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Topological materials are a new family of quantum materials with distinct electronic and optical properties in a protected surface region. In three dimensional topological materials, the bulk of the material displays an insulating characteristics whereas the surface states (SSs) exhibit a conducting nature with massless Dirac-type band structure. Due to the substantial overlapping of wavefunctions of the top and bottom SSs, the hybridization energy gap DH emerges at the Dirac points which can be tuned by controlling the thickness of TI thin films. However in real space, controlling the thickness for tunning the hybridization energy gap is challenging and economically non-viable. One of the most exciting phenomena in quantum materials is the topological quantum phase transition (TQPT). A TI thin film may be driven through several TQPTs from topologically non-trivial to a semi metallic state and further to a band insulating state subjected to perpendicular magnetic field to its surface.

When a polarized beam of light with finite width experiences total internal reflection at an interface of two different dielectric media, there is a longitudinal (parallel to the plane of incidence) shift with respect to the position predicted by the geometric optics. This beam displacement phenomena is called the Goos–Hanchen (GH) shift. The potential applications of GH shift are in biosensors, optical measurement and optical heterodyne sensors. Recently, W. Wu et. al theoretically predicted a giant quantized GH shift on the surface of graphene in the quantum Hall regime. Similarly, T. Tang et. al proposed an experimental scheme based on a prism graphene coupling structure for magneto-optical (MO) tunable GH effect. More recently, the GH shift on the surface of staggered 2D materials, TMDCs and Weyl semimetals has also been investigated. The present study details the impact of magnetic field modulation of the surface state-polarized GH shifts in TI thin film.

thin film.

Magnetically Tunable Goos-Hanchen Shifts in

Topological Quantum Materials

Muzamil Shah*, Mudasir Shah and Ali Akbar

Department of Physics, School of Science and Engineering,

Lahore University of Management Sciences (LUMS), 54792 Lahore, Pakistan

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Introduction

Goos-Hänchen shifts

Dispersion Relation

Results

Hybridized 3D topological insulator thin films Hamiltonian

References

[1] M. Shah, A. Akbar, M. Sajid, and M. S. Anwar, "Transitional Faraday and Kerr effect in hybridized topological insulator thin films," Opt. Mater. Express 11, 525-538 (2021).

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[2] A. A. Zyuzin and A. A. Burkov, “Thin topological insulator film in a perpendicular magnetic field,” Phys. Rev. B 83(19), 195413 (2011).

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[3] M. Shah and M. S. Anwar, “Magneto-optic modulation of lateral and angular shifts in spin-orbit coupled

e graphene family,” OSA Continuum 3(4), 878–892 (2020).

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[4] M. Shah and M. S. Anwar, “Magneto-optical effects in the Landau level 374 manifold of 2D lattices with spin-orbit interaction,” Opt. Express 27(16), 375 23217–23233 (2019).

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[5] M. Shah and M. S. Anwar, “Quantized and Topological Photonic Spin Hall 380 Effects in the Graphene Family in the Presence of Magnetic Fields,” Optical 381 Sensors OSA, STu3D–5 (2020).

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The effective Hamiltonian for an ultra-thin film TI system in the presence of a perpendicular magnetic field can be described by the 2D Dirac-type Hamiltonian

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The eigen-energy and eigenvectors are given as

where

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denotes top/bottom surface

Magneto-optical conductivity of topological insulator

The Kubo formula is given as

The real and imaginary parts of the zero temperature longitudinal magneto-optical conductivities are

The real and imaginary parts of the transverse Hall conductivities are

and

Fresnel’s Coefficients:

The reflection and transmission coefficients given as

and

where

Solving Maxwell’s equations and imposing the appropriate boundary conditions on the interface at z =0

For p polarized incident beam the angular and spatial GH shifts can be define as

where

where

and

and

and

Acknowledgements

The authors would like to acknowledge financial support from the National Research program for Universities (NRPU), scheme number 10375 funded by the Higher Education Commission of Pakistan.