1.Schoolof Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 2.Department of Physics, Renmin University of China, Beijing 100876, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 21961132023, 11774026)
Received Date:25 August 2020
Accepted Date:10 October 2020
Available Online:04 February 2021
Published Online:20 February 2021
Abstract:It is of ongoing interest to uncover energy and charge transfer processes in molecular systems, which are essentially important for photovoltaic cells or light emitting diodes. The exciton-exciton annihilation is one of the important aspects in excitation energy transfer in molecular aggregations, so it is important to study its dynamics of exciton-exciton annihilation, and to compare the theoretical parameters with the related transient absorption signal. Upon the excitation of laser pulses, multiple excitons can be produced in molecular aggregations, and its annihilation process is composed of two steps. The first step is that two excitations existing in the first excited state of the molecules move together so that their excitation energy can be used to create a high excited state in one molecule, called exciton fussion. The second step is that an ultrafast internal conversion process brings the molecule which is in the higher excited state back to the first excited state. This paper uses the scheme of classical rate equation in the approximation of weak coupling among molecules to describe the dynamics of exciton-exciton annihilation. With the parameters of squaraine, the effects of external or internal parameters such as the intensity of external field, the dipole configuration in aggregations, the decay rate of molecules on the annihilation process are studied. The relationship between the relaxation time of exciton in the first excited state and the high excited state, between their times of coherent charge transfer, and between their times of exciton and annihilation are studied. These conclusions are suitable to the aggregations with their single molecule having an energy level of ${E_{\rm fm}} \approx 2{E_{\rm em}}$. It is found that the J-aggregate has a higher rate of annihilation than the H-aggregate because its coherent energy transfer time is shorter than H-aggregate’s. The high-intensity external field makes high exciton-exciton annihilation rate. The dipole configuration and the decay rate of higher excited state of molecules have strong effects on the annihilation, so one can adjust these factors to control the exciton-exciton annihilation in molecular aggregations. Keywords:exciton-exciton annihilation/ rate equation/ singlet state annihilation/ excitation of femtosecond laser pulse
表1率方程的输入参数 Table1.Input parameters of the rate equation.
文献[9]中已经讨论了聚集体内单体数量对分子激发态动力学过程的影响, 分子数目越多, 湮灭现象越明显, 对应的${\bar P^M}$随时间衰变越快, 由此可以看到激子-激子湮灭在聚集体激发态动力学中的明显影响. 为了讨论高阶激发态的衰变率对激子-激子湮灭过程的影响, 在图2中给出了4个分子组成的聚集体在不同的r下第一激发态和高阶激发态电子占据数的动力学过程. 由图2可以看到, 在J型聚集体中${\bar P^4}$的衰变随着r的增大而变慢, 同时${\bar N^4}$的峰值下降, 说明高阶激发态的衰变率高会在一定程度上抑制激子-激子湮灭过程. 这可以从激子-激子湮灭过程来分析. 湮灭过程实际由两部分组成, 首先是融合过程, 然后是高阶激发态的内转换而导致的湮灭过程, 融合过程与激发态能量转移密切相关. 这里的最近邻相互作用 $J_{mn}^{\rm{J}} = - 37.6\;{\rm{meV}}$, 其对应的相干转移时间(${{\pi \hbar }}/{J}$)约为55 fs; 融合对应的耦合相互作用$K_{mn}^{\rm{J}} = - 1.3\;{\rm{meV}}$, 对应的相干转移时间约为1.6 ps. 高阶激发态的衰变时间远快于激子融合的相干能量转移时间, 这样改变衰变率r在100飞秒附近就使第一激发态和高阶激发态动力学产生显著的变化. 本研究组还比较了H型聚集体的湮灭动力学行为, 其随r变化的行为与J型聚集体类似(文中没有给出), H型聚集体高阶激发态的峰值低于J聚集体. 前面提到J型聚集体偶极矩排列方向与分子链方向平行, 分子间库仑力相互吸引, H型聚集体偶极矩的排列方向与分子链方向垂直, 分子间库仑力相互排斥, 这也决定其与J型聚集体相比有较小的能量转移矩阵元, 在能量表象上J型和H型聚集体有不同的激子能带, 对于H型聚集体电子优先占据高能量激子能级, 对于J型聚集体优先占据低能量激子能级. 为了比较J型聚集体和H型聚集体在激子湮灭动力学上的影响, 图3给出了r = 15 ps–1的J型和H型4分子聚集体内第一激发态和高阶激发态占据数的动力学曲线, 可以看到, 对于第一激发态占据数动力学, J型的衰变快于H型; 对于高激发态占据数, J型的峰值显著大于H型, 同时H型峰值对应的时间与J型相比延迟了大约0.1 ps的时间. J型聚集体在同等情况下比H型聚集体有较强的激发耦合, 相干能量转移时间更短, 可以实现较快速的激子融合过程和湮灭过程, 使得高阶激发态上能实现较高的电荷占据和有效的激子-激子湮灭过程. 图 2 不同高阶激发态衰变率r下J型分子聚集体的第一激发态和高阶激发态占据数动力学 (a)第一激发态; (b)高阶激发态 Figure2. The population dynamics of the first excited state and the higher excited state of the J-type molecular aggregate with different decay rate r: (a) The first excited state; (b) the higher excited state.
