The channel of modern nano-scale MOSFETs is made of crystalline material, shaped in bulk or thin film layers. Since most of the basic electronic properties of crystals can be understood by considering the quantum mechanical behavior of electrons in an infinite periodic arrangement of atoms (bulk crystal), it is reasonable to begin the technical part of this book with a description of the electronic properties of bulk semiconductors.

To this purpose, we start with a short introduction to the basic notions regarding crystal structures and electrons in a strictly periodic potential. The concept of band structure is thus briefly developed. The reader can refer to excellent textbooks for a more detailed treatment of these basic topics [1–3]. We then describe a few methodologies to compute the band structure of electrons and holes in semiconductors and the simplest analytical approximations commonly used to represent the energy relation in proximity to the band edges.

The last sections of the chapter illustrate the effective mass approximation and the foundations of the semi-classical model of carrier transport; namely, the motion of wave-packets in slowly varying potentials and the basics of scattering by rapidly fluctuating potentials.

The chapter sets the stage for the detailed treatment of situations where additional built-in or external potentials cause non-negligible quantum mechanical confinement of the carriers in at least one physical space direction, as, for instance, in the case of the MOSFET inversion layer.

The semi-classical transport model for inversion layers developed in the previous chapters finds a natural field of application in the analysis of advanced nano-scale MOSFETs.

In this chapter we illustrate the ability of this model to describe low and high field transport in unstrained (001) silicon. To this purpose, extensive simulations are presented for the low field effective mobility of long channel devices and for the on-current of short bulk and SOI transistors. The first half of the chapter describes how the multiple and complex dependencies of the mobility on the bias, channel doping, silicon film thickness and temperature relate to the physical ingredients of the inversion layer transport model, namely the discrete energy levels, the occupation probability and the scattering rates in the subbands.

The second half of the chapter covers high field transport in uniform silicon slabs and in short channel MOSFETs. The quasi-ballistic transport model outlined in Chapter 5 guides the interpretation of the transistor simulations.

The results proposed in this chapter set a reference for the analysis of more complex cases of interest for advanced CMOS technologies. In particular, the impact on carrier transport of technology boosters such as crystal orientation, strain and alternative materials are analyzed in Chapters 8, 9, and 10, respectively.

The historical CMOS scaling scenario

Complementary Metal Oxide Semiconductor (CMOS) technology is nowadays the backbone of the semiconductor industry worldwide and the enabler of the impressive number of electronic applications that continue to revolutionize our daily life. The pace of growth of CMOS technology in the last 40 years is clearly shown in the so-called Moore's plot (see Fig.1.1 [1]), reporting the historical trend in the number of transistors per chip, as well as in the trends of many other circuit performance metrics and economic indicators.

Key to the success of CMOS technology is the extraordinary scalability of the Metal Oxide Semiconductor Field Effect Transistor (MOSFET). The word scaling denotes the possibility, illustrated in Fig.1.2 and Table1.1, of fabricating functional devices with equally good or even improved performance metrics but smaller physical dimensions. The design of scaled transistors starting from an existing technology has been driven initially by simple similarity laws aimed to maintain essentially unaltered either the maximum internal electric field (hence, to a first approximation, the device reliability) or the supply voltage (hence the system integration capability) [2].

According to these two scaling strategies, defined in Table1.1, all the lateral (primarily the gate width, W, and length, LG) and the vertical physical dimensions (the thickness of the gate dielectric, tox, and the junction depth, xj) should decrease from one technology generation to the next by a factor α, thus yielding an increase of the number of transistors per unit chip area by a factor of α2.

In this chapter we introduce the Boltzmann transport equation (BTE), which is the basis for the semi-classical description of carrier transport in electron devices. We begin with a brief reminder of the key assumptions behind the formulation of the BTE for a free carrier gas. Since this topic is extensively discussed in many textbooks [1, 2], we go on to focus on use of the BTE for the description of transport in inversion layers.

In particular, Section 5.4 explains how the carrier mobility in inversion layers can be computed solving the BTE in the Momentum Relaxation Time approximation, once the scattering rates introduced in Chapter 4 are known.

Section 5.5 reviews the methodology to solve the BTE in the limiting cases of near equilibrium transport through the derivation of balance equations and of the widely used Drift-Diffusion model.

At the end of the chapter, Section 5.6 overviews the modeling of the far from equilibrium ballistic transport and Section 5.7 illustrates the quasi-ballistic transport regime. Expressions for the MOSFET on-current are derived in all cases from Drift-Diffusion to purely ballistic transport. These equations become useful in Chapter 7 to interpret the results of numerical simulations.

The BTE for the free-carrier gas

As discussed in Section 2.5, the dynamics of electrons in crystals can be described in terms of classical point charges provided that the extension of the wave-packet in real and momentum space is assumed to be negligible.

