Lattice Vibration

12 Key excerpts on "Lattice Vibration"

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  Modern Condensed Matter Physics

  • Steven M. Girvin, Kun Yang(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Dynamics of Lattice Vibrations Now that we have warmed up in the previous chapter with a study of the simple 1D harmonic oscilla-tor, let us attempt a more realistic model of the Lattice Vibrations in a 3D crystal. We begin here with a classical analysis and later, in Chapter 6 , we will study the underlying quanta of sound waves known as phonons. These Lattice Vibrations carry energy and contribute signicantly to the heat capacity and the thermal conductivity of many solids. Particularly in insulators, where the electronic contributions are nearly negligible, Lattice Vibrations dominate the thermodynamic properties. It may seem complicated to have to deal with the enormous number of motional degrees of freedom (3 N in a crystal of N atoms), but we will see in this chapter that the translation symmetry of the crystalline lattice allows vast simplification of the description of the dynamics of the particles. To get started, we will first ignore the discrete atomic structure of the crystal and treat it as an elastic continuum with complete translation symmetry. After studying elasticity in the continuum, we will move on to a more realistic approximation that takes into account the detailed microscopic structure of the atomic lattice. This is will lead to significant differences in the collective mode properties at short wavelengths, where the modes are sensitive to the underlying discrete atomic structure of the solid. 5.1 Elasticity and Sound Modes in Continuous Media Elastic distortions of the crystal displace the atoms from their nominal equilibrium positions. The displacement vector u j was defined in Eq. ( 4.9 ). The elastic continuum approximation is appropriate if u j = u ( R j ) varies slowly with R j , in which case we can ignore the discrete nature of R j , and treat it as a continuous variable r . In this continuum limit u j = u ( R j ) is replaced by a displacement field u ( r ) describing the elastic deformations.

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  Introduction to the Physics of the Earth's Interior

  • Jean-Paul Poirier(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Lattice Vibrations 3.1 Generalities In a crystal at temperatures above the absolute zero, atoms vibrate about their equilibrium positions. The crystal can therefore be considered as a collection of oscillators, whose global properties can be calculated. In particular, it will be interesting to determine: (i) The normal modes of vibration of the crystal. (ii) The dispersion relation, i.e. the relation f ( k ) between the fre-quency and the wave vector k . (iii) The vibrational energy. The vibrational approach is especially fruitful since it allows a synthesis between the thermal and elastic properties and gives a physical basis to thermoelastic coupling. This is due to the fact that the low-frequency, long-wavelength part of the vibrational spectrum corresponds to elastic waves, whereas the high-frequency part corresponds to thermal vibrations. In finite crystals, the Lattice Vibrations are quantized and behave as quasi-particles: the phonons . In the following section, we will give the elementary basis of the calcula-tions in the simple case of a monatomic lattice. This will be sufficient to introduce the concepts and formulas needed for our purpose. For a more complete and still elementary treatment, the reader is referred to the standard textbooks by Kittel (1967) and Ziman (1965). 3.2 Vibrations of a monatomic lattice 3.2.1 Dispersion curve of an in fi nite lattice Let us consider an infinite crystalline lattice formed of only one kind of atoms. Furthermore, let us assume that the lattice is a very simple one and 27 Figure 3.1 Parallel identical lattice planes of an infinite crystalline lattice. The displacement of plane n i with respect to plane n is u u . can be described as an infinite stacking of identical, equally spaced, lattice planes. Each atomic plane, of mass M , is labeled by an index n and is connected to all the other planes n p ( p positive or negative can become infinite) by a symmetrical pair-interaction potential V (Fig. 3.1).

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  Electronic and Optoelectronic Properties of Semiconductor Structures

  • Jasprit Singh(Author)
  • 2007(Publication Date)
  • Cambridge University Press

    (Publisher)

