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6717 results in Condensed Matter Physics, Nanoscience and Mesoscopic Physics

Page 77 of 336

  • 12 - Superconductor Quantum Critical Points

  • Mucio Continentino, Centro Brasileiro de Pesquisas Físicas, Brazil
  • Book:
    Quantum Scaling in Many-Body Systems
    Published online:
    04 May 2017
    Print publication:
    17 April 2017, pp 158-176

  • Summary

    Introduction

    In this chapter we examine a special type of quantum critical point related to superconducting zero-temperature instabilities in many-body systems. We consider the case where the superconductor order parameter is inhomogeneous and characterised by a wave vector qC. Inhomogeneous ground states also appear in other systems like magnetic materials, in the form of spin-density waves or helicoidal ground states, in charge density wave systems and in excitonic insulators. They share the existence of a characteristic wave vector that determines the spatial modulation of the order parameter. For superconductors, we consider two cases.

    First, the problem, treated independently by Fulde and Ferrell (1964)and Larkin and Ovchinnikov (1965), of an s-wave, singlet superconductor in a homogeneous external magnetic field in the absence of vortices or orbital effects. The predicted ground state is generally known as the FFLO superconductor and is described by an order parameter that oscillates in space like ΔeiqC·r. It arises from the competition between the pairing energy of electrons with opposite spins and the Zeemann energy due to the external magnetic field that forces the alignment of the spins. For sufficiently large magnetic fields the Zeemann energy always wins and the ground state of the system is a spin-polarised, normal fermionic gas. However, as the magnetic field decreases, there is a quantum phase transition to a modulated superconducting state (Samokhin and Marénko, 2006; Caldas and Continentino, 2012) whose nature we identify. We also determine the universality class of this superconducting quantum phase transition. The characteristic wave vector qC of the resulting modulated superconducting phase is related to the difference between the Fermi wave vectors of the antiparallel spin bands that in turn is proportional to the magnetic field. A related question we also investigate is the fate of the uniform superconductor when the magnetic field is increased in the absence of orbital effects. We show that these two extreme approaches, increasing the field in the homogeneous superconductor and decreasing the field starting from the polarised metal do not merge smoothly into one another.

  • 8 - Metal and Superfluid–Insulator Transitions

  • Mucio Continentino, Centro Brasileiro de Pesquisas Físicas, Brazil
  • Book:
    Quantum Scaling in Many-Body Systems
    Published online:
    04 May 2017
    Print publication:
    17 April 2017, pp 102-114

  • Summary

    Conductivity and Charge Stiffness

    In this chapter, we will consider initially metal–insulator (MI) transitions which occur in pure systems without disorder. In the final section we consider a superfluid–insulator transition induced by some special type of disorder. We will show that the quantities which characterise the phase transitions in both problems obey similar scaling laws and are governed by similar exponents.

    Differently from the magnetic transitions treated before metal–insulator (MI) transitions have no obvious order parameter to distinguish the metallic from the insulating phase. This precludes a power expansion of the free energy in terms of a small quantity as for magnetic phase transitions. In spite of that, the concepts of phase transitions and critical phenomena turn out to be very useful to describe metal–insulator transitions. In case this problem is approached using the renormalisation group, the localisation transition is associated with an unstable fixed point and the flow of the RG equations to the different attractors is sufficient to distinguish the nature of the phases. Then, in general, we can associate a set of critical exponents with the MI transitions. Besides, these exponents are not independent but obey scaling relations. Since we are concerned with zero temperature instabilities, the quantum hyperscaling law, Eq. 1.18, plays a crucial role in this problem.

    We will distinguish here two different types of metal–insulator transitions. Those due to a competition between parameters of the relevant Hamiltonian and metal–insulator transitions arising by varying the number of electrons or chemical potential. An example of the former is the Mott transition due to the competition between kinetic energy and the local Coulomb repulsion as described by the half-filled band Hubbard model. The latter will be referred as density-driven metal– insulator transitions in analogy with the superfluid–insulator transition in bosonic systems with varying particle number (Fisher et al., 1989). These two types of transitions may occur in the same system or Hamiltonian. They in general belong to different universality classes and consequently have different critical exponents.

  • 1 - Scaling Theory of Quantum Critical Phenomena

  • Mucio Continentino, Centro Brasileiro de Pesquisas Físicas, Brazil
  • Book:
    Quantum Scaling in Many-Body Systems
    Published online:
    04 May 2017
    Print publication:
    17 April 2017, pp 1-24

  • Summary

    Quantum Phase Transitions

    Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases.

    If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory, where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject.

    We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.

