513.803 Advanced Solid State Physics
2.7.2014

Problem 1
All observable properties of a metal can be calculated from the $E$ vs. $k$ dispersion relation.

(a) How could the specific heat be calculated from the dispersion relation?

(b) How could the electrical conductivity be calculated from the dispersion relation?

(b) How could the dielectric function be calculated from the dispersion relation?

Solution

<hr />

(a) How could the specific heat be calculated from the dispersion relation?

From the dispersion relation you could calculate the density of states $D(E)$ by
counting how many states there are for every energy $E$.
You know that electrons are fermions. Moreover,
for metals you can use the model of free non-interacting electrons
that's why you can use the fermi-function $f(E)$.
Therefore, you only need to calculate the following two formulas to
get the specific heat:

\begin{align}
 u = \int_{-\infty}^{\infty} \! E \, D(E) \, f(E) \, \mathrm{d}E
\end{align}
\begin{align}
 c_v = \left( \frac{\partial u} {\partial T} \right)_V
\end{align}

(b) How could the electrical conductivity be calculated from the dispersion
 relation?

If you know the dispersion relation, you can use the Boltzmann equation to calculate the electrical conductivity.

\begin{equation}
\vec{j}_{\text{elec}}=  -e\int \vec{v}_{\vec{k}}D(\vec{k})f(\vec{k})d^3k =   -\frac{e}{4\pi^3\hbar}\int \nabla_{\vec{k}}E(\vec{k})\,f(\vec{k})d^3k.
\end{equation}

If the high frequency response of the electrical conductivity is desired, it can be calucated if the dielectric function is known (see c). You can calculate the
 susceptibility from the dielectric function by subtracting 1. Then you can use the following formula to
 get the electrical conductivity:

\begin{align}
 \chi_E(\omega) = \epsilon_r(\omega) -1
 \end{align}
 \begin{align}
 \sigma = \chi_E(\omega) \cdot i \omega \epsilon_0
 \end{align}

(c) How could the dielectric function be calculated from the dispersion
 relation?

From the band structure (dispersion relation) you can calculate the optical absorption
 spectrum. The absorption rate is calcuated using Fermi's golden rule for a transition from an occupied electron state into an empty state. The conservation of energy requires that the two electrons states have the same $\vec{k}$.
 The absorption coefficient is related to the extinction coefficient,
 which again is the imaginary part of the square root of the dielectric function.
 With the Kramer-Kronig relations you can calculate the real part of the square
 root of the dielectric function. To get the dielectric function you simply
 square the combined result.

\begin{align}
    \alpha = \frac{2 \omega K} {c}
  \end{align}

\begin{align}
    \sqrt{\epsilon_r} = n + iK
  \end{align}

$\alpha$ ... absorption coefficent, $\omega$ ... angular frequency, c ...
 speed of light in the medium, K ... extinction coefficent,
 $\epsilon_r$ ... dielectric function, n ... index of refraction

Problem 2
Describe the quantum Hall effect.

Solution

Problem 3
Describe electron screening. How can it be responsible for a metal-insulator transition?

Problem 4
An Electron Energy Loss Spectroscopy (EELS) experiment is performed on a metal and there are peaks in the energy loss at 10 eV, 20 eV, 30 eV, and 40 eV. You suspect that these are due to certain quasiparticles in the metal. You perform an ellipsometry measurement to test this idea. What can the ellipsometry measurement tell you about these quasiparticles?

Solution

<hr />

With an EELS you can measure quasiparticles, because the electrons loose energy by creating these quasiparticles. It's very easy to measure plasmons (quasiparticles of the electron density waves) since electrons interact a lot with them. The measured peaks suggest a quasiparticle-energy of 10 eV (difference of two peaks; first peak: generation of one quasipartile, second peak: generation of two quasiparticles etc.), which would fit to the idea of plasmons.
Ellipsometry is used to measure the polarizability of the material. The measurement can be used to calculate the dielectric function. If you know the dielectric function, you know the plasma frequency of the material (the frequency, at which the real part of the dielectric function crosses the frequency-axis). Since plasmons oscillate with the plasma frequency, you can check, if the measured frequency of the EELS measurement is the same as the plasma frequency you get with the ellipsometry measurement. If this is true, you have measured plasmons.

513.803 Advanced Solid State Physics2.7.2014

513.803 Advanced Solid State Physics
2.7.2014