# Generalized Kinetic Monte Carlo Framework for Organic Electronics

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## Abstract

**:**

## 1. Introduction

## 2. Kinetic Monte Carlo Method

- Initialize the system, and define all parameters that are required for the transition rates of all processes in the kMC. Define termination conditions such as the maximum simulation time ${t}_{\mathrm{max}}$.
- Identify the activated transitions for the current system state i and calculate the transition rates $\left\{{a}_{ij}\right\}$ of all physical processes that can occur. The rates ${a}_{ij}$ mirror the probability of the occurrence of the transition to state j. The sum $a={\sum}_{j}{a}_{ij}$ of all rates is calculated.
- Two uniformly-distributed random numbers ${r}_{1}\in \left(0,a\right)$ and ${r}_{2}\in \left(0,1\right)$ are chosen. The first number ${r}_{1}$ chooses the transition $\mu $ from the set of events $\left\{{a}_{ij}\right\}$, which fulfills the condition:$$\sum _{j=1}^{\mu -1}{a}_{ij}<{r}_{1}\le \sum _{j=1}^{\mu}{a}_{ij}\phantom{\rule{0.166667em}{0ex}}.$$The second number ${r}_{2}$ gives the time passed before the transition is performed, given by:$$\tau =-\frac{ln\left({r}_{2}\right)}{a}\phantom{\rule{0.166667em}{0ex}}.$$
- The simulation time t is advanced by $\tau $, and the system configuration is updated following the transition $\mu $.
- If the simulation time exceeds ${t}_{\mathrm{max}}$, the simulation stops, and all particle trajectories and the counters tracking the events are evaluated. If $t<{t}_{\mathrm{max}}$ , the kMC loop (Step 2–4) is repeated.

## 3. kMC Framework for Organic Materials and Devices

#### 3.1. Modeling of the Organic Material

#### 3.1.1. Voronoi Tessellation

#### 3.1.2. Polymer Chains

#### 3.1.3. Phosphorescent Dopants

#### 3.2. Energetic Landscape

- Starting from the randomly Gaussian-distributed energetic landscape, we compute the weighted average of the energies ${E}_{k}$ of all sites k with ${r}_{ik}=\left|{\mathbf{r}}_{i}-{\mathbf{r}}_{k}\right|\le {r}_{\mathrm{corr}}$ for every site i using:$${E}_{i}^{*}=\frac{1}{{\sum}_{k}\frac{1}{\left|{r}_{ik}\right|}}\sum _{k}{E}_{k}\frac{1}{\left|{r}_{ik}\right|}\phantom{\rule{0.166667em}{0ex}}.$$Note that the average is only performed over sites k representing the same material as site i. The average energies ${E}_{i}^{*}$ are stored in an intermediate vector.
- Replace the site energies by the average energies ${E}_{i}\to {E}_{i}^{*}$.
- Scaling of the site energies: The averaging procedure reduces the energetic disorder. To keep the initially chosen $\sigma $, we compute the energetic disorder ${\sigma}^{*}$ after the averaging process. Then, the site energies are scaled by:$${E}_{i}\to {E}_{i}\sqrt{\frac{\sigma}{{\sigma}^{*}}}-\left(\sqrt{\frac{\sigma}{{\sigma}^{*}}}-1\right){E}_{i}^{0}\phantom{\rule{0.166667em}{0ex}}.$$
- The level of correlation can be varied by either changing ${r}_{\mathrm{corr}}$ or repeating Steps 1–3.

#### 3.3. Particles and Transitions

#### 3.3.1. Charge Carriers

#### 3.3.2. Excitons

## 4. Applications

#### 4.1. Charge Carrier Transport through Polymer Chains

#### 4.2. Impact of Dopant Volume Concentration on Exciton Dynamics

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Coarse-graining of the system dynamics and illustration of long-term system states; (

**b**) transition from initial state i to final states j as employed in the kinetic Monte Carlo (kMC) algorithm.

**Figure 2.**Visualization of the next neighbor generation in a cubic lattice (left) and a Voronoi set (right). For most physical processes, only the nearest-neighbors, highlighted in red color, are considered. The blue lattice sites do not define the Voronoi cell; thus, they are not labeled as next neighbors.

**Figure 3.**Chain morphologies on a two-dimensional $20\times 20$ grid with an average polymer chain length of six monomers. Single material with Gaussian chains $\eta =0$ (left), worm-like chains with preferred orientation $\eta =0.9$ (middle) and chains in a BHJ where only the blue material is a polymer and the red material is small molecules (right).

**Figure 4.**Correlation of site energies inside a blend. The black and red dots represent lattice sites of the small molecule PCBM and the polymer P3HT, respectively. A weighted average is performed over all sites of the same material type within the radius of ${r}_{\mathrm{corr},i}$.

**Figure 5.**(Left) Bulk-heterojunction including dopant sites with a volume concentration of $0.8\phantom{\rule{0.166667em}{0ex}}\mathrm{vol}\phantom{\rule{3.33333pt}{0ex}}\%$. (Right) Jablonski diagram illustrating the exciton transitions. All processes considering triplet excitons are visualized in red color, while singlet transitions are highlighted in blue. The diagram shows the lowest excited singlet (${S}_{1}$) and lowest excited triplet state (${T}_{1}$), the singlet ground state (${S}_{0}$) and charge transfer state ($CT$). Our excitonic model consists of the optical singlet exciton generation (${a}_{\mathrm{gen}}^{s}$) upon the absorption of an incident photon $\gamma $, singlet/triplet exciton separation (${a}_{\mathrm{exs}}^{s/t}$), electrical generation (${a}_{\mathrm{exg}}^{s/t}$) and decay (${a}_{\mathrm{exd}}^{s/t}$), triplet-triplet annihilation (${a}_{\mathrm{TTA}}$) and intersystem crossing (${a}_{\mathrm{ISC}}$). Two triplet excitons can annihilate with a rate ${a}_{\mathrm{TTA}}$ forming an intermediate state $T{T}^{*}$, which relaxes to either the lowest singlet ${S}_{1}$ or lowest triplet state ${T}_{1}$.

