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Chapter 5: Phonons II – Thermal Properties. What is a Phonon ?. We’ve seen that the physics of lattice vibrations in a crystalline solid Reduces to a CLASSICAL normal mode problem . The goal of the entire discussion so far has been to find the normal mode vibrational frequencies of
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Chapter 5: Phonons II – Thermal Properties
What is a Phonon? • We’ve seen that the physicsof lattice vibrationsin a crystalline solid • Reduces to a CLASSICAL normal mode problem. • Thegoalof theentire discussionso far has been to • find the normal mode vibrational frequencies of • the 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.
These quantized normal modes of vibration are called • PHONONS • PHONONSare massless quantum mechanical “particles” which have no classical analogue. • They behave like particles in momentum space or k space.
These quantized normal modes of vibration are called • PHONONS • PHONONSare 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”
These quantized normal modes of vibration are called • PHONONS • PHONONSare 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” • Examples of otherQuasiparticles: • Photons: Quantized Normal Modes of electromagnetic waves.
These quantized normal modes of vibration are called • PHONONS • PHONONSare 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” • Examples of otherQuasiparticles: • 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
These quantized normal modes of vibration are called • PHONONS • PHONONSare 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” • Examples of otherQuasiparticles: • 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!!!
These quantized normal modes of vibration are called • PHONONS • PHONONSare 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” • Examples of otherQuasiparticles: • 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 • PHONONS • Quantized normal modes of lattice vibrations. The energies & momentaof 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 • 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!
Often, we consider Enas 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
A knowledge of the thermal energyis fundamental to obtaining anunderstanding 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 is this thermal energy calculated? 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.
How is this thermal energy calculated? 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. Most important, the thermal energy plays a fundamental role in determining the • Thermal (Thermodynamic) • Properties of a Solid • Knowledge of how the thermal energy changes wih temperature gives an understanding of heat energy necessary to raise the temperature of the material. An important, measureable property of a solid is its • Specific Heat or Heat Capacity
Lattice Vibrational Contribution to the Heat Capacity The Thermal Energyis the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Other contributions: Conduction Electronsin metals & semiconductors. Magnetic ordering in magnetic materials. The 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
Historical Background • In 1819, using room temperature data, • Dulong and Petit empirically found that • the molar heat capacityfor solids is • approximately Here, R is the gas constant. This relationship is now known as the Dulong-Petit “Law”
The Molar Heat Capacity • Assume that the heat supplied to a solid is • transformed into the vibrational kinetic & • potential energies of the lattice. • To explain the Dulong-Petit • “Law” theoretically, using • Classical, Maxwell- • Boltzmann Statistical • Mechanics, a knowledge of how the • heat is divided up among the degrees • of freedom of the solid is needed.
The Molar Heat Capacity The Molar Energy of a Solid • The Dulong-Petit “Law” can be explained using • Classical Maxwell-Boltzmann statistical physics. • Specifically, • The Equipartition Theorem can be used. • This theorem states that, for a system in thermal equilibrium • with a heat reservoir at temperature T, • The thermal average energy per • degree of freedom is (½)kT • If each atom has 6 degrees of • freedom: 3 translational & • 3 vibrational, then R = NA k
The Molar Heat Capacity Heat Capacity at Constant Volume By definition, the heat capacityof a substance at constant volumeis Classical physics therefore predicts: A value independent of temperature
The Molar Heat Capacity Experimentally, the Dulong-Petit Law, however, is found to be valid only at high temperatures.
Einstein Model of a Vibrating Solid In 1907, Einstein extended Planck’s ideas to matter: he proposed that the energy values of atoms are quantized and proposed the following simple model of a vibrating solid: Each atom is independent Each vibrates in 3-dimensions Each vibrational normal mode has energy:
In effect, Einstein modeled one mole of a solid as an assembly of 3NAdistinguishable oscillators. He used the Canonical Ensemble to calculate the average energy of an oscillator in this model.
To compute the average, note that it can be written as Z has the form: Here, b [1/(kT)] Z is called The Partition Function
With b [1/(kT)] and En = ne, , the partition function Z for the Einstein Modelis This follows from the geometric series result
Differentiating Z with respect to b gives: Multiplying by –1/Z gives: This is the Einstein Model Resultfor the average thermal energy of an oscillator. The total vibrational energy of the solid is just 3NA times this result.
