We now describe the properties of the SU(>K) chain in more algebraic terms. The relevant symmetry is the SU(K) Yangian, which is usually referred to as Y(slK). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y(gl2). The difference between Y(slK) and Y(glK) is that the charge degrees of freedom U(1) is absent in Y(slK), while the spin degrees of freedom is included in both Y(slK) and Y(slK). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y(slK), which makes the relation between Y(glK) and Y(slK) clearer.

In Section 9.1, we begin with the construction of the SU(K) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU(K) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU(K) Yangian.

In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU(K) Yangian states.

“What are the electrons really doing in molecules?” This famous question was posed by R. A. Mulliken over a half-century ago. Accurate quantitative answers to this question would allow us, in principle, to know all there is to know about the properties and interactions of molecules. Achieving this goal, however, requires a very accurate solution of the quantum-mechanical equations, primarily the Schrödinger equation, a task that was not possible for most of the past half-century. This situation has now changed, primarily due to the development of numerically accurate many-body methods and the emergence of powerful supercomputers.

Today it is well known that the many-body instantaneous interactions of the electrons in molecules tend to keep electrons apart; this is manifested as a correlation of their motions. Hence a correct description of electron correlation has been the focal point of atomic, molecular and solid state theory for over 50 years. In the last two decades the most prominent methods for providing accurate quantum chemical wave functions and using them to describe molecular structure and spectra are many-body perturbation theory (MBPT) and its coupled-cluster (CC) generalizations. These approaches have become the methods of choice in quantum chemistry, owing to their accuracy and their correct scaling with the number of electrons, a property known as extensivity (or size-extensivity). This property distinguishes many-body methods from the configuration-interaction (CI) tools that have commonly been used for many years. However, maintaining extensivity – a critical rationale for all such methods – requires many-body methods that employ quite different mathematical tools for their development than those that have been customary in quantum chemistry.

It has been fifty years since the invention of the bipolar transistor, more than forty years since the invention of the integrated-circuit (IC) technology, and more than thirty-five years since the invention of the MOSFET. During this time, there has been a tremendous and steady progress in the development of the IC technology with a rapid expansion of the IC industry. One distinct characteristic in the evolution of the IC technology is that the physical feature sizes of the transistors are reduced continually over time as the lithography technologies used to define these features become available. For almost thirty years now, the minimum lithography feature size used in IC manufacturing has been reduced at a rate of 0.7× every three years. In 1997, the leading-edge IC products have a minimum feature size of 0.25 μm.

The basic operating principles of large and small transistors are the same. However, the relative importance of the various device parameters and performance factors for transistors of the 1- μm and smaller generations is quite different from those for transistors of larger-dimension generations. For example, in the case of CMOS, the power-supply voltage was lowered from the standard 5 V, starting with the 0.6- to 0.8- μm generation. Since then CMOS power supply voltage has been lowered in steps once every few years as the device physical dimensions are reduced. At the same time, many physical phenomena, such as short-channel effect and velocity saturation, which are negligible in large-dimension MOSFETs, are becoming more and more important in determining the behavior of MOSFETs of deep-submicron dimensions. In the case of bipolar devices, breakdown voltage and base-widening effects are limiting their performance, and power dissipation is limiting their level of integration on a chip. Also, the advent of SiGe-base bipolar technology has extended the frequency capability of small-dimension bipolar transistors into the range previously reserved for GaAs and other compound-semiconductor devices.

This chapter reviews the basic concepts of semiconductor device physics. Starting with electrons and holes and their transport in silicon, we focus on the most elementary types of devices in VLSI technology: p–n junction, metal–oxide–semiconductor (MOS) capacitor, and metal-semiconductor contacts. The rest of the chapter deals with subjects of importance to VLSI device reliability: high-field effects, the Si–SiO2 system, and dielectric breakdown.

Electrons and Holes in Silicon

The first section covers energy bands in silicon, Fermi level, n-type and p-type silicon, electrostatic potential, drift and diffusion current transport, and basic equations governing VLSI device operation. These will serve as the basis for understanding the more advanced device concepts discussed in the rest of the book.

