#### 3.3.1. Charge Carriers

In disordered organic materials, charge carriers are spatially localized due to the lack of long-range order. Charge transport is modeled as a hopping process from one localized state to a neighboring localized state. This state can be seen as local energy valleys where charges locate in a long-term, which e.g., represent a small molecule or a few monomers along a polymer chain.

Miller and Abrahams describe hopping as a combination of thermal activation and tunneling [

74]. The hopping rate between states

i and

j reads:

where

${a}_{0}$ is the attempt-to-escape frequency,

$\gamma $ is the inverse localization length,

${k}_{B}$ is the Boltzmann constant,

T gives the temperature,

${r}_{ij}$ the distance and

$\Delta {E}_{ij}={E}_{i}-{E}_{j}$ the energy difference between the initial configuration

i and the final configuration

j, before and after a hopping process, respectively. When hopping upwards in energy, the Boltzmann term accounts for thermal activation [

74]. This rate model is usually used when the coupling between electrons and phonons is weaker [

75,

76].

Marcus has introduced a model that accounts for stronger coupling between lattice and charges [

77]. Because of the low permittivity of organic materials, charges lead to local polarizations of the lattice, which are described as quasi-particles called polarons. The formula for polaron transitions reads:

with the reduced Planck’s constant

ℏ, transfer integral

${I}_{ij}$ and the reorganization energy

$\lambda $ describing the reconfiguration of the molecule due to the deformation by the charge. The effect of a spatial disorder in an irregular Voronoi set is included via the site to site distance

${r}_{ij}$ in Miller–Abrahams rates (Equation (

11)) and via the distance-dependent transfer integral

${I}_{ij}$ in Marcus rates (Equation (

12)). The hopping rates are calculated for every particle and all free neighbors. If a particle already occupies a neighboring site, the hopping rate is zero. When a hopping transition occurs, the occupation information of origin and destination sites and the particle position have to be updated.

To calculate hopping rates, it is necessary to compute the electric energy

${E}_{i}$ for every site

i. The energy at site

i is given by:

with

${E}_{i}^{0}$ being the energy level of the molecular orbital and the energetic disorder

${E}_{i}^{\sigma}$. The energy arising from the external applied field is referred to as

${E}_{i}^{F}$ and

${E}_{i}^{C}$ represents the Coulomb energy. The Coulomb interaction between all particles is calculated using the Ewald sum [

12,

78]. The Ewald sum makes use of the principle of image charges. As the potential at the contact layers has to be constant, all the field vectors have to end perpendicular to contacts. This can be achieved by removing the contact and introducing an image charge of opposite sign and the same distance behind the contact. Since an OSC device has two contacts, every image charge behind one contact also induces an image charge behind the other contact, leading to a periodic scheme. The electrostatic potential at node

i can be calculated by:

where the outer sum corresponds to the periodic replica in the

x-,

y- and

z-direction and the inner one sums over all charged particles

${q}_{j}$ inside the box

$\overline{n}$. The prime in the inner sum notes that self-interactions are excluded. Here,

${\u03f5}_{0}$ is the vacuum permittivity and

${\u03f5}_{r}$ is the permittivity of the material.

The Miller–Abrahams rate provides a generic model for the hopping process modeling the contribution of tunneling and thermal activation by the parameters

${a}_{0}$,

$\gamma $ and the Boltzmann term, respectively. It is used if hopping is mainly caused by single phonon-assisted tunneling between sites of energy

${E}_{i}$ and

${E}_{j}$. Using Marcus theory, one implicitly accounts for molecular details including

${I}_{ij}$ and

$\lambda $. It is frequently used in multiscale studies, as for given organic molecules, the molecular positions and transfer integrals need to be calculated by molecular dynamics and quantum chemical calculations, respectively [

42,

43,

79,

80]. Marcus theory accounts for multiphonon hopping processes.

Recombination of charge carriers is possible when an electron and a hole are located at neighboring sites. Charge recombination is one of the primary loss processes in OSCs besides exciton decay. When recombination occurs, the two particles are removed, and the sites are free again.

