Theoretical study of exciton-exciton annihilation dynamics in the approximation of weak coupling

Fan Xu-Yang; Chen Han-Chao; Wang Lu-Xia

doi:10.7498/aps.70.20211242

Abstract

Authors and contacts

FullText HTML

1. 引　言
2. 理论和模型
3. 结果与讨论
3.1 参数的选择
3.2 全激发下的激子态动力学
3.3 光激发作用下激子湮灭过程的量子波包动力学模拟
4. 结　论

References

Cited By

Abstract

Dynamics of exciton-exciton annihilation (EEA) in molecular aggregates is closely related to its luminescence characteristics and energy transfer. It is meaningful to uncover energy and charge transfer process in molecular systems. Therefore, studying the dynamics of exciton is important for simulating photosynthesis in nature and analyzing the transport process of photocarriers. In this paper the weak coupling approximation is adopted to obtain the rate equation in the framework of density matrix theory. The relation among the intermolecular distance, exciton state density, excited state dipole moment and exciton-exciton annihilation dynamics is studied by the rate equations. It is found that the decrease of intermolecular distance leads the generation rate of higher-order excited states to increase, resulting in the obvious S-shaped decay characteristics. Moreover, the dipole moment of the higher-order excited state is the key factor of the exciton fusion process, and the greater the exciton density, the more easily the exciton fusion process occurs. Therefore, the reduction of intermolecular distance and the increase of the dipole moment of the higher-order excited state make the nearest neighbor molecules have a strong coupling, resulting in a high generation rate of the higher-order excited state. It is found that the evolution processes of the first excited state in different exciton densities are consistent with the experimental results of the excitation of OPPV7 monomer (PPV oligomers of 7) at a low excitation energy, and the excitation of OPPV7 aggregates at different excitation energy levels. It can be observed that the exciton decay rate is faster under the excitation of the strong external field. Using the quantum wave packet under optical excitation as the initial state, the excited state dynamics is simulated at different exciton energy levels. It is found that the exciton state can maintain good locality within a few hundreds of femtoseconds, which shows that the exciton state is a coherent superposition state, and its local characteristics are related to the excitation energy level. These conclusions are applicable to the aggregations whose single molecule has an energy level of

${E_{mf}} \approx 2{E_{me}}$ , and also provide a reasonable reference for the exciton-exciton annihilation process under optical excitation.

Keywords:

PACS:

73.63.-b	(Electronic transport in nanoscale materials and structures)
78.20.Bh	(Theory, models, and numerical simulation)
71.35.Aa	(Frenkel excitons and self-trapped excitons)

Authors and contacts

Physics Department, Institute of Theoretical Physics, University of Science and Technology Beijing, Beijing 100083, China

Corresponding author: Wang Lu-Xia, luxiawang@sas.ustb.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 21961132023, 11774026)

FullText HTML

1. 引　言

染料分子聚集体、有机半导体材料和分子聚合物材料中的激子动力学研究在有机光电材料领域中一直是热点问题^[1-7]. 这是由于激子的产生和动力学过程直接与有机材料中的能量转移和发光特性相关, 研究染料分子聚集体中的激子动力学过程及相关的能量转移过程对模拟自然界中的光合作用以及制作光合作用材料具有重要的意义.

激子-激子湮灭过程是分子聚集体或异质结构中的光载流子输运的关键因素^[8]. 在实验上瞬态吸收谱是检测激子-激子湮灭过程的主要手段, 例如应用超快光谱技术可以研究光合色素-蛋白质复合物的能量和电荷转移动力学^[9], 应用瞬间吸收光谱实验可以检测方酸-方酸聚合物中的激子动力学过程, 发现其瞬间吸收信号有具有明显的S型, 说明聚合物中出现了多激子态及相应的激子-激子湮灭过程^[10]. 应用激子-激子湮灭原理可以解释孤立单体链和聚集体波段重叠区的重叠光谱^[11].

另一方面, 近年来光纳科学的发展对实验研究分子聚集体中的多激子激发过程提供了更多的机会, 在光学微腔中或金属纳米粒子周围的分子聚集体受到局域强电场的激发可以产生多激子态, 其动力学过程出现更为复杂的行为^[12,13]. 在理论研究方面, 单激子态在分子中或分子界面的动力学过程已经有较多的研究^[14], 但分子系统中多激子态在光激发作用下的能量转移和动力学过程相比于半导体量子点材料要少得多^[15-18]. 一方面是由于分子聚集体中的多个激子态在不同分子上排列组成大量的多激子位形, 若考虑激子-激子湮灭过程需要考虑分子的高阶激发态, 将构成更为复杂的激发态位形, 在半导体量子点材料中应用的方法在分子聚集体中不再适用; 另一方面分子聚集体中分子间的相互作用在能量表象上形成激子能带, 不同的激光脉冲会激发不同能带上的激子态, 也将产生不同的非线性动力学过程, 增加了研究其动力学过程的复杂性, 也增加了相关动力学理论准确描述的难度.

对于光合作用材料来讲, 光俘获过程是一个多光子的吸收过程, 也是多激子激发过程. 以染料分子聚集体为模型的光激发动力学研究对光俘获相关的光吸收、能量转移及激子湮灭及发光过程都有重要意义, 在理论上也迫切需要包括光激发过程和激子-激子湮灭过程的多激子动力学的描述. 目前已有不同方法描述与激子-激子湮灭和多激子的动力学过程的相关理论, 如在密度矩阵理论框架下用二能级模型描述的非局域多激子动力学过程的研究^[19,20], 通过费米黄金法则在微观模型上描述激子-激子湮灭过程, 研究干涉双激子波函数的相位关系来分析聚集体的不同构型对激子湮灭率的影响^[21].

本文从分子聚集体激子态的量子描述出发, 假设分子间的量子相干过程在较为稳定的半经典近似下来描述多激子态的激子-激子湮灭过程. 在分子三能级近似下研究相关动力学过程, 同时模拟体系受到光激发条件下的能量转移和多激子动力学过程, 对进一步深入理解和解释分子聚集体内的多激子非线性动力学过程将是有益的尝试, 也将对研究更为复杂的聚集体材料的光激发下的激子-激子湮灭过程提供合理的参考.

2. 理论和模型

在分子的三能级近似下, 分子聚集体的哈密顿量可以写为

$H = \sum\limits_m^{} {{H_m} + \frac{1}{2}} \sum\limits_{m,n}^{} {{V_{mn}}} \text{, }$

(1)

式中, ${H_m}$ 是单分子的哈密顿量, $m(n)$ 表示分子聚集体中的分子编号, ${V_{mn}}$ 表示聚集体中不同分子间的库仑相互作用. 本文不考虑分子的振动态, 若考虑激子-激子湮灭过程, 至少要考虑分子的三个电子态: 基态 ${\varphi _{mg}}$ , 第一激发态 ${\varphi _{me}}$ , 以及高阶激发态 ${\varphi _{mf}}$ , 对应的能量分别为 ${E_{mg}}$ , ${E_{me}}$ 和 ${E_{mf}}$ , 且高阶激发态能量约为第一激发态能量的2倍( ${E_{mf}} \approx$ $2{E_{me}}$ ). 引入单分子跃迁算符 $B_m^ +=| {{\varphi _{me}}} \rangle \langle {{\varphi _{mg}}} |$ , $B_m=$ $\left| {{\varphi _{mg}}} \right\rangle \left\langle {{\varphi _{me}}} \right|$ , $D_m^{\text{ + }}{\text{ = }}| {{\varphi _{mf}}} \rangle \langle {{\varphi _{me}}} |$ 和 $D_m^{}{\text{ = }}| {{\varphi _{me}}}\rangle \langle {{\varphi _{mf}}} |$ , 这样与激子-激子湮灭过程相关的哈密顿量的算符表示为