Energy Bands in Silicon

The starting material used in the fabrication of VLSI devices is silicon in the crystalline form. The silicon wafers are cut parallel to either the 〈111〉or 〈100〉planes (Sze, 1981), with 〈100〉material being the most commonly used. This is largely due to the fact that 〈100〉wafers, during processing, produce the lowest charges at the oxide–silicon interface as well as higher mobility (Balk et al., 1965). In a silicon crystal each atom has four valence electrons to share with its four nearest neighboring atoms. The valence electrons are shared in a paired configuration called a covalent bond. The most important result of the application of quantum mechanics to the description of electrons in a solid is that the allowed energy levels of electrons are grouped into bands (Kittel, 1976). The bands are separated by regions of energy that the electrons in the solid cannot possess: forbidden gaps . The highest energy band that is completely filled by electrons at 0 K is called the valence band. The next higher energy band, separated by a forbidden gap from the valence band, is called the conduction band, as shown in Fig. 2.1.

The Sutherland model has a number of variants. One of them is the U(K) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L, and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U(K) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U(KB, KF) Sutherland model [177] consists of bosons having KB possible colors and fermions having KF colors.

The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/r2 supersymmetric t–J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models.

In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.

The scale length model described in Appendix 9 is called the “one-region” model. It replaces the gate oxide with an equivalent region of the same dielectric constant as silicon, but with a thickness equal to (εsi /εox)tox or 3tox . As pointed out earlier, this treats the normal field (ℰ x ) correctly, but the tangential field (ℰ y ) incorrectly. The one-region approximation is valid only if the gate oxide is much thinner than the scale length λ, in which case the oxide field is dominated by its normal component. In 1998, a generalized scale length model was published (Frank et al., 1998) which extended the one-region model to two- and three-regions with arbitrary dielectric constants and thicknesses. It considers the different boundary conditions of the normal and tangential fields separately at the dielectric interfaces. These relations then lead to an eigenvalue equation that can be solved for the scale length λ for such general structures. The generalized scale length model is particularly important for high-κ gate dielectrics which can be physically thick (Section 3.2.1.5), as well as for SOI and double-gate MOSFETs (Section 10.3.2).

Two-Region Scale Length Equation

In this appendix, the derivation of a generalized MOSFET scale length is described. Consider the two-region MOSFET model depicted in Fig. A10.1. The gate insulator region is assumed to have a permittivity ε 1 and thickness t 1. The depletion region in the semiconductor has a permittivity ε 2 and thickness t 2. Note that the bottom boundary of the depletion region is simplified to a straight line in the same manner as in Fig. A9.1. In subthreshold, there are negligible mobile carriers in the channel. The electric potential is solved from the 2-D Poisson equation applied to the rectangular region (lightly shaded) in Fig. A10.1. This is a boundary value problem in which the potential on the four conductor sides of the rectangle, left (source), top (gate), right (drain), bottom (body), is specified. There are actually two small gaps not enclosed by conductors: on the top left between the gate and source and on the top right between the gate and drain. When these gaps are not excessively large (compared with, e.g., λ), it is a good approximation to assign potential values by linear interpolation between the gate and source potentials for the left gap and between the gate and drain potentials for the right gap.

When Ge is added to Si, the energy-band edges of the resulting SiGe alloy are shifted relative to those of Si such that the energy bandgap of the SiGe alloy is smaller than that of Si. Figure A17.1 is a schematic showing the shifts in the energy-band edges of SiGe relative to the energy-band edges of Si for two Si1−xGex compositions (People, 1986). It shows that the bandgap narrowing in SiGe is caused mostly by shifts in the valence-band edge. Shifts in the conduction-band edge are relatively small.

To understand the operation of a SiGe-base bipolar transistor, we should first establish its energy-band diagram. Since the currents in a bipolar transistor in normal operation are controlled by the emitter–base diode, we need to consider only the energy-band diagram of the emitter–base diode. Therefore, we want to consider an n+–p Si–SiGe diode. For simplicity, we assume that the bandgap narrowing in the SiGe is due entirely to shifts in the valence-band edge. This is a good approximation since shifts in the conduction-band edge are relatively small.