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Dirac and Weyl semimetal states in Na 3 Bi from first principles
Patrick Buhl, Stefan Blügel, Yuriy Mokrousov
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich
Those points, called Weyl points (WP) not only form Dirac point in reciprocal space, but also carry the
helicity .
It is necessary to break either time-reversal or
inversion symmetry to ensure a non-degeneracy and therefore the possibility of a linear dispersion.
One can show, that the WPs correspond to sources of the Berry curvature, such that the integral of the Berry curvature over small spheres around the WP yields the quantized . This is often related to magnetic
monopoles in the reciprocal space comparing the Berry curvature with the magnetic field. The integral corresponds to the Chern number.
This ensures to some extend the stability of the WPs, as the quantization has to persist such that WPs can only annihilate as they approach their counterpart of opposite helicity and thereby cancel the quantization of the integral around both.
An even count of WPs is ensured, as the overall
integral over the Brillouin zone has to be zero and WP can only be created or annihilated, if two WPs of
different helicity meet.
Theory of Weyl semimetals
Another property arising form the WP’s are the Fermi arc surface states. Coming from the 2 dimensional integrals and their quantization it is rather intuitive to view the 2 dimensional spheres as topological
insulators.
Assuming one has 2 WP’s of different helicity
separated in kz-direction, then one can calculate the Chern number of the kx-ky planes in between and see, that they will be nonzero. Like before one can identify these layers with topological insulators.
As each of these layers exhibits surface states Fermi arcs emerge, which are surface states connecting the surface projection of both WP and merge into the bulk by reaching the projected WP. Further reading in ref. 1.
Weyl semimetals are 3 dimensional topological semimetals obeying the Weyl equation (linear
dispersion) at an even number of points at the Fermi surface.
Considering the Berry connection A and the Berry
curvature B one can explore the topological property of the WP.
Picture taken from Ref. 1
The Bi atoms and 2 Na atoms(Na(1) in the picture) form two hexagonal lattices and the other 4 Na
atoms are positioned kz shifted above or below the Bi atoms.
It was predicted to exhibit a Weyl semi-metallic phase through DFT calculations. There it is
calculated, that for an effective low energy model of this system upon applying an exchange field, to
break the time reversal symmetry, WPs emerge from the former symmetry protected Dirac point between A and Γ.[2]
The Dirac points were experimentally measured by an ARPES experiment and proofed robust against in-situ surface modifications[3].
Na
3Bi
Na3Bi
crystallizes into the hexagonal P63/mmc with inversion
symmetry.
It has 8 atoms
per unit cell. Picture taken from Ref. 2
The FLEUR[4] all-electron DFT code was used to
create a description of the system. The PBE functional and muffin tin radii of 2,8 aB were chosen. The
calculations were performed with and without spin orbit coupling (SOC) and yield comparable band structures to reference 2.
Band touching point at Fermi level if SOC and no Dirac point without SOC.
Large atomic number of Bi → necessary to include SOC, so that from now on only this case will be
pursued. It would be very interesting to investigate the effect of varying SOC strength, which will be subject of further studies.
The Dirac point is yet due to TR and INV symmetry 4 times degenerate and therefore cannot be a WP, but through applying an external exchange-field there is a decent chance to discover Weyl physics.
DFT band structure
Including SOC Without SOC
Wannier basis
Wannier90 [5] code → maximally localized Wannier functions description near the Fermi surface.
SOC DFT description→ a more applicable basis of 24 bands. Band structure in well agreement with initial
ones → work in this basis → create real space
hopping Hamiltonian and spin expectation values.
An exchange field is added to the Wannier function Hamiltonian through a Zeeman like term. The band
structure is plotted for 0.7 eV exchange field with color coded spin.
Calculating the Chern number
For calculating the Chern number a code by Hongbin Zhang is used. [6]
For layers of constant kz the Hamiltonian is transformed back in reciprocals space and
diagonalized. With these eigenvectors the Berry curvature B is constructed.
The summation over all k points in the
plane yields the Chern number for each band.
Calculation of the
whole Chern number is difficult, as one has to perform a sum over all occupied state. As some bands cross the Fermi energy and
have some
degeneracies this is not well defined.
Investigate topological properties through Chern numbers: kx- ky grid of 800x800 k-points is used.
Bands 1-4 and 5-6 sum up to 0. Bands 7 to 11 and 13 to 16 are summed up, as they have mixing points. At the kz of the suspected WP the Chern number
changes, which supports the WP character.
Several surface states are visible. The most noticable is in between the highest occupied bunch of bands
and the bunch below have a 4 times degenerate state at the projected Γ and splits at increasing exchange field. To see the Fermi arcs one has to investigate another surface which will be the subject of further studies.
The surface projected band structure is compared with a layers calculation done with the hopping
Hamiltonian. The layering is done in [0,0,1] direction.
The localization of each band is calculated by adding the layers up, weighted with the projection of the
eigenvector onto the layer. Here plots with 10 layers and the bulk Fermi energy are shown with 0eV and 0,07eV exchange field.
Surface properties
[1] A. M. Turner, arXiv:1301.0330 ‘13
[2] Zhijun Wang, et al., PRB 85, 195320 ‘12 [3] Z.K.Liu, B.Zhou, et al., arXiv:1310.0391 ‘13 [4] www.flapw.de
[5] Comput. Phys. Commun. 178, 685 ‘08
[6] Hongbin Zhang, et al., PR B86, 035104 ‘12 Picture taken from ref. 1
B = 0.07 eV B = 0 eV
References