$\begin{split} {H_{}} =\;& \sum\limits_m^{} {{E_m}B_m^ + {B_m}} + \sum\limits_m^{} {{\varepsilon _m}D_m^ + {D_m}} \\ & + \sum\limits_{m,n}^{} {{J_{mn}}B_m^ + {B_n}}+ \sum\limits_{m,n}^{} {{\cal{J}}_{mn}}D_m^ + {D_n} \\ & + \sum\limits_{m,n}^{} {\left( {{K_{mn}}D_m^ + {B_n} + K_{mn}^*D_m^{}B_n^ + } \right)} \text{, }\end{split}$

(2)

式中 ${E_m} = {E_{me}} - {E_{mg}}$ , ${{\cal{E}}_m} = {E_{mf}} - {E_{mg}}$ , 设分子系统中基态能 ${E_{mg}}$ 为0, 其中 ${J_{mn}} = \dfrac{{{\kappa _{mn}}{d_m}{d_n}}}{{{{\left| {{R_{mn}}} \right|}^3}}}$ , ${{\cal{J}}_{mn}}=\dfrac{{{\kappa _{mn}}{m_m}{m_n}}}{{{{\left| {{R_{mn}}} \right|}^3}}}$ , ${K_{mn}} = \dfrac{{{\kappa _{mn}}{d_m}{m_n}}}{{{{\left| {{R_{mn}}} \right|}^3}}}$ 是(1)式中 ${V_{mn}}$ 的不同分子态之间的相互作用的具体形式, 分别为第一激发态间、高阶激发态间和融合过程相关的库仑耦合矩阵元. 其中 ${{\boldsymbol{d}}_{m\left( n \right)}}{\text{ = }}{d_{m\left( n \right)}}{{\boldsymbol{e}}_{m(n)}}$ 和 ${{\boldsymbol{m}}_{m\left( n \right)}}{\text{ = }}{m_{m\left( n \right)}}{{\boldsymbol{e}}_{m(n)}}$ 分别表示分子 $m\left( n \right)$ 的第一激发态和高阶激发态跃迁偶极矩, ${{\boldsymbol{R}}_{mn}}{\text{ = }}{R_{mn}}{{\boldsymbol{n}}_{mn}}$ 代表 $m$ 和 $n$ 分子间的位移矢量( ${{\boldsymbol{e}}_{m(n)}}$ 和 ${{\boldsymbol{e}}_{m(n)}}$ 是单位向量), ${\kappa _{mn}} = {{\boldsymbol{e}}_m} \cdot {{\boldsymbol{e}}_n} - 3\left( {{{\boldsymbol{e}}_m} \cdot {{\boldsymbol{n}}_{mn}}} \right)\left( {{{\boldsymbol{n}}_{mn}} \cdot {{\boldsymbol{e}}_n}} \right)$ .

用密度矩阵理论描述分子系统的多激子动力学过程时, 需要考虑分子聚集体中多个激子态的情况. 多个激子态分布在不同的分子上形成不同的多激子态位形, 多激子态的位形对应于不同的密度矩阵元, 矩阵元的数目随分子聚集体中分子数量的增大而指数增大. 这里关心的物理量用其相关算符的算术平均值表示^[22], 其思想是只关注某个分子上的平均产生率, 不考虑其具体位形, 在这个近似下对由多个分子组成的聚集体而言计算量大幅减少, 且不影响对激子-激子湮灭过程的相关基本物理量的分析. 比如分子的第一激发态占据数用其数学期望值 ${P_m}(t) = \left\langle {B_m^ + B_m^{}} \right\rangle$ 来表示, 高阶激发态占据数用 ${N_m}(t) = \left\langle {D_m^ + D_m^{}} \right\rangle$ 表示, 在考虑的三能级体系中, 电子在三个能级的占据数之和为1, 因此分子基态占据数表示为 $\varPi _m^{}(t) = \left\langle {B_m^{}B_m^ + } \right\rangle = 1 - P_m^{}(t) - N_m^{}(t)$ , 任意算符 $\hat O\left( t \right)$ 的算术平均值满足的运动方程:

$\frac{\partial }{{\partial t}}\big\langle {\hat O} \big\rangle = \frac{i}{\hbar }\Big\langle {{{\big[ {H,\hat O} \big]}_ - }} \Big\rangle - \big\langle {\widetilde {\cal D}\hat O} \big\rangle ,$

(3)

其中 $\widetilde{{\cal{D}}}$ 为耗散算符, 耗散部分包括高阶激发态到第一激发态的内部转换产生的耗散 ${\widetilde{{\cal{D}}}_{{\text{IC}}}}$ 以及失相耗散 ${\widetilde{{\cal{D}}}_{{\text{pd}}}}$ . 对于激发态占据数 $\left\langle {B_m^ + B_m^{}} \right\rangle$ 和 $\left\langle {D_m^ + D_m^{}} \right\rangle$ 得到了期望值的运动方程:

$\begin{split} &\frac{\partial }{{\partial t}}\left\langle {B_m^ + B_m^{}} \right\rangle \\ =\;& - k\left\langle {B_m^ + B_m^{}} \right\rangle + r\left\langle {D_m^ + D_m^{}} \right\rangle\\ &- \frac{{\rm{i}}}{\hbar }\sum\limits_n ({J_{mn}}\left\langle {B_m^ + B_n^{}} \right\rangle - {J_{nm}}\left\langle {B_n^ + B_m^{}} \right\rangle \\ & - {\cal{J}}_{mn}^{}\left\langle {D_m^ + D_n^{}} \right\rangle + {\cal{J}}_{nm}^{}\left\langle {D_n^ + D_m^{}} \right\rangle \\ &{\text{ + }}K_{mn}^*\left\langle {B_n^ + D_m^{}} \right\rangle - {K_{mn}}\left\langle {D_m^ + B_n^{}} \right\rangle \\ &+ K_{nm}^*\left\langle {B_m^ + D_n^{}} \right\rangle - {K_{nm}}\left\langle {D_n^ + B_m^{}} \right\rangle ) \text{, } \end{split}$

(4)

$\begin{split} &\frac{\partial }{{\partial t}}\left\langle {D_m^ + D_m^{}} \right\rangle = - r\left\langle {D_m^ + D_m^{}} \right\rangle\\ &+ \frac{i}{\hbar }\sum\limits_n {( - {\cal{J}}_{mn}^{}\left\langle {D_m^ + D_n^{}} \right\rangle + {\cal{J}}_{nm}^{}\left\langle {D_n^ + D_m^{}} \right\rangle }\\ &{\text{ + }}K_{mn}^*\left\langle {B_n^ + D_m^{}} \right\rangle - {K_{mn}}\left\langle {D_m^ + B_n^{}} \right\rangle ) \text{, }\end{split}$

(5)

其中, $k$ 和 $r$ 分别是第一激发态到基态和高阶激发态到第一激发态的衰变率, 此外, 在方程(4)和(5)中出现了分子间耦合密度矩阵元的算术平均, 将其定义为 $W_{mn}^{} = \left\langle {B_m^ + {B_n}} \right\rangle$ , $Z_{mn}^{} = \left\langle {D_m^ + {D_n}} \right\rangle$ 和 $X_{mn}^{} =$ $\left\langle {D_m^ + {B_n}} \right\rangle$ $\left( {m \ne n} \right)$ , 这些非对角矩阵元代表不同分子间的相干跃迁, 进一步简化方程(4)和(5)得到:

