Supplementary Information
Quantum spin Hall insulators and quantum valley Hall
insulators of BiX/SbX (X = H, F, Cl, and Br) monolayers
with a record bulk band gap
Zhigang Song1†, Cheng-Cheng Liu2†, Jinbo Yang1,3*, Jingzhi Han1, Meng Ye1, Botao
Fu2, Yingchang Yang1, Qian Niu3, 4, Jing Lu1,3 * & Yugui Yao2 *
1
State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University, Beijing
100871, China
2
School of Physics, Beijing Institute of Technology, Beijing 100081, China
3
Collaborative Innovation Center of Quantum Matter, Beijing, China
4
International Center for Quantum Materials, Peking University, Beijing 10087, China
S1
Free electron gas in BiH monolayer
A perfect free electron gas could be observed in BiH monolayer based on our work.
If a vertical electronic field is applied on BiH monolayer, the exchange interaction is
modified, leading to a large spin-split. A spin-polarized band projecting on free
electrons is formed. The dispersion curve is a perfect parabola and can be described
by
F
2
k2
 aeE  C ,
2m
(1)
where F is free energy of free electron gas, e and E are electron charge and electric
field strength, respectively; a and C are constant values of a (= 10.88 ) and C (= 7.636)
can be obtained by fitting first-principles band structure. If E d increases, the
parabolic band will move to the Fermi level. In a range of (0.61-0.67 V/Å), the
parabolic band enters into the original gap and even cross the Fermi level making BiH
monolayer a metal under a critical electric field of 0.7 V/Å. The parabolic band is
spin-polarized and strongly dependent of the vertical electric field (or gate voltage),
and the spin field effect transistor (SFET) device made of BiH monolayer TI can be
designed. The bands of BiH monolayer under different vertical electric fields are plot
in Fig S4.
Chemical stability test
We have chosen oxygen as one example of the gas molecules to evaluate how
active of these monolayers to the gas adsorbates. First, we carried out a geometry
optimization including SOC interaction with two different types of configurations: (1)
one oxygen molecule above one Bi atom; (2) one oxygen molecule above one
hydrogen atom. The atomic positions were relaxed until the maximal force on each
relaxed atom was smaller than 0.003 eV/Å. After the geometry optimization, it was
S2
found that the oxygen molecule is repelled from the BiH monolayer. Therefore, BiH
monolayer is not affected by the oxygen molecule absorption at 0 K. This suggests
that extra energy has to be provided to overcome an energy barrier in order to form a
chemical bond between BiH monolayer and O2 at 0 K.
Secondly, we carried out ab-initio MD simulation under an extremely high oxygen
concentration (ratio of Bi and O atoms is 1: 1 in a cell) at the room temperature (RT).
A small portion (about 10 %) of oxygen molecule will decompose to oxygen atoms
after 1500 steps. One oxygen atom is located at the position above the center of the
Bi-atom ring, and the other forms a covalent bond with Bi in a hydroxide fashion.
There is a small local deformation around the absorbed O atom. This suggests that
BiX monolayer can survive at very low temperature and need to be protected at RT by
vacuum or using inert gases or an anti-oxidization layer such as two-dimensional
graphene, BN, or MoS2.
S3
(a) SbH
(e) BiF
(b) SbF
(c) SbCl
(f) BiCl
(d) SbBr
(g) BiBr
Figure S1: (a-g) Snapshots of the crystal structures from the MD simulation after
2.25 ps for SbH (a), SbF (b), and SbCl (c) monolayers at 400 K, for SbBr monolayer
at 300 K, and for BiF (e), BiCl (f), and BiBr (g) monolayers at 600 K. A supercell
with 3×3 unit cell is used.
Figure S2: Snapshots from the MD simulation of the structure for BiH monolayer
at 600 K after 2.25 ps. Pink balls: Bi atoms; green balls: H atoms. A supercell with
4  4 unit cell is used.
S4
(a) BiCl
(b) BiBr
(c) SbH
(d) SbCl
(f) SbBr
Figure S3: Band structures of the Bi(Sb)X monolayers without (gray) and with
(red) SOC. The Fermi level is set to zero.
S5
(b) 2
Energy (eV)
Energy (eV)
(a) 2
1
0
-1
1
0
-1


(d)
Band gap (eV)
Energy (eV)
(c)1.0
0.5
0.0
-0.5
0.9
0.6
0.3
0.0
-1.0
0.2
0.4
0.6
Electric field V/Å

Figure S4: Band structures of BiH monolayer under vertical electric fields of 0.65
V/Å (a) and 0.7 V/Å (b). The green dot line is a standard parabola from Eq. 1. (c)
Band structure of BiH monolayer based on a crystal structure from MD simulation at
600 K after 2.25 ps (See Fig. 2 in the main text). (d) Band gap versus electric field.
Figure S5: Energy gap and total energy as a function of biaxial strain from 0 to 12%
for BiH monolayer. The calculated total energy increases continually with the applied
biaxial strains, suggesting that the tensile deformation is within the elastic range.
S6