  Chapter 6 Lattice VibrationS: PHONON SCATTERING In a crystalline material atoms vibrate about the rigid lattice sites and one of the most important scattering mechanisms for mobile carriers in semiconductors is due to these vibrations. In our discussions for the bandstructure we assumed that the background potential is periodic, and does not have any time dependence. In actual materials the background ions forming the crystal are not fixed rigidly but vibrate. The vibration increases as the temperature is increased. To understand the properties of electrons in a vibrating structure we use an approach shown schematically in Fig. 6.1. Scattering will occur due to the potential disturbances by the Lattice Vibration. Before we can answer the question regarding how Lattice Vibrations cause scattering, we must understand some basic properties of these vibrations. Once we understand the nature of the Lattice Vibrations we can begin to examine how the potential fluctuations arising from these vibrations cause scattering. 6.1 Lattice VibrationS In Chapter 1 we have discussed how atoms are arranged in a crystalline material. The reason a particular crystal structure is chosen by a material has to do with the minimum energy of the system. As atoms are brought together to form a crystal, there is an attractive potential that tends to bring the atoms closer and a repulsive potential which tends to keep them apart. The attractive interaction is due to a variety of different causes including Van der Waals forces (resulting from the dipole moment created when 218 Chapter 6. Lattice Vibrations: Phonon Scattering Perfect crystal E vs k relation k -state maintained Lattice Vibrations − phonons E vs k picture still valid, but k -state is not maintained, i.e., scattering can occur Potential variations Figure 6.1 : The effect of imperfections caused by either Lattice Vibrations or other potential fluctuations lead to scattering of electrons.

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  Applications of Group Theory to Atoms, Molecules, and Solids

  • Thomas Wolfram, Şinasi Ellialtıoğlu(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  11 Applications of space-group theory: Lattice Vibrations The atoms of a crystal execute small, oscillatory motions about their equilib- rium positions called Lattice Vibrations. These vibrations are stimulated by thermal energy or by external agents such as electromagnetic and mechanical forces. As with molecular vibrations, the atomic motions of the lattice can be expressed as linear combinations of the normal modes of motion. Classically, the energy con- tained in a given normal mode is unrestricted. In quantum theory the energy in a normal mode is quantized in discrete units of ω. A quantum (ω) of energy in a normal mode of vibration is called a phonon. More loosely, the Lattice Vibration wave in a crystal is also called a phonon. Because of the translation symmetry of an (infinite) crystal the normal modes are characterized by a wavevector, k. In the case of Lattice Vibrations we associate a vector with the physical displacement of each atom from its equilibrium posi- tion. The Cartesian components of displacements transform in the same way as the p-orbitals and therefore the application of space-group theory to lattice vibra- tions is analogous to finding the tight-binding energy bands of a crystal with only p-orbitals on each atom. 1 The method of analysis of Lattice Vibrations is the same as that employed in Chapter 10 for tight-binding energy bands. Instead of energy bands we obtain “phonon branches”. There are, however, a couple of major differences between the treatment of Lattice Vibration and that of tight-binding energy bands. In the case of lattice vibra- tions there is a requirement that three of the branches have a zero eigenvalue at  (k = 0) in the Brillouin zone (for the translation modes). In addition, in forming eigenvectors, a vibrational symmetry function does not mix with the translation symmetry function even if these two functions belong to the same row of the same IR.

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  Atomic Mechanics of Solids

  • A.K. Macpherson(Author)
  • 2012(Publication Date)
  • North Holland

    (Publisher)

  CHAPTER 2 LATTICE DYNAMICS 2.1 Introduction T h e concept of treating a solid as a t o m s vibrating a b o u t their lattice sites u n d e r the influence of interparticle force laws has been investigated since the eighteenth century. An account of the development of the subject is given in [ 1 , 2 ] . Traditionally a d v a n t a g e has been t a k e n of the translational invariance of the lattice to simplify the problem. As a result, calculations generally have been restricted to the Lattice Vibrations of perfect crystals. In addition, as the solution involves the m a n y -b o d y problem, it has been necessary to m a k e further a p p r o x i m a t i o n s depending on the c o m p u t a t i o n a l facilities available at the time w h e n the research was u n d e r t a k e n . T h e development of large high-speed c o m p u t e r s has m a d e it possible to reduce the n u m b e r of approxi-m a t i o n s required. Lattice dynamics was p r o b a b l y the earliest a t t e m p t to calculate m a c r o -scopic properties from an a t o m i c m o d e l of a solid. T h e m e t h o d has been applied in m a n y areas a n d a comprehensive review of the whole field would be b e y o n d the scope of this b o o k . In acco rd an ce with the aim of this work, the survey will be restricted to those applications where material properties have been calculated. Solutions have only been obtained when the problems are simplified. In this introduction, the various a p p r o x i m a t i o n s are outlined to indicate the relationships which exist between the results obtained. T h e models generally have been of a classical n a t u r e with q u a n t u m -m e c h a n i c a l overtones. T h e models by Einstein a n d Debye, a l t h o u g h classical, d o use a n energy-packet theory. In the simplest form, a solid can be modelled by a one-dimensional cyclic chain of identical a t o m s interacting t h r o u g h nearest ne ighbour forces only.

