Tasks for completing the curse

2DEG constriction

  • Calculate the transmission coefficient trough a constriction in 2DEG.
  • Perform calculations for different geometries.
  • Recover the conductance quantization.
  • Use a square lattice disretization for the whole system.
  • Set $\mu_L$ in such a manner that the number of open channels is high in the leads.
  • Scan with $\mu_S$
  • Some theoretical background can be found here.

2DEG constriction in a magnetic field

  • Calculate the transmission coefficient for a fixed geometry constriction in the presence of a magnetic field.
  • In order to take the magnetic field in to account use Peierls substitution.
  • Set chemical potential the same everywhere.
  • Use a square lattice discretization for the whole system.

2DEG kink in a magnetic field

  • Calculate the transmission coefficient for a kink geometry as the function of the magnetic field.
  • In order to take the magnetic field in to account use Peierls substitution.
  • Set chemical potential the same everywhere.
  • Use a square lattice discretization for the whole system.

Graphene instead of 2DEG

  • Any of the above three projects can be done with graphen instead of 2DEG.
  • Use a hexagonal lattice instead of a square lattice.

2DEG Rashba weak localizaton

  • Introduce spin-orbit coupling in a 2DEG and some impurities and external, orbital magnetic field.
  • Calculate the transmission coefficient as the function of the magnetic field.
  • Some theoretical background can be found here.

Graphene minimal conductivity

  • Calculate the transmission coefficient of a clean graphene ribbon for different aspect ratios.
  • Show that for short and wide samples the conductivity is universal.
  • Use a honeycomb lattice for the discretization to see Dirac physics.
  • Set the chemical potential $\mu_S$ to a value cloes to zero.
  • Set $\mu_L$ such that the number of open channels in the lead should be high.
  • Some additional details in this paper.

Graphene focusing

  • Look for inspiration in this paper !
  • Calculate the focusing effect of a graphene sample in a pn junction.
  • You have to perform calculations for transverse k-ponts and then Fourier transform.
  • It is enogh to look at the appropriate Green's function elements.
  • Use a honeycomb lattice for the discretization to see Dirac physics.

QWZ model dirty edge local density

  • Calculate the site resolved local density of states for a QWZ wire with a scatterer located at the edge of the sample.
  • $T_{x/y}=\frac{\sigma_z+i\sigma_{x/y}}{2}$ and $U=\Delta\sigma_z$
  • Tune the parameter $\Delta$

QWZ model dirty edge conductivity

  • Calculate the transmission coefficient for a QWZ wire with a scatterer located at the edge of the sample.
  • $T_{x/y}=\frac{\sigma_z+i\sigma_{x/y}}{2}$ and $U=\Delta\sigma_z$
  • Tune the parameter $\Delta$

BHZ model dirty edge local density

$$ H_{BHZ}= \begin{bmatrix} H_{QWZ} & 0\\ 0 & H_{QWZ}^* \end{bmatrix} $$

  • Calculate the site resolved local density of states for a BHZ wire with a scatterer located at the edge of the sample.
  • Take simple potetial scattering i.e. $V\propto I_{4}$ .
  • Take completely random scatterer.

BHZ model dirty edge conductivity

  • Calculate the transmission coefficient for a BHZ wire with a scatterer located at the edge of the sample.
  • Take simple potetial scattering i.e. $V\propto I_{4}$ .
  • Take completely random scatterer.

Topological Anderson Insulator

  • Reproduce some of the results of this paper.