# Bandstructure of topological insulating Bi2Se3

Topological insulators are a special class of material that is insulating in the
bulk, however exhibit conducting states in the surface.
Bi_{2}Se_{3} is such a material. Spin orbit coupling and
breaking of the inversion symmetry at the surface of the crystal is crucial to
the existence of the Dirac surface state. Here we will calculate the
bandstructure step by step: first for the bulk, next including SOC, and finally
for the slab. Please check the respective input files. I followed the
following steps:

`# SCF calculation for bulk`

mpirun -np 24 pw.x -i pw.scf.bi2se3_01.in > pw.scf.bi2se3_01.out

# bands calculation for bulk

mpirun -np 24 pw.x -i pw.bands.bi2se3_01.in > pw.bands.bi2se3_01.out

# post processing for bulk bands

mpirun -np 24 bands.x -i pp.bands.bi2se3_01.in > pp.bands.bi2se3_01.out

# for bulk with SOC

mpirun -np 24 pw.x -i pw.scf.bi2se3_02.in > pw.scf.bi2se3_02.out

mpirun -np 24 pw.x -i pw.bands.bi2se3_02.in > pw.bands.bi2se3_02.out

mpirun -np 24 bands.x -i pp.bands.bi2se3_02.in > pp.bands.bi2se3_02.out

# slab calculation

mpirun -np 24 pw.x -i pw.scf.bi2se3_03.in > pw.scf.bi2se3_03.out

mpirun -np 24 pw.x -i pw.bands.bi2se3_03.in > pw.bands.bi2se3_03.out

mpirun -np 24 bands.x -i pp.bands.bi2se3_03.in > pp.bands.bi2se3_03.out

# DOS

mpirun -np 24 pw.x -i pw.nscf.bi2se3_04.in > pw.nscf.bi2se3_04.out

mpirun -np 24 dos.x -i pp.dos.bi2se3_04.in > pp.dos.bi2se3_04.out

For the slab calculation the periodicity of the lattice was broken along the
c-axis to artificially add 10 Å vacuum. In above calculation electronic spin
was not considered (meaning the states are degenerate with spin up and down).
If `starting_magnetization`

is set to zero (or not given) the code makes a
spin-orbit calculation without spin magnetization. It assumes that time reversal
symmetry holds and it does not calculate the magnetization. The states are
still two-component spinors but the total magnetization is zero.

Notice that for the Dirac surface states the gap did not completely close at the Fermi energy. This is possibly due to finite size effect. We could repeat the calculation with larger vacuum, and see what happens. Also the Fermi energy estimation seems incorrect.

In order to sample the $\Gamma$ point for our DOS calculation, an odd k-grid mesh (25✕25✕5) was used. The signature of Dirac cone is evident from the DOS figure.