Computers, Materials & Continua DOI:10.32604/cmc.2021.017439 | |

Article |

A Fractional Drift Diffusion Model for Organic Semiconductor Devices

School of Engineering Technology, Purdue University, West Lafayette, 47906, USA

*Corresponding Author: Yi Yang. Email: yang1087@purdue.edu

Received: 30 January 2021; Accepted: 8 March 2021

Abstract: 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.

Keywords: Fractional drift diffusion model; Gummel’s iteration; Caputo’s fractional-order ordinary differential equation; organic field effect transistor

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

where current densities are given by

where

where

where

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 [9], which implies that our Fr-DD model solver has wider applicability than the DD model solver based on SG method. Finally, we design three numerical examples to demonstrate the high accuracy and computational performance of the Fr-DD model solver, and experimentally validate the Fr-DD model for a steady-state OFET.

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 [36]. In Section 7, we show the conclusions of this work.

2.1 Definition of Fractional Operators

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

Lemma 2.1

Proof. See [35].

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

Lemma 2.2

Proof. Eqs. (13) and (14) can be inferred from RL and Caputo’s definitions, and the proof for Eq. (15) is given in [35]. From Lemma 2.1, we have

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.

Lemma 2.3

Proof. See [35].

One important formula relating the Laplace transform and two-parameter Mittag–Leffler function is given in Eq. (19), and its proof can be found in [37].

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

2.2 Analytic Solution of Caputo’s Linear Fractional-Order ODE

If we let

Theorem 2.4 Consider the Caputo’s linear fractional-order ODE defined in a discrete 1D space domain with

Then, its solution is given by

Proof. Apply Laplace transform on both sides of Eq. (20), it gives

Rearrange both sides of Eq. (22) and take the inverse Laplace transform, we have

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

Let us assume that the input function

We notice that the second term on the RHS of Eq. (24) involves a fractional integral of order

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

2.3 Discrete Approximation of Fractional Integrals 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.

Lemma 2.5

Proof. See [39].

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

Lemma 2.6

Proof. The discrete approximation in Eq. (29) can be constructed by applying the piecewise quadrature, the error estimates can be derived as follows,

3 Derivation of the Computational Scheme

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 x direction be

3.1 Discretization of Fr-DD Model in Transient State

For Poisson equation, we directly apply the second-order finite central difference on the Laplace operator, then Eq. (7) becomes

where the generalized dielectric coefficients are given by

where

where

For the electron continuity equation, the diffusion coefficient Dp, Dn and carrier mobility

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 k + 1):

Referring to Eq. (24) and Theorem 2.4, the solutions to these two fractional-order ODEs can be obtained as follows:

where the generalized state transition functions are defined by

In Eq. (8), the gradient of current density is approximated by

where

where the generalized state transition functions

For the left Riemann–Liouville integrals appearing in Eqs. (41)–(45), the three-point Simpson’s rule inferred by Lemma 2.5 can be applied to obtain their approximated values to order of

The other Riemann–Liouville integrals

The hole continuity equation in Eq. (9) can also be discretized with a similar procedure, and its discrete form is obtained in Eq. (49),

where coefficients

where we denote the generalized reversed state transition functions by

As remarkably similar in format to Eqs. (40), (49) can also be transformed into a compact matrix equation

where fn, fp are the predefined functions in Dirichlet boundary conditions, and gn, gp are the functions derived from the Neumann or Robin boundary conditions. Furthermore, the initial value conditions are specified in Eq. (61). Given the discrete form of the transient-state Fr-DD model in Eqs. (31), (40) and (49) and the consistency between the initial value and boundary conditions, we propose Algorithm 1 to solve the unknowns

3.2 Discretization of Fr-DD Model in Steady State

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 Eqs. (31), (62) and (63).

The boundary conditions for Poisson’s equation and the carrier continuity equations are specified in Eqs. (32), (33), (59) and (60). By rearranging Eqs. (31), (62) and (63), three matrix equations, i.e.,

3.3 Special Case when

When

where the new coefficients are defined as

4 Consistency and Convergence Analysis

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 [40]. Furthermore, it can be inferred from Lemma 2.5 and Lemma 2.6 that the truncation error of the discretized carrier continuity equations in Eqs. (40) and (49) will vanish as the spatial and time step sizes shrink to zero. Nevertheless, Eq. (16) hints that an additional truncation error can be generated by composing Caputo’s fractional derivative terms in the current density with an integer-order gradient operator on the left side of the equation. To test the influence of this truncation error on the consistency of Eqs. (40) and (49), we propose Theorem 4.1, which gives the shrinking order of this truncation error with the spatial step sizes.

Theorem 4.1 Consider the two-dimensional divergence terms

Proof. Observing that both equations are similar in structure and symmetric in x and y directions, it is sufficient to prove Eq. (66) in only the x direction. From Eq. (12), we obtain

If we take the first derivative of both sides of Eq. (68), we can obtain

According to Eq. (12),

Substituting Eq. (70) into Eq. (69) and observing that

Then we can see Eq. (66) as a corollary to Eq. (71). This completes the proof.

In the derivation of Eqs. (40) and (49), we treat current I and J as constants and solve Caputo’s linear fractional-order ODE within two consecutive grid points. Therefore, from Theorem 4.1, we can get

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

Theorem 4.2 The Gummel mapping

Proof. substitute

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.

Example 5.1 Consider the following single-carrier transport problem with fractional derivatives in both time and space for

where

Example 5.1 is a benchmark problem constructed by the method of manufactured solutions [41]. The ground truth is known with its solutions at t = 0.02 s sketched in Fig. 2. The ground truth is compared to the numerical solutions to evaluate the convergence order of our algorithms. The error in this example is calculated by a variant form of the Frobenius norm acting on the error matrix, i.e.,

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

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

Example 5.2 Consider the following steady-state single-carrier transport problem in a 2D p-type organic field effect transistor (OFET).

where the effective hole mobility

Eqs. (84) and (85) are subject to proper boundary conditions for

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

Example 5.3 Consider the following single-carrier transport problem in a 2D p-type solar cell in both time and space for

where

In Example 5.3, we let the spatial step size be 1e −8 m and the temporal step size be 1e −6 s. First, we fix

6 Experimental Validation of the Fractional Drift-Diffusion OFET Model

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

Consider that the length of the source electrode LS and the drain electrode LD are both

where Jy is the y-component of the continuous current density at the Au-DNTT interface, iD is the discrete grid index in the x-direction and jD is the discrete grid y index at the Au-DNTT interface. In our case, the spatial step sizes

The

When

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

Funding Statement: This work was supported in part by the National Science Foundation through Grant CNS-1726865 and by the USDA under Grant 2019-67021-28990.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

1This was initially a simplified Fr-DD model with only time-derivative fractionalized, the order of spatial derivatives remained integer.

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