Phonons: The Quantum Mechanics of Lattice Vibrations

The Following Material is Partially Borrowed from the course Physics 4309/5304 “Solid State Physics” Taught in the Fall of every odd numbered year!

find the normal mode vibrational In any Solid State Physics course, it is shown that the (classical) physics of lattice vibrations in a crystalline solid Reduces to a CLASSICAL Normal Mode Problem. A goal of much of the discussion in the vibrational properties chapter in solid state physics is to find the normal mode vibrational frequencies of the crystalline solid.

Note! The Debye Model of the Vibrational Heat Capacity is discussed in Chapter 10 Sections 1 & 2 of the book by Reif

it is necessary to QUANTIZE The CLASSICAL Normal Mode Problem. 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.

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

“Quasiparticles” Some Examples:

“Quasiparticles” Some Examples: Lattice Vibrational Waves. Phonons: Quantized Normal Modes of Lattice Vibrational Waves.

“Quasiparticles” Some Examples: Phonons: Quantized Normal Modes of Lattice Vibrational Waves. Photons: Quantized Normal Modes of Electromagnetic Waves.

“Quasiparticles” Some Examples: Phonons: Quantized Normal Modes of Lattice Vibrational Waves. Photons: Quantized Normal Modes of Electromagnetic Waves. Rotons: Quantized Normal Modes of Molecular Rotational Excitations.

“Quasiparticles” Some Examples: Phonons: Quantized Normal Modes of Lattice Vibrational Waves. Photons: Quantized Normal Modes of Electromagnetic Waves. Rotons: Quantized Normal Modes of Molecular Rotational Excitations. Magnons: Quantized Normal Modes of Magnetic Excitations in Solids.

“Quasiparticles” Some Examples: Phonons: Quantized Normal Modes of Lattice Vibrational Waves. Photons: Quantized Normal Modes of Electromagnetic Waves. Rotons: Quantized Normal Modes of Molecular Rotational Excitations. Magnons: Quantized Normal Modes of Magnetic Excitations in Solids. Excitons: Quantized Normal Modes of Electron-Hole Pairs.

“Quasiparticles” Some Examples: Phonons: Quantized Normal Modes of Lattice Vibrational Waves. Photons: Quantized Normal Modes of Electromagnetic Waves. Rotons: Quantized Normal Modes of Molecular Rotational Excitations. Magnons: Quantized Normal Modes of Magnetic Excitations in 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 electromagnetic waves. The energies & momenta of photons are quantized Photon Wavelength: λphoton ≈ 10-6 m (visible)

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 Photon Wavelength: λphoton ≈ 10-6 m (visible) Phonon Wavelength: λphonon ≈ a ≈ 10-10 m

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

The number of phonons is NOT conserved. Often, we consider En as being constructed by adding n excitation quanta of energy ħ to the ground state. Ground State (or “zero point”) Energy of the Oscillator. 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 ħ. Phonon Absorption or Emission ΔE = (n – n΄) n & n ΄ = integers 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 is discussed in detail in any solid state physics course, 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 18

Examples: Heat Capacity, Entropy, Helmholtz Free Energy, Knowledge of the thermal energy is fundamental to obtaining an understanding many properties of solids. Examples: Heat Capacity, Entropy, Helmholtz Free Energy, Equation of State, etc. A relevant question is how is this thermal energy calculated? For example, we might 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 heat energy necessary to raise the temperature of the material. An important, measureable property of a solid is it’s Specific Heat or Heat Capacity

Lattice Vibrational Contribution In non-magnetic insulators, it is to the Heat Capacity The Thermal Energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Some other contributions: Conduction Electrons in metals & semiconductors. Magnetic ordering in magnetic materials.

1. Evaluation of the contribution of a single vibrational mode. Calculation of the vibrational contribution to the thermal energy & heat capacity of a solid has 2 parts: 1. Evaluation of the contribution of a single vibrational mode. 2. Summation over the frequency distribution of the modes.

Vibrational Specific Heat of Solids cp Data at T = 298 K

Classical Theory of Heat Capacity of Solids We briefly discussed this model in the last class! Summary: Each atom is bound to its site by a harmonic force. When the solid is heated, atoms vibrate around their equilibrium sites like a coupled set of harmonic oscillators. By the Equipartition Theorem, the thermal average energy for a 1D oscillator is kT. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3kT. So, the energy per mole is E = 3RT R is the gas constant. The heat capacity per mole is given by Cv  (dE/dT)V . This clearly gives:

Thermal Energy & Heat Capacity: Einstein Model We already briefly discussed the Einstein Model! The following makes use of the Canonical Ensemble We’ve already seen that the Quantized Energy solution to the Schrodinger Equation for a single oscillator is: n = 0,1,2,3,.. If the oscillator interacts with a heat reservoir at absolute temperature T, the probability Pn of it being in level n is proportional to:

Quantized Energy of a Single Oscillator: The probability of the oscillator being in level n has the form: Pn  In the Canonical Ensemble, the average energy of the harmonic oscillator & therefore of a lattice normal mode of angular frequency ω at temperature T is:

Straightforward but tedious math manipulation! Thermal Average Energy: Putting in the explicit form gives: The denominator is the Partition Function Z.

