Section 2: Emergent Behavior in Crystalline Solids
Figure 3: Nanowires are crystalline fibers with emergent behaviors expected to be used for nanoscale applications.
Source: © Deli Wang Laboratory at UCSD. More info
Crystalline solids provide familiar examples of emergent behavior. This section will outline the theoretical steps that have revealed the fundamental nature of solids and the ways in which such critical ideas as quantum statistics, excitations, energy bands, and collective modes have enabled theorists to understand how solids exhibit emergent behavior.
At high enough temperatures, any form of quantum electronic matter becomes a plasma—a gas of ions and electrons linked via their mutual electromagnetic interaction. As it cools down, a plasma will first become liquid and then, as the temperature falls further, a crystalline solid. For metals, that solid will contain a stable periodic array of ions along with electrons that are comparatively free to move under the application of external electric and magnetic fields.
Figure 4: Left: Diagram showing experimental set-up for measurement of the energy loss spectrum of neutrons that are inelastically scattered by a crystal. Right: A typical phonon spectrum obtained through an inelastic neutron scattering (INS) experiment.
Source: © Top: Wikimedia Commons, Public Domain. Author: Joachim Wuttke, 22 February 2007. Bottom: Courtesy of Joe Wong, Lawrence Livermore National Laboratory. More info
The crystallization process has broken a basic symmetry of the plasma: there are no preferred directions for the motion of its electrons. Broken symmetry is a key organizing concept in our understanding of quantum matter. The crystalline confinement of the ions to regular positions in the crystal leads to a quantum description of their behavior in terms of the ionic elementary excitations, called "phonons," which describe their vibrations about these equilibrium positions. Physicists can study phonons in detail through inelastic neutron scattering experiments (Figure 4) that fire neutrons at solid samples and measure the neutrons' energies after their collisions with the vibrating ions in the solid. Moreover, the solids' low-energy, long-wavelength behavior is protected: It is independent of details and describable in terms of a small number of parameters—in this case, the longitudinal and transverse sound velocities of their collective excitations, the quantized phonons.
Independent electrons in solids
What of the electrons? The interactions between the closely packed atoms in a periodic array in a crystal cause their outermost electrons to form the energy bands depicted in Figure 5. Here, the behavior of the electrons is characterized by both a momentum and a band index, and their corresponding physical state depends to a first approximation on the valency of the ions and may be characterized by the material's response to an external electric field. Thus, the solid can take any one of three forms. It may be a metal in which the electrons can move in response to an external electric field; an insulator in which the electrons are localized and no current arises in response to the applied electric field; or a semi-conductor in which the valence band is sufficiently close to a conduction band that a small group of electrons near the top of the band are easily excited by heating the material and can move in response to an applied electric field.
Figure 5: Comparison of the electronic band structures of metals, semiconductors, and insulators.
Source: © Wikimedia Commons, Public Domain. Author: Pieter Kuiper, 6 June 2007. More info
Physicists for many years used a simple model to describe conduction electrons in metals: the free electron gas or, taking the periodic field of ions into account through band theory, the independent electron model. The model is based on Wolfgang Pauli's exclusion principle that we met in Unit 2. Because electrons have an intrinsic spin of 1/2, no two can occupy the same quantum state. In the absence of the periodic field of ions, the quantum states of each electron can be labeled by its momentum, p, and its spin.
Figure 6: The Fermi surface reveals how the energy varies with momentum for the highest-energy electrons—those that have the Fermi energy.
Source: © Los Alamos National Laboratory. More info
Since the electrons can carry momentum in each of three independent spatial directions, it's useful to imagine a 3D coordinate system (which we call "momentum space") that characterizes the x, y, and z components of momentum.
The ground state of the electrons moving in a uniform background of positive charge would then be a simple sphere in momentum space bounded by its Fermi surface, a concept derived from the statistical work of Enrico Fermi and P.A.M. Dirac that defines the energies of electrons in a metal. When the uniform positive charge is replaced by the actual periodic array of the ions in a metallic lattice, the simple sphere becomes a more complex geometric structure that reflects the nature of the underlying ionic periodic structure, an example of which is shown in Figure 6.
To calculate the ground state wavefunction of the electrons, physicists first applied a simple approach called the Hartree-Fock approximation. This neglects the influence of the electrostatic interaction between electrons on the electronic wavefunction, but takes into account the Pauli principle. The energy per electron consists of two terms: the electrons' average kinetic energy and an attractive exchange energy arising from the Pauli principle which keeps electrons of parallel spin apart.
