Scharfetter-Gummel Scheme Research Articles

Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

Combining the hybrid mimetic mixed method with the Scharfetter-Gummel scheme for magnetised transport in plasmas

In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant of the Scharfetter-Gummel (SG) scheme for advection. The proposed hybrid scheme can be implemented very efficiently via static condensation. Numerical tests are then performed to show the applicability of the combined HMM-SG scheme, even for highly anisotropic magnetic fields.

Importance of satisfying thermodynamic consistency in optoelectronic device simulations for high carrier densities

We show the importance of using a thermodynamically consistent flux discretization when describing drift–diffusion processes within light emitting diode simulations. Using the classical Scharfetter–Gummel scheme with Fermi–Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show an unphysical hump within the quantum well regions. This result originates from neglecting diffusion enhancement associated with Fermi–Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the humps in the Fermi levels disappear. We show that thermodynamic inconsistency has far reaching implications on the current–voltage curves and recombination rates.

Energy dissipative and positivity preserving schemes for large-convection ion transport with steric and solvation effects

Ion transport plays a crucial role in biophysical and electrochemical applications. While being modeled by the classical Poisson–Nernst–Planck (PNP) equations, many ionic features are ignored in the course-graining modeling of ionic interactions. In this work, we study ion transport by modified PNP equations including effects arising from steric interactions and the Born solvation, which both could cause strong convection. The modified NP equations are first rewritten in a gradient-flow reformulation that involves mobility and chemical potential terms. The mobility is then approximated by upwind fluxes with various flux limiters on the half-grid points, and the chemical potential terms are then implicitly approximated by central differencing. The resulting first-order in time scheme is proved to be energy dissipative unconditionally. Furthermore, the unconditional existence of positive numerical concentration is proved by making use of the singularity of the entropy term, which prevents the concentration from approaching zero. The flux with the Minmod or Osher limiter is reduced to a second-order upwind flux in the strong convective limit. Numerical tests are performed to confirm the expected order of accuracy and the properties of the developed scheme at discrete level. In addition, our numerical experiments demonstrate that the developed schemes with limiters can achieve higher accuracy than the Scharfetter-Gummel scheme and classical upwind scheme in tests with strong convection. Finally, the developed schemes are applied to numerically assess the effects of Coulombic interaction, steric interaction, and Born solvation on the rectifying transport of ions through nanopores.

Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator

We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.

The Scharfetter–Gummel scheme for aggregation–diffusion equations

Abstract In this paper we propose a finite-volume scheme for aggregation–diffusion equations based on a Scharfetter–Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structure-preserving features. More specifically, it is shown that the discrete solutions satisfy a free-energy dissipation relation analogous to the continuous model. Consequently, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization. Numerical experiments complement the study.

Implicit and coupled fluid plasma solver with adaptive Cartesian mesh and its applications to non-equilibrium gas discharges

We present a new fluid plasma solver with adaptive Cartesian mesh (ACM) based on a full-Newton (nonlinear, implicit) scheme for non-equilibrium gas discharge plasma. The electrons and ions are described using drift–diffusion approximation coupled to Poisson equation for the electric field. The electron-energy transport equation is solved to account for electron thermal conductivity, Joule heating, and energy loss of electrons in collisions with neutral species. The rate of electron-induced ionization is a function of electron temperature and could also depend on electron density (important for plasma stratification). The ion and gas temperature are kept constant. The transport equations are discretized using a non-isothermal Scharfetter–Gummel scheme to resolve possible large temperature gradients in the sheaths. We demonstrate the new solver for simulations of direct current (DC) and radiofrequency (RF) discharges. The implicit treatment of the coupled equations allows using large time steps. The full-Newton method (FNM) enables fast nonlinear convergence at each time step, offering significantly improved simulation efficiency. We discuss the selection of time steps for solving different plasma problems. The new solver enables solving several problems we could not solve before with existing software: two- and three-dimensional structures of the entire DC discharges including cathode and anode regions, electric field reversals and double-layer formation, the normal cathode spot and an anode ring, moving striations in diffuse and constricted DC discharges, and standing striations in RF discharges. The developed FNM-ACM technique offers many benefits for tackling the disparity of gas discharge plasma systems' time scales and nonlinearity.

