QMP

In this project we are going to analyze the phenomenon of Hall conductivity quantization in the so-called Hall bar structure. This effect is a clear manifestation of the quantized Landau levels creation for electrons in the presence of the external magnetic field as well as the appearance of the so-called edge states.

Conclusions

In this project we have succesfully calculated longitudinal and transverse conductivity. Using a magnetic field perpendicular to the system.

interpretations

Through plots we could have observed, how as the magnetic field increased, the electron flow patterns changed and the transverse conductivity increased.

The conductivities exhibited the characteristic plateaus in $\sigma_{xy}$ and corresponding drops in $\sigma_{xx}$, which are signatures of the Quantum Hall Effect.

Effect of impurities

Introducing impurities into the system changed the behavior of the conductivities. When we increase the level of impurities, the conductance matrix and the resulting $\sigma_{xx}$ and $\sigma_{xy}$ values were affected. Despite the impurities causing more electron scattering, the overall quantized nature of the Hall conductivity was preserved. This shows that the Quantum Hall Effect can withstand some level of disorder.

Summary

The quantization of Hall conductivity, the resilience of edge states, and the impact of impurities were key observations that aligned well with theoretical expectations and experimental results documented in the project description and literature.