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6943 results in Condensed Matter Physics, Nanoscience and Mesoscopic Physics

Page 177 of 348

  • 8 - Yang–Baxter relations and orthogonal eigenbasis

  • from Part II - Mathematics related to 1/r2 systems
  • Yoshio Kuramoto, Tohoku University, Japan, Yusuke Kato, University of Tokyo
  • Book:
    Dynamics of One-Dimensional Quantum Systems
    Published online:
    12 January 2010
    Print publication:
    06 August 2009, pp 391-421

  • Summary

    The spectrum of the Sutherland model for fermions with spin has a degeneracy. The orthogonal set of the basis of the Fock space of degenerate states is usually related to a Lie algebra such as su2 and su3. In our case, however, the degeneracy extends to states with different values of total spin, and the ordinary Lie algebra is insufficient. The relevant algebra is called Yangian [43, 44], and is the algebra controlling those operators satisfying the Yang–Baxter relation [118, 178]. In this chapter we describe how the Yangian naturally emerges in the many-particle states described by Jack polynomials, and constructs an orthogonal basis of degenerate states. We concentrate here on the simplest case of the gl2 Yangian, corresponding to spin 1/2 particles. More general cases will be discussed in Chapter 9.

    We introduce a variant of the R-matrix in Section 8.1, and its standard form in Section 8.2. We then introduce a monodromy matrix and discuss its algebraic properties, such as the Yang–Baxter relation and the intertwining property. In Section 8.4, we consider the action of the monodromy matrix on the set of states constituting the degenerate eigenstates. We will see that an element of the monodromy matrix plays the role of a creation or annihilation operator of orthogonal eigenfunctions. This means that all eigenfunctions can be generated algebraically from the highest-weight state, for example, by successive action of an element of the monodromy matrix. Thus, the degeneracy beyond the Lie algebra is accounted for by the symmetry of the monodromy matrix, which is nothing but the Yangian.

    A demonstration of this construction will be given in Section 8.5 for the two-particle and three-particle systems.

  • 2 - Formal perturbation theory

  • Isaiah Shavitt, University of Illinois, Urbana-Champaign, Rodney J. Bartlett, University of Florida
  • Book:
    Many-Body Methods in Chemistry and Physics
    Published online:
    06 January 2010
    Print publication:
    06 August 2009, pp 18-53

  • Summary

    Background

    There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development, carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development, where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory.

    Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways.

    Classical derivation of Rayleigh–Schrödinger perturbation theory

    The perturbation Ansatz

    We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).

  • Afterword

  • Yoshio Kuramoto, Tohoku University, Japan, Yusuke Kato, University of Tokyo
  • Book:
    Dynamics of One-Dimensional Quantum Systems
    Published online:
    12 January 2010
    Print publication:
    06 August 2009, pp 455-457

  • Summary

    We have discussed the physical and mathematical aspects of one-dimensional quantum particles with1/r2 interactions. On the physics side, we began with the simplest two-body system, and culminated in the exact dynamics of a supersymmetric electronic model. In this way we have tried to draw a comprehensive picture of one-dimensional quantum systems, taking the canonical systems with 1/r2 interactions. On the mathematical side, the most important subjects are first the Jack polynomials, and second the Yangians. These two subjects were originally developed independently, but are actually closely interrelated. Our basic viewpoint is that the 1/r2 systems provide the most natural working model to synthesize these two subjects, and to sharpen the concepts further.

    There are many topics, however, which we could not discuss in this book. This is firstly because of our insufficient understanding of the topics, and secondly because of our intention of making a reasonably sized book. We shall mention some of the omitted topics, hoping that they will be covered on another occasion.

    Examples of other canonical systems, which have escaped our discussion in this book, include the Calogero–Sutherland model with harmonic confinement. The spectrum of the model is equidistant as in the simple harmonic oscillator. Because of this feature the model can be compared with the chiral conformal field theory (CFT) with right-going particles only. We refer to a recent review [88] for the systems. In relation to the CFT, there are active studies to investigate the quasi-particle structure for the general case of the internal symmetry.

