Physics
Phonon
A phonon is a collective excitation or quantized mode of vibration in a periodic, elastic arrangement of atoms or molecules in a solid material. It is a fundamental concept in condensed matter physics, describing the vibrational and thermal properties of materials. Phonons play a crucial role in phenomena such as heat conduction, thermal expansion, and superconductivity.
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12 Key excerpts on "Phonon"
eBook - PDF- Gerald D. Mahan(Author)
- 2010(Publication Date)
- Princeton University Press(Publisher)
7 Phonons 7.1 Phonon Dispersion Phonons are among the most important excitations in condensed matter. The ions can vibrate, and a crystal of ions has collective vibrations. The description of the vibrations can be done classically or using quantum mechanics. In the quantum mechanical descrip-tion, these collective motions are called Phonons. Phonons have the same frequency as the classical vibrations. Classical physics, using Newton’s law of motion, is used to find the frequencies. Quantum mechanics is used to determine the amplitude of vibrations. It is the amplitude that is quantized. The initial section of this chapter employs classical physics to find force constants. Phonon energies are reported in several different units. The frequency ( ) q ~ m has the dimensions of radians per second. The related frequency ( ) ( ) q q 2 o r ~ = m m (7.1) has the units of cycles per second, which is hertz. A natural unit is terahertz (THz). For example, the optical Phonon frequency in silicon at zero wave vector is 15 THz. This unit is used by those measuring Phonons throughout the Brillouin zone, which can be done by either neutron or x-ray scattering from crystals. Electron energies are usually reported in electron volts. Similarly, the Phonon energy ( ) q ' ~ m is often reported in milli-electron volts. The optical Phonon energy in silicon is at 62 meV. Some long-wavelength Phonons can be observed in Raman scattering. These op-tical experiments usually report the frequency in a unit that looks like a wave vector, but is actually the inverse of the wavelength of the photon with the same energy as the Phonon: ( ) k c q 1 m o = = m u (7.2) Phonons | 177 The silicon optical Phonon has a frequency of 500 cm -1 .
eBook - PDF- G.K. Horton, Alexei A. Maradudin(Authors)
- 1995(Publication Date)
- North Holland(Publisher)
One of the classic textbooks on the matter, the work of Born and Huang (1954) was drafted over four decades ago, and yet is useful even today. One might ask why the physics of Phonons is not completely understood, and how it can possibily be at the leading edge of condensed matter research today, after so many years. The answer, of course, is that the research has dealt with increasingly complex phenomena and has passed through as least three phases. Early work focussed on ther- mal and elastic phenomena, and the long-wavelength properties of matter, for example, elastic constants and the Debye theory of specific heats. The second phase dealt with all wavelengths of vibrational excitations in har- monic crystals, and began in the 1960's, when thermal-neutron scattering and infrared and Raman (light scattering) spectroscopy became widely used tools for studying the vibrational excitations of solids. The initial experi- mental studies with those tools were of the simplest crystals: monatomic or diatomic solids. Neutron and Raman spectroscopies focussed attention on the Phonon dispersion relations S2(k, A), which could be determined directly from neutron spectroscopy and indirectly from infrared reflectivity or Raman data. (Here k is the wave-vector and A is the polarization index.) Optical spectroscopies had the disadvantage that they probed only states near k = 0 (because, on the scale of the Brillouin zone, namely ~ TriaL, where aL is the lattice constant, the wave-vector of light is nearly zero), but laser light is an inexpensive probe compared with a nuclear reactor, and so infrared and Raman measurements probably provided more information about the vibra- tional excitations of solids than neutron scattering data. Today we are in the third phase of Phonon physics, which deals with the vibrational excitations of anharmonic crystals and harmonic non-crystalline solids, with the latter being the subject of this article.
