Atomistic Simulation of Quantum Transport in Nanoelectronic Devices

The following sections are included:

• Dedication
• Foreword
• Preface
• Acknowledgments
• Contents

Approaching the end of Moore's law, we are facing a revolution from traditional microelectronics to a new field called nanoelectronics where all the characteristic lengths are of the order of nanometers. According to the International Technology Roadmap for Semiconductors [1], the device size is expected to shrink continuously from 22 nm (2012) to 14 nm (2014), 10 nm (2016), 7 nm (2018), and 5 nm (2020). The continuous shrinking of the device size results in discontinuous changes in the device physics: At the nanometer scale, electrons behave more like waves than particles, and the well-established semi-classical transport theory needs to be updated to a quantum version; At the nanometer scale, material is no longer continuous, and atomic details may have great impact on the transport properties; At the nanometer scale, the impurities and defects generate considerable randomness, such that disorder effects on the device performance and reliability deserve careful analysis. This chapter aims to provide a physics background and to elaborate some features of nanoelectronic devices.

This chapter aims to present the theory of quantum transport in disordered open systems. The theory is built upon the nonequilibrium coherent potential approximation (NECPA) which is a generalization of the coherent potential approximation (CPA) developed in disordered bulk systems. The generalization is carried out by applying the Langreth theorem to a contour ordered CPA equation in the formalism of nonequilibrium Green's function (NEGF). Section 2.2,2.3,2.4 introduce some background knowledge of the NEGF; Section 2.1,2.5,2.6,2.7 derive the NEGF formalism for two-probe systems; Section 2.8 presents the NECPA theory which is the heart of this chapter; Section 2.9 discusses the dephasing effect due to disorder scattering; and Section 2.10 illustrates the application of the NECPA theory with a toy model.

In Chapter 2, we have developed the NECPA theory for the quantum transport in disordered two-probe systems. In the derivation of the formalism, it is assumed that the Hamiltonian is known a priori. For example, in the toy model of Section 2.9, the parameters are chosen as ε₀ = 0 and t₀ = 1. But those numbers are somehow arbitrary and just for the purpose of illustration. How do we include the atomic information in the Hamiltonian, e.g., assigning this site to be a Co atom and that site to be a Cu atom? This topic will be the focus of this chapter where an implementation of DFT, the LMTO method, is combined with the NECPA theory to solve a two-probe Hamiltonian self-consistently. Consequently the NECPA-DFT theory is reduced to the NECPA-LMTO method in this specific implementation. We shall see that the Hamiltonian elements in Eq. (2.1) are replaced by matrix blocks to represent different atoms. At the end of the chapter, a flowchart and a complete set of formulas will be provided, serving as a blueprint for the numerical implementations.

NanoDsim, short for nanoelectronic device simulator, is a modeling package for atomistic simulation of quantum transport and electronic properties of nanostructures. The theoretical foundation of NanoDsim has been presented in Chapter 2 to 3. The goal of Chapter 4 to 7 is to convert the formalism of Section 3.11 and 3.12 into a software package. Therefore Chapter 2 to 3 read more like a physics book while Chapter 4 to 7 read more like a computer programming book. Particularly in this chapter we shall explain the programming language, the programming style, and the design of NanoDsim.

In this chapter, we shall discuss how to implement dsim-classes, dsim-solvers, and dsim-calculators for bulk systems. Although the theoretical formulas have been derived in Chapter 3, one cannot write computer programs by translating the formulas directly. One needs to design the classes and solvers, and choose proper algorithms to implement the formulas. Section 5.1 and 5.2 will be devoted to the design of dsim-classes and dsim-solver for bulk systems. Section 5.3 to 5.10 will be devoted to various algorithms used in the self-consistent and post-analysis calculations.

In this chapter, we shall discuss how to implement dsim-classes, dsim-solvers, and dsim-calculators for two-probe systems. Some algorithms in two-probe systems are similar to those of bulk systems, such as calculating the structure constant, solving the radial equation, and evaluating the complex contour integral. For a discussion of these algorithms, we refer readers to Chapter 5. Other algorithms are unique to two-probe systems, such as calculating the lead self-energies and evaluating the real-axis integral. These algorithms are the focus of this chapter. Section 6.1 and 6.2 will be devoted to the design of dsim-classes and dsim-solvers for two-probe systems. Section 6.3 to 6.9 will be devoted to various algorithms used in both self-consistent and post-analysis calculations. Section 6.10 will be devoted to the verification of the algorithms implemented for two-probe systems.

By using the algorithms of Chapter 5 and Chapter 6, one can develop a preliminary version of the NanoDsim package, which is capable of simulating two-probe systems with 10 to 10² atoms on a single PC. However, realistic device simulations often involve much more atoms. Nowadays supercomputers have become standard computing tools, and the power of parallel computing makes it possible to carry out quantum transport simulation of large atomic systems. The goal of this chapter is to parallelize and optimize the NanoDsim package so that it is capable of handling two-probe systems with 10³ to 10⁴ atoms on a moderate supercomputer.

Confucius said “it is a great pleasure to learn something and apply it to solve realistic problems”. In previous chapters, we have learned the theory and the algorithms of the NanoDsim package. Now it is time to apply the package to study the quantum transport in nanoelectronic devices. We are especially interested in how disorder scattering at the surfaces, interfaces, dopant sites, and structural defects affect the electronic structures and transport properties.

This appendix is a collection of various “subroutines” and “functions”. These “subroutines” and “functions” have no logical relation to each other. Instead they are “called” independently by previous chapters.