述的一个解。
where 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) can be found in our previous publication ^(30){ }^{30}. As argued previously by us, the lineshape of the steady-state luminescence spectrum of localized states, given by N(E,T)//tau_(r)N(E, T) / \tau_{r}, is essentially described by 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} 其中 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) 的显式表达式可以在我们之前的出版物 ^(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)//tau_(r)N(E, T) / \tau_{r} 给出,基本上由 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} 找到。
where, E_(0)E_{0} and sigma\sigma are the parameters derived from 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} 和 sigma\sigma 是从 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}. 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}. 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} 。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} 。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 gamma_(c)N^(')(rho(E))/(Lambda)\gamma_{c} N^{\prime} \frac{\rho(E)}{\Lambda}, where 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}. 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}. 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 gamma_(c)N^(')(rho(E))/(Lambda)\gamma_{c} N^{\prime} \frac{\rho(E)}{\Lambda} 描述,其中 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} 。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} 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