图 3 在高级衰变率r = 15 ps–1时J型与H型分子聚集体第一激发态和高阶激发态占据数动力学 (a)第一激发态; (b)高阶激发态 Figure3. The first excited state and higher excited state population dynamics of J-type and H-type molecular aggregates at r = 15 ps–1. (a) The first excited state; (b) the higher excited state.
聚集体中的多激子态一般是在超快脉冲场激发作用下产生的, 本文采用的模型中没有考虑光激发强弱对激子湮灭过程的影响, 而是将激发态占据按照初始时刻进行归一化研究其动力学过程(见公式12), 图4展示了不同${P_m}(0)$初始第一激发态占据数对第一激发态占据数和高激发态占据数动力学的影响, 其光激发效应通过设定第一激发态占据数的初始值来定性判断. 为了研究光激发强弱对激子湮灭过程的影响, 设定${P_m}\left( 0 \right) = 0.1, \;0.3$和0.7分别模拟弱场强、中场强和高场强下激子湮灭过程的影响. 可以看到, 随着场强的增加, 第一激发态随时间的衰变过程也伴随${P_m}(0)$的升高而变快, 同时高阶激发态占据数峰值上升, 在高场强激发下, 激子-激子湮灭效率较高. 图 4 不同${P_m}(0)$下的占据数动力学 (a)第一激发态占据数; (b)高阶激发态占据数 Figure4. population dynamics under different ${P_m}(0)$ (a) First excited state population; (b) higher excited state population.
前面假设${P_m}\left( 0 \right)$为一个固定值, 聚集体内分子数越多, 激子数就越多, 若考虑分子聚集体内只有两个激子产生, 那么激子态在整个分子链内巡游, 产生离域的激子态. 图5(a)给出了分子链内共有6个激子时H型聚集体和J型聚集体内的湮灭率随聚集体分子数的变化曲线. 这里的湮灭率为$(1 - {P_m}(\tau ))/\tau $, τ为第一激发态占据数衰变为初始值的$ 1/e $(0.367879)时所需时间. 可以看到, 随着分子数的增加, J型聚集体和H型聚集体的激子湮灭率都减小, 但越来越平缓, 有意思的是J型聚集体的湮灭率不论分子数多少总是高于H型聚集体. 图5(b)给出了在聚集体中共有6个非局域激子时J型和H型湮灭率的比值随分子数的变化, 可以看到, 如果忽略边界效应, J型和H型湮灭率的比值随分子数目的增加趋于收敛, 最后达到3.5附近, 说明激子-激子湮灭在J型聚集体中比H型聚集体更容易产生[18]. 从激子融合和湮灭的角度考虑是因为在本文考虑的参数范围内, J型聚集体的耦合强度强, 有较快的相干转移时间, 这个时间小于第一激发态的寿命但远大于高阶激发态的寿命, 这样要实现对激子-激子湮灭过程的控制, 激子融合过程尤为关键. 图5(c)给出分子链内共有2, 4, 6个激子的J型聚集体内的湮灭率随聚集体分子数的变化曲线, 可以看到, 聚集体中激子密度越高, 湮灭率越大, 但随着密度的增加, 湮灭率的变化越不明显, 随聚集体分子数增加的变化也趋于平缓. 由此可以看到, 实验上若想得到较高的激子湮灭率, 应该选取J型聚集体和相对高的激子密度. 图 5 J型与H型分子聚集体的激子-激子湮灭率随分子数变化曲线. (a) $P(0) = \displaystyle\sum\nolimits_m {{P_m}} (0){{ = 6}}$时J型与H型湮灭率曲线; (b) $P(0){{ = 6}}$时J型与H型湮灭率的比值; (c) 不同$P(0)$下J型湮灭率曲线 Figure5. The curve of exciton-exciton annihilation rate of J-type and H-type molecular aggregates with the number of molecules. (a) J-type and H-type annihilation rate curve when $P(0) = \displaystyle\sum\nolimits_m {{P_m}} (0){{ = 6}}$; (b) The ratio of J-type and H-type annihilation rate when $P(0){{ = 6}}$; (c) J-type annihilation rate curve under different $P(0)$.