$\begin{split} \frac{\partial }{{\partial t}}{P_m} = \;& - k_{}^{}P_m^{}{\text{ + }}r_{}^{}N_m^{} + \frac{2}{\hbar }{Im} \sum\limits_n {{J_{mn}}} {W_{mn}} \\ &- \frac{2}{\hbar }{Im} \sum\limits_n {{\cal{J}}_{mn}^{}} {Z_{mn}} \\ &- \frac{2}{\hbar }{Im} \sum\limits_n {{K_{mn}}} {X_{mn}} \\ &- \frac{2}{\hbar }{Im} \sum\limits_n {{K_{nm}}} {X_{nm}}\text{, } \end{split}$

(6)

$\begin{split} \frac{\partial }{{\partial t}}{N_m} =\;& - r_{}^{}N_m^{} + \frac{2}{\hbar }{Im} \sum\limits_n {{\cal{J}}_{mn}^{}} {Z_{mn}}\\ &+ \frac{2}{\hbar }{Im} \sum\limits_n {{K_{mn}}{X_{mn}}} .\end{split}$

(7)

若分子间间距在1 nm以上, 分子间的库仑相互作用较弱, 符合弱耦合近似条件, 在弱耦合近似下非对角元的相干转移变化较小, 忽略 $W_{mn}^{}$ , $Z_{mn}^{}$ 和 $X_{mn}^{}$ 随时间的变化, 令 $\dfrac{\partial }{{\partial t}}W_{mn}^{} = 0$ , $\dfrac{\partial }{{\partial t}}Z_{mn}^{}{\text{ = }}0$ 和 $\dfrac{\partial }{{\partial t}}X_{mn}^{}{\text{ = }}0$ , 只考虑分子间库仑相互作用下的准静态过程, 得到:

$W_{mn}^{} = \frac{{ - {J_{nm}}\left[ {\left( {{E_m} - {E_n}} \right) - {\rm{i}}\hbar k} \right]}}{{{{\left( {{E_m} - {E_n}} \right)}^2} + {\hbar ^2}{k^2}_{}}}\left[ {(\varPi _m^{} - P_m^{})P_n^{} - P_m^{}(\varPi _n^{} - P_n^{})} \right]\text{, }$

(8)

${Z}_{mn}^{}=\frac{-{{\cal{J}}}_{nm}^{}\left[\left({\cal{E}}_{m}-{E}_{m}\right)-\left({\cal{E}}_{n}-{E}_{n}\right)-\text{i}\hslash \left(k\text{+ }r\right)\right]}{{\left[\left({\cal{E}}_{m}-{E}_{m}\right)-\left({\cal{E}}_{n}-{E}_{n}\right)\right]}^{2}+{\hslash }^{2}{\left(k\text{+ }r\right)}^{2}}\left[({P}_{m}^{}-{N}_{m}^{}){N}_{n}^{}-{N}_{m}^{}({P}_{n}^{}-{N}_{n}^{})\right]\text{, }$

(9)

$X_{mn}^{} = \dfrac{{ - K_{mn}^*\left[ {\left( {{{\cal{E}}_m} - {E_m} - {E_n}} \right) - {\text{i}}\hbar \left( {k{\text{ + }}\dfrac{r}{2}} \right)} \right]}}{{{{\left( {{{\cal{E}}_m} - {E_m} - {E_n}} \right)}^2} + {\hbar ^2}{{\left( {k{\text{ + }}\dfrac{r}{2}} \right)}^2}}}\left[ {(P_m^{} - N_m^{})P_n^{} - N_m^{}(\varPi _n^{} - P_n^{})} \right]{\text{.}}$

(10)

将(8)式—(10)式的虚部代入方程(6)和(7)中, 最终得到一组率方程:

$\begin{split} \frac{\partial }{{\partial t}}P_m =\;& - k P_m + r N_m - P_m \sum\limits_n {{J'_{mn}}(\varPi _n - P_n ) + (\varPi _m - P_m )} \sum\limits_n {{J'_{mn}}P_n }+ N_m \sum\limits_n {{{\cal{J}}'_{mn}}}(P_n - N_n ) \\ &- (P_m - N_m )\sum\limits_n {{{\cal{J}}'_{mn}}}N_n - P_m \sum\limits_n {{K'_{nm}}(P_n - N_n )} + (\varPi _m - P_m )\sum\limits_n {{K'_{nm}}} N_n \\ &-({P}_{m} -{N}_{m} ){\displaystyle \sum _{n} K^{\prime}_{mn}{P}_{n} +{N}_{m} {\displaystyle \sum _{n} K'_{mn}}({\varPi }_{n} -{P}_{n} )}， \end{split}$

(11)

$\begin{split} \frac{\partial }{{\partial t}}N_m =\;& - r_{} N_m - N_m \sum\limits_n {{{{\cal{J}}'_{mn}}}(P_n - N_n ) + (P_m - N_m )\sum\limits_n {{{{\cal{J}}'_{mn}}}N_n } } \\ & -{N}_{m} {\displaystyle \sum _{n} K‘_{mn}({\varPi }_{n} -{P}_{n} )}+({P}_{m} -{N}_{m} ){\displaystyle \sum _{n} K’_{mn}{P}_{n}}, \end{split}$

(12)

式中 ,

$\begin{split} &{J}'_{mn}=2\left| J_{nm}^{{}} \right|_{{}}^{2}k/\left( \Delta {{E}_{J}}^{2}+\hbar _{{}}^{2}{{k}^{2}} \right), \\ & {{\cal{J}}}'_{mn} = 2\left| {\cal{J}}_{nm}^{{}} \right|_{{}}^{2}\left( k+r \right)/\left( \Delta {{E}_{{\cal{J}}}}^{2}+\hbar Z _{{}}^{2}(k_{{}}^{{}}+r_{{}}^{{}})_{{}}^{2} \right),\\ & K'_{mn} = 2\left| {K_{mn}^{}} \right|_{}^2\left( {k + r/2} \right)/\big( {\Delta {E_k}^2 + \hbar _{}^2{{\left( {k + r/2} \right)}^2}} \big)\end{split}$

分别代表了第一激发态, 高阶激发态和激子融合过程相关的能量转移率. 为了保证体系的弱耦合近似, 在这里引入分子间的能量差 $\Delta {E_J}{\text{ = }}{E_m} - {E_n}$ , $\Delta {E_{\cal{J}}}=({{\cal{E}}_m} - E_m^{}) - ({{\cal{E}}_n} - E_n^{})$ 和 $\Delta {E_K}={{\cal{E}}_m} - E_m^{} -$ $E_n^{}$ . 在下面的计算模拟中分别用平均第一激发态

${\bar P_N}( t ) =\frac{1}{N}\displaystyle\sum\limits_{m = 1}^N {{P_m}} (t),$

平均高阶激发态

${\bar N_N}( t ) = \dfrac{1}{N}\displaystyle\sum\limits_{m = 1}^N {{N_m}} (t),$

总第一激发态

${P_{{\text{tot}}}}( t ) = \displaystyle\sum\limits_{m = 1}^N{{P_m}(t)} ,$

总高阶激发态

${N_{{\text{tot}}}}\left( t \right) = \displaystyle\sum\limits_{m = 1}^N {{P_m}(t)},$

和归一化的第一激发态

$P\left( t \right) = \dfrac{1}{{N \times {P_m}\left( {t = 0} \right)}}\times \displaystyle\sum\limits_{m = 1}^N {{P_m}(t)}$

来表征激子-激子湮灭的动力学过程.