The Boltzmann Transport Equation (BTE) is an integro-differential equation where the unknown occupation function f depends on up to seven independent variables for a 3D carrier gas and five for a 2D carrier gas: the real and momentum space coordinates and the time. The equation is also non-linear due to the (1 - f) terms in the collision integral (Eq.5.9). Symmetries in the simulated device structure can reduce the number of variables in real-space, but the problem still remains very hard to solve with standard numerical methods [1].

In the previous chapter we examined ways to solve the BTE under various simplifying assumptions: absence of scattering (that is, ballistic transport), momentum-relaxation-time approximation for near equilibrium conduction in a low and uniform electric field, solutions via the moments of the BTE as in the Drift–Diffusion model.

Modern MOS transistors operate in a regime of transport where the number of scattering events suffered by the carriers traveling along the channel is largely reduced compared to long-channel devices, leading to quasi-ballistic transport, as described in Section 5.7. In this situation the distribution function is quite different from an equilibrium distribution and the simulation approaches relying on the moments of the BTE become inadequate [2, 3], thus demanding an exact solution of the BTE.

The Monte Carlo (MC) method is a powerful technique for solving exactly the semi classical BTE without a-priori assumptions on the carrier distribution function [4–7].

The traditional geometrical scaling of the CMOS technologies has recently evolved in a generalized scaling scenario where material innovations for different intrinsic regions of MOS transistors as well as new device architectures are considered as the main routes toward further performance improvements. In this regard, high-κ dielectrics are used to reduce the gate leakage with respect to the SiO2 for a given drive capacitance, while the on-current of the MOS transistors is improved by using strained silicon and possibly with the introduction of alternative channel materials. Moreover, the ultra-thin body Silicon-On-Insulator (SOI) device architecture shows an excellent scalability even with a very lightly doped silicon film, while non-planar FinFETs are also of particular interest, because they are a viable way to obtain double-gate SOI MOSFETs and to realize in the same fabrication process n-MOS and p-MOS devices with different crystal orientations.

Given the large number of technology options, physically based device simulations will play an important role in indicating the most promising strategies for forthcoming CMOS technologies. In particular, most of the device architecture and material options discussed above are expected to affect the performance of the transistors through the band structure and the scattering rates of the carriers in the device channel. Hence microscopic modeling is necessary in order to gain a physical insight and develop a quantitative description of the carrier transport in advanced CMOS technologies.

In this context, our book illustrates semi-classical transport modeling for both n-MOS and p-MOS transistors, extending from the theoretical foundations to the challenges and opportunities related to the most recent developments in nanometric CMOS technologies.

The working principle of MOS transistors requires formation of an extremely thin layer of carriers at the semiconductor-oxide interface. This is obtained by means of an appropriate gate bias, which produces a narrow minimum of the carrier potential energy at such an interface, resulting in an inversion layer with significant quantization effects.

The carriers in such inversion layers are essentially free to move only in the plane parallel to the silicon-dielectric interface. The allowed energy states stemming from either the conduction or the valence band of the underlying crystal form subbands, inside which the energy depends on a quasi-continuum two-dimensional wave-vector k, so that the carriers are said to form a two-dimensional (2D) gas. The energy dispersion in the 2D subbands is very important to describe the electrostatics and the transport in MOS transistors and it depends both on the confining potential and on the characteristics of the underlying crystal.

This chapter presents the fundamental concepts and the models to determine the energy relation and the carrier densities at equilibrium in the inversion layer of both n-type and p-type MOS transistors.

After introducing the basic concepts related to subband quantization in Section 3.1 by using a pedagogical example, Section 3.2 discusses the application to an electron inversion layer of the effective mass approximation (EMA) approach (see Section 2.4). Then Section 3.3 presents the extension to hole inversion layers of the k·p model introduced in Section 2.2.2; a simplified semi-analytical model for the hole energy relation is also discussed.

In the previous chapters we considered devices featuring a silicon channel and a (001) transport plane. These assumptions simplify calculation of the electron subband structure with the EMA approach, since one of the principal axes of the constant energy ellipsoids of the Δ valleys is aligned to the quantization direction. Furthermore, we have considered the [100] transport direction, which is again aligned to one of the principal axes of the constant energy ellipses in the transport plane.

In this chapter we show that semi-classical transport modeling in the frame of multi subband theory can be supplemented to describe arbitrary orientations by extending quite naturally most of the theoretical concepts explained in the previous chapters. In particular we generalize to arbitrary materials and crystal orientations the EMA (for electron inversion layers) and the k·p model (for hole inversion layers) discussed in Chapter 3.

We demonstrate the application of the theory by comparing results for silicon MOSFETs with non-conventional crystal orientations, namely (110) and (111), to the ones for the conventional (001) orientation.

The general theory is also used in Chapter 10 to analyze some alternative channel materials such as germanium and gallium arsenide.