- Radi A. Jishi(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
11 Phonons, photons, and electrons When the sky is illumined with crystal Then gladden my road and broaden my path And clothe me in light. –From “The Book of the Dead,” Ancient Egypt Translated by Robert Hillyer In this chapter we turn to Phonons, photons, and their interactions with electrons. These interactions play an important role in condensed matter physics. At room temperature, the resistivity of metals results mainly from electron–Phonon inter- action. At low temperature, this interaction is responsible for the superconducting properties of many metals. On the other hand, the electron–photon interaction plays a dominant role in light scattering by solids, from which we derive a great deal of information about excitation modes in solids. Much of our knowledge about energy bands in crystals has been obtained through optical absorption experiments, whose interpretation relies on an understanding of how electrons and photons interact. We begin by discussing lattice vibrations in crystals and show that, upon quanti- zation, the vibrational modes are described in terms of Phonons, which are particle- like excitations that carry energy and momentum. We will see that the effect of lattice vibrations on electronic states is to cause scattering, whereby electrons change their states by emitting or absorbing Phonons. Similarly, the interaction of electrons with an electromagnetic field will be represented as scattering processes in which electrons emit or absorb photons. A discussion of lattice vibrations in the general case of a three-dimensional crystal with a basis of more than one atom is somewhat complicated. To keep the presentation simple, we consider in detail the simplest case, a one-dimensional crystal with only one atom per unit cell. Next, we consider a diatomic chain, and then indicate briefly how things look in three dimensions.
eBook - PDF- Bipin R. Desai(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
38 Interactions of electrons and Phonons in condensed matter A condensed matter system in its most basic form can be viewed in terms of two components. One corresponds to electrons which, as discussed in Chapter 5, arrange themselves into energy levels as a consequence of Fermi statistics, with each level containing no more than two electrons with opposite spin orientations. These are the Fermi levels and the energy of the electrons in the topmost level is called the Fermi energy. Thus, in metals, where electrons can move about freely, even near absolute zero temperatures the electrons possess kinetic energy. The second component corresponds to ions, or atomic lattices, the vibrational modes of which are described in terms of simple harmonic motion. The normal modes of these oscillations, when quantized, have particle-like properties and are called Phonons. Since electrons contribute to the vibrations of the ions, the Phonons interact with electrons, which in turn leads to interaction between electrons themselves through Phonon exchange. We elaborate on all of this below. 38.1 Fermi energy 38.1.1 One dimension We repeat here the discussion in Chapter 5. Let us consider the ground state consisting of N noninteracting electrons confined in one dimension of length L . Each electron will be described by a free wavefunction u ( x ) = 1 √ L e ikx χ λ (38.1) where the χ λ ’s designate the spin-up and spin-down states, χ + = 1 0 , χ − = 0 1 , (38.2) which are normalized according to χ † λ 1 χ λ 2 = δ λ 1 λ 2 . (38.3) 700 Interactions of electrons and Phonons in condensed matter We assume, as we have done before, that the wavefunction satisfies the periodic boundary condition u ( x + L ) = u ( x ) , (38.4) which implies that the momentum vector can take only discrete values given by k n L = 2 n π . (38.5) The energy eigenvalues will then be E n = 2 k 2 n 2 m = 2 n 2 π 2 2 mL 2 .
eBook - ePubModeling, Characterization, and Production of Nanomaterials
Electronics, Photonics, and Energy Applications
- Vinod Tewary, Yong Zhang(Authors)
- 2022(Publication Date)
- Woodhead Publishing(Publisher)
Section 6.4 .6.2: Phonons, constraints, and symmetries
6.2.1: Phonons
The concept of the Phonon was first introduced by Debye [10] and has since become a cornerstone of condensed matter physics having provided a foundation for understanding measured vibrational frequencies [11 ,12] , heat capacity [10] , thermal expansion [13 ,14] , and lattice thermal conductivity [15] in crystalline solids. Phonons are built from the equations of motion of the atoms described as simple harmonic oscillators combined to construct the so-called dynamical matrix [16 ,17] :D αβkk ′q →=1m kmk ′∑l ′Φ αβ0 k ,l ′k ′eiq →⋅R →l ′(6.1)for a particular wavevectorq →and diagonalized to determine the Phonon frequencies,ω, for each polarization j. These depend on the atomic masses, mq →jk, where k labels a unit cell atom, and on the harmonic interatomic force constants (IFCs), Φαβ0k, l′k′= ∂2 V/∂ uα0k∂ uβl′k′, which describe how the atoms interact. These are second-order derivatives of the interatomic potential, V, with respect to small atomic displacement uαlkof the kth atom in the l = 0 unit cell in the αth direction and the k′th atom in the l′th