.. _lecture16:
Lecture 16: Phonons
+++++++++++++++++++
.. note::
*If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.* -- Nikola Tesla
.. warning::
This lecture corresponds to Chapter 24 of the textbook.
Summary
-------
.. attention::
In the previous lecture (:ref:`lecture15`) we focused on photons,
which is one type of particles we investigated using what we learned
regarding "a gas of particles". This has been the recurring subject of most
of our course so far. In this chapter, we focus on another type of
particles: phonons. Phonons are technically "quasi-particles" in the
sense that they do not correspond to particles as we usually
imagine. However, we are able to use the machinery developed so far
and, as result, find a number of temperature-dependent properties
of a "gas of phonons".
Most students are familiar with the concept of phonons: these are
elementary vibrations of a periodic system (*e.g.*, a crystal). The
great result is that phonons are actually obtained as elementary
vibrations of a system and those vibrations (or normal modes) are,
in fact, independent (they are, mathematically,
orthogonal to one another). Since they are independent, we can
safely apply everything we learned about independent particles! One
thing to remember: in a solid, the vibration of a single particle
will affect other particles as well. However, students will
remember that phonons are *collective* excitations, by construction.
The main topic of interest in this chapter is to find the heat
capacity of a solid. There are two main mechanisms for a solid to
store thermal energy: through electrons and through
phonons. However, phonons are the main vehicles though which a
solid store thermal energy, especially at high enough temperature.
Phonons are quantized excitations and they can be seen as the
combination of simple harmonic oscillators, which we have already
encountered in the course.
.. image:: _images/phonons_BLG.png
:width: 350
:align: left
In strong contrast to photons (for which the dispersion relation is
fairly simple: :math:`\omega= c k`), phonons feature a more complex
(solid-dependent) dispersion relation. The figure on the left
shows the phonon band-structure (*i.e.*, dispersion) in bilayer
graphene (calculated by Lamparski for his thesis in Meunier's
group). Clearly, there is no simple mathematical formula to relate
:math:`\omega` and :math:`k`. To address this, we will examine two
main approximations. Before we move on, a note must be made
regarding notations: conventions dictate the momentum for phonons
is designated with the letter :math:`q`, rather than with the
letter :math:`k` as we do with light or with electrons. Do not
worry about it too much!
1. **Einstein approximation**: In this model, each vibrational mode
of the solid is modeled as having the same frequency
:math:`\omega_{\rm E}` , chosen to be the average frequency of
the actual set of frequencies. This looks like a crude
approximation but the results obtained with it are actually
instructive!
Because the phonons are independent, the partition function of
the :math:`3N` vibrations is simply:
.. math::
Z=\prod_{k=1}^{3 N} Z_{k},
where
.. math::
Z_{k}=\sum_{n=0}^{\infty} e ^{-\left(n+\frac{1}{2}\right) \hbar \omega_{ E } \beta}=\frac{ e ^{-\frac{1}{2} \hbar \omega_{ E } \beta}}{1- e ^{-\hbar \omega_{ E } \beta}}
is the partition function for a system with a single
frequency. The sum comes from the fact that different modes can
be populated with a larger and larger number of quanta of
vibrations (just like we did for the simple harmonic
oscillator!).
We can thus calculate the internal energy (per mole) using equations we found in :ref:`lecture12`:
.. math::
U=-\left(\frac{\partial \ln Z}{\partial \beta}\right)=3 R \Theta_{ E }\left[\frac{1}{2}+\frac{1}{ e ^{\Theta_{ E } / T}-1}\right]
where we have introduced the :index:`Einstein temperature`
:math:`\Theta_{ E }=\hbar\omega_E/k_B`. This equation tends to
the equipartition result at high temperature (this is, frankly,
quite reassuring but not surprising).
The heat capacity is:
.. math::
C=\left(\frac{\partial U}{\partial T}\right)= 3 R \Theta_{ E } \frac{-1}{\left( e ^{\Theta_{ E } / T}-1\right)^{2}} e ^{\Theta_{ E } / T}\left[-\frac{\Theta_{ E }}{T^{2}}\right].
At the limit of very high temperature, we recover the Dulong-Petit
result. At low temperature, we find that the heat capacity falls
off very fast (in fact, too fast compared to experiment -- see below).
2. **Debye approximation**: In this model, we approximate the dispersion relation as
.. math::
\omega=v_{ s } q,
that is: all phonons travel with the same velocity
:math:`v_s`. This is the sound velocity of the mode (in fact,
this model is quite good for the so-called acoustic modes, those
modes that carry sound in a solid). The maximal frequency is
called the Debye frequency :math:`\omega_{ D
}=\left(\frac{6 N \pi^{2} v_{ s }^{3}}{V}\right)^{1 / 3}` and is
chosen so that the total number of modes integrates to
:math:`3N`. We define the :index:`Debye temperature` as
:math:`\Theta_{ D }=\frac{\hbar \omega_{ D }}{k_{ B }}` (this
can be seen as a simple change of units between temperature and
frequency using two constants: the Planck and Boltzmann
constants).
