Simulation Testing

Learning Physics with Interactive Simulations

This particular page is maintained by Thushari Jayasekera, as a part of her ongoing educational project, "Physics and FUN with Clicks",

Go to the following Link:

[1] to open the interactive simulation to look at atomic displacement for different values of $k$.

This particular page is aimed at learning Solid State Physics and Materials Science with Interactive Simulations. Pl. Note that this is only a sample lecture (first version) from Undergraduate Solid State Physics Class. All notes will be uploaded as time permits. This particular lecture contains one interactive simulations to understand the properties of phonons.....)

       - Please do not only play with the simulation. Work through the Math and use the simulation only to visualize the concept

       - As of now, when you click on the simulation title, it will be re-directed to a different page. This will be fixed eventually. 

Solid State Physics Lecture 10 - Understanding Phonons

Phonons are quantized Lattice Vibrations. Understanding the properties of phonons is important, as phonons govern the thermal properties of semiconductor materials. In this lecture, we will discuss the concept of phonons via atomic vibrations of the simplest possible solid: One-Dimensional (1D) Monoatomic Chain, using the first order model; Hooke's Law.

One-Dimensional Atomic Chain

The system is in equilibrium for a given atomic positions. That we have a set of equilibrium coordinates for the atomic configuration. The total energy of system is a function of equilibrium coordinates, $u_{i}$.

$\epsilon =\epsilon (u_{1},u_{2},u_{3},........,u_{N})$

The total energy of the system is a function of atomic coordinates. The equilibrium energy is the minimum energy configuration. We can expand the total energy in terms of the atomic displacements.

$\epsilon =\epsilon _{0}+\sum _{\alpha ,l}{\frac {\partial \epsilon }{\partial u_{\alpha }^{l}}}du_{\alpha }^{l}+{\frac {1}{2}}\sum _{\alpha ,\beta ,l,l'}{\frac {\partial ^{2}\epsilon }{\partial u_{\alpha }^{l}\partial u_{\beta }^{l'}}}du_{\alpha }^{l}du_{\beta }^{l'}+\theta u^{3}+\cdots$

We can choose $\epsilon _{0}$ to be zero. The term ${\frac {\partial \epsilon }{\partial u_{\alpha }^{l}}}=0$, and taking higher order terms to be zero.

$\epsilon ={\frac {1}{2}}\sum _{\alpha ,\beta ,l,l'}{\frac {\partial ^{2}\epsilon }{\partial u_{\alpha }^{l}\partial u_{\beta }^{l'}}}du_{\alpha }^{l}du_{\beta }^{l'}$ is therefore the total energy in the Harmonic Approximation.

So the elastic energy stored can be written as $\epsilon ={\frac {1}{2}}kx^{2}$ in the Harmonic Approximation (where the third order and harmonic terms are neglected).

In the Harmonic Approximation, the system can be thought of as the atomic spheres connected by springs.

The spring constant is thus related to the second derivative of the corresponding Elastic Energy.

$k={\frac {1}{2}}\sum _{\alpha ,\beta ,l,l'}{\frac {\partial ^{2}\epsilon }{\partial u_{\alpha }^{l}\partial u_{\beta }^{l'}}}$

Notice that the harmonic spring constant has 2 indices for the atoms ($l,l'$) , and two other indices for direction $(\alpha ,\beta )$

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alt EquilibriumPotential

The above figure shows the interatomic potential as a function of atomic separation. Let's look at the Lattice Vibrations of a one-dimensional atomic chain.

When an atom is shifted from its equilibrium position it tends to restore. That makes the system vibrates. Quantized such vibrations are called Phonons. Here we are going to understand the normal modes oflattice vibrations in the first order approximation, where, the interatomic potential is considered in the Harmonic approximation. (explained above). We are going to first study this problem classically, where the inter atomic force in explained by Hooke's Law (Simple Spring Mass Model)

One-Dimensional Lattice Vibrations

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1D Chain

Let's imagine for the moment that the atoms are vibrating in the direction of the wire. Let's consider the force on the $j^{th}$ atom.

