Title: ETH_Sem5 / Solid_State_Intro / Woche_07

Q: What is the Fermi wave number?
A: \(k_F = \sqrt{2m_e\varepsilon_F}/\hbar\)
This corresponds to the radius of the Fermi sphere

Q: What is the relation between electron density and Fermi wave number?
A: \(n_e = \frac{k_F^3}{3\pi^2}\)
Where \(n_e := N_e/V\)

Q: What is the specific heat of a metal due to electrons?
A: Mathematically:
\(C_V/V = \frac{\pi^2}{3}\nu(\varepsilon_F)k_B^2 T\) = \(\gamma T\)
with \(\gamma\) the sommerfeld coefficient, which is a measure of the DOS at \(\varepsilon_F\)

Q: Why can we assume the electrons in a metal to be an ideal gas? Why would this assumption be unintuitive at first?
A: Due to fermion nature we know that scattering is only allowed if the final state is unoccupied. In fact only electrons on the boundary can collide, as those are the only electrons that exist and have matching energy levels to scatter into.
This assumption is unintuitive because the strong and wide ranging coulomb force makes the non interaction part of the ideal gas seem unreasonable.

Q: What is the Bloch Theorem?
A: Eigenstates for particles in a periodic potential are plane waves, whose amplitude is modulated with the periodicity of the underlying potential.
Mathematically:
\(\psi_k(r) = u_k(r) \exp(ikr)\)
With \(u_k(r) \) a function with the periodicity of the lattice.

Q: Group velocity of electrons:
A: \(v_n = \frac{1}{\hbar} \nabla_k\varepsilon_n(k)\)
With \(\varepsilon_n \) the electron dispersion relation for the \(n\)-th band.

Q: What is an almost-free electron? How can we use it to understand metals?
A: It's an electron confined to a periodic potential, but the potential is way weaker than the kinetic energy of the electron.
This assumption is useful, because we can keep our old dispersion relation of the free electron
\(\varepsilon_0 = \frac{\hbar^2 k^2}{2m_e}\)
We can follow from this that we will get a "periodic form" of the free electron.
Using the points where the assumption breaks the perturbation we can "cut" the parable and rearrange.

Q: What is ARPES, how does it work?
A: Angle Resolved Photo Emission Spectroscopy
We ionize a sample using photons. By detecting the direction and energy of the electrons we can deduce the dispersion relation using conservation of momentum.

Q: How does the potential in the tight binding model look?
A: In the tight binding model we consider the ionic cores of the atoms
The potential takes the form \(\sum_j U_o(r-r_j)\)

Q: What is the trick to solving the tight binding model, how do we obtain the band structure?
A: We consider far apart atoms and perturb the local solution with the other potentials. We then consider the overlap integrals (to predict the jumping probability) between nearest neighbours.
The band structure is given by the combination of MO's into Bands.

Q: Why is it better to talk about quasi-electrons and quasi-holes in a material?
A: The momentum of an excited state is not related to the energy of the excitation, because the excited electron could have an almost arbitrary \(\Delta E\)
This stops us from having to redefine a dispersion relation.

Q: What is a quasi-electron/quasi-hole?
A: QE: We consider N-1 electrons in ground state +1 excited electron
QH: We consider N+1 electrons in ground state -1 electron from the fermi sea.

Q: What is the chemical potential of quasiparticles?
A: It's always zero, because both the chemical potential as well as the ground state for quasiparticle (no excited states) are the same.