Introducing a 1D numerical model for the simulation of PN junctions of varying spectral material properties and operating conditions

During the last years, innovative concepts of solid-state devices, such as multi-junction solar cells and thermophotovoltaic converters, have emerged as efficient means of direct electricity production. The accurate estimation of the conversion efficiency of such devices using simple yet effective numerical tools is a necessity to optimize their performance. This work presents an in-house code, which is based on the Drift Diffusion Model to simulate p-n junction diodes, under equilibrium (bias voltage application) and non-equilibrium (bias voltage and device illumination) conditions. Under non-equilibrium, illumination can originate from either solar radiation (conventional photovoltaic cell) or a thermally heated emitter (thermophotovoltaic operation). The drift-diffusion and Poisson’s equations are solved using a one-dimensional (1D) finite element Petrov-Galerkin method based on piecewise nonlinear interpolants of second-order accuracy, while the total current is evaluated in a post-process manner using the Scharfetter-Gummel scheme. Initially, the model is verified against the freeware SimWindows. Later on, a parametric analysis on the photovoltaic cell design and operating conditions reveals that its efficiency is highly affected by its total length, the n-type sub-region width, the doping levels of both p and n regions, the semiconductor material type, and, the device’s operating temperature. In contrast to other solvers, this one takes into account the model parameters’ dependence on temperature and electromagnetic spectrum, while it can be extended to incorporate the thermally stimulated electron emission in thermionic-based devices and 2D spatial effects. Finally, the calculated conversion efficiencies can be used to build a Reduced Order Model that can be further coupled with a computational fluid dynamics model to evaluate the overall thermo-electric performance of a solid-state device.

Development of a Massively Parallelized Fluid-Based Plasma Simulation Code With a Finite-Volume Method on an Unstructured Grid

A massively parallelized unstructured-grid plasma simulation code based on a multifluid plasma model has been developed and validated. The code uses a collocated cell-centered finite-volume method and is designed to simulate intermediate low-pressure and atmospheric-pressure (AP) plasma sources with complex machine geometries. One of the novel features of this code is that it is implemented in a highly flexible computational platform named ultrafast massively parallel processing (ultraMPP), which allows straightforward addition and integration of different partial differential equation (PDE) solvers in a self-consistent manner. As to the numerical methods, the Scharfetter–Gummel scheme is adopted to handle the drift-diffusion flux of electrons, whereas the Harten–Lax-van Leer (HLL)-type approximate Riemann solver is used to handle the convection terms of ion momentum equations. For discretization of the diffusion terms in an unstructured grid, the Taylor expansion is used to deal with the effects of nonorthogonality of the cells, and the cell-center gradient is calculated using a least-squares method. The simulation code with a local-field approximation (LFA) was validated for two cases of AP plasmas as well as a case of an argon capacitively coupled plasma generated in a Gaseous Electronic Conference (GEC) reference cell with local mean energy approximation (LMEA) at intermediate low pressure. The simulation results were found to be in good agreement with the previously published experimental and simulation data.

Drift\u2013diffusion simulation of S-shaped current\u2013voltage relations for organic semiconductor devices

We present an electrothermal drift–diffusion model for organic semiconductor devices with Gauss–Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature-dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter–Gummel scheme. Using path-following techniques, we demonstrate that the model exhibits S-shaped current–voltage curves with regions of negative differential resistance, which were only recently observed experimentally.

