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.