The Heat Capacity in the Einstein Model is given by: Do the derivative & define TEe/k. TEis called The Einstein Temperature Finally, in the Einstein Model, CV has the form:
The Einstein Model of a Vibrating Solid Einstein, Annalen der Physik 22 (4), 180 (1907) CV for Diamond
Thermal Energy & Heat Capacity: The Einstein Model: Another Derivation • The following assumes that you know enough • statistical physics to have seen the Cannonical • Ensemble & the Boltzmann Distribution! • The Quantized Energy of a SingleOscillatorhas • the form: • If the oscillator interacts with a heat reservoir at • absolute temperature T, the probability Pnof it • being in quantum level n is proportional to the • BoltzmannFactor: • Pn
Quantized Energy of a Single Oscillator: • In the Cannonical Ensemble, a formal expression for the • average energy of the harmonic oscillator &therefore of • a lattice normal mode of angularfrequencyωat • temperature Tis given by: • The probability Pnof the oscillator being in quantum • level n has the form: • Pn [exp (-β)/Z] • where the partition function Z is given by:
Now, some straightforward math manipulation! Average Energy: Putting in the explicit form gives: According to the Binomial expansion, for x << 1 where
The equation for εcan be rewritten: Finally, the result is:
(1) • This is the Mean Phonon Energy.The first term in • (1) is called the Zero-Point Energy.Asmentioned • before, even at0ºKthe atoms vibratein thecrystal & • have a zero-pointenergy. This isthe minimumenergy • of thesystem. • The thermal average number of phonons n(ω) at • Temperature Tis given by The Bose-Einstein • Distribution, & the denominator of the second • term in (1) is often written:
(2) (1) • By using (2) in (1), (1) can be rewritten: • <> = ћω[n() + ½] • In this form, the mean energy <> looks analogous to a • quantum mechanical energy level for a simple harmonic • oscillator. That is, it looks similar to: • So the second term in the mean energy (1)is interpreted as • The number of phonons at temperature • T & frequency ω.
Temperature dependence of the 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).
Temperature dependence of the mean energy <> of a quantum harmonic oscillator. LowTemperature Limit: ħω > > kBT “Zero Point Energy” 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
Einstein Heat Capacity CV The heat capacity CVis found by differentiating the average phonon energy: Let
EinsteinHeat Capacity CV The specific heatCV in this approximationvanishes exponentially at lowT& tends to the classical value at highT.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. where Area=
The specific heatat constant volume Cvdepends • qualitatively ontemperature Tas shown in the • figurebelow. For hightemperatures,Cv(per mole) is • close to 3R(R= universal gasconstant. R 2 cal/K- mole). • So, at high temperaturesCv6 cal/K-mole The figure shows that Cv= 3R at high temperatures for all substances.This is the classicalDulong-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 SolidCv vs T for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) Points: Experiment Curve: Einstein Model Prediction
Einstein Model of Heat Capacity of Solids The Einstein Modelwas the first quantum theory of lattice vibrations in solids. He made the assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency,so that the whole solid had a heat capacity 3N times • In this model, the atoms are treated as independent oscillators, but the energies of the oscillators are the quantum mechanical energies. This assumes that the atoms are each isolated oscillators, which is not at all realistic. In reality, they are a huge number of coupled oscillators. 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 heatof6 cal/K per mole; they require 6 calories per mole to raise their temperature 1 K.This arrangement between observation and classical theory breaks down if the temperature is not high.Observations show thatat room temperatures and belowthe specific heat of crystalline solidsis not a universal constant. In each of these materials (Pb,Al, Si,and Diamond) specific heat approaches a constant value asymptotically at high T. But at low T, the specific heat decreases towards zero which is in a complete contradiction with the above classical result.
The Einstein model also gives correctly a specific heat tending to zero at absolute zero, but the temperature dependence near T=0 doesnot agree with experiment. Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be accounted using one dimensional model of monoatomic lattice
Density of States According to Quantum Mechanics if a particle is constrained; the energy of particle can only have special discrete energy values. it cannot increase infinitely from one value to another. it has to go up in steps. Thermal Energy & Heat Capacity Debye Model
These steps can be so small depending on the system that the energy can be considered as continuous. This is the case of classical mechanics. But on atomic scale the energy can only jump by a discrete amount from one value to another. Definite energy levels Steps get small Energy is continuous
In some cases, each particular energy level can be associated with more than one different state (or wavefunction ) This energy level is said to be degenerate. The density of states is the number of discrete states per unit energy interval, and so that the number of states between and will be .
There are two sets of waves for solution; Running waves Standing waves Running waves: These allowed k wavenumbers correspondto the running waves; all positive and negative values of k are allowed. By means of periodic boundary condition an integer Length of the 1D chain These allowed wavenumbers are uniformly distibuted in k at a density of between k and k+dk. running waves
Standing waves: In some cases it is more suitable to use standing waves,i.e. chain with fixed ends. Therefore we will have an integral number of half wavelengths in the chain; These are the allowed wavenumbers for standing waves; only positive values are allowed. for running waves for standing waves
These allowed k’s are uniformly distributed between k and k+dk at a density of DOS of standing wave DOS of running wave The density of standing wave states is twice that of the running waves. However in the case of standing waves only positive values are allowed Then the total number of states for both running and standing waves will be the same in a range dk of the magnitude k The standing waveshave the same dispersion relation as running waves, and for a chain containing N atoms there are exactly N distinct states with k values in the range 0 to .
The density of states per unit frequency range g(): The number of modes with frequencies & +d will be g()d. g() can be written in terms of S(k) and R(k). modes with frequency from to +d corresponds to modes with wavenumber from k to k+dk