3. 结果与讨论

首先用三能级的双分子模型来简述激子-激子湮灭过程. 如图1所示, ${S_0}$ 表示分子的基态, ${S_1}$ 表示分子的第一激发态, 产生激子-激子湮灭的条件是存在某个高阶激发态, 其能量是第一激发态能量的二倍, 把这个高阶激发态记为 ${S_n}$ . 设两个分子在光激发条件下形成两个激子, 相互靠近的激子单体进行能量交换, 一个激子吸收另外一个激子态回落至基态的能量跃迁至其高阶激发态, 称其为融合过程. 处于高阶激发态的分子不稳定, 衰变率更高, 在很短的时间内通过非辐射过程快速弛豫至第一激发态, 称为湮灭过程. 第一激发态上分子的寿命一般在纳秒量级, 在所讨论的皮秒动力学过程中保持稳定. 将包括激子的融合过程和湮灭过程的整个过程称为激子-激子湮灭过程.

$图 1 激子-激子湮灭过程的能级示意图\r\nFig. 1. Schematic diagram of the energy levels of the exciton-exciton annihilation (EEA) process.$

图 1 激子-激子湮灭过程的能级示意图

Fig. 1. Schematic diagram of the energy levels of the exciton-exciton annihilation (EEA) process.

下载: 全尺寸图片幻灯片

3.1 参数的选择

在本文的计算中将不针对某个具体的分子系统, 考虑分子链组成的聚集体模型, 将研究分子体系中多激子的能量转移和动力学过程, 其结果对相似结构的分子有参考意义. 本文按照分子偶极矩一致的方向形成J型( ${\kappa _{mn}}$ = –2, 分子偶极矩头尾相连)聚集体结构, 设聚集体内有20个分子( $N=20$ ). 为了保证系统的弱耦合近似, 分子间的能量差设为1 meV. 染料分子中基态和第一激发态之间的跃迁偶极矩 ${d_{m\left( n \right)}}$ 一般有较大的数值, 设其为 ${d_{m\left( n \right)}}=8~\rm D$ ( $\rm D$ 为德拜). 一般来说高阶激发态的跃迁偶极矩( ${m_{m\left( n \right)}}$ )比第一激发态的跃迁偶极矩( ${d_{m\left( n \right)}}$ )小, 设其在 $0.1~\rm D$ 到 $8.0~\rm D$ 之间变化. 第一激发态上的激子寿命一般在纳秒量级, 设 $k_{}^{}=$ ${10^{ - 3}}~{\text{p}}{{\text{s}}^{{{ - 1}}}}$ , 高阶激发态上的高阶激子态寿命没有相关的实验数据, 但一般来说比第一激发态上的激子寿命小得多, 其数值在50—100 fs之间变化(r = 10—20 ps^–1). 分子的第一激发态和基态间的能级差为 ${E_m}= 2.5\;{\text{eV}}$ , 高阶激发态和基态间的能级差为 ${{\cal{E}}_m}$ = 5.0 eV. 分子间的相互作用是库仑耦合, 其距离在1.2—2.5 nm之间, 具体参数为: N = 20, d_m(n)= 8 D, m_m(n)= 0.1 D, 0.8 D, 8 D, Δ_{m, m±1}= 1.2, 1.5, 2.5 nm, E_m= 2.5 eV, 1/k = 1000 ps, 1/r = 0.1, 0.005 ps, $\Delta {E_J}^{}\Delta {E_\mathcal{J}}^{}\Delta {E_K}$ = 1 meV.

3.2 全激发下的激子态动力学

在弱耦合近似下的率方程可以计算分子链中的分子全部被激发至其第一激发态时激子的融合过程和湮灭过程. 图2给出了分子间距离 ${\varDelta _{m, m \pm 1}}$ 分别为1.2 nm, 1.5 nm和2.5 nm, 初始第一激发态占据数 ${P_m}\left( {t = 0} \right)$ 为1.0时对应的第一激发态和高阶激发态的电子占据随时间的变化. 当分子间距离为1.2 nm时, 第一激发态随时间的变化出现了明显的S型, 说明第一激发态的衰变过程受到激子融合过程的影响而变缓. 这里先理想假设所有分子的第一激发态被激发, 在分子间耦合较强的情况下( ${\varDelta _{m, m \pm 1}}$ = 1.2 nm和1.5 nm), 由于融合耦合强度 ${K_{mn}}$ 较大(–4.63 meV和–2.37 meV), 一部分高阶激子在10 ps内由激子融合过程产生(见图2中的插图), 高阶激发态的产生率也较高(大于20%). 若分子间耦合较小( ${\varDelta _{m, m \pm 1}}$ = 2.5 nm), 激子的融合过程在0.1—1 ps内产生, 且高阶激子态产生率很低. 激子融合过程虽然不依赖于波函数的交叠, 但分子间距离大, 相互作用小是高阶激子态产生低的主要原因. 说明当分子间距离较小时, 分子间相互作用较强, 提高了高阶激发态的产生率.

$图 2 不同分子间距离${\varDelta _{m, m \pm 1}}$下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程(${m_{m\left( n \right)}}= $$ 0.8~\rm D$).插图: 前100 fs的J型分子聚集体的平均激发态占据数和时间的线性关系图\r\nFig. 2. Population of the average first excited state and higher excited state of the J-type molecular aggregates with the distance between molecules${\varDelta _{m, m \pm 1}}$ (${m_{m\left( n \right)}}= $$ 0.8~\rm D$). Inset: Linear graph of the population of the average first excited state and higher excited state by the J-type molecular aggregates in the first 100 fs versus time.$

图 2 不同分子间距离

${\varDelta _{m, m \pm 1}}$

下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程(

${m_{m\left( n \right)}}=$

$0.8~\rm D$

).插图: 前100 fs的J型分子聚集体的平均激发态占据数和时间的线性关系图

Fig. 2. Population of the average first excited state and higher excited state of the J-type molecular aggregates with the distance between molecules

${\varDelta _{m, m \pm 1}}$

(

${m_{m\left( n \right)}}=$

$0.8~\rm D$

). Inset: Linear graph of the population of the average first excited state and higher excited state by the J-type molecular aggregates in the first 100 fs versus time.

下载: 全尺寸图片幻灯片

图3给出了分子间距离为1.2 nm, 第一激发态到高阶激发态间的跃迁偶极矩 ${m_{{{m}}\left( n \right)}}$ 为0.1 $~\rm D$ , 0.8 $~\rm D$ 和8.0 $~\rm D$ 对应的高阶激发态和第一激发态的动力学过程图. 当 ${m_{{ {m}}\left( n \right)}} \ll {d_{{ {m}}\left( n \right)}}$ 时, 高阶激发态的产生率只有5%左右. 若令 ${m_{m\left( n \right)}}={d_{{ {m}}\left( n \right)}}$ 时, 在5 fs的时间内激子融合过程即刻完成, 激子融合的相干时间远小于随之发生的湮灭过程的湮灭时间, 出现了更明显的S型特征, 说明高阶激发态的偶极矩是激子融合过程的关键因素. 在 ${m_{m\left( n \right)}}={d_{{ {m}}\left( n \right)}}$ (蓝线)的极端条件下分子的基态、第一激发态和高阶激发态的产生率在10 fs时间内各占1/3, 达到瞬间平衡. 与图2的黑线比较可以看到, 在最近邻分子间间距较小( ${\varDelta _{m, m \pm 1}}$ = 1.2 nm)的条件下, 激子的融合主要依赖于高阶激发态的偶极矩 ${m_{{ {m}}\left( n \right)}}$ , 偶极矩越大, 融合过程越快, 相应的湮灭过程也较快, 当高阶激发态占据率相同时湮灭率趋于一致. 在率方程近似下计算的激子动力学过程可以得到与相关实验一致的结果, 其他关于高阶激发态衰减率r和聚集体湮灭率的讨论参见文献[23].

