A Semi-Implicit Scheme for Simultaneously Solving NEGF-Poisson Equations in Quantum Transport Simulation With Consideration of Carrier-Phonon Scattering Effects

In this article, a semi-implicit scheme for simultaneously solving nonequilibrium Green’s function (NEGF)-Poisson equations is proposed for quantum transport simulation with consideration of carrier-phonon scattering effect. First, by employing the Dyson equation and Keldysh equation, the NEGF is expanded to the first-order term. Subsequently, the Green’s function terms are classified as explicit if related to scattering self-energy and implicit otherwise. The explicit terms are assembled into column vectors in the system matrix equation, while the derivative of implicit terms with respect to potential is calculated and coupled with the Poisson equation to form the Jacobian matrix. These two parts are solved simultaneously by employing the Newton–Raphson method. In addition, a method to calculate the derivative of lead self-energy with respect to potential is proposed by employing the surface Green’s function. The semi-implicit scheme can also greatly improve the convergence of simulation procedure compared to the typical Gummel scheme with predictor–corrector. Specifically, for the simulation case in this article, the number of iterations required to achieve an absolute potential error of less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-{10}}$</tex-math> </inline-formula> V is reduced from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$50\sim 60$</tex-math> </inline-formula> to a dozen. Besides, the proposed semi-implicit simulation scheme can significantly reduce the size of system matrix compared to the fully coupled scheme, as the Jacobian matrix construction no longer needs to consider the coupling relationship between Green’s functions in different energy levels induced by inelastic scattering. Consequently, the size of the Jacobian matrix becomes independent of the number of energy levels.