Since the invention of the bipolar transistor in 1947, there has been an unprecedented growth of the semiconductor industry, with an enormous impact on the way people work and live. In the last thirty years or so, by far the strongest growth area of the semiconductor industry has been in silicon very-large-scale-integration (VLSI) technology. The sustained growth in VLSI technology is fueled by the continued shrinking of transistors to ever smaller dimensions. The benefits of miniaturization – higher packing densities, higher circuit speeds, and lower power dissipation – have been key in the evolutionary progress leading to today’s computers, wireless units, and communication systems that offer superior performance, dramatically reduced cost per function, and much reduced physical size, in comparison with their predecessors. On the economic side, the integrated-circuit (IC) business has grown worldwide in sales from $1 billion in 1970 to $20 billion in 1984 and has reached $250 billion in 2007. The electronics industry is now among the largest industries in terms of output as well as employment in many nations. The importance of microelectronics in economic, social, and even political development throughout the world will no doubt continue to ascend. The large worldwide investment in VLSI technology constitutes a formidable driving force that will all but guarantee the continued progress in IC integration density and speed, for as long as physical principles will allow.

This chapter examines the key device design issues in a modern CMOS VLSI technology. It begins with an extensive review of the concept of MOSFET scaling. Two important CMOS device design parameters, threshold voltage and channel length, are then discussed in detail.

MOSFET Scaling

CMOS technology evolution in the past thirty years has followed the path of device scaling for achieving density, speed, and power improvements. MOSFET scaling was propelled by the rapid advancement of lithographic techniques for delineating fine lines of 1 μm width and below. In Section 3.2.1, we discussed that reducing the source-to-drain spacing, i.e., the channel length of a MOSFET, led to short-channel effects. For digital applications, the most undesirable short-channel effect is a reduction in the gate threshold voltage at which the device turns on, especially at high drain voltages. Full realization of the benefits of the new high-resolution lithographic techniques therefore requires the development of new device designs, technologies, and structures which can be optimized to keep short-channel effects under control at very small dimensions. Another necessary technological advancement for device scaling is in ion implantation, which not only allows the formation of very shallow source and drain regions but also is capable of accurately introducing a sharply profiled, low concentration of doping atoms for optimum channel profile design.

The previous chapters have considered the operation of CMOS and bipolar devices mainly in the context of logic circuits. This chapter addresses another basic functional block in modern VLSI chips – memory. A predominant majority of the VLSI devices produced today are in various forms of random-access memory (RAM).

Viewed from the operation standpoint, a RAM functional unit is usually organized into an array of memory cells (or bits) together with its supporting circuits for selecting, writing, and reading the memory cells. In an array, the bits on the same row are selected by a word signal. A schematic block diagram of a RAM unit is shown in Fig. 9.1. The array consists of W words with B bits each, for a total memory capacity of W × B bits. A random bit in the array can be accessed through signals applied to its wordline and bitline.

Depending on the retention of information in the cells of a memory array, random-access memories can be classified into three categories: static random-access memory (SRAM), dynamic random-access memory (DRAM), and nonvolatile random-access memory (NVRAM). NVRAM is often referred to as nonvolatile memory for short. SRAMs have fast access times. They retain data as long as they are connected to the power supply. Practically every VLSI chip contains a certain amount of SRAM which is usually built using basically the same devices as in the logic circuits. DRAMs have relatively slow access times. They require periodic refresh in order to prevent loss of data. On a per-bit basis, DRAMs have a much lower cost than SRAMs because a DRAM cell is typically only about one tenth the size of an SRAM cell. For systems that require much more SRAM than can be contained on the logic chip, stand-alone SRAM chips are often used to meet the need. However, in order to reduce system cost and size, designers often use stand-alone DRAM chips instead of stand-alone SRAM chips. In that case, some form of memory-hierarchy architecture is usually employed to minimize the impact of the relatively slow DRAM on the system performance. Both SRAMs and DRAMs are volatile in that data are lost once the power supply to the chip is disconnected.

Scope

The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT), particularly in its diagrammatic representation (referred to as many-body perturbation theory, or MBPT), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods.

The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).

This chapter addresses several more subtle but nevertheless important aspects of coupled-cluster MBPT theory.

Spin summations and computational considerations

The formalism described in the previous sections was presented in terms of spinorbitals, without regard to integration over spin coordinates. Even in the case of unrestricted Hartree–Fock (UHF) reference functions, in which the spatial orbitals for α and β spin are different, integration over spin is absolutely necessary to eliminate many integrals and to allow the introduction of constraints over the summation indices, achieving a computational effort of no more than three times that of comparable RHF calculations. Furthermore, all amplitudes in which the number of α and β spinorbitals is different for the hole and particle indices vanish, preserving the MS, but not the S, quantum number. In the restricted closed-shell Hartree–Fock (RHF) case, spin integration is used to combine contributions from α and β spinorbitals, deriving expressions in terms of spatial orbitals only and thus reducing the range of all indices by about a factor 2 (see Section 7.3). Restricted open-shell Hartree–Fock (ROHF) calculations are usually performed as UHF, despite double occupancy, because the most effective algorithms are still of the spin-integrated, spatial-orbital, form. The double occupancy cannot be exploited further without special effort.

The incorporation of spin integration can be done algebraically or, in some cases, diagrammatically. As an example of the diagrammatic treatment of spin summation in coupled-cluster calculations we shall consider the case of the CCD equation with an RHF reference function. The diagrammatic representation of this equation in a spinorbital basis was given in Fig. 9.2 in terms of antisymmetrized Goldstone diagrams.

Although most microelectronics products are now made of CMOS transistors, bipolar transistors remain important in microelectronics because of their superior characteristics for analog circuit applications. There are two types of bipolar devices: the n–p–n type which has a p-type base and n-type emitter and collector, and the p–n–p type which has an n-type base and p-type emitter and collector. Commonly used bipolar devices are either lateral transistors, where the active device regions are arranged horizontally adjacent to one another and the active currents flow laterally, or vertical transistors, where the active device regions are arranged vertically one on top of another and the active currents flow vertically. Practically all bipolar transistors used in modern VLSI applications are of the vertical n–p–n type.

For simplicity, only vertical n–p–n bipolar transistors will be considered explicitly here. The equations derived for vertical transistors apply to horizontal transistors as well, provided that the device parameter values are adjusted accordingly. Also, the equations for an n–p–n transistor can be extended to a p–n–p transistor simply by reversing the voltage and dopant polarities and using the appropriate device parameter values.

Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension.

Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].

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.