Because charge carriers of many organic semiconductors (OSCs) exhibit fractional drift diffusion (Fr-DD) transport properties, the need to develop a Fr-DD model solver becomes more apparent. However, the current research on solving the governing equations of the Fr-DD model is practically nonexistent. In this paper, an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices. The Fr-DD model is composed of two fractional-order carriers (i.e., electrons and holes) continuity equations coupled with Poisson’s equation. By treating the current density as constants within each pair of consecutive grid nodes, a linear Caputo’s fractional-order ordinary differential equation (FrODE) can be produced, and its analytic solution gives an approximation to the carrier concentration. The convergence of the solver is guaranteed by implementing a successive over-relaxation (SOR) mechanism on each loop of Gummel’s iteration. Based on our derivations, it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme. In addition, the consistency and convergence of the two core algorithms are proved, with three numerical examples designed to demonstrate the accuracy and computational performance of this solver. Finally, we validate the Fr-DD model for a steady-state organic field effect transistor (OFET) by fitting the simulated transconductance and output curves to the experimental data.

The mathematical modeling of the electrons and holes transports in an inorganic semiconductor (ISC) is established by a system of coupled partial differential equations (PDEs), which are formulated by Gauss’ law applied to the electrical potential _{n}_{p}_{p}_{n}

where current densities are given by _{T}_{n}_{p}_{t}_{t}

where _{0}. The 1D continuity equation for free charge carriers in p-type OSCs was also derived by References [

where _{f}_{t}

where _{p}^{1}

This was initially a simplified Fr-DD model with only time-derivative fractionalized, the order of spatial derivatives remained integer.

A discretization scheme, which discretizes the time-fractional derivative with backward finite difference method and the integer-order spatial derivatives with finite center difference method, was proposed to solve the 1D Fr-DD model [where

Here, we set up a general-form Fr-DD model to simulate the anomalous transport behavior of charge carriers in OSCs. Equipped with proper initial values and boundary conditions, the Fr-DD model can handle the transient or steady-state dynamics of any-type OSC device. In addition, we develop an iterative solver for the Fr-DD model based on two novel algorithms and propose Theorem 4.2 to show the convergence of the model solver. It can be shown that the discretized DD model equation via our discretization scheme coincides with the discrete-form Fr-DD equations derived from the Scharfetter–Gummel (SG) discretization method [

The remainder of the paper is organized as follows. Section 2 presents preliminaries in fractional calculus. Section 3 develops the solver in detail. Section 4 discusses the consistency and convergence analysis of the algorithms. Three numerical examples are provided in Section 5 to support our theoretical analysis and demonstrate the computational performance of our method. In Section 6, we adjust the parameters in the Fr-DD model to fit the experimental characteristic curves measured from a DNTT-based OFET [

The Riemann–Liouville (RL) fractional derivative with order

where

Both RL and Caputo’s fractional derivatives can be considered as interpolation to integer-order derivatives, which means

By directly employing the definitions, the composition rules for fractional derivatives can be given as the following lemma.

It can be observed that both RL’s and Caputo’s fractional derivatives can be composed with an integer-order derivative from two sides, but the composition is not commutative. Next, let us give the Laplace transformation on RL and Caputo’s fractional derivatives as the following lemma.

One important formula relating the Laplace transform and two-parameter Mittag–Leffler function is given in

Subsequently, we will present the analytic solution for Caputo’s fractional linear time-invariant (LTI) state equation.

If we let

Rearrange both sides of

The theorem we presented above establishes the precise relationship between states on two consecutive grid points in a 1D discrete space _{i}

Let us assume that the input function

We notice that the second term on the RHS of

In the next section, we present discrete approximation formula for the left Riemann–Liouville integral and fractional derivatives.

The fractional integral of the generalized state transition function cannot be evaluated through an analytic formula. The following lemma gives the composite Simpson’s rule for approximating a left Riemann–Liouville integral.

