Topological Insulators- The New Physics

We have all heard of conductors and insulators. And indeed some of us are more familiar with magnets and semiconductors or even superconductors; they are all manifestations of electronic band structure. But what about the topological insulator? They conduct on the outside but insulate on the inside, much like a plastic wire wrapped with a metallic layer. Weird enough, they create a 'spin-current', where the conducted electrons themselves into spin-down electrons moving in one direction and spin-up moving in the other. Such a topological insulator is an exotic state resulting from quantum mechanics: the spin-orbit interaction and invariance (symmetry) under time reversal. What's more, the topological insulator has topologically protected surface state which is free of impairment of impurities. So how can we understand this 'new' physics? The insulating state has a conductivity of exactly zero around a temperature of absolute zero due to the energy gap segregating the vacant and occupied electron states. The quantum Hall state (QHS) near absolute zero has a quantised Hall conductance (ratio of current to voltage orthogonal to flow of current), unlike other materials like ferromagnet which have order arising from a broken symmetry, topologically ordered states are made distinct by wound up quantum states of electrons (and this protects the surface state). The QHS (the most basic topologically ordered state) happens when electrons trapped to a 2-D interface in between a pair of semiconductors encounters a strong magnetic field. This field causes the electrons to 'feel' an orthogonal Lorentz force, making them move around in circles (like electrons confined to an atom). Quantum mechanics substitutes these circular movements with discrete energies, causing a energy gap to segregate the vacant and occupied states like in an insulator.

However, at the boundary of the interface, the electron's circular motion can rebound off the edge, creating so called 'skipping orbits'. At the quantum scale, such skipping orbits create electronic states that spread across the boundary in a one-way manner with energies that are not discrete; this state can conduct owing to the lack of an energy gap. In addition, the flow in one-direction creates perfect electric transport (electrons have no other option but to move forward because there are no backward-motion modes). Dispationless transport emerges because the electrons don't scatter and hence no energy or work is lost (it also explains the discrete transport). But topological insulators happen without a magnetic field, unlike the quantum hall effect; the job of the magnetic field is taken over by spin-orbit coupling (interplay between orbital motion of electrons via space and the electron's spin). Relativistic electrons arise in atoms with high atomic numbers and thus produce strong spin-orbit forces; so any particle will experience a strong spin-momentum reliant force that plays the part of the magnetic field (when spin changes, its direction changes). Such a comparison between a  spin-reliant magnetic field and spin-orbit coupling allows us to introduce the most basic 2-D topological insulator; the quantum Hall spin state. This happens when both the spin-up and spin-down electrons experience equal but opposite 'magnetic fields'.

Just as in a regular insulator, there exists an energy gap but there are edge states where the spin-up and spin-down electrons propagate in opposition to another. Time-reversal invariance exchanges both the direction of spin and propagation; hence swapping the two oppositely-propagating modes. But the 3-D topological insulator can't be explained by a spin-dependant magnetic field. The surface state of 3-D topological insulators promotes the movement of electrons in any direction, but the direction of electronic motion decides the spin direction. The relation between momentum and energy has a Dirac cone structure like in graphene.