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 (ℰx) correctly, but the tangential field (ℰy) 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 ε1 and thickness t1. The depletion region in the semiconductor has a permittivity ε2 and thickness t2. 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.

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).