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  Introduction to Thermodynamics and Kinetic Theory of Matter

  • Anatoly I. Burshtein(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  Having arisen in one molecule, a 3.3 PHONON GAS 165 vibration is transferred to its nearest neighbors, and successively to the whole crystal. The nature of this phenomenon is somewhat different from that of collective motions of the lattice discussed above. The transfer of vibration from one pendulum to another can serve as an illustration. If the pendulums are in resonance, a slight interaction is sufficient to transfer energy from one to another. It is exactly this resonance transfer that is responsible for the propa- gation of intramolecular excitations throughout the crystal. Along with vibrations, a molecule as an entire unit or a separate group of its constituent atoms can also execute rotational motions. However, as a rule, close packing of particles in a crystal structure prevents free rotation due to steric difficulties. In this case, limited rotation (libration) is observed, in which a molecule or a group of atoms exhibits partial rotation in both directions within the limits of a backlash. Such librations are typical for molecular crys- tals and polymers. 3.3 PHONON GAS Acoustic vibrations of the lattice are the only type of thermal motion common to all crystals. These are longitudinal and transverse waves of different fre- quency and intensity which propagate with the velocity of sound in all direc- tions, reflecting from the crystal boundaries. At low temperatures, when optical oscillations are frozen out, this is the only possible type of motion. An analogy with radiation suggests itself. The fact that the material struc- ture of a crystal and not a vacuum is the agent transferring acoustic vibrations affects only their spectrum, which is bounded above by the limiting frequency v,. Nevertheless, we can consider harmonic vibrations of a crystal in the same way as any other oscillations, be they vibrations of molecules or the vacuum. Oscillations of the crystal lattice, just as any other, should be quantized.

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  Introductory Solid State Physics

  • H.P. Myers(Author)
  • 1997(Publication Date)
  • CRC Press

    (Publisher)

  The quanta of Lattice Vibrations are known as phonons. A discrete vibrational state or mode has a well defined wave vector k and an application of de Broglie’s principle implies a linear momentum . However, the only true mechanical momentum that can arise is a rigid motion of the whole specimen. The 140 Introductory Solid State Physics vibrational modes transport energy, but they are completely described by the relative motion of the individual atoms, whose average displacement is zero; they cannot therefore contain a net momentum and the quantity is not to be identified with the conventional linear momentum. Figure 5.18 The quantum of Lattice Vibration is called the phonon. On the other hand, vibrational modes interact and this interaction demands that not only energy be conserved but also the quantity —the latter is therefore often called the crys- tal momentum or pseudomomentum. Two (or more) interacting modes, or phonons as we shall now call them, obey the following conservation laws (Fig. 5.19): (5.39) (The presence of is explained in the next section.) 5.10 Brillouin Zones So far, our discussion of phonons has been couched in terms of the Wigner-Seitz cell of the reciprocal lattice. When considering the propagation of waves in periodic structures, it is more customary to call this cell the first Brillouin zone, and it is the only zone of physical significance for Lattice Vibrations. In later work concerning electrons in periodic potentials, we shall consider higher Brillouin zones, and reference to the second zone is often made in the case of phonons. We must return to Fig. 5.6 and the periodic variation of ω with k. Suppose, contrary to what we said in Section 5.4, that we choose an arbitrary zero in the reciprocal lattice and let k vary monotonically, taking higher and higher values. Our significant interval is still 2π/a and k space is divided into a series of regions centred around our chosen origin. Brillouin suggested that this division be made as in Fig.

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  Thermophysical Properties of Materials

  • G. Grimvall(Author)
  • 1999(Publication Date)
  • North Holland

    (Publisher)

  CHAPTER 7

  THERMAL PROPERTIES OF HARMONIC Lattice VibrationS

  1 Introduction

  Lattice Vibrations give the key to many temperature dependent properties of solids. If we know the quantum mechanical energy eigenvalues of these vibrations, we can easily model thermodynamic quantities such as the Gibbs energy and the heat capacity. If we also know the eigenfunctions corresponding to the eigenvalues, we can calculate, e.g. the vibrational displacements of the atoms and connect that to properties such as the electrical resistivity. In applications to materials science the temperature is often so high that a classical description agrees very well with the more fundamental quantum mechanical approach. We therefore start with a brief comparison of these two descriptions.