$F_{j}=C\left(U_{j+1}-u_{j}\right)-C\left(u_{j}-u_{j-1}\right)$

$m{\frac {d^{2}u_{j}}{dt^{2}}}=c\left[u_{j+1}+u_{j-1}-2u_{j}\right]$

These equations are written for the $j^{th}$ atom. The dynamics of each atoms can be written by exactly similar equations.

Now we seek the solutions of the form:

$u(x,t)~u_{0}e^{i(kx-\omega t}$

$u(ja,t)~u_{0}e^{i(kja-\omega t)}$

Let's substitute this form of the solution in the above dynamical equation.

$m{\frac {d^{2}u_{j}}{dt^{2}}}=c\left[u_{j+1}+u_{j-1}-2u_{j}\right]$

$M(-i\omega )(-i\omega )u_{0}e^{i(kja-\omega t)}=C\left[e^{ika}+e^{-ika}-2\right]e^{i(kja-\omega t)}$

$-M\omega ^{2}=C\left[e^{ika}+e^{-ika}-2\right]$

$M\omega ^{2}=C\left[2-2Coska\right]$

$\omega ^{2}={\frac {2C}{M}}\left(1-Coska\right)$

This equation tells you the allowable lattice vibration frequencies for a give propagation wave vector. This is called the phonon-dispersion relation. (Yet, we have not talked about the quantum mechanical nature of the lattice vibrations, however used the name phonons. The quantum mechanical nature of lattice vibrations will be discussed later)

Let's plot the phonon dispersion for a 1D atomic Chain.

$\omega ^{2}={\frac {2C}{M}}\left(1-Coska\right)$

At $k=\pm {\frac {\pi }{a}}$, we get $\omega ^{2}={\frac {4C}{M}}$, and the slope of $\omega -k$ curve is zero.

We can also re-wrtie the above as

$\omega ^{2}={\frac {4C}{M}}Sin^{2}{\frac {ka}{2}}$

We can plot this as follows:

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1D Chain

What range of $k$ is physically important?

Let's look at the solution $u=u_{0}e^{i(kja-\omega t)}$

What is the relative displacement between two neighboring atoms:

${\frac {u_{j+1}}{u_{j}}}=e^{ika}$

That says, if we cover $ka$ ranges from $-\pi$ to $\pi$ , it covers all possible relative displacements.

The range of independent $k$ values are therefore defined by:

$-\pi <ka<\pi$

$-{\frac {\pi }{a}}<k<{\frac {\pi }{a}}$

The area enclosed by these limits are called the First Brillouin Zone. (as Marked in the above Phonon Dispersion Diagram)

We may treat a $k$ value out side the first Brillouin Zone by substracting an integer multiply of reciprocal wave vector, in this 1D case equals to ${\frac {2\pi }{a}}$.

It is therefore enough to plot the phonon dispersion in the first Brillouin Zone.

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alt EquilibriumPotential

Understanding Atomic Displacements for DIfferent values of </math> \omega - k </math> Combination using Interactive Simulations

Go to the following Link: [2] to open the interactive simulation to look at atomic displacement for different values of $k$.

In the simulation, you will be able to select a wave vector $k$ in the first or second Brillouin Zone.

Using the simulation, Answer the following questions.

(a) Select a very small wave vector, (that is a long wave length) and observe the atomic displacements. What can you tell about the wave propagation in the long wave length limit? Use the dispersion relation and evaluate the dispersion relation in the long wave length limit. How does it compare to the long wave length approximation we discussed in the last lecture.?

(b) Select the wave vector equal to the edge of the Brillouin Zone, Explain the atomic displacement. What can you say about them. Use the expression for the atomic displacement above and explain the atomic displacement you observe from the simulation and what you obtain from the mathematical expressions.

(c) When you pick a higher order BZ, the simulation will show the atomic displacements corresponding to the wave vector in the first BZ, and the related wave vector in the second BZ (i.e. $k+G$. For two wave vectors $k$ and $k+G$, how do the atomic displacements relate to each other.

(d) In this calculation we have done for the 1D monoatomic chain, for a given value of $k$, there is only one possible $\omega$. Explain the reasoning? Name two scenarios(with clear reasoning) where you can expect more than one possible frequencies?