  Topological insulators (TIs) are characterized by possessing metallic
(gapless) surface states and a finite band-gap state in the bulk. As the
thickness of a TI layer decreases down to a few nanometer, hybridization
between the top and bottom surfaces takes place due to quantum tunneling,
consequently at a critical thickness a crossover from a 3D-TI to a 2D insulator
occurs. Although such a crossover is generally accessible by scanning tunneling
microscopy, or by angle-resolved photoemission spectroscopy, such measurements
require clean surfaces. Here, we demonstrate that a cascading nonlinear
magneto-optical effect induced via strong spin-orbit coupling can examine such
crossovers. The helicity dependence of the time-resolved Kerr rotation exhibits
a robust change in periodicity at a critical thickness, from which it is
possible to predict the formation of a Dirac cone in a film several quintuple
layers thick. This method enables prediction of a Dirac cone using a
fundamental nonlinear optical effect that can be applied to a wide range of TIs
and related 2D materials.