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson–Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter–Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter–Gummel scheme for degenerate semiconductors (Fermi–Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

Numerical simulation of air ionization processes in plasma-actuator electric field

Mathematical model based on drift-diffusion approximation of hydrodynamic method of plasma description with two-temperature approximation was carried out for air ionization case in order to model process of interaction between dielectric barrier discharge plasma actuator and gas flow. Electron-induced ionization-recombination mechanism was used. Air was considered as mixture of molecular oxygen and nitrogen in proportion at normal conditions. Process kinetics was described by electron-induced reactions with molecular nitrogen and oxygen, ions and electron components. Equations system consisted of transport equations for concentrations of all mixture components, electron temperature and equation for electric potential was solved. Extended Scharfetter-Gummel scheme for large quantity of transport equations was developed. It was shown, that main influence on charge distribution has electron temperature. This model could be used for volume and surface charge density estimations necessary for volume force calculation in the vicinity of electrodes.

Highly accurate quadrature-based Scharfetter–Gummel schemes for charge transport in degenerate semiconductors

We introduce a family of two point flux expressions for charge carrier transport described by drift–diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Voronoï finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newton’s method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter–Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed.

Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi–Dirac and Gauss–Fermi statistics

We compare three thermodynamically consistent Scharfetter–Gummel schemes for different distribution functions for the carrier densities, including the Fermi–Dirac integral of order 1/2 and the Gauss–Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter–Gummel scheme requires the solution of an integral equation. Since one cannot solve this integral equation analytically, several modified Scharfetter–Gummel schemes have been proposed, yielding explicit flux approximations to the implicit generalized flux. The two state-of-the-art modified fluxes used in device simulation software are the diffusion-enhanced flux and the inverse activity coefficient averaging flux. We would like to study which of these two modified schemes approximates the implicit flux better. To achieve this, we propose a new method to solve the integral equation numerically based on Gauss quadrature and Newton’s method. This numerical procedure provides a highly accurate reference flux, enabling us to compare the quality of the two modified Scharfetter–Gummel schemes. We extend previous results (Farrell in J Comput Phys 346:497–513, 2017a) showing that the diffusion-enhanced ansatz leads to considerably lower flux errors for the Blakemore approximation to the physically more relevant Fermi–Dirac and Gauss–Fermi statistics.

Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics

We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter–Gummel scheme to non-Boltzmann (e.g. Fermi–Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

Accurate numerical simulation of moisture front in porous material

When comparing measurements to numerical simulations of moisture transfer through porous materials a rush of the experimental moisture front is commonly observed in several works shown in the literature, with transient models that consider only the diffusion process. Thus, to overcome the discrepancies between the experimental and the numerical results, this paper proposes to include the moisture advection transfer in the governing equation. To solve the advection-diffusion or the so-called convection differential equation, it is first proposed two efficient numerical schemes whose efficiencies are investigated for both linear and nonlinear cases. The first scheme, Scharfetter–Gummel, presents a Courant-Friedrichs-Lewy (CFL) condition but it is more accurate and faster than the second one, the well-known Crank–Nicolson approach. Furthermore, the Scharfetter–Gummel scheme has the advantages of being well-balanced and asymptotically preserved. Then, to conclude, results of the convective moisture transfer problem obtained by means of the Scharfetter–Gummel numerical scheme are compared to experimental data from the literature. The inclusion of an advective term in the model may clearly lead to better results than purely diffusive models.

Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems

AbstractIn this paper, we study the large-time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws.We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform in time

On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement

Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter–Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter–Gummel schemes.

Discretization scheme for drift-diffusion equations with strong diffusion enhancement

Inspired by organic semiconductor models based on hopping transport introducing Gauss-Fermi integrals a nonlinear generalization of the classical Scharfetter–Gummel scheme is derived for the distribution function $$\mathcal F_\gamma (\eta ) = 1/(\exp (-\eta )+\gamma )$$ . This function provides an approximation of the Fermi-Dirac integrals of different order and restricted argument ranges. The scheme requires the solution of a nonlinear equation per edge and continuity equation to calculate the edge currents. In the current formula the density-dependent diffusion enhancement factor, resulting from the generalized Einstein relation, shows up as a weighting factor. Additionally the current modifies the argument of the Bernoulli functions.

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.