#### 3.3.2. Excitons

An exciton is an intramolecular excited state between a Coulomb-bound electron-hole pair [

64]. Compared to the weakly-bound Wannier–Mott excitons present in inorganic semiconductors, strongly-bound Frenkel excitons are formed in organic materials due to the low dielectric constant [

3,

81]. The excited state is a singlet (triplet) state, if the spins of the electron and the hole are anti-parallel (parallel). We will now describe all physical rates modeled for both singlets and triplets, as well as the conversion rates between these excitons. For a detailed discussion on the photophysical processes, please refer to [

64,

82]. A schematic overview of the rates and the corresponding energetic levels are depicted in

Figure 5. The singlet (blue) and triplet excitons (red), residing at the energetic levels

${S}_{1}$ and

${T}_{1}$, respectively, as well as the separated electron-hole pair forming a charge-transfer (CT) state and the ground state

${S}_{0}$ are visualized.

Singlet excitons can be generated upon light absorption. Absorption of a photon excites one electron, resulting in a singlet exciton. The optical generation profile of singlets is modeled using a transfer matrix model (TMM) [

83,

84,

85]. The TMM is used to calculate the generation profile in the active layer under an AM1.5 illumination spectrum. For every layer in penetration direction

z, a constant generation rate

${a}_{\mathrm{gen}}^{s}$ is obtained. As the TMM does not provide the generation profile perpendicular to the penetration direction, the

x and

y position of the generated singlet is chosen randomly [

10,

13].

The generation of both singlets and triplets can occur by electrical excitation. The rate

${a}_{\mathrm{exg}}^{s/t}$ for electrical generation of a singlet/triplet is enabled if two charges of different polarity reside at neighboring sites and if the respective spins, a constant property of the charge carriers, show a singlet/triplet configuration. Depending on the origin of charges, we assign their spin states following a distinguished procedure. If a charge is injected, we allocate the spin following the expected singlet to triplet ratio of 1:3. If an exciton dissociates, the spin states of the charges reflect the spins of the singlet or triplet excitation. We use the convention that holes always have a spin with eigenvalue

$-\hslash /2$ (spin-down), while the spin of the electrons is assigned such that the above criteria are fulfilled. Following Mesta et al. [

23], the electrical generation rate can be calculated as a hopping process of the electron on a hole-occupied neighboring site using the Miller–Abrahams equation (Equation (

11)) [

74]. In comparison to Mesta et al., our model only considers electron hops for the electrical exciton generation. The energy difference for the singlet/triplet generation is given by:

where

${E}_{\mathrm{B}}^{s/t}$ denotes the exciton binding energy.

${E}_{\mathrm{CT},\mathrm{B}}$ gives the charge transfer binding energy, corresponding to the Coulomb binding energy of an electron-hole pair residing at neighboring sites, and

${E}_{\mathrm{LUMO}}$ defines the LUMO energy level difference of the neighboring sites. Coulomb interactions with the surrounding charge carriers are omitted. If an electrical generation event is chosen, the electron and hole are removed, and a singlet/triplet exciton is generated at the site the former hole was located.

All excited states have a certain lifetime before they decay to the ground state. The lifetime of triplets exceeds that of singlets due to the spin selection rules by far. The rate for the exciton decay is chosen as the inverse of average lifetime ${a}_{\mathrm{exd}}^{s/t}=1{\tau}^{s/t}$. Due to the spin-orbit coupling of a phosphorescent dopant, triplets may decay due to the lifted spin-selection rules. Thus, the triplet lifetime is reduced within the radius ${r}_{\mathrm{ISC}}$ around dopant sites.

Excitons are quasi-particles with a net charge of zero. Thus, they diffuse through the organic semiconductor by energy transfer between sites. Existing kMC models account for exciton diffusion using either Förster resonance energy transfer (FRET), Dexter type transfer or are based on a random walk [

19,

86,

87,

88]. The diffusion of singlets and triplets can be modeled using FRET [

89] or by a random walk based on the Einstein–Smoluchowksi model for the Brownian motion with a diffusion coefficient

D [

90,

91]. Förster described a non-radiative energy transfer between an exciton donor and acceptor as a dipole-dipole interaction [

89]. As the Coulomb interaction causes this energy transfer, FRET is a long-distance mechanism. For the random walk (RW), the hopping rate is computed as the inverse of the mean time interval for a hopping process

${\tau}_{\mathrm{hop}}$ between neighboring sites, which gives the rate:

where

l is the next-neighbor distance. Diffusion rates by FRET are modeled with:

where

${r}_{F}$ is the Förster radius,

$\Gamma $ is the total decay rate,

$\Delta {E}_{ij}={E}_{i}-{E}_{j}$ is the exciton energy difference between the initial and final configuration and

${r}_{ij}$ denotes the distance between two sites. The computational effort of random walk is typically 2–3 orders of magnitude smaller compared to FRET as the number of sites considered for the FRET diffusion increases with

${r}_{F}$, while RW diffusion only considers next-neighbors [

19]. Especially for a low energetic disorder, RW and FRET give similar results [

19], with rising energetic disorder FRET needing to be considered.