  • Appendix 6 - Diffusion Capacitance of a p–n Diode

  • Yuan Taur, University of California, San Diego, Tak H. Ning
  • Book:
    Fundamentals of Modern VLSI Devices
    Published online:
    28 May 2018
    Print publication:
    06 August 2009, pp 562-568

  • Summary

    In a quasisteady approximation, the mobile carriers in a forward-biased diode are assumed to follow the applied voltage instantaneously. This assumption is a good one for calculating the depletion-layer capacitance where the majority carriers are able to respond to the applied voltage virtually instantaneously. (See Section 2.1.4.8 for a discussion on majority-carrier response time.) However, the redistribution of minority carriers is through diffusion and recombination processes. These processes do not occur instantaneously, but on a time scale related to the minority-carrier lifetime or transit time. As a consequence, when a small signal is applied to a diode, the changes in the minority-carrier densities at different locations in the diode have different phases and cannot be lumped together and treated as a single entity. In this appendix, the diffusion capacitance is derived from a small-signal analysis of the current through a diode starting from the differential equations governing the transport of minority carriers (Shockley, 1949; Lindmayer and Wrigley 1965; Pritchard, 1967). The diffusion capacitance can also be obtained from a transmission-line analysis of a diode equivalent circuit (Bulucea, 1968).

    Consider an n+–p diode with a time-dependent forward-bias voltage vBE (t) applied across it. The emitter is assumed to be wide, i.e., LpE << WE, where LpE is the hole diffusion length in the emitter and WE is the thickness of the emitter. The base is assumed to be narrow, i.e., LnB >> WB, where LnB is the electron diffusion length in the base and WB is the base width. (This diode represents the emitter–base diode of an n–p–n bipolar transistor.) We assume that vBE (t) consists of a small-signal voltage vbe (t) in series with a dc voltage VBE , i.e., vBE (t) = VBE + vbe (t). For simplicity, we assume parasitic resistances are negligible so that vBE (t) is the same as the emitter–base junction voltage. The current flows, including the displacement current, are shown schematically in Figure A6.1, where iE (t) is the emitter terminal current, iB (t) is the base terminal current, and CdBE,tot is the depletion-layer capacitance of the emitter–base junction. Overall charge neutrality or Kirchoff’s law requires that

    iE(t) + iB(t) = 0.

  • Appendix 5 - Generation and Recombination Processes and Space-Charge-Region Current

  • Yuan Taur, University of California, San Diego, Tak H. Ning
  • Book:
    Fundamentals of Modern VLSI Devices
    Published online:
    28 May 2018
    Print publication:
    06 August 2009, pp 553-561

  • Summary

    Electron and hole generation and recombination processes play an important role in the operation of many silicon devices and in determining their current–voltage characteristics. Electron and hole generation and recombination can take place directly between the valence band and the conduction band, or indirectly via trap centers in the energy gap. Direct transitions involve energies larger than the bandgap energy Eg . Indirect transitions via trap centers involve energies smaller than Eg . As a result, indirect transitions are much more efficient than direct transitions. In this appendix, we derive the expressions for the generation and recombination rates due to indirect transitions via trap centers and examine the characteristics of generation and recombination currents.

    Capture and Emission at a Trap Center

    Consider a piece of silicon having in it a concentration of Nt trap centers per unit volume. For simplicity, we assume all of the trap centers to be identical and located at energy Et in the bandgap. Also, we assume that each trap center can exist in one of two charge states, namely neutral when it is not occupied by an electron and negatively charged when it is occupied by an electron. Each unoccupied center can capture an electron from the conduction band (electron capture). An electron in an occupied center can be emitted into the conduction band (electron emission). Similarly, each occupied center can capture a hole from the valence band (hole capture), and an unoccupied center can emit a hole into the valence band (hole emission). These four capture and emission processes are illustrated in Fig. A5.1. Note that in hole capture, the center turns from a negatively charged (occupied) state into a neutral (unoccupied) state. Hole capture from the valence band is equivalent to electron emission into the valence band. In hole emission, the center turns from a neutral (unoccupied) state into a negatively charged (occupied) state. Hole emission is equivalent to electron capture from the valence band.

  • 5 - CMOS Performance Factors

  • Yuan Taur, University of California, San Diego, Tak H. Ning
  • Book:
    Fundamentals of Modern VLSI Devices
    Published online:
    28 May 2018
    Print publication:
    06 August 2009, pp 256-317

  • Summary

    The performance of a CMOS VLSI chip is measured by its integration density, switching speed, and power dissipation. CMOS circuits have the unique characteristic of practically zero standby power, which enables higher integration levels and makes them the technology of choice for most VLSI applications. This chapter examines the various factors that determine the switching speed of basic CMOS circuit elements.

    Basic CMOS Circuit Elements

    In a modern CMOS VLSI chip, the most important function components are CMOS static gates. In gate array circuits, CMOS static gates are used almost exclusively. In microprocessors and supporting circuits of memory chips, most of the control interface logic is implemented using CMOS static gates. Static logic gates are the most widely used CMOS circuit because of their simplicity and noise immunity. This section describes basic static CMOS circuit elements and their switching characteristics.

