The electronic band topology of monolayer β-Sb (antimonene) is studied from the flat honeycomb to the equilibrium buckled structure using first-principles calculations and analyzed using a tight-binding model and low energy Hamiltonians. In flat monolayer Sb, the Fermi level occurs near the intersection of two warped Dirac cones, one associated with the pz-orbitals, and one with the {px,py}-orbitals. The differently oriented threefold warping of these two cones leads to an unusually shaped nodal line, which leads to anisotropic in-plane transport properties and goniopolarity. A slight buckling opens a gap along the nodal line except at six remaining Dirac points, protected by symmetry. Under increasing buckling, pairs of Dirac points of opposite winding number annihilate at a critical buckling angle. At a second critical angle, the remaining Dirac points disappear when the band structure becomes a trivial semiconductor. Spin-orbit coupling and edge states are discussed.

1. Topological band transitions of monolayer-Sb
as a function of buckling
From Nodal line to unpinned Dirac cones.
e- h+
e-
e-
h+
h+
2D Goniopolarity Merging Dirac fermions Surface states
Santosh Kumar Radha, Walter RL Lambrecht
Department of physics
Case Western Reserve University
arXiv:1912.03755

2. Structure
https://doi.org/10.1021/acsami.5b02441
Flat
Low Buckled
Buckled
Energetically Stable
Meta stable
Unstable
• Study using DFT
• Use Tight binding
Hamiltonian to
understand symmetry
• Construct low energy
Hamiltonian to further
probe the physics

3. Band structures
Flat Low Buckled Buckled
−12
−8
−4
0
4
8
12
Γ K M Γ
p- total
Metal Metal Insulator

4. Flat Sb - band structure
pz px,y
s
Total
• Different from graphene, s level is much bellow p => no sp hybridization
• But one extra e- than graphene so Fermi level is in-between pz-pxy
• Low Energy physics captured by px-py-pz in honeycomb

5. Flat Sb - Story of double Dirac cone
px − py − pz
TB Hameltonian
pz
px + ipy
??
Use TB Hameltonian to study and understand the
wavefunctions and bands and symmetries
In honeycomb

6. pz
px + ipy
Nodal line ?
• 2 Dirac cones intersecting
• Similar to AA stacked bilayer graphene
AA stacked Bi-layer Graphene

7. Lissajous Nodal ring
• But literally with a twist
• Intersects at k>>O(k), need O(2) terms including trigonal warping
• It can be shown that warping of pz is opposite to px-py forming
lissajous nodal ring
DFTLowenergy

8. 2-Dimensional Goniopolarity
He, B., Wang, Y., Arguilla, M.Q. et al. The Fermi surface geometrical origin of axis-dependent conduction
polarity in layered materials. Nat. Mater. 18, 568–572 (2019) doi:10.1038/s41563-019-0309-4
Change in in-plane carrier type
from e- to hole every 30∘

9. Buckling
K
M
G
log(Ec
− Ev
)
buckling
• Buckling breaks plane mirror symmetry and allows pz-(px,py) interaction and can open the gaps
• But bands in M-K and K-G are protected by C2 symmetry
• Thus nodal ring breaks up into 6 Dirac cones
Gap opens up
Dirac crossing

10. Buckling
3D Band plot around M showing the merging physics
• Dirac crossing in graphene is forced to happen because of TRS and sublattice symmetry
• In Sb, crossing are not forced to happen, but if it happens, are protected by C2
symmetry
• Thus these can be moved in BZ by any transformation that preserved C2 (ex, further
buckling)
• Each dirac cone has a pair with opposite winding number because of TRS
• Dirac fermions with opp. winding numbers can come together to annihilate
• First merging at M around
• Second margining at G around
7∘
33.3∘

11. Dirac Cones - So What?
Red- surface bands
Pz
Pxy at K
• Because of Dirac cones at 6 points in BZ, surface states
in almost all directions of nano ribbon

12. SOC
Arsenic
•SOC strong enough to open the Gap
• Quantum spin hall insulator
•Unique shape of Dirac cone, system is still Semimetal,
leading to Gapless TI
•Example of spin hall insulator with big band gap
•Arsenic, small SOC metallic spin hall insulator

13. Dirac Cones - So What?
1. Nodal Ring
2. Nested Fermi surface
3. In plane conductance anisotropy
4. In plane positive and negative seebak coefficient
2D Sb Flat system
2D Sb low buckled system 1. 6 un-pinned Dirac cones around K
2. Unique shape of Dirac cones
3. Dirac cone merging at critical angle
4. Edge states in almost all directions
5. With SOC system is a QSHI
Reference - arXiv:1912.03755
Interactive plots -
www.santoshkumarradha.me
Santosh Kumar Radha, Walter RL Lambrecht
Department of physics
Case Western Reserve University