$图 3 不同高阶激发态偶极矩${m_{m\left( n \right)}}$下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程; 插图: 前100 fs的J型分子聚集体平均激发态占据数和时间的线性关系图\r\nFig. 3. Population of the average first excited state and higher excited state of the J-type molecular aggregateswith different dipole moment of higher excited state${m_{m\left( n \right)}}$.Inset: Linear graph of the population of the average first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.$

图 3 不同高阶激发态偶极矩

${m_{m\left( n \right)}}$

下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程; 插图: 前100 fs的J型分子聚集体平均激发态占据数和时间的线性关系图

Fig. 3. Population of the average first excited state and higher excited state of the J-type molecular aggregateswith different dipole moment of higher excited state

${m_{m\left( n \right)}}$

.Inset: Linear graph of the population of the average first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.

下载: 全尺寸图片幻灯片

图4给出了分子聚集体中的激子数分别为1, 5, 10, 20 (对应的初始第一激发态占据数 ${P_m}\left( {t = 0} \right)$ = 1.0, 0.5, 0.25, 0.05)用率方程模拟的总的激发态占据数随时间变化的动力学过程. 从图4中小插图中能够看出, 激子密度越大, 激子融合过程越容易发生. 并且随着分子聚集体中激子密度的增大, 第一激发态占据数随时间的变化出现了明显的S型特征曲线, 这是因为高阶激发态占据数的增大导致随后的弛豫过程中大量的高阶激子弛豫到第一激发态, 同时高阶激发态的弛豫时间要比第一激发态快得多从而使第一激发态的湮灭过程减慢.

$图 4 不同激子数下J型分子聚集体的总第一激发态和总高阶激发态占据数动力学过程(${m_{m\left( n \right)}}=0.8~\rm D$); 插图: 前100 fs的J型分子聚集体总第一激发态占据数和总高阶激发态占据数和时间的线性关系图\r\nFig. 4. Thepopulation of the total first excited state and higher excited state of the J-type molecular aggregates with different numbers of excitons(${m_{m\left( n \right)}}=0.8~\rm D$). Inset: Linear graph of the population of the total first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.$

图 4 不同激子数下J型分子聚集体的总第一激发态和总高阶激发态占据数动力学过程(

${m_{m\left( n \right)}}=0.8~\rm D$

); 插图: 前100 fs的J型分子聚集体总第一激发态占据数和总高阶激发态占据数和时间的线性关系图

Fig. 4. Thepopulation of the total first excited state and higher excited state of the J-type molecular aggregates with different numbers of excitons(

${m_{m\left( n \right)}}=0.8~\rm D$

). Inset: Linear graph of the population of the total first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.

下载: 全尺寸图片幻灯片

文献[11]中给出了用光谱监测OPPV7(聚对苯乙炔类)单体和在不同激发功率下监测OPPV7 (溶剂: 四氢呋喃(THF))聚集体的信号衰减速率随时间的变化曲线. 如图5(a)所示, 外场的激发功率越强, 衰减速率越大, 表明分子聚集体内的多激子产生率就越高. 图5(b)是20个分子组成的分子聚集体中有不同激子数时, 对应的归一化的第一激发态随时间演变的动力学过程. 比较图5(a)和5(b)可以发现, 本文理论计算结果与文献[11]的实验结果非常一致, 在25 μW功率下激发OPPV7单体的信号衰减速率与理论计算的分子链中初始态有一个激子的第一激发态随时间的湮灭率一致. 分子链中初始态有5个激子、10个激子和20个激子的第一激发态随时间的湮灭率与在10 μW, 50 μW和600 μW功率下激发OPPV7聚集体的信号衰减曲线相比一致. 在激子单体模型中由于没有融合过程, 动力学过程由分子第一激发态的内转换过程决定. 外场越强, 聚集体内的激子数越多, 融合过程越明显, 在纳秒量级内的衰变就越快, 激子-激子湮灭过程越容易发生.

$图 5 (a)OPPV7单体和OPPV7(THF: water)聚集体的发射衰减速率[11]; (b)不同激子数下J型分子聚集体激发时的第一激发态湮灭过程(${m_{m\left( n \right)}}=0.1~\rm D$, ${r^{}}{\text{ = 200 p}}{{\text{s}}^{{{ - 1}}}}$)\r\nFig. 5. (a) Emission decays of OPPV7 monomer and OPPV7 (THF: water) Aggregates [11]; (b) annihilation process of the first excited state when the J-type molecular aggregates are excited in different excitons(${m_{m\left( n \right)}}=0.1~\rm D$, ${r^{}}{{ = 200~ {\rm{p}}}}{{\text{s}}^{{{ - 1}}}}$).$

图 5 (a)OPPV7单体和OPPV7(THF: water)聚集体的发射衰减速率^[11]; (b)不同激子数下J型分子聚集体激发时的第一激发态湮灭过程(

${m_{m\left( n \right)}}=0.1~\rm D$

${r^{}}{\text{ = 200 p}}{{\text{s}}^{{{ - 1}}}}$

)

Fig. 5. (a) Emission decays of OPPV7 monomer and OPPV7 (THF: water) Aggregates ^[11]; (b) annihilation process of the first excited state when the J-type molecular aggregates are excited in different excitons(

${m_{m\left( n \right)}}=0.1~\rm D$

${r^{}}{{ = 200~ {\rm{p}}}}{{\text{s}}^{{{ - 1}}}}$

下载: 全尺寸图片幻灯片

3.3 光激发作用下激子湮灭过程的量子波包动力学模拟

激子态是在光激发作用下产生的, 由于分子间的相互作用, 激子态在激发后不是局域在某个分子上, 而是在分子聚集体内巡游, 形成非局域激子态, 激子-激子湮灭过程实质上是非局域激子态的激发湮灭过程. 分子聚集体的非局域激子态在能量表象上形成一个有宽度的能带, 外场对分子聚集体的激发其实质是对这个能带上能级共振的有效激发. 激子态的激发效率直接与外场的激发频率以及脉冲的宽度和强度有关, 在率方程近似下原则上不能体现分子聚集体的能带效应, 也无法模拟非局域激子的湮灭过程. 在非局域激子态的研究中发现在共振激光激发作用下, 激子态在不同能级的电子占据呈现明显的量子波包形态. 在光激发作用下量子波包的量子动力学过程可以反映相关激子态的动力学演变情况. 在分子聚集体中模拟光激发作用下激子的动力学过程需要考虑分子聚集体内激子态的不同位形和激子相互作用, 将涉及更大的计算量, 本文将系统的哈密顿量对角化可以得到分子聚集体内的激子态能级以及相应的波包分布概率. 图6给出了有20个分子的J型聚集体对应的激子能级上的波包分布概率, 对应的能级分别是第一、三、五、七能级(能量由低到高), 偶数能级上的波函数为奇函数, 对应的偶极跃迁矩阵元为0, 使得相应的偶数能级为暗能级, 这里不再讨论. 奇数能级上的电子占据分布出现0, 2, 4, 6个节点波包.

