INTRODUCTION

As transistor dimensions continue to shrink into the nanometer regime, the demand for accurate simulation tools capable of handling quantum effects at this scale has significantly increased. This study focuses on modeling ballistic electron transport in a double-gate metal-oxide-semiconductor field-effect transistor (MOSFET) using the nonequilibrium Green’s function (NEGF) framework [1], [2]. The NEGF approach provides a general and powerful method for investigating quantum transport phenomena in nanoscale devices.

Although the Schrödinger equation serves as the foundational equation for quantum transport theory, solving it directly for many-body systems remains a major computational challenge. To overcome this, various methods have been developed-among which, the Green’s function formalism is particularly well-suited for nanoelectronic applications. The NEGF method, widely used in quantum device simulations, was originally developed in the 1960s by researchers including Martin, Schwinger, Kadanoff, Baym, and Keldysh [3],[ 4].

Traditional formulations of NEGF are rooted in many-body perturbation theory (MBPT), which requires a deep understanding of advanced quantum mechanics. However, in this work, we adopt a simplified version of the formalism introduced by Supruyo Datta and others,

which emphasizes physical intuition over mathematical complexity.

The core of the NEGF method involves a self-consistent loop between the electrostatic (Poisson) equation and the quantum transport equation. The transport equation yields the electron density corresponding to a given potential, while the Poisson equation updates the electrostatic potential based on that density. These equations are iteratively solved until both quantities reach mutual convergence. To efficiently handle the two-dimensional device structure, an uncoupled mode space approximation is employed.

In this work, we implement a self-consistent numerical approach to model quantum transport, consisting of iterative steps described as follows:

1. Starting with an initial guess for the electrostatic potential, the Schrödinger equation is solved along the confinement direction to extract quantized subband energy levels.
2. These subband energies are then utilized to form the Hamiltonian matrix governing the transport characteristics of the device.
3. Using the constructed Hamiltonian, we compute the retarded Green’s function, which in turn enables the evaluation of electron density and current flowing through the device terminals.
4. The calculated carrier density is then fed into the Poisson equation to update the potential profile, completing one iteration of the self-consistency loop.

Unlike prior works that commonly apply finite difference techniques for numerical discretization, this study adopts the finite element method (FEM). FEM provides notable advantages, such as its adaptability to irregular geometries and the ease of handling diverse boundary conditions. These features are particularly beneficial in quantum device simulations, where precise boundary representation is crucial.

In our Poisson equation formulation, homogeneous Neumann boundary conditions are imposed at both source and drain terminals. A notable strength of FEM is that it naturally accommodates such boundary conditions-often referred to as floating boundaries-without requiring additional modifications or constraints in the model.

The simulation outputs include the self-consistently computed electrostatic potential, spatially resolved carrier density, and the integrated Fermi-Dirac distribution at the contacts.

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