  • Preface

  • Sandeep Kumar, Santanu Kumar Pal
  • Book:
    Liquid Crystal Dimers
    Published online:
    23 July 2017
    Print publication:
    06 January 2017, pp ix-x

  • Summary

    Liquid crystals (LCs) are unique functional soft materials which combine both order and mobility on a molecular, supramolecular and macroscopic level. Hierarchical self-assembly in LCs offers a powerful strategy for producing nanostructured mesophases. Molecular shape, microsegregation of incompatible parts, specific molecular interaction, self-assembly and self-organization are important factors that drive the formation of various LC phases. LCs are accepted as the fourth state of matter after solid, liquid and gas. LCs form a state of matter intermediate between the solid and the liquid state. For this reason, they are referred to as intermediate phases or mesophases. This is a true thermodynamic stable state of matter. The constituents of the mesophase are called mesogens. Since the discovery of LCs in 1888 by F. Reinitzer, it was assumed that LC molecules are mainly composed of mesogenic core attached to which are one or more alkyl chains. However, during 1980s a new class of LCs attracted particular attention acknowledged as the LC dimers. An LC dimer is composed of molecules containing two mesogenic units linked via a flexible alkyl spacer, most commonly an alkyl chain. Thus, LC dimers contravened the accepted structure–property relationships for low molar mass mesogens by consisting of molecules having a highly flexible core rather than a semirigid central unit. In these respect, therefore, these molecules actually represented an inversion of the conventional molecular design for low molar mass mesogens. Although this class of compounds has been discovered by Vorlander long back in 1927, these dimers did not achieve considerable attention until their rediscovery by Griffin and Britt in 1980s. Subsequently, several classes of dimeric LC compounds have been prepared and studied extensively.

    As of now, no book exists on this topic. However, a chapter on LC dimers can be seen in many LC-related books. While a number of books are available on LCs, no exclusive book describing the basic design principles, transitional properties, device fabrication and applications of dimeric LCs is available. Researchers working in the field of LC dimers and device fabrication need to have an upto- date source of reference material to establish a solid foundation of understanding. It is extremely important that students and researchers in this field have ready access to what is known and what has already been accomplished in the field. This book contains all the recent literature up to 2015 and covers the physics, chemistry, electronics, and materials-related properties.

Liquid Crystal Dimers
  • Sandeep Kumar, Santanu Kumar Pal
  • Published online:
    23 July 2017
    Print publication:
    06 January 2017
  • This book covers in-depth discussion of design principles, synthesis and thermal behavior of all types of liquid crystal (LC) dimers. The text presents recent advances in the field of LC dimers consisting of different mesogenic units such as calamitic, discotic and bent-core molecules. It starts with a chapter on the introduction of liquid crystal dimers, including their odd-even behavior, basic classification of dimers and common mesophases in dimers. The text shows how the molecular architectures are being used to develop new materials to study a range of interesting phenomena such as the biaxial nematic phase containing rod-like and disc-like mesogenic units. Finally, the text presents perspectives related to technological relevance of these dimers such as dopants in LC display mixtures exhibiting faster relaxation time, strong flexoelectric coupling and others to effect control over the properties of these materials.

  • 2 - Calamitic-Calamitic LC Dimers

  • Sandeep Kumar, Santanu Kumar Pal
  • Book:
    Liquid Crystal Dimers
    Published online:
    23 July 2017
    Print publication:
    06 January 2017, pp 10-58

  • Summary

    INTRODUCTION

    Scientific community has done a lot of efforts for controlling molecular ordering of polymer chains through various supramolecular approaches. Among them, a famous approach is keeping the LC molecules a repeating unit of polymer chain. The self-assembling tendency of LC molecules tends to assemble polymer chains into well-ordered structures. A variety of liquid crystalline polymers (LCPs) have been synthesized since the discovery of LCPs in 1923 by Vorlander et al. to achieve the highly ordered polymers.1 The understanding of self-assembly in condensed phases is challenged by these polymers whose interpretation at a molecular level requires experimental investigations as well as the development of new molecular theories. However, the structural heterogeneity inherent in a polymeric system complicates these tasks. An alternative approach is the use of monodisperse low molar mass compounds whose behaviour encapsulates the essential physics of polymeric system for developing a molecular understanding of polymers.

    Catalyzed with the motivation to understand LCPs at a molecular level, a wide variety of nonconventional low molar mass compounds were synthesized and shown to support liquid crystallinity. The fact that these compounds were LCs was very surprising because at that time most of the known low molar mass LCs were composed of molecules consisting of a single semirigid anisometric core with alkyl chains. Indeed, it had been widely assumed for many years that such a molecular structure was a prerequisite for the observation of liquid crystallinity. We now know that 1980s evidenced the beginning of the discovery of a rich diversity of structures capable of supporting mesogenic behaviour that has continued to the present day.

    Of all these new low molar mass LC discovered during the 1980s, one class that attracted particular attention and which still remains the focus of much research are the so-called LC dimers. An LC dimer is composed of molecules containing two conventional mesogenic groups linked by a flexible spacer. Thus, LC dimers contravened the accepted structure–property relationships for low molar mass mesogens by consisting of molecules having a highly flexible core rather than a semirigid central unit. In this respect, LC dimers represent an inversion of the conventional molecular design for low molar mass mesogens. Several names have been used to refer to these materials including dimesogens or Siamese twins but these have all now been superseded by the preferred term LC dimer.