$图 6 J型分子聚集体哈密顿量对角化的前四个明能级对应的波包分布概率${\left| {{C_m}} \right|^2}$对应率方程的4种初始激发位形　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级\r\nFig. 6. Four initial excitation configurations of the wave packet distribution corresponding (${\left| {{C_m}} \right|^2}$) to the first four bright energy levels corresponding to the diagonalization of the Hamiltonian of the J-type molecular aggregates: (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.$

图 6 J型分子聚集体哈密顿量对角化的前四个明能级对应的波包分布概率

${\left| {{C_m}} \right|^2}$

对应率方程的4种初始激发位形　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Fig. 6. Four initial excitation configurations of the wave packet distribution corresponding (

${\left| {{C_m}} \right|^2}$

) to the first four bright energy levels corresponding to the diagonalization of the Hamiltonian of the J-type molecular aggregates: (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

下载: 全尺寸图片幻灯片

图7(a)—7(d)和图8(a)—8(d)给出了图6(a)—6(d)对应的波包分布下用率方程模拟的J型聚集体的第一激发态和高阶激发态的动力学演变过程图. 设分子链中有10个激子同时被激发, 由图8可以看到, 相应激子能级上的电子占据在 ${\varDelta _{m, m \pm 1}}$ = 1.2 nm的条件下在几百飞秒时间内保持很好的局域性, 在0.02—1 ps之间高阶激发态经历产生和衰变的过程. 高阶激发态与第一激发态的波包局域性保持一致, 都出现相同节点波包分布. 比较分子链中多个激子以及单激子态(在文中未给出)的动力学过程, 发现其随时间的演变规律大致相同. 除了多激子态的占据数较大以外, 激子的融合过程很类似, 在演变过程中保持其原有的局域性, 说明分子聚集体中的激子态是相干叠加态, 其局域特点与所在的激发能级有关.

$图 7 J型分子聚集体(10个激子)激发时的第一激发态占据数演变(${m_{m\left( n \right)}}=0.8~\rm D$, ${\varDelta _{m, m \pm 1}}$ = 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级\r\nFig. 7. Population evolution of the first excited state when J-type molecular aggregates are excited (${m_{m\left( n \right)}}=0.8~\rm D$, ${\varDelta _{m, m \pm 1}}$ = 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.$

图 7 J型分子聚集体(10个激子)激发时的第一激发态占据数演变(

${m_{m\left( n \right)}}=0.8~\rm D$

${\varDelta _{m, m \pm 1}}$

= 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Fig. 7. Population evolution of the first excited state when J-type molecular aggregates are excited (

${m_{m\left( n \right)}}=0.8~\rm D$

${\varDelta _{m, m \pm 1}}$

= 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

下载: 全尺寸图片幻灯片

$图 8 J型分子聚集体(10个激子)激发时的高阶激发态占据数演变(${m_{m\left( n \right)}}=0.8~\rm D$, ${\varDelta _{m, m \pm 1}}$ = 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级\r\nFig. 8. Population evolution of the higher excited state when J-type molecular aggregates are excited(${m_{m\left( n \right)}}=0.8~\rm D$, ${\varDelta _{m, m \pm 1}}$ = 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.$

图 8 J型分子聚集体(10个激子)激发时的高阶激发态占据数演变(

${m_{m\left( n \right)}}=0.8~\rm D$

${\varDelta _{m, m \pm 1}}$

= 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Fig. 8. Population evolution of the higher excited state when J-type molecular aggregates are excited(

${m_{m\left( n \right)}}=0.8~\rm D$

${\varDelta _{m, m \pm 1}}$

= 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

下载: 全尺寸图片幻灯片

4. 结　论

分子聚集体中的分子间的间距大于1 nm, 分子间的库仑相互作用可以看作弱耦合, 密度矩阵理论下的非对角元的相干转移不随时间变化, 可以得到表征激子-激子湮灭过程的率方程. 应用率方程计算了不同分子间的间距、高阶激发态偶极矩以及不同激子浓度下的第一激发态和高阶激发态的动力学过程. 研究发现分子间的间距较小和高阶激发态偶极矩较大时, 高阶激子态产生率较高. 第一激发态衰变呈S型特征曲线, 说明第一激发态的衰变过程受到激子融合过程的影响而变缓. 通过对比实验上的OPPV7聚集体在不同激发功率下的信号衰减随时间的变化与理论计算上初始态为一个或多个激子态的归一化动力学数值结果, 发现理论模拟结果与实验非常一致, 说明在强场激发下分子聚集体内产生多激子并发生明显的激子湮灭过程. 在激子能带理论下模拟了不同能级上J型分子聚集体的第一激发态和高阶激发态的动力学演变过程图, 发现第一激发态和高阶激发态在几百飞秒内保持很好的局域性, 说明聚集体中的激子态是相干叠加态, 其局域特点与所在的激发能级有关. 在后续的工作中将考虑外场真实激发作用下分子聚集体的激子-激子湮灭过程, 研究相干能量转移过程等因素对湮灭过程的影响.

References

[1]	Cook S, Liyuan H, Furube A, Katoh R 2010 J. Phys. Chem. C 114 10962 Google Scholar
[2]	Peckus D, Devizis A, Hertel D, Meerholz K, Gulbinas V 2012 Chem. Phys. 404 42 Google Scholar
[3]	Ginas S, Kirkpatrick J, Howard I A, Johnson K, Wilson M W B, Friend R H, Silva C 2013 J. Phys. Chem. B 117 4649 Google Scholar
[4]	Dai D C, Monkman A P 2013 Phys. Rev. B 87 045308 Google Scholar
[5]	Völker S F, Schmiedel A, Holzapfel M, Renziehausen K, Engel V, Lambert C 2014 J. Phys. Chem. C 111 17467
[6]	Fennel F, Lochbrunner S 2015 Phys. Rev. B 92 140301 Google Scholar
[7]	Engel E, Leo K, Hoffmann M 2006 Chem. Phys. 325 170 Google Scholar
[8]	Linardy E, Yadav D, Vella D, Verzhbitskiy I A, Watanabe K, Taniguchi T, Pauly F, Trushin M 2020 Nano Lett. 20 1647 Google Scholar
[9]	Kühn O, Renger T, May V, Voigt J, Pullerits T, Sundström V 1997 Trends in Photochem. Photobiol. 4 213
[10]	Hader K, May V, Lambert, Engel V 2016 Phys. Chem. Chem. Phys. 18 13368 Google Scholar
[11]	Peteanu L A, Chowdhury S, Wildeman J, Sfeir M Y 2017 J. Phys. Chem. B 121 1707 Google Scholar
[12]	Feist J, Garcia-Vidal F J 2015 Phys. Rev. Lett. 114 196402 Google Scholar
[13]	Schachenmayer J, Genes C, Tignone E, Pupillo G 2015 Phys. Rev. Lett. 114 196403 Google Scholar
[14]	Xu L, Ji Y, Shi X, Wang L, Gao K 2020 Org. Electron. 85 105886 Google Scholar
[15]	Spano F C 1992 Phys. Rev. B 46 13017 Google Scholar
[16]	Juzeliunas G, Knoester J 2000 J. Chem. Phys. 112 2325 Google Scholar
[17]	Mukamel S, Berman O 2003 J. Chem. Phys. 119 12194 Google Scholar
[18]	Mukamel S, Abramavicius D 2004 Chem. Rev. 104 2073 Google Scholar
[19]	Wang L, Plehn T, May V 2020 Phys. Rev. B 102 075401 Google Scholar
[20]	Wang L, May V 2016 Phys. Rev. B 94 195413 Google Scholar
[21]	Tempelaar R, Jansen T L C, Knoester J 2017 J. Phys. Chem. Lett. 8 6113 Google Scholar
[22]	May V 2014 J. Chem. Phys. 140 054103 Google Scholar
[23]	茅江奇, 范旭阳, 路彦珍, 王鹿霞 2021 物理学报 70 047302 Mao J Q, Fan X Y, Lu Y Z, Wang L X 2021 Acta Phys. Sin. 70 047302

Cited By

期刊类型引用(0)

其他类型引用(2)

图 1 激子-激子湮灭过程的能级示意图

Figure 1. Schematic diagram of the energy levels of the exciton-exciton annihilation (EEA) process.

