Channel analysis of high electron mobility transistors: a pathway from the density matrix to the quantum drift-diffusion approach

The problem of incrementing the computational power and memory capacity of digital systems is tightly related to that of scaling the transistor size to improve the transistor density per unit area. As a result, MOS-based technologies have been subjected to extremely strong scaling, with their gate size going from 10 μm to the modern attempts at 3nm technological nodes. This scaling phenomenon has led to approaching sizes at which quantum effects often dominate the behavior of the devices. Rigorous modelling of electronic devices at nanoscale should be based on genuine quantum kinetic approaches, such as the non-equilibrium Green's functions approach self-consistently coupled with Poisson's equation. However, these techniques are extremely computationally intensive and are often applicable to one-dimensional structures only. An alternative and more viable route when these rigorous models are not applicable is to restart from BTE-like equations and augment them with a description of quantum transport, obtaining quantum-corrected models (among which the quantum drift-diffusion model) that are trade-offs between computational efficiency and rigour. The work initiated by Wigner in 1932 with the introduction of an entirely new framework for quantum-mechanics has led through multiple assumptions and simplifications to the formulation of multiple quantum-corrected models that involve only simple modifications to the semiclassical ones, the most well-known and successfully applied being the quantum drift-diffusion model (or density-gradient model). However, the usage of this model has mostly been limited to silicon MOSFET and FinFET devices in recent years, leaving the possibilities offered by its application to heterostructure devices (in which quantum effects are becoming more and more important) mostly unexplored. The main motivation behind this thesis work is then to explore the applicability of the density-gradient model beyond silicon and, in particular, to III-V semiconductors (opening countless possibilities in optoelectronic and high-speed electronic device simulation). This work has multiple goals, the first one being to provide a derivation of the quantum drift-diffusion model, filling the gap that is often found between theoretical physics and the implementation of this model. The second goal is then to test different discretization techniques for the density-gradient model, with the goal of understanding which is the most stable and efficient implementation for the case of III-V heterostructure devices. In this regard, the choice that was made is to employ a Scharfetter-Gummel scheme for the continuity equations and a finite-box scheme for Poisson's equation and the density gradient equations. The starting point for this was a in-house one-dimensional drift-diffusion simulator, which has been modified and augmented with quantum corrections. Finally, high electron mobility transistors are taken as case studies for determining the accuracy and performance of this method, which is compared against the more rigorous Poisson-Schrödinger formalism. The results of the analysis of this case study are extremely encouraging, as the density gradient model appears to compare well against the results of a self-consistent Poisson-Schrödinger solver, with errors in the charge estimate of the same order of those obtained on silicon devices. Simulations using the density-gradient model are however quite faster than those performed with the Poisson- Schrödinger solver.