# Topological-Insulator nanocylinders

## Abstract:

Nanostructures, such a quantum dots or nanoparticles, made of three-dimensional topological insulators (3DTIs) [1-5] have been recently attracting increasing interest, especially for their optical properties. We present results for the energy spectrum, the surface states and the dipole matrix elements for optical transitions with in-plane polarisation of 3DTI nanocylinders of finite height L and radius R. We first derive an effective 2D Hamiltonian by exploiting the cylindrical symmetry of the problem. We develop two approaches: The first one is an exact numerical tight-binding model obtained by discretising the Hamiltonian; The second one, which allows us to obtain analytical results, is an approximated model based on a large-R expansion and on an effective boundary condition to account for the finite height of the nanocylinder. We find that the agreement between the two models, as far as eigenenergies and eigenfunctions are concerned, is excellent for the lowest absolute value of the longitudinal component of the angular momentum. Finally, we derive analytical expressions for the dipole matrix elements by first considering the lateral surface alone and the bases alone, and then for the whole nanocylinder. In particular, we focus on the two limiting cases of tall and squat nanocylinders. The latter case is compared with the numerical results finding a good agreement.

[1] L. Fu, C. L. Kane and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007).

[2] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).

[3] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).

[4] M. Z. Hasan and J. E. Moore, Three-dimensional topological insulators, Annual Review of Condensed Matter Physics 2(1), 55 (2011).

[5] Y. Ando, Topological insulator materials, Journal of the Physical Society of Japan 82(10), 102001 (2013).