The denominator is the Partition Function Z The denominator is the Partition Function Z. Evaluate it using the Binomial expansion for x << 1:

The equation for ε can be rewritten: The Final Result is:

The Zero Point Energy is the minimum energy of the system. (1) This is the Thermal Average Phonon Energy. The first term in the above equation is called “The Zero-Point Energy”. It’s physical interpretation is that, even at T = 0 K the atoms vibrate in the crystal & have a zero-point energy. The Zero Point Energy is the minimum energy of the system.

Thermal Average Phonon Energy: (1) The first term in (1) is the Zero Point Energy. The denominator of second term in (1) is often written: (2) (2) is interpreted as the thermal average number of phonons n(ω) at temperature T & frequency ω. In modern terminology, (2) is called The Bose-Einstein Distribution: or The Planck Distribution.

High Temperature Limit: Temperature dependence of mean energy of a quantum harmonic oscillator. Taylor’s series expansion of ex for x << 1 High Temperature Limit: ħω << kBT At high T, <> is independent of ω. This high T limit is equivalent to the classical limit, (the energy steps are small compared to the total energy). So, in this case, <> is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem).

Low Temperature Limit: The temperature dependence of the mean energy of a quantum harmonic oscillator. “Zero Point Energy” Low Temperature Limit: ħω >> kBT At low T, the exponential in the denominator of the 2nd term gets larger as T gets smaller. At small enough T, neglect 1 in the denominator. Then, the 2nd term is e-x, x = (ħω/(kBT). At very small T, e-x  0. So, in this case, <> is independent of T: <>  (½)ħω

Heat Capacity C (at constant volume) The heat capacity C (for one oscillator) is found by differentiating the thermal average vibrational energy Let

The Einstein Approximation where The specific heat in this form Vanishes exponentially at low T & tends to the classical value at high T. These features are common to all quantum systems: The energy tends to the zero-point-energy at low T & to the classical value at high T. The Einstein Approximation Starts with this form. Area =

The specific heat at constant volume Cv depends qualitatively on temperature T as shown in the figure below. For high temperatures, Cv (per mole) is close to 3R (R = universal gas constant. R  2 cal/K-mole). So, at high temperatures Cv  6 cal/K-mole The figure shows that Cv = 3R At high temperatures for all substances. This is called the “Dulong-Petit Law”. This states that specific heat of a given number of atoms of any solid is independent of temperature & is the same for all materials!

Einstein Model for Lattice Vibrations in a Solid Cv vs T for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) Points: Experiment Curve: Einstein Model Prediction

Einstein’s Model of Heat Capacity of Solids The Einstein Model was the first application of quantum theory to solids. He made the (absurd & unphysical) assumption the each of 3N vibrational modes of a solid of N atoms has the same frequency, so that the whole solid has a heat capacity 3N times the heat capacity of one mode.

Einstein’s Model The whole solid has a vibrational heat capacity equal to 3N times the heat capacity of one mode. In this model, the atoms are treated as independent oscillators, but their energies are quantum mechanical. It assumes that the atoms are each isolated oscillators, which is not at all realistic. In reality, they are a huge number of coupled oscillators. But, even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

At high temperatures, all crystalline solids have a vibrational specific heat of 3R = 6 cal/K per mole; they require 6 calories per mole to raise their temperature 1 K. This agreement between observation and classical theory breaks down if the temperature is not high. Observations show that at room temperatures and below the specific heat of crystalline solids is not a universal constant.

The Einstein model correctly gives a specific heat tending to zero at absolute zero, but the temperature dependence near T= 0 does not agree with experiment. However, a model which takes into account the actual distribution of vibration frequencies in a solid is needed in order to understand the observed temperature dependence of Cv at low temperatures: CV = AT3 A= constant

Debye Model Vibrational Heat Capacity: Brief Discussion In general, the thermal average vibrational energy of a solid has the form: The term in parenthesis is the mean thermal energy for one mode as before: so with this we have the density of states The function g() is called the density of modes. Formally, it is the number of modes between  & d Its form depends on the details of the (k).

In the Debye Model, it is assumed that every (k) is an acoustic mode with It can be shown that this results in a density of modes g() which has the form: so with this we have the density of states

Use this form & do math manipulation:

More math manipulation & assume low temperatures kBT << ħ Finally,

The Debye model for the heat capacity low temperature, kBT << ħ at low T, x is big. We can integrate to infinity, then we differentiate and get the desired T^3 law. for one mole

Lattice heat capacity due to Debye interpolation scheme Figure shows the heat capacity between the two limits of high and low T as predicted by the Debye interpolation formula. 1 Because it is exact in both high and low T limits the Debye formula gives quite a good representation of the heat capacity of most solids, even though the actual phonon-density of states curve may differ appreciably from the Debye assumption. Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1 Debye frequency and Debye temperature scale with the velocity of sound in the solid. So solids with low densities and large elastic moduli have high . Values of for various solids is given in table. Debye energy can be used to estimate the maximum phonon energy in a solid. Solid Ar Na Cs Fe Cu Pb C KCl 93 158 38 457 343 105 2230 235

The Debye Model Vibrating Solids

Limits of the Debye model phonon density of states from copper. Not very realistic but good at low T