In case of the triplet diffusion, FRET is a spin-forbidden process and can only occur between phosphorescent dopants within the Förster radius

${r}_{F}$. A second mechanism for the diffusion comes into play. In contrast to the Förster rate, the Dexter type energy transfer rises due to the exchange interaction and may occur between next neighbors [

92]. Following [

22], the Dexter type hopping is modeled as a Miller–Abrahams hopping defined by:

with the exciton energy difference

$\Delta {E}_{ij}$ of initial state

i and final state

j, the Dexter hop rate

${a}_{\mathrm{D}}$ in the zero distance limit and the inverse localization length

$\gamma $ of donor and acceptor sites [

64]. The triplet binding energy

${E}_{\mathrm{B}}^{t}$ is usually assumed to be equal for every site, but can be modified by a disorder following a Gaussian distribution. The Boltzmann factor vanishes in the case of no energetic disorder in the binding energy. Both transfer mechanisms are shown schematically in

Figure 6.

To extract electrical energy from the absorbed photons in organic solar cells, singlets need to be dissociated. This process is only possible if singlets reach an acceptor-donor interface. The dissociation process is reported to happen on extremely fast timescales

${\tau}_{\mathrm{exs}}^{s}$ in BHJs [

93]. A separation rate

${a}_{\mathrm{exs}}^{s}=1/{\tau}_{\mathrm{exs}}^{s}$ is enabled if the singlet is located at an interface. Dissociation of a triplet exciton is interpreted as an electron hopping from its current position towards an unoccupied neighboring site [

21,

22,

23]. The separation rate

${a}_{\mathrm{exs}}^{t}$ is calculated with the Miller–Abrahams rate using an energy difference of

$\Delta E=\Delta {E}_{\mathrm{LUMO}}+{E}_{\mathrm{CT},\mathrm{B}}-{E}_{\mathrm{B}}^{t}$. The dissociation is probable if the state following a dissociation process is energetically lower than the triplet binding energy.

Regarding high spin-orbit coupling, intersystem crossing from a singlet to a triplet state is possible. The reverse process is not considered here. In our model, a transition from singlets to triplets is only possible within a certain radius

${r}_{\mathrm{ISC}}$ around a dopant molecule. The radius of intersystem crossing is a material dependent property. If an exciton resides at a site within

${r}_{\mathrm{ISC}}$, the intersystem crossing rate

${a}_{\mathrm{ISC}}$ is enabled. The intersystem crossing rate is given by:

with an intersystem crossing frequency

${a}_{\mathrm{ISC}}^{0}$. The intersystem crossing decreases exponentially with the distance

${r}_{i\to d}$ to a dopant site

d. If the intersystem crossing occurs, the singlet is converted into a triplet exciton, residing at the corresponding site.

With rising concentration of triplets, another undesired process occurs, the triplet-triplet annihilation (TTA). If two triplet excitons meet, they either create a singlet or a triplet exciton [

20,

21,

94,

95]. The spin state of the resulting excitons depends on the spin angular momentum of the two involved triplet excitons. If a triplet exciton resides at a neighboring site or two involved triplet excitons reside at two dopant sites within the Förster radius, TTA is enabled. As explained before, TTA is interpreted as a triplet exciton hopping on a site occupied by a second triplet. If this transition is performed, at least one triplet exciton is removed. If the TTA process is Dexter-mediated, with a ratio of 1:3, the particle at the final site is either a singlet or a triplet, respectively. In the case of the Förster mediated TTA, one triplet is excited to a higher triplet level

${T}_{\mathrm{n}}$, followed by a relaxation to the lowest triplet exciton level

${T}_{1}$. We assume the relaxation to happen immediately. The second triplet decays to the singlet ground state.