Final discussion post.

Here is a place to post questions, comments, etc regarding things we have covered this quarter and what you can focus on for our upcoming exam.

Topics:
dispersion relations for Bloch states
effective mass
Fermi energy
Fermi boundaries in 1 and 2D,
the nature of Bloch states
the nature and importance of sp2 bonding (involving 2s, 2px and 2py atomic wave-functions)
conduction and valence bands of graphene (from 2pz atomic wave function)
dispersion relations for lattice vibrations
also, calculating actual numbers.

I'll post more notes here on each of these topics over the next few days. Maybe there are also other topics we could consider? Please feel free to comment on these topics here. Which ones do you like? not like?

Let's start with dispersion relations for Bloch states:
There are two primary aspects to Bloch states. The most important is the energy of the state as a function of k. Energy vs k relationships form the cornerstone of solid state physics. Bands of states, in crystals, tend to form from particular atomic functions. The energy of these states as a function of k is called a dispersion relation. From that dispersion relation one can obtain things like electron velocity and, near the bottom or top of a band, effective mass.

You will want to:
understand the concept of k space.
understand the concept of filling states up to a particular energy and having empty states above that energy.
understand the difference between states and electrons.

You will want to know how to:
calculate effective mass,
find Fermi boundaries in 2D, (assuming Fermi energy is given)
estimate and sketch Fermi boundaries in 2D for isotropic and anisotropic dispersion relationships,
calculate electron state velocities at particular k values.

The second aspect of Bloch states is the nature of the wave-function itself. Wave functions have real and imaginary parts. You will want to know how to plot a Bloch wave-function and maybe have a sense of how nodes are related to energy.

Fermi boundaries in 1D and 2D. Well, in 1D the 1st Brillouin zone (BZ) is just a line from -pi/a to +pi/a, and, for a given value of the Fermi energy, the Fermi boundary consists of two dots along that line. Where they are depends on the Fermi energy given. In 2 dimensions (2D) the 1st BZ for a simple square lattice structure is a square with sides of length 2pi/a, and for a rectangular lattice with different lattice constants for the x and y directions, the first BZ is a rectangle.  Within the 1st BZ a typical Fermi boundary will be a line, or sometimes several lines, that separates regions of occupied states from regions of unoccupied states. You should have a sense and intuition for how to sketch approximate Fermi boundaries for square or rectangular lattices in 2D, as well as an sense for and understanding of the relationship between Fermi energy and Fermi boundaries.

sp2 bonding. You will want to know how to construct sp2 states and how to calculate expectation values with them. Not every detail of every integral is critical, but knowing which integrals are zero is valuable, and knowing the value, in very simple terms, e.g., "3a", of non-zero integrals is worthwhile as well. Understanding and being able to discuss the nature, importance, underlying physics etc. of sp2 states is important. How are they different from the \(\Psi_{2,l,m}\) states?

Conduction and valence bands of graphene. You will want to be familiar with the 1st BZ for graphene, though you don't need to be able to derive it. Also, know the special high-symmetry points, where they are in k-space, and that dispersion is essentially the same around each of the six K points and M points. Familiarity with the matrix and dispersion relations is also important. You will want to know how to plot dispersion relations, how to calculate electron velocity at different points, and how and when you can use simple one-variable derivatives to calculate velocity. It probably doesn't hurt to understand something about the two-component eigenvectors as well.

Lattice vibrations: We model the lattice as a network of masses connected by springs. Our focus is on a simple 1D lattice with atoms separated by a distance a.  In this case one gets a single phonon band (or branch) with a dispersion relation for \(\omega\) vs k that goes from \(k = -\pi/a\) to \(k = +\pi/a\). Salient features include the zone boundary lattice vibration (phonon) frequency and the sound velocity associated with the linear part of the dispersion relation. What is the nature of the eigenvector at the zone boundary? Given a spring constant and mass, can you calculate the speed of sound and the zone boundary phonon frequency?

calculating actual numbers. In my experience students trying to calculate numbers in mks units on tests hardly ever get correct results. You have values for \( \hbar c\), \( \hbar^2/m\), \( e^2/(4 \pi \epsilon_o)\), etc. in eV nm units. It is probably a good idea to practice using them to get correct numbers.  (I think for a proton mc^2 is about 10^9 eV. Does that seem about right? (comment below).

