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

Abstract

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.

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

“…The resulting discretized current density expressions feature exponential terms that reflect the characteristics of the doping profile and allow for numerically stable calculations. Over the last decades, several generalizations of the Scharfetter-Gummel method have been proposed for either degenerate semiconductors [52][53][54][55][56][57] or non-isothermal carrier transport with included thermoelectric cross effects [44][45][46][47][48][49][50][51].…”

“…Following Refs. [54,56], we solve the BVP (28) by freezing the degeneracy factor g (n/N c (T )) → g n,K,L to a carefully chosen average. The resulting problem has the same structure as in the non-degenerate case (see above), but with a modified thermal voltage k B T K,L /q → k B T K,L g n,K,L /q along the edge, which takes the temperature variation and the degeneration of the electron gas into account.…”

“…where T K,L is the logarithmic mean temperature (29). In order to ensure the consistency with the thermodynamic equilibrium and boundedness g n,K ≤ g n,K,L ≤ g n,L (for η n,K ≤ η n,L or with K ↔ L else), the edge-averaged degeneracy factor is taken as [54,56] g n,K,L = η n,L − η n,K log F η n,L /F η n,…”

“…For constant temperature T L = T K = T , the scheme reduces to the modified Scharfetter-Gummel scheme discussed in Refs. [54,56]. It can thus be regarded as a non-isothermal generalization of this approach.…”

“…The problem has been solved by Scharfetter and Gummel for the case of non-degenerate semiconductors under isothermal conditions [43]. Since then, several adaptations of the method have been developed to account for more general situations (non-isothermal conditions [44][45][46][47][48][49][50][51], degeneration effects [52][53][54][55][56][57]). The Kelvin formula for the Seebeck coefficients allows for a straightforward generalization of the Scharfetter-Gummel approach to the non-isothermal case.…”