  Consider a one-dimensional oscillator in classical mechanics. If the restoring force constant is k , and the mass is M , the frequency is

  (7.1)

  The potential energy E pot is

  (7.2)

  The equipartition theorem says that, in thermal equilibrium, the potential energy and and the kinetic energy are equal;

  (7.3)

  Here “ − ” denotes thermal average. It follows that the thermally averaged squared displacement is

  (7.4)

  The heat capacity C = ∂E/∂T is

  (7.5)

  The absolute value of the Helmholtz (free) energy F = E − TS is undefined since classical physics only deals with entropy differences and not their absolute values.

  These classical properties can be contrasted with the quantum mechanical description. There the Helmholtz energy can be obtained as

  (7.6)

  Z is the partition function, containing a sum over all quantum states i with energies

  Ei

  ;

  (7.7)

  In our case with a single oscillator, the quantum energies are with n = 0, 1, 2, 3, …, and Z takes a simple closed form;

  (7.8)

  which yields

  (7.9)

  From a knowledge of F = E − TS , ordinary thermodynamics gives, e.g. the energy E and the entropy S. The third law of thermodynamics implies that S = 0 at T = 0 K, and the entropy has a well-defined value. There is also an “equipartition” theorem for each quantum state of a harmonic oscillator, saying that the expectation values and of the potential and kinetic parts of the Hamiltonian H

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  Solid State Physics

  • Mircea S. Rogalski, Stuart B. Palmer(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 4 L attice D ynam ics 4.1. LINEAR Lattice VibrationS 4.2. PHONONS 4.3. LATTICE THERMAL PROPERTIES 4.1. L in ear L attice V ibrations Crystal lattice dynamics can be discussed in the frame of the adiabatic approximation, as introduced in Chapter 1, by solving Eq.(1.7) for the motion of the ions about their equilibrium position R0 which defines the lattice spatial distribution. It is found that either the three-dimensional or the quantum mechanical aspects are irrelevant for most of the interesting features of lattice dynamics, and therefore we first consider a vibrating lattice in one-dimension , described by the Hamiltonian (1.19), which reduces to: where N identical ions of mass M and position coordinates R have been assumed. The motion is described by the canonical equations which assume the form: H = ^ ' L Pl + E(R (4.1) p dRp _ dH dPp (4.2) dt dPp ’ dt 121 122 Solid State Physics The equilibrium position R p0 of the pth atom may be expressed in terms of the equilibrium spacing a of the atoms as R p0 = pa. The actual position Rp can then be written in terms of the sp displacement from equilibrium: R p ~ R pO + S p (4.3) Assuming that the interaction between atoms is conservative, the potential energy takes the form (1.8), so that the Hamiltonian (4.1) becomes: h = — y p2„ +-y y e i r -r j 2M 1 2 A ^ 1V1 p ^ p q*p (4.4) The potential energy can be expanded in terms of the small sp , s about the equilibrium separation R p0 -R q0, up to the second order: R p q ( R p R q ^ ~ R p q ( R p O R q o ) + S f y dRP J RpO iRqO ' K 2 X dR« J tpO’KqO 1 2 + — si dRl 1 2 ' £ V , d 2E pq ' d R P d R « J R lin,R „„ -E pq(Rp0-R (lo) +-3 d Sp dR„ +Sq dR„ (4.5) where the first order derivatives in the displacement are zero, because of the equilibrium conditions: KdRP j = 0 The second order derivatives of the potential energy: xr — K — pq qp d E pq( R p -R q ) dRpdRq (4.6)

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  Bonds and Bands in Semiconductors