cell in the βth direction at equilibrium. Fig. 6.1A gives the calculated Phonon dispersion for monolayer WS2 using density functional theory (DFT) [23 ,24] . Specific details of the WS2 calculations are presented in the caption of Fig. 6.1 . The three lowest frequency curves correspond to vibrations for which the unit cell atoms oscillate in-phase and are called “acoustic” modes as they are responsible for sound propagation—at low frequency. These Phonon branches (i.e., polarizations) haveasω→ 0q →jcorresponding to translational invariance of the crystal in three dimensions (these will be discussed in more detail in the next section). The higher frequency curves correspond to vibrations for which the unit cell atoms oscillate out-of-phase for the zone center Phonons and are called “optic” modes as they are excited by light—particularly as measured by Raman and infrared measurements. The number of polarizations corresponds to the number of degrees of freedom (N unit cell atoms times three dimensions). Here, WS2 has three atoms in its unit cell, each allowed to vibrate in x, y, and zq →→ 0
eBook - PDF- Marvin L. Cohen, Steven G. Louie(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
PART I BASIC CONCEPTS: ELECTRONS AND PhononS 1 Concept of a solid: qualitative introduction and overview 1.1 Classification of solids Condensed matter physics and solid state physics usually refer to the same area of physics, but in principle the former title is broader. Condensed matter is meant to include solids, liquids, liquid crystals, and some plasmas in or near solids. This is the largest branch of physics at this time, and it covers a broad scope of physical phenomena. Topics range from studies of the most fundamental aspects of physics to applied problems related to technology. The focus of this book will be primarily on the quantum theory of solids. To begin, it is useful to start with the concept of a solid and then describe the two commonly used models that form the basis for modern research in this area. The word “solid” evokes a familiar visual picture well described by the definition in the Oxford Dictionary: “Of stable shape, not liquid or fluid, having some rigidity.” It is the property of rigidity that is basic to the early studies of solids. These studies focused on the mechanical properties of solids. As a result, until the nineteenth century the most common classification of solids involved their rigidity or mechanical properties. The Mohs hardness scale (talc – 1; calcite – 3; quartz – 7; diamond – 10) is a typical example. This is a useful but limited approach for classifying solids. The advent of atomic theory brought more microscopic concepts about solids. Solids were viewed as collections of more or less strongly interacting atoms. From the point of view of atomic theory, a gas is described in terms of a collection of almost independent atoms, while a liquid is formed by atoms that are weakly interacting. This picture leads to a description of the formation of solids, under pressure or by freezing, in which the dis-tances between atoms are reduced and, in turn, this causes them to interact more strongly.
eBook - PDF- Xiao-feng Pang, Yuan-ping Feng(Authors)
- 2005(Publication Date)
- World Scientific(Publisher)
Chapter 9 Nonlinear Quantum-Mechanical Properties of Excitons and Phonons In the remaining two chapters, we will continue to discuss applications of the non-linear quantum mechanics. We will discuss nonlinear properties of exciton, Phonon, proton, polaron and magnon. Through these discussions, and what we already know about electrons and helium atom from Chapter 2, we can get further understanding on properties and motion of microscopic particles in these nonlinear systems. These discussions will also establish the nonlinear quantum mechanics theory presented in Chapter 3 as the correct theory for describing properties and motion of microscopic particles in nonlinear systems. We will begin with motion of exciton and Phonon in a molecular crystal. 9.1 Excitons in Molecular Crystals Let's first consider a particular molecular crystal, the acetanilide (CH3COHNC6 Hs)^ or ACN. Two close chains of hydrogen-bonded amide-I groups which consists of atoms of carbon, oxygen, nitrogen and hydrogen (CONH) run through the ac-etanilide crystal. Its crystal structure has been determined and a unit cell of ACN is shown in Fig. 9.1. The space group is D (Pbca) and the unit cell or factor group is D 2 h for this crystal. The average lattice constants are a = 1.9640 nm, b — 0.9483 nm, and c = 0.7979 nm. There are eight molecules in an unit cell and at the amide-I frequency, each of these has one degree of freedom (d.f.). Thus, there are three infrared-active modes {B u , B 2u , and B 3u ), four Raman-active modes (A g , Bi g , Big-, and B 3g ), and one inactive mode (A u ). However, at low frequency (< 200 cm 1 ), each molecule exhibits 6 d.f. (three translations and three rotations. This gives 48 low-frequency modes: 24 Raman active modes (6A g +6Bi 9 +6B 2g +6B 3g ), 18 infrared-active modes (6Bi u + 6B2 U +6B 3u ) and six (A u ) modes corresponding to the acoustic modes of translation and rotation). All of these active modes are seen in infrared absorption and Raman experiments.