_{2n},

For the transient-state Fr-DD model, the discretization of the time-fractional derivative is necessary, the following lemma gives a first-order approximation for Caputo’s fractional time derivative of order

_{k+1},

Without loss of the generality, we implement the discretization schemes in the two-dimensional spatial domain, the equations and algorithms derived afterward can be extended to one-dimensional and three-dimensional scenarios. Let the spatial step size in the

For Poisson equation, we directly apply the second-order finite central difference on the Laplace operator, then

where the generalized dielectric coefficients are given by _{k}

where

where

For the electron continuity equation, the diffusion coefficient _{p}_{n}

For Caputo’s space-fractional gradient operator in the electron continuity equation, firstly we will treat the current density flowing through the interval of two consecutive normal grids as a constant, which can result in two Caputo’s linear fractional-order ODEs (assume that the time step is at

Referring to

where the generalized state transition functions are defined by

In

where

where the generalized state transition functions

For the left Riemann–Liouville integrals appearing in

The other Riemann–Liouville integrals ^{(k+1)} is the unknown interior electron concentrations at the (

The hole continuity equation in

where coefficients

where we denote the generalized reversed state transition functions by

As remarkably similar in format to

where _{n}_{p}_{n}_{p}

Since Caputo’s fractional derivative of any constant is zero, the time-derivative term with Caputo’s fractional derivatives vanishes in steady state. In contrast to the transient-state Fr-DD model, the discretized steady-state Fr-DD model is formed by

The boundary conditions for Poisson’s equation and the carrier continuity equations are specified in _{n}n_{n}_{p}p_{p}

When

where the new coefficients are defined as

The proposed discretization scheme is consistent if the truncation error terms can be made to vanish as the mesh and time step size is reduced to zero. First of all, the consistency of the finite center difference scheme applied to the Poisson equation can be easily proved [

If we take the first derivative of both sides of

According to

Substituting

Then we can see

In the derivation of

where

For convenience, the convergence analysis is only performed on Algorithm 2, but the conclusions of the analysis also apply to Algorithm 1 due to its structural similarity to Algorithm 2 within each step of time advancement. Let us begin our analysis by setting up a finite dimensional vector space

where

By damping the intermediate results, we can get

Taking quotient on both sides and applying triangular inequality yield

Since the relative sizes of

In this section, we consider three numerical examples to evaluate the accuracy and demonstrate the computational performance of our Fr-DD model solver. All the numerical computations below are based on a MATLAB (R2019b) subroutine and performed on a laptop (MacBook Pro 2019) with Intel Core i9 CPU and 16 GB of RAM.

where

Example 5.1 is a benchmark problem constructed by the method of manufactured solutions [

To verify the convergence order in time, we make the spatial step size

Error | Order | CPU time | Error | Order | CPU time | Error | Order | CPU time | |
---|---|---|---|---|---|---|---|---|---|

1/100 | 1.118e −4 | 15.8 | 7.443e −5 | 15.6 | 4.443e −5 | 16.1 | |||

1/200 | 5.416e −5 | 1.046 | 26.4 | 3.085e −5 | 1.271 | 26.2 | 1.277e −5 | 1.799 | 27.3 |

1/400 | 2.300e −5 | 1.236 | 49.0 | 8.391e −6 | 1.878 | 48.9 | 3.033e −6 | 2.074 | 49.6 |

1/800 | 6.886e −6 | 1.740 | 89.5 | 3.141e −6 | 1.418 | 88.7 | 1.103e −6 | 1.459 | 91.4 |

1/1600 | 2.005e −6 | 1.780 | 156.8 | 9.358e −7 | 1.747 | 155.2 | 3.946e −7 | 1.483 | 162.8 |

Error | Order | CPU time | Error | Order | CPU time | Error | Order | CPU time | |
---|---|---|---|---|---|---|---|---|---|