    Circuit symbols for nMOSFETs and pMOSFETs are defined in Fig. 5.1. A MOSFET is a four-terminal device, although usually only three are shown. Unless specified, the body (p-substrate) terminal of an nMOSFET is connected to the ground (lowest voltage), while the body terminal (n-well) of a pMOSFET is connected to the power supply Vdd (highest voltage).

  • 10 - Uglov's theory

  • from Part II - Mathematics related to 1/r2 systems
  • Yoshio Kuramoto, Tohoku University, Japan, Yusuke Kato, University of Tokyo
  • Book:
    Dynamics of One-Dimensional Quantum Systems
    Published online:
    12 January 2010
    Print publication:
    06 August 2009, pp 441-454

  • Summary

    In this chapter, we give an example of how the Yangian theory works in practice in the calculation of physical quantities. Using the Yangian representation theory, calculation of the dynamical correlation function of the Sutherland model with glK symmetry can be performed in the same way as (a modified version of) the single-component Sutherland model [189]. Namely, Uglov [189] showed that the spin and the spatial momentum can be unified as a fictitious momentum. This spin–momentum unification is particularly powerful for exact derivation of spin correlation functions [189, 197]. Further development for the SU(K) chain has also been achieved [198, 199]. Uglov's theory is outlined in this chapter.

    In Section 10.1 we introduce as a prelude the symmetric Macdonald polynomials, which include the symmetric Jack polynomials as a special case. Then in Section 10.2, Uglov symmetric polynomials are introduced as another limit of symmetric Macdonald polynomials. We explain in Section 10.3 an isomorphism between the Fock space of the Sutherland model with SU(2) internal symmetry and the space of Laurent symmetric polynomials. The isomorphism by which the Yangian Gelfand–Zetlin basis is mapped onto the Uglov polynomials preserves the inner product. By this isomorphism, density and spin-density operators in the U(2) model find their correspondence in the single-component model. In this way, the calculations of dynamical density and spin-density correlation functions in the U(2) Sutherland model reduce to those of a modified version of the single-component Sutherland model.

    Macdonald symmetric polynomials

    We have seen in Section 2.5 that the Jack symmetric polynomials can be defined as the homogeneous symmetric polynomials of z = (z1, …, zN) satisfying conditions of triangularity (2.177) and orthogonality (2.196).

  • Appendix 1 - CMOS Process Flow

  • Yuan Taur, University of California, San Diego, Tak H. Ning
  • Book:
    Fundamentals of Modern VLSI Devices
    Published online:
    28 May 2018
    Print publication:
    06 August 2009, pp 538-541

  • Summary

    A more or less generic CMOS process flow is described below. It features shallow trench isolation (STI) (Davari et al., 1988, 1989), dual n+/p+ polysilicon gates (Wong et al., 1988; Sun et al., 1989), and self-aligned silicide (Ting et al., 1982). The front-end-of-the-line process consists of six or seven masking levels and is suitable for sub-0.5-µm generations of VLSI logic and SRAM technology.

  • Starting material p-type substrate or p− epi on p+ substrate for latch-up prevention (Taur et al., 1984).

  • Grow pad oxide. Deposit CVD (Chemical Vapor Deposition) nitride. (See Fig. A1.1.)

  • Lithography to cover the active region with photoresist.

  • Reactive ion etching (RIE) nitride and oxide in the field region.

  • RIE shallow trench in silicon. (See Fig. A1.2.)

  • Grow pad oxide. Deposit thick CVD oxide. (See Fig. A1.3.)

  • Chemical–mechanical polishing planarization. (See Fig. A1.4.)

  • n-well lithography and implant (also channel doping).

  • p-well lithography and implant (also channel doping). (See Fig. A1.5.)

  • Grow gate oxide.

  • Deposit polysilicon film. (See Fig. A1.6.)

  • Gate lithography.

  • RIE polysilicon gate. (See Fig. A1.7.)

  • Sidewall reoxidation.

  • n+ source–drain lithography and implant (also dope n+ polysilicon gate).

  • p+ source–drain lithography and implant (also dope p+ polysilicon gate). (See Fig. A1.8.)

  • Oxide (or nitride) spacer formation by CVD and RIE.

  • Source–drain anneal. (See Fig. A1.9.)

  • Self-aligned silicide process. (See Fig. A1.10.)

  • Back-end-of-the-line process.

Page 177 of 348