513.803 Advanced Solid State Physics
9.5.2014

Problem 1
(a) What are polaritons?

(b) How would you calculate the dispersion relation ($\omega$ vs. $k$) for polaritons?

(c) Make a plot of the polariton dispersion relation. Indicate roughly the values of $\omega$ (in Hz) and $k$ in (1/m) that you expect the polaritons will have.

(d) How could you measure polaritons?

(e) Do you observe polaritons in metals or insulators? Why?

Problem 2
When using the Boltzmann equation to calculate currents, a function $f$ is defined, which gives the probability that state $\vec{k}$ is occupied at position $\vec{r}$ and time $t$.

(a) Give an expression for the electrical current density in terms of $f$ and the density of states $D(\vec{k})$.

(b) Write down the Boltzmann equation that must be solved to find $f$. (Hint: Take the total derivative of $f$.)

Solution

<hr />

(a) Give an expression for the electrical current density in terms of $f$ and the density of states $D(\vec{k})$.

\begin{align}
 \vec{j}_{elec} = - e \int \vec{v}_{\vec{k}} D(\vec{k}) f(\vec{k}) d^3 k
 \end{align}

(b) Write down the Boltzmann equation that must be solved to find $f$. (Hint: Take the total derivative of $f$.)

From 
 \begin{align}
 \frac{d}{dt} f(\vec{r},\vec{k},t)=0
 \end{align}
one gets
 \begin{align}
 \frac{d}{dt} f(\vec{r},\vec{k},t) = \frac{d \vec{r}}{d t} \cdot \bigtriangledown_{\vec{k}} f + \frac{d \vec{k}}{d t} \cdot \bigtriangledown_{\vec{r}}f + \frac{\partial f}{\partial t} = 0
 \end{align}
and with a crystal momentum of $\vec{F}_{ext} = \hbar \frac{d \vec{k}}{d t}$ and the addition of a collision term this becomes
 \begin{align}
 \frac{\partial f}{\partial t} = - \frac{1}{\hbar} \vec{F}_{ext} \cdot \bigtriangledown_{\vec{k}} f - \vec{v} \cdot \bigtriangledown_{\vec{r}}f + \frac{\partial f}{\partial t} \Bigr|_{\substack{collision}}
 \end{align}

(c) What is the meaning of the collision term in the Boltzmann equation?

The collision term represents the transition between different $k$-states.

Problem 3
Ferromagnetism can be described by Landau's theory of second order phase transitions.

(a) What is the order parameter for ferromagnetism? Sketch the temperature dependence of the order parameter.

(b) How could you observe this phase transition in an experiment?

(c) How could you calculate the magnetic susceptibility from Landau theory? Sketch the temperature dependence of the magnetic susceptibility.

Solution

<hr />

(a) What is the order parameter for ferromagnetism? Sketch the temperature dependence of the order parameter.

The order parameter is the magnetization.
<img src="magnetization.png" />

(b) How could you observe this phase transition in an experiment?

As the magnetization can be measured (albeit indirectly) by letting the sample vibrate in an induction coil and measuring the induced current, one could continuously measure the magnetization while continuously decreasing the temperature from a point above the critical temperature. When one reaches the critical temperature, one should measure a magnetization, which was not present before.

(c) How could you calculate the magnetic susceptibility from Landau theory? Sketch the temperature dependence of the magnetic susceptibility.

When using the expansion of $f$ (up to the fourth order, neglecting terms of odd order), to calculate the magnetization $m$ one adds a term for the energy of a magnetic field: $m B$. For small $m$ one can thus calculate the magnetization to be
 \begin{align}
 m = \frac{B}{2 \alpha_0 (T-T_c)}.
 \end{align}
The magnetic susceptibility can now be calculated as such:
 \begin{align}
 \chi = \frac{\partial m}{\partial B} = \frac{1}{2 \alpha_0 (T-T_c)}.
 \end{align}
The resulting susceptibility follows the Curie-Weiss-law.
<img src="curie.png" />

Problem 4
Describe a Mott transition and a Peierls transition. What properties would a material need to have to exhibit a Mott transition or a Peierls transition?

513.803 Advanced Solid State Physics9.5.2014

513.803 Advanced Solid State Physics
9.5.2014