1. Phonons: The Quantum Mechanics of Lattice Vibrations

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

3. 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.

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

5. 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.

6. “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”

7. “Quasiparticles” Some Examples:

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

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

10. “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.

11. “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.

12. “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.

13. “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!!!

14. Comparison of Phonons & Photons
Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized
Photon Wavelength:
λphoton ≈ 10-6 m (visible)

15. 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 ≈ m

16. 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!

17. 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.

18. 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

19. 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.

20. 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

21. 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.

22. 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.

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

24. 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:

25. 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:

26. 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:

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

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

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

30. 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.

31. 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.

32. 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).

33. 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

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

35. 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 =

36. 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!

37. 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

38. 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.

39. 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.

40. 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.

41. 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

42. 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).

43. 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

44. Use this form & do math manipulation:

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

46. 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

47. 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

48. The Debye Model Vibrating Solids

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