Abstract
When light is incident on a medium with spatially disordered index of refraction, interference effects lead to near-perfect reflection when the number of dielectric interfaces is large, so that the medium becomes a "transparent mirror." We investigate the analog of this effect for electrons in twisted bilayer graphene (TBG), for which local fluctuations of the twist angle give rise to a spatially random Fermi velocity. In a description that includes only spatial variation of Fermi velocity, we derive the incident-angle-dependent localization length for the case of quasi-one-dimensional disorder by mapping this problem onto one dimensional Anderson localization. The localization length diverges at normal incidence as a consequence of Klein tunneling, leading to a power-law decay of the transmission when averaged over incidence angle. In a minimal model of TBG, the modulation of twist angle also shifts the location of the Dirac cones in momentum space in a way that can be described by a random gauge field, and thus Klein tunneling is inexact. However, when the Dirac electron's incident momentum is large compared to these shifts, the primary effect of twist disorder is only to shift the incident angle associated with perfect transmission away from zero. These results suggest a mechanism for disorder-induced collimation, valley filtration, and energy filtration of Dirac electron beams, so that TBG offers a promising new platform for Dirac fermion optics.
Highlights
Transmission of light through a medium with random refractive index is a well studied problem in optics [1,2,3]
We assume steplike changes in velocity, which corresponds to the limit where D is long compared to the electron wavelength and the thickness of the domain wall is small compared to the wavelength. (Such steplike changes are seen in experiments that make a spatial map of local twist angle across a twisted bilayer graphene (TBG) sample [13,14].) Within this setup we consider two scenarios: I
I we show that the transmission of two-dimensional massless Dirac fermions through a medium with quasi-onedimensional random Fermi velocity can be exactly mapped onto the problem of conventional Anderson localization in one dimension
Summary
1 φc Ttypical ≡ π dφ0 Ttypical(φ0 ) This quantity corresponds to the average intensity of Dirac electrons transmitted across a sample, given an a initial wave packet of varying initial angle of incidence. For a given incident angle, the mapping to Anderson localization implies an exponential decrease of the transmission with distance, the divergence of the localization length for small angles gives an angle-averaged typical transmission that decays only as a slow power-law N−1/2. The corresponding calculations are discussed in Appendix A and the references therein
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