Broken Mirrors
The breakdown of the law of reflection in bilayer graphene
Rutger Engelberts (TU Delft - Applied Sciences)
Anton Akhmerov – Mentor (TU Delft - Applied Sciences)
Johan Dubbeldam – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Jos Thijssen – Graduation committee member (TU Delft - Applied Sciences)
Gioia Carinci – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Tomas Rosdahl – Graduation committee member (TU Delft - QRD/Wimmer Group)
Kim Pöyhönen – Graduation committee member (TU Delft - QRD/Wimmer Group)
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Abstract
As long as a wave has a large enough wavelength, it should reflect off of smooth surfaces specularly: that is what the law of reflection states. This phenomenon is widely known, and used in for instance sonars. Electrons that reflect off of boundaries within conductors should abide this law as well, due to the particle-wave duality. However, it was recently discovered that for disordered graphene boundaries electrons scatter diffusively even when their wavelengths should be long enough not to. Graphene is a 2-dimensional semiconductor with linear dispersion at low energies, making it a widely researched topic as its applications in future electronic parts are promising. Its bilayer counterpart is made by connecting two layers on top of each other, which in contrast has a quadratic dispersion at low energies. In this work, the specularity of electron reflections within bilayer graphene were studied. This was done by constructing a tight-binding model, where random potentials on the outermost atoms represented imperfections of the lattice. Subsequently the variance of the scattering angle and the scattering phase was analysed, which can be derived using the scattering matrix of themodel. This was done numerically for both, and analytically for the phase as well. It was discovered that the variance of the scattering angle increases cubically with the Fermi wavelength of the electrons. This happens no matter what the distance between the Fermi energy and the disorder mean is, as long as the disorder strength is nonzero. The behaviour of these reflections is the exact opposite of what the law of reflection states. Furthermore, the variance of the scattering phase remains constant when the Fermi energy is close to the disorder mean, and changes inversely proportional to the boundary periodicity if the Fermi energy is further away. Consequently, the variance of the scattering phase versus the Fermi wavelength and the variance of the scattering phase versus the boundary periodicity do not share the same trend in bilayer graphene. More research is needed to verify this phenomenon, which could be done using the continuum description or a magnetic focusing device.