Chapter 5 - Phonons II: Quantum Mechanics of Lattice Vibrations

NOTE!! If you took statistical physics from me, much of the discussion on the topics that follow will be essentially the same as you heard in that course when thermal properties of materials was being discussed!!

the normal mode frequencies of the What is a Phonon? We’ve seen that the physics of lattice vibrations in a crystalline solid Reduces to a CLASSICAL normal mode problem for a system of coupled oscillators. The goal of the entire discussion of lattice vibrations so far has been to find the normal mode frequencies of the vibrating crystalline solid.

It is necessary to QUANTIZE The goal of the entire discussion of lattice vibrations in Ch. 4 was to find the normal mode frequencies of the vibrating crystalline solid. In the harmonic approximation, this is achieved by first writing the solid’s vibrational energy as a system of coupled simple harmonic oscillators & then finding the classical normal mode frequencies & ion displacements for that system. Given the results of the classical normal mode calculation for the lattice vibrations, in order to treat some properties of the solid, It is necessary to QUANTIZE these normal modes.

Phonons “Quasiparticles” These quantized normal modes of vibration are called Phonons Phonons are massless quantum mechanical “particles” which have No Classical Analogue! They behave like particles in momentum space or k space. Phonons are only one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles”

Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles”

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves.

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations.

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs Polaritons: Quantized Normal Modes of electric polarization excitations in solids

Examples of other Quasiparticles: Phonons are one example of many in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs Polaritons: Quantized Normal Modes of electric polarization excitations in solids + Many Others!!!

Comparison of Phonons & Photons Quantized normal modes of lattice vibrations. The energies & momenta of phonons are quantized PHOTONS Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized Phonon Wavelength: λphonon ≈ a ≈ 10-10 m (visible) Photon Wavelength: λphoton ≈ 10-6 m >> a

Quantum Mechanical Simple Harmonic Oscillator The energy is quantized. Quantum mechanical results for a simple harmonic oscillator with classical frequency ω: The energy is quantized. n = 0,1,2,3,.. En E The energy levels are equally spaced!

(or “zero point”) Energy. The number of phonons is NOT conserved. Often, we consider En as being constructed by adding n excitation quanta of energy to the ground state. Oscillator Ground State (or “zero point”) Energy. E0 = If the system makes a transition from a lower energy level to a higher energy level, it is always true that the change in energy is an integer multiple of ΔE = (n – n΄) n & n΄ = integers Phonon Absorption or Emission In complicated processes, such as phonons interacting with electrons or photons, it is known that The number of phonons is NOT conserved. That is, phonons can be created & destroyed during such interactions.

Thermal Energy & Lattice Vibrations As was already discussed in detail, the atoms in a crystal vibrate about their equilibrium positions. This motion produces vibrational waves. The amplitude of this vibrational motion increases as the temperature increases. In a solid, the energy associated with these vibrations is called the Thermal Energy 16

A knowledge of the thermal energy is fundamental to obtaining an understanding many of the basic properties (thermodynamic properties & others!) of solids. Examples Heat Capacity, Entropy, Helmholtz Free Energy, Equation of State, etc.... A relevant question is how can this thermal energy be calculated? As only one example, suppose we would like to know how much thermal energy is available to scatter a conduction electron in a metal or a semiconductor. This is important because this scattering contributes to electrical resistance & other transport properties.

Thermal (Thermodynamic) Specific Heat or Heat Capacity Most importantly, the thermal energy plays a fundamental role in determining the Thermal (Thermodynamic) Properties of a Solid Knowledge of how the thermal energy changes with temperature gives an understanding of the heat energy necessary to raise the temperature of the material. An important, measureable property of a solid is its Specific Heat or Heat Capacity