  • 5 - Bent-Core LC Dimers

  • Sandeep Kumar, Santanu Kumar Pal
  • Book:
    Liquid Crystal Dimers
    Published online:
    23 July 2017
    Print publication:
    06 January 2017, pp 185-224

  • Summary

    Over the past two decades, bent-core liquid crystals (BLCs) have provided fascinating results to the scientific arena of LC research. The sharp bend within the linkage of the core group leads to the unique physical properties having no complement in conventional calamitic LCs such as occurrence of chiral phases and/ or polar phases even though the constituent molecules are achiral. Specially the bent (angular) molecular architecture leads to the variation in self-assembly processes that demonstrate as the unconventional phase structures. BLCs themselves are primarily comprised of three parts: bent core, rigid arm and the flexible tail chain. Up to now, a rich variety of BLCs with different structures and unique functions have been developed and their phase structures are also even more enriched. LC dimers obtained from monomeric BLC units, interconnecting two mesogenic units through flexible spacers (such as alkylene, siloxane, carbosilane, oxyethylene) are relatively new. The inherent motivation behind the preparation of bent-core dimers was to use the microsegregation to manipulate the phase structure of bent-core molecules. Based on the adjoined mesogenic units, the bent-core dimers can be categorically divided into three sub units:

  • 5.1 Symmetrical bent-core LC dimers

  • 5.2 Nonsymmetrical bent-core LC dimers

  • 5.3 Unconventional bent-core LC dimmers

    • SYMMETRICAL BENT-CORE LC DIMERS

    Introduction

    In symmetrical bent-core dimers, both the two mesogenic units joined via flexible spacer are BLC unit. The possible different types of arrangements possible for the bent-core dimers that can be achieved by connecting through a flexible spacer with possible combinations of the bent-core mesogenic units are presented in Figure 5.1.

    Structure–Property Relationships

    Dimers Based on Siloxane and Alkylene Spacer

    The first attempt of bent-core LC dimers consisting of two BLC units connected by flexible oligosiloxane spacer (dimethylsiloxane) units, shown in Figure 5.2, was reported almost a decade ago by G. Dantlgraber and his coauthors.1 The motivation behind the work was initiated by the observation that decoupling of the layers by the microsegregated oligosiloxane units disfavours an antiferroelectric (AF) assembly of the bentcore molecules. Therefore, antiferroelectricity is not solely arising by the compensation of the layer polarization but interlayer fluctuations of the molecules also play a vital role. Hence anticlinic layer arrangement or ferroelectricity can be induced by suppressing of the layer fluctuation. The dimers were synthesized by hydrosilylation of terminally unsaturated bentcore mesogens with hexamethyltrisiloxane (1) or octamethyltetrasiloxane (2) using Karstedt's catalyst.

  • 1 - Introduction

  • Sandeep Kumar, Santanu Kumar Pal
  • Book:
    Liquid Crystal Dimers
    Published online:
    23 July 2017
    Print publication:
    06 January 2017, pp 1-9

  • Summary

    LIQUID CRYSTALS

    Liquid crystals (LCs) are distinctive functional soft materials with a combination of order and mobility on a molecular, supramolecular and macroscopic level. Hierarchical self-assembly in LCs offers a powerful strategy for producing nanostructured mesophases. Molecular shape, microsegregation of incompatible parts, specific molecular interaction, self-assembly and selforganization are important factors that lead to the formation of various LC phases. LCs are accepted as the fourth state of matter after solid, liquid and gas. This fourth state of matter is intermediate between the solid and the liquid state. For this reason, they are referred to as intermediate phases or mesophases. This is a real thermodynamic stable state of matter where the constituents of a mesophase are called mesogens. A rigid core of the mesogen (which often consists of aromatic rings) induces structural order whereas the flexible parts (e.g., alkyl chains) provide the necessary mobility within the LC phase.

    The unique feature of LCs is the presence of both order and high degree of mobility in the mesophase that leads to the self-healing, adaptive and stimuli-responsive behaviour of these supramolecular systems and because of this, LCs have become the quintessential self-assembling molecular materials of the modern era. LCs have made huge impact on the development of the human societies. LCs are the advanced technological material found in lowpower- consuming LC displays (LCDs) which are being used in the last decades for the development of mobile data processing and communication tools. It is quite possible that LCDs might be replaced by other technologies in the future but, the fundamental knowledge gained with LCs can be used for the self-assembly of a huge variety of other materials.

    In 1888, Friedrich Reinitzer, Professor of Botany and Technical Microscopy at the German Technical University in Prague found that the compound cholesteryl benzoate which he had extracted from carrots exhibited two melting points, one at 145.5 °C and other at 178.5 °C. Between these two melting points, there was a milky liquid phase. Above 178.5 °C, the phase became clear. He observed distinct violet and blue colour phenomena at both these two different melting points under polarizing optical microscope. After having similar observations with a further derivative cholesteryl acetate that has a monotropic cholesteric phase, he contacted physicist Otto Lehmann.

Page 77 of 336