  • J Phillips(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  4 {Lattice Vibrations The sound waves discussed in the preceding chapter represent special cases of harmonic waves in crystals, namely the limit in which the wave-length —> <*> or the wave number k —> 0. In this limit the frequency is linearly related to the velocity of sound, = ck. When is reduced to the point that it becomes comparable to the lattice constant a, dispersion sets in and usually ω < ck. For example, one might have for k along a symmetry axis, = uosin(cfc/coo). (4.1) It follows that knowledge of (&), for k~ir/a, can give us more in-formation on interatomic forces. In particular tha elastic constants and the velocity of sound involve certain weighted averages of short-range and long-range forces. As decreases towards a, the effect of the long-range forces tends to cancel out because of the oscillatory character of wave motion. Thus (&) at large k (small ) gives more information about short-range forces. 77 78 4* Lattice Vibrations BRILLOUIN ZONES Because of the periodicity of the crystal lattice, apart from a phase factor the amplitudes of one type of Lattice Vibration or one type of one-electron wave function are the same in each unit cell. The difference be-tween the phase factor for the unit cell centered at R m and the one centered at R n is exp[tk· (R m — R n ) ]. Just as one can construct unit cells in real space, so there are unit cells in k space. One may also introduce p = fik as the crystal momentum. If R is a primitive lattice vector in real space, then G is a primitive lattice vector in k space, sometimes called reciprocal space. For each R one defines G by G R = 2 . (4.2) The smallest unit cell in reciprocal space is called the first Brillouin zone. All k lying inside the first Brillouin zone are inequivalent in the sense that if ki lies inside the first zone, then subtracting any G from ki does not give a k 2 which is also inside the zone.

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  Understanding Solid State Physics

  • Sharon Ann Holgate(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Wright. In the same way that light can be regarded as either a wave or a stream of particles known as photons, Lattice Vibrations in solids can be thought of as either sound waves or packages of vibratory energy called phonons. Different crystal structures bring phonons to a focus in different directions, creating a range of patterns that can be predicted theoretically. Since phonons also become scattered when they reach defects, deviations from the predicted pat-terns highlight impurities and structural faults (see Plate 5.4). “The ability to directly visualize the propagation of the surface acoustic waves can reveal information about the shape of a defective region that may not be possible to obtain by other techniques,” explains Prof. Wright. Since the technique allows real-time imaging of thin films on opaque sub-strates (which form the basis of electronic devices), it should prove useful to the microelectronics industry not only for detecting defects, but also for evaluating SAW devices. There is currently a lot of interest in improving the performance of SAW filters—that use surface acoustic waves to remove unwanted frequencies from electrical signals—because of their widespread use in mobile phones and cable-based communications networks. In, Out, Shake It All About 147 and tends towards zero as the temperature is decreased. This partial failure is due to the fact that deriving the Dulong-Petit law mathematically requires the use of a clas-sical model that considers the specific heat to be due to the vibrations of individual atoms that are not coupled together, and which we now know is incorrect. Looking again at the low-temperature region in Figure 5.11, you can see the spe-cific heat tending to zero at absolute zero and increasing in value between this point and the flattened-off high-temperature region of the graph.

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  Quantum Theory of Materials

  • Efthimios Kaxiras, John D. Joannopoulos(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  This implies that the vibrational total energy per unit cell, due to the phonon excitations, must be given by: E vib s =  k,l  n (l) k,s + 1 2  ¯ hω (l) k (7.43) where n (l) k,s is the number of phonons of frequency ω (l) k that have been excited in a particular state (denoted by s) of the system. This expression is appropriate for quantum harmonic oscillators, with n (l) k,s allowed to take any non-negative integer value. One interesting aspect of this expression is that, even in the ground state of the system when none of the phonon modes is excited, that is, n (l) k0 = 0, there is a certain amount of energy in the system due to the so-called zero-point motion associated with quantum harmonic oscillators; this arises from the factors 1 2 which are added to the phonon occupation numbers in Eq. (7.43). We will see below that this has measurable consequences (see Chapter 8). In practice, if the atomic displacements are not too large, the harmonic approximation to phonon excitations is reasonable. For large displacements, anharmonic terms become 343 7.5 Application: Specific Heat of Crystals increasingly important, and a more elaborate description is necessary which takes into account phonon–phonon interactions arising from the anharmonic terms. Evidently, this places a limit on the number of phonons that can be excited before the harmonic approximation breaks down. 7.5 Application: Specific Heat of Crystals We can use the concept of phonons to determine the thermal properties of crystals, and in particular their specific heat. This is especially interesting at low temperatures, where the quantum nature of excitations becomes important, and gives behavior drastically different from the classical result. We discuss this topic next, beginning with a brief review of the result of the classical theory.

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