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Pascal
Kordt
*^{a},
Sven
Stodtmann
^{b},
Alexander
Badinski
^{b},
Mustapha
Al Helwi
^{cd},
Christian
Lennartz
^{b} and
Denis
Andrienko
*^{a}
^{a}Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany. E-mail: kordt@mpip-mainz.mpg.de; denis.andrienko@mpip-mainz.mpg.de
^{b}BASF SE, Scientific Computing Group, 67056 Ludwigshafen, Germany
^{c}BASF SE, GVE/M-B009, 67056 Ludwigshafen, Germany
^{d}IHF Institut, Technische Universität Braunschweig, Braunschweig, Germany

Received
22nd June 2015
, Accepted 31st July 2015

First published on 3rd August 2015

Continuous drift–diffusion models are routinely used to optimize organic semiconducting devices. Material properties are incorporated into these models via dependencies of diffusion constants, mobilities, and injection barriers on temperature, charge density, and external field. The respective expressions are often provided by the generic Gaussian disorder models, parametrized on experimental data. We show that this approach is limited by the fixed range of applicability of analytic expressions as well as approximations inherent to lattice models. To overcome these limitations we propose a scheme which first tabulates simulation results performed on small-scale off-lattice models, corrects for finite size effects, and then uses the tabulated mobility values to solve the drift–diffusion equations. The scheme is tested on DPBIC, a state of the art hole conductor for organic light emitting diodes. We find a good agreement between simulated and experimentally measured current–voltage characteristics for different film thicknesses and temperatures.

The aforementioned approach has become a standard tool for analyzing experimental data.^{13} It has, however, several issues: (i) Gaussian disorder models are parametrized only for materials with moderate energetic disorder, σ < 0.15 eV at room temperature, while many amorphous materials have a higher σ. (ii) The spatial correlation of site energies in the ECDM is material-independent and has an (approximate) 1/r decay, where r is the intermolecular distance, but recent studies show that this decay may be different.^{14} (iii) Due to the non-Gaussian shape of the density of states,^{15} the energetic disorder and the lattice constant are different from those provided by microscopic calculations,^{14} thus making them merely fitting parameters without a comprehensive link between macroscopic properties and the chemical composition of the material.

In this paper we propose an approach which does not have these limitations. In a nutshell, the mobility dependence on charge density, field, and temperature is first tabulated by combining quantum mechanical, classical atomistic and coarse-grained stochastic models for charge transfer and transport. These tables, corrected for finite-size effects, are then used to solve the drift–diffusion equations.

To illustrate the advantages of the method, we compare it to the ECDM and the Mott–Gurney model^{16} as well as to experimental measurements performed on amorphous layers of Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]Ir (DPBIC), a hole-conducting material used in organic light emitting diodes (OLEDs)^{17} and organic photovoltaic cells (OPVs).^{18}

The paper is organized as follows. In the Methods section we describe the coarse-grained, off-lattice transport model, the procedure used to tabulate the charge carrier mobility, the algorithm used to solve drift–diffusion equations, and the parametrization of the extended correlated Gaussian disorder model. The entire workflow is summarized in Fig. 1. We also recapitulate the main results of the Mott–Gurney model and provide details of experimental measurements. The I–V curves, electrostatic potential, and charge density profiles are then compared in Section 3, where we also validate the transferability of the method by studying different layer thicknesses and temperatures. A short summary concludes the paper.

The charge transport network is then generated as follows. A list of links is constructed from all molecules with adjacent conjugated segments closer than 0.7 nm. For each link a charge transfer rate is calculated using Marcus theory, i.e., in the high-temperature limit of the non-adiabatic charge transfer theory,^{22}

(1) |

Electronic couplings J_{ij} are evaluated for each dimer by using the dimer projection method,^{23} the PBE functional and the def2-TZVP basis set. These calculations were performed using the TURBOMOLE package.^{24} Note that the values of electronic couplings can deviate by up to 50%, depending on the functional and the basis set size.^{25,26} This deviation is, however, systematic and will result in a constant prefactor for the mobility, i.e., we do not expect any changes in functional dependencies on the external field, charge density, or temperature.

The hole reorganization energy,^{27}λ_{ij} = 0.068 eV, was evaluated in the gas phase using the B3LYP functional and 6-311g(d,p) basis set. Site energy differences, ΔE_{ij} = E_{i} − E_{j} were evaluated using a perturbative scheme^{28} with the molecular environment modeled by a polarizable force-field, parametrized specifically for these calculations. In this approach, the site energy E_{i} = E^{int}_{i} + E^{el}_{i} + E^{pol}_{i} + qF·r_{i} is the sum of the gas phase ionization potential, E^{int}_{i} = 5.87 eV, an electrostatic part, E^{el}_{i}, an induction contribution, E^{pol}_{i}, and the contribution due to an external electric field, qF·r_{i}. The mean value of these energies gives an ionization potential of E^{IP} = 5.28 eV.

The electrostatic contribution was evaluated using the Ewald summation technique^{29,30} adapted for charged, semi-periodic systems^{31,32} and distributed multipole expansions.^{33,34} Note that using an interaction cutoff would yield a shifted energetic landscape with an underestimated spatial correlation of energies.^{35}

The induction contribution, E^{pol}_{i}, was calculated self-consistently using the Thole model^{36,37} with a 3 nm interaction range. Note that the set of Thole polarizabilities were scaled in order to match the volume of the polarizability ellipsoid calculated using the B3LYP functional and 6-311g(d,p) basis set. This step is required to account for larger polarizabilities of conjugated, as compared to biological, molecules.

