Calculate electron transmission probability through planar barriers in 3D crystals by applying momentum conservation principles. Use when analyzing tunneling in three-dimensional lattices, when perpendicular momentum conservation must be accounted for, or when converting 1D tunneling results to 3D scenarios involving crystal structures.
Apply this skill when:
Recognize that in a 3D lattice, momentum components perpendicular to the tunneling direction are conserved. Only the momentum component in the tunneling direction decreases exponentially through the barrier.
Compute the energy associated with momentum perpendicular to tunneling:
E_perp = (ℏ² × k_perp²) / (2m)