Emergent concepts for a quantum plasma
The Russian physicist Lev Landau famously said, "You cannot repeal Coulomb's law." But until 1950, it appeared that the best way to deal with it was to ignore it, because microscopic attempts to include it had led to inconsistencies, or worse yet, divergent results. The breakthrough came with work carried out between 1949 and 1953 by quantum theorist David Bohm and myself, his Ph.D. student. Our research focused on the quantum plasma—electrons moving in a uniform background of positive charge, an idealized state of matter that solid-state physicist Conyers Herring called "jellium." Bohm and I discovered that when we viewed particle interactions as a coupling between density fluctuations, we could show, within an approximation we called "the random phase approximation (RPA)," that the major consequence of the long range electrostatic interaction between electrons was to produce an emergent collective mode: a plasma oscillation at a frequency, , where N is the electron density and m its mass, whose quantized modes are known as plasmons. Once these had been introduced explicitly, we argued what was left was an effective short-range interaction between electrons that could be treated using perturbation-theoretic methods. See the math
The plasma oscillation is an example of a "collisionless" collective mode, in which the restoring force is an effective field brought about by particle interaction; in this case, the fluctuations in density produce a fluctuating internal electric field. This is the first of many examples we will consider in which effective fields produced by particle interaction are responsible for emergent behavior. As we shall see later in this unit, the zero sound mode of 3He furnishes another example, as does the existence of phonons in the normal state of liquid 4He. All such modes are distinct from the "emergent," but familiar, sound modes in ordinary liquids, in which the restoring forces originate in the frequent collisions between particles that make possible a "protected" long wavelength description using the familiar laws of hydrodynamics.
The importance of plasmons
Following their predicted existence, plasmons were identified as the causes of peaks that experimentalists had already seen in the inelastic scattering of fast electrons passing through or reflected from thin solid films. We now know that they are present in nearly all solids. Thus, plasmons have joined electrons, phonons, and magnons (collective waves of magnetization in ferromagnets) in the family of basic elementary excitations in solids. They are as well defined an elementary excitation for an insulator like silicon as for a metal like aluminum.
By the mid 1950s, it was possible to show that the explicit introduction of plasmons in a collective description of electron interactions resolved the difficulties that had arisen in previous efforts to deal in a consistent fashion with electron interaction in metals. After taking the zero point energy of plasmons into account in a calculation of the ground state energy, what remained was a screened electrostatic interaction between electrons of comparatively short range, which could be dealt with using perturbation theory. This work provided a microscopic justification of the independent electron model for metal, in which the effects of electron interaction on "single" electron properties had been neglected to first approximation. It also proved possible to include their influence on the cohesive energy of jellium with results that agreed well with earlier efforts by Eugene Wigner to estimate this quantity, and on the exchange and correlation corrections to the Pauli spin susceptibility, with results that agreed with its subsequent direct measurement by my Illinois colleague, C.P. Slichter.
It subsequently proved possible to establish that both plasma oscillations and screening in both quantum and classical plasmas are not simply mathematical artifacts of using the random phase approximation, but represent protected emergent behavior. Thus, in the limit of long wavelengths, plasma oscillations at are found at any density or temperature in a plasma, while the effective interaction at any temperature or density is always screened, with a screening length given by , where s is the isothermal sound velocity. Put another way, electrons in metals are never seen in isolation but always as "quasielectrons," each consisting of a bare electron and its accompanying screening cloud (a region in space in which there is an absence of other electrons). It is these quasielectrons that interact via a short range screened electrostatic interaction. For many metals, the behavior of these quasielectrons is likewise protected, in this case by the adiabatic continuity that enables them to behave like the Landau Fermi liquids we will consider in the next section.
From plasmons to plasmonics
Figure 8: Harvard School of Engineering and Applied Sciences' researchers Federico Capasso (red shirt) and Nanfang Yu working on their nanoscale quantum clusters for plasmonic applications.
Source: © Eliza Grinnell. More info
When plasmons were proposed and subsequently identified, there seemed scant possibility that these would become a subject of practical interest. Unexpectedly, plasmons found at the surface of a metal, or at an interface between two solids, turn out to be sufficiently important in electronic applications at the nanoscale, that there now exists a distinct sub-field that marks an important intersection of physics and nanotechnology called "plasmonics." Indeed, beginning in 2006, there have been bi-annual Gordon research conferences devoted to the topic. To quote from the description of the founding conference: "Since 2001, there has been an explosive growth of scientific interest in the role of plasmons in optical phenomena, including guided-wave propagation and imaging at the subwavelength scale, nonlinear spectroscopy, and 'negative index' metamaterials. The unusual dispersion properties of metals near the plasmon resonance enables excitation of surface modes and resonant modes in nanostructures that access a very large range of wave vectors over a narrow frequency range, and, accordingly, resonant plasmon excitation allows for light localization in ultra-small volumes. This feature constitutes a critical design principle for light localization below the free space wavelength and opens the path to truly nanoscale plasmonic optical devices. This principle, combined with quantitative electromagnetic simulation methods and a broad portfolio of established and emerging nanofabrication methods, creates the conditions for dramatic scientific progress and a new class of subwavelength optical components." A description of the third such conference began with a description by Federico Capasso (Figure 8) of his bottom-up work on using self-assembled nanoclusters for plasmonic applications.