and LUMO energies are assumed to be uncorrelated. We choose a relatively large injection barrier (0.4 eV) in order to suppress the unipolar Coulomb correlation effect. This effect can in principle be included, but this is outside the scope of this work. For simplicity we take = = σ . The simulations typically include 10 disorder configurations to obtain sufficient statistics. The lateral dimensions of the devices simulated are typically 50 nrn x 50 nm. In realistic devices (e.g. OLEDs), a three-layer structure is commonly used, consisting of a hole transport layer (HTL), an emissive layer (EML), and an electron transport layer (ETL). In Fig. 3A, the schematic band diagram for a simple symmetric three-layer HTL (20 nm) /EML (60 nm) /ETL (20 nm) device is shown. In Fig. 3B, the simulated current density-voltage characteristics of this device for different energetic disorder σ is plotted. The agreement between the ME (curves) and KMC (symbols) results is very good for low and moderate voltages. Deviations are observed for large voltages at large disorder. 

In Fig. 4, the electron and hole concentration profile, the recombination pro- hle, and the electric held profile are shown for the device shown in Fig. 3A with σ = 0.10 eV. The agreement between the 3D-ME (curves) and 3D-KMC (symbols) simulations is very good at low voltages (V = 2 V and 4 V). At at large voltages (V = 10 V), the 3D-ME simulations do not reproduce the 3D-KMC results well. We suspect that this is because our model for recombination (Eqs. (8) (14)) is based on the low electric field regime, which cannot be directly applied to high electric fields (i.e. eVa/L σ ). Such high electric fields add complexity to the recombination process. The 3D-ME model in this case underestimates the local recombination rate, leading to an overestimation of the carrier concentration and current density. This is consistent with the current density-voltage simulation results in Fig. 3. We note that at these large voltages 3D-KMC simulations are computationally as efficient or even more efficient than 3D-ME simulations. Therefore, we have not extended the 3D-ME model to include high-held cases. 

To further validate the 3D bipolar ME model, we have compared the current density-voltage characteristics, charge carrier profile, and recombination profile of 3D-ME and 3D-KMC simulations for 32 additional devices. The following device parameters are varied: width of the energetic disorder ( σ = 0.05, 0.075, 0.10, or 0.12 eV), disorder energies for HOMO and LUMO ( e u , for the combination of 0.05/0.075 eV, 0.05/0.10 eV, or 0.075/0.10 eV), disorder energies σ in different layers 