The resulting charge transport network is used to parametrize the coarse-grained model, by matching characteristic morphological and transport properties of the system, such as the radial distribution function of molecular positions, the list of neighboring molecules, the site energy distribution and spatial correlation, and the distance-dependent distribution of transfer integrals.^{14,38} The coarse-grained model allows to study larger systems, here of 4 × 10^{4} and 4 × 10^{5} sites, which are required to perform simulations at low charge carrier densities, in our case from 0.025 down to 10^{−5} carriers per site.

Charge transport is modeled using the kinetic Monte Carlo (KMC) algorithm. Note that charge carriers interact only via the exclusion principle, i.e., a double occupation of a molecule is forbidden. Charge mobility is evaluated by averaging the carrier velocity along the field, μ = 〈v〉·F/F^{2}. KMC simulations are repeated for eight different temperature values, from 220 K to 992 K, and twelve field values, in the range of 2.5–30 × 10^{7} V m^{−1}.

To avoid finite size effects, an extrapolation procedure^{39,40} is used for small charge carrier densities. The mobility is simulated at a range of higher temperatures, where mobilities are non-dispersive and hence system-size independent. The extrapolation to lower temperatures is performed by parametrizing the analytic mobility versus temperature dependence available for one-dimensional systems^{8} or, alternatively, using the box-size scaling relation.^{39}

The tabulated mobility is finally interpolated and smoothed by the scattered data interpolation method using radial base functions,^{41} which can treat many-dimensional, unstructured data.

J_{n/p} = ±ρ_{n/p}μ_{n/p} ∇ψ − D_{n/p} ∇ρ_{n/p}, | (2) |

(3) |

(4) |

Since charge carriers occupy energetic levels according to Fermi–Dirac statistics, the carrier density is related to the quasi-Fermi level, η, as

(5) |

(6) |

Eqn (2)–(6) are solved using an iterative scheme, until a self-consistent solution for electrostatic potential, ψ, density, ρ, and current, I, is found.^{43} First the equations are rescaled to ensure numerical stability, which is necessary since carrier density and electrostatic potential vary by several orders of magnitude. Then they are discretized according to a scheme proposed by Scharfetter and Gummel,^{44} linearized,^{45} and solved by using the Gummel iteration method,^{46} adapted to organic semiconductors at finite carrier density. This method is less sensitive to the initial value than a Newton algorithm and thus is the method of choice despite its slower convergence^{47} in terms of iteration steps. The tabulated mobility values, μ(F,ρ,T), computed in Section 2.1, are used while solving eqn (2)–(6).

We use Dirichlet boundary conditions for the electrostatic potential, ψ, by setting the potential difference at the boundaries to ψ_{eff} = V_{app} − V_{int}, where V_{app} is the applied potential and V_{int} the built-in potential, defined as the difference of the materials' work functions. For ITO and Aluminum we use experimental values: the work function of ITO is reported to lie in the range from 4.15 eV to 5.3 eV,^{48–51} and for Aluminum from 4.06 eV to 4.26 eV.^{52} Here we assume average values of 4.73 eV for ITO and 4.16 eV for Aluminum. In combination with the calculated DPBIC solid-state ionization potential (IP) of 5.28 eV, which is the mean value of the site energies, E_{i}, that are calculated as described before, this yields injection barriers of ΔE_{ITO} = 0.55 eV and ΔE_{Al} = 1.12 eV.

The charge density at the electrodes is fixed to the density resulting from inserting ΔE_{ITO/Al} into eqn (5). To model the doped interlayers (see Section 2.5) within a five nanometer range from both electrodes, an additional charge concentration of 3 × 10^{−4} carriers per site, estimated from previous calculation^{53} is added in these regions when solving the Poisson eqn (4), leading to high hole densities in the doped regions even without space-charge limited effects.

The fit to the ECDM model yields a lattice constant of a = 0.44 nm, an energetic disorder of σ = 0.211 eV, and a zero-field zero-density mobility of μ_{0}(300 K) = 1.8 × 10^{−13} m^{2} V^{−1} s^{−1}.^{20} These values serve mainly for providing a fitting and extrapolation function as they differ from the values observed in microscopic simulations (a = 1.06 nm, σ = 0.176 eV, μ_{0}(300 K) = 3.4 × 10^{−12} m^{2} V^{−1} s^{−1}).

(7) |

(8) |

(9) |

All films were fabricated by vacuum thermal evaporation of DPBIC on a glass substrate, patterned with the ITO layer. Thicknesses were determined by optical ellipsometry after a simultaneous deposition of the same amount of DPBIC on a silicon wafer.

To illustrate the transferability of the proposed method we also compare current–voltage characteristics for different temperatures and different film thicknesses. Fig. 3 shows that for high temperatures the agreement between theory and experiment is excellent. At 233 K deviations are significant and can be attributed to the breakdown of the drift–diffusion description, since at low temperature and large energetic disorder charge transport becomes dispersive, showing anomalous diffusion.^{53} Its description using equilibrium distributions, mobility and diffusion constant cannot be justified in this situation. Moreover, Marcus theory only applies to sufficiently high temperatures. The crossover temperature below which Miller–Abrahams rates^{54} become a more appropriate description has been estimated to be about 250 K.^{55}

Fig. 3 Current–voltage characteristics for different temperatures and slab thicknesses simulated using tabulated mobilities (lines) and measured (symbols). |

Using this scheme, we have simulated I–V characteristics of a single-layer device, and found them to be in a good agreement with the experimentally measured I–V curves, whereas significant deviations have been observed for the ECDM and Mott–Gurney models.

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