eBook - PDF- Charles Kittel(Author)
- 2018(Publication Date)
- Wiley(Publisher)
5 Phonons II. Thermal Properties 125 Thermal Resistivity of Phonon Gas The Phonon mean free path is determined principally by two processes, geometrical scattering and scattering by other Phonons. If the forces between atoms were purely harmonic, there would be no mechanism for collisions be-tween different Phonons, and the mean free path would be limited solely by collisions of a Phonon with the crystal boundary, and by lattice imperfections. There are situations where these effects are dominant. With anharmonic lattice interactions, there is a coupling between differ-ent Phonons which limits the value of the mean free path. The exact states of the anharmonic system are no longer like pure Phonons. The theory of the effect of anharmonic coupling on thermal resistivity pre-dicts that is proportional to at high temperatures, in agreement with many experiments. We can understand this dependence in terms of the num-ber of Phonons with which a given Phonon can interact: at high temperature the total number of excited Phonons is proportional to T . The collision fre-quency of a given Phonon should be proportional to the number of Phonons with which it can collide, whence . To define a thermal conductivity there must exist mechanisms in the crys-tal whereby the distribution of Phonons may be brought locally into thermal equilibrium. Without such mechanisms we may not speak of the Phonons at one end of the crystal as being in thermal equilibrium at a temperature and those at the other end in equilibrium at . It is not sufficient to have only a way of limiting the mean free path, but there must also be a way of establishing a local thermal equilibrium distribu-tion of Phonons. Phonon collisions with a static imperfection or a crystal boundary will not by themselves establish thermal equilibrium, because such collisions do not change the energy of individual Phonons: the frequency of the scattered Phonon is equal to the frequency of the incident Phonon.
eBook - PDFCourse of Theoretical Physics
Physical Kinetics
- L. P. Pitaevskii, E.M. Lifshitz, J. B. Sykes(Authors)
- 2017(Publication Date)
- Pergamon(Publisher)
C H A P T E R VII INSULATORS §66. Interaction of Phonons T H E physical nature of transport processes such as thermal and electrical conduc-tion in gases consists in transfer by the thermal motion of the gas particles; in solids, the particles are replaced by quasi-particles. In going on to study these processes, we shall begin with thermal conduction in non-magnetic insulators. The relative simplicity of the physical picture here, as compared with transport proces-ses in solids of other kinds, arises from the presence of quasi-particles of only one sort, namely Phonons. The concept of free Phonons is the result of quantization of the vibrational motion of atoms in the crystal lattice in the harmonic approximation, i.e. with only the quadratic terms (in the displacements of the atoms) included in the Hamil-tonian; see SP 1, §72. The various Phonon interaction processes result when terms of higher orders of smallness are considered: the anharmonic terms of the third and subsequent orders in the displacements. t The first anharmonic (the cubic) terms in the classical lattice energy are H ( 3 ) = i Σ Λ ^ 3 ( η ι -η 3 , η 2 -η 3 ) 1 7 5 ι α ( η 1 ) ΐ ; 5 2 β ( η 2 ) 1 / 5 3 γ ( η 3 ) . (66.1) Ό (n, s) Here U s (n) are the atomic displacement vectors in the lattice; α, β, y are vector suffixes taking the values x, y, z; s u s 2 , s 3 number the atoms in the unit cell; ni, n 2 , n 3 are integral vectors giving the position of the cell in the lattice; (n, s) under the summation sign denotes summation over all η and s. Because the crystal is homogeneous, the functions Λ depend only on the relative positions ni -n 3 , n 2 - n 3 of the cells, not on their absolute positions in the lattice.
