Title: ETH_Sem5 / Solid_State_Intro / Woche_08

Q: Which assumptions are made for the conductivity of a Drude metal?
A: Electrons are classical charged particles bouncing of imobile ionic cores.
Electron movement is can be sparated in two components:
- fast component: chaotic Brownian-motion (always)
- slow drift motion: position averaged over a very large number of collisions (only with external field)

Q: Equation for the drift motion under an applied field. (Drude Model)
A: Equivalent to the movement of a particle in a viscous medium.
\(m\dot{v}_\text{d} + \frac{m{v_\text{d}}}{\tau} = -eE \;\;\;\;\;\;\;\;\;\;\;\; v_\text{d} = -\frac{e\tau}{m}E \)
\(v_\text{d}\) is the drift velocity averaged over many random collisions.
\(\tau\) characterizes the "viscosity" of the metal.
Pluging in number density \(n_e\) this results in a current density:
\(j = n_e\frac{e^2\tau}{m}E\)

Q: What can you conclude if a metal has a positive Hall coefficient
A: The charge carriers in that metal must be charged positively.
This is one of the key failings of the simplistic classical approach.

Q: Why are Bloch wavefunctions an unsuitable basis set for transport processes?
Which different set is chosen instead?
A: The position of the electron in a Bloch wavefunction is totally undefined. Transport processes are described by a particle at a certain position initially and somewhere else later.
Typically one uses wave packets instead.

Q: What is a wave packet?
A: A wave packet is a sum of Bloch waves in some quasimomentum interval  \(\hbar\Delta\vec{k}\) arround a given central value \(\hbar\vec{k}\).
\(\Phi_\vec{k}(\vec{r}, t) = \int_{|\vec{k} - \vec{k'}| \; < \; \Delta\vec{k}} \text{d}\vec{k'} \; u_\vec{k'}(\vec{r}) \exp( i\vec{k'}\vec{r} \; - \; i\varepsilon(\vec{k})t/\hbar )\)
In by assuming \(u_\vec{k'}(\vec{r})\) does not change with \(\vec{r}\) and a taylor approximation the wavepacket is given by:
\(\Phi_\vec{k}(\vec{r}, t) \approx u_\vec{k}(\vec{r}) \exp( i\vec{k}\vec{r} - i\varepsilon(\vec{k}) t / \hbar )\int_{|\delta\vec{k}| < \Delta\vec{k}} \text{d}\delta\vec{k} \; \exp(i\delta\vec{k} \;[\vec{r} - \nabla_\vec{k}\varepsilon t / \hbar ])\)

Q: Explain the terms in the following wave packet:
\(\Phi_\vec{k}(\vec{r}, t) \approx u_\vec{k}(\vec{r}) \exp( i\vec{k}\vec{r} - i\varepsilon(\vec{k}) t / \hbar )\int_{|\delta\vec{k}| < \Delta\vec{k}} \text{d}\delta\vec{k} \; \exp(i\delta\vec{k} \;[\vec{r} - \nabla_\vec{k}\varepsilon t / \hbar ])\)
What does the quantity \(\nabla_\vec{k}\varepsilon /\hbar\) represent?
A: In front of the integral is a Bloch wave uniformly spread out over all unit cells.
The integral gives the wave packet its localized nature ( envelope function ).
The gradient of the dispersion relation over hbar represents the actual movement of charge with a velocity \(\vec{v} = \nabla_\vec{k} \varepsilon / \hbar\)

Q: The effective mass tensor is defined as:
[...]
Unlike the mass of a free electron, the effective mass of a band electron is [...]
A: \(m^*_{\alpha\beta} = \hbar^2 \big( \frac{\partial^2 \varepsilon}{\partial k_\alpha \partial k_\beta} \big)^{-1}\); wave vector dependent.

Q: Explain how current flows in metal.
A: a) Electrons in the conduction band are at rest.
b) Field is applied and filled part of the band moves right.
c) Electron hits an impurity, other electrons or emits a phonon. Therefore the electron looses energy and scatters to a low-energy state (red arrow). This results in a net current to the right side.