After some calculations, we find that the internal energy is:
.. math::
U=\frac{9}{8} N \hbar \omega_{ D }+\frac{9 N \hbar}{\omega_{ D }^{3}} \int_{0}^{\omega_{ D }} \frac{\omega^{3} d \omega}{ e ^{\hbar \omega \beta}-1},
and the heat capacity (per mole) is:
.. math::
C=\frac{9 R}{x_{ D }^{3}} \int_{0}^{x_{ D }} \frac{x^{4} e ^{x} d x}{\left( e ^{x}-1\right)^{2}} \quad {\rm with}\quad x=\hbar \beta \omega.
Again, the Debye model matches the Dulong-Petit (equipartition)
result at high temperature. This is not surprising since we made
sure to include the correct number of degrees of freedom (*i.e.*,
modes) in the treatment. Even though the frequency values are approximated, we
remember that for the equipartition theorem, only the *number* of
modes matters!
.. image:: _images/debye.png
:align: left
:width: 300
The behavior at low temperature is markedly different than
Einstein's result. In fact, the Debye result is much closer to
experiment: this makes sense since at low temperature, only the
acoustic modes are excited and those are well represented by
Debye's model:
.. math::
C_D({\rm low-T})=3 R \times \frac{4 \pi^{4}}{5}\left(\frac{T}{\Theta_{ D }}\right)^{3}.
We conclude that Einstein is a good approximation for optical modes
and Debye is better for acoustic modes (at low-T). The blue box
below provides an quick overview of what an optical mode is.
.. admonition:: Acoustic and :index:`Optical modes`
There are two main types of vibrations in a solid. (1) Those that
"start at zero": these are the acoustic modes and (2) those that do
not start at zero and have a flatter appearance. The acoustic modes
correspond to the transport of sound in the solid. The optical
modes got their names from the fact they can be excited with light
(or an electromagnetic-wave in general). The reason for this is
that an optical mode presents a variation in electrical dipole
moment, which, in turn, couples with the electric field of the
electromagnetic excitation.
Because a picture is often worth a thousand words, look at the small animation below from physics-animations.com to check the difference:
.. raw:: html
.. raw:: latex
Check out the animation \href{https://www.youtube.com/watch?v=M4WQs_U1nmU}{here}.
One thing to remember is that while all structures feature acoustic
modes, you need at least two atoms per unit cell to have an optical
mode. This is logical: you can't create a net dipole moment with a
single atom!
.. note:: **Comment on tricks of the trade**
Physics is sometimes scary because it seems to be leaning
heavily on a strong knowledge of mathematics. Since the
latter is already hard enough on its own right, people get
intimidated with pursuing physics! However, it does not have
to be so. Thermodynamics is a tough subject as it requires
knowledge in many fields such as quantum mechanics and
electro-magnetic theory!
.. image:: _images/exp.png
:align: left
:width: 300
One thing I find useful is to rely on a book of tricks and a future version of this site will have a special book of tricks section included somewhere. Among those tricks, it is good to rely on some useful identities:
.. math::
e^{x} \approx 1+x
.. math::
\sin(x) \approx x
.. math::
\sqrt (1+x) \approx 1+\frac{x}{2}
in all cases, for small enough :math:`x`. One trick is to
remember the plot of those functions where the
approximation, as shown in the figure. Another trick to make
sense of trigonometry and complex number is to remember the
unit circle. This is just a preview of a more substantial
page to be available soon!
Key Definitions
---------------
.. note::
:index:`Phonon`:
A phonon is a collective excitation in a periodic, elastic
arrangement of particles in condensed matter. These phonons
behave like particles and are often called
**quasi-particles**.
A full list of terms, including the ones provided here, can be
found in the :ref:`genindex`.
Learning Material
-----------------
Copy of Slides
~~~~~~~~~~~~~~
The slides for Lecture 16 are available in pdf format here: :download:`pdf <_pdfs/slides/lecture16.pdf>`
Screencast
~~~~~~~~~~
.. raw:: html
.. raw:: latex
This lecture is available as a YouTube recording at : \href{https://www.youtube.com/embed/-SIIpSL-bFU}{chapter 24}.
Test your knowledge
-------------------
1. What is a phonon?
A. It is a fundamental quantized elementary vibration of a material.
B. It is a quantum of phony physics.
C. It is a quantum of light.
D. It is a certain type of fermion
2. What is a dispersion relation?
A. It is a measure on how fast gas molecules travel during a Joule expansion experiment.
B. It is a relation providing a link between the partition function and the temperature.
C. It is a function that provides a relationship between a momentum and a frequency.
D. It is a way to quantize vibrations as a function of the volume of the material.
3. The Debye and Einstein models provide a description of the heat capacity of a collection of harmonic oscillators. What best describes those models?
A. Debye's model is superior to Einstein's at low temperature as it provides an accurate description of the speed of sound in the material.
B. Einstein's model is always superior to Debye's at high temperature.
C. Debye's model is built on the use of an average frequency to describe the system of phonons.
D. Einstein's model allows for an evaluation of the maximal frequency of the collection of phonons.
4. In the Einstein model for the harmonic vibrations in a crystal, we define the Einstein temperature as:
A. The temperature at which the effects of special relativity start to influence the vibrational frequencies.
B. The temperature at which the crystal freezes.
C. The temperature where the Brownian motion of the atomic species is most pronounced.
D. The average temperature of the phonons, treated so that the phonon density of states is approximated by a Dirac delta distribution.
.. hint::
Find the answer keys on this page: :ref:`answerkeys`. Don't cheat! Try solving the problems on your own first!
Homework Assignment
-------------------
Solve the following problems from the textbook: