[Paper Review] Modeling of Nanoscale Devices

M. P. Anantram, Mark Lundstrom+1 more|arXiv (Cornell University)|Oct 10, 2006

Advancements in Semiconductor Devices and Circuit Design14 references4 citations

TL;DR

This paper presents the non-equilibrium Green's function (NEGF) method as a fundamental framework for modeling quantum transport in nanoscale electronic devices, where semiclassical approximations fail. It details the NEGF formalism for atomistic, quantum mechanical simulation of carriers in ballistic to diffusive regimes, with efficient recursive algorithms for layered structures like nanotransistors and nanowires.

ABSTRACT

We aim to provide engineers with an introduction to the non-equilibrium Green's function (NEGF) approach, which provides a powerful conceptual tool and a practical analysis method to treat small electronic devices quantum mechanically and atomistically. We first review the basis for the traditional, semiclassical description of carriers that has served device engineers for more than 50 years. We then describe why this traditional approach loses validity at the nanoscale. Next, we describe semiclassical ballistic transport and the Landauer-Buttiker approach to phase coherent quantum transport. Realistic devices include interactions that break quantum mechanical phase and also cause energy relaxation. As a result, transport in nanodevices are between diffusive and phase coherent. We introduce the non equilbrium Green's function (NEGF) approach, which can be used to model devices all the way from ballistic to diffusive limits. This is followed by a summary of equations that are used to model a large class of layered structures such as nanotransistors, carbon nanotubes and nanowires. An application of the NEGF method in the ballistic and scattering limits to silicon nanotransistors is discussed.

Motivation & Objective

• Address the limitations of semiclassical transport models (e.g., drift-diffusion, Boltzmann transport) in nanoscale devices where quantum effects and atomic-scale structure dominate.
• Provide a comprehensive introduction to the non-equilibrium Green's function (NEGF) formalism as a unified framework for quantum transport in nanodevices.
• Enable device engineers to model carrier transport from ballistic to diffusive regimes using a single, consistent quantum mechanical approach.
• Develop and present efficient recursive algorithms for computing Green's functions in layered nanostructures such as nanotransistors, carbon nanotubes, and nanowires.
• Bridge the gap between continuum-based device modeling and atomistic, quantum mechanical simulation for next-generation nanoelectronics.

Proposed method

• Formulate the NEGF approach as a many-body quantum transport method that treats electron transport in open quantum systems with non-equilibrium conditions.
• Derive the Dyson equation for the retarded Green's function and use self-energy corrections to account for contacts and scattering processes.
• Implement a recursive algorithm to compute the Green's function for layered structures by solving tridiagonal matrix equations block-wise along the device axis.
• Use the recursive algorithm to compute the retarded, advanced, and lesser Green's functions, enabling calculation of current and density of states.
• Introduce the electron and hole Green's functions via $ G^n $ and $ G^p $, with $ G^p = i(G - G^\dagger) - G^n $, to describe particle and hole coherence.
• Apply the method to silicon nanotransistors in both ballistic and scattering-limited regimes, validating its accuracy and versatility.

Experimental results

Research questions

• RQ1 How can quantum transport in nanoscale devices be accurately modeled when semiclassical approximations break down due to quantum confinement and atomic-scale disorder?
• RQ2 What is the role of phase coherence and inelastic scattering in determining the transport characteristics of nanoscale transistors and nanostructures?
• RQ3 How can the non-equilibrium Green's function formalism be efficiently implemented for layered nanostructures such as nanowires and nanotransistors?
• RQ4 What are the key differences in transport behavior between ballistic and diffusive limits in nanoscale devices, and how can NEGF capture both?
• RQ5 How can recursive algorithms be used to compute Green's functions efficiently without full matrix inversion, enabling practical device simulations?

Key findings

• The NEGF formalism provides a consistent and rigorous framework for modeling quantum transport across the full range from ballistic to diffusive regimes in nanoscale devices.
• The recursive algorithm efficiently computes the Green's function for layered structures by solving tridiagonal matrix equations block-wise, significantly reducing computational cost.
• The method successfully models both ballistic and scattering-limited transport in silicon nanotransistors, demonstrating its versatility and accuracy.
• The algorithm computes the diagonal blocks of $ G^n $, $ G^p $, and $ G $ using iterative updates, with $ G^{n}_{q,q+1} = (G^{n}_{q+1,q})^\dagger $, ensuring Hermitian symmetry.
• The code implementation in MATLAB (recursealg3d.m) demonstrates the practical feasibility of the algorithm for simulating realistic nanodevice structures.
• The method correctly handles self-energies for in- and out-scattering, enabling accurate modeling of contact effects and inelastic scattering in nanoscale systems.