Electron inversion layers

We appreciated in Section 3.2.1 the usefulness of the effective mass approximation (EMA) for analysis of the electron inversion layer, particularly when the dispersion relationship is assumed to be parabolic and the constant energy surfaces of the bulk crystal are ellipsoids.

Starting from the 90 nm technology node, several semiconductor companies have introduced strain as an important booster for the performance of MOS transistors; among them we can mention IBM [1], Intel [2], Texas Instruments [3], and Freescale [4]. This consideration explains the decision to devote an entire chapter of the book to transport in strained MOS devices.

Strain affects the characteristics of MOS transistors in several respects. In fact, besides its impact on carrier transport in the device channel, strain induces shifts of the band edges affecting the threshold voltage of the transistors [5], the leakage of the source and drain junctions [6], the energy barrier to the gate dielectric and consequently the gate leakage current [7], and also the transistor reliability [8]. The present chapter, however, is essentially focused on the methodologies and the models necessary to account for the strain effects on transport in MOS transistors, more precisely on the low field mobility and the drain current IDS.

The chapter is organized as follows. After a concise introduction to the fabrication techniques used for strain engineering in Section 9.1, all the relevant definitions related to stress and strain in cubic crystals are described in Section 9.2. Correct evaluation of the strain tensor in the crystal coordinate system is the first step necessary to model the effects of strain on the band structure of n-type and p-type MOS transistors, which are described respectively in Section 9.3 and 9.4.

Carbon nanotubes have come a long way since their modern rediscovery in 1991. This time period has afforded a great many scholars across the globe to conduct a vast amount of research investigating their fundamental properties and ensuing applications. Finally, after two decades, the knowledge and understanding obtained, once only accessible to select scholars, is now sufficiently widespread and accepted that the time is ripe for a textbook on this matter. This textbook develops the basic solid-state and device physics of carbon nanotubes and to a lesser extent graphene. The lesser coverage of graphene is simply due to its relative infancy, with a good deal of the device physics still in its formative stage.

The technical discourse starts with the solid-state physics of graphene, subsequently warping into the solid-state physics of nanotubes, which serves as the foundation of the device physics of metallic and semiconducting nanotubes. An elementary and limited introduction to the device physics of graphene nanoribbons and graphene are also developed. This textbook is suitable for senior undergraduates and graduate students with prior exposure to semiconductor devices. Students with a background in solid-state physics will find this book dovetails with their physics background and extends their knowledge into a new material that can potentially have an enormous impact in society. Scholars in the fields of materials, devices, and circuits and researchers exploring ideas and applications of nanoscience and nanotechnology will also find the book appealing as a reference or to learn something new about an old soul (carbon).

Introduction

The goal of this chapter is to explore the excitation and motion of electron waves under ideal conditions in a metallic conductor. By ideal conditions, we mean that electrons can be excited and transported without any scattering or collision involved. The excitation of electrons can be achieved by applying an external potential to energize the electron waves to oscillate more frequently, which can result in a net electron motion in the presence of a driving electric field, say between two ends of a metallic conductor. It is advisable to commit to memory that the absence of electron scattering is technically called ballistic transport; as such, the metallic conductor in this case would be referred to as a ballistic conductor.

Electrically, the ideal excitation and motion of electrons in low dimensions, such as in 1D space, is manifest in the form of a quantum conductance, quantum capacitance, and kinetic inductance, which represents a different paradigm from our classical electrostatic and magnetostatic ideas. The conductance and inductance reflect the electrical properties of traveling electron waves which lead to charge transport and energy storage, while the quantum capacitance accounts for the intrinsic charge storage that comes about from exciting electrons with an electric potential. In macroscopic bulk metals, the quantum electrical properties are not readily observable or accessible owing to the large number of mobile electrons at hand and the frequent collisions involved.

Introduction

The objective of this chapter is to describe the physical and electronic structure of graphene. Familiarity with concepts such as the crystal lattice and Schrödinger's quantum mechanical wave equation discussed in Chapter2 will be useful. The electronic band structure of graphene is of primary importance because (i) it is the starting point for the understanding of graphene's solid-state properties and analysis of graphene devices and (ii) it is also the starting point for the understanding and derivation of the band structure of CNTs. We begin by broadly discussing carbon and then swiftly focus on graphene, including its crystal lattice and band structure. This chapter concludes on the contemporary topic of GNRs.

Carbon is a Group IV element that is very active in producing many molecular compounds and crystalline solids. Carbon has four valence electrons, which tend to interact with each other to produce the various types of carbon allotrope. In elemental form, the four valence electrons occupy the 2s and 2p orbitals, as illustrated in Figure 3.1a. When carbon atoms come together to form a crystal, one of the 2s electrons is excited to the 2pz orbital from energy gained from neighboring nuclei, which has the net effect of lowering the overall energy of the system. Interactions or bonding subsequently follow between the 2s and 2p orbitals of neighboring carbon atoms.