The Dirac equation as a model of topological insulators

ABSTRACT

The Dirac equation with a spatially dependent mass can be used as a simple, exactly soluble, continuum model of a three-dimensional Topological Insulator. For a bulk system, the sign of the mass determines the parity at the only time-reversal point ($\underset{\_}{k}=0$) and, hence, leads to the designation of the bulk as being topologically trivial or non-trivial. Since the mass changes sign at the interface between a topologically trivial and non-trivial materials, topological surface states appear on that boundary. We propose that electron scattering experiments may provide an alternate probe of the topological character of the surface states. For infinitely thick slabs, the states on the opposite sides of the slab decouple. The spatial decoupling results in the surface states become gapless, non-degenerate and, due to the Rashba spin–orbit coupling generated by the loss of inversion symmetry, exhibit spin-momentum locking. We review several characteristic properties of the topological surface states which are dependent on the topological quantum numbers and show that, using this model, they can be calculated exactly using simple methods.

KEYWORDS:

• Topological insulators
• Dirac equation
• Weyl cone
• berry phase
• Chern number
• experimental manifestations
• electron diffraction

Acknowledgments

Part of this work was presented as an undergraduate Thesis at Drexel University. The work is dedicated to Prof. M.B. Maple on the occasion of his 80th birthday.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work at Temple was supported by the U.S. Department of Energy, Office of Basic Energy Science, Materials Science, through the award DE-FG02-01ER45872.