DownLoad: Full-Size Img PowerPoint

图 2 不同分子间距离 ${\varDelta _{m, m \pm 1}}$ 下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程( ${m_{m\left( n \right)}}=$ $0.8~\rm D$ ).插图: 前100 fs的J型分子聚集体的平均激发态占据数和时间的线性关系图

Figure 2. Population of the average first excited state and higher excited state of the J-type molecular aggregates with the distance between molecules ${\varDelta _{m, m \pm 1}}$ ( ${m_{m\left( n \right)}}=$ $0.8~\rm D$ ). Inset: Linear graph of the population of the average first excited state and higher excited state by the J-type molecular aggregates in the first 100 fs versus time.

DownLoad: Full-Size Img PowerPoint

图 3 不同高阶激发态偶极矩 ${m_{m\left( n \right)}}$ 下的J型分子聚集体的平均第一激发态和高阶激发态占据数动力学过程; 插图: 前100 fs的J型分子聚集体平均激发态占据数和时间的线性关系图

Figure 3. Population of the average first excited state and higher excited state of the J-type molecular aggregateswith different dipole moment of higher excited state ${m_{m\left( n \right)}}$ .Inset: Linear graph of the population of the average first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.

DownLoad: Full-Size Img PowerPoint

图 4 不同激子数下J型分子聚集体的总第一激发态和总高阶激发态占据数动力学过程( ${m_{m\left( n \right)}}=0.8~\rm D$ ); 插图: 前100 fs的J型分子聚集体总第一激发态占据数和总高阶激发态占据数和时间的线性关系图

Figure 4. Thepopulation of the total first excited state and higher excited state of the J-type molecular aggregates with different numbers of excitons( ${m_{m\left( n \right)}}=0.8~\rm D$ ). Inset: Linear graph of the population of the total first excited state and higher excited state of the J-type molecular aggregates in the first 100 fs versus time.

DownLoad: Full-Size Img PowerPoint

图 5 (a)OPPV7单体和OPPV7(THF: water)聚集体的发射衰减速率^[11]; (b)不同激子数下J型分子聚集体激发时的第一激发态湮灭过程( ${m_{m\left( n \right)}}=0.1~\rm D$ , ${r^{}}{\text{ = 200 p}}{{\text{s}}^{{{ - 1}}}}$ )

Figure 5. (a) Emission decays of OPPV7 monomer and OPPV7 (THF: water) Aggregates ^[11]; (b) annihilation process of the first excited state when the J-type molecular aggregates are excited in different excitons( ${m_{m\left( n \right)}}=0.1~\rm D$ , ${r^{}}{{ = 200~ {\rm{p}}}}{{\text{s}}^{{{ - 1}}}}$ ).

DownLoad: Full-Size Img PowerPoint

图 6 J型分子聚集体哈密顿量对角化的前四个明能级对应的波包分布概率 ${\left| {{C_m}} \right|^2}$ 对应率方程的4种初始激发位形　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Figure 6. Four initial excitation configurations of the wave packet distribution corresponding ( ${\left| {{C_m}} \right|^2}$ ) to the first four bright energy levels corresponding to the diagonalization of the Hamiltonian of the J-type molecular aggregates: (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

DownLoad: Full-Size Img PowerPoint

图 7 J型分子聚集体(10个激子)激发时的第一激发态占据数演变( ${m_{m\left( n \right)}}=0.8~\rm D$ , ${\varDelta _{m, m \pm 1}}$ = 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Figure 7. Population evolution of the first excited state when J-type molecular aggregates are excited ( ${m_{m\left( n \right)}}=0.8~\rm D$ , ${\varDelta _{m, m \pm 1}}$ = 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

DownLoad: Full-Size Img PowerPoint

图 8 J型分子聚集体(10个激子)激发时的高阶激发态占据数演变( ${m_{m\left( n \right)}}=0.8~\rm D$ , ${\varDelta _{m, m \pm 1}}$ = 1.2 nm)　(a)第一能级; (b)第三能级; (c)第五能级; (d)第七能级

Figure 8. Population evolution of the higher excited state when J-type molecular aggregates are excited( ${m_{m\left( n \right)}}=0.8~\rm D$ , ${\varDelta _{m, m \pm 1}}$ = 1.2 nm): (a) The first energy level; (b) the third energy level; (c) the fifth energy level; (d) the seventh energy level.

DownLoad: Full-Size Img PowerPoint

[1]	Cook S, Liyuan H, Furube A, Katoh R 2010 J. Phys. Chem. C 114 10962 Google Scholar
[2]	Peckus D, Devizis A, Hertel D, Meerholz K, Gulbinas V 2012 Chem. Phys. 404 42 Google Scholar
[3]	Ginas S, Kirkpatrick J, Howard I A, Johnson K, Wilson M W B, Friend R H, Silva C 2013 J. Phys. Chem. B 117 4649 Google Scholar
[4]	Dai D C, Monkman A P 2013 Phys. Rev. B 87 045308 Google Scholar
[5]	Völker S F, Schmiedel A, Holzapfel M, Renziehausen K, Engel V, Lambert C 2014 J. Phys. Chem. C 111 17467
[6]	Fennel F, Lochbrunner S 2015 Phys. Rev. B 92 140301 Google Scholar
[7]	Engel E, Leo K, Hoffmann M 2006 Chem. Phys. 325 170 Google Scholar
[8]	Linardy E, Yadav D, Vella D, Verzhbitskiy I A, Watanabe K, Taniguchi T, Pauly F, Trushin M 2020 Nano Lett. 20 1647 Google Scholar
[9]	Kühn O, Renger T, May V, Voigt J, Pullerits T, Sundström V 1997 Trends in Photochem. Photobiol. 4 213
[10]	Hader K, May V, Lambert, Engel V 2016 Phys. Chem. Chem. Phys. 18 13368 Google Scholar
[11]	Peteanu L A, Chowdhury S, Wildeman J, Sfeir M Y 2017 J. Phys. Chem. B 121 1707 Google Scholar
[12]	Feist J, Garcia-Vidal F J 2015 Phys. Rev. Lett. 114 196402 Google Scholar
[13]	Schachenmayer J, Genes C, Tignone E, Pupillo G 2015 Phys. Rev. Lett. 114 196403 Google Scholar
[14]	Xu L, Ji Y, Shi X, Wang L, Gao K 2020 Org. Electron. 85 105886 Google Scholar
[15]	Spano F C 1992 Phys. Rev. B 46 13017 Google Scholar
[16]	Juzeliunas G, Knoester J 2000 J. Chem. Phys. 112 2325 Google Scholar
[17]	Mukamel S, Berman O 2003 J. Chem. Phys. 119 12194 Google Scholar
[18]	Mukamel S, Abramavicius D 2004 Chem. Rev. 104 2073 Google Scholar
[19]	Wang L, Plehn T, May V 2020 Phys. Rev. B 102 075401 Google Scholar
[20]	Wang L, May V 2016 Phys. Rev. B 94 195413 Google Scholar
[21]	Tempelaar R, Jansen T L C, Knoester J 2017 J. Phys. Chem. Lett. 8 6113 Google Scholar
[22]	May V 2014 J. Chem. Phys. 140 054103 Google Scholar
[23]	茅江奇, 范旭阳, 路彦珍, 王鹿霞 2021 物理学报 70 047302 Mao J Q, Fan X Y, Lu Y Z, Wang L X 2021 Acta Phys. Sin. 70 047302

