Title: ETH_Sem5 / Solid_State_Intro / Woche_09

Q: How does Band Theory explain the negative Hall effect?
A: Due to the effective mass-tensor being of the form \(\dede{^2 \varepsilon}{ k^2} \) we can get negative effective masses whenever the dispersion curve is concave.

Q: How do we combine relaxation times for more complex scattering phenomena?
A: We can add the inverses:
\(1/\tau = 1/\tau_a + 1/\tau_b + \ldots\)
This means also that effects on resisitivity just add.
\(\rho = \rho_a + \rho_b + \ldots\)

Q: How does the lattice resistivity behave as \(T \gg \theta_D\)? Why?
A: \(\rho_{lattice} \propto \tau_{lattice}^{-1} \propto T\)
Bose-Einstein scales approx \(n(\hbar\omega) \approx T/(\hbar\omega)\)
and because scattering is proportional to number of phonons.

Q: How does lattice scattering behave as \(T \ll \theta_D\)? Why?
A: \(\rho_{lattice} \propto \tau_{lattice}^{-1} \propto T^5\)
Number of phonons scales with \(T^3\) but it takes \(N^2\) collisions to actually transfer enough momentum. Thus we get \(T^5\)

Q: How does defect resistivity scale with temperature? Why?
A: \(\rho_{defects} \approx \frac{3n_dQ}{e^2 v_F \nu(\varepsilon_F)} = const\)
We modeled defect scattering as colision with rigid spheres. This is temperature independant.

Q: How does electron-electron scattering resistivity scale? Why?
A: \(\rho_{e-e} \propto T^2\)
It is a thermal effect which has a scattering rate \(\tau^{-1} \propto T^2\)
Note that for most cases \(e-e\) scattering is very weak

Q: How do we model a temperature gradient in a solid?
A: We cut up the solid into small pieces with a local temperature.

Draw the picture of thermal transport within solids

Q: What is the Wiedemann-Franz law?
A: Empirical relation between thermal conductivity and electrical conductivity
\(\kappa = \frac{\pi^2}{3}\frac{k_B^2}{e^2}T\sigma\)