A 3D topological insulator (Bi₂Se₃, Bi₂Te₃, …) has an insulating bulk but hosts a single, gapless 2D Dirac cone of metallic states on its surface, protected by time-reversal symmetry. Near the Dirac point the surface Hamiltonian is
H(k) = ħv_F(σx k_y − σy k_x) + m σz
E(k) = ±√[(ħv_F k)² + m²]
The first term locks each electron's spin (σ) perpendicular to its momentum k — the surface "spin-momentum locking" seen in ARPES spin-texture maps. The second term, m, is a mass gap that only opens if time-reversal symmetry is broken — e.g. by doping the surface with magnetic atoms (Cr, Mn, Fe) or an external Zeeman field. With m = 0 the eigenspinors at k and −k are exactly orthogonal, so elastic scattering off any non-magnetic (time-reversal-symmetric) impurity has zero backscattering amplitude, however strong the impurity potential — a hallmark of topological protection reported in STM quasiparticle-interference experiments. Turning on m tilts the spin out of the surface plane by a canting angle
cos η = m / E(k), η = canting from the surface plane
Backscattering probability ≈ (m / E_F)²
so a finite gap partially restores backscattering — the mechanism behind gap-opening and the onset of the quantum anomalous Hall effect when a topological insulator is magnetically doped.
- EF — Fermi level, sets the radius of the constant-energy (Fermi) circle on the Dirac cone.
- vF — surface Fermi velocity, a material-dependent cone steepness.
- Magnetic gap m — 0 reproduces the ideal protected surface; increasing it cants the spin texture and switches on backscattering.
- Fire electron — launches a wavepacket around the Fermi circle toward the fixed impurity (white marker) and resolves transmission vs. backscattering by sampling the theoretical probability above.