[1]	Xu Zhuo, Guo Jing-Yuan, Xiong Zheng-Ye, Tang Qiang, Gao Mu. Luminescence spectra and energy transfer of Tm³⁺ and Tb³⁺ doped in LiMgPO₄ phosphors. Acta Physica Sinica, 2021, 70(16): 167801. doi: 10.7498/aps.70.20210357
[2]	Mao Jiang-Qi, Fan Xu-Yang, Lu Yan-Zhen, Wang Lu-Xia. Exciton-exciton annihilation in molecular aggregations. Acta Physica Sinica, 2021, 70(4): 047302. doi: 10.7498/aps.70.20201399
[3]	Qin Ya-Qiang, Chen Rui-Yun, Shi Ying, Zhou Hai-Tao, Zhang Guo-Feng, Qin Cheng-Bing, Gao Yan, Xiao Lian-Tuan, Jia Suo-Tang. The role of chain conformation in energy transfer properties of single conjugated polymer molecule. Acta Physica Sinica, 2017, 66(24): 248201. doi: 10.7498/aps.66.248201
[4]	Li Mu-Ye, Li Fang, Wei Lai, He Zhi-Cong, Zhang Jun-Pei, Han Jun-Bo, Lu Pei-Xiang. Fluorescence resonance energy transfer in a aqueous system of CdTe quantum dots and Rhodamine B with two-photon excitation. Acta Physica Sinica, 2015, 64(10): 108201. doi: 10.7498/aps.64.108201
[5]	Mo Ya-Xiao, Piao Sheng-Chun, Zhang Hai-Gang, Li Li. Mode coupling and energy transfer in a range-dependent waveguide. Acta Physica Sinica, 2014, 63(21): 214302. doi: 10.7498/aps.63.214302
[6]	Ning Cheng, Feng Zhi-Xing, Xue Chuang. Basic characteristics of kinetic energy transfer in the dynamic hohlraums of Z-pinch. Acta Physica Sinica, 2014, 63(12): 125208. doi: 10.7498/aps.63.125208
[7]	He Yue-Di, Xu Zheng, Zhao Su-Ling, Liu Zhi-Min, Gao Song, Xu Xu-Rong. Electroluminescent energy transfer of hybrid quantum dotsdevice. Acta Physica Sinica, 2014, 63(17): 177301. doi: 10.7498/aps.63.177301
[8]	Wu Qing-Yang, Xie Guo-Hua, Zhang Zhen-Song, Yue Shou-Zhen, Wang Peng, Chen Yu, Guo Run-Da, Zhao Yi, Liu Shi-Yong. Highly efficient all fluorescent white organic light-emitting devices made by sequential doping. Acta Physica Sinica, 2013, 62(19): 197204. doi: 10.7498/aps.62.197204
[9]	Yuan Qiang, Wei Xiao-Feng, Zhang Xiao-Min, Zhang Xin, Zhao Jun-Pu, Huang Wen-Hui, Hu Dong-Xia. Study on stimulated Brillouin scatting energy transfer to amplify laser pulses for shock ignition in laser fusion facilities. Acta Physica Sinica, 2012, 61(11): 114207. doi: 10.7498/aps.61.114207
[10]	Liu Shi-Bing, Liu Yuan-Xing, He Run, Chen Tao. Instantaneous characteristics of excited atom state 5s' 4D7/2 in the copper plasma induced by laser. Acta Physica Sinica, 2010, 59(8): 5382-5386. doi: 10.7498/aps.59.5382
[11]	Wang Jun-Zhuan, Shi Zhuo-Qiong, Lou Hao-Nan, Zhang Xin-Luan, Zuo Ze-Wen, Pu Lin, Ma En, Zhang Rong, Zheng You-Liao, Lu Fang, Shi Yi. Influence of Si crystallization evolution on 1.54 μm luminescence in Er-doped Si/Al2O3 multilayer. Acta Physica Sinica, 2009, 58(6): 4243-4248. doi: 10.7498/aps.58.4243
[12]	Xu Deng. Stimulated emission properties of an organic salt-doped polymer film in microcavity. Acta Physica Sinica, 2009, 58(4): 2781-2784. doi: 10.7498/aps.58.2781
[13]	Shen Han, Liu Jie, Chen Zhi-Feng, Huang Jin-Wang, Shen Yong, Wang Hui, Ji Liang-Nian. Ultrafast energy transfer of a porphyrin-polypyridyl ruthenium (Ⅱ) hybrid linked by a butyl chain. Acta Physica Sinica, 2008, 57(11): 7354-7359. doi: 10.7498/aps.57.7354
[14]	Xu Deng, Ye Li-Hua, Cui Yi-Ping, Xi Jun, Li Li, Wang Qiong. Study of photoluminescence and energy transfer properties of an organic dye salt doped thin films. Acta Physica Sinica, 2008, 57(5): 3267-3270. doi: 10.7498/aps.57.3267
[15]	Wu Chun-Hong, Liu Peng-Yi, Hou Lin-Tao, Li Yan-Wu. The energy transfer in phosphorescent dye PtOEP doped organic molecule Alq. Acta Physica Sinica, 2008, 57(11): 7317-7321. doi: 10.7498/aps.57.7317
[16]	Song Shu-Fang, Zhao De-Wei, Xu Zheng, Xu Xu-Rong. Energy transfer in organic quantum well structures. Acta Physica Sinica, 2007, 56(6): 3499-3503. doi: 10.7498/aps.56.3499
[17]	Wang Si-Sheng, Kong Rui-Hong, Tian Zhen-Yu, Shan Xiao-Bin, Zhang Yun-Wu, Sheng Liu-Si, Wang Zhen-Ya, Hao Li-Qing, Zhou Shi-Kang. Research on photoionization of Ar·NO cluster using synchrotron radiation. Acta Physica Sinica, 2006, 55(7): 3433-3437. doi: 10.7498/aps.55.3433
[18]	Zhang Peng, Zhou Yin-Hua, Liu Xiu-Fen, Tian Wen-Jing, Li Min, Zhang Guo. Study on the energy transfer and luminescent properties in PVK:DBVP blend system. Acta Physica Sinica, 2006, 55(10): 5494-5498. doi: 10.7498/aps.55.5494
[19]	Yu Chun-Lei, Dai Shi-Xun, Zhou Gang, Zhang Jun-Jie, Hu Li-Li, Jiang Zhong-Hong. Concentration quenching mechanism in erbium-doped tellurite glass. Acta Physica Sinica, 2005, 54(8): 3894-3899. doi: 10.7498/aps.54.3894
[20]	Feng Zhi-Fang, Wang Yi-Quan, Xu Xing-Sheng, Jiang Shao-Lin, Hao Wei, Cheng Bing-Ying, Zhang Dao-Zhong. Energy transfer between two continuous channels in photonic crystals. Acta Physica Sinica, 2004, 53(1): 62-65. doi: 10.7498/aps.53.62

期刊类型引用(0)
其他类型引用(2)

Catalog

Vol. 70, Issue 22 - 2021-11-20

Metrics

Abstract views: 6145
PDF Downloads: 112
Cited By: 2

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板