Brillouin Zone for Graphene

So the Brillouin Zone can be described as the set of points that are closer to the origin than any other lattice point in reciprocal space. So one way to get the shape of the Brillouin Zone would be to first find the reciprocal lattice. Using the Bravais Lattice generating vectors we found before: \(  \vec{b_{1}} = ( \frac{3}{2}b , \frac{ \sqrt{3}}{2} b) \)  and  \( \vec{b_{2}} = ( \frac{3}{2}b , -\frac{ \sqrt{3}}{2} b) \) ,  we can find the reciprocal vectors \( \vec{a_{1}} \)  and \( \vec{a_{2}}\) by the relation \( \vec{b_{1}} \cdot \vec{a_{1}} = \vec{b_{2}} \cdot \vec{a_{2}} = 2\pi  \) . More simply we can find the reciprocal vectors by:
 \( \vec{a_1} = 2 \pi \frac{R \vec{b_2}}{\vec{b_1} \cdot R \vec{b_2}} \) and \( \vec{a_2} = 2 \pi \frac{R \vec{b_1}}{\vec{b_2} \cdot R \vec{b_1}} \) where \( R\) is the  \( 90^{\circ} \) rotation matrix \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)

Following this through yields \( \vec{a_1} = ( \frac{ 2 \pi} { 3 b} , \frac{ 2 \pi \sqrt{3}}{3b} ) \) and \( \vec{a_2} = ( \frac{ 2 \pi} { 3 b} , -\frac{ 2 \pi \sqrt{3}}{3b} ) \).

So our reciprocal lattice looks like:

To get the Brillouin Zone from the earlier definition, we'll draw in the perpendicular bisectors of the reciprocal vectors, along with the vertical line through the midpoint between the nearests neighbor on the  \( k_x \)-axis to get something like:

Through some geometry you can find this boundary intersects the \( k_y \) -axis at \( k_y = \frac{4 \pi }{3 \sqrt{3}b} \) and the \( k_x \) - axis at \( k_x = \frac{2 \pi}{3b} \).

HW 8. Lattice vibrations

1. For a linear chain of atoms and gamma = 2 meV/nm^2, 
oops, let's change that to: (see comments)
 \(\gamma = 40 \: eV/nm^2\),

Also, let's use the mass of a proton, (the mc^2 for that) and a =0.2 nm.  How does that sound?
a) Plot the lattice vibration frequency as a function of k.
b) what is the frequency of the zone boundary, (\(k = \pi/a\)), lattice vibration mode?
c) what is the characteristic velocity associated with sound propagation in this crystal*?
d) do you need another parameter to answer some of these? what is a reasonable value for it?
e) extra credit. what is the energy, in meV, of a zone boundary acoustic phonon in this crystal?

* extra credit. How does this speed compare with the speed associated with electron dispersion near the K point in graphene?

Lattice vibrations.

I am thinking that we could study lattice vibrations this week. These are the vibrations of the nuclei. These vibrations involve small deviations away from the equilibrium positions of the nuclei in the lattice.

When sound travels through a solid, it travels via lattice vibrations. Lattice vibrations involve small deviations from equilibrium which occur in a collective, wave-like manner. Some aspects of lattice vibration mathematics are similar to the mathematical formalism of Bloch states, but they also differ in critical and interesting ways!...

Here is a warm-up problem, a pre-lecture problem, to do if you have time:
1. Consider two equal masses, of mass m, connected by a spring of spring constant k.
mass==k==mass      <--(this is a "picture", not an equation.)
a) Find the normal modes of this two mass system.
b) Express the equations of motion for this system using a 2x2 matrix.

Here is another warm-up problem. Basically the same thing but with 3 masses.
2. Consider three equal masses in a line, connected by a 2 springs of spring constant k.
mass==k==mass==k==mass       <--(this is a "picture", not an equation.)
a) Express the equations of motion for this system using a 3x3 matrix.
b) Find the normal modes of this three mass system.

2x2 matrices (hermitian)

(from Anikeya)
---
Also, here is a video about 2x2 matrices:

Hydrogen atom energies.

--This post is related to things we will cover in the near future, not the current HW..
To understand the origin and nature of chemical bonding, it is important to first understand that role of kinetic and potential energy in the formation of an atom.  The reason is: in a molecule or  covalently bonded crystal such as graphene, 2 or more atoms have decided it is preferable to be together than to be apart. Why is that? What is their motivation? The answer lies in energy. The energy is lower. Before we attempt to understand that, we need to understand the energy of one atom, for example, a hydrogen atom.

Video related to March 1 class and HW 7.

Here is a video on graphene dispersion:

HW 7b

5. Calculate the characteristic speed associated with the massless Dirac fermions near the K point in graphene. That is, use the relationship, \(v_k = (1/\hbar)\) times the derivative of E vs k.
(hint: You can make this easier by doing it along the ky axis.)
a) What is the speed in the K to Gamma direction?
b) What is the speed in the K to M direction?
c) How does this speed compare to the speed of light?

Here is a possibly interesting article:
https://www.nature.com/articles/nature04233