Formal quasidegenerate perturbation theory (QDPT)

As is well known, ordinary Rayleigh–Schrödinger perturbation theory breaks down when applied to a state that is degenerate in zero order, unless spin or symmetry restrictions eliminate all but one of the degenerate determinants from the expansion. The breakdown is due to singularities arising from the vanishing of denominators involving differences in energy between the reference determinant and determinants that are degenerate with it. Even when exact zero-order degeneracies are not present but two or more closelying zero-order states contribute strongly to the wave function, as is the case for many excited states or in situations involving bond breaking, the RSPT expansions tend either to diverge or to converge very slowly.

These problems commonly arise in the case of open-shell states because different distributions of the open-shell electrons among the open-shell orbitals, all with the same or very similar total zero-order energies, are possible. Many open-shell high-spin states can be treated effectively with singlereference- determinant methods using either unrestricted or restricted openshell HF reference determinants because the spin restrictions exclude alternative assignment of the electrons to the open-shell orbitals; however, low-spin states, such as open-shell singlets, require alternative approaches.

Several common series-extrapolation techniques can be used to speed up the convergence of a perturbation expansion or to obtain an approximate limit of a divergent series. The results of such an extrapolation usually improve as more of the early terms of the series become available. Approaches based on Padé approximants (closely related to continued fractions) have been applied in some studies (e.g. Reid 1967, Goscinski 1967, Brändas and Goscinski 1970, Bartlett and Brändas 1972, Bartlett and Shavitt 1977b, Swain 1977).

The last chapter of this book deals with silicon-on-insulator (SOI) devices, which include SOI CMOS (Lim and Fossum, 1983; Colinge et al., 1986), SOI bipolar, and double-gate MOSFETs. They are not the mainstream VLSI technologies at present, but have the potential of playing a significant role in future generations.

There are three main types of SOI materials: SIMOX, BESOI, and Smart Cut (Auberton-Hervé, 1996). SIMOX stands for synthesis by implanted oxygen. It is formed by first implanting a high dose of high energy oxygen ions into a silicon substrate. A high temperature anneal subsequently drives the chemical reaction which forms a stochiometric oxide layer buried in the silicon wafer. The anneal also regenerates the crystalline quality of the silicon layer remaining over the buried oxide. The main advantage of this technique is the thickness uniformity of the thin SOI layer. The main drawback is the high defect densities in the regrown silicon and in the buried oxide. BESOI stands for bond and etch back. It starts with two silicon wafers. After oxidation, the two wafers are bonded together by heating them to high temperatures. One of them is then etched back until only a thin film of silicon remains over the oxide and the other wafer. The crystalline quality of BESOI material is in principle as good as that of bulk silicon wafer. But there are usually significant thickness variations within the wafer. They can be tolerated for thick film SOI devices. For thin film SOI, some kind of etch stop technique needs to be employed in the etch back step to obtain better thickness uniformity. In the Smart Cut® process, both ion implantation and bonding are used. Before bonding, a high dose hydrogen implantation is made to wafer A to weaken the silicon bond strength at the implanted depth. After oxidation and bonding of wafers A and B, they are pulled apart mechanically. They break apart at the weakened cleavage plane, thus leaving a thin silicon film of A over the oxide and wafer B. The rest of A can be reused to save the cost. High temperature anneal and chemical-mechanical polish steps are done to the SOI wafer before device fabrication.

Since the publication of the first edition of Fundamentals of Modern VLSI Devices by Cambridge University Press in 1998, we received much praise and many encouraging reviews on the book. It has been adopted as a textbook for first-year graduate courses on microelectronics in many major universities in the United States and worldwide. The first edition was translated into Japanese by a team led by Professor Shibahara of Hiroshima University in 2002.

During the past 10 years, the evolution and scaling of VLSI (very-large-scale-integration) technology has continued. Now, sixty years after the first invention of the transistor, the number of transistors per chip for both microprocessors and DRAM (dynamic random access memory) has increased to over one billion, and the highest clock frequency of microprocessors has reached 5 GHz. In 2007, the worldwide IC (integrated circuits) sales grew to $250 billion. In 2008, the IC industry reached the 45-nm generation, meaning that the leading-edge IC products employ a minimum lithography feature size of 45 nm. As bulk CMOS (complementary metal–oxide–semiconductor field-effect transistor) technologies are scaled to dimensions below 100 nm, the very factor that makes CMOS technology the technology of choice for digital VLSI circuits, namely, its low standby power, can no longer be taken for granted. Not only has the off-state current gone up with the power supply voltage down scaled to the 1 V level, the gate leakage has also increased exponentially from quantum mechanical tunneling through gate oxides only a few atomic layers thick. Power management, both active and standby, has become a key challenge to continued increase of clock frequency and transistor count in microprocessors. New materials and device structures are being explored to replace conventional bulk CMOS in order to extend scaling to 10 nm.

In this chapter, we discuss the mathematical properties of Jack polynomials. Since the content is a little intricate, we first explain the scope of the chapter, and the relationship between different kinds of Jack polynomials.

In deriving dynamics of the single-component Sutherland model, the symmetric Jack polynomial Jk(z) plays a fundamental role since each eigenfunction is a product of a Jack polynomial and a power of the Vandermonde determinant. Here z = (z1, …, zN) represent complex coordinates, and k is a partition specifying a set of momenta for particles. The product of a Vandermonde determinant Δ(z) and a Jack polynomial is an antisymmetric polynomial, which is called an antisymmetric Jack polynomial J k(−)(z). Thus the fermionic eigenfunctions of the Sutherland model are constructed as the product of an antisymmetric Jack polynomial and an even power of the Vandermonde determinant. The Yangian highest-weight states (YHWS) of the HaldaneShastry spin chain are also expressed in terms of the symmetric Jack polynomials with the particular value λ = 2 of the repulsion parameter.

In the multi-component Sutherland model, on the other hand, proper eigenfunctions must be symmetric or antisymmetric against exchange of coordinates with the same internal quantum number. Such eigenfunctions can be constructed from non-symmetric Jack polynomials Eη(z), which do not have any symmetry against exchange of coordinates, but which are eigenfunctions of the Hamiltonian. Here η is a composition specifying a set of momenta for particles without ordering of magnitudes. For example, with the YHWS in the supersymmetric t–J model, eigenfunctions are constructed from Jack polynomials which are odd against exchange of hole coordinates, and even against exchange of magnon coordinates.

A one-dimensional electron has both spin and charge degrees of freedom. In the limit of strong on-site repulsion, double occupation of a site can be neglected since it has a large energy cost. If the number Ne of electrons is less than the number N of lattice sites, vacant sites appear and the system acquires both spin and charge degrees of freedom. The simplest model to incorporate these degrees of freedom is called the t–J model, which is discussed in this chapter.

We begin by reviewing the SU(2,1) supersymmetry (SUSY) as an extension of the SU(2) symmetry in the spin chain. With the supersymmetry, the hopping and exchange in the t–J model can be treated in a unified way as the graded permutation. Then the ground state of the SUSY t–J model is derived in Section 6.2 by the coordinate representation of wave functions by using the fully polarized state as the reference. The static structure factor at zero temperature is completely determined by the wave function at the ground state. The spin and charge components are derived using the determinant and its generalizations in Section 6.3.

In Section 6.4, we proceed to derive the spectrum of the SUSY t–J model. The elementary excitations consist of spinons, holons, and their antiparticles. It is proved in Section 6.5 that these excitations span the complete set. The argument relies on a generalization of Young diagrams, which are called ribbon diagrams. At the same time, the degeneracy of energy levels beyond the global SU(2) symmetry is ascribed to the presence of a Yangian supersymmetry Y (sl2|1).

Introduction

As in the case of quasidegenerate perturbation theory (Chapter 8), multireference coupled-cluster (MRCC) theory is designed to deal with electronic states for which a zero-order description in terms of a single Slater determinant does not provide an adequate starting point for calculating the electron correlation effects. As already discussed in Chapters 8 and 13, these situations arise primarily for certain open-shell systems that are not adequately described by a high-spin single determinant (such as transitionmetal atoms), for excited states in general and for studies of bond breaking on potential-energy surfaces; they arise usually because of the degeneracy or quasidegeneracy of the reference determinants. While single-reference coupled-cluster (SRCC) methods are very effective in treating dynamic electron correlation, the conditions discussed here involve nondynamic correlation effects that are not described well by truncated SRCC at practical levels of treatment.

As shown in Section 13.4, many open-shell and multireference states can be treated by EOM-CC methods, including a single excitation from a closed shell state to an open-shell singlet state, which normally requires two equally weighted determinants in its zero-order description. Furthermore, doubleionization and double-electron-attachment EOM-CC, as well as spin-flip CC (Krylov 2001), allow the treatment of many inherently multireference target states. These methods have the advantage of being operationally of single-reference form, since then the only choices that need to be made are of the basis set and the level of correlation treatment. Although, they require an SRCC solution for an initial state (not necessarily the ground state) to initiate the procedure, once initiated multireference target states are available by the diagonalization of an effective Hamiltonian matrix in a determinantal representation.

In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation.

There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.

A ballistic MOSFET is a hypothetical device in which the mobile carriers suffer no collisions in the channel. This may happen, in principle, when the channel length is shorter than the mean free path, the average distance carriers travel between collisions. In an ordinary MOSFET, carriers moving from the source to the drain under the influence of the applied field (Vds ) collide with the silicon lattice, impurity (dopant) atoms, and surfaces. These collisions limit the velocity they can acquire from the field (Appendix 3), resulting in a reduced drain current. Under low field conditions, the effect of these collisions is lumped into a mobility factor proportional to the mean free time between successive collisions (Appendix 3). For long-channel MOSFETs, the drain current is proportional to the mobility (Section 3.1.2). For short-channel MOSFETs under high drain bias conditions, high-field scattering becomes important. This is usually modeled by velocity saturation (Section 3.2.2). In the absence of any scattering, carriers entering the channel from the source are accelerated by the applied field ballistically toward the drain. They can attain very high speeds especially in the high-field region near the drain. However, such high speeds (velocity overshoot) do not necessarily translate into large currents. Since current must be continuous from source to drain, it is bounded by the rate at which carriers are injected from the source. In a ballistic MOSFET then, the bottleneck is near the source where carriers move into the channel at relatively low velocities (before field acceleration). Current continuity is satisfied by a decreased carrier density near the drain such that the product of carrier density and velocity at the drain is the same as that at the source (see Fig. 3.31). This appendix describes the drain current model for a ballistic MOSFET published by Natori in 1994 (Natori, 1994).

This book is concerned primarily with the exact dynamical properties of one-dimensional quantum systems. As a crucial property of exactly soluble models, we assume that the interaction decays as the inverse square of the distance. The family of these models is called the inverse-square interaction (1/r2) models. In the one-dimensional continuum space, the model is often referred to as the Calogero–Sutherland model. In the one-dimensional lattice, on the other hand, the first 1/r2 models appeared as a spin model, which is now called the Haldane–Shastry model. Soon after the discovery of the Haldane–Shastry model, it was recognized that the imposition of supersymmetry allows the model to acquire the charge degrees of freedom, while keeping the exactly soluble nature. The resultant one-dimensional electron model is called the supersymmetric t–J model. Various generalizations of these models have been proposed.

Recent experimental progress in quasi-one-dimensional electron systems, especially by neutron scattering and photoemission spectroscopy, has enhanced the theoretical motivation for exploring the dynamics over a wide frequency and momentum range. The 1/r2 models are ideally suited to meet this situation, since the model allows derivation of exact dynamical information most easily and transparently. In spite of the special appearance of the 1/r2 models, the intuition thus obtained contributes greatly to understanding low-dimensional physics in general. This kind of approach to dynamics is complementary to another powerful approach using the bosonization and conformal field theory. The latter is especially suitable to asymptotics of correlation functions at long spatial and temporal distances.

In Chapter 7, the design of the individual regions and parameters of a bipolar transistor was discussed. It was noted that, during device operation, an individual device region is not isolated from and independent of the other device regions. Optimization of one device parameter often adversely affects the other device parameters. Thus, optimization of the design of a bipolar transistor is a tradeoff process . This design tradeoff should be done at the circuit and/or chip level, for the optimum design of a transistor is a function of its application and environment. In this Chapter, we will first discuss some figures of merit for evaluating a bipolar transistor for typical digital and analog circuit applications, and then discuss the tradeoffs in the design of a bipolar transistor for these applications.

When we consider the performance of a circuit, the wires connecting the transistors and elements that make up the circuit and connecting the output of the circuit to the input of another circuit must be included. The resistance and capacitance as well as the signal propagation delays associated with the interconnect wires have been discussed in Section 5.2.4 in connection with CMOS circuits. The reader is referred to that subsection for details. In this Chapter, the wire capacitance which acts as a load on a bipolar circuit is included when we consider the performance and optimization of bipolar transistors and circuits.