1/100 | 5.936e −4 | 18.2 | 8.205e −4 | 17.8 | 8.828e −4 | 18.7 | |||

1/200 | 5.639e −4 | 0.074 | 27.9 | 7.976e −4 | 0.040 | 28.7 | 8.590e −4 | 0.039 | 31.7 |

1/400 | 5.485e −4 | 0.040 | 55.8 | 7.860e −4 | 0.021 | 53.4 | 8.472e −4 | 0.020 | 58.4 |

1/800 | 5.407e −4 | 0.020 | 111.5 | 7.802e −4 | 0.011 | 104.2 | 8.413e −4 | 0.010 | 109.1 |

1/1600 | 5.368e −4 | 0.010 | 181.4 | 7.773e −4 | 0.005 | 194.5 | 8.384e −4 | 0.005 | 201.5 |

To check the spatial convergence order, we take a sufficiently small temporal step size

Error | Order | CPU time | Error | Order | CPU time | Error | Order | CPU time | |
---|---|---|---|---|---|---|---|---|---|

0.2 | 1.605e −3 | 43.1 | 1.764e −3 | 41.2 | 2.037e −3 | 41.9 | |||

0.1 | 1.050e −4 | 3.934 | 106.5 | 8.078e −5 | 4.449 | 108.5 | 7.632e −5 | 4.738 | 116.7 |

0.05 | 1.084e −5 | 3.276 | 305.9 | 3.446e −5 | 1.229 | 298.7 | 5.169e −5 | 0.562 | 307.6 |

Error | Order | CPU time | Error | Order | CPU time | Error | Order | CPU time | |
---|---|---|---|---|---|---|---|---|---|

0.2 | 9.323e −3 | 48.3 | 1.346e −2 | 56.8 | 1.522e −2 | 52.2 | |||

0.1 | 4.146e −3 | 1.169 | 158.3 | 6.313e −3 | 1.092 | 170.4 | 7.220e −3 | 1.076 | 175.8 |

0.05 | 2.182e −3 | 0.926 | 343.2 | 3.296e −3 | 0.938 | 363.7 | 3.716e −3 | 0.958 | 396.4 |

where the effective hole mobility _{p}

^{2}/Vs) |
_{T} |
|||
---|---|---|---|---|

Basic electric charge | Relative permittivity for OSC | Relative permittivity for dielectric | Hole mobility for OSC | Thermal voltage |

1.60217646e −19 | 3.0 | 3.9 | 4.5e −5 | 0.0255 |

Applying the above boundary conditions and Algorithm 2, we can obtain the simulated steady-state electric potentials and hole concentrations within the solution domain for

The current density contains drift and diffusive components, i.e.,

Since it is challenging in this example to find an initial value condition consistent with the boundary conditions, even if we can get the steady-state solution for the OFET, it is almost impossible to obtain a transient solution that approaches the steady-state solution over time. To explore the effects of time-derivative order

where

^{2}/Vs) |
_{0} ( |
||||
---|---|---|---|---|---|

The side length of the domain | Vacuum permittivity | Relative permittivity for OSC | Hole mobility for OSC | Initial hole concentrations agitated by the light pulse | |

1e–6 | 8.854e −12 | 3.0 | 4.5e −5 | 1e22 |

In Example 5.3, we let the spatial step size be 1

In Example 5.2, we simplify boundary conditions to better discuss the influence of

Consider that the length of the source electrode _{S}_{D}

where _{y}_{D}_{D}_{D}

The _{gs}_{ds}

When _{ds}_{gs}_{ds}_{gs}_{ds}_{gs}_{gs}_{ds}

This work aimed to develop a Fr-DD model solver for simulating the anomalous dynamics of OSC devices. Two algorithms based on a novel discretization scheme and successively over-relaxed Gummel’s iteration are proposed here to solve the transient and steady-state Fr-DD model equations. This study has identified the consistency of the two algorithms by showing that the truncation error from the discretized divergence of current density functions will vanish with the spatial step size to a positive fractional order of