**Figure 6.**Schematic diagram illustrating the hopping processes of singlet (blue spheres) and triplet (red spheres) excitons. The black dots represent the organic material, while gray dots label dopant sites. FRET is shown for a triplet exciton hopping between dopant sites within the Förster radius ${r}_{F}$. The Dexter type energy transfer is possible between all adjacent sites. The intersystem crossing is visualized for a singlet exciton in proximity to the dopant as a conversion of a singlet to a triplet exciton.

**Figure 7.**Carrier mobility $\mu $ versus electric field F for different morphologies with chain length ${l}_{c}=20\mathrm{nm}$. (

**a**) Fixed morphology with Gaussian chains $\eta =0$ and varying ${a}_{\mathrm{chain}}$ for intrachain transport; (

**b**) morphologies with varying orientation for fixed ${a}_{\mathrm{chain}}=100\times {a}_{0}$.

**Figure 8.**Impact of the dopant volume concentration c on the average exciton lifetime $\tau $ (solid curves) and the effective diffusion length ${L}_{\mathrm{diff},\mathrm{eff}}$ (dashed curves) for singlet excitons (

**a**) and triplet excitons (

**b**).

**Figure 9.**Impact of the dopant volume concentration c on (

**a**) the effective exciton lifetime and (

**b**) intersystem crossing and triplet-triplet annihilation. The red bars show the proportion of converted triplets over the amount of optically-generated singlets, and the black bars represent the percentage of TTA over the amount of converted/generated triplets.

Description | Parameter | Value |
---|---|---|

Gaussian energetic disorder | $\sigma $ | $0.1$ eV |

Attempt-to-escape frequency | ${a}_{0}$ | 3 × 10^{12} s^{−1} |

Inverse localization length | $\gamma $ | 2 nm^{−1} |

Charge density | n | 3.2 × 10^{16} cm^{−3} |

Temperature | T | 298 K |

Simulation time | ${t}_{\mathrm{sim}}$ | 6 μs |

Applied field | F | 5 × 10^{5} − 5 × 10^{6} V cm^{−1} |

Dielectric constant | ${\u03f5}_{r}$ | 3.5 |

Intersite distance | a | 1 nm |

Description | Parameter | Value | Ref. |
---|---|---|---|

Common Parameters | |||

Phosphorescent dopant concentration | c | $0\phantom{\rule{0.166667em}{0ex}}\mathrm{vol}\%-10\phantom{\rule{0.166667em}{0ex}}\mathrm{vol}$% | - |

Energetic disorder excitons | ${\sigma}^{s/t}$ | $0\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ | - |

Inverse localization length | $\gamma $ | $2\phantom{\rule{0.166667em}{0ex}}{\mathrm{nm}}^{-1}$ | [11,12] |

Simulation time | ${t}_{\mathrm{sim}}$ | $1\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ | - |

Exciton parameters | |||

Optical generation rate | ${a}_{\mathrm{gen}}^{s}$ | $7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{nm}}^{-3}$ | - |

Hopping rate (RW) | ${a}_{\mathrm{exh},\mathrm{RW}}^{s}$ | $2.0\times {10}^{10}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | [13] |

Singlet decay rate | ${a}_{\mathrm{exd}}^{s}$ | $2.0\times {10}^{9}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | [12,17,97] |

Singlet binding energy | ${E}_{\mathrm{B}}^{s}$ | $0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ | - |

ISC radius phosphorescent dopant | ${r}_{\mathrm{ISC}}$ | $2\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ | - |

ISC rate | ${a}_{\mathrm{ISC}}^{0}$ | $1.0\times {10}^{12}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | - |

Förster radius dopant | ${r}_{\mathrm{F},\mathrm{dopant}}$ | $1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ | [98] |

Dexter rate | ${a}_{\mathrm{D}}$ | $2.9\times {10}^{8}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | [22] |

Decay rate dopant | ${a}_{\mathrm{exd},\mathrm{dopant}}^{t}$ | $1.0\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | - |

Decay rate host material | ${a}_{\mathrm{exd},\mathrm{host}}^{t}$ | $1.0\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | [24,63] |

Triplet binding energy | ${E}_{\mathrm{B}}^{t}$ | $1.0\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ | - |

TTA rate Dexter type | ${a}_{\mathrm{D},\mathrm{TTA}}$ | $2.9\times {10}^{8}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | [22] |

TTA Förster radius | ${r}_{\mathrm{F},\mathrm{TTA}}$ | $1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ | [98] |

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## Share and Cite

**MDPI and ACS Style**

Kaiser, W.; Popp, J.; Rinderle, M.; Albes, T.; Gagliardi, A. Generalized Kinetic Monte Carlo Framework for Organic Electronics. *Algorithms* **2018**, *11*, 37.
https://doi.org/10.3390/a11040037

**AMA Style**

Kaiser W, Popp J, Rinderle M, Albes T, Gagliardi A. Generalized Kinetic Monte Carlo Framework for Organic Electronics. *Algorithms*. 2018; 11(4):37.
https://doi.org/10.3390/a11040037

**Chicago/Turabian Style**

Kaiser, Waldemar, Johannes Popp, Michael Rinderle, Tim Albes, and Alessio Gagliardi. 2018. "Generalized Kinetic Monte Carlo Framework for Organic Electronics" *Algorithms* 11, no. 4: 37.
https://doi.org/10.3390/a11040037