eBook - ePub- Timothy S Fisher(Author)
- 2013(Publication Date)
- WSPC(Publisher)
Chapter 1 Lattice Structure, Phonons, and Electrons 1.1 Introduction Guessing the technical background of students in a course or readers of a book is always a hazardous enterprise for an instructor, yet one must start a book or a course somewhere on the landscape of knowledge. Here, we begin with some essential concepts from condensed-matter physics and statistical mechanics. The definition of essential, too, is questionable and is presently intended to be information that recurs too frequently in the later parts of the text to leave the requisite information to the many excellent reference sources on these subjects. Our overarching objective is to develop the tools required to predict thermal transport in structures such as the one shown in Fig. 1.1. Arguably the most important thermal characteristic of an object is its thermal conductivity () defined as: For roughly a century, thermal conductivity was considered a basic material property in the engineering sense (e.g., with minor accommodation for variations in temperature), and therefore, the effects of the geometric terms in Eq. (1.1) were assumed to normalize with the others such that the final property was independent of size and shape. However, with the advent of microscale fabrication (and later nanoscale fabrication), the technical community was able to create tiny materials that exhibited deviations from the size-independent property assumption. In such circumstances, knowledge of not only a material’s size and shape becomes crucial but also the details of the atomic-scale carriers of thermal energy (Chen, 2005)
- Javier E. Hasbun, Trinanjan Datta(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
→ .Inelastic scattering of neutrons by Phonons can be used to obtain information about the frequency dispersion relation in crystals. Neutrons interact with a crystal by the scattering from the atomic nuclei and thus carry information about the vibrational properties of the crystal. If we letk →be the wavevector of the initial neutron,the wavevector of the scattered neutron, then by momentum conservationk →′k →+G →=±k →′K →,(4.4.48) so that the inelastic interaction of the neutron with the crystal (G →) gives rise to the creation or destruction of a Phonon (K →). By energy conservation we expectℏω=K →−ℏ 2k 22m n.ℏ 2k ′22m n(4.4.49) Therefore, by detecting the neutron properties, from Equation 4.4.48 one obtains±; i.e., the Phonon wavevector, and from Equation 4.4.49 the dispersion relation value at the corresponding wavevector is obtained.K →=k →+G →−k →′4.5 Phonon Heat CapacityThe change in the internal energy (U) per unit temperature is referred to as the heat capacity. Theoretically, the heat capacity is studied at constant volume,C V= ( ∂ U / ∂ T) V,(4.50) while experimentally the heat capacity is obtained at constant pressure, CP= (∂U/∂T)P. The two are related through the expression CV− CP= 9α2 BVT, where α is the coefficient of linear expansion, V is the crystal volume, B is the bulk modulus, and T is the temperature. For small enough temperatures, assuming α and B remain constant, the difference between CVand CPis negligibly small. Here we work with the heat capacity at constant volume, Equation 4.5.50 , which is also referred to as the lattice heat capacity. We will let the energy associated with the temperature be τ ≡ kBT, with kB= 1.38065 × 1023 J/K the Boltzmann constant, so that at a given temperature the total energy due to the Phonons in a crystal is a sum of the energies over all modes, indexed by wavevectorK →and polarization p; where, as mentioned before, there are three polarization modes, two transverse and one longitudinal. We also let UK,pbe the average energy associated with a Phonon of wavevectorK →and polarization mode p
eBook - ePub- G. Rickayzen(Author)
- 2013(Publication Date)
- Dover Publications(Publisher)
Chapter 7Electrons and Phonons§7.1 PhononsIn this chapter we consider a number of problems concerning Phonons and their interaction with electrons where Green’s functions have been used with success. Indeed, because the inertia of the ions is important, the interaction between the electrons which is mediated by the Phonons is not instantaneous but retarded. This makes Green’s functions a particularly useful vehicle for describing them.In §§7.1 -7.3 we consider the Phonons alone and look at their scattering of thermal neutrons, a prime method for investigating the spectrum of low lying excitations of a condensed system. The observations are expressed in terms of the structure factor S(k, ω), which in turn is related to density-density Green’s functions. We derive some general relations which the structure factor must satisfy.In succeeding sections we investigate the interaction between electrons and Phonons in a normal metal. We show that the interactions can be studied by a perturbation theory which is a simple generalization of that already given for a two-body interaction in Chapter 3 . We show also that, because of the smallness of the ratio of electron mass to ion mass, Phonon corrections to the electron-Phonon vertex can be ignored. This simplifies applications enormously. Finally we calculate the renormalization of the effective mass of the electrons due to the Phonons. In the next chapter we study the effect of the electron-Phonon interaction on superconductivity.FIG. 7.1. Position vectors used in the description of the motion of the lattice. is the equilibrium position vector of an ion of type α, Rjα is the corresponding vector for the ion in motion and ujα is its displacement from its equilibrium position.We assume that the reader has a working knowledge of lattice vibrations and Phonons in a crystalline lattice and refer him to standard texts such as Ashcroft and Mermin (1976) or Ziman (1960) for further information. We shall denote by Rjα the position of the atom of type α in the jth cell at any time and by the equilibrium position of the same atom. The equilibrium position of one of